An in-depth guide to fixed-point theorems 9781536195651, 1536195650

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MATHEMATICS RESEARCH DEVELOPMENTS

AN IN-DEPTH GUIDE TO FIXED-POINT THEOREMS

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MATHEMATICS RESEARCH DEVELOPMENTS

AN IN-DEPTH GUIDE TO FIXED-POINT THEOREMS

RAJINDER SHARMA AND

VISHAL GUPTA EDITORS

Copyright © 2021 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the Publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Names: Sharma, Rajinder (Mathematics), editor. Title: An in-depth guide to fixed-point theorems / Rajinder Sharma, Faculty, University of Technology and Applied Sciences-Sohar (Formerly College of Applied Sciences-Sohar), Wilayat of Shinas in the Governorate of Batinah, Oman, Vishal Gupta, Professor, Maharishi Markandeshwar, Mullana, India, editors. Identifiers: LCCN 2021023440 (print) | LCCN 2021023441 (ebook) | ISBN 9781536195651 (hardcover) | ISBN 9781536197303 (adobe pdf) Subjects: LCSH: Fixed point theory. Classification: LCC QA329.9 .I53 2021 (print) | LCC QA329.9 (ebook) | DDC 515/.7248--dc23 LC record available at https://lccn.loc.gov/2021023440 LC ebook record available at https://lccn.loc.gov/2021023441

Published by Nova Science Publishers, Inc. † New York

This book is dedicated to my parents, Mr. Om Parkash Sharma and Mrs. Vidaya Sharma

CONTENTS Preface

ix

Acknowledgments

xi

Chapter 1

Topological Properties of Tvs-Metric Cone Spaces and Applications to Fixed Point Theory Raúl Fierro

1

Chapter 2

Fixed Points of Some Mixed Iterated Function Systems Bhagwati Prasad and Ritu Sahni

27

Chapter 3

Random Iteration Scheme Leading to a Random Fixed Point Theorem and Its Application Debashis Dey

51

Chapter 4

Some Common Fixed Point Theorems for Self-Mappings Satisfying Rational Inequalities Contraction in Complex Valued Metric Spaces and Applications Khaled Berrah, Oussaeif Taki Eddine and Abdelkrim Aliouche

81

Chapter 5

Best Proximity Point Theorems Using Simulation Functions Sanjay Mishra, Rashmi Sharma and Manoj Kumar

107

Contents

viii

Chapter 6

On B - Metric Spaces and Their Completion Stefan Czerwik

121

Chapter 7

On Banach Contraction Principle in Generalized B-Metric Spaces Stefan Czerwik

127

Chapter 8

Metric Fixed Point Theory in Context of Cyclic Contractions Ashish Kumar

137

Chapter 9

An Investigation of the Fixed Point Analysis and Practices Ö. Özer and A. Shatarah

203

About the Editors

247

Index

251

PREFACE Fixed point theory is an attractive area of Functional Analysis. Several major branches of Mathematics and Engineering including set theory, general topology, algebraic topology, robotic analysis provides natural setting for fixed point theorems. An application of fixed-point theorems encompasses diverse disciplines of mathematics, statistics, engineering, biology, and economics. Using fixed point theory techniques, it is possible to analyze several concrete problems from science and engineering, where one is concerned with a system of differential/integral/functional equations. This approach is also useful in dealing with certain problems of control systems and theory of elasticity. Fixed Point theorems are the most important tools for proving the existence and the uniqueness of the solutions to various mathematical models (differential, integral, PDE and variational inequalities etc.) representing phenomena arising in the different fields such as steady state temperature distribution, chemical reactions, neuron transport theories, economic theories, epidemic and flow of fluids. Fixed point Theory also provide mathematical basis to carry out asymptotic complexity analysis of algorithms. An In-depth Guide to Fixed Point Theorems is devoted to the recent developments in the said area evolved in different directions. The primary goal of this text is readers’ comprehension. One of the primary objectives of this book is to give the readers substantial experience in applications of Fixed-Point Theory. There are nine chapters in this book. First chapter deals with the topological properties of TVS-metric space cone space and its

