Applied Mathematics Letters 24 (2011) 97–102

Contents lists available at ScienceDirect

Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

Common fixed points of two multivalued nonexpansive mappings byone-step iterative schemeMujahid Abbas a, Safeer Hussain Khan b, Abdul Rahim Khan c, Ravi P. Agarwal d,∗a Department of Mathematics, Lahore University of Management Sciences, 54792-Lahore, Pakistanb Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatarc Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabiad Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

a r t i c l e i n f o

Article history:Received 13 July 2010Accepted 11 August 2010

Keywords:One-step iterative schemeNonexpansive mappingCommon fixed pointCondition (A′)Weak and strong convergence

a b s t r a c t

In this paper, we introduce a new one-step iterative process to approximate commonfixed points of twomultivalued nonexpansivemappings in a real uniformly convex Banachspace.We establishweak and strong convergence theorems for the proposed process undersome basic boundary conditions.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Let X be a real Banach space. A subset K of X is called proximinal if for each x ∈ X , there exists an element k ∈ K suchthat

d(x, K) = inf{‖x − y‖ : y ∈ K} = d(x, k).

It is known that weakly compact convex subsets of a Banach space and closed convex subsets of a uniformly convex Banachspace are proximinal. We shall denote the family of nonempty bounded proximinal subsets of X by P(X) and the family ofnonempty compact subsets of X by C(X). Consistent with [1], let CB(X) be the class of all nonempty bounded and closedsubsets of X . Let H be the Hausdorff metric induced by the metric d of X and given by

H(A, B) = maxsupx∈A

d(x, B), supy∈B

d(y, A)

for A, B ∈ CB(X). It is obvious that P(K) ⊆ CB(X).

A point x ∈ K is called a fixed point of a multivalued mapping T if x ∈ Tx. We denote the set of all fixed points of T byF(T ). A multivalued mapping T : K −→ CB(X) is said to be

(i) contraction if there exists a constant k ∈ [0, 1) such that for any x, y ∈ K ,

H(Tx, Ty) ≤ k ‖x − y‖ ,

∗ Corresponding author.E-mail addresses:mujahid@lums.edu.pk (M. Abbas), safeer@qu.edu.qa, safeerhussain5@yahoo.com (S.H. Khan), arahim@kfupm.edu.sa (A.R. Khan),

agarwal@fit.edu (R.P. Agarwal).

0893-9659/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.aml.2010.08.025

98 M. Abbas et al. / Applied Mathematics Letters 24 (2011) 97–102

(ii) nonexpansive if

H(Tx, Ty) ≤ ‖x − y‖

for all x, y ∈ K .(iii) quasi nonexpansive if F(T ) = φ and

H(Tx, Tp) ≤ ‖x − p‖

for every p ∈ F(T ). It is known that a multivalued nonexpansive mapping T : K −→ C(X) has a fixed point if K isnonempty closed bounded convex subset of a uniformly convex Banach space X [2].

Throughout the paper N denotes the set of positive integers.The study of fixed points for multivalued contractions and nonexpansive mappings by using the Hausdorff metric was

initiated by Markin [3] (see also [1]). Later, an interesting and rich fixed point theory for such maps was developed whichhas applications in control theory, convex optimization, differential inclusion and economics (see [4] and references citedtherein).

The theory of multivalued nonexpansive mappings is harder than the corresponding theory of single valuednonexpansive mappings. Different iterative processes have been used to approximate fixed points of multivaluednonexpansive mappings; in particular, Sastry and Babu [5] considered the following:

Let T : X → P(X) be a mapping with p ∈ Tp.(I) Define a sequence by x = x1 ∈ X,

xn+1 = (1 − an)xn + anyn,

where yn ∈ Txn is such that ‖yn − p‖ = d(p, Txn), and {an} is a sequence in (0, 1) satisfying∑

an = ∞.(II) The sequence of Ishikawa iterates is defined by

x1 = x ∈ X,

yn = (1 − bn)xn + bnznxn+1 = (1 − an)xn + anun

where zn ∈ Txn, un ∈ Tyn are such that ‖zn − p‖ = d(p, Txn) and ‖un − p‖ = d(p, Tyn), and {an},{bn} are real sequencesof numbers with 0 ≤ an, bn < 1 satisfying limn→∞ bn = 0 and

∑anbn = ∞.

Recently, Panyanak [6] obtained the following result.

Theorem 1 ([6, Theorem3.1]). Let K be a nonempty compact convex subset of a uniformly convex Banach space X. Suppose thatT : K → P(K) is a nonexpansive map satisfying Ty = {y} for all y ∈ F(T ). Let {xn} be the sequence of Ishikawa iterates definedby (II). Then {xn} converges to a fixed point of T .

Following useful lemma is due to Nadler [1].

Lemma 1. Let A, B ∈ CB(X) and a ∈ A. If η > 0, then there exists b ∈ B such that d(a, b) ≤ H(A, B) + η.

Based on the above lemma, Song andWang [7]modified the iteration scheme used in [6] and improved the results presentedtherein. This scheme reads as follows:

(III) The sequence of Ishikawa iterates is defined by

x1 = x ∈ X,

yn = (1 − bn)xn + bnznxn+1 = (1 − an)xn + anun

where zn ∈ Txn, un ∈ Tyn are such that ‖zn − un‖ ≤ H(Txn, Tyn) + ηn and ‖zn+1 − un‖ ≤ H(Txn+1, Tyn) + ηn, and {an},{bn} are real sequences of numbers with 0 ≤ an, bn < 1 satisfying limn→∞ bn = 0 and

∑anbn = ∞.

On the other hand, Hu et al. [8] obtained common fixed point of two nonexpansive mappings T , S : K → CB(K) satisfyingcertain contractive condition. To achieve this, they employed the following iterative process:

x1 = x ∈ K ,

yn = (1 − bn)xn + bnyn, yn ∈ Txnxn+1 = (1 − an)xn + anun, un ∈ Syn, n ∈ N

where {an} and {bn} are in [0, 1] satisfying certain conditions.In this paper, we introduce a new one-step iterative process to compute common fixed points of two multivalued

nonexpansive mappings. Let S, T : K → CB(K) be two multivalued nonexpansive mappings. Our process reads as follows:x1 ∈ K ,xn+1 = anxn + bnyn + cnzn, n ∈ N (1)

M. Abbas et al. / Applied Mathematics Letters 24 (2011) 97–102 99

where yn ∈ Txn and zn ∈ Sxn such that ‖yn − p‖ ≤ d(p, Sxn) and ‖zn − p‖ ≤ d(p, Txn) whenever p is fixed point of any oneof the mappings S and T , and {an}, {bn} and {cn} are sequence of numbers in (0, 1) satisfying an + bn + cn ≤ 1. Inspiredby the work in [9], here we provide a simple and unified approach to Mann [10] and Ishikawa processes of multivaluednonexpansive mappings. From computational point of view, our scheme is simpler than Ishikawa iteration process.

Let us recall some definitions.A Banach space X is said to satisfy Opial’s condition [11] if for any sequence {xn} in X , xn ⇀ x (⇀ denotes weak

convergence) implies that lim supn→∞ ‖xn − x‖ < lim supn→∞ ‖xn − y‖ for all y ∈ X with y = x. Examples of Banach spacessatisfying this condition are Hilbert spaces and all lp spaces (1 < p < ∞). On the other hand, Lp[0, 2π ] with 1 < p = 2 failto satisfy Opial’s condition.

We need the following useful lemma.

Lemma 2 ([12]). Suppose that X is a uniformly convex Banach space and 0 < p ≤ tn ≤ q < 1 for all positive integersn. Also suppose that {xn} and {yn} are two sequences of X such that lim supn→∞ ‖xn‖ ≤ r, lim supn→∞ ‖yn‖ ≤ r andlimn→∞ ‖tnxn + (1 − tn)yn‖ = r hold for some r ≥ 0. Then limn→∞ ‖xn − yn‖ = 0.

2. Main results

In the sequel, F = F(S)∩F(T ) is the set of all common fixed points of themappings S and T . We begin with the followingtechnical result.

Lemma 3. Let X be a uniformly convex Banach space and K a nonempty closed convex subset of X. Let S, T : K → C(K)be multivalued nonexpansive mappings and F = φ. Then for the sequence {xn} in (1), we have limn→∞ d(xn, Sxn) = 0 =

limn→∞ d(xn, Txn).

Proof. Let p ∈ F . It follows from (1) that

‖xn+1 − p‖ = ‖anxn + bnyn + cnzn − p‖= ‖an(xn − p) + bn (yn − p) + cn(zn − p)‖≤ an ‖xn − p‖ + bn ‖yn − p‖ + cn ‖zn − p‖≤ an ‖xn − p‖ + bnd(Sxn, p) + cnd(p, Txn)≤ an ‖xn − p‖ + µH(Sp, Sxn) + cnH(Tp, Txn)≤ an ‖xn − p‖ + bn ‖xn − p‖ + cn ‖xn − p‖= ‖xn − p‖ . (2)

Thus, limn→∞ ‖xn − p‖ exists for each p ∈ F . We suppose that limn→∞ ‖xn − p‖ = c for some c ≥ 0. Then

limn→∞

‖xn+1 − p‖ = limn→∞

‖an(xn − p) + bn (yn − p) + cn(zn − p)‖

= limn→∞

(1 − cn)[

an1 − cn

(xn − p) +bn

1 − cn(yn − p)

]+ cn(zn − p)

= c. (3)

Since S, T are nonexpansive mappings and F = ∅, we have ‖yn − p‖ ≤ d(Sxn, p) ≤ H(Sxn, Sp) ≤ ‖xn − p‖ for each p ∈ F .Taking lim sup on both sides, we obtain

lim supn→∞

‖yn − p‖ ≤ c. (4)

Similarly,

lim supn→∞

‖zn − p‖ ≤ c. (5)

Next,

lim supn→∞

an1 − cn

(xn − p) +bn

1 − cn(yn − p)

≤ lim supn→∞

[an

1 − cn‖xn − p‖ +

bn1 − cn

‖yn − p‖]

≤ lim supn→∞

[an

1 − cn‖xn − p‖ +

bn1 − cn

‖xn − p‖]

= lim supn→∞

an + bn1 − cn

‖xn − p‖

=

an + bn1 − cn

c

≤ c. (6)

100 M. Abbas et al. / Applied Mathematics Letters 24 (2011) 97–102

Using (3)–(6) and Lemma 2, we get

limn→∞

an1 − cn

(xn − p) +bn

1 − cn(yn − p) − (zn − p)

= 0.

This yields

0 = limn→∞

an1 − cn

xn +bn

1 − cnyn − zn

= lim

n→∞

1

1 − cn

‖λxn + µyn − (1 − cn) zn‖

= limn→∞

1

1 − cn

‖λxn + µyn + νzn − zn‖

= limn→∞

1

1 − cn

‖xn+1 − zn‖.

That is,

limn→∞

‖xn+1 − zn‖ = 0.

In a similar way, we can show that

limn→∞

‖xn+1 − yn‖ = 0 and limn→∞

‖xn+1 − xn‖ = 0.

Also, ‖xn − zn‖ ≤ ‖xn − xn+1‖ + ‖xn+1 − zn‖ and ‖xn − yn‖ ≤ ‖xn − xn+1‖ + ‖xn+1 − yn‖ imply that

limn→∞

‖xn − zn‖ = 0 and limn→∞

‖xn − yn‖ = 0.

Now

d(xn, Txn) ≤ d(xn, zn)

and

d(xn, Sxn) ≤ d(xn, yn)

give d(xn, Txn) → 0 and d(xn, Sxn) → 0 as n → ∞. �

We now approximate common fixed points of mappings S and T through weak convergence of the sequence {xn} in (1) asfollows:

Theorem 2. Let X be a uniformly convex Banach space satisfying Opial’s condition. Let K be a nonempty closed convex subset ofX and S, T and {xn} be as in Lemma 3. Then {xn} converges weakly to a common fixed point of S and T .

Proof. As in the proof of Lemma 3, limn→∞ ‖xn − p‖ exists for all p ∈ F and therefore {xn} is bounded. Since X is uniformlyconvex, there exists a subsequence {xnk} of {xn} such that xnk ⇀ q for some q ∈ K as k → ∞. First we show that q ∈ Sq.Since Sq is compact, for each xnk in X there exists yk in Sq such that

d(xnk , yk) = d(xnk , Sq).

Now let {zk} be the subsequence of {yk} such that zk → z. Obviously, z ∈ Sq. Now we show that z = q. If not, then we have:

lim supk→∞

‖xnk − z‖ ≤ lim supk→∞

‖xnk − zk‖ + lim supk→∞

‖zk − z‖

= lim supk→∞

‖xnk − zk‖

= lim supk→∞

d(xnk , Sq)

≤ lim supk→∞

d(xnk , Sxnk) + lim supk→∞

H(Sxnk , Sq)

≤ lim supk→∞

‖xnk − q‖ < lim supk→∞

‖xnk − z‖

which gives a contradiction and hence q ∈ Sq. Similarly, it can be shown that q ∈ Tq. Now we prove that {xn} has aunique weak subsequential limit in F . To prove this, let z1 and z2 be weak limits of the subsequences {xni} and {xnj} of

M. Abbas et al. / Applied Mathematics Letters 24 (2011) 97–102 101

{xn}, respectively and z1 = z2. Then by Opial’s condition,

limn→∞

‖xn − z1‖ = limni→∞

‖xni − z1‖

< limni→∞

‖xni − z2‖

= limn→∞

‖xn − z2‖

= limnj→∞

‖xnj − z2‖

< limnj→∞

‖xnj − z1‖

= limn→∞

‖xn − z1‖.

This is a contradiction. Hence {xn} converges weakly to a point in F . �

The following result gives a necessary and sufficient condition for strong convergence of the sequence in (1) to a commonfixed point of two mappings on a real Banach space.

Theorem 3. Let K be a closed convex subset of a real Banach space X, S and T assume bounded values and {xn} be as in Lemma 3.Then {xn} converges strongly to a common fixed point of S and T if and only if lim infn→∞ d(xn, F) = 0.

Proof. The necessity is obvious. Conversely, suppose that lim infn→∞ d(xn, F) = 0. By (2) we have,

‖xn+1 − p‖ ≤ ‖xn − p‖.

This gives

d(xn+1, F) ≤ d(xn, F).

Hence limn→∞ d(xn, F) exists. By hypothesis, lim infn→∞ d(xn, F) = 0 so we must have limn→∞ d(xn, F) = 0.Next we show that {xn} is a Cauchy sequence in K . Let ε > 0 be arbitrarily chosen. Since limn→∞ d(xn, F) = 0, there

exists n0 such that for all n ≥ n0, we have

d(xn, F) <ε

4.

In particular, inf{xn0 − p

: p ∈ F} < ε4 ; so there must exist a p∗

∈ F such thatxn0 − p∗ <

ε

2.

Now form, n ≥ n0, we have

‖xn+m − xn‖ ≤xn+m − p∗

+xn − p∗

≤ 2

xn0 − p∗

< 2ε

2

= ε.

Hence {xn} is a Cauchy sequence in a closed subset K of a Banach space X , and therefore it must converge in K . Letlimn→∞ xn = q. Now

d(q, Tq) ≤ d(q, xn) + d(xn, Txn) + H(Txn, Tq)≤ d(q, xn) + d(xn, zn) + d(xn, q)→ 0 as n → ∞

gives that d(q, Tq) = 0 which implies that q ∈ Tq. Similarly,

d(q, Sq) ≤ d(q, xn) + d(xn, Sxn) + H(Sxn, Sq)≤ d(q, xn) + d(xn, yn) + d(xn, q)→ 0 as n → ∞

implies that q ∈ Sq. Consequently, q ∈ F . �

Khan and Fukhar-ud-din [13] introduced the so-called condition (A′) and gave an improved version in [14]. A multivaluedversion of condition (A′) which is weaker than compactness of the domain, is given as follows.

Two multivalued nonexpansive mappings S, T : K → CB(K) are said to satisfy condition (A′) if there exists anondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for all r ∈ (0, ∞) such that either d(x, Tx) ≥

f (d(x, F)) or d(x, Sx) ≥ f (d(x, F)) for all x ∈ K .

102 M. Abbas et al. / Applied Mathematics Letters 24 (2011) 97–102

We use the condition (A′) to study strong convergence of {xn} in (1). In our next theorem, we assume that S, T : K →

CB(K) satisfy condition (A′). Here we do not impose the condition of proximinality on F and hence our result improves andgeneralizes ([6, Theorem 3.1] and [5, Theorem 6]).

Theorem 4. Let X be a uniformly convex Banach space and K , {xn} be as in Lemma 3. Let S, T : K → C(K) be two nonexpansivemappings satisfying condition (A′). Then {xn} converges strongly to a common fixed point of S and T .

Proof. As earlier, limn→∞ d(xn, F) exists. So by condition(A′), either

limn→∞

f (d(xn, F)) ≤ limn→∞

d(xn, Txn) = 0

or

limn→∞

f (d(xn, F)) ≤ limn→∞

d(xn, Sxn) = 0.

In both the cases,

limn→∞

f (d(xn, F)) = 0.

Since f is a nondecreasing function and f (0) = 0, limn→∞ d(xn, F) = 0. An application of Theorem 3 now gives the desiredresult. �

Remark 1. (a) Theorem 2 gives multivalued analogue of [9, Theorem 3.3(a)] and [13, Theorem 1] for two maps.(b) Theorem 3 extends [14, Theorem 1] for multivalued mappings.(c) Theorem 4 generalizes [9, Theorem 3.3(b)] and [7, Theorem 2] to the case of twomultivalued nonexpansivemappings.

Acknowledgements

The author A.R. Khan is grateful to King Fahd University of Petroleum & Minerals and SABIC for supporting researchproject SB100012.

References

[1] S.B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475–488.[2] T.C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80 (1974)

1123–1126.[3] J.T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc. 38 (1973) 545–547.[4] L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer Academic Pub., Dordrecht, Netherlands, 1999.[5] K.P.R. Sastry, G.V.R. Babu, Convergence of Ishikawa iterates for a multivalued mapping with a fixed point, Czechoslovak Math. J. 55 (2005) 817–826.[6] B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comp. Math. Appl. 54 (2007) 872–877.[7] Y. Song, H. Wang, Erratum to ‘‘Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces’’, Comput. Math. Appl. 55 (2008)

2999–3002.[8] T. Hu, J.C. Huang, B.E. Rhoades, A general principle for Ishikawa iterations for multivalued mappings, Indian J. Pure Appl. Math. 28 (8) (1997)

1091–1098.[9] A.R. Khan, On modified Noor iterations for asymptotically nonexpansive mappings, Bull. Belg. Math. Soc. Simon Stevin 17 (2010) 127–140.

[10] W.R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (1953) 506–510.[11] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597.[12] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991) 153–159.[13] S.H. Khan, H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 8 (2005)

1295–1301.[14] H. Fukhar-ud-din, S.H. Khan, Convergence of iterates with errors of asymptotically quasi- nonexpansive mappings and applications, J. Math. Anal.

Appl. 328 (2007) 821–829.