1

CHAPTER 1

INTRODUCTION

The work reported in this thesis is based on fixed points, common fixed points, approximate

fixed points, end points and approximate best proximity points of a variety of maps satisfying

different contractive conditions. The stability of different iterative procedures is investigated

in a new setting. Several existing results concerning single valued and multi valued maps are

extended to the theory of iterated function systems (IFS) and iterated multifunction systems

(IMS). Further, some minimax and saddle point theorems are also established.

We first provide a brief historical development of the fixed point theory.

1.1 HISTORICAL OUTLINE

The origin of fixed point theory lies in the method of successive approximations used for

proving existence of solutions of differential equations introduced independently by Joseph

Liouville [1] in 1837 and Charles Emile Picard [2] in 1890. But formally it was started in the

beginning of twentieth century as an important part of analysis. The abstraction of this

classical theory is the pioneering work of the great Polish mathematician Stefan Banach [3]

published in 1922 which provides a constructive method to find the fixed points of a map.

However, on historical point of view, the major classical result in fixed point theory is due to

2

L. E. J. Brouwer [4] given in 1912 (also see Zeidler [5], Kirk and Sims [6] and Granas and

Dugundji [7]).

The celebrated Banach contraction principle (BCP) states that “a contraction mapping

on a complete metric space has a unique fixed point”. Banach used the idea of shrinking map

to obtain this fundamental result. The Brouwer fixed point theorem is of great importance in

the numerical treatment of equations. It exactly states that “a continuous map on a closed unit

ball in Rn has a fixed point”. An important extension of this is the Schauder’s fixed point

theorem [8] of 1930 stating “a continuous map on a convex compact subspace of a Banach

space has a fixed point”. These celebrated results have been used, generalized and extended

in various ways by several mathematicians, scientists, economists for single valued and

multivalued mappings under different contractive conditions in various spaces. Kannan [9]

proved a fixed point theorem for the maps not necessarily continuous. This was another

important development in fixed point theory (see also Chatterjea [10]). Thus various results

pertaining to fixed points, common fixed points, coincidence points, etc. have been

investigated for maps satisfying different contractive conditions in different settings, see

among others, Tychonoff [11], Lefschetz [12], Kakutani [13], Tarski [14], Edelstein [15],

Zamfirescu [16], Ćirić [17]-[18], Mishra [19], Singh et al [20], Singh and Mishra [21],

Browder [22], Göhde [23], Suzuki [24]-[25], Suzuki and Kikkawa [26] and references

therein. For a fundamental comparison and development of various contractive conditions,

one may refer Rhoades [27]-[30], Collaco and Silva [31], Murthy [32], Singh and Tomar [33]

and Pant et al [34].

The study of fixed points of multivalued mappings was initiated by Nadler [35]-[37]

and nonexpansive mapping by Markin [38]-[39] using the concept of Hausdorff metric.

Indeed, Nadler [37] proved that a multivalued contraction has a fixed point in a complete

metric space. Ćirić [40] generalized Nadler’s result to multivalued quasi-contraction maps.

Subsequently, it received great attention in applicable mathematics and was extended and

generalized on various settings. Further, hybrid fixed point theory for nonlinear single-valued

and multivalued maps is a new development in the domain of multivalued analysis (see, for

instance, Krasnoselskii [41], Corley [42], Mishra et al [43], Singh and Arora [44], Singh and

Prasad [45], Dhage [46], Singh and Mishra [47]-[50], Singh et al [51]-[52] and references

thereof). For a historical development of the hybrid fixed point theory, one may refer to Singh

and Mishra [47] and Singh et al [51]-[52].

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Thus fixed point theory has been extensively studied, generalized and enriched in

different approaches such as, metric, topological and order-theoretic (see Goebel and Kirk

[53], Kirk and Sims [54], Brown [55], Carl [56], Andres and Gorniewicz [57], Wegrzyk [58]

and Smart [59]). This advancement in fixed point theory diversified the applications of

various fixed point results in various areas such as the existence theory of differential and

integral equations, dynamic programming, fractal and chaos theory, discrete dynamics,

population dynamics, differential inclusions, system analysis, interval arithmetic, optimization

and game theory, variational inequalities and control theory, elasticity and plasticity theory

and other diverse disciplines of mathematical sciences (see, for instance, Zeidler [5], Corley

[42], Brown [55], Andres and Górniewicz [57], Wegrzyk [58], Smart [59] and Agarwal et al

[60]).

In the following, we give a brief sketch of the development of the topics studied in this thesis.

1.1.1.

In many situations of practical utility, the mapping under consideration may not have an exact

fixed point due to some tight restrictions on the space or the map. Further, there may arrise

some practical situations where the existence of a fixed point is not strictly required but an

approximate fixed point is more than enough. The theory of approximate fixed points plays an

important role in such situations. Approximate fixed point property for various types of

mappings has been a prominent area of research for the last few decades. A classical best

approximation theorem was introduced by Fan [61] in 1969. Afterward, several authors,

including Reich [62], Prolla [63], Sehgal and Singh [64]-[65], have derived extensions of

Fan’s theorem in many directions. Approximate fixed points of continuous maps have been

studied by Gajek et al [66], Rafi and Salami [67], Hadzic [68]-[69] and many others. Tijs et al

[70] studied approximate fixed point theorems for contraction and non-expansive maps by

weakening the conditions on the spaces. Branzei et al [71] further extended these results to

multifunctions in Banach spaces. Singh and Prasad [72] proved an approximate fixed point

theorem for quasi contraction in metric spaces. Recently Berinde [73] obtained some

approximate fixed point theorems for operators satisfying Kannan, Chatterjea and Zamfirescu

type of conditions on metric spaces. P˘acurar and P˘acurar [74] obtained approximate fixed

point results for almost or weak contraction maps.

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If T is a non-self-mapping, it is probable that the fixed point equation xTx = has no solution.

In such a case best approximation theorems explore the existence of an approximate solution

whereas best proximity point theorems analyze the existence of an optimal approximate

solution. In 2003, Kirk et al [75] introduced 2-cyclic contraction, which becomes a relevant

research area in search of best proximity points. Eldred and Veeramani [76] defined a new

contraction which is more abstract formulation than 2-cyclic contraction. This definition is

more general than the notion of cyclical maps, in the sense that if the sets intersect, then every

point is a best proximity point. It turns out that many of the contractive-type conditions which

have been investigated for fixed points ensure the existence of best proximity points. Some

results of this kind are obtained in Bari et al [77], Al-Thagafi and Shahzad [78], Karpagam

and Agrawal [79], Petric [80], Mishra and Pant [81], etc. Recently, Mohsenalhosseini et al

[82] also defined a stronger concept of approximate best proximity points and obtained some

interesting existence results.

1.1.2.

It is a well known that the celebrated Banach contraction principle (BCP) is one of the

main tools for both the theoretical and the computational aspects in mathematical sciences.

This theorem has witnessed numerous generalizations and extensions in the literature due to

its simplicity and constructive approach. Kannan [9] by taking an entirely different condition,

proved a fixed point theorem for operators, which need not be contraction or contractive.

Indeed, he was the first to propose a fixed point theorem for a discontinuous map. Jungck [83]

obtained an important generalization of (BCP) in the form of common fixed point theorem for

commuting pair of maps. Sessa [84] introduced the concept of weakly commuting maps. This

concept was further improved by Jungck and Rhoades [85] with the notion of weakly

compatible mappings. Pant [86] studied the common fixed point theorem of noncommuting

maps. Singh and Mishra [49] studied coincidence and fixed points of reciprocally continuous

and compatible hybrid maps. Aamri and Moutawakil [87] defined property E. A., Liu et al

[88] generalized it as common property E. A. and obtained interesting results. Branciari [89]

studied contractive conditions of integral type and obtained an integral version of the Banach

contraction principle. A number of papers appeared for the maps satisfying the integral type

conditions in different settings, see for example, Rhoades [90], Vijayaraju et al [91], Aliouche

[92], Djoudi and Aliouche [93], Pathak et al [94], Pathak and Verma [95]-[96], Day et al [97]

and many others.

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After the initiation of fuzzy sets by Zadeh [98], fuzzy metric spaces are introduced by many

authors such as Kramosil and Michalek [99], George and Veeramani [100], Gregori and

Sapena [101], Kaleva and Seikkala [102], Park [103], Adibi et al [104], Saadati and Park

[105]-[106] in different ways. Grabiec [107] first extended BCP and Edelstein fixed point

theorem to fuzzy metric space Thereafter, may authors worked on this line (see, George and

Veeramani [100], Gregori and Sapena [101], Schweizer et al [108], Song [109], Mihet [110],

Mishra et al [111]-[113], Jha [114] etc). Gogeun [115] generalized fuzzy set to −L fuzzy set

in 1967. Another extension includes intuitionistic fuzzy sets proposed by Atanassov [116] in

1986. Saadati et al [117], Saadati [118] generalized the notions of fuzzy metric spaces due to

George and Veeramani [100] and intuitionistic fuzzy metric spaces due to Saadati and Park

[105] by introducing the notion of −L fuzzy metric spaces with the help of continuous t-

norms. These developments enthused the fuzzyfication of fixed point theory and authors

contributed profusely in this line, see for example, [100]-[101], [107], [119]-[128] and

references thereof.

1.1.3.

Fixed point theory has always been exciting in itself and its applications in new areas.

Currently, it has found new and hot areas of activity. The initiation of fixed point theory in

computer science by Tarski [14] and Scarf [129] enhances its applicability in different

domains. Due to the advent of speedy and fast computational tools, a new horizon has been

provided to fixed point theory. The fixed point equations are solved by means of some

iterative procedures. In view of their concrete applications, it is of great interest to know

whether these iterative procedures are numerically stable or not. The study of stability of

iterative procedures enjoys a celebrated place in applicable mathematics due to chaotic

behavior of functions in discrete dynamics, fractal graphics and various other numerical

computations where computer programming is involved. This kind of problem for real-valued

functions was first discussed by M. Urabe [130] in 1956. The first result on the stability of

iterative procedures on metric spaces is due to Alexander M. Ostrowski [131]. This result was

extended to multivalued operators by Singh and Chadha [132]. Czerwik et al [133]-[134]

extended this to the setting of generalized metric spaces.

The stability of iterative schemes in various settings has thus been studied during the

last three decades by a number of authors, see for instance, Agarwal et al [60], Czerwik et al

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[133]-[134], Ahmed [135], Berinde [136]-[137], Ćirić and Ume [138], Chidume et al [139],

Goebel and Kirk [140], Harder and Hicks [141]-[142], Osilike [143]-[144], Rhoades [145]-

[146], Singh et al [147]-[149], Olatinwo and Imoru [150], Osilike and Udomene [151], Singh

and Prasad [152], Mishra and Kalinde [153], Mishra et al [154] and several references thereof.

Recently Timis and Berinde [155] and Timis [156] respectively studied weak stability and

weak −2w stability of an operator in metric spaces.

1.1.4.

The study of fixed point problems for multivalued maps was initiated by Kakutani

[13] in the year 1941 in finite dimensional spaces by generalizing the Brouwer’s fixed point

theorem [4]. This was the beginning of the fixed point theory of multimaps having a vital

connection with the minimax theory in games. Kakutani used his results to obtain a simple

proof of von Neumann’s minimax theorem [157]. This was extended to infinite dimensional

Banach spaces by Bohnenblust and Karlin [158]. Further, Sion [159] generalized von

Neumann results on the basis of Knaster, Kuratowski and Mazurkiewicz (KKM) theorem

[160]. At the beginning, the results of the KKM theory were established for convex subsets of

topological vector spaces mainly by Ky Fan [161]-[164]. Horvath [165]-[167] extended most

of the Fan’s results in the KKM theory to H-spaces by replacing the convexity condition by

contractibility. Park [168] in 1992 established new versions of KKM theorems and minimax

inequalities on H-spaces which were extended by Park and Kim [169]-[173]. Thereafter more

general spaces such as abstract convex space and KKM spaces were introduced by Park in

[174]-[178]. In this way, KKM theory has been continuously upgraded and enriched to

enhance its applicability to the wider range of problems, such as minimax theorems, saddle

point theorems in game theory, economics and optimization theory, variational inequalities

etc., (see for instance [5], [13], [158]-[159], [162]-[163], [165], [168]-[195]).

1.1.5.

One of the recent advancement of the fixed pint theory of single-valued and multivalued

mappings is in the areas of iterated function systems (IFS), iterated multifunction systems

(IMS) and fractals. Fractal theory is a new domain initiated by Mandelbrot [196] in which

Banach [3] and Nadler [37] fixed point theorems and their variants are of vital importance.

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Michael Barnsley and Steven Demko [197] popularized the theory of IFS after Hutchinson

[198] gave a formal definition of it in 1981. An IFS usually consists of a complete metric

space together with a finite set of contraction mappings. It was born as an application of the

theory of discrete dynamical systems and is a useful tool to build fractal and other similar sets

(see, for instance, [196], [197]-[229] and references thereof). Indeed, self similar sets are

special class of fractals and there are no objects in nature which have exact structures of self

similar sets. These sets are perhaps the simplest and the most basic structures in the theory of

fractals which should give us much information on what would happen in the general case of

fractals (cf. Kigami [209]-[211]).

This basic notion of IFS has been extended and enriched to more general settings by changing

the condition on mappings or the space by various authors (see for instance, [197]-[199],

[212]-[229] ). In Jain and Fiser [212] and Fiser [215] contraction maps are replaced by weakly

contractive or non-expansive maps. Mate [219] and Rus and Triff [229] replaced contraction

constant by a comparison function to obtain their results. In [230]-[232] the formulations of

the contraction due to Meir and Keeler have been used to generalize the IFS theory. Recently

Mihail and Miculescu [221] introduced the notion of generalized iterated function system

(GIFS), which is a family of functions in a complete metric space and obtained GIFS to be a

natural generalization of the notion of IFS (see [220]-[224]). Further, Llorens-Fuster et al

[218] defined mixed iterated function system by taking more general conditions and obtained

a mixed iterated function system theory for contraction and Meir Keeler contraction maps.

Thus there has been a huge development in the metric as well as topological fixed point

theory. The recent advancement of the computational tools has tremendously enhanced the

scope of the applications of it. The entire theory of contractions and multivalued contractions

is expected to revolutionize not only various branches of mathematics but also diverse

dimensions of knowledge in engineering and sciences.

In the following section, we recall the basic concepts required for our study in the subsequent

chapters regarding fixed point theory for single valued, multivalued and hybrid maps.

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1.2 BASIC CONCEPTS

We start with the basic concepts related to metric spaces, fixed points and different

contraction conditions.

Definition 1.2.1 [233]. Let X be a non-empty set together with a distance function

.: +→× RXXd The function d is said to be a metric iff for all ,,, Xzyx ∈ the following

conditions are satisfied

(i) ( , ) 0d x y ≥ and ,0),( yxiffyxd ==

(ii) ),,(),( xydyxd =

(iii) ).,(),(),( zydyxdzxd +≤

The pair ),( dX is called a metric space.

Definition 1.2.2 [6]. A sequence in a metric space is a Cauchy sequence if for every ,0>ε

there exists Nn ∈0 such that ,),( ε<mn xxd for all ., 0nmn >

Definition 1.2.3 [6]. A metric space ),( dX is called complete, if every Cauchy sequence

converges in it.

Definition 1.2.4 [6]. The diameter of a set A denoted by )(Aδ is defined as

}.,:),({sup)( AbabadA ∈=δ It means that diameter of the set is the least upper bound of

the distances between the points of the set .A If the diameter is finite, i.e., ,)( ∞<Aδ then A

is bounded.

Definition 1.2.5 [6]. Let ),( dX be a metric space and : .T X X→ Then T has a fixed point if

there is an Xx∈ such that .Tx x= The point x is called a fixed point of T.

Definition 1.2.6 [6]. Let ),( dX be a metric space and .:, XXST → Then x is called a

coincidence (respectively, common fixed) point of T and S, if Xx∈ such that Tx = Sx

(respectively, x = Tx = Sx).

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In some situations an approximate solution of the problem is more than enough, so we have to

find approximate fixed point instead of fixed point.

Definition 1.2.7 [73]. Let ),( dX be a metric space and : .T X X→ An element 0x X∈ is

called an approximate fixed point (or −ε fixed point) of T if 0 0( , ) ,d Tx x ε< when 0.ε >

Approximate fixed point of the function exists if the defined mapping has approximate fixed

point property.

Definition 1.2.8 [73]. A map T is said to satisfy approximate fixed point property (AFPP)

if for every ,0>ε ,)( φε ≠TFix where )(TFixε is the set of all approximate fixed point of T.

The following condition guarantees the existence of approximate fixed points.

Definition 1.2.9 ([234]). A map XXT →: is said to be asymptotically regular if for any

,Xx∈

.0),(lim 1 =+

∞→xTxTd nn

n

Definition 1.2.10 [137]. Let ),( dX be a metric space. A mapping XXT →: is called

(i) Lipschitzian (or −L Lipschitzian) if there exists 0>L such that for all ,, Xyx ∈

),,(),( yxdLTyTxd ⋅≤ (lm)

(ii) Banach contraction if T is −α Lipschitzian, with [0, 1)α∈ and for all ,, Xyx ∈

( , ) ( , ),d Tx Ty d x yα≤ ⋅ (bc)

(iii) nonexpansive if T is 1-Lipschitzian and for all ,, Xyx ∈

( , ) ( , ),d Tx Ty d x y≤ (nm)

(iv) contractive if ),,(),( yxdTyTxd < for all ,, Xyx ∈ ,yx ≠ (cm)

(v) isometry if ),,(),( yxdTyTxd = for all ., Xyx ∈ (im)

Examples 1.2.1 [137]. (i) Let ],2,2/1[]2,2/1[: →T be defined as ,/1)( xxT = then T is

4-Lipschitzian with Fix (T) = {1}, where ( )Fix T denotes fixed point of the mapping T.

(ii) ,: RRT → ,32/)( += xxT .x R∈ Obviously T is a Banach contraction and ( ) {6}.Fix T =

10

(iii) 1 ,Tx x x R= − ∈ is nonexpansive and ( ) {1/ 2}.Fix T =

(iv) ],,1[],1[: ∞→∞T ,/1)( xxxT += is contractive and ( ) .Fix T φ=

(v) ,2)( += xxT then ( )Fix T φ= is isometry.

The celebrated Banach contraction principle (BCP) is the simplest and one of the most

versatile and fundamental results in the fixed point theory. It not only produces approximation

of any desired accuracy but also determines a-priori and a-posteriori error estimates.

Theorem 1.2.1 [3]. Let ),( dX be a complete metric space and XXT →: satisfies (bc).

Then T has a unique fixed point ,p i.e., Tp = p and lim .n

nT x p

→∞=

Moreover, we have a-priori error estimate

( , ) ( , )1

nnd T x p d Tx xα

α≤

−

and the a-posteriori error estimate

1( , ) ( , ).1

n n nd T x p d T x T xαα

−≤−

)).(),((1

)),(( 1 xTxTdpxTd nnn −

−≤

αα

There are various generalizations of the celebrated contraction mapping principle. These

generalizations are obtained either by weakening the contraction condition of the map by

giving a sufficiently rich structure to the space in order to compensate the relaxation of the

contraction condition or by extending the structure of the space or sometime combining both

the approaches.

Kannan [9] was the first to propose a fixed point theorem for a discontinuous map. He exactly

proved the following.

Theorem 1.2.2 [9] (Kannan Contraction Theorem). Let T be a self map on a complete

metric space (X, d) satisfying the following condition (usually called Kannan contraction),

for all Xyx ∈, and some [0,1/ 2),β ∈

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[ ].),(),(),( TyydTxxdTyTxd +≤ β (kc)

Then T has a unique fixed point.

Kannan’s theorem motivated numerous extensions and generalizations of the BCP and his

own fixed point theorem on various settings. Chatterjea [10] proved a fixed point theorem for

discontinuous mapping satisfying a condition which is actually a kind of dual of Kannan

mapping.

Definition 1.2.11 [10]. Let ),( dX be a metric space. A mapping XXT →: is called

Chatterjea contraction if for all Xyx ∈, and some [0,1/ 2),γ ∈

[ ].),(),(),( TxydTyxdTyTxd +≤ γ (cc)

Zamfirescu [16] obtained some interesting results by combining Banach, Kannan and

Chatterjea contraction conditions.

Definition 1.2.12 [16]. Let ),( dX be a metric space. A mapping XXT →: is called

Zamfirescu contraction if for all Xyx ∈, and some [0, 1),α ∈ 1, [0, ),2

β γ ∈ satisfies at

least one of the following conditions.

(i) ),(),( yxdTyTxd α≤

(ii) [ ]),(),(),( TyydTxxdTyTxd +≤ β (zc)

(iii) [ ].),(),(),( TxydTyxdTyTxd +≤ γ

Among various generalizations of the Banach contraction, the Ćirić [17]-[18] contraction

(also called as quasi-contraction) is considered to be the most general one.

Definition 1.2.13. Let ),( dX be a metric space. A mapping XXT →: is called quasi-

contraction (Ćirić [17]-[18]) if

)},,(),,(),,(),,(),,(max{.),( TxydTyxdTyydTxxdyxdkTyTxd ≤ (qc)

for some 10 <≤ k and all yx, in .X

12

Definition 1.2.14 [235]. A mapping XXT →: is called weak or almost contraction if there

exist )1,0(∈α and 0≥L such that for all ,, Xyx ∈

).,(),(),( TxydLyxdTyTxd +≤α (ac)

It is important to note that any mapping satisfying Banach, Kannan, Chatterjea, Zamfirescu, or

Ciric (with constant k in )2/1,0( ) type conditions are a weak or almost contraction, see [71].

Jungck [83] extended Banach contraction in the following manner.

Definition 1.2.15 [83]. Let ( , )X d be a metric space and , : .T S X X→ The following

condition is known as Jungck contraction.

),,(),( SySxdTyTxd ⋅≤α (jc)

for some [0, 1)α∈ and all ., Xyx ∈

Although this condition was known to Singh and Kulshrestha [236] but Jungck was credited

for presenting constructive proof regarding the existence of a common fixed point of

commuting maps. Further various fixed point and common fixed point results are investigated

in the literature for the maps satisfying different conditions. We recall some of them as

follows.

Definition 1.2.16. Let ( , )X d be a metric space and .:, XXST → The mappings T and S are

(i) commuting [83], if TSx = STx for all ,Xx∈

(ii) weakly commuting [84], if ),(),( SxTxdSTxTSxd ≤ for all ,Xx∈

(iii) compatible [237, 238], if ,0),(lim =∞→ nnn

STxTSxd whenever }{ nx is a sequence in X such

that

,limlim tSxTx nnnn==

∞→∞→ for some ,Xt∈

(iv) weakly compatible [85], if they commute at their coincidence points; i.e., if Tu = Su for

some u in X, then TSu = STu,

(v) satisfying the property (E. A.) [87], if there exists a sequence }{ nx such that

,limlim tSxTx nnnn==

∞→∞→ for some .Xt∈

13

Notice that weakly commuting mappings are compatible and compatible mappings are weakly

compatible but the converse need not be true (see [87], [237]-[239]). It is easy to see that two

noncompatible mappings satisfy the property (E. A.).

Fixed point theory for multivalued mappings is a natural generalization of the theory of single

valued mappings. Most of the results including the well known fixed point theorems of

Banach and others have been extensively studied for the multivalued cases, see, for instance

[35]-[41], [55], [57]-[59], [63], [85] and several references therein. These multivalued fixed

point theorems have applications in control theory, convex optimization, differential

equations, economics, etc. (see also [178], [180]).

In the following, we discuss the basic concepts and preliminaries regarding multivalued fixed

point theory.

Definition 1.2.17 [59]. If T is a multivalued map, i.e., from X to the collection of nonempty

subsets of .X Then a point p in X is called a fixed point of T if .Tpp∈

Definition 1.2.18 [59]. Let XXS →: and ).(: XCBXT → A point Xp∈ is a coincidence

(respectively, common fixed) point of S and T if TpSp∈ (respectively, TpSpp ∈= ).

Example 1.2.2. Consider ),0[ ∞=X with the usual metric. Define XXS →: and

)(: XCBXT → as

∞∈∈

=),1[3)1,0[0

xifxxif

Sx and

∞∈+∈

=),1[]31,1[)1,0[}{

xifxxifx

Tx

We have ,1]4,1[31 TS =∈= that is 1=x is a coincidence point of S and T and 0 is the

common fixed point of S and T.

Definition 1.2.19 [240]. Let A and B be the nonempty compact subsets of ,X then the

distance between a point x and set A is defined as ),(min),( yxdAxdAy∈

= .

The distance from set A to set B is given as ),(max),( BxdBAdAx∈

= .

14

Note that d is not a metric, since d is not symmetric. That is, ),(),( ABdBAd ≠ in general.

The Hausdorff distance ),,( BAh between Aand B is ( , ) max{ ( , ), ( , )}.h A B d A B d B A=

Definition 1.2.20 [240]. Let ),( dX be a metric space and K(X) be the nonempty compact

subsets of X together with a distance function h. Then for all

, , ( ),A B C K X∈ ( , )h A B satisfies following properties.

(i) ( , ) 0h A B ≥ and ( , ) 0 ,h A B iff A B= =

(iii) ( , ) ( , ),h A B h B A=

(iv) ( , ) ( , ) ( , ).h A B h A C h C B≤ +

The pair ( , )K h is called a Hausdorff metric space.

Definition 1.2.21 [137]. Let ),( dX be a metric space and ( )P X is the family of nonempty

subsets of .X Let : ( )T X P X→ satisfies the following condition.

( , ) ( , ),h Tx Ty a d x y≤ ⋅ (mc)

for some )1,0[∈a and all ., Xyx ∈ Then T is called a multivalued contraction.

Further, T is called

(i) nonexpansive, if it is 1-Lipschitzian,

(ii) contractive if ( , ) ( , ),h Tx Ty d x y< for all ,, Xyx ∈ ,yx ≠

(iii) isometry if ( , ) ( , ),h Tx Ty d x y= for all ., Xyx ∈

Definition 1.2.22. Let :T X X→ and : ( ).S X CB X→ Then mappings T and S are

(i) weakly commuting [84], if ( )TSx CB X∈ for all x X∈ and ( , ) ( , ),h TSx STx d Tx Sx≤

(iii) compatible [237, 238], if lim ( , ) 0,n nnh TSx STx

→∞= whenever }{ nx is a sequence in X such

that

lim ( )nnSx A CB X

→∞= ∈ and lim ,nn

Tx t A→∞

= ∈

(iv) weakly compatible [85], if they commute at their coincidence points; i.e., if Tu Su∈ for

some u in X, then TSu = STu,

(v) R-weakly commutativity [86], if ( )TSx CB X∈ for all x X∈ and there exists a real

number 0>R such that

( , ) ( , ),h STx TSx R d Tx Sx≤

15

(v) property (E . A.) [87], if there exists a sequence }{ nx such that

lim ( )nnSx A CB X

→∞= ∈ and .lim AtTxnn

∈=∞→

Liu et al [86] defined a more general condition then property (E. A.) as follows.

Definition 1.2.23 [88]. Let XXgf →:, and ).(:, XCBXGF → The pairs ),( Ff and

),( Gg are said to satisfy the common property (E. A.) if there exists two sequences

}{},{ nn yx in X, some t in X, and A, B in CB(X) such that

,lim AFxnn=

∞→ ,lim BGynn

=∞→

.limlim BAtgxfx nnnnI∈==

∞→∞→

Nadler [37] extended the BCP to multivalued maps in the following manner.

Theorem 1.2.3 [37]. Let ),( dX be a complete metric space and ( )CB X denotes the

collection of all nonempty closed and bounded subsets of X with the Hausdorff metric h.

Suppose : ( , ) ( ( ), )T X d CB X h→ satisfies (mc). Then T has a fixed point in .X

For approximating fixed point of the corresponding contraction operator of a given

nonlinear equation, we need some iterative methods. There are many iterative procedures

given in the literature which can be used in different conditions (see, [2], [147], [241]-[244],

etc.), i.e. if one iterative scheme fails in a given situation other may be applied or some time

we may choose them according to their speed of convergence.

The most popular iterative procedure, called Picard iteration is defined as follows.

Definition 1.2.24 [2]. Let ( , )X d be a metric space and : .T X X→ Choose 0x X∈ and

define 1 0 2 1, ,...x Tx x Tx= = and obtain a relation

...,2,1,0,1 ==+ nTxx nn . (PI)

Here nx is the thn Picard iterate of 0x in X.

16

Banach fixed point theorem uses Picard iteration for the convergence towards a fixed point.

But in many cases Picard iterations may not converge to the fixed point of the map, some

other iterative scheme may be used in such cases.

In 1953, Mann [242] defined an iteration scheme in the following manner.

Definition 1.2.25 [242]. Choose an initial point 0x in a given set and let }{ nα with

,10 <≤ nα be a sequence of real numbers. Then Mann iteration is defined by

...,2,1,0,)1(1 =+−=+ nTxxx nnnnn αα . (MI)

If T is a continuous map and the Mann iterative process converges, then it converges to a

fixed point of .T But if T is not continuous, then there is no guarantee that it will converge to

a fixed point of ,T even if the Mann process converges. This is shown by the following

example.

Example 1.2.3 [137]. Let ]1,0[]1,0[: →T be given by 010 == TT and .10,1 <<= xTx

Then ( ) {0},Fix T = where ( )Fix T is the set of all fixed points of T and the Mann iteration

with 1,1≥= n

nnα converges to 1, which is not a fixed point of .T

Some other scheme may be used in such a case. Ishikawa [243] defined the following iterative

scheme.

Definition 1.2.26 [243]. Choose an initial point 0x in a given set and let }{},{ nn βα be the

sequences of real numbers with .10 <≤≤ nn βα Then

...,2,1,0,)1(

)1(1

=+−=+−=+

nTxxyTyxx

nnnnn

nnnnn

ββαα

(II)

In the similar way, Noor [244] defined a three-step iteration scheme as follows.

17

Definition 1.2.27 [244]. Choose an initial point 0x in a given set and let

)1,0[}{},{},{ ⊂nnn γβα be the sequences of real numbers. Then

...,2,1,0,)1(

)1()1(1

=+−=+−=+−=+

nTxxzTzxyTyxx

nnnnn

nnnnn

nnnnn

γγββαα

(NI)

For a detailed development of different iteration scheme, one may refer Berinde [137].

Jungck [237, 238] generalized BCP, by replacing the identity map with two continuous maps

and obtained a common fixed point theorem. He defined a new iterative scheme using two

mappings in the following way.

Definition 1.2.28 [238]. Let XYTS →:, and ).()( YSYT ⊆ For any ,0 Yx ∈ consider

...,1,0),,(1 ==+ nxTfSx nn .

This procedure was essentially introduced by Jungck, and it becomes the Picard iterative

procedure when XY = and =S identity map.

Notice that, If we take ,),( nn TxxTf = then the method is called Jungck-Picard iteration

(JPI) and if ,)1(),( nnnnn TxSxxTf αα +−= where ,10 <≤ nα then it is called Jungck-Mann

iteration (JMI) (see [147] & [245]).

Definition 1.2.29 [150]. Let XXS →: and ).()( XSXT ⊆ Define

...,1,0,)1(

)1(1 =

+−=+−=+ n

TxSxSzTzSxSx

nnnnn

nnnnn

ββαα

(JII)

where }{ nα and }{ nβ satisfies

∑ ∏∑+==

+−∞=

≥≤≤=n

jiii

n

jn

nn

aiviii

niii

10

0

}1{)(,)(

,0,1,0)(,1)(

ααα

βαα

converges. It is called Jungck-Ishikawa iteration.

Notice that when =S identity map, it is called Ishikawa iteration.

18

These fixed point iterative procedures have tremendous applications in the problems of

solving nonlinear equations. An iterative scheme is said to be stable if small perturbations

during computations, produce small changes in approximate value of the fixed point

computed by means of these itearions. Formally, Harder and Hicks [142] defined the stability

of an iteration procedure as follows.

Definition 1.2.30 [142]. Let ),( dX be a complete metric space and : .T X X→ Let

Xx nn ⊂∞=0}{ be the sequence generated by an iteration procedure involving T which is

defined by ,...,1,0),,(1 ==+ nxTfx nn where Xx ∈0 is the initial approximation and f is

some function. Suppose ∞=0}{ nnx converges to a fixed point p of .T Let Xy nn ⊂∞

=0}{ and set

...,2,1,0)),,(,( 1 == + nyTfyd nnnε Then, the iteration procedure is said to be −T stable or

stable with respect to T if and only if 0lim =∞→ nnε implies .lim pynn

=∞→

Singh et al [147] first defined stability for Jungck type iterative procedures in the following

manner.

Definition 1.2.31 [147]. Let ,:, XYTS → )()( YSYT ⊆ and z be a coincidence point of T

and ,S that is, ,pTzSz == say. For any ,0 Yx ∈ let the sequence },{ nSx generated by the

iterative procedure ...,1,0),,(1 ==+ nxTfSx nn , converges to .p Let XSyn ⊂}{ be an arbitrary

sequence, and set ...,2,1,0)),,(,( 1 == + nyTfSyd nnnε . Then the iterative procedure

),( nxTf will be called −),( TS stable if and only if 0lim =∞→ nnε implies that .lim pSynn

=∞→

Timis and Berinde [155] defined weak stability using the concept of approximate sequence,

which is defined as follows.

Definition 1.2.32 [137]. Let ),( dX be a metric space and Xx nn ⊂∞=1}{ be a given sequence.

We shall say that Xy nn ∈∞=0}{ is an approximate sequence of }{ nx if, for any ,Nk ∈ there

exists )(kηη = such that

,),( η≤nn yxd for all .kn ≥

19

Definition 1.2.33 [137]. Let ),( dX be a metric space and : .T X X→ Let }{ nx be an iteration

procedure defined by Xx ∈0 and

0),,(1 ≥=+ nxTfx nn .

Suppose that }{ nx converges to a fixed point p of .T If for any approximate sequence

Xyn ⊂}{ of },{ nx 0)),(,(lim 1 =+∞→ nnnyTfyd implies ,lim pynn

=∞→

then we shall say that

iterative procedure is weakly −T stable or weakly stable with respect to .T

Further Timis [156] defined weak −2w stability using the following concept of equivalent

sequences

Definition 1.2.34 [156]. Two sequences { }∞=0nnx and { }∞=0nny are equivalent sequences if

0),( →nn yxd as .∞→n

Remark 1.2.1. Any equivalent sequence is an approximate sequence but the converse may

not be true.

The following example illustrates it.

Example 1.2.4. Let ∞=0}{ nnx be a sequence with .2nxn = First, we take an equivalent sequence

of ∞=0}{ nnx to be ∞

=0}{ nny where .12

nnyn += In this case, we have ,01),( →=

nxyd nn .∞→n

Now, take an approximate sequence of ∞=0}{ nnx to be ,}{ 0

∞=nny where .

122

++=

nnnyn Then,

.,021

12),( ∞→>→

+= n

nnxyd nn

Definition 1.2.35 [156]. Let ),( dX be a metric space and XXT →: be a map. Let }{ nx be an

iteration procedure defined by Xx ∈0 and

0),,(1 ≥=+ nxTfx nn .

20

Suppose that }{ nx converges to a fixed point p of .T If for any equivalent sequence }{ ny of

},{ nx 0)),(,(lim 1 =+∞→ nnnyTfyd implies ,lim pynn

=∞→

then we shall say that iterative scheme is

weak −2w stable with respect to .T

1.3 SOME GENERAL SPACES

In this section we define some general spaces used in the subsequent work reported in this

thesis.

1.3.1 b - METRIC SPACE

First we define a b- metric space introduced by Czerwik [246]. The basic concepts of

convergence, compactness, closedness and completeness are also defined in such space.

Definition 1.3.1.1 [246]. Let X be a non empty set and 1≥b be a given real number. A

function +ℜ→× XXd : is said to be a −b metric iff for all ,,, Xzyx ∈ the following

conditions are satisfied

(i) ,0),( yxiffyxd ==

(ii) ),,(),( xydyxd =

(iii) )].,(),([),( zydyxdbzxd +≤

The pair ),( dX is called a b-metric space.

As noted in [246], mathematical problems such as the problem of metrization of convergence

with respect to measure, lead to the generalization of metric, so called b-metric. The class of

b-metric spaces is effectively larger than that of metric spaces, since a b-metric space is a

metric space when s = 1 in the above condition (iii). The following example of Singh and

Prasad [152] shows that a b-metric on X need not be a metric on X (see also [246, p. 264]).

Example 1.3.1.1 [152]. Let },,,{ 4321 xxxxX = and ,2),( 21 ≥= kxxd

),(),( 4131 xxdxxd = ),(),( 4232 xxdxxd == ,1),( 43 == xxd ),(),( ijji xxdxxd = for all

4,3,2,1, =ji and .4,3,2,1,0),( == ixxd ii Then

21

[ ]),(),(2

),( jnniji xxdxxdkxxd +≤ for 4,3,2,1,, =jin

and if ,2>k the ordinary triangle inequality does not hold.

Definition 1.3.1.2 [214]. Let ),( dX be a −b metric space. Then a sequence Nnnx ∈}{ in X is

called

(a) convergent if and only if there exists Xx∈ such that 0),( →xxd n as .∞→n In this

case, we write ,lim xxnn=

∞→

(b) Cauchy if and only if 0),( →mn xxd as ., ∞→nm

Remark 1.3.1.1 [214].

In a b-metric space ),( dX the following assertions hold

(i) a convergent sequence has a unique limit,

(ii) each convergent sequence is Cauchy,

(iii) in general, a −b metric is not continuous.

Definition 1.3.1.3 [214]. Let ),( dX be a −b metric space. If Y is a nonempty subset of .X

Then the closure Y of Y is the set of limits of all convergent sequences of points in ,Y i.e.,

}.limthatsuch}{sequenceaexiststhere:{ xxxXxY nnNnn =∈=∞→∈

Definition 1.3.1.4 [214]. Let ),( dX be a −b metric space. Then a subset XY ⊂ is called:

(a) closed if and only if for each sequence Nnnx ∈}{ in Y which converges to an element ,x we

have Yx∈ ,

(b) compact if and only if for every sequence of elements of Y there exists a subsequence

that converges to an element of Y,

(c) bounded if and only if .},:),(sup{)( ∞<∈= YbabadYδ

Definition 1.3.1.5 [214]. The −b metric space ),( dX is complete iff every Cauchy sequence

in X converges.

22

In the following section we recall some definitions and preliminaries regarding fuzzy metric

spaces required for our results. We follow R. P. Pant [86], George and Veeramani [100],

Grabiec [107], Mishra et al [111], Schweizer and Sklar [120], Subramanyam [247], Vasuki

[248], V. Pant [249], Singh and Jain [250] and Pathak et al [251] for notations and

preliminaries.

1.3.2 FUZZY METRIC SPACE

Definition 1.3.2.1 [120]. A binary operation ]1,0[]1,0[]1,0[: →×T is a −t norm if T

satisfies following conditions for all ].1,0[,,, ∈dcba

(i) ),,(),( abba TT =

(ii) )),,(,()),,(( cbacba TT =

(iii) ),(),( dcba TT ≤ whenever ca ≤ and ,db ≤ for all ],1,0[,,, ∈dcba

(iv) ,)1,( aa =T for all ].1,0[∈a

Definition 1.3.2.2 [100]. The triplet ( , , )X M T is said to be fuzzy metric space, if X is an

arbitrary set, T is a continuous −t norm and M is a fuzzy set on ),0(2 ∞×X satisfying the

following conditions.

(i) ( , , ) 0,x y t >M

(ii) ( , , ) 1 ,x y t iff x y= =M

(iii) ( ( , , ), ( , , ) ( , , ), , 0, , , ,x y t y z s x z t s t s x y z X≤ + > ∀ ∈T M M M

(iv) ( , , .) : (0, ) [0, 1]x y ∞ →M is continuous.

Definition 1.3.2.3 [107]. Let ( , , )X M T be a fuzzy metric space. A sequence }{ nx in X is

said to be

(i) convergent to a point Xx∈ if 1),,(lim =∞→

txxnnM , for all ,0>t

(ii) a Cauchy sequence if 1),,(lim =+∞→txx npnn

M , for all ,0>t ,0>p

(iii) a complete fuzzy metric space in which every Cauchy sequence converges to a point in it.

23

The concept of commuting mappings in the setting of fuzzy metric space is given by

Subramanyam [247] as follows.

Definition 1.3.2.4 [247]. Mappings A and S of a fuzzy metric space ( , , )X TM into itself are

said to be commuting if ( , , ) 1ASx SAx t =M for all .Xx∈

After this several other weaker conditions of commutativity are defined in fuzzy metric space,

some of them are described as: weakly commuting mappings, compatible mappings, R-weakly

commuting maps, R-weakly commutativity of type )( gA , R-weakly commutativity of type

)( fA , etc., (see [111], [247]-[248]).

Definition 1.3.2.5 [86]. Mappings A and S of a fuzzy metric space ( , , )X TM into itself is

said to be weakly commuting if ( , , ) ( , , )ASx SAx t Ax Sx t≥M M for each Xx∈ and .0>t

In 1994, Mishra et al [111] introduced the concept of compatible mapping as follows:

Definition 1.3.2.6 [111]. Mappings A and S of a fuzzy metric space ( , , )X TM into itself is

said to be compatible if 1),,( =∞→

tSAxASx nn for all ,0>t whenever }{ nx is a sequence in

X such that uSxAx nnnn==

∞→∞→limlim

for some .Xu∈

Vasuki [248] extended the notion of R-weakly commuting maps introduced by Pant [86] in

metric space to fuzzy metric spaces.

Definition 1.3.2.7 [248]. Mappings A and S of a fuzzy metric space ( , , )X M T into itself are

−R weakly commuting provided there exists some positive real number R such that

( , , ) ( , , / )ASx SAx t Ax Sx t R≥M M for each Xx∈ and .0>t

Pathak et al [251] improved the notion of R-weakly commuting mappings in metric spaces to

the notions of R-weakly commutativity of type )( gA and type ).( fA V. Pant [249] extended it

to fuzzy metric space.

24

Definition 1.3.2.8 [249]. Mappings A and S of a fuzzy metric space ( , , )X M T into itself are

−R weakly commuting of type )( fA (or of type )( gA ) provided there exists some positive

real number R, for each Xx∈ and 0>t such that

( , , ) ( , , / )SSx ASx t Sx Ax t R≥M M (or ( , , ) ( , , / )AAx SAx t Ax Sx t R≥M M ).

Jungck and Rhoades [85] introduced the concept of weakly compatible maps which were

found to be more generalized than compatible maps. Weak compatibility in fuzzy metric

space is given by Singh and Jain [250].

Definition 1.3.2.9 [250]. Mappings A and S of a fuzzy metric space ( , , )X M T into itself are

said to be weakly compatible if they commute at the coincidence points, i.e., if Au = Su for

some ,Xu∈ then ASu = SAu.

Many authors defined fuzzy metric spaces in different ways (see for instance [99]-[106],

[252]-[253]). Using the idea of L -fuzzy set [112], Saadati et al [115] introduced the notion

of L -fuzzy metric spaces with the help of continuous t-norms as a generalization of fuzzy

metric space due to George and Veeramani [100] and intuitionistic fuzzy metric space due to

Saadati and Park [105]. The following section devoted to basic concepts related to L -fuzzy

metric spaces.

1.3.3 L -FUZZY METRIC SPACE

Definition 1.3.3.1 [115]. Let ( , )LL= ≤L be a complete lattice, and U be a nonempty set

called a universe. An L -fuzzy set on U is defined as a mapping .: LU → For each

u in ,U )(u represents the degree ( in L ) to which u satisfies .

Lemma 1.3.3.1 [254, 255]. Consider the set *L and the operation *L≤ defined by

},1and]1,0[),(:),{( 212

2121* ≤+∈= xxxxxxL

25

1121*21 ),(),( yxyyxxL

≤⇔≤ and ,22 yx ≥ for every .),(),,( *2121 Lyyxx ∈ Then ),( *

*

LL ≤

is a complete lattice.

Classically, a triangular norm on )],1,0([ ≤ is defined as an increasing, commutative,

associative mapping ]1,0[]1,0[: 2 → satisfying ,),1( xx = for all ].1,0[∈x These

definitions can be straightforwardly extended to any lattice ( , ).LL= ≤L Also 0 inf L=L and

1 sup L=L

Definition 1.3.3.2 [120]. A triangular norm ( −t norm) on L is a mapping 2: L L→T

satisfying the following conditions.

(i) ( )( ( , 1 ) );x L x x∀ ∈ =LT

(ii) 2( ( , ) )( ( , ) ( , ));x y L x y y x∀ ∈ =T T

(iii) 3( ( , , ) ) ( ( , ( , )) ( ( , ), );x y z L x y z x y z∀ ∈ =T T T T

(iv) 4( ( , , , ) ) ( ( , ) ( , )).L L Lx x y y L x x and y y x y x y′ ′ ′ ′ ′ ′∀ ∈ ≤ ≤ ⇒ ≤T T

A −t norm T on L is said to be continuous if for any ,x y∈L and any sequences }{ nx and

}{ ny which converge to x and y, we have

lim ( , ) ( , )n nnx y x y

→∞=T T .

Definition 1.3.3.3 [254]. A negation on L is any decreasing mapping : L L→N satisfying

(0 ) 1=L LN and (1 ) 0 .=L LN If ( ( )) ,x x=N N for all ,Lx∈ then N is called an involutive

negation.

Definition 1.3.3.4 [118]. The 3-triplet ( , , )X M T is said to be an −L fuzzy metric space, if

X is an arbitrary (non-empty) set, is a continuous t-norm on L and M is an −L fuzzy set

on ),0(2 ∞×X satisfying the following conditions for every zyx ,, in X and st, in ),0( ∞ .

(i) ( , , ) 0Lx y t >M L ,

(ii) ( , , ) 1 0,x y t for all t iff x y= > =M L ,

(iii) ( , , ) ( , , ),x y t y x t=M M

(iv) ( ( , , ), ( , , ) ( , , ),Lx y t y z s x z t s≤ +T M M M

26

(v) ( , , .) : (0, )x y L∞ →M is continuous and lim ( , , ) 1 .t

x y t→∞

=M L

In this case M is called an −L fuzzy metric. If ,M N=M M is an intuitionistic fuzzy set,

then the 3-tuple ,( , , )M NX M T is said to be an intuitionistic fuzzy metric space, defined by

Park [103] in 2004, using the concept of intuitionistic fuzzy set [116].

Definition 1.3.3.5 A sequence Nnnx ∈}{ in an −L fuzzy metric space ( , , )X M T is called a

Cauchy sequence, if for each \{0 }Lε ∈ L and ,0>t there exists Nn ∈0 such that for all

,0nnm ≥≥

( , , ) ( )m n Lx x t ε>M N .

The sequence Nnnx ∈}{ is said to be convergent to Xx∈ in −L fuzzy metric space

( , , ),X M T if ( , , ) ( , , ) 1 ,n nx x t x x t= → LM M whenever ,∞→n for every .0>t −L fuzzy metric

space is said to be complete if and only if every Cauchy sequence converges to a point in it.

1.4 THESIS OUTLINE

This thesis is organized as follows.

In Chapter 2, we obtain some basic approximate fixed point results in generalized

metric spaces. Further some existence results concerning approximate fixed points, endpoints

and approximate endpoints of multivalued contractions are also derived. Some quantitative

estimates of the sets of approximate fixed points and approximate endpoints for set-valued

contractions in generalized metric space are developed. These results extend some recent

results in the literature. When the mapping under consideration is a nonself mapping, the fixed

point equation in such cases may not have a solution. Best proximity pair theorems deal with

these problems and provide not only approximate solution but also an optimal approximate

solution. Some approximate best proximity pair theorems are also obtained with application to

Hammerstein integral equation.

The intent of Chapter 3 is to obtain some common fixed point theorems in general

settings. We study the common fixed points of hybrid pair of maps consisting of single and

27

multivalued maps satisfying an integral type contractive condition in b-metric spaces. An

application of the results to functional equation of dynamic programming problem is also

given. Further we define the notion of θ − −L fuzzy metric space and obtain some common

fixed point theorems for maps satisfying integral type conditions in such spaces.

In Chapter 4, we study the stability results of various iterative schemes of maps

satisfying some general condition. Weak and −2w weak stability results are also provided for

these iterative schemes. Our results extend and improve several recent results. A comparative

study of the different iterative schemes is also presented for some examples reported in the

literature.

A generalized version of the KKM theorems by using the concept of Chang and Zhang

[182] is studied in Chapter 5. Our results generalize some of the recent result of Chang and

Zhang [182] and Ansari et al [179]. As applications of the results to game theory, minimax

theorem and saddle point theorem for two-person-zero-sum game are established.

Consequently, a saddle point theorem for two person zero sum parametric game is also

proved.

In Chapter 6, we first define some contraction conditions of more general nature and

establish a generalized iterated function and multi function system theory using those

contraction conditions. Correspondingly some existence and uniqueness results are also

obtained. This theory extends several recent results and enhances the scope of IFS and IMS.

28

ONE SHOULD STUDY MATHEMATICS SIMPLY

BECAUSE IT HELPS TO ARRANGE ONE’S IDEAS

M. W. Lomonossow