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Rajinder Sharma and Vishal Gupta

applications to fixed point theory. Second chapter show case the establishment of fixed-point theorems for some generalized mixed type of iterated function systems defined on b-metric spaces. Third chapter comprises of a random iteration scheme leading to a random fixed point theorem and its application to analyze the existence of a solution of a nonlinear stochastic integral equation of the Hammerstein type. In chapter four, some common fixed-point theorems for self-mappings satisfying rational inequalities contraction in complex valued metric space with applications validating the main result are presented. Chapter five presents best proximity point theorems using simulation functions. Chapters six and seven are devoted to b-metric spaces and Banach Contraction principle in generalized b-metric spaces. Eighth Chapter deals with some metric fixedpoint results in the context of cyclic contractions. The last chapter is devoted to an investigation of fixed-point analysis and practices. The goal throughout this project has been to offer an in-depth guide to fixed point theory and its applications. Every effort is made to make this a “readable” fixed point theory text and to enhance learning and understanding in the aforesaid area.

ACKNOWLEDGMENTS None of this project would have been completed without the inspiration, invaluable guidance, steadfast encouragement, marathon discussions, indigenous support and help of several individuals who in any respect contributed and extended their valuable assistance in the preparation and completion of the same. First and foremost, we are heartily thankful to Dr. Awadh Ali Al Mamari, Dean, UTAS-Sohar whose kind concern, understanding and personal guidance acted as an inspiration to cross all the obstacles in the completion of this book. We wish to express our warm and sincere thanks to Dr. Naktel Al Kharusi, Head, CR Department, UTAS-Sohar for his valuable and extensive discussions related to advances and recent developments in the field of fixed point theory and its applications. Finally, the Editors are thankful to the reviewers and Management of the Nova Publishers, USA for their continuous support throughout the publication process of the Book.

In: An In-Depth Guide to Fixed-Point Theorems ISBN: 978-1-53619-565-1 c 2021 Nova Science Publishers, Inc. Editors: R. Sharma and V. Gupta

Chapter 1

T OPOLOGICAL P ROPERTIES OF T VS -M ETRIC C ONE S PACES AND A PPLICATIONS TO F IXED P OINT T HEORY Rau´ l Fierro∗ Instituto de Matem´aticas, Universidad de Valpara´ıso, Valpara´ıso, Chile

Abstract Let E be a topological vector space with θ as zero and suppose that a cone P is defined on E. By denoting by  the order on E defined by P , a conical metric on a set X is defined as a function d : X × X → E satisfying the following two conditions: (i) for all x, y ∈ X, d(x, y) = θ, if and only if, x = y, and (ii) for all x, y, z ∈ X, d(x, y)  d(x, z) + d(y, z). By means of this metric and the local basis of neighborhoods of E, a quite natural topology on X is defined. With this topology, X turns out a uniform topological space. A fundamental system of entourages is stated and topological properties are proved, which depends on the properties of E. In particular, X satisfies the first countability axiom, whenever E is metrizable. Some topological properties of X are studied and a noncompactness measure is defined, which allows us to define condensing functions and correspondences. Moreover, a classical result by Kuratowsky is obtained in the framework used in this chapter. The mentioned extension allows us to define different types of contractions for functions and correspondences on spaces that are not metric. ∗

Corresponding Author’s Email: [email protected].

2

Ra´ul Fierro In particular, a kind of Nadler result is obtained in our context. Moreover, existence theorems for fixed and approximate fixed points are stated and a Brønsted type order is defined on (X, d) in order to prove Bishop-Phelps and Caristi type theorems. The content of this chapter is based primarily on some articles of the author, but the preliminary facts about cones in topological vector spaces have been taken from the book Cones and Duality by Aliprantis and Tourky.

Keywords: cone metric spaces; contractions; noncompactness measure; ordering in topological vector spaces; uniform topology AMS Subject Classification: 53D, 37C, 65P

1.

INTRODUCTION

It has been proven by Huang and Zhang in [1] that, the fixed point theory based on cone metric spaces is not a banal extension of the classical theory based on metric spaces. Indeed, these authors introduced an example of contraction, on a cone metric space, which is not contraction in a standard metric space. As in [2, 3, 4, 5, 6, 7, 8, 9, 10], this has led some authors to publish numerous papers both on fixed point theory as on properties of cone metric spaces. However, it has not been paid attention enough to the topology of these spaces. Indeed, by making use of sequences, some concepts such as closed set and the completeness of these spaces are defined. In general, the results of these works are obtained by means of sequential properties of the cone metric, but without reference to the topology of the corresponding space, which is often not explicitly defined. One of the aims of this chapter is to provide a natural topology for a cone metric space. Indeed, we prove that this topology is generated by a uniformity and, reciprocally, it is proved that the topology of any uniform space is generated by a cone metric. Of course, the topology of a cone metric space depends strongly on the topological vector spaces where the cone metric takes its values. For instance, when this latter topology is locally convex, the topology of the cone metric space so is. One of the advantages of having a topology for the spaces under our study is that the precompact sets can be characterized by means of their cone metrics. Indeed, in this paper, a noncompactness measure is defined for these spaces, and precompactness of sets is characterized by this measure, which also enables us to define condensing mappings and condensing

Topological Properties of Tvs-Metric Cone Spaces

3

set-valued mappings. Originally, this type of measure was introduced by Kuratowski in [11] allowing to characterize relatively compact sets for complete (classical) metric spaces. Other authors such as [12, 13] have characterized relatively compact sets in locally convex spaces. In [14], a noncompactness measure is defined for cone metric spaces with normal cone in a Banach space. However, these authors do not prove that the noncompactness measure of a compact set equals zero. The Banach contraction theorem allows proving that a non-expansive self-mapping defined on a bounded, closed, and convex subset of a Banach space has an approximate fixed point. Furthermore, if this subset is compact, then the non-expansive self-mapping has a fixed point. Proof of these results is given, for instance, in Chapter 3 of [15]. By making use of a general result, introduced in this work, and a version of the Nadler theorem in the context of cone metric spaces, we prove that non-expansive set-valued mappings with respect to cone metric, have also approximate fixed points. Moreover, when a set-valued mapping has a compact domain, it has a fixed point. Another application, of the main results of this paper, is an extension of a fixed point theorem by Darbo in [16], which is proved in the context of cone metric spaces. The second aim of this chapter is to provide results for set-valued mappings defined on cone metric spaces, whose metric takes values in a general topological vector space since it is only assumed this space is order complete. In [2] (see also, [3]), Agarwal and Khamsi proved a version of Caristi’s theorem based in a Bishop-Phelps type result for a cone metric taking values in a Banach space. In this paper, we extend this result, which enables us to prove a more general version of Caristi’s theorem for cone metric spaces. Natural consequences are deduced from this fact and, as an application, we prove the existence of a fixed point for an analogous weak contraction of a set-valued mapping defined by Berinde and Berinde in [17], which, in our case, is defined according to a pseudo-Hausdorff cone metric. The content of this chapter is based primarily on some articles by the author [18, 19], but the preliminary facts about cones in topological vector spaces have been taken from the book Cones and Duality by Aliprantis and Tourky [20]. The chapter is organized as follows. In Section 2, elementary concepts of ordering are established and the essential of ordered topological vector space is presented in Section 3. The topology of a cone metric space is introduced in Section 4. Also in this section, it is proved that cone metric spaces are uniform topological spaces and the noncompactness measure on these spaces is introduced. Results on fixed points and approximated fixed points, both for single

4

Ra´ul Fierro

and multivalued functions are presented in Section 5. Finally, in Section 6, extensions of fixed point theorems existing in the classical theory of contractions in metric spaces are developed in the context of conical metrics.

2.

O RDERINGS

We start with some elementary concepts. Definitions 2.1. Let X be a nonempty set. A relation in X is any nonempty subset of X × X. If R is a relation in X and (x, y) ∈ R, we denote xRy. Let R be a relation in X. This relation R is said to be (i) reflexive, if for all x ∈ X, xRx, (ii) symmetric, if xRy implies yRx, (iii) anti-symmetric, if xRy and yRx imply x = y, and (iv) transitive, if xRy and yRz imply xRz. We say R is an equivalence relation, whenever R is reflexive, symmetric and transitive, R is said to be a pre-ordering, whenever R is reflexive and transitive, and R is a partial ordering, whenever R is a pre-ordering and anti-symmetric. If R is partially ordered and for any x, y ∈ X, either xRy or yRx, we say that R is totally ordered. Let R be a partial ordering on X and Y be a nonempty subset of X. Then, Y naturally inherits the order properties of X. If Y is totally ordered, we say that Y is a chain. Let  be a pre ordering on X, Y be a nonempty subