Fixed-Point Theory, Variational Inequalities, and Its Approximation … · 2019. 8. 7. ·...

International Journal of Mathematics and Mathematical Sciences

Guest Editors: Giuseppe Marino, Vittorio Colao, Yonghong Yao, Genaro Lopez, and Enrique Llorens-Fuster

Fixed-Point Theory, Variational Inequalities, and Its Approximation Algorithms

Fixed-Point Theory, VariationalInequalities, and Its ApproximationAlgorithms

International Journal of Mathematics andMathematical Sciences

Fixed-Point Theory, VariationalInequalities, and Its ApproximationAlgorithms

Guest Editors: Giuseppe Marino, Vittorio Colao, Yonghong Yao,

Genaro Lopez, and Enrique Llorens-Fuster

Copyright q 2011 Hindawi Publishing Corporation. All rights reserved.

This is an issue published in volume 2011 of “International Journal of Mathematics and Mathematical Sciences.” All ar-ticles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is properly cited.

Editorial Board

Asao Arai, JapanErik J. Balder, The NetherlandsA. Ballester-Bolinches, SpainMartino Bardi, ItalyPeter Basarab-Horwath, SwedenPeter W. Bates, USAHeinrich Begehr, GermanyHoward E. Bell, CanadaKenneth S. Berenhaut, USAOscar Blasco, SpainMartin Bohner, USAHuseyin Bor, TurkeyTomasz Brzezinski, UKTeodor Bulboaca, RomaniaStefaan Caenepeel, BelgiumWolfgang zu Castell, GermanyAlberto Cavicchioli, ItalyShih Sen Chang, ChinaDer Chen Chang, USACharles E. Chidume, ItalyHi Jun Choe, Republic of KoreaColin Christopher, UKChristian Corda, ItalyRodica D. Costin, USAM.-E. Craioveanu, RomaniaRaul E. Curto, USAPrabir Daripa, USAH. V. De Snoo, The NetherlandsLokenath Debnath, USAAndreas Defant, GermanyDavid E. Dobbs, USAS. S. Dragomir, AustraliaJewgeni Dshalalow, USAJ. Dydak, USAMessoud A. Efendiev, GermanyHans Engler, USARicardo Estrada, USAB. Forster-Heinlein, GermanyDalibor Froncek, USAXianguo Geng, China

Attila Gilanyi, HungaryJerome A. Goldstein, USASiegfried Gottwald, GermanyN. K. Govil, USAR. H. J. Grimshaw, UKHeinz Peter Gumm, GermanyS. M. Gusein-Zade, RussiaSeppo Hassi, FinlandPentti Haukkanen, FinlandJoseph Hilbe, USAHelge Holden, NorwayHenryk Hudzik, PolandPetru Jebelean, RomaniaPalle E. Jorgensen, USAShyam L. Kalla, KuwaitV. R. Khalilov, RussiaH. M. Kim, Republic of KoreaT. Kim, Republic of KoreaEvgeny Korotyaev, GermanyAloys Krieg, GermanyWolfgang Kuhnel, GermanyIrena Lasiecka, USAYuri Latushkin, USABao Qin Li, USASongxiao Li, ChinaNoel G. Lloyd, UKR. Lowen, BelgiumAnil Maheshwari, CanadaRaul F. Manasevich, ChileB. N. Mandal, IndiaEnzo Luigi Mitidieri, ItalyVladimir Mityushev, PolandManfred Moller, South AfricaV. Nistor, USAEnrico Obrecht, ItalyChia-ven Pao, USAWen L. Pearn, TaiwanGelu Popescu, USAMihai Putinar, USAFeng Qi, China

Hernando Quevedo, MexicoJean Michel Rakotoson, FranceRobert H. Redfield, USAB. E. Rhoades, USAPaolo E. Ricci, ItalyFrederic Robert, FranceAlexander Rosa, CanadaAndrew Rosalsky, USAMisha Rudnev, UKStefan Samko, PortugalGideon Schechtman, IsraelNaseer Shahzad, Saudi ArabiaN. Shanmugalingam, USAZhongmin Shen, USAMarianna A. Shubov, USAHarvinder S. Sidhu, AustraliaTheodore E. Simos, GreeceAndrzej Skowron, PolandFrank Sommen, BelgiumLinda R. Sons, USAFrits C. R. Spieksma, BelgiumIlya M. Spitkovsky, USAMarco Squassina, ItalyH. M. Srivastava, CanadaYucai Su, ChinaPeter Takac, GermanyChun-Lei Tang, ChinaMichael M. Tom, USARam U. Verma, USAAndrei I. Volodin, CanadaLuc Vrancken, FranceDorothy I. Wallace, USAFrank Werner, GermanyRichard G. Wilson, MexicoIngo Witt, GermanyPei Yuan Wu, TaiwanSiamak Yassemi, IranA. Zayed, USAKaiming Zhao, CanadaYuxi Zheng, USA

Contents

Fixed-Point Theory, Variational Inequalities, and Its Approximation Algorithms,Giuseppe Marino, Vittorio Colao, Yonghong Yao, Genaro Lopez, and Enrique Llorens-FusterVolume 2011, Article ID 182024, 2 page

A Penalization-Gradient Algorithm for Variational Inequalities,Abdellatif Moudafi and Eman Al-ShemasVolume 2011, Article ID 305856, 12 page

Hybrid Proximal-Point Methods for Zeros of Maximal Monotone Operators, VariationalInequalities and Mixed Equilibrium Problems, Kriengsak Wattanawitoon and Poom KumamVolume 2011, Article ID 174796, 31 page

A New Hybrid Algorithm for Solving a System of Generalized Mixed Equilibrium Problems,Solving a Family of Quasi-φ-Asymptotically Nonexpansive Mappings, and ObtainingCommon Fixed Points in Banach Space, J. F. Tan and S. S. ChangVolume 2011, Article ID 106323, 16 page

A New Hybrid Iterative Scheme for Countable Families of Relatively Quasi-NonexpansiveMappings and System of Equilibrium Problems, Yekini ShehuVolume 2011, Article ID 131890, 23 page

A New Hybrid Algorithm for a Pair of Quasi-φ-Asymptotically Nonexpansive Mappings andGeneralized Mixed Equilibrium Problems in Banach Spaces, Jinhua Zhu and Shih-Sen ChangVolume 2011, Article ID 956852, 22 page

A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces,Pongsakorn Sunthrayuth and Poom KumamVolume 2011, Article ID 560671, 18 page

Strong Convergence Theorems of the General Iterative Methods for NonexpansiveSemigroups in Banach Spaces, Rattanaporn WangkeereeVolume 2011, Article ID 643740, 21 page

Strong and Weak Convergence Theorems for an Infinite Family of LipschitzianPseudocontraction Mappings in Banach Spaces, Shih-sen Chang, Xiong Rui Wang,H. W. Joseph Lee, and Chi Kin ChanVolume 2011, Article ID 409898, 10 page

Strong Convergence Theorems of Modified Ishikawa Iterative Method for an Infinite Familyof Strict Pseudocontractions in Banach Spaces, Phayap Katchang, Wiyada Kumam,Usa Humphries, and Poom KumamVolume 2011, Article ID 549364, 18 page

Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space,Zhanfei ZuoVolume 2011, Article ID 459085, 9 page

A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and VariationalInclusions Approach to Variational Inequalities, Thanyarat Jitpeera and Poom KumamVolume 2011, Article ID 619813, 25 page

On Common Solutions for Fixed-Point Problems of Two Infinite Families of StrictlyPseudocontractive Mappings and the System of Cocoercive Quasivariational InclusionsProblems in Hilbert Spaces, Pattanapong TianchaiVolume 2011, Article ID 691839, 32 page

Approximation of Fixed Points of Weak Bregman Relatively Nonexpansive Mappings inBanach Spaces, Jiawei Chen, Zhongping Wan, Liuyang Yuan, and Yue ZhengVolume 2011, Article ID 420192, 23 page

Comparison between Certain Equivalent Norms Regarding Some Familiar PropertiesImplying WFPP, Helga Fetter and Berta Gamboa de BuenVolume 2011, Article ID 287145, 12 page

A Suzuki Type Fixed-Point Theorem, Ishak Altun and Ali ErduranVolume 2011, Article ID 736063, 9 page

Fixed-Point Theory on a Frechet Topological Vector Space, Afif Ben Amar,Mohamed Amine Cherif, and Maher MnifVolume 2011, Article ID 390720, 9 page

Some Coupled Fixed Point Results on Partial Metric Spaces, Hassen AydiVolume 2011, Article ID 647091, 11 page

Optimal Selling Rule in a Regime Switching Levy Market, Moustapha PemyVolume 2011, Article ID 264603, 28 page

Semiconservative Systems of Integral Equations with Two Kernels,N. B. Yengibaryan and A. G. BarseghyanVolume 2011, Article ID 917951, 11 page

The Bolzano-Poincare Type Theorems, Przemysław Tkacz and Marian TurzanskiVolume 2011, Article ID 793848, 9 page

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2011, Article ID 182024, 2 pagesdoi:10.1155/2011/182024

EditorialFixed-Point Theory, Variational Inequalities, andIts Approximation Algorithms

Giuseppe Marino,1 Vittorio Colao,1 Yonghong Yao,2Genaro Lopez,3 and Enrique Llorens-Fuster3

1 Dipartimento di Matematica, Universita della Calabria, 87036 Arcavacata di Rende (CS), Italy2 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China3 Departemento de Analisis Matematico, Universidad de Sevilla, 41004 Seville, Spain

Correspondence should be addressed to Giuseppe Marino, [email protected]

Received 26 September 2011; Accepted 26 September 2011

Copyright q 2011 Giuseppe Marino et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The study of variational inequalities, fixed points and approximation algorithms constituteda topic of intensive research efforts, especially within the past 30 years. As of today, thisremains one of the most active fields in mathematics, and its ground of application variesfrom game theory, economics, engineering, and natural sciences, among others. On the otherhand, the nature of many practical problems suggests an iterative approach to the solution.

The aim of this special issue had been to present newest and extended coverage ofthe fundamental ideas, concepts, and important results on at least one the following topics:iterative schemes to approximate fixed points of nonexpansive-type mappings, iterativeapproximations of zeros of accretive-type operators, iterative approximations of solutionsof variational inequalities problems, iterative approximations of solutions of equilibriumproblems, and iterative approximations of common fixed points (and/or common zeros) offamilies of these mappings.

In the first and very interesting paper by A. Moudafi and E. Al-Shemas, a forwardbackward algorithm with penalization parameters is introduced and studied for solvingvariational inequalities in Hilbert spaces. K. Wattanawitoon and P. Kumam studied the weakand strong convergence of iterativemethods to a solution of a variational inequality involvingan α-inverse strongly monotone operator, which is also a zero of a maximal monotoneoperator and the mixed-equilibrium point of a bifunction Θ. New hybrid algorithms hadbeen analyzed in the papers by J. F. Tan and S. S. Chang, by Y. Shehu, and by J. Zhu and S. S.Chang for solutions of generalized equilibrium problems and fixed points of nonexpansive-typemappings. As for semigroups of nonexpansive mappings, P. Sunthrayuth and P. Kumamintroduced a composite iterative scheme for finding a common fixed point in the frameworkof Banach spaces which admit a weakly continuous duality mapping, while in the paper by

2 International Journal of Mathematics and Mathematical Sciences

R. Wangkeeree, a weakened viscosity algorithm is studied for these semigroups. For generalfamilies of mappings, iterative procedures had been examinated by S. Chang et al. for familiesof Lipschitz pseudocontractions, by P. Katchang et al., for strict pseudocontractions, and by Z.Zuo for families of multivalued nonexpansive mappings. Viscosity methods are also studiedin the papers by T. Jitpeera and P. Kumam, here used to approximate the unique solution ofa minimum problem, and by P. Tianchai for finding solutions of systems of quasivariationalinclusions.

J. Chen et al. introduced the concept of Bregman relatively nonexpansive mappingsand proved convergence results for the most common iterative schemes. H. Fetter and B. G.de Buen made a comparative study of certain geometric properties of a generic Banach spacewith basis, endowed with its standard norm as well as with an equivalent norm, similar tothe one studied by Lin. Several papers introduce new results about fixed points, variationalinequalities, and solutions of equations. Such results may surely lead to new approachesby means of iterative approximations. In particular and for fixed-point theory, I. Altun andA. Erduran presented a fixed-point theorem for a single-valued map in a complete metricspace using implicit relations. A. B. Amar et al. proved versions of a fixed-point theorem in aFrechet topological vector space. H. Aydi gave some coupled fixed-point results for mappingssatisfying different contractive conditions on complete partial metric spaces. The paper of M.Pemy concerns the optimal selling rule problem and its connection with some variationalinequalities. N. B. Yengibaryan and A. G. Barseghyan dealt with solvability and properties ofsolutions of some homogeneous and nonhomogeneous systems of integral equations. Finally,in the paper by P. Tkacz and M. Turzanski are examined Bolzano-Poincare type theorems.

Acknowledgments

The editors would like to thank the authors for their interesting contributions, as well as theStaff and the Editorial Office of the journal for their valuable support.

Giuseppe MarinoVittorio ColaoYonghong YaoGenaro Lopez

Enrique Llorens-Fuster


Research ArticleA Penalization-Gradient Algorithm forVariational Inequalities

Abdellatif Moudafi1 and Eman Al-Shemas2

1 Departement Scientifique Interfacultaires, Universite des Antilles et de la Guyane, CEREGMIA,97275 Schoelcher, Martinique, France

2 Department of Mathematics, College of Basic Education, PAAET Main Campus-Shamiya, Kuwait

Correspondence should be addressed to Abdellatif Moudafi,[email protected]

Received 11 February 2011; Accepted 5 April 2011

Academic Editor: Giuseppe Marino

Copyright q 2011 A. Moudafi and E. Al-Shemas. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

This paper is concerned with the study of a penalization-gradient algorithm for solving variationalinequalities, namely, find x ∈ C such that 〈Ax, y − x〉 ≥ 0 for all y ∈ C, where A : H → His a single-valued operator, C is a closed convex set of a real Hilbert space H. Given Ψ : H →� ∪ {+∞} which acts as a penalization function with respect to the constraint x ∈ C, and apenalization parameter βk, we consider an algorithmwhich alternates a proximal step with respectto ∂Ψ and a gradient step with respect to A and reads as xk = (I + λkβk∂Ψ)−1(xk−1 − λkAxk−1).Under mild hypotheses, we obtain weak convergence for an inverse strongly monotone operatorand strong convergence for a Lipschitz continuous and strongly monotone operator. Applicationsto hierarchical minimization and fixed-point problems are also given and the multivalued case isreached by replacing themultivalued operator by its Yosida approximatewhich is always Lipschitzcontinuous.

1. Introduction

LetH be a real Hilbert space,A : H → H a monotone operator, and let C be a closed convexset in H , we are interested in the study of a gradient-penalization algorithm for solving theproblem of finding x ∈ C such that

⟨Ax, y − x⟩ ≥ 0 ∀y ∈ C, (1.1)

or equivalently

Ax +NC(x) 0, (1.2)


where NC is the normal cone to a closed convex set C. The above problem is a variationalinequality, initiated by Stampacchia [1], and this field is now a well-known branch of pureand applied mathematics, and many important problems can be cast in this framework.

In [2], Attouch et al., based on seminal work by Passty [3], solve this problem with amultivalued operator by using splitting proximal methods. A drawback is the fact that theconvergence in general is only ergodic. Motivated by [2, 4] and by [5]where penalty methodsfor variational inequalities with single-valued monotone maps are given, we will prove thatour proposed forward-backward penalization-gradient method (1.9) enjoys good asymptoticconvergence properties. We will provide some applications to hierarchical fixed-point andoptimization problems and also propose an idea to reach monotone variational inclusions.

To begin with, see, for instance [6], let us recall that an operator with domain D(T)and range R(T) is said to be monotone if

⟨u − v, x − y⟩ ≥ 0 whenever u ∈ T(x), v ∈ T(y). (1.3)

It is said to be maximal monotone if, in addition, its graph, gphT := {(x, y) ∈ H ×H : y ∈T(x)}, is not properly contained in the graph of any other monotone operator. An operatorsequence Tk is said to be graph convergent to T if (gph(Tk)) converges to gph(T) in theKuratowski-Painleve’s sense, that is, lim supk gph(Tk) ⊂ gph(T) ⊂ lim infk gph(Tk). It is well-known that for each x ∈ H and λ > 0 there is a unique z ∈ H such that x ∈ (I + λT)z.The single-valued operator JTλ := (I + λT)−1 is called the resolvent of T of parameter λ. It is anonexpansive mappingwhich is everywhere defined and is related to its Yosida approximate,namely Tλ(x) := (x − JT

λ(x))/λ, by the relation Tλ(x) ∈ T(JT

λ(x)). The latter is 1/λ-Lipschitz

continuous and satisfies (Tλ)μ = Tλ+μ. Recall that the inverse T−1 of T is the operator definedby x ∈ T−1(y) ⇔ y ∈ T(x) and that, for all x, y ∈ H , we have the following key inequality

∥∥∥JTλ (x) − JTλ

(y)∥∥∥2≤ ∥∥x − y∥∥2 +

∥∥∥(I − JTλ

)(x) −

(I − JTλ

)(y)∥∥∥2. (1.4)

Observe that the relation (Tλ)μ(x) = Tλ+μ(x) leads to

JTλμ (x) =λ

λ + μx +

(1 − λ

λ + μ

)JTλ+μ(x). (1.5)

Now, given a proper lower semicontinuous convex function f : H → � ∪ {+∞}, thesubdifferential of f at x is the set

∂f(x) ={u ∈ H : f

(y) ≥ f(x) + ⟨

u, y − x⟩ ∀y ∈ H}. (1.6)

Its Moreau-Yosida approximate and proximal mapping fλ and proxλf are given, respectively,by

fλ(x) = infy∈H

{f(y)+

12λ

∥∥y − x∥∥2}, proxλf(x) = argmin

y∈H

{f(y)+

12λ

∥∥y − x∥∥2}. (1.7)

International Journal of Mathematics and Mathematical Sciences 3

We have the following interesting relation (∂f)λ = ∇fλ. Finally, given a nonempty closedconvex set C ⊂ H , its indicator function is defined as δC(x) = 0 if x ∈ C and +∞ otherwise.The projection onto C at a point u is PC(u) = infc∈C‖u − c‖. The normal cone to C at x is

NC(x) = {u ∈ H : 〈u, c − x〉 ≤ 0 ∀c ∈ C} (1.8)

if x ∈ C and ∅ otherwise. Observe that ∂δC = NC, proxλf = J∂fλ, and JNC

λ= PC.

Given some xk−1 ∈ H , the current approximation to a solution of (1.2), we study thepenalization-gradient iteration which will generate, for parameters λk > 0, βk → +∞, xk asthe solution of the regularized subproblem

1λk

(xk − xk−1) +Axk−1 + βk∂Ψ(xk) 0, (1.9)

which can be rewritten as

xk =(I + λkβk∂Ψ

)−1(xk−1 − λkAxk−1). (1.10)

Having in view a large range of applications, we shall not assume any particular structure orregularity on the penalization function Ψ. Instead, we just suppose that Ψ is convex, lowersemicontinuous and C = argminΨ/= ∅. We will denote by VI(A,C) the solution set of (1.2).

The following lemmas will be needed in our analysis, see for example [6, 7],respectively.

Lemma 1.1. Let T be a maximal monotone operator, then (βkT) graph converges toNT−1(0) as βk →+∞ provided that T−1(0)/= ∅.

Lemma 1.2. Assume that αk and δk are two sequences of nonnegative real numbers such that

αk+1 ≤ αk + δk. (1.11)

If limk→+∞δk = 0, then there exists a subsequence of (αk) which converges. Furthermore, if∑∞k=0 δk < +∞, then limk→+∞αk exists.

2. Main Results

2.1. Weak Convergence

Theorem 2.1. Assume that VI(A,C)/= ∅, A is inverse strongly monotone, namely

⟨Ax −Ay, x − y⟩ ≥ 1

L

∥∥Ax −Ay∥∥2 ∀x, y ∈ H, for some L > 0. (2.1)


If

∞∑

k=0

∥∥∥x − Jβk∂Ψ

λk(x − λkAx)

∥∥∥ < +∞ ∀x ∈ VI(A,C), (2.2)

and λk ∈]ε, 2/L − ε[ (where ε > 0 is a small enough constant), then the sequence (xk)k∈� generatedby algorithm (1.9) converges weakly to a solution of Problem (1.2).

Proof. Let x be a solution of (1.2), observe that x solves (1.2) if and only if x = (I+λkNC)−1(x−

λkAx) = PC(x − λkAx). Set xk = (I + λkβk∂Ψ)−1(x − λkAx), by the triangular inequality, wecan write

‖xk − x‖ ≤ ‖xk − xk‖ + ‖xk − x‖. (2.3)

On the other hand, by virtue of (1.4) and (2.1), we successively have

‖xk − xk‖2 ≤ ‖xk−1 − x − λk(Axk−1 −Ax)‖2 − ‖xk−1 − xk − λk(Axk−1 −Ax) + xk − x‖2

≤ ‖xk−1 − x‖2 − λk(2L− λk

)‖Axk−1 −Ax‖2

− ‖xk−1 − xk − λk(Axk−1 −Ax) + xk − x‖2.

(2.4)

Hence

‖xk − x‖ <√‖xk−1 − x‖2 − ε2‖Axk−1 −Ax‖2 − ‖xk−1 − xk − λk(Axk−1 −Ax) + xk − x‖2

+ ‖x − xk‖.(2.5)

The later implies, by Lemma 1.2 and the fact that (2.2) insures limk→+∞‖x − xk‖ = 0,that the positive real sequence (‖xk − x‖2)k∈� converges to some limit l(x), that is,

l(x) = limk→+∞

‖xk − x‖2 < +∞, (2.6)

and also assures that

limk→+∞

‖Axk−1 −Ax‖2 = 0,

limk→+∞

‖xk−1 − xk − λk(Axk−1 −Ax) + xk − x‖2 = 0.(2.7)

Combining the two latter equalities, we infer that

limk→+∞

‖xk−1 − xk‖2 = 0. (2.8)


Now, (1.9) can be written equivalently as

xk−1 − xkλk

+Axk −Axk−1 ∈(A + βk∂Ψ

)(xk). (2.9)

By virtue of Lemma 1.1, we have (βk∂Ψ) graph converges toNargminΨ because

(∂Ψ)−1(0) = ∂Ψ∗(0) = argminΨ. (2.10)

Furthermore, the Lipschitz continuity of A (see, e.g., [8]) clearly ensures that the sequence(A + βk∂Ψ)graph converges in turn to A +NargminΨ.

Now, let x∗ be a cluster point of {xk}. Passing to the limit in (2.9), on a subsequencestill denoted by {xk}, and taking into account the fact that the graph of a maximal monotoneoperator is weakly strongly closed inH ×H , we then conclude that

0 ∈ (A +NC)x∗, (2.11)

because A is Lipschitz continuous, (xk) is asymptotically regular thanks to (2.8), and (λk) isbounded away from zero.

It remains to prove that there is no more than one cluster point, our argument isclassical and is presented here for completeness.

Let x be another cluster of {xk}, we will show that x = x∗. This is a consequence of(2.6). Indeed,

l(x∗) = limk→+∞

‖xk − x∗‖2, l(x) = limk→+∞

‖xk − x‖2, (2.12)

from

‖xk − x‖2 = ‖xk − x∗‖2 + ‖x∗ − x‖2 + 2〈xk − x∗, x∗ − x〉, (2.13)

we see that the limit of 〈xk − x∗, x∗ − x〉 as k → +∞ must exists. This limit has to be zerobecause x∗ is a cluster point of {xk}. Hence at the limit, we obtain

l(x) = l(x∗) + ‖x∗ − x‖2. (2.14)

Reversing the role of x and x∗, we also have

l(x∗) = l(x) + ‖x∗ − x‖2. (2.15)

That is x = x∗, which completes the proof.

Remark 2.2. (i) Note that, we can remove condition (2.2), but in this case we obtain that thereexists a subsequence of (xk) such that every weak cluster point is a solution of problem(1.2). This follows by Lemma 1.2 combined with the fact that x = J∂δCλ∗ (x − λ∗Ax) and that


(βk∂Ψ) graph converges to ∂δC. The later is equivalent, see for example [6], to the pointwiseconvergence of Jβk∂Ψλk

to J∂δCλ∗ and therefore ensures that

limk→+∞

∥∥∥x − Jβk∂Ψλk

(x − λkAx)∥∥∥ = 0. (2.16)

(ii) In the special case Ψ(x) = (1/2)dist(x, C)2, (2.2) reduces to∑∞

k=0 1/βk < +∞, seeApplication (2) of Section 3.

Suppose now that Ψ(x) = dist(x, C), it well-known that proxγΨ(x) = PC(x) ifdist(x, C) ≤ γ . Consequently,

Jβk∂Ψλk

(x) = PC(x) if dist (x, C) ≤ λkβk, (2.17)

which is the case for all k ≥ κ for some κ ∈ � because (λk) is bounded and limk→+∞βk = +∞.Hence limk→+∞‖x − Jβk∂Ψλk

(x − λkAx)‖ = 0, for all k ≥ κ, and thus (2.2) is clearly satisfied.The particular case Ψ = 0 corresponds to the unconstrained case, namely, C = H . In

this context the resolvent associated to βk∂Ψ is the identity, and condition (2.2) is triviallysatisfied.

2.2. Strong Convergence

Now, we would like to stress that we can guarantee strong convergence by reinforcingassumptions on A.

Proposition 2.3. Assume that A is strong monotone with constant α > 0, that is,

⟨Ax −Ay, x − y⟩ ≥ α∥∥x − y∥∥2 ∀x, y ∈ H, for some α > 0, (2.18)

and Lipschitz continuous with constant L > 0, that is,

∥∥Ax −Ay∥∥ ≤ L∥∥x − y∥∥ ∀x, y ∈ H, for some L > 0. (2.19)

If λk ∈ ]ε, 2α/L2 − ε[ (where ε > 0 is a small enough constant) and limk→+∞λk = λ∗ > 0, then thesequence generated by (1.9) strongly converges to the unique solution of (1.2).

Proof. Indeed, by replacing inverse strong monotonicity of A by strong monotonicity andLipschitz continuity, it is easy to see from the first part of the proof of Theorem 2.1 that theoperator of I − λkA satisfies

∥∥(I − λkA)(x) − (I − λkA)

(y)∥∥2 ≤

(1 − 2λkα + λ2kL

2)∥∥x − y∥∥2

. (2.20)


Following the arguments in the proof of Theorem 2.1 to obtain

‖xk − x‖ ≤√1 − 2λkα + λ2kL

2‖xk−1 − x‖ + δk(x) with δk(x) :=∥∥∥x − Jβk∂Ψλk

(x − λkAx)∥∥∥.

(2.21)

Now, by setting Θ(λ) =√1 − 2λα + λ2L2, we can check that 0 < Θ(λ) < 1 if and only

if λk ∈ ]0, 2α/L2[, and a simple computation shows that 0 < Θ(λk) ≤ Θ∗ < 1 withΘ∗ = max{Θ(ε),Θ(2α/L2 − ε)}. Hence,

‖xk − x‖ ≤ (Θ∗)k‖x0 − x‖ +k−1∑

j=0

(Θ∗)jδk−j(x). (2.22)

The result follows fromOrtega and Rheinboldt [9, page 338] and the fact that limk→+∞δk(x) =0. The later follows thanks to the equivalence between graph convergence of the sequence ofoperators (βk∂Ψ) to ∂δC and the pointwise convergence of their resolvent operators combinedwith the fact that limk→+∞λk = λ∗.

3. Applications

(1) Hierarchical Convex Minimization Problems

Having in mind the connection between monotone operators and convex functions, we mayconsider the special case A = ∇Φ, Φ being a proper lower semicontinuous differentiableconvex function. Differentiability of Φ ensures that ∇Φ+NargminΨ = ∂(Φ + δargminΨ) and (1.2)reads as

minx∈argminΨ

Φ(x). (3.1)

Using definition of the Moreau-Yosida approximate, algorithm (1.9) reads as

xk = argminy∈H

{f(y)+

12λk

∥∥y − (I − λkA)xk−1∥∥2}. (3.2)

In this case, it is well-known that the assumption (2.1) of inverse strong monotonicity of∇Φ is equivalent to its L-Lipschitz continuity. If further we assume

∑∞k=1 δk(x) < +∞ for

all x ∈ VI(∇Φ, C) and λk ∈ ]ε, 2/L − ε[, then by Theorem 2.1 we obtain weak convergence


of algorithm (3.2) to a solution of (3.1). The strong convergence is obtained, thanks toProposition 2.3, if in additionΨ is strongly convex (i.e., there is α > 0;

(1 − μ)Ψ(x1) + μΨ(x2) ≥ Ψ

((1 − μ)x1 + μx2

)+α

2μ(1 − μ)‖x1 − x2‖2 (3.3)

for all μ ∈ [0, 1], all x1, x2 ∈ H) and (λk) a convergent sequence with λk ∈ ]ε, 2α/L2 − ε[. Notethat strong convexity of Ψ is equivalent to α-strong monotonicity of its gradient. A concreteexample in signal recovery is the Projected Land weber problem, namely,

minx∈C

Φ(x) :=12‖Lx − z‖2, (3.4)

L being a linear-bounded operator. SetA(x) := ∇Φ(x) = L∗(Lx − z). Consequently,

∀x, y ∈H ∥∥A(x) −A(

y)∥∥ =

∥∥L∗L

(x − y)∥∥ ≤ ‖L‖2∥∥x − y∥∥, (3.5)

and A is therefore Lipschitz continuous with constant ‖L‖2. Now, it is well-known that theproblem possesses exactly one solution if L is bounded below, that is,

∃κ > 0 ∀x ∈H ‖L(x)‖ ≥ κ‖x‖. (3.6)

In this case, A is strongly monotone. Indeed, it is easily seen that f is strongly convex:consider x, y ∈ H and μ ∈ ]0, 1[, one has

∥∥μ(Lx − z) + (1 − μ)(Ly − z)∥∥2

2≤ μ‖Lx − z‖2

2+

(1 − μ)∥∥Ly − z∥∥2

2− κ2μ

(1 − μ)∥∥x − y∥∥2

2.

(3.7)

(2) Classical Penalization

In the special case where Ψ(x) = (1/2)dist(x, C)2, we have

∂Ψ(x) = x − ProjC(x), (3.8)

which is nothing but the classical penalization operator, see [10]. In this context, taking intoaccount the fact that

((∂f

)λ

)μ= ∇fλ+μ, J

∂f

λ = I − λ(∂f)λ = I − λ∇fλ, (δC)λ =1λΨ, (3.9)


and that x solves (1.2), and thus x = PC(x − λkAx), we successively have

‖xk − x‖ =∥∥∥J

βk∂Ψλk

(x − λkAx) − JNC

λk(x − λkAx)

∥∥∥

= λk∥∥∥(βk∂Ψ

)λk(x − λkAx) − (NC)λk(x − λkAx)

∥∥∥

= λk∥∥∥βk(∂Ψ)λkβk (x − λkAx) − ∇(δC)λk(x − λkAx)

∥∥∥

= λk∥∥∥βk(∂(δC)1)λkβk (x − λkAx) − ∇(δC)λk(x − λkAx)

∥∥∥

= λk∥∥∥βk∇(δC)1+λkβk (x − λkAx) − ∇(δC)λk (x − λkAx)

∥∥∥

= λk(

1λk

− βk1 + λkβk

)‖(x − λkAx) − PC(x − λkAx)‖

=1

1 + λkβk‖λkAx‖ ≤ 1

βk‖Ax‖.

(3.10)

So condition on the parameters reduces to∑∞

k=1 1/βk < +∞, and algorithm (1.9) is nothingbut a relaxed projection-gradient method. Indeed, using (1.5) and the fact that JNC

λ = PC, weobtain

xk =(

11 + λkβk

I +λkβk

1 + λkβkPC

)(I − λkA)xk−1. (3.11)

An inspection of the proof of Theorem 2.1 shows that the weak converges is assured withλk ∈ ]ε, 2/L − ε[.

(3) A Hierarchical Fixed-Point Problem

Having in mind the connection between inverse strongly monotone operators andnonexpansive mappings, we may consider the following fixed-point problem:

(I − P)x +NC(x) 0, (3.12)

with P a nonexpansive mapping, namely, ‖Px − Py‖ ≤ ‖x − y‖.It is well-known that A = I − P is inverse strongly monotone with L = 2. Indeed, by

definition of P , we have

∥∥(I −A)x − (I −A)y

∥∥ ≤ ∥

∥x − y∥∥. (3.13)

On the other hand

∥∥(I −A)x − (I −A)y

∥∥2 =

∥∥x − y∥∥2 +

∥∥Ax −Ay∥∥2 − 2

⟨x − y,Ax −Ay⟩. (3.14)


Combining the two last inequalities, we obtain

⟨x − y,Ax −Ay⟩ ≥ 1

2∥∥Ax −Ay∥∥2

. (3.15)

Therefore, by Theorem 2.1 we get the weak convergence of the sequence (xk) generated bythe following algorithm:

xk = proxβkΨ((I − λk)xk−1 + λkPxk−1) (3.16)

to a solution of (3.12) provided that∑∞

k=1 δk(x) < +∞ for all x ∈ VI(I −P, C) and λk ∈]ε, 1−ε[.The strong convergence of (1.9) is obtained, by applying Proposition 2.3, for P a contractionmapping, namely, ‖Px−Py‖ ≤ γ‖x−y‖ for 0 < γ < 1 which is equivalent to the (1− γ)-strongmonotonicity of (I − P), and (λk) is a convergent sequence with λk ∈ ]ε, 2(1− γ)/(1 + γ)2 − ε[.It is easily seen that in this case I − P is (1 + γ)-Lipschitz continuous.

4. Towards the Multivalued Case

Now, we are interested in (1.2) when A : H → 2H is a multi-valued maximal monotoneoperator.With the help of the Yosida approximatewhich is always inverse stronglymonotone(and thus single-valued), we consider the following partial regularized version of (1.2):

Aγx∗γ +NC

(x∗γ

) 0, (4.1)

where Aγ stands for the Yosida approximate of A.It is well-known thatAγ is inverse strongly monotone. More precisely, we have

⟨Aγx −Aγy, x − y⟩ ≥ γ∥∥Aγx −Aγy

∥∥2. (4.2)

Using definition of the Yosida approximate, algorithm (1.9) applied to (4.1) reads as

xγ

k =(I + λkβk∂Ψ

)−1((

1 − λkγ

)xγ

k−1 +λkγJAγ

(xγ

k−1))

. (4.3)

From Theorem 2.1, we infer that xγkconverges weakly to a solution xγ provided that λk ∈

]ε, 2γ − ε[. Furthermore, it is worth mentioning that if A is strongly monotone, Aγ is alsostrongly monotone, and thus (4.1) has a unique solution xγ . By a result in [8, page 35], wehave the following estimate:

∥∥x − xγ∥∥ ≤ o(√γ). (4.4)

Consequently, (4.3) provides approximate solutions to the variational inclusion (1.2) for smallvalues of γ . Furthermore, when A = ∇Φ, we have

(∂Φ)γ(x) +NC(x) = ∇Φγ(x) +NC(x) = ∂(Φγ + δC

)(x), (4.5)


and thus (4.1) reduces to

minx∈C

Φγ(x). (4.6)

If (3.1) and (4.1) are solvable, by ([11] Theorem 3.3), we have for all γ > 0

0 ≤ minx∈C

Φ(x) −minx∈C

Φγ(x) ≤ γ∥∥y

∥∥2, (4.7)

where y = ∇Φ(y)(∈ −NC(x)) with x a solution of (3.1). The value of (3.1) is thus close tothose of (4.1) for small values of γ , and hence, this confirmed the pertinence of the proposedapproximation idea to reach the multi-valued case. Observe that in this context, algorithm(4.3) reads as

xγ

k = proxβkΨ((

1 − λkγ

)xγ

k−1 +λkγproxγΦ

(xγ

k−1))

. (4.8)

5. Conclusion

The authors have introduced a forward-backward penalization-gradient algorithm forsolving variational inequalities and studied their asymptotic convergence properties. Wehave provided some applications to hierarchical fixed-point and optimization problems andalso proposed an idea to reach monotone variational inclusions.

Acknowledgment

We gratefully acknowledge the constructive comments of the anonymous referees whichhelped them to improve the first version of this paper.

References

[1] G. Stampacchia, “Formes bilineaires coercitives sur les ensembles convexes,” Comptes Rendus del’Academie des Sciences de Paris, vol. 258, pp. 4413–4416, 1964.

[2] H. Attouch, M. O. Czarnecki, and J. Peypouquet, “Prox-penalization and splitting methods forconstrained variational problems,” SIAM Journal on Optimization, vol. 21, no. 1, pp. 149–173, 2011.

[3] G. B. Passty, “Ergodic convergence to a zero of the sum of monotone operators in Hilbert space,”Journal of Mathematical Analysis and Applications, vol. 72, no. 2, pp. 383–390, 1979.

[4] B. Lemaire, “Coupling optimization methods and variational convergence,” in Trends in MathematicalOptimization, vol. 84 of International Series of Numerical Mathematics, pp. 163–179, Birkhauser, Basel,Switzerland, 1988.

[5] J. Gwinner, “On the penalty method for constrained variational inequalities,” in Optimization: Theoryand Algorithms, vol. 86 of Lecture Notes in Pure and Applied Mathematics, pp. 197–211, Dekker, NewYork, NY, USA, 1983.

[6] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317, Springer, Berlin, Germany, 1998.[7] P. L. Martinet, Algorithmes pour la resolution de problemes d’optimisation et de minimax, thesis, Universite

de Grenoble, 1972.[8] H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,

North-Holland Mathematics Studies, no. 5, North-Holland, Amsterdam, The Netherlands, 1973.


[9] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables,Academic Press, New York, NY, USA, 1970.

[10] D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Martinus Nijhoff, The Hague, TheNetherlands, 1978.

[11] N. Lehdili, Methodes proximales de selection et de decomposition pour les inclusions monotones, thesis,Universite de Montpellier, 1996.


Research ArticleHybrid Proximal-Point Methods forZeros of Maximal Monotone Operators, VariationalInequalities and Mixed Equilibrium Problems

Kriengsak Wattanawitoon1, 2 and Poom Kumam2, 3

1 Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology,Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

2 Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand3 Department of Mathematics, Faculty of Science, King Mongkut’s University ofTechnology Thonburi (KMUTT), Bang Mod, Bangkok 10140, Thailand

Correspondence should be addressed to Poom Kumam, [email protected]

Received 15 November 2010; Accepted 29 December 2010

Academic Editor: Yonghong Yao

Copyright q 2011 K. Wattanawitoon and P. Kumam. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We prove strong and weak convergence theorems of modified hybrid proximal-point algorithmsfor finding a common element of the zero point of a maximal monotone operator, the set ofsolutions of equilibrium problems, and the set of solution of the variational inequality operators ofan inverse stronglymonotone in a Banach space under different conditions. Moreover, applicationsto complementarity problems are given. Our results modify and improve the recently announcedones by Li and Song (2008) and many authors.

1. Introduction

Let E be a Banach space with norm ‖·‖,C a nonempty closed convex subset of E, let E∗ denotethe dual of E and < ·, · > is the pairing between E and E∗.

Consider the problem of finding

v ∈ E such that 0 ∈ T(v), (1.1)

where T is an operator from E into E∗. Such v ∈ E is called a zero point of T . When T is amaximal monotone operator, a well-known method for solving (1.1) in a Hilbert space H isthe proximal point algorithm x1 = x ∈ H and

xn+1 = Jrnxn, n = 1, 2, 3, . . . , (1.2)


where {rn} ⊂ (0,∞) and Jrn = (I + rnT)−1, then Rockafellar [1] proved that the sequence {xn}

converges weakly to an element of T−1(0).In 2000, Kamimura and Takahashi [2] proved the following strong convergence

theorem in Hilbert spaces, by the following algorithm:

xn+1 = αnx + (1 − αn)Jrnxn, n = 1, 2, 3, . . . , (1.3)

where Jr = (I + rT)−1J , then the sequence {xn} converges strongly to PT−10(x), where PT−10

is the projection from H onto T−1(0). These results were extended to more general Banachspaces see [3, 4].

In 2004, Kohsaka and Takahashi [4] introduced the following iterative sequence for amaximal monotone operator T in a smooth and uniformly convex Banach space: x1 = x ∈ Eand

xn+1 = J−1(αnJx + (1 − αn)J(Jrnxn)), n = 1, 2, 3, . . . , (1.4)

where J is the duality mapping from E into E∗ and Jr = (I + rT)−1J .Recently, Li and Song [5] proved a strong convergence theorem in a Banach space, by

the following algorithm: x1 = x ∈ E and

yn = J−1(βnJ(xn) +

(1 − βn

)J(Jrnxn)

),

xn+1 = J−1(αnJx1 + (1 − αn)J

(yn)),

(1.5)

with the coefficient sequences {αn}, {βn} ⊂ [0, 1] and {rn} ⊂ (0,∞) satisfying limn→∞αn = 0,∑∞n=1 αn = ∞, limn→∞βn = 0, and limn→∞rn = ∞. Where J is the duality mapping from E into

E∗ and Jr = (I + rT)−1J . Then, they proved that the sequence {xn} converges strongly toΠCx,where ΠC is the generalized projection from E onto C.

Let C be a nonempty closed convex subset of E, and let A be a monotone operator of Cinto E∗. The variational inequality problem is to find a point x∗ ∈ C such that

〈v − x∗, Ax∗〉 ≥ 0, ∀v ∈ C. (1.6)

The set of solutions of the variational inequality problem is denoted by VI(C,A). Such aproblem is connected with the convex minimization problem, the complementarity problem,the problem of finding a point u ∈ E satisfying 0 = Au, and so on. An operator A of C into E∗

is said to be inverse-strongly monotone if there exists a positive real number α such that

⟨x − y,Ax −Ay⟩ ≥ α∥∥Ax −Ay∥∥2, (1.7)

for all x, y ∈ C. In such a case, A is said to be α-inverse-strongly monotone. If an operator A ofC into E∗ is α-inverse-strongly monotone, then A is Lipschitz continuous, that is, ‖Ax −Ay‖ ≤(1/α)‖x − y‖ for all x, y ∈ C.


In a Hilbert space H, Iiduka et al. [6] proved that the sequence {xn} defined by: x1 =x ∈ C and

xn+1 = PC(xn − λnAxn), (1.8)

where PC is the metric projection ofH ontoC and {λn} is a sequence of positive real numbers,converges weakly to some element of VI(C,A).

In 2008, Iiduka and Takahashi [7] introduced the folowing iterative scheme for findinga solution of the variational inequality problem for an inverse-strongly monotone operatorAin a Banach space x1 = x ∈ C and

xn+1 = ΠCJ−1(Jxn − λnAxn), (1.9)

for every n = 1, 2, 3, . . ., where ΠC is the generalized metric projection from E onto C, J is theduality mapping from E into E∗ and {λn} is a sequence of positive real numbers. They provedthat the sequence {xn} generated by (1.9) converges weakly to some element of VI(C,A).

Let Θ be a bifunction of C ×C into R and ϕ : C → R a real-valued function. The mixedequilibrium problem, denoted by MEP(Θ, ϕ), is to find x ∈ C such that

Θ(x, y)+ ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.10)

If ϕ ≡ 0, the problem (1.10) reduces into the equilibrium problem for Θ, denoted by EP(Θ), is tofind x ∈ C such that

Θ(x, y) ≥ 0, ∀y ∈ C. (1.11)

If Θ ≡ 0, the problem (1.10) reduces into the minimize problem, denoted by Argmin(ϕ), is tofind x ∈ C such that

ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.12)

The above formulation (1.11)was shown in [8] to covermonotone inclusion problems, saddlepoint problems, variational inequality problems, minimization problems, optimizationproblems, variational inequality problems, vector equilibrium problems, and Nash equilibriain noncooperative games. In addition, there are several other problems, for example, thecomplementarity problem, fixed point problem, and optimization problem, which can alsobe written in the form of an EP(Θ). In other words, the EP(Θ) is an unifying model forseveral problems arising in physics, engineering, science, optimization, economics, and soforth. In the last two decades, many papers have appeared in the literature on the existenceof solutions of EP(Θ); see, for example, [8–11] and references therein. Some solution methodshave been proposed to solve the EP(Θ); see, for example, [9, 11–21] and references therein.In 2005, Combettes and Hirstoaga [12] introduced an iterative scheme of finding the bestapproximation to the initial data when EP(Θ) is nonempty and they also proved a strongconvergence theorem.


Recall, a mapping S : C → C is said to be nonexpansive if

∥∥Sx − Sy∥∥ ≤ ∥∥x − y∥∥, (1.13)

for all x, y ∈ C. We denote by F(S) the set of fixed points of S. If C is bounded closedconvex and S is a nonexpansive mapping of C into itself, then F(S) is nonempty (see [22]).A mapping S is said to be quasi-nonexpansive if F(S)/= ∅ and ‖Sx − y‖ ≤ ‖x − y‖ for allx ∈ C and y ∈ F(S). It is easy to see that if S is nonexpansive with F(S)/= ∅, then it is quasi-nonexpansive. We write xn → x(xn ⇀ x, resp.) if {xn} converges (weakly, resp.) to x. Let Ebe a real Banach space with norm ‖ · ‖ and let J be the normalized duality mapping from E into2E

∗given by

Jx = {x∗ ∈ E∗ : 〈x, x∗〉 = ‖x‖‖x∗‖, ‖x‖ = ‖x∗‖}, (1.14)

for all x ∈ E, where E∗ denotes the dual space of E and 〈·, ·〉 the generalized duality pairingbetween E and E∗. It is well known that if E∗ is uniformly convex, then J is uniformlycontinuous on bounded subsets of E.

Let C be a closed convex subset of E, and let S be a mapping from C into itself. A pointp in C is said to be an asymptotic fixed point of S [23] if C contains a sequence {xn} whichconverges weakly to p such that limn→∞‖xn − Sxn‖ = 0. The set of asymptotic fixed points ofSwill be denoted by F(S). A mapping S from C into itself is said to be relatively nonexpansive[24–26] if F(S) = F(S) and φ(p, Sx) ≤ φ(p, x) for all x ∈ C and p ∈ F(S). The asymptoticbehavior of a relatively nonexpansive mapping was studied in [27, 28]. S is said to be φ-nonexpansive, if φ(Sx, Sy) ≤ φ(x, y) for x, y ∈ C. S is said to be relatively quasi-nonexpansiveif F(S)/= ∅ and φ(p, Sx) ≤ φ(p, x) for x ∈ C and p ∈ F(S).

In 2009, Takahashi and Zembayashi [29] introduced the following shrinking projectionmethod of closed relatively nonexpansive mappings as follows:

x0 = x ∈ C, C0 = C,

yn = J−1(αnJ(xn) + (1 − αn)JS(xn)),

un ∈ C such that Θ(un, y

)+

1rn

⟨y − un, Jun − Jyn

⟩ ≥ 0, ∀y ∈ C,

Cn+1 ={z ∈ Cn : φ(z, un) ≤ φ(z, xn)

},

xn+1 = ΠCn+1x,

(1.15)

for every n ∈ N ∪ {0}, where J is the duality mapping on E, {αn} ⊂ [0, 1] satisfieslim infn→∞αn(1 − αn) > 0 and {rn} ⊂ [a,∞) for some a > 0. Then, they proved that thesequence {xn} converges strongly to ΠF(S)∩EP(Θ)x.


In 2009, Qin et al. [30]modified the Halpern-type iteration algorithm for closed quasi-φ-nonexpansive mappings (or relatively quasi-nonexpansive) defined by

x0 ∈ E chosen arbitrarily,

C1 = C,

x1 = ΠC1x0,

yn = J−1(αnJ(x1) + (1 − αn)JT(xn)),Cn+1 =

{z ∈ Cn : φ

(z, yn

) ≤ αnφ(z, x1) + (1 − αn)φ(z, xn)},

xn+1 = ΠCn+1x1, ∀n ≥ 1.

(1.16)

Then, they proved that under appropriate control conditions the sequence {xn} convergesstrongly to ΠF(T)x1.

Recently, Ceng et al. [31] proved the following strong convergence theorem for findinga common element of the set of solutions for an equilibrium and the set of a zero point for amaximal monotone operator T in a Banach space E

yn = J−1(αnJ(x0) + (1 − αn)

(βnJxn +

(1 − βn

)JJrn(xn)

)),

Hn ={z ∈ C : φ

(z, Trnyn

) ≤ αnφ(z, x0) + (1 − αn)φ(z, xn)},

Wn = {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0},xn+1 = ΠHn∩Wnx0.

(1.17)

Then, the sequence {xn} converges strongly to ΠT−10∩EP(Θ)x0, where ΠT−10∩EP(Θ) is thegeneralized projection of E onto T−10 ∩ EP(Θ).

In this paper, motivated and inspired by Li and Song [5], Iiduka and Takahashi [7],Takahashi and Zembayashi [29], Ceng et al. [31] and Qin et al. [30], we introduce thefollowing new hybrid proximal-point algorithms defined by x1 = x ∈ C:

wn = ΠCJ−1(Jxn − λnAxn),

zn = J−1(βnJ(xn) +

(1 − βn

)J(Jrnwn)

),

yn = J−1(αnJ(x1) + (1 − αn)J(zn)),


)+ ϕ(y) − ϕ(un) + 1

rn


⟩ ≥ 0, ∀y ∈ C,

Cn+1 ={z ∈ Cn : φ(z, un) ≤ αnφ(z, x1) + (1 − αn)φ(z, xn)

},

xn+1 = ΠCn+1x

(1.18)


and


)+ ϕ(y) − ϕ(un) + 1

rn


⟩ ≥ 0, ∀y ∈ C,

zn = ΠCJ−1(Jun − λnAun),


(1 − βn

)J(Jrnzn)

),

xn+1 = ΠCJ−1(αnJ(x1) + (1 − αn)J

(yn)).

(1.19)

Under appropriate conditions, we will prove that the sequence {xn} generated by algorithms(1.18) and (1.19) converges strongly to the point ΠVI(C,A)∩T−1(0)∩MEP(Θ,ϕ)x and convergesweakly to the point limn→∞ΠVI(C,A)∩T−1(0)∩MEP(Θ,ϕ)xn, respectively. The results presented inthis paper extend and improve the corresponding ones announced by Li and Song [5] andmany authors in the literature.

2. Preliminaries

A Banach space E is said to be strictly convex if ‖(x + y)/2‖ < 1 for all x, y ∈ E with ‖x‖ =‖y‖ = 1 and x /=y. Let U = {x ∈ E : ‖x‖ = 1} be the unit sphere of E. Then, the Banach spaceE is said to be smooth provided

limt→ 0

∥∥x + ty∥∥ − ‖x‖t

(2.1)

exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformlyfor x, y ∈ E. The modulus of convexity of E is the function δ : [0, 2] → [0, 1] defined by

δ(ε) = inf{1 −∥∥∥∥x + y2

∥∥∥∥ : x, y ∈ E, ‖x‖ =∥∥y∥∥ = 1,

∥∥x − y∥∥ ≥ ε}. (2.2)

A Banach space E is uniformly convex if and only if δ(ε) > 0 for all ε ∈ (0, 2]. Let p be a fixedreal number with p ≥ 2. A Banach space E is said to be p-uniformly convex if there exists aconstant c > 0 such that δ(ε) ≥ cεp for all ε ∈ [0, 2]; see [32, 33] for more details. Observethat every p-uniform convex is uniformly convex. One should note that no Banach space isp-uniform convex for 1 < p < 2. It is well known that a Hilbert space is 2-uniformly convexand uniformly smooth. For each p > 1, the generalized duality mapping Jp : E → 2E

∗is defined

by

Jp(x) ={x∗ ∈ E∗ : 〈x, x∗〉 = ‖x‖p, ‖x∗‖ = ‖x‖p−1

}, (2.3)

for all x ∈ E. In particular, J = J2 is called the normalized duality mapping. If E is a Hilbert space,then J = I, where I is the identity mapping. It is also known that if E is uniformly smooth,then J is uniformly norm-to-norm continuous on each bounded subset of E.


We know the following (see [34]):

(1) if E is smooth, then J is single-valued,

(2) if E is strictly convex, then J is one-to-one and 〈x − y, x∗ − y∗〉 > 0 holds for all(x, x∗), (y, y∗) ∈ J with x /=y,

(3) if E is reflexive, then J is surjective,

(4) if E is uniformly convex, then it is reflexive,

(5) if E∗ is uniformly convex, then J is uniformly norm-to-norm continuous on eachbounded subset of E.

The duality J from a smooth Banach space E into E∗ is said to be weakly sequentiallycontinuous [35] if xn ⇀ x implies Jxn ⇀∗ Jx, where⇀∗ implies the weak∗ convergence.

Lemma 2.1 (see [36, 37]). If E be a 2-uniformly convex Banach space. Then, for all x, y ∈ E one has

∥∥x − y∥∥ ≤ 2c2∥∥Jx − Jy∥∥, (2.4)

where J is the normalized duality mapping of E and 0 < c ≤ 1.

The best constant 1/c in Lemma is called the 2-uniformly convex constant of E; see[32].

Lemma 2.2 (see [36, 38]). If E a p-uniformly convex Banach space and let p be a given real numberwith p ≥ 2. Then, for all x, y ∈ E,Jx ∈ Jp(x) and Jy ∈ Jp(y)

⟨x − y, Jx − Jy⟩ ≥ cp

2p−2p

∥∥x − y∥∥p, (2.5)

where Jp is the generalized duality mapping of E and 1/c is the p-uniformly convexity constant of E.

Lemma 2.3 (see Xu [37]). Let E be a uniformly convex Banach space. Then, for each r > 0, thereexists a strictly increasing, continuous, and convex functionK : [0,∞) → [0,∞) such thatK(0) = 0and

∥∥λx +(1 − λy)∥∥2 ≤ λ‖x‖2 + (1 − λ)∥∥y∥∥2 − λ(1 − λ)K(∥∥x − y∥∥), (2.6)

for all x, y ∈ {z ∈ E : ‖z‖ ≤ r} and λ ∈ [0, 1].

Let E be a smooth, strictly convex, and reflexive Banach space and letC be a nonemptyclosed convex subset of E. Throughout this paper, we denote by φ the function defined by

φ(x, y)= ‖x‖2 − 2

⟨x, Jy

⟩+∥∥y∥∥2, for x, y ∈ E. (2.7)


Following Alber [39], the generalized projection ΠC : E → C is a map that assigns to anarbitrary point x ∈ E the minimum point of the functional φ(x, y), that is, ΠCx = x, where xis the solution to the minimization problem

φ(x, x) = infy∈C

φ(y, x)

(2.8)

existence and uniqueness of the operator ΠC follows from the properties of the functionalφ(x, y) and strict monotonicity of the mapping J . It is obvious from the definition of functionφ that (see [39])

(∥∥y∥∥ − ‖x‖)2 ≤ φ(y, x) ≤ (∥∥y∥∥ + ‖x‖)2, ∀x, y ∈ E. (2.9)

If E is a Hilbert space, then φ(x, y) = ‖x − y‖2.If E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E, φ(x, y) =

0 if and only if x = y. It is sufficient to show that if φ(x, y) = 0, then x = y. From (2.9), wehave ‖x‖ = ‖y‖. This implies that 〈x, Jy〉 = ‖x‖2 = ‖Jy‖2. From the definition of J , one hasJx = Jy. Therefore, we have x = y; see [34, 40] for more details.

Lemma 2.4 (see Kamimura and Takahashi [3]). Let E be a uniformly convex and smooth realBanach space and let {xn}, {yn} be two sequences of E. If φ(xn, yn) → 0 and either {xn} or {yn} isbounded, then ‖xn − yn‖ → 0.

Lemma 2.5 (see Alber [39]). Let C be a nonempty, closed, convex subset of a smooth Banach spaceE and x ∈ E. Then, x0 = ΠCx if and only if

⟨x0 − y, Jx − Jx0

⟩ ≥ 0, ∀y ∈ C. (2.10)

Lemma 2.6 (see Alber [39]). Let E be a reflexive, strictly convex, and smooth Banach space, let Cbe a nonempty closed convex subset of E and let x ∈ E. Then,

φ(y,ΠCx

)+ φ(ΠCx, x) ≤ φ

(y, x), ∀y ∈ C. (2.11)

Let E be a strictly convex, smooth, and reflexive Banach space, let J be the dualitymapping from E into E∗. Then, J−1 is also single-valued, one-to-one, and surjective, and it isthe duality mapping from E∗ into E. Define a function V : E × E∗ → R as follows (see [4]):

V (x, x∗) = ‖x‖2 − 2〈x, x∗〉 + ‖x∗‖2, (2.12)

for all x ∈ E, x ∈ E and x∗ ∈ E∗. Then, it is obvious that V (x, x∗) = φ(x, J−1(x∗)) andV (x, J(y)) = φ(x, y).


Lemma 2.7 (see Kohsaka and Takahashi [4, Lemma 3.2]). Let E be a strictly convex, smooth, andreflexive Banach space, and let V be as in (2.12). Then,

V (x, x∗) + 2⟨J−1(x∗) − x, y∗

⟩≤ V (x, x∗ + y∗), (2.13)

for all x ∈ E and x∗, y∗ ∈ E∗.

Let E be a reflexive, strictly convex, and smooth Banach space. Let C be a closedconvex subset of E. Because φ(x, y) is strictly convex and coercive in the first variable,we know that the minimization problem infy∈Cφ(x, y) has a unique solution. The operatorΠCx := argminy∈Cφ(x, y) is said to be the generalized projection of x on C.

A set-valued mapping T : E → E∗ with domain D(T) = {x ∈ E : T(x)/= ∅} and rangeR(T) = {x∗ ∈ E∗ : x∗ ∈ T(x), x ∈ D(T)} is said to be monotone if 〈x − y, x∗ − y∗〉 ≥ 0 for allx∗ ∈ T(x), y∗ ∈ T(y). We denote the set {s ∈ E : 0 ∈ Tx} by T−10. T is maximal monotone ifits graph G(T) is not properly contained in the graph of any other monotone operator. If T ismaximal monotone, then the solution set T−10 is closed and convex.

Let E be a reflexive, strictly convex, and smooth Banach space, it is known that T is amaximal monotone if and only if R(J + rT) = E∗ for all r > 0.

Define the resolvent of T by Jrx = xr . In other words, Jr = (J + rT)−1J for all r > 0. Jr isa single-valued mapping from E toD(T). Also, T−1(0) = F(Jr) for all r > 0, where F(Jr) is theset of all fixed points of Jr . Define, for r > 0, the Yosida approximation of T by Ar = (J − JJr)/r.We know that Arx ∈ T(Jrx) for all r > 0 and x ∈ E.

Lemma 2.8 (see Kohsaka and Takahashi [4, Lemma 3.1]). Let E be a smooth, strictly convex,and reflexive Banach space, T ⊂ E × E∗ a maximal monotone operator with T−10/= ∅, r > 0 andJr = (J + rT)−1J . Then,

φ(x, Jry

)+ φ(Jry, y

) ≤ φ(x, y), (2.14)

for all x ∈ T−10 and y ∈ E.

Let A be an inverse-strongly monotone mapping of C into E∗ which is said to behemicontinuous if for all x, y ∈ C, the mapping F of [0, 1] into E∗, defined by F(t) =A(tx + (1 − t)y), is continuous with respect to the weak∗ topology of E∗. We define byNC(v)the normal cone for C at a point v ∈ C, that is,

NC(v) ={x∗ ∈ E∗ :

⟨v − y, x∗⟩ ≥ 0, ∀y ∈ C}. (2.15)

Theorem 2.9 (see Rockafellar [1]). Let C be a nonempty, closed, convex subset of a Banach spaceE and A a monotone, hemicontinuous operator of C into E∗. Let T ⊂ E × E∗ be an operator defined asfollows:

Tv =

⎧⎨

⎩

Av +NC(v), v ∈ C,∅, otherwise.

(2.16)

Then, T is maximal monotone and T−10 = VI(C,A).


Lemma 2.10 (see Tan and Xu [41]). Let {an} and {bn} be two sequence of nonnegative real numberssatisfying the inequality

an+1 = an + bn, ∀n ≥ 0. (2.17)

If∑∞

n=1 bn <∞, then limn→∞an exists.

For solving the mixed equilibrium problem, let us assume that the bifunction Θ :C × C → R and ϕ : C → R is convex and lower semicontinuous satisfies the followingconditions:

(A1) Θ(x, x) = 0 for all x ∈ C,(A2) Θ is monotone, that is, Θ(x, y) + Θ(y, x) ≤ 0 for all x, y ∈ C,(A3) for each x, y, z ∈ C,

lim supt↓0

Θ(tz + (1 − t)x, y) ≤ Θ

(x, y), (2.18)

(A4) for each x ∈ C, y �→ Θ(x, y) is convex and lower semicontinuous.

Motivated by Blum and Oettli [8], Takahashi and Zembayashi [29, Lemma 2.7]obtained the following lemmas.

Lemma 2.11 (see [29, Lemma 2.7]). Let C be a nonempty closed convex subset of a smooth, strictlyconvex, and reflexive Banach space E, let θ be a bifunction from C × C to R satisfying (A1)–(A4), letr > 0, and let x ∈ E. Then, there exists z ∈ C such that

Θ(z, y)+1r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C. (2.19)

Lemma 2.12 (see Takahashi and Zembayashi [29]). LetC be a closed convex subset of a uniformlysmooth, strictly convex, and reflexive Banach space E and let Θ be a bifunction from C × C to R

satisfying (A1)–(A4). For all r > 0 and x ∈ E, define a mapping Tr : E → C as follows:

Trx ={z ∈ C : Θ

(z, y)+1r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C

}, (2.20)

for all x ∈ E. Then, the followings hold:(1) Tr is single-valued,

(2) Tr is a firmly nonexpansive-type mapping, that is, for all x, y ∈ E,⟨Trx − Try, JTrx − JTry

⟩ ≤ ⟨Trx − Try, Jx − Jy⟩, (2.21)

(3) F(Tr) = EP(Θ),

(4) EP(Θ) is closed and convex.


Lemma 2.13 (see Takahashi and Zembayashi [29]). Let C be a closed, convex subset of a smooth,strictly convex, and reflexive Banach space E, letΘ a bifunction from C×C to R satisfying (A1)–(A4)and let r > 0. Then, for x ∈ E and q ∈ F(Tr),

φ(q, Trx

)+ φ(Trx, x) ≤ φ

(q, x). (2.22)

Lemma 2.14. Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach spaceE. Let ϕ : C → R is convex and lower semicontinuous and Θ be a bifunction from C × C to R

satisfying (A1)–(A4). For r > 0 and x ∈ E, then there exists u ∈ C such that

Θ(u, y)+ ϕ(y) − ϕ(u) + 1

r

⟨y − u, Ju − Jx⟩. (2.23)

Define a mapping Kr : E → C as follows:

Kr(x) ={u ∈ C : Θ

(u, y)+ ϕ(y) − ϕ(u) + 1

r

⟨y − u, Ju − Jx⟩ ≥ 0, ∀y ∈ C

}(2.24)

for all x ∈ E. Then, the followings hold:

(1) Kr is single-valued,

(2) Kr is firmly nonexpansive, that is, for all x, y ∈ E, 〈Krx −Kry, JKrx − JKry〉 ≤ 〈Krx −Kry, Jx − Jy〉,

(3) F(Kr) = MEP(Θ, ϕ),

(4) MEP(Θ, ϕ) is closed and convex.

Proof. Define a bifunction F : C × C → R as follows:

F(u, y)= Θ(u, y)+ ϕ(y) − ϕ(u), ∀u, y ∈ C. (2.25)

It is easily seen that F satisfies (A1)–(A4). Therefore,Kr in Lemma 2.14 can be obtainedfrom Lemma 2.12 immediately.

3. Strong Convergence Theorem

In this section, we prove a strong convergence theorem for finding a common element of thezero point of a maximal monotone operator, the set of solutions of equilibrium problems, andthe set of solution of the variational inequality operators of an inverse strongly monotone ina Banach space by using the shrinking hybrid projection method.

Theorem 3.1. Let E be a 2-uniformly convex and uniformly smooth Banach space and let C be anonempty closed convex subset of E. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4) letϕ : C → R be a lower semicontinuous and convex function, and let T : E → E∗ be a maximalmonotone operator. Let Jr = (J + rT)−1J for r > 0 and let A be an α-inverse-strongly monotone


operator of C into E∗ with F := VI(C,A) ∩ T−1(0) ∩MEP(Θ, ϕ)/= ∅ and ‖Ay‖ ≤ ‖Ay −Au‖ for ally ∈ C and u ∈ F. Let {xn} be a sequence generated by x0 ∈ E with x1 = ΠC1x0 and C1 = C,

wn = ΠCJ−1(Jxn − λnAxn),


(1 − βn

)J(Jrnwn)

),

yn = J−1(αnJ(x1) + (1 − αn)J(zn)),


)+ ϕ(y) − ϕ(un) + 1

rn


⟩ ≥ 0, ∀y C,


},

xn+1 = ΠCn+1x0,

(3.1)

for n ∈ N, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. Thecoefficient sequence {αn}, {βn} ⊂ (0, 1), {rn} ⊂ (0,∞) satisfying limn→∞αn = 0, lim supn→∞βn <1, lim infn→∞rn > 0, and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, 1/c is the 2-uniformlyconvexity constant of E. Then, the sequence {xn} converges strongly toΠFx0.

Proof. We first show that {xn} is bounded. Put vn = J−1(Jxn − λnAxn), let p ∈ F := VI(C,A) ∩T−1(0) ∩MEP(Θ, ϕ), and let {Krn} be a sequence of mapping define as Lemma 2.14 and un =Krnyn. By (3.1) and Lemma 2.7, the convexity of the function V in the second variable, wehave

φ(p,wn

)= φ(p,ΠCvn

)

≤ φ(p, vn)= φ(p, J−1(Jxn − λnAxn)

)

≤ V (p, Jxn − λnAxn + λnAxn) − 2

⟨J−1(Jxn − λnAxn) − p, λnAxn

⟩

= V(p, Jxn

) − 2λn⟨vn − p,Axn

⟩

= φ(p, xn

) − 2λn⟨xn − p,Axn

⟩+ 2〈vn − xn,−λnAxn〉.

(3.2)

Since p ∈ VI(A,C) and A is α-inverse-strongly monotone, we have

−2λn⟨xn − p,Axn

⟩= −2λn

⟨xn − p,Axn −Ap

⟩ − 2λn⟨xn − p,Ap

⟩

≤ −2αλn∥∥Axn −Ap

∥∥2,(3.3)


and by Lemma 2.1, we obtain

2〈vn − xn,−λnAxn〉 = 2⟨J−1(Jxn − λnAxn) − xn,−λnAxn

⟩

≤ 2∥∥∥J−1(Jxn − λnAxn) − xn

∥∥∥‖λnAxn‖

≤ 4c2‖Jxn − λnAxn − Jxn‖‖λnAxn‖

=4c2λ2n‖Axn‖2 ≤

4c2λ2n∥∥Axn −Ap

∥∥2.

(3.4)

Substituting (3.3) and (3.4) into (3.2), we get

φ(p,wn

) ≤ φ(p, xn) − 2αλn

∥∥Axn −Ap∥∥2 +

4c2λ2n∥∥Axn −Ap

∥∥2

≤ φ(p, xn)+ 2λn

(2c2λn − α

)∥∥Axn −Ap∥∥2

≤ φ(p, xn).

(3.5)

By Lemmas 2.7, 2.8 and (3.5), we have

φ(p, zn

)= φ(p, J−1

(βnJ(xn) +

(1 − βn

)J(Jrnwn)

))

= V(p, βnJ(xn) +

(1 − βn

)J(Jrnwn)

)

≤ βnV(p, J(xn)

)+(1 − βn

)V(p, J(Jrnwn)

)

= βnφ(p, xn

)+(1 − βn

)φ(p, Jrnwn

)

≤ βnφ(p, xn

)+(1 − βn

)(φ(p,wn

) − φ(Jrnwn,wn))

≤ βnφ(p, xn

)+(1 − βn

)φ(p,wn

)

≤ βnφ(p, xn

)+(1 − βn

)φ(p, xn

)

= φ(p, xn

).

(3.6)

It follows that

φ(p, yn

)= φ(p, J−1(αnJ(x1) + (1 − αn)J(zn))

)

= V(p, αnJ(x1) + (1 − αn)J(zn)

) ≤ αnV(p, J(x1)

)+ (1 − αn)V

(p, J(zn)

)

= αnφ(p, x1

)+ (1 − αn)φ

(p, zn

) ≤ αnφ(p, x1

)+ (1 − αn)φ

(p, xn

).

(3.7)


From (3.1) and (3.7), we obtain

φ(p, un

)= φ(p,Krnyn

) ≤ φ(p, yn) ≤ φ(p, x1

)+ (1 − αn)φ

(p, xn

). (3.8)

So, we have p ∈ Cn+1. This implies that F ⊂ Cn, for all n ∈ N.From Lemma 2.5 and xn = ΠCnx0, we have

〈xn − z, Jx0 − Jxn〉 ≥ 0, ∀z ∈ Cn,⟨xn − p, Jx0 − Jxn

⟩ ≥ 0, ∀p ∈ F. (3.9)

From Lemma 2.6, one has

φ(xn, x0) = φ(ΠCnx0, x0) ≤ φ(p, x0

) − φ(p, xn) ≤ φ(p, x0

), (3.10)

for all p ∈ F ⊂ Cn and n ≥ 1. Then, the sequence {φ(xn, x0)} is bounded. Since xn = ΠCnx0and xn+1 = ΠCn+1x0 ∈ Cn+1 ⊂ Cn, we have

φ(xn, x0) ≤ φ(xn+1, x0), ∀n ∈ N. (3.11)

Therefore, {φ(xn, x0)} is nondecreasing. Hence, the limit of {φ(xn, x0)} exists. By theconstruction of Cn, one has that Cm ⊂ Cn and xm = ΠCmx0 ∈ Cn for any positive integerm ≥ n. It follows that

φ(xm, xn) = φ(xm,ΠCnx0) ≤ φ(xm, x0) − φ(ΠCnx0, x0) = φ(xm, x0) − φ(xn, x0). (3.12)

Letting m,n → ∞ in (3.12), we get φ(xm, xn) → 0. It follows from Lemma 2.4, that ‖xm −xn‖ → 0 asm,n → ∞, that is, {xn} is a Cauchy sequence. Since E is a Banach space and C isclosed and convex, we can assume that xn → u ∈ C, as n → ∞. Since

φ(xn+1, xn) = φ(xn+1,ΠCnx0) ≤ φ(xn+1, x0) − φ(ΠCnx0, x0) = φ(xn+1, x0) − φ(xn, x0), (3.13)

for all n ∈ N, we also have limn→∞φ(xn+1, xn) = 0. From Lemma 2.4, we get limn→∞‖xn+1 −xn‖ = 0. Since xn+1 = ΠCn+1x0 ∈ Cn+1 and by definition of Cn+1, we have

φ(xn+1, un) ≤ αnφ(xn+1, x1) + (1 − αn)φ(xn+1, xn). (3.14)

Noticing the conditions limn→∞αn = 0 and limn→∞φ(xn+1, xn) = 0, we obtain

limn→∞

φ(xn+1, un) = 0. (3.15)

From again Lemma 2.4,

limn→∞

‖xn+1 − xn‖ = limn→∞

‖xn+1 − un‖ = 0. (3.16)


So, by the triangle inequality, we get

limn→∞

‖xn − un‖ = 0. (3.17)

Since J is uniformly norm-to-norm continuous on bounded sets, we have

limn→∞

‖Jxn − Jun‖ = 0. (3.18)

On the other hand, we observe that

φ(p, xn

) − φ(p, un)= ‖xn‖2 − ‖un‖2 − 2

⟨p, Jxn − Jun

⟩

≤ ‖xn − un‖(‖xn‖ + ‖un‖) + 2∥∥p∥∥‖Jxn − Jun‖.

(3.19)

It follows that

φ(p, xn

) − φ(p, un) −→ 0, as n −→ ∞. (3.20)

From (3.1), (3.5), (3.6), (3.7), and (3.8), we have

φ(p, un

) ≤ φ(p, yn) ≤ αnφ

(p, x1

)+ (1 − αn)φ

(p, zn

)

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)(φ(p,wn

) − φ(Jrnwn,wn))]

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)(φ(p, xn

) − φ(Jrnwn,wn))]

≤ αnφ(p, x1

)+ (1 − αn)φ

(p, xn

) − (1 − αn)(1 − βn

)φ(Jrnwn,wn)

(3.21)

and then

(1 − αn)(1 − βn

)φ(Jrnwn,wn) ≤ αnφ

(p, x1

)+ (1 − αn)φ

(p, xn

) − φ(p, un). (3.22)

From conditions limn→∞αn = 0, lim supn→∞βn < 1 and (3.20), we obtain

limn→∞

φ(Jrnwn,wn) = 0. (3.23)

By again Lemma 2.4, we have limn→∞‖Jrnwn −wn‖ = 0.Since J is uniformly norm-to-norm continuous on bounded sets, we obtain

limn→∞

‖J(Jrnwn) − J(wn)‖ = 0. (3.24)


Applying (3.5) and (3.6), we observe that

φ(p, un

) ≤ φ(p, yn) ≤ αnφ

(p, x1

)+ (1 − αn)φ

(p, zn

)

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)φ(p,wn

)] ≤ αnφ(p, x1

)

+ (1 − αn)[βnφ(p, xn

)+(1 − βn

)[φ(p, xn

) − 2λn(α − 2

c2λn

)∥∥Axn −Ap

∥∥2]]

≤ αnφ(p, x1

)+ (1 − αn)φ

(p, xn

) − (1 − αn)(1 − βn

)2λn(α − 2

c2λn

)∥∥Axn −Ap

∥∥2

(3.25)

and, hence,

2λn(α − 2

c2λn

)∥∥Axn −Ap∥∥2 ≤ 1

(1 − αn)(1 − βn

)(αnφ(p, x1

)+ (1 − αn)φ

(p, xn

) − φ(p, un)),

(3.26)

for all n ∈ N. Since 0 < a ≤ λn ≤ b < c2α/2, limn→∞αn = 0, lim supn→∞βn < 1 and (3.20), wehave

limn→∞

∥∥Axn −Ap∥∥ = 0. (3.27)

From Lemmas 2.6, 2.7, and (3.4), we get

φ(xn,wn) = φ(xn,ΠCvn) ≤ φ(xn, vn) = φ(xn, J

−1(Jxn − λnAxn))= V (xn, Jxn − λnAxn)

≤ V (xn, (Jxn − λnAxn) + λnAxn)

− 2⟨J−1(Jxn − λnAxn) − xn, λnAxn

⟩

= φ(xn, xn) + 2〈vn − xn,−λnAxn〉

= 2〈vn − xn,−λnAxn〉 ≤ 4λ2nc2∥∥Axn −Ap

∥∥2.

(3.28)

From Lemma 2.4 and (3.27), we have

limn→∞

‖xn −wn‖ = 0. (3.29)

Since J is uniformly norm-to-norm continuous on bounded sets, we obtain

limn→∞

‖J(xn) − J(wn)‖ = 0. (3.30)


Since {xn} is bounded, there exists a subsequence {xni} of {xn} such that xni ⇀ u ∈ E. Sincexn −wn → 0, then we get wni ⇀ u as i → ∞.

Now,we claim that u ∈ F. First, we show that u ∈ T−10. Indeed, since lim infn→∞rn > 0,it follows from (3.24) that

limn→∞

‖Arnwn‖ = limn→∞

1rn‖Jwn − J(Jrnwn)‖ = 0. (3.31)

If (z, z∗) ∈ T , then it holds from the monotonicity of T that

⟨z −wni , z

∗ −Arniwni

⟩≥ 0, (3.32)

for all i ∈ N. Letting i → ∞, we get 〈z−u, z∗〉 ≥ 0. Then, the maximality of T implies u ∈ T−10.Next, we show that u ∈ VI(C,A). Let B ⊂ E × E∗ be an operator as follows:

Bv =

⎧⎨

⎩


(3.33)

By Theorem 2.9, B is maximal monotone and B−10 = VI(A,C). Let (v,w) ∈ G(B). Since w ∈Bv = Av +NC(v), we get w −Av ∈NC(v). From wn ∈ C, we have

〈v −wn,w −Av〉 ≥ 0. (3.34)

On the other hand, since wn = ΠCJ−1(Jxn − λnAxn), then by Lemma 2.5, we have

〈v −wn, Jwn − (Jxn − λnAxn)〉 ≥ 0. (3.35)

Thus,

⟨v −wn,

Jxn − Jwn

λn−Axn

⟩≤ 0. (3.36)


It follows from (3.34) and (3.36) that

〈v −wn,w〉 ≥ 〈v −wn,Av〉 ≥ 〈v −wn,Av〉 +⟨v −wn,

Jxn − Jwn

λn−Axn

⟩

= 〈v −wn,Av −Axn〉 +⟨v −wn,

Jxn − Jwn

λn

⟩

= 〈v −wn,Av −Awn〉 + 〈v −wn,Awn −Axn〉 +⟨v −wn,

Jxn − Jwn

λn

⟩

≥ −‖v −wn‖‖wn − xn‖α

− ‖v −wn‖‖Jxn − Jwn‖a

≥ −M(‖wn − xn‖

α+‖Jxn − Jwn‖

a

),

(3.37)

where M = supn≥1{‖v − wn‖}. From (3.29) and (3.30), we obtain 〈v − u,w〉 ≥ 0. By themaximality of B, we have u ∈ B−10 and, hence, u ∈ VI(C,A).

Next, we show that u ∈ MEP(Θ, ϕ). Since un = Krnyn. From Lemmas 2.13 and 2.14, wehave

φ(un, yn

)= φ(Krnyn, yn

) ≤ φ(u, yn) − φ(u,Krnyn

) ≤ φ(u, xn) − φ(u, un). (3.38)

Similarly by (3.20),

limn→∞

φ(un, yn

)= 0, (3.39)

and so

limn→∞

∥∥un − yn∥∥ = 0. (3.40)


limn→∞

∥∥Jun − Jyn∥∥ = 0. (3.41)

From (3.1) and (A2), we also have

ϕ(y) − ϕ(un) + 1

rn


⟩ ≥ Θ(y, un

), ∀y ∈ C. (3.42)

Hence,

ϕ(y) − ϕ(uni) +

⟨y − uni ,

Juni − Jynirni

⟩≥ Θ(y, uni

), ∀y ∈ C. (3.43)


From ‖xn − un‖ → 0, ‖xn −wn‖ → 0, we get uni ⇀ u. Since (Juni − Jyni/rni) → 0, it followsby (A4) and the weak, lower semicontinuous of ϕ that

Θ(y, u)+ ϕ(u) − ϕ(y) ≤ 0, ∀y ∈ C. (3.44)

For t with 0 < t ≤ 1 and y ∈ C, let yt = ty + (1 − t)u. Since y ∈ C and u ∈ C, we have yt ∈ Cand hence Θ(yt, u) + ϕ(u) − ϕ(yt) ≤ 0. So, from (A1), (A4), and the convexity of ϕ, we have

0 = Θ(yt, yt

)+ ϕ(yt) − ϕ(yt

) ≤ tΘ(yt, y)+ (1 − t)Θ(yt, u

)+ tϕ(y)+ (1 − t)ϕ(y) − ϕ(yt

)

≤ t(Θ(yt, y)+ ϕ(y) − ϕ(yt

)).

(3.45)

Dividing by t, we get Θ(yt, y) + ϕ(y) − ϕ(yt) ≥ 0. From (A3) and the weakly lowersemicontinuity of ϕ, we have Θ(u, y) + ϕ(y) − ϕ(u) ≥ 0 for all y ∈ C implies u ∈ MEP(Θ, ϕ).Hence, u ∈ F := VI(C,A) ∩ T−1(0) ∩MEP(Θ, ϕ).

Finally, we show that u = ΠFx. Indeed, from xn = ΠCnx and Lemma 2.5, we have

〈Jx − Jxn, xn − z〉 ≥ 0, ∀z ∈ Cn. (3.46)

Since F ⊂ Cn, we also have

⟨Jx − Jxn, xn − p

⟩ ≥ 0, ∀p ∈ F. (3.47)

Taking limit n → ∞, we have

⟨Jx − Ju, u − p⟩ ≥ 0, ∀p ∈ F. (3.48)

By again Lemma 2.5, we can conclude that u = ΠFx0. This completes the proof.

Corollary 3.2. Let E be a 2-uniformly convex and uniformly smooth Banach space, let C be anonempty, closed, convex subset of E. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4)let ϕ : C → R be a lower semicontinuous and convex function, and let T : E → E∗ be a maximalmonotone operator. Let Jr = (J + rT)−1J for r > 0 with F := T−1(0) ∩MEP(Θ, ϕ)/= ∅. Let {xn} be asequence generated by x0 ∈ E with x1 = ΠC1x0 and C1 = C,


(1 − βn

)J(Jrnxn)

),

yn = J−1(αnJ(x1) + (1 − αn)J(zn)),


)+ ϕ(y) − ϕ(un) + 1

rn


⟩ ≥ 0, ∀y ∈ C,


},

xn+1 = ΠCn+1x0,

(3.49)


for n ∈ N, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. Thecoefficient sequence {αn}, {βn} ⊂ (0, 1), {rn} ⊂ (0,∞) satisfying limn→∞αn = 0, lim supn→∞βn < 1and lim infn→∞rn > 0. Then, the sequence {xn} converges strongly toΠFx0.

Proof. In Theorem 3.1 if A ≡ 0, then (3.1) reduced to (3.49).

4. Weak Convergence Theorem

In this section, we first prove the following strong convergence theorem by using the idea ofPlubtieng and Sriprad [42].

Theorem 4.1. Let E be a 2-uniformly convex and uniformly smooth Banach space whose dualitymapping J is weak sequentially continuous. Let T : E → E∗ be a maximal monotone operator and letJr = (J + rT)−1J for r > 0. Let C be a nonempty, closed, convex subset of E such that D(T) ⊂ C ⊂J−1(⋂r>0 R(J + rT)), let Θ be a bifunction from C ×C to R satisfying (A1)–(A4), let ϕ : C → R be a

lower semicontinuous and convex function, and let A be an α-inverse-strongly monotone operator ofC into E∗ with F := VI(C,A) ∩ T−1(0) ∩MEP(Θ, ϕ)/= ∅ and ‖Ay‖ ≤ ‖Ay −Au‖ for all y ∈ C andu ∈ F. Let {xn} be a sequence generated by x1 = x ∈ C and

un = Krnxn,



(1 − βn

)J(Jrnzn)

),

xn+1 = ΠCJ−1(αnJ(x1) + (1 − αn)J

(yn)),

(4.1)

for n ∈ N ∪ {0}, where ΠC is the generalized projection from E onto C, J is the duality mappingon E. The coefficient sequence {αn}, {βn} ⊂ [0, 1], {rn} ⊂ (0,∞) satisfying

∑∞n=0 αn < ∞,

lim supn→∞βn < 1 lim infn→∞rn > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2,1/c is the 2-uniformly convexity constant of E. Then, the sequence {ΠFxn} converges strongly to anelement of F, which is a unique element v ∈ F such that

limn→∞

φ(v, xn) = miny∈F

limn→∞

φ(y, xn

), (4.2)

whereΠF is the generalized projection from C onto F.

Proof. Put vn = J−1(Jun − λnAun). Let p ∈ F := VI(C,A) ∩ T−1(0) ∩MEP(Θ, ϕ), by Lemma 2.14and nonexpansiveness of Kr , we have

φ(p, un

)= φ(p,Krnxn

) ≤ φ(p, xn). (4.3)


By (4.1) and Lemma 2.7, the convexity of the function V in the second variable, we obtain

φ(p, zn

)= φ(p,ΠCvn

) ≤ φ(p, vn)= φ(p, J−1(Jun − λnAun)

)

≤ V (p, Jun − λnAun + λnAun) − 2

⟨J−1(Jun − λnAun) − p, λnAun

⟩

= V(p, Jxun

) − 2λn⟨vn − p,Aun

⟩

= φ(p, un

) − 2λn⟨un − p,Aun

⟩+ 2〈vn − un,−λnAun〉.

(4.4)

Since p ∈ VI(A,C) and A is α-inverse-strongly monotone, we also have

−2λn⟨un − p,Aun

⟩= −2λn

⟨un − p,Aun −Ap

⟩ − 2λn⟨un − p,Ap

⟩ ≤ −2αλn∥∥Aun −Ap

∥∥2,

(4.5)

2〈vn − un,−λnAun〉 = 2⟨J−1(Jun − λnAun) − xn,−λnAun

⟩

≤ 2∥∥∥J−1(Jun − λnAun) − xn

∥∥∥‖λnAun‖

≤ 4c2‖Jun − λnAun − Jun‖‖λnAun‖ ≤ 4

c2λ2n∥∥Aun −Ap

∥∥2.

(4.6)

Substituting (4.5) and (4.6) into (4.4) and (4.3), we get

φ(p, zn

) ≤ φ(p, un) − 2αλn

∥∥Aun −Ap∥∥2 +

4c2λ2n∥∥Aun −Ap

∥∥2

≤ φ(p, un) − 2λn

(α − 2

c2λn

)∥∥Aun −Ap∥∥2 ≤ φ(p, un

) ≤ φ(p, xn).

(4.7)

By Lemmas 2.7, 2.8, (4.7), and using the same argument in Theorem 3.1, (3.6), we obtain

φ(p, yn

) ≤ φ(p, xn), (4.8)

and hence by Lemma 2.6 and (4.7), we note that

φ(p, xn+1

)= φ(p, J−1

(αnJ(x1) + (1 − αn)J

(yn)))

= V(p, αnJ(x1) + (1 − αn)J

(yn)) ≤ αnV

(p, J(x1)

)+ (1 − αn)V

(p, J(yn))

= αnφ(p, x1

)+ (1 − αn)φ

(p, yn

) ≤ αnφ(p, x1

)+ (1 − αn)φ

(p, xn

),

(4.9)

for all n ≥ 0. So, from∑∞

n=0 αn < ∞ and Lemma 2.10, we deduce that limn→∞φ(p, xn) exists.This implies that {φ(p, xn)} is bounded. It implies that {xn}, {yn}, {zn}, and {Jrnzn} arebounded. Define a function g : F → [0,∞) as follows:

g(p)= lim

n→∞φ(p, xn

), ∀p ∈ F. (4.10)


Then, by the same argument as in proof of [43, Theorem 3.1], we obtain g is a continuousconvex function and if ‖zn‖ → ∞, then g(zn) → ∞. Hence, by [34, Theorem 1.3.11], thereexists a point v ∈ F such that

g(v) = miny∈F

g(y)(:= l). (4.11)

Putwn = ΠFxn for all n ≥ 0. We next prove thatwn → v as n → ∞. Suppose on the contrarythat there exists ε0 > 0 such that, for each n ∈ N, there is n′ ≥ n satisfying ‖wn′ − v‖ ≥ ε0. Sincev ∈ F, we have

φ(wn, xn) = φ(ΠFxn, xn) ≤ φ(v,ΠFxn) + φ(ΠFxn, xn) ≤ φ(v, xn), (4.12)

for all n ≥ 0. This implies that

lim supn→∞

φ(wn, xn) ≤ limn→∞

φ(v, xn) = l. (4.13)

Since (‖v‖ − ‖ΠFxn‖)2 ≤ φ(v,wn) ≤ φ(v, xn) for all n ≥ 0 and {xn} is bounded, {wn} isbounded. By Lemma 2.3, there exists a stricly increasing, continuous, and convex functionK : [0,∞) → [0,∞) such that K(0) = 0 and

∥∥∥wn + v

2

∥∥∥2≤ 1

2‖wn‖2 + 1

2‖v‖2 − 1

4K(‖wn − v‖), (4.14)

for all n ≥ 0. Now, choose σ satisfying 0 < σ < (1/4)K(ε0). Hence, there exists n0 ∈ N suchthat

φ(wn, xn) ≤ l + σ, φ(v, xn) ≤ l + σ, (4.15)

for all n ≥ 0. Thus, there exists k ≥ n0 satisfying the following:

φ(wk, xk) ≤ l + σ, φ(v, xk) ≤ l + σ, ‖wk − v‖ ≥ ε0. (4.16)

From (4.9), (4.14), and (4.16), we obtain

φ(wk + v

2, xn+k

)≤ φ(wk + v

2, xk)=∥∥∥wk + v

2

∥∥∥2− 2⟨wk + v

2, Jxk

⟩+ ‖xk‖2

≤ 12‖wk‖2 + 1

2‖v‖2 − 1

4K(‖wk − v‖) − 〈wk + v, Jxk〉 + ‖xk‖2

=12φ(wk, xk) +

12φ(v, xk) − 1

4K(‖wk − v‖) ≤ l + σ − 1

4K(ε0),

(4.17)

for all n ≥ 0. Hence,

l ≤ limn→∞

φ(wk + v

2, xn)= lim

n→∞φ(wk + v

2, xn+k

)≤ l + σ − 1

4K(ε0) < l + σ − σ = l. (4.18)


This is a contradiction. So, {wn} converges strongly to v ∈ F := VI(C,A)∩ T−1(0)∩MEP(Θ, ϕ).Consequently, v ∈ F is the unique element of F such that

limn→∞


limn→∞

φ(y, xn

). (4.19)

This completes the proof.

Now, we prove a weak convergence theorem for the algorithm (4.20) below underdifferent condition on data.

Theorem 4.2. Let E be a 2-uniformly convex and uniformly smooth Banach space whose dualitymapping J is weakly sequentially continuous. Let T : E → E∗ be a maximal monotone operator andlet Jr = (J + rT)−1J for r > 0. Let C be a nonempty closed convex subset of E such that D(T) ⊂ C ⊂J−1(⋂r>0 R(J + rT)), let Θ be a bifunction from C ×C to R satisfying (A1)–(A4), let ϕ : C → R be a

lower semicontinuous and convex function, and let A be an α-inverse-strongly monotone operator ofC into E∗ with F := VI(C,A) ∩ T−1(0) ∩MEP(Θ, ϕ)/= ∅ and ‖Ay‖ ≤ ‖Ay −Au‖ for all y ∈ C andu ∈ F. Let {xn} be a sequence generated by x1 = x ∈ C and

un = Krnxn,



(1 − βn

)J(Jrnzn)

),

xn+1 = ΠCJ−1(αnJ(x1) + (1 − αn)J

(yn)),

(4.20)

for n ∈ N ∪ {0}, where ΠC is the generalized projection from E onto C, J is the duality mappingon E. The coefficient sequence {αn}, {βn} ⊂ [0, 1], {rn} ⊂ (0,∞) satisfying

∑∞n=0 αn < ∞,

lim supn→∞βn < 1 lim infn→∞rn > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, 1/cis the 2-uniformly convexity constant of E. Then, the sequence {xn} converges weakly to an element vof F, where v = limn→∞ΠFxn.

Proof. By Theorem 4.1, we have {xn} is bounded and so are {zn}, {Jrnzn}.From (4.9), we obtain

φ(p, xn+1

) ≤ αnφ(p, x1

)+ (1 − αn)φ

(p, yn

)

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)(φ(p, zn

) − φ(Jrnzn, zn))]

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)(φ(p, xn

) − φ(Jrnzn, zn))]

≤ αnφ(p, x1

)+(1 − αnβn

)φ(p, xn

) − (1 − αn)(1 − βn

)φ(Jrnzn, zn),

(4.21)

and then

(1 − αn)(1 − βn

)φ(Jrnzn, zn) ≤ αnφ

(p, x1

)+(1 − αnβn

)φ(p, xn

) − φ(p, xn+1). (4.22)


Since limn→∞αn = 0, lim supn→∞βn < 1 and {φ(p, xn)} exists, then we have

limn→∞

φ(Jrnzn, zn) = 0. (4.23)

By again Lemma 2.4, we have limn→∞‖Jrnzn − zn‖ = 0. Since J is uniformly norm-to-normcontinuous on bounded sets, we obtain

limn→∞

‖J(Jrnzn) − J(zn)‖ = 0. (4.24)

Apply (4.7), (4.8), and (4.9), we observe that

φ(p, xn+1

)

≤ αnφ(p, x1

)+ (1 − αn)φ

(p, yn

) ≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)φ(p, zn

)]

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)[φ(p, un

) − 2λn(α − 2

c2λn

)∥∥Aun −Ap∥∥2]]

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)[φ(p, xn

) − 2λn(α − 2

c2λn

)∥∥Aun −Ap∥∥2]]

≤ αnφ(p, x1

)+ (1 − αn)φ

(p, xn

) − (1 − αn)(1 − βn

)2λn(α − 2

c2λn

)∥∥Aun −Ap∥∥2,

(4.25)

and hence

2λn(α − 2

c2λn

)∥∥Aun −Ap∥∥2 ≤ 1

(1 − αn)(1 − βn

)(αnφ(p, x1

)+ (1 − αn)φ

(p, xn

) − φ(p, xn+1)),

(4.26)

for all n ∈ N. Since 0 < a ≤ λn ≤ b < c2α/2, limn→∞αn = 0 and lim supn→∞βn < 1, we have

limn→∞

∥∥Aun −Ap∥∥ = 0. (4.27)

From Lemmas 2.6, 2.7, and (4.7), we get

φ(un, zn) = φ(un,ΠCvn) ≤ φ(un, vn) = φ(un, J

−1(Jun − λnAun))= V (un, Jun − λnAun)

≤ V (un, (Jun − λnAun) + λnAun) − 2⟨J−1(Jun − λnAun) − xn, λnAun

⟩

= φ(un, un) + 2〈vn − un, λnAun〉 = 2〈vn − un, λnAun〉

≤ 4λ2nc2∥∥Aun −Ap

∥∥2.

(4.28)


From Lemma 2.4 and (4.27), we have

limn→∞

‖un − zn‖ = 0. (4.29)


limn→∞

‖J(un) − J(zn)‖ = 0. (4.30)

Since {zn} is bounded, there exists a subsequence {zni} of {zn} such that zni ⇀ u ∈ C.It follows that Jrni zni ⇀ u and uni ⇀ u ∈ C as i → ∞.

Now, we claim that u ∈ F. First, we show that u ∈ T−10. Indeed, since lim infn→∞rn > 0,it follows that

limn→∞

‖Arnzn‖ = limn→∞

1rn‖Jzn − J(Jrnzn)‖ = 0. (4.31)

If (z, z∗) ∈ T , then it holds from the monotonicity of T that

⟨z − Jrni zni , z∗ −Arni

zni

⟩≥ 0, (4.32)

for all i ∈ N. Letting i → ∞, we get 〈z−u, z∗〉 ≥ 0. Then, the maximality of T implies u ∈ T−10.Next, we show that u ∈ VI(C,A). Let B ⊂ E × E∗ be an operator as follows:

Bv =

⎧⎨

⎩


(4.33)

By Theorem 2.9, B is maximal monotone and B−10 = VI(A,C). Let (v,w) ∈ G(B). Since w ∈Bv = Av +NC(v), we get w −Av ∈NC(v). From zn ∈ C, we have

〈v − zn,w −Av〉 ≥ 0. (4.34)

On the other hand, since zn = ΠCJ−1(Jun − λnAun). Then, by Lemma 2.5, we have

〈v − zn, Jwn − (Jun − λnAun)〉 ≥ 0. (4.35)

Thus,

⟨v − zn, Jun − Jzn

λn−Aun

⟩≤ 0. (4.36)


It follows from (4.34) and (4.36) that

〈v − zn,w〉 ≥ 〈v − zn,Av〉 ≥ 〈v − zn,Av〉 +⟨v − zn, Jun − Jzn

λn−Axn

⟩

= 〈v − zn,Av −Aun〉 +⟨v − zn, Jun − Jzn

λn

⟩

= 〈v − zn,Av −Azn〉 + 〈v − zn,Azn −Aun〉 +⟨v − zn, Jun − Jzn

λn

⟩

≥ −‖v − zn‖‖zn − un‖α

− ‖v − zn‖‖Jun − Jzn‖a

≥ −M(‖zn − un‖

α+‖Jun − Jzn‖

a

),

(4.37)

where M = supn≥1{‖v − zn‖}. From (4.29) and (4.30), we obtain 〈v − u,w〉 ≥ 0. By themaximality of B, we have u ∈ B−10 and hence u ∈ VI(C,A).

Next, we show u ∈ MEP(f) = F(Kr). From un = Krnxn. It follows from (4.7), (4.8), and(4.9) that

φ(p, xn+1

) ≤ αnφ(p, x1

)+ (1 − αn)φ

(p, yn

)

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)φ(p, zn

)]

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)φ(p, un

)]

≤ αnφ(p, x1

)+ (1 − αn)

[βnφ(p, xn

)+(1 − βn

)φ(p, xn

)],

(4.38)

or, equivalently,

φ(p, xn+1

) − αnφ(p, x1

) ≤ (1 − αn)[βnφ(p, xn

)+(1 − βn

)φ(p, un

)] ≤ (1 − αn)φ(p, xn

),(4.39)

with limn→∞αn = 0 and lim supn→∞βn < 1, yield that limn→∞φ(p, un) = limn→∞φ(p, xn).From Lemmas 2.13 and 2.14, for p ∈ F,

φ(un, xn) ≤ φ(p, xn

) − φ(p, un). (4.40)

This implies that limn→∞φ(un, xn) = 0. Noticing Lemma 2.4, we get

‖un − xn‖ −→ 0, as n −→ ∞. (4.41)


limn→∞

‖Jun − Jxn‖ = 0. (4.42)


From (4.20) and (A2), we also have

ϕ(y) − ϕ(un) + 1

rn

⟨y − un, Jun − Jxn

⟩ ≥ Θ(y, un

), ∀y ∈ C. (4.43)

Hence,

ϕ(y) − ϕ(uni) +

⟨y − uni ,

Juni − Jxnirni

⟩≥ Θ(y, uni

), ∀y ∈ C. (4.44)

From ‖un − zn‖ → 0, we get uni ⇀ u. Since (Juni − Jxni/rni) → 0, it follows by (A4) and theweakly lower semicontinuous of ϕ that

Θ(y, u)+ ϕ(u) − ϕ(y) ≤ 0, ∀y ∈ C. (4.45)

For t with 0 < t ≤ 1 and y ∈ C, let yt = ty + (1 − t)u. Since y ∈ C and u ∈ C, we have yt ∈ Cand hence Θ(yt, u) + ϕ(u) − ϕ(yt) ≤ 0. So, from (A1), (A4), and the convexity of ϕ, we have

0 = Θ(yt, yt

)+ ϕ(yt) − ϕ(yt

)

≤ tΘ(yt, y)+ (1 − t)Θ(yt, u

)+ tϕ(y)+ (1 − t)ϕ(y) − ϕ(yt

)

≤ t(Θ(yt, y)+ ϕ(y) − ϕ(yt

)).

(4.46)

Dividing by t, we get Θ(yt, y) + ϕ(y) − ϕ(yt) ≥ 0. From (A3) and the weakly lowersemicontinuity of ϕ, we have Θ(u, y) + ϕ(y) − ϕ(u) ≥ 0 for all y ∈ C implies u ∈ MEP(Θ, ϕ).Hence, u ∈ F := VI(C,A) ∩ T−1(0) ∩MEP(Θ, ϕ).

By Theorem 4.1, the {ΠFxn} converges strongly to a point v ∈ F which is a uniqueelement of F such that

limn→∞


limn→∞

φ(y, xn

). (4.47)

By the uniform smoothness of E, we also have limn→∞‖JΠFxni − Jv‖ = 0.Finally, we prove u = v. From Lemma 2.5 and u ∈ F, we have

〈ΠFxni − u, Jxni − JΠFxni〉 ≥ 0. (4.48)

Since J is weakly sequentially continuous, uni ⇀ u and un − xn → 0. Then,

〈v − u, Ju − Jv〉 ≥ 0. (4.49)

On the other hand, since J is monotone, we have

〈v − u, Ju − Jv〉 ≤ 0. (4.50)


Hence,

〈v − u, Ju − Jv〉 = 0. (4.51)

Since E is strict convexity, it follows that u = v. Therefore, the sequence {xn} convergesweakly to v = limn→∞ΠFxn. This completes the proof.

5. Application to Complementarity Problems

Let C be a nonempty, closed convex cone in E and A an operator of C into E∗. We define itspolar in E∗ to be the set

K∗ ={y∗ ∈ E∗ :

⟨x, y∗⟩ ≥ 0, ∀x ∈ C}. (5.1)

Then, the element u ∈ C is called a solution of the complementarity problem if

Au ∈ K∗, 〈u,Au〉 = 0. (5.2)

The set of solutions of the complementarity problem is denoted by CP(K,A); see [34], formore detial.

Theorem 5.1. Let E be a 2-uniformly convex and uniformly smooth Banach space and let K be anonempty closed convex subset of E. Let Θ be a bifunction from K × K to R satisfying (A1)–(A4)let ϕ : K → R be a lower semicontinuous and convex function, and let T : E → E∗ be a maximalmonotone operator. Let Jr = (J + rT)−1J for r > 0 and let A be an α-inverse-strongly monotoneoperator of K into E∗ with F := T−1(0) ∩ CP(K,A) ∩MEP(Θ, ϕ)/= ∅ and ‖Ay‖ ≤ ‖Ay −Au‖ forall y ∈ K and u ∈ F. For an initial point x0 ∈ E with x1 = ΠC1x0 and K1 = K,

wn = ΠKJ−1(Jxn − λnAxn),


(1 − βn

)J(Jrnwn)

),

yn = J−1(αnJ(x1) + (1 − αn)J(zn)),

un ∈ K such that Θ(un, y

)+ ϕ(y) − ϕ(un) + 1

rn


⟩ ≥ 0, ∀y ∈ K,

Kn+1 ={z ∈ Kn : φ(z, un) ≤ αnφ(z, x1) + (1 − αn)φ(z, xn)

},

xn+1 = ΠKn+1x0,

(5.3)

for n ∈ N, where ΠK is the generalized projection from E onto K and J is the duality mappingon E. The coefficient sequence {αn}, {βn} ⊂ (0, 1), {rn} ⊂ (0,∞) satisfying limn→∞αn = 0,lim supn→∞βn < 1, lim infn→∞rn > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, 1/cis the 2-uniformly convexity constant of E. Then, the sequence {xn} converges strongly toΠFx0.

Proof. As in the proof Lemma 7.1.1 of Takahashi in [44], we have VI(C,A) = CP(K,A). So, weobtain the desired result.


Theorem 5.2. Let E be a 2-uniformly convex and uniformly smooth Banach space whose dualitymapping J is weakly sequentially continuous. Let T : E → E∗ be a maximal monotone operator andlet Jr = (J + rT)−1J for r > 0. Let K be a nonempty closed convex subset of E such that D(T) ⊂ K ⊂J−1(⋂r>0 R(J + rT)), let Θ be a bifunction from K × K to R satisfying (A1)–(A4), let ϕ : K → R

be a proper lower semicontinuous and convex function, and let A be an α-inverse-strongly monotoneoperator ofK into E∗ with F := CP(K,A)∩ T−1(0)∩MEP(Θ, ϕ)/= ∅ and ‖Ay‖ ≤ ‖Ay −Au‖ for ally ∈ K and u ∈ F. Let {xn} be a sequence generated by x1 = x ∈ K and

un = Krnxn,

zn = ΠKJ−1(Jun − λnAun),


(1 − βn

)J(Jrnzn)

),

xn+1 = ΠKJ−1(αnJ(x1) + (1 − αn)J

(yn)),

(5.4)

for n ∈ N ∪ {0}, where ΠK is the generalized projection from E onto K, J is the duality mappingon E. The coefficient sequence {αn}, {βn} ⊂ [0, 1], {rn} ⊂ (0,∞) satisfying

∑∞n=0 αn < ∞,

lim supn→∞βn < 1 lim infn→∞rn > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, 1/cis the 2-uniformly convexity constant of E. Then, the sequence {xn} converges weakly to an element vof F, where v = limn→∞ΠFxn.

Proof. It follows by Lemma 7.1.1 of Takahashi in [44], we have VI(C,A) = CP(K,A). Hence,Theorem 4.2, {xn} converges weakly to an element v of F, where v = limn→∞ΠFxn.

Acknowledgments

The authors would like to thank the referees for their careful readings and valuablesuggestions to improve the writing of this paper. This research is supported by the Centreof Excellence in Mathematics, the Commission on Higher Education, Thailand.

References

[1] R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Controland Optimization, vol. 14, no. 5, pp. 877–898, 1976.

[2] S. Kamimura andW. Takahashi, “Approximating solutions of maximal monotone operators in Hilbertspaces,” Journal of Approximation Theory, vol. 106, no. 2, pp. 226–240, 2000.

[3] S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banachspace,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.

[4] F. Kohsaka and W. Takahashi, “Strong convergence of an iterative sequence for maximal monotoneoperators in a Banach space,” Abstract and Applied Analysis, no. 3, pp. 239–249, 2004.

[5] L. Li and W. Song, “Modified proximal-point algorithm for maximal monotone operators in Banachspaces,” Journal of Optimization Theory and Applications, vol. 138, no. 1, pp. 45–64, 2008.

[6] H. Iiduka, W. Takahashi, and M. Toyoda, “Approximation of solutions of variational inequalities formonotone mappings,” Panamerican Mathematical Journal, vol. 14, no. 2, pp. 49–61, 2004.

[7] H. Iiduka andW. Takahashi, “Weak convergence of a projection algorithm for variational inequalitiesin a Banach space,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 668–679, 2008.

[8] E. Blum andW. Oettli, “From optimization and variational inequalities to equilibrium problems,” TheMathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

[9] S. D. Flam and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,”Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.


[10] A. Moudafi and M. Thera, “Proximal and dynamical approaches to equilibrium problems,” inIll-Posed Variational Problems and Regularization Techniques, vol. 477, pp. 187–201, Springer, Berlin,Germany, 1999.

[11] S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems andfixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no.1, pp. 506–515, 2007.

[12] P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal ofNonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

[13] C. Jaiboon, “The hybrid steepest descent method for addressing fixed point problems and system ofequilibrium problems,” Thai Journal of Mathematics, vol. 8, no. 2, pp. 275–292, 2010.

[14] C. Jaiboon and P. Kumam, “A general iterative method for addressing mixed equilibrium problemsand optimization problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 5, pp.1180–1202, 2010.

[15] T. Jitpeera and P. Kumam, “An extragradient type method for a system of equilibrium problems,variational inequality problems and fixed points of finitely many nonexpansive mappings,” Journalof Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 71–91, 2010.

[16] S. Saewan and P. Kumam, “Modified hybrid block iterative algorithm for convex feasibilityproblems and generalized equilibrium problems for uniformly quasi-φ-asymptotically nonexpansivemappings,” Abstract and Applied Analysis, vol. 2010, Article ID 357120, 22 pages, 2010.

[17] S. Saewan and P. Kumam, “A hybrid iterative scheme for a maximal monotone operator and twocountable families of relatively quasi-nonexpansive mappings for generalizedmixed equilibrium andvariational inequality problems,” Abstract and Applied Analysis, vol. 2010, Article ID 123027, 31 pages,2010.

[18] S. Saewan, P. Kumam, and K. Wattanawitoon, “Convergence theorem based on a new hybridprojection method for finding a common solution of generalized equilibrium and variationalinequality problems in Banach spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 734126,25 pages, 2010.

[19] A. Tada and W. Takahashi, “Strong convergence theorem for an equilibrium problem and anonexpansive mapping,” inNonlinear Analysis and Convex Analysis, W. Takahashi and T. Tanaka, Eds.,pp. 609–617, Yokohama Publications, Yokohama, Japan, 2007.

[20] A. Tada andW. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping andan equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370,2007.

[21] K. Wattanawitoon, P. Kumam, and U. W. Humphries, “Strong convergence theorem by the shrinkingprojection method for hemi-relatively nonexpansive mappings,” Thai Journal of Mathematics, vol. 7,no. 2, pp. 329–337, 2009.

[22] W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The AmericanMathematical Monthly, vol. 72, pp. 1004–1006, 1965.

[23] S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” inTheory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol.178, pp. 313–318, Marcel Dekker, New York, NY, USA, 1996.

[24] W. Nilsrakoo and S. t. Saejung, “Strong convergence to common fixed points of countable relativelyquasi-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 312454, 19pages, 2008.

[25] Y. Su, D.Wang, andM. Shang, “Strong convergence of monotone hybrid algorithm for hemi-relativelynonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 284613, 8 pages,2008.

[26] H. Zegeye and N. Shahzad, “Strong convergence theorems for monotone mappings and relativelyweak nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 7, pp.2707–2716, 2009.

[27] D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operatorsin Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001.

[28] Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators withapplications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323–339, 1996.

[29] W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problemsand relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods &Applications, vol. 70, no. 1, pp. 45–57, 2009.


[30] X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, “Convergence of a modified Halpern-type iterationalgorithm for quasi-φ-nonexpansive mappings,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1051–1055, 2009.

[31] L. C. Ceng, G. Mastroeni, and J. C. Yao, “Hybrid proximal-point methods for common solutions ofequilibrium problems and zeros of maximal monotone operators,” Journal of Optimization Theory andApplications, vol. 142, no. 3, pp. 431–449, 2009.

[32] K. Ball, E. A. Carlen, and E. H. Lieb, “Sharp uniform convexity and smoothness inequalities for tracenorms,” Inventiones Mathematicae, vol. 115, no. 3, pp. 463–482, 1994.

[33] Y. Takahashi, K. Hashimoto, andM. Kato, “On sharp uniform convexity, smoothness, and strong type,cotype inequalities,” Journal of Nonlinear and Convex Analysis, vol. 3, no. 2, pp. 267–281, 2002.

[34] W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Application, YokohamaPublishers, Yokohama, Japan, 2000.

[35] J. Diestel, Geometry of Banach spaces—Selected Topics, Lecture Notes in Mathematics, Springer, Berlin,Germany, 1975.

[36] B. Beauzamy, Introduction to Banach Spaces, and Their Geometry, North-Holland, Amsterdam, TheNetherlands, 2nd edition, 1995.

[37] H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis. Theory, Methods &Applications, vol. 16, no. 12, pp. 1127–1138, 1991.

[38] C. Zalinescu, “On uniformly convex functions,” Journal of Mathematical Analysis and Applications, vol.95, no. 2, pp. 344–374, 1983.

[39] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties andapplications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol.178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA,1996.

[40] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.

[41] K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawaiteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.

[42] S. Plubtieng and W. Sriprad, “An extragradient method and proximal point algorithm for inversestrongly monotone operators and maximal monotone operators in Banach spaces,” Fixed Point Theoryand Applications, vol. 2009, Article ID 591874, 16 pages, 2009.

[43] S. Kamimura, F. Kohsaka, and W. Takahashi, “Weak and strong convergence theorems for maximalmonotone operators in a Banach space,” Set-Valued Analysis, vol. 12, no. 4, pp. 417–429, 2004.

[44] W. Takahashi,Convex Analysis and Approximation Fixed points, vol. 2, Yokohama Publishers, Yokohama,Japan, 2000.


Research ArticleA New Hybrid Algorithm for Solving a System ofGeneralized Mixed Equilibrium Problems, Solvinga Family of Quasi-φ-Asymptotically NonexpansiveMappings, and Obtaining Common Fixed Points inBanach Space

J. F. Tan and S. S. Chang

Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Correspondence should be addressed to J. F. Tan, [email protected] S. S. Chang, [email protected]


Academic Editor: Vittorio Colao

Copyright q 2011 J. F. Tan and S. S. Chang. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Themain purpose of this paper is to introduce a new hybrid iterative scheme for finding a commonelement of set of solutions for a system of generalized mixed equilibrium problems, set of commonfixed points of a family of quasi-φ-asymptotically nonexpansivemappings, and null spaces of finitefamily of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smoothreal Banach space. The results presented in the paper improve and extend the correspondingresults announced by some authors.

1. Introduction

Throughout this paper, we assume that E is a real Banach space with a dual E∗, C is anonempty closed convex subset of E, and 〈·, ·〉 is the duality pairing between members ofE and E∗. The mapping J : E → 2E

∗defined by

J(x) ={f∗ ∈ E∗ :

⟨x, f∗⟩ = ‖x‖2;∥∥f∗∥∥ = ‖x‖

}, x ∈ E (1.1)

is called the normalized duality mapping.Let F : C × C → R be a bifunction, let B : C → E∗ be a nonlinear mapping, and

let Φ : C → R be a proper extended real-valued function. The “so-called” generalized mixedequilibrium problem for F, B, Φ is to find x∗ ∈ C such that

F(x∗, y

)+⟨y − x∗, Bx∗⟩ + Φ

(y) −Φ(x∗) ≥ 0, ∀y ∈ C. (1.2)


The set of solutions of (1.2) is denoted by GMEP(F, B,Φ), that is,

GMEP(F, B,Φ) ={x ∈ C : F

(x∗, y

)+⟨y − x∗, Bx∗⟩ + Φ

(y) −Φ(x∗) ≥ 0, ∀y ∈ C}. (1.3)

Special Examples

(1) If Φ ≡ 0, then the problem (1.2) is reduced to the generalized equilibrium problem(GEP), and the set of its solutions is denoted by

GEP(F, B) ={x ∈ C : F

(x∗, y

)+⟨y − x∗, Bx∗⟩ ≥ 0, ∀y ∈ C}. (1.4)

(2) If B ≡ 0, then the problem (1.2) is reduced to the mixed equilibrium problem (MEP),and the set of its solutions is denoted by

MEP(F, B) ={x ∈ C : F

(x∗, y

)+ Φ

(y) −Φ(x∗) ≥ 0, ∀y ∈ C}. (1.5)

These show that the problem (1.2) is very general in the sense that numerous problemsin physics, optimization, and economics reduce to finding a solution of (1.2). Recently, somemethods have been proposed for the generalizedmixed equilibrium problem in Banach space(see, e.g., [1–3]).

Let E be a smooth, strictly convex, and reflexive Banach space, and letC be a nonemptyclosed convex subset of E. Throughout this paper, the Lyapunov function φ : E × E → R

+ isdefined by

φ(x, y

)= ‖x‖2 − 2〈x, Jy〉 + ∥∥y

∥∥2, ∀x, y ∈ E. (1.6)

Following Alber [4], the generalized projection ΠC : E → C is defined by

ΠC(x) = argminy∈C

φ(y, x

), ∀x ∈ E. (1.7)

Let C be a nonempty closed convex subset of E, let S : C → C be a mapping, and letF(S) be the set of fixed points of S. A point p ∈ C is said to be an asymptotic fixed point of Tif there exists a sequence {xn} ⊂ C such that xn ⇀ p and ||xn − Sxn|| → 0. We denoted theset of all asymptotic fixed points of S by F(S). A point p ∈ C is said to be a strong asymptoticfixed point of S if there exists a sequence {xn} ⊂ C such that xn → p and ||xn − Sxn|| → 0. Wedenoted the set of all strongly asymptotic fixed points of S by F(S).

A mapping S : C → C is said to be nonexpansive if

∥∥Sx − Sy∥∥ ≤ ∥∥x − y∥∥, ∀x, y ∈ C. (1.8)

Amapping S : C → C is said to be relatively nonexpansive [5] if F(S)/= ∅, F(S) = F(S)and

φ(p, Sx

) ≤ φ(p, x), ∀x ∈ C, p ∈ F(S). (1.9)


A mapping S : C → C is said to be weak relatively nonexpansive [6] if F(S)/= ∅, F(S) =F(S) and

φ(p, Sx

) ≤ φ(p, x), ∀x ∈ C, p ∈ F(S). (1.10)

A mapping S : C → C is said to be closed if for any sequence {xn} ⊂ C with xn → xand Sxn → y, then Sx = y.

A mapping S : C → C is said to be quasi-φ-nonexpansive if F(S)/= ∅ and

φ(p, Sx

) ≤ φ(p, x), ∀x ∈ C, p ∈ F(S). (1.11)

A mapping S : C → C is said to be quasi-φ-asymptotically nonexpansive, if F(S)/= ∅ andthere exists a real sequence {kn} ⊂ [1,∞) with kn → 1 such that

φ(p, Snx

) ≤ knφ(p, x

), ∀n ≥ 1, x ∈ C, p ∈ F(S). (1.12)

From the definition, it is easy to know that each relatively nonexpansive mapping isclosed. The class of quasi-φ-asymptotically nonexpansive mappings contains properly theclass of quasi-φ-nonexpansive mappings as a subclass. The class of quasi-φ-nonexpansivemappings contains properly the class of weak relatively nonexpansive mappings as asubclass, and the class of weak relatively nonexpansive mappings contains properly the classof relatively nonexpansive mappings as a subclass, but the converse may be not true.

A mapping A : C → E∗ is said to be α-inverse strongly monotone if there exists α > 0such that

〈x − y,Ax −Ay〉 ≥ α∥∥Ax −Ay∥∥2. (1.13)

If A is an α-inverse strongly monotone mapping, then it is 1/α-Lipschitzian.Iterative approximation of fixed points for relatively nonexpansive mappings in the

setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushitaand Takahashi [5] obtained weak and strong convergence theorems to approximate a fixedpoint of a single relatively nonexpansive mapping. Recently, Su et al. [6, 7], Zegeyeand Shahzad [8], Wattanawitoon and Kumam [9], and Zhang [10] extend the notionfrom relatively nonexpansive mappings or quasi-φ-nonexpansive mappings to quasi-φ-asymptotically nonexpansive mappings and also prove some convergence theoremsto approximate a common fixed point of quasi-φ-nonexpansive mappings or quasi-φ-asymptotically nonexpansive mappings.

Motivated and inspired by these facts, the purpose of this paper is to introduce ahybrid iterative scheme for finding a common element of null spaces of finite family ofinverse strongly monotone mappings, set of common fixed points of an infinite familyof quasi-φ-asymptotically nonexpansive mappings, and the set of solutions of generalizedmixed equilibrium problem.


2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will beneeded in proving our main results.

A Banach space E is said to be strictly convex if ||x + y||/2 < 1 for all x, y ∈ U = {z ∈E : ||z|| = 1} with x /=y. It is said to be uniformly convex if for each ε ∈ (0, 2], there exists δ > 0such that ||x + y||/2 ≤ 1 − δ for all x, y ∈ U with ||x − y|| ≥ ε. The convexity modulus of E is thefunction δE : (0, 2] → [0, 1] defined by

δE(ε) = inf{1 −

∥∥∥∥12(x + y

)∥∥∥∥ : x, y ∈ U,∥∥x − y∥∥ ≥ ε

}, (2.1)

for all ε ∈ (0, 2]. It is well known that δE(ε) is a strictly increasing and continuous functionwith δE(0) = 0, and δE(ε)/ε is nondecreasing for all ε ∈ (0, 2]. Let p > 1, then E is said to bep-uniformly convex if there exists a constant c > 0 such that δE(ε) ≥ cεp, for all ε ∈ (0, 2]. Thespace E is said to be smooth if the limit

limt→ 0

∥∥x + ty∥∥ − ‖x‖t

(2.2)

exists for all x, y ∈ U. And E is said to be uniformly smooth if the limit exists uniformly inx, y ∈ U.

In the sequel, we will make use of the following lemmas.

Lemma 2.1 (see [11]). Let E be a 2-uniformly convex real Banach space, then for all x, y ∈ E, theinequality ||x − y|| ≤ (2/c2)||Jx − Jy|| holds, where 0 < c ≤ 1, and c is called the 2-uniformly convexconstant of E.

Lemma 2.2 (see [12]). Let E be a smooth, strict convex, and reflexive Banach space, and let C be anonempty closed convex subset of E, then the following conclusions hold:

(i) φ(x,ΠCy) + φ(ΠCy, y) ≤ φ(x, y), for all x ∈ C, y ∈ E,(ii) let x ∈ E and z ∈ C, then

z = ΠCx ⇐⇒ 〈z − y, Jx − Jz〉 ≥ 0, ∀y ∈ C. (2.3)

Lemma 2.3 (see [12]). Let E be a uniformly convex and smooth Banach space, and let {xn}, {yn} besequences of E. If φ(xn, yn) → 0 (as n → ∞) and either {xn} or {yn} is bounded, then xn − yn →0(as n → ∞).

Lemma 2.4 (see [10]). Let E be a uniformly convex Banach space, let r be a positive number, andlet Br(0) be a closed ball of E. For any given points {x1, x2, . . . , xn, . . .} ⊂ Br(0) and for any givenpositive numbers {λ1, λ2, . . .} with

∑∞n=1 λn = 1, there exists a continuous, strictly increasing, and

convex function g : [0, 2r) → [0,∞) with g(0) = 0 such that for any i, j ∈ {1, 2, . . .}, i < j,∥∥∥∥∥

∞∑

n=1

λnxn

∥∥∥∥∥

2

≤∞∑

n=1

λn‖xn‖2 − λiλjg(∥∥xi − xj

∥∥). (2.4)


For solving the generalized mixed equilibrium problem, let us assume that thebifunction F : C × C → R satisfies the following conditions:

(A1) F(x, x) = 0 for all x ∈ C,(A2) F is monotone, that is, F(x, y) + F(y, x) ≤ 0, for all x, y ∈ C,(A3) lim supt↓0F(x + t(z − x), y) ≤ F(x, y), for all x, y, z ∈ C,(A4) the function y �→ F(x, y) is convex and lower semicontinuous.

Lemma 2.5 (see [13]). Let E be a smooth, strict convex, and reflexive Banach space, and let C bea nonempty closed convex subset of E. Let F : C × C → R be a bifunction satisfying conditions(A1)–(A4). Let r > 0 and x ∈ E, then there exists z ∈ C such that

F(z, y

)+1r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C. (2.5)

By the same way as given in the proofs of [14, Lemmas 2.8 and 2.9], we can prove thatthe bifunction

Γ(x, y

)= F

(x, y

)+ Φ

(y) −Φ(x) +

⟨y − x, Bx⟩, ∀x, y ∈ C (2.6)

satisfies conditions (A1)–(A4) and the following conclusion holds.

Lemma 2.6. Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonemptyclosed convex subset of E. Let F : C ×C → R be a bifunction satisfying conditions (A1)–(A4), let B :C → E∗ be a β-inverse strongly monotone mapping, and let Φ : C → R be a lower semicontinuousand convex function. For given r > 0 and x ∈ E, define a mapping KΓ

r : E → C by

KΓr (x) =

{z ∈ C : F

(z, y

)+ Φ

(y) −Φ(z) +

⟨y − z, Bz⟩ + 1

r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C

},

(2.7)

then the following hold:

(i) KΓr is single valued,

(ii) KΓr is a firmly nonexpansive-type mapping, that is, for all x, y,∈ E,⟨KΓr (x) −KΓ

r

(y), JKΓ

r (x) − JKΓr

(y)⟩ ≤

⟨KΓr (x) −KΓ

r

(y), Jx − Jy

⟩, (2.8)

(iii) F(KΓr ) = GMEP(F,Φ, B),

(iv) GMEP(F,Φ, B) is closed and convex,

(v) φ(p,KΓr (x)) + φ(K

Γr (x), x) ≤ φ(p, x), for all p ∈ F(KΓ

r ).

In the sequel, we make use of the function V : E × E∗ → R defined by

V (x, x∗) = ‖x‖2 − 2〈x, x∗〉 + ‖x∗‖2, (2.9)

for all x ∈ E and x∗ ∈ E∗. Observe that V (x, x∗) = φ(x, J−1x∗) for all x ∈ E and x∗ ∈ E∗. Thefollowing lemma is well known.


Lemma 2.7 (see [4]). Let E be a smooth, strict convex, and reflexive Banach space with E∗ as itsdual, then

V (x, x∗) + 2⟨J−1x∗ − x, y∗

⟩≤ V (

x, x∗ + y∗), (2.10)

for all x ∈ E and x∗, y∗ ∈ E∗.

3. Main Results

In this section, wewill propose the following new iterative scheme {xn} for finding a commonelement of set of solutions for a system of generalized mixed equilibrium problems, the set ofcommon fixed points of a family of quasi-φ-asymptotically nonexpansive mappings, and nullspaces of finite family of γ-inverse strongly monotone mappings in the setting of 2-uniformlyconvex and uniformly smooth real Banach spaces:

x0 ∈ C0 = C,

yn = ΠCJ−1(Jxn − λAn+1xn),

zn = J−1(

αn,0Jxn +∞∑

i=1

αn,iJTni yn

)

,

un = KΓMrM,n

KΓM−1rM−1,n · · ·KΓ2

r2,nKΓ1r1,nzn,

Cn+1 ={v ∈ Cn : φ(v, un) ≤ φ(v, xn) + ξn

},

xn+1 = ΠCn+1x0, n ≥ 0,

(3.1)

where KΓkrk,n : C → C, k = 1, 2, . . . ,M is the mapping defined by (2.7), An = An(modN), rk,n ∈

[d,∞) for some d > 0 and 0 < λ < c2γ/2, where c is the 2-uniformly convex constant of E, foreach n ≥ 1, αn,0 +

∑∞i=1 αn,i = 1 and for each j ≥ 1, lim infn→∞αn,0αn,j > 0.

Definition 3.1. A countable family of mappings {Ti : C → C} is said to be uniformly quasi-ϕ-asymptotically nonexpansive mappings if there exists a sequence {kn} ⊂ [1,∞) with kn → 1such that for each i ≥ 1

∥∥Tni x − Tni y∥∥ ≤ kn

∥∥x − y∥∥, ∀x, y ∈ C, and for each n ≥ 1. (3.2)

Theorem 3.2. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformlysmooth real Banach space E with a dual E∗. Let {Ti : C → C}∞i=1 be a countable family of closed anduniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞) such thatkn → 1. Suppose further that for each i ≥ 1, Ti is uniformly Li-Lipschitzian. Let An : C → E∗,n = 1, 2, . . . ,N be a finite family of γn-inverse strongly monotone mappings, and let γ = min{γn, n =1, 2, . . . ,N}. Let {Fm : C × C → R, m = 1, 2, . . . ,M} be a finite family of equilibrium functionssatisfying conditions (A1)–(A4), and {Φm : C → R, m = 1, 2, . . . ,M} be a finite family of lower


semicontinuous convex function, and let {Bm : C → E∗, m = 1, 2, . . . ,M} be a finite family ofβm-inverse strongly monotone mappings. If

Ω =∞⋂

i=1

F(Ti) ∩N⋂

n=1

A−1n (0) ∩

M⋂

m=1

GMEP(Fm, Bm,Φm) (3.3)

is a nonempty and bounded subset in C and ξn = supp∈Ω(kn − 1)φ(p, xn), then the sequence {xn}defined by (3.1) converges strongly to some point x∗ ∈ Ω.

Proof. We divide the proof of Theorem 3.2 into five steps.

(I) Sequences {xn}, {yn}, and {Tni yn} all are bounded.In fact, since xn = ΠCnx0, for any p ∈ Ω, from Lemma 2.2, we have

φ(xn, x0) = φ(ΠCnx0, x0) ≤ φ(p, x0

) − φ(p, xn) ≤ φ(p, x0

). (3.4)

This implies that the sequence {φ(xn, x0)} is bounded, and so {xn} is bounded.On the other hand, by Lemmas 2.1 and 2.7, we have that

φ(p, yn

)= φ

(p,ΠCJ

−1(Jxn − λAn+1xn))

≤ φ(p, J−1(Jxn − λAn+1xn)

)

= V(p, Jxn − λAn+1xn

)

≤ V (p, (Jxn − λAn+1xn) + λAn+1xn

) − 2⟨J−1(Jxn − λAn+1xn) − p, λAn+1xn

⟩

= V(p, Jxn

) − 2λ⟨J−1(Jxn − λAn+1xn) − p,An+1xn

⟩

= φ(p, xn

) − 2λ〈xn − p,An+1xn〉 − 2λ〈J−1(Jxn − λAn+1xn) − xn,An+1xn〉= φ

(p, xn

) − 2λ〈xn − p,An+1xn −An+1p〉(since Anp = 0, ∀n ≥ 1

)

− 2λ⟨J−1(Jxn − λAn+1xn) − xn,An+1xn

⟩

≤ φ(p, xn) − 2λγ‖An+1xn‖2 + 2λ

∥∥∥J−1(Jxn − λAn+1xn) − J−1Jxn∥∥∥ × ‖An+1xn‖

≤ φ(p, xn) − 2λγ‖An+1xn‖2 + 4λ2

c2‖An+1xn‖2

(by Lemma 2.1

)

≤ φ(p, xn)+ 2λ

(2λc2

− γ)‖An+1xn‖2.

(3.5)


Thus, using the fact that λ ≤ (c2/2)γ , we have that

φ(p, yn

) ≤ φ(p, xn). (3.6)

Moreover, by the assumption that {Ti : C → C}∞i=1 is a countable family of uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn,i} ⊂ [1,∞) such that kn =supi≥1kn,i → 1 (n → ∞), hence for any given p ∈ Ω, from (3.6)we have that

ϕ(p, Tni yn

) ≤ knϕ(p, yn

) ≤ knϕ(p, xn

), ∀n ≥ 1, i ≥ 1. (3.7)

Hence, for each i ≥ 1, {Tni yn} is also bounded, denoted by

M = supn≥0, i≥1

{‖xn‖,∥∥yn

∥∥,∥∥Tni yn

∥∥} <∞. (3.8)

By the way, from the definition of {ξn}, it is easy to see that

ξn = supp∈Ω

(kn − 1)φ(p, xn

) ≤ supp∈Ω

(kn − 1)(∥∥p

∥∥ +M)2 −→ 0 (n −→ ∞). (3.9)

(II) For each n ≥ 0, Cn is a closed and convex subset of C and Ω ⊂ Cn.

It is obvious that C0 = C is closed and convex. Suppose that Cn is closed and convex for somen ≥ 1. Since the inequality φ(v, un) ≤ φ(v, xn) + ξn is equivalent to

2〈v, Jxn − Jun〉 ≤ ‖xn‖2 − ‖un‖2 + ξn, (3.10)

therefore, we have

Cn+1 ={v ∈ Cn : 2〈v, Jxn − Jun〉 ≤ ‖xn‖2 − ‖un‖2 + ξn

}. (3.11)

This implies that Cn+1 is closed and convex. Thus, for each n ≥ 0, Cn is a closed and convexsubset of C.


Next, we prove that Ω ⊂ Cn for all n ≥ 0. Indeed, it is obvious that Ω ⊂ C0 = C.Suppose that Ω ⊂ Cn for some n ≥ 1. Since E is uniformly smooth, E∗ is uniformly convex.For any given p ∈ Ω ⊂ Cn and for any positive integer j > 0, from Lemma 2.4, we have

φ(p, un

)= φ

(p,KΓM

rM,nKΓM−1rM−1,n · · ·KΓ2

r2,nKΓ1r1,nzn

)

≤ φ(p, zn)= φ

(

p, J−1(

αn,0Jxn +∞∑

i=1

αn,iJTni yn

))

=∥∥p

∥∥2 − 2

⟨

p, αn,0Jxn +∞∑

i=1

αn,iJTni yn

⟩

+

∥∥∥∥∥αn,0Jxn +

∞∑

i=1

αn,iJTni yn

∥∥∥∥∥

2

≤ ∥∥p∥∥2 − 2αn,0〈p, Jxn〉 − 2

∞∑

i=1

αn,i⟨p, JTni yn

⟩

+ αn,0‖xn‖2 +∞∑

i=1

αn,i∥∥Tni yn

∥∥2 − αn,0αn,jg(∥∥∥Jxn − JTnj yn

∥∥∥)

= αn,0φ(p, xn

)+

∞∑

i=1

αn,iφ(p, Tni yn

) − αn,0αn,jg(∥∥∥Jxn − JTnj yn

∥∥∥)

≤ αn,0φ(p, xn

)+

∞∑

i=1

αn,iknφ(p, yn


∥∥∥).

(3.12)

Having this together with (3.6), we have

φ(p, un

) ≤ φ(p, zn)

≤ αn,0φ(p, xn

)+

∞∑

i=1

αn,iknφ(p, xn


∥∥∥)

≤ knφ(p, xn


∥∥∥)

≤ φ(p, xn)+ sup

z∈Ω(kn − 1)φ(z, xn) − αn,0αn,jg

(∥∥∥Jxn − JTnj yn∥∥∥)

= φ(p, xn

)+ ξn − αn,0αn,jg

(∥∥∥Jxn − JTnj yn∥∥∥)

≤ φ(p, xn)+ ξn.

(3.13)

Hence, p ∈ Cn+1 and Ω ⊂ Cn for all n ≥ 0.

(III) {xn} is a Cauchy sequence.Since xn = ΠCnx0 and xn+1 = ΠCn+1x0 ∈ Cn+1 ⊂ Cn, we have that

φ(xn, x0) ≤ φ(xn+1, x0), (3.14)


which implies that the sequence {φ(xn, x0)} is nondecreasing and bounded, and solimn→∞φ(xn, x0) exists. Hence, for any positive integerm, using Lemma 2.2 we have

φ(xn+m, xn) = φ(xn+m,ΠCnx0) ≤ φ(xn+m, x0) − φ(xn, x0), (3.15)

for all n ≥ 0. Since limn→∞φ(xn, x0) exists, we obtain that

φ(xn+m, xn) −→ 0 (n −→ ∞), ∀m ∈ Z+. (3.16)

Thus, by Lemma 2.3, we have that ‖xn+m−xn‖ → 0 as n → ∞. This implies that the sequence{xn} is a Cauchy sequence in C. Since C is a nonempty closed subset of Banach space E, it iscomplete. Hence, there exists an x∗ in C such that

xn −→ x∗ (n −→ ∞). (3.17)

(IV) We show that x∗ ∈ ⋂∞i=1 F(Ti).

Since xn+1 ∈ Cn+1 by the structure of Cn+1, we have that

φ(xn+1, un) ≤ φ(xn+1, xn) + ξn. (3.18)

Again by (3.16) and Lemma 2.3, we get that limn→∞||xn+1 − un|| = 0. But

‖xn − un‖ ≤ ‖xn − xn+1‖ + ‖xn+1 − un‖. (3.19)

Thus,

limn→∞

‖xn − un‖ = 0. (3.20)

This implies that un → x∗ as n → ∞. Since J is norm-to-norm uniformly continuous onbounded subsets of E, we have that

limn→∞

‖Jxn − Jun‖ = 0. (3.21)

From (3.13), (3.20), and (3.21), we have that

αn,0αn,jg(∥∥∥Jxn − JTnj yn

∥∥∥)≤ φ(p, xn

) − φ(p, un)+ ξn

= ‖xn‖2 − ‖un‖2 + 2⟨p, Jun − Jxn

⟩+ ξn

≤ ‖xn − un‖(‖un‖ + ‖xn‖) + 2〈p, Jun − Jxn〉 + ξn −→ 0 (n −→ ∞).(3.22)


In view of condition lim infn→∞αn,0αn,j > 0, we see that

g(∥∥∥Jxn − JTnj yn

∥∥∥)−→ 0 (n −→ ∞). (3.23)

It follows from the property of g that

∥∥∥Jxn − JTnj yn

∥∥∥ −→ 0 (n −→ ∞). (3.24)

Since xn → x∗ and J is uniformly continuous, it yields Jxn → Jx∗. Hence, from (3.24), wehave

JTnj yn −→ Jx∗ (n −→ ∞). (3.25)

Since E∗ is uniformly smooth and J−1 is uniformly continuous, it follows that

Tnj yn −→ x∗ (n −→ ∞), ∀j ≥ 1. (3.26)

Moreover, using inequalities (3.12) and (3.5), we obtain that

φ(p, un

) ≤ αn,0φ(p, xn

)+

∞∑

i=1

αn,iknφ(p, xn

)+

∞∑

i=1

αn,ikn2λ(

2c2λ − γ

)‖An+1xn‖2

≤ φ(p, xn)+ ξn + kn2λ

(2c2λ − γ

)‖An+1xn‖2, ∀p ∈ Ω.

(3.27)

This implies that

kn2λ(γ − 2

c2λ

)‖An+1xn‖2 ≤ φ

(p, xn

) − φ(p, un)+ ξn, (3.28)

that is,

limn→∞

‖An+1xn‖2 = 0. (3.29)

It follows from (3.1) and (3.29) that we have

limn→∞

∥∥yn − x∗∥∥ = limn→∞

∥∥∥ΠCJ−1(Jxn − λAn+1xn) − x∗

∥∥∥

≤ limn→∞

∥∥∥J−1(Jxn − λAn+1xn) − x∗∥∥∥ = 0.

(3.30)


Furthermore, by the assumption that for each j ≥ 1, Tj is uniformly Li-Lipschitz continuous,hence, we have

∥∥∥Tn+1j yn − Tnj yn

∥∥∥ ≤

∥∥∥Tn+1j yn − Tn+1j yn+1

∥∥∥ +

∥∥∥Tn+1j yn+1 − yn+1

∥∥∥ +

∥∥yn+1 − yn

∥∥ +

∥∥∥yn − Tnj yn

∥∥∥

≤ (Lj + 1

)∥∥yn+1 − yn∥∥ +

∥∥∥Tn+1j yn+1 − yn+1

∥∥∥ +

∥∥∥yn − Tnj yn

∥∥∥.

(3.31)

This together with (3.26) and (3.30) yields

limn→∞

∥∥∥Tn+1j yn − Tnj yn

∥∥∥ = 0. (3.32)

Hence, from (3.26), we have

limn→∞

Tn+1j yn = x∗, (3.33)

that is,

limn→∞

TjTnj yn = x∗. (3.34)

In view of (3.26) and the closeness of Tj , it yields that Tjx∗ = x∗ for all j ≥ 1. This implies thatx∗ ∈ ⋂∞

j=1 F(Tj).

(IV) Now, we prove that x∗ ∈ ⋂Nn=1A

−1n (0).

It follows from (3.29) that

limn→∞

‖An+1xn‖ = 0. (3.35)

Since limn→∞xn = x∗, we have that for every subsequence {xnj}j≥1 of {xn}n≥0, limj→∞xnj = x∗

and

limj→∞

Anj+1xnj = 0. (3.36)

Let {nq}q≥1 ⊂ N be an increasing sequence of natural numbers such that Anq+1 = A1, for allq ∈ N, then limp→∞||xnq − x∗|| = 0 and

0 = limq→∞

Anq+1xnq = limq→∞

A1xnq . (3.37)

Since A1 is γ-inverse strongly monotone, it is Lipschitz continuous, and thus

A1x∗ = A1

(limq→∞

xnq

)= lim

q→∞A1xnq = 0. (3.38)


Hence,

x∗ ∈ A−11 (0). (3.39)

Continuing this process, we obtain that x∗ ∈ A−1i (0), for all i = 1, 2, . . . ,N. Hence,

x∗ ∈N⋂

n=1

A−1n (0). (3.40)

(V) Next, we prove that x∗ ∈ ⋂Mm=1 GMEP(Fm, Bm,Φm).

Putting Smn = KΓm

rm,nKΓm−1rm−1,n · · ·KΓ2

r2,nKΓ1r1,n for m ∈ {1, 2, . . . ,M} and S0

n = I for all n ∈ N. For anyp ∈ Ω, we have

φ(Smn zn,Sm−1

n zn)≤ φ

(p,Sm−1

n zn)− φ(p,Sm

n zn)

≤ φ(p, zn) − φ(p,Sm

n zn)

≤ φ(p, xn)+ ξn − φ

(p,Sm

n zn) (

by (3.13))

= φ(p, xn

)+ ξn − φ

(p, un

).

(3.41)

It follows from (3.22) that limn→∞φ(Smn zn,Sm−1

n zn) = 0. Since E is 2-uniformly convex anduniformly smooth Banach space and {zn} is bounded, we have that

limn→∞

∥∥∥Smn zn − Sm−1

n zn∥∥∥ = 0, m = 1, 2, . . . ,M. (3.42)

Since xn → x∗ and un → x∗, now we prove that for each m = 1, 2, . . . ,M, Smn zn → x∗ as

n → ∞. In fact, ifm =M, then we have

limn→∞

∥∥∥SMn zn − SM−1

n zn∥∥∥ = lim

n→∞

∥∥∥un − SM−1n zn

∥∥∥ = 0, (3.43)

that is, SM−1n zn → x∗. By induction, the conclusion can be obtained. Since J is norm-to-norm

uniformly continuous on bounded subsets of E, we get

limn→∞

∥∥∥JSmn zn − JSm−1

n zn∥∥∥ = 0, (3.44)

and since rk,n ∈ [d,∞) for some d > 0, we have that

limn→∞

∥∥JSmn zn − JSm−1

n zn∥∥

rm,n= 0. (3.45)


Next, since Γm(Smn zn, y)+ (1/rm,n)〈y−Sm

n zn, JSmn zn − JSm−1

n zn〉 ≥ 0, for all y ∈ C, this impliesthat

1rm,n

⟨y − Sm


n zn⟩≥ −Γm

(Smn zn, y

) ≥ Γm(y,Sm

n zn), ∀y ∈ C. (3.46)

This implies that

Γm(y,Sm

n zn) ≤ 1

rm,n

⟨y − Sm


n zn⟩

≤ (M1 +

∥∥y

∥∥)

∥∥JSm

n zn − JSm−1n zn

∥∥

rm,n,

(3.47)

for someM1 ≥ 0. Since y �→ Γm(x, y) is a convex and lower semicontinuous, we obtain from(3.45) and (3.47) that

Γm(y, x∗) ≤ lim inf

n→∞Γm

(y,Sm

n zn) ≤ 0, ∀y ∈ C. (3.48)

For any t ∈ (0, 1] and y ∈ C, then yt = ty + (1 − t)x∗ ∈ C. Since Γm satisfies conditions (A1)and (A4), from (3.48), we have

0 = Γm(yt, yt

) ≤ tΓm(yt, y

)+ (1 − t)Γm

(yt, x

∗)

≤ tΓm(yt, y

), ∀m = 1, 2, . . .M.

(3.49)

Delete t, and then let t → 0, by condition (A3), we have

0 ≤ Γm(x∗, y

), ∀y ∈ C, ∀m = 1, 2, . . .M, (3.50)

that is, for eachm = 1, 2, . . . ,M, we have

Fm(x∗, y

)+ 〈y − x∗, Bmx∗〉 + Ψm

(y) −Ψm(x∗) ≥ 0, ∀y ∈ C. (3.51)

Therefore, we have that

x∗ ∈M⋂

m=1

GMEP(Fm, Bm,Φm). (3.52)


Remark 3.3. (1) Theorem 3.2 not only improves and extends the main results in [3, 6–10] butalso improves and extends the corresponding results of Chang et al. [1, 15], Wang et al. [16],Su et al. [17], and Kang et al. [18].

(2)It should be pointed out that the results presented in the paper can be used directlyto study the existence problems and approximal problems of solutions to optimization


problems, monotone variational inequality problems, variational inclusion problems, andequilibrium problems in some Banach spaces. For saving space, we will give them in anotherpaper.

Acknowledgment

The authors would like to express their thanks to the referees for their helpful comments andsuggestions.

References

[1] S.S. Chang, H. W. Lee, and C. K. Chan, “A new hybrid method for solving a generalized equilibriumproblem, solving a variational inequality problem and obtaining common fixed points in Banachspaces, with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 7, pp. 2260–2270, 2010.

[2] L. C. Ceng and J. C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed pointproblems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.

[3] J. F. Tang and S. S. Chang, “Strong convergence theorem for a generalized mixed equilibriumproblem and fixed point problem for a family of infinitely nonexpansive mappings in Hilbert spaces,”Panamerican American Mathematical Journal, vol. 19, no. 2, pp. 75–86, 2009.

[4] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties andapplications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G.Kartosator, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, NewYork, NY, USA, 1996.

[5] S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansivemappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

[6] Y. F. Su, J. Y. Gao, and H. Y. Zhou, “Monotone CQ algorithm of fixed points for weak relativelynonexpansive mappings and applications,” Journal of Mathematical Research and Exposition, vol. 28,no. 4, pp. 957–967, 2008.

[7] H. Zhang and Y. F. Su, “Strong convergence of modified hybrid algorithm for quasi-φ-asymptoticallynonexpansive mappings,” Communications of the Korean Mathematical Society, vol. 24, no. 4, pp. 539–551, 2009.

[8] H. Zegeye and N. Shahzad, “Strong convergence theorems for monotone mappings and relativelyweak nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp.2707–2716, 2009.

[9] K. Wattanawitoon and P. Kumam, “Strong convergence theorems by a new hybrid projectionalgorithm for fixed point problems and equilibrium problems of two relatively quasi-φ-nonexpansivemappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 1, pp. 11–20, 2009.

[10] S. S. Zhang, “The generalized mixed equilibrium problem in Banach spaces,” Applied Mathematics andMechanics, vol. 30, no. 9, pp. 1105–1112, 2009.

[11] H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods &Applications, vol. 16, no. 12, pp. 1127–1138, 1991.



[14] W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problemsand relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods &Applications, vol. 70, no. 1, pp. 45–57, 2009.

[15] S. S. Chang, C. K. Chan, and H. W. Joseph Lee, “Modified block iterative algorithm for quasi-φ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces,” AppliedMathematics and Computation, vol. 217, no. 18, pp. 7520–7530, 2011.

[16] Z. M. Wang, Y. F. Su, D. X. Wang, and Y. C. Dong, “A modified halpern-type iteration algorithm fora family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banachspaces,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2364–2371, 2011.


[17] Y. F. Su, H. K. Xu, and X. Zhang, “Strong convergence theorems for two countable families ofweak relatively nonexpansive mappings and applications,” Nonlinear Analysis: Theory, Methods &Applications, vol. 73, no. 12, pp. 3890–3906, 2010.

[18] J. Kang, Y. F. Su, and X. Zhang, “Hybrid algorithm for fixed points of weak relatively nonexpansivemappings and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 4, no. 4, pp. 755–765, 2010.


Research ArticleA New Hybrid Iterative Scheme for CountableFamilies of Relatively Quasi-NonexpansiveMappings and System of Equilibrium Problems

Yekini Shehu1, 2

1 Mathematics Institute, African University of Science and Technology, Abuja, Nigeria2 Department of Mathematics, University of Nigeria, Nsukka, Nigeria

Correspondence should be addressed to Yekini Shehu, [email protected]



Copyright q 2011 Yekini Shehu. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We construct a new iterative scheme by hybrid methods and prove strong convergence theoremfor approximation of a common fixed point of two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in auniformly smooth and strictly convex real Banach space with Kadec-Klee property using theproperties of generalized f -projection operator. Using this result, we discuss strong convergencetheorem concerning variational inequality and convex minimization problems in Banach spaces.Our results extend many known recent results in the literature.

1. Introduction

Let E be a real Banach space with dual E∗ and C a nonempty, closed, and convex subset of E.A mapping T : C → C is called nonexpansive if

∥∥Tx − Ty∥∥ ≤ ∥

∥x − y∥∥, ∀x, y ∈ C. (1.1)

A point x ∈ C is called a fixed point of T if Tx = x. The set of fixed points of T is denoted byF(T) := {x ∈ C : Tx = x}.

We denote by J the normalized duality mapping from E to 2E∗defined by

J(x) ={f ∈ E∗ :

⟨x, f

⟩= ‖x‖2 = ∥∥f

∥∥2}. (1.2)


The following properties of J are well known (the reader can consult [1–3] for more details).

(1) If E is uniformly smooth, then J is norm-to-norm uniformly continuous on eachbounded subset of E.

(2) J(x)/= ∅, x ∈ E.(3) If E is reflexive, then J is a mapping from E onto E∗.

(4) If E is smooth, then J is single valued.

Throughout this paper, we denote by φ the functional on E × E defined by

φ(x, y

)= ‖x‖2 − 2

⟨x, J

(y)⟩

+∥∥y

∥∥2, ∀x, y ∈ E. (1.3)

It is obvious from (1.3) that

(‖x‖ − ∥∥y

∥∥)2 ≤ φ(x, y) ≤ (‖x‖ + ∥

∥y∥∥)2, ∀x, y ∈ E. (1.4)

Definition 1.1. LetC be a nonempty subset of E, and let T be a mapping fromC into E. A pointp ∈ C is said to be an asymptotic fixed point of T if C contains a sequence {xn}∞n=0 whichconverges weakly to p and limn→∞‖xn − Txn‖ = 0. The set of asymptotic fixed points of T isdenoted by F(T). We say that a mapping T is relatively nonexpansive (see, e.g., [4–9]) if thefollowing conditions are satisfied:

(R1) F(T)/= ∅,(R2) φ(p, Tx) ≤ φ(p, x), for all x ∈ C, p ∈ F(T),(R3) F(T) = F(T).

If T satisfies (R1) and (R2), then T is said to be relatively quasi-nonexpansive. It is easy tosee that the class of relatively quasi-nonexpansive mappings contains the class of relativelynonexpansive mappings. Many authors have studied the methods of approximating the fixedpoints of relatively quasi-nonexpansive mappings (see, e.g., [10–12] and the references citedtherein). Clearly, in Hilbert space H , relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same, for φ(x, y) = ‖x−y‖2, for all x, y ∈ H , and this impliesthat

φ(p, Tx

) ≤ φ(p, x) ⇐⇒ ∥∥Tx − p∥∥ ≤ ∥

∥x − p∥∥, ∀x ∈ C, p ∈ F(T). (1.5)

The examples of relatively quasi-nonexpansive mappings are given in [11].Let F be a bifunction of C ×C into �. The equilibrium problem (see, e.g., [13–25]) is to

find x∗ ∈ C such that

F(x∗, y

) ≥ 0, (1.6)

for all y ∈ C. We will denote the solutions set of (1.6) by EP(F). Numerous problems inphysics, optimization, and economics reduce to find a solution of problem (1.6). The equi-librium problems include fixed point problems, optimization problems, and variationalinequality problems as special cases (see, e.g., [26]).


In [7], Matsushita and Takahashi introduced a hybrid iterative scheme for approxima-tion of fixed points of relatively nonexpansive mapping in a uniformly convex real Banachspace which is also uniformly smooth: x0 ∈ C,

yn = J−1(αnJxn + (1 − αn)JTxn),Hn =

{w ∈ C : φ

(w, yn

) ≤ φ(w, xn)},

Wn = {w ∈ C : 〈xn −w, Jx0 − Jxn〉 ≥ 0},xn+1 = ΠHn∩Wnx0, n ≥ 0.

(1.7)

They proved that {xn}∞n=0 converges strongly toΠF(T)x0, where F(T)/= ∅.In [27], Plubtieng and Ungchittrakool introduced the following hybrid projection

algorithm for a pair of relatively nonexpansive mappings: x0 ∈ C,

zn = J−1(β(1)n Jxn + β

(2)n JTxn + β

(3)n JSxn

),

yn = J−1(αnJx0 + (1 − αn)Jzn),

Cn ={z ∈ C : φ

(z, yn

) ≤ φ(z, xn) + αn(‖x0‖2 + 2〈w, Jxn − Jx0〉

)},

Qn = {z ∈ C : 〈xn − z, Jxn − Jx0〉 ≤ 0},xn+1 = PCn∩Qnx0,

(1.8)

where {αn}, {β(1)n }, {β(2)n }, and {β(3)n } are sequences in (0, 1) satisfying β(1)n + β(2)n + β(3)n = 1 andT and S are relatively nonexpansive mappings and J is the single-valued duality mapping onE. They proved under the appropriate conditions on the parameters that the sequence {xn}generated by (1.8) converges strongly to a common fixed point of T and S.

In [9], Takahashi and Zembayashi introduced the following hybrid iterative schemefor approximation of fixed point of relatively nonexpansive mapping which is also a solutionto an equilibrium problem in a uniformly convex real Banach space which is also uniformlysmooth: x0 ∈ C, C1 = C, x1 = ΠC1x0,

yn = J−1(αnJxn + (1 − αn)JTxn),

F(un, y

)+

1rn


⟩ ≥ 0, ∀y ∈ C,

Cn+1 ={w ∈ Cn : φ(w, un) ≤ φ(w, xn)

},

xn+1 = ΠCn+1x0, n ≥ 1,

(1.9)

where J is the duality mapping on E. Then, they proved that {xn}∞n=0 converges strongly toΠFx0, where F = EP(F) ∩ F(T)/= ∅.


Furthermore, in [28], Qin et al. introduced the following hybrid iterative algorithm forapproximation of common fixed point of two countable families of closed relatively quasi-nonexpansive mappings in a uniformly convex and uniform smooth real Banach space:

zi,n = J−1(β(1)n,i Jxn + β

(2)n,i JTixn + β

(3)n,i JSixn

),

yi,n = J−1(αn,iJx0 + (1 − αn,i)Jzi,n),

Cn,i ={z ∈ C : φ

(z, yi,n

) ≤ φ(z, xn) + αn,i(‖x0‖2 + 2〈z, Jxn − Jx0〉

},

Cn =⋂

i∈ICn,i,

Q0 = C,

Qn = {z ∈ Qn−1 : 〈xn − z, Jx0 − Jxn〉 ≥ 0},

xn+1 = ΠCn∩Qnx0, n ≥ 0.

(1.10)

They proved that the sequence {xn} converges strongly to a common fixed point of the count-able families {Ti} and {Si} of closed relatively quasi-nonexpansive mappings in a uniformlyconvex and uniformly smooth Banach space under some appropriate conditions on {β(1)n,i },{β(2)n,i }, {β

(3)n,i }, and {αn,i}.

Recently, Li et al. [29] introduced the following hybrid iterative scheme for approxima-tion of fixed points of a relatively nonexpansive mapping using the properties of generalizedf-projection operator in a uniformly smooth real Banach space which is also uniformlyconvex: x0 ∈ C, C0 = C,

yn = J−1(αnJxn + (1 − αn)JTxn),

Cn+1 ={w ∈ Cn : G

(w, Jyn

) ≤ G(w, Jxn)},

xn+1 = Πf

Cn+1x0, n ≥ 0.

(1.11)

They proved a strong convergence theorem for finding an element in the fixed points set ofT . We remark here that the results of Li et al. [29] extended and improved on the results ofMatsushita and Takahashi [7].

Quite recently, motivated by the results of Takahashi and Zembayashi [9], Cholamjiakand Suantai [30] proved the following strong convergence theorem by hybrid iterativescheme for approximation of common fixed point of a countable family of closed relativelyquasi-nonexpansive mappings which is also a solution to a system of equilibrium problemsin uniformly convex and uniformly smooth Banach space.

Theorem 1.2. Let E be a uniformly convex real Banach space which is also uniformly smooth, and letC be a nonempty, closed, and convex subset of E. For each k = 1, 2, . . . , m, let Fk be a bifunction from


C × C satisfying (A1)–(A4). Suppose {Ti}∞i=1 is an infinitely countable family of closed and relativelyquasi-nonexpansive mappings ofC into itself such thatΩ :=

⋂mk=1 EP(Fk)∩(

⋂∞i=1 F(Ti))/= ∅. Suppose

{xn}∞n=0 is iteratively generated by x0 ∈ C, C0 = C,

yi,n = J−1(αnJxn + (1 − αn)JTixn),

ui,n = TFmrm,nTFm−1rm−1,n · · ·TF2

r2 ,nTF1r1,nyi,n,

Cn+1 =

{

z ∈ Cn : supi≥1

φ(z, ui,n) ≤ φ(z, xn)}

,

xn+1 = ΠCn+1x0, n ≥ 0.

(1.12)

Assume that {αn}∞n=1 and {rk,n}∞n=1 (k = 1, 2, . . . , m) are sequences which satisfy the followingconditions:

(i) lim supn→∞αn < 1,

(ii) lim infn→∞rk,n > 0 (k = 1, 2, . . . , m).

Then, {xn}∞n=0 converges strongly toΠΩx0.

Motivated by the above-mentioned results and the on-going research, it is our purposein this paper to prove a strong convergence theorem for two countable families of closedrelatively quasi-nonexpansive mappings which is also a solution to a system of equilibriumproblems in a uniformly smooth and strictly convex real Banach space with Kadec-Kleeproperty using the properties of generalized f-projection operator. Our results extend theresults of Matsushita and Takahashi [7], Takahashi and Zembayashi [9], Qin et al. [28],Cholamjiak and Suantai [30], Li et al. [29], and many other recent known results in theliterature.

2. Preliminaries

Let E be a real Banach space. The modulus of smoothness of E is the function ρE : [0,∞) →[0,∞) defined by

ρE(t) := sup{12(∥∥x + y

∥∥ +

∥∥x − y∥∥) − 1 : ‖x‖ ≤ 1, ‖y‖ ≤ t

}. (2.1)

E is uniformly smooth if and only if

limt→ 0

ρE(t)t

= 0. (2.2)

Let dimE ≥ 2. The modulus of convexity of E is the function δE : (0, 2] → [0, 1] defined by

δE(ε) := inf{1 −

∥∥∥∥x + y2

∥∥∥∥ : ‖x‖ =

∥∥y

∥∥ = 1; ε =

∥∥x − y∥∥

}. (2.3)


E is uniformly convex if, for any ε ∈ (0, 2], there exists a δ = δ(ε) > 0 such that if x, y ∈ E with‖x‖ ≤ 1, ‖y‖ ≤ 1, and ‖x − y‖ ≥ ε, then ‖(1/2)(x + y)‖ ≤ 1 − δ. Equivalently, E is uniformlyconvex if and only if δE(ε) > 0 for all ε ∈ (0, 2]. A normed space E is called strictly convex iffor all x, y ∈ E, x /=y, ‖x‖ = ‖y‖ = 1, we have ‖λx + (1 − λ)y‖ < 1, for all λ ∈ (0, 1).

Let E be a smooth, strictly convex, and reflexive real Banach space, and let C bea nonempty, closed, and convex subset of E. Following Alber [31], the generalized projectionΠC from E onto C is defined by

ΠC(x) := argminy∈C

φ(y, x

), ∀x ∈ E. (2.4)

The existence and uniqueness of ΠC follows from the property of the functional φ(x, y) andstrict monotonicity of the mapping J (see, e.g., [3, 31–34]). If E is a Hilbert space, then ΠC isthe metric projection ofH onto C.

Next, we recall the concept of generalized f-projector operator, together with itsproperties. Let G : C × E∗ → � ∪ {+∞} be a functional defined as follows:

G(ξ, ϕ

)= ‖ξ‖2 − 2

⟨ξ, ϕ

⟩+∥∥ϕ

∥∥2 + 2ρf(ξ), (2.5)

where ξ ∈ C, ϕ ∈ E∗, ρ is a positive number, and f : C → � ∪ {+∞} is proper, convex,and lower semicontinuous. From the definitions of G and f , it is easy to see the followingproperties:

(i) G(ξ, ϕ) is convex and continuous with respect to ϕwhen ξ is fixed,

(ii) G(ξ, ϕ) is convex and lower semicontinuous with respect to ξ when ϕ is fixed.

Definition 2.1 (see Wu and Huang [35]). Let E be a real Banach space with its dual E∗. Let Cbe a nonempty, closed, and convex subset of E. We say that Πf

C : E∗ → 2C is a generalizedf-projection operator if

Πf

Cϕ ={u ∈ C : G

(u, ϕ

)= inf

ξ∈CG(ξ, ϕ

)}, ∀ϕ ∈ E∗. (2.6)

For the generalized f-projection operator, Wu and Huang [35] proved the followingtheorem basic properties.

Lemma 2.2 (see Wu and Huang [35]). Let E be a real reflexive Banach space with its dual E∗. LetC be a nonempty, closed, and convex subset of E. Then, the following statements hold:

(i) Πf

C is a nonempty closed convex subset of C for all ϕ ∈ E∗,

(ii) if E is smooth, then, for all ϕ ∈ E∗, x ∈ Πf

C if and only if

⟨x − y, ϕ − Jx⟩ + ρf(y) − ρf(x) ≥ 0, ∀y ∈ C, (2.7)

(iii) if E is strictly convex and f : C → �∪{+∞} is positive homogeneous (i.e., f(tx) = tf(x)for all t > 0 such that tx ∈ C where x ∈ C), then Πf

C is a single-valued mapping.


Fan et al. [36] showed that the condition f is positive homogeneous which appearedin Lemma 2.2 can be removed.

Lemma 2.3 (see Fan et al. [36]). Let E be a real reflexive Banach space with its dual E∗ and Ca nonempty, closed, and convex subset of E. Then, if E is strictly convex, then Πf

C is a single-valuedmapping.

Recall that J is a single-valuedmappingwhen E is a smooth Banach space. There existsa unique element ϕ ∈ E∗ such that ϕ = Jx for each x ∈ E. This substitution in (2.5) gives

G(ξ, Jx) = ‖ξ‖2 − 2〈ξ, Jx〉 + ‖x‖2 + 2ρf(ξ). (2.8)

Now, we consider the second generalized f-projection operator in a Banach space.

Definition 2.4. Let E be a real Banach space and C a nonempty, closed, and convex subset ofE. We say that Πf

C : E → 2C is a generalized f-projection operator if

Πf

Cx ={u ∈ C : G(u, Jx) = inf

ξ∈CG(ξ, Jx)

}, ∀x ∈ E. (2.9)

Obviously, the definition of T : C → C is a relatively quasi-nonexpansive mappingand is equivalent to

(R′1) F(T)/= ∅,(R′2) G(p, JTx) ≤ G(p, Jx), for all x ∈ C, p ∈ F(T).

Lemma 2.5 (see Li et al. [29]). Let E be a Banach space, and let f : E → � ∪ {+∞} be a lowersemicontinuous convex functional. Then, there exists x∗ ∈ E∗ and α ∈ � such that

f(x) ≥ 〈x, x∗〉 + α, ∀x ∈ E. (2.10)

We know that the following lemmas hold for operatorΠf

C.

Lemma 2.6 (see Li et al. [29]). Let C be a nonempty, closed, and convex subset of a smooth andreflexive Banach space E. Then, the following statements hold:

(i) Πf

Cx is a nonempty closed and convex subset of C for all x ∈ E,(ii) for all x ∈ E, x ∈ Πf

Cx if and only if

⟨x − y, Jx − Jx⟩ + ρf(y) − ρf(x) ≥ 0, ∀y ∈ C, (2.11)

(iii) if E is strictly convex, thenΠf

Cx is a single-valued mapping.

Lemma 2.7 (see Li et al. [29]). Let C be a nonempty, closed, and convex subset of a smooth andreflexive Banach space E. Let x ∈ E and x ∈ Πf

Cx. Then,

φ(y, x) +G(x, Jx) ≤ G(y, Jx), ∀y ∈ C. (2.12)


The fixed points set F(T) of a relatively quasi-nonexpansive mapping is closed andconvex as given in the following lemma.

Lemma 2.8 (see Chang et al. [37]). Let C be a nonempty, closed, and convex subset of a uniformlysmooth and strictly convex real Banach space E which also has Kadec-Klee property. Let T be a closedrelatively quasi-nonexpansive mapping of C into itself. Then, F(T) is closed and convex.

Also, this following lemma will be used in the sequel.

Lemma 2.9 (see Cho et al. [38]). Let E be a uniformly convex real Banach space. For arbitraryr > 0, let Br(0) := {x ∈ E : ‖x‖ ≤ r} and λ, μ, γ ∈ [0, 1] such that λ + μ + γ = 1. Then, there existsa continuous strictly increasing convex function

g : [0, 2r] −→ �, g(0) = 0, (2.13)

such that, for every x, y, z ∈ Br(0), the following inequality holds:

∥∥λx + μy + γz∥∥2 ≤ λ‖x‖2 + μ∥∥y∥∥2 − λμg(∥∥x − y∥∥). (2.14)

For solving the equilibrium problem for a bifunction F : C × C → �, let us assumethat F satisfies the following conditions:

(A1) F(x, x) = 0 for all x ∈ C,(A2) F is monotone, that is, F(x, y) + F(y, x) ≤ 0 for all x, y ∈ C,(A3) for each x, y ∈ C, limt→ 0F(tz + (1 − t)x, y) ≤ F(x, y),(A4) for each x ∈ C, y �→ F(x, y) is convex and lower semicontinuous.

Lemma 2.10 (see Blum and Oettli [26]). Let C be a nonempty closed convex subset of a smooth,strictly convex, and reflexive Banach space E, and let F be a bifunction of C × C into � satisfying(A1)–(A4). Let r > 0 and x ∈ E. Then, there exists z ∈ C such that

F(z, y

)+1r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ K. (2.15)

Lemma 2.11 (see Takahashi and Zembayashi [39]). Let C be a nonempty closed convex subsetof a smooth, strictly convex, and reflexive Banach space E. Assume that F : C × C → � satisfies(A1)–(A4). For r > 0 and x ∈ E, define a mapping TFr : E → C as follows:

TFr (x) ={z ∈ C : F

(z, y

)+1r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C

}(2.16)

for all z ∈ E. Then, the following hold:(1) TFr is singlevalued,

(2) TFr is firmly nonexpansive-type mapping, that is, for any x, y ∈ E,⟨TFr x − TFr y, JTFr x − JTFr y

⟩≤⟨TFr x − TFr y, Jx − Jy

⟩, (2.17)


(3) F(TFr ) = EP(F),

(4) EP(F) is closed and convex.

Lemma 2.12 (see Takahashi and Zembayashi [39]). Let C be a nonempty closed convex subsetof a smooth, strictly convex, and reflexive Banach space E. Assume that F : C × C → � satisfies(A1)–(A4), and let r > 0. Then, for each x ∈ E and q ∈ F(TFr ),

φ(q, TFr x

)+ φ

(TFr x, x

)≤ φ(q, x). (2.18)

For the rest of this paper, the sequence {xn}∞n=0 converges strongly to p and will bedenoted by xn → p as n → ∞, {xn}∞n=0 converges weakly to p and will be denoted by xn ⇀ p

and we will assume that β(1)n,i , β(2)n,i , β

(3)n,i ∈ [0, 1], for all i = 1, 2, 3, . . . such that β(1)n,i + β

(2)n,i + β

(3)n,i =

1, for all n ≥ 0.We recall that a Banach space E has Kadec-Klee property if, for any sequence {xn}∞n=0 ⊂ E

and x ∈ E with xn ⇀ x and ‖xn‖ → ‖x‖, xn → x as n → ∞. We note that every uniformlyconvex Banach space has the Kadec-Klee property. For more details on Kadec-Klee property,the reader is referred to [2, 33].

Lemma 2.13 (see Li et al. [29]). Let E be a Banach space and y ∈ E. Let f : E → � ∪ {+∞} be aproper, convex, and lower semicontinuous mapping with convex domainD(f). If {xn} is a sequence inD(f) such that xn ⇀ x ∈ int(D(f)) and limn→∞G(xn, Jy) = G(x, Jy), then limn→∞‖xn‖ = ‖x‖.

3. Main Results

Theorem 3.1. Let E be a uniformly smooth and strictly convex real Banach space which also hasKadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k = 1, 2, . . . , m,let Fk be a bifunction from C×C satisfying (A1)–(A4). Suppose {Ti}∞i=1 and {Si}∞i=1 are two countablefamilies of closed relatively quasi-nonexpansive mappings ofC into itself such thatΩ :=

⋂mk=1 EP(Fk)∩

(⋂∞i=1 F(Ti)) ∩ (

⋂∞i=1 F(Si))/= ∅. Let f : E → � be a convex and lower semicontinuous mapping

with C ⊂ int(D(f)), and suppose {xn}∞n=0 is iteratively generated by x0 ∈ C, C1,i = C, C1 =∩∞i=1C1,i, x1 = Πf

C1x0,

zn,i = J−1(β(1)n,i Jxn + β

(2)n,i JTixn + β

(3)n,i JSixn

),

yn,i = J−1(αn,iJxn + (1 − αn,i)Jzn,i),

un,i = TFmrm,nT

Fm−1rm−1,n · · ·TF2

r2 ,nTF1r1,nyn,i,

Cn+1,i = {z ∈ Cn,i : G(z, Jun,i) ≤ G(z, Jxn)},

Cn+1 =∞⋂

i=1

Cn+1,i,

xn+1 = Πf

Cn+1x0, n ≥ 1,

(3.1)


with the conditions

(i) lim infn→∞β(1)n,i β

(2)n,i > 0,

(ii) lim infn→∞β(1)n,i β

(3)n,i > 0,

(iii) 0 ≤ αn,i ≤ α < 1 for some α ∈ (0, 1),

(iv) {rk,n}∞n=1 ⊂ (0,∞) (k = 1, 2, . . . , m) satisfying lim infn→∞rk,n > 0 (k = 1, 2, . . . , m).

Then, {xn}∞n=0 converges strongly toΠf

Ωx0.

Proof. We first show that Cn, for all n ≥ 1 is closed and convex. It is obvious that C1,i = C isclosed and convex. Suppose Ck,i is closed and convex for some k > 1. For each z ∈ Ck,i, wesee that G(z, Juk,i) ≤ G(z, Jxk) is equivalent to

2(〈z, Jxk〉 − 〈z, Juk,i〉) ≤ ‖xk‖2 − ‖uk,i‖2. (3.2)

By the construction of the set Ck+1,i, we see that Ck+1,i is closed and convex. Therefore, Ck+1 =⋂∞i=1 Ck+1,i is also closed and convex. Hence, Cn, for all n ≥ 1 is closed and convex.

By taking θkn = TFkrk ,nTFk−1rk−1 ,n · · ·TF2

r2 ,nTF1r1,n, k = 1, 2, . . . , m and θ0n = I for all n ≥ 1, we obtain

un,i = θmn yn,i.We next show that Ω ⊂ Cn, for all n ≥ 1. For n = 1, we have Ω ⊂ C = C1. Then, for

each x∗ ∈ Ω, we obtain

G(x∗, Jun,i) = G(x∗, Jθmn yn,i

) ≤ G(x∗, Jyn,i)

= G(x∗, (αn,iJxn + (1 − αn,i)Jzn,i))

= ‖x∗‖2−2αn,i〈x∗, Jxn〉−2(1 − αn,i)〈x∗, Jzn,i〉+‖αn,iJxn+(1 − αn,i)Jzn,i‖2+2ρf(x∗)

≤ ‖x∗‖2−2αn,i〈x∗, Jxn〉−2(1 − αn,i)〈x∗, Jzn,i〉+αn,i‖xn‖2+(1−αn,i)‖zn,i‖2+2ρf(x∗)= αn,iG(x∗, Jxn) + (1 − αn,i)G(x∗, Jzn,i)

= αn,iG(x∗, Jxn) + (1 − αn,i)G(x∗,

(β(1)n,i Jxn + β

(2)n,i JTixn + β

(3)n,i JSixn

))

≤ αn,iG(x∗, Jxn) + (1 − αn,i)(‖x∗‖2 − 2β(1)n,i 〈x∗, Jxn〉

− 2β(2)n,i 〈x∗, JTixn〉 − 2β(3)n,i 〈x∗, JSixn〉 + β(1)n,i ‖xn‖2

+β(2)n,i ‖Tixn‖2 + β(3)n,i ‖Sixn‖2 + 2ρf(x∗)

)

= αn,iG(x∗, Jxn) + (1 − αn,i)(β(1)n,i G(x

∗, Jxn) + β(2)n,i G(x

∗, JTixn) + β(3)n,i G(x

∗, JSixn))

≤ G(x∗, Jxn).(3.3)

So, x∗ ∈ Cn. This implies that Ω ⊂ Cn, for all n ≥ 1. Therefore, {xn} is well defined.


We now show that limn→∞G(xn, Jx0) exists. Since f : E → � is convex and lowersemicontinuous, applying Lemma 2.5, we see that there exists u∗ ∈ E∗ and α ∈ � such that

f(y) ≥ ⟨

y, u∗⟩+ α, ∀y ∈ E. (3.4)

It follows that

G(xn, Jx0) = ‖xn‖2 − 2〈xn, Jx0〉 + ‖x0‖2 + 2ρf(xn)

≥ ‖xn‖2 − 2〈xn, Jx0〉 + ‖x0‖2 + 2ρ〈xn, u∗〉 + 2ρα

= ‖xn‖2 − 2⟨xn, Jx0 − ρu∗

⟩+ ‖x0‖2 + 2ρα

≥ ‖xn‖2 − 2‖xn‖∥∥Jx0 − ρu∗

∥∥ + ‖x0‖2 + 2ρα

=(‖xn‖ −

∥∥Jx0 − ρu∗∥∥)2 + ‖x0‖2 −

∥∥Jx0 − ρu∗∥∥2 + 2ρα.

(3.5)

Since xn = Πf

Cnx0, it follows from (3.5) that

G(x∗, Jx0) ≥ G(xn, Jx0) ≥(‖xn‖ −

∥∥Jx0 − ρu∗∥∥)2 + ‖x0‖2 −

∥∥Jx0 − ρu∗∥∥2 + 2ρα (3.6)

for each x∗ ∈ F. This implies that {xn}∞n=0 is bounded and so is {G(xn, Jx0)}∞n=0. By theconstruction of Cn, we have that Cn+1 ⊂ Cn and xn+1 = Πf

Cn+1x0 ∈ Cn. It then follows from

Lemma 2.7 that

φ(xn+1, xn) +G(xn, Jx0) ≤ G(xn+1, Jx0). (3.7)

It is obvious that

φ(xn+1, xn) ≥ (‖xn+1‖ − ‖xn‖)2 ≥ 0, (3.8)

and so {G(xn, Jx0)}∞n=0 is nondecreasing. It follows that the limit of {G(xn, Jx0)}∞n=0 exists.Now since {xn}∞n=0 is bounded in C and E is reflexive, we may assume that xn ⇀ p,

and since Cn is closed and convex for each n ≥ 1, it is easy to see that p ∈ Cn for each n ≥ 1.Again since xn = Πf

Cnx0, from the definition of Πf

Cn, we obtain

G(xn, Jx0) ≤ G(p, Jx0

), ∀n ≥ 1. (3.9)

Since

lim infn→∞

G(xn, Jx0) = lim infn→∞

{‖xn‖2 − 2〈xn, Jx0〉 + ‖x0‖2 + 2ρf(xn)

}

≥ ∥∥p

∥∥2 − 2

⟨p, Jx0

⟩+ ‖x0‖2 + 2ρf

(p)= G

(p, Jx0

),

(3.10)


then we obtain

G(p, Jx0

) ≤ lim infn→∞

G(xn, Jx0) ≤ lim supn→∞

G(xn, Jx0) ≤ G(p, Jx0

). (3.11)

This implies that limn→∞G(xn, Jx0) = G(p, Jx0). By Lemma 2.13, we obtain limn→∞‖xn‖ =‖p‖. In view of Kadec-Klee property of E, we have that limn→∞xn = p.

We next show that p ∈ ⋂mk=1 EP(Fk) ∩ (

⋂∞i=1 F(Ti)) ∩ (

⋂∞i=1 F(Si)). We first show that

p ∈ ⋂∞i=1 F(Ti)) ∩ (

⋂∞i=1 F(Si). By the fact that Cn+1 ⊂ Cn and xn+1 = Πf

Cn+1x0 ∈ Cn, we obtain

φ(xn+1, un,i) ≤ φ(xn+1, xn). (3.12)

Now, (3.7) implies that

φ(xn+1, un,i) ≤ φ(xn+1, xn) ≤ G(xn+1, Jx0) −G(xn, Jx0). (3.13)

Taking the limit as n → ∞ in (3.13), we obtain

limn→∞

φ(xn+1, xn) = 0. (3.14)

Therefore,

limn→∞

φ(xn+1, un,i) = 0, ∀i ≥ 1. (3.15)

It then yields that limn→∞(‖xn+1‖ − ‖un,i‖) = 0, for all i ≥ 1. Since limn→∞‖xn+1‖ = ‖p‖, wehave

limn→∞

‖un,i‖ =∥∥p

∥∥, ∀i ≥ 1. (3.16)

Hence,

limn→∞

‖Jun,i‖ =∥∥Jp

∥∥, ∀i ≥ 1. (3.17)

This implies that {‖Jun,i‖}∞n=0, i ≥ 1 is bounded in E∗. Since E is reflexive, and so E∗ isreflexive, we can then assume that Jun,i ⇀ f0 ∈ E∗, for all i ≥ 1. In view of reflexivity ofE, we see that J(E) = E∗. Hence, there exists x ∈ E such that Jx = f0. Since

φ(xn+1, un,i) = ‖xn+1‖2 − 2〈xn+1, Jun,i〉 + ‖un,i‖2

= ‖xn+1‖2 − 2〈xn+1, Jun,i〉 + ‖Jun,i‖2,(3.18)


taking the limit inferior of both sides of (3.18) and in view of weak lower semicontinuity of‖ · ‖, we have

0 ≥ ∥∥p∥∥2 − 2

⟨p, f0

⟩+∥∥f0

∥∥2 =∥∥p

∥∥2 − 2⟨p, Jx

⟩+ ‖Jx‖2

=∥∥p

∥∥2 − 2

⟨p, Jx

⟩+ ‖x‖2 = φ(p, x),

(3.19)

that is, p = x. This implies that f0 = Jp and so Jun,i ⇀ Jp, for all i ≥ 1. It follows fromlimn→∞‖Jun,i‖ = ‖Jp‖, for all i ≥ 1 and Kadec-Klee property of E∗ that Jun,i → Jp, for all i ≥1. Note that J−1 : E∗ → E is hemicontinuous; it yields that un,i ⇀ p, for all i ≥ 1. Itthen follows from limn→∞‖un,i‖ = ‖p‖, for all i ≥ 1 and Kadec-Klee property of E thatlimn→∞un,i = p, for all i ≥ 1. Hence,

limn→∞

‖xn − un,i‖ = 0, ∀i ≥ 1. (3.20)

Since J is uniformly norm-to-norm continuous on bounded sets and limn→∞‖xn − un,i‖ =0, for all i ≥ 1, we obtain

limn→∞

‖Jxn − Jun,i‖ = 0, ∀i ≥ 1. (3.21)

Since {xn} is bounded, so are {zn,i}, {JTixn}, and {JSixn}. Also, since E is uniformly smooth,E∗ is uniformly convex. Then, from Lemma 2.9, we have

G(x∗, Jun,i) = G(x∗, Jθmn yn,i

) ≤ G(x∗, Jyn,i)

= G(x∗, (αn,iJxn + (1 − αn,i)Jzn,i))

= ‖x∗‖2−2αn,i〈x∗, Jxn〉−2(1−αn,i)〈x∗, Jzn,i〉+‖αn,iJxn+(1 − αn,i)Jzn,i‖2+2ρf(x∗)

≤ ‖x∗‖2−2αn,i〈x∗, Jxn〉−2(1−αn,i)〈x∗, Jzn,i〉+αn,i‖xn‖2+(1 − αn,i)‖zn,i‖2+2ρf(x∗)

= αn,iG(x∗, Jxn) + (1 − αn,i)G(x∗, Jzn,i)

= αn,iG(x∗, Jxn) + (1 − αn,i)G(x∗,

(β(1)n,i Jxn + β

(2)n,i JTixn + β

(3)n,i JSixn

))

≤αn,iG(x∗, Jxn)+ (1−αn,i)(‖x∗‖2−2β(1)n,i 〈x∗, Jxn〉 − 2β(2)n,i 〈x∗, JTixn〉

−2β(3)n,i 〈x∗, JSixn〉 + β(1)n,i ‖xn‖2+β

(2)n,i ‖Tixn‖2+β

(3)n,i ‖Sixn‖2

−β(1)n,i β(2)n,i g(‖Jxn − JTixn‖) + 2ρf(x∗)

)


= αn,iG(x∗, Jxn) + (1 − αn,i)(β(1)n,i G(x

∗, Jxn) + β(2)n,i G(x

∗, JTixn)

+β(3)n,i G(x∗, JSixn) − β(1)n,i β

(2)n,i g(‖Jxn − JTixn‖)

)

≤ αn,iG(x∗, Jxn) + (1 − αn,i)(β(1)n,i G(x

∗, Jxn) + β(2)n,i G(x

∗, Jxn)

+β(3)n,i G(x∗, Jxn) − β(1)n,i β


)

= αn,iG(x∗, xn) + (1 − αn,i)(G(x∗, Jxn) − β(1)n,i β


)

≤ G(x∗, Jxn) − (1 − αn,i)β(1)n,i β(2)n,i g(‖Jxn − JTixn‖).

(3.22)

It then follows that

(1 − α)β(1)n,i β(2)n,i g(‖Jxn − JTixn‖) ≤ (1 − αn,i)β(1)n,i β


≤ G(x∗, Jxn) −G(x∗, Jun,i).(3.23)

But

G(x∗, Jxn) −G(x∗, Jun,i) = ‖xn‖2 − ‖un,i‖2 − 2〈x∗, Jxn − Jun,i〉

≤∣∣∣‖xn‖2 − ‖un,i‖2

∣∣∣ + 2

∣∣∣〈x∗, Jxn − Jun,i〉

∣∣∣

≤ |‖xn‖ − ‖un,i‖|(‖xn‖ + ‖un,i‖) + 2‖x∗‖‖Jxn − Jun,i‖

≤ ‖xn − un,i‖(‖xn‖ + ‖un,i‖) + 2‖x∗‖‖Jxn − Jun,i‖.

(3.24)

From limn→∞‖xn − un,i‖ = 0 and limn→∞‖Jxn − Jun,i‖ = 0, we obtain

G(x∗, Jxn) −G(x∗, Jun,i) −→ 0, n −→ ∞. (3.25)

Using the condition lim infn→∞β(1)n,i β

(2)n,i > 0, we have

limn→∞

g(‖Jxn − JTixn‖) = 0, ∀i ≥ 1. (3.26)

By property of g, we have limn→∞‖Jxn − JTixn‖ = 0, for all i ≥ 1. Since J−1 is also uniformlynorm-to-norm continuous on bounded sets, we have

limn→∞

‖xn − Tixn‖ = 0, ∀i ≥ 1. (3.27)


Similarly, we can show that

limn→∞

‖xn − Sixn‖ = 0, ∀i ≥ 1. (3.28)

Since xn → p and Ti, Si are closed, we have p ∈ (⋂∞i=1 F(Ti)) ∩ (

⋂∞i=1 F(Si)).

Next, we show that p ∈ ⋂mk=1 EP(Fk). Now, by Lemma 2.12, we obtain

φ(un,i, yn,i

)= φ

(θmn yn,i, yn,i

)

≤ φ(x∗, yn,i) − φ(x∗, θmn yn,i

)

≤ φ(x∗, xn) − φ(x∗, un,i) −→ 0, n −→ ∞.

(3.29)

It then yields that limn→∞(‖un,i‖ − ‖yn,i‖) = 0. Since limn→∞‖un,i‖ = ‖p‖, i ≥ 1, we have

limn→∞

∥∥yn,i∥∥ =

∥∥p∥∥, i ≥ 1. (3.30)

Hence,

limn→∞

∥∥Jyn,i∥∥ =

∥∥Jp∥∥, i ≥ 1. (3.31)

This implies that {‖Jyn,i‖}∞n=0 is bounded in E∗. Since E is reflexive, and so E∗ is reflexive, wecan then assume that Jyn,i ⇀ f1 ∈ E∗. In view of reflexivity of E, we see that J(E) = E∗. Hence,there exists z ∈ E such that Jz = f1. Since

φ(un,i, yn,i

)= ‖un,i‖2 − 2

⟨un,i, Jyn,i

⟩+∥∥yn,i

∥∥2

= ‖un,i‖2 − 2⟨un,i, Jyn,i

⟩+∥∥Jyn,i

∥∥2,

(3.32)


0 ≥ ∥∥p

∥∥2 − 2

⟨p, f1

⟩+∥∥f1

∥∥2 =

∥∥p

∥∥2 − 2

⟨p, Jz

⟩+ ‖Jz‖2

=∥∥p

∥∥2 − 2

⟨p, Jz

⟩+ ‖z‖2 = φ(p, z),

(3.33)

that is, p = z. This implies that f1 = Jp and so Jyn,i ⇀ Jp. It follows from limn→∞‖Jyn,i‖ =‖Jp‖ and Kadec-Klee property of E∗ that Jyn,i → Jp. Note that J−1 : E∗ → E is hem-icontinuous; it yields that yn,i ⇀ p. It then follows from limn→∞‖yn,i‖ = ‖p‖ and Kadec-Klee property of E that limn→∞yn,i = p, i ≥ 1. By the fact that θkn, k = 1, 2, . . . , m is relativelynonexpansive and using Lemma 2.12 again, we have that

φ(θknyn,i, yn,i

)≤ φ(x∗, yn,i

) − φ(x∗, θknyn,i

)

≤ φ(x∗, xn) − φ(x∗, θknyn,i

).

(3.34)


Observe that

φ(x∗, un,i) = φ(x∗, θmn yn,i

)

= φ(x∗, TFmrm,nT

Fm−1rm−1,n · · ·TFkrk,nTFk−1rk−1 ,n · · ·TF2

r2,nTF1r1 ,nyn,i

)

= φ(x∗, TFmrm,nT

Fm−1rm−1,n · · ·θknyn,i

)

≤ φ(x∗, θknyn,i

).

(3.35)

Using (3.35) in (3.34), we obtain

φ(θknyn,i, yn,i

)≤ φ(x∗, xn) − φ(x∗, un,i) −→ 0, n −→ ∞. (3.36)

It then yields that limn→∞(‖θknyn,i‖ − ‖yn,i‖) = 0. Since limn→∞‖yn,i‖ = ‖p‖, we have

limn→∞

∥∥∥θknyn,i∥∥∥ =

∥∥p

∥∥, k = 1, 2, . . . , m. (3.37)

This implies that {‖θknyn,i‖}∞n=0 is bounded in E. Since E is reflexive, we can then assume thatθknyn,i ⇀ w ∈ E. Since

φ(θknyn,i, yn,i

)=∥∥∥θknyn,i

∥∥∥2− 2

⟨θknyn,i, Jyn,i

⟩+∥∥yn,i

∥∥2

=∥∥∥θknyn,i

∥∥∥2− 2

⟨θknyn,i, Jyn,i

⟩+∥∥Jyn,i

∥∥2,

(3.38)


0 ≥ ‖w‖2 − 2⟨w, Jp

⟩+∥∥p

∥∥2 = ‖w‖2 − 2⟨w, Jp

⟩+∥∥Jp

∥∥2

= φ(w, p

),

(3.39)

that is, p = w. This implies that θknyn,i ⇀ p. It follows from limn→∞‖θknyn,i‖ = ‖p‖ and Kadec-Klee property of E that

θknyn,i −→ p, n −→ ∞, k = 1, 2, . . . , m. (3.40)

Similarly, limn→∞‖p − θk−1n yn,i‖ = 0, k = 1, 2, . . . , m. This further implies that

limn→∞

∥∥∥θknyn,i − θk−1n yn,i

∥∥∥ = 0, i ≥ 1. (3.41)


Also, since J is uniformly norm-to-norm continuous on bounded sets and using (3.41), weobtain

limn→∞

∥∥∥Jθknyn,i − Jθk−1n yn,i∥∥∥ = 0, i ≥ 1. (3.42)

Since lim infn→∞rk,n > 0 (k = 1, 2, . . . , m),

limn→∞

∥∥Jθknyn,i − Jθk−1n yn,i∥∥

rk,n= 0. (3.43)

By Lemma 2.11, we have that for each k = 1, 2, . . . , m

Fk(θknyn,i, y

)+

1rk,n

⟨y − θknyn,i, Jθknyn,i − Jθk−1n yn,i

⟩≥ 0, ∀y ∈ C. (3.44)

Furthermore, using (A2), we obtain

1rk,n

⟨y − θknyn,i, Jθknyn,i − Jθk−1n yn,i

⟩≥ Fk

(y, θknyn,i

). (3.45)

By (A4), (3.43), and θknyn,i → p, we have for each k = 1, 2, . . . , m

Fk(y, p

) ≤ 0, ∀y ∈ C. (3.46)

For fixed y ∈ C, let zt,y := ty + (1 − t)p for all t ∈ (0, 1]. This implies that zt,y ∈ C. This yieldsthat Fk(zt,y, p) ≤ 0. It follows from (A1) and (A4) that

0 = Fk(zt,y, zt,y

) ≤ tFk(zt,y, y

)+ (1 − t)Fk

(zt,y, p

)

≤ tFk(zt,y, y

),

(3.47)

and hence

0 ≤ Fk(zt,y, y

). (3.48)

From condition (A3), we obtain

Fk(p, y

) ≥ 0, ∀y ∈ C. (3.49)

This implies that p ∈ EP(Fk), k = 1, 2, . . . , m. Thus, p ∈ ⋂mk=1 EP(Fk). Hence, we have p ∈ Ω =

⋂mk=1 EP(Fk) ∩ (

⋂∞n=0 F(Ti)) ∩ (

⋂∞i=1 F(Si)).


Finally, we show that p = Πf

Ωx0. Since Ω =⋂mk=1 EP(Fk) ∩ (

⋂∞n=0 F(Ti)) ∩ (

⋂∞i=1 F(Si))

is a closed and convex set, from Lemma 2.6, we know that Πf

Fx0 is single valued and denote

w = Πf

Ωx0. Since xn = Πf

Cnx0 andw ∈ Ω ⊂ Cn, we have

G(xn, Jx0) ≤ G(w, Jx0), ∀n ≥ 1. (3.50)

We know that G(ξ, Jϕ) is convex and lower semicontinuous with respect to ξ when ϕ is fixed.This implies that

G(p, Jx0

) ≤ lim infn→∞

G(xn, Jx0) ≤ lim supn→∞

G(xn, Jx0) ≤ G(w, Jx0). (3.51)

From the definition of Πf

Ωx0 and p ∈ Ω, we see that p = w. This completes the proof.

Take f(x) = 0 for all x ∈ E in Theorem 3.1, then G(ξ, Jx) = φ(ξ, x) and Πf

Cx0 = ΠCx0.Then we obtain the following corollary.

Corollary 3.2. Let E be a uniformly smooth and strictly convex real Banach space which also hasKadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k = 1, 2, . . . , m,let Fk be a bifunction from C×C satisfying (A1)–(A4). Suppose {Ti}∞i=1 and {Si}∞i=1 are two countablefamilies of closed relatively quasi-nonexpansive mappings ofC into itself such thatΩ :=

⋂mk=1 EP(Fk)∩

(⋂∞n=1 F(Ti))∩ (

⋂∞n=1 F(Si))/= ∅. Suppose {xn}∞n=0 is iteratively generated by x0 ∈ C, C1,i = C, C1 =

∩∞i=1C1,i, x1 = ΠC1x0,


(2)n,i JTixn + β

(3)n,i JSixn

),


un,i = TFmrm,nT

Fm−1rm−1,n · · ·TF2

r2 ,nTF1r1,nyn,i,

Cn+1,i ={z ∈ Cn,i : φ(z, un,i) ≤ φ(z, xn)

},

Cn+1 =∞⋂

i=1

Cn+1,i,

xn+1 = ΠCn+1x0, n ≥ 1,

(3.52)

with the conditions


(2)n,i > 0,


(3)n,i > 0,



Then, {xn}∞n=0 converges strongly toΠΩx0.


Corollary 3.3 (see Li et al. [29]). Let E be a uniformly convex real Banach space which is alsouniformly smooth. Let C be a nonempty, closed, and convex subset of E. Suppose T is a relativelynonexpansive mapping of C into itself such that Ω := F(T)/= ∅. Let f : E → � be a convex andlower semicontinuous mapping with C ⊂ int(D(f)), and suppose {xn}∞n=0 is iteratively generated byx0 ∈ C, C0 = C,

yn = J−1(αnJxn + (1 − αn)JTxn),Cn+1 =

{w ∈ Cn : G

(w, Jyn

) ≤ G(w, Jxn)},

xn+1 = Πf

Cn+1x0, n ≥ 0.

(3.53)

Suppose {αn}∞n=1 is a sequence in (0, 1) such that lim supn→∞αn < 1. Then, {xn}∞n=0 convergesstrongly toΠΩx0.

Corollary 3.4 (see Takahashi and Zembayashi [9]). Let E be a uniformly convex real Banachspace which is also uniformly smooth. Let C be a nonempty, closed, and convex subset of E. Let F bea bifunction from C × C satisfying (A1)–(A4). Suppose T is a relatively nonexpansive mapping of Cinto itself such that Ω := EP(F) ∩ F(T)/= ∅. Let {xn}∞n=0 be iteratively generated by x0 ∈ C, C1 = C,x1 = ΠC1x0,

yn = J−1(αn,iJxn + (1 − αn,i)JTxn),

F(un, y

)+

1rn


⟩ ≥ 0, ∀y ∈ C,

Cn+1 ={w ∈ Cn : φ(w, un) ≤ φ(w, xn)

},

xn+1 = ΠCn+1x0, n ≥ 1,

(3.54)

where J is the duality mapping on E. Suppose {αn,i}∞n=1 is a sequence in (0, 1) such thatlim infn→∞αn,i(1 − αn,i) > 0 and {rn}∞n=1 ⊂ (0,∞) satisfying lim infn→∞rn > 0. Then, {xn}∞n=0converges strongly to ΠΩx0.

4. Applications

Let A be a monotone operator from C into E∗, the classical variational inequality is to findx∗ ∈ C such that

⟨y − x,Ax∗⟩ ≥ 0, ∀y ∈ C. (4.1)

The set of solutions of (4.1) is denoted by VI(C,A).Let ϕ : C → � be a real-valued function. The convex minimization problem is to find

x∗ ∈ C such that

ϕ(x∗) ≤ ϕ(y), ∀y ∈ C. (4.2)

The set of solutions of (4.2) is denoted by CMP(ϕ).


The following lemmas are special cases of Lemmas 2.8 and Lemma 2.9 of [39].

Lemma 4.1. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexiveBanach space E. Assume that A : C → E∗ is a continuous and monotone operator. For r > 0 andx ∈ E, define a mapping TAr : E → C as follows:

TAr (x) ={z ∈ C :

⟨Az, y − z⟩ + 1

r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C

}. (4.3)

Then, the following hold:

(1) TAr is singlevalued,

(2) F(TAr ) = VI(C,A),

(3) VI(C,A) is closed and convex,

(4) φ(q, TAr x) + φ(TAr x, x) ≤ φ(q, x), for all q ∈ F(TAr ).

Lemma 4.2. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexiveBanach space E. Assume that ϕ : C → � is lower semicontinuous and convex. For r > 0 and x ∈ E,define a mapping Tϕr : E → C as follows:

Tϕr (x) =

{z ∈ C : ϕ

(y)+1r

⟨y − z, Jz − Jx⟩ ≥ ϕ(z), ∀y ∈ C

}. (4.4)

Then, the following hold:

(1) Tϕr is single valued,

(2) F(Tϕr ) = CMP(ϕ),

(3) CMP(ϕ) is closed and convex,

(4) φ(q, Tϕr x) + φ(Tϕr x, x) ≤ φ(q, x), for all q ∈ F(Tϕr ).

Then we obtain the following theorems from Theorem 3.1.

Theorem 4.3. Let E be a uniformly smooth and strictly convex real Banach space which also hasKadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k = 1, 2, . . . , m,let Ak be a continuous and monotone operator from C into E∗. Suppose {Ti}∞i=1 and {Si}∞i=1 aretwo countable families of closed relatively quasi-nonexpansive mappings of C into itself such thatΩ :=

⋂mk=1 VI(C,Ak) ∩ (

⋂∞i=1 F(Ti)) ∩ (

⋂∞i=1 F(Si))/= ∅. Let f : E → � be a convex and lower


semicontinuous mapping with C ⊂ int(D(f)), and suppose {xn}∞n=0 is iteratively generated byx0 ∈ C, C1,i = C, C1 = ∩∞

i=1C1,i, x1 = Πf

C1x0,


(2)n,i JTixn + β

(3)n,i JSixn

),


un,i = TAmrm,nT

Am−1rm−1,n · · ·TA2

r2 ,nTA1r1,nyn,i,


Cn+1 =∞⋂

i=1

Cn+1,i,

xn+1 = Πf

Cn+1x0, n ≥ 1,

(4.5)

with the conditions


(2)n,i > 0,


(3)n,i > 0,




Ωx0.

Theorem 4.4. Let E be a uniformly smooth and strictly convex real Banach space which also hasKadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k = 1, 2, . . . , m,let ϕk : C → � be lower semicontinuous and convex. Suppose {Ti}∞i=1 and {Si}∞i=1 are twocountable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω :=⋂mk=1 CMP(ϕk) ∩ (

⋂∞i=1 F(Ti)) ∩ (

⋂∞i=1 F(Si))/= ∅. Let f : E → � be a convex and lower semi-

continuous mapping with C ⊂ int(D(f)), and suppose {xn}∞n=0 is iteratively generated by x0 ∈ C,

C1,i = C, C1 =⋂∞i=1 C1,i, x1 = Πf

C1x0,


(2)n,i JTixn + β

(3)n,i JSixn

),


un,i = Tϕmrm,nT

ϕm−1rm−1,n · · ·T

ϕ2r2 ,nT

ϕ1r1,nyn,i,


Cn+1 =∞⋂

i=1

Cn+1,i,

xn+1 = Πf

Cn+1x0, n ≥ 1,

(4.6)

with the conditions



(2)n,i > 0,


(3)n,i > 0,




Ωx0.

References

[1] C. E. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notesin Mathematics, Springer, London, UK, 2009.

[2] W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and Applications, Yokohama Publishers,Yokohama, Japan, 2000.

[3] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.[4] D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators

in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001.[5] D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in

reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489–508,2003.

[6] Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with appli-cations to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323–339, 1996.

[7] S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansivemappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

[8] X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banachspace,” Nonlinear Analysis, vol. 67, no. 6, pp. 1958–1965, 2007.

[9] W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equi-librium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol.2008, Article ID 528476, 11 pages, 2008.

[10] W. Nilsrakoo and S. Saejung, “Strong convergence to common fixed points of countable relativelyquasi-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 312454, 19pages, 2008.

[11] X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium prob-lems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics,vol. 225, no. 1, pp. 20–30, 2009.

[12] K. Wattanawitoon and P. Kumam, “Strong convergence theorems by a new hybrid projectionalgorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansivemappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 1, pp. 11–20, 2009.

[13] A. Chinchuluun, P. Pardalos, A. Migdalas, and L. Pitsoulis, Eds., Pareto optimality, Game Theory andEquilibria, vol. 17, Springer, New York, NY, USA, 2008.

[14] P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Non-linear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

[15] C. A. Floudas, A. Christodoulos, and P. M. Pardalos, Eds., Encyclopedia of Optimization, Springer, NewYork, NY, USA, 2nd edition, 2009.

[16] F. Giannessi, A. Maugeri, and P. M. Pardalos, Eds., Equilibrium Problems: Nonsmooth Optimization andVariational Inequality Models, vol. 58, Springer, New York, NY, USA, 2002.

[17] L. Lin, “System of generalized vector quasi-equilibrium problems with applications to fixed pointtheorems for a family of nonexpansive multivalued mappings,” Journal of Global Optimization, vol. 34,no. 1, pp. 15–32, 2006.

[18] Y. Liu, “A general iterativemethod for equilibriumproblems and strict pseudo-contractions in Hilbertspaces,” Nonlinear Analysis, vol. 71, no. 10, pp. 4852–4861, 2009.

[19] A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,”Journal of Nonlinear and Convex Analysis, vol. 9, no. 1, pp. 37–43, 2008.

[20] P. M. Pardalos, T. M. Rassias, and A. A. Khan, Eds., Nonlinear Analysis and Variational Problems, vol. 35of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2010.


[21] S. Plubtieng and R. Punpaeng, “A new iterative method for equilibrium problems and fixed pointproblems of nonexpansive mappings and monotone mappings,” Applied Mathematics and Computa-tion, vol. 197, no. 2, pp. 548–558, 2008.

[22] X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibriumproblems and variational inequality problems,”Mathematical and Computer Modelling, vol. 48, no. 7-8,pp. 1033–1046, 2008.

[23] Y. Su, M. Shang, and X. Qin, “An iterative method of solution for equilibrium and optimizationproblems,” Nonlinear Analysis, vol. 69, no. 8, pp. 2709–2719, 2008.


[25] R. Wangkeeree, “An extragradient approximation method for equilibrium problems and fixed pointproblems of a countable family of nonexpansive mappings,” Journal of Fixed Point Theory andApplications, vol. 2008, Article ID 134148, 17 pages, 2008.


[27] S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of tworelatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2,pp. 103–115, 2007.

[28] X. L. Qin, Y. J. Cho, S. M. Kang, and H. Y. Zhou, “Convergence of a hybrid projection algorithm inBanach spaces,” Acta Applicandae Mathematicae, vol. 108, no. 2, pp. 299–313, 2009.

[29] X. Li, N. Huang, and D. O’Regan, “Strong convergence theorems for relatively nonexpansivemappings in Banach spaces with applications,” Computers & Mathematics with Applications, vol. 60,no. 5, pp. 1322–1331, 2010.

[30] P. Cholamjiak and S. Suantai, “Convergence analysis for a system of equilibrium problems and acountable family of relatively quasi-nonexpansive mappings in Banach spaces,” Abstract and AppliedAnalysis, vol. 2010, Article ID 141376, 17 pages, 2010.

[31] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applica-tions,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 ofLecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996.

[32] Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations inBanach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994.

[33] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 ofMathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1990.


[35] K. Q. Wu and N. J. Huang, “The generalised f -projection operator with an application,” Bulletin of theAustralian Mathematical Society, vol. 73, no. 2, pp. 307–317, 2006.

[36] J. Fan, X. Liu, and J. Li, “Iterative schemes for approximating solutions of generalized variationalinequalities in Banach spaces,” Nonlinear Analysis, vol. 70, no. 11, pp. 3997–4007, 2009.

[37] S. Chang, J. K. Kim, and X. R.Wang, “Modified block iterative algorithm for solving convex feasibilityproblems in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 869684, 14pages, 2010.

[38] Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterationswith errors for asymptotically nonexpansive mappings,” Computers & Mathematics with Applications,vol. 47, no. 4-5, pp. 707–717, 2004.

[39] W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problemsand relatively nonexpansive mappings in Banach spaces,”Nonlinear Analysis, vol. 70, no. 1, pp. 45–57,2009.


Research ArticleA New Hybrid Algorithm for a Pair ofQuasi-φ-Asymptotically NonexpansiveMappings and Generalized MixedEquilibrium Problems in Banach Spaces

Jinhua Zhu and Shih-Sen Chang

Department of Mathematics, Yibin University, Yibin 644007, China

Correspondence should be addressed to Shih-Sen Chang, [email protected]

Received 22 February 2011; Accepted 20 May 2011


Copyright q 2011 J. Zhu and S.-S. Chang. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The purpose of this paper is, by using a new hybrid method, to prove a strong convergencetheorem for finding a common element of the set of solutions for a generalized mixed equilibriumproblem, the set of solutions for a variational inequality problem, and the set of common fixedpoints for a pair of quasi-φ-asymptotically nonexpansive mappings. Under suitable conditionssome strong convergence theorems are established in a uniformly smooth and strictly convexBanach space with Kadec-Klee property. The results presented in the paper improve and extendsome recent results.

1. Introduction

Throughout this paper, we denote by N and R the sets of positive integers and real numbers,respectively. We also assume that E is a real Banach space, E∗ is the dual space of E, C is anonempty closed convex subset of E, and 〈·, ·〉 is the pairing between E and E∗.

Let ψ : C → R be a real-valued function,Θ : C×C → R a bifunction, andA : C → E∗

a nonlinear mapping. The “so-called” generalizedmixed equilibrium problem is to find u ∈ Csuch that

Θ(u, y

)+⟨Au, y − u⟩ + ψ(y) − ψ(u) ≥ 0, ∀y ∈ C. (1.1)

The set of solutions for (1.1) is denoted by Ω, that is,

Ω ={u ∈ C : Θ

(u, y

)+⟨Au, y − u⟩ + ψ(y) − ψ(u) ≥ 0, ∀y ∈ C}. (1.2)


Special examples are as follows.

(I) If ψ = 0, the problem (1.1) is equivalent to finding u ∈ C such that

Θ(u, y

)+⟨Au, y − u⟩ ≥ 0, ∀y ∈ C, (1.3)

which is called the generalized equilibrium problem. The set of solutions for (1.3)is denoted by GEP.

(II) If A = 0, the problem (1.1) is equivalent to finding u ∈ C such that

Θ(u, y

)+ ψ

(y) − ψ(u) ≥ 0, ∀y ∈ C, (1.4)

which is called the mixed equilibrium problem (MEP) [1]. The set of solutions for(1.4) is denoted by MEP.

(III) If Θ = 0, the problem (1.1) is equivalent to finding u ∈ C such that

⟨Au, y − u⟩ + ψ(y) − ψ(u) ≥ 0, ∀y ∈ C, (1.5)

which is called the mixed variational inequality of Browder type (VI) [2]. The setof solutions for (1.5) is denoted by VI(C,A, ψ).

(IV) If ψ = 0 and A = 0, the problem (1.1) is equivalent to finding u ∈ C such that

Θ(u, y

) ≥ 0, ∀y ∈ C, (1.6)

which is called the equilibrium problem. The set of solutions for (1.6) is denoted byEP(Θ).

(V) If ψ = 0 and Θ = 0, the problem (1.1) is equivalent to finding u ∈ C such that

⟨Au, y − u⟩ ≥ 0, ∀y ∈ C, (1.7)

which is called the variational inequality of Browder type. The set of solutions for(1.7) is denoted by VI(C,A).

The problem (1.1) is very general in the sense that numerous problems in physics,optimiztion and economics reduce to finding a solution for (1.1). Some methods have beenproposed for solving the generalized equilibrium problem and the equilibrium problem inHilbert space (see, e.g., [3–6]).

A mapping S : C → E is called nonexpansive if

∥∥Sx − Sy∥∥ ≤ ∥∥x − y∥∥, ∀x, y ∈ C. (1.8)

We denote the fixed point set of S by F(S).In 2008, S. Takahashi and W. Takahashi [6] proved some strong convergence theorems

for finding an element or a common element of EP, EP(f) ∩ F(S) or VI(C,A) ∩ F(S),respectively, in a Hilbert space.


Recently, Takahashi and Zembayashi [7, 8] proved someweak and strong convergencetheorems for finding a common element of the set of solutions for equilibrium (1.6) and theset of fixed points of a relatively nonexpansive mapping in a Banach space.

In 2010, Chang et al. [9] proved a strong convergence theorem for finding a commonelement of the set of solutions for a generalized equilibrium problem (1.3) and the set ofcommon fixed points of a pair of relatively nonexpansive mappings in a Banach space.

Motivated and inspired by [4–9], we intend in this paper, by using a new hybridmethod, to prove a strong convergence theorem for finding a common element of the setof solutions for a generalized mixed equilibrium problem (1.1) and the set of common fixedpoints of a pair of quasi-φ-asymptotically nonexpansive mappings in a uniformly smoothand strictly convex Banach space with the Kadec-Klee property.

2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will beneeded in proving our main results.

The mapping J : E → 2E∗defined by

J(x) = {x∗ ∈ E∗ : 〈x, x∗〉 = ‖x‖ = ‖x∗‖}, x ∈ E, (2.1)

is called the normalized duality mapping. By the Hahn-Banach theorem, J(x)/= ∅ for eachx ∈ E.

In the sequel, we denote the strong convergence and weak convergence of a sequence{xn} by xn → x and xn ⇀ x, respectively.

A Banach space E is said to be strictly convex if ‖x+y‖/2 < 1 for all x, y ∈ U = {z ∈ E :‖z‖ = 1} with x /=y. E is said to be uniformly convex if, for each ε ∈ (0, 2], there exists δ > 0such that ‖x + y‖/2 < 1 − δ for all x, y ∈ U with ||x − y|| ≥ ε. E is said to be smooth if the limit

limt→ 0

∥∥x + ty∥∥ − ‖x‖t

(2.2)

exists for all x, y ∈ U. E is said to be uniformly smooth if the above limit exists uniformly inx, y ∈ U.

Remark 2.1. The following basic properties can be found in Cioranescu [10].

(i) If E is a uniformly smooth Banach space, then J is uniformly continuous on eachbounded subset of E.

(ii) If E is a reflexive and strictly convex Banach space, then J−1 is hemicontinuous.

(iii) If E is a smooth, strictly convex, and reflexive Banach space, then J is singlevalued,one-to-one and onto.

(iv) A Banach space E is uniformly smooth if and only if E∗ is uniformly convex.

(v) Each uniformly convex Banach space E has the Kadec-Klee property, that is, for anysequence {xn} ⊂ E, if xn ⇀ x ∈ E and ||xn|| → ||x||, then xn → x.


Next we assume that E is a smooth, strictly convex, and reflexive Banach space and Cis a nonempty closed convex subset of E. In the sequel, we always use φ : E × E → R

+ todenote the Lyapunov functional defined by

φ(x, y

)= ‖x‖2 − 2

⟨x, Jy

⟩+∥∥y

∥∥2, ∀x, y ∈ E. (2.3)

It is obvious from the definition of φ that

(‖x‖ − ∥∥y

∥∥)2 ≤ φ(x, y) ≤ (‖x‖ + ∥

∥y∥∥)2, ∀x, y ∈ E. (2.4)

Following Alber [11], the generalized projection ΠC : E → C is defined by

ΠC(x) = arg infy∈C

φ(y, x

), ∀x ∈ E. (2.5)

Lemma 2.2 (see [11, 12]). Let E be a smooth, strictly convex, and reflexive Banach space and C anonempty closed convex subset of E. Then, the following conclusions hold:

(a) φ(x,ΠCy) + φ(ΠCy, y) ≤ φ(x, y) for all x ∈ C and y ∈ E;(b) if x ∈ E and z ∈ C, then

z = ΠCx ⇐⇒ ⟨z − y, Jx − Jz⟩ ≥ 0, ∀y ∈ C; (2.6)

(c) for x, y ∈ E, φ(x, y) = 0 if and only x = y.

Remark 2.3. If E is a real Hilbert space H, then φ(x, y) = ||x − y||2 and ΠC is the metricprojection PC ofH onto C.

Let E be a smooth, strictly, convex and reflexive Banach space, C a nonempty closedconvex subset of E, T : C → C a mapping, and F(T) the set of fixed points of T . A pointp ∈ C is said to be an asymptotic fixed point of T if there exists a sequence {xn} ⊂ C such thatxn ⇀ p and ||xn − Txn|| → 0. We denoted the set of all asymptotic fixed points of T by F(T).

Definition 2.4 (see [13]). (1) A mapping T : C → C is said to be relatively nonexpansive ifF(T)/= ∅, F(T) = F(T), and

φ(p, Tx

) ≤ φ(p, x), ∀x ∈ C, p ∈ F(T). (2.7)

(2) A mapping T : C → C is said to be closed if, for any sequence {xn} ⊂ C withxn → x and Txn → y, Tx = y.

Definition 2.5 (see [14]). (1) A mapping T : C → C is said to be quasi-φ-nonexpansive ifF(T)/= ∅ and

φ(p, Tx

) ≤ φ(p, x), ∀x ∈ C, p ∈ F(T). (2.8)


(2) A mapping T : C → C is said to be quasi-φ-asymptotically nonexpansive ifF(T)/= ∅ and there exists a real sequence {kn} ⊂ [1,∞)with kn → 1 such that

φ(p, Tnx

) ≤ knφ(p, x

), ∀n ≥ 1, x ∈ C, p ∈ F(T). (2.9)

(3) A pair of mappings T1, T2 : C → C is said to be uniformly quasi-φ-asymptoticallynonexpansive if F(T1)

⋂F(T2)/= ∅ and there exists a real sequence {kn} ⊂ [1,∞) with kn → 1

such that for i = 1, 2

φ(p, Tni x

) ≤ knφ(p, x

), ∀n ≥ 1, x ∈ C, p ∈ F(T1) ∩ F(T2). (2.10)

(4)Amapping T : C → C is said to be uniformly L-Lipschitz continuous if there existsa constant L > 0 such that

∥∥Tnx − Tny∥∥ ≤ L∥∥x − y∥∥, ∀x, y ∈ C. (2.11)

Remark 2.6. (1) From the definition, it is easy to know that each relatively nonexpansivemapping is closed.

(2) The class of quasi-φ-asymptotically nonexpansive mappings contains properly theclass of quasi-φ-nonexpansive mappings as a subclass, and the class of quasi-φ-nonexpansivemappings contains properly the class of relatively nonexpansive mappings as a subclass, butthe converse may be not true.

Lemma 2.7 (see [15]). Let E be a uniformly convex Banach space, r > 0 a positive number, andBr(0) a closed ball of E. Then, for any given subset {x1, x2, . . . , xN} ⊂ Br(0) and for any positivenumbers {λ1, λ2, . . . , λN} with ∑N

i=1 λi = 1, there exists a continuous, strictly increasing, and convexfunction g : [0, 2r) → [0,∞) with g(0) = 0 such that, for any i, j ∈ {1, 2, . . . ,N} with i < j,

∥∥∥∥∥

N∑

n=1

λnxn

∥∥∥∥∥

2

≤N∑

n=1

λn‖xn‖2 − λiλjg(∥∥xi − xj

∥∥). (2.12)

Lemma 2.8 (see [15]). Let E be a real uniformly smooth and strictly convex Banach space with theKadec-Klee property and C a nonempty closed convex subset of E. Let T : C → C be a closed andquasi-φ-asymptotically nonexpansive mapping with a sequence {kn} ⊂ [1,∞), kn → 1. Then F(T)is a closed convex subset of C.

For solving the generalized mixed equilibrium problem (1.1), let us assume that thefunction ψ : C → R is convex and lower semicontinuous, the nonlinear mapping A : C →E∗ is continuous and monotone, and the bifunction Θ : C × C → R satisfies the followingconditions:

(A1) Θ(x, x) = 0, for all x ∈ C,(A2) Θ is monotone, that is, Θ(x, y) + Θ(y, x) ≤ 0, ∀x, y ∈ C,(A3) limsupt↓0Θ(x + t(z − x), y) ≤ Θ(x, y) ∀x, z, y ∈ C,(A4) the function y �→ Θ(x, y) is convex and lower semicontinuous.


Lemma 2.9. Let E be a smooth, strictly convex, and reflexive Banach space and C a nonempty closedconvex subset of E. Let Θ : C × C → R a bifunction satisfying the conditions (A1)–(A4). Let r > 0and x ∈ E. Then, the followings hold.

(i) (Blum and Oettli [3]) there exists z ∈ C such that

Θ(z, y

)+1r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C. (2.13)

(ii) (Takahashi and Zembayashi [8]) Define a mapping Tr : E → C by

Tr(x) ={z ∈ C : Θ

(z, y

)+1r

⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C

}, x ∈ E. (2.14)

Then, the following conclusions hold:

(a) Tr is single-valued,

(b) Tr is a firmly nonexpansive-type mapping, that is, ∀z, y ∈ E,

⟨Trz − Try, JTrz − JTry

⟩ ≤ ⟨Trz − Try, Jz − Jy

⟩, (2.15)

(c) F(Tr) = EP(Θ) = F(Tr),

(d) EP(Θ) is closed and convex,

(e) φ(q, Trx) + φ(Trx, x) ≤ φ(q, x), ∀q ∈ F(Tr).

Lemma 2.10 (see [16]). Let E be a smooth, strictly convex, and reflexive Banach space, and C anonempty closed convex subset of E. Let A : C → E∗ be a continuous and monotone mapping,ψ : C → R a lower semicontinuous and convex function, andΘ : C×C → R a bifunction satisfyingconditions (A1)–(A4). Let r > 0 be any given number and x ∈ E any given point. Then, the followinghold.

(i) There exists u ∈ C such that

Θ(u, y

)+ 〈Au, y − u〉 + ψ(y) − ψ(u) + 1

r〈y − u, Ju − Jx〉 ≥ 0, ∀y ∈ C. (2.16)

(ii) If we define a mapping Kr : C → C by

Kr(x) ={u ∈ C : Θ

(u, y

)+⟨Au, y − u⟩ + ψ(y) − ψ(u)

+1r

⟨y − u, Ju − Jx⟩ ≥ 0, ∀y ∈ C

}, ∀x ∈ C.

(2.17)


Then, the mapping Kr has the following properties:

(a) Kr is single valued,

(b) Kr is a firmly nonexpansive-type mapping, that is,

⟨Krz −Kry, JKrz − JKry

⟩ ≤ ⟨Krz −Kry, Jz − Jy

⟩, ∀z, y ∈ E, (2.18)

(c) F(Kr) = Ω = F(Kr),

(d) Ω is closed and convex,

(e)

φ(q,Krz

)+ φ(Krz, z) ≤ φ

(q, z

), ∀q ∈ F(Kr), z ∈ E. (2.19)

Remark 2.11. It follows from Lemma 2.9 that the mapping Kr is a relatively nonexpansivemapping. Thus, it is quasi-φ-nonexpansive.

3. Main Results

In this section, we will prove a strong convergence theorem for finding a common elementof the set of solutions for the generalized mixed equilibrium problem (1.1) and the set ofcommon fixed points for a pair of quasi-φ-asymptotically nonexpansive mappings in Banachspaces.

Theorem 3.1. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. LetA : C → E∗ be a continuous and monotonemapping, ψ : C → R a lower semicontinuous and convex, function, and Θ : C × C → R abifunction satisfying conditions (A1)–(A4). Let S, T : C → C be two closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞) and kn → 1. Suppose thatS and T are uniformly L-Lipschitz continuous and that G = F(T)

⋂F(S)

⋂Ω is a nonempty and

bounded subset in C. Let {xn} be the sequence generated by

x0 ∈ C, C0 = C, Q0 = C,

zn = J−1(αnJxn + (1 − αn)JTnxn),

yn = J−1(βnJxn +

(1 − βn

)JSnzn

),

un ∈ C such that, ∀y ∈ C,

Θ(un, y

)+⟨Aun, y − un

⟩+ ψ

(y) − ψ(un) + 1

rn


⟩ ≥ 0,

Cn ={v ∈ Cn−1 : φ(v, zn) ≤ φ(v, xn) + ξn, φ(v, un) ≤ φ(v, xn) + (1 + kn)

(1 − βn

)ξn},

Qn = {z ∈ Qn−1 : 〈xn − z, Jx0 − Jxn〉 ≥ 0},xn+1 = ΠCn∩Qnx0, ∀n ≥ 0,

(3.1)


where J : E → E∗ is the normalized duality mapping, {αn} and {βn} are sequences in [0, 1] and{rn} ⊂ [a,∞) for some a > 0, ξn = supu∈G(kn − 1)φ(u, xn). Suppose that the following conditionsare satisfied:

(i) lim infn→∞αn(1 − αn) > 0,

(ii) lim infn→∞βn(1 − βn) > 0.

Then {xn} converges strongly to ΠF(S)∩F(T)∩Ωx0, where ΠF(S)∩F(T)∩Ω is the generalized projection ofE onto F(S) ∩ F(T) ∩Ω.

Proof. Firstly, we define two functionsH : C × C → R and Kr : C → C by

H(x, y

)= Θ

(x, y

)+⟨Ax, y − x⟩ + ψ(y) − ψ(x), ∀x, y ∈ C,

Kr(x) ={u ∈ C : H

(u, y

)+1r〈y − u, Ju − Jx〉 ≥ 0, ∀y ∈ C

}, x ∈ C.

(3.2)

By Lemma 2.10, we know that the function H satisfies conditions (A1)–(A4) and Kr hasproperties (a)–(e). Therefore, (3.1) is equivalent to

x0 ∈ C, C0 = C, Q0 = C,


yn = J−1(βnJxn +

(1 − βn

)JSnzn

),


H(un, y

)+

1rn〈y − un, Jun − Jyn〉 ≥ 0,


(1 − βn

)ξn},

Qn = {z ∈ Qn−1 : 〈xn − z, Jx0 − Jxn〉 ≥ 0},

xn+1 = ΠCn∩Qnx0, ∀n ≥ 0.

(3.3)

We divide the proof of Theorem 3.1 into five steps.(I) First we prove that Cn andQn are both closed and convex subsets of C for all n ≥ 0.In fact, it is obvious that Qn is closed and convex for all n ≥ 0. Again we have that

φ(v, zn) ≤ φ(v, xn) + ξn ⇐⇒ 2〈v, Jxn − Jzn〉 ≤ ‖xn‖2 − ‖zn‖2 + ξn,

φ(v, un) ≤ φ(v, xn) + (1 + kn)(1 − βn

)ξn ⇐⇒ 2〈v, Jxn − Jun〉

≤ ‖xn‖2 − ‖un‖2 + (1 + kn)(1 − βn

)ξn.

(3.4)

Hence Cn, ∀n ≥ 0, is closed and convex, and so Cn ∩Qn is closed and convex for all n ≥ 0.


(II) Next we prove that F(T) ∩ F(S) ∩Ω ⊂ Cn ∩Qn, ∀n ≥ 0.Putting un = Krnyn, ∀n ≥ 0, by Lemma 2.10 and Remark 2.11, Krn is relatively

nonexpansive. Again since S and T are quasi-φ-asymptotically nonexpansive, for any givenu ∈ F(S) ∩ F(T) ∩Ω, we have that

φ(u, zn) = φ(u, J−1(αnJxn + (1 − αn)JTnxn)

)

= ‖u‖2 − 2〈u, αnJxn + (1 − αn)JTnxn〉 + ‖αnJxn + (1 − αn)JTnxn‖2

≤ ‖u‖2 − 2αn〈u, Jxn〉 − 2(1 − αn)〈u, JTnxn〉 + αn‖xn‖2

+ (1 − αn)‖Tnxn‖2 − αn(1 − αn)g(‖Jxn − JTnxn‖)= αnφ(u, xn) + (1 − αn)φ(u, Tnxn) − αn(1 − αn)g(‖Jxn − JTnxn‖)≤ αnφ(u, xn) + (1 − αn)knφ(u, xn) − αn(1 − αn)g(‖Jxn − JTnxn‖)≤ knφ(u, xn) − αn(1 − αn)g(‖Jxn − JTnxn‖)≤ φ(u, xn) + sup

p∈G(kn − 1)φ

(p, xn

) − αn(1 − αn)g(‖Jxn − JTnxn‖)

= φ(u, xn) + ξn − αn(1 − αn)g(‖Jxn − JTnxn‖)≤ φ(u, xn) + ξn.

(3.5)

From (3.5)we have that

φ(u, un) = φ(u,Krnyn

) ≤ φ(u, yn)

≤ φ(u, J−1

(βnJxn +

(1 − βn

)JSnzn

))

= ‖u‖2 − 2⟨u, βnJxn +

(1 − βn

)JSnzn

⟩+∥∥βnJxn +

(1 − βn

)JSnzn

∥∥2

≤ ‖u‖2 − 2βn〈u, Jxn〉 − 2(1 − βn

)〈u, JSnzn〉 + βn‖xn‖2

+(1 − β)‖Snzn‖2 − βn

(1 − βn

)g(‖Jxn − JSnzn‖)

= βnφ(u, xn) +(1 − βn

)φ(u, Snzn) − βn

(1 − βn


≤ βnφ(u, xn) +(1 − βn

)knφ(u, zn) − βn

(1 − βn


≤ βnφ(u, xn) +(1 − βn

)kn

(φ(u, xn) + ξn

) − βn(1 − βn


≤ βnφ(u, xn) +(1 − βn

)(φ(u, xn) + ξn

)+(1 − βn

)knξn − βn

(1 − βn


≤ φ(u, xn) +(1 − βn

)ξn +

(1 − βn

)knξn − βn

(1 − βn


≤ φ(u, xn) + (1 + kn)(1 − βn

)ξn − βn

(1 − βn


≤ φ(u, xn) + (1 + kn)(1 − βn

)ξn ∀n ≥ 0.

(3.6)

This implies that u ∈ Cn, ∀n ≥ 0, and so F(T) ∩ F(S) ∩Ω ⊂ Cn, ∀n ≥ 0.


Now we prove that F(T) ∩ F(S) ∩Ω ⊂ Cn ∩Qn, ∀n ≥ 0.In fact, from Q0 = C, we have that F(T) ∩ F(S) ∩ Ω ⊂ C0 ∩ Q0. Suppose that F(T) ∩

F(S)∩Ω ⊂ Ck ∩Qk, for some k ≥ 0. Now we prove that F(T)∩F(S)∩Ω ⊂ Ck+1 ∩Qk+1. In fact,since xk+1 = ΠCk∩Qkx0, we have that

〈xk+1 − z, Jx0 − Jxk+1〉 ≥ 0, ∀z ∈ Ck ∩Qk. (3.7)

Since F(T) ∩ F(S) ∩Ω ⊂ Ck ∩Qk, for any z ∈ F(T) ∩ F(S) ∩Ω, we have that

〈xk+1 − z, Jx0 − Jxk+1〉 ≥ 0. (3.8)

This shows that z ∈ Qk+1, and so F(T) ∩ F(S) ∩Ω ⊂ Qk+1. The conclusion is proved.(III) Now we prove that {xn} is bounded.From the definition of Qn, we have that xn = ΠQnx0, ∀n ≥ 0. Hence, from

Lemma 2.2(1),

φ(xn, x0) = φ(ΠQnx0, x0

) ≤ φ(u, x0) − φ(u,ΠQnx0

)

≤ φ(u, x0), ∀u ∈ F(T) ∩ F(S) ∩Ω ⊂ Qn, ∀n ≥ 0.(3.9)

This implies that {φ(xn, x0)} is bounded. By virtue of (2.4), {xn} is bounded. Denote

M = supn≥0

{‖xn‖} <∞. (3.10)

Since xn+1 = ΠCn∩Qnx0 ∈ Cn ∩ Qn ⊂ Qn and xn = ΠQnx0, from the definition of ΠQn , we havethat

φ(xn, x0) ≤ φ(xn+1, x0) ≤ (M + ‖x0‖)2, ∀n ≥ 0. (3.11)

This implies that {φ(xn, x0)} is nondecreasing, and so the limit limn→∞φ(xn, x0) exists.Without loss of generality, we can assume that

limn→∞

φ(xn, x0) = r ≥ 0. (3.12)

By the way, from the definition of {ξn}, (2.4), and (3.10), it is easy to see that

ξn = supu∈G

(kn − 1)φ(u, xn) ≤ supu∈G

(kn − 1)(‖u‖ +M)2 −→ 0 (as n −→ ∞). (3.13)

(IV)Now,we prove that {xn} converges strongly to some point p ∈ G = F(T)∩F(S)∩Ω.In fact, since {xn} is bounded in C and E is reflexive, there exists a subsequence {xni} ⊂

{xn} such that xni ⇀ p. Again since Qn is closed and convex for each n ≥ 0, it is weakly


closed, and so p ∈ Qn for each n ≥ 0. Since xn = ΠQnx0, from the defintion of ΠQn , we havethat

φ(xni , x0) ≤ φ(p, x0

), n ≥ 0. (3.14)

Since

lim infni →∞

φ(xni , x0) = lim infni →∞

{‖xni‖2 − 2〈xni , Jx0〉 + ‖x0‖2

}

≥ ∥∥p

∥∥2 − 2

⟨p, Jx0

⟩+ ‖x0‖2 = φ

(p, x0

),

(3.15)

we have that

φ(p, x0

) ≤ lim infni →∞

φ(xni , x0) ≤ lim supni →∞

φ(xni , x0) ≤ φ(p, x0

). (3.16)

This implies that limni →∞φ(xni , x0) = φ(p, x0), that is, ||xni || → ||p||. In view of the Kadec-Kleeproperty of E, we obtain that limn→∞xni = p.

Now we first prove that xn → p (n → ∞). In fact, if there exists a subsequence{xnj} ⊂ {xn} such that xnj → q, then we have that

φ(p, q

)= lim

ni →∞,nj →∞φ(xni , xnj

)≤ lim

ni →∞,nj →∞φ(xni , x0) − φ

(ΠQnj

x0, x0)

= limni →∞,nj →∞

φ(xni , x0) − φ(xnj , x0

)= 0

(by (3.12)

).

(3.17)

Therefore we have that p = q. This implies that

limn→∞

xn = p. (3.18)

Now we first prove that p ∈ F(T) ∩ F(S). In fact, by the construction of Qn, we havethat xn = ΠQnx0. Therefore, by Lemma 2.2(a) we have that

φ(xn+1, xn) = φ(xn+1,ΠQnx0

) ≤ φ(xn+1, x0) − φ(ΠQnx0, x0

)

= φ(xn+1, x0) − φ(xn, x0) −→ 0 (as n −→ ∞).(3.19)

In view of xn+1 ∈ Cn ∩Qn ⊂ Cn and noting the construction of Cn we obtain

φ(xn+1, zn) ≤ φ(xn+1, xn) + ξn,φ(xn+1, un) ≤ φ(xn+1, xn) + (1 + kn)

(1 − βn

)ξn.

(3.20)

From (3.13) and (3.19), we have that

limn→∞

φ(xn+1, un) = 0, limn→∞

φ(xn+1, zn) = 0. (3.21)


From (2.4) it yields that (||xn+1||−||un||)2 → 0 and (||xn+1||−||zn||)2 → 0. Since ||xn+1|| →||p||, we have that

‖un‖ −→ ∥∥p

∥∥, ‖zn‖ −→ ∥

∥p∥∥ (as n −→ ∞). (3.22)

Hence, we have that

‖Jun‖ −→ ∥∥Jp∥∥, ‖Jzn‖ −→ ∥∥Jp

∥∥ (as n −→ ∞). (3.23)

This implies that {Jzn} is bounded in E∗. Since E is reflexive, and so E∗ is reflexive,there exists a subsequence {Jzni} ⊂ {Jzn} such that Jzni ⇀ p0 ∈ E∗. In view of the reflexive-ness of E, we see that J(E) = E∗. Hence, there exists x ∈ E such that Jx = p0. Since

φ(xni+1, zni) = ‖xni+1‖2 − 2〈xni+1, Jzni〉 + ‖zni‖2 = ‖xni+1‖2 − 2〈xni+1, Jzni〉 + ‖Jzni‖2, (3.24)

taking lim infn→∞ on both sides of the equality above and in view of the weak lower semi-continuity of norm || · ||, it yields that

0 ≥ ∥∥p∥∥2 − 2

⟨p, p0

⟩+∥∥p0

∥∥2 =∥∥p

∥∥2 − 2⟨p, Jx

⟩+ ‖Jx‖2

=∥∥p

∥∥2 − 2⟨p, Jx

⟩+ ‖x‖2 = φ(p, x),

(3.25)

that is, p = x. This implies that p0 = Jp, and so Jzn ⇀ Jp. It follows from (3.23) and theKadec-Klee property of E∗ that Jzni → Jp (as n → ∞). Noting that J−1 : E∗ → E is hemicon-tinuous, it yields that zni ⇀ p. It follows from (3.22) and the Kadec-Klee property of E thatlimni →∞zni = p.

By the same way as given in the proof of (3.18), we can also prove that

limn→∞

zn = p. (3.26)

From (3.18) and (3.26), we have that

‖xn − zn‖ −→ 0 (as n −→ ∞). (3.27)

Since J is uniformly continuous on any bounded subset of E, we have that

‖Jxn − Jzn‖ −→ 0 (as n −→ ∞). (3.28)

For any u ∈ F(T)⋂F(S)⋂Ω, it follows from (3.5) that

αn(1 − αn)g(‖Jxn − Tnxn‖) ≤ φ(u, xn) − φ(u, zn) + ξn. (3.29)


Since

φ(u, xn) − φ(u, zn) = ‖xn‖2 − ‖zn‖2 − 2〈u, Jxn − Jzn〉

≤∣∣∣‖xn‖2 − ‖zn‖2

∣∣∣ + 2‖u‖ · ‖Jxn − Jzn‖

≤ ‖xn − zn‖(‖xn‖ + ‖zn‖) + 2‖u‖ · ‖Jxn − Jzn‖,

(3.30)

From (3.27) and (3.28), it follows that

φ(u, xn) − φ(u, zn) −→ 0 (as n −→ ∞). (3.31)

In view of condition (i) and lim infn→∞αn(1 − αn) > 0, we see that

g(‖Jxn − JTnxn‖) −→ 0 (as n −→ ∞). (3.32)


‖Jxn − JTnxn‖ −→ 0 (as n −→ ∞). (3.33)

Since xn → p and J is uniformly continuous, it yields that Jxn → Jp. Hence from (3.33) wehave that

JTnxn −→ Jp (as n −→ ∞). (3.34)

Since J−1 : E∗ → E is hemicontinuous, it follows that

Tnxn ⇀ p. (3.35)

On the other hand, we have that

∣∣‖Tnxn‖ −∥∥p

∥∥∣∣ =∣∣‖J(Tnxn)‖ −

∥∥Jp∥∥∣∣ ≤ ∥∥JTnxn − Jp

∥∥ −→ 0 (as n −→ ∞). (3.36)

This together with (3.35) shows that

Tnxn −→ p. (3.37)

Furthermore, by the assumption that T is uniformly L-Lipschitz continuous, we havethat

∥∥∥Tn+1xn − Tnxn∥∥∥ ≤

∥∥∥Tn+1xn − Tn+1xn+1∥∥∥ +

∥∥∥Tn+1xn+1 − xn+1∥∥∥ + ‖xn+1 − xn‖ + ‖xn − Tnxn‖

≤ (L + 1)‖xn+1 − xn‖ +∥∥∥Tn+1xn+1 − xn+1

∥∥∥ + ‖xn − Tnxn‖.(3.38)


This together with (3.18) and (3.37), yields ||Tn+1xn − Tnxn|| → 0 (as n → ∞). Hencefrom (3.37)we have that Tn+1xn → p, that is, TTnxn → p. In view of (3.37) and the closenessof T , it yields that Tp = p. This implies that p ∈ F(T).

By the same way as given in the proof of (3.23) to (3.31), we can also prove that

limn→∞

un = p, φ(u, xn) − φ(u, un) −→ 0 (as n −→ ∞). (3.39)

Since un = Krnyn, from (2.19), (3.6), (3.13), and (3.39), we have that

φ(un, yn

)= φ

(Krnyn, yn

) ≤ φ(u, yn) − φ(u, un)

≤ φ(u, xn) − φ(u, un) + (1 + kn)(1 − βn

)ξn −→ 0 (as n −→ ∞).

(3.40)

From (2.4) it yields that (||un|| − ||yn||)2 → 0. Since ||un|| → ||p||, we have that

∥∥yn∥∥ −→ ∥∥p

∥∥ (as n −→ ∞). (3.41)

Hence we have that

∥∥Jyn∥∥ −→ ∥∥Jp

∥∥ (as n −→ ∞). (3.42)

By the same way as given in the proof of (3.26), we can also prove that

limn→∞

yn = p. (3.43)

From (3.39) and (3.43) we have that

∥∥un − yn∥∥ −→ 0 (as n −→ ∞). (3.44)

Since J is uniformly continuous on any bounded subset of E, we have that

∥∥Jun − Jyn∥∥ −→ 0 (as n −→ ∞). (3.45)

For any u ∈ F(T)⋂F(S)⋂Ω, it follows from (3.6), (3.13), and (3.39) that

βn(1 − βn

)g(‖Jxn − Snzn‖) ≤ φ(u, xn) − φ(u, un) + (1 + kn)

(1 − βn

)ξn −→ 0. (3.46)

In view of condition (ii) and lim infn→∞βn(1 − βn) > 0, we see that

g(‖Jxn − JSnzn‖) −→ 0 (as n −→ ∞). (3.47)


‖Jxn − JSnzn‖ −→ 0 (as n −→ ∞). (3.48)


Since xn → p and J is uniformly continuous, it yields, Jxn → Jp. Hence from (3.48)we havethat

JSnzn −→ Jp (as n −→ ∞). (3.49)

Since J−1 : E∗ → E is hemicontinuous, it follows that

Snzn ⇀ p. (3.50)

On the other hand, we have that

∣∣‖Snzn‖ −

∥∥p

∥∥∣∣ =

∣∣‖J(Snzn)‖ −

∥∥Jp

∥∥∣∣ ≤ ∥

∥JSnzn − Jp∥∥ −→ 0 (as n −→ ∞). (3.51)

This together with (3.50) shows that

Snzn −→ p. (3.52)

Furthermore, by the assumption that S is uniformly L-Lipschitz continuous, we havethat

∥∥∥Sn+1zn − Snzn∥∥∥ ≤

∥∥∥Sn+1zn − Sn+1zn+1∥∥∥ +

∥∥∥Sn+1zn+1 − zn+1∥∥∥ + ‖zn+1 − zn‖ + ‖zn − Snzn‖

≤ (L + 1)‖zn+1 − zn‖ +∥∥∥Sn+1zn+1 − zn+1

∥∥∥ + ‖zn − Snzn‖.(3.53)

This together with (3.26) and (3.52), yields that ||Sn+1zn − Snzn|| → 0 (as n → ∞).Hence from (3.52) we have that Sn+1zn → p, that is, SSnzn → p. In view of (3.52) and thecloseness of T , it yields that Sp = p. This implies that p ∈ F(S).

Next we prove that p ∈ Ω. From (3.45) and the assumption that rn ≥ a, ∀n ≥ 0, wehave that

limn→∞

∥∥Jun − Jyn∥∥

rn= 0. (3.54)

Since un = Krnyn, we have that

H(un, y

)+

1rn


⟩ ≥ 0, ∀y ∈ C. (3.55)

Replacing n by nk in (3.55), from condition (A2), we have that

1rnk

⟨y − unk , Junk − Jynk

⟩ ≥ −H(unk , y

) ≥ H(y, unk

), ∀y ∈ C. (3.56)


By the assumption that y �→ H(x, y) is convex and lower semicontinuous, it is also weaklylower semicontinuous. Letting nk → ∞ in (3.55), from (3.54) and condition (A4), we havethatH(y, p) ≤ 0, ∀y ∈ C.

For t ∈ (0, 1] and y ∈ C, letting yt = ty + (1 − t)p, there are yt ∈ C andH(yt, p) ≤ 0. Byconditions (A1) and (A4), we have that

0 = H(yt, yt

) ≤ tH(yt, y

)+ (1 − t)H(

yt, p) ≤ tH(

yt, y). (3.57)

Dividing both sides of the above equation by t, we have that H(yt, y) ≥ 0, ∀y ∈ C. Lettingt ↓ 0, from condition (A3), we have that H(p, y) ≥ 0, ∀y ∈ C, that is, Θ(p, y) + 〈Ap, y − p〉 +ψ(y) − ψ(p) ≥ 0, ∀y ∈ C. Therefore p ∈ Ω, and so p ∈ F(T)⋂F(S)

⋂Ω.

(V) Finally, we prove that xn → ΠF(T)⋂F(S)

⋂Ωx0.

Let w = ΠF(T)⋂F(S)

⋂Ωx0. From w ∈ F(T)⋂F(S)

⋂Ω ⊂ Cn ∩Qn, and xn+1 = ΠCn∩Qnx0,

we have that

φ(xn+1, x0) ≤ φ(w,x0), ∀n ≥ 0. (3.58)

Since the norm is weakly lower semicontinuous, this implies that

φ(p, x0

)=∥∥p

∥∥2 − 2⟨p, Jx0

⟩+ ‖x0‖2 ≤ lim

nk →∞inf

{‖xnk‖2 − 2〈xnk , Jx0〉 + ‖x0‖2

}

≤ limnk →∞

infφ(xnk , x0) ≤ limnk →∞

supφ(xnk , x0) ≤ φ(w,x0).(3.59)

It follows from the definition of ΠF(T)⋂F(S)

⋂Ωx0 and (3.59) that we have p = w. Therefore,

xn → ΠF(T)⋂F(S)

⋂Ωx0. This completes the proof of Theorem 3.1.

Remark 3.2. Theorem 3.1 improves and extends the corresponding results in [7–9].

(a) For the framework of spaces, we extend the space from a uniformly smooth anduniformly convex Banach space to a uniformly smooth and strictly convex Banachspace with the Kadec-Klee property(note that each uniformly convex Banach spacemust have the Kadec-Klee property).

(b) For the mappings, we extend the mappings from nonexpansive mappings,relatively nonexpansive mappings, or weak relatively nonexpansive mappings to apair of quasi-φ-asymptotically nonexpansive mappings.

(c) For the equilibrium problem, we extend the generalized equilibrium problem to thegeneralized mixed equilibrium problem.

The following theorems can be obtained from Theorem 3.1 immediately.

Theorem 3.3. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. Let A : C → E∗ be a continuous andmonotone mapping and Θ : C × C → R a bifunction satisfying conditions (A1)–(A4). Let S, T :C → C be two closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence{kn} ⊂ [1,∞) and kn → 1. Suppose that S and T are uniformly L-Lipschitz continuous and that


G = F(T)⋂F(S)

⋂GEP is a nonempty and bounded subset in C. Let {xn} be the sequence generated

by

x0 ∈ C, C0 = C, Q0 = C,

zn = J−1(αnJxn + (1 − αn)JTnxn),yn = J−1

(βnJxn +

(1 − βn

)JSnzn

),


Θ(un, y

)+⟨Aun, y − un

⟩+

1rn


⟩ ≥ 0,


(1 − βn

)ξn},


(3.60)

where J : E → E∗ is the normalized duality mapping, {αn} and {βn} are sequences in [0, 1], and{rn} ⊂ [a,∞) for some a > 0, ξn = supu∈G(kn − 1)φ(u, xn). If {αn} and {βn} satisfy conditions(i)-(ii) in Theorem 3.1, then {xn} converges strongly toΠF(S)∩F(T)∩GEPx0, whereGEP is the set for thesolutions of generalized equilibrium problem (1.3).

Proof. Putting ψ = 0 in Theorem 3.1, the conclusion of Theorem 3.3 can be obtained fromTheorem 3.1.

Theorem 3.4. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. Let ψ : C → R be a lower semicontinuousand convex function and Θ : C × C → R a bifunction satisfying conditions (A1)–(A4). Let S, T :C → C be two closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence{kn} ⊂ [1,∞) and kn → 1. Suppose that S and T are uniformly L-Lipschitz continuous and thatG =F(T)

⋂F(S)

⋂MEP is a nonempty and bounded subset in C. Let {xn} be the sequence generated by

x0 ∈ C, C0 = C, Q0 = C,


(βnJxn +

(1 − βn

)JSnzn

),


Θ(un, y

)+ ψ

(y) − ψ(un) + 1

rn


⟩ ≥ 0,


(1 − βn

)ξn},


(3.61)

where J : E → E∗ is the normalized duality mapping, {αn} and {βn} are sequences in [0, 1], and{rn} ⊂ [a,∞) for some a > 0, ξn = supu∈G(kn − 1)φ(u, xn). If {αn} and {βn} satisfy conditions


(i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩MEPx0, where MEP is the set ofsolutions for mixed equilibrium problem (1.4).

Proof. Putting A = 0 in Theorem 3.1, the conclusion of Theorem 3.4 can be obtained fromTheorem 3.1.

Theorem 3.5. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. Let A : C → E∗ be a continuous andmonotone mapping and ψ : C → R a lower semicontinuous and convex function. Let S, T : C → Cbe two closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence{kn} ⊂ [1,∞) and kn → 1. Suppose that S and T are uniformly L-Lipschitz continuous and thatG = F(T)

⋂F(S)

⋂VI(C,A, ψ) is a nonempty and bounded subset in C. Let {xn} be the sequence

generated by

x0 ∈ C, C0 = C, Q0 = C,


(βnJxn +

(1 − βn

)JSnzn

),

un ∈ C such that, ∀y ∈ C,⟨Aun, y − un

⟩+ ψ

(y) − ψ(un) + 1

rn


⟩ ≥ 0,


(1 − βn

)ξn},


(3.62)

where J : E → E∗ is the normalized duality mapping, {αn} and {βn} are sequences in [0, 1], and{rn} ⊂ [a,∞) for some a > 0, ξn = supu∈G(kn − 1)φ(u, xn). If {αn} and {βn} satisfy conditions(i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩VI(C,A,ψ)x0, where VI(C,A, ψ) isthe set of solutions for the mixed variational inequality (1.5).

Proof. Putting Θ = 0 in Theorem 3.1, the conclusion of Theorem 3.5 can be obtained fromTheorem 3.1.

Theorem 3.6. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. Let Θ : C × C → R be a bifunction satisfyingconditions (A1)–(A4). Let S, T : C → C be two closed and uniformly quasi-φ-asymptoticallynonexpansive mappings with a sequence {kn} ⊂ [1,∞) and kn → 1. Suppose that S and T areuniformly L -Lipschitz continuous and that G = F(T)

⋂F(S)

⋂EP(Θ) is a nonempty and bounded

subset in C. Let {xn} be the sequence generated by

x0 ∈ C, C0 = C, Q0 = C,


yn = J−1(βnJxn +

(1 − βn

)JSnzn

),



Θ(un, y

)+

1rn


⟩ ≥ 0,


(1 − βn

)ξn},

Qn = {z ∈ Qn−1 : 〈xn − z, Jx0 − Jxn〉 ≥ 0},xn+1 = ΠCn∩Qn

x0, ∀n ≥ 0,

(3.63)

where J : E → E∗ is the normalized duality mapping, {αn} and {βn} are sequences in [0, 1], and{rn} ⊂ [a,∞) for some a > 0, ξn = supu∈G(kn − 1)φ(u, xn). If {αn} and {βn} satisfy conditions(i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩EP(Θ)x0, where EP(Θ) is the set ofsolutions for the equilibrium problem (1.6).

Proof. Putting ψ = 0 andA = 0 in Theorem 3.1, the conclusion of Theorem 3.6 can be obtainedfrom Theorem 3.1.

Theorem 3.7. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. LetA : C → E∗ be a continuous and monotonemapping and S, T : C → C two closed and uniformly quasi-φ-asymptotically nonexpansivemappings with a sequence {kn} ⊂ [1,∞) and kn → 1. Suppose that S and T are uniformly L-Lipschitz continuous and that G = F(T)

⋂F(S)

⋂VI(C,A) is a nonempty and bounded subset in C.

Let {xn} be the sequence generated by

x0 ∈ C, C0 = C, Q0 = C,


(βnJxn +

(1 − βn

)JSnzn

),

un ∈ C such that, ∀y ∈ C,⟨Aun, y − un

⟩+

1rn


⟩ ≥ 0,


(1 − βn

)ξn},


(3.64)

where J : E → E∗ is the normalized duality mapping, {αn} and {βn} are sequences in [0, 1], and{rn} ⊂ [a,∞) for some a > 0, ξn = supu∈G(kn − 1)φ(u, xn). If {αn} and {βn} satisfy conditions(i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩VI(C,A)x0, where VI(C,A) is theset of solutions for the variational inequality (1.7)

Proof. Putting ψ = 0 andΘ = 0 in Theorem 3.1, the conclusion of Theorem 3.7 can be obtainedfrom Theorem 3.1.

Theorem 3.8. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. LetA : C → E∗ be a continuous and monotone


mapping, ψ : C → R a lower semicontinuous and convex function, and Θ : C ×C → R a bifunctionsatisfying conditions (A1)–(A4). Let S : C → C be a closed and quasi-φ-asymptotically nonexpansivemappings with a sequence {kn} ⊂ [1,∞) and kn → 1. Suppose that S is uniformly L-Lipschitzcontinuous and that F(S)

⋂Ω is a nonempty and bounded subset in C. Let {xn} be the sequence

generated by

x0 ∈ C, C0 = C, Q0 = C,

yn = J−1(βnJxn +

(1 − βn

)JSnxn

),


Θ(un, y

)+⟨Aun, y − un

⟩+ ψ

(y) − ψ(un) + 1

rn


⟩ ≥ 0,

Cn ={v ∈ Cn−1 : φ(v, un) ≤ φ(v, xn) + ξn

},


(3.65)

where J : E → E∗ is the normalized duality mapping, {αn} and {βn} are sequences in [0, 1], and{rn} ⊂ [a,∞) for some a > 0, ξn = supu∈F(S)∩Ω(kn − 1)φ(u, xn). If {βn} satisfy condition (ii) inTheorem 3.1, then {xn} converges strongly toΠF(S)∩Ωx0.

Proof. Taking T = I in Theorem 3.1, we have that zn = xn, ∀n ≥ 0. Hence, the conclusion ofTheorem 3.8 is obtained.

Theorem 3.9. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. LetA : C → E∗ be a continuous and monotonemapping, ψ : C → R a lower semicontinuous and convex function, and Θ : C ×C → R a bifunctionsatisfying conditions (A1)–(A4). Suppose that Ω is a nonempty subset in C. Let {xn} be the sequencegenerated by

x0 ∈ C, C0 = C, Q0 = C,


Θ(un, y

)+⟨Aun, y − un

⟩+ ψ

(y) − ψ(un) + 1

rn

⟨y − un, Jun − Jxn

⟩ ≥ 0,

Cn ={v ∈ Cn−1 : φ(v, un) ≤ φ(v, xn)

},


(3.66)

where {rn} ⊂ [a,∞) for some a > 0. Then {xn} converges strongly toΠΩx0.

Proof. Taking T = S = I in Theorem 3.1, the conclusion is obtained.


Theorem 3.10. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Kleeproperty and C a nonempty closed convex subset of E. Let S, T : C → C be two closed and uniformlyquasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞) and kn → 1. Supposethat S and T are uniformly L-Lipschitz continuous and that F(T)

⋂F(S) is a nonempty and bounded

subset in C. Let {xn} be the sequence generated by

x0 ∈ C, C0 = C, Q0 = C,


yn = J−1(βnJxn +

(1 − βn

)JSnzn

),

un = ΠCyn,


(1 − βn

)ξn},


(3.67)

where J : E → E∗ is the normalized duality mapping, {αn} and {βn} are sequences in [0, 1], andξn = supu∈F(S)∩F(T)(kn − 1)φ(u, xn). If {αn} and {βn} satisfy conditions (i)-(ii) in Theorem 3.1, then{xn} converges strongly toΠF(S)∩F(T)x0.

Proof. Taking A = Θ = 0 and rn = 1, ∀n ≥ 0 in Theorem 3.1, the conclusion of Theorem 3.10 isobtained.

References

[1] L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed pointproblems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.

[2] F. E. Browder, “Existence and approximation of solutions of nonlinear variational inequalities,”Proceedings of the National Academy of Sciences of the United States of America, vol. 56, pp. 1080–1086,1966.



[5] A. Moudafi, “Second-order differential proximal methods for equilibrium problems,” Journal ofInequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 18, 2003.

[6] S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problemand a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications,vol. 69, no. 3, pp. 1025–1033, 2008.

[7] W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method forequilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications,vol. 2008, Article ID 528476, 11 pages, 2008.

[8] W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problemsand relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods &Applications, vol. 70, no. 1, pp. 45–57, 2009.

[9] S.-S. Chang, H. W. J. Lee, and C. K. Chan, “A new hybrid method for solving a generalizedequilibrium problem, solving a variational inequality problem and obtaining common fixed pointsin Banach spaces, with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 7,pp. 2260–2270, 2010.


[10] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.

[11] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties andapplications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178of Lecture Notes in Pure and Appl. Math., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.


[13] S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansivemappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

[14] H. Zhou, G. Gao, and B. Tan, “Convergence theorems of a modified hybrid algorithm for a familyof quasi-ϕ-asymptotically nonexpansive mappings,” Journal of Applied Mathematics and Computing,vol. 32, no. 2, pp. 453–464, 2010.

[15] S.-S. Chang, J. K. Kim, and X. R. Wang, “Modified block iterative algorithm for solving convexfeasibility problems in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID869684, 14 pages, 2010.

[16] S.-S. Zhang, “Generalized mixed equilibrium problem in Banach spaces,” Applied Mathematics andMechanics (English Edition), vol. 30, no. 9, pp. 1105–1112, 2009.


Research ArticleA New Composite General Iterative Scheme forNonexpansive Semigroups in Banach Spaces

Pongsakorn Sunthrayuth and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi(KMUTT), Bangmod, Bangkok 10140, Thailand


Received 1 February 2011; Accepted 19 March 2011


Copyright q 2011 P. Sunthrayuth and P. Kumam. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We introduce a new general composite iterative scheme for finding a common fixed point ofnonexpansive semigroups in the framework of Banach spaces which admit a weakly continuousduality mapping. A strong convergence theorem of the purposed iterative approximation methodis established under some certain control conditions. Our results improve and extend announcedby many others.

1. Introduction

Throughout this paper we denoted by � and �+ the set of all positive integers and all positivereal numbers, respectively. Let X be a real Banach space, and let C be a nonempty closedconvex subset ofX. Amapping T ofC into itself is said to be nonexpansive if ‖Tx−Ty‖ ≤ ‖x−y‖for each x, y ∈ C. We denote by F(T) the set of fixed points of T . We know that F(T) isnonempty ifC is bounded; for more detail see [1]. A one-parameter familyS = {T(t) : t ∈ �+}from C of X into itself is said to be a nonexpansive semigroup on C if it satisfies the followingconditions:

(i) T(0)x = x for all x ∈ C;(ii) T(s + t) = T(s) ◦ T(t) for all s, t ∈ �+ ;

(iii) for each x ∈ C the mapping t �→ T(t)x is continuous;

(iv) ‖T(t)x − T(t)y‖ ≤ ‖x − y‖ for all x, y ∈ C and t ∈ �+ .

We denote by F(S) the set of all common fixed points of S, that is, F(S) := ∩t∈�+F(T(t)) ={x ∈ C : T(t)x = x}. We know that F(S) is nonempty if C is bounded; see [2]. Recall that


a self-mapping f : C → C is a contraction if there exists a constant α ∈ (0, 1) such that‖f(x) − f(y)‖ ≤ α‖x − y‖ for each x, y ∈ C. As in [3], we use the notation

∏C to denote the

collection of all contractions on C, that is,∏

C = {f : C → C a contraction}. Note that eachf ∈ ∏

C has a unique fixed point in C.In the last ten years, the iterative methods for nonexpansive mappings have recently

been applied to solve convex minimization problems; see, for example, [3–5]. LetH be a realHilbert space, whose inner product and norm are denoted by 〈·, ·〉 and ‖ · ‖, respectively. LetA be a strongly positive bounded linear operator onH : that is, there is a constant γ > 0 withproperty

〈Ax, x〉 ≥ γ‖x‖2 ∀x ∈ H. (1.1)

A typical problem is to minimize a quadratic function over the set of the fixed points of anonexpansive mapping on a real Hilbert spaceH :

minx∈F

12〈Ax, x〉 − 〈x, b〉, (1.2)

where C is the fixed point set of a nonexpansive mapping T onH and b is a given point inH .In 2003, Xu [3] proved that the sequence {xn} generated by

x0 ∈ C chosen arbitrarily,

xn+1 = (I − αnA)Txn + αnu, ∀n ≥ 0,(1.3)

converges strongly to the unique solution of the minimization problem (1.2) provided thatthe sequence {αn} satisfies certain conditions. Using the viscosity approximation method,Moudafi [6] introduced the iterative process for nonexpansive mappings (see [3, 7] forfurther developments in both Hilbert and Banach spaces) and proved that if H is a realHilbert space, the sequence {xn} generated by the following algorithm:


xn+1 = αnf(xn) + (1 − αn)Txn, ∀n ≥ 0,(1.4)

where f : C → C is a contraction mapping with constant α ∈ (0, 1) and {αn} ⊂ (0, 1) satisfiescertain conditions, converges strongly to a fixed point of T in C which is unique solution x∗

of the variational inequality:

⟨(f − I)x∗, y − x∗⟩ ≤ 0, ∀y ∈ F(T). (1.5)

In 2006, Marino and Xu [8] combined the iterative method (1.3) with the viscosityapproximation method (1.4) considering the following general iterative process:


xn+1 = αnγf(xn) + (I − αnA)Txn, ∀n ≥ 0,(1.6)


where 0 < γ < γ/α. They proved that the sequence {xn} generated by (1.6) converges stronglyto a unique solution x∗ of the variational inequality:

⟨(γf −A)

x∗, y − x∗⟩ ≤ 0, ∀y ∈ F(T), (1.7)

which is the optimality condition for the minimization problem:

minx∈C

12〈Ax, x〉 − h(x), (1.8)

where C is the fixed point set of a nonexpansive mapping T and h is a potential function forγf (i.e., h′(x) = γf(x) for x ∈ H). Kim and Xu [9] studied the sequence generated by thefollowing algorithm:


yn = αnxn + (1 − αn)Txn,xn+1 = βnu +

(1 − βn

)yn, ∀n ≥ 0,

(1.9)

and proved strong convergence of scheme (1.9) in the framework of uniformly smoothBanach spaces. Later, yao, et al. [10] introduced a new iteration process by combining themodified Mann iteration [9] and the viscosity approximation method introduced byMoudafi[6]. Let C be a closed convex subset of a Banach space, and let T : C → C be a nonexpansivemapping such that F(T)/= ∅ and f ∈ ∏

C. Define {xn} in the following way:

x1 ∈ C chosen arbitrarily;

yn = αnxn + (1 − αn)Txn;xn+1 = βnf(xn) +

(1 − βn

)yn, ∀n ≥ 0,

(1.10)

where {αn} and {βn} are two sequences in (0, 1). They proved under certain different controlconditions on the sequences {αn} and {βn} that {xn} converges strongly to a fixed point of T .Recently, Chen and Song [11] studied the sequence generated by the algorithm in a uniformlyconvex Banach space, as follows:


xn+1 = αnf(xn) + (1 − αn) 1tn

∫ tn

0T(s)xnds, ∀n ∈ �,

(1.11)

and they proved that the sequence {xn} defined by (1.11) converges strongly to the uniquesolution of the variational inequality:

⟨(f − I)x∗, J(y − x∗)⟩ ≤ 0, ∀y ∈ F(T). (1.12)


In 2010, Sunthrayuth and Kumam [12] introduced the a general iterative scheme generatedby


xn+1 = αnγf(xn) + βnxn +((1 − βn

)I − αnA

) 1tn

∫ tn

0T(s)xnds, ∀n ≥ 0,

(1.13)

for the approximation of common fixed point of a one-parameter nonexpansive semigroup ina Banach space under some appropriate control conditions. They proved strong convergencetheorems of the iterative scheme which solve some variational inequality. Very recently,Kumam and Wattanawitoon [13] studied and introduced a new composite explicit viscosityiteration method of fixed point solutions of variational inequalities for nonexpansivesemigroups in Hilbert spaces. They proved strong convergence theorems of the compositeiterative schemes which solve some variational inequalities under some appropriate con-ditions. In the same year, Sunthrayuth et al. [14] introduced a general composite iterativescheme for nonexpansive semigroups in Banach spaces. They established some strong con-vergence theorems of the general iteration scheme under different control conditions.

In this paper, motivated by Yao et al. [10], Sunthrayuth, and Kumam [12] and Kumamand Wattanawitoon [13] we introduce a new general iterative algorithm (3.23) for findinga common point of the set of solution of some variational inequality for nonexpansivesemigroups in Banach spaces which admit a weakly continuous duality mapping and thenproved the strong convergence theorem generated by the proposed iterative scheme. Theresults presented in this paper improve and extend some others fromHilbert spaces to Banachspaces and some others as special cases.

2. Preliminaries

Throughout this paper, we write xn ⇀ x (resp., xn⇀∗x) to indicate that the sequence {xn}weakly (resp., weak∗) converges to x; as usual xn → x will symbolize strong convergence;also, a mapping I denote the identity mapping. Let X be a real Banach space, and let X∗ beits dual space. Let U = {x ∈ X : ‖x‖ = 1}. A Banach space X is said to be uniformly convexif, for each ε ∈ (0, 2], there exists a δ > 0 such that for each x, y ∈ U, ‖x − y‖ ≥ ε implies‖x + y‖/2 ≤ 1 − δ. It is known that a uniformly convex Banach space is reflexive and strictlyconvex (see also [15]). A Banach space is said to be smooth if the limit limt→ 0‖x + ty‖ − ‖x‖/texists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformlyfor x, y ∈ U.

Let ϕ : [0,∞) := �+ → �

+ be a continuous strictly increasing function such thatϕ(0) = 0 and ϕ(t) → ∞ as t → ∞. This function ϕ is called a gauge function . The dualitymapping Jϕ : X → 2X

∗associated with a gauge function ϕ is defined by

Jϕ(x) ={f∗ ∈ X∗ :

⟨x, f∗⟩ = ‖x‖ϕ(‖x‖),∥∥f∗∥∥ = ϕ(‖x‖), ∀x ∈ X}

, (2.1)

where 〈·, ·〉 denotes the generalized duality paring. In particular, the duality mapping withthe gauge function ϕ(t) = t, denoted by J , is referred to as the normalized duality mapping.Clearly, there holds the relation Jϕ(x) = (ϕ(‖x‖)/‖x‖)J(x) for each x /= 0 (see [16]).


Browder [16] initiated the study of certain classes of nonlinear operators by means ofthe duality mapping Jϕ. Following Browder [16], we say that Banach space X has a weaklycontinuous duality mapping if there exists a gauge function ϕ for which the duality mappingJϕ(x) is single-valued and continuous from the weak topology to the weak∗ topology; thatis, for each {xn} with xn ⇀ x, the sequence {J(xn)} converges weakly∗ to Jϕ(x). It is knownthat lp has a weakly continuous duality mapping with a gauge function ϕ(t) = tp−1 for all1 < p < ∞. Set Φ(t) =

∫ t0 ϕ(τ)dτ , for all t ≥ 0; then Jϕ(x) = ∂Φ(‖x‖), where ∂ denotes the

subdifferential in the sense of convex analysis (recall that the subdifferential of the convexfunction φ : X → � at x ∈ X is the set ∂φ(x) = {x∗ ∈ X;φ(y) ≥ φ(x) + 〈x∗, y − x〉, for all y ∈X}).

In a Banach space having a weakly continuous duality mapping Jϕ with a gaugefunction ϕ, we defined an operator A is to be strongly positive (see [17]) if there exists aconstant γ > 0 with the property

⟨Ax, Jϕ(x)

⟩ ≥ γ‖x‖ϕ(‖x‖), (2.2)

‖aI − bA‖ = sup‖x‖≤1

∣∣⟨(aI − bA)x, Jϕ(x)⟩∣∣, a ∈ [0, 1], b ∈ [−1, 1]. (2.3)

If X := H is a real Hilbert space, then the inequality (2.2) reduces to (1.1).The first part of the next lemma is an immediate consequence of the subdifferential

inequality and the proof of the second part can be found in [18].

Lemma 2.1 (see [18]). Assume that a Banach space X has a weakly continuous duality mapping Jϕwith gauge ϕ.

(i) For all x, y ∈ X, the following inequality holds:

Φ(∥∥x + y

∥∥) ≤ Φ(‖x‖) + ⟨y, Jϕ

(x + y

)⟩. (2.4)

In particular, for all x, y ∈ X,

∥∥x + y∥∥2 ≤ ‖x‖2 + 2

⟨y, J

(x + y

)⟩. (2.5)

(ii) Assume that a sequence {xn} in X converges weakly to a point x ∈ X. Then the followingidentity holds:

lim supn→∞

Φ(∥∥xn − y

∥∥) = lim supn→∞

Φ(‖xn − x‖) + Φ(∥∥y − x∥∥), ∀x, y ∈ X. (2.6)

Lemma 2.2 (see [17]). Assume that a Banach space X has a weakly continuous duality mapping Jϕwith gauge ϕ. Let A be a strongly positive linear bounded operator on X with a coefficient γ > 0 and0 < ρ ≤ ϕ(1)‖A‖−1. Then ‖I − ρA‖ ≤ ϕ(1)(1 − ργ).


Lemma 2.3 (see [11]). Let C be a closed convex subset of a uniformly convex Banach space X andlet S = {T(t) : t ∈ �

+} be a nonexpansive semigroup on C such that F(S)/= ∅. Then, for each r > 0and h ≥ 0,

limt→∞

supx∈C∩Br

∥∥∥∥∥1t

∫ t

0T(s)x ds − T(h)

(1t

∫ t

0T(s)x ds

)∥∥∥∥∥= 0. (2.7)

Lemma 2.4 (see [19]). Assume that {an} is a sequence of nonnegative real numbers such that

an+1 ≤(1 − μn

)an + δn, (2.8)

where {μn} is a sequence in (0, 1) and {δn} is a sequence in � such that

(i)∑∞

n=0 μn = ∞;

(ii) lim supn→∞(δn/μn) ≤ 0 or∑∞

n=0 |δn| <∞.

Then, limn→∞an = 0.

3. Main Results

Let X be a Banach space which admits a weakly continuous duality mapping Jϕ with gaugeϕ such that ϕ is invariant on [0, 1], and let C be a nonempty closed convex subset of X suchthat C ± C ⊂ C. Let S = {T(t) : t ∈ �

+} be a nonexpansive semigroup from C into itself, letf be a contraction mapping with a coefficient α ∈ (0, 1), let A be a strongly positive linearbounded operator with a coefficient γ > 0 such that 0 < γ < γϕ(1)/α, and let t ∈ (0, 1) suchthat t ≤ ϕ(1)‖A‖−1 which satisfies t → 0. Define the mapping Tft : C → C by

Tft := tγf + (I − tA)

1λt

∫λt

0T(s)ds (3.1)

to be a contraction mapping. Indeed, for each x, y ∈ C,

∥∥∥T

ft x − Tft y

∥∥∥ =

∥∥∥∥∥tγ(f(x) − f(y)) + (I − tA)

(1λt

∫λt

0

(T(s)x − T(s)y)ds

)∥∥∥∥∥

≤ tγ∥∥f(x) − f(y)∥∥ + ‖I − tA‖(

1λt

∫λt

0

∥∥T(s)x − T(s)y∥∥ds)

≤ tγα∥∥x − y∥∥ + ϕ(1)(1 − tγ)∥∥x − y∥∥

≤ (1 − t(ϕ(1)γ − γα))∥∥x − y∥∥.

(3.2)

Thus, by Banach contraction mapping principle, there exists a unique fixed point xt ∈ C, thatis,

xt = tγf(xt) + (I − tA)1λt

∫λt

0T(s)xt ds. (3.3)


Remark 3.1. We note that space lp has a weakly continuous duality mapping with a gaugefunction ϕ(t) = tp−1 for all 1 < p <∞. This shows that ϕ is invariant on [0, 1].

Lemma 3.2. Let X be a uniformly convex Banach space which admits a weakly continuous dualitymapping Jϕ with gauge ϕ such that ϕ is invariant on [0, 1], and let C be a nonempty closed convexsubset of X such that C ± C ⊂ C. Let S = {T(t) : t ∈ �

+} be a nonexpansive semigroup from Cinto itself such that F(S)/= ∅, let f be a contraction mapping with a coefficient α ∈ (0, 1), let A be astrongly positive linear bounded operator with a coefficient γ > 0 such that 0 < γ < γϕ(1)/α, and lett ∈ (0, 1) such that t ≤ ϕ(1)‖A‖−1 which satisfies t → 0. Then the net {xt} defined by (3.3) with{λt}0<t<1 is a positive real divergent sequence, converges strongly as t → 0 to a common fixed pointx∗, in which x∗ ∈ F(S), and is the unique solution of the variational inequality:

⟨γf(x∗) −Ax∗, Jϕ(x − x∗)⟩ ≤ 0, ∀x ∈ F(S). (3.4)

Proof. Firstly, we show the uniqueness of a solution of the variational inequality (3.4).Suppose that x, x∗ ∈ F(S) are solutions of (3.4); then

⟨γf(x∗) −Ax∗, Jϕ(x − x∗)⟩ ≤ 0,⟨γf(x) −Ax, Jϕ(x∗ − x)

⟩ ≤ 0.(3.5)

Adding up (3.5), we obtain

0 ≥ ⟨(γf(x∗) −Ax∗) − (

γf(x) −Ax), Jϕ(x − x∗)⟩

=⟨A(x − x∗), Jϕ(x − x∗)⟩ − γ⟨f(x) − f(x∗), Jϕ(x − x∗)⟩

≥ γ‖x − x∗‖ϕ‖x − x∗‖ − γ∥∥f(x) − f(x∗)∥∥∥∥Jϕ(x − x∗)∥∥

≥ γΦ(‖x − x∗‖) − γαΦ(‖x − x∗‖)=(γ − γα)Φ(‖x − x∗‖)

≥ (ϕ(1)γ − γα)Φ(‖x − x∗‖),

(3.6)

which is a contradiction, we must have x = x∗, and the uniqueness is proved. Here in after,we use x to denote the unique solution of the variational inequality (3.4).

Next, we show that {xt} is bounded. Indeed, for each p ∈ F(S), we have

∥∥xt − p

∥∥ =

∥∥∥∥∥t(γf(xt) −Ap

)+ (I − tA)

(1λt

∫λt

0

(T(s)xt − p

)ds

)∥∥∥∥∥

≤ t∥∥γf(xt) −Ap∥∥ + ‖I − tA‖ 1

λt

∫λt

0

∥∥T(s)xt − p∥∥ds

≤ tγ∥∥f(xt) − f(p)∥∥ + t

∥∥γf

(p) −Ap∥∥ + ϕ(1)

(1 − tγ)∥∥xt − p

∥∥

≤ (1 − t(ϕ(1)γ − γα))∥∥xt − p

∥∥ + t∥∥γf

(p) −Ap∥∥.

(3.7)


It follows that

∥∥xt − p∥∥ ≤ 1

ϕ(1)γ − γα∥∥γf

(p) −Ap∥∥. (3.8)

Hence, {xt} is bounded, so are {f(xt)} and {A((1/λt)∫λt0 T(s)xtds)}.

Next, we show that ‖xt − T(h)xt‖ → 0 as t → 0. We note that

∥∥∥∥∥xt − 1

λt

∫λt

0T(s)xt ds

∥∥∥∥∥= t

∥∥∥∥∥γf(xt) −A

(1λt

∫λt

0T(s)xt ds

)∥∥∥∥∥. (3.9)

Moreover, we note that

‖xt − T(h)xt‖ ≤∥∥∥∥∥xt − 1

λt

∫λt

0T(s)xt ds

∥∥∥∥∥+

∥∥∥∥∥1λt

∫λt

0T(s)xt ds − T(h)

(1λt

∫λt

0T(s)xt ds

)∥∥∥∥∥

+

∥∥∥∥∥T(h)

(1λt

∫λt

0T(s)xt ds

)

− T(h)xt∥∥∥∥∥

≤ 2

∥∥∥∥∥xt − 1

λt

∫λt

0T(s)xt ds

∥∥∥∥∥+

∥∥∥∥∥1λt

∫λt

0T(s)xt ds − T(h)

(1λt

∫λt

0T(s)xt ds

)∥∥∥∥∥,

(3.10)

for all h ≥ 0. Define the set K = {z ∈ C : ‖z − p‖ ≤ ‖γf(p) − Ap‖/(ϕ(1)γ − γα)}; then K is anonempty bounded closed convex subset of C which is T(s)-invariant for each h ≥ 0. Since{xt} ⊂ K and K is bounded, there exists r > 0 such that K ⊂ Br , and it follows by Lemma 2.3that

limλt→∞

∥∥∥∥∥1λt

∫λt

0T(s)xt ds − T(h)

(1λt

∫λt

0T(s)xt ds

)∥∥∥∥∥= 0, (3.11)

for each h ≥ 0. From (3.9)-(3.10), letting t → 0 and noting (3.11) then, for each h ≥ 0, weobtain

‖xt − T(h)xt‖ −→ 0. (3.12)

Assume that {tn}∞n=1 ⊂ (0, 1) is such that tn → 0 as n → ∞. Put xn := xtn and λn := λtn . Wewill show that {xn} contains a subsequence converging strongly to x ∈ F(S). Since {xn} isbounded sequence and Banach space X is a uniformly convex, hence it is reflexive, and thereexists a subsequence {xnj} of {xn} which converges weakly to some x ∈ C as j → ∞. Again,since Jϕ is weakly sequentially continuous, we have by Lemma 2.1 that

lim supj→∞

Φ(∥∥∥xnj − z

∥∥∥)= lim sup

j→∞Φ(∥∥∥xnj − x

∥∥∥)+ Φ(‖z − x‖). (3.13)

LetH(z) = lim supj→∞Φ(‖xnj − z‖), for all z ∈ C.


It follows thatH(z) = H(x) + Φ(‖z − x‖), for all z ∈ C. From (3.12), we have

H(T(h)x) = lim supj→∞

Φ(∥∥∥xnj − T(h)x

∥∥∥)

= lim supj→∞

Φ(∥∥∥T(h)xnj − T(h)x

∥∥∥)

≤ lim supj→∞

Φ(∥∥∥xnj − x

∥∥∥)= H(x).

(3.14)

On the other hand, we note that



∥∥∥)+ Φ(‖T(h)x − x‖)

= H(x) + Φ(‖T(h)x − x‖).(3.15)

Combining (3.14) with (3.15), we obtain Φ(‖T(h)x − x‖) ≤ 0. This implies that T(h)x = x,that is, x ∈ F(S). In fact, since Φ(t) =

∫ t0 ϕ(τ)dτ , for all t ≥ 0 and ϕ : �+ → �

+ is the gaugefunction, then for 1 ≥ k ≥ 0, ϕ(ky) ≤ ϕ(y) and

Φ(kt) =∫kt

0ϕ(τ)dτ = k

∫ t

0ϕ(ky

)dy ≤ k

∫ t

0ϕ(y)dy = kΦ(t). (3.16)

By Lemma 2.1, we have

Φ(‖xn − x‖) = Φ

(∥∥∥∥∥tn(γf(xn) −Ax

)+ (I − tA)

(1λn

∫λt

0T(s)xn ds − x

)∥∥∥∥∥

)

≤ Φ

(∥∥∥∥∥(I − tnA)

(1λn

∫λn

0(T(s)xn − x)ds

)∥∥∥∥∥

)

+ tn⟨γf(xn) −Ax, Jϕ(xn − x)

⟩

≤ Φ(ϕ(1)

(1 − tnγ

)(‖xn − x‖)

)+ tn

⟨γf(xn) − γf(x), Jϕ(xn − x)

⟩

+ tn⟨γf(x) −Ax, Jϕ(xn − x)

⟩

≤ ϕ(1)(1 − tnγ)Φ(‖xn − x‖) + tnγα‖xn − x‖

∥∥Jϕ(xn − x)∥∥

+ tn⟨γf(x) −Ax, Jϕ(xn − x)

⟩

= ϕ(1)(1 − tnγ

)Φ(‖xn − x‖) + tnγαΦ(‖xn − x‖) + tn

⟨γf(x) −Ax, Jϕ(xn − x)

⟩

≤ (1 − tn

(ϕ(1)γ − γα))Φ(‖xn − x‖) + tn


⟩.

(3.17)


This implies that

Φ(‖xn − x‖) ≤ 1ϕ(1)γ − γα


⟩. (3.18)

In particular, we have


∥∥∥)≤ 1ϕ(1)γ − γα

⟨γf(x) −Ax, Jϕ

(xnj − x

)⟩. (3.19)

Since the mapping Jϕ is single-valued and weakly continuous, it follows from (3.19) thatΦ(‖xnj − x‖) → 0 as j → ∞. This implies that xnj → x as j → ∞.

Next, we show that x solves the variational inequality (3.4), for each x ∈ F(S). From(3.3), we derive that

(γf −A)

xt = −1t(I − tA)

(1λt

∫λt

0T(s)xt ds − xt

)

. (3.20)

Now, we observe that

⟨1λt

∫λt

0(I − T(s))xds − 1

λt

∫λt

0(I − T(s))xt ds, Jϕ(x − xt)

⟩

=⟨x − xt, Jϕ(x − xt)

⟩ −⟨

1λt

∫λt

0(T(s)x − T(s)xt)ds, Jϕ(x − xt)

⟩

≥ ‖x − xt‖∥∥Jϕ(x − xt)

∥∥ − 1λt

∫λt

0‖T(s) x − T(s)xt‖ds

∥∥Jϕ(x − xt)∥∥

≥ Φ(‖x − xt‖) − ‖x − xt‖∥∥Jϕ(x − xt)

∥∥

= Φ(‖x − xt‖) −Φ(‖x − xt‖) = 0.

(3.21)



⟨(γf −A)

xt, Jϕ(x − xt)⟩= −1

t

⟨

(I − tA)

(1λt

∫λt

0T(s)xt ds − xt

)

, Jϕ(x − xt)⟩

= −1t

⟨

(I − tA)

(1λt

∫λt

0T(s)xt ds − 1

λt

∫λt

0xt ds

)

, Jϕ(x − xt)⟩

= −1t

⟨1λt

∫λt

0(I − T(s))xds − 1

λt

∫λt

0(I − T(s))xt ds, Jϕ(x − xt)

⟩

+

⟨

A

(1λt

∫λt

0(T(s) − I)xt ds

)

, Jϕ(x − xt)⟩

≤⟨

A

(1λt

∫λt

0(T(s) − I))xt ds

)

, Jϕ(x − xt)⟩

.

(3.22)

Now, replacing t and λt with tnj and λnj , respectively, in (3.22), and letting j → ∞, and wenotice that (T(s)−I)xnj → (T(s)−I)x = 0 for x ∈ F(S), we obtain that 〈(γf−A)x, Jϕ(x−x)〉 ≤0. That is, x is a solution of the variational inequality (3.4). By uniqueness, as x = x∗, we haveshown that each cluster point of the net {xt} is equal to x∗. Then, we conclude that xt → x∗

as t → 0. This proof is complete.

Theorem 3.3. Let X be a uniformly convex Banach space which admits a weakly continuous dualitymapping Jϕ with the gauge function ϕ such thatϕ is invariant in [0, 1], and letC be a nonempty closedconvex subset of X such that C ± C ⊂ C. Let S = {T(t) : t ∈ �+} be a nonexpansive semigroup fromC into itself such that F(S)/= ∅, let f be a contraction mapping with a coefficient α ∈ (0, 1), and letA be a strongly positive linear bounded operator with a coefficient γ > 0 such that 0 < γ < γϕ(1)/α.Let {αn}∞n=0, {βn}∞n=0, {γn}∞n=0 be the sequences in (0, 1) and let {tn}∞n=0 be a positive real divergentsequence. Assume that the following conditions hold:

(C1) limn→∞α n = 0 and∑∞

n=0 αn = ∞,

(C2) limn→∞γn = 0,

(C3) βn = o(αn),

Then the sequence {xn} defined by


zn = γnxn +(1 − γn

) 1tn

∫ tn

0T(s)xn ds;

yn = αnγf(zn) + (I − αnA)zn;

xn+1 = βnxn +(1 − βn

)yn, ∀n ≥ 0,

(3.23)

converges strongly to the common fixed point x∗ that is obtained in Lemma 3.2.


Proof. From the condition (C1), we may assume, with no loss of generality, that αn ≤ϕ(1)‖A‖−1 for each n ≥ 0. From Lemma 2.2, we have ‖I − αnA‖ ≤ ϕ(1)(1 − αnγ).

Firstly, we show that {xn} is bounded. Let p ∈ F(S); we get

∥∥zn − p∥∥ =

∥∥∥∥∥γn(xn − p

)+(1 − γn

) 1tn

∫ tn

0

(T(s)xn − p

)ds

∥∥∥∥∥

≤ γn∥∥xn − p

∥∥ +

(1 − γn

) 1tn

∫ tn

0

∥∥T(s)xn − p

∥∥ds


∥∥ +(1 − γn

)∥∥xn − p∥∥

=∥∥xn − p

∥∥,∥∥yn − p

∥∥ =∥∥αn

(γf(zn) −Ap

)+ (I − αnA)

(zn − p

)∥∥

≤ αnγ∥∥f(zn) − f

(p)∥∥ + αn

∥∥γf

(p) −Ap∥∥ + ‖I − αnA‖∥∥zn − p

∥∥

≤ (1 − αn

(ϕ(1)γ − γα))∥∥xn − p

∥∥ + αn∥∥γf

(p) −Ap∥∥.

(3.24)

It follows that

∥∥xn+1 − p∥∥ =

∥∥βn(xn − p

)+(1 − βn

)(yn − p

)∥∥

≤ βn∥∥xn − p

∥∥ +

(1 − βn

)∥∥yn − p∥∥

≤ βn∥∥xn − p

∥∥ +(1 − βn

)((1 − αn

(ϕ(1)γ − γα))∥∥xn − p

∥∥ + αn∥∥γf

(p) −Ap∥∥)

=(1 − αn

(ϕ(1)γ − γα)(1 − βn

))∥∥xn − p∥∥ + αn

(ϕ(1)γ − γα)(1 − βn

)∥∥γf

(p) −Ap∥∥

ϕ(1)γ − γα

≤ max

{∥∥xn − p

∥∥,

∥∥γf

(p) −Ap∥∥

ϕ(1)γ − γα

}

.

(3.25)

By induction on n, we have

∥∥xn − p

∥∥ ≤ max

{∥∥x0 − p

∥∥,

∥∥γf(p) −Ap∥∥

ϕ(1)γ − γα

}

, ∀n ≥ 0. (3.26)

Thus, {xn} is bounded. Since {xn} is bounded, then ‖(1/tn)∫ tn0 T(s)xn ds − p‖ ≤ ‖xn − p‖ and

{A((1/tn)∫ tn0 T(s)xn ds)} and {f(zn)} are also bounded.


Next, we show that limn→∞‖xn − T(h)xn‖ = 0, for all h ≥ 0. From (3.23), we note that

∥∥∥∥∥xn+1 − 1

tn

∫ tn

0T(s)xn ds

∥∥∥∥∥≤ ∥∥xn+1 − yn

∥∥ +∥∥yn − zn

∥∥ +

∥∥∥∥∥zn − 1

tn

∫ tn

0T(s)xn ds

∥∥∥∥∥

≤ βn∥∥xn − yn

∥∥ + αn

∥∥γf(zn) −Azn

∥∥ + γn

∥∥∥∥∥xn − 1

tn

∫ tn

0T(s)xn ds

∥∥∥∥∥.

(3.27)

By the conditions (C1)–(C3), then (3.27), we obtain

limn→∞

∥∥∥∥∥xn+1 − 1

tn

∫ tn

0T(s)xn ds

∥∥∥∥∥= 0. (3.28)

Moreover, we note that

‖xn+1 − T(h)xn+1‖ ≤∥∥∥∥∥xn+1 − 1

tn

∫ tn

0T(s)xn ds

∥∥∥∥∥+

∥∥∥∥∥1tn

∫ tn

0T(s)xn ds − T(h)

(1tn

∫ tn

0T(s)xn ds

)∥∥∥∥∥

+

∥∥∥∥∥T(h)

(1tn

∫ tn

0T(s)xn ds

)

− T(h)xn+1∥∥∥∥∥

≤ 2

∥∥∥∥∥xn+1− 1

tn

∫ tn

0T(s)xn ds

∥∥∥∥∥+

∥∥∥∥∥1tn

∫ tn

0T(s)xn ds−T(h)

(1tn

∫ tn

0T(s)xn ds

)∥∥∥∥∥.

(3.29)

Define the set K = {z ∈ C : ‖z − p‖ ≤ ‖x0 − p‖ + ‖γf(p) − Ap‖/(ϕ(1)γ − γα)}. Then K is anonempty bounded closed convex subset of C, which is T(s)—invariant for each s ≥ 0 andcontains {xn}; it follows from Lemma 2.3 that

limn→∞

∥∥∥∥∥1tn

∫ tn

0T(s)xn ds − T(h)

(1tn

∫ tn

0T(s)xn ds

)∥∥∥∥∥= 0, ∀h ≥ 0. (3.30)

Then, for all h ≥ 0, from (3.28) and (3.30), into (3.29), we obtain limn→∞‖xn+1 −T(h)xn+1‖ = 0,and hence

limn→∞

‖xn − T(h)xn‖ = 0, ∀h ≥ 0. (3.31)

Next, we show that lim supn→∞〈γf(x∗) − Ax∗, Jϕ(xn − x∗)〉 ≤ 0. We can take subsequence{xnj } ⊂ {xn} such that

limj→∞

⟨γf(x∗) −Ax∗, Jϕ

(xnj − x∗

)⟩= lim sup

n→∞

⟨γf(x∗) −Ax∗, Jϕ(xn − x∗)

⟩. (3.32)


By the assumption thatX is uniformly convex, hence it is reflexive and {xn} is bounded; thenthere exists a subsequence {xnj} which converges weakly to some x ∈ C as j → ∞. Since Jϕis weakly continuous, from Lemma 2.1, we have

lim supj→∞

Φ(∥∥∥xnj − z

∥∥∥)= lim sup

j→∞Φ(∥∥∥xnj − x

∥∥∥)+ Φ(‖z − x‖), ∀z ∈ C. (3.33)

LetH(z) = lim supj→∞Φ(‖xnj − z‖), for all z ∈ C.It follows thatH(z) = H(x) + Φ(‖z − x‖), for all z ∈ C.From (3.31), we have


Φ(∥∥∥xnj − T(h)x

∥∥∥)

= lim supj→∞

Φ(∥∥∥T(h)xnj − T(h)x

∥∥∥)

≤ lim supj→∞


∥∥∥)= H(x).

(3.34)




∥∥∥)+ Φ(‖T(h)x − x‖)

= H(x) + Φ(‖T(h)x − x‖).(3.35)

Combining (3.34) with (3.35), we obtain Φ(‖T(h)x − x‖) ≤ 0.This implies that T(h)x = x; that is, x ∈ F(S).Since the duality map Jϕ is single-valued and weakly continuous, we get that

lim supn→∞

⟨γf(x∗) −Ax∗, Jϕ(xn − x∗)

⟩= lim

j→∞


(xnj − x∗

)⟩

=⟨γf(x∗) −Ax∗, Jϕ(x − x∗)⟩ ≤ 0,

(3.36)

as required. Hence,

lim supn→∞

⟨γf(x∗) −Ax∗, Jϕ(xn+1 − x∗)

⟩ ≤ 0. (3.37)

Since ‖xn+1 − yn‖ = βn‖xn − yn‖, by condition (C3), we obtain that limn→∞‖xn+1 − yn‖ = 0. Itfollows from (3.37), that

lim supn→∞


(yn − x∗

)⟩ ≤ 0. (3.38)


Finally, we show that xn → x∗ as n → ∞. Now, from Lemma 2.1, we have

Φ(∥∥yn − x∗

∥∥) = Φ

(∥∥αn(γf(zn) −Ax∗

)+ (I − αnA)(zn − x∗)

∥∥)

= Φ(∥∥αn

(γf(zn) − γf(x∗)

)+ αn

(γf(x∗) −Ax∗) + (I − αnA)(zn − x∗)

∥∥)

≤ Φ(∥∥αn

(γf(zn) − γf(x∗)

)+ (I − αnA)(zn − x∗)

∥∥)

+ αn⟨γf(x∗) −Ax∗, Jϕ

(yn − x∗

)⟩

≤ (1 − αn

(ϕ(1)γ − γα))Φ(‖xn − x∗‖) + αn


(yn − x∗

)⟩.

(3.39)


Φ(‖xn+1 − x∗‖) = Φ(∥∥βn(xn − x∗) +

(1 − βn

)(yn − x∗

)∥∥)

≤ (1 − βn

)Φ(∥∥yn − x∗

∥∥) + βn⟨xn − x∗, Jϕ(xn+1 − x∗)

⟩.

(3.40)


Φ(‖xn+1 − x∗‖) ≤(1 − αn

(ϕ(1)γ − γα))Φ(‖xn − x∗‖) + αn


(yn − x∗

)⟩

+ βn⟨xn − x∗, Jϕ(xn+1 − x∗)

⟩

≤ (1−αn

(ϕ(1)γ−γα))Φ(‖xn−x∗‖)+αn

[⟨γf(x∗)−Ax∗, Jϕ

(yn−x∗

)⟩+βnαnM

],

(3.41)

whereM = supn≥0{‖xn − x∗‖ϕ(‖xn+1 − x∗‖)}.Put μn := αn(ϕ(1)γ − γα) and δn := αn[〈γf(x∗) − Ax∗, Jϕ(yn − x∗)〉 + (βn/αn)M].

Then (3.41) reduces to formula Φ(‖xn+1 − x∗‖) ≤ (1 − μn)Φ(‖xn − x∗‖) + δn. By conditions(C1) and (C3) and noting (3.38), it is easy to see that

∑∞n=0 μn = ∞ and lim supn→∞(δn/μn) =

lim supn→∞(1/ϕ(1)γ−γα)[〈γf(x∗)−Ax∗, Jϕ(yn−x∗)〉+(βn/αn)M] ≤ 0. Applying Lemma 2.4,we obtain Φ(‖xn − x∗‖) → 0 as n → ∞ this implies that xn → x∗ as n → ∞. This completesthe proof.

Taking γn = 0 in (3.23), we can get the following corollary easily.

Corollary 3.4. Let X be a uniformly convex Banach space which admits a weakly continuous dualitymapping Jϕ with the gauge function ϕ such that ϕ invariant in [0, 1], C be a nonempty closed convexsubset of X such that C ± C ⊂ C. Let S = {T(t) : t ∈ �

+} be a nonexpansive semigroup from Cinto itself such that F(S)/= ∅, f be a contraction mapping with a coefficient α ∈ (0, 1) and A be astrongly positive linear bounded operator with a coefficient γ > 0 such that 0 < γ < γϕ(1)/α. Let{αn}∞n=0, {βn}∞n=0 be the sequences in (0, 1) and {tn}∞n=0 be a positive real divergent sequence. Assumethe following conditions are hold:

(C1) limn→∞αn = 0 and∑∞

n=0 αn = ∞;

(C2) βn = o(αn).




yn = αnγf

(1tn

∫ tn

0T(s)xn ds

)

+ (I − αnA)1tn

∫ tn

0T(s)xn ds,


)yn, ∀n ≥ 0,

(3.42)

converges strongly to the common fixed point x∗, in which x∗ ∈ F(S) is the unique solution of thevariational inequality:

⟨γf(x∗) −Ax∗, Jϕ(x − x∗)⟩ ≤ 0, ∀x ∈ F(S). (3.43)

A strong mean convergence theorem for nonexpansive mapping was first establishedby Baillon [20] and it was generalized to that for nonlinear semigroups by Reich et al. [21–23]. It is clear that Theorem 3.3 are valid for nonexpansive mappings. Thus, we have thefollowing mean ergodic theorem of viscosity iteration process for nonexpansive mappings inHilbert spaces.

Corollary 3.5. Let H be a real Hilbert space, and letC be a nonempty closed convex subset of Hsuch that C ± C ⊂ C. Let T be a nonexpansive mapping from C into itself such that F(T)/= ∅, f bea contraction mapping with a coefficient α ∈ (0, 1), and let A be a strongly positive linear boundedoperator with a coefficient γ > 0 such that 0 < γ < γϕ(1)/α. Let {αn}∞n=0, {βn}∞n=0, and{γn}∞n=0 bethe sequences in (0, 1) and let {tn}∞n=0 be a positive real divergent sequence. Assume that the followingconditions are hold:


n=0 αn = ∞;

(C2) limn→∞γn = 0;

(C3) βn = o(αn).




) 1n + 1

n∑

j=0

Tjxn,

yn = αnγf(zn) + (I − αnA)zn,


)yn, ∀n ≥ 0,

(3.44)

converges strongly to the common fixed point x∗, in which x∗ ∈ F(T) is the unique solution of thevariational inequality:

〈γf(x∗) −Ax∗, x − x∗〉 ≤ 0, ∀x ∈ F(T). (3.45)


Acknowledgments

The authors are grateful for the reviewers for the careful reading of the paper and for thesuggestions which improved the quality of this work. They would like to thank the NationalResearch University Project of Thailand’s Office of the Higher Education Commission forfinancial support under NRU-CSEC project no. 54000267.

References

[1] F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of theNational Academy of Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965.

[2] F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the NationalAcademy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.

[3] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of MathematicalAnalysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.

[4] F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixedpoint sets of nonexpansive mappings,”Numerical Functional Analysis and Optimization. An InternationalJournal, vol. 19, no. 1-2, pp. 33–56, 1998.

[5] H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory andApplications, vol. 116, no. 3, pp. 659–678, 2003.

[6] A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of MathematicalAnalysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.

[7] S. Plubtieng and T. Thammathiwat, “A viscosity approximation method for equilibrium problems,fixed point problems of nonexpansive mappings and a general system of variational inequalities,”Journal of Global Optimization, vol. 46, no. 3, pp. 447–464, 2010.

[8] G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.

[9] T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis:Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005.

[10] Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Manniteration,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1687–1693, 2008.

[11] R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,” Journal ofComputational and Applied Mathematics, vol. 200, no. 2, pp. 566–575, 2007.

[12] P. Sunthrayuth and P. Kumam, “A general iterative algorithm for the solution of variationalinequalities for a nonexpansive semigroup in Banach spaces,” Journal of Nonlinear Analysis andOptimization: Theory and Applications, vol. 1, no. 1, pp. 139–150, 2010.

[13] P. Kumam and K. Wattanawitoon, “A general composite explicit iterative scheme of fixed pointsolutions of variational inequalities for nonexpansive semigroups,” Mathematical and ComputerModelling, vol. 53, no. 5-6, pp. 998–1006, 2011.

[14] P. Sunthrayuth, K. Wattanawitoon, and P. Kumam, “Convergence theorems of a general compositeiterative method for nonexpansive semigroups in Banach spaces,” Mathematical Analysis, vol. 2011,Article ID 576135, 24 pages, 2011.

[15] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Application, YokohamaPublishers, Yokohama, Japan, 2000.

[16] F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,”Mathematische Zeitschrift, vol. 100, pp. 201–225, 1967.

[17] R. Wangkeeree, N. Petrot, and R. Wangkeeree, “The general iterative methods for nonexpansivemappings in Banach spaces,” Journal of Global Optimization. In press.

[18] T.-C. Lim and H. K. Xu, “Fixed point theorems for asymptotically nonexpansive mappings,”NonlinearAnalysis: Theory, Methods & Applications, vol. 22, no. 11, pp. 1345–1355, 1994.

[19] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society,vol. 66, no. 1, pp. 240–256, 2002.

[20] J.-B. Baillon, “Un theoreme de type ergodique pour les contractions non lineaires dans un espace deHilbert,” vol. 280, no. 22, pp. A1511–A1514, 1975.


[21] W. Kaczor, T. Kuczumow, and S. Reich, “A mean ergodic theorem for nonlinear semigroups whichare asymptotically nonexpansive in the intermediate sense,” Journal of Mathematical Analysis andApplications, vol. 246, no. 1, pp. 1–27, 2000.

[22] S. Reich, “Almost convergence and nonlinear ergodic theorems,” Journal of Approximation Theory, vol.24, no. 4, pp. 269–272, 1978.

[23] S. Reich, “A note on the mean ergodic theorem for nonlinear semigroups,” Journal of MathematicalAnalysis and Applications, vol. 91, no. 2, pp. 547–551, 1983.


Research ArticleStrong Convergence Theorems of the GeneralIterative Methods for Nonexpansive Semigroups inBanach Spaces

Rattanaporn Wangkeeree1, 2

1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand2 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Rattanaporn Wangkeeree, [email protected]

Received 4 February 2011; Accepted 22 March 2011


Copyright q 2011 Rattanaporn Wangkeeree. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Let E be a real reflexive Banach space which admits a weakly sequentially continuous dualitymapping from E to E∗. Let S = {T(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on E suchthat Fix(S) :=

⋂t≥0Fix(T(t))/= ∅, and f is a contraction on E with coefficient 0 < α < 1. Let

F be δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1 and γ a positivereal number such that γ < 1/α(1 −

√1 − δ/λ). When the sequences of real numbers {αn} and

{tn} satisfy some appropriate conditions, the three iterative processes given as follows: xn+1 =αnγf(xn) + (I − αnF)T(tn)xn, n ≥ 0, yn+1 = αnγf(T(tn)yn) + (I − αnF)T(tn)yn, n ≥ 0, andzn+1 = T(tn)(αnγf(zn) + (I − αnF)zn), n ≥ 0 converge strongly to x, where x is the unique solutionin Fix(S) of the variational inequality 〈(F − γf)x, j(x − x)〉 ≥ 0, x ∈ Fix(S). Our results extend andimprove corresponding ones of Li et al. (2009) Chen and He (2007), and many others.

1. Introduction

Let E be a real Banach space. A mapping T of E into itself is said to be nonexpansive if ‖Tx −Ty‖ ≤ ‖x − y‖ for each x, y ∈ E. We denote by Fix(T) the set of fixed points of T . A mappingf : E → E is called α-contraction if there exists a constant 0 < α < 1 such that ‖f(x) − f(y)‖ ≤α‖x − y‖ for all x, y ∈ E. A family S = {T(t) : 0 ≤ t < ∞} of mappings of E into itself is calleda nonexpansive semigroup on E if it satisfies the following conditions:

(i) T(0)x = x for all x ∈ E;

(ii) T(s + t) = T(s)T(t) for all s, t ≥ 0;


(iii) ‖T(t)x − T(t)y‖ ≤ ‖x − y‖ for all x, y ∈ E and t ≥ 0;

(iv) for all x ∈ E, the mapping t �→ T(t)x is continuous.

We denote by Fix(S) the set of all common fixed points of S, that is,

Fix(S) := {x ∈ E : T(t)x = x, 0 ≤ t <∞} =⋂

t≥0Fix(T(t)). (1.1)

In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbertspace

xn = αnx + (1 − αn) 1tn

∫ tn

0T(s)xnds, ∀n ∈ �, (1.2)

where {αn} is a sequence in (0, 1) and {tn} is a sequence of positive real numbers whichdiverges to ∞. Under certain restrictions on the sequence {αn}, Shioji and Takahashi [1]proved strong convergence of the sequence {xn} to a member of F(S). In [2], Shimizu andTakahashi studied the strong convergence of the sequence {xn} defined by

xn+1 = αnx + (1 − αn) 1tn

∫ tn

0T(s)xnds, ∀n ∈ � (1.3)

in a real Hilbert space where {T(t) : t ≥ 0} is a strongly continuous semigroup of nonex-pansive mappings on a closed convex subset C of a Banach space E and limn→∞ tn = ∞.Using viscosity method, Chen and Song [3] studied the strong convergence of the followingiterative method for a nonexpansive semigroup {T(t) : t ≥ 0} with Fix(S)/= ∅ in a Banachspace:

xn+1 = αnf(x) + (1 − αn) 1tn

∫ tn

0T(s)xnds, ∀n ∈ �, (1.4)

where f is a contraction. Note however that their iterate xn at step n is constructed throughthe average of the semigroup over the interval (0, t). Suzuki [4] was the first to introduceagain in a Hilbert space the following implicit iteration process:

xn = αnu + (1 − αn)T(tn)xn, ∀n ∈ �, (1.5)

for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smoothBanach space, showed that if S satisfies an asymptotic regularity condition and {αn} fulfillsthe control conditions limn→∞αn = 0,

∑∞n=1 αn = ∞, and limn→∞αn/αn+1 = 0, then both the

implicit iteration process (1.5) and the explicit iteration process (1.6),

xn+1 = αnu + (1 − αn)T(tn)xn, ∀n ∈ �, (1.6)

converge to a same point of F(S). In 2005, Xu [6] studied the strong convergence of theimplicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a


weakly sequentially continuous duality mapping. Recently, Chen and He [7] introduced theviscosity approximation process:

xn+1 = αnf(xn) +(1 − βn

)T(tn)xn, ∀n ∈ �, (1.7)

where f is a contraction and {αn} is a sequence in (0, 1) and a nonexpansive semigroup {T(t) :t ≥ 0}. The strong convergence theorem of {xn} is proved in a reflexive Banach space whichadmits a weakly sequentially continuous duality mapping. In [8], Chen et al. introduced andstudied modified Mann iteration for nonexpansive mapping in a uniformly convex Banachspace.

On the other hand, iterative approximation methods for nonexpansive mappings haverecently been applied to solve convex minimization problems; see, for example, [9–11] andthe references therein. Let H be a real Hilbert space, whose inner product and norm aredenoted by 〈·, ·〉 and ‖ · ‖, respectively. Let A be a strongly positive bounded linear operatoronH ; that is, there is a constant γ > 0 with property

〈Ax, x〉 ≥ γ‖x‖2 ∀x ∈ H. (1.8)

A typical problem is to minimize a quadratic function over the set of the fixed points of anonexpansive mapping on a real Hilbert spaceH :

minx∈C

12〈Ax, x〉 − 〈x, b〉, (1.9)

where C is the fixed point set of a nonexpansive mapping T onH and b is a given point inH .In 2003, Xu [10] proved that the sequence {xn} defined by the iterative method below, withthe initial guess x0 ∈ H chosen arbitrarily,

xn+1 = (I − αnA)Txn + αnu, n ≥ 0, (1.10)

converges strongly to the unique solution of the minimization problem (1.9) providedthe sequence {αn} satisfies certain conditions. Using the viscosity approximation method,Moudafi [12] introduced the following iterative process for nonexpansive mappings (see [13]for further developments in both Hilbert and Banach spaces). Let f be a contraction on H .Starting with an arbitrary initial x0 ∈ H , define a sequence {xn} recursively by

xn+1 = (1 − αn)Txn + αnf(xn), n ≥ 0, (1.11)

where {αn} is a sequence in (0, 1). It is proved [12, 13] that, under certain appropriate con-ditions imposed on {αn}, the sequence {xn} generated by (1.11) strongly converges to theunique solution x∗ in C of the variational inequality

⟨(I − f)x∗, x − x∗⟩ ≥ 0, x ∈ H. (1.12)


Recently, Marino and Xu [14]mixed the iterative method (1.10) and the viscosity approxima-tion method (1.11) and considered the following general iterative method:

xn+1 = (I − αnA)Txn + αnγf(xn), n ≥ 0, (1.13)

where A is a strongly positive bounded linear operator on H . They proved that if thesequence {αn} of parameters satisfies the certain conditions, then the sequence {xn} generatedby (1.13) converges strongly to the unique solution x∗ inH of the variational inequality

⟨(A − γf)x∗, x − x∗⟩ ≥ 0, x ∈ H (1.14)

which is the optimality condition for the minimization problem, minx∈C(1/2)〈Ax, x〉 − h(x),where h is a potential function for γf (i.e., h′(x) = γf(x) for x ∈ H).

Very recently, Li et al. [15] introduced the following iterative procedures for theapproximation of common fixed points of a one-parameter nonexpansive semigroup on aHilbert spaceH :

x0 = x ∈ H, xn+1 = (I − αnA)1tn

∫ tn

0T(s)xnds + αnγf(xn), n ≥ 0, (1.15)

where A is a strongly positive bounded linear operator onH .Let δ and λ be two positive real numbers such that δ, λ < 1. Recall that a mapping F

with domain D(F) and range R(F) in E is called δ-strongly accretive if, for each x, y ∈ D(F),there exists j(x − y) ∈ J(x − y) such that

⟨Fx − Fy, j(x − y)⟩ ≥ δ∥∥x − y∥∥2

, (1.16)

where J is the normalized duality mapping from E into the dual space E∗. Recall also that amapping F is called λ-strictly pseudocontractive if, for each x, y ∈ D(F), there exists j(x − y) ∈J(x − y) such that

⟨Fx − Fy, j(x − y)⟩ ≤ ∥

∥x − y∥∥2 − λ∥∥(x − y) − (

Fx − Fy)∥∥2. (1.17)

It is easy to see that (1.17) can be rewritten as

⟨(I − F)x − (I − F)y, j(x − y)⟩ ≥ λ∥∥(I − F)x − (I − F)y∥∥2

, (1.18)

see [16].In this paper, motivated by the above results, we introduce and study the strong con-

vergence theorems of the general iterative scheme {xn} defined by (1.19) in the framework ofa reflexive Banach space E which admits a weakly sequentially continuous duality mapping:

x0 = x ∈ E, xn+1 = αnγf(xn) + (I − αnF)T(tn)xn, n ≥ 0, (1.19)


where F is δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, f is a con-traction on E with coefficient 0 < α < 1, γ is a positive real number such that γ < (1/α)(1 −√(1 − δ)/λ), and S = {T(t) : 0 ≤ t < ∞} is a nonexpansive semigroup on E. The strong

convergence theorems are proved under some appropriate control conditions on parameters{αn} and {tn}. Furthermore, by using these results, we obtain strong convergence theoremsof the following new general iterative schemes {yn} and {zn} defined by

y0 = y ∈ E, yn+1 = αnγf(T(tn)yn

)+ (I − αnF)T(tn)yn, n ≥ 0, (1.20)

z0 = z ∈ E, zn+1 = T(tn)(αnγf(zn) + (I − αnF)zn

), n ≥ 0. (1.21)

The results presented in this paper extend and improve the main results in Li et al. [15], Chenand He [7], and many others.

2. Preliminaries

Throughout this paper, it is assumed that E is a real Banach space with norm ‖ · ‖ and let Jdenote the normalized duality mapping from E into E∗ given by

J(x) ={f ∈ E∗ :

⟨x, f

⟩= ‖x‖2 = ∥

∥f∥∥2

}(2.1)

for each x ∈ E, where E∗ denotes the dual space of E, 〈·, ·〉 denotes the generalized dualitypairing, and � denotes the set of all positive integers. In the sequel, we will denote thesingle-valued duality mapping by j, and consider F(T) = {x ∈ C : Tx = x}. When {xn}is a sequence in E, then xn → x (resp., xn ⇀ x, xn

∗⇀ x ) will denote strong (resp.,

weak, weak∗) convergence of the sequence {xn} to x. In a Banach space E, the followingresult (the subdifferential inequality) is well known [17, Theorem 4.2.1]: for all x, y ∈ E, for allj(x + y) ∈ J(x + y), for all j(x) ∈ J(x),

‖x‖2 + 2⟨y, j(x)

⟩ ≤ ∥∥x + y∥∥2 ≤ ‖x‖2 + ⟨

y, j(x + y

)⟩. (2.2)

A real Banach space E is said to be strictly convex if ‖x + y‖/2 < 1 for all x, y ∈ E with‖x‖ = ‖y‖ = 1 and x/=y. It is said to be uniformly convex if, for all ε ∈ [0, 2], there exits δε > 0such that

‖x‖ =∥∥y

∥∥ = 1 with

∥∥x − y∥∥ ≥ ε implies

∥∥x + y∥∥

2< 1 − δε. (2.3)

The following results are well known and can be founded in [17]:

(i) a uniformly convex Banach space E is reflexive and strictly convex [17, Theorems4.2.1 and 4.1.6],

(ii) if E is a strictly convex Banach space and T : E → E is a nonexpansive mapping,then fixed point set F(T) of T is a closed convex subset of E [17, Theorem 4.5.3].


If a Banach space E admits a sequentially continuous duality mapping J from weaktopology to weak star topology, then from Lemma 1 of [18], it follows that the dualitymapping J is single-valued and also E is smooth. In this case, duality mapping J is also saidto be weakly sequentially continuous, that is, for each {xn} ⊂ E with xn ⇀ x, then J(xn)

∗⇀ J(x)

(see [18, 19]).In the sequel, we will denote the single-valued duality mapping by j. A Banach space

E is said to satisfy Opial’s condition if, for any sequence {xn} in E, xn ⇀ x as n → ∞ implies

lim supn→∞

‖xn − x‖ < lim supn→∞

∥∥xn − y∥∥ ∀y ∈ E with x/=y. (2.4)

By Theorem 1 of [18], we know that if E admits a weakly sequentially continuous dualitymapping, then E satisfies Opial’s condition and E is smooth; for the details, see [18].

Now, we present the concept of uniformly asymptotically regular semigroup (also see[20, 21]). Let C be a nonempty closed convex subset of a Banach space E, S = {T(t) : 0 ≤t < ∞} a continuous operator semigroup on C. Then, S is said to be uniformly asymptoticallyregular (in short, u.a.r.) on C if, for all h ≥ 0 and any bounded subset D of C,

limt→∞

supx∈D

‖T(h)(T(t)x) − T(t)x‖ = 0. (2.5)

The nonexpansive semigroup {σt : t > 0} defined by the following lemma is an example ofu.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in [20,Examples 17 and 18].

Lemma 2.1 (see [3, Lemma 2.7]). Let C be a nonempty closed convex subset of a uniformly convexBanach space E,D a bounded closed convex subset of C, and S = {T(s) : 0 ≤ s <∞} a nonexpansivesemigroup on C such that F(S)/= ∅. For each h > 0, set σt(x) = (1/t)

∫ t0 T(s)xds, then

limt→∞

supx∈D

‖σt(x) − T(h)σt(x)‖ = 0. (2.6)

Example 2.2. The set {σt : t > 0} defined by Lemma 2.1 is u.a.r. nonexpansive semigroup. Infact, it is obvious that {σt : t > 0} is a nonexpansive semigroup. For each h > 0, we have

‖σt(x) − σhσt(x)‖ =

∥∥∥∥∥σt(x) − 1

h

∫h

0T(s)σt(x)ds

∥∥∥∥∥

=

∥∥∥∥∥1h

∫h

0(σt(x) − T(s)σt(x))ds

∥∥∥∥∥

≤ 1h

∫h

0‖σt(x) − T(s)σt(x)‖ds.

(2.7)


Applying Lemma 2.1, we have

limt→∞

supx∈D

‖σt(x) − σhσt(x)‖ ≤ 1h

∫h

0limt→∞

supx∈D

‖σt(x) − T(s)σt(x)‖ds = 0. (2.8)

Let C be a nonempty closed and convex subset of a Banach space E andD a nonemptysubset of C. A mapping Q : C → D is said to be sunny if

Q(Qx + t(x −Qx)) = Qx, (2.9)

wheneverQx+ t(x−Qx) ∈ C for x ∈ C and t = 0. A mappingQ : C → D is called a retractionif Qx = x for all x ∈ D. Furthermore, Q is a sunny nonexpansive retraction from C onto Dif Q is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C iscalled a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retractionfrom C onto D. The following lemma concerns the sunny nonexpansive retraction.

Lemma 2.3 (see [22, 23]). Let C be a closed convex subset of a smooth Banach space E. Let D be anonempty subset of C and Q : C → D be a retraction. Then, Q is sunny and nonexpansive if andonly if

⟨u −Qu, j(y −Qu)⟩ ≤ 0 (2.10)

for all u ∈ C and y ∈ D.

Lemma 2.4 (see [24, Lemma 2.3]). Let {an} be a sequence of nonnegative real numbers satisfyingthe property

an+1 ≤ (1 − tn)an + tncn + bn, ∀n ≥ 0, (2.11)

where {tn}, {bn}, and {cn} satisfy the restrictions(i)

∑∞n=1 tn = ∞;

(ii)∑∞

n=1 bn < ∞;

(iii) lim supn→∞cn ≤ 0.


The following lemma will be frequently used throughout the paper and can be foundin [25].

Lemma 2.5 (see [25, Lemma 2.7]). Let E be a real smooth Banach space and F : E → E a mapping.

(i) If F is δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, then I − F iscontractive with constant

√(1 − δ)/λ.

(i) If F is δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, then, for anyfixed number τ ∈ (0, 1), I − τF is contractive with constant 1 − τ(1 −

√(1 − δ)/λ).


3. Main Results

Now, we are in a position to state and prove our main results.

Theorem 3.1. Let E be a reflexive Banach space which admits a weakly sequentially continuousduality mapping J . Let S = {T(t) : 0 ≤ t < ∞} be a u.a.r. nonexpansive semigroup on E suchthat Fix(S)/= ∅. Suppose that the real sequences {αn} ⊂ [0, 1], {tn} ⊂ (0,∞) satisfy the conditions

limn→∞

αn = 0,∞∑

n=0αn = ∞, lim

n→∞tn = ∞. (3.1)

Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, f : E → E a con-traction mapping with coefficient α ∈ (0, 1), and γ a positive real number such that γ < (1/α)(1 −√(1 − δ)/λ). Then, the sequence {xn} defined by (1.19) converges strongly to x, where x is the

unique solution in Fix(S) of the variational inequality

⟨(F − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.2)

or equivalently x = QFix(S)(I − F + γf)x, where QFix(S) is the sunny nonexpansive retraction of Eonto Fix(S).

Proof. Note that Fix(S) is a nonempty closed convex set. We first show that {xn} is bounded.Let q ∈ Fix(S). Thus, by Lemma 2.5, we have

∥∥xn+1 − q∥∥ =

∥∥αnγf(xn) + (I − αnF)T(tn)xn − (I − αnF)q − αnFq∥∥

≤ αn∥∥γf(xn) − Fq

∥∥ + ‖I − αnF‖∥∥T(tn)xn − q

∥∥

≤ αnγ∥∥f(xn) − f

(q)∥∥ + αn

∥∥γf

(q) − Fq∥∥ + ‖I − αnF‖

∥∥xn − q

∥∥

≤ αnαγ∥∥xn − q

∥∥ + αn∥∥γf

(q) − Fq∥∥

+

⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠∥∥xn − q

∥∥

=

⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

− αγ⎞

⎠

⎞

⎠∥∥xn − q

∥∥

+ αn

⎛

⎝1 −√

1 − δλ

− αγ⎞

⎠∥∥γf

(q) − Fq∥∥

1 −√(1 − δ)/λ − αγ

≤ max

{∥∥xn − q

∥∥,1

1 −√(1 − δ)/λ − αγ

∥∥γf(q) − Fq∥∥

}

, ∀n ≥ 0.

(3.3)


By induction, we get

∥∥xn − q∥∥ ≤ max

{∥∥x0 − q

∥∥,1

1 − √(1 − δ)/λ − αγ

∥∥γf(q) − Fq∥∥

}

, n ≥ 0. (3.4)

This implies that {xn} is bounded and, hence, so are {f(xn)} and {FT(tn)xn}. This impliesthat

limn→∞

‖xn+1 − T(tn)xn‖ = limn→∞

αn∥∥γf(xn) − FT(tn)xn

∥∥ = 0. (3.5)

Since {T(t)} is a u.a.r. nonexpansive semigroup and limn→∞ tn = ∞, we have, for all h > 0,

limn→∞

‖T(h)(T(tn)xn) − T(tn)xn‖ ≤ limn→∞

supx∈{xn}

‖T(h)(T(tn)x) − T(tn)x‖ = 0. (3.6)

Hence, for all h > 0,

‖xn+1 − T(h)xn+1‖ ≤ ‖xn+1 − T(tn)xn‖ + ‖T(tn)xn − T(h)T(tn)xn‖ + ‖T(h)T(tn)xn − T(h)xn+1‖≤ 2‖xn+1 − T(tn)xn‖ + ‖T(tn)xn − T(h)T(tn)xn‖ −→ 0.

(3.7)

That is, for all h > 0,

limn→∞

‖xn − T(h)xn‖ = 0. (3.8)

LetΦ = QFix(S). Then, Φ(I −F − γf) is a contraction on E. In fact, from Lemma 2.5(i), we have

∥∥Φ

(I − F − γf)x −Φ

(I − F − γf)y∥∥ ≤ ∥

∥(I − F − γf)x − (

I − F − γf)y∥∥

≤ ∥∥(I − F)x − (I − F)y∥∥ + γ∥∥f(x) − f(y)∥∥

≤√

1 − δλ

∥∥x − y∥∥ + αγ

∥∥x − y∥∥

=

⎛

⎝

√1 − δλ

+ αγ

⎞

⎠∥∥x − y∥∥, ∀x, y ∈ E.

(3.9)

Therefore,Φ(I−F−γf) is a contraction on E due to (√(1 − δ)/λ+αγ) ∈ (0, 1). Thus, by Banach

contraction principle, QFix(S)(I − F − γf) has a unique fixed point x. Then, using Lemma 2.3,x is the unique solution in Fix(S) of the variational inequality (3.2). Next, we show that

lim supn→∞

⟨γf(x) − Fx, j(xn − x)

⟩ ≤ 0. (3.10)


Indeed, we can take a subsequence {xnk} of {xn} such that

lim supn→∞


⟩= lim

k→∞⟨γf(x) − Fx, j(xnk − x)

⟩. (3.11)

Wemay assume that xnk ⇀ p ∈ E as k → ∞, since a Banach spaceE has aweakly sequentiallycontinuous duality mapping J satisfying Opial’s condition [13]. We will prove that p ∈Fix(S). Suppose the contrary, p /∈ Fix(S), that is, T(h0)p /= p for some h0 > 0. It follows from(3.8) and Opial’s condition that

lim infk→∞

∥∥xnk − p∥∥ < lim inf

k→∞

∥∥xnk − T(h0)p∥∥

≤ lim infk→∞

{‖xnk − T(h0)xnk‖ +∥∥T(h0)xnk − T(h0)p

∥∥}

≤ lim infk→∞

{‖xnk − T(h0)xnk‖ +∥∥xnk − p

∥∥}

= lim infk→∞

∥∥xnk − p∥∥.

(3.12)

This is a contradiction, which shows that p ∈ F(T(h)) for all h > 0, that is, p ∈ Fix(S). Inview of the variational inequality (3.2) and the assumption that duality mapping J is weaklysequentially continuous, we conclude

lim supn→∞


⟩= lim


⟩

≤ ⟨γf(x) − Fx, j(p − x)⟩ ≤ 0.

(3.13)

Finally, we will show that xn → x. For each n ≥ 0, we have

‖xn+1 − x‖2 =∥∥αnγf(xn) + (I − αnF)T(tn)xn − (I − αnF)x − αnFx

∥∥2

≤ ∥∥αnγf(xn) − αnFx + (I − αnF)T(tn)xn − (I − αnF)x∥∥2

= ‖(I − αnF)T(tn)xn − (I − αnF)x‖2 + 2αn⟨γf(xn) − Fx, j(xn+1 − x)

⟩

≤⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠

2

‖xn − x‖2 + 2αn⟨γf(xn) − γf(x), j(xn+1 − x)

⟩

+ 2αn⟨γf(x) − Fx, j(xn+1 − x)

⟩.

(3.14)


On the other hand,

⟨γf(xn) − γf(x), j(xn+1 − x)

⟩

≤ γα‖xn − x‖‖xn+1 − x‖

≤ γα‖xn − x‖

⎡

⎢⎢⎣

√√√√√

⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠

2

‖xn − x‖2 + 2αn∣∣⟨γf(xn) − Fx, j(xn+1 − x)

⟩∣∣

⎤

⎥⎥⎦

≤ γα⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠‖xn − x‖2

+ γα‖xn − x‖√2∣∣⟨γf(xn) − Fx, j(xn+1 − x)

⟩∣∣√αn

≤ γα⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠‖xn − x‖2 +√αnM0,

(3.15)

whereM0 is a constant satisfyingM0 ≥ γα‖xn − x‖√2|〈γf(xn) − Fx, j(xn+1 − x)〉|. Substitut-

ing (3.15) in (3.14), we obtain

‖xn+1 − x‖2 ≤⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠

2

‖xn − x‖2 + 2αnγα

⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠

× ‖xn − x‖2 + 2αn√αnM0 + 2αn

⟨γf(x) − Fx, j(xn+1 − x)

⟩

=

⎛

⎜⎝1 − 2αn

⎛

⎝1 −√

1 − δλ

⎞

⎠ + α2n

⎛

⎝1 −√

1 − δλ

⎞

⎠

2⎞

⎟⎠‖xn − x‖2

+ 2αnγα

⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠‖xn − x‖2

+ 2αn√αnM0 + 2αn

⟨γf(x) − Fx, j(xn+1 − x)

⟩

=

⎛

⎝1 − 2αn

⎡

⎣

⎛

⎝1 −√

1 − δλ

⎞

⎠ − αγ + αnγα⎛

⎝1 −√

1 − δλ

⎞

⎠

⎤

⎦

⎞

⎠‖xn − x‖2

+ αn

⎡

⎢⎣αn

⎛

⎝1 −√

1 − δλ

⎞

⎠

2

‖xn − x‖2 + 2M0√αn + 2

⟨γf(x) − Fx, j(xn+1 − x)

⟩

⎤

⎥⎦

=(1 − αnγn

)‖xn − x‖2 + αnγnβn

γn,

(3.16)


where

γn = 2

⎡

⎣

⎛

⎝1 −√

1 − δλ

⎞

⎠ − αγ + αnγα⎛

⎝1 −√

1 − δλ

⎞

⎠

⎤

⎦,

βn =

⎡

⎢⎣αn

⎛

⎝1 −√

1 − δλ

⎞

⎠

2

‖xn − x‖2 + 2M0√αn + 2

⟨γf(x) − Fx, j(xn+1 − x)

⟩

⎤

⎥⎦.

(3.17)

It is easily seen that∑∞

n=1 αnγn = ∞. Since {xn} is bounded and limn→∞ αn = 0, by (3.46), weobtain lim supn→∞ βn/γn ≤ 0, applying Lemma 2.4 to (3.16) to conclude xn → x as n → ∞.This completes the proof.

Using Theorem 3.1, we obtain the following two strong convergence theorems of newiterative approximation methods for a nonexpansive semigroup {T(t) : 0 ≤ t <∞}.

Corollary 3.2. Let E be a reflexive Banach space which admits a weakly sequentially continuousduality mapping J . Let S = {T(t) : 0 ≤ t < ∞} be a u.a.r. nonexpansive semigroup on E such thatFix(S)/= ∅. Suppose that the real sequences {αn} ⊂ [0, 1], {tn} ⊂ (0,∞) satisfy the conditions

limn→∞

αn = 0,∞∑

n=0

αn = ∞, limn→∞

tn = ∞. (3.18)

Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, f : E → E a con-traction mapping with coefficient α ∈ (0, 1), and γ a positive real number such that γ < (1/α)(1 −√(1 − δ)/λ). Then, the sequence {yn} defined by (1.20) converges strongly to x, where x is the

unique solution in Fix(S) of the variational inequality⟨(F − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.19)


Proof. Let {xn} be the sequence given by x0 = y0 and

xn+1 = αnγf(xn) + (I − αnF)T(tn)xn, ∀n ≥ 0. (3.20)

Form Theorem 3.1, xn → x. We claim that yn → x. Indeed, we estimate

∥∥xn+1 − yn+1∥∥

≤ αnγ∥∥f

(T(tn)yn

) − f(xn)∥∥ + ‖I − αnF‖

∥∥T(tn)xn − T(tn)yn∥∥

≤ αnγα∥∥T(tn)yn − xn

∥∥ +

⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠∥∥xn − yn∥∥


≤ αnγα∥∥T(tn)yn − T(tn)x

∥∥ + αnγα‖T(tn)x − xn‖ +⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠∥∥xn − yn

∥∥

≤ αnγα∥∥yn − x

∥∥ + αnγα‖x − xn‖ +⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠∥∥xn − yn∥∥

≤ αnγα∥∥yn − xn

∥∥ + αnγα‖xn − x‖ + αnγα‖x − xn‖ +⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠∥∥xn − yn

∥∥

=

⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

− γα⎞

⎠

⎞

⎠∥∥xn − yn

∥∥

+ αn

⎛

⎝1 −√

1 − δλ

− γα⎞

⎠ 2αγ(1 − √

(1 − δ)/λ − γα)‖x − xn‖.

(3.21)

It follows from∑∞

n=1 αn = ∞, limn→∞‖xn − x‖ = 0, and Lemma 2.4 that ‖xn − yn‖ → 0.Consequently, yn → x as required.

Corollary 3.3. Let E be a reflexive Banach space which admits a weakly sequentially continuousduality mapping J . Let S = {T(t) : 0 ≤ t < ∞} be a u.a.r. nonexpansive semigroup on E such thatFix(S)/= ∅. Suppose that the real sequences {αn} ⊂ [0, 1], {tn} ⊂ (0,∞) satisfy the conditions

limn→∞

αn = 0,∞∑

n=0


tn = ∞. (3.22)

Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, f : E → E acon-traction mapping with coefficient α ∈ (0, 1), and γ a positive real number such that γ < (1/α)(1 −√(1 − δ)/λ). Then, the sequence {zn} defined by (1.21) converges strongly to x, where x is the unique

solution in Fix(S) of the variational inequality

⟨(F − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.23)


Proof. Define the sequences {yn} and {βn} by

yn = αnγf(zn) + (I − αnF)zn, βn = αn+1 ∀n ∈ �. (3.24)


Taking p ∈ Fix(S), we have

∥∥zn+1 − p∥∥ =

∥∥T(tn)yn − T(tn)p∥∥ ≤ ∥∥yn − p

∥∥

=∥∥αnγf(zn) + (I − αnF)zn − (I − αnF)p − αnFp

∥∥

≤⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠∥∥zn − p

∥∥ + αn

∥∥γf(zn) − F

(p)∥∥

=

⎛

⎝1 − αn⎛

⎝1 −√

1 − δλ

⎞

⎠

⎞

⎠∥∥zn − p

∥∥ + αn

⎛

⎝1 −√

1 − δλ

⎞

⎠∥∥γf(zn) − F

(p)∥∥

(1 −√

(1 − δ)/λ) .

(3.25)

It follows from induction that

∥∥zn+1 − p∥∥ ≤ max

{∥∥z0 − p

∥∥,

∥∥γf(z0) − F

(p)∥∥

1 −√(1 − δ)/λ

}

, n ≥ 0. (3.26)

Thus, both {zn} and {yn} are bounded. We observe that

yn+1 = αn+1γf(zn+1) + (I − αn+1F)zn+1 = βnγf(T(tn)yn

)+

(I − βnF

)T(tn)yn. (3.27)

Thus, Corollary 3.2 implies that {yn} converges strongly to some point x. In this case, we alsohave

‖zn − x‖ ≤ ∥∥zn − yn∥∥ +

∥∥yn − x∥∥ = αn

∥∥γf(zn) − Fzn∥∥ +

∥∥yn − x∥∥ −→ 0. (3.28)

Hence, the sequence {zn} converges strongly to some point x. This complete the proof.

Using Theorem 3.1, Lemma 2.1, and Example 2.2, we have the following result.

Corollary 3.4. Let E be a uniformly convex Banach space which admits a weakly sequentiallycontinuous duality mapping J . Let S = {T(t) : 0 ≤ t < ∞} be a nonexpansive semigroup on E suchthat Fix(S)/= ∅. Suppose that the real sequences {αn} ⊂ [0, 1], {tn} ⊂ (0,∞) satisfy the conditions

limn→∞

αn = 0,∞∑

n=0


tn = ∞. (3.29)

Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, f : E → E a con-traction mapping with coefficient α ∈ (0, 1), and γ a positive real number such that γ < (1/α)(1 −√(1 − δ)/λ). Then, the sequence {xn} defined by

x0 = x ∈ E,

xn+1 = αnγf(xn) + (I − αnF) 1tn

∫ tn

0T(t)xnds, n ≥ 0

(3.30)


converges strongly to x, where x is the unique solution in Fix(S) of the variational inequality

⟨(F − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.31)

or equivalently x = QFix(S)((I − F + γf)x), where QFix(S) is the sunny nonexpansive retraction of Eonto Fix(S).

Corollary 3.5. Let H be a real Hilbert space. Let S = {T(t) : 0 ≤ t < ∞} be a nonexpansivesemigroup on H such that Fix(S)/= ∅. Suppose that the real sequences {αn} ⊂ [0, 1], {tn} ⊂ (0,∞)satisfy the conditions

limn→∞

αn = 0,∞∑

n=0


tn = ∞. (3.32)

Let f : E → E be a contraction mapping with coefficient α ∈ (0, 1) andA a strongly positive bounded

linear operator with coefficient γ > 1/2 and 0 < γ < (1 −√2 − 2γ)/α. Then, the sequence {xn}

defined by

x0 = x ∈ E,

xn+1 = αnγf(xn) + (I − αnA)1tn

∫ tn

0T(t)xnds, n ≥ 0

(3.33)


⟨(A − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.34)

or equivalently x = QFix(S)((I −A + γf)x), where QFix(S) is the sunny nonexpansive retraction of Eonto Fix(S).

Proof. Since A is a strongly positive bounded linear operator with coefficient γ , we have

⟨Ax −Ay, x − y⟩ ≥ γ∥∥x − y∥∥2

. (3.35)

Therefore,A is γ-strongly accretive. On the other hand,

∥∥(I −A)x − (I −A)y∥∥2 =

⟨(x − y) − (

Ax −Ay), (x − y) − (Ax −Ay)⟩

=⟨x − y, x − y⟩ − 2

⟨Ax −Ay, x − y⟩ + ⟨

Ax −Ay,Ax −Ay⟩

=∥∥x − y∥∥2 − 2

⟨Ax −Ay, x − y⟩ + ∥∥Ax −Ay∥∥2

≤ ∥∥x − y∥∥2 − 2

⟨Ax −Ay, x − y⟩ + ‖A‖2∥∥x − y∥∥2

.

(3.36)


Since A is strongly positive if and only if (1/‖A‖)A is strongly positive, we may assume,without loss of generality, that ‖A‖ = 1, so that

⟨Ax −Ay, x − y⟩ ≤ ∥

∥x − y∥∥2 − 12∥∥(I −A)x − (I −A)y

∥∥2

=∥∥x − y∥∥2 − 1

2∥∥(x − y) − (

Ax −Ay)∥∥2.

(3.37)

Hence, A is 12-strongly pseudocontractive. Applying Corollary 3.4, we conclude the result.

Theorem 3.6. Let E be a reflexive Banach space which admits a weakly sequentially continuousduality mapping J . Let S = {T(t) : 0 < t < ∞} be a u.a.r. nonexpansive semigroup on E suchthat Fix(S)/= ∅. Let {αn} and {tn} be sequences of real number satisfying

0 < αn < 1,∞∑

n=0

αn = ∞, tn > 0, limn→∞

αn = limn→∞

αntn

= 0. (3.38)


x0 = x ∈ E,

xn+1 = αnγf(xn) + (I − αnF)T(tn)xn, n ≥ 0(3.39)

converges strongly to x, where x is the unique solution in Fix(S) of the variational inequality⟨(F − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.40)


Proof. By the same argument as in the proof of Theorem 3.1, we can obtain that {xn}, {f(xn)},and {FT(tn)xn} are bounded and QFix(S)(I − F − γf) is a contraction on E. Thus, by Banachcontraction principle, QFix(S)(I − F − γf) has a unique fixed point x. Then, using Lemma 2.3,x is the unique solution in Fix(S) of the variational inequality (3.40). Next, we show that

lim supn→∞


⟩ ≤ 0. (3.41)

Indeed, we can take a subsequence {xnk} of {xn} such that

lim supn→∞


⟩= lim


⟩. (3.42)


We may assume that xnk ⇀ p ∈ E as k → ∞. Now, we show that p ∈ Fix(S). Put

xk = xnk , αk = αnk sk = tnk ∀k ∈ �. (3.43)

Fix t > 0, then we have

∥∥xk − T(t)p∥∥ =

[t/si]−1∑

i=0‖T((i + 1)sk)xk − T(isk)xk‖

+∥∥∥∥T

([t

sk

]sk

)xk − T

([t

sk

]sk

)p

∥∥∥∥ +

∥∥∥∥T

([t

sk

]sk

)p − T(t)p

∥∥∥∥

≤[t

sk

]‖T(sk)xk − xk+1‖ +

∥∥xk+1 − p∥∥ +

∥∥∥∥T

(t −

[t

sk

]sk

)p − p

∥∥∥∥

≤[t

sk

]αk

∥∥FT(sk)xk − f(xk)

∥∥ +

∥∥xk+1 − p

∥∥ +

∥∥∥∥T

(t −

[t

sk

]sk

)p − p

∥∥∥∥

≤(tαksk

)∥∥FT(sk)xk − f(xk)∥∥ +

∥∥xk+1 − p∥∥ +max

{∥∥T(s)p − p∥∥ : 0 ≤ s ≤ sk}.

(3.44)

Thus, for all k ∈ �, we obtain

lim supk→∞

∥∥xk − T(t)p

∥∥ ≤ lim sup

k→∞

∥∥xk+1 − p

∥∥ = lim sup

k→∞

∥∥xk − p

∥∥. (3.45)

Since Banach space E has a weakly sequentially continuous duality mapping satisfyingOpial’s condition [13], we can conclude that T(t)p = p for all t > 0, that is, p ∈ Fix(S). Inview of the variational inequality (3.2) and the assumption that duality mapping J is weaklysequentially continuous, we conclude

lim supn→∞


⟩= lim


⟩

≤ ⟨γf(x) − Fx, J(p − x)⟩ ≤ 0.

(3.46)

By the same argument as in the proof of Theorem 3.1, we conclude that xn → x as n → ∞.This completes the proof.

Using Theorem 3.6 and the method as in the proof of Corollary 3.7, we have thefollowing result.

Corollary 3.7. Let E be a reflexive Banach space which admits a weakly sequentially continuousduality mapping J . Let S = {T(t) : 0 < t < ∞} be a u.a.r. nonexpansive semigroup on E such thatFix(S)/= ∅. Let {αn} and {tn} be sequences of real number satisfying

0 < αn < 1,∞∑

n=0

αn = ∞, tn > 0, limn→∞

αn = limn→∞

αntn

= 0. (3.47)


Let F be a δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, f : E → E acontraction mapping with coefficient α ∈ (0, 1), and γ is a positive real number such that γ < 1/α(1−√(1 − δ)/λ). Then, the sequence {yn} defined by

y0 = y ∈ E,

yn+1 = αnγf(T(tn)yn

)+ (I − αnF)T(tn)yn, n ≥ 0

(3.48)


⟨(F − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.49)


Using Theorem 3.6 and the method as in the proof of Corollary 3.8, we have thefollowing result.

Corollary 3.8. Let E be a reflexive Banach space which admits a weakly sequentially continuousduality mapping J . Let S = {T(t) : 0 < t < ∞} be a u.a.r. nonexpansive semigroup on E such thatFix(S)/= ∅. Let {αn} and {tn} be sequences of real number satisfying

0 < αn < 1,∞∑

n=0

αn = ∞, tn > 0, limn→∞

αn = limn→∞

αntn

= 0. (3.50)

Let F be a δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1, f : E → E a con-traction mapping with coefficient α ∈ (0, 1), and γ is a positive real number such that γ < (1/α)(1 −√(1 − δ)/λ). Then, the sequence {zn} defined by

z0 = z ∈ E,

zn+1 = T(tn)(αnγf(zn) + (I − αnF)zn

), n ≥ 0

(3.51)


⟨(F − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.52)


Using Theorem 3.6, Lemma 2.1, and Example 2.2, we have the following result.


Corollary 3.9. Let E be a uniformly convex Banach space which admits a weakly sequentiallycontinuous duality mapping J . Let S = {T(t) : 0 < t < ∞} be a nonexpansive semigroup on E suchthat Fix(S)/= ∅. Let {αn} and {tn} be sequences of real numbers satisfying

0 < αn < 1,∞∑

n=0

αn = ∞, tn > 0, limn→∞

αn = limn→∞

αntn

= 0. (3.53)


x0 = x ∈ E,

xn+1 = αnγf(xn) + (I − αnF) 1tn

∫ tn

0T(t)xnds, n ≥ 0

(3.54)

converges strongly to x, where x is the unique solution in Fix(S) of the variational inequality⟨(F − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.55)


Corollary 3.10. Let H be a real Hilbert space. Let S = {T(t) : 0 ≤ t < ∞} be a nonexpansivesemigroup on H such that Fix(S)/= ∅. Suppose that the real sequences {αn} ⊂ [0, 1], {tn} ⊂ (0,∞)satisfy the conditions

0 < αn < 1,∞∑

n=0

αn = ∞, tn > 0, limn→∞

αn = limn→∞

αntn

= 0. (3.56)

Let f : E → E be a contraction mapping with coefficient α ∈ (0, 1) andA a strongly positive bounded

linear operator with coefficient γ > 1/2 and 0 < γ < (1 −√2 − 2γ)/α. Then, the sequence {xn}

defined by

x0 = x ∈ E,

xn+1 = αnγf(xn) + (I − αnA)1tn

∫ tn

0T(t)xnds, n ≥ 0

(3.57)

converges strongly to x, where x is the unique solution in Fix(S) of the variational inequality⟨(A − γf)x, j(x − x)⟩ ≥ 0, x ∈ Fix(S) (3.58)

or equivalently x = QFix(S)((I −A + γf)x), where QFix(S) is the sunny nonexpansive retraction of Eonto Fix(S).


Acknowledgment

The project was supported by the “Centre of Excellence in Mathematics” under the Commis-sion on Higher Education, Ministry of Education, Thailand.

References

[1] N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semi-groups in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 1, pp. 87–99,1998.

[2] T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families ofnonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83,1997.

[3] R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,” Journal ofComputational and Applied Mathematics, vol. 200, no. 2, pp. 566–575, 2007.

[4] T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbertspaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003.

[5] T. D. Benavides, G. Lopez Acedo, and H.-K. Xu, “Construction of sunny nonexpansive retractions inBanach spaces,” Bulletin of the Australian Mathematical Society, vol. 66, no. 1, pp. 9–16, 2002.

[6] H.-K. Xu, “A strong convergence theorem for contraction semigroups in Banach spaces,” Bulletin ofthe Australian Mathematical Society, vol. 72, no. 3, pp. 371–379, 2005.

[7] R. Chen and H. He, “Viscosity approximation of common fixed points of nonexpansive semigroupsin Banach space,” Applied Mathematics Letters, vol. 20, no. 7, pp. 751–757, 2007.

[8] R. D. Chen, H. M. He, and M. A. Noor, “Modified Mann iterations for nonexpansive semigroups inBanach space,” Acta Mathematica Sinica, vol. 26, no. 1, pp. 193–202, 2010.

[9] F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixedpoint sets of nonexpansive mappings,”Numerical Functional Analysis and Optimization. An InternationalJournal, vol. 19, no. 1-2, pp. 33–56, 1998.


[11] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society.Second Series, vol. 66, no. 1, pp. 240–256, 2002.


[13] H.-K. Xu, “Approximations to fixed points of contraction semigroups in Hilbert spaces,” NumericalFunctional Analysis and Optimization, vol. 19, no. 1-2, pp. 157–163, 1998.


[15] S. Li, L. Li, and Y. Su, “General iterative methods for a one-parameter nonexpansive semigroup inHilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3065–3071, 2009.

[16] E. Zeidler, Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization,Springer, New York, NY, USA, 1985.

[17] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, YokohamaPublishers, Yokohama, Japan, 2000.

[18] J.-P. Gossez and E. Lami Dozo, “Some geometric properties related to the fixed point theory fornonexpansive mappings,” Pacific Journal of Mathematics, vol. 40, pp. 565–573, 1972.

[19] J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banachspaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509–520, 2005.

[20] A. Aleyner and Y. Censor, “Best approximation to common fixed points of a semigroup ofnonexpansive operators,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 137–151, 2005.

[21] A. Aleyner and S. Reich, “An explicit construction of sunny nonexpansive retractions in Banachspaces,” Fixed Point Theory and Applications, no. 3, pp. 295–305, 2005.

[22] R. E. Bruck Jr., “Nonexpansive retracts of Banach spaces,” Bulletin of the American Mathematical Society,vol. 76, pp. 384–386, 1970.

[23] S. Reich, “Asymptotic behavior of contractions in Banach spaces,” Journal of Mathematical Analysis andApplications, vol. 44, pp. 57–70, 1973.


[24] K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of acountable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods& Applications, vol. 67, no. 8, pp. 2350–2360, 2007.

[25] H. Piri and H. Vaezi, “Strong convergence of a generalized iterative method for semigroups ofnonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID907275, 16 pages, 2010.


Review ArticleStrong and Weak Convergence Theorems for anInfinite Family of Lipschitzian PseudocontractionMappings in Banach Spaces

Shih-sen Chang,1 Xiong Rui Wang,1 H. W. Joseph Lee,2and Chi Kin Chan2

1 Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom,Kowloon, Hong Kong

Correspondence should be addressed to Shih-sen Chang, [email protected]

Received 16 December 2010; Accepted 9 February 2011


Copyright q 2011 Shih-sen Chang et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The purpose of this paper is to study the strong and weak convergence theorems of the implicititeration processes for an infinite family of Lipschitzian pseudocontractive mappings in Banachspaces.

1. Introduction and Preliminaries

Throughout this paper, we assume that E is a real Banach space, E∗ is the dual space of E,C is a nonempty closed convex subset of E, R+ is the set of nonnegative real numbers, andJ : E → 2E

∗is the normalized duality mapping defined by

J(x) ={f ∈ E∗ :

⟨x, f

⟩= ‖x‖ · ∥∥f∥∥, ‖x‖ =

∥∥f∥∥}, x ∈ E. (1.1)

Let T : C → C be a mapping. We use F(T) to denote the set of fixed points of T . We also use“→ ” to stand for strong convergence and “⇀” for weak convergence. For a given sequence{xn} ⊂ C, letWω(xn) denote the weak ω-limit set, that is,

Wω(xn) ={z ∈ C : there exists a subsequence {xni} ⊂ {xn} such that xni ⇀ z

}. (1.2)


Definition 1.1. (1)Amapping T : C → C is said to be pseudocontraction [1], if for any x, y ∈ C,there exists j(x − y) ∈ J(x − y) such that

⟨Tx − Ty, j(x − y)⟩ ≤ ∥

∥x − y∥∥2. (1.3)

It is well known that [1] the condition (1.3) is equivalent to the following:

∥∥x − y∥∥ ≤ ∥

∥x − y + s[(I − Tx) − (

I − Ty)]∥∥, (1.4)

for all s > 0 and all x, y ∈ C.(2) T : C → C is said to be strongly pseudocontractive, if there exists k ∈ (0, 1) such that

⟨Tx − Ty, j(x − y)⟩ ≤ k∥∥x − y∥∥2

, (1.5)

for each x, y ∈ C and for some j(x − y) ∈ J(x − y).(3) T : C → C is said to be strictly pseudocontractive in the terminology of Browder and

Petryshyn [1], if there exists λ > 0 such that

⟨Tx − Ty, j(x − y)⟩ ≤ ∥∥x − y∥∥2 − λ∥∥(I − T)x − (I − T)y∥∥2

, (1.6)

for every x, y ∈ C and for some j(x − y) ∈ J(x − y).In this case, we say T is a λ-strictly pseudocontractive mapping.(4) T : C → C is said to be L-Lipschitzian, if there exists L > 0 such that

∥∥Tx − Ty∥∥ ≤ L∥∥x − y∥∥, ∀x, y ∈ C. (1.7)

Remark 1.2. It is easy to see that if T : C → C is a λ-strictly pseudocontractive mapping, thenit is a (1 + λ)/λ-Lipschitzian mapping.

In fact, it follows from (1.6) that for any x, y ∈ C,

λ∥∥(I − T)x − (I − T)y∥∥2 ≤ ⟨

(I − T)x − (I − T)y, j(x − y)⟩

≤ ∥∥(I − T)x − (I − T)y∥∥∥∥x − y∥∥.(1.8)

Simplifying it, we have

∥∥(I − T)x − (I − T)y∥∥ ≤ 1λ

∥∥x − y∥∥, (1.9)

that is,

∥∥Tx − Ty∥∥ ≤ 1 + λλ

∥∥x − y∥∥, ∀x, y ∈ C. (1.10)


Lemma 1.3 (see [2, Theorem 13.1] or [3]). Let E be a real Banach space, C be a nonempty closedconvex subset of E, and T : C → C be a continuous strongly pseudocontractive mapping. Then T hasa unique fixed point in C.

Remark 1.4. Let E be a real Banach space, C be a nonempty closed convex subset of E and T :C → C be a Lipschitzian pseudocontraction mapping. For every given u ∈ C and s ∈ (0, 1),define a mappingUs : C → C by

Usx = su + (1 − s)Tx, x ∈ C. (1.11)

It is easy to see that Us is a continuous strongly pseudocontraction mapping. By usingLemma 1.3, there exists a unique fixed point xs ∈ C ofUs such that

xs = su + (1 − s)Txs. (1.12)

The concept of pseudocontractive mappings is closely related to accretive operators. Itis known that T is pseudocontractive if and only if I − T is accretive, where I is the identitymapping. The importance of accretive mappings is from their connection with theory ofsolutions for nonlinear evolution equations in Banach spaces. Many kinds of equations, forexample, Heat, wave, or Schrodinger equations can be modeled in terms of an initial valueproblem:

du

dt= Tu − u, u(0) = u0, (1.13)

where T is a pseudocontractive mapping in an appropriate Banach space.In order to approximate a fixed point of Lipschitzian pseudocontractive mapping, in

1974, Ishikawa introduced a new iteration (it is called Ishikawa iteration). Since then, a questionof whether or not the Ishikawa iteration can be replaced by the simpler Mann iteration hasremained open. Recently Chidume andMutangadura [4] solved this problem by constructingan example of a Lipschitzian pseudocontractive mapping with a unique fixed point for whichevery Mann-type iteration fails to converge.

Inspired by the implicit iteration introduced by Xu and Ori [5] for a finite familyof nonexpansive mappings in a Hilbert space, Osilike [6], Chen et al. [7], Zhou [8] andBoonchari and Saejung [9] proposed and studied convergence theorems for an implicititeration process for a finite or infinite family of continuous pseudocontractive mappings.

The purpose of this paper is to study the strong andweak convergence problems of theimplicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappingsin Banach spaces. The results presented in this paper extend and improve some recent resultsof Xu and Ori [5], Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9].

For this purpose, we first recall some concepts and conclusions.A Banach space E is said to be uniformly convex, if for each ε > 0, there exists a δ > 0

such that for any x, y ∈ E with ‖x‖, ‖y‖ ≤ 1 and ‖x − y‖ ≥ ε, ‖x + y‖ ≤ 2(1 − δ) holds. Themodulus of convexity of E is defined by

δE(ε) = inf{1 −

∥∥∥∥x + y2

∥∥∥∥ : ‖x‖, ∥∥y∥∥ ≤ 1,∥∥x − y∥∥ ≥ ε

}, ∀ε ∈ [0, 2]. (1.14)


Concerning the modulus of convexity of E, Goebel and Kirk [10] proved the followingresult.

Lemma 1.5 (see [10, Lemma 10.1]). Let E be a uniformly convex Banach space with a modulus ofconvexity δE. Then δE : [0, 2] → [0, 1] is continuous, increasing, δE(0) = 0, δE(t) > 0 for t ∈ (0, 2]and

‖cu + (1 − c)v‖ ≤ 1 − 2min{c, 1 − c}δE(‖u − v‖), (1.15)

for all c ∈ [0, 1], and u, v ∈ E with ‖u‖, ‖v‖ ≤ 1.A Banach space E is said to satisfy the Opial condition, if for any sequence {xn} ⊂ E with

xn ⇀ x, then the following inequality holds:

lim supn→∞

‖xn − x‖ < lim supn→∞

∥∥xn − y

∥∥, (1.16)

for any y ∈ E with y /=x.

Lemma 1.6 (Zhou [8]). Let E be a real reflexive Banach space with Opial condition. Let C be anonempty closed convex subset of E and T : C → C be a continuous pseudocontractive mapping.Then I − T is demiclosed at zero, that is, for any sequence {xn} ⊂ E, if xn ⇀ y and ‖(I − T)xn‖ → 0,then (I − T)y = 0.

Lemma 1.7 (Chang [11]). Let J : E → 2E∗be the normalized duality mapping, then for any

x, y ∈ E,∥∥x + y

∥∥2 ≤ ‖x‖2 + 2⟨y, j

(x + y

)⟩, ∀j(x + y

) ∈ J(x + y). (1.17)

Definition 1.8 (see [12]). Let {Tn} : C → E be a family of mappings with⋂∞n=1 F(Tn)/= ∅. We

say {Tn} satisfies the AKTT-condition, if for each bounded subset B of C the following holds:

∞∑

n=1

supz∈B

‖Tn+1z − Tnz‖ <∞. (1.18)

Lemma 1.9 (see [12]). Suppose that the family of mappings {Tn} : C → C satisfies the AKTT-condition. Then for each y ∈ C, {Tny} converges strongly to a point in C. Moreover, let T : C → Cbe the mapping defined by

Ty = limn→∞

Tny, ∀y ∈ C. (1.19)

Then, for each bounded subset B ⊂ C, limn→∞supz∈B‖Tz − Tnz‖ = 0.

2. Main Results

Theorem 2.1. Let E be a uniformly convex Banach space with a modulus of convexity δE, and Cbe a nonempty closed convex subset of E. Let {Tn} : C → C be a family of Ln-Lipschitzian and


pseudocontractive mappings with L := supn≥1Ln < ∞ and F :=⋂n≥1 F(Tn)/= ∅. Let {xn} be the

sequence defined by

x1 ∈ C,xn = αnxn−1 + (1 − αn)Tnxn, n ≥ 1,

(2.1)

where {αn} is a sequence in [0, 1]. If the following conditions are satisfied:

(i) lim supn→∞αn < 1;

(ii) there exists a compact subsetK ⊂ E such that⋃∞n=1 Tn(C) ⊂ K;

(iii) {Tn} satisfies the AKTT-condition, and F(T) ⊂ F, where T : C → C is the mappingdefined by (1.19).

Then xn converges strongly to some point p ∈ F

Proof. First, we note that, by Remark 1.4, the method is well defined. So, we can divide theproof in three steps.

(I) For each p ∈ F the limit limn→∞‖xn − p‖ exists.In fact, since {Tn} is pseudocontractive, for each p ∈ F, we have

∥∥xn − p∥∥2 =

⟨xn − p, j

(xn − p

)⟩

= αn〈xn−1 − p, j(xn − p

)〉 + (1 − αn)⟨Tnxn − p, j

(xn − p

)⟩

≤ αn∥∥xn−1 − p

∥∥∥∥xn − p∥∥ + (1 − αn)

∥∥xn − p∥∥2, ∀n ≥ 1.

(2.2)

Simplifying, we have that

∥∥xn − p∥∥ ≤ ∥∥xn−1 − p

∥∥, ∀n ≥ 1. (2.3)

Consequently, the limit limn→∞‖xn−p‖ exists, and so the sequence {xn} is bounded.(II) Now, we prove that limn→∞‖xn − Tnxn‖ = 0.

In fact, by virtue of (2.1) and (1.4), we have

∥∥xn − p∥∥ ≤

∥∥∥∥xn − p +1 − αn2αn

(xn − Tnxn)∥∥∥∥

=∥∥∥∥xn − p +

1 − αn2

(xn−1 − Tnxn)∥∥∥∥

=∥∥∥∥αnxn−1 + (1 − αn)Tnxn − p + 1 − αn

2(xn−1 − Tnxn)

∥∥∥∥

=∥∥∥xn−1 + xn

2− p

∥∥∥

=∥∥xn−1 − p

∥∥ ·∥∥∥∥∥

xn−1 − p2∥∥xn−1 − p

∥∥ +xn − p

2∥∥xn−1 − p

∥∥

∥∥∥∥∥.

(2.4)


Letting u = (xn−1 − p)/‖xn−1 − p‖ and v = (xn − p)/‖xn−1 − p‖, from (2.3), we knowthat ‖u‖ = 1, ‖v‖ ≤ 1. It follows from (2.4) and Lemma 1.5 that

∥∥xn − p

∥∥ ≤ ∥

∥xn−1 − p∥∥{

1 − δE(

‖xn−1 − xn‖∥∥xn−1 − p

∥∥

)}

. (2.5)

Simplifying, we have that

∥∥xn−1 − p

∥∥{

δE

(‖xn−1 − xn‖∥∥xn−1 − p

∥∥

)}

≤ ∥∥xn−1 − p

∥∥ − ∥

∥xn − p∥∥. (2.6)

This implies that

∞∑

n=1

∥∥xn−1 − p∥∥{

δE

(‖xn−1 − xn‖∥∥xn−1 − p

∥∥

)}

≤ ∥∥x0 − p∥∥. (2.7)

Letting limn→∞‖xn − p‖ = r, if r = 0, the conclusion of Theorem 2.1 is proved. Ifr > 0, it follows from the property of modulus of convexity δE that ‖xn−1 − xn‖ →0 (n → ∞). Therefore, from (2.1) and the condition (i), we have that

‖xn−1 − Tnxn‖ =1

1 − αn ‖xn − xn−1‖ −→ 0 (as n −→ ∞). (2.8)

In view of (2.1) and (2.8), we have

‖xn − Tnxn‖ = αn‖xn−1 − Tnxn‖ −→ 0 (as n −→ ∞). (2.9)

(III) Now, we prove that {xn} converges strongly to some point in F.In fact, it follows from (2.9) and condition (ii) that there exists a subsequence {xni} ⊂{xn} such that ‖xni−Tnixni‖ → 0 (as ni → ∞), Tnixni → p and xni → p (some pointin C). Furthermore, by Lemma 1.9, we have Tnip → Tp. consequently, we have

∥∥p − Tp∥∥ ≤ ∥∥p − xni∥∥ +

∥∥xni − Tnip∥∥ +

∥∥Tnip − Tp∥∥

≤ ∥∥p − xni∥∥ + ‖xni − Tnixni‖ +

∥∥Tnixni − Tnip∥∥ +

∥∥Tnip − Tp∥∥

≤ (1 + L)∥∥p − xni

∥∥ + ‖xni − Tnixni‖ +∥∥Tnip − Tp

∥∥ −→ 0.

(2.10)

This implies that p = Tp, that is, p ∈ F(T) ⊂ F. Since xni → p and the limitlimn→∞‖xn − p‖ exists, we have xn → p.

This completes the proof of Theorem 2.1.

Theorem 2.2. Let E be a uniformly convex Banach space satisfying the Opial condition. Let Cbe a nonempty closed convex subset of E and {Tn} : C → C be a family of Ln-Lipschitzian


pseudocontractive mappings with L := supn≥1Ln < ∞ and F :=⋂n≥1 F(Tn)/= ∅. Let {xn} be the

sequence defined by (2.1) and {αn} be a sequence in (0, 1). If the following conditions are satisfied:

(i) lim supn→∞αn < 1,

(ii) for any bounded subset B of C

limn→∞

supz∈B

‖TmTnz − Tnz‖ = 0, for each m ≥ 1. (2.11)

Then the sequence {xn} converges weakly to some point u ∈ F.

Proof. By the same method as given in the proof of Theorem 2.1, we can prove that thesequence {xn} is bounded and

limn→∞

∥∥xn − p

∥∥ exists, for each p ∈ F;

limn→∞

‖xn − Tnxn‖ = 0.(2.12)

Now, we prove that

limn→∞

‖Tmxn − xn‖ = 0, for each m ≥ 1. (2.13)

Indeed, for eachm ≥ 1, we have

‖Tmxn − xn‖ ≤ ‖Tmxn − TmTnxn‖ + ‖TmTnxn − Tnxn‖ + ‖Tnxn − xn‖≤ (1 + L)‖Tnxn − xn‖ + ‖TmTnxn − Tnxn‖≤ (1 + L)‖Tnxn − xn‖ + sup

z∈{xn}‖TmTnz − Tnz‖.

(2.14)

By (2.12) and condition (ii), we have

limn→∞

‖Tmxn − xn‖ = 0, for each m ≥ 1. (2.15)

The conclusion of (2.13) is proved.Finally, we prove that {xn} converges weakly to some point u ∈ F.In fact, since E is uniformly convex, and so it is reflexive. Again since {xn} ⊂ C is

bounded, there exists a subsequence {xni} ⊂ {xn} such that xni ⇀ u. Hence from (2.13), foranym > 1, we have

‖Tmxni − xni‖ −→ 0 (as ni −→ ∞). (2.16)

By virtue of Lemma 1.6, u ∈ F(Tm), for allm ≥ 1. This implies that

u ∈ F :=⋂

n≥F(Tn) ∩Wω(xn). (2.17)


Next, we prove thatWω(xn) is a singleton. Let us suppose, to the contrary, that if thereexists a subsequence {xnj} ⊂ {xn} such that xnj ⇀ q ∈Wω(xn) and q /=u. By the same methodas given above we can also prove that q ∈ F :=

⋂n≥1 F(Tn) ∩Wω(xn). Taking p = u and p = q

in (2.12). We know that the following limits

limn→∞

‖xn − u‖, limn→∞

∥∥xn − q

∥∥ (2.18)

exist. Since E satisfies the Opial condition, we have

limn→∞

‖xn − u‖ = lim supni →∞

‖xni − u‖ < lim supni →∞

∥∥xni − q

∥∥

= limn→∞

∥∥xn − q

∥∥ = lim sup

nj →∞

∥∥∥xnj − q

∥∥∥

< lim supnj →∞

∥∥∥xnj − u∥∥∥ = lim

n→∞‖xn − u‖.

(2.19)

This is a contradiction, which shows that q = u. Hence,

Wω(xn) = {u} ⊂ F :=⋂

n≥1F(Tn). (2.20)

This implies that xn ⇀ u.This completes the proof of Theorem 2.2.

In the next lemma, we propose a sequence of mappings that satisfy condition (iii) inTheorem 2.1. Moreover, we apply this lemma to obtain a corollary of our main Theorem 2.1.

Let E be a Banach space and C be a nonempty closed convex subset of E. FromDefinition 1.1(3), we know that if T : C → C is a λ-strictly pseudocontractive mapping,then it is a ((1 + λ)/λ)-Lipschitzian pseudocontractive mapping.

On the other hand, by the same proof as given in [12] we can prove the followingresult.

Lemma 2.3 (see [12] or [9]). Let E be a smooth Banach space, C be a closed convex subset of E. Let{Sn} : C → C be a family of λn-strictly pseudocontractive mappings with F :=

⋂∞n=1 F(Sn)/= ∅ and

λ := infn≥1λn > 0. For each n ≥ 1 define a mapping Tn : C → C by:

Tnx =n∑

k=1

βknSkx, x ∈ C, n ≥ 1, (2.21)

where {βkn} is sequence of nonnegative real numbers satisfying the following conditions:

(i)∑n

k=1 βkn = 1, for all n ≥ 1;

(ii) βk := limn→∞βkn > 0, for all k ≥ 1;

(iii)∑∞

n=1∑n

k=1 |βkn+1 − βkn| <∞.


Then,

(1) each Tn, n ≥ 1 is a λ-strictly pseudocontractive mapping;

(2) {Tn} satisfies the AKTT-condition;(3) if T : C → C is the mapping defined by

Tx =∞∑

k=1

βkSkx, x ∈ C. (2.22)

Then Tx = limn→∞Tnx and F(T) =⋂∞k=1 F(Tn) = F :=

⋂∞n=1 F(Sn).

The following result can be obtained from Theorem 2.1 and Lemma 2.3 immediately.

Theorem 2.4. Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E.Let {Sn} : C → C be a family of λn-strictly pseudocontractive mappings with F :=

⋂∞n=1 F(Sn)/= ∅

and λ := infn≥1λn > 0. For each n ≥ 1 define a mapping Tn : C → C by

Tnx =n∑

k=1

βknSkx, x ∈ C, n ≥ 1, (2.23)

where {βkn} is a sequence of nonnegative real numbers satisfying the following conditions:

(i)∑n

k=1 βkn = 1, for all n ≥ 1;

(ii) βk := limn→∞βkn > 0, for all k ≥ 1;

(iii)∑∞

n=1∑n

k=1 |βkn+1 − βkn| <∞.

Let {xn} be the sequence defined by

x1 ∈ C,xn = αnxn−1 + (1 − αn)Tnxn, n ≥ 1,

(2.24)

where {αn} is a sequence in [0, 1]. If the following conditions are satisfied:

(i) lim supn→∞αn < 1;

(ii) there exists a compact subset K ⊂ E such that⋃∞n=1 Sn(C) ⊂ K. Then, {xn} converges

strongly to some point p ∈ F.

Proof. Since {Sn} : C → C is a family of λn-strictly pseudocontractive mappings withλ := infn≥1λn > 0. Therefore, {Sn} is a family of λ-strictly pseudocontractive mappings.By Remark 1.2, {Sn} is a family of (1 + λ)/λ-Lipschitzian and strictly pseudocontractivemappings. Hence, by Lemma 2.3, {Tn} defined by (2.21) is a family of (1+ λ)/λ-Lipschitzian,strictly pseudocontractive mappings with

⋂∞n=1 F(Tn) = F :=

⋂∞n=1 F(Sn)/= ∅ and it has also

the following properties:

(1) {Tn} satisfies the AKTT-condition;


(2) if T : C → C is the mapping defined by (2.22), then Tx = limn→∞Tnx, x ∈ C andF(T) = F :=

⋂∞k=1 F(Sn) =

⋂∞n=1 F(Tn). Hence, by Definition 1.1, {Tn} is also a family

of (1 + λ)/λ-Lipschitzian and pseudocontractive mappings having the properties(1) and (2) and F :=

⋂∞n=1 F(Tn)/= ∅. Therefore, {Tn} satisfies all the conditions in

Theorem 2.1. By Theorem 2.1, the sequence {xn} converges strongly to some pointp ∈ F :

⋂∞k=1 F(Sn) =

⋂∞n=1 F(Tn).

This completes the proof of Theorem 2.4.

Acknowledgment

This paper was supported by the Natural Science Foundation of Yibin University (no.2009Z01).

References

[1] F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbertspace,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.

[2] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.[3] S. Chang, Y. J. Cho, and H. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces,

Nova Science, Huntington, NY, USA, 2002.[4] C. E. Chidume and S. A. Mutangadura, “An example of the Mann iteration method for Lipschitz

pseudocontractions,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2359–2363,2001.

[5] H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” NumericalFunctional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001.

[6] M. O. Osilike, “Implicit iteration process for common fixed points of a finite family of strictlypseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 73–81,2004.

[7] R. Chen, Y. Song, and H. Zhou, “Convergence theorems for implicit iteration process for a finitefamily of continuous pseudocontractive mappings,” Journal of Mathematical Analysis and Applications,vol. 314, no. 2, pp. 701–709, 2006.

[8] H. Zhou, “Convergence theorems of common fixed points for a finite family of Lipschitzpseudocontractions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no.10, pp. 2977–2983, 2008.

[9] D. Boonchari and S. Saejung, “Construction of common fixed points of a countable family of λ-demicontractive mappings in arbitrary Banach spaces,”AppliedMathematics and Computation, vol. 216,no. 1, pp. 173–178, 2010.

[10] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in AdvancedMathematics, Cambridge University Press, Cambridge, UK, 1990.

[11] S.-S. Chang, “Some problems and results in the study of nonlinear analysis,” Nonlinear Analysis:Theory, Methods & Applications, vol. 30, no. 7, pp. 4197–4208, 1997.

[12] K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of acountable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods& Applications, vol. 67, no. 8, pp. 2350–2360, 2007.


Research ArticleStrong Convergence Theorems of ModifiedIshikawa Iterative Method for an Infinite Family ofStrict Pseudocontractions in Banach Spaces

Phayap Katchang,1 Wiyada Kumam,2 Usa Humphries,2and Poom Kumam2

1 Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology,Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

2 Department of Mathematics, Faculty of Science, King Mongkut’s University ofTechnology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand


Received 15 December 2010; Accepted 14 March 2011


Copyright q 2011 Phayap Katchang et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We introduce a new modified Ishikawa iterative process and a new W-mapping for comput-ing fixed points of an infinite family of strict pseudocontractions mapping in the framework ofq-uniformly smooth Banach spaces. Then, we establish the strong convergence theorem of theproposed iterative scheme under some mild conditions. The results obtained in this paper extendand improve the recent results of Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011and many others in the literature.

1. Introduction

Let E be a real Banach space with norm ‖ · ‖ and C a nonempty closed convex subset of E. LetE∗ be the dual space of E, and let 〈·, ·〉 denotes the generalized duality pairing between E andE∗. For q > 1, the generalized duality mapping Jq : E → 2E

∗is defined by

Jq(x) ={f ∈ E∗ :

⟨x, f

⟩= ‖x‖q,∥∥f∥∥ = ‖x‖q−1

}, (1.1)

for all x ∈ E. In particular, if q = 2, the mapping J = J2 is called the normalized duality mappingand Jq(x) = ‖x‖q−2J2(x) for x /= 0. It is well known that if E is smooth, then Jq is single-valued,which is denoted by jq.


A mapping T : C → C is called nonexpansive if

∥∥Tx − Ty∥∥ ≤ ∥

∥x − y∥∥, ∀x, y ∈ C. (1.2)

We use F(T) to denote the set of fixed points of T ; that is, F(T) = {x ∈ C : Tx = x}.T is said to be a λ-strict pseudocontraction in the terminology of Browder and Petryshyn

[1] if there exists a constant λ > 0 and for some jq(x − y) ∈ Jq(x − y) such that

⟨Tx − Ty, jq

(x − y)⟩ ≤ ∥

∥x − y∥∥q − λ∥∥(I − T)x − (I − T)y∥∥q, ∀x, y ∈ C. (1.3)

T is said to be a strong pseudocontraction if there exists k ∈ (0, 1) such that

⟨Tx − Ty, jq

(x − y)⟩ ≤ k‖x − y‖q, ∀x, y ∈ C. (1.4)

Remark 1.1 (see [2]). Let T be a λ-strict pseudocontraction in a Banach space. Let x ∈ C andp ∈ F(T). Then,

∥∥Tx − p∥∥ ≤(1 +

1λ1/q−1

)∥∥x − p∥∥. (1.5)

Recall that a self mapping f : C → C is contraction on C if there exists a constantα ∈ (0, 1) and x, y ∈ C such that

∥∥f(x) − f(y)∥∥ ≤ α∥∥x − y∥∥. (1.6)

We use ΠC to denote the collection of all contractions on C. That is, ΠC = {f | f : C →C a contraction}. Note that each f ∈ ΠC has a unique fixed point in C.

Very recently, Cai and Hu [3] also proved the strong convergence theorem in Banachspaces. They considered the following iterative algorithm:

x1 = x ∈ C chosen arbitrarily,

yn = PC

[

βnxn +(1 − βn

) N∑

i=i

η(n)i Tixn

]

,

xn+1 = αnγf(xn) + γnxn +((1 − γn

)I − αnA

)yn, ∀n ≥ 1,

(1.7)

where Ti is a non-self-λi-strictly pseudocontraction, f is a contraction, and A is a stronglypositive linear bounded operator.


Dong et al. [2] proved the sequence {xn} converges strongly in Banach spaces undercertain appropriate assumptions and used the Wn mapping defined by (1.11). Let thesequences {xn} be generated by


yn = δnxn + (1 − δn)Wnxn,

xn+1 = αnf(xn) + βnxn + γnWnyn, ∀n ≥ 0.

(1.8)

On the other hand, Katchang and Kumam [4, 5] introduced the following newmodified Ishikawa iterative process for computing fixed points of an infinite familynonexpansive mapping in the framework of Banach spaces; let the sequences {xn} begenerated by



)Wnxn,

yn = βnxn +(1 − βn

)Wnzn,

xn+1 = αnγf(x) + (I − αnA)yn, ∀n ≥ 0,

(1.9)

where f is a contraction, A is a strongly positive linear bounded self-adjoint operator, andWn mapping (see [6, 7]) is defined by

Un,n+1 = I,

Un,n = λnTnUn,n+1 + (1 − λn)I,Un,n−1 = λn−1Tn−1Un,n + (1 − λn−1)I,

...

Un,k = λkTkUn,k+1 + (1 − λk)I,Un,k−1 = λk−1Tk−1Un,k + (1 − λk−1)I,

...

Un,2 = λ2T2Un,3 + (1 − λ2)I,Wn = Un,1 = λ1T1Un,2 + (1 − λ1)I,

(1.10)

where T1, T2, . . . is an infinite family of nonexpansive mappings of C into itself and λ1, λ2, . . .is real numbers such that 0 ≤ λn ≤ 1 for every n ∈ N. In 2010, Cho [8] considered andproved the strong convergence of the implicit iterative process for an infinite family of strictpseudocontractions in an arbitrary real Banach space.


In this paper, motivated and inspired by Cai and Hu [9], we consider the mappingWn

defined by

Un,n+1 = I,

Un,n = tnTn,nUn,n+1 + (1 − tn)I,...

Un,k = tkTn,kUn,k+1 + (1 − tk)I,Un,k−1 = tk−1Tn,k−1Un,k + (1 − tk−1)I,

...

Un,2 = t2Tn,2Un,3 + (1 − t2)I,Wn = Un,1 = t1Tn,1Un,2 + (1 − t1)I,

(1.11)

where t1, t2, . . . are real numbers such that 0 ≤ tn ≤ 1. Tn,k = θn,kSk + (1 − θn,k)I, where Sk is aλk-strict pseudocontraction of C into itself and θn,k ∈ (0, μ], μ = min{1, {qλ/Cq}1/q−1}, whereλ = inf λk for all k ∈ N. By Lemma 2.3, we know that Tn,k is a nonexpansive mapping, andtherefore, Wn is a nonexpansive mapping. We note that the W-mapping (1.10) is a specialcase of aW-mapping (1.11)when θn,k = θk is constant for all n ≥ 1.

Throughout this paper, we will assume that inf λi > 0, 0 < tn ≤ b < 1 for all n ∈ N and{θn,k} satisfies

(H1) θn,k ∈ (0, μ], μ = min{1, infi{qλi/Cq}1/q−1} for all k ∈ N,

(H2) |θn+1,k − θn,k| ≤ an for all n ∈ N and 1 ≤ k ≤ n, where {an} satisfies∑∞

n=1 an <∞.

The hypothesis (H2) secures the existence of limn→∞θn,k for all k ∈ N. Set θ1,k :=limn→∞θn,k for all k ∈ N. Furthermore, we assume that

(H3) θ1,k > 0 for all k ∈ N.

It is obvious that θ1,k satisfy (H1). Using condition (H3), from Tn,k = θn,kSk+(1−θn,k)I,we define mappings T1,kx := limn→∞Tn,kx = θ1,kSkx + (1 − θ1,k)x for all x ∈ C.

Our results improve and extend the recent ones announced by Cai and Hu [3],Dong et al. [2], Katchang and Kumam [4, 5], and many others.

2. Preliminaries

Recall that U = {x ∈ E : ‖x‖ = 1}. A Banach space E is said to be uniformly convex if, for anyε ∈ (0, 2], there exists δ > 0 such that for any x, y ∈ U, ‖x −y‖ ≥ ε implies ‖(x +y)/2‖ ≤ 1− δ.It is known that a uniformly convex Banach space is reflexive and strictly convex (see also[10]). A Banach space E is said to be smooth if the limit limt→ 0(‖x + ty‖ − ‖x‖)/t exists for allx, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U.


In a smooth Banach space, we define an operator A as strongly positive if there existsa constant γ > 0 with the property

〈Ax, J(x)〉 ≥ γ‖x‖2, ‖aI − bA‖ = sup‖x‖≤1

|〈(aI − bA)x, J(x)〉| a ∈ [0, 1], b ∈ [−1, 1], (2.1)

where I is the identity mapping and J is the normalized duality mapping.If C and D are nonempty subsets of a Banach space E such that C is a nonempty

closed convex and D ⊂ C, then a mapping Q : C → D is sunny [11, 12] provided thatQ(x + t(x −Q(x))) = Q(x) for all x ∈ C and t ≥ 0 whenever x + t(x −Q(x)) ∈ C. A mappingQ : C → C is called a retraction if Q2 = Q. If a mapping Q : C → C is a retraction, thenQz = z for all z in the range of Q. A subset D of C is said to be a sunny nonexpansive retractof C if there exists a sunny nonexpansive retraction Q of C onto D. A sunny nonexpansiveretraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractionsplay an important role in our argument. They are characterized as follows [11, 12]: if E is asmooth Banach space, then Q : C → D is a sunny nonexpansive retraction if and only if thereholds the inequality

⟨x −Qx, J(y −Qx)⟩ ≤ 0, ∀x ∈ C, y ∈ D. (2.2)

We need the following lemmas for proving our main results.

Lemma 2.1 (see [13]). In a Banach space E, the following holds:

∥∥x + y∥∥2 ≤ ‖x‖2 + 2

⟨y, j

(x + y

)⟩, ∀x, y ∈ E, (2.3)

where j(x + y) ∈ J(x + y).

Lemma 2.2 (see [14]). Let E be a real q-uniformly smooth Banach space, then there exists a constantCq > 0 such that

∥∥x + y

∥∥q ≤ ‖x‖q + q⟨y, jqx⟩+ Cq

∥∥y

∥∥q, ∀x, y ∈ E. (2.4)

In particular, if E be a real 2-uniformly smooth Banach space with the best smooth constant K, thenthe following inequality holds:

∥∥x + y∥∥2 ≤ ‖x‖2 + 2

⟨y, jx

⟩+ 2

∥∥Ky∥∥2, ∀x, y ∈ E. (2.5)

The relation between the λ-strict pseudocontraction and the nonexpansive mappingcan be obtained from the following lemma.

Lemma 2.3 (see [15]). LetC be a nonempty convex subset of a real q-uniformly smooth Banach spaceE and S : C → C a λ-strict pseudocontraction. For α ∈ (0, 1), one defines Tx = (1 − α)x + αSx.Then, as α ∈ (0, μ], μ = min{1, {qλ/Cq}1/q−1}, T : C → C is nonexpansive such that F(T) = F(S),where Cq is the constant in Lemma 2.2.


ConcerningWn, we have the following lemmas, which are important to prove themainresults.

Lemma 2.4 (see [2]). Let C be a nonempty closed convex subset of a q-uniformly smooth and strictlyconvex Banach space E. Let Si, i = 1, 2, . . ., be a λi-strict pseudocontraction from C into itself such that∩∞n=1F(Sn)/= ∅, and let inf λi > 0. Let tn, n = 1, 2, . . ., be real numbers such that 0 < tn ≤ b < 1 for

any n ≥ 1. Assume that the sequence {θn,k} satisfies (H1) and (H2). Then, for every x ∈ C and k ∈ N,the limit limn→∞Un,kx exists.

Using Lemma 2.4, we define the mappingsU1,k andW : C → C as follows:

U1,kx := limn→∞

Un,kx,

Wx := limn→∞

Wnx = limn→∞

Un,1x,(2.6)

for all x ∈ C. Such W is called the W-mapping generated by S1, S2, . . . , t1, t2, . . . andθn,k, for all n ∈ N and 1 ≤ k ≤ n.

Lemma 2.5 (see [2]). Let {xn} be a bounded sequence in a q-uniformly smooth and strictly convexBanach space E. Under the assumptions of Lemma 2.4, it holds

limn→∞

‖Wnxn −Wxn‖ = 0. (2.7)

Lemma 2.6 (see [2]). Let C be a nonempty closed convex subset of a q-uniformly smooth and strictlyconvex Banach space E. Let Si, i = 1, 2, . . ., be a λi-strict pseudocontraction from C into itself suchthat ∩∞

n=1F(Sn)/= ∅, and let inf λi > 0. Let tn, n = 1, 2, . . ., be real numbers such that 0 < tn ≤ b < 1for any n ≥ 1. Assume that the sequence {θn,k} satisfies (H1)–(H3). Then, F(W) = ∩∞

n=1F(Sn).


an+1 ≤ (1 − αn)an + δn, n ≥ 0, (2.8)

where {αn} is a sequence in (0, 1) and {δn} is a sequence in R such that

(i)∑∞

n=1 αn = ∞,

(ii) lim supn→∞δn/αn ≤ 0 or∑∞

n=1 |δn| <∞.


Lemma 2.8 (see [17]). Let {xn} and {yn} be bounded sequences in a Banach spaceX, and let {βn} bea sequence in [0, 1] with 0 < lim infn→∞βn ≤ lim supn→∞βn < 1. Suppose that xn+1 = (1 − βn)yn +βnxn for all integers n ≥ 0 and lim supn→∞(‖yn+1−yn‖−‖xn+1−xn‖) ≤ 0. Then, limn→∞‖yn−xn‖ =0.

Lemma 2.9 (see [3]). Assume thatA is a strong positive linear bounded operator on a smooth Banachspace E with coefficient γ > 0 and 0 < ρ ≤ ‖A‖−1. Then, ‖I − ρA‖ ≤ 1 − ργ .


3. Main Results

In this section, we prove a strong convergence theorem.

Theorem 3.1. Let E be a real q-uniformly smooth and strictly convex Banach space which admitsa weakly sequentially continuous duality mapping J from E to E∗. Let C be a nonempty closed andconvex subset of E which is also a sunny nonexpansive retraction of E such that C + C ⊂ C. Let Abe a strongly positive linear bounded operator on E with coefficient γ > 0 such that 0 < γ < γ/α,and let f be a contraction of C into itself with coefficient α ∈ (0, 1). Let Si, i = 1, 2, . . ., be λi-strict pseudocontractions from C into itself such that ∩∞

n=1F(Sn)/= ∅ and inf λi > 0. Assume that thesequences {αn}, {βn}, {γn}, and {δn} in (0, 1) satisfy the following conditions:

(i)∑∞

n=0 αn = ∞; and limn→∞αn = 0,

(ii) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1,

(iii) limn→∞|γn+1 − γn| = 0,

(iv) limn→∞|δn+1 − δn| = 0,

(v) δn(1 + γn) − 2γn > a for some a ∈ (0, 1),

and the sequence {θn,k} satisfies (H1)–(H3). Then, the sequence {xn} generated by


zn = δnxn + (1 − δn)Wnxn,

yn = γnxn +(1 − γn

)Wnzn,


)I − αnA

)yn, ∀n ≥ 0

(3.1)

converges strongly to x∗ ∈ ∩∞n=1F(Sn), which solves the following variational inequality:

⟨γf(x∗) −Ax∗, J

(p − x∗)⟩ ≤ 0, ∀f ∈ ΠC, p ∈

∞⋂

n=1

F(Sn). (3.2)

Proof. By (i), we may assume, without loss of generality, that αn ≤ (1 − βn)‖A‖−1 for all n.Since A is a strongly positive bounded linear operator on E and by (2.1), we have

‖A‖ = sup{|〈Ax, J(x)〉| : x ∈ E, ‖x‖ = 1}. (3.3)

Observe that

⟨((1 − βn

)I − αnA

)x, J(x)

⟩= 1 − βn − αn〈Ax, J(x)〉≥ 1 − βn − αn‖A‖≥ 0, ∀x ∈ E.

(3.4)


This shows that (1 − βn)I − αnA is positive. It follows that

∥∥(1 − βn

)I − αnA

∥∥ = sup

{∣∣⟨((1 − βn)I − αnA

)x, J(x)

⟩∣∣ : x ∈ E, ‖x‖ = 1}

= sup{1 − βn − αn〈Ax, J(x)〉 : x ∈ E, ‖x‖ = 1

}

≤ 1 − βn − αnγ.(3.5)

First, we show that {xn} is bounded. Let p ∈ ∩∞n=1F(Sn). By the definition of {zn}, {yn},

and {xn}, we have

∥∥zn − p

∥∥ =

∥∥δnxn + (1 − δn)Wnxn − p

∥∥

≤ δn∥∥xn − p

∥∥ + (1 − δn)

∥∥Wnxn − p

∥∥

≤ δn∥∥xn − p

∥∥ + (1 − δn)∥∥xn − p

∥∥

=∥∥xn − p

∥∥,

(3.6)

and from this, we have

∥∥yn − p∥∥ =

∥∥γnxn +(1 − γn

)Wnzn − p

∥∥


∥∥ +(1 − γn

)∥∥Wnzn − p∥∥


∥∥ +(1 − γn

)∥∥zn − p∥∥


∥∥ +(1 − γn

)∥∥xn − p∥∥

=∥∥xn − p

∥∥.

(3.7)

It follows that

∥∥xn+1 − p

∥∥ =∥∥αnγf(xn) + βnxn +

((1 − βn

)I − αnA

)yn − p

∥∥

=∥∥αn

(γf(xn) −Ap

)+ βn

(xn − p

)+((1 − βn

)I − αnA

)(yn − p

)∥∥

≤ αn∥∥γf(xn) −Ap

∥∥ + βn∥∥xn − p

∥∥ +(1 − βn − αnγ

)∥∥yn − p∥∥


∥∥ + βn∥∥xn − p

∥∥ +(1 − βn − αnγ

)∥∥xn − p∥∥

≤ αn∥∥γf(xn) − γf

(p)∥∥ + αn

∥∥γf(p) −Ap∥∥ +

(1 − αnγ

)∥∥xn − p∥∥

≤ αnγα∥∥xn − p

∥∥ + αn∥∥γf

(p) −Ap∥∥ +

(1 − αnγ

)∥∥xn − p∥∥

=(1 − (

γ − γα)αn)∥∥xn − p

∥∥ +(γ − γα)αn

∥∥γf(p) −Ap∥∥

γ − γα

≤ max

{∥∥x1 − p

∥∥,

∥∥γf(p) −Ap∥∥

γ − γα

}

.

(3.8)


By induction on n, we obtain ‖xn − p‖ ≤ max{‖x0 − p‖, ‖γf(p) −Ap‖/γ − γα‖} for every n ≥ 0and x0 ∈ C, then {xn} is bounded. So, {yn}, {zn}, {Ayn}, {Wnxn}, {Wnzn}, and {f(xn)} arealso bounded.

Next, we claim that ‖xn+1 − xn‖ → 0 as n → ∞. Let x ∈ C and p ∈ ∩∞n=1F(Sn).

Fix k ∈ N for any n ∈ N with n ≥ k, and since Tn,k and Un,k are nonexpansive, we have‖Tn,kx − p‖ ≤ ‖x − p‖ and ‖Un,kx − p‖ ≤ ‖x − p‖, respectively. From (1.5), it follows that‖Skx − p‖ ≤ (1 + (1/λ1/q−1k ))supn‖x − p‖. We can set

M1 = infi

⎛

⎝2 +1

λ1/q−1i

⎞

⎠supn

∥∥xn − p

∥∥ <∞,

M2 = infi

⎛

⎝2 +1

λ1/q−1i

⎞

⎠supn

∥∥zn − p∥∥ <∞.

(3.9)

From (1.11), we have

‖Wn+1xn −Wnxn‖ = ‖Un+1,1xn −Un,1xn‖= ‖t1Tn+1,1Un+1,2xn + (1 − t1)xn − t1Tn,1Un,2xn − (1 − t1)xn‖= t1‖Tn+1,1Un+1,2xn − Tn,1Un,2xn‖= t1‖(θn+1,1S1 + (1 − θn+1,1))Un+1,2xn − Tn,1Un,2xn‖= t1‖(θn,1S1 + (1 − θn,1))Un+1,2xn

−Tn,1Un,2xn + (θn+1,1 − θn,1)(S1Un+1,2xn −Un+1,2xn)‖≤ t1‖Tn,1Un+1,2xn − Tn,1Un,2xn‖ + t1|θn+1,1 − θn,1|‖S1Un+1,2xn −Un+1,2xn‖≤ t1‖Un+1,2xn −Un,2xn‖ + t1|θn+1,1 − θn,1|M1

≤ t1‖Un+1,2xn −Un,2xn‖ + t1anM1

...

≤n∏

i=1

ti‖Un+1,n+1xn −Un,n+1xn‖ + anM1

n∑

j=1

j∏

i=1

ti

≤n∏

i=1

ti‖tn+1Tn+1,n+1xn + (1 − tn+1)xn − xn‖ + anM1b

1 − b

≤n+1∏

i=1

ti‖Tn+1,n+1xn − xn‖ + anM1b

1 − b

≤(bn+1 + an

b

1 − b)M1,

(3.10)


for all n ≥ 0. Similarly, we also have ‖Wn+1zn −Wnzn‖ ≤ (bn+1 +an(b/(1−b)))M2 for all n ≥ 0.We compute that

‖zn+1 − zn‖ = ‖(δn+1xn+1 + (1 − δn+1)Wn+1xn+1) − (δnxn + (1 − δn)Wnxn)‖

≤ (1 − δn+1)‖Wn+1xn+1 −Wn+1xn‖ + |δn+1 − δn|‖xn −Wn+1xn‖ + δn+1‖xn+1 − xn‖

+ (1 − δn)‖Wn+1xn −Wnxn‖

≤ (1 − δn+1)‖xn+1 − xn‖ + |δn+1 − δn|‖Wn+1xn − xn‖ + δn+1‖xn+1 − xn‖

+ (1 − δn)‖Wn+1xn −Wnxn‖

= ‖xn+1 − xn‖ + |δn+1 − δn|‖Wn+1xn − xn‖ + (1 − δn)‖Wn+1xn −Wnxn‖

≤ ‖xn+1 − xn‖ + |δn+1 − δn|‖Wn+1xn − xn‖ + ‖Wn+1xn −Wnxn‖

≤ ‖xn+1 − xn‖ + |δn+1 − δn|‖Wn+1xn − xn‖ +(bn+1 + an

b

1 − b)M1,

(3.11)

and∥∥yn+1 − yn

∥∥ =∥∥(γn+1xn+1 +

(1 − γn+1

)Wn+1zn+1

) − (γnxn +

(1 − γn

)Wnzn

)∥∥

=∥∥γn+1xn+1 +

(1 − γn+1

)Wn+1zn+1 −

(1 − γn+1

)Wn+1zn +

(1 − γn+1

)Wn+1zn

−γnxn −(1 − γn

)Wnzn −

(1 − γn

)Wn+1zn +

(1 − γn

)Wn+1zn − γn+1xn + γn+1xn

∥∥

=∥∥(1 − γn+1

)(Wn+1zn+1 −Wn+1zn) +

(γn − γn+1

)Wn+1zn + γn+1(xn+1 − xn)

+(γn+1 − γn

)xn +

(1 − γn

)(Wn+1zn −Wnzn)

∥∥

≤ (1 − γn+1

)‖Wn+1zn+1 −Wn+1zn‖ +∣∣γn+1 − γn

∣∣‖xn −Wn+1zn‖ + γn+1‖xn+1 − xn‖

+(1 − γn

)‖Wn+1zn −Wnzn‖≤ (

1 − γn+1)‖zn+1 − zn‖ +

∣∣γn+1 − γn∣∣‖Wn+1zn − xn‖ + γn+1‖xn+1 − xn‖

+ ‖Wn+1zn −Wnzn‖

≤ (1 − γn+1

)(‖xn+1 − xn‖ + |δn+1 − δn|‖Wn+1xn − xn‖ +

(bn+1 + an

b

1 − b)M1

)

+∣∣γn+1 − γn

∣∣‖Wn+1zn − xn‖ + γn+1‖xn+1 − xn‖ +(bn+1 + an

b

1 − b)M2

≤ ‖xn+1 − xn‖ + |δn+1 − δn|‖Wn+1xn − xn‖ +∣∣γn+1 − γn

∣∣‖Wn+1zn − xn‖

+ 2(bn+1 + an

b

1 − b)M,

(3.12)


where M = infi(2 + (1/λ1/q−1i ))supn(‖xn − p‖ + ‖zn − p‖) < ∞. Observe that we put ln =(xn+1 − βnxn)/1 − βn, then

xn+1 =(1 − βn

)ln + βnxn, ∀n ≥ 0. (3.13)

Now, we have

‖ln+1 − ln‖ =

∥∥∥∥∥αn+1γf(xn+1) +

((1 − βn+1

)I − αn+1A

)yn+1

1 − βn+1 − αnγf(xn) +((1 − βn

)I − αnA

)yn

1 − βn

∥∥∥∥∥

=

∥∥∥∥∥αn+1γf(xn+1)

1 − βn+1 +

(1 − βn+1

)yn+1

1 − βn+1 − αn+1Ayn+11 − βn+1 −αnγf(xn)

1 − βn −(1 − βn

)yn

1 − βn +αnAyn1 − βn

∥∥∥∥∥

=∥∥∥∥

αn+11 − βn+1

(γf(xn+1) −Ayn+1

)+

αn1 − βn

(Ayn − γf(xn)

)+ yn+1 − yn

∥∥∥∥

≤ αn+11 − βn+1

∥∥γf(xn+1) −Ayn+1∥∥ +

αn1 − βn

∥∥Ayn − γf(xn)∥∥ +

∥∥yn+1 − yn∥∥

≤ αn+11 − βn+1

∥∥γf(xn+1) −Ayn+1∥∥ +

αn1 − βn

∥∥Ayn − γf(xn)∥∥

+ ‖xn+1 − xn‖ + |δn+1 − δn|‖Wn+1xn − xn‖ +∣∣γn+1 − γn


+ 2(bn+1 + an

b

1 − b)M.

(3.14)

Therefore, we have

‖ln+1 − ln‖ − ‖xn+1 − xn‖ ≤ αn+11 − βn+1

∥∥γf(xn+1) −Ayn+1∥∥ +

αn1 − βn

∥∥Ayn − γf(xn)∥∥

+ |δn+1 − δn|‖Wn+1xn − xn‖ +∣∣γn+1 − γn


+ 2(bn+1 + an

b

1 − b)M.

(3.15)

From the conditions (i)–(iv), (H2), 0 < b < 1 and the boundedness of {xn}, {f(xn)}, {Ayn},{Wnxn}, and {Wnzn}, we obtain

lim supn→∞

(‖ln+1 − ln‖ − ‖xn+1 − xn‖) ≤ 0. (3.16)

It follows from Lemma 2.8 that limn→∞‖ln − xn‖ = 0. Noting (3.13), we see that

‖xn+1 − xn‖ =(1 − βn

)‖ln − xn‖ −→ 0, (3.17)


as n → ∞. Therefore, we have

limn→∞

‖xn+1 − xn‖ = 0. (3.18)

We also have ‖yn+1 − yn‖ → 0 and ‖zn+1 − zn‖ → 0 as n → ∞. Observing that

∥∥xn − yn

∥∥ ≤ ‖xn − xn+1‖ +

∥∥xn+1 − yn

∥∥

≤ ‖xn − xn+1‖ + αn∥∥γf(xn) −Ayn

∥∥ + βn

∥∥xn − yn

∥∥,

(3.19)

it follows that

(1 − βn

)∥∥xn − yn∥∥ ≤ ‖xn − xn+1‖ + αn

∥∥γf(xn) −Ayn

∥∥. (3.20)

By the conditions (i), (ii), (3.18), and the boundedness of {xn}, {f(xn)}, and {Ayn}, we obtain

limn→∞

∥∥xn − yn∥∥ = 0. (3.21)

Consider∥∥yn −Wnzn

∥∥ =∥∥γnxn +

(1 − γn

)Wnzn −Wnzn

∥∥ = γn‖xn −Wnzn‖,‖zn − xn‖ = ‖δnxn + (1 − δn)Wnxn − xn‖ = (1 − δn)‖Wnxn − xn‖.

(3.22)

It follows that

‖xn −Wnxn‖ ≤ ∥∥xn − yn∥∥ +

∥∥yn −Wnzn∥∥ + ‖Wnzn −Wnxn‖

≤ ∥∥xn − yn∥∥ + γn‖xn −Wnzn‖ + ‖zn − xn‖

≤ ∥∥xn − yn∥∥ + γn‖xn −Wnxn‖ + γn‖Wnxn −Wnzn‖ + ‖zn − xn‖

≤ ∥∥xn − yn∥∥ + γn‖xn −Wnxn‖ +

(1 + γn

)‖zn − xn‖=

∥∥xn − yn∥∥ + γn‖xn −Wnxn‖ +

(1 + γn

)(1 − δn)‖Wnxn − xn‖.

(3.23)

This implies that

(δn

(1 + γn

) − 2γn)‖Wnxn − xn‖ ≤ ∥∥xn − yn

∥∥. (3.24)

From the condition (v) and (3.21), we get

limn→∞

‖Wnxn − xn‖ = 0. (3.25)

On the other hand,

‖Wxn − xn‖ ≤ ‖Wxn −Wnxn‖ + ‖Wnxn − xn‖. (3.26)


From the boundedness of {xn} and using (2.7), we have ‖Wxn −Wnxn‖ → 0 as n → ∞. Itfollows that

limn→∞

‖Wxn − xn‖ = 0. (3.27)

Next, we prove that

lim supn→∞

⟨γf(x∗) −Ax∗, J(xn − x∗)

⟩ ≤ 0, (3.28)

where x∗ = limt→ 0xt with xt being the fixed point of contraction x �→ tγf(x) + (1 − tA)Wx.Noticing that xt solves the fixed point equation xt = tγf(xt) + (1 − tA)Wxt, it follows that

‖xt − xn‖ =∥∥(I − tA)(Wxt − xn) + t

(γf(xt) −Axn

)∥∥. (3.29)

It follows from Lemma 2.1 that

‖xt − xn‖2 =∥∥(I − tA)(Wxt − xn) + t

(γf(xt) −Axn

)∥∥2

≤ (1 − γt)2‖Wxt − xn‖2 + 2t

⟨γf(xt) −Axn, J(xt − xn)

⟩

=(1 − γt)2‖Wxt −Wxn +Wxn − xn‖2 + 2t


⟩

≤ (1 − γt)2

[‖Wxt −Wxn‖2 + 2〈Wxn − xn, J(Wxt − xn)〉

]

+ 2t⟨γf(xt) −Axn, J(xt − xn)

⟩

≤ (1 − γt)2

[‖xt − xn‖2 + 2‖Wxn − xn‖‖Wxt − xn‖

]+ 2t


⟩

≤ (1 − γt)2

[‖xt − xn‖2 + 2‖Wxn − xn‖(‖xt − xn‖ + ‖Wxn − xn‖)

]

+ 2t⟨γf(xt) −Axn, J(xt − xn)

⟩

=(1 − 2γt +

(γt

)2)‖xt − xn‖2 + 2(1 − γt)2‖Wxn − xn‖(‖xt − xn‖ + ‖Wxn − xn‖)

+ 2t⟨γf(xt) −Axt, J(xt − xn)

⟩+ 2t〈Axt −Axn, J(xt − xn)〉,

(3.30)

where

fn(t) = 2(1 − γt)2‖Wxn − xn‖(‖xt − xn‖ + ‖Wxn − xn‖) −→ 0 as n −→ ∞. (3.31)

Since A is linearly strong and positive and using (2.1), we have

〈Axt −Axn, J(xt − xn)〉 = 〈A(xt − xn), J(xt − xn)〉 ≥ γ‖xt − xn‖2. (3.32)


Substituting (3.32) in (3.30), we have

⟨Axt − γf(xt), J(xt − xn)

⟩ ≤(γ2t2

− γ)

‖xt − xn‖2 + 12tfn(t) + 〈Axt −Axn, J(xt − xn)〉

≤(γt

2− 1

)〈Axt −Axn, J(xt − xn)〉

+12tfn(t) + 〈Axt −Axn, J(xt − xn)〉

=γt

2〈Axt −Axn, J(xt − xn)〉 + 1

2tfn(t).

(3.33)

Letting n → ∞ in (3.33) and noting (3.31) yield that

lim supn→∞


⟩ ≤ t

2M3, (3.34)

whereM3 > 0 is a constant such thatM3 ≥ γ〈Axt −Axn, J(xt −xn)〉 for all t ∈ (0, 1) and n ≥ 0.Taking t → 0 from (3.34), we have

lim supt→ 0

lim supn→∞


⟩ ≤ 0. (3.35)

On the other hand, we have

⟨γf(x∗) −Ax∗, J(xn − x∗)

⟩=

⟨γf(x∗) −Ax∗, J(xn − x∗)

⟩

− ⟨γf(x∗) −Ax∗, J(xn − xt)

⟩+⟨γf(x∗) −Ax∗, J(xn − xt)

⟩

− ⟨γf(x∗) −Axt, J(xn − xt)

⟩+⟨γf(x∗) −Axt, J(xn − xt)

⟩

− ⟨γf(xt) −Axt, J(xn − xt)

⟩+⟨γf(xt) −Axt, J(xn − xt)

⟩

=⟨γf(x∗) −Ax∗, J(xn − x∗) − J(xn − xt)

⟩

+ 〈Axt −Ax∗, J(xn − xt)〉+⟨γf(x∗) − γf(xt), J(xn − xt)

⟩+⟨γf(xt) −Axt, J(xn − xt)

⟩,

(3.36)

which implies that

lim supn→∞

⟨γf(x∗) −Ax∗, J(xn − x∗)

⟩ ≤ lim supn→∞

⟨γf(x∗) −Ax∗, J(xn − x∗) − J(xn − xt)

⟩

+ ‖A‖‖xt − x∗‖lim supn→∞

‖xn − xt‖

+ γα‖x∗ − xt‖lim supn→∞

‖xn − xt‖

+ lim supn→∞

⟨γf(xt) −Axt, J(xn − xt)

⟩.

(3.37)


Noticing that J is norm-to-norm uniformly continuous on bounded subsets of C, itfollows from (3.35) that

lim supn→∞

⟨γf(x∗) −Ax∗, J(xn − x∗)

⟩= lim sup

t→ 0lim supn→∞

⟨γf(x∗) −Ax∗, J(xn − x∗)

⟩ ≤ 0. (3.38)

Therefore, we obtain that (3.28) holds.Finally, we prove that xn → x∗ as n → ∞. Now, from Lemma 2.1, we have

‖xn+1 − x∗‖2 = ∥∥αnγf(xn) + βnxn +

[(1 − βn

)I − αnA

]yn − x∗∥∥2

=∥∥[(1 − βn

)I − αnA

](yn − x∗) + αn

(γf(xn) −Ax∗) + βn(xn − x∗)

∥∥2

≤ (1 − βn − αnγ

)2∥∥yn − x∗∥∥2 + 2⟨αn

(γf(xn) −Ax∗) + βn(xn − x∗), J(xn+1 − x∗)

⟩

=(1 − βn − αnγ

)2∥∥yn − x∗∥∥2 + 2⟨αn

(γf(xn) −Ax∗), J(xn+1 − x∗)

⟩

+ 2βn〈xn − x∗, J(xn+1 − x∗)〉

=(1 − βn − αnγ

)2∥∥yn − x∗∥∥2 + 2αnγ⟨f(xn) − f(x∗), J(xn+1 − x∗)

⟩

+ 2αn⟨γf(x∗) −Ax∗, J(xn+1 − x∗)

⟩+ 2βn〈xn − x∗, J(xn+1 − x∗)〉

≤ (1 − βn − αnγ

)2‖xn − x∗‖2 + αnγα(‖xn+1 − x∗‖2 + ‖xn − x∗‖2

)

+ 2αn⟨γf(x∗) −Ax∗, J(xn+1 − x∗)

⟩+ βn

(‖xn+1 − x∗‖2 + ‖xn − x∗‖2

)

=[(1 − βn − αnγ

)2 + αnγα + βn]‖xn − x∗‖2 + (

αnγα + βn)‖xn+1 − x∗‖2

+ 2αn⟨γf(x∗) −Ax∗, J(xn+1 − x∗)

⟩,

(3.39)

and consequently,

‖xn+1 − x∗‖2 ≤(1 − βn − αnγ

)2 + αnγα + βn1 − αnγα − βn ‖xn − x∗‖2

+2αn

1 − αnγα − βn⟨γf(x∗) −Ax∗, J(xn+1 − x∗)

⟩

=

[

1 − 2αn(γ − γα)

1 − αnγα − βn

]

‖xn − x∗‖2 + β2n + 2βnαnγ + α2nγ2

1 − αnγα − βn ‖xn − x∗‖2

+2αn

1 − αnγα − βn⟨γf(x∗) −Ax∗, J(xn+1 − x∗)

⟩

=

[

1 − 2αn(γ − γα)


]

‖xn − x∗‖2 + 2αn(γ − γα)


×[β2n + 2βnαnγ + α2nγ

2

2αn(γ − γα) M4 +

1γ − γα

⟨γf(x∗) −Ax∗, J(xn+1 − x∗)

⟩]

,

(3.40)


where M4 is an appropriate constant such that M4 ≥ supn≥0‖xn − x∗‖2. Setting cn = 2αn(γ −γα)/(1 − αnγα − βn) and bn = (β2n + 2βnαnγ + α2nγ

2)/(2αn(γ − γα))M4 + (1/(γ − γα))〈γf(x∗) −Ax∗, J(xn+1 − x∗)〉, then we have

‖xn+1 − x∗‖2 ≤ (1 − cn)‖xn − x∗‖2 + cnbn. (3.41)

By (3.28), (i) and applying Lemma 2.7 to (3.41), we have xn → x∗ as n → ∞. This completesthe proof.

Corollary 3.2. Let E be a real q-uniformly smooth and strictly convex Banach space which admitsa weakly sequentially continuous duality mapping J from E to E∗. Let C be a nonempty closed andconvex subset of E which is also a sunny nonexpansive retraction of E such that C + C ⊂ C. Let Abe a strongly positive linear bounded operator on E with coefficient γ > 0 such that 0 < γ < γ/α,and let f be a contraction of C into itself with coefficient α ∈ (0, 1). Let Si, i = 1, 2, . . ., be λi-strict pseudocontractions from C into itself such that ∩∞


(i)∑∞

n=0 αn = ∞; and limn→∞ αn = 0,


(iii) limn→∞|γn+1 − γn| = 0,

(iv) limn→∞|δn+1 − δn| = 0,

(v) δn(1 + γn) − 2γn > a for some a ∈ (0, 1),

and the sequence {θn} satisfies (H1). Then, the sequence {xn} generated by




)Wnzn,


)I − αnA

)yn, ∀n ≥ 0

(3.42)



(p − x∗)⟩ ≤ 0, ∀f ∈ ΠC, p ∈

∞⋂

n=1

F(Sn). (3.43)

Corollary 3.3. Let E be a real q-uniformly smooth and strictly convex Banach space which admitsa weakly sequentially continuous duality mapping J from E to E∗. Let C be a nonempty closed andconvex subset of E which is also a sunny nonexpansive retraction of E such that C + C ⊂ C. LetA be a strongly positive linear bounded operator on E with coefficient γ > 0 such that 0 < γ <γ/α, and let f be a contraction of C into itself with coefficient α ∈ (0, 1). Let Si, i = 1, 2, . . ., be anonexpansive mapping from C into itself such that ∩∞


(i)∑∞

n=0 αn = ∞; and limn→∞ αn = 0,



(iii) limn→∞|γn+1 − γn| = 0,

(iv) limn→∞|δn+1 − δn| = 0,

(v) δn(1 + γn) − 2γn > a for some a ∈ (0, 1).

Then, the sequence {xn} generated by




)Wnzn,


)I − αnA

)yn, ∀n ≥ 0

(3.44)



(p − x∗)⟩ ≤ 0, ∀f ∈ ΠC, p ∈

∞⋂

n=1

F(Sn). (3.45)

Remark 3.4. Theorem 3.1, Corollaries 3.2, and 3.3, improve and extend the correspondingresults of Cai and Hu [3], Dong et al. [2], and Katchang and Kumam [4, 5] in the followingsenses.

(i) For the mappings, we extend the mappings from an infinite family of nonexpansivemappings to an infinite family of strict pseudocontraction mappings.

(ii) For the algorithms, we propose newmodified Ishikawa iterative algorithms, whichare different from the ones given in [2–5] and others.

Acknowledgments

The authors would like to thank The National Research Council of Thailand (NRCT)and the Faculty of Science KMUTT for financial support. Furthermore, they also wouldlike to thank the National Research University Project of Thailand’s Office of the HigherEducation Commission for financial support (NRU-CSEC Project no. 54000267). Finally, theyare grateful for the reviewers for the careful reading of the paper and for the suggestionswhich improved the quality of this work.

References


[2] Q.-L. Dong, S. He, and F. Su, “Strong convergence of an iterative algorithm for an infinite family ofstrict pseudo-contractions in Banach spaces,” Applied Mathematics and Computation, vol. 216, no. 3, pp.959–969, 2010.

[3] G. Cai and C. S. Hu, “Strong convergence theorems of a general iterative process for a finite familyof λi-strict pseudo-contractions in q-uniformly smooth Banach spaces,” Computers &Mathematics withApplications, vol. 59, no. 1, pp. 149–160, 2010.


[4] P. Katchang and P. Kumam, “Strong convergence of the modified Ishikawa iterative methodfor infinitely many nonexpansive mappings in Banach spaces,” Computers & Mathematics withApplications, vol. 59, no. 4, pp. 1473–1483, 2010.

[5] P. Katchang and P. Kumam, “Corrigendum to “Strong convergence of the modified Ishikawa iterativemethod for infinitely many nonexpansive mappings in Banach spaces” [Comput. Math. Appl. 59(2010) 1473-1483],” Computers & Mathematics with Applications, vol. 61, no. 1, p. 148, 2011.

[6] K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansivemappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001.

[7] W. Takahashi, “Weak and strong convergence theorems for families of nonexpansive mappings andtheir applications,” Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 51, no. 2, pp. 277–292,1997.

[8] Y. J. Cho, S. M. Kang, and X. Qin, “Strong convergence of an implicit iterative process for an infinitefamily of strict pseudocontractions,” Bulletin of the KoreanMathematical Society, vol. 47, no. 6, pp. 1259–1268, 2010.

[9] G. Cai and C. S. Hu, “Strong convergence theorems of modified Ishikawa iterative process with errorsfor an infinite family of strict pseudo-contractions,”Nonlinear Analysis: Theory, Methods & Applications,vol. 71, no. 12, pp. 6044–6053, 2009.

[10] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, YokohamaPublishers, Yokohama, Japan, 2000.

[11] R. E. Bruck Jr., “Nonexpansive projections on subsets of Banach spaces,” Pacific Journal of Mathematics,vol. 47, pp. 341–355, 1973.

[12] S. Reich, “Asymptotic behavior of contractions in Banach spaces,” Journal of Mathematical Analysis andApplications, vol. 44, pp. 57–70, 1973.

[13] S.-S. Chang, “On Chidume’s open questions and approximate solutions of multivalued stronglyaccretive mapping equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol.216, no. 1, pp. 94–111, 1997.

[14] H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods &Applications, vol. 16, no. 12, pp. 1127–1138, 1991.

[15] H. Zhang and Y. Su, “Strong convergence theorems for strict pseudo-contractions in q-uniformlysmooth Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3236–3242, 2009.


[17] T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non-expansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications,vol. 305, no. 1, pp. 227–239, 2005.


Research ArticleCommon Fixed-Point Problem for a FamilyMultivalued Mapping in Banach Space

Zhanfei Zuo

Department of Mathematics and Computer Science, Chongqing Three Gorges University,Wanzhou 404000, China

Correspondence should be addressed to Zhanfei Zuo, [email protected]

Received 11 December 2010; Revised 15 March 2011; Accepted 17 March 2011


Copyright q 2011 Zhanfei Zuo. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

It is our purpose in this paper to prove two convergents of viscosity approximation scheme toa common fixed point of a family of multivalued nonexpansive mappings in Banach spaces.Moreover, it is the unique solution in F to a certain variational inequality, where F := ∩∞

n=0F(Tn)stands for the common fixed-point set of the family of multivalued nonexpansive mapping {Tn}.

1. Introduction

LetX be a Banach space with dualX∗, and letK be a nonempty subset ofX. A gauge functionis a continuous strictly increasing function ϕ : R+ → R+ such that ϕ(0) = 0 and limt→∞ϕ(t) =∞. The duality mapping Jϕ : X → X∗ associated with a gauge function ϕ is defined byJϕ(x) := {f ∈ X∗ : 〈x, f〉 = ‖x‖‖f‖, ‖f‖ = ϕ(‖x‖)}, x ∈ X, where 〈·, ·〉 denotes the generalizedduality pairing. In the particular case ϕ(t) = t, the duality map J = Jϕ is called the normalizedduality map. We note that Jϕ(x) = (ϕ(‖x‖)/‖x‖)J(x). It is known that if X is smooth, then Jϕis single valued and norm to weak∗ continuous (see [1]). When {xn} is a sequence in X, thenxn → x(xn ⇀ x, xn ⇁ x)will denote strong (weak, weak∗) convergence of the sequence {xn}to x. s

Following Browder [2], we say that a Banach spaceX has the weakly continuous dual-ity mapping if there exists a gauge function ϕ for which the duality map Jϕ is single valuedand weak to weak∗ sequentially continuous, that is, if {xn} is a sequence in X weakly conver-gent to a point x, then the sequence {Jϕ(xn)} converges weak∗ to Jϕ(x). It is known that lp(1 <p < 1) spaces have a weakly continuous duality mapping Jϕ with a gauge ϕ(t) = tp−1. Setting

Φ(t) =∫ t

0ϕ(τ)dτ, t ≥ 0, (1.1)


it is easy to see that Φ(t) is a convex function and Jϕ(x) = ∂Φ(‖x‖), for x ∈ X, where ∂denotes the subdifferential in the sense of convex analysis. We will denote by 2X the familyof all subsets ofX, by CB(X) the family of all nonempty closed bounded subsets ofX, and byC(X) the family of all nonempty compact subsets of X. A multivalued mapping T : K → 2X

is said to be nonexpansive (resp., contractive) if

H(Tx, Ty

) ≤ ∥∥x − y∥∥, x, y ∈ K,(resp., H

(Tx, Ty

) ≤ k∥∥x − y∥∥, for some k ∈ (0, 1)),

(1.2)

whereH(·, ·) denotes the Hausdorff metric on CB(X) defined by

H(A,B) := max

{

supx∈A

infy∈B

∥∥x − y∥∥, sup

y∈Binfx∈A

∥∥x − y∥∥

}

, A, B ∈ CB(X). (1.3)

Since Banach’s contraction mapping principle was extended nicely to multivalued mappingsbyNadler in 1969 (see [3]), many authors have studied the fixed-point theory formultivaluedmappings.

In this paper, we construct two viscosity approximation sequences for a family ofmultivalued nonexpansive mappings in Banach spaces. Let K be a nonempty closed convexsubset of Banach space X and let Tn : K → C(K), n = 1, 2 . . . be a family of multivaluednonexpansive mapping, f : K → K is a contraction mapping with constant α ∈ (0, 1). Letαn ∈ (0, 1), βn ∈ (0, 1). For any given x0 ∈ K, let y0 ∈ T0x0 such that

x1 = α0f(x0) + (1 − α0)y0. (1.4)

From Nadler Theorem (see [3]), we can choose y1 ∈ T1x1 such that

∥∥y0 − y1∥∥ ≤ H(T0x0, T1x1). (1.5)

Inductively, we can get the sequence {xn} as follows:

xn+1 = αnf(xn) + (1 − αn)yn, ∀n ∈N, (1.6)

where, for each n ∈N, yn ∈ Tnxn such that

∥∥yn+1 − yn∥∥ ≤ H(Tn+1xn+1, Tnxn). (1.7)

Similarly, we also have the following multivalued version of the modified Mann iteration:

xn+1 = βnf(xn) + αnxn +(1 − αn − βn

)yn, (1.8)

and yn ∈ Tnxn such that ‖yn+1 − yn‖ ≤ H(Tn+1xn+1, Tnxn). Then, {xn} is said to satisfyCondition (A′) if for any subsequence xnk ⇀ x and d(xn+1, Tn(xn)) → 0 implies that x ∈ F,


where F := ∩∞i=0F(Tn)/= ∅ is the common fixed-point set of the family of multivalued mapping

{Tn}. We give an example of a family of multivalued nonexpansive mappings with Condition(A′) as follows.

Example 1.1. Take X = R and Tn = T (for all n ≥ 0), where T is defined by

T(x) =

⎧⎪⎨

⎪⎩

{0}, x ≤ 1,{x − 1

2,12− x}, otherwise.

(1.9)

Let f : R → {0} and αn = 1/n, n ≥ 2, then F = {0} and the iteration (1.6), reduced to

xn+1 =(1 − 1

n + 2

)yn, ∀n ≥ 0, (1.10)

where yn ∈ Txn, and it satisfies Condition (A′). In fact, if x0 ≤ 1, then (for all n ∈ N, n > 0)xn =0 and Condition (A′) is automatically satisfied. If x0 > 1, then there exists an integer p ≥ 2,such that

x0 ∈(p(p + 1

)

4− 12,

(p + 1

)(p + 2

)

4− 12

]

, xp−1 =1p

(

x0 −p(p − 1

)

4

)

. (1.11)

Then, yp ∈ Txp−1 = {0}; hence, xn = 0 (for all n ≥ p), from which we deduce that Condition(A′) is satisfied.

2. Preliminaries

Let K ⊂ X be a closed convex and Q a mapping of X onto K, then Q is said to be sunny ifQ(Q(x) + t(x − Q(x))) = Q(x) for all x ∈ X and t ≥ 0. A mapping Q of X into X is said tobe a retraction if Q2 = Q. A subset K of X is said to be a sunny nonexpansive retract of X ifthere exists a sunny nonexpansive retraction of X onto K, and it is said to be a nonexpansiveretract of X. If X = H, the metric projection P is a sunny nonexpansive retraction fromH toany closed convex subset ofH. The following Lemmas will be useful in this paper.

Lemma 2.1 (see [4]). Let K be a nonempty convex subset of a smooth Banach space X, let J : X →X∗ be the (normalized) duality mapping of X, and let Q : X → K be a retraction, then the followingare equivalent:

(1) 〈x − Px, j(y − Px)〉 ≤ 0 for all x ∈ X and y ∈ K,

(2) Q is both sunny and nonexpansive.

We note that Lemma 2.1 still holds if the normalized duality map J is replaced with the general dualitymap Jϕ, where ϕ is a gauge function.

Lemma 2.2 (see [5]). Let {an} be a sequence of nonnegative real numbers satisfying the property

an+1 ≤(1 − γn

)an + βn, n ≥ 0, (2.1)


where {γn} ⊂ (0, 1) and {βn} is a real number sequence such that

(i)∑∞

n=0 γn = ∞,

(ii) either lim supn→∞(βn/γn) ≤ 0 or∑∞

n=0 |βn| <∞,

then {an} converges to zero, as n → ∞.

Lemma 2.3 (see [1]). Let X be a real Banach space, then for all x, y ∈ X, one gets that

Φ(‖x + y‖) ≤ Φ(‖x‖) + 〈y, jϕ

(x + y

)〉, ∀jϕ ∈ Jϕ. (2.2)

Lemma 2.4 (see [6]). Let {xn} and {yn} be bounded sequences in a Banach space X such that

xn+1 = γnxn +(1 − γn

)yn, n ≥ 0, (2.3)

where {γn} is a sequence in [0, 1] such that

0 < lim infn→∞

γn ≤ lim supn→∞

γn < 1. (2.4)

Assume that lim supn→∞(‖yn+1 − yn‖ − ‖xn+1 − xn‖) ≤ 0, then limn→∞‖yn − xn‖ = 0.

3. Main Results

Theorem 3.1. LetX be a reflexive Banach space with weakly sequentially continuous duality mappingJϕ for some gauge ϕ, let K be a nonempty closed convex subset of X, and let Tn : K → C(K), n =0, 1, 2 . . ., be a family of multivalued nonexpansive mappings with F /= ∅ which is sunny nonexpansiveretract of K with Q a nonexpansive retraction. Furthermore, Tn(p) = {p} for any fixed-point p ∈ F,{xn} is defined by (1.6), and αn ∈ (0, 1) satisfies the following conditions:

(1) αn → 0 as n → ∞,

(2)∑∞

n=0 αn = ∞,

(3) {xn} satisfies Condition (A′).

Then, {xn} converges strongly to a common fixed-point x = Q(f(x)) of a family Tn, n = 0, 1, 2 . . ., asn → ∞. Moreover, x is the unique solution in F to the variational inequality

⟨f(x) − x, jϕ

(y − x)⟩ ≤ 0, ∀y ∈ F. (3.1)

Proof. First, we show the uniqueness of the solution to the variational inequality (3.1) in X.In fact, let y ∈ F be another solution of (3.1) in F, then we have

⟨f(x) − x, jϕ

(y − x)⟩ ≤ 0,

⟨f(y) − y, jϕ

(x − y)⟩ ≤ 0. (3.2)

From (3.2), we have that

(1 − α)ϕ(∥∥x − y∥∥)∥∥x − y∥∥ ≤ 0. (3.3)


We must have x = y and the uniqueness is proved. Let p ∈ F, then, from iteration (1.6), weobtain that

∥∥xn+1 − p

∥∥ ≤ ∥∥xn+1 − αnf

(p) − (1 − αn)p

∥∥ +∥∥αnf

(p)+ (1 − αn)p − p

∥∥

=∥∥αn(f(xn) − f

(p))

+ (1 − αn)(yn − p

)∥∥ + αn∥∥f(p) − p∥∥

≤ αnα∥∥xn − p

∥∥ + (1 − αn)H

(Tnxn, Tnp

)+ αn

∥∥f(p) − p∥∥

≤ (1 − (1 − α)αn)∥∥xn − p

∥∥ + αn

∥∥f(p) − p∥∥.

(3.4)

Using an induction, we obtain ‖xn −p‖ ≤ max{‖x0 −p‖, (1/(1−α))‖f(p)−p‖}, for all integersn, thus, {xn} is bounded and so are {Tnxn} and {f(xn)}. This implies that

d(xn+1, Tn(xn)) ≤∥∥xn+1 − yn

∥∥ = αn

∥∥f(xn) − yn

∥∥ −→ 0 as n −→ ∞. (3.5)

Next, we will show that

lim supn→∞

〈f(x) − x, jϕ(xn+1 − x)〉 ≤ 0. (3.6)

Since X is reflexive and {xn} is bounded, we may assume that xnk ⇀ q such that

lim supn→∞

⟨f(x) − x, jϕ(xn+1 − x)

⟩= lim sup

k→∞

⟨f(x) − x, jϕ(xnk − x)

⟩. (3.7)

From (3.5) and {xn} satisfying Condition (A′), we obtain that q ∈ F. On the other hand, wenotice that the assumption that the duality mapping Jϕ is weakly continuous implies that Xis smooth; from Lemma 2.1, we have

lim supn→∞

⟨f(x) − x, jϕ(xn+1 − x)

⟩= lim sup

k→∞

⟨f(x) − x, jϕ(xnk − x)

⟩

= 〈f(x) − x, jϕ(q − x)〉

= 〈Q(x) − x, jϕ(q − x)〉 ≤ 0.

(3.8)

Finally, we will show that xn → x as n → ∞. From iteration (1.6) and Lemma 2.3, we getthat

Φ(‖xn+1 − x‖) ≤ Φ(∥∥αn

(f(xn) − f(x)

)+ (1 − αn)

(yn − x

)∥∥) + αn⟨f(x) − x, jϕ(xn+1 − x)

⟩

≤ Φ(αnα‖xn − x‖ + (1 − αn)H(Tnxn, Tnx)) + αn⟨f(x) − x, jϕ(xn+1 − x)

⟩

≤ (1 − αn(1 − α))Φ(‖xn − x‖) + αn⟨f(x) − x, jϕ(xn+1 − x)

⟩.

(3.9)

Lemma 2.2 gives that xn → x as n → ∞. Moreover, x satisfying the variational inequalityfollows from the property of Q.


Let f ≡ u ∈ K in iteration (1.6) be a constant mapping, then x = Qu. In fact, we havethe following corollary.

Corollary 3.2. Let {xn} and Tn be as in Theorem 3.1, f ≡ u ∈ K, then {xn} converges strongly to acommon fixed-point x = Q(u) of a family Tn, n = 0, 1, 2 . . ., as n → ∞. Moreover, x is the uniquesolution in F to the variational inequality

⟨u −Q(u), jϕ

(y −Q(u)

)⟩ ≤ 0, ∀y ∈ F. (3.10)

If X = H, then the condition that F is a sunny nonexpansive retract of K in Theorem 3.1 is notnecessary, and one has the following Corollary.

Corollary 3.3. LetH be a Hilbert space with weakly sequentially continuous duality mapping Jϕ forsome gauge ϕ, and let {xn} and Tn be as in Theorem 3.1, then {xn} converges strongly to a commonfixed-point x = PFf(x) of a family of Tn, n = 0, 1, 2 . . ., where PF is the metric projection from Konto F.

Proof. It is well known thatH is reflexive; by Propositions 2.3 and 2.6(ii) of [7], we get that Fis closed and convex, and hence the projection mapping PF is sunny nonexpansive retractionmapping, and the result follows from Theorem 3.1.

Corollary 3.4. LetX be a real smooth Banach space, letK be a nonempty compact subset ofX, and letTn and {xn} be as in Theorem 3.1, then {xn} converges strongly to a common fixed-point x = Q(f(x))of a family of Tn, n = 0, 1, 2 . . ., as n → ∞. Moreover, x is the unique solution in F to the variationalinequality

⟨f(x) − x, jϕ

(y − x)⟩ ≤ 0, ∀y ∈ F. (3.11)

Proof. Following the method of the proof of Theorem 3.1, we get that

d(xn+1, Tn(xn)) ≤∥∥xn+1 − yn

∥∥ = αn∥∥f(xn) − yn

∥∥ −→ 0 as n −→ ∞. (3.12)

Next, we will show that

lim supn→∞

⟨f(x) − x, jϕ(xn+1 − x)

⟩ ≤ 0. (3.13)

SinceK is compact and {xn} is bounded, we can assume that there exists a subsequence {xnk}of {xn} such that xnk → q ∈ K,

lim supn→∞

⟨f(x) − x, jϕ(xn+1 − x)

⟩= lim

k→∞⟨f(x) − x, jϕ(xnk − x)

⟩. (3.14)


From (3.12) and {xn} satisfying Condition (A′), we obtain that q ∈ F. On the other hand,from the fact thatX is smooth, the duality being norm to weak∗ continuous, and the standardcharacterization of retraction on F, we obtain that

lim supn→∞

⟨f(x) − x, jϕ(xn+1 − x)

⟩= lim

k→∞〈f(x) − x, jϕ(xnk − x)〉

= 〈f(x) − x, jϕ(q − x)〉

= 〈Q(x) − x, jϕ(q − x)〉 ≤ 0.

(3.15)

Now, following the method of the proof of Theorem 3.1, we get the required result.

Theorem 3.5. LetX be a reflexive Banach space with weakly sequentially continuous duality mappingJϕ for some gauge ϕ, let K be a nonempty closed convex subset of X, and let Tn : K → C(K), n =0, 1, 2 . . ., be a family of multivalued nonexpansive mappings with F /= ∅ which is sunny nonexpansiveretract of K with Q a nonexpansive retraction. H(Tn+1x, Tny) ≤ ‖x − y‖ for arbitrary n ∈ N.Furthermore, Tn(p) = {p} for any fixed-point p ∈ F. {xn} is defined by (1.8) and αn, βn satisfy thefollowing conditions:

(i) βn → 0 as n → ∞,

(ii)∑∞

n=0 βn = ∞,

(iii) 0 < liminfn→∞αn ≤ lim supn→∞αn < 1.

If {xn} satisfies Condition (A′), then {xn} converges strongly to a common fixed-point x = Q(f(x))of a family of Tn, n = 0, 1, 2 . . ., as n → ∞. Moreover, x is the unique solution in F to the variationalinequality

〈f(x) − x, jϕ(y − x)〉 ≤ 0, ∀y ∈ F. (3.16)

Proof. We first show that the sequence {xn} defined by (1.8) is bounded. In fact, take p ∈ F,noting that Tn(p) = {p}, we have

∥∥xn+1 − p∥∥ =(1 − αn − βn

)∥∥yn − p∥∥ + αn

∥∥xn − p∥∥ + βn

∥∥f(xn) − p∥∥

=(1 − αn − βn

)∥∥yn − Tnp∥∥ + αn

∥∥xn − p∥∥ + βn

∥∥f(xn) − p∥∥

≤ (1 − αn − βn)H(Tnxn, Tnp

)+ αn

∥∥xn − p∥∥ + βn

∥∥f(xn) − p∥∥

≤ (1 − αn − βn)∥∥xn − p

∥∥ + αn∥∥xn − p

∥∥ + βn∥∥f(xn) − p

∥∥

≤ (1 − βn)∥∥xn − p

∥∥ + βn(α∥∥xn − p

∥∥ +∥∥f(p) − p∥∥)

≤ (1 + (α − 1)βn)∥∥xn − p

∥∥ + βn(1 − α)∥∥f(p) − p∥∥

1 − α .

(3.17)


∥∥xn − p∥∥ ≤ max

{∥∥x0 − p

∥∥,

∥∥f(p) − p∥∥

1 − α

}

, (3.18)


so are {yn} and {f(xn)}. Thus, we have that

limn→∞

βn∥∥f(xn) − yn

∥∥ = 0. (3.19)

Next, we show that

limn→∞

d(xn+1, Tn(xn)) = 0. (3.20)

Let λn = βn/(1 − αn) and zn = λnf(xn) + (1 − λn)yn, then

limn→∞

λn = 0, xn+1 = αnxn + (1 − αn)zn. (3.21)

Therefore, we have for some appropriate constantM > 0 that the following inequality:

‖zn+1 − zn‖ =∥∥λn+1f(xn+1) + (1 − λn+1)yn+1 −

(λnf(xn) + (1 − λn)yn

)∥∥

≤ |λn+1 − λn|∥∥f(xn+1) − f(xn)

∥∥ +∥∥yn+1 − yn

∥∥ + λn∥∥yn∥∥ + λn+1

∥∥yn+1∥∥

≤ |λn+1 − λn|∥∥f(xn+1) − f(xn)

∥∥ +H(Tn+1xn+1, Tnxn) + (λn + λn+1)M

≤ |λn+1 − λn|∥∥f(xn+1) − f(xn)

∥∥ + ‖xn+1 − xn‖ + (λn + λn+1)M

(3.22)

holds. Thus, lim supn→∞(‖zn+1 − zn‖ − ‖xn+1 − xn‖) ≤ limn→∞(|λn+1 − λn|‖f(xn+1) − f(xn)‖ +(λn + λn+1)M) = 0. By Lemma 2.4, we obtain

limn→∞

‖xn − zn‖ = 0,

∥∥xn − yn∥∥ ≤ ‖xn − zn‖ +

∥∥zn − yn∥∥ = ‖xn − zn‖ + λn

∥∥f(xn) − yn∥∥ −→ 0.

(3.23)

Therefore, we have

d(xn+1, Tn(xn)) ≤∥∥xn+1 − yn

∥∥ ≤ βn∥∥f(xn) − yn

∥∥ + αn∥∥xn − yn

∥∥ −→ 0. (3.24)

Using (3.20) and {xn} satisfying Condition (A′), we can use the same argumentation asTheorem 3.1 proves that x ∈ F and

lim supn→∞

⟨f(x) − x, jϕ(xn+1 − x)

⟩ ≤ 0. (3.25)


Finally, we show that xn → x as n → ∞. In fact, from iteration (1.8) and Lemma 2.3, we have

Φ(‖xn+1 − x‖) = Φ(∥∥βnf(xn) + αnxn +

(1 − αn − βn

)yn − x

∥∥)

= Φ(∥∥αn(xn − x) +

(1 − αn − βn

)(yn − x

)+ βn

(f(xn) − f(x)

)+ βn

(f(x) − x)∥∥)

≤ Φ(‖αn(xn − x)‖ +

(1 − αn − βn

)H(Tnxn, Tnx) + αβn‖xn − x‖

)

+ βn⟨f(x) − x, jϕ(xn+1 − x)

⟩

≤ [1 − (1 − α)βn]Φ(‖xn − x‖) + βn

⟨f(x) − x, jϕ(xn+1 − x)

⟩.

(3.26)

From (ii) and (3.25), it then follows that

∞∑

n=0(1 − α)βn = ∞, lim sup

n

⟨f(x) − x, jϕ(xn+1 − x)

⟩

1 − α ≤ 0. (3.27)

Apply Lemma 2.2 to conclude that xn → x.

References

[1] I. Cioranescu,Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 ofMathematicsand Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.

[2] F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,”Mathematische Zeitschrift, vol. 100, pp. 201–225, 1967.

[3] S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.

[4] R. E. Bruck Jr., “Nonexpansive projections on subsets of Banach spaces,” Pacific Journal of Mathematics,vol. 47, pp. 341–355, 1973.


[6] T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in generalBanach spaces,” Fixed Point Theory and Applications, vol. 1, pp. 103–123, 2005.

[7] H. H. Bauschke and P. L. Combettes, “A weak-to-strong convergence principle for Fejer-monotonemethods in Hilbert spaces,” Mathematics of Operations Research, vol. 26, no. 2, pp. 248–264, 2001.


Research ArticleA General Iterative Algorithm for GeneralizedMixed Equilibrium Problems and VariationalInclusions Approach to Variational Inequalities

Thanyarat Jitpeera and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University ofTechnology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand


Received 1 December 2010; Revised 28 January 2011; Accepted 17 February 2011


Copyright q 2011 T. Jitpeera and P. Kumam. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We introduce a new general iterative method for finding a common element of the set of solutionsof fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibriumproblems, and the set of solutions of the variational inclusion for a β-inverse-strongly monotonemapping in a real Hilbert space. We prove that the sequence converges strongly to a commonelement of the above three sets under some mild conditions. Our results improve and extend thecorresponding results of Marino and Xu (2006), Su et al. (2008), Klin-eam and Suantai (2009), Tanand Chang (2011), and some other authors.

1. Introduction

Let C be a closed convex subset of a real Hilbert spaceH with the inner product 〈·, ·〉 and thenorm ‖·‖. Let F be a bifunction of C×C intoR, whereR is the set of real numbers,Ψ : C → Ha mapping, and ϕ : C → R a real-valued function. The generalized mixed equilibrium problem isfor finding x ∈ C such that

F(x, y)+⟨Ψx, y − x⟩ + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.1)

The set of solutions of (1.1) is denoted by GMEP(F, ϕ,Ψ), that is,

GMEP(F, ϕ,Ψ

)={x ∈ C : F

(x, y)+⟨Ψx, y − x⟩ + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C}. (1.2)


If F ≡ 0, the problem (1.1) is reduced into the mixed variational inequality of Browder type [1]for finding x ∈ C such that

⟨Ψx, y − x⟩ + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.3)

The set of solutions of (1.3) is denoted by MVI(C, ϕ,Ψ).If Ψ ≡ 0 and ϕ ≡ 0, the problem (1.1) is reduced into the equilibrium problem [2] for

finding x ∈ C such that

F(x, y) ≥ 0, ∀y ∈ C. (1.4)

The set of solutions of (1.4) is denoted by EP(F). This problem contains fixed point problemsand includes as special cases numerous problems in physics, optimization, and economics.Some methods have been proposed to solve the equilibrium problem; see [3–5].

If F ≡ 0 and ϕ ≡ 0, the problem (1.1) is reduced into the Hartmann-Stampacchiavariational inequality [6] for finding x ∈ C such that

⟨Ψx, y − x⟩ ≥ 0, ∀y ∈ C. (1.5)

The set of solutions of (1.5) is denoted by VI(C,Ψ). The variational inequality has beenextensively studied in the literature [7].

If F ≡ 0 and Ψ ≡ 0, the problem (1.1) is reduced into the minimize problem for findingx ∈ C such that

ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.6)

The set of solutions of (1.6) is denoted by Arg min(ϕ).Iterative methods for nonexpansive mappings have recently been applied to solve

convex minimization problems. Convex minimization problems have a great impact andinfluence on the development of almost all branches of pure and applied sciences. A typicalproblem is to minimize a quadratic function over the set of the fixed points of a nonexpansivemapping on a real Hilbert spaceH:

θ(x) =12〈Ax, x〉 − ⟨x, y⟩, ∀x ∈ F(S), (1.7)

where A is a linear bounded operator, F(S) is the fixed point set of a nonexpansive mappingS, and y is a given point inH [8].

Recall that a mapping S : C → C is said to be nonexpansive if

∥∥Sx − Sy∥∥ ≤ ∥∥x − y∥∥, (1.8)


for all x, y ∈ C. If C is bounded closed convex and S is a nonexpansive mapping of C intoitself, then F(S) is nonempty [9]. We denote weak convergence and strong convergence bynotations⇀ and → , respectively. A mapping A of C intoH is called monotone if

⟨Ax −Ay, x − y⟩ ≥ 0, (1.9)

for all x, y ∈ C. A mapping A of C intoH is called α-inverse-strongly monotone if there exists apositive real number α such that

⟨Ax −Ay, x − y⟩ ≥ α∥∥Ax −Ay∥∥2, (1.10)

for all x, y ∈ C. It is obvious that any α-inverse-strongly monotone mapping A is monotoneand Lipschitz continuous mapping. A linear bounded operator A is strongly positive if thereexists a constant γ > 0 with the property

〈Ax, x〉 ≥ γ‖x‖2, (1.11)

for all x ∈ H. A self mapping f : C → C is a contractions on C if there exists a constantα ∈ (0, 1) such that

∥∥f(x) − f(y)∥∥ ≤ α∥∥x − y∥∥, (1.12)

for all x, y ∈ C. We use∏

C to denote the collection of all contraction on C. Note that eachf ∈∏C has a unique fixed point in C.

Let B : H → H be a single-valued nonlinear mapping andM : H → 2H a set-valuedmapping. The variational inclusion problem is to find x ∈ H such that

θ ∈ B(x) +M(x), (1.13)

where θ is the zero vector inH. The set of solutions of problem (1.13) is denoted by I(B,M).The variational inclusion has been extensively studied in the literature, see, for example, [10–13] and the reference therein.

A set-valued mapping M : H → 2H is called monotone if for all x, y ∈ H, f ∈ M(x),and g ∈ M(y) imply 〈x − y, f − g〉 ≥ 0. A monotone mapping M is maximal if its graphG(M) := {(f, x) ∈ H ×H : f ∈ M(x)} of M is not properly contained in the graph of anyother monotone mapping. It is known that a monotone mapping M is maximal if and onlyif, for (x, f) ∈ H ×H, 〈x − y, f − g〉 ≥ 0 for all (y, g) ∈ G(M) imply f ∈M(x).

Let B be an inverse-strongly monotone mapping of C into H, and let NCv be normalcone to C at v ∈ C, that is,NCv = {w ∈ H : 〈v − u,w〉 ≥ 0, ∀u ∈ C}, and define

Tv =

⎧⎨

⎩

Bv +NCv, if v ∈ C,∅, if v /∈ C.

(1.14)

Then, T is a maximal monotone and θ ∈ Tv if and only if v ∈ VI(C,B) [14].


LetM : H → 2H be a set-valued maximal monotone mapping; then the single-valuedmapping JM,λ : H → H defined by

JM,λ(x) = (I + λM)−1(x), x ∈ H (1.15)

is called the resolvent operator associated with M, where λ is any positive number and I isthe identity mapping. It is worth mentioning that the resolvent operator is nonexpansive,1-inverse-strongly monotone and that a solution of problem (1.13) is a fixed point of theoperator JM,λ(I − λB) for all λ > 0 [15].

In 2000, Moudafi [16] introduced the viscosity approximation method for nonexpan-sive mapping and proved that, ifH is a real Hilbert space, the sequence {xn} defined by theiterative method below, with the initial guess x0 ∈ C, is chosen arbitrarily,

xn+1 = αnf(xn) + (1 − αn)Sxn, n ≥ 0, (1.16)

where {αn} ⊂ (0, 1) satisfies certain conditions and converges strongly to a fixed point of S(say x ∈ C) which is the unique solution of the following variational inequality:

⟨(I − f)x, x − x⟩ ≥ 0, ∀x ∈ F(S). (1.17)

In 2006, Marino and Xu [8] introduced a general iterative method for nonexpansivemapping. They defined the sequence {xn} generated by the algorithm x0 ∈ C:

xn+1 = αnγf(xn) + (I − αnA)Sxn, n ≥ 0, (1.18)

where {αn} ⊂ (0, 1) and A is a strongly positive linear bounded operator. They proved that,if C = H and the sequence {αn} satisfies appropriate conditions, then the sequence {xn}generated by (1.18) converges strongly to a fixed point of S (say x ∈ H) which is the uniquesolution of the following variational inequality:

〈(A − γf)x, x − x〉 ≥ 0, ∀x ∈ F(S), (1.19)

which is the optimality condition for the minimization problem

minx∈F(S)∩EP(F)

12〈Ax, x〉 − h(x), (1.20)

where h is a potential function for γf (i.e., h′(x) = γf(x) for x ∈ H).


For finding a common element of the set of fixed points of nonexpansive mappingsand the set of solution of the variational inequalities, let PC be the projection ofH onto C. In2005, Iiduka and Takahashi [17] introduced following iterative process for x0 ∈ C:

xn+1 = αnu + (1 − αn)SPC(xn − λnAxn), ∀n ≥ 0, (1.21)

where u ∈ C, {αn} ⊂ (0, 1), and {λn} ⊂ [a, b] for some a, b with 0 < a < b < 2β. Theyproved that under certain appropriate conditions imposed on {αn} and {λn}, the sequence{xn} generated by (1.21) converges strongly to a common element of the set of fixed points ofnonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say x ∈ C) which solve some variational inequality

〈x − u, x − x〉 ≥ 0, ∀x ∈ F(S) ∩ VI(C,A). (1.22)

In 2008, Su et al. [18] introduced the following iterative scheme by the viscosityapproximation method in a real Hilbert space: x1, un ∈ H

F(un, y

)+

1rn〈y − un, un − xn〉 ≥ 0, ∀y ∈ C,

xn+1 = αnf(xn) + (1 − αn)SPC(un − λnAun),(1.23)

for all n ∈ N, where {αn} ⊂ [0, 1) and {rn} ⊂ (0,∞) satisfy some appropriate conditions.Furthermore, they proved that {xn} and {un} converge strongly to the same point z ∈ F(S) ∩VI(C,A) ∩ EP(F), where z = PF(S)∩VI(C,A)∩EP(F)f(z).

In 2011, Tan and Chang [12] introduced following iterative process for {Tn : C → C}which is a sequence of nonexpansive mappings. Let {xn} be the sequence defined by

xn+1 = αnxn + (1 − αn)(SPC((1 − tn)JM,λ(I − λA)Tμ

(I − μB))xn

), ∀n ≥ 0, (1.24)

where {αn} ⊂ (0, 1), λ ∈ (0, 2α], and μ ∈ (0, 2β]. The sequence {xn} defined by (1.24)converges strongly to a common element of the set of fixed points of nonexpansive mapping,the set of solutions of the variational inequality, and the generalized equilibrium problem.

In this paper, we modify the iterative methods (1.18), (1.23), and (1.24) by proposingthe following new general viscosity iterative method: x0, un ∈ C,

F(un, y

)+ ϕ(y) − ϕ(un) +

⟨Ψxn, y − un

⟩+

1rn

⟨y − un, un − xn

⟩ ≥ 0, ∀y ∈ C,

xn+1 = PC[αnγf(xn) + (I − αnA)SJM,λ(un − λBun)

],

(1.25)

for all n ∈ N, where {αn} ⊂ (0, 1), {rn} ⊂ (0, 2σ), and λ ∈ (0, 2β) satisfy some appropriateconditions. The purpose of this paper is to show that under some control conditionsthe sequence {xn} strongly converges to a common element of the set of fixed points ofnonexpansive mapping, the solution of the generalized mixed equilibrium problems, andthe set of solutions of the variational inclusion in a real Hilbert space.


2. Preliminaries

Let H be a real Hilbert space and C a nonempty closed convex subset of H. Recall that the(nearest point) projection PC from H onto C assigns to each x ∈ H the unique point inPCx ∈ C satisfying the property

‖x − PCx‖ = miny∈C

∥∥x − y∥∥. (2.1)

The following characterizes the projection PC. We recall some lemmas which will be neededin the rest of this paper.

Lemma 2.1. The function u ∈ C is a solution of the variational inequality (1.5) if and only if u ∈ Csatisfies the relation u = PC(u − λΨu) for all λ > 0.

Lemma 2.2. For a given z ∈ H, u ∈ C, u = PCz⇔ 〈u − z, v − u〉 ≥ 0, for all v ∈ C.It is well known that PC is a firmly nonexpansive mapping ofH onto C and satisfies

‖PCx − PCy‖2 ≤⟨PCx − PCy, x − y⟩, ∀x, y ∈ H. (2.2)

Moreover, PCx is characterized by the following properties: PCx ∈ C and, for all x ∈ H, y ∈ C,⟨x − PCx, y − PCx

⟩ ≤ 0. (2.3)

Lemma 2.3 (see [19]). LetM : H → 2H be a maximal monotone mapping, and let B : H → H bea monotone and Lipshitz continuous mapping. Then the mapping L = M+B : H → 2H is a maximalmonotone mapping.

Lemma 2.4 (see [20]). Each Hilbert space H satisfies Opial’s condition, that is, for any sequence{xn} ⊂ H with xn ⇀ x, the inequality lim infn→∞‖xn − x‖ < lim infn→∞‖xn − y‖ holds for eachy ∈ H with y /=x.


an+1 ≤(1 − γn

)an + δn, ∀n ≥ 0, (2.4)

where {γn} ⊂ (0, 1) and {δn} is a sequence in R such that

(i)∑∞

n=1 γn = ∞,

(ii) lim supn→∞(δn/γn) ≤ 0 or∑∞

n=1 |δn| <∞.

Then limn→∞an = 0.

Lemma 2.6 (see [22]). Let C be a closed convex subset of a real Hilbert spaceH, and let T : C → Cbe a nonexpansive mapping. Then I − T is demiclosed at zero, that is,

xn ⇀ x, xn − Txn −→ 0 (2.5)

implies x = Tx.


For solving the generalized mixed equilibrium problem, let us assume that thebifunction F : C × C → R, the nonlinear mapping Ψ : C → H is continuous monotone,and ϕ : C → R satisfies the following conditions:

(A1) F(x, x) = 0 for all x ∈ C;(A2) F is monotone, that is, F(x, y) + F(y, x) ≤ 0 for any x, y ∈ C;(A3) for each fixed y ∈ C, x �→ F(x, y) is weakly upper semicontinuous;

(A4) for each fixed x ∈ C, y �→ F(x, y) is convex and lower semicontinuous;

(B1) for each x ∈ C and r > 0, there exist a bounded subsetDx ⊆ C and yx ∈ C such that,for any z ∈ C \Dx,

F(z, yx

)+ ϕ(yx) − ϕ(z) + 1

r

⟨yx − z, z − x

⟩< 0; (2.6)

(B2) C is a bounded set.

Lemma 2.7 (see [23]). Let C be a nonempty closed convex subset of a real Hilbert space H. LetF : C × C → R be a bifunction mapping satisfying (A1)–(A4), and let ϕ : C → R be convex andlower semicontinuous such that C ∩ domϕ/= ∅. Assume that either (B1) or (B2) holds. For r > 0 andx ∈ H, there exists u ∈ C such that

F(u, y)+ ϕ(y) − ϕ(u) + 1

r

⟨y − u, u − x⟩. (2.7)

Define a mapping Kr : H → C as follows:

Kr(x) ={u ∈ C : F

(u, y)+ ϕ(y) − ϕ(u) + 1

r

⟨y − u, u − x⟩ ≥ 0, ∀y ∈ C

}(2.8)

for all x ∈ H. Then, the following hold:

(i) Kr is single valued;

(ii) Kr is firmly nonexpansive, that is, for any x, y ∈ H, ‖Krx −Kry‖2 ≤ 〈Krx−Kry, x−y〉;(iii) F(Kr) = MEP(F, ϕ);

(iv) MEP(F, ϕ) is closed and convex.

Lemma 2.8 (see [8]). Assume that A is a strongly positive linear bounded operator on a HilbertspaceH with coefficient γ > 0 and 0 < ρ ≤ ‖A‖−1; then ‖I − ρA‖ ≤ 1 − ργ .

3. Strong Convergence Theorems

In this section, we show a strong convergence theorem which solves the problem of findinga common element of F(S), GMEP(F, ϕ,Ψ), and I(B,M) of an inverse-strongly monotonemappings in a Hilbert space.


Theorem 3.1. Let H be a real Hilbert space, C a closed convex subset of H, B, Ψ : C → Hbe β, σ-inverse-strongly monotone mappings, respectively. Let ϕ : C → R be a convex and lowersemicontinuous function, f : C → C a contraction with coefficient α (0 < α < 1),M : H → 2H amaximal monotone mapping, and A a strongly positive linear bounded operator of H into itself withcoefficient γ > 0. Assume that 0 < γ < γ/α. Let S be a nonexpansive mapping of C into itself such that

Ω := F(S) ∩ GMEP(F, ϕ,Ψ

) ∩ I(B,M)/= ∅. (3.1)

Suppose that {xn} is a sequences generated by the following algorithm for x0 ∈ C arbitrarily:

F(un, y

)+ ϕ(y) − ϕ(un) +

⟨Ψxn, y − un

⟩+

1rn


⟩ ≥ 0, ∀y ∈ C,

xn+1 = PC[αnγf(xn) + (I − αnA)SJM,λ(un − λBun)

](3.2)

for all n = 0, 1, 2, . . ., where

(C1) {αn} ⊂ (0, 1), limn→ 0αn = 0,∑∞

n=1 αn = ∞, and∑∞

n=1 |αn+1 − αn| <∞,

(C2) {rn} ⊂ [c, d] with c, d ∈ (0, 2σ) and∑∞

n=1 |rn+1 − rn| <∞,

(C3) λ ∈ (0, 2β).

Then {xn} converges strongly to q ∈ Ω, where q = PΩ(γf + I − A)(q) which solves thefollowing variational inequality:

⟨(γf −A)q, p − q⟩ ≤ 0, ∀p ∈ Ω (3.3)


minq∈Ω

12〈Aq, q〉 − h(q), (3.4)

where h is a potential function for γf (i.e., h′(q) = γf(q) for q ∈ H).

Proof. Due to condition (C1), we may assume without loss of generality, then, that αn ∈(0, ‖A‖−1) for all n. By Lemma 2.8, we have that ‖I − αnA‖ ≤ 1 − αnγ . Next, we will assumethat ‖I −A‖ ≤ ‖1 − γ‖.

Next, we will divide the proof into six steps.

Step 1. We will show that {xn}, {un} are bounded.Since B, Ψ are β, σ-inverse-strongly monotone mappings, we have that

∥∥(I − λB)x − (I − λB)y∥∥2 = ∥∥(x − y) − λ(Bx − By)∥∥2

=∥∥x − y∥∥2 − 2λ〈x − y, Bx − By〉 + λ2∥∥Bx − By∥∥2

≤ ∥∥x − y∥∥2 + λ(λ − 2β)∥∥Bx − By∥∥2

≤ ∥∥x − y∥∥2.

(3.5)


In a similar way, we can obtain

∥∥(I − rnΨ)x − (I − rnΨ)y

∥∥2 ≤ ∥∥x − y∥∥2. (3.6)

It is clear that if 0 < λ < 2β, 0 < rn < 2σ, then I − λB, I − rnΨ are all nonexpansive.Put yn = JM,λ(un − λBun), n ≥ 0. It follows that

∥∥yn − q

∥∥ =∥∥JM,λ(un − λBun) − JM,λ

(q − λBq)∥∥

≤ ∥∥un − q∥∥.

(3.7)

By Lemma 2.7, we have that un = Krn(xn − rnΨxn) for all n ≥ 0. Then, we have that

∥∥un − q∥∥2 =

∥∥Krn(xn − rnΨxn) −Krn(q − rnΨq)∥∥2

≤ ∥∥(xn − rnΨxn) − (q − rnΨq)∥∥2

≤ ∥∥xn − q∥∥2 + rn(rn − 2σ)

∥∥Ψxn −Ψq∥∥2

=∥∥xn − q

∥∥2.

(3.8)

Hence, we have that

∥∥yn − q∥∥ ≤ ∥∥xn − q

∥∥. (3.9)

From (3.2), we deduce that

∥∥xn+1 − q

∥∥ =∥∥PC(αnγf(xn) + (I − αnA)Syn

) − PC(q)∥∥

≤ ∥∥αn(γf(xn) −Aq

)+ (I − αnA)

(Syn − q

)∥∥

≤ αn∥∥γf(xn) −Aq

∥∥ +(1 − αnγ

)∥∥yn − q∥∥

≤ αn∥∥γf(xn) − γf

(q)∥∥ + αn

∥∥γf(q) −Aq∥∥ + (1 − αnγ

)∥∥yn − q∥∥

≤ αγαn∥∥xn − q

∥∥ + αn∥∥γf(q) −Aq∥∥ + (1 − αnγ

)∥∥xn − q∥∥

=(1 − (γ − γα)αn

)∥∥xn − q∥∥ + αn

∥∥γf(q) −Aq∥∥

=(1 − (γ − γα)αn

)∥∥xn − q∥∥ +(γ − γα)αn

∥∥γf(q) −Aq∥∥

γ − γα

≤ max

{∥∥xn − q

∥∥,

∥∥γf(q) −Aq∥∥

γ − γα

}

.

(3.10)


It follows by induction that

∥∥xn − q

∥∥ ≤ max

{∥∥x0 − q

∥∥,

∥∥γf(q) −Aq∥∥

γ − γα

}

, n ≥ 0. (3.11)

Therefore {xn} is bounded, so are {yn}, {Syn}, {Bun}, {f(xn)}, and {ASyn}.

Step 2. We claim that limn→∞‖xn+1 − xn‖ = 0. From (3.2), we have that

‖xn+1 − xn‖ =∥∥PC(αnγf(xn) + (I − αnA)Syn

) − PC(αn−1γf(xn−1) + (I − αn−1A)Syn−1

)∥∥

≤ ∥∥(I − αnA)(Syn − Syn−1

) − (αn − αn−1)ASyn−1

+γαn(f(xn) − f(xn−1)

)+ γ(αn − αn−1)f(xn−1)

∥∥

≤ (1 − αnγ)∥∥yn − yn−1

∥∥ + |αn − αn−1|∥∥ASyn−1

∥∥

+ γααn‖xn − xn−1‖ + γ |αn − αn−1|∥∥f(xn−1)

∥∥.(3.12)

Since I − λB are nonexpansive, we also have that

‖yn − yn−1‖ = ‖JM,λ(un − λBun) − JM,λ(un−1 − λBun−1)‖

≤ ‖(un − λBun) − (un−1 − λBun−1)‖

≤ ‖un − un−1‖.

(3.13)

On the other hand, from un−1 = Krn−1(xn−1 − rn−1Ψxn−1) and un = Krn(xn − rnΨxn), it followsthat

F(un−1, y

)+⟨Ψxn−1, y − un−1

⟩+ ϕ(y) − ϕ(un−1) + 1

rn−1

⟨y − un−1, un−1 − xn−1

⟩ ≥ 0, ∀y ∈ C,(3.14)

F(un, y

)+ 〈Ψxn, y − un〉 + ϕ

(y) − ϕ(un) + 1

rn〈y − un, un − xn〉 ≥ 0, ∀y ∈ C. (3.15)

Substituting y = un into (3.14) and y = un−1 into (3.15), we get

F(un−1, un) + 〈Ψxn−1, un − un−1〉 + ϕ(un) − ϕ(un−1) + 1rn−1

〈un − un−1, un−1 − xn−1〉 ≥ 0,

F(un, un−1) + 〈Ψxn, un−1 − un〉 + ϕ(un−1) − ϕ(un) + 1rn〈un−1 − un, un − xn〉 ≥ 0.

(3.16)


From (A2), we obtain

⟨un − un−1,Ψxn−1 −Ψxn +

un−1 − xn−1rn−1

− un − xnrn

⟩≥ 0, (3.17)

and then

⟨un − un−1, rn−1(Ψxn−1 −Ψxn) + un−1 − xn−1 − rn−1

rn(un − xn)

⟩≥ 0. (3.18)

So

〈un − un−1, rn−1Ψxn−1 − rn−1Ψxn + un−1 − un + un − xn−1 − rn−1rn

(un − xn)〉 ≥ 0. (3.19)

It follows that

⟨un − un−1, (I − rn−1Ψ)xn − (I − rn−1Ψ)xn−1 + un−1 − un + un − xn − rn−1

rn(un − xn)

⟩≥ 0,

〈un − un−1, un−1 − un〉 + 〈un − un−1, xn − xn−1 +(1 − rn−1

rn

)(un − xn)〉 ≥ 0.

(3.20)

Without loss of generality, let us assume that there exists a real number c such that rn−1>c >0,for all n ∈ N. Then, we have that

‖un − un−1‖2 ≤⟨un − un−1, xn − xn−1 +

(1 − rn−1

rn

)(un − xn)

⟩

≤ ‖un − un−1‖{‖xn − xn−1‖ +

∣∣∣∣1 −rn−1rn

∣∣∣∣‖un − xn‖},

(3.21)

and hence

‖un − un−1‖ ≤ ‖xn − xn−1‖ + 1rn|rn − rn−1|‖un − xn‖

≤ ‖xn − xn−1‖ + M1

c|rn − rn−1|,

(3.22)


whereM1 = sup{‖un − xn‖ : n ∈ N}. Substituting (3.22) into (3.13), we have that

∥∥yn − yn−1

∥∥ ≤ ‖xn − xn−1‖ + M1

c|rn − rn−1|. (3.23)

Substituting (3.23) into (3.12), we get

‖xn+1 − xn‖ ≤ (1 − αnγ)(‖xn − xn−1‖ + M1

c|rn − rn−1|

)+ |αn − αn−1|

∥∥ASyn−1

∥∥


∥∥

=(1 − αnγ

)‖xn − xn−1‖ +(1 − αnγ

)M1

c|rn − rn−1| + |αn − αn−1|

∥∥ASyn−1∥∥


∥∥

≤ (1 − (γ − γα)αn)‖xn − xn−1‖ + M1

c|rn − rn−1| + |αn − αn−1|

∥∥ASyn−1∥∥

+ γ |αn − αn−1|∥∥f(xn−1)

∥∥

≤ (1 − (γ − γα)αn)‖xn − xn−1‖ + M1

c|rn − rn−1| +M2|αn − αn−1|,

(3.24)

where M2 = sup{max{‖ASyn−1‖, ‖f(xn−1)‖ : n ∈ N}}. By conditions (C1)-(C2) and Lemma2.5, we have that ‖xn+1 − xn‖ → 0 as n → ∞. From (3.23), we also have that ‖yn+1 − yn‖ → 0as n → ∞.

Step 3. We show the following:

(i) limn→∞‖Bun − Bq‖ = 0;

(ii) limn→∞‖Ψxn −Ψq‖ = 0.

For q ∈ Ω and q = JM,λ(q − λBq), by (3.5) and (3.8), we get

∥∥yn − q∥∥2 =

∥∥JM,λ(un − λBun) − JM,λ(q − λBq)∥∥2

≤ ∥∥(un − λBun) − (q − λBq)∥∥2

≤ ∥∥un − q∥∥2 + λ

(λ − 2β

)∥∥Bun − Bq∥∥2

≤ ∥∥xn − q∥∥2 + λ

(λ − 2β

)∥∥Bun − Bq∥∥2.

(3.25)


It follows that

∥∥xn+1 − q

∥∥2 =

∥∥PC(αnγf(xn) + (I − αnA)Syn) − PC(q)

∥∥2

≤ ∥∥αn(γf(xn) −Aq) + (I − αnA)(Syn − q)∥∥2

≤ (αn∥∥γf(xn) −Aq

∥∥ +(1 − αnγ

)∥∥yn − q∥∥)2


∥∥2 +

(1 − αnγ

)∥∥yn − q∥∥2

+ 2αn(1 − αnγ

)∥∥γf(xn) −Aq∥∥∥∥yn − q

∥∥


∥∥2 + 2αn

(1 − αnγ

)∥∥γf(xn) −Aq∥∥∥∥yn − q

∥∥

+(1 − αnγ

)(‖xn − q‖2 + λ(λ − 2β

)∥∥Bun − Bq∥∥2)


∥∥2 + 2αn(1 − αnγ

)∥∥γf(xn) −Aq∥∥∥∥yn − q

∥∥

+ ‖xn − q‖2 +(1 − αnγ

)λ(λ − 2β

)‖Bun − Bq‖2.

(3.26)

So, we obtain

(1 − αnγ

)λ(2β − λ)∥∥Bun − Bq

∥∥2


∥∥2 + ‖xn − xn+1‖(∥∥xn − q

∥∥ +∥∥xn+1 − q

∥∥) + εn,(3.27)

where εn = 2αn(1−αnγ)‖γf(xn)−Aq‖‖yn−q‖. By conditions (C1) and (C3) and limn→∞‖xn+1−xn‖ = 0, we obtain that ‖Bun − Bq‖ → 0 as n → ∞.


∥∥yn − q∥∥2 ≤ ∥∥un − q

∥∥2 + λ(λ − 2β

)∥∥Bun − Bq∥∥2

≤(∥∥xn − q

∥∥2 + rn(rn − 2σ)∥∥Ψxn −Ψq

∥∥2)+ λ(λ − 2β

)∥∥Bun − Bq∥∥2.

(3.28)


∥∥xn+1 − q∥∥2 ≤ αn

∥∥γf(xn) −Aq∥∥2 + 2αn

(1 − αnγ

)∥∥γf(xn) −Aq∥∥∥∥yn − q

∥∥

+(1 − αnγ

)(∥∥xn − q∥∥2 + rn(rn − 2σ)

∥∥Ψxn −Ψq∥∥2 + λ

(λ − 2β

)∥∥Bun − Bq∥∥2)


∥∥2 + 2αn(1 − αnγ

)∥∥γf(xn) −Aq∥∥∥∥yn − q

∥∥ +∥∥xn − q

∥∥2

+(1 − αnγ

)rn(rn − 2σ)

∥∥Ψxn −Ψq∥∥2 +

(1 − αnγ

)λ(λ − 2β

)∥∥Bun − Bq∥∥2.

(3.29)


So, we also have that

(1 − αnγ

)rn(2σ − rn)‖Ψxn −Ψq‖2


∥∥2 + ‖xn − xn+1‖

(∥∥xn − q∥∥ +∥∥xn+1 − q

∥∥)

+ εn +(1 − αnγ

)λ(λ − 2β

)∥∥Bun − Bq∥∥2,

(3.30)

where εn = 2αn(1−αnγ)‖γf(xn)−Aq‖‖yn−q‖. By conditions (C1)–(C3), limn→∞‖xn+1−xn‖ = 0and limn→∞‖Bun − Bq‖ = 0, we obtain that ‖Ψxn −Ψq‖ → 0 as n → ∞.

Step 4. We show the following:

(i) limn→∞‖xn − un‖ = 0;

(ii) limn→∞‖un − yn‖ = 0;

(iii) limn→∞‖yn − Syn‖ = 0.

Since Krn is firmly nonexpansive and by (2.2), we observe that

∥∥un − q∥∥2 =

∥∥Krn(xn − rnΨxn) −Krn(q − rnΨq)∥∥2

≤ ⟨(xn − rnΨxn) −(q − rnΨq

), un − q

⟩

=12

(∥∥(xn − rnΨxn) − (q − rnΨq)∥∥2 +

∥∥un − q∥∥2

−∥∥(xn − rnΨxn) − (q − rnΨq) − (un − q)∥∥2)

≤ 12

(∥∥xn − q∥∥2 +

∥∥un − q∥∥2 − ∥∥(xn − un) − rn(Ψxn −Ψq)

∥∥2)

=12

(∥∥xn − q∥∥2 +

∥∥un − q∥∥2 − ‖xn − un‖2

+2rn⟨Ψxn −Ψq, xn − un

⟩ − r2n∥∥Ψxn −Ψq

∥∥2).

(3.31)

Hence, we have that

∥∥un − q∥∥2 ≤ ∥∥xn − q

∥∥2 − ‖xn − un‖2 + 2rn∥∥Ψxn −Ψq

∥∥‖xn − un‖. (3.32)


Since JM,λ is 1-inverse-strongly monotone and by (2.2), we compute

∥∥yn − q

∥∥2 =

∥∥JM,λ(un − λBun) − JM,λ(q − λBq)

∥∥2

≤ ⟨(un − λBun) −(q − λBq), yn − q

⟩

=12

(∥∥(un − λBun) −

(q − λBq)∥∥2 + ∥∥yn − q

∥∥2

−∥∥(un − λBun) −(q − λBq) − (yn − q

)∥∥2)

≤ 12

(∥∥un − q

∥∥2 +

∥∥yn − q

∥∥2 − ∥∥(un − yn

) − λ(Bun − Bq)∥∥2)

=12

(∥∥un − q

∥∥2 +

∥∥yn − q

∥∥2 − ∥∥un − yn

∥∥2

+2λ⟨un − yn, Bun − Bq

⟩ − λ2∥∥Bun − Bq∥∥2),

(3.33)

which implies that

∥∥yn − q∥∥2 ≤ ∥∥un − q

∥∥2 − ∥∥un − yn∥∥2 + 2λ

∥∥un − yn∥∥∥∥Bun − Bq

∥∥. (3.34)

Substituting (3.32) into (3.34), we have that

∥∥yn − q∥∥2 ≤

(∥∥xn − q∥∥2 − ‖xn − un‖2 + 2rn

∥∥Ψxn −Ψq∥∥‖xn − un‖

)

− ∥∥un − yn∥∥2 + 2λ


∥∥.(3.35)


∥∥xn+1 − q∥∥2 ≤ αn

∥∥γf(xn) −Aq∥∥2 +

∥∥yn − q∥∥2 + 2αn

(1 − αnγ

)∥∥γf(xn) −Aq∥∥∥∥yn − q

∥∥


∥∥2 +(∥∥xn − q

∥∥2 − ‖xn − un‖2 + 2rn∥∥Ψxn −Ψq

∥∥‖xn − un‖

−∥∥un − yn∥∥2 + 2λn


∥∥)

+ 2αn(1 − αnγ

)∥∥γf(xn) −Aq∥∥∥∥yn − q

∥∥.(3.36)


Then, we derive

‖xn − un‖2 +∥∥un − yn

∥∥2 ≤ αn

∥∥γf(xn) −Aq

∥∥2 +

∥∥xn − q

∥∥2 − ∥∥xn+1 − q

∥∥2

+ 2rn∥∥Ψxn −Ψq

∥∥‖xn − un‖ + 2λ

∥∥un − yn

∥∥∥∥Bun − Bq

∥∥

+ 2αn(1 − αnγ

)∥∥γf(xn) −Aq∥∥∥∥yn − q

∥∥

= αn∥∥γf(xn) −Aq

∥∥2 + ‖xn − xn+1‖

(∥∥xn − q∥∥ +∥∥xn+1 − q

∥∥)

+ 2rn∥∥Ψxn −Ψq

∥∥‖xn − un‖ + 2λ

∥∥un − yn

∥∥∥∥Bun − Bq

∥∥

+ 2αn(1 − αnγ

)‖γf(xn) −Aq‖‖yn − q‖.

(3.37)

By condition (C1), limn→∞‖xn−xn+1‖ = 0, limn→∞‖Ψxn−Ψq‖ = 0, and limn→∞‖Bun−Bq‖ = 0.So, we have that ‖xn − un‖ → 0, ‖un − yn‖ → 0 as n → ∞. It follows that

∥∥xn − yn∥∥ ≤ ‖xn − un‖ +

∥∥un − yn∥∥ −→ 0, as n −→ ∞. (3.38)


∥∥xn − Syn∥∥ ≤ ∥∥xn − Syn−1

∥∥ +∥∥Syn−1 − Syn

∥∥

≤ ∥∥PC(αn−1γf(xn−1) + (I − αn−1A)Syn−1

) − PC(Syn−1

)∥∥ +∥∥yn−1 − yn

∥∥

≤ αn−1∥∥γfxn−1 −ASyn−1

∥∥ +∥∥yn−1 − yn

∥∥.

(3.39)

By condition (C1) and limn→∞‖yn−1 − yn‖ = 0, we obtain that ‖xn − Syn‖ → 0 as n → ∞.Next, we observe that

∥∥xn+1 − Syn∥∥ =∥∥PC(αnγf(xn) + (I − αnA)Syn

) − PC(Syn)∥∥

≤ ∥∥αnγf(xn) + (I − αnA)Syn − Syn∥∥

= αn∥∥γf(xn) −ASyn

∥∥.

(3.40)

Since {ASyn} is bounded and by condition (C1), we have that ‖xn+1 − Syn‖ → 0 as n → ∞,and

∥∥xn − Syn∥∥ ≤ ‖xn − xn+1‖ +

∥∥xn+1 − Syn∥∥. (3.41)

Since limn→∞‖xn − xn+1‖ = 0 and limn→∞‖xn+1 − Syn‖ = 0, it implies that ‖xn − Syn‖ → 0 asn → ∞. Hence, we have that

‖xn − Sxn‖ ≤ ∥∥xn − Syn∥∥ +∥∥Syn − Sxn

∥∥

≤ ∥∥xn − Syn∥∥ +∥∥yn − xn

∥∥.(3.42)


By (3.38) and limn→∞‖xn − Syn‖ = 0, we obtain ‖xn − Sxn‖ → 0 as n → ∞. Moreover, wealso have that

∥∥yn − Syn∥∥ ≤ ∥∥yn − xn

∥∥ +∥∥xn − Syn

∥∥. (3.43)

By (3.38) and limn→∞‖xn − Syn‖ = 0, we obtain ‖yn − Syn‖ → 0 as n → ∞.

Step 5. We show that q ∈ Ω := F(S) ∩ GMEP(F, ϕ,Ψ) ∩ I(B,M) and lim supn→∞〈(γf −A)q, Syn − q〉 ≤ 0. It is easy to see that PΩ(γf + (I − A)) is a contraction of H into itself.Indeed, since 0 < γ < γ/α, we have that

∥∥PΩ(γf + (I −A)

)x − PΩ

(γf + (I −A)

)y∥∥ ≤ ∥∥γf + (I −A)x − γf − (I −A)y

∥∥

≤ γ∥∥f(x) − f(y)∥∥ + ‖I −A‖∥∥x − y∥∥

≤ γα∥∥x − y∥∥ + (1 − γ)∥∥x − y∥∥

≤ (1 − γ + γα)∥∥x − y∥∥.

(3.44)

HenceH is complete, and there exists a unique fixed point q ∈ H such that q = PΩ(γf + (I −A))(q). By Lemma 2.2, we obtain that 〈(γf −A)q,w − q〉 ≤ 0 for all w ∈ Ω.

Next, we show that lim supn→∞〈(γf −A)q, Syn − q〉 ≤ 0, where q = PΩ(γf + I −A)(q)is the unique solution of the variational inequality 〈(γf −A)q, p−q〉 ≥ 0, for all p ∈ Ω. We canchoose a subsequence {yni} of {yn} such that

lim supn→∞

⟨(γf −A)q, Syn − q

⟩= lim

i→∞⟨(γf −A)q, Syni − q

⟩. (3.45)

As {yni} is bounded, there exists a subsequence {ynij } of {yni}which converges weakly tow.We may assume without loss of generality that yni ⇀ w.

We claim that w ∈ Ω. Since ‖yn − Syn‖ → 0, ‖xn − Sxn‖ → 0, and ‖xn − yn‖ → 0 andby Lemma 2.6, we have that w ∈ F(S).

Next, we show that w ∈ GMEP(F, ϕ,Ψ). Since un = Krn(xn − rnΨxn), we know that

F(un, y

)+ ϕ(y) − ϕ(un) + 〈Ψxn, y − un〉 + 1

rn〈y − un, un − xn〉 ≥ 0, ∀y ∈ C. (3.46)

It follows by (A2) that

ϕ(y) − ϕ(un) + 〈Ψxn, y − un〉 + 1

rn〈y − un, un − xn〉 ≥ F(y, un

), ∀y ∈ C. (3.47)

Hence,

ϕ(y) − ϕ(uni) +

⟨Ψxni , y − uni

⟩+

1rni

⟨y − uni , uni − xni

⟩ ≥ F(y, uni), ∀y ∈ C. (3.48)


For t ∈ (0, 1] and y ∈ H, let yt = ty + (1 − t)w. From (3.48), we have that

〈yt − uni ,Ψyt〉 ≥ 〈yt − uni ,Ψyt〉 − ϕ(yt)+ ϕ(uni) − 〈Ψxni , yt − uni〉

− 1rni

〈yt − uni , uni − xni〉 + F(yt, uni

)

=⟨yt − uni ,Ψyt −Ψuni

⟩+⟨yt − uni ,Ψuni −Ψxni

⟩ − ϕ(yt)+ ϕ(uni)

− 1rni

⟨yt − uni , uni − xni

⟩+ F(yt, uni

).

(3.49)

From ‖uni − xni‖ → 0, we have that ‖Ψuni − Ψxni‖ → 0. Further, from (A4) and the weaklylower semicontinuity of ϕ,(uni − xni)/rni → 0 and uni ⇀ w, we have that

⟨yt −w,Ψyt

⟩ ≥ −ϕ(yt)+ ϕ(w) + F

(yt,w

). (3.50)

From (A1), (A4), and (3.50), we have that

0 = F(yt, yt

) − ϕ(yt)+ ϕ(yt)

≤ tF(yt, y)+ (1 − t)F(yt,w

)+ tϕ(y)+ (1 − t)ϕ(w) − ϕ(yt

)

= t[F(yt, y

)+ ϕ(y) − ϕ(yt

)]+ (1 − t)[F(yt,w

)+ ϕ(w) − ϕ(yt

)]

≤ t[F(yt, y)+ ϕ(y) − ϕ(yt

)]+ (1 − t)〈yt −w,Ψyt〉

= t[F(yt, y

)+ ϕ(y) − ϕ(yt

)]+ (1 − t)t〈y −w,Ψyt〉,

(3.51)

and hence

0 ≤ F(yt, y)+ ϕ(y) − ϕ(yt

)+ (1 − t)⟨y −w,Ψyt

⟩. (3.52)

Letting t → 0, we have, for each y ∈ C, that

F(w,y

)+ ϕ(y) − ϕ(w) + 〈y −w,Ψw〉 ≥ 0. (3.53)

This implies that w ∈ GMEP(F, ϕ,Ψ).Lastly, we show that w ∈ I(B,M). In fact, since B is a β-inverse-strongly monotone, B

is monotone and Lipschitz continuous mapping. It follows from Lemma 2.3 that M + B is amaximal monotone. Let (v, g) ∈ G(M + B), since g − Bv ∈M(v). Again since yni = JM,λ(uni −λBuni), we have that uni −λBuni ∈ (I +λM)(yni), that is, (1/λ)(uni −yni −λBuni) ∈M(yni). Byvirtue of the maximal monotonicity ofM + B, we have that

⟨v − yni , g − Bv − 1

λ

(uni − yni − λBuni

)⟩

≥ 0, (3.54)


and hence

⟨v − yni , g

⟩ ≥⟨v − yni , Bv +

1λ

(uni − yni − λBuni

)⟩

=⟨v − yni , Bv − Byni

⟩+⟨v − yni , Byni − Buni

⟩

+⟨v − yni ,

1λ

(uni − yni

)⟩.

(3.55)

It follows from limn→∞‖un − yn‖ = 0, limn→∞‖Bun − Byn‖ = 0, and yni ⇀ w that

lim supn→∞

⟨v − yn, g

⟩=⟨v −w, g⟩ ≥ 0. (3.56)

It follows from the maximal monotonicity of B+M that θ ∈ (M+B)(w), that is,w ∈ I(B,M).Therefore, w ∈ Ω. It follows that

lim supn→∞

⟨(γf −A)q, Syn − q

⟩= lim

i→∞⟨(γf −A)q, Syni − q

⟩=⟨(γf −A)q,w − q⟩ ≤ 0. (3.57)

Step 6. We prove that xn → q. By using (3.2) and together with Schwarz inequality, we havethat

∥∥xn+1 − q∥∥2 =

∥∥PC(αnγf(xn) + (I − αnA)Syn

) − PC(q)∥∥2

≤ ∥∥αn(γf(xn) −Aq) + (I − αnA)(Syn − q

)∥∥2

≤ (I − αnA)2∥∥(Syn − q

)∥∥2 + α2n∥∥γf(xn) −Aq

∥∥2

+ 2αn⟨(I − αnA)

(Syn − q

), γf(xn) −Aq

⟩

≤ (1 − αnγ)2∥∥yn − q

∥∥2 + α2n∥∥γf(xn) −Aq

∥∥2

+ 2αn⟨Syn − q, γf(xn) −Aq

⟩ − 2α2n⟨A(Syn − q

), γf(xn) −Aq

⟩

≤ (1 − αnγ)2∥∥xn − q

∥∥2 + α2n∥∥γf(xn) −Aq

∥∥2 + 2αn〈Syn − q, γf(xn) − γf(q)〉

+ 2αn〈Syn − q, γf(q) −Aq〉 − 2α2n〈A

(Syn − q

), γf(xn) −Aq〉

≤ (1 − αnγ)2∥∥xn − q

∥∥2 + α2n∥∥γf(xn) −Aq

∥∥2 + 2αn∥∥Syn − q

∥∥∥∥γf(xn) − γf(q)∥∥

+ 2αn⟨Syn − q, γf

(q) −Aq⟩ − 2α2n〈A

(Syn − q

), γf(xn) −Aq〉

≤ (1 − αnγ)2∥∥xn − q

∥∥2 + α2n∥∥γf(xn) −Aq

∥∥2 + 2γααn∥∥yn − q

∥∥∥∥xn − q∥∥

+ 2αn⟨Syn − q, γf

(q) −Aq⟩ − 2α2n〈A

(Syn − q

), γf(xn) −Aq〉

≤ (1 − αnγ)2∥∥xn − q

∥∥2 + α2n∥∥γf(xn) −Aq

∥∥2 + 2γααn∥∥xn − q

∥∥2

+ 2αn〈Syn − q, γf(q) −Aq〉 − 2α2n

⟨A(Syn − q

), γf(xn) −Aq

⟩


≤((

1 − αnγ)2 + 2γααn

)∥∥xn − q

∥∥2

+ αn{αn∥∥γf(xn) −Aq

∥∥2 + 2

⟨Syn − q, γf

(q) −Aq⟩

−2αn∥∥A(Syn − q

)∥∥∥∥γf(xn) −Aq

∥∥}

=(1 − 2

(γ − γα)αn

)∥∥xn − q∥∥2

+ αn{αn∥∥γf(xn) −Aq

∥∥2 + 2

⟨Syn − q, γf

(q) −Aq⟩

−2αn∥∥A(Syn − q

)∥∥∥∥γf(xn) −Aq

∥∥ + αnγ

2∥∥xn − q∥∥2}.

(3.58)

Since {xn} is bounded, where η ≥ ‖γf(xn) −Aq‖2 − 2‖A(Syn − q)‖‖γf(xn) − Aq‖ +γ2‖xn − q‖2 for all n ≥ 0, it follows that

∥∥xn+1 − q∥∥2 ≤ (1 − 2

(γ − γα)αn

)∥∥xn − q∥∥2 + αnςn, (3.59)

where ςn = 2〈Syn − q, γf(q) − Aq〉 + ηαn. By lim supn→∞〈(γf − A)q, Syn − q〉 ≤ 0, we getlim supn→∞ςn ≤ 0. Applying Lemma 2.5, we can conclude that xn → q. This completes theproof.

Corollary 3.2. Let H be a real Hilbert space and C a closed convex subset ofH. Let B,Ψ : C → Hbe β, σ-inverse-strongly monotone mappings and ϕ : C → R a convex and lower semicontinuousfunction. Let f : C → C be a contraction with coefficient α (0 < α < 1),M : H → 2H a maximalmonotone mapping, and S a nonexpansive mapping of C into itself such that


) ∩ I(B,M)/= ∅. (3.60)

Suppose that {xn} is a sequence generated by the following algorithm for x0, un ∈ C arbitrarily:

F(un, y

)+ ϕ(y) − ϕ(un) + 〈Ψxn, y − un〉 + 1

rn〈y − un, un − xn〉 ≥ 0, ∀y ∈ C,

xn+1 = PC[αnf(xn) + (1 − αn)SJM,λ(un − λBun)

](3.61)

for all n = 0, 1, 2, . . ., by (C1)–(C3) in Theorem 3.1.Then {xn} converges strongly to q ∈ Ω, where q = PΩ(f + I)(q) which solves the following

variational inequality:

⟨(f − I)q, p − q⟩ ≤ 0, ∀p ∈ Ω. (3.62)

Proof. Putting A ≡ I and γ ≡ 1 in Theorem 3.1, we can obtain the desired conclusion immedi-ately.


Corollary 3.3. LetH be a real Hilbert space andC a closed convex subset ofH. Let B,Ψ : C → H beβ, σ-inverse-strongly monotone mappings, ϕ : C → R a convex and lower semicontinuous function,andM : H → 2H a maximal monotone mapping. Let S be a nonexpansive mapping of C into itselfsuch that


) ∩ I(B,M)/= ∅. (3.63)

Suppose that {xn} is a sequence generated by the following algorithm for x0, u ∈ C and un ∈ C:

F(un, y

)+ ϕ(y) − ϕ(un) + 〈Ψxn, y − un〉 + 1

rn〈y − un, un − xn〉 ≥ 0, ∀y ∈ C,

xn+1 = PC[αnu + (1 − αn)SJM,λ(un − λBun)](3.64)

for all n = 0, 1, 2, . . ., by (C1)–(C3) in Theorem 3.1.Then {xn} converges strongly to q ∈ Ω, where q = PΩ(q) which solves the following


〈u − q, p − q〉 ≤ 0, ∀p ∈ Ω. (3.65)

Proof. Putting f(x) ≡ u, for all x ∈ C, in Corollary 3.2, we can obtain the desired conclusionimmediately.

Corollary 3.4. Let H be a real Hilbert space, C a closed convex subset of H, B : C → H be β-inverse-strongly monotone mappings, and A a strongly positive linear bounded operator of H intoitself with coefficient γ > 0. Assume that 0 < γ < γ/α. Let f : C → C be a contraction withcoefficient α(0 < α < 1) and S a nonexpansive mapping of C into itself such that

Ω := F(S) ∩ VI(C,B)/= ∅. (3.66)

Suppose that {xn} is a sequence generated by the following algorithm for x0 ∈ C arbitrarily:

xn+1 = PC[αnγf(xn) + (I − αnA)SPC(xn − λBxn)

](3.67)

for all n = 0, 1, 2, . . ., by (C1)–(C3) in Theorem 3.1.Then {xn} converges strongly to q ∈ Ω, where q = PΩ(γf + I − A)(q) which solves the

following variational inequality:

⟨(γf −A)q, p − q⟩ ≤ 0, ∀p ∈ Ω. (3.68)

Proof. Taking F ≡ 0, Ψ ≡ 0, ϕ ≡ 0, un = xn, and JM,λ = PC in Theorem 3.1, we can obtain thedesired conclusion immediately.

Remark 3.5. In Corollary 3.4 we generalize and improve the result of Klin-eam and Suantai[24].


4. Applications

In this section, we apply the iterative scheme (1.25) for finding a common fixed point ofnonexpansive mapping and strictly pseudocontractive mapping and also apply Theorem 3.1for finding a common fixed point of nonexpansive mappings and inverse-strongly monotonemappings.

Definition 4.1. A mapping T : C → C is called strictly pseudocontraction if there exists aconstant 0 ≤ κ < 1 such that

∥∥Tx − Ty∥∥2 ≤ ∥∥x − y∥∥2 + κ∥∥(I − T)x − (I − T)y∥∥2, ∀x, y ∈ C. (4.1)

If κ = 0, then S is nonexpansive. In this case, we say that T : C → C is a κ-strictlypseudocontraction. Putting B = I − T . Then, we have that

∥∥(I − B)x − (I − B)y∥∥2 ≤ ∥∥x − y∥∥2 + κ∥∥Bx − By∥∥2, ∀x, y ∈ C. (4.2)

Observe that

∥∥(I − B)x − (I − B)y∥∥2 = ∥∥x − y∥∥2 + ∥∥Bx − By∥∥2 − 2⟨x − y, Bx − By⟩, ∀x, y ∈ C. (4.3)

Hence, we obtain

⟨x − y, Bx − By⟩ ≥ 1 − κ

2∥∥Bx − By∥∥2, ∀x, y ∈ C. (4.4)

Then, B is ((1 − κ)/2)-inverse-strongly monotone mapping.

Using Theorem 3.1, we first prove a strong convergence theorem for finding a commonfixed point of a nonexpansive mapping and a strict pseudocontraction.

Theorem 4.2. Let H be a real Hilbert space, C a closed convex subset of H, B, Ψ : C → H be β,σ-inverse-strongly monotone mappings, ϕ : C → R a convex and lower semicontinuous function,f : C → C a contraction with coefficient α (0 < α < 1), and A a strongly positive linear boundedoperator of H into itself with coefficient γ > 0. Assume that 0 < γ < γ/α. Let S be a nonexpansivemapping of C into itself, and let T be a κ-strictly pseudocontraction of C into itself such that

Ω := F(S) ∩ F(T) ∩ GMEP(F, ϕ,Ψ

)/= ∅. (4.5)

Suppose that {xn} is a sequence generated by the following algorithm for x0, un ∈ C arbitrarily:

F(un, y

)+ ϕ(y) − ϕ(un) + 〈Ψxn, y − un〉 + 1

rn


⟩ ≥ 0, ∀y ∈ C,

xn+1 = PC[αnγf(xn) + (I − αnA)S((1 − λ)un + λTun)

](4.6)

for all n = 0, 1, 2, . . ., by (C1)–(C3) in Theorem 3.1.


Then {xn} converges strongly to q ∈ Ω, where q = PΩ(γf+I−A)(q)which solves the followingvariational inequality:

⟨(γf −A)q, p − q⟩ ≤ 0, ∀p ∈ Ω (4.7)


minq∈Ω

12⟨Aq, q

⟩ − h(q), (4.8)


Proof. Put B ≡ I − T , then B is ((1 − κ)/2)-inverse-strongly monotone, F(T) = I(B,M), andJM,λ(xn − λBxn) = (1 − λ)xn + λTxn. So by Theorem 3.1, we obtain the desired result.

Corollary 4.3. LetH be a real Hilbert space, C a closed convex subset ofH, B, Ψ : C → H be β, σ-inverse-strongly monotone mappings, and ϕ : C → R a convex and lower semicontinuous function.Let f : C → C be a contraction with coefficient α (0 < α < 1) and S a nonexpansive mapping of Cinto itself, and let T be a κ-strictly pseudocontraction of C into itself such that

Ω := F(S) ∩ F(T) ∩ GMEP(F, ϕ,Ψ

)/= ∅. (4.9)

Suppose that {xn} is a sequence generated by the following algorithm for x0 ∈ C arbitrarily:

F(un, y

)+ ϕ(y) − ϕ(un) + 〈Ψxn, y − un〉 + 1

rn〈y − un, un − xn〉 ≥ 0, ∀y ∈ C,

xn+1 = PC[αnf(xn) + (I − αn)S((1 − λ)un + λTun)

](4.10)

for all n = 0, 1, 2, . . ., by (C1)–(C3) in Theorem 3.1.Then {xn} converges strongly to q ∈ Ω, where q = PΩ(f + I)(q) which solves the following


〈(f − I)q, p − q〉 ≤ 0, ∀p ∈ Ω (4.11)


minq∈Ω

12〈Aq, q〉 − h(q), (4.12)


Proof. Putting A ≡ I and γ ≡ 1 in Theorem 4.2, we obtain the desired result.


Acknowledgments

The authors would like to thank the National Research University Project of Thailand’s Officeof the Higher Education Commission for financial support under the project NRU-CSECno. 54000267. Furthermore, they also would like to thank the Faculty of Science (KMUTT)and the National Research Council of Thailand. Finally, the authors would like to thankProfessor Vittorio Colao and the referees for reading this paper carefully, providing valuablesuggestions and comments, and pointing out a major error in the original version of thispaper.

References

[1] F. E. Browder, “Existence and approximation of solutions of nonlinear variational inequalities,”Proceedings of the National Academy of Sciences of the United States of America, vol. 56, pp. 1080–1086,1966.



[4] S. D. Flam and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,”Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.


[6] P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,” ActaMathematica, vol. 115, pp. 271–310, 1966.

[7] J.-C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,”in Handbook of Generalized Convexity and Generalized Monotonicity, vol. 76 of Nonconvex Optim. Appl.,pp. 501–558, Springer, New York, NY, USA, 2005.


[9] W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The AmericanMathematical Monthly, vol. 72, pp. 1004–1006, 1965.

[10] Y. Hao, “Some results of variational inclusion problems and fixed point problems with applications,”Applied Mathematics and Mechanics. English Edition, vol. 30, no. 12, pp. 1589–1596, 2009.

[11] M. Liu, S. S. Chang, and P. Zuo, “An algorithm for finding a common solution for a system of mixedequilibrium problem, quasivariational inclusion problem, and fixed point problem of nonexpansivesemigroup,” Journal of Inequalities and Applications, vol. 2010, Article ID 895907, 23 pages, 2010.

[12] J. F. Tan and S. S. Chang, “Iterative algorithms for finding common solutions to variational inclusionequilibrium and fixed point problems,” Fixed Point Theory and Applications, vol. 2011, Article ID915629, 17 pages, 2011.

[13] S.-S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variationalinclusion and fixed point problems,” Applied Mathematics and Mechanics. English Edition, vol. 29, no. 5,pp. 571–581, 2008.

[14] R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of theAmerican Mathematical Society, vol. 149, pp. 75–88, 1970.

[15] B. Lemaire, “Which fixed point does the iteration method select?” in Recent Advances in Optimization(Trier, 1996), vol. 452 of Lecture Notes in Econom. and Math. Systems, pp. 154–167, Springer, Berlin,Germany, 1997.


[17] H. Iiduka andW. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 3, pp.341–350, 2005.


[18] Y. Su, M. Shang, and X. Qin, “An iterative method of solution for equilibrium and optimizationproblems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2709–2719, 2008.

[19] H. Brezis, “Operateur maximaux monotones,” in Mathematics Studies, vol. 5, North-Holland,Amsterdam, The Netherlands, 1973.

[20] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansivemappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.

[21] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society.Second Series, vol. 66, no. 1, pp. 240–256, 2002.

[22] F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” inNonlinear Functional Analysis, pp. 1–308, Amer. Math. Soc., Providence, RI, USA, 1976.

[23] J.-W. Peng, Y.-C. Liou, and J.-C. Yao, “An iterative algorithm combining viscositymethodwith parallelmethod for a generalized equilibrium problem and strict pseudocontractions,” Fixed Point Theory andApplications, vol. 2009, Article ID 794178, 21 pages, 2009.

[24] C. Klin-eam and S. Suantai, “A new approximation method for solving variational inequalities andfixed points of nonexpansive mappings,” Journal of Inequalities and Applications, vol. 2009, Article ID520301, 16 pages, 2009.


Research ArticleOn Common Solutions for Fixed-PointProblems of Two Infinite Families ofStrictly Pseudocontractive Mappingsand the System of Cocoercive QuasivariationalInclusions Problems in Hilbert Spaces

Pattanapong Tianchai

Faculty of Science, Maejo University, Chiangmai 50290, Thailand

Correspondence should be addressed to Pattanapong Tianchai, [email protected]

Received 24 February 2011; Accepted 14 June 2011


Copyright q 2011 Pattanapong Tianchai. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

This paper is concerned with a common element of the set of common fixed points for twoinfinite families of strictly pseudocontractive mappings and the set of solutions of a systemof cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergencetheorem for the above two sets is obtained by a novel general iterative scheme based on theviscosity approximation method, and applicability of the results has shown difference with theresults of many others existing in the current literature.

1. Introduction

Throughout this paper, we always assume that C is a nonempty closed-convex subset of areal Hilbert spaceH with inner product and norm denoted by 〈·, ·〉 and ‖ · ‖, respectively, and2H denotes the family of all the nonempty subsets ofH.

Let B : H → H be a single-valued nonlinear mapping andM : H → 2H a set-valuedmapping. We consider the following quasivariational inclusion problem, which is to find a pointx ∈ H such that

θ ∈ Bx +Mx, (1.1)

where θ is the zero vector in H. The set of solutions of the problem (1.1) is denoted byVI(H,B,M). As special cases of the problem (1.1), we have the following.


(i) If M = ∂φ : H → 2H , where φ : H → R ∪ {+∞} is a proper convex lowersemicontinuous function such that R is the set of real numbers, and ∂φ is thesubdifferential of φ, then the quasivariational inclusion problem (1.1) is equivalentto find x ∈ H such that

〈Bx, v − x〉 + φ(y) − φ(x) ≥ 0, ∀v, y ∈ H, (1.2)

which is called the mixed quasivariational inequality problem (see [1]).

(ii) IfM = ∂δC, where δC : H → {0,+∞} is the indicator function of C, that is,

δC(x) =

⎧⎨

⎩

0, x ∈ C,+∞, x /∈ C,

(1.3)

then the quasivariational inclusion (1.1) is equivalent to find x ∈ C such that

〈Bx, v − x〉 ≥ 0, ∀v ∈ C, (1.4)

which is called Hartman-Stampacchia variational inequality problem (see [2–4]).

Recall that PC is the metric projection of H onto C, that is, for each x ∈ H, there existsthe unique point in PCx ∈ C such that

‖x − PCx‖ = miny∈C

∥∥x − y∥∥. (1.5)

A mapping T : C → C is called nonexpansive if

∥∥Tx − Ty∥∥ ≤ ∥∥x − y∥∥, ∀x, y ∈ C, (1.6)

and the mapping f : C → C is called a contraction if there exists a constant α ∈ (0, 1) suchthat

∥∥f(x) − f(y)∥∥ ≤ α∥∥x − y∥∥, ∀x, y ∈ C. (1.7)

A point x ∈ C is a fixed point of T provided Tx = x. We denote by F(T) the set of fixedpoints of T , that is, F(T) = {x ∈ C : Tx = x}. If C ⊂ H is bounded, closed, and convex andT is a nonexpansive mapping of C into itself, then F(T) is nonempty (see [5]). Recall that amapping A : C → C is said to be

(i) monotone if

⟨Ax −Ay, x − y⟩ ≥ 0, ∀x, y ∈ C, (1.8)


(ii) k-Lipschitz continuous if there exists a constant k > 0 such that

∥∥Ax −Ay∥∥ ≤ k∥∥x − y∥∥, ∀x, y ∈ C, (1.9)

if k = 1, then A is a nonexpansive,

(iii) pseudocontractive if

∥∥Ax −Ay∥∥2 ≤ ∥∥x − y∥∥2 + ∥∥(I −A)x − (I −A)y

∥∥2, ∀x, y ∈ C, (1.10)

(iv) k-strictly pseudocontractive if there exists a constant k ∈ [0, 1) such that

∥∥Ax −Ay∥∥2 ≤ ∥∥x − y∥∥2 + k∥∥(I −A)x − (I −A)y∥∥2, ∀x, y ∈ C, (1.11)

and it is obvious that A is a nonexpansive if and only if A is a 0-strictlypseudocontractive,

(v) α-strongly monotone if there exists a constant α > 0 such that

⟨Ax −Ay, x − y⟩ ≥ α∥∥x − y∥∥2, ∀x, y ∈ C, (1.12)

(vi) α-inverse-strongly monotone (or α-cocoercive) if there exists a constant α > 0 such that

⟨Ax −Ay, x − y⟩ ≥ α∥∥Ax −Ay∥∥2, ∀x, y ∈ C, (1.13)

if α = 1, then A is called that firmly nonexpansive; it is obvious that any α-inverse-strongly monotone mapping A is monotone and (1/α)-Lipschitz continuous,

(vii) relaxed α-cocoercive if there exists a constant α > 0 such that

⟨Ax −Ay, x − y⟩ ≥ (−α)∥∥Ax −Ay∥∥2, ∀x, y ∈ C, (1.14)

(viii) relaxed (α, r)-cocoercive if there exists two constants α, r > 0 such that

⟨Ax −Ay, x − y⟩ ≥ (−α)∥∥Ax −Ay∥∥2 + r∥∥x − y∥∥2, ∀x, y ∈ C, (1.15)

and it is obvious that any r-strongly monotonicity implies to the relaxed (α, r)-cocoercivity.

The existence common fixed points for a finite family of nonexpansive mappings havebeen considered by many authors (see [6–9] and the references therein).


In this paper, we study the mappingWn defined by

Un,n+1 = I,

Un,n = μnSnUn,n+1 +(1 − μn

)I,

Un,n−1 = μn−1Sn−1Un,n +(1 − μn−1

)I,

...

Un,k = μkSkUn,k+1 +(1 − μk

)I,

Un,k−1 = μk−1Sk−1Un,k +(1 − μk−1

)I,

...

Un,2 = μ2S2Un,3 +(1 − μ2

)I,

Wn = Un,1 = μ1S1Un,2 +(1 − μ1

)I,

(1.16)

where {μi} is nonnegative real sequence in (0, 1), for all i ∈ N, S1, S2, . . . from a family ofinfinitely nonexpansive mappings of C into itself. It is obvious that Wn is a nonexpansiveof C into itself, such a mapping Wn is called a W-mapping generated by S1, S2, . . . , Sn andμ1, μ2, . . . , μn.

A typical problem is to minimize a quadratic function over the set of fixed points of anonexpansive mapping in a real Hilbert spaceH,

minx∈Ω

{12〈Ax, x〉 − 〈x, b〉

}, (1.17)

where A is a bounded linear operator on H, Ω is the fixed-point set of a nonexpansivemapping S on H, and b is a given point in H. Recall that A is a strongly positive boundedlinear operator onH if there exists a constant γ > 0 such that

〈Ax, x〉 ≥ γ‖x‖2, ∀x ∈ H. (1.18)

Marino and Xu [10] introduced the following iterative scheme based on the viscosityapproximation method introduced by Moudafi [11]:

xn+1 = αnγf(xn) + (I − αnA)Sxn, ∀n ∈ N, (1.19)

where x1 ∈ H, A is a strongly positive bounded linear operator on H, f is a contraction onH, and S is a nonexpansive onH. They proved that under some appropriateness conditionsimposed on the parameters, if F(S)/= ∅, then the sequence {xn} generated by (1.19) convergesstrongly to the unique solution z = PF(S)(I −A + γf)z of the variational inequality

⟨(A − γf)z, x − z⟩ ≥ 0, ∀x ∈ F(S), (1.20)



minx∈F(S)

{12〈Ax, x〉 − h(x)

}, (1.21)

where h is a potential function for γf (i.e., h′(x) = γf(x) for x ∈ H).Iiduka and Takahashi [12] introduced an iterative scheme for finding a common

element of the set of fixed points of a nonexpansive mapping and the set of solutions ofthe variational inequality (1.4) as in the following theorem.

Theorem IT. Let C be a nonempty closed-convex subset of a real Hilbert space H. Let B be an α-inverse-strongly monotone mapping of C intoH, and let S be a nonexpansive mapping of C into itselfsuch that F(S) ∩ VI(C,B)/= ∅. Suppose that x1 = x ∈ C and {xn} is the sequence defined by

xn+1 = αnx + (1 − αn)SPC(xn − λnBxn), (1.22)

for all n ∈ N, where {αn} ⊂ (0, 1) and {λn} ⊂ [a, b] such that 0 < a < b < 2α satisfying the followingconditions:


n=1 αn = ∞,

(C2)∑∞

n=1 |αn+1 − αn| <∞ and∑∞

n=1 |λn+1 − λn| <∞,

then {xn} converges strongly to PF(S)∩VI(C,B)x.

Definition 1.1 (see [13]). Let M : H → 2H be a multivalued maximal monotone mapping,then the single-valued mapping JM,λ : H → H defined by JM,λ(u) = (I + λM)−1(u), for allu ∈ H, is called the resolvent operator associated withM, where λ is any positive number, and Iis the identity mapping.

Recently, Zhang et al. [13] considered the problem (1.1). To be more precise, theyproved the following theorem.

Theorem ZLC. Let H be a real Hilbert space, let B : H → H be an α-inverse-strongly monotonemapping, letM : H → 2H be a maximal monotone mapping, and let S : H → H be a nonexpansivemapping. Suppose that the set F(S) ∩ VI(H,B,M)/= ∅, where VI(H,B,M) is the set of solutions ofquasivariational inclusion (1.1). Suppose that x1 = x ∈ H and {xn} is the sequence defined by

yn = JM,λ(xn − λBxn),xn+1 = αnx + (1 − αn)Syn,

(1.23)

for all n ∈ N, where λ ∈ (0, 2α) and {αn} ⊂ (0, 1) satisfying the following conditions:


n=1 αn = ∞,

(C2)∑∞

n=1 |αn+1 − αn| <∞,

then {xn} converges strongly to PF(S)∩VI(H,B,M)x.


Peng et al. [14] introduced an iterative scheme

Φ(un, y)+

1rn


⟩ ≥ 0, ∀y ∈ H,

yn = JM,λ(un − λBun),xn+1 = αnf(xn) + (1 − αn)Syn,

(1.24)

for all n ∈ N, where x1 ∈ H, B is an α-cocoercive mapping on H, f is a contraction onH, S is anonexpansive onH,M is a maximal monotone mapping ofH into 2H , and Φ is a bifunctionfromH ×H into R.

We note that their iteration is well defined if we let C = H, and the appropriateness ofthe control conditions αn and λ of their iteration should be {αn} ⊂ (0, 1) and λ ∈ (0, 2α) (seeTheorem 3.1 in [14]). They proved that under some appropriateness imposed on the otherparameters, if Ω = F(S) ∩ VI(H,B,M) ∩ EP(Φ)/= ∅, then the sequences {xn}, {yn}, and {un}generated by (1.24) converge strongly to z = PΩf(z) of the variational inequality

⟨z − f(z), x − z⟩ ≥ 0, ∀x ∈ Ω, (1.25)

where EP(Φ) is the set of solutions of equilibrium problem defined by

EP(Φ) ={x ∈ H : Φ

(x, y) ≥ 0, ∀y ∈ H}. (1.26)

Moreover, Plubtieng and Sriprad [15] introduced an iterative scheme

Φ(un, y)+

1rn


⟩ ≥ 0, ∀y ∈ H,

yn = JM,λ(un − λBun),xn+1 = αnγf(xn) + (I − αnA)Snyn,

(1.27)

for all n ∈ N, where x1 ∈ H,A is a strongly bounded linear operator onH, B is an α-cocoercivemapping onH, f is a contraction onH, Sn is a nonexpansive onH,M is a maximal monotonemapping ofH into 2H , and Φ is a bifunction fromH ×H into R.

We note that the appropriateness of the control conditions αn and λ of their iterationshould be {αn} ⊂ (0, 1) and λ ∈ (0, 2α) (see Theorem 3.2 in [15]). They proved that undersome appropriateness imposed on the other parameters, if Ω =

⋂∞n=1 F(Sn) ∩ VI(H,B,M) ∩

EP(Φ)/= ∅, then the sequences {xn}, {yn}, and {un} generated by (1.27) converge strongly toz = PΩ(I −A + γf)z.

On the other hand, Li and Wu [16] introduced an iterative scheme for finding acommon element of the set of fixed points of a k-strictly pseudocontractive mapping with


a fixed point and the set of solutions of relaxed cocoercive quasivariational inclusions asfollows:

yn = JM,λ(xn − λBxn),xn+1 = αnγf(xn) + βnxn +

((1 − βn

)I − αnA

)(μSkxn +

(1 − μ)yn

),

(1.28)

for all n ∈ N, where x1 ∈ H, A is a strongly positive bounded linear operator on H, f is acontraction onH, Sk is amapping onH defined by Skx = kx+(1−k)Sx for all x ∈ H, such thatS is a k-strictly pseudocontractive mapping on H with a fixed point, B is relaxed cocoerciveand Lipschitz continuous mappings onH, andM is a maximal monotone mapping ofH into2H .

They proved that under the missing condition of μ, which should be 0 < μ < 1 (seeTheorem 2.1 in [16]) and some appropriateness imposed on the other parameters, if Ω =F(S) ∩ VI(H,B,M)/= ∅, then the sequence {xn} generated by (1.28) converges strongly toz = PΩ(I −A + γf)z.

Very recently, Tianchai and Wangkeeree [17] introduced an implicit iterative schemefor finding a common element of the set of common fixed points of an infinite family ofa kn-strictly pseudocontractive mapping and the set of solutions of the system of generalizedrelaxed cocoercive quasivariational inclusions as follows:

zn = JM2,λ2(xn − λ2(B2 + C2)xn),

yn = JM1,λ1(zn − λ1(B1 + C1)zn),

xn+1 = αnγf(Wnxn) + βnxn +((1 − βn

)I − αnA

)(γnWnxn +

(1 − γn

)yn),

(1.29)

for all n ∈ N, where x1 ∈ H, A is a strongly positive bounded linear operator on H, f is acontraction on H, Wn is a W-mapping on H generated by {Sn} and {μn} such that Snx =δnx + (1 − δn)Tnx for all x ∈ H, Tn is a kn-strictly pseudocontractive mapping on H with afixed point,Mi is a maximal monotone mapping ofH into 2H , and Bi, Ci are two mappingsof relaxed cocoercive and Lipschitz continuous mappings onH for each i = 1, 2.

They proved that under some appropriateness imposed on the parameters, if Ω =⋂∞n=1 F(Tn) ∩ F(D)/= ∅ such that the mapping D : H → H defined by

Dx = JM1,λ1((I − λ1(B1 + C1))JM2,λ2(I − λ2(B2 + C2))x), ∀x ∈ H, (1.30)

then the sequence {xn} generated by (1.29) converges strongly to z = PΩ(I −A + γf)z.In this paper, we introduce a novel general iterative scheme (1.32) below by the

viscosity approximation method to find a common element of the set of common fixed pointsfor two infinite families of strictly pseudocontractive mappings and the set of solutions ofa system of cocoercive quasivariational inclusions problems in Hilbert spaces. Firstly, weintroduce a mappingWn, whereWn is aW-mapping generated by {Rn} and {μn} for solvinga common fixed point for two infinite families of strictly pseudocontractive mappings byiteration such that the mapping Rn : H → H defined by

Rnx = αx + (1 − α)(αSnx + (1 − α)Tnx), ∀x ∈ H, (1.31)


for all n ∈ N, where {Sn : H → H} and {Tn : H → H} are two infinite families of k1 andk2-strictly pseudocontractive mappings with a fixed point, respectively, and {μn} ⊂ (0, μ] forsome μ ∈ (0, 1). It follows that a linear general iterative scheme of the mappings Wn andJMi,λi(I − λiCi) is obtained as follows:

yn = γnWnxn +(1 − γn

) N∑

i=1

ρiJMi,λi(xn − λiCixn),

xn+1 = αnγf(xn) + βnBxn +((1 − εn)I − βnB − αnA

)yn,

(1.32)

for all n ∈ N, where x1 = u ∈ H,Mi : H → 2H is a maximal monotonemapping,Ci : H → His a cocoercive mapping for each i = 1, 2, . . . ,N, f : H → H is a contraction mapping, andA,B : H → H are twomappings of the strongly positive linear bounded self-adjoint operatormappings.

As special cases of the iterative scheme (1.32), we have the following.

(i) If εn = 0 for all n ∈ N, then (1.32) is reduced to the iterative scheme


) N∑

i=1


xn+1 = αnγf(xn) + βnBxn +(I − βnB − αnA

)yn, ∀n ∈ N.

(1.33)

(ii) If B ≡ I, then (1.32) is reduced to the iterative scheme


) N∑

i=1


xn+1 = αnγf(xn) + βnxn +((1 − εn − βn

)I − αnA

)yn, ∀n ∈ N.

(1.34)

(iii) If εn = 0 for all n ∈ N, then (1.34) is reduced to the iterative scheme


) N∑

i=1



)I − αnA

)yn, ∀n ∈ N.

(1.35)

(iv) If βn = 0 for all n ∈ N, then (1.34) is reduced to the iterative scheme


) N∑

i=1


xn+1 = αnγf(xn) + ((1 − εn)I − αnA)yn, ∀n ∈ N.

(1.36)


(v) If εn = 0 for all n ∈ N, then (1.36) is reduced to the iterative scheme


) N∑

i=1


xn+1 = αnγf(xn) + (I − αnA)yn, ∀n ∈ N.

(1.37)

(vi) If γ = 1 and A ≡ I, then (1.37) is reduced to the iterative scheme


) N∑

i=1


xn+1 = αnf(xn) + (1 − αn)yn, ∀n ∈ N.

(1.38)

(vii) IfMi ≡ Ci ≡ 0 for each i = 1, 2, . . . ,N and∑N

i=1 ρi = 1, then (1.32) is reduced to theiterative scheme


)xn,


)yn, ∀n ∈ N.

(1.39)

Furthermore, if Sn ≡ Tn for all n ∈ N, then the mapping Rn : H → H in (1.31) isreduced to

Rnx = αx + (1 − α)Tnx, ∀x ∈ H, (1.40)

for all n ∈ N. It follows that the iterative scheme (1.32) is reduced to find a common element ofthe set of common fixed points for an infinite family of strictly pseudocontractive mappingsand the set of solutions of a system of cocoercive quasivariational inclusions problems inHilbert spaces.

It is well known that the class of strictly pseudocontractive mappings contains theclass of nonexpansive mappings; it follows that if the mapping Rn is defined as (1.31) andk1 = k2 = 0, then the iterative scheme (1.32) is reduced to find a common element of theset of common fixed points for two infinite families of nonexpansive mappings and the setof solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces,and if the mapping Rn is defined as (1.40) and k1 = k2 = 0, then the iterative scheme (1.32)is reduced to find a common element of the set of common fixed points for an infinite familyof nonexpansive mappings and the set of solutions of a system of cocoercive quasivariationalinclusions problems in Hilbert spaces.

We suggest and analyze the iterative scheme (1.32) above under some appropriatenessconditions imposed on the parameters, the strong convergence theorem for the above two setsis obtained, and applicability of the results has shown difference with the results of manyothers existing in the current literature.


2. Preliminaries

We collect the following lemmas which are used in the proof for the main results in the nextsection.

Lemma 2.1. Let C be a nonempty closed-convex subset of a Hilbert space H then the followinginequalities hold:

(1) 〈x − PCx, PCx − y〉 ≥ 0, ∀x ∈ H,y ∈ C,(2) ‖x + y‖2 ≤ ‖x‖2 + 2〈y, x + y〉, ∀x, y ∈ H.

Lemma 2.2 (see [10]). Let H be a Hilbert space, let f : H → H be a contraction with coefficient0 < α < 1, and let A : H → H be a strongly positive linear bounded operator with coefficient γ > 0,then

(1) if 0 < γ < γ/α, then

⟨x − y, (A − γf)x − (A − γf)y⟩ ≥ (γ − γα)∥∥x − y∥∥2, ∀x, y ∈ H, (2.1)

(2) if 0 < ρ ≤ ‖A‖−1, then ‖I − ρA‖ ≤ 1 − ργ .


an+1 ≤(1 − ηn

)an + δn, n ≥ 1, (2.2)

where {ηn} is a sequence in (0, 1) and {δn} is a sequence in R such that

(1) limn→∞ηn = 0 and∑∞

n=1 ηn = ∞,

(2) lim supn→∞(δn/ηn) ≤ 0 or∑∞

n=1 |δn| <∞,

then limn→∞an = 0.

Lemma 2.4 (see [9]). LetC be a nonempty closed-convex subset of a Hilbert spaceH, define mappingWn as (1.16), let Si : C → C be a family of infinitely nonexpansive mappings with

⋂∞i=1 F(Si)/= ∅,

and let {μi} be a sequence such that 0 < μi ≤ μ < 1, for all i ≥ 1, then

(1) Wn is nonexpansive and F(Wn) =⋂ni=1 F(Si) for each n ≥ 1,

(2) for each x ∈ C and for each positive integer k, limn→∞Un,kx exists,

(3) the mappingW : C → C defined by

Wx := limn→∞

Wnx = limn→∞

Un,1x, x ∈ C, (2.3)

is a nonexpansive mapping satisfying F(W) =⋂∞i=1 F(Si), and it is called theW-mapping

generated by S1, S2, . . . and μ1, μ2, . . ..

Lemma 2.5 (see [13]). The resolvent operator JM,λ associated with M is single-valued andnonexpansive for all λ > 0.


Lemma 2.6 (see [13]). u ∈ H is a solution of quasivariational inclusion (1.1) if and only if u =JM,λ(u − λBu), for all λ > 0, that is,

VI(H,B,M) = F(JM,λ(I − λB)), ∀λ > 0. (2.4)

Lemma 2.7 (see [19]). Let C be a nonempty closed-convex subset of a strictly convex Banach spaceX. Let {Tn : n ∈ N} be a sequence of nonexpansive mappings on C. Suppose that

⋂∞n=1 F(Tn)/= ∅.

Let {αn} be a sequence of positive real numbers such that∑∞

n=1 αn = 1, then a mapping S on Cdefined by

Sx =∞∑

n=1

αnTnx, (2.5)

for x ∈ C, is well defined, nonexpansive, and F(S) = ⋂∞n=1 F(Tn) holds.

Lemma 2.8 (see [2]). LetC be a nonempty closed-convex subset of a Hilbert spaceH and S : C → Ca nonexpansive mapping, then I − S is demiclosed at zero. That is, whenever {xn} is a sequence inC weakly converging to some x ∈ C and the sequence {(I − S)xn} strongly converges to some y, itfollows that (I − S)x = y.

Lemma 2.9 (see [20]). Let C be a nonempty closed-convex subset of a real Hilbert space H andT : C → C a k-strict pseudocontraction. Define S : C → C by Sx = αx + (1 − α)Tx for each x ∈ C,then, as α ∈ [k, 1), S is a nonexpansive such that F(S) = F(T).

3. Main Results

Lemma 3.1. Let C be a nonempty closed-convex subset of a real Hilbert spaceH, and let S, T : C →C be two mappings of k1 and k2-strictly pseudocontractive mappings with a fixed point, respectively.Suppose that F(S) ∩ F(T)/= ∅ and define a mapping R : C → C by

Rx = αx + (1 − α)(αSx + (1 − α)Tx), ∀x ∈ C, (3.1)

where α ∈ [k, 1) \ {0} such that k = max{k1, k2}, then R is well defined, nonexpansive, and F(R) =F(S) ∩ F(T).

Proof. Define the mappings S1, T1 : C → C as follows:

S1x = αx + (1 − α)Sx, T1x = αx + (1 − α)Tx, (3.2)

for all x ∈ C. By Lemma 2.9, we have S1 and T1 as nonexpansive such that F(S1) = F(S) andF(T1) = F(T). Therefore, for all x ∈ C, we have

Rx = αx + (1 − α)(αSx + (1 − α)Tx)

= αx + α(1 − α)Sx + (1 − α)2Tx


= α2x + α(1 − α)Sx + (1 − α)αx + (1 − α)2Tx= α(αx + (1 − α)Sx) + (1 − α)(αx + (1 − α)Tx)= αS1x + (1 − α)T1x.

(3.3)

It follows from Lemma 2.7 that R is well defined, nonexpansive, and F(R) = F(S1) ∩ F(T1) =F(S) ∩ F(T).

Theorem 3.2. Let H be a real Hilbert space, let Mi : H → 2H be a maximal monotone mapping,and let Ci : H → H be a ξi-cocoercive mapping for each i = 1, 2, . . . ,N. Let A,B : H → H betwo mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficientsδ, β ∈ (0, 1] such that δ ≤ ‖A‖ ≤ 1 and ‖B‖ = β, respectively, and let f : H → H be a contractionmapping with coefficient δ ∈ (0, 1). Let {Sn : H → H} and {Tn : H → H} be two infinitefamilies of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1, k2 ∈ [0, 1),respectively. Define a mapping Rn : H → H by


for all n ∈ N, where α ∈ [k, 1) \ {0} such that k = max{k1, k2}. Let Wn : H → H be a W-mapping generated by {Rn} and {μn} such that {μn} ⊂ (0, μ], for some μ ∈ (0, 1). Assume thatΩ := (

⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn)) ∩ (

⋂Ni=1 VI(H,Ci,Mi))/= ∅ and 0 < γ < δ/δ. For x1 = u ∈ H,

suppose that {xn} is generated iteratively by


) N∑

i=1



)yn,

(3.5)

for all n ∈ N, where {αn}, {γn} ⊂ (0, 1), {βn}, {εn} ⊂ [0, 1) such that εn ≤ αn, ρi ∈ (0, 1), andλi ∈ (0, 2ξi] for each i = 1, 2, . . . ,N satisfying the following conditions:

(C1) limn→∞αn = limn→∞(εn/αn) = 0,

(C2) 0 < limn→∞γn < 1 and lim supn→∞βn < 1,

(C3)∑∞

n=1 αn = ∞ and∑N

i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞,∑∞

n=1 |βn+1 − βn| <∞, and∑∞

n=1 |εn+1 − εn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,

then the sequences {xn} and {yn} converge strongly to w ∈ Ω where w = PΩ(I − A + γf)w is aunique solution of the variational inequality

⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (3.6)

Proof. From ‖B‖ = β ∈ (0, 1], εn ≤ αn for all n ∈ N, (C1) and (C2), we have αn → 0, εn → 0as n → ∞ and lim supn→∞βn < 1. Thus, we may assume without loss of generality that


αn < (1 − εn − βn‖B‖)‖A‖−1 for all n ∈ N. For any x, y ∈ H and for each i = 1, 2, . . . ,N, by theξi-cocoercivity of Ci, we have

∥∥(I − λiCi)x − (I − λiCi)y

∥∥2 =

∥∥(x − y) − λi

(Cix − Ciy

)∥∥2

=∥∥x − y∥∥2 − 2λi

⟨x − y,Cix − Ciy

⟩+ λ2i∥∥Cix − Ciy

∥∥2

≤ ∥∥x − y∥∥2 − (2ξi − λi)λi∥∥Cix − Ciy

∥∥2

≤ ∥∥x − y∥∥2,

(3.7)

which implies that I − λiCi is a nonexpansive. Since A and B are two mappings of the linearbounded self-adjoint operators, we have

‖A‖ = sup{|〈Ax, x〉| : x ∈ H, ‖x‖ = 1},

‖B‖ = sup{|〈Bx, x〉| : x ∈ H, ‖x‖ = 1}.(3.8)

Observe that

⟨((1 − εn)I − βnB − αnA

)x, x⟩= (1 − εn)〈x, x〉 − βn〈Bx, x〉 − αn〈Ax, x〉

≥ 1 − εn − βn‖B‖ − αn‖A‖

> 0.

(3.9)

Therefore, we obtain that (1 − εn)I − βnB − αnA is positive. Thus, by the strong positivity of Aand B, we get

∥∥(1 − εn)I − βnB − αnA∥∥ = sup

{⟨((1 − εn)I − βnB − αnA

)x, x⟩: x ∈ H, ‖x‖ = 1

}

= sup{(1 − εn)〈x, x〉 − βn〈Bx, x〉 − αn〈Ax, x〉 : x ∈ H, ‖x‖ = 1

}

≤ 1 − εn − βnβ − αnδ

≤ 1 − βnβ − αnδ.(3.10)

Define the sequences of mappings {Pn : H → H} and {Qn : H → H} as follows:

Pnx = αnγf(x) + βnBx +((1 − εn)I − βnB − αnA

)Qnx, ∀x ∈ H,

Qnx = γnWnx +(1 − γn

) N∑

i=1

ρiJMi,λi(I − λiCi)x, ∀x ∈ H,(3.11)


for all n ∈ N. Firstly, we prove that Pn has a unique fixed point inH. Note that for all x, y ∈ H,by (3.11), (C3), the nonexpansiveness ofWn, JMi,λi , and I − λiCi, we have

∥∥Qnx −Qny

∥∥ ≤ γn

∥∥Wnx −Wny

∥∥

+(1 − γn

) N∑

i=1

ρi∥∥JMi,λi(I − λiCi)x − JMi,λi(I − λiCi)y

∥∥

≤ γn∥∥x − y∥∥ + (1 − γn

) N∑

i=1

ρi∥∥(I − λiCi)x − (I − λiCi)y

∥∥

≤ γn∥∥x − y∥∥ + (1 − γn

)(

N∑

i=1

ρi

)∥∥x − y∥∥

=∥∥x − y∥∥.

(3.12)

Therefore, Qn is a nonexpansive. It follows from (3.10), (3.11), (3.12), the contraction of f ,and the linearity of A and B that

∥∥Pnx − Pny∥∥ ≤ αnγ

∥∥f(x) − f(y)∥∥ + βn‖B‖∥∥x − y∥∥

+∥∥(1 − εn)I − βnB − αnA

∥∥∥∥Qnx −Qny∥∥

≤ αnγδ∥∥x − y∥∥ + βnβ

∥∥x − y∥∥ +(1 − βnβ − αnδ

)∥∥x − y∥∥

=(1 −(δ − γδ

)αn)∥∥x − y∥∥.

(3.13)

Hence, Pn is a contraction with coefficient 1−(δ−γδ)αn ∈ (0, 1). Therefore, Banach contractionprinciple guarantees that Pn has a unique fixed point in H, and so the iteration (3.5) is welldefined.

Next, we prove that {xn} is bounded. Pick p ∈ Ω. Therefore, by Lemma 2.6, we have

p = JMi,λi(I − λiCi)p, (3.14)

for each i = 1, 2, . . . ,N. By (3.14), the nonexpansiveness of JMi,λi , and I − λiCi, we have

∥∥JMi,λi(xn − λiCixn) − p∥∥ =∥∥JMi,λi(xn − λiCixn) − JMi,λi

(p − λiCip

)∥∥

≤ ∥∥(xn − λiCixn) −(p − λiCip

)∥∥

≤ ∥∥xn − p∥∥.

(3.15)


Let tn =∑N

i=1 ρiJMi,λi(xn − λiCixn). By (3.14), (C3), the nonexpansiveness of JMi,λi , andI − λiCi, we have

∥∥tn − p

∥∥ =

∥∥∥∥∥

N∑

i=1

ρiJMi,λi(xn − λiCixn) −N∑

i=1

ρip

∥∥∥∥∥

=

∥∥∥∥∥

N∑

i=1

ρiJMi,λi(xn − λiCixn) −N∑

i=1

ρiJMi,λi

(p − λiCip

)∥∥∥∥∥

≤N∑

i=1

ρi∥∥JMi,λi(xn − λiCixn) − JMi,λi

(p − λiCip

)∥∥

≤N∑

i=1

ρi∥∥(xn − λiCixn) −

(p − λiCip

)∥∥

≤(

N∑

i=1

ρi

)∥∥xn − p

∥∥

=∥∥xn − p

∥∥.

(3.16)

Since Rnx = αx + (1 − α)(αSnx + (1 − α)Tnx), where α ∈ [k, 1) \ {0}, {Sn} and {Tn} aretwo infinite families of k1 and k2-strict pseudocontractions with a fixed point, respectively,such that k = max{k1, k2}; therefore, by Lemma 3.1, we have that Rn is a nonexpansive andF(Rn) = F(Sn) ∩ F(Tn) for all n ∈ N. It follows from Lemma 2.4(1) that we get F(Wn) =⋂ni=1 F(Ri) = (

⋂ni=1 F(Si)) ∩ (

⋂ni=1 F(Ti)), which implies thatWnp = p. Hence, by (3.16) and the

nonexpansiveness ofWn, we have

∥∥yn − p∥∥ =∥∥γnWnxn +

(1 − γn

)tn − p

∥∥

=∥∥γn(Wnxn − p

)+(1 − γn

)(tn − p

)∥∥

≤ γn∥∥Wnxn −Wnp

∥∥ +(1 − γn

)∥∥tn − p∥∥


∥∥ +(1 − γn

)∥∥xn − p∥∥

=∥∥xn − p

∥∥.

(3.17)

By (3.10), (3.17), the contraction of f , and the linearity of A and B, we have

∥∥xn+1 − p∥∥ =∥∥αnγf(xn) + βnBxn +

((1 − εn)I − βnB − αnA

)yn − p

∥∥

=∥∥αn(γf(xn) −Ap

)+ βnB

(xn − p

)

+((1 − εn)I − βnB − αnA

)(yn − p

) − εnp∥∥


∥∥ + βn‖B‖∥∥xn − p

∥∥

+∥∥(1 − εn)I − βnB − αnA

∥∥∥∥yn − p∥∥ + εn

∥∥p∥∥


≤ αnγ∥∥f(xn) − f

(p)∥∥ + αn

∥∥γf(p) −Ap∥∥ + βnβ

∥∥xn − p

∥∥

+(1 − βnβ − αnδ

)∥∥xn − p

∥∥ + αn

∥∥p∥∥

≤(1 −(δ − γδ

)αn)∥∥xn − p

∥∥ + αn(∥∥γf

(p) −Ap∥∥ + ∥∥p∥∥)

≤ max

{∥∥xn − p

∥∥,

∥∥γf(p) −Ap∥∥ + ∥∥p∥∥δ − γδ

}

.

(3.18)


∥∥xn+1 − p∥∥ ≤ max

{∥∥x1 − p

∥∥,

∥∥γf(p) −Ap∥∥ + ∥∥p∥∥δ − γδ

}

, (3.19)

for all n ∈ N. Hence, {xn} is bounded, and so are {yn}, {Wnxn}, {tn}, {f(xn)}, {Ayn}, {Bxn},and {Byn}.

Next, we prove that ‖xn+1 − xn‖ → 0 as n → ∞. By (C3), the nonexpansiveness ofJMi,λi , and I − λiCi, we have

‖tn+1 − tn‖ =

∥∥∥∥∥

N∑

i=1

ρiJMi,λi(xn+1 − λiCixn+1) −N∑

i=1

ρiJMi,λi(xn − λiCixn)

∥∥∥∥∥

≤N∑

i=1

ρi‖JMi,λi(xn+1 − λiCixn+1) − JMi,λi(xn − λiCixn)‖

≤N∑

i=1

ρi‖(xn+1 − λiCixn+1) − (xn − λiCixn)‖

≤(

N∑

i=1

ρi

)

‖xn+1 − xn‖

= ‖xn+1 − xn‖.

(3.20)

By the nonexpansiveness of Ri andUn,i, we have

‖Wn+1xn −Wnxn‖ = ‖Un+1,1xn −Un,1xn‖=∥∥μ1R1Un+1,2xn +

(1 − μ1

)xn −(μ1R1Un,2xn +

(1 − μ1

)xn)∥∥

≤ μ1‖Un+1,2xn −Un,2xn‖= μ1∥∥μ2R2Un+1,3xn +

(1 − μ2

)xn −(μ2R2Un,3xn +

(1 − μ2

)xn)∥∥


≤ μ1μ2‖Un+1,3xn −Un,3xn‖...

≤(

n∏

i=1

μi

)

‖Un+1,n+1xn −Un,n+1xn‖

≤Mn∏

i=1

μi,

(3.21)

for some constant M such that M ≥ ‖Un+1,n+1xn − Un,n+1xn‖ ≥ 0. Therefore, from (3.21), bythe nonexpansiveness ofWn+1, we have

‖Wn+1xn+1 −Wnxn‖ ≤ ‖Wn+1xn+1 −Wn+1xn‖ + ‖Wn+1xn −Wnxn‖

≤ ‖xn+1 − xn‖ +Mn∏

i=1

μi.(3.22)

Since

yn+1 − yn =(γn+1Wn+1xn+1 +

(1 − γn+1

)tn+1) − (γnWnxn +

(1 − γn

)tn)

= γn+1(Wn+1xn+1 −Wnxn) +(γn+1 − γn

)(Wnxn − tn)

+(1 − γn+1

)(tn+1 − tn),

(3.23)

combining (3.20), (3.22), and (3.23), we have

∥∥yn+1 − yn∥∥ ≤ γn+1‖Wn+1xn+1 −Wnxn‖ +

∣∣γn+1 − γn∣∣‖Wnxn − tn‖

+(1 − γn+1

)‖tn+1 − tn‖

≤ γn+1(

‖xn+1 − xn‖ +Mn∏

i=1

μi

)

+∣∣γn+1 − γn

∣∣‖Wnxn − tn‖

+(1 − γn+1

)‖xn+1 − xn‖

≤ ‖xn+1 − xn‖ +Mn∏

i=1

μi +∣∣γn+1 − γn

∣∣‖Wnxn − tn‖.

(3.24)

By the linearity of A and B, we have

xn+2 − xn+1 =(αn+1γf(xn+1) + βn+1Bxn+1 +

((1 − εn+1)I − βn+1B − αn+1A

)yn+1)

− (αnγf(xn) + βnBxn +((1 − εn)I − βnB − αnA

)yn)

=((1 − εn+1)I − βn+1B − αn+1A

)(yn+1 − yn

)+(βn − βn+1

)Byn


+ (αn − αn+1)Ayn + (εn − εn+1)yn + αn+1γ(f(xn+1) − f(xn)

)

+ γ(αn+1 − αn)f(xn) + βn+1B(xn+1 − xn)

+(βn+1 − βn

)Bxn.

(3.25)

Therefore, by (3.10), (3.24), (3.25), and the contraction of f , we have

‖xn+2 − xn+1‖ ≤ ∥∥(1 − εn+1)I − βn+1B − αn+1A∥∥∥∥yn+1 − yn

∥∥ +∣∣βn − βn+1

∣∣∥∥Byn

∥∥

+ |αn − αn+1|∥∥Ayn

∥∥ + |εn − εn+1|

∥∥yn∥∥ + αn+1γ

∥∥f(xn+1) − f(xn)

∥∥

+ γ |αn+1 − αn|∥∥f(xn)

∥∥ + βn+1‖B‖‖xn+1 − xn‖ +

∣∣βn+1 − βn

∣∣‖Bxn‖

≤(1 − βn+1β − αn+1δ

)∥∥yn+1 − yn∥∥ +∣∣βn − βn+1

∣∣∥∥Byn∥∥

+ |αn − αn+1|∥∥Ayn

∥∥ + |εn − εn+1|∥∥yn∥∥ + αn+1γδ‖xn+1 − xn‖

+ γ |αn+1 − αn|∥∥f(xn)

∥∥ + βn+1β‖xn+1 − xn‖ +∣∣βn+1 − βn

∣∣‖Bxn‖≤ (1 − ηn

)‖xn+1 − xn‖ + δn,

(3.26)

where ηn := (δ − γδ)αn+1 ∈ (0, 1) and

δn :=Mn∏

i=1

μi +N(∣∣γn − γn+1

∣∣ + |εn − εn+1| +∣∣βn − βn+1

∣∣ + |αn − αn+1|), (3.27)

such that

N = max

{

supn≥1

‖Wnxn − tn‖, supn≥1

(∥∥Byn∥∥ + ‖Bxn‖

), sup

n≥1

∥∥yn∥∥, sup

n≥1

(∥∥Ayn∥∥ + γ

∥∥f(xn)∥∥)}

.

(3.28)

By (C1), (C3), (C4), and (C5), we can find that limn→∞ηn = 0,∑∞

n=1 ηn = ∞, and∑∞

n=1 δn <∞;therefore, by (3.26) and Lemma 2.3, we obtain

‖xn+1 − xn‖ −→ 0 as n −→ ∞. (3.29)

Next, we prove that ‖xn − yn‖ → 0 as n → ∞. By the linearity of B, we have

∥∥xn+1 − yn∥∥ =∥∥αnγf(xn) + βnBxn +

((1 − εn)I − βnB − αnA

)yn − yn

∥∥

=∥∥αn(γf(xn) −Ayn

)+ βnB(xn − xn+1) + βnB

(xn+1 − yn

) − εnyn∥∥

≤ αn∥∥γf(xn) −Ayn

∥∥ + βn‖B‖‖xn − xn+1‖ + βn‖B‖∥∥xn+1 − yn

∥∥ + εn∥∥yn∥∥

≤ αn(∥∥γf(xn) −Ayn

∥∥ +∥∥yn∥∥) + βnβ‖xn − xn+1‖ + βnβ

∥∥xn+1 − yn∥∥.

(3.30)


It follows that

(1 − βnβ

)∥∥xn+1 − yn

∥∥ ≤ αn

(∥∥γf(xn) −Ayn∥∥ +∥∥yn∥∥) + βnβ‖xn − xn+1‖. (3.31)

Hence, by (C1), (C2), (3.29), and (3.31), we have

∥∥xn+1 − yn

∥∥ −→ 0 as n −→ ∞. (3.32)

Since

∥∥xn − yn

∥∥ ≤ ‖xn − xn+1‖ +

∥∥xn+1 − yn

∥∥, (3.33)

therefore, by (3.29) and (3.32), we obtain

∥∥xn − yn∥∥ −→ 0 as n −→ ∞. (3.34)

For all x, y ∈ H, by Lemma 2.2(2), the nonexpansiveness of PΩ, the contraction of f ,and the linearity of A, we have

∥∥PΩ(I −A + γf

)x − PΩ

(I −A + γf

)y∥∥ ≤ ∥∥(I −A + γf

)x − (I −A + γf

)y∥∥

≤ γ∥∥f(x) − f(y)∥∥ + ‖I −A‖∥∥x − y∥∥

≤ γδ∥∥x − y∥∥ +(1 − δ)∥∥x − y∥∥

=(1 −(δ − γδ

))∥∥x − y∥∥.

(3.35)

Therefore, PΩ(I−A+γf) is a contractionwith coefficient 1−(δ−γδ) ∈ (0, 1); Banach contractionprinciple guarantees that PΩ(I − A + γf) has a unique fixed point, say w ∈ H, that is, w =PΩ(I −A + γf)w. Hence, by Lemma 2.1(1), we obtain

⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (3.36)

Next, we claim that

lim supn→∞

⟨γf(w) −Aw,xn −w

⟩ ≤ 0. (3.37)

To show this inequality, we choose a subsequence {xni} of {xn} such that

lim supn→∞


⟩= lim

i→∞⟨γf(w) −Aw,xni −w

⟩. (3.38)

Since {xni} is bounded, there exists a subsequence {xnij } of {xni} which converges weakly tow. Without loss of generality, we can assume that xni ⇀ w as i → ∞.


Next, we prove that w ∈ Ω. Define the sequence of mappings {Qn : H → H} and themapping Q : H → H by

Qnx = γnWnx +(1 − γn

) N∑

i=1

ρiJMi,λi(I − λiCi)x, ∀x ∈ H,

Qx = limn→∞

Qnx,

(3.39)

for all n ∈ N. Therefore, by (C2) and Lemma 2.4(3), we have

Qx = aWx + (1 − a)N∑

i=1

ρiJMi,λi(I − λiCi)x, ∀x ∈ H, (3.40)

where 0 < a = limn→∞γn < 1. From (C3), Lemma 2.4(3), we have thatW and∑N

i=1 ρiJMi,λi(I −λiCi) are nonexpansive. Therefore, by (C3), Lemmas 2.4(3), 2.6, 2.7, and 3.1, we have

F(Q) = F(W) ∩ F(

N∑

i=1

ρiJMi,λi(I − λiCi)

)

=

( ∞⋂

i=1

F(Ri)

)

∩(

N⋂

i=1

F(JMi,λi(I − λiCi))

)

=

( ∞⋂

i=1

F(Si)

)

∩( ∞⋂

i=1

F(Ti)

)

∩(

N⋂

i=1

VI(H,Ci,Mi)

)

,

(3.41)

that is, F(Q) = Ω. From (3.34), we have ‖yni − xni‖ → 0 as i → ∞. Thus, from (3.5) and(3.39), we get ‖Qxni − xni‖ → 0 as i → ∞. It follows from xni ⇀ w and by Lemma 2.8 thatw ∈ F(Q), that is, w ∈ Ω. Therefore, from (3.36) and (3.38), we obtain

lim supn→∞


⟩= lim

i→∞⟨γf(w) −Aw,xni −w

⟩

=⟨(γf −A)w,w −w⟩ ≤ 0.

(3.42)

Next, we prove that xn → w as n → ∞. Since w ∈ Ω, the same as in (3.17), we have

∥∥yn −w∥∥ ≤ ‖xn −w‖. (3.43)

Therefore, by (3.10), (3.43), Lemma 2.1(2), the contraction of f , and the linearity of A and B,we have

‖xn+1 −w‖2 = ∥∥αnγf(xn) + βnBxn +((1 − εn)I − βnB − αnA

)yn −w

∥∥2

=∥∥αn(γf(xn) −Aw

)+ βnB(xn −w)

+((1 − εn)I − βnB − αnA

)(yn −w

) − εnw∥∥2


≤ ∥∥βnB(xn −w) +((1 − εn)I − βnB − αnA

)(yn −w

)∥∥2

+ 2⟨αn(γf(xn) −Aw

) − εnw, xn+1 −w⟩

≤ (βn‖B‖‖xn −w‖ + ∥∥(1 − εn)I − βnB − αnA∥∥∥∥yn −w

∥∥)2

+ 2αnγ⟨f(xn) − f(w), xn+1 −w

⟩

+ 2αn⟨γf(w) −Aw,xn+1 −w

⟩ − 2εn〈w,xn+1 −w〉

≤(βnβ‖xn −w‖ +

(1 − βnβ − αnδ

)‖xn −w‖

)2

+ 2αnγδ‖xn −w‖‖xn+1 −w‖+ 2αn

⟨γf(w) −Aw,xn+1 −w

⟩ − 2εn〈w,xn+1 −w〉

≤(1 − αnδ

)2‖xn −w‖2 + αnγδ

(‖xn −w‖2 + ‖xn+1 −w‖2

)

+ 2αn⟨γf(w) −Aw,xn+1 −w

⟩ − 2εn〈w,xn+1 −w〉.(3.44)

If follows that

‖xn+1 −w‖2 ≤ 1 − 2αnδ + αnγδ1 − αnγδ ‖xn −w‖2 + δ′n

=

⎛

⎜⎝1 −

2(δ − γδ

)αn

1 − αnγδ

⎞

⎟⎠‖xn −w‖2 + δ′n

≤ (1 − η′n)‖xn −w‖2 + δ′n,

(3.45)

where η′n := (δ − γδ)αn/(1 − αnγδ) ∈ (0, 1) and

δ′n :=1

1 − αnγδ(α2nδ

2‖xn −w‖2 + 2αn⟨γf(w) −Aw,xn+1 −w

⟩ − 2εn〈w,xn+1 −w〉). (3.46)

By (3.29), (3.42), (C1), and (C3), we can found that limn→∞η′n = 0,∑∞

n=1 η′n = ∞, and

lim supn→∞(δ′n/η

′n) ≤ 0. Therefore, by Lemma 2.3, we obtain that {xn} converges strongly

to w, and so is {yn}. This completes the proof.

Remark 3.3. The iteration (3.5) is the difference with many others as follows.

(1) TwomappingsA andB of the strongly positive linear bounded self-adjoint operatormappings are used in the iteration of {xn}, which used only one mapping A bymany others.

(2) Three parameters αn, βn, and εn are used in the iteration of {xn}, which used onlytwo parameters αn and βn by many others.


(3) The parameter βn can be chosen to be βn = 0 for all n ∈ N, because the conditionlim infn→∞βn > 0 of Suzuki’s Lemma (see [21]) is ignored in the control conditionsof the iteration, which is used by many others.

(4) A solving of a common fixed point for two infinite families of strictly pseudocon-tractive mappings by iteration is obtained by the mapping Wn, where Wn is a W-mapping generated by {Rn} and {μn} such that Rn is defined as in Theorem 3.2.

4. Applications

Theorem 4.1. Let H be a real Hilbert space, let Mi : H → 2H be a maximal monotone mapping,and let Ci : H → H be a ξi-cocoercive mapping for each i = 1, 2, . . . ,N. Let A,B : H → H betwo mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficientsδ, β ∈ (0, 1] such that δ ≤ ‖A‖ ≤ 1 and ‖B‖ = β, respectively, and let f : H → H be a contractionmapping with coefficient δ ∈ (0, 1). Let {Sn : H → H} and {Tn : H → H} be two infinitefamilies of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1, k2 ∈ [0, 1),respectively. Define a mapping Rn : H → H by



⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn)) ∩ (




) N∑

i=1


xn+1 = αnγf(xn) + βnBxn +(I − βnB − αnA

)yn,

(4.2)

for all n ∈ N, where {αn}, {γn} ⊂ (0, 1), {βn} ⊂ [0, 1), ρi ∈ (0, 1), and λi ∈ (0, 2ξi] for eachi = 1, 2, . . . ,N satisfying the following conditions:

(C1) limn→∞αn = 0,


(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞ and∑∞

n=1 |βn+1 − βn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,


⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.3)

Proof. It is concluded from Theorem 3.2 immediately, by putting εn = 0 for all n ∈ N.


Theorem 4.2. Let H be a real Hilbert space, let Mi : H → 2H be a maximal monotone mapping,and let Ci : H → H be a ξi-cocoercive mapping for each i = 1, 2, . . . ,N. Let A : H → H be astrongly positive linear bounded self-adjoint operator mapping with coefficient δ ∈ (0, 1] such thatδ ≤ ‖A‖ ≤ 1, and let f : H → H be a contraction mapping with coefficient δ ∈ (0, 1). Let{Sn : H → H} and {Tn : H → H} be two infinite families of k1 and k2-strictly pseudocontractivemappings with a fixed point such that k1, k2 ∈ [0, 1), respectively. Define a mapping Rn : H → Hby



⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn)) ∩ (




) N∑

i=1


xn+1 = αnγf(xn) + βnxn +((1 − εn − βn

)I − αnA

)yn,

(4.5)

for all n ∈ N, where {αn}, {γn} ⊂ (0, 1), {βn}, {εn} ⊂ [0, 1) such that εn ≤ αn, ρi ∈ (0, 1), andλi ∈ (0, 2ξi] for each i = 1, 2, . . . ,N satisfying the following conditions:



(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞,∑∞

n=1 |βn+1 − βn| <∞, and∑∞

n=1 |εn+1 − εn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,


⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.6)

Proof. It is concluded from Theorem 3.2 immediately, by putting B ≡ I.





⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn)) ∩ (




) N∑

i=1



)I − αnA

)yn,

(4.8)

for all n ∈ N, where {αn}, {γn} ⊂ (0, 1), {βn} ⊂ [0, 1), ρi ∈ (0, 1), and λi ∈ (0, 2ξi] for eachi = 1, 2, . . . ,N satisfying the following conditions:

(C1) limn→∞αn = 0,


(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞ and∑∞

n=1 |βn+1 − βn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,


⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.9)





⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn)) ∩ (




) N∑

i=1


xn+1 = αnγf(xn) + ((1 − εn)I − αnA)yn,

(4.11)


for all n ∈ N, where {αn}, {γn} ⊂ (0, 1), {εn} ⊂ [0, 1) such that εn ≤ αn, ρi ∈ (0, 1), and λi ∈ (0, 2ξi]for each i = 1, 2, . . . ,N satisfying the following conditions:


(C2) 0 < limn→∞γn < 1,

(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞ and∑∞

n=1 |εn+1 − εn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,


⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.12)

Proof. It is concluded from Theorem 4.2 immediately, by putting βn = 0 for all n ∈ N.




⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn)) ∩ (




)N∑

i=1ρiJMi,λi(xn − λiCixn),

xn+1 = αnγf(xn) + (I − αnA)yn,(4.14)

for all n ∈ N, where {αn}, {γn} ⊂ (0, 1), ρi ∈ (0, 1), and λi ∈ (0, 2ξi] for each i = 1, 2, . . . ,Nsatisfying the following conditions:

(C1) limn→∞αn = 0,

(C2) 0 < limn→∞γn < 1,

(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,



⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.15)


Theorem 4.6. LetH be a real Hilbert space, letMi : H → 2H be a maximal monotone mapping, andlet Ci : H → H be a ξi-cocoercive mapping for each i = 1, 2, . . . ,N. Let f : H → H be a contractionmapping with coefficient δ ∈ (0, 1), and let {Sn : H → H} and {Tn : H → H} be two infinitefamilies of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1, k2 ∈ [0, 1),respectively. Define a mapping Rn : H → H by



⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn)) ∩ (

⋂Ni=1 VI(H,Ci,Mi))/= ∅. For x1 = u ∈ H, suppose that {xn}

is generated iteratively by


) N∑

i=1


xn+1 = αnf(xn) + (1 − αn)yn,(4.17)

for all n ∈ N, where {αn}, {γn} ⊂ (0, 1), ρi ∈ (0, 1), and λi ∈ (0, 2ξi] for each i = 1, 2, . . . ,Nsatisfying the following conditions:

(C1) limn→∞αn = 0,

(C2) 0 < limn→∞γn < 1,

(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,

then the sequences {xn} and {yn} converge strongly to w ∈ Ω where w = PΩf(w) is a uniquesolution of the variational inequality

⟨(I − f)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.18)

Proof. It is concluded from Theorem 4.5 immediately, by putting γ = δ = 1 and A ≡ I.

Theorem 4.7. Let H be a real Hilbert space. Let A,B : H → H be two mappings of the stronglypositive linear bounded self-adjoint operator mappings with coefficients δ, β ∈ (0, 1] such that δ ≤‖A‖ ≤ 1 and ‖B‖ = β, respectively, and let f : H → H be a contraction mapping with coefficientδ ∈ (0, 1). Let {Sn : H → H} and {Tn : H → H} be two infinite families of k1 and k2-strictly


pseudocontractive mappings with a fixed point such that k1, k2 ∈ [0, 1), respectively. Define a mappingRn : H → H by



⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn))/= ∅ and 0 < γ < δ/δ. For x1 = u ∈ H, suppose that {xn} is

generated iteratively by


)xn,


)yn,

(4.20)

for all n ∈ N, where {αn}, {γn} ⊂ (0, 1) and {βn}, {εn} ⊂ [0, 1) such that εn ≤ αn satisfying thefollowing conditions:



(C3)∑∞

n=1 αn = ∞,

(C4)∑∞

n=1 |αn+1 − αn| <∞,∑∞

n=1 |βn+1 − βn| <∞, and∑∞

n=1 |εn+1 − εn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,

then the sequences {xn} and {yn} converge strongly to w ∈ Ω where w = PΩ(I − A + γf)w isa unique solution of the variational inequality

⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.21)

Proof. It is concluded from Theorem 3.2 immediately, by putting Mi ≡ Ci ≡ 0 for each i =1, 2, . . . ,N.

Theorem 4.8. Let H be a real Hilbert space, let Mi : H → 2H be a maximal monotone mapping,and let Ci : H → H be a ξi-cocoercive mapping for each i = 1, 2, . . . ,N. Let A,B : H → H be twomappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients δ, β ∈(0, 1] such that δ ≤ ‖A‖ ≤ 1 and ‖B‖ = β, respectively, and let f : H → H be a contraction mappingwith coefficient δ ∈ (0, 1). Let {Tn : H → H} be an infinite family of k-strictly pseudocontractivemappings with a fixed point such that k ∈ [0, 1). Define a mapping Rn : H → H by

Rnx = αx + (1 − α)Tnx, ∀x ∈ H, (4.22)


for all n ∈ N, where α ∈ [k, 1). LetWn : H → H be aW-mapping generated by {Rn} and {μn} suchthat {μn} ⊂ (0, μ], for some μ ∈ (0, 1). Assume that Ω := (

⋂∞n=1 F(Tn)) ∩ (

⋂Ni=1 VI(H,Ci,Mi))/= ∅

and 0 < γ < δ/δ. For x1 = u ∈ H, suppose that {xn} is generated iteratively by


) N∑

i=1



)yn,

(4.23)

for all n ∈ N, where {αn}, {γn} ⊂ (0, 1) and {βn}, {εn} ⊂ [0, 1) such that εn ≤ αn, ρi ∈ (0, 1), andλi ∈ (0, 2ξi] for each i = 1, 2, . . . ,N satisfying the following conditions:



(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞,∑∞

n=1 |βn+1 − βn| <∞, and∑∞

n=1 |εn+1 − εn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,


⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.24)

Proof. It is concluded from Theorem 3.2 immediately, by putting Sn ≡ Tn for all n ∈ N, andnote that α ∈ [k, 1) by Lemma 2.9.

Theorem 4.9. Let H be a real Hilbert space, let Mi : H → 2H be a maximal monotone mapping,and let Ci : H → H be a ξi-cocoercive mapping for each i = 1, 2, . . . ,N. Let A,B : H → H betwo mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficientsδ, β ∈ (0, 1] such that δ ≤ ‖A‖ ≤ 1 and ‖B‖ = β, respectively, and let f : H → H be a contractionmapping with coefficient δ ∈ (0, 1). Let {Sn : H → H} and {Tn : H → H} be two infinite familiesof nonexpansive mappings. Define a mapping Rn : H → H by


for all n ∈ N, where α ∈ (0, 1). Let Wn : H → H be a W-mapping generated by {Rn} and {μn}such that {μn} ⊂ (0, μ], for some μ ∈ (0, 1). Assume that Ω := (

⋂∞n=1 F(Sn)) ∩ (

⋂∞n=1 F(Tn)) ∩

(⋂Ni=1 VI(H,Ci,Mi))/= ∅ and 0 < γ < δ/δ. For x1 = u ∈ H, suppose that {xn} is generated

iteratively by


) N∑

i=1



)yn,

(4.26)





(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞,∑∞

n=1 |βn+1 − βn| <∞, and∑∞

n=1 |εn+1 − εn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,


⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.27)

Proof. It is concluded from Theorem 3.2 immediately, by putting k1 = k2 = 0.

Theorem 4.10. Let H be a real Hilbert space, letMi : H → 2H be a maximal monotone mapping,and let Ci : H → H be a ξi-cocoercive mapping for each i = 1, 2, . . . ,N. Let A,B : H → H betwo mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficientsδ, β ∈ (0, 1] such that δ ≤ ‖A‖ ≤ 1 and ‖B‖ = β, respectively, and let f : H → H be a contractionmapping with coefficient δ ∈ (0, 1). Let {Tn : H → H} be an infinite family of nonexpansivemappings. Define a mapping Rn : H → H by

Rnx = αx + (1 − α)Tnx, ∀x ∈ H, (4.28)

for all n ∈ N, where α ∈ [0, 1). LetWn : H → H be aW-mapping generated by {Rn} and {μn} suchthat {μn} ⊂ (0, μ], for some μ ∈ (0, 1). Assume that Ω := (

⋂∞n=1 F(Tn)) ∩ (

⋂Ni=1 VI(H,Ci,Mi))/= ∅

and 0 < γ < δ/δ. For x1 = u ∈ H, suppose that {xn} is generated iteratively by


) N∑

i=1



)yn,

(4.29)




(C3)∑∞


i=1 ρi = 1,

(C4)∑∞

n=1 |αn+1 − αn| <∞,∑∞

n=1 |βn+1 − βn| <∞, and∑∞

n=1 |εn+1 − εn| <∞,

(C5)∑∞

n=1 |γn+1 − γn| <∞ and∑∞

n=1∏n

i=1μi <∞,



⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ Ω. (4.30)

Proof. It is concluded from Theorem 4.8 immediately, by putting k = 0.

Theorem 4.11. Let H be a real Hilbert space. Let A,B : H → H be two mappings of the stronglypositive linear bounded self-adjoint operator mappings with coefficients δ, β ∈ (0, 1] such that δ ≤‖A‖ ≤ 1 and ‖B‖ = β, respectively, and let f : H → H be a contraction mapping with coefficientδ ∈ (0, 1). Let T : H → H be a nonexpansive mapping. Assume that F(T)/= ∅ and 0 < γ < δ/δ. Forx1 = u ∈ H, suppose that {xn} is generated iteratively by


)(σnTxn + (1 − σn)xn), (4.31)

for all n ∈ N, where {αn}, {σn} ⊂ (0, 1) and {βn}, {εn} ⊂ [0, 1) such that εn ≤ αn satisfying thefollowing conditions:


(C2) 0 < limn→∞σn < 1 and lim supn→∞βn < 1,

(C3)∑∞

n=1 αn = ∞,

(C4)∑∞

n=1 |αn+1 − αn| <∞ and∑∞

n=1 |βn+1 − βn| <∞,

(C5)∑∞

n=1 |εn+1 − εn| <∞ and∑∞

n=1 |σn+1 − σn| <∞,

then the sequences {xn} and {yn} converge strongly to w ∈ F(T) where w = PF(T)(I −A + γf)w isa unique solution of the variational inequality

⟨(A − γf)w,y −w⟩ ≥ 0, ∀y ∈ F(T). (4.32)

Proof. From Theorem 4.10, putting α = 0 andMi ≡ Ci ≡ 0 for all i = 1, 2, . . . ,N. Setting T1 ≡ T ,Tn ≡ I for all n = 2, 3, . . ., and let μn ⊂ (0, μ] for some μ ∈ (0, 1) such that

∑∞n=1∏n

i=1μi < ∞.Therefore, from the definition of Rn in Theorem 4.10, we have R1 = T1 = T and Rn = Ifor all n = 2, 3, . . .. Since Wn is a W-mapping generated by {Rn} and {μn}, therefore by thedefinition of Un,i and Wn in (1.16), we have Un,i = I for all i = 2, 3, . . . and Wn = Un,1 =μ1R1Un,2 + (1 − μ1)I = μ1T + (1 − μ1)I. Hence, by Theorem 4.10, we obtain


) N∑

i=1

ρiJMi,λi(xn − λiCixn)

= γn(μ1Txn +

(1 − μ1

)xn)+(1 − γn

)(

N∑

i=1

ρi

)

xn

= γn(μ1Txn +

(1 − μ1

)xn)+(1 − γn

)xn

= σnTxn + (1 − σn)xn,

(4.33)

where σn := γnμ1. This completes the proof.


Acknowledgment

The author would like to thank the Faculty of Science, Maejo University for its financialsupport.

References

[1] M. A. Noor, “Generalized set-valued variational inclusions and resolvent equations,” Journal ofMathematical Analysis & Applications, vol. 228, no. 1, pp. 206–220, 1998.

[2] F. E. Browder, “Nonlinear monotone operators and convex sets in Banach spaces,” Bulletin of theAmerican Mathematical Society, vol. 71, pp. 780–785, 1965.

[3] P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,” ActaMathematica, vol. 115, pp. 271–310, 1966.

[4] J.-L. Lions and G. Stampacchia, “Variational inequalities,” Communications on Pure & AppliedMathematics, vol. 20, pp. 493–519, 1967.

[5] W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, YokohamaPublishers, Yokohama, Japan, 2000.

[6] K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points ofa countable family of nonexpansive mappings in a Banach space,”Nonlinear Analysis: Theory, Methods& Applications, vol. 67, no. 8, pp. 2350–2360, 2007.

[7] H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings inHilbert space,” Journal of Mathematical Analysis & Applications, vol. 202, no. 1, pp. 150–159, 1996.

[8] M. Shang, Y. Su, and X. Qin, “Strong convergence theorems for a finite family of nonexpansivemappings,” Fixed Point Theory & Applications, vol. 2007, Article ID 76971, 9 pages, 2007.

[9] K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansivemappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001.

[10] G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”Journal of Mathematical Analysis & Applications, vol. 318, no. 1, pp. 43–52, 2006.

[11] A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of MathematicalAnalysis & Applications, vol. 241, no. 1, pp. 46–55, 2000.

[12] H. Iiduka andW. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 3, pp.341–350, 2005.

[13] S. S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variationalinclusion and fixed point problems,” Applied Mathematics & Mechanics, vol. 29, no. 5, pp. 571–581,2008.

[14] J.-W. Peng, Y. Wang, D. S. Shyu, and J.-C. Yao, “Common solutions of an iterative scheme forvariational inclusions, equilibrium problems, and fixed point problems,” Journal of Inequalities &Applications, vol. 2008, Article ID 720371, 15 pages, 2008.

[15] S. Plubtieng and W. Sriprad, “A viscosity approximation method for finding common solutions ofvariational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces,” Fixed PointTheory & Applications, vol. 2009, Article ID 567147, 20 pages, 2009.

[16] Y. Li and C. Wu, “On the convergence for an iterative method for quasivariational inclusions,” FixedPoint Theory & Applications, vol. 2010, Article ID 278973, 11 pages, 2010.

[17] P. Tianchai and R. Wangkeeree, “An iterative method for solving the generalized system of relaxedcocoercive quasivariational inclusions and fixed point problems of an infinite family of strictlypseudocontractive mappings,” International Journal of Mathematics andMathematical Sciences, vol. 2010,Article ID 683584, 23 pages, 2010.

[18] T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families ofnonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83,1997.

[19] R. E. Bruck Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,”Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973.



[21] T. Suzuki, “A sufficient and necessary condition for Halpern-type strong convergence to fixed pointsof nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 135, no. 1, pp. 99–106, 2007.


Research ArticleApproximation of Fixed Points of WeakBregman Relatively Nonexpansive Mappings inBanach Spaces

Jiawei Chen, Zhongping Wan, Liuyang Yuan, and Yue Zheng

School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, China

Correspondence should be addressed to Jiawei Chen, [email protected]

Received 7 April 2011; Accepted 19 May 2011


Copyright q 2011 Jiawei Chen et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We introduce a concept of weak Bregman relatively nonexpansive mapping which is distinct fromBregman relatively nonexpansive mapping. By using projection techniques, we construct severalmodification of Mann type iterative algorithms with errors and Halpern-type iterative algorithmswith errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregmanrelatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weakBregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings arederived under some suitable assumptions. The main results in this paper develop, extend, andimprove the corresponding results of Matsushita and Takahashi (2005) and Qin and Su (2007).

1. Introduction

Throughout this paper, without other specifications, we denote by R the set of real numbers.Let E be a real reflexive Banach space with the dual space E∗. The norm and the dual pairbetween E∗ and E are denoted by ‖ · ‖ and 〈·, ·〉, respectively. Let f : E → R∪{+∞} be properconvex and lower semicontinuous. The Fenchel conjugate of f is the function f∗ : E∗ →(−∞,+∞] defined by

f∗(ξ) = sup{〈ξ, x〉 − f(x) : x ∈ E}. (1.1)

We denote by dom f the domain of f , that is, dom f = {x ∈ E : f(x) < +∞}. Let C bea nonempty closed and convex subset of E and T : C → C a nonlinear mapping. Denoteby F(T) = {x ∈ C : Tx = x}, the set of fixed points of T . T is said to be nonexpansive if‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ C.


In 1967, Bregman [1] discovered an elegant and effective technique for the using ofthe so-called Bregman distance function Df (see, Section 2, Definition 2.1) in the processof designing and analyzing feasibility and optimization algorithms. This opened a growingarea of research in which Bregman’s technique is applied in various ways in order to designand analyze iterative algorithms for solving not only feasibility and optimization problems,but also algorithms for solving variational inequalities, for approximating equilibria, forcomputing fixed points of nonlinear mappings, and so on (see, e.g., [1–25], and the referencestherein).

Nakajo and Takahashi [26] introduced the following modification of the Manniteration method for a nonexpansive mapping T : C → C in a Hilbert spaceH as follows:

x0 ∈ C,

yn = αnxn + (1 − αn)Txn,

Cn ={z ∈ C :

∥∥z − yn∥∥ ≤ ‖z − xn‖

},

Qn = {z ∈ C : 〈xn − z, xn − x0〉 ≤ 0},

xn+1 = PCn∩Qnx0, ∀n ≥ 0,

(1.2)

where {αn} ⊂ [0, 1] and PC is the metric projection from H onto a closed and convex subsetC of H. They proved that {xn} generated by (1.2) converges strongly to a fixed point of Tunder some suitable assumptions. Motivated by Nakajo and Takahashi [26], Matsushita andTakahashi [27] introduced the following modification of the Mann iteration method for arelatively nonexpansive mapping T : C → C in a Banach space E as follows:

x0 ∈ C,

yn = J−1(αnJ(xn) + (1 − αn)J(Txn)),

Cn ={z ∈ C : φ

(z, yn

) ≤ φ(z, xn)},

Qn = {z ∈ C : 〈J(xn) − J(x0), xn − z〉 ≤ 0},

xn+1 = ΠCn∩Qnx0, ∀n ≥ 0,

(1.3)

where {αn} ⊂ [0, 1], φ(y, x) = ‖y‖2−2〈y, J(x)〉+‖x‖2 for all x, y ∈ E, J is the duality mappingof E andΠC is the generalized projection (see, e.g., [2, 3, 28]) from E onto a closed and convexsubset C of E. They also proved that {xn} generated by (1.3) converges strongly to a fixed


point of T under some suitable assumptions. Martinez-Yanes and Xu [29] gave a Halpern-type iterative algorithm for a nonexpansive mapping T : C → C as follows:

x0 ∈ C,

yn = βnx0 +(1 − βn

)Txn,

Cn ={z ∈ C :

∥∥z − yn

∥∥2 ≤ ‖z − xn‖2 + βn

(‖x0‖2 + 2〈xn − x0, z〉

)},

Qn = {z ∈ C : 〈xn − z, xn − x0〉 ≤ 0},

xn+1 = PCn∩Qnx0, ∀n ≥ 0,

(1.4)

where {βn} ⊂ [0, 1]. They derived that {xn} generated by (1.3) converges strongly to a fixedpoint of T under some suitable assumptions. Qin and Su [30] generalized the results ofMartinez-Yanes and Xu [29] to a uniformly convex and uniformly smooth Banach space fora relatively nonexpansive mapping and proposed the following iterative algorithm:

x0 ∈ C,

yn = J−1(βnJ(x0) +

(1 − βn

)J(Txn)

),

Cn ={z ∈ C : φ

(z, yn

) ≤ βnφ(z, x0) +(1 − βn

)φ(z, xn)

},

Qn = {z ∈ C : 〈J(xn) − J(x0), xn − z〉 ≤ 0},

xn+1 = ΠCn∩Qnx0, ∀n ≥ 0,

(1.5)

where {βn} ⊂ [0, 1], ΠC is the generalized projection (see, e.g., [2, 3, 28]) from E ontoa closed and convex subset C of E. They also obtained that {xn} generated by (1.5) convergesstrongly to a fixed point of T under some suitable assumptions. In 2003, Butnariu et al. [13]studied several notions of convex analysis: uniformly convexity at a point, total convexityat a point, uniformly convexity on bounded sets, and sequential consistency, which areuseful in establishing convergence properties for fixed point and optimization algorithmsin infinite dimensional Banach spaces. They established connections between these conceptsand used these relations in order to obtain improved convergence results concerning the outerBregman projection algorithm for solving convex feasibility problems and the generalizedproximal point algorithm for optimization in Banach spaces. In 2006, Butnariu and Resmerita[14] presented a Bregman-type iterative algorithms and studied the convergence of theBregman-type iterative method of solving operator equations. Resmerita [19] investigatedthe existence of totally convex functions in Banach spaces and, further, established continuityand stability properties of Bregman projections. Very recently, by using Bregman projection,Reich and Sabach [21] presented the following algorithms for finding common zeroes of


maximal monotone operators Ai : E → 2E∗(i = 1, 2, . . . ,N) in reflexive Banach space E as

follows:

x0 ∈ E,

yin = Resfλin

(xn + ein

),

Cin ={z ∈ E : Df

(z, yin

)≤ Df

(z, xn + ein

)},

Cn =N⋂

i=1

Cin,

Qn ={z ∈ E :

⟨∇f(x0) − ∇f(xn), z − xn⟩ ≤ 0

},

xn+1 = projfCn∩Qnx0, ∀n ≥ 0,

(1.6)

x0 ∈ E,

ηin = ξin +1λin

(∇f(yin

)− ∇f(xn)

), ξin ∈ Aiy

in,

ωin = ∇f∗

(λinη

in +∇f(xn)

),

Cin ={z ∈ E : Df

(z, yin

)≤ Df

(z,ωi

n

)},

Cn =N⋂

i=1

Cin,

Qn ={z ∈ E :

⟨∇f(x0) − ∇f(xn), z − xn⟩ ≤ 0

},

xn+1 = projfCn∩Qnx0, ∀n ≥ 0.

(1.7)

Further, under some suitable conditions, they obtained two strong convergence theorems ofmaximal monotone operators in reflexive Banach spaces. Reich and Sabach [22] studied theconvergence of two iterative algorithms for finitely many Bregman strongly nonexpansiveoperators in Banach spaces and obtained two strong convergence theorems for finitely manyBregman strongly nonexpansive operators under some assumptions. In [24], Reich andSabach proposed the following algorithms for finding common fixed points of finitely manyBregman firmly nonexpansive operators Ti : C → C (i = 1, 2, . . . ,N) in reflexive Banachspace E as follows: if

⋂Ni=1 F(Ti)/= ∅

x0 ∈ E,

Qi0 = E, i = 1, 2, . . . ,N,

yin = Ti(xn + ein

),


Qin+1 =

{z ∈ Qi

n :⟨∇f(xn + ein

)− ∇f

(yin

), z − yin

⟩≤ 0},

Qn =N⋂

i=1

Qin,

xn+1 = projfQn+1x0, ∀n ≥ 0.

(1.8)

Under some suitable conditions, they proved that the sequence {xn} generated by (1.8) con-verges strongly to

⋂Ni=1 F(Ti) and applied it to the solution of convex feasibility and equilib-

rium problems.Inspired andmotivated by the works, we introduce the concept of weak Bregman rela-

tively nonexpansive mappings in reflexive Banach space and give an example to illustrate theexistence of weak Bregman relatively nonexpansive mapping and the difference betweenweak Bregman relatively nonexpansive mapping and Bregman relatively nonexpansive map-ping. Secondly, by using the conception of the Bregman projection (see, e.g., [1, 13, 14]), weconstruct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively non-expansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. Thestrong convergence theorems for weak Bregman relatively nonexpansivemappings and Breg-man relatively nonexpansive mappings are derived under some suitable assumptions. More-over, the convergence rate of our algorithms is faster than that of Matsushita and Takahashi[27] and Qin and Su [30]. The main results in this paper develop, extend, and improve thecorresponding results in the literature.

2. Preliminaries

Let C be a nonempty closed convex subset of a real reflexive Banach space E, and let T : C →C be a nonlinear mapping. A point ω ∈ C is called an asymptotic fixed point of T (see, e.g.,[2, 3]) if C contains a sequence {xn} which converges weakly to ω such that limn→∞‖Txn −xn‖ = 0. A point ω ∈ C is called an strong asymptotic fixed point of T (see, e.g., [2, 3]) if Ccontains a sequence {xn}which converges strongly to ω such that limn→∞‖Txn −xn‖ = 0. Wedenote the sets of asymptotic fixed points and strong asymptotic fixed points of T by F(T)and F(T), respectively. When {xn} is a sequence in E, we denote strong convergence of {xn}to x ∈ E by xn → x. For any x ∈ int(dom f) and y ∈ E, the right-hand derivative of f at x inthe direction y defined by

f0(x, y):= lim

t↘0

f(x + ty

) − f(x)t

. (2.1)

f is called Gateaux differentiable at x if, for all y ∈ E, limt↘0(f(x + ty) − f(x))/t exists. Inthis case, f0(x, y) coincides with ∇f(x), the value of the gradient of f at x. f is calledGateaux differentiable if it is Gateaux differentiable for any x ∈ int(dom f). f is called Frechetdifferentiable at x if this limit is attained uniformly for ‖y‖ = 1. We say f is uniformly Frechetdifferentiable on a subset C of E if the limit is attained uniformly for x ∈ C and ‖y‖ = 1.


Legendre function f : E → (−∞,+∞] is defined in [7]. From [7], if E is a reflexiveBanach space, then f is Legendre if and only if it satisfies the following conditions (L1) and(L2):

(L1) the interior of the domain of f , int(dom f), is nonempty, f is Gateaux differentiableon int(dom f), and dom f = int(dom f),

(L2) the interior of the domain of f∗, int(dom f∗), is nonempty, f∗ is Gateaux differen-tiable on int(dom f∗), and dom f∗ = int(dom f∗).

Since E is reflexive, we know that (∂f)−1 = ∂f∗ (see, e.g., [31]). This, by (L1) and (L2),implies

∇f =(∇f∗)−1, ran∇f = dom ∇f∗ = int

(dom f∗),

ran∇f∗ = dom ∇f = int(dom f

).

(2.2)

By Theorem 5.4 [7], conditions (L1) and (L2) also yield that the functions f and f∗ are strictlyconvex on the interior of their respective domains. From now on, we assume that the convexfunction f : E → (−∞,+∞] is Legendre.

We first recall some definitions and lemmas which are needed in our main results.

Definition 2.1 (see [1, 13]). Let f : E → (−∞,+∞] be a Gateaux differentiable and convexfunction. The function Df : dom f × int(dom f) → [0,+∞), defined by

Df

(y, x):= f(y) − f(x) − ⟨∇f(x), y − x⟩ (2.3)

is called the Bregman distancewith respect to f .

Remark 2.2 (see [24]). The Bregman distance has the following properties:

(i) the three point identity, for any x ∈ dom f and y, z ∈ int(dom f),

Df

(x, y)+Df

(y, z) −Df(x, z) =

⟨∇f(z) − ∇f(y), x − y⟩, (2.4)

(ii) the four point identity, for any y,ω ∈ dom f and x, z ∈ int(dom f),

Df

(y, x) −Df

(y, z) −Df(ω, x) +Df(ω, z) =

⟨∇f(z) − ∇f(x), y −ω⟩. (2.5)

Definition 2.3 (see [1]). Let f : E → (−∞,+∞] be a Gateaux differentiable and convexfunction. The Bregman projection of x ∈ int(dom f) onto the nonempty closed and convexset C ⊂ dom f is the necessarily unique vector projfC(x) ∈ C satisfying

Df

(projfC(x), x

)= inf

{Df

(y, x): y ∈ C}. (2.6)


Remark 2.4. (i) If E is a Hilbert space and f(y) = (1/2)‖x‖2 for all x ∈ E, then the Bregmanprojection projfC(x) is reduced to the metric projection of x onto C.

(ii) If E is a smooth Banach space and f(y) = (1/2)‖x‖2 for all x ∈ E, then the Bregmanprojection projfC(x) is reduced to the generalized projection ΠC(x) (see, e.g. [3]) whichdefined by

φ(ΠC(x), x) = miny∈C

φ(y, x), (2.7)

where φ(y, x) = ‖y‖2 − 2〈y, J(x)〉 + ‖x‖2, J is the normalized duality mapping from E to 2E∗.

Definition 2.5 (see[12, 21]). Let C be a nonempty closed and convex set of dom f . Theoperator T : C → int(dom f) with F(T)/= ∅ is called:

(i) quasi-Bregman nonexpansive if

Df(u, Tx) ≤ Df(u, x), ∀x ∈ C, u ∈ F(T); (2.8)

(ii) Bregman relatively nonexpansive if

Df(u, Tx) ≤ Df(u, x), ∀x ∈ C, u ∈ F(T), (2.9)

and F(T) = F(T),

(iii) Bregman firmly nonexpansive if

⟨∇f(Tx) − ∇f(Ty), Tx − Ty⟩ ≤ ⟨∇f(x) − ∇f(y), Tx − Ty⟩, ∀x, y ∈ C, (2.10)

or equivalently

Df

(Tx, Ty

)+Df

(Ty, Tx

)+Df(Tx, x) +Df

(Ty, y

)

≤ Df

(Tx, y

)+Df

(Ty, x

), ∀x, y ∈ C. (2.11)

Definition 2.6. Let C be a nonempty closed and convex set of dom f . The operator T : C →int(dom f) with F(T)/= ∅ is called weak Bregman relatively nonexpansive if F(T) = F(T) and

Df(u, Tx) ≤ Df(u, x), ∀x ∈ C, u ∈ F(T). (2.12)

Remark 2.7. It is easy to see that each nonexpansive mapping T is quasi-Bregman non-expansive mapping with respect to f(x) = (1/2)‖x‖2 for all x ∈ E. Moreover, everyrelatively nonexpansive mapping T also is Bregman relatively nonexpansive mapping, where


T is called relatively nonexpansive mapping (see, e.g., [32]) if the following conditionsare satisfied:

F(T) = F(T)/= ∅, φ(u, Tx) ≤ φ(u, x), ∀x ∈ C, u ∈ F(T). (2.13)

Now, we give an example which is weak Bregman relatively nonexpansive mappingbut not Bregman relatively nonexpansive mapping.

Example 2.8. Let E = l2, f(x) = (1/2)‖x‖2 for all x ∈ E, where

l2 =

{

ξ = (ξ1, ξ2, . . . , ξn, . . .) :∞∑

n=1

|ξn|2 <∞}

, ‖ξ‖ =

( ∞∑

n=1

|ξn|2)1/2

, ∀ξ ∈ l2, (2.14)

and for any ξ = (ξ1, ξ2, . . . , ξn, . . .), μ = (μ1, μ2, . . . , μn, . . .) ∈ E, 〈ξ, μ〉 =∑∞

n=1 ξnμn. It is wellknown that l2 is a Hilbert space. Let {xn} ⊂ E be a sequence defined by x0 = (1, 0, 0,0, . . .), x1 = (1, 1, 0, 0, . . .), x2 = (1, 0, 1, 0, . . .), . . . , xn = (ξn,1, . . . , ξn,k, . . .), . . ., where

ξn,k =

⎧⎨

⎩

1, if k = 1, n + 1,

0, otherwise,(2.15)

for all n ≥ 0.Define a mapping T : E → E by

T(x) =

⎧⎨

⎩

nxnn + 1

, if x = xn (∃n ≥ 1),

−x, if x /=xn (∀n ≥ 1),(2.16)

for all n ≥ 0. It is easy to see that F(T) = {0}, and so, {xn} converges weakly to x0. Indeed, forany g = (ζ1, ζ2 . . . , ζk, . . .) ∈ E, we have

g(xn − x0) =⟨g, xn − x0

⟩=

∞∑

k=2

ζkξn,k = ζn+1. (2.17)

From∑∞

n=1 |ζn|2 <∞, it shows that limn→∞ζn+1 = 0. Moreover,

limn→∞

g(xn − x0) = limn→∞

ζn+1 = 0. (2.18)

Next, for anym/=n, one has ‖xn−xm‖ =√2/= 0; that is, {xn} is not a Cauchy sequence. Owing

to ‖Txn − xn‖ = ‖xn‖/(n + 1), we obtain

limn→∞

‖Txn − xn‖ = 0. (2.19)


Then, x0 is an asymptotic fixed point of T , but x0 /∈ F(T) = {0}. So, T is not Bregman relativelynonexpansive mapping.

For any strong convergent sequence {yn} ⊂ l2 such that yn → y0 and ‖Tyn − yn‖ → 0as n → ∞. Then, there exists a sufficiently large nature numberM such that yn /=xm for anyn,m > M. Thus, Tyn = −yn for n > M, which implies that 2yn → 0 and yn → y0 = 0 asn → ∞. That is, y0 = 0 is a strong asymptotic fixed point of T , and so, F(T) = F(T) = {0}.Since

Df(0, Tx) = f(0) − f(Tx) −⟨∇f(Tx), 0 − Tx⟩ = −1

2‖Tx‖2 + 〈Tx, Tx〉 =

12‖Tx‖2

≤ 12‖x‖2 = f(0) − f(x) − ⟨∇f(x), 0 − x⟩ = Df(0, x), x ∈ E.

(2.20)

Therefore, T is a weak Bregman relatively nonexpansive mapping.

Definition 2.9 (see [12]). Let f : E → (−∞,+∞] be a convex and Gateaux differentiable func-tion. f is called:

(i) totally convex at x ∈ int(dom f) if its modulus of total convexity at x; that is, thefunction νf : int(dom f) × [0,+∞) → [0,+∞) defined by

νf(x, t) := inf{Df

(y, x): y ∈ dom f,

∥∥y − x∥∥ = t}

(2.21)

is positive whenever t > 0,

(ii) totally convex if, it is totally convex at every point x ∈ int(dom f),

(iii) totally convex on bounded sets if νf(B, t) is positive for any nonempty bounded subsetB of E and t > 0, where the modulus of total convexity of the function f on the setB is the function νf : int(dom f) × [0,+∞) → [0,+∞) defined by

νf(B, t) := inf{νf(x, t) : x ∈ B ∩ dom f

}. (2.22)

Definition 2.10 (see [12, 21]). The function f : E → (−∞,+∞] is called:

(i) cofinite if dom f∗ = E∗,

(ii) sequentially consistent if, for any two sequences {xn} and {yn} in E such that the firstis bounded, and

limn→∞

Df

(yn, xn

)= 0 =⇒ lim

n→∞∥∥yn − xn

∥∥ = 0. (2.23)

Lemma 2.11 (see [21, Proposition 2.3]). If f : E → (−∞,+∞] is Frechet differentiable and totallyconvex, then f is cofinite.


Lemma 2.12 (see [14, Theorem 2.10]). Let f : E → (−∞,+∞] be a convex function whose domaincontains at least two points. Then, the following statements hold:

(i) f is sequentially consistent if and only if it is totally convex on bounded sets,

(ii) if f is lower semicontinuous, then f is sequentially consistent if and only if it is uniformlyconvex on bounded sets,

(iii) if f is uniformly strictly convex on bounded sets, then it is sequentially consistent and theconverse implication holds when f is lower semicontinuous, Frechet differentiable on itsdomain, and the Frechet derivative f ′ is uniformly continuous on bounded sets.

Lemma 2.13 (see [20, Proposition 2.1]). Let f : E → R be a uniformly Frechet differentiable andbounded on bounded subsets of E. Then, ∇f is uniformly continuous on bounded subsets of E fromthe strong topology of E to the strong topology of E∗.

Lemma 2.14 (see [21, Lemma 3.1]). Let f : E → R be a Gateaux differentiable and totally convexfunction. If x0 ∈ E and the sequence {Df(xn, x0)}∞n=1 is bounded, then the sequence {xn}∞n=1 is alsobounded.

Lemma 2.15 (see [21, Proposition 2.2]). Let f : E → R be a Gateaux differentiable and totallyconvex function, x0 ∈ E, and let C be a nonempty closed convex subset of E. Suppose that the sequence{xn}∞n=1 is bounded and any weak subsequential limit of {xn}∞n=1 belongs to C. If Df(xn, x0) ≤Df(proj

f

C(x0), x0) for any n ∈N, then {xn}∞n=1 converges strongly to projfC(x0).

In [23], Reich and Sabach proved the following result.

Lemma 2.16 (see [23, Lemma 15.5]). Let f : E → (−∞,+∞] be a Legendre function. Let C bea nonempty closed convex subset of int(dom f) and T : C → C a Bregman firmly nonexpansivemapping with respect to f . Then, F(T) is closed and convex.

Motivated by Lemma 2.16, we get the similar result for quasi-Bregman nonexpansivemapping.

Proposition 2.17. Let f : E → (−∞,+∞] be a Legendre function. Let C be a nonempty closedconvex subset of int(dom f) and T : C → C a quasi-Bregman nonexpansive mapping with respectto f . Then, F(T) is closed and convex.

Proof. Without loss of generality, set F(T) is nonempty. Firstly, we show that F(T) is closed.Let {xn}∞n=0 be a sequence in F(T) such that xn → x. By the definition of quasi-Bregmannonexpansive mapping, we have

Df(xn, Tx) ≤ Df(xn, x), n ≥ 0. (2.24)

Since f : E → (−∞,+∞] is a Legendre function, f is continuous at x ∈ C ⊂ int(dom f). Then,from the definition of Bregman distance,

limn→∞

Df(xn, Tx) = Df(x, Tx),

limn→∞

Df(xn, x) = Df(x, x) = 0.(2.25)


From (2.24) and (2.25), it follows that Df(x, Tx) = 0, and so, from [7, Lemma 7.3(vi), page642], Tx = x. Therefore, x ∈ F(T), and so, F(T) is closed.

We now show that F(T) is convex. For any x, y ∈ F(T) and t ∈ (0, 1), it yields thatz = tx+(1− t)y ∈ C. From the definition of quasi-Bregman nonexpansive mapping, it followsthat

Df(z, Tz) = f(z) − f(Tz) −⟨∇f(Tz), tx + (1 − t)y − T(tx + (1 − t)y)⟩

= f(z) + tDf(x, Tz) + (1 − t)Df

(y, Tz

) − tf(x) − (1 − t)f(y)

≤ f(z) + tDf(x, z) + (1 − t)Df

(y, z) − tf(x) − (1 − t)f(y)

=⟨∇f(z), z − tx − (1 − t)y⟩ = 0.

(2.26)

Again, from [7, Lemma 7.3(vi), page 642], we get Tz = z. Therefore, F(T) is convex. Thiscompletes the proof.

From the definitions of Bregman distance and the Fenchel conjugate of f , we have thefollowing result.

Lemma 2.18. Let f : E → (−∞,+∞] be a Gateaux differentiable and proper convex lower semi-continuous. Then, for all z ∈ E,

Df

(

z,∇f∗(

N∑

i=1

ti∇f(xi)))

≤N∑

i=1

tiDf(z, xi), (2.27)

where {xi}Ni=1 ⊂ E and {ti}Ni=1 ⊂ (0, 1) with∑N

i=1 ti = 1.

Lemma 2.19 (see [14, Corollary 4.4]). Let f : E → (−∞,+∞] be a Gateaux differentiable andtotally convex on int(dom f). Let x ∈ int(dom f) and C ⊂ int(dom f) a nonempty closed convexset. If x ∈ C, then the following statements are equivalent:

(i) the vector x is the Bregman projection of x onto C with respect to f ,

(ii) the vector x is the unique solution of the variational inequality

⟨∇f(x) − ∇f(z), z − y⟩ ≥ 0, ∀y ∈ C, (2.28)

(iii) the vector x is the unique solution of the inequality

Df

(y, z)+Df(z, x) ≤ Df

(y, x), ∀y ∈ C. (2.29)

3. Main Results

In this section, we introduce several modification of Mann-type iterative algorithms witherrors andHalpern-type iterative algorithmswith errors to find fixed points of weak Bregman


relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Ba-nach spaces. The strong convergence theorems for weak Bregman relatively nonexpansivemappings and Bregman relatively nonexpansive mappings are proved under some suitableconditions.

Theorem 3.1. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f :E → R a Legendre function which is bounded, uniformly Frechet differentiable, and totally convex onbounded subset of E, and let T : C → C be a weak Bregman relatively nonexpansive mapping suchthat F(T)/= ∅. Define a sequence {xn} in C by the following algorithm:

x0 ∈ C, Q0 = C,

zn = ∇f∗(βn∇f(T(xn + en)) +(1 − βn

)∇f(xn + en)),

yn = ∇f∗(αn∇f(xn + en) + (1 − αn)∇f(zn)),

Cn ={z ∈ Cn−1 ∩Qn−1 : Df

(z, yn

) ≤ Df(z, xn + en)},

C0 ={z ∈ C : Df

(z, y0

) ≤ Df(z, x0)},

Qn ={z ∈ Cn−1 ∩Qn−1 :

⟨∇f(x0) − ∇f(xn), z − xn⟩ ≤ 0

},


(3.1)

where {αn}, {βn} ⊂ [0, 1] such that lim infn→∞(1 − αn)βn > 0, and {en} is an error sequence in Ewith en → 0 as n → ∞. Then, the sequences {xn} and {yn} converge strongly to the pointprojf

F(T)(x0), where projf

F(T)(x0) is the Bregman projection of C onto F(T).

Proof. By Proposition 2.17, it follows that F(T) is a nonempty closed and convex subset of E. Itis easy to verify thatC0, C1, Q0, andQ1 are closed and convex. Suppose thatCk andQk (k ≥ 1)are closed and convex. Then, Ck ∩Qk is closed and convex. For any z ∈ Ck+1, y ∈ Qk+1,

Df

(z, yk+1

) ≤ Df(z, xk+1 + ek+1)

⇐⇒ f(z) − f(yk+1) − ⟨∇f(yk+1

), z − yk+1

⟩

≤ f(z) − f(xk+1 + ek+1) −⟨∇f(xk+1 + ek+1), z − (xk+1 + ek+1)

⟩

⇐⇒ ⟨∇f(xk+1 + ek+1), z − (xk+1 + ek+1)⟩ − ⟨∇f(yk+1

), z − yk+1

⟩

≤ f(yk+1) − f(xk+1 + ek+1)

⇐⇒ ⟨∇f(xk+1 + ek+1) − ∇f(yk+1), z − yk+1

⟩

≤ f(yk+1) − f(xk+1 + ek+1) −

⟨∇f(xk+1 + ek+1), yk+1 − (xk+1 + ek+1)⟩

⇐⇒ ⟨∇f(xk+1 + ek+1) − ∇f(yk+1), z − yk+1

⟩ ≤ Df

(yk+1, xk+1 + ek+1

),

⟨∇f(x0) − ∇f(xk), y − xk⟩ ≤ 0,

(3.2)


which implies that Ck+1 and Qk+1 are closed and convex. As a consequence, Cn and Qn areclosed and convex for all n ≥ 0. Taking p ∈ F(T) arbitrarily,

Df

(p, yn

)

= Df

(p,∇f∗(αn∇f(xn + en) + (1 − αn)∇f(zn)

))

≤ αnDf

(p, xn + en

)+ (1 − αn)Df

(p, zn

)

= αnDf

(p, xn + en

)+ (1 − αn)Df

(p,∇f∗(βn∇f(T(xn + en)) +

(1 − βn

)∇f(xn + en)))

≤ αnDf

(p, xn + en

)+ (1 − αn)

[(1 − βn

)Df

(p, xn + en

)+ βnDf

(p, T(xn + en)

)]

≤ αnDf

(p, xn + en

)+ (1 − αn)

[(1 − βn

)Df

(p, xn + en

)+ βnDf

(p, xn + en

)]

= Df

(p, xn + en

),

(3.3)

that is, p ∈ Cn, and so, F(T) ⊂ Cn for all n ≥ 0. We now show that F(T) ⊂ Qn for all n ≥ 0.Clearly, F(T) ⊂ Q0 = C. Assume that F(T) ⊂ Qk for all k ≥ 0. Note that xk+1 = projfCk∩Qk

(x0),and we have

⟨∇f(x0) − ∇f(xk+1), xk+1 − z⟩ ≥ 0, z ∈ Ck ∩Qk. (3.4)

Therefore,

⟨∇f(x0) − ∇f(xk+1), xk+1 − p⟩ ≥ 0, p ∈ F ⊂ Ck ∩Qk, (3.5)

which yields that p ∈ Qk+1. Then, F(T) ⊂ Qn for all n ≥ 0. Consequently, F(T) ⊂ Cn ∩Qn andCn ∩Qn is nonempty closed and convex for all n ≥ 0. Moreover, {xn} is well defined.

Secondly, we show that {xn} is a Cauchy sequence and bounded. Since

⟨∇f(x0) − ∇f(xn), z − xn⟩ ≤ 0, ∀z ∈ Qn, (3.6)

it follows that xn = projfQn(x0). Therefore, by xn+1 = projfCn∩Qn

(x0) ∈ Qn,

Df(xn, x0) ≤ Df(xn+1, x0). (3.7)

Taking p ∈ F(T) arbitrarily. From Lemma 2.19, it yields that

Df

(p,projfQn

(x0))+Df

(projfQn

(x0), x0)≤ Df

(p, x0

). (3.8)

Moreover, one has

Df(xn, x0) ≤ Df

(p, x0

) −Df

(p, xn

) ≤ Df

(p, x0

). (3.9)


Hence, {Df(xn, x0)} is bounded and so {xn}, {yn}, and {zn} are also bounded. From (3.7),it shows that limn→∞Df(xn, x0) exists. In the light of xm ∈ Qm−1 ⊂ Qn for any m > n, byLemma 2.19,

Df

(xm,proj

f

Qn(x0))+Df

(projfQn

(x0), x0)≤ Df(xm, x0), (3.10)

that is,

Df(xm, xn) ≤ Df(xm, x0) −Df(xn, x0). (3.11)

Consequently, one has

limn→∞

Df(xm, xn) = 0. (3.12)

Since f is totally convex on bounded subsets of E, by Lemma 2.12 and (3.12), we have

limn→∞

‖xm − xn‖ = 0. (3.13)

Thus, {xn} is a Cauchy sequence, and so,

limn→∞

‖xn+1 − xn‖ = 0. (3.14)

Since en → 0 as n → ∞, one has

limn→∞

‖(xn+1 + en+1) − (xn + en)‖ = 0, limn→∞

‖xn+1 − (xn + en)‖ = 0. (3.15)

Let xn → ω ∈ C. Then, xn + en → ω.Thirdly, we show that {xn} converges strongly to a point of F(T). Since f is uniformly

Frechet differentiable on bounded subsets of E, from Lemma 2.12, ∇f is norm-to-normuniformly continuous on bounded subsets of E. So, by (3.15),

limn→∞

∥∥∇f(xn+1) − ∇f(xn + en)∥∥ = 0. (3.16)

It follows from xn+1 ∈ Cn that

Df

(xn+1, yn

) ≤ Df(xn+1, xn + en). (3.17)

By the uniformly Frechet differentiable of f on bounded subsets of E, f is also uniformlycontinuous on bounded subsets of E. Hence, from (3.12) and limn→∞en = 0,

limn→∞

Df(xn+1, xn + en) = limn→∞

(f(xn+1) − f(xn + en) −

⟨∇f(xn + en), xn+1 − (xn + en)⟩)

= 0.

(3.18)


As a consequence, limn→∞Df(xn+1, yn) = 0 and so, limn→∞‖xn+1 −yn‖ = 0. Moreover, one has

limn→∞

∥∥∇f(xn+1) − ∇f(yn

)∥∥ = 0. (3.19)

Since ‖xn − yn‖ ≤ ‖xn − xn+1‖ + ‖xn+1 − yn‖, ‖xn − yn‖ → 0 and yn → ω as n → ∞. Noticingthat

∥∥∇f(xn+1) − ∇f(yn

)∥∥

=∥∥∇f(xn+1) −

(αn∇f(xn + en) + (1 − αn)∇f(zn)

)∥∥

≥ (1 − αn)∥∥∇f(xn+1) − ∇f(zn)

∥∥ − αn

∥∥∇f(xn+1) − ∇f(xn + en)

∥∥

= (1 − αn)∥∥∇f(xn+1) −

(βn∇f(T(xn + en)) +

(1 − βn

)∇f(xn + en))∥∥

− αn∥∥∇f(xn+1) − ∇f(xn + en)

∥∥

≥ −αn∥∥∇f(xn+1) − ∇f(xn + en)

∥∥ + (1 − αn)βn∥∥∇f(xn+1) − ∇f(T(xn + en))

∥∥

− (1 − αn)(1 − βn

)∥∥∇f(xn+1) − ∇f(xn + en)∥∥.

(3.20)

Therefore,

(1 − αn)βn‖∇f(xn+1) − ∇f(T(xn + en))‖≤ ‖∇f(xn+1) − ∇f(yn

)‖ + αn‖∇f(xn+1) − ∇f(xn + en)‖+ (1 − αn)

(1 − βn

)‖∇f(xn+1) − ∇f(xn + en)‖.(3.21)

In view of lim infn→∞(1 − αn)βn > 0 and from both (3.16) and (3.19), one has

limn→∞

∥∥∇f(xn+1) − ∇f(T(xn + en))∥∥ = 0. (3.22)

Furthermore, we have

limn→∞

‖xn+1 − T(xn + en)‖ = 0, (3.23)

and so, by (3.14),

limn→∞

‖(xn + en) − T(xn + en)‖ = 0. (3.24)

Since xn → ω and en → 0, we get ω ∈ F(T) = F(T).Finally, we show ω = projf

F(T)(x0). Since projf

F(T)(x0) ∈ F(T) ⊂ Cn ∩Qn, it follows from

xn+1 = projf(Cn∩Qn)(x0) that Df(xn+1, x0) ≤ Df(proj

f

F(T)(x0), x0). By Lemma 2.15, xn →projf

F(T)(x0) as n → ∞. Therefore, {xn} and {yn} converge strongly to projfF(T)(x0). This

completes the proof.


Theorem 3.2. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f :E → R a Legendre function which is bounded, uniformly Frechet differentiable, and totally convexon bounded subset of E, and let T : C → C be a Bregman relatively nonexpansive mapping suchthat F(T)/= ∅. Assume that {αn}, {βn} ⊂ [0, 1] such that lim infn→∞(1 − αn)βn > 0, and {en} is anerror sequence in E with en → 0 as n → ∞. Then, the sequences {xn} and {yn} generated by (3.1)converge strongly to the point projf


F(T)(x0) is the Bregman projection of C ontoF(T).

Proof. As in the proof of Theorem 3.1, we know that the sequences {xn} and {yn} convergestrongly to ω ∈ C, and so,

limn→∞

‖(xn + en) − T(xn + en)‖ = 0. (3.25)

Then, for any subsequence {xnk} of {xn} converges weakly to ω,

limk→∞

‖(xnk + enk) − T(xnk + enk)‖ = 0. (3.26)

Therefore,ω ∈ F(T) = F(T). By the similar proof of Theorem 3.2, the sequences {xn} and {yn}converge strongly to projf

F(T)(x0). This completes the proof.

If αn ≡ 0, en ≡ 0, and f(x) = (1/2)‖x‖2 for all x ∈ E, n ≥ 0, then from Remark 2.4 andTheorem 3.1, we have the following result.

Corollary 3.3. Let C be a nonempty closed convex subset of a real reflexive, smooth, and strictlyconvex Banach space E, and let T : C → C be a relatively nonexpansive mapping such that F(T)/= ∅.Define a sequence {xn} in C by the following algorithm:

x0 ∈ C, Q0 = C,

yn = J−1(βnJ(T(xn)) +

(1 − βn

)J(xn)

),

Cn ={z ∈ Cn−1 ∩Qn−1 : φ

(z, yn

) ≤ φ(z, xn)},

C0 ={z ∈ C : φ

(z, y0

) ≤ φ(z, x0)},

Qn = {z ∈ Cn−1 ∩Qn−1 : 〈J(x0) − J(xn), z − xn〉 ≤ 0},xn+1 = ΠCn∩Qnx0, ∀n ≥ 0,

(3.27)

where J is the duality mapping on E, {βn} ⊂ [0, 1] such that lim infn→∞βn > 0. Then, the sequences{xn} and {yn} converge strongly to the pointΠF(T)(x0), whereΠF(T)(x0) is the generalized projection(see, e.g., [2, 3, 28]) of C onto F(T).

In [27], Matsushita and Takahashi proved the following result.

TheoremMT (see [27, Theorem 3.1]). LetC be a nonempty closed convex subset of a real uniformlyconvex and uniformly smooth Banach space E, and let T : C → C be a relatively nonexpansivemapping such that F(T)/= ∅. Assume that {αn} is a sequence of real numbers such that 0 ≤ αn < 1


and lim supn→∞αn < 1. Then, the sequence {xn} generated by (1.3) converges strongly to the pointΠF(T)(x0), where ΠF(T)(x0) is the generalized projection (see, e.g., [2, 3, 28]) of C onto F(T).

Remark 3.4. Corollary 3.3 extends Theorem MT [27] from uniformly convex and uniformlysmooth Banach spaces to reflexive, smooth, and strictly convex Banach space.

Now, we investigate convergence theorems for Halpern-type iterative algorithms witherrors.

Theorem 3.5. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f :E → R a Legendre function which is bounded, uniformly Frechet differentiable, and totally convex onbounded subset of E, and let T : C → C be a weak Bregman relatively nonexpansive mapping suchthat F(T)/= ∅. Define a sequence {xn} in C by the following algorithm:

x0 ∈ C, Q0 = C,

zn = ∇f∗(βn∇f(x0) +(1 − βn

)∇f(T(xn + en))),

yn = ∇f∗(αn∇f(zn) + (1 − αn)∇f(xn + en)),

Cn ={z ∈ Cn−1 ∩Qn−1 : Df

(z, yn

) ≤ (1 − αnβn)Df(z, xn + en) + αnβnDf(z, x0)

},

C0 ={z ∈ C : Df

(z, y0

) ≤ Df(z, x0)},

Qn ={z ∈ Cn−1 ∩Qn−1 :

⟨∇f(x0) − ∇f(xn), z − xn⟩ ≤ 0

},

xn+1 = projf(Cn∩Qn)x0, ∀n ≥ 0,

(3.28)

where {αn}, {βn} ⊂ [0, 1] such that lim infn→∞αn > 0 and limn→∞βn = 0, and {en} is an errorsequence in E with en → 0 as n → ∞. Then, the sequences {xn} and {yn} converge strongly to thepoint projf



Proof. By Proposition 2.17, it follows that F(T) is a nonempty closed and convex subset of E.It is easy to see that Cn is closed and Qn is closed and convex for all n ≥ 0. For any z ∈ Cn,n ≥ 1,

Df

(z, yn


⇐⇒ f(z) − f(yn) − ⟨∇f(yn

), z − yn

⟩

≤ (1 − αnβn)(f(z) − f(xn + en) −

⟨∇f(xn + en), z − xn − en⟩)

+ αnβn(f(z) − f(x0) −

⟨∇f(x0), z − x0⟩)

⇐⇒ (1 − αnβn)f(xn + en) + αnβnf(x0) − f

(yn)

≤ ⟨∇f(yn), z − yn

⟩ − (1 − αnβn)⟨∇f(xn + en), z − xn − en

⟩ − αnβn⟨∇f(x0), z − x0

⟩,

(3.29)


which implies that Cn is closed and convex for all n ≥ 1. Since, for any z ∈ C0,

Df

(z, y0

) ≤ Df(z, x0)

⇐⇒ f(z) − f(y0) − ⟨∇f(y0

), z − y0

⟩ ≤ f(z) − f(x0) −⟨∇f(x0), z − x0

⟩

⇐⇒ ⟨∇f(x0), z − x0⟩ − ⟨∇f(y0

), z − y0

⟩ ≤ f(y0) − f(x0)

⇐⇒ ⟨∇f(x0), z − x0⟩ − ⟨∇f(y0

), z − x0 + x0 − y0

⟩ ≤ f(y0) − f(x0)

⇐⇒ ⟨∇f(x0) − ∇f(y0), z − x0

⟩+Df

(x0, y0

) ≤ 0,

(3.30)

which shows that C0 is closed and convex. As a consequence, Cn is closed and convex for alln ≥ 0. Taking p ∈ F(T) arbitrarily, by Lemma 2.18,

Df

(p, yn

)= Df

(p,∇f∗(αn∇f(zn) + (1 − αn)∇f(xn + en)

))

≤ αnDf

(p, zn

)+ (1 − αn)Df

(p, xn + en

)

= (1 − αn)Df

(p, xn + en

)+ αnDf

(p,∇f∗(βn∇f(x0) +

(1 − βn

)∇f(T(xn + en))))

≤ (1 − αn)Df

(p, xn + en

)+ αnβnDf

(p, x0

)+ αn

(1 − βn

)Df

(p, T(xn + en)

)

≤ (1 − αn)Df

(p, xn + en

)+ αnβnDf

(p, x0

)+ αn

(1 − βn

)Df

(p, xn + en

)

=(1 − αnβn

)Df

(p, xn + en

)+ αnβnDf

(p, x0

),

(3.31)

that is, p ∈ Cn, and so, F(T) ⊂ Cn for all n ≥ 0. As in the proof of Theorem 3.1, we getF(T) ⊂ Qn for all n ≥ 0, {xn} is a Cauchy sequence, {xn}, {yn}, and {zn} are also bounded,and thus,

limn→∞

Df(xn+1, xn + en) = 0, limn→∞

‖(xn+1 + en+1) − (xn + en)‖ = 0, (3.32)

limn→∞

‖xn+1 − (xn + en)‖ = 0. (3.33)

Consequently, F(T) ⊂ Cn ∩ Qn and Cn ∩ Qn is nonempty closed and convex for all n ≥ 0.Moreover, {xn} is well defined. Set xn → ω ∈ C.

Secondly, we show that {xn} converges strongly to a point of F(T). Since f is uniformlyFrechet differentiable on bounded subsets of E, from Lemma 2.12, ∇f is norm-to-normuniformly continuous on bounded subsets of E. So, by (3.33),

limn→∞

∥∥∇f(xn+1) − ∇f(xn + en)∥∥ = 0. (3.34)

In view of xn+1 = projf(Cn∩Qn)(x0) ∈ Cn, we have

Df

(xn+1, yn

) ≤ (1 − αnβn)Df(xn+1, xn + en) + αnβnDf(xn+1, x0). (3.35)


Due to limn→∞βn = 0, from (3.32), one has

limn→∞

Df

(xn+1, yn

)= 0. (3.36)

Therefore, limn→∞‖xn+1 − yn‖ = 0. Moreover, one has

limn→∞

∥∥∇f(xn+1) − ∇f(yn

)∥∥ = 0. (3.37)

Since ‖xn − yn‖ ≤ ‖xn − xn+1‖ + ‖xn+1 − yn‖, by (3.32) and (3.33),

limn→∞

∥∥xn − yn

∥∥ = 0, (3.38)

and thus, yn → ω as n → ∞. Noticing that

∥∥∇f(xn+1) − ∇f(yn)∥∥

=∥∥∇f(xn+1) −

(αn∇f(zn) + (1 − αn)∇f(xn + en)

)∥∥

≥ αn∥∥∇f(xn+1) − ∇f(zn)

∥∥ − (1 − αn)∥∥∇f(xn+1) − ∇f(xn + en)

∥∥

= αn∥∥∇f(xn+1) −

(βn∇f(x0) +

(1 − βn

)∇f(Tn(xn + en)))∥∥

− (1 − αn)∥∥∇f(xn+1) − ∇f(xn + en)

∥∥

≥ αn(1 − βn

)∥∥∇f(xn+1) − ∇f(Tn(xn + en))∥∥ − αnβn

∥∥∇f(xn+1) − ∇f(x0)∥∥

− (1 − αn)∥∥∇f(xn+1) − ∇f(xn + en)

∥∥.

(3.39)

That is,

αn(1 − βn

)‖∇f(xn+1) − ∇f(T(xn + en))‖≤ ∥∥∇f(xn+1) − ∇f(yn

)∥∥ + αnβn∥∥∇f(xn+1) − ∇f(x0)

∥∥

+ (1 − αn)∥∥∇f(xn+1) − ∇f(xn + en)

∥∥.

(3.40)

Together with lim infn→∞αn > 0, limn→∞βn = 0, and (3.37), this yields that

limn→∞

∥∥∇f(xn+1) − ∇f(T(xn + en))∥∥ = 0. (3.41)

Since f is uniformly Frechet differentiable on bounded subsets of E, from Lemma 2.12, ∇f isnorm-to-norm uniformly continuous on bounded subsets of E and so is ∇f∗. Then, by (3.41),we get

limn→∞

‖xn+1 − T(xn + en)‖ = 0. (3.42)


From ‖(xn+en)−T(xn+en)‖ ≤ ‖(xn+en)−(xn+1+en+1)‖+‖(xn+1+en+1)−T(xn+en)‖, it followsthat limn→∞‖(xn + en) − T(xn+1 + en+1)‖ = 0. So, ω ∈ F(T) = F(T). By the same argument ofTheorem 3.1, we know that {xn} and {yn} converge strongly to projf

F(T)(x0). This completesthe proof.

If αn ≡ 1 and en ≡ 0 for all n ≥ 0, then from Theorem 3.5, we have the following result.

Corollary 3.6. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f :E → R a Legendre function which is bounded, uniformly Frechet differentiable, and totally convex onbounded subset of E, and let T be a weak Bregman relatively nonexpansive mapping from C into itselfsuch that F(T)/= ∅. Define a sequence {xn} in C by the following algorithm:

x0 ∈ C, Q0 = C,

yn = ∇f∗(βn∇f(x0) +(1 − βn

)∇f(Txn)),

Cn ={z ∈ Cn−1 ∩Qn−1 : Df

(z, yn

) ≤ (1 − βn)Df(z, xn) + βnDf(z, x0)

},

C0 ={z ∈ C : Df

(z, y0

) ≤ Df(z, x0)},

Qn ={z ∈ Cn−1 ∩Qn−1 :

⟨∇f(x0) − ∇f(xn), z − xn⟩ ≤ 0

},


(3.43)

where {βn} ⊂ [0, 1] such that limn→∞βn = 0 and {en} is an error sequence in E with en → 0 asn → ∞. Then, the sequences {xn} and {yn} converges strongly to the point projf

F(T)(x0), where

projfF(T)(x0) is the Bregman projection of C onto F(T).

Now, we develop a strong convergence theorem for a Bregman relatively nonexpan-sive mapping.

Theorem 3.7. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f :E → R a Legendre function which is bounded, uniformly Frechet differentiable, and totally convexon bounded subset of E, and let T : C → C be a Bregman relatively nonexpansive mapping such thatF(T)/= ∅. Define a sequence {xn} in C by the following algorithm:

x0 ∈ C, Q0 = C,

zn = ∇f∗(βn∇f(x0) +(1 − βn

)∇f(T(xn + en))),

yn = ∇f∗(αn∇f(zn) + (1 − αn)∇f(xn + en)),

Cn ={z ∈ Cn−1 ∩Qn−1 : Df

(z, yn


},

C0 ={z ∈ C : Df

(z, y0

) ≤ Df(z, x0)},

Qn ={z ∈ Cn−1 ∩Qn−1 :

⟨∇f(x0) − ∇f(xn), z − xn⟩ ≤ 0

},


(3.44)


where {αn}, {βn} ⊂ [0, 1] such that lim infn→∞αn > 0 and limn→∞βn = 0 and {en} is an errorsequence in E with en → 0 as n → ∞. Then, the sequences {xn} and {yn} converges strongly to thepoint projf



Proof. The proof is similar to Theorem 3.5 and so is omitted. This completes the proof.

If αn ≡ 1 and en ≡ 0 for all n ≥ 0, then from Theorem 3.7, we get the following corollary.

Corollary 3.8. Let E be a real reflexive Banach space and f : E → R a Legendre function whichis bounded, uniformly Frechet differentiable, and totally convex on bounded subset of E, and let T :E → E be a Bregman relatively nonexpansive mapping such that F(T)/= ∅. Assume that {βn} is a realsequence in [0, 1] such that limn→∞βn = 0. Define a sequence {xn} by the following algorithm:

x0 ∈ C, Q0 = C,

yn = ∇f∗(βn∇f(x0) +(1 − βn

)∇f(Txn)),

Cn ={z ∈ Cn−1 ∩Qn−1 : Df

(z, yn

) ≤ (1 − αnβn)Df(z, xn) + αnβnDf(z, x0)

},

C0 ={z ∈ C : Df

(z, y0

) ≤ Df(z, x0)},

Qn ={z ∈ Cn−1 ∩Qn−1 :

⟨∇f(x0) − ∇f(xn), z − xn⟩ ≤ 0

},

xn+1 = projfCn∩Qnx0, ∀n ≥ 0.

(3.45)

Then, the sequences {xn} and {yn} converge strongly to the point projfF(T)(x0), where projf

F(T)(x0) isthe Bregman projection of C onto F(T).

In [30], Qin and Su obtained the following.

Theorem QS (see [30, Theorem 2.2]). Let C be a nonempty closed convex subset of a uniformlyconvex and uniformly smooth Banach space E, and let T : C → C be a relatively nonexpansive map-ping such that F(T)/= ∅. Assume that {βn} is a real sequence in [0, 1) such that limn→∞βn = 0. Then,the sequence {xn} generated by (1.5) converges strongly to ΠF(T)x0, where ΠF(T) is the generalizedprojection (see, e.g., [2, 3]) from E onto F(T).

Remark 3.9. Corollary 3.8 extends Theorems QS [30] from uniformly convex and uniformlysmooth Banach spaces to reflexive Banach spaces.

4. Conclusions

In this paper, we introduce a conception of weak Bregman relatively nonexpansive mappingin reflexive Banach space and give an example to illustrate the existence of weak Bregmanrelatively nonexpansive mapping and the difference between weak Bregman relatively non-expansive mapping and Bregman relatively nonexpansive mapping which enlarge the Breg-man operator theory. Secondly, by using projection techniques, we construct several modifi-cation of Mann-type iterative algorithms with errors and Halpern-type iterative algorithmswith errors to find fixed points of weak Bregman relatively nonexpansive mappings andBregman relatively nonexpansive mappings in Banach spaces. Thirdly, strong convergence


theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively non-expansive mappings are derived under some suitable assumptions. By further research, onthe one hand, we may apply our algorithms to find zeros of finite families of maximalmonotone operators, solutions of system of convex minization problems, solutions of systemof variational inequalities, equilibrium, and equation operators (see, e.g., [24]). On the otherhand, onemay give some numerical experiments to verify the theoretical assertions and showhow to compute the generalized projections. These topics will be done in the future.

Acknowledgments

The authors would like to thank anonymous referees for their constructive review and usefulcomments on an earlier version of the work and express gratitude to Professor SimenonReich, Department of Mathematics, The Technion-Israel Institute of Technology, Israel, andProfessor Yeol Je Cho, Department of Mathematics Education and the RINS, GyeongsangNational University, Chinju 660-701, Korea, for providing their nice works. This research issupported by the Natural Science Foundation of China (nos. 70771080, 60804065) and theFundamental Research Fund for the Central Universities (201120102020004).

References

[1] L. M. Bregman, “A relaxation method of finding a common point of convex sets and its application tothe solution of problems in convex programming,”USSR Computational Mathematics and MathematicalPhysics, vol. 7, no. 3, pp. 200–217, 1967.

[2] Ya. I. Alber, “Generalized projection operators in Banach spaces: properties and applications,” inFunctional-Differential Equations, vol. 1 of Proceedings of the Israel Seminar Ariel, Israel, Function Dif-ferential Equation, pp. 1–21, The College of Judea and Samaria, Ariel, Israel, 1994.

[3] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applica-tions,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 ofLecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996.

[4] Y. Alber and D. Butnariu, “Convergence of Bregman projectionmethods for solving consistent convexfeasibility problems in reflexive Banach spaces,” Journal of Optimization Theory and Applications, vol. 92,no. 1, pp. 33–61, 1997.

[5] H. H. Bauschke and J. M. Borwein, “Legendre functions and the method of random Bregman projec-tions,” Journal of Convex Analysis, vol. 4, no. 1, pp. 27–67, 1997.

[6] H. H. Bauschke and A. S. Lewis, “Dykstra’s algorithm with Bregman projections: a convergenceproof,” Optimization, vol. 48, no. 4, pp. 409–427, 2000.

[7] H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Essential smoothness, essential strict convexity,and Legendre functions in Banach spaces,” Communications in Contemporary Mathematics, vol. 3, no. 4,pp. 615–647, 2001.

[8] H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Bregman monotone optimization algorithms,”SIAM Journal on Control and Optimization, vol. 42, no. 2, pp. 596–636, 2003.

[9] H. H. Bauschke and P. L. Combettes, “Construction of best Bregman approximations in reflexiveBanach spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 12, pp. 3757–3766, 2003.

[10] R. S. Burachik, Generalized proximal point methods for the variational inequality problem, Ph.D. thesis,Instituto de Mathematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil, 1995.

[11] R. S. Burachik and S. Scheimberg, “A proximal point method for the variational inequality problemin Banach spaces,” SIAM Journal on Control and Optimization, vol. 39, no. 5, pp. 1633–1649, 2000.

[12] D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimen-sional Optimization, vol. 40 of Applied Optimization, Kluwer Academic Publishers, Dordrecht, TheNetherlands, 2000.

[13] D. Butnariu, A. N. Iusem, and C. Zalinescu, “On uniform convexity, total convexity and convergenceof the proximal point and outer Bregman projection algorithms in Banach spaces,” Journal of ConvexAnalysis, vol. 10, no. 1, pp. 35–61, 2003.


[14] D. Butnariu and E. Resmerita, “Bregman distances, totally convex functions, and amethod for solvingoperator equations in Banach spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 84919, 39pages, 2006.

[15] J. W. Chen, Y. J. Cho, J. K. Kim, and J. Li, “Multiobjective optimization problems with modifiedobjective functions and cone constraints and applications,” Journal of Global Optimization, vol. 49, no.1, pp. 137–147, 2011.

[16] J. Eckstein, “Nonlinear proximal point algorithms using Bregman functions, with applications toconvex programming,”Mathematics of Operations Research, vol. 18, no. 1, pp. 202–226, 1993.

[17] K. C. Kiwiel, “Proximal minimization methods with generalized Bregman functions,” SIAM Journalon Control and Optimization, vol. 35, no. 4, pp. 1142–1168, 1997.

[18] S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” inTheory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notesin Pure and Applied Mathematics, pp. 313–318, Dekker, New York, NY, USA, 1996.

[19] E. Resmerita, “On total convexity, Bregman projections and stability in Banach spaces,” Journal ofConvex Analysis, vol. 11, no. 1, pp. 1–16, 2004.

[20] S. Reich and S. Sabach, “A strong convergence theorem for a proximal-type algorithm in reflexiveBanach spaces,” Journal of Nonlinear and Convex Analysis, vol. 10, no. 3, pp. 471–485, 2009.

[21] S. Reich and S. Sabach, “Two strong convergence theorems for a proximal method in reflexive Banachspaces,” Numerical Functional Analysis and Optimization, vol. 31, no. 1–3, pp. 22–44, 2010.

[22] S. Reich and S. Sabach, “Two strong convergence theorems for Bregman strongly nonexpansive oper-ators in reflexive Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 1, pp.122–135, 2010.

[23] S. Reich and S. Sabach, “Existence and approximation of fixed points of Bregman firmly nonexpansivemappings in reflexive Banach spaces,” in Fixed-Point Algorithms for Inverse Problems in Science and En-gineering, Springer, New York, NY, USA, 2011.

[24] S. Reich and S. Sabach, “A projection method for solving nonlinear problems in reflexive Banachspaces,” Journal of Fixed Point Theory and Applications. In press.

[25] M. V. Solodov and B. F. Svaiter, “An inexact hybrid generalized proximal point algorithm and somenew results on the theory of Bregman functions,”Mathematics of Operations Research, vol. 25, no. 2, pp.214–230, 2000.

[26] K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and non-expansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379,2003.

[27] S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive map-pings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

[28] J. W. Chen and Y. Z. Zou, “Existence of solutions of F-implicit variational inequality problems withextended projection operators,” Acta Mathematica Sinica. Chinese Series, vol. 53, no. 2, pp. 375–384,2010.

[29] C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iterationprocesses,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006.

[30] X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banachspace,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1958–1965, 2007.

[31] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Oper-ations Research, Springer, New York, NY, USA, 2000.

[32] S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of tworelatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2,pp. 103–115, 2007.


Research ArticleComparison between Certain Equivalent NormsRegarding Some Familiar Properties ImplyingWFPP

Helga Fetter and Berta Gamboa de Buen

Centro de Investigacion en Matematicas (CIMAT), Apartado Postal 402, 36000 Guanajuato, GTO, Mexico

Correspondence should be addressed to Helga Fetter, [email protected]

Received 13 December 2010; Accepted 1 March 2011

Academic Editor: Enrique Llorens-Fuster

Copyright q 2011 H. Fetter and B. Gamboa de Buen. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

In a Banach space with a basis we define a similar norm to the norm shown by Lin to make l1into a space with FPP and make a comparative study of certain geometric properties such as theOpial property,WNS, and uniform nonsquareness of the original space and the space with the newnorm.

1. Introduction

Dowling et al. in [1] defined a norm in l1 which was used by Lin [2] to exhibit an equivalentnorm which makes l1 into a space with the fixed point property (FPP). A similar norm can bedefined in every Banach spaceX with a basis. Since l1 with its usual norm does not have FPP,we asked ourselves if this norm in these spaces would also improve properties that implythe weak fixed point property (WFPP). We found out that in some instances it does, in somecases the original norm has better properties, and in some cases you cannot compare them.We give several examples to illustrate our assertions.

2. The Γ Norm in a Banach Space

We start by giving the definition of the generalization of the norm used by Lin in a Banachspace with a basis.

Definition 2.1. Let (X, ‖ · ‖) be a Banach space with a basis {en}. Let x =∑∞

i=1 xiei ∈ X andQn : X → X be the projectionQn(

∑∞i=1 xiei) =

∑∞i=n xiei. The basis {en} is called premonotone


if ‖Qnx‖ ≥ ‖Qn+1x‖ and monotone if ‖Pnx‖ ≤ ‖Pn+1x‖ where Pnx = (I − Qn+1)x for everyx ∈ X and for every n ∈ �.

Definition 2.2. Let (X, ‖ · ‖) be a Banach space with a basis {en} and Γ = {γn} ⊂ � with 0 < γn <γn+1 and limnγn = 1. Let x =

∑∞i=1 xiei ∈ X. Then if

|‖x‖| = supnγn‖Qnx‖, (2.1)

|‖ · ‖| is a norm in X which we will call Γ-norm.Clearly

γ1‖x‖ ≤ |‖x‖| ≤(supn

‖Qn‖)‖x‖. (2.2)

Observe that, if {en} is a basis in (X, ‖ · ‖), then it is always premonotone in (X, |‖ · ‖|)and, if {en} is monotone in (X, ‖ · ‖), then it is also monotone in (X, |‖ · ‖|). Also observethat since for every x ∈ X we have that limnγn‖Qnx‖ = 0, there exists n0 such that |‖x‖| =γn0‖Qn0x‖.

Next we define the properties related to wfpp we are going to analyze. The definitionof GGLD is not the original one found in [3], but an equivalent one found in [4].

Definition 2.3. Let Y be a Banach space.

(1) Y has the Opial property if for every weakly null sequence {xn} ⊂ Y and for everyx ∈ Y , x/= 0,

lim supn

‖xn‖ < lim supn

‖xn − x‖. (2.3)

(2) If Y has a basis, it has the generalized Gossez-Lami Dozo property (GGLD) [4] if,for every weakly null normalized block basic sequence {yn}, we have that

limn

supi,j≥n

∥∥yi − yj

∥∥ > 1. (2.4)

(3) A bounded sequence {yn} ⊂ Y is called diametral if

limnd(yn+1, conv

{yi}ni=1

)= diam

{yn}∞n=1. (2.5)

Y has weak normal structure (WNS) if there is no weakly null diametral nonzerosequence in Y .

(4) The coefficient J(Y), related to uniform nonsquareness, since J(Y) < 2 if and onlyif Y is uniformly nonsquare, is given by

J(Y) = sup{min(∥∥x + y

∥∥,∥∥x − y∥∥) : x, y ∈ BY

}. (2.6)


(5) The coefficient R(Y) [5] is defined by

R(Y) = sup{lim inf

n‖xn + x‖ : {x}, {xn} ⊂ BY , xn −→

w0}. (2.7)

(6) Coefficients RW(a, Y) and MW(Y) [6] are defined as follows: for each a > 0

RW(a, Y) = sup{min(lim inf

n‖xn + x‖, lim inf

n‖xn − x‖

): ‖x‖ ≤ a, {xn} ⊂ BY , xn −→

w0},

MW(Y) = sup{

1 + aRW(a, Y)

: a > 0}.

(2.8)

It is known that GGLD ⇒ WNS ⇒ wfpp and the Opial property implies wfpp. Also

R(Y) < 2 =⇒ MW(Y) > 1 =⇒ wfpp, (2.9)

J(Y) < 2 =⇒ MW(Y) > 1 =⇒ wfpp. (2.10)

First we will show that the Opial property is inherited from (X, |‖ · ‖|) to (X, ‖ · ‖) andthat (X, ‖ · ‖) has GGLD if and only if (X, |‖ · ‖|) has GGLD. In order to achieve this, we needthe following result shown in [7].

Lemma 2.4. Let (X, ‖ · ‖) be a Banach space with a premonotone basis {en}. Then(1) if {xn} converges weakly to x, limn‖xn − x‖ = a if and only if limn|‖xn − x‖| = a,(2) if {xn} converges weakly to 0 and limnlimr‖xn − xr‖ = a, there exists a subsequence {yn}

of {xn} such that limnlimr |‖yn − yr‖| = a.

Lemma 2.5. Let (X, ‖·‖) be a Banach space with a premonotone basis {en}. If (X, |‖ ·‖|) has the Opialproperty, then (X, ‖ · ‖) also has the Opial property, but the converse is false.

Proof. Let {xn} be weakly null in X and x ∈ X, x /= 0. Then, by Lemma 2.4 and by (2.2),

lim supn

‖xn‖ = lim supn

|‖xn‖| < lim supn

|‖xn − x‖| ≤ lim supn

‖xn − x‖. (2.11)

It is known that, for 1 < p < ∞, (lp, ‖ · ‖) has the Opial property. Consider any Γ-norm

|‖ · ‖| in lp with the canonical basis {en}, and let δ > 0 be such that δ < ((γp2 − γp1 )/γp

1 )1/p

.Then, for n ≥ 2,

|‖δe1 + en‖| = max[γ1(δp + 1)1/p, γn

]= γn,

limn|‖en‖| = lim

nγn = lim

n|‖δe1 + en‖| = 1.

(2.12)

Thus, (lp, |‖ · ‖|) does not have the Opial property.


Lemma 2.6. Let (X, ‖·‖) be a Banach space with a premonotone basis {en}. Then, (X, ‖·‖) has GGLDif and only if (X, |‖ · ‖|) has GGLD.

Proof. Let {yn} be a weakly null normalized block basic sequence. By Lemma 2.4, limn‖yn‖exists if and only if limn|‖yn‖| exists and in this case limn‖yn‖ = limn|‖yn‖|. Also

limn

supi,j≥n

∥∥yi − yj∥∥ = lim

nsupi,j≥n

∣∣∥∥yi − yj∥∥∣∣. (2.13)

The above equality follows immediately from the following inequality, for i, j ≥ n :

∣∣∥∥yi − yj

∥∥∣∣ ≤ ∥∥yi − yj

∥∥ ≤ 1

γn

∣∣∥∥yi − yj

∥∥∣∣. (2.14)

This proves the lemma.

Now we will show that there exists a space with WNS such that with the Γ-norm itdoes not have WNS.

Lemma 2.7. Let X be the space c0 with the norm ‖x‖ = sup |bi| +∑∞

i=1 εi|bi|, where x =∑∞

i=1 biei,εi > 0 and

∑∞i=1 εi < ∞. Then X has WNS.

Proof. Let {xn} ⊂ X be a weakly null nonzero sequence. We may assume that x1 /= 0 and thatthere exists a block basic sequence {un} ⊂ Xwith ‖un−xn‖ →

n→∞0. Suppose that un =

∑qni=pn

aiei

with pn ≤ qn < pn+1 for n ∈ � and that x1 =∑∞

i=1 biei. Let k be such that∑k

i=1 εi|bi| = δ /= 0. Letε < δ/2, s > k with ‖Qsx1‖ < ε and n > s. Then,

‖x1 − un‖ + ε ≥ ‖Psx1 − un‖ = max(‖Psx1‖∞, ‖un‖∞) +s∑

i=1

εi|bi| +qn∑

i=pn

εi|ai| ≥ ‖un‖ + δ. (2.15)

Hence, lim supn‖x1 − xn‖ ≥ lim supn‖xn‖ + δ/2 and {xn} cannot be a diametral sequence,since for a diametral sequence {xn} it is true that limn‖x − xn‖ = diam{xn} for every x ∈conv{xn}.

Lemma 2.8. Let Γ = {γn} ⊂ (0, 1) be an increasing sequence with limnγn = 1. Then there is a space(X, ‖ · ‖) with WNS, such that X with the Γ-norm |‖ · ‖| does not have WNS.

Proof. Let {γnj}j be a subsequence of {γn} such that, if εnj = (1/3)(1/γnj −1), then∑∞

i=1 εni <∞.Observe that εnj < εnj+1 . Let X be the space c0 with the norm ‖x‖ = sup |ai| +

∑∞i=1 εni |ani | +∑∞

i=1,i /=nj (|ai|/2i), where x =∑∞

i=1 aiei. By Lemma 2.7, since∑∞

i=1(εni + 1/2i) < ∞, we knowthat X has WNS. Now let |‖ · ‖| be the Γ-norm in X with respect to {γi}. Let uj = enj ; thenγnj = 1/(1 + 3εnj ), and thus

∣∣∥∥uj∥∥∣∣ = γnj

(1 + εnj

)< 1, (2.16)


and, if j < m,

∣∣∥∥uj − um∥∥∣∣ = max

(γnj

(1 + εnj + εnm

), γnm(1 + εnm)

)

≤ max(γnj

(1 + 2εnj

), γnm(1 + εnm)

)< 1.

(2.17)

Therefore, since 0 ∈ conv{un} and limn|‖un‖| = 1, we have that diam|‖·‖|{un} = 1. Also, if0 ≤ λi,

∑n−1i=1 λi = 1,

|‖un‖| ≤∣∣∣∣∣

∥∥∥∥∥

n−1∑

i=1

λiui − un∥∥∥∥∥

∣∣∣∣∣≤ 1. (2.18)

Hence, {un} is diametral in (X, |‖ · ‖|).

The above example is another proof of the fact that for every ε > 0 there are Banachspaces X and Y with d(X, Y) < 1 + ε so that X has WNS but Y does not.

With regard to the coefficient MW(X), we will see that, if X has a premonotone basisand MW(X, ‖ · ‖) > 1, then MW(X, |‖ · ‖|) > 1 and we will show a sufficient condition for thereverse implication. For this we need the following lemma.

Lemma 2.9. Let X be a Banach space with a basis. If

RW1(a,X) = sup{min(lim inf

∥∥un + y∥∥, lim inf

∥∥un − y∥∥) : un −→

w0,

{un} ⊂ BX is a block basic sequence,

∥∥y∥∥ ≤ a and support of y is finite

},

(2.19)

then RW(a,X) = RW1(a,X).

Proof. It is clear that RW1(a,X) ≤ RW(a,X).Now let ε > 0, x ∈ X, ‖x‖ ≤ a, and {xn} ⊂ BX , with xn →

w0 such that min(limn‖xn +

x‖, limn‖xn − x‖) > RW(a,X) − ε. By passing to a subsequence, we may assume that thereexist a block basic sequence {un} ⊂ BX with ‖xn − un‖ < ε andm ∈ � such that ‖x − Pmx‖ < εand ‖Pmx‖ < (1 + ε)‖x‖ ≤ a(1 + ε). Then,

∥∥∥∥Pmx

1 + ε+ un

∥∥∥∥ ≥ ‖xn + x‖ − ‖xn − un‖ − ‖x − Pmx‖ − ‖Pmx‖ ε

1 + ε,

∥∥∥∥Pmx

1 + ε− un

∥∥∥∥ ≥ ‖xn − x‖ − ‖xn − un‖ − ‖x − Pmx‖ − ‖Pmx‖ ε

1 + ε.

(2.20)

Let y = Pmx/(1 + ε); then ‖y‖ ≤ a, and we conclude that RW1(a,X) ≥ RW(a,X) − (3 + a)ε,thus proving the assertion.


Similarly one can prove that, if X is a space with a basis,

R(Y) = sup{lim inf

n‖un + x‖ : {x}, {un} ⊂ BY , un −→

w0 ,

{un} is a block basic sequence and support of x is finite}.

(2.21)

It is known (see [6, 8]) that, if X, Y are Banach spaces, then

MW(X) ≤ MW(Y)d(X, Y), J(X) ≤ J(Y)d(X, Y). (2.22)

So, if X is a Banach space with a basis, Γ = {γn} with 0 ≤ γn ≤ γn+1 ≤ 1 and γ1 > 1/MW(X),and Y is X with the Γ-norm, then MW(Y) ≥ MW(X)/d(X, Y) ≥ γ1MW(X) > 1, and if γ1 >1/MW(Y), MW(X) > 1. Similarly, if γ1 > J(X)/2, then J(Y) < 2 and, if γ1 > J(Y)/2, thenJ(X) < 2. But the next proposition shows that in fact MW(X) > 1 always implies MW(Y) > 1.For the coefficient J , in general neither J(X) < 2 implies J(Y) < 2 nor the other way round, aswe will see in Examples 2.16 and 2.17.

Proposition 2.10. Suppose that X is a Banach space with a premonotone basis {en}. Then MW =MW(X, ‖ · ‖) > 1 implies that MW1 = MW(X, |‖ · ‖|) > 1.

Proof. Let RW(a, (X, ‖·‖)) = R(a) and RW(a, (X, |‖·‖|)) = R1(a). Suppose that MW1 = 1. Then,R1(a) = 1 + a for every a > 0. Let a > 0 and 0 < ε < a, y ∈ X with finite support, |‖y‖| ≤ a, andlet {un} be aweakly null block basic sequence with |‖un‖| ≤ 1 such that limn|‖un+y‖| > 1+a−εand limn|‖un − y‖| > 1 + a − ε. Then we may suppose that for every n, |‖un + y‖| > 1 + a − εand |‖un − y‖| > 1 + a − ε. Hence,

|‖un‖| ≥ 1 − ε, ∣∣∥∥y∥∥∣∣ ≥ a − ε. (2.23)

We may also assume that the supports of y and un are disjoint. Let un =∑rn

i=lnaiei and y =

∑ri=1 biei.

Suppose that |‖un +y‖| = γmn‖∑rn

i=mn(ai + bi)ei‖ for somemn ≤ rn. It is not possible that

mn > r, because this would mean that

1 + a − ε ≤ ∣∣∥∥un + y∥∥∣∣ = γmn

∥∥∥∥∥

rn∑

mn

aiei

∥∥∥∥∥≤ |‖un‖| ≤ 1. (2.24)

So, by passing to a subsequence if necessary, we may assume that for every n we have thatmn = i0 ≤ r. Thus,

1 + a − ε ≤ ∣∣∥∥un + y∥∥∣∣ = γi0

∥∥∥∥∥

rn∑

i=i0

(ai + bi)ei

∥∥∥∥∥

≤ γi0(∥∥∥∥∥

rn∑

i=ln

aiei

∥∥∥∥∥+

∥∥∥∥∥

r∑

i=i0

biei

∥∥∥∥∥

)

≤ γi0γln

+ a.

(2.25)


Since limnγln = 1, by passing to the limit, we obtain that

γi0 ≥ 1 − ε. (2.26)

Similarly, there exists γi1 ≥ 1 − ε with

1 + a − ε ≤ ∣∣∥∥un − y∥∥∣∣ ≤ γi1

γln+ a. (2.27)

Suppose that i1 ≥ i0, and let y0 =∑r

i=i0 biei and y1 =∑r

i=i1 biei. Then, since the basis ispremonotone, 1 + a − ε ≤ |‖un + y‖| = |‖un + y0‖| ≤ ‖un + y0‖ and 1 + a − ε ≤ |‖un − y‖| =|‖un − y1‖| ≤ ‖un − y1‖ ≤ ‖un − y0‖; thus,

a − ε ≤ ∣∣∥∥y0∥∥∣∣ ≤ ∣∣∥∥y∥∥∣∣ ≤ a. (2.28)

Therefore,

a − ε ≤ ∣∣∥∥y0∥∥∣∣ ≤ ∥∥y0

∥∥ ≤ 1

γi0

∣∣∥∥y0∥∥∣∣ ≤ a

γi0≤ a

1 − ε (2.29)

and |‖y0‖ − a| ≤ max{ε, aε/(1 − ε)}. Further, since γi0 ≤ γln ,

1 − ε ≤ |‖un‖| ≤ ‖un‖ ≤ 1γln

|‖un‖| ≤ 1γln

≤ 11 − ε (2.30)

and |‖un‖ − 1| ≤ ε/(1 − ε).Hence,

∥∥∥∥∥un‖un‖ +

y0∥∥y0∥∥a

∥∥∥∥∥≥ ∥∥un + y0

∥∥ − |‖un‖ − 1| − ∣∣∥∥y0∥∥ − a∣∣

≥ 1 + a − ε − ε

1 − ε −max{ε,

aε

1 − ε}.

(2.31)

Similarly,

∥∥∥∥∥un‖un‖ − y0∥

∥y0∥∥a

∥∥∥∥∥≥ 1 + a − ε − ε

1 − ε −max{ε,

aε

1 − ε}. (2.32)

We deduce that min(lim infn‖(un/‖un‖) + (y0/‖y0‖)a‖, lim infn‖(un/‖un‖) − (y0/‖y0‖)a‖) ≥1 +a− ε − ε/(1 − ε)−max{ε, aε/(1 − ε)}, and letting ε tend to zero we obtain R(a) = 1+ a andMW = 1.

Examples 2.14 and 2.17 exhibit spaces in which MW(X, ‖ · ‖) = 1 andMW(X, |‖ · ‖|) > 1.There is however a special case in which MW(X, ‖ · ‖) ≥ MW(X, |‖ · ‖|).


Recall that a basis {en} of a Banach space X is 1-spreading if, whenever x =∑∞

i=1 aiei ∈X and {eni}i is a subsequence of {en},

∑∞i=1 aieni ∈ X and ‖∑∞

i=1 aiei‖ = ‖∑∞i=1 aieni‖.

Proposition 2.11. IfX is a Banach space with a premonotone 1-spreading basis, thenMW(X, ‖·‖) ≥MW(X, |‖ · ‖|).

Proof. Let Tm : X → X be the translation given by Tm∑∞

i=1 aiei =∑∞

i=1 aiei+m, and let a >0,y ∈ X, ‖y‖ ≤ a, {un} ⊂ BX , where y has finite support and {un} is a weakly null block basicsequence such that min(‖y + un‖, ‖y − un‖) > RW(a, (X, ‖ · ‖)) − ε for every n. Let m ∈ �;then there exists N ∈ � such that for n > N the supports of Tmy and un are disjoint, andthus, since the basis is 1-spreading, ‖y − un‖ = ‖Tmy − un‖ and ‖y + un‖ = ‖Tmy + un‖. Then,|‖Tmy‖| ≤ ‖Tmy‖ ≤ a, |‖un‖| ≤ ‖un‖ ≤ 1 and for n > N

∣∣∥∥Tmy − un∥∥∣∣ ≥ γm

∥∥Tmy − un∥∥ = γm

∥∥y − un∥∥,

∣∣∥∥Tmy + un

∥∥∣∣ ≥ γm

∥∥Tmy + un

∥∥ = γm

∥∥y + un

∥∥.

(2.33)

Hence, RW(a, (X, |‖ · ‖|)) ≥ γmRW(a, (X, ‖ · ‖)), and, by passing to the limit as m tends toinfinity, we get RW(a, (X, |‖ · ‖|)) ≥ RW(a, (X, ‖ · ‖)) and thus the desired result.

Similarly to Propositions 2.10 and 2.11 we can prove the following.

Proposition 2.12. Suppose that X is a Banach space with a premonotone basis {en}. Then R =R(X, ‖ · ‖) < 2 implies that R1 = R(X, |‖ · ‖|) < 2 and, if the basis is premonotone and 1-spreading,then R = R(X, ‖ · ‖) ≤ R(X, |‖ · ‖|) = R1.

Corollary 2.13. If X is a Banach space with a premonotone 1-spreading basis, thenMW1 > 1 if andonly ifMW > 1; also R1 < 2 if and only if R < 2.

Next we will show an example of a space without a 1-spreading basis, such that R = 2,MW = 1 but R1 < 2 and thus MW1 > 1.

Example 2.14. Let X be c0 with the following norm:

‖{an}‖ = |a1| +maxi≥2

|ai|. (2.34)

Let {en} denote the canonical basis of c0. Then for every a > 0, ‖ae1 + en‖ = ‖ae1 − en‖ = 1 + a,and thus R = 2 and MW = 1. On the other hand let x ∈ X with finite support and |‖x‖| ≤ 1and suppose that {un} is a block basic sequence with |‖un‖| ≤ 1 for n ∈ N, un =

∑rni=mn

aieiwithmn ≤ rn < mn+1 and the support of x does not intersect the support of un. Then, for everym ≥ 2 and n ≥ m, γm‖Qmx + un‖ = max(γm‖Qmx‖c0 , γm‖un‖c0) ≤ 1,

γ1‖x + un‖ ≤ 1 + γ1‖un‖c0 ≤ 1 +γ1γmn

. (2.35)

Thus, lim inf |‖x + un‖| ≤ 1 + γ1. Hence R1 ≤ 1 + γ1 < 2 and, by (2.9), MW1 > 1.

With regard to the coefficient J we have the following which is proved similarly toProposition 2.11.


Proposition 2.15. Suppose that X is a Banach space with a premonotone 1-spreading basis{en}. Then J(X, ‖ · ‖) ≤ J(X, |‖ · ‖|).

In general neither J(X, ‖ · ‖) < 2 implies J(X, |‖ · ‖|) < 2 nor the other way round, as thefollowing examples show.

Example 2.16. Let 1 > μ ≥ 1/√2 and X = �

2μ

⊕2l2, where �2

μ = (�2 , ‖ · ‖(μ)) and ‖(x1, x2)‖(μ) =max(|x1|, |x2|, μ(|x1| + |x2|)). Then, if μ ≤ γ2/(γ1 + γ2), J(X) < 2 but for every Γ, J(X, |‖ · ‖|) = 2.

Since μ ≥ 1/√2, it is easy to see that for x = (x1, x2) ∈ �2 ,

μ√1 − 2μ + 2μ2

‖x‖2 ≤ ‖x‖(μ) ≤ μ√2‖x‖2. (2.36)

Thus, d(l2, X) ≤ d(�2 ,�2μ) =

√2√1 − 2μ + 2μ2 <

√2, and by (2.22), since J(l2) =

√2, we obtain

that J(X) < 2.Now let Γ = {γn}, μ ≤ γ2/(γ1 + γ2), x = (1/γ1, 1/γ2, 0, 0, . . .), and y =

(−1/γ1, 1/γ2, 0, 0, . . .). Then |‖x‖| = |‖y‖| = 1 but |‖x + y‖| = |‖x − y‖| = 2.

This last example is another proof of the known fact that for every ε > 0 there areBanach spacesX and Y with d(X, Y) < 1 + ε, J(X) < 2 but J(Y) = 2. In the following examplewe exhibit a space X with J(X) = 2 such that J(X, |‖ · ‖|) < 2.

Example 2.17. Let X = (l2, ‖ · ‖), where, for x = (an) ∈ l2, ‖x‖ = |a1| + (∑∞

i=2 a2i )

1/2. ThenJ(X) = 2,MW(X) = 1, and, if Γ is such that γ2 > 1/

√2 and γ1/γ2 < 1/

√2, J(X, |‖ · ‖|) < 2 and

thus MW(X, |‖ · ‖|) > 1.Obviously if {en} is the canonical basis of l2, and a > 0, then ‖ae1 + en‖ = ‖ae1 − en‖ =

1 + a for n > 1 and thus J(X) = 2 and MW(X) = 1.Suppose now that J(Y) = J(X, |‖ · ‖|) = 2. Then there exist sequences {xn},{yn} ⊂ BY

such that limn|‖xn + yn‖| = limn|‖xn − yn‖| = 2 and {mn}, {ln} ⊂ � so that

∣∣∥∥xn + yn∥∥∣∣ = γmn

∥∥Qmn

(xn + yn

)∥∥,∣∣∥∥xn − yn

∥∥∣∣ = γln∥∥Qln

(xn − yn

)∥∥.(2.37)

By passing to a subsequence, wemay assume that, for every n ∈ �, e∗1(xn) ≥ 0, e∗1(yn) ≥0 and e∗1(xn) ≥ e∗1(yn), and that the subsequence satisfies one of the next three cases:

(1) mn > 1 and ln > 1 for every n,

(2) mn = ln = 1 for every n,

(3) mn = 1 and ln > 1 for every n.

Observe that

γm∥∥Qm

(x + y

)∥∥ ≥ 2 − ε implies γm‖Qm(x)‖ ≥ 1 − ε and γm∥∥Qm

(y)∥∥ ≥ 1 − ε. (2.38)


Case 1. Let Z = (l2, |‖ · ‖|Γ1), where |‖ · ‖|Γ1 is the Γ1 = {γi}∞i=2-norm. Then

|‖Q2xn‖|Γ1 ≤ 1,∣∣∥∥Q2yn

∥∥∣∣Γ1

≤ 1,∣∣∥∥Q2

(xn + yn

)∥∥∣∣Γ1

≥ 2 − ε, ∣∣∥∥Q2(xn − yn

)∥∥∣∣Γ1

≥ 2 − ε.(2.39)

Since 1/γ2 <√2, then, by (2.2), d(Z, l2) <

√2 and, by (2.22), since J(l2) =

√2, J(Z) < 2 and

this is a contradiction.

Case 2. Let ε > 0. Suppose that x =∑∞

i=1 aiei, y =∑∞

i=1 biei ∈ BY , a1 > 0, b1 > 0, a1 > b1 and

γ1

( ∞∑

i=2(ai + bi)2

)1/2

≥ 2 − ε − γ1(a1 + b1),

γ1

( ∞∑

i=2(ai − bi)2

)1/2

≥ 2 − ε − γ1(a1 − b1).

(2.40)

Squaring both inequalities and adding them, since γ2(∑∞

i=2 a2i )

1/2 ≤ |‖x‖|, we get

2γ21γ22

≥ γ21∞∑

i=2

(a2i + b

2i

)≥ (2 − ε − γ1a1

)2 + γ21b21 ≥(2 − ε − γ1a1

)2. (2.41)

By passing to the limit as ε → 0, since γ1a1 ≤ 1, we obtain that√2(γ1/γ2) ≥ 2 − γ1a1 ≥ 1, and

this contradicts γ1/γ2 < 1/√2.

Case 3. Let 2 > ε > 0. Suppose that x =∑∞

i=1 aiei, y =∑∞

i=1 biei ∈ BY , a1 > 0, b1 ≥ 0, a1 > b1 and

|‖x + y‖| = γ1(|a1 + b1| + (∑∞

i=2 (ai + bi)2)

1/2) ≥ 2 − ε and |‖x − y‖| = γm(

∑∞i=m (ai − bi)2)1/2 ≥

2 − ε. Then, by (2.38),

γm

( ∞∑

i=m

a2i

)1/2

≥ 1 − ε. (2.42)


Since in l2 if u, v ∈ Bl2 , one has that ‖u − v‖ ≥ δ implies ‖u + v‖ ≤ 2√1 − (δ/2)2, then

γm(∑∞

i=m (ai + bi)2)1/2 ≤

√4ε − ε2. Also

2 − ε ≤ γ1⎛

⎝|a1 + b1| +(

m−1∑

i=2

(ai + bi)2)1/2

+

( ∞∑

i=m

(ai + bi)2)1/2

⎞

⎠

≤ γ1|a1 + b1| + γ1(

m−1∑

i=2

(ai + bi)2)1/2

+γ1γm

√4ε − ε2

≤ γ1|a1 + b1| + γ1(

m−1∑

i=2(ai + bi)2

)1/2

+√4ε − ε2.

(2.43)

Let φ = ε +√4ε − ε2; then

2 − φ ≤ γ1|a1 + b1| + γ1(

m−1∑

i=2

(ai + bi)2)1/2

. (2.44)

By (2.38), γ1|a1| + γ1(∑m−1

i=2 a2i )1/2 ≥ 1 − φ, and since, for A,B > 0,

(A + B)1/2 −A1/2 =B

(A + B)1/2 +A1/2, (2.45)

we have, using (2.42), that

1 ≥ γ1|a1| + γ1( ∞∑

i=2

a2i

)1/2

≥ γ1|a1| + γ1(

m−1∑

i=2

a2i +(1 − ε)2γ2m

)1/2

= γ1|a1| + γ1(

m−1∑

i=2

a2i

)1/2

+γ1(1 − ε)2

γ2m

1

(A + B)1/2 +A1/2

≥ 1 − φ +γ1(1 − ε)2

γ2m

1

(A + B)1/2 +A1/2,

(2.46)

where A = (∑m−1

i=2 a2i )1/2

and B = (1 − ε)2/γ2m. But

(A + B)1/2 +A1/2 ≤(

1γ22

+(1 − ε)2γ2m

)1/2

+1γ2

≤ 3γ2; (2.47)


therefore,

γ1 <γ1

γ2m≤ φ

(1 − ε)23γ2, (2.48)

and, taking the limit as ε → 0, we get γ1 = 0 which is a contradiction.Hence, J(X, |‖ · ‖|) < 2, and, by (2.10), we have that MW(X, |‖ · ‖|) > 1.

Acknowledgments

The authors thank the referees for their detailed review and valuable observations. This workwas partially funded by Grant SEP CONACYT 102380.

References

[1] P. N. Dowling, C. J. Lennard, and B. Turett, “Renormings of l1 and c0 and fixed point properties,”in Handbook of Metric Fixed Point Theory, pp. 269–297, Kluwer Academic Publishers, Dordrecht, TheNetherlands, 2001.

[2] P.-K. Lin, “There is an equivalent norm on l1 that has the fixed point property,” Nonlinear Analysis:Theory, Methods & Applications, vol. 68, no. 8, pp. 2303–2308, 2008.

[3] A. Jimenez-Melado, “Stability of weak normal structure in James quasi reflexive space,” Bulletin of theAustralian Mathematical Society, vol. 46, no. 3, pp. 367–372, 1992.

[4] H. Fetter and B. Gamboa de Buen, “Geometric properties related to the fixed point property in Banachspaces,” Revista de la Real Academia de Ciencias Exactas, Fısicas y Naturales, vol. 94, no. 4, pp. 431–436,2000.

[5] J. Garcıa Falset, “The fixed point property in Banach spaces with the NUS-property,” Journal ofMathematical Analysis and Applications, vol. 215, no. 2, pp. 532–542, 1997.

[6] J. Garcıa Falset, E. Llorens-Fuster, and E. M.Mazcunan-Navarro, “Uniformly nonsquare Banach spaceshave the fixed point property for nonexpansive mappings,” Journal of Functional Analysis, vol. 233, no.2, pp. 494–514, 2006.

[7] H. Fetter and B. Gamboa de Buen, “Banach spaces with a basis that are hereditarily asymptoticallyisometric to l1 and the fixed point property,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71,no. 10, pp. 4598–4608, 2009.

[8] M. Kato, L. Maligranda, and Y. Takahashi, “On James and Jordan-von Neumann constants and thenormal structure coefficient of Banach spaces,” Studia Mathematica, vol. 144, no. 3, pp. 275–295, 2001.


Research ArticleA Suzuki Type Fixed-Point Theorem

Ishak Altun and Ali Erduran

Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan,71450 Kirikkale, Turkey

Correspondence should be addressed to Ishak Altun, [email protected]

Received 16 December 2010; Accepted 7 February 2011

Academic Editor: Genaro Lopez

Copyright q 2011 I. Altun and A. Erduran. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We present a fixed-point theorem for a single-valuedmap in a complete metric space using implicitrelation, which is a generalization of several previously stated results including that of Suziki(2008).

1. Introduction

There are a lot of generalizations of Banach fixed-point principle in the literature. See [1–5]. One of the most interesting generalizations is that given by Suzuki [6]. This interestingfixed-point result is as follows.

Theorem 1.1. Let (X, d) be a complete metric space, and let T be a mapping on X. Define a non-increasing function θ from [0, 1) into (1/2, 1] by

θ(r) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, 0 ≤ r ≤√5 − 12

,

1 − rr2

,

√5 − 12

≤ r ≤ 1√2,

11 + r

,1√2≤ r < 1.

(1.1)

Assume that there exists r ∈ [0, 1), such that

θ(r)d(x, Tx) ≤ d(x, y) implies d(Tx, Ty

) ≤ rd(x, y), (1.2)

for all x, y ∈ X, then there exists a unique fixed-point z of T . Moreover, limnTnx = z for all x ∈ X.


Like other generalizations mentioned above in this paper, the Banach contractionprinciple does not characterize the metric completeness of X. However, Theorem 1.1 doescharacterize the metric completeness as follows.

Theorem 1.2. Define a nonincreasing function θ as in Theorem 1.1, then for a metric space (X, d)the following are equivalent:

(i) X is complete,

(ii) Every mapping T on X satisfying (1.2) has a fixed point.

In addition to the above results, Kikkawa and Suzuki [7] provide a Kannan typeversion of the theorems mentioned before. In [8], it is provided a Chatterjea type version.Popescu [9] gives a Ciric type version. Recently, Kikkawa and Suzuki also providemultivalued versions which can be found in [10, 11]. Some fixed-point theorems related toTheorems 1.1 and 1.2 have also been proven in [12, 13].

The aim of this paper is to generalize the above results using the implicit relationtechnique in such a way that

F(d(Tx, Ty

), d(x, y), d(x, Tx), d

(y, Ty

), d(x, Ty

), d(y, Tx

)) ≤ 0, (1.3)

for x, y ∈ X, where F : [0,∞)6 → R is a function as given in Section 2.

2. Implicit Relation

Implicit relations on metric spaces have been used in many papers. See [1, 14–16].Let R+ denote the nonnegative real numbers, and let Ψ be the set of all continuous

functions F : [0,∞)6 → R satisfying the following conditions:

F1: F(t1, . . . , t6) is nonincreasing in variables t2, . . . , t6,

F2: there exists r ∈ [0, 1), such that

F(u, v, v, u, u + v, 0) ≤ 0 (2.1)

or

F(u, v, 0, u + v, u, v) ≤ 0 (2.2)

or

F(u, v, v, v, v, v) ≤ 0 (2.3)

implies u ≤ rv,F3: F(u, 0, 0, u, u, 0) > 0, for all u > 0.

Example 2.1. F(t1, . . . , t6) = t1 − rt2, where r ∈ [0, 1). It is clear that F ∈ Ψ.


Example 2.2. F(t1, . . . , t6) = t1 − α[t3 + t4], where α ∈ [0, 1/2).Let F(u, v, v, u, u+v, 0) = u−α[u+v] ≤ 0, then we have u ≤ (α/(1−α))v. Similarly, let

F(u, v, 0, u + v, u, v) ≤ 0, then we have u ≤ (α/(1 − α))v. Again, let F(u, v, v, v, v, v) ≤ 0, thenu ≤ 2αv. Since α/(1−α) ≤ 2α < 1, F2 is satisfied with r = 2α. Also F(u, 0, 0, u, u, 0) = (1−α)u >0, for all u > 0. Therefore, F ∈ Ψ.

Example 2.3. F(t1, . . . , t6) = t1 − αmax{t3, t4}, where α ∈ [0, 1/2).Let F(u, v, v, u, u + v, 0) = u − αmax{u, v} ≤ 0, then we have u ≤ αv ≤ (α/(1 −

α))v. Similarly, let F(u, v, 0, u + v, u, v) ≤ 0, then we have u ≤ (α/(1 − α))v. Again, letF(u, v, v, v, v, v) ≤ 0, then u ≤ αv ≤ (α/(1 − α))v. Thus, F2 is satisfied with r = α/(1 − α).Also F(u, 0, 0, u, u, 0) = (1 − α)u > 0, for all u > 0. Therefore, F ∈ Ψ.

Example 2.4. F(t1, . . . , t6) = t1 − α[t5 + t6], where α ∈ [0, 1/2).Let F(u, v, v, u, u+v, 0) = u−α[u+v] ≤ 0, then we have u ≤ (α/(1−α))v. Similarly, let

F(u, v, 0, u + v, u, v) ≤ 0, then we have u ≤ (α/(1 − α))v. Again, let F(u, v, v, v, v, v) ≤ 0, thenu ≤ 2αv. Since α/(1−α) ≤ 2α < 1, F2 is satisfied with r = 2α. Also F(u, 0, 0, u, u, 0) = (1−α)u >0, for all u > 0. Therefore, F ∈ Ψ.

Example 2.5. F(t1, . . . , t6) = t1 − at3 − bt4, where a, b ∈ [0, 1/2).Let F(u, v, v, u, u + v, 0) = u − av − bu ≤ 0, then we have u ≤ (a/(1 − b))v. Similarly,

let F(u, v, 0, u + v, u, v) ≤ 0, then we have u ≤ (b/(1 − b))v. Again, let F(u, v, v, v, v, v) ≤0, then u ≤ (a + b)v. Thus, F2 is satisfied with r = max{a/(1 − b), b/(1 − b), a + b}. AlsoF(u, 0, 0, u, u, 0) = (1 − b)u > 0, for all u > 0. Therefore, F ∈ Ψ.

3. Main Result

Theorem 3.1. Let (X, d) be a complete metric space, and let T be a mapping on X. Define anonincreasing function θ from [0, 1) into (1/2, 1] as in Theorem 1.1. Assume that there exists F ∈ Ψ,such that θ(r)d(x, Tx) ≤ d(x, y) implies

F(d(Tx, Ty

), d(x, y), d(x, Tx), d

(y, Ty

), d(x, Ty

), d(y, Tx

)) ≤ 0, (3.1)

for all x, y ∈ X, then T has a unique fixed-point z and limnTnx = z holds for every x ∈ X.

Proof. Since θ(r) ≤ 1, θ(r)d(x, Tx) ≤ d(x, Tx) holds for every x ∈ X, by hypotheses, we have

F(d(Tx, T2x

), d(x, Tx), d(x, Tx), d

(Tx, T2x

), d(x, T2x

), 0)≤ 0, (3.2)

and so from (F1),

F(d(Tx, T2x

), d(x, Tx), d(x, Tx), d

(Tx, T2x

), d(x, Tx) + d

(Tx, T2x

), 0)≤ 0. (3.3)

By (F2), we have

d(Tx, T2x

)≤ rd(x, Tx), (3.4)


for all x ∈ X. Now fix u ∈ X and define a sequence {un} in X by un = Tnu. Then from (3.4),we have

d(un, un+1) = d(Tun−1, T2un−1

)≤ rd(un−1, Tun−1) ≤ · · · ≤ rnd(u, Tu). (3.5)

This shows that∑∞

n=1 d(un, un+1) < ∞, that is, {un} is Cauchy sequence. Since X is complete,{un} converges to some point z ∈ X. Now, we show that

d(Tx, z) ≤ rd(x, z) ∀x ∈ X \ {z}. (3.6)

For x ∈ X \ {z}, there exists n0 ∈ N, such that d(un, z) ≤ d(x, z)/3 for all n ≥ n0. Then, wehave

θ(r)d(un, Tun) ≤ d(un, Tun) = d(un, un+1)≤ d(un, z) + d(z, un+1)

≤ 23d(x, z) = d(x, z) − d(x, z)

3

≤ d(x, z) − d(un, z) ≤ d(un, x).

(3.7)

Hence, by hypotheses, we have

F(d(Tun, Tx), d(un, x), d(un, Tun), d(x, Tx), d(un, Tx), d(x, Tun)) ≤ 0, (3.8)

and so

F(d(un+1, Tx), d(un, x), d(un, un+1), d(x, Tx), d(un, Tx), d(x, un+1)) ≤ 0. (3.9)

Letting n → ∞, we have

F(d(z, Tx), d(z, x), 0, d(x, Tx), d(z, Tx), d(x, z)) ≤ 0, (3.10)

and so

F(d(z, Tx), d(z, x), 0, d(x, z) + d(z, Tx), d(z, Tx), d(x, z)) ≤ 0. (3.11)

By (F2), we have

d(z, Tx) ≤ rd(x, z), (3.12)

and this shows that (3.6) is true.


Now, we assume that Tmz/= z for allm ∈ N, then from (3.6), we have

d(Tm+1z, z

)≤ rmd(Tz, z), (3.13)

for allm ∈ N.

Case 1. Let 0 ≤ r ≤ (√5 − 1)/2. In this case, θ(r) = 1. Now, we show by induction that

d(Tnz, Tz) ≤ rd(z, Tz), (3.14)

for n ≥ 2. From (3.4), (3.14) holds for n = 2. Assume that (3.14) holds for some n with n ≥ 2.Since

d(z, Tz) ≤ d(z, Tnz) + d(Tnz, Tz)≤ d(z, Tnz) + rd(z, Tz),

(3.15)

we have

d(z, Tz) ≤ 11 − r d(z, T

nz), (3.16)

and so

θ(r)d(Tnz, Tn+1z

)= d(Tnz, Tn+1z

)≤ rnd(z, Tz)

≤ rn

1 − r d(z, Tnz) ≤ r2

1 − r d(z, Tnz)

≤ d(z, Tnz).

(3.17)

Therefore, by hypotheses, we have

F(d(Tn+1z, Tz

), d(Tnz, z), d

(Tnz, Tn+1z

), d(z, Tz), d(Tnz, Tz), d

(z, Tn+1z

))≤ 0, (3.18)

and so

F(d(Tn+1z, Tz

), rn−1d(Tz, z), rnd(z, Tz), d(z, Tz), rd(z, Tz), rnd(z, Tz)

)≤ 0, (3.19)

then

F(d(Tn+1z, Tz

), d(Tz, z), d(z, Tz), d(z, Tz), d(z, Tz), d(z, Tz)

)≤ 0, (3.20)


and by (F2), we have

d(Tn+1z, Tz

)≤ rd(Tz, z). (3.21)

Therefore, (3.14) holds.

Now, from (3.6), we have

d(Tn+1z, z

)≤ rd(Tnz, z) ≤ rnd(Tz, z). (3.22)

This shows that Tnz → z, which contradicts (3.14).

Case 2. Let (√5− 1)/2 ≤ r ≤ √

2/2. In this case, θ(r) = (1− r)/r2. Again we want to show that(3.14) is true for n ≥ 2. From (3.4), (3.14) holds for n = 2. Assume that (3.14) holds for somenwith n ≥ 2. Since

d(z, Tz) ≤ d(z, Tnz) + d(Tnz, Tz)≤ d(z, Tnz) + rd(z, Tz),

(3.23)

we have

d(z, Tz) ≤ 11 − r d(z, T

nz), (3.24)

and so

θ(r)d(Tnz, Tn+1z

)=

1 − rr2

d(Tnz, Tn+1z

)≤ 1 − r

rnd(Tnz, Tn+1z

)

≤ (1 − r)d(z, Tz) ≤ d(z, Tnz).(3.25)

Therefore, as in the previous case, we can prove that (3.14) is true for n ≥ 2. Again from (3.6),we have

d(Tn+1z, z

)≤ rd(Tnz, z) ≤ rnd(Tz, z). (3.26)

This shows that Tnz → z, which contradicts (3.14).

Case 3. Let√2/2 ≤ r < 1. In this case, θ(r) = 1/(1 + r). Note that for x, y ∈ X, either

θ(r)d(x, Tx) ≤ d(x, y) (3.27)

or

θ(r)d(Tx, T2x

)≤ d(Tx, y) (3.28)


holds. Indeed, if

θ(r)d(x, Tx) > d(x, y),

θ(r)d(Tx, T2x

)> d(Tx, y

),

(3.29)

then we have

d(x, Tx) ≤ d(x, y) + d(Tx, y) < θ(r)[d(x, Tx) + d

(Tx, T2x

)]

≤ θ(r)[d(x, Tx) + rd(x, Tx)] = d(x, Tx),(3.30)

which is a contradiction. Therefore, either

θ(r)d(u2n, Tu2n) ≤ d(u2n, z) (3.31)

or

θ(r)d(u2n+1, Tu2n+1) ≤ d(u2n+1, z) (3.32)

holds for every n ∈ N. If

θ(r)d(u2n, Tu2n) ≤ d(u2n, z) (3.33)

holds, then by hypotheses we have

F(d(Tu2n, Tz), d(u2n, z), d(u2n, Tu2n), d(z, Tz), d(u2n, Tz), d(z, Tu2n)) ≤ 0, (3.34)

and so

F(d(u2n+1, Tz), d(u2n, z), d(u2n, u2n+1), d(z, Tz), d(u2n, Tz), d(z, u2n+1)) ≤ 0. (3.35)


F(d(z, Tz), 0, 0, d(z, Tz), d(z, Tz), 0) ≤ 0, (3.36)

which contradicts (F3). If

θ(r)d(u2n+1, Tu2n+1) ≤ d(u2n+1, z) (3.37)

holds, then by hypotheses we have

F(d(Tu2n+1, Tz), d(u2n+1, z), d(u2n+1, Tu2n+1), d(z, Tz), d(u2n+1, Tz), d(z, Tu2n+1))≤0,(3.38)


and so

F(d(u2n+2, Tz), d(u2n+1, z), d(u2n+1, u2n+2), d(z, Tz), d(u2n+1, Tz), d(z, u2n+2)) ≤ 0. (3.39)


F(d(z, Tz), 0, 0, d(z, Tz), d(z, Tz), 0) ≤ 0, (3.40)

which contradicts (F3).

Therefore, in all the cases, there existsm ∈ N, such that Tmz = z. Since {Tnz} is Cauchysequence, we obtain Tz = z. That is, z is a fixed point of T . The uniqueness of fixed pointfollows easily from (3.6).

Remark 3.2. If we combine Theorem 3.1 with Examples 2.1, 2.2, 2.3, and 2.4, we have Theorem2 of [6], Theorem 2.2 of [7], Theorem 3.1 of [7], and Theorem 4 of [8], respectively.

Using Example 2.5, we obtain the following result.

Corollary 3.3. Let (X, d) be a complete metric space, and let T be a mapping on X. Define anonincreasing function θ from [0, 1) into (1/2, 1] as in Theorem 1.1. Assume that

θ(r)d(x, Tx) ≤ d(x, y) (3.41)

implies

d(Tx, Ty

) ≤ ad(x, Tx) + bd(y, Ty), (3.42)

for all x, y ∈ X, where a, b ∈ [0, 1/2), then there exists a unique fixed point of T .

Remark 3.4. We obtain some new results, if we combine Theorem 3.1 with some examplesof F.

References

[1] A. Aliouche and V. Popa, “General common fixed point theorems for occasionally weakly compatiblehybrid mappings and applications,” Novi Sad Journal of Mathematics, vol. 39, no. 1, pp. 89–109, 2009.

[2] S. K. Chatterjea, “Fixed-point theorems,” Comptes Rendus de l’Academie Bulgare des Sciences, vol. 25, pp.727–730, 1972.

[3] Lj. B. Ciric, “Generalized contractions and fixed-point theorems,” Publications de l’InstitutMathematique, vol. 12(26), pp. 19–26, 1971.

[4] R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp.71–76, 1968.

[5] T. Suzuki and M. Kikkawa, “Some remarks on a recent generalization of the Banach contractionprinciple,” in Fixed Point Theory and Its Applications, pp. 151–161, Yokohama Publ., Yokohama, Japan,2008.

[6] T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,”Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.


[7] M. Kikkawa and T. Suzuki, “Some similarity between contractions and Kannan mappings,” FixedPoint Theory and Applications, vol. 2008, Article ID 649749, 8 pages, 2008.

[8] O. Popescu, “Fixed point theorem in metric spaces,” Bulletin of the Transilvania University of Brasov,vol. 1(50), pp. 479–482, 2008.

[9] O. Popescu, “Two fixed point theorems for generalized contractions with constants in completemetricspace,” Central European Journal of Mathematics, vol. 7, no. 3, pp. 529–538, 2009.

[10] M. Kikkawa and T. Suzuki, “Some notes on fixed point theorems with constants,” Bulletin of theKyushu Institute of Technology. Pure and Applied Mathematics, no. 56, pp. 11–18, 2009.

[11] M. Kikkawa and T. Suzuki, “Three fixed point theorems for generalized contractions with constantsin complete metric spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2942–2949, 2008.

[12] Y. Enjouji, M. Nakanishi, and T. Suzuki, “A generalization of Kannan’s fixed point theorem,” FixedPoint Theory and Applications, vol. 2009, Article ID 192872, 10 pages, 2009.

[13] M.Kikkawa and T. Suzuki, “Some similarity between contractions andKannanmappings. II,” Bulletinof the Kyushu Institute of Technology. Pure and Applied Mathematics, no. 55, pp. 1–13, 2008.

[14] I. Altun and D. Turkoglu, “Some fixed point theorems for weakly compatible mappings satisfying animplicit relation,” Taiwanese Journal of Mathematics, vol. 13, no. 4, pp. 1291–1304, 2009.

[15] M. Imdad and J. Ali, “Common fixed point theorems in symmetric spaces employing a new implicitfunction and common property (E.A),” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol.16, no. 3, pp. 421–433, 2009.

[16] V. Popa, M. Imdad, and J. Ali, “Using implicit relations to prove unified fixed point theorems inmetricand 2-metric spaces,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 33, no. 1, pp. 105–120,2010.


Research ArticleFixed-Point Theory on a Frechet TopologicalVector Space

Afif Ben Amar,1 Mohamed Amine Cherif,2 and Maher Mnif2

1 Departement de Mathematiques, Faculte des Sciences de Gafsa, Universite de Gafsa,Cite Universitaire Zarrouk, Gafsa 2112, Tunisia

2 Departement de Mathematiques, Faculte des Sciences de Sfax, Universite de Sfax,Route de Soukra Km 3.5, B.P.1171, Sfax 3000, Tunisia

Correspondence should be addressed to Maher Mnif, [email protected]

Received 6 December 2010; Revised 14 February 2011; Accepted 15 February 2011

Academic Editor: Genaro Lopez

Copyright q 2011 Afif Ben Amar et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We establish some versions of fixed-point theorem in a Frechet topological vector space E. Themain result is that every map A = BC (where B is a continuous map and C is a continuouslinear weakly compact operator) from a closed convex subset of a Frechet topological vector spacehaving the Dunford-Pettis property into itself has fixed-point. Based on this result, we presenttwo versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-knownKrasnoselskii’s fixed-point theorem for U-contractions and weakly compact mappings, while thesecond one, by assuming that the family {T(·, y) : y ∈ C(M) where M ⊂ E and C : M → E acompact operator} is nonlinear ϕ equicontractive, we give a fixed-point theorem for the operatorof the form Ex := T(x,C(x)).

1. Introduction

Fixed-point theorems are very important in mathematical analysis. They are an interestingway to show that something exists without setting it out, which sometimes is very hard,or even impossible to do. Several algebraic and topological settings in the theory andapplications of nonlinear operator equations lead naturally to the investigation of fixed-points of a sum of two nonlinear operators, or more generally, fixed-points of mappings onthe cartesian product E × E into E, where E is some appropriate space.

Fixed-point theorems in topology and nonlinear functional analysis are usually basedon certain properties (such as complete continuity, monotonicity, contractiveness, etc.) thatthe operator, considered as a single entity must satisfy. We recall for instance the Banachfixed-point theorem, which asserts that a strict contraction on a complete metric space into


itself has unique fixed-point, and the Schauder principle, which asserts that a continuousmapping A on a closed convex setM in Hausdorff locally convex topological vector space Einto M such that A(M) is contained in a compact set, has a fixed-point. In many problemsof analysis, one encounters operators which may be split in the form T = A + B, where Ais a contraction in some sense, and B is completely continuous, and T itself has neither ofthese properties (see [1–3]). Thus neither the Schauder fixed-point theorem nor the Banachfixed-point theorem applies directly in this case, and it becomes desirable to develop fixed-point theorems for such situations. An early theorem of this type was given by Krasnosel’skiı[4]: “Let E be a Banach space, M be a bounded closed convex subset of E, and A,B beoperators on M into E such that Ax + By ∈ M for every pair x, y ∈ M. If A is a strictcontraction and B is continuous and compact, then the equation Ax + Bx = x has a solutionin M.” This result has been extended to locally convex spaces in 1971 by Cain and Nashed[5]. There is also another theorem of this type which was given by Amar et al. [6] in 2005and which extended the Schauder and Krasnoselskii fixed-point theorems in Dunford-Pettisspaces to weakly compact operators. Also in 2010, Amar and Mnif [7] established some newvariants of Leray-Schauder type fixed-point theorems for weakly sequentially continuousoperators.

In this paper, we give also a generalization of Krasnoselskii fixed-point theorems notin Dunford-Pettis Banach spaces but in Dunford-Pettis Frechet spaces. More precisely, letE be a Frechet topological vector space having the property of Dunford-Pettis, M a closedbounded convex subset of E, and A = BC (where B is a continuous map and C is a linearweakly compact operator). If A leaves M invariant then A has a fixed-point in M (seeProposition 3.1). In addition, if B is a ϕ-contraction map of M into E, for each x, y ∈ Mwith Bx+Ay /∈ M, there is a z ∈ (x, Bx+Ay)∩M such that Bz+Ay ∈M and (I −B)−1A(M)is relatively weakly compact, then A + B has a fixed-point inM (see Proposition 3.3).

Based on our results and other theorems which was given by Sehgal and Singh in1976 ([8]), we give also an extension of the Krasnoselskii fixed-point theorem: Let E be aFrechet topological vector space having the property of Dunford-Pettis (DP), M ⊆ E, C :M → C(M) ⊆ E a compact operator (An operator C : M → E is said to be compact ifit is continuous and maps bounded sets into precompact.) and T a map defined on the setM × C(M) and having range in E. By assuming that the family {T(·, y) : y ∈ C(M)} isnonlinear ϕ equicontractive we prove the existence of a point x ∈M such that

x = T(x,C(x)). (1.1)

Our paper is organized as follows. In Section 2, we give some important definitionsand preliminaries which will be used in this paper. Among this preliminaries we citedefinition of Dunford-Pettis space, the theorems of Schauder-Tychonoff, Krein-Smulian.The Section 3 is devoted to the generalization of the Krasnoselskii fixed-point theorem inDunford-Pettis Frechet spaces where our proofs of our two results (Proposition 3.3 andTheorem 3.5) in this section are based on the theorem of Sehgal and Singh and the mainresult (Proposition 3.1).

2. Preliminaries

In this section, we give the following well-known definitions and results which will be usedin this paper.


Definition 2.1. Suppose that E and F are locally convex spaces. A continuous linear operatorA from E into F is said to be weakly compact if A(B) is relatively weakly compact subset ofF whenever B is a bounded subset of E.

Theorem 2.2 (Eberlein-Smulian, see [9]). Let E be a metrizable locally convex topological vectorspace, (xn)n a weakly relatively compact sequence in E. Then from (xn)n may be extracted a weaklyconvergent subsequence.

Definition 2.3. A subset C in a vector space E is called balanced if for all x ∈ C, λx ∈ C if|λ| ≤ 1.

Definition 2.4 (see [9, 10]). A locally convex topological vector space E is said to have theDunford-Pettis (DP) property if any continuous linear map of E into a complete locallyconvex topological vector space F, which transforms bounded sets into weakly relativelycompact sets, transforms each balanced and weakly compact subset of E into a relativelycompact subset of F.

Remark 2.5 (see [9]). If E is complete, we replace in the precedent definition each balancedand weakly compact subset of E by each weakly compact subset of E.

Theorem 2.6 (see [11]). Let E be a locally convex topological vector space andM a convex subset ofE. ThenM is closed if and only if it is weakly closed.

Theorem 2.7 (Krein-Smulian). Let E be a metrizable and complete locally convex topological vectorspace andM ⊂ E weakly compact. Then the closed convex hull ofM is weakly compact.

Theorem 2.8 (Schauder-Tychonoff [12]). LetM be a closed and convex subset of a locally convextopological vector space E and A : M → M a continuous mapping such that the range A(M) iscontained in a compact set. Then A has a fixed-point.

In the remainder of this section, E denotes a Frechet topological vector space havingthe Dunford-Pettis (DP) property and ϑ is a neighborhood basis of the origin consisting ofabsolutely convex open subsets of E. Let for eachU ∈ ϑ, pU the Minkowski’s functional of ϑ.

LetM be a nonempty subset of E. A mapping A : M → E is a U-contraction (U ∈ ϑ)if for each ε > 0 there is a δ > 0 such that if x, y ∈M and if

x − y ∈ (ε + δ)U, then A(x) −A(y) ∈ εU. (2.1)

If A :M → E is aU-contraction for eachU ∈ ϑ, then A is a ϑ-contraction.Note that if A is a ϑ-contraction, then A is continuous. (For a related definition of

ϑ-contraction, see Taylor [13].)

Lemma 2.9 (see [8]). Let A : M → E be a ϑ-contraction, then A is ϑ-contractive, that is for eachU ∈ ϑ, pU(A(x) −A(y)) < pU(x − y) if pU(x − y)/= 0 and 0, otherwise.

Theorem 2.10 (Theorem of Sehgal and Singh [8]). Let M be a sequentially complete subset of acomplete separated locally convex topological vector space F andA :M → F be a ϑ-contraction. IfA


satisfies the condition:

for each x ∈M with A(x) /∈ M,

there is a z ∈ (x,A(x)) ∩M such that A(z) ∈M.(2.2)

Then A has a unique fixed-point inM.

Definition 2.11. Let T : M × E → E be a map such that M be a nonempty subset of E. Thefamily {T(·, y) : y ∈ E} is called U-equicontractive (U ∈ ϑ) if for each ε > 0 there is a δ > 0such that if (x1, y), (x2, y) in the domain of T and if

x1 − x2 ∈ (ε + δ)U, then T(x1, y

) − T(x2, y) ∈ εU. (2.3)

If {T(·, y) : y} is a U-equicontractive for each U ∈ ϑ, then the family {T(·, y) : y}is a ϑ-equicontraction. Note that if the family {T(·, y) : y} is a ϑ-equicontraction, then theoperator x → T(x, y) is a ϑ-contraction for all y.

Definition 2.12. let ϕ = {p = pU for some U ∈ ϑ}, R+ the nonnegative reals and ψ a family

of mapping defined as ψ = {Φ : R+ → R

+ such that Φ is continuous and Φ(t) < t if t > 0}.A mapping A : M → E is a nonlinear ϕ contraction (see [14]) if for each p ∈ ϕ, there is aΦp ∈ ψ such that p(A(x) −A(y)) ≤ Φp(p(x − y)) for all x, y ∈M. If this inequality holds withΦp(t) = αpt such that 0 < αp < 1, then A is called ϕ-contraction (see [5]).

Since a nonlinear ϕ contraction is a ϑ-contraction, the following result immediatelyfollows by Theorem 2.10 and provides an extension of a result in [5]:

Theorem 2.13 (see [8]). Let M be a sequentially complete subset of a complete separated locallyconvex topological vector space F and A : M → F be a nonlinear ϕ contraction. If A satisfies (2.2)then A has a unique fixed-point inM.

Definition 2.14. The family {T(·, y) : y ∈ E} is called nonlinear ϕ equicontractive if for eachp ∈ ϕ, there is a Φp ∈ ψ such that if (x1, y), (x2, y) in the domain of T , then

p(T(x1, y

) − T(x2, y)) ≤ Φp

(p(x1 − x2)

). (2.4)

Remark 2.15. Since any nonlinear ϕ contraction is a ϑ-contraction then any nonlinear ϕequicontraction is a ϑ-equicontraction.

3. Krasnoselskii’s Type Theorems

In this section, we will give some new fixed-point results for the sum of two operators whereE is a Frechet topological vector space having the Dunford-Pettis property. Firstly, we givethe following proposition which is a generalization of Theorem 2.1 in [6].


Proposition 3.1. Let E be a Frechet topological vector space having the Dunford-Pettis property,Ma closed, bounded and convex subset of E and B,C two operators such that:

(i) B : E �→ E a continuous map;

(ii) C : E �→ E a linear weakly compact operator on E;

(iii) B(C(M)) is relatively weakly compact;

(iv) A(M) ⊂M.

Then A = BC has a fixed-point inM.

Proof. We denote byN = co(A(M)), the closed convex hull ofA(M). Firstly, we show thatNis a weakly compact subset of E. Indeed, we haveA(M) ⊂ B(C(M)). This implies thatA(M)is relatively weakly compact and therefore A(M) is weakly compact. We have

A(M) ⊂ A(M) =⇒N = co(A(M)) ⊂ co(A(M)

)(3.1)

and since A(M) is weakly compact, then by Krein-Smulian’s theorem co(A(M)) is alsoweakly compact. SinceN is a closed convex subset of E, therefore it is weakly closed and thisimplies that N is a weakly closed subset of a weakly compact. Consequently, N is weaklycompact.

Now, we show that C(N) is relatively compact. We have N is a weakly compact setin E and C is a weakly compact operator on E and since E is a Frechet topological vectorspace having the Dunford-Pettis property, then by Definition 2.4, we haveC(N) is a relativelycompact set in E. Since B is a continuous map, then BC(N) is a relatively compact set in E.

Moreover, we have

A(M) ⊂M so co(A(M)) ⊂ co(M). (3.2)

Therefore

N = co(A(M)) ⊂M (3.3)

and this implies that

A(N) ⊂ A(M) ⊂ co(A(M)) =N, (3.4)

where N is a closed convex and A(N) = BC(N) is a relatively compact set. Since C is aweakly compact oprator on E, then by Definition 2.1 C is continuous and so A : N → N iscontinuous. Finally, the use of Schauder-Tychonoff’s fixed-point theorem shows thatA has atleast one fixed-point inN ⊂M.

Lemma 3.2. Let E be a Frechet topological vector space,M a sequentially complete subset of E and B :M �→ E a nonlinear ϕ contraction. Suppose that for y ∈ Ewe have: for each x ∈M withBx +y /∈ M,


there is a z ∈ (x, Bx + y) ∩M such that Bz + y ∈ M. Then, there exists a unique u(y) ∈ M withB(u(y)) + y = u(y), that is (I − B)−1y = u(y) ∈M.

Proof. Consequence of Theorem 2.13 (see [8]).

The following proposition is a generalization of Theorem 2.2 in [6].

Proposition 3.3. Let E be a Frechet topological vector space having the Dunford-Pettis property,Ma closed, bounded and convex subset of E and A,B two operators such that:

(i) A : E �→ E a linear weakly compact operator on E;

(ii) B :M �→ E be a ϕ-contraction;

(iii) For each x, y ∈ M with Bx + Ay /∈ M, there is a z ∈ (x, Bx + Ay) ∩ M such thatBz +Ay ∈M;

(iv) (I − B)−1A(M) is relatively weakly compact.

Then there exists y inM such that Ay + By = y

Proof. Firstly, we have B is a ϕ-contraction then B is a continuous function and for any x, y ∈M we have

pU((I − B)x − (I − B)y) ≥ pU

(x − y) − pU

(Bx − By) ≥ (

1 − αp)pU

(x − y) (3.5)

with αp ∈ (0, 1) which gives the continuity of (I − B)−1.Now, by Lemma 3.2 equation z = Bz +Ay has a unique solution z ∈ M for all y ∈ M.

It follows, that

z = (I − B)−1Ay ∈M, (3.6)

so

(I − B)−1A(M) ⊂M. (3.7)

For conclusion, we have (I − B)−1 is a continuous mapping, A a linear weakly compactoperator on E and (I − B)−1A(M) is relatively weakly compact on E where (I − B)−1A(M) ⊂M. So, by Proposition 3.1, we prove that (I − B)−1A has a fixed-point inM and this impliesthat, there exists y ∈M such that Ay + By = y.

We will now take C :M → C(M) ⊆ E a compact operator and T a map defined on thesetM ×C(M) and having range in E. We are interested to the existence of a point x ∈M ⊂ Esuch that

x = T(x,C(x)). (H)


Proposition 3.4. Let E be a Frechet topological vector space, M a bounded sequentially completesubset of E and

T :M × E −→ E (3.8)

a map such that the family {T(·, y) : y ∈ E} is nonlinear ϕ equicontractive, for all x ∈ M, y →T(x, y) is continuous and which satisfies the condition: for each (x, y) ∈ M × E with T(x, y) /∈ M,there is a

z ∈ (x, T

(x, y

)) ∩M such that T(z, y

) ∈M. (3.9)

Then there exists a continuous map FT : E → M such that

T(FT

(y), y

)= FT

(y). (3.10)

Proof. We start from an arbitrary point y ∈ E. Since the family {T(·, y) : y ∈ E} is a nonlinearϕ equicontractive then the operator

x −→ T(x, y

):M −→ E is a nonlinear ϕ contraction (3.11)

which satisfy for each x ∈M with T(x, y) /∈ M, there is a

z ∈ (x, T

(x, y


) ∈M. (3.12)

Then by Theorem 2.13, there is a unique point x = FT (y) ∈ M that satisfies the operatorequation:

T(FT

(y), y

)= FT

(y). (3.13)

We will show that the mapping y �→ FT (y) : E → M is continuous. To do this we let (yn) bea sequence in E, with limyn = y0 ∈ E. We suppose that FT (yn) does not converge to FT (y0).Then there exist p ∈ ϕ, an ε > 0 and (n) such that

p(FT

(y(n)

), FT

(y0))

> ε ∀n ∈ N. (3.14)


Since {p(FT (y(n)), FT (y0)) > ε, n ∈ N} is a bounded real subsequence, it has a subsequence{p(FT(y(1(n))), FT (y0)), n ∈ N} → r ≥ 0. However, we have

p(FT

(y(1(n))

) − FT(y0))

= p(T(FT

(y(1(n))

), y(1(n))

) − T(FT(y0), y0

))

≤ p(T(FT(y(1(n))

), y(1(n))

) − T(FT(y0), y((n))

))

+ p(T(FT

(y0), y(1(n))

) − T(FT(y0), y0

))

≤ Φp

(p(FT

(y(1(n))

) − FT(y0)))

+ p(T(FT

(y0), y(1(n))

) − T(FT(y0), y0

))

(3.15)

which implies that r = 0. This contradicts (3.14) and consequently FT is continuous.

In what follows, we give also another result of Krasnoselskii type.

Theorem 3.5. Let M be a closed, bounded and convex subset of a Frechet topological vector spacehaving the Dunford-Pettis property E and C : M → E a linear weakly compact operator such thatthe image of C(M) by any continuous mapping is contained in a weakly compact subset of E. Let

T :M × C(M) −→ E (3.16)

be a map such that the family {T(·, y) : y ∈ C(M)} is nonlinear ϕ equicontractive, for all x ∈ M,y �→ T(x, y) is continuous on C(M) and which satisfies that for each (x, y) ∈ M × C(M) withT(x, y) /∈ M, there is a

z ∈ (x, T

(x, y


) ∈M. (3.17)

Then (H) admits a solution inM.

Proof. We start from an arbitrary point y ∈ C(M). By Proposition 3.4 we prove that thereexists a unique point x = FT (y) ∈M that satisfies the operator equation

T(FT

(y), y

)= FT

(y), (3.18)

where the mapping y �→ FT (y) : C(M) → M is continuous. Then the operator FTC mapsthe setM into itself. We have by hypothesis that FT (C(M)) is contained in a weakly compactsubset of E. Therefore, by Proposition 3.1, we prove the existence of a point x ∈ M such thatFT (C(x)) = x. This means that

T(x,C(x)) = T(FT (C(x)), C(x)) = FT (C(x)) = x. (3.19)


References

[1] T. A. Burton, “A fixed-point theorem of Krasnoselskii,” Applied Mathematics Letters, vol. 11, no. 1, pp.85–88, 1998.

[2] T. A. Burton and C. Kirk, “A fixed point theorem of Krasnoselskii-Schaefer type,” MathematischeNachrichten, vol. 189, pp. 23–31, 1998.

[3] B. C. Dhage, “Local fixed point theory for the sum of two operators in Banach spaces,” InternationalJournal on Fixed Point Theory, Computation and Applications, vol. 4, no. 1, pp. 49–60, 2003.

[4] M. A. Krasnosel’skiı, “Two remarks on the method of successive approximations,” UspekhiMatematicheskikh Nauk, vol. 10, no. 1(63), pp. 123–127, 1955.

[5] G. L. Cain, Jr. and M. Z. Nashed, “Fixed points and stability for a sum of two operators in locallyconvex spaces,” Pacific Journal of Mathematics, vol. 39, pp. 581–592, 1971.

[6] A. B. Amar, A. Jeribi, and M. Mnif, “On a generalization of the Schauder and Krasnosel’skii fixedpoints theorems on Dunford-Pettis spaces and applications,” Mathematical Methods in the AppliedSciences, vol. 28, no. 14, pp. 1737–1756, 2005.

[7] A. B. Amar andM. Mnif, “Leray-Schauder alternatives for weakly sequentially continuous mappingsand application to transport equation,”Mathematical Methods in the Applied Sciences, vol. 33, no. 1, pp.80–90, 2010.

[8] V. M. Sehgal and S. P. Singh, “On a fixed point theorem of Krasnoselskii for locally convex spaces,”Pacific Journal of Mathematics, vol. 62, no. 2, pp. 561–567, 1976.

[9] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York,NY, USA, 1965.

[10] J.-L. Clerc and Y. C. De Verdiere, “Compacite faible dans les espaces localement convexes; applicationsaux espaces C(K) et L1(μ),” Seminaire Choquet. Initiation a l’Analys, vol. 7, no. 2, 1967-1968.

[11] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, Cataloging-in-Publication Data. Library of Congress Control Number: 2006921177.

[12] A. Tychonoff, “Ein Fixpunktsatz,”Mathematische Annalen, vol. 111, no. 1, pp. 767–776, 1935.[13] W.W. Taylor, “Fixed-point theorems for nonexpansive mappings in linear topological spaces,” Journal

of Mathematical Analysis and Applications, vol. 40, pp. 164–173, 1972.[14] D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical

Society, vol. 20, pp. 458–464, 1969.


Research ArticleSome Coupled Fixed Point Results onPartial Metric Spaces

Hassen Aydi

Institut Superieur d’Informatique de Mahdia, Universite de Monastir,route de Rejiche, Km 4, BP 35, Mahdia 5121, Tunisia

Correspondence should be addressed to Hassen Aydi, [email protected]

Received 20 December 2010; Revised 19 February 2011; Accepted 19 March 2011

Academic Editor: Enrique Llorens-Fuster

Copyright q 2011 Hassen Aydi. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We give some coupled fixed point results for mappings satisfying different contractive conditionson complete partial metric spaces.

1. Introduction and Preliminaries

For a given partially ordered set X, Bhaskar and Lakshmikantham in [1] introduced theconcept of coupled fixed point of a mapping F : X × X → X. Later in [2], Cırıc andLakshmikantham investigated some more coupled fixed point theorems in partially orderedsets. The following is the corresponding definition of a coupled fixed point.

Definition 1.1 (see [3]). An element (x, y) ∈ X × X is said to be a coupled fixed point of themapping F : X ×X → X if F(x, y) = x and F(y, x) = y.

Sabetghadam et al. [4] obtained the following.

Theorem 1.2. Let (X, d) be a complete cone metric space. Suppose that the mapping F : X ×X → Xsatisfies the following contractive condition for all x, y, u, v ∈ X

d(F(x, y

), F(u, v)

) ≤ kd(x, u) + ld(y, v), (1.1)

where k, l are nonnegative constants with k + l < 1. Then, F has a unique coupled fixed point.


In this paper, we give the analogous of this result (and some others in [4]) on partialmetric spaces, and we establish some coupled fixed point results.

The concept of partial metric space (X, p) was introduced by Matthews in 1994. Insuch spaces, the distance of a point in the self may not be zero. First, we start with somepreliminaries definitions on the partial metric spaces [3, 5–13].

Definition 1.3 (see ([6–8])). A partial metric on a nonempty setX is a function p : X×X −→ �+

such that for all x, y, z ∈ X:

(p1) x = y ⇐⇒ p(x, x) = p(x, y) = p(y, y),

(p2) p(x, x) ≤ p(x, y),(p3) p(x, y) = p(y, x),

(p4) p(x, y) ≤ p(x, z) + p(z, y) − p(z, z).

A partial metric space is a pair (X, p) such that X is a nonempty set and p is a partialmetric on X.

Remark 1.4. It is clear that if p(x, y) = 0, then from (p1), (p2), and (p3), x = y. But if x = y,p(x, y) may not be 0.

If p is a partial metric on X, then the function ps : X ×X −→ R+ given by

ps(x, y

)= 2p

(x, y

) − p(x, x) − p(y, y), (1.2)

is a metric on X.

Definition 1.5 (see ([6–8])). Let (X, p) be a partial metric space. Then,

(i) a sequence {xn} in a partial metric space (X, p) converges to a point x ∈ X if andonly if p(x, x) = limn→+∞p(x, xn);

(ii) a sequence {xn} in a partial metric space (X, p) is called a Cauchy sequence if thereexists (and is finite) limn,m→+∞p(xn, xm);

(iii) a partial metric space (X, p) is said to be complete if every Cauchy sequence {xn}in X converges to a point x ∈ X, that is, p(x, x) = limn,m→+∞p(xn, xm).

Lemma 1.6 (see ([6, 7, 9])). Let (X, p) be a partial metric space;

(a) {xn} is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequence in the metricspace (X, ps),

(b) a partial metric space (X, p) is complete if and only if the metric space (X, ps) is complete;furthermore, limn→+∞ps(xn, x) = 0 if and only if

p(x, x) = limn→+∞

p(xn, x) = limn,m→+∞

p(xn, xm). (1.3)

2. Main Results

Our first main result is the following.


Theorem 2.1. Let (X, p) be a complete partial metric space. Suppose that the mapping F : X ×X →X satisfies the following contractive condition for all x, y, u, v ∈ X

p(F(x, y

), F(u, v)

) ≤ kp(x, u) + lp(y, v), (2.1)


Proof. Choose x0, y0 ∈ X and set x1 = F(x0, y0) and y1 = F(y0, x0). Repeating this process, setxn+1 = F(xn, yn) and yn+1 = F(yn, xn). Then, by (2.1), we have

p(xn, xn+1) = p(F(xn−1, yn−1

), F

(xn, yn

))

≤ kp(xn−1, xn) + lp(yn−1, yn

),

(2.2)

and similarly

p(yn, yn+1

)= p

(F(yn−1, xn−1

), F

(yn, xn

))

≤ kp(yn−1, yn)+ lp(xn−1, xn).

(2.3)

Therefore, by letting

dn = p(xn, xn+1) + p(yn, yn+1

), (2.4)

we have

dn = p(xn, xn+1) + p(yn, yn+1

)

≤ kp(xn−1, xn) + lp(yn−1, yn

)+ kp

(yn−1, yn

)+ lp(xn−1, xn)

= (k + l)[p(yn−1, yn

)+ p(xn−1, xn)

]

= (k + l)dn−1.

(2.5)

Consequently, if we set δ = k + l, then, for each n ∈ �, we have

dn ≤ δdn−1 ≤ δ2dn−2 ≤ · · · ≤ δnd0. (2.6)

If d0 = 0 then p(x0, x1)+p(y0, y1) = 0. Hence, from Remark 1.4, we get x0 = x1 = F(x0, y0) andy0 = y1 = F(y0, x0), meaning that (x0, y0) is a coupled fixed point of F. Now, let d0 > 0. Foreach n ≥ m, we have, in view of the condition (p4)

p(xn, xm) ≤ p(xn, xn−1) + p(xn−1, xn−2) − p(xn−1, xn−1)+ p(xn−2, xn−3) + p(xn−3, xn−4) − p(xn−3, xn−3)+ · · · + p(xm+2, xm+1) + p(xm+1, xm) − p(xm+1, xm+1)

≤ p(xn, xn−1) + p(xn−1, xn−2) + · · · + p(xm+1, xm).

(2.7)


Similarly, we have

p(yn, ym

) ≤ p(yn, yn−1)+ p

(yn−1, yn−2

)+ · · · + p(ym+1, ym

). (2.8)

Thus,

p(xn, xm) + p(yn, ym

) ≤ dn−1 + dn−2 + · · · + dm

≤(δn−1 + δn−2 + · · · + δm

)d0

≤ δm

1 − δd0.

(2.9)

By definition of ps, we have ps(x, y) ≤ 2p(x, y), so, for any n ≥ m

ps(xn, xm) + ps(yn, ym

) ≤ 2p(xn, xm) + 2p(yn, ym

) ≤ 2δm

1 − δd0, (2.10)

which implies that {xn} and {yn} are Cauchy sequences in (X, ps) because of 0 ≤ δ = k+ l < 1.Since the partial metric space (X, p) is complete, hence thanks to Lemma 1.6, the metric space(X, ps) is complete, so there exist u∗, v∗ ∈ X such that

limn→+∞

ps(xn, u∗) = limn→+∞

ps(yn, v

∗) = 0. (2.11)

Again, from Lemma 1.6, we get

p(u∗, u∗) = limn→+∞

p(xn, u∗) = limn→+∞

p(xn, xn),

p(v∗, v∗) = limn→+∞

p(yn, v

∗) = limn→+∞

p(yn, yn

).

(2.12)

But, from condition (p2) and (2.6),

p(xn, xn) ≤ p(xn, xn+1) ≤ dn ≤ δnd0, (2.13)

so since δ ∈ [0, 1[, hence letting n → +∞, we get limn→+∞p(xn, xn) = 0. It follows that

p(u∗, u∗) = limn→+∞


p(xn, xn) = 0. (2.14)

Similarly, we get

p(v∗, v∗) = limn→+∞

p(yn, v

∗) = limn→+∞

p(yn, yn

)= 0. (2.15)


Therefore, we have, using (2.1),

p(F(u∗, v∗), u∗) ≤ p(F(u∗, v∗), xn+1) + p(xn+1, u∗) − p(xn+1, xn+1), by (p4)

≤ p(F(u∗, v∗), F(xn, yn))

+ p(xn+1, u∗)

≤ kp(xn, u∗) + lp(yn, v

∗) + p(xn+1, u∗),

(2.16)

and letting n → +∞, then from (2.14) and (2.15), we obtain p(F(u∗, v∗), u∗)) = 0, soF(u∗, v∗) = u∗. Similarly, we have F(v∗, u∗) = v∗, meaning that (u∗, v∗) is a coupled fixedpoint of F.

Now, if (u′, v′) is another coupled fixed point of F, then

p(u′, u∗

)= p

(F(u′, v′

), F(u∗, v∗)

) ≤ kp(u′, u∗) + lp(v′, v∗),p(v′, v∗

)= p

(F(v′, u′

), F(v∗, u∗)

) ≤ kp(v′, v∗) + lp(u′, u∗).(2.17)

It follows that

p(u′, u∗

)+ p

(v′, v∗

) ≤ (k + l)[p(u′, u∗

)+ p

(v′, v∗

)]. (2.18)

In view of k + l < 1, this implies that p(u′, u∗) + p(v′, v∗) = 0, so u∗ = u′ and v∗ = v′. The proofof Theorem 2.1 is completed.

It is worth noting that when the constants in Theorem 2.1 are equal, we have thefollowing corollary

Corollary 2.2. Let (X, p) be a complete partial metric space. Suppose that the mapping F : X ×X →X satisfies the following contractive condition for all x, y, u, v ∈ X

p(F(x, y

), F(u, v)

) ≤ k

2(p(x, u) + p

(y, v

)), (2.19)

where 0 ≤ k < 1. Then, F has a unique coupled fixed point.

Example 2.3. LetX = [0,+∞[ endowed with the usual partial metric p defined by p : X×X →[0,+∞[ with p(x, y) = max{x, y}. The partial metric space (X, p) is complete because (X, ps)is complete. Indeed, for any x, y ∈ X,

ps(x, y

)= 2p

(x, y

) − p(x, x) − p(y, y) = 2max{x, y

} − (x + y

)=∣∣x − y∣∣, (2.20)

Thus, (X, ps) is the Euclidean metric space which is complete. Consider the mapping F :X ×X → X defined by F(x, y) = (x + y)/6. For any x, y, u, v ∈ X, we have

p(F(x, y

), F(u, v)

)=16max

{x + y, u + v

} ≤ 16[max{x, u} +max

{y, v

}]

=16[p(x, u) + p

(y, v

)].

(2.21)


which is the contractive condition (2.19) for k = 1/3. Therefore, by Corollary 2.2, F has aunique coupled fixed point, which is (0, 0). Note that if the mapping F : X ×X → X is givenby F(x, y) = (x + y)/2, then F satisfies the contractive condition (2.19) for k = 1, that is,

p(F(x, y

), F(u, v)

)=12max

{x + y, u + v

} ≤ 12[max{x, u} +max

{y, v

}]

=12[p(x, u) + p

(y, v

)].

(2.22)

In this case, (0, 0) and (1, 1) are both coupled fixed points of F, and, hence, the coupled fixedpoint of F is not unique. This shows that the condition k < 1 in Corollary 2.2, and hencek + l < 1 in Theorem 2.1 cannot be omitted in the statement of the aforesaid results.


p(F(x, y

), F(u, v)

) ≤ kp(F(x, y), x) + lp(F(u, v), u), (2.23)


Proof. We take the same sequences {xn} and {yn} given in the proof of Theorem 2.1 by

xn+1 = F(xn, yn

), yn+1 = F

(yn, xn

)for any n ∈ �. (2.24)

Applying (2.23), we get

p(xn, xn+1) ≤ δp(xn−1, xn) (2.25)

p(yn, yn+1

) ≤ δp(yn−1, yn)

(2.26)

where δ = k/(1 − l). By definition of ps, we have

ps(xn, xn+1) ≤ 2p(xn, xn+1) ≤ 2δnp(x1, x0) , (2.27)

ps(yn, yn+1

) ≤ 2p(yn, yn+1

) ≤ 2δnp(y1, y0

). (2.28)

Since k+l < 1, hence δ < 1, so the sequences {xn} and {yn} are Cauchy sequences in the metricspace (X, ps). The partial metric space (X, p) is complete, hence from Lemma 1.6, (X, ps) iscomplete, so there exist u∗, v∗ ∈ X such that

limn→+∞


ps(yn, v

∗) = 0. (2.29)


From Lemma 1.6, we get

p(u∗, u∗) = limn→+∞


p(xn, xn),

p(v∗, v∗) = limn→+∞

p(yn, v

∗) = limn→+∞

p(yn, yn

).

(2.30)

By the condition (p2) and (2.25), we have

p(xn, xn) ≤ p(xn, xn+1) ≤ δnp(x1, x0), (2.31)

so limn→+∞p(xn, xn) = 0. It follows that

p(u∗, u∗) = limn→+∞


p(xn, xn) = 0. (2.32)

Similarly, we find

p(v∗, v∗) = limn→+∞

p(yn, v

∗) = limn→+∞

p(yn, yn

)= 0. (2.33)

Therefore, by (2.23),

p(F(u∗, v∗), u∗) ≤ p(F(u∗, v∗), xn+1) + p(xn+1, u∗)= p

(F(u∗, v∗), F

(xn, yn

))+ p(xn+1, u∗)

≤ kp(F(u∗, v∗), u∗) + lp(F(xn, yn), xn

)+ p(xn+1, u∗)

= kp(F(u∗, v∗), u∗) + lp(xn+1, xn) + p(xn+1, u∗),

(2.34)

and letting n → +∞, then from (2.27)–(2.32), we obtain

p(F(u∗, v∗), u∗) ≤ kp(F(u∗, v∗), u∗). (2.35)

From the preceding inequality, we can deduce a contradiction if we assume thatp(F(u∗, v∗), u∗)/= 0, because in that case we conclude that 1 ≤ k and now this inequality is,in fact, a contradiction, so p(F(u∗, v∗), u∗) = 0, that is, F(u∗, v∗) = u∗. Similarly, we haveF(v∗, u∗) = v∗, meaning that (u∗, v∗) is a coupled fixed point of F. Now, if (u′, v′) is anothercoupled fixed point of F, then, in view of (2.23),

p(u′, u∗

)= p

(F(u′, v′

), F(u∗, v∗)

)

≤ kp(F(u′, v′), u′) + lp(F(u∗, v∗), u∗)= kp

(u′, u′

)+ lp(u∗, u∗)

≤ kp(u′, u∗) + lp(u′, u∗) = (k + l)p(u′, u∗

), using (p2),

(2.36)


that is, p(u′, u∗) = 0 since (k + l) < 1. It follows that u∗ = u′. Similarly, we can have v∗ = v′, andthe proof of Theorem 2.4 is completed.


p(F(x, y

), F(u, v)

) ≤ kp(F(x, y), u) + lp(F(u, v), x), (2.37)

where k, l are nonnegative constants with k + 2l < 1. Then, F has a unique coupled fixed point.

Proof. Since, k + 2l < 1, hence k + l < 1, and as a consequence the proof of the uniquenessin this theorem is as trivial as in the other results. To prove the existence of the fixed point,choose the sequences {xn} and {yn} like in the proof of Theorem 2.1, that is

xn+1 = F(xn, yn

), yn+1 = F

(yn, xn

), for any n ∈ �. (2.38)

Applying again (2.37), we have

p(xn, xn+1) = p(F(xn−1, yn−1

), F

(xn, yn

))

≤ kp(F(xn−1, yn−1), xn

)+ lp

(F(xn, yn

), xn−1

)

= kp(xn, xn) + lp(xn+1, xn−1)

≤ kp(xn+1, xn) + lp(xn+1, xn−1)], by (p2)

≤ kp(xn+1, xn) + lp(xn+1, xn) + lp(xn, xn−1) − lp(xn, xn), using (p4)

≤ (k + l)p(xn, xn+1) + lp(xn−1, xn).

(2.39)

It follows that for any n ∈ �∗

p(xn, xn+1) ≤ l

1 − l − kp(xn−1, xn). (2.40)

Let us take δ = l/(1 − l − k). Hence, we deduce

ps(xn, xn+1) ≤ 2p(xn, xn+1) ≤ 2δnp(x0, x1). (2.41)

Under the condition 0 ≤ k + 2l < 1, we get 0 ≤ δ < 1. From this fact, we immediately obtainthat {xn} is Cauchy in the complete metric space (X, ps). Of course, similar arguments applyto the case of the sequence {yn} in order to prove that

ps(yn, yn+1

) ≤ 2p(yn, yn+1

) ≤ 2δnp(y0, y1

), (2.42)


and, thus, that the sequence {yn} is Cauchy in (X, ps). Therefore, there exist u∗, v∗ ∈ X suchthat

limn→+∞


ps(yn, v

∗) = 0. (2.43)

Thanks to Lemma 1.6, we have

limn→+∞


p(xn, xn) = p(u∗, u∗),

limn→+∞

p(yn, v

∗) = limn→+∞

p(yn, yn

)= p(v∗, v∗).

(2.44)

The condition (p2) together with (2.41) yield that

p(xn, xn) ≤ p(xn, xn+1) ≤ δnp(x0, x1), (2.45)

hence letting n → +∞, we get limn→+∞p(xn, xn) = 0. It follows that

p(u∗, u∗) = limn→+∞


p(xn, xn) = 0. (2.46)

Similarly, we have

p(v∗, v∗) = limn→+∞

p(yn, v

∗) = limn→+∞

p(yn, yn

)= 0. (2.47)

Therefore, we have, using (2.37),

p(F(u∗, v∗), u∗) ≤ p(F(u∗, v∗), xn+1) + p(xn+1, u∗)= p

(F(u∗, v∗), F

(xn, yn

))+ p(xn+1, u∗)

≤ kp(F(u∗, v∗), xn) + lp(F(xn, yn

), u∗

)+ p(xn+1, u∗)

= kp(F(u∗, v∗), xn) + lp(xn+1, u∗) + p(xn+1, u∗)

≤ kp(F(u∗, v∗), u∗) + kp(u∗, xn) + lp(xn+1, u∗) + p(xn+1, u∗), using (p4).(2.48)

Letting n → +∞ yields, using (2.46),

p(F(u∗, v∗), u∗) ≤ kp(F(u∗, v∗), u∗), (2.49)

and since k < 1, we have p(F(u∗, v∗), u∗) = 0, that is, F(u∗, v∗) = u∗. Similarly, thanks to (2.47),we get F(v∗, u∗) = v∗, and hence (u∗, v∗) is a coupled fixed point of F.

When the constants in Theorems 2.4 and 2.5 are equal, we get the following corollaries.



p(F(x, y

), F(u, v)

) ≤ k

2(p(F(x, y

), x

)+ p(F(u, v), u)

), (2.50)

where 0 ≤ k < 1. Then, F has a unique coupled fixed point.


p(F(x, y

), F(u, v)

) ≤ k

2(p(F(x, y

), u

)+ p(F(u, v), x)

), (2.51)

where 0 ≤ k < 2/3. Then, F has a unique coupled fixed point.

Proof. The condition 0 ≤ k < 2/3 follows from the hypothesis on k and l given in Theorem 2.5.

Remark 2.8. (i) Theorem 2.1 extends the Theorem 2.2 of [4] on the class of partial metricspaces.

(ii) Theorem 2.4 extends the Theorem 2.5 of [4] on the class of partial metric spaces.

Remark 2.9. Note that in Theorem 2.4, if the mapping F : X ×X → X satisfies the contractivecondition (2.23) for all x, y, u, v ∈ X, then F also satisfies the following contractive condition:

p(F(x, y

), F(u, v)

)= p

(F(u, v), F

(x, y

))

≤ kp(F(u, v), u) + lp(F(x, y), x).(2.52)

Consequently, by adding (2.23) and (2.52), F also satisfies the following:

p(F(x, y

), F(u, v)

) ≤ k + l2

p(F(u, v), u) +k + l2

p(F(x, y

), x

), (2.53)

which is a contractive condition of the type (2.50) in Corollary 2.6 with equal constants.Therefore, one can also reduce the proof of general case (2.23) in Theorem 2.4 to the specialcase of equal constants. A similar argument is valid for the contractive conditions (2.37) inTheorem 2.5 and (2.51) in Corollary 2.7.

Acknowledgment

The author thanks the editor and the referees for their kind comments and suggestions toimprove this paper.


References

[1] T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spacesand applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393,2006.

[2] L. Cırıc and V. Lakshmikantham, “Coupled fixed point theorems for nonlinear contractions inpartially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12,pp. 4341–4349, 2009.

[3] I. Altun, F. Sola, and H. Simsek, “Generalized contractions on partial metric spaces,” Topology and ItsApplications, vol. 157, no. 18, pp. 2778–2785, 2010.

[4] F. Sabetghadam, H. P. Masiha, and A. H. Sanatpour, “Some coupled fixed point theorems in conemetric spaces,” Fixed Point Theory and Applications, Article ID 125426, 8 pages, 2009.

[5] H. Aydi, “Some fixed point results in ordered partial metric spaces,” The Journal of Nonlinear Sciencesand Applications. In press.

[6] S. G. Matthews, “Partial metric topology, in: Proc. 8th Summer Conference on General Topology andApplications,” in Annals of the New York Academy of Sciences, vol. 728, pp. 183–197, 1994.

[7] S. J. O’Neill, “Two topologies are better than one,” Tech. Rep., University of Warwick, Coventry, UK,1995, http://www.dcs.warwick.ac.uk/reports/283.html .

[8] S. J. O’Neill, “Partial metrics, valuations and domain theory, in: Proc. 11th Summer Conference onGeneral Topology and Applications,” in Annals of the New York Academy of Sciences, vol. 806, pp. 304–315, 1996.

[9] S. Oltra and O. Valero, “Banach’s fixed point theorem for partial metric spaces,” Rendiconti dell’Istitutodi Matematica dell’Universita di Trieste, vol. 36, no. 1-2, pp. 17–26, 2004.

[10] S. Romaguera and M. Schellekens, “Partial metric monoids and semivaluation spaces,” Topology andIts Applications, vol. 153, no. 5-6, pp. 948–962, 2005.

[11] S. Romaguera and O. Valero, “A quantitative computational model for complete partial metric spacesvia formal balls,” Mathematical Structures in Computer Science, vol. 19, no. 3, pp. 541–563, 2009.

[12] M. P. Schellekens, “The correspondence between partial metrics and semivaluations,” TheoreticalComputer Science, vol. 315, no. 1, pp. 135–149, 2004.

[13] O. Valero, “On Banach fixed point theorems for partial metric spaces,” Applied General Topology, vol.6, no. 2, pp. 229–240, 2005.


Research ArticleOptimal Selling Rule in a RegimeSwitching Levy Market

Moustapha Pemy

Department of Mathematics, Towson University, Towson, MD 21252-0001, USA

Correspondence should be addressed to Moustapha Pemy, [email protected]

Received 5 March 2011; Accepted 5 April 2011


Copyright q 2011 Moustapha Pemy. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper is concerned with a finite-horizon optimal selling rule problem when the underlyingstock price movements are modeled by a Markov switching Levy process. Assuming that thetransaction fee of the selling operation is a function of the underlying stock price, the optimalselling rule can be obtained by solving an optimal stopping problem. The corresponding valuefunction is shown to be the unique viscosity solution to the associated HJB variational inequalities.A numerical example is presented to illustrate the results.

1. Introduction

One of the major decision investors have to make on a daily basis is to identify the best timeto sell or buy a particular stock. Usually if the right decision is not taken at the right time,this will generally result in large losses for the investor. Such decisions are mainly affected byvarious macro- andmicro-economical parameters. One of the main factors that affect decisionmaking in the marketplace is the trend of the stock market. In this paper, we study tradingdecision making when we assume that market trends are subject to change and that thesefluctuations can be captured by a combination of a latent Markov chain and a jump process.In fact, we model the stock price dynamics with a regime switching Levy process. Regimeswitching Levy processes are obtained by combining a finite number of geometric Levyprocesses modulated by a finite-state Markov chain. This type of processes clearly capturethe main features of a wide variety of stock such as energy stock and commodities whichusually display a lot of spikes and seasonality. Selling rule problems in general have beenintensively studied in the literature, and most of the work have been done when the stockprice follows a geometric Brownian motion or a simple Markov switching process. Amongmany others, we can cite the work of Zhang [1]; in this paper, a selling rule is determinedby two threshold levels, and a target price and a stop-loss limit are considered. One makesa selling decision whenever the price reaches either the target price or the stop-loss limit.


The objective is to choose these threshold levels to maximize an expected return function.In [1], such optimal threshold levels are obtained by solving a set of two-point boundaryvalue problems. Recently Pemy and Zhang [2] studied a similar problem in the case wherethere is no jump process associated and the underlying dynamics is just a traditional Markovswitching process built by coupling a set of geometric Brownian motions.

In this paper, we extend the result of Pemy and Zhang [2], we consider an optimalselling rule among the class of almost all stopping times under a regime switching Levymodel. We study the case when the stock has to be sold within a prespecified time limit.Given a transaction cost which is a function of the underlying stock price, the objective is tochoose a stopping time so as to maximize an expected return. The optimal stopping problemwas studied by McKean [3] back to the 1960s when there is no switching; see also Samuelson[4] in connection with derivative pricing and Øksendal [5] for optimal stopping in general.In models with regime switching, Guo and Zhang [6] considered the model with a two-stateMarkov chain. Using a smooth-fit technique, they were able to convert the optimal stoppingproblem to a set of algebraic equations under certain smoothness conditions. Closed-formsolutions were obtained in these cases. However, it can be shown with extensive numericaltests that the associated algebraic equations may have no solutions. This suggests that thesmoothness (C2) assumption may not hold in these cases. Moreover, the results in [5, 6] areestablished on an infinite time horizon setup. However, in practice, an investor often has tosell his stock holdings by a certain date due to various nonprice-related considerations suchas year-end tax deduction or the need for raising cash for major purchases. In these cases, it isnecessary to consider the corresponding optimal sellingwith a finite horizon. It is the purposeof this paper to treat the underlying finite horizon optimization problem with possiblenonsmoothness of the solutions to the associated HJB variational inequalities. We resort tothe concept of viscosity solutions and show that the corresponding value function is indeedthe only viscosity solution to the HJB variational inequalities. We clearly prove that the valuefunction of this optimal stopping time problem is the unique viscosity solution to the associ-atedHJB variational inequalities, which enables us to run some numerical schemes in order toapproximate the value function and derive the both the continuation region and the stoppingregion. It is well known that the optimal stopping rule can be determined by the correspond-ing value function; see, for example, Krylov [7] and Øksendal [5] for diffusions, Pham [8] forjump diffusions, and Guo and Zhang [6] and [9] for regime switching diffusions.

The paper is organized as follows. In the next section, we formulate the problemunder consideration and then present the associated HJB inequalities and their viscositysolutions. In Section 3, we obtain the continuity property of the value function and showthat it is the only viscosity solution to the HJB equations. In Section 4, we give a numericalexample in order to illustrate our results. To better present the results without undue technicaldifficulties, all proofs are moved to the appendix placed at the end of the paper.

2. Problem Formulation

Given an integer m ≥ 2, let α(t) ∈ M = {1, 2, . . . , m} denote a Markov chain with an m × mmatrix generator Q = (qij)m,m, that is, qij ≥ 0 for i /= j and Σm

j=1qij = 0 for i ∈ M and a Levyprocess (ηt)t. LetN be the Poisson random measure of (ηt)t, then it is defined as follows: forany Borel setU ⊂ R,

N(t,U) =∑

0<s≤t1U(ηs − ηs−

). (2.1)


The differential form of N is denoted by N(dt, dz). Let ν be the Levy measure of (ηt)t; wehave ν(U) = E[N(1, U)] for any Borel setU ⊂ R. We define the differential formN(dt, dz) asfollows:

N(dt, dz) =

⎧⎨

⎩

N(dt, dz) − ν(dz)dt, if |z| < 1,

N(dt, dz), if |z| ≥ 1.(2.2)

From Levy-Khintchine formula, we have

∫

R

min(|z|2, 1

)ν(dz) <∞. (2.3)

In our regime switching Levy market model, the stock price denoted by X(t) satisfies thefollowing Levy stochastic differential equation

dX(t) = X(t)(μ(α(t))dt + σ(α(t))dW(t) +

∫

R

γ(α(t))zN(dt, dz)),

X(s) = x, s ≤ t ≤ T,(2.4)

where x is the initial price and T is a finite time. For each state i ∈ M, μ(i) the rate of return,σ(i) the volatility and γ(i) the jump intensity are known and satisfied the linear growthcondition. There exists a constant C > 0 such that for all x ∈ R and all t ∈ [0, T], we have

x2(μ(α(t))2 + σ(α(t))2 +

∫

R

∣∣γ(α(t))∣∣2z2ν(dz)

)≤ C(1 + x2). (2.5)

W(t) is the standard Weiner process, and N(dt, dz) represents the differential form of thejump measure of ηt. The processes W(·), α(·), and η(·) are defined on a probability space(Ω,F, P) and are independent of each other.

In this paper, we consider the optimal selling rule with a finite horizon T . We assumethat the transaction cost function a(·) > 0 is the function of the stock price itself. In this case,we take into account all costs associated with the selling operation. The main objective of thisselling problem is to sell the stock by time T so as to maximize the quantity E[e−r(τ−s)(X(τ) −a(X(τ)))], where r > 0 is a discount rate.

Let Ft = σ{α(s),W(s), η(s); s ≤ t} and let Λs,T denote the set of Ft-stopping times suchthat s ≤ τ ≤ T a.s. The value function can be written as follows:

v(s, x, i) = supτ∈Λs,T

E[e−r(τ−s)(X(τ) − a(X(τ))) | X(s) = x, α(s) = i

]. (2.6)

Given the value function v(s, x, i), it is typical that an optimal stopping time τ∗ can bedetermined by the following continuation region:

D = {(t, x, i) ∈ [0, T) × R ×M; v(t, x, i) > x − a(x)}, (2.7)


as follows:

τ∗ = inf{t > 0; (t, X(t), α(t)) /∈ D}. (2.8)

It can be proved that if τ∗ < +∞, then

v(s, x, i) = Es,x,i[e−r(τ

∗−s)(X(τ∗) − a(x))]. (2.9)

Thus, τ∗ is the optimal stopping time; see [9].The process (X(t), α(t)) is a Markov process with generator A defined as follows:

(Af)(s, x, i) =

12x2σ2(i)

∂2f(s, x, i)∂x2

+ xμ(i)∂f(s, x, i)

∂x+Qf(s, x, ·)(i)

+∫

R

(f(s, x + γ(i)xz

) − f(s, x) − γ(i)xz1{|z|<1}(z)∂f

∂x

)ν(dz),

(2.10)

where

Qf(s, x, ·)(i) =∑

j /= i

qij(f(s, x, j

) − f(s, x, i)). (2.11)

The corresponding Hamiltonian has the following form:

H(i, s, x, u,Dsu,Dxu,D2

xu)

= min[ru(s, x, i) − ∂u(s, x, i)

∂s− (Au)(s, x, i), u(s, x, i) − (x − a(x))

]

= 0.

(2.12)

Note that X(t) > 0 for all t. Let R+ = (0,∞). Formally, the value function v(s, x, i) satisfies the

HJB equation

H(i, s, x, v,Dsv,Dxv,D2

xv)= 0, for (s, x, i) ∈ [0, T) × R

+ ×M,

v(T, x, α(T)) = (x − a(x)).(2.13)

In order to study the possibility of existence and uniqueness of a solution of (2.12), we use anotion of viscosity solution introduced by Crandall et al. [10].

Definition 2.1. We say that f(s, x, i) is a viscosity solution of

H(

i, s, x, f,∂f

∂s,∂f

∂x,∂2f

∂x2

)

= 0, for i ∈ M, s ∈ [0, T), x ∈ R+,

f(T, x, α(T)) = (x − a(x)).(2.14)


If

(1) for all x ∈ R+f(T, x, α(T)) = (x − a(x)), and for each i ∈ M, f(s, x, i) is continuous

in (s, x), moreover, there exist constants K and κ such that

f(s, x, i) ≤ K(1 + |x|κ), (2.15)

(2) for each i ∈ M,

H(

i, s0, x0, f,∂φ

∂s,∂φ

∂x,∂2φ

∂x2

)

≤ 0 (2.16)

whenever φ(s, x) ∈ C2 such that f(s, x, i) − φ(s, x) has local maximum at (s, x) =(s0, x0),

(3) and for each i ∈ M,

H(

i, s0, x0, f,∂ψ

∂s,∂ψ

∂x,∂2ψ

∂x2

)

≥ 0 (2.17)

whenever ψ(s, x) ∈ C2 such that f(s, x, i) − ψ(s, x) has local minimum at (s, x) =(s0, x0).

Let f be a function that satisfies (2.3). It is a viscosity subsolution (resp. supersolution) if itsatisfies (2.4) (resp. (2.5)).

3. Properties of Value Functions

In this section, we study the continuity of the value function, show that it satisfies theassociated HJB equation as a viscosity solution, and establish the uniqueness. We first showthe continuity property.

Lemma 3.1. For each i ∈ M, the value function v(s, x, i) is continuous in (s, x). Moreover, it has atmost linear growth rate, that is, there exists a constant C such that |v(s, x, i)| ≤ C(1 + |x|).

The continuity of the value function and its at most linear growth will be very helpfulin deriving the maximum principle which itself guarantees the uniqueness of the valuefunction. The following lemma is a simple version of the dynamic programming principlein optimal stopping. A similar result has been proven in Pemy [9]. For general dynamicprogramming principle, see Krylov [7] for diffusions, Pham [8] for jump diffusions, and Yinand Zhang [11] for dynamic models with regime switching.

Definition 3.2. For each ε > 0, a stopping time τε ∈ Λs,T is said to be ε-optimal if

0 ≤ v(s, x, i) − E[e−r(τε−s)v(τε, X(τε), α(τε))

]≤ ε. (3.1)


Lemma 3.3. (1) Let β, γ ∈ Λs,T two stopping time such that s ≤ β ≤ γ a.s., then one has

E[e−r(β−s)v

(β,X(β), α(β))] ≥ E

[e−r(γ−s)v

(γ,X(γ), α(γ))]

. (3.2)

In particular for any stopping time θ ∈ Λs,T , one has

v(s, x, i) ≥ Es,x,i[e−r(θ−s)v(θ,X(θ), α(θ))

]. (3.3)

(2) Let θ ∈ Λs,T such that s ≤ θ ≤ τε for any ε > 0, where τε an ε-optimal stopping time. Then,one has

v(s, x, i) = Es,x,i[e−r(θ−s)v(θ,X(θ), α(θ))

]. (3.4)

With Lemma 3.3 at our hand, we proceed and show that the value function v(s, x, i) is aviscosity solution of the variational inequality (2.13).

Theorem 3.4. The value function v(s, x, i) is a viscosity solution of (2.13).

3.1. Uniqueness of the Viscosity Solution

In this subsection, wewill prove that the value function defined in (2.6) is the unique viscositysolution of the HJB equation (2.13). We begin by recalling the definition of parabolic superjetand subjet.

Definition 3.5. Let f(s, x, i) : [0, T] × R ×M → R. Define the parabolic superjet by

P2,+f(s, x, i) ={(p, q,M

) ∈ R × R : f(t, y, i

) ≤ f(s, x, i) + p(t − s) + q(y − x)

+12(y − x)2M + o

(∣∣y − x∣∣2)as(t, y) −→ (s, x)

},

(3.5)

and its closure is

P2,+f(s, x, i) =

{(p, q,M

)= lim

n→∞(pn, qn,Mn

)

with(pn, qn,Mn

) ∈ P2,+f(sn, xn, i) and

limn→∞

(sn, xn, f(sn, xn, i)

)=(x, f(s, x, i)

)}.

(3.6)

Similarly, we define the parabolic subjet P2,−f(s, x, i) = −P2,+(−f)(s, x, i) and its closure

P2,−f(s, x, i) = −P2,+

(−f)(s, x, i)


We have the following result.

Lemma 3.6. P2,+f(s, x, i) (resp. P2,−f(s, x, i)) consist of the set of (∂φ(s, x)/∂s, ∂φ(s, x)/∂x, ∂2φ(s, x)/∂x2) where φ ∈ C2([0, T] × R) and f − φ has a global maximum (resp. minimum)at (s, x).

A proof can be found in Fleming and Soner [12].The following result from Crandall et al. [10] is crucial for the proof of the uniqueness.

Theorem 3.7 (Crandall et al. [10]). For i = 1, 2, let Ωi be locally compact subsets of R, and Ω =

Ω1 ×Ω2, and let ui be upper semicontinuous in [0, T] ×Ωi, and P2,+Ωiui(t, x) the parabolic superjet of

ui(t, x), and φ twice continuously differentiable in a neighborhood of [0, T] ×Ω.Set

w(t, x1, x2) = u1(t, x1) + u2(t, x2) (3.7)

for (t, x1, x2) ∈ [0, T] ×Ω, and suppose (t, x1, x2) ∈ [0, T] ×Ω is a local maximum of w − φ relativeto [0, T] ×Ω. Moreover, let us assume that there is an r > 0 such that for everyM > 0 there exists aC such that for i = 1, 2

bi ≤ C whenever(bi, qi, Xi

) ∈ P2,+Ωiui(t, xi),

|xi − xi| +∣∣∣t − t

∣∣∣ ≤ r, |ui(t, xi)| +∣∣qi∣∣ + ‖Xi‖ ≤M.

(3.8)

Then, for each ε > 0, there exists Xi ∈ S(1) = R such that

(1)

(bi,Dxiφ

(t, x), Xi

)∈ P2,+

Ωiui(t, xi)

for i = 1, 2, (3.9)

(2)

−(1ε+∥∥∥D2φ(x)

∥∥∥)I ≤(X1 0

0 X2

)

≤ D2φ(x) + ε(D2φ(x)

)2, (3.10)

(3)

b1 + b2 =∂φ(t, x, y

)

∂t. (3.11)

We have the following maximum principle.

Theorem 3.8 (Comparison Principle). If v1(t, x, i) and v2(t, x, i) are continuous in (t, x) and are,respectively, viscosity subsolution and supersolution of (2.13) with at most a linear growth. Then,

v1(t, x, i) ≤ v2(t, x, i) ∀(t, x, i) ∈ [0, T] × R+ ×M. (3.12)


The following lemma is be very useful in derivation of the maximum principle.

Lemma 3.9. Let CLip([0, T] × R × M) be the set of functions v(s, x, i) on [0, T] × R × M whichare continuous with respect to s and Lipschitz continuous with respect to the variable x. For fixedtδ ∈ [0, T] and α0 ∈ M, let the operator F on R×CLip([0, T]×R×M)×R×R be defined as follows:

F(x, v, β,X) = −12x2σ2(α0)X − xμ(α0)β −Qv(tδ, x, ·)(α0)

−∫

R

(v(tδ, x + γ(α0)xz, α0

) − v(tδ, x, α0) − γ(α0)xz1{|z|<1}(z)β)ν (dz).

(3.13)

Then, there exits a constant C > 0 such that

F(y,w, a(x − y) − by, Y) − F(x, v, a(x − y) + bx,X)

≤ C(x2X − y2Y

)+ Ca

∣∣x − y∣∣2

+ Cb(1 +∣∣∣x2∣∣∣)+Qv(tδ, x, ·)(α0) −Qw

(tδ, y, ·

)(α0)

+∫

Rn

(v(tδ, x + γ(α0)zx, α0

) −w(tδ, y + γ(α0)zy, α0)

−v(tδ, x, α0) +w(tδ, y, α0

))ν(dz),

(3.14)

for any v, w ∈ CLip([0, T] × R ×M) and x, y, a, b, X, Y ∈ R.

Remark 3.10. Theorem 3.8 obviously implies the uniqueness of the viscosity solution of thevariational inequality (2.13). If we assume that (2.13) has two solutions v1 and v2 with lineargrowth, then they are both viscosity subsolution and supersolution of (2.13). Therefore, usingthe fact that v1 is subsolution and v2 is supersolution, Theorem 3.8 implies that v1(t, x, i) ≤v2(t, x, i) for all (t, x, i) ∈ [0, T] × R

+ ×M. And conversely, we also have v2(t, x, i) ≤ v1(t, x, i)for all (t, x, i) ∈ [0, T]×R

+ ×M, since v2 is subsolution and v1 is supersolution. Consequently,we have v1(t, x, i) = v2(t, x, i) for all (t, x, i) ∈ [0, T] × R

+ × M, which confirms the fact thevalue function defined on (2.6) is the unique solution of the variational inequality (2.13).

4. Numerical Example

This example is for a stock which share price roughly around $55 in average; in this examplewe assume that the market has two main movements: an uptrend and a downtrend. Thusthe Markov chain α takes two states M = {1, 2}, where α(t) = 1 denotes the uptrend andα(t) = 2 denotes the downtrend. The transaction fee a = 0.5, the discount rate r = 0.05, thereturn vector is μ = (0.01,−0.01), the volatility vector is σ = (0.4, 0.2), the intensity vector isλ = (0.25, 0.5), the time T = 0.35 (in year), and the generator of the Markov chain is

Q =

(−0.023 0.023

1.023 −1.023

)

. (4.1)


00.10.2

0.3

xt 50 55 60 65 70 75

50

60

70

v(t,x,1) uptrend value function

(a)

50 55 60 65 70 75

50

60

70

00.10.2

0.3

t x

v(t,x,2) downtrend value function

(b)

Figure 1: Value functions v(t, x, 1) for the uptrend and v(t, x, 2) for the downtrend.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

50

55

60

t

x

Continuation for the uptrend

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

50

55

60

t

x

Continuation for the downtrend

(b)

Figure 2: Free boundary curves.

Figures 1 and 2 represent the value function v(t, x, i) computed by solving thenonlinear system of equations (2.13) and the free boundary curves in both trends. Thesecurves divide the plane in two regions. The region below the free boundary curve is the


continuation region, and the region above the free boundary curve is the stopping region. Inother terms, whenever the share price at any given time within the interval [0, T] is belowthe free boundary curve, it is optimal for the investor to hold the stock. And whenever theshare price is above the free boundary, it is optimal for the investor to sell. This selling rule isobviously easy to implement. This example clearly shows that the selling rule derived by ourmethod can be very attractive to practitioners in an automated trading setting.

Appendix

A. Proofs of Results

A.1. Proof of Lemma 3.3

We have

v(β,X(β), α(β))

= supτ∈Λβ,T

E[e−r(τ−β)g(α(τ), X(τ))

],

E[e−r(β−s)v

(β,X(β), α(β))] ≥ sup

τ∈Λβ,T

E[e−r(β−s)e−r(τ−β)g(α(τ), X(τ))

]

≥ supτ∈Λβ,T

E[e−r(τ−s)g(α(τ), X(τ))

]

≥ supτ∈Λγ,T

E[e−r(γ−s)e−r(τ−γ)g(α(τ), X(τ))

]

= E[e−r(γ−s)v

(γ,X(γ), α(γ))]

.

(A.1)

This proves (3.2).Now let us prove (3.4). Since θ ≤ τε for any ε > 0, using (3.2), we have

E[e−r(θ−s)v(θ,X(θ), α(θ))

]≥ E[e−r(τε−s)v(τε, X(τε), α(τε))

]. (A.2)

So using (3.1) and (3.3), we obtain

0 ≤ v(s, x, i) − E[e−r(θ−s)v(θ,X(θ), α(θ))

]≤ v(s, x, i) − E

[e−r(τε−s)v(τε, X(τε), α(τε))

]≤ ε,(A.3)

for any ε > 0. Thus, letting ε → 0, we obtain (3.4).



Given x1 and x2, let X1 and X2 be two solutions of (2.4) with X1(s) = x1 and X2(s) = x2,respectively. For each t ∈ [s, T], we have

X1(t) −X2(t) = x1 − x2 +∫ t

s

(X1(ξ) −X2(ξ))μ(α(ξ))dξ +∫ t

s

(X1(ξ) −X2(ξ))σ(α(ξ))dW(ξ)

+∫ t

s

∫

R

(X1(ξ) −X2(ξ))γ(α(ξ))zN(dξ, dz).

(A.4)

Using the Ito-Levy isometry, we have

E(X1(t) −X2(t))2 ≤ C0|x1 − x2|2 + C1

∫ t

s

E(X1(ξ) −X2(ξ))2dξ + C2

∫ t

s

E(X1(ξ) −X2(ξ))2dξ

+ C3

∫ t

s

∫

R

E(X1(ξ) −X2(ξ))2z2ν (dz)dξ

≤ C0|x1 − x2|2 +max(C1, C2)∫ t

s

E(X1(ξ) −X2(ξ))2dξ

+ C3

∫ t

s

E(X1(ξ) −X2(ξ))2dξ

∫

R

z2ν (dz).

(A.5)

Taking into account (2.5), we can find K < ∞ such that C3∫Rz2ν (dz) < K. The inequality

(A.5) becomes

E(X1(t) −X2(t))2 ≤ C0|x1 − x2|2 +max(C1, C2)∫ t

s

E(X1(ξ) −X2(ξ))2dξ

+K∫ t

s

E(X1(ξ) −X2(ξ))2dξ.

(A.6)

Let C = max(C0, C1, C2, K), then (A.7) becomes

E(X1(t) −X2(t))2 ≤ C|x1 − x2|2 + C∫ t

s

E(X1(ξ) −X2(ξ))2dξ. (A.7)


Applying Gronwall’s inequality, we have

E|X1(t) −X2(t)|2 ≤ C|x1 − x2|2eCt. (A.8)

This implies, in view of Cauchy-Schwarz inequality, that

E|X1(t) −X2(t)| ≤ C|x1 − x2|eCt. (A.9)

Using this inequality, we have

v(s, x1, i) − v(s, x2, i) ≤ supτ∈Λs,T

E[e−r(τ−s)|(X1(τ) − a) − (X2(τ) − a)|

]

≤ supτ∈Λs,T

E[|X1(τ) −X2(τ)|]

≤ C|x1 − x2|eCT .

(A.10)

This implies the (uniform) continuity of v(s, x, i)with respect to x.We next show the continuity of v(s, x, i) with respect to s. Let Xt be the solution of

(2.4) that starts at t = swith X(s) = x and α(s) = i. Let 0 ≤ s ≤ s′ ≤ T , we define

X′(t) = X(t − (s′ − s)),

α′(t) = α(t − (s′ − s)).

(A.11)

It is easy to show that

E(X(t) −X′(t)

)2 ≤ C(s′ − s) for some constant C > 0. (A.12)

Given τ ∈ Λs,T , let τ ′ = τ + (s′ − s). Then, τ ′ ≥ s′ and P(τ ′ > T) → 0 as s′ − s → 0.Let g(t, x) = e−rt(x − a(x)). Then, v(s, x, i) = erssupτ∈Γs,T Eg(τ,X(τ)). It is easy to show

that

∣∣g(s, x) − g(s′, x′)∣∣ ≤ ∣∣x − x′∣∣ + C∣∣x′ − a(x′)∣∣∣∣s − s′∣∣, (A.13)

for some constant C.We define

J(s, x, i, τ) = ersEg(τ,X(τ)). (A.14)


We have

J(s, x, i, τ) =ersEg(τ ′ − (s′ − s), X′(τ ′

))

=ers′Eg(τ ′, X′(τ ′

))+ o(1)

=ers′Eg(τ ′, X′(τ ′

))I{τ ′≤T} + ers

′Eg(τ ′, X′(τ ′

))I{τ ′>T} + o(1)

=J(s′, x, i, τ ′ ∧ T) + o(1),

(A.15)

where o(1) → 0 as s′ − s → 0. It follows that

∣∣v(s′, x, i

) − v(s, x, i)∣∣ ≤ supτ∈Λs,T

∣∣J(s′, x, i, τ ′

) − J(s, x, i, τ)∣∣ −→ 0. (A.16)

Therefore, we have

lims′−s→ 0

∣∣v(s′, x, i

) − v(s, x, i)∣∣ = 0. (A.17)

This gives the continuity of v with respect to s.The joint continuity of v follows from (A.10) and (A.17). Finally, the linear growth

inequality follows from (A.10) and

|v(s, x, i)| ≤ |x| + |v(s, 0, i)| ≤ C(1 + |x|). (A.18)


A.3. Proof of Theorem 3.4

First we prove that v(s, x, i) is a viscosity supersolution of (2.13). Given (s, xs) ∈ [0, T]×R+, let

ψ ∈ C2([0, T]×R+) such that v(t, x, α)−ψ(t, x) has local minimum at (s, xs) in a neighborhood

N(s, xs). We define a function

ϕ(t, x, i) =

⎧⎨

⎩

ψ(t, x) + v(s, xs, αs) − ψ(s, xs) if i = αs,

v(t, x, i) if i /=αs.(A.19)


Let γ ≥ s be the first jump time of α(·) from the initial state αs, and let θ ∈ [s, γ] be such that(t, X(t)) starts at (s, xs) and stays inN(s, xs) for s ≤ t ≤ θ. Moreover, α(t) = αs, for s ≤ t ≤ θ.Using Dynkin’s formula, we have

Es,xs,αse−r(θ−s)ϕ(θ,X(θ), αs) − ϕ(s, xs, αs)

= Es,xs,αs∫θ

s

e−r(t−s)(

− rϕ(t, X(t), αs) +∂ϕ(t, X(t), αs)

∂t+12X2t σ

2(αs)∂2ϕ(t, X(t), αs)

∂x2

+Xtμ(αs)∂ϕ(t, X(t), αs)

∂x+Qϕ(t, X(t), ·)(αs)

+∫

R

(ϕ(t, X(t) + γ(i)X(t)z

) − ϕ(t, X(t)) − γ(i)X(t)z1{|z|<1}(z)∂ϕ

∂x

)

×ν(dz))dt.

(A.20)

Recall that (s, xs) is the minimum of v(t, x, αs) − ψ(t, x) inN(s, xs). For s ≤ t ≤ θ, we have

v(t, Xt, αs) ≥ ψ(t, Xt) + v(s, xs, αs) − ψ(s, xs) = ϕ(t, Xt, αs). (A.21)

Using (A.19) and (A.21), we have

Es,xs,αse−r(θ−s)v(θ,Xθ, αs) − v(s, xs, αs)

≥ Es,xs,αs∫θ

s

e−r(t−s)(− rv(t, X(t), αs)

+∂ψ(t, X(t))

∂t+12X2t σ

2(αs)∂2ψ(t, Xt)

∂x2

+Xtμ(αs)∂ψ(t, Xt)

∂x+Qϕ(t, Xt, ·)(αs)

×∫

R

(ϕ(t, X(t) + γ(i)X(t)z, αs

) − ϕ(t, X(t), αs)

−γ(i)X(t)z1{|z|<1}(z)∂ψ(t, X(t))

∂x

)ν(dz)

)dt.

(A.22)

Moreover, we have

Qϕ(t, Xt, ·)(αs) =∑

β /=αs

qαsβ(ϕ(t, Xt, β

) − ϕ(t, Xt, αs))

≥∑

β /=αs

qαsβ(v(t, Xt, β

) − v(t, Xt, αs))

≥ Qv(t, Xt, ·)(αs).

(A.23)


Combining (A.22) and (A.23), we have

Es,xs,αse−r(θ−s)v(θ,Xθ, αs) − v(s, xs, αs)

≥ Es,xs,αs∫θ

s

e−r(t−s){

− rv(t, X(t), αs) +∂ψ(t, X(t))

∂t+12X2t σ

2(αs)∂2ψ(t, Xt)

∂x2

+Xtμ(αs)∂ψ(t, Xt)

∂x+Qv(t, Xt, ·)(αs)

+∫

R


) − ϕ(t, X(t), αs)

−γ(i)X(t)z1{|z|<1}(z)∂ψ(t, X(t))

∂x

)ν(dz)

}

dt.

(A.24)

It follows from Lemma 3.3 that

Es,xs,αs∫θ

s

e−r(t−s)(

−rv(t, X(t), αs) +∂ψ(t, X(t))

∂t+12X2(t)σ2(αs)

∂2ψ(t, X(t))∂x2

+X(t)μ(αs)∂ψ(t, X(t))

∂x+Qv(t, Xt, ·)(αs)

+∫

R


) − ϕ(t, X(t), αs)

−γ(i)X(t)z1{|z|<1}(z)∂ψ(t, X(t))

∂x

)ν(dz)

)dt ≤ 0.

(A.25)

Dividing both sides by Eθ > 0 and sending θ → s lead to

− rv(s, xs, αs) +∂ψ(s, xs)

∂t+12x2sσ

2(αs)∂2ψ(s, xs)

∂x2+ xsμ(αs)

∂ψ(s, xs)∂x

+Qv(s, xs, ·)(αs)

+∫

R

(v(s, xs + γ(i)xsz, αs

) − v(s, xs, αs) − γ(i)xsz1{|z|<1}(z)∂ψ(s, xs)

∂x

)ν(dz) ≤ 0.

(A.26)

By definition, v(s, x, i) ≥ x − a(x). The supersolution inequality follows from this inequalityand (A.27).

Now, let us prove the subsolution inequality, namely, that let φ ∈ C1,2([s, T] × R+) and

v(t, x, αs) − φ(t, x) has a local maximum at (s, xs) ∈ [s, T] × R+, then we can assume without

loss of generality that v(s, xs, αs) − φ(s, xs) = 0.Define

Φ(t, x, i) =

⎧⎨

⎩

φ(t, x) if i = αs,

v(t, x, i) if i /=αs.(A.27)


Let γ be the first jump time of α(·) from the state αs, and let θ0 ∈ [s, γ] be such that (t, X(t))starts at (s, xs) and stays in N(s, xs) for s ≤ t ≤ θ0. Since θ0 ≤ γ we have α(t) = αs, fors ≤ t ≤ θ0, and let τD be the optimal stopping time, and for s ≤ θ ≤ min(τD, θ0), we have fromLemma 3.3 (The appendix)

v(s, xs, αs) ≤ Es,xs,αs[e−r(θ−s)v(θ,X(θ), α(θ))

]. (A.28)

Moreover, since v(s, xs, αs) − φ(s, xs) = 0 and attains its maximum at (s, xs) inN(s, xs), then

v(θ,X(θ), α(θ)) ≤ φ(θ,X(θ)). (A.29)

Thus, we also have

v(θ,X(θ), α(θ)) ≤ Φ(θ,X(θ), α(θ)). (A.30)

This implies, using Dynkin’s formula, that

Es,xs,αse−r(θ−s)v(θ,X(θ), αs)

≤ Es,xs,αse−r(θ−s)Φ(θ,X(θ), αs)

= Φ(s, xs, αs) + Es,xs,αs∫θ

s

e−r(t−s)[∂φ(t, X(t))

∂t− rΦ(t, X(t), α(t))

+X(t)μ(αs)∂φ(t, X(t))

∂x+QΦ(t, X(t), ·)(αs)

+12X(t)2σ2(αs)

∂2φ(t, X(t))∂x2

+∫

R

(Φ(t, X(t) + γ(i)X(t)z, αs

) −Φ(t, X(t), αs)

−γ(i)X(t)z1{|z|<1}(z)∂φ(t, X(t))

∂x

)ν(dz)

]dt.

(A.31)

Note that

QΦ(t, X(t), ·)(αs) =∑

β /=αsqαsβ(v(t, X(t), β

) − φ(t, X(t)))

≤∑

β /=αs

qαsβ(v(t, X(t), β

) − v(t, X(t), αs))

≤ Qv(t, X(t), ·)(αs).

(A.32)


Using (A.27) and (A.32), we obtain

Es,xs,αse−r(θ−s)v(θ,X(θ), αs)

≤ Es,xs,αse−rθ−sΦ(θ,X(θ), αs)

= φ(s, xs) + Es,xs,αs∫θ

s

e−r(t−s)[∂φ(t, X(t))

∂t+X(t)μ(αs)

∂φ(t, X(t))∂x

− rv(t, X(t), αs)

+12X(t)2σ2(αs)

∂2φ(t, X(t))∂x2

+Qv(t, X(t), ·)(αs)

+∫

R


) −Φ(t, X(t), αs)

−γ(i)X(t)z1{|z|<1}(z)∂φ(t, X(t))

∂x

)ν(dz)

]dt.

(A.33)

Recall that, v(s, xs, αs) = φ(s, xs) by assumption. From (A.28), we deduce

0 ≤ E,xs,αse−r(θ−s)v(θ,X(θ), αs) − φ(s, xs)

≤ Es,xs,αs∫θ

s

e−rt[

− rv(t, X(t), αs) +∂φ(t, X(t))

∂t+12X(t)2σ2(αs)

∂2φ(X(t), αs)∂x2

+X(t)μ(αs)∂φ(X(t), αs)

∂x+Qv(t, X(t), ·)(αs)

+∫

R


)

−Φ(t, X(t), αs) − γ(i)X(t)z1{|z|<1}(z)∂φ(t, X(t))

∂x

)ν(dz)

]

dt.

(A.34)

Dividing the last inequality by Eθ > 0 and sending θ ↓ s give

rv(s, xs, αs) −∂φ(s, xs)

∂t− 12x2sσ

2(αs)∂2φ(xs, αs)

∂x2− xsμ(αs)

∂φ(xs, αs)∂x

−Qv(s, xs, ·)(αs)

−∫

R

(v(s, xs + γ(i)xsz, αs

) − v(s, xs, αs) − γ(i)xsz1{|z|<1}(z)∂φ(s, xs)

∂x

)ν(dz) ≤ 0.

(A.35)

This gives the subsolution inequality. Therefore, v(t, x, α) is a viscosity solution of (2.13).



Let v,w ∈ CLip([0, T] × R ×M) and x, y, λ, a, b, X, Y ∈ R, then we have


= −12y2σ2(α0)Y − yμ(α0)

(a(x − y) − by) −Qw(tδ, y, ·

)(α0)

−∫

R

(w(tδ, y + γ(α0)yz, α0

) −w(tδ, y, α0) − γ(α0)yz1{|z|<1}(z)

[a(x − y) − by])ν(dz)

+12x2σ2(α0)X + xμ(α0)

(a(x − y) + bx) +Qv(tδ, x, ·)(α0)

+∫

R


) − v(tδ, x, α0) − γ(α0)xz1{|z|<1}(z)[a(x − y) + bx])ν(dz)

=12σ2(α0)

(x2X−y2Y

)+μ(α0)

(a(x−y)2+b

(x2+y2

))+Qv(tδ, x, ·)(α0)−Qw

(tδ, y, ·

)(α0)

+∫

R

([v(tδ, x + γ(α0)xz, α0

) −w(tδ, y + γ(α0)yz, α0)]

−[v(tδ, x, α0) −w(tδ, y, α0

)] − zγ(α0)1{|z|<1}(z)[a(x − y)2 + b

(x2 + y2

)])ν(dz)

≤ μ(α0)a∣∣x−y∣∣2+μ(α0)b

(x2+y2

)+12σ2(α0)

(x2X−y2Y

)+Qv(tδ, x, ·)(α0)−Qw

(tδ, y, ·

)(α0)

+∫

R

([v(tδ, x + γ(α0)xz, α0

) −w(tδ, y + γ(α0)yz, α0)]

−[v(tδ, x, α0) −w(tδ, y, α0

)]+∣∣zγ(α0)1{|z|<1}(z)

∣∣[∣∣∣a(x − y)2|+ |b

(x2 + y2

)∣∣∣])ν(dz).

(A.36)

Note that from the Levy-Khintchine inequality (2.3), one can prove∫{|z|<1} |z|ν(dz) < ∞;

therefore, there exists a constant C > 0 such that


≤ C(a∣∣x − y∣∣2 + b

(x2 + y2

)+(x2X − y2Y

))+Qv(tδ, x, ·)(α0) −Qw

(tδ, y, ·

)(α0)

+∫

R

([v(tδ, x+γ(α0)xz, α0

)−w(tδ, y+γ(α0)yz, α0)]−[v(tδ, x, α0)−w

(tδ, y, α0

)])ν(dz).

(A.37)

This proves (3.14).Now, let us proof the theorem.


A.5. Proof of Theorem 3.8

For any 0 < δ < 1 and 0 < η < 1, we define

Φ(t, x, y, i

)= v1(t, x, i) − v2

(t, y, i

) − 1δ

∣∣x − y∣∣2 − ηe(T−t)

(x2 + y2

),

φ(t, x, y

)=

1δ

∣∣x − y∣∣2 + ηe(T−t)

(x2 + y2

).

(A.38)

Note that v1(t, x, i) and v2(t, x, i) satisfy the linear growth. Then, we have for each i ∈ M

lim|x|+|y|→∞

Φ(t, x, y, i

)= −∞ (A.39)

and since Φ is a continuous in (t, x, y), therefore it has a global maximum at a point(tδ, xδ, yδ, α0). Observe that

Φ(tδ, xδ, xδ, α0) + Φ(tδ, yδ, yδ, α0

) ≤ 2Φ(tδ, xδ, yδ, α0

). (A.40)

It implies

v1(tδ, xδ, α0) − v2(tδ, xδ, α0) − 2ηe(T−tδ)(x2δ

)+ v1(tδ, yδ, α0

) − v2(tδ, yδ, α0

) − 2ηe(T−tδ)(y2δ

)

≤ 2v1(tδ, xδ, α0) − 2v2(tδ, yδ, α0

) − 2δ

∣∣xδ − yδ∣∣2 − 2ηe(T−tδ)

(x2δ + y

2δ

).

(A.41)

Then,

− v2(tδ, yδ, α0

) − 2e(T−tδ)η(x2δ

)+ v1(tδ, xδ, α0) − 2ηe(T−tδ)

(y2δ

)

≤ v1(tδ, xδ, α0) − v2(tδ, yδ, α0

) − 2δ

∣∣xδ − yδ∣∣2 − 2ηe(T−tδ)

(x2δ + y

2δ

).

(A.42)

Consequently, we have

2δ

∣∣xδ − yδ∣∣2 ≤ (v1(tδ, xδ, α0) − v1

(tδ, yδ, α0

))+(v2(tδ, xδ, α0) − v2

(tδ, yδ, α0

)). (A.43)

By the linear growth condition, we know that there exist K1, K2 such that v1(t, x, i) ≤ K1(1 +|x|) and v2(t, x, i) ≤ K2(1 + |x|). Therefore, there exists C such that we have

2δ

∣∣xδ − yδ∣∣2 ≤ C(1 + |xδ| +

∣∣yδ∣∣). (A.44)


So,

∣∣∣x0

δ1 − x0δ2

∣∣∣2 ≤ δC

(1 +∣∣∣x0

δ1

∣∣∣κ1+∣∣∣x0

δ1

∣∣∣κ2)

. (A.45)

We also have Φ(s, 0, 0, α0) ≤ Φ(tδ, xδ, yδ, α0) and |Φ(s, 0, 0, α0)| ≤ K(1 + |xδ| + |yδ|). This leadsto

ηe(−tδ)(x2δ + y

2δ

)≤ v1(tδ, xδ, α0) − v2

(tδ, yδ, α0

) − 1δ

∣∣xδ − yδ

∣∣2 −Φ(s, 0, 0, α0)

≤ 3C(1 + |xδ| +

∣∣yδ∣∣).

(A.46)

It comes that

ηe(T−tδ)(x2δ+ y2

δ

)

(1 + |xδ| +

∣∣yδ∣∣) ≤ 3C, (A.47)

therefore there exists Cη such that

|xδ| +∣∣yδ∣∣ ≤ Cη, tδ ∈ [s, T]. (A.48)

The inequality (A.48) implies the sets {xδ, δ > 0} and {yδ, δ > 0} are bounded by Cη

independent of δ, so we can extract convergent subsequences that we also denote (xδ)δ, (yδ)δ,and (tδ)δ. Moreover, from the inequality (A.45), it comes that the exists x0 such that

limδ→ 0

xδ = x0 = limδ→ 0

yδ, limδ→ 0

tδ = t0. (A.49)

Using (A.43) and the previous limit, we obtain

limδ→ 0

2δ

∣∣xδ − yδ

∣∣2 = 0. (A.50)

Φ achieves its maximum at (tδ, xδ, yδ, α0), so by the Theorem 3.7 for each ε > 0, there existb1δ, b2δ, Xδ, and Yδ such that

(b1δ,

2δ

(xδ − yδ

)+ 2ηe(T−t)xδ,Xδ

)∈ P2,+

v1(tδ, xδ, α0), (A.51)

(−b2δ,− 2

δ

(xδ − yδ

)+ 2ηe(T−t)yδ,−Yδ

)∈ P2,+(−v2

(tδ, yδ, α0

)). (A.52)

But, we know that

P2,+(−v2(tδ, yδ, α0

))= −P2,−

v2(tδ, yδ, α0

). (A.53)


Therefore, we obtain

(b2δ,

2δ

(xδ − yδ

) − 2ηe(T−t)yδ, Yδ)

∈ P2,−v2(tδ, yδ, α0

). (A.54)

Equation (A.51) implies by the definition of the viscosity solution that

min[rv1(tδ, xδ, α0)−b1δ− 1

2x2δσ

2(α0)Xδ−xδμ(α0)(2δ

(xδ−yδ

)+2ηe(T−t)xδ

)−Qv1(tδ, xδ, ·)(α0)

−∫

R

(v1(tδ, xδ + γ(α0)xδz, α0

) − v1(tδ, xδ, α0) − γ(α0)xδz1{|z|<1}(z)

×(2δ

(xδ − yδ

)+ 2ηe(T−t)xδ

))ν(dz), v1(tδ, xδ, α0) − (xδ − a)

]≤ 0.

(A.55)

Consequently, we have two cases; either

v1(tδ, xδ, α0) − (xδ − a) ≤ 0 (A.56)

or

rv1(tδ, xδ, α0) − b1δ − 12x2δσ

2(α0)Xδ − xδμ(α0)(2δ

(xδ − yδ

)+ 2ηe(T−t)xδ

)−Qf(s, xδ, ·)(α0)

−∫

R

(v1(tδ, xδ + γ(α0)xδz

) − v1(tδ, xδ)

−γ(α0)xδz1{|z|<1}(z) ×(2δ

(xδ − yδ

)+ 2ηe(T−t)xδ

))ν(dz) ≤ 0.

(A.57)

First of all, we assume that v1(tδ, xδ, α0) − (xδ − a) ≤ 0. And similarly, (A.54) implies by thedefinition of the viscosity solution that

min[rv2(tδ, yδ, α0

)−b2δ− 12y2δσ

2(α0)Yδ−yδμ(α0)(2δ

(xδ−yδ

)−2ηe(T−t)yδ)−Qv2

(tδ, yδ, ·

)(α0)

−∫

R

(v2(tδ, yδ + γ(α0)yδz, α0

) − v2(tδ, yδ, α0

) − γ(α0)yδz1{|z|<1}(z)

×(2δ

(xδ − yδ

) − 2ηe(T−t)yδ))

ν(dz), v2(tδ, yδ, α0

) − (yδ − a)]≥ 0.

(A.58)

Therefore, we have

v2(tδ, yδ, α0

) − (yδ − a) ≥ 0. (A.59)


It comes that

v1(tδ, xδ, α0) − v2(tδ, yδ, α0

) − (xδ − yδ) ≤ 0. (A.60)

Letting δ → 0, we obtain

v1(t0, x0, α0) − v2(t0, x0, α0) ≤ 0. (A.61)

Note that the function Φ reaches its maximum at (tδ, xδ, yδ, α0). It follows that for all x ∈ R,t ∈ [s, T], and i ∈ M, we have

v1(t, x, i) − v2(t, x, i) − 2ηe(T−t)x2 = Φ(x, x, i) ≤ Φ(tδ, xδ, yδ, α0

)

≤ v1(tδ, xδ, α0) − v2(tδ, yδ, α0

)

− ηe(T−tδ)(x2δ + y

2δ

).

(A.62)

Again letting δ → 0 and using (A.61), we obtain

v1(t, x, i) − v2(t, x, i) − 2ηe(T−t)x2 ≤ v1(t0, x0, α0) − v2(t0, x0, α0) − 2ηe(T−t0)(x0)2 ≤ 0. (A.63)

so, we have

v1(, x, i) − v2(t, x, i) ≤ 2ηe(T−t)x2. (A.64)

Second of all, let assume that

rv1(tδ, xδ, α0) − b1δ − 12x2δσ

2(α0)Xδ − xδμ(α0)(2δ

(xδ − yδ

)+ 2ηe(T−t)xδ

)−Qf(s, xδ, ·)(α0)

−∫

R

(v1(tδ, xδ + γ(α0)xδz

) − v1(tδ, xδ) − γ(α0)xδz1{|z|<1}(z)(2δ

(xδ − yδ

)+ 2ηe(T−t)xδ

))

× ν(dz) ≤ 0,(A.65)

and, moreover, we have

rv2(tδ, yδ, α0

) − b2δ − 12y2δσ

2(α0)Yδ − yδμ(α0)(2δ

(xδ − yδ

) − 2ηe(T−tδ)yδ)−Qv2

(tδ, yδ, ·

)(α0)

−∫

R

(v2(tδ, yδ + γ(α0)yδz, α0

) − v2(tδ, yδ, α0

) − γ(α0)yδz1{|z|<1}(z)

×(2δ

(xδ − yδ

) − 2ηe(T−t)yδ))

ν(dz) ≥ 0.

(A.66)


Let us use the operator F(x, u, β,M) defined on the Lemma 3.9; thus,

F(x, v, β,M) = −12x2σ2(α0)M − xμ(α0)β −Qv(tδ, x, ·)(α0)

−∫

R


) − v(tδ, x, α0) − γ(α0)xzβ)ν(dz).

(A.67)

Using the operator F, thus (A.65) becomes

rv1(tδ, xδ, α0) + F(xδ, v1,

2δ

(xδ − yδ

)+ 2xδηe(T−tδ), Xδ

)− b1δ ≤ 0, (A.68)

and (A.66) becomes

rv2(tδ, yδ, α0

)+ F(yδ, v2,

2δ

(xδ − yδ

) − 2yδηe(T−tδ), Yδ)− b2δ ≥ 0. (A.69)

Combining the last two inequalities, we obtain

r(v1(tδ, xδ, α0) − v2

(tδ, yδ, α0

)) ≤ F(yδ, v2,

2δ

(xδ − yδ

) − 2yδηe(T−tδ), Yδ)

− F(xδ, v1,

2δ

(xδ − yδ


)+ b2δ − b1δ.

(A.70)

From Lemma 3.9, there exists a constant C > 0 such that

F(yδ, v2,

2δ

(xδ − yδ

) − 2yδηe(T−tδ), Yδ)− F(xδ, v1,

2δ

(xδ − yδ


)

≤ C(2δ

∣∣xδ − yδ∣∣2 + 2ηe(T−tδ)

(x2δ + y

2δ

)+(x2δXδ − y2

δYδ))

+Qv(tδ, x, ·)(α0)

−Qw(tδ, y, ·)(α0) +

∫

R

([v(tδ, x + γ(α0)xz, α0

) −w(tδ, y + γ(α0)yz, α0)]

−[v(tδ, x, α0) −w(tδ, y, α0

)])ν(dz).

(A.71)

Recall that Qv(t, x, ·)(i) =∑j /= i v(t, x, j) − v(t, x, i), so we have

Qv1(tδ, xδ, ·)(α0) −Qv2(tδ, yδ, ·

)(α0) =

∑

i /=αδ

[v1(tδ, xδ, i) − v1(tδ, xδ, α0)]

− [v2(tδ, yδ, i

) − v2(tδ, yδ, α0

)].

(A.72)


Since Φ attains its maximum at (tδ, xδ, yδ, α0), we have Φ(tδ, xδ, yδ, i) ≤ Φ(tδ, xδ, yδ, α0);therefore

v1(tδ, xδ, i) − v2(tδ, yδ, i

) − 1δ

∣∣xδ − yδ

∣∣2 + ηeT−tδ

(|xδ|2 +

∣∣yδ∣∣2)

≤ v1(tδ, xδ, α0) − v2(tδ, yδ, α0

) − 1δ

∣∣xδ − yδ

∣∣2 − ηeT−tδ

(|xδ|2 +

∣∣yδ∣∣2).

(A.73)

Thus, we have

v1(tδ, xδ, i) − v2(tδ, yδ, i

) ≤ v1(tδ, xδ, α0) − v2(tδ, yδ, α0

). (A.74)

Consequently, from (A.74) it comes that (A.72) implies

Qv1(tδ, xδ, ·)(α0) −Qv2(tδ, yδ, ·

)(α0) ≤ 0. (A.75)

Note that from (3.11), we have

b1δ − b2δ =∂φ(tδ, xδ, yδ

)

∂t= ηe(T−tδ)

((xδ)2 +

(yδ)2)

. (A.76)

Therefore, we have

r(v1(tδ, xδ, α0) − v2

(tδ, yδ, α0

)) ≤ C(2δ

∣∣xδ − yδ∣∣2 + 2ηe(T−tδ)

(x2δ + y

2δ

)+(x2δXδ − y2

δYδ))

+∫

R

([v1(tδ, xδ + γ(α0)xδz, α0

) − v2(tδ, yδ + γ(α0)yδz, α0

)]

−[v1(tδ, xδ, α0) − v2(tδ, yδ, α0

)])ν(dz).

(A.77)

Similarly, for any z ∈ R, we have

Φ(tδ, xδ + γ(α0)xδz, yδ + γ(α0)yδz, α0

) ≤ Φ(tδ, xδ, yδ, α0

); (A.78)


consequently, we derive that

∫

R

[v1(tδ, xδ + γ(α0)xδz, α0

) − v2(tδ, yδ + γ(α0)yδz, α0

)]ν(dz)

≤∫

R

[v1(tδ, xδ, αδ) − v2

(tδ, yδ, αδ

)+1δ

∣∣(xδ + γ(α0)xδz

) − (yδ + γ(α0)yδz)∣∣2

+ηeT−tδ(∣∣xδ + γ(α0)xδz

∣∣2 − ∣∣yδ + γ(α0)yδz

∣∣2)]ν(dz)

≤∫

R

[v1(tδ, xδ, α0) − v2

(tδ, yδ, α0

)+1δ

(∣∣xδ − yδ

∣∣2 +∣∣γ(α0)xδz − γ(α0)yδz

∣∣2)

+ηeT−tδ(|xδ|2 +

∣∣γ(α0)xδz

∣∣2 +∣∣yδ∣∣2 +∣∣γ(α0)yδz

∣∣2)]ν(dz).

(A.79)

Therefore, using (2.3), we obtain

∫

R

([v1(tδ, xδ + γ(α0)xδz, α0

) − v2(tδ, yδ + γ(α0)yδz, α0

)]

−[v1(tδ, xδ, α0) − v2(tδ, yδ, α0

)])ν(dz)

≤∫

R

[1δ

(∣∣xδ − yδ∣∣2 +∣∣γ(α0)xδz − γ(α0)yδz

∣∣2)

+ηeT−tδ(|xδ|2 +

∣∣γ(α0)xδz∣∣2 +∣∣yδ∣∣2 +∣∣γ(α0)yδz

∣∣2)]ν(dz)

≤ C(1δ

∣∣xδ − yδ∣∣2 + ηeT−tδ

(1 + |xδ|2 +

∣∣yδ∣∣2))

,

(A.80)

for some constant C > 0. Taking into account (A.75) and (A.80), thus (A.77) becomes

r(v1(tδ, xδ, α0) − v2

(tδ, yδ, α0

)) ≤ C(2δ

∣∣xδ − yδ∣∣2 + 2ηe(T−tδ)

(x2δ + y

2δ

)+∣∣∣x2

δXδ − y2δYδ∣∣∣).

(A.81)

We know from the Maximum principle that

−(1ε+∥∥∥∥D

2(x,y)φ

(tδ, xδ, yδ

)∥∥∥∥

)I ≤(Xδ 0

0 −Yδ

)

≤ D2(x,y)φ

(tδ, xδ, yδ

)+ ε(D2(x,y)φ

(tδ, xδ, yδ

))2

.

(A.82)


Moreover,

D2(x,y)φ

(tδ, xδ, yδ

)=

2δ

(1 −1−1 1

)

+ 2ηe(T−tδ)(1 0

0 1

)

,

(D2(x,y)φ

(tδ, xδ, yδ

))2

=8δ2

(1 −1−1 1

)

+8ηe(T−tδ)

δ

(1 −1−1 1

)

+ 4η2e2(T−tδ)(1 0

0 1

)

=8 + 8ηδe(T−tδ)

δ2

(1 −1−1 1

)

+ 4η2e2(T−tδ)(1 0

0 1

)

.

(A.83)

Note that

(xδ)2Xδ −(yδ)2Yδ =

(xδ, yδ

)(Xδ 0

0 −Yδ

)(xδ

yδ

)

≤ (xδ, yδ)[2δ

(1 −1−1 1

)

+(2ηe(T−tδ) + 4εη2e2(T−tδ)

)(1 0

0 1

)

+ε8 + 8ηδe(T−tδ)

δ2

(1 −1−1 1

)](xδ

yδ

)

.

(A.84)

Letting η → 0, we obtain

(xδ)2Xδ −(yδ)2Yδ ≤ (xδ, yδ

)[(

2δ+ ε

8δ2

)( 1 −1−1 1

)](xδ

yδ

)

. (A.85)

Take ε = δ/4, this leads to

(xδ)2Xδ −(yδ)2Yδ ≤ (xδ, yδ

)[4δ

(1 −1−1 1

)](xδ

yδ

)

=4δ

(xδ − yδ

)2. (A.86)

Using (A.50), we obtain

lim supδ↓0

(xδ)2Xδ −(yδ)2Yδ ≤ lim sup

δ↓0

(xδ, yδ

)[4δ

(1 −1−1 1

)](xδ

yδ

)

= lim supδ↓0

4δ

(xδ − yδ

)2 = 0.

(A.87)

Letting η → 0 in (A.81), we have

r(v1(tδ, xδ, α0) − v2

(tδ, yδ, α0

)) ≤ C(2δ

∣∣xδ − yδ∣∣2 +(x2δXδ − y2

δYδ))

(A.88)


and taking the lim sup as δ goes to zero and using (A.87), we obtain

r(v1(t0, x0, α0) − v2(t0, x0, α0)) ≤ 0. (A.89)

Recall that (tδ, xδ, yδ, α0) is maximum of Φ. Then, for all x ∈ R, t ∈ [s, T], and for all i ∈ M,we have

Φ(t, x, x, i) ≤ Φ(tδ, xδ, yδ, α0

). (A.90)

It comes that

v1(t, x, i) − v2(t, x, i) − 2ηe(T−t)x2 ≤ v1(tδ, xδ, α0) − v2(tδ, yδ, α0

) − 2ηe(T−tδ)(x2δ + y

2δ

).

(A.91)

Letting δ → 0, we obtain

v1(t, x, i) − v2(t, x, i) − 2ηe(T−t)x2 ≤ v1(t0, x0, α0) − v2(t0, x0, α0) − 2ηe(T−t)x20. (A.92)

Using (A.89), we have

v1(t0, x0, α0) − v2(t0, x0, α0)≤ 0. (A.93)

Therefore, using (A.92), we conclude that

v1(t, x, i) − v2(t, x, i) − 2ηe(T−t)x2 ≤ v1(t0, x0, α0) − v2(t0, x0, α0) − 2ηe(T−t0)x20 ≤ 0. (A.94)

Letting η → 0 in (A.64) and the previous inequality, we have

v1(t, x, i) ≤ v2(t, x, i). (A.95)

This completes the proof of the theorem.

Acknowledgment

The author is grateful to the editor and the referee for their helpful suggestions andcomments.

References

[1] Q. Zhang, “Stock trading: an optimal selling rule,” SIAM Journal on Control and Optimization, vol. 40,no. 1, pp. 64–87, 2001.

[2] M. Pemy and Q. Zhang, “Optimal stock liquidation in a regime switching model with finite timehorizon,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 537–552, 2006.


[3] H. P. McKean, “A free boundary problem for the heat equation arising from a problem inmathematical economics,” Industrial Management Review, vol. 60, pp. 32–39, 1960.

[4] P. A. Samuelson, “Rational theory of warrant pricing,” Industrial Management Review, vol. 6, pp. 13–32,1995.

[5] B. ØKsendal, Stochastic Differential Equations, Springer, New York, NY, USA, 1998.[6] X. Guo and Q. Zhang, “Optimal selling rules in a regime switching model,” Institute of Electrical and

Electronics Engineers Transactions on Automatic Control, vol. 50, no. 9, pp. 1450–1455, 2005.[7] N. V. Krylov, Controlled Diffusion Processes, vol. 14 of Applications of Mathematics, Springer, Berlin,

Germany, 1980, Translated from the Russian by A. B. Arie.[8] H. Pham, “Optimal stopping of controlled jump diffusion processes: a viscosity solution approach,”

Journal of Mathematical Systems, Estimation, and Control, vol. 8, no. 1, pp. 1–27, 1998.[9] M. Pemy, Option pricing under regime witching, Ph.D. thesis, University of Georgia, Athens, Ga, USA,

2005.[10] M. G. Crandall, H. Ishii, and P.-L. Lions, “User’s guide to viscosity solutions of second order partial

differential equations,” Bulletin of the American Mathematical Society, vol. 27, no. 1, pp. 1–67, 1992.[11] G. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation

Approach, vol. 37, Springer, New York, NY, USA, 1998.[12] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, vol. 25 of Stochastic

Modelling and Applied Probability, Springer, New York, NY, USA, 2nd edition, 2006.


Research ArticleSemiconservative Systems of IntegralEquations with Two Kernels

N. B. Yengibaryan and A. G. Barseghyan

Institute of Mathematics of National Academy of Sciences of Armenia,24b Marshal Baghramian Avenue, 0019 Yerevan, Armenia

Correspondence should be addressed to N. B. Yengibaryan, [email protected]

Received 3 February 2011; Revised 23 April 2011; Accepted 30 May 2011


Copyright q 2011 N. B. Yengibaryan and A. G. Barseghyan. This is an open access articledistributed under the Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.

The solvability and the properties of solutions of nonhomogeneous and homogeneous vectorintegral equation f(x) = g(x) +

∫∞0 k(x − t)f(t)dt +

∫0−∞ T(x − t)f(t)dt, where K, T are n × n

matrix valued functions, n ≥ 1, with nonnegative integrable elements, are considered in onesemiconservative (singular) case, where the matrix A =

∫∞−∞K(x)dx is stochastic one and

the matrix B =∫∞−∞ T(x)dx is substochastic one. It is shown that in certain conditions the

nonhomogeneous equation simultaneously with the corresponding homogeneous one possessespositive solutions.

1. Introduction: Problem Statement

Consider the scalar or vector integral equations on the whole line with two kernels (see [1–4]):

f(x) = g(x) +∫∞

0K(x − t)f(t)dt +

∫0

−∞T(x − t)f(t)dt , −∞ < x <∞ , (1.1)

where the kernel-functions K(x), T(x) are matrix-valued functions with nonnegative ele-ments; g and f are the given and sought-for column vectors (vectorfunctions); respectively.Assume that

K, T ∈ Ln×n, g ∈ Ln, K, T, g ≥ O. (1.2)

Here Ln×n is the space of n × n- (n ≥ 1) order matrix-valued functions, and Ln is the spaceof column vectors, with components in Lebesgue space L ≡ L1(−∞,∞). The zero vector or


matrix is denoted by O. The inequalities between the matrices or vectors, the operation ofintegration, and some other operations shall be treated componentwise.

Denote by ς the following n-dimensional row vector:

ς = (1, 1, . . . , 1). (1.3)

Let C ≥ O be an n × nmatrix. If

ςC = ς, (1.4)

then the matrix C is a stochastic one (accurate within transpose, see [5]). If

ςC ≤ ς, (1.5)

then the matrix C is substochastic to a wide extent. We shall call the matrix C reallysubstochastic, if ςC ≤ ς, ςC /= ς and uniform substochastic if there exist μ ∈ [0, 1) such that

ςC ≤ μς, 0 ≤ μ < 1. (1.6)

Let us introduce the following n × nmatrices A,B ≥ O, related with the equation (1.1):

A =∫∞

−∞K(x)dx, B =

∫∞

−∞T(x)dx. (1.7)

We shall call the kernelK conservative, dissipative, or uniform dissipative if thematrixA is stochastic, really substochastic, or uniform substochastic, respectively. We shall useanalogous names to the kernel T .

We shall call (1.1) semiconservative, if one of the kernels K,T is conservative and theother is dissipative. Without the loss of generality, one can assume that

ςA = ς, ςB ≤ ς, ςB /= ς. (1.8)

In the uniform semiconservative case of (1.1) we have

ςA = ς, ςB ≤ μς, 0 ≤ μ < 1, (1.9)

whereas in the conservative case, both of the kernels K, T are assumed to be conservative.If T = O, then (1.1) is reduced to the well-known Wiener-Hopf integral equation:

ϕ(x) = h(x) +∫∞

0K(x − t)ϕ(t)dt, x > 0, (1.10)

Here ϕ = f |[0,∞) and h = g|[0,∞) are restrictions on [0,∞) of f and g, respectively.


The theory of the scalar and vector conservative Wiener-Hopf equation (1.10) (whereK is the conservative one) passed a long way of development. Many (conservative) physicalprocesses in homogeneous half-space are described by such equations. They are of essentialinterest in the radiative transfer (RT), kinetic theory of Gases (see [6, 7]), in the mathematicaltheory of stochastic processes, and so forth.

In the RT, the conservative equation (1.10) corresponds to the absence of losses of theradiation inside media (case of pure scattering). However, such losses occur as a result of anexit of radiation from media. In case of the dissipative one, there are losses inside media aswell.

Equation (1.1) with two kernels arises in some more general (and more complicated)problems, where the physical processes occur in the infinite media, consisting of two adjacenthomogeneous half-spaces (see [7]). In each of these half-spaces, the processes may bedissipative or conservative. Another area of applications is connected with the RT in theatmosphere-ocean system.

In the theory of RT, the free term g in (1.1) plays the role of initial sources of radiation.The conservative and semiconservative cases belong to the singular cases of (1.1). In thesecases, the unique solvability of (1.1) in the “standard” functional spaces Lnp (1 ≤ p ≤ ∞) isviolated.

A number of results concerning to the scalar conservative equation (1.1) have beenobtained by Arabadzhyan [3]. The systems of conservative or semiconservative equationswith two kernels have not ever been treated.

The present paper is devoted to the solvability and the properties of the solutions of thenonhomogeneous and homogeneous vector equation (1.1). The main attention will be paidto the uniform semiconservative case (1.9). It will be shown that in certain conditions boththe nonhomogeneous equation (1.1) and the corresponding homogeneous equation possesspositive locally integrable solutions.

2. Auxiliary Propositions

2.1. Integral Operators

Let (a, b) ⊂ (−∞,∞). Consider Banach space (B-space) L(a, b) ≡ L1(a, b) and thecorresponding B-space Ln(a, b) of vector-valued functions (vector columns) f = (f1, . . . , fn)

T .Here T is a sign of the transpose. The norm in Ln(a, b) is defined by

∥∥f∥∥ =

n∑

k=1

∥∥fk∥∥L(a,b) = ς

∫b

a

∣∣f(x)∣∣dx. (2.1)

Consider the linear topological space LLoc[0,∞) of the functions, which are integrableon each finite interval (0, r), r < ∞.The space LnLoc[0,∞) possesses the topology of thecomponentwise convergence.

The unit operator in each of spaces introduced above is denoted by I. Let Ωn be thefollowing class of matrix convolution operators on the whole line: if U ∈ Ωn, then

ϕ(x) = Uf(x) =∫∞

−∞U(x − t)f(t)dt, U ∈ Ln×n. (2.2)

The operator U ∈ Ωn acts in the spaces Ln, Lnp (1 ≤ p ≤ ∞), and in some other spaces ofvector-valued functions.


The class Ωn is an algebra where the kernel function of the operator product is theconvolution of the kernel functions of the factors.

Let us estimate the norm of operator U ∈ Ωn in the B-space Ln. Let C ≥ O be thefollowing n × n matrix: C = (ckm) =

∫∞−∞ |U(x)|dx. Taking the (componentwise) modulus in

(2.2) and integrating on (−∞,∞), we come to the following inequality:

∫∞

−∞

∣∣ϕ(x)

∣∣dx ≤ C

∫∞

−∞

∣∣f(t)

∣∣dt. (2.3)

Multiplying this inequality on the left by the vector ς, we come to the followinginequality:

∥∥∥Uf

∥∥∥ ≤ γ∥∥f∥∥, where γ = max

k

n∑

m=1

ckm. (2.4)

From here the estimate follows:

∥∥∥U∥∥∥ ≤ γ. (2.5)

Let us introduce the projectors (projection operators) P±, acting in the spaces ofsummable or locally summable functions on (−∞,∞) by the equalities:

P+f(x) = f(x)ϑ(x), P−f(x) = f(x)ϑ(−x). (2.6)

Here ϑ is the Heaviside function of the unit jump. In each of the spaces Lp (1 ≤ p ≤ ∞),we have

‖P±‖ = 1. (2.7)

Denote by K, T ∈ Ωn the following operators, whose kernel functions K, T participatein (1.1):

Kf(x) =∫∞

−∞K(x − t)f(t)dt, Tf(x) =

∫∞

−∞T(x − t)f(t)dt. (2.8)

Equation (1.1) admits the following operator entry

f = g + Wf, (2.9)

where W = KP+ + T P−.The projectors P± are the diagonal matrices of the operators with the diagonal

elements P±.


The operator W is an Integral operator:

Wf(x) =∫∞

−∞W(x, t)f(t)dt, where W(x, t) = K(x − t)ϑ(t) + T(x − t)ϑ(−t). (2.10)

Here ϑ(x) is the diagonal matrix with the diagonal elements ϑ(x).

2.2. On the Invertibility of the Operator I − W in Ln

Let us estimate the norm of W in Ln. Assume at first that the kernel functions K, T arearbitrary elements of Ln×n. Let ϕ = Wf , f ∈ Ln. One can obtain the following inequality(similar to (2.3)):

∫∞

−∞

∣∣ϕ(x)∣∣dx ≤ A

∫∞

0

∣∣f(t)∣∣dt + B

∫0

−∞

∣∣f(t)∣∣dt. (2.11)

Here A = (akm) =∫∞−∞ |K(x)|dx, B = (bkm) =

∫∞−∞ |T(x)|dx.

We have ςA ≤ λς, ςB ≤ μς, where

λ = maxm

n∑

k=1

akm, μ = maxm

n∑

k=1

bkm. (2.12)

Multiplying (2.11) on the left by the vector ς, we get

∥∥ϕ∥∥ ≤ λς

∫∞

0

∣∣f(t)∣∣dt + μς

∫0

−∞

∣∣f(t)∣∣dt ≤ max

(λ, μ

)∥∥f∥∥. (2.13)

Thus, we proved the following.

Lemma 2.1. The following estimate for the norm of the operator W in Ln is valid:

∥∥∥W∥∥∥ ≤ q = max

(λ, μ

). (2.14)

If q < 1, then the operator W is contracting in Ln, hence the operator I−W is invertible,and (1.1) with g ∈ Ln has a unique solution f ∈ Ln. If therewith K, T, g ≥ O, then f ≥ O.

In accordance with the general theory of the integral equations with two kernels (see[1, 2]), for the invertibility of the operator I − W in Ln, it is necessary the fulfilment of thefollowing conditions of nondegeneration:

det[J −K(s)

]/= 0, det

[J − T(s)

]/= 0, −∞ < s < +∞. (2.15)

Here J is the unit n × n matrix; the matrices K(s) and T(s) are the (elementwise) Fouriertransforms of K and T , respectively. For example, K(s) =

∫∞−∞K(x)eisxdx, −∞ < s < +∞.


In the semiconservative case (1.9), we have: ς[J −K(0)] = ς[J −A] = O. Hence det(J −A) = 0, that is, the symbol J −K(s), is degenerated in the point s = 0. In the conservative case(where A and B are stochastic matrices), both of the conditions (2.15) are violated. Thus, theoperator I − W is noninvertible in Ln in the semiconservative and conservative cases.

3. Semiconservative Nonhomogeneous Equation

In this section, we shall consider the question of the solvability of the uniform semiconserva-tive nonhomogeneous equation (1.1), (1.9) under the following additional assumption: thereexists a strong positive vector-column η such thatAη = η, η > O. In accordance with Perron-Frobenius theorem (see [8]), the existence of such vector η is secured if the stochastic matrixA is an irreducible one.

3.1. One Auxiliary Equation

At the outset, consider the auxiliary conservative Wiener-Hopf equation (1.10), where

O ≤ h ∈ Ln(0,∞), x > 0, ςA = ς, Aη = η (3.1)

with the conservative kernel K participating in (1.1).The following lemma follows from the results [9]:

Lemma A. Equation (1.10), (3.1) possesses the minimal solution ϕ ≥ O which is locally integrableon [0,∞) (see [9]). The following asymptotics holds

∫x

0ϕ(t)dt = o

(x2), x −→ ∞. (3.2)

This asymptotics admits an adjustment subject to additional assumptions on kernelK and free term h(see [9]).

Denote by ν the following matrix the first moments of matrix-function K:

ν =∫∞

−∞xK(x)dx, (3.3)

with the assumption of componentwise absolute convergence of this integral. Let

σ = ςνη, −∞ < σ < +∞. (3.4)

The number σ plays a principal role in the classification of the conservative equation (1.10)(see [9]). If σ < 0, then

∫x

0ϕ(t)dt = o(x), x −→ ∞. (3.5)


If therewith the free term h has a finite first moment:∫∞0 th(t)dt <∞, then ϕ ∈ Ln(0,∞).

Consider the simple iterations for (1.10):

ϕ(m+1)(x) = h(x) +∫∞

0K(x − t)ϕ(m)(t)dt, ϕ(0) = O, m = 0, 1, 2, . . . . (3.6)

The sequence ϕ(m) possesses the following properties: O ≤ ϕ(m) ∈ Ln(0,∞). It is easyto show that the sequence ϕ(m) is monotonic. Indeed we have

ϕ(m+1)(x) − ϕ(m)(x) =∫∞

0K(x − t)

(ϕ(m)(t) − ϕ(m−1)(t)

)dt, ϕ(0) = O, m = 0, 1, 2, . . . . (3.7)

Using the induction by m, we obtain that ϕ(m+1)(x) − ϕ(m)(x) ≥ O, which implies themonotonicity of the sequence ϕ(m). The sequence ϕ(m) converges monotonically by thetopology of LnLoc[0,∞) to the minimal solution ϕ of (1.10):

ϕ(m) ↑ ϕ in LnLoc[0,∞). (3.8)

3.2. One Existence Theorem for (1.1)

Consider now (1.1) under conditions

ςA = ς, Aη = η, ςB ≤ μς, 0 ≤ μ < 1. (3.9)

Let us consider the following iterations for (1.1):

f (m+1)(x) = g(x) +∫∞

0K(x − t)f (m)(t)dt +

∫0

−∞T(x − t)f (m)(t)dt, (3.10)

f (0) = O, m = 0, 1, 2, . . . . (3.11)

We have

f (m) ∈ Ln, m = 0, 1, . . . , O ≤ f (m)(x) ↑ by m. (3.12)

Let f ≥ O be any positive solution of (1.1), (3.9):

f(x) = g(x) +∫∞

0K(x − t)f(t)dt +

∫0

−∞T(x − t)f(t)dt. (3.13)

It is easy to verify by induction that f (m) ≤ f , for eachm ≥ 0. Hence, if the sequence f (m) → f

converges by the topology of LnLoc(−∞,∞), then f ≤ f .


Remark 3.1. If the sequence f (m) → f converges by the topology of LnLoc(−∞,∞), then onecan take the limit in (3.10), and f ≥ O will be the minimal positive solution of (1.1).

This fact is proved using the monotonicity of f (m) and the two-sided inequalities (see[10] Item 2).

Let us introduce the restrictions of the functions f (m) on (0,∞) and (−∞, 0):

ω(m) = f (m)∣∣[0,∞) , ψ(m) = f (m)∣∣(−∞,0) ∈ Ln(−∞, 0). (3.14)

Theorem 3.2. Let the conditions (3.9) hold. Then (1.1) has the minimal positive solution f ∈LnLoc(−∞,∞) with f |(−∞,0) ∈ Ln(−∞, 0) and

∫x

0f(t)dt = o

(x2), x −→ +∞. (3.15)

If ∃ σ < 0, then∫x0 f(t)dt = o(x), x → ∞.

Proof. After the integration of (3.10) over x on (−∞,∞), we shall have

a(m+1) + b(m+1) = γ +Aa(m) + Bb(m), (3.16)

where a(m) =∫∞0 ω(m)(x)dx, b(m) =

∫0−∞ ψ(m)(x)dx, γ =

∫∞−∞ g(x)dx.

Multiplying (3.16) on the left by the vector ς and taking into account (3.9), we obtainthe inequality

ςa(m+1) + ςb(m+1) ≤ ςγ + ςa(m) + μςb(m), (3.17)

whence it follows, with due regard for the monotony of sequences a(m), b(m), that (1 −μ) ςb(m) ≤ ςγ . We arrive at the following estimate:

ςb(m) =∥∥∥ψ(m)

∥∥∥ ≤ (1 − μ)−1ςγ. (3.18)

It follows from B. Levy well-known theorem that the monotonous and bounded by normsequence ψ(m) converges in Ln(−∞, 0):

O ≤ ψ(m) ↑ ψ ∈ Ln(−∞, 0). (3.19)

Now compare relations (3.10) for x > 0 with iterations (3.6), in which h(x) = g(x) +∫0−∞ T(x−

t)ψ(t)dt, x > 0 (ψ is determined according to (3.19)). In virtue of ψ(m) ≤ ψ, we have theinequality ω(m)(x) ≤ ϕ(m)(x), x > 0, m = 0, 1, . . .. Hence ω(m)(x) ≤ ϕ(x), x > 0, m = 0, 1, . . ..According to the Lebesgue theorem, the monotonic sequenceω(m) converges by the topologyLnLoc[0,∞):

O ≤ ω(m) ↑ ω ∈ LnLoc[0,∞). (3.20)


We have obtained that the narrowing of the monotonic iteration sequence f (m) to the negativesemiaxis is convergent in Ln(−∞, 0), and the narrowing of f (m) to the positive semiaxis isconvergent in LnLoc[0,∞). If we denote f(x) =

{ω(x), x>0ψ(x), x<0 , then f

(m) → f in LnLoc[−∞,∞) (i.e.,in Ln(−∞, r), for all r < +∞). Taking limit in (3.10) (see Remark 3.1), we obtain that thevector function f satisfies (1.1),(3.9), and thereby, it is its minimal solution. The Theorem isproved.

Observe that, under the assumptions of Theorem 3.2, the existence of the locallyintegrable solution of (1.1) could be proved using the fixed point principle of the paper [10].Anyway, with this method, one cannot obtain the properties f |(−∞,0) ∈ Ln(−∞, 0) and (3.15).

4. The Homogeneous Semiconservative Equation

The homogeneous system (1.1) under the conditions (3.9) will be considered in the presentsection:

G(x) =∫∞

0K(x − t)G(t)dt +

∫0

−∞T(x − t)G(t)dt. (4.1)

Consider at first the corresponding conservative homogeneous system of Wiener-Hopfequations:

S(x) =∫∞

0K(x − t)S(t)dt. (4.2)

Let us formulate some results on the existence of positive solutions of the system (4.1) (see[9]).

Theorem A. Let K satisfy the conditions ς A = ς, Aη = η (see (3.9)), and one of the followingconditions (a) or (b):

(a) the property of symmetry (here T is the sign of transpose):

K(−x) = KT (x), (4.3)

(b) the kernel K has a finite first moment ν (see (3.3)) and that

σ ≤ 0, (4.4)

where σ is determined by (3.4).

Then the equation (4.2) has a positive solution S(x) > O. The vector function S is absolutelycontinuous and monotone increasing. The following asymptotics holds

S(x) = O(x), x −→ ∞. (4.5)


Let us (in conditions of the Theorem A) continue the vector function S to all the realaxis in accordance with the equality (4.2). Then the equality (4.2) takes place on the wholereal axis.

The convergence of the following integral is necessary and sufficient in order that Shas a integrable extension on the negative semiaxis

∫∞

0S(t)dt

∫−t

−∞K(x)dx < +∞. (4.6)

If (4.6) holds, then we will have S ∈ LnLoc[−∞,∞), S(x) > O.It follows from the asymptotics (4.5) that for the fulfilment of the requirement (4.6), it

is sufficient that the kernel functionK has the (componentwise) finite second moment on thenegative semiaxis, that is,

∫∞

0K(−x)x2dx < +∞. (4.7)

Now consider, uniform semiconservative (4.1).

Theorem 4.1. Let the homogeneous equation (4.2) satisfy the conditions (3.9), (4.7) and either of theconditions (4.3) or (4.4). Then there exists a solution G > O, G ∈ LnLoc[−∞,∞) of this equation. Thefollowing asymptotics hold:

∫x

−∞G(t)dt = O

(x2), x −→ ∞. (4.8)

Proof. In accordance with Theorem A, there exists a solution S > O of (4.1). The inequality(4.6) follows from the condition (4.7) and from the asymptotics (4.5); hence, S ∈ Ln(−∞, 0).

Let us introduce a new sought-for vector function f ≥ O in (4.1) by means of therelation:

G(x) = f(x) + S(x), −∞ < x < +∞. (4.9)

Substituting (4.9) into (4.1)with due regard for (4.2), we obtain an inhomogeneous equationof the type (1.1)with respect to f , in which

g(x) =∫0

−∞T(x − t)S(t)dt, x ∈ R. (4.10)

Because of S ∈ Ln(−∞, 0), we have g ∈ Ln. In accordance with Theorem 3.2, there exists a(minimal) solution of (1.1) with a free term (4.10) that implies the existence of the strongpositive solution of the form (4.9) of the homogeneous equation (4.1).

The asymptotics (4.8) follow immediately from the properties of f and S, included inTheorem 3.2 and Theorem A. The Theorem is proved.


It is remarkable that under the conditions of Theorem 4.1, both the nonhomogeneousequation (1.1) (with g ∈ Ln) and the homogeneous equation (4.1) simultaneously havepositive solutions.

References

[1] I. C. Gohberg and I. A. Feldman, Convolution Equations and Projection Methods for Their Solution, vol.41 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA,1974.

[2] S. Prossdorf, Einige Klassen Singularer Gleichungen, Akademie-Verlag, Berlin, Germany, 1974.[3] L. G. Arabadzhyan, “On a conservative integral equation with two kernels,” Mathematical Notes, vol.

62, no. 3, pp. 271–277, 1997.[4] A. G. Barsegyan, “Transfer equation in adjacent half-spaces,” Contemporary Mathematical Analysis, vol.

41, no. 6, pp. 8–19, 2006.[5] W. Feller,An Introduction of Probability Theory and Its Applications, vol. 2, JohnWiley & Sons, New York,

NY, USA, 2nd edition, 1971.[6] C. Cercignani, Theory and Application of the Boltzmann Equation, Elsevier, New York, NY, USA, 1975.[7] B. Davison, Neutron Transport Theory , Oxford Clarendon Press, Oxford, UK, 1957.[8] P. Lancaster, Theory of Matrices, Academic Press, London, UK, 1969.[9] N. B. Engibaryan, “Conservative systems of integral convolution equations on the half-line and the

whole line,” Sbornik: Mathematics, vol. 193, no. 6, pp. 847–867, 2002.[10] N. B. Engibaryan, “On the fixed points of monotonic operators in the critical case,” Izvestiya RAN,

Series Mathematics, vol. 70, no. 5, pp. 931–947, 2006.


Research ArticleThe Bolzano-Poincare Type Theorems

Przemysław Tkacz and Marian Turzanski

College of Science, Cardinal Stefan Wyszynski, University in Warsaw, ul. Dewajtis 5,01-815 Warszawa, Poland

Correspondence should be addressed to Marian Turzanski, [email protected]

Received 24 February 2011; Revised 7 May 2011; Accepted 30 May 2011


Copyright q 2011 P. Tkacz and M. Turzanski. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

In 1883–1884, Henri Poincare announced the result about the structure of the set of zeros of functionf : In → Rn, or alternatively the existence of solutions of the equation f(x) = 0. In the case n = 1the Poincare Theorem is well known Bolzano Theorem. In 1940Miranda rediscovered the PoincareTheorem. Except for few isolated results it is essentially a non-algorithmic theory. The aim of thisarticle is to introduce an algorithmical proof of the Theorem “On the existence of a chain” andfor n = 3 an algorithmical proof of the Bolzano-Poincare Theorem and to show the equivalence ofPoincare, Brouwer and “On the existence of a chain” theorems.

1. Introduction

It is well known how influential topology was for the development of many other branchesof mathematics and economics. Among many others, let us recall significant place of fixedpoint theorems of Brouwer and Banach which served as a main tool in solving problemsin differential equations, theory of fractals and problems of market equilibrium. Some ofthese applications raised a question of computability of the fixed points. In [1, 2] Steinhauspresented following conjecture: Let some segments of the chessboard be mined. Assume that theking cannot go across the chessboard from the left edge to the right one without meeting a minedsquare. Then the rook can go from upper edge to the lower one moving exclusively on mined segments.

According to Surowka [3] several proofs of the Steinhaus Chessboard Theorem seemto be incomplete or use induction on the size of the chessboard [4].

The simple proof of the Steinhaus Chessboard Theorem was presented in [5]. In [6]the following generalization of the Steinhaus Chessboard Theorem was published: Theorem[On the existence of a chain] For an arbitrary decomposition of n-dimensional cube In onto kn cubesand an arbitrary coloring function F : T(k) → {1, . . . , n} for some natural number i ∈ {1, . . . , n}there exists an ith colored chain P1, . . . , Pr such that P1 ∩ I+i /= ∅ and Pr ∩ I−i /= ∅.

This theorem was the main tool in the proof (see [6]) of the Bolzano-Poincare theorem(see [7, 8]). In the first part of our paper an algorithm of finding the chain will be presented


and will be shown that the theorem “on the existence of a chain”, the Bolzano-Poincaretheorem, and the Brouwer fixed point theorem are equivalent (for more informations see[9, 10]).

2. Theorems

Let In := [0, 1]n be the n-dimensional cube in Rn.Its ith opposite faces are defined as follows:

I−i := {x ∈ In : x(i) = 0}, I+i := {x ∈ In : x(i) = 1}. (2.1)

Let

∂In :=n⋃

i=1

(I−i ∪ I+i

)(2.2)

be the boundary of the cube In.Let k be an arbitrary natural number.We call the family

T(k) :={[

i1k,i1 + 1k

]× · · · ×

[ink,in + 1k

]: ij ∈ {0, . . . , k − 1}

}(2.3)

the decomposition of In into kn cubes.The map F : T(k) → {1, . . . , n} is said to be a coloring function of the decomposition

T(k).The sequence P1, . . . , Pr where Pl ∈ T(k) for l = 1, . . . , r is said to be an ith colored chain,

if for all l ∈ {1, . . . , r} F(Pl) = i and Pj ∩ Pj+1 /= ∅ for j = 1, . . . , r − 1.The set C = {−1/2k, 1/2k, . . . , 1 + 1/2k}n is said to be the n-dimensional combinatorial

cube.Its ith opposite faces are defined as follows:

C−i ={z ∈ C : z(i) = − 1

2k

},

C+i ={z ∈ C : z(i) = 1 +

12k

}.

(2.4)

Let

∂C =n⋃

i=1

C−i ∪ C+

i (2.5)

be the boundary of the n-dimensional combinatorial cube.Let ei = (0, . . . , 0, 1/k, 0, . . . , 0), ei(i) = 1/k be the ith basic vector.An ordered set S = [z0, . . . , zn] ⊂ C is said to be an n-simplex if there exists permutation

α of set {1, . . . , n} such that z1 = z0 + eα(1) · · · zn = zn−1 + eα(n).


Any subset [z0, . . . , zi−1, zi+1, . . . , zn] ⊂ S, i = 0, . . . , n is said to be an (n−1)-face of then-simplex S.

Every map Φ : C → {1, . . . , n} is said to be a coloring map of C.The set A ⊂ C we call n’-colored if Φ(A) = {1, . . . , n′}.

Observation 1. Let S = [z0, . . . , zn] ⊂ C be an n-simplex. Then for each zi ∈ S if[z0, . . . , zi−1, zi+1, . . . , zn] /⊂ Cε

p for each p ∈ {1, . . . , n}, ε ∈ {+,−} then there exists exactly onen-simplex S[i] ⊂ C such that S ∩ S[i] = [z0, . . . , zi−1, zi+1, . . . , zn] else there does not exist suchS[i] ⊂ C.Observation 2. Any (n−1)-face of an n-simplex S ⊂ C is an (n−1)-face of exactly one or oftwo n-simplexes from C depending on whether or not it lies on Cε

p for some p ∈ {1, . . . , n},ε ∈ {+,−}.Observation 3. Each n-colored n-simplex has exactly two n-colored (n−1)-faces.

Theorem 2.1 (on the existence of a chain). For an arbitrary decomposition of n- dimensional cubeIn onto kn cubes and an arbitrary coloring function F : T(k) → {1, . . . , n} for some natural numberi ∈ {1, . . . , n} there exists an ith colored chain P1, . . . , Pr such that P1 ∩ I+i /= ∅ and Pr ∩ I−i /= ∅.

The algorithm (based on the proof from [6]) is as follows.

Step 1. Let us define the coloring map Φ : C → {1, . . . , n}:

Φ(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

F(t) for z ∈ C \ ∂C and z ∈ t1 for z ∈ C−

1 ∪ C+2

j for z ∈(C−j ∪ C+

j+1

)\(

j−1⋃

l=1

(C−l ∪ C+

l+1

))

, j = 2, . . . , n − 1

n for z ∈ (C−n ∪ C+

1

) \(

n−1⋃

l=1

(C−l ∪ C+

l+1

))

.

(2.6)

Step 2. Let us take n-colored n-simplex S1 = [z10, . . . , z1n−1, z

1n] where z10 = (−1/2k,−1/2k, . . . ,

−1/2k),z11 = z

10 + e1, . . . , z

1n−1 = z

1n−2 + en−1,

z1n =(

12k,12k, . . . ,

12k

)= z1n−1 + en.

(2.7)

We say that the n-colored (n−1)-face [z10, . . . , z1n−1] is “used”.Let S = S1.

Step 3. Take “unused” n-colored (n−1)-face of the n-simplex S.If this face is contained in Cε

p for some p ∈ {1, . . . , n}, ε ∈ {+,−} then go to Step 5. Elsethis is (n−1)-face of exactly one n-simplex S′ different to S.

Since that moment this (n−1)-face is said to be “used”. Go to the Step 4.

Step 4. Let us create the sequence of n-simplexes S1, . . . , Sl, S′.

Let S = S′.Go to Step 3.


Step 5. After finitely many iterations we obtain the sequence S1, . . . , Sm ⊂ C such that Φ(Sl ∩Sl+1) = {1, . . . , n} for l = 1, . . . , m − 1. And the n-simplex Sm has the n-colored (n−1)-facewhich is a subset of Cε

p for some p ∈ {1, . . . , n}, ε ∈ {+,−}. Hence Sm = [zm0 , zm1 , . . . , z

mn ]where

zm0 = (1 − 1/2k, 1 − 1/2k, . . . , 1 − 1/2k), zm1 = zm0 + e1, zm2 = zm1 + e2, . . . , zmn = zmn−1 + en.Let us take the smallest index l1 ∈ {1, 2, . . . , m} such that Sl1 ∩ C+

i /= ∅ for some i ∈{1, . . . , n}, then let us find the biggest index l2 ∈ {1, 2, . . . , l1} such that Sl2 ∩ C−

i /= ∅.

Step 6. Then from the chain Sl2+1, . . . , Sl1 choose successively points z1, z2, . . . , zr in the waythat Φ(zj) = i for j = 1, 2, . . . , r and zj /= zj+1 for j = 1, 2, . . . , r − 1, z1 ∈ C \ ∂C and z1 − ei ∈ C−

i ,zr ∈ Sl1 .

Step 7. For the sequence z1, . . . , zr we have the chain P1, . . . , Pr where Pj ∈ T(k) and zj ∈ Pjfor j = 1, . . . , r.

END

Theorem 2.2 (Bolzano-Poincare). Let f : In → Rn, f(x) = (f1(x), . . . , fn(x)) be a continuousmap such that fi(I−i ) ⊂ (−∞, 0] and fi(I+i ) ⊂ [0,∞) for i = 1, . . . , n then there exists x0 ∈ In suchthat f(x0) = (0, . . . , 0).

Theorem 2.3 (Brouwer fixed point theorem). Let g : In → In, g(x) = (g1(x), . . . , gn(x)) be acontinuous map then there exists x0 ∈ In such that g(x0) = x0.

Theorem 2.4. The following theorems are equivalent:

(1) Theorem on the existence of a chain

(2) Bolzano-Poincare theorem

(3) Brouwer fixed point theorem.

Proof. “(1)⇒(2)” let us assume that for each x ∈ In f(x)/= (0, . . . , 0). Let us define sets:Ui = {x ∈ In : fi(x)/= 0} for i = 1, . . . , n, each setUi is open.We have In = U1 ∪ · · · ∪Un.Let us consider the space Rn with the metric δ(x, y) = max{|xi−yi| : i = 1, . . . , n}. From

the Lebesgue lemma of covering it follows that there exists λ > 0 such that for every k ∈ Nand 1/k < λwe have for every t ∈ T(k) there exist j ∈ {1, . . . , n} such that t ⊂ Uj .

Let us define coloring function F : T(k) → {1, . . . , n}:

F(t) := min{j : t ⊂ Uj

}. (2.8)

From theorem “on the existing of a chain” there exists ith colored sequence P1(k), . . . , Pr(k)(k)connecting ith opposite faces of the cube In.

The setW :=⋃r(k)l=1 Pl(k) is closed and connected.

The intersectionsW ∩ I−i /= ∅/=W ∩ I+i , Hence there exists x, y ∈ W such that fi(x) < 0and fi(y) > 0. Since f(x) is the continuous map, hence fi(W) is connected in R. Hence the setfi(W) is an interval containing [fi(x), fi(y)]. From the Bolzano theorem there exists c ∈ Wsuch that fi(c) = 0.

Contradiction.“(2)⇒(3)” let f(x) = x−g(x). The function f(x) fulfills the assumptions of the Bolzano-

Poincare theorem. hence there exist c ∈ In such that f(c) = 0.So g(c) = c.


“(3)⇒(1)” let us assume that there exists decomposition of n-dimensional cube In ontokn cubes and a coloring function F : T(k) → {1, . . . , n} such that for each i ∈ {1, . . . , n} thereis no ith colored chain connecting I−i and I+i .

Let Ci = {t ∈ T(k) : F(t) = i}.Let Li be the family of components of

⋃Ci ⊂ In.

C−i ={l ∈ Li : l ∩ I−i /= ∅},

C+i ={l ∈ Li : l ∩ I+i /= ∅},

C0i ={l ∈ Li : l ∩

(I−i ∪ I+i

)= ∅}.

(2.9)

The subsets of In:

Ai =⋃C−i ∪⋃C0i ∪{x ∈ In : x(i) ∈

[0,

12k

]},

Bi =⋃C+i ∪{x ∈ In : x(i) ∈

[1 − 1

2k, 1]} (2.10)

are closed and disjoint.In with the Euclidean metric is a normal space, hence there exists a continuous map

fi : In → [−1/2k, 1/2k] such that fi(Ai) = 1/2k and fi(Bi) = −1/2k.For each x ∈ In let us define the map g(x) := x + f(x)where f(x) = (f1(x), . . . , fn(x)).Observe that g : In → In is continuous map. Take an arbitrary x ∈ In.There exists t ∈ T(k) such that x ∈ t. The cube t is a subset of Ai or Bi for some

i ∈ {1, . . . , n}. We have gi(x) = x(i) + 1/2k or gi(x) = x(i) − 1/2k.Hence the function g(x) has no fixed point. Contradiction.

3. Poincare Theorem for n = 3

3.1. The Basic Algorithm

Let k be an arbitrary natural number.We have the decomposition of I3 into k3 cubes.Assume w.l.o.g. that fi(I−i ) ⊂ (−∞, 0) and fi(I+i ) ⊂ (0,∞) for i = 1, 2, 3. Let d : I3 × I3 →

R be the Euclidean metric.Observe that there exist ε∗ > 0 such that for each x ∈ I3, d(x, I−i ) < ε∗ and for each

y ∈ I3, d(y, I+i ) < ε∗ we have fi(x) < 0, fi(y) > 0, i = 1, 2, 3.

3.1.1. Surface

Let k be a natural number, such that 1/k < ε∗.The center of each t ∈ T(k), t = [i1/k, (i1 + 1)/k] × · · · × [i3/k, (i3 + 1)/k] is defined as

follows:

tc =(i1k+

12k,i2k+

12k,i3k+

12k

). (3.1)


Let us define coloring map φ1 : T(k) → {0, 1}

φ1(t) =

⎧⎨

⎩

0 f1(tc) ≤ 0

1 f1(tc) > 0.(3.2)

Algorithm for surface is as follows.

Step 1. Let

A0 ={t ∈ T(k) : t ∩ I−1 /= ∅},

A1 ={t ∈ T(k) : t ∩A0 /= ∅, φ1(t) = 1

},

B ={t ∩ t′ : t ∈ A0, t

′ ∈ A1}.

(3.3)

Step 2. If C = {t ∈ T(k) \A0 : dim[t ∩A0] = 2, φ1(t) = 0} = ∅ then END.Otherwise do Step 3.

Step 3. Add elements of the set C to A0.Next

A1 ={t ∈ T(k) : t ∩A0 /= ∅, φ1(t) = 1

},

B ={t ∩ t′ : t ∈ A0, t

′ ∈ A1, t ∩ t′ /= ∅}(3.4)

and go to Step 2.Since T(k) is finite, hence after finitely many steps setC is empty (the procedure ends).Let us consider the family B. Wemay assume that B is closed under finite intersections.The elements b ∈ B, such that dim[b] = 2, dim[b] = 1, dim[b] = 0 are called squares,

edges, and vertices.

Observation 4. The⋃B separates cube I3 between I−1 and I+1 .

Observation 5. Each edge b ∈ B if b ⊂ ∂I3 it is an edge of exactly 1 square, else it is an edge of2 or 4 squares.

3.1.2. Modification of B

Let us divide each element of {a ∈ A0 : a ∩⋃B /= ∅} onto 27 cubes (in the natural way).Denote the set consisting of all this cubes by T ′.Create coloring map φ′

1 : T′ → {0, 1} as follows:

φ′1

(t′)=

⎧⎨

⎩

0 t′ ∩A1 = ∅1 t′ ∩A1 /= ∅.

(3.5)

NowB′ = {t ∩ t′ : t, t′ ∈ T ′, φ′1(t) = 0, φ′

1(t′) = 1, t ∩ t′ /= ∅}.


Observation 6. Any edge of B′ is an edge of exactly one or of two squares from B′ dependingon whether or not it lies on ∂I3.

Let us define coloring φ2 : {t ∈ B′ : t is a square} → {0, 1}:

φ2(t) =

⎧⎨

⎩

0 f2(tc) ≤ 0

1 f2(tc) > 0,(3.6)

where tc is the center of square t.The edge t ∈ B′ is said to be 2-coloured if there exists squares s, s′ ∈ B′ such that s∩s′ = t

and φ2({s, s′}) = {0, 1}.

Observation 7. The vertex of 2-coloured edge is a subset of exactly one or even number of2-coloured edges depending on whether or not it lies on ∂I3.

Observation 8. The components of⋃B′ ∩ ∂I3 are broken lines without self-cutting.

Observation 9. The number of broken lines lying on I−3 and connecting I−2 and I+2 is odd.

Lemma 3.1. The number of 2-coloured edges from B′, which one of vertices lies on I−3 is odd.

Proof. Let us consider components of the set⋃B′ ∩ I−3 .

We have odd number of broken lines connecting I−2 and I+2 and the number of the restcomponents is arbitrary.

Let us see that⋃B′ ⊂ I3 \ I−1 ∪ I+1 .

So, the number of 2-coloured edges from B′, which one of vertices lies on I−3 is odd if itlies on broken line connecting I−2 and I+2 else it is even (using the definition of φ2).

According to Observation 9 this ends the proof.

3.1.3. Broken Line Connecting I−3 and I+3

Step 1. Let E0 = {t ∈ B′ : t is a 2-coloured edge, t ∩ I−3 /= ∅},

E1 = ∅. (3.7)

Step 2. Take e ∈ E0 \ E1.Add e to E1.The vertex v ∈ e ∩ I−3 is said to be used.Go to Step 3.

Step 3. Take unused vertex u of the last added edge to the set E1.If u ∈ I+3 END.Otherwise,If u ∈ I−3 go to Step 2.Else go to Step 4.

Step 4. Take unused vertex u of the last added edge to the set E1.Next take 2-coloured edge e ∈ B′ \ E1 such that v ∈ e.


Now vertice v is said to be used.Add e to the set E1.Go to Step 3.

First of all the number of 2-coloured edges from B′, which one of vertices lies on I−3 isodd (Lemma 3.1).

The second each vertex of 2-coloured edge is a subset of exactly one or even numberof 2-coloured edges depending on whether or not it lies on ∂I3 (Observation 7).

This arguments allows one to say that procedure is well defined.Now our broken line connecting I+3 and I−3 is created as follows:let e1 be the last added element to E1.If ei ∩ I−3 = ∅ then ei+1 is previous added element to E1

else Stop.We obtained the sequence of edges {e1, e2, . . . , em} ⊂ B′. Let us define coloring φ3 : {t ∈

B′ : t is an edge of ei} → {0, 1} where i ∈ {1, . . . , m}:

φ3(t) =

⎧⎨

⎩

0 f3(t) ≤ 0

1 f3(t) > 0.(3.8)

It is easy to see that φ3(e1) = {1} and φ3(em) = {0}.So starting from e1 we search with order the first edge ek ∈ {e1, e2, . . . , em} such that

φ3(ek) = {0, 1}.

3.2. Topological Part

For each k ∈N, 1/k < ε∗ we have

(i) vk, v′k ∈ ek such that f3(vk) ≤ 0 and f3(v′

k) > 0,

(ii) uk, u′k ∈ tu ∪ t′u such that f2(uk) ≤ 0 and f2(u′k) > 0 where tu, t′u are squares from Band ek ∩ tu /= ∅/= ek ∩ t′u,

(iii) wk,w′k ∈ tw ∪ t′w such that f1(wk) ≤ 0 and f1(w′

k) > 0 where tw is a cube fromA0, t′wis a cube from A1 and ek ∩ tw /= ∅/= ek ∩ t′w.

Define the setsWk := conv{vk, v′k, uk, u

′k,wk,w

′k}.

For eachWk there exist c1k, c2k, c

3k ∈Wk such that

f1(c1k

)= f2(c2k

)= f3(c3k

)= 0. (3.9)

Without loss of generality we can assume that limk→∞c1k = c.Moreover, limk→∞ diam[Wk] = 0. So for each c′

k∈ Wk the fact d(c, c′

k) ≤ d(c, c1

k) +

d(c1k, c′

k) yields

limk→∞

c1k = limk→∞

c2k = limk→∞

c3k = c. (3.10)

So f(c) = 0 ends proof.


References

[1] H. Steinhaus, Kalejdoskop Matematyczny, PWN, Warszawa, Poland, 1954.[2] H. Steinhaus,Mathematical Snapshot, Oxford University Press, New York, NY, USA, 1969.[3] W. Surowka, “A discrete form of Jordan curve theorem,” Annales Mathematicae Silesianae, no. 7, pp.

57–61, 1993.[4] Y. A. Shaskin, Fixed Points, vol. 2 ofMathematical World, AMS-MSA, 1991.[5] W. Kulpa, L. Socha, and M. Turzanski, “Steinhaus chessboard theorem,” Acta Universitatis Carolinae,

vol. 41, no. 2, pp. 47–50, 2000.[6] P. Tkacz and M. Turzanski, “An n-dimensional version of Steinhaus’ chessboard theorem,” Topology

and Its Applications, vol. 155, no. 4, pp. 354–361, 2008.[7] H. Poincare, “Sur certaines solutions particulieres du probleme des trois corps,” Comptes Rendus de

l’Academie des Sciences, vol. 97, pp. 251–252, 1883.[8] H. Poincare, “Sur certaines solutions particulieres du probleme des trois corps,” Bulletin Astronomique,

vol. 1, pp. 63–74, 1884.[9] C.Miranda, “Un’ osservazione su una teorema di Brouwer,” Bollettino della Unione Matematica Italiana,

p. 527, 1940.[10] W. Kulpa, “The Poincare-Miranda theorem,” The American Mathematical Monthly, vol. 104, no. 6, pp.

545–550, 1997.

Fixed-Point Theory, Variational Inequalities, and Its Approximation … · 2019. 8. 7. ·...

Documents

Transcript of Fixed-Point Theory, Variational Inequalities, and Its Approximation … · 2019. 8. 7. ·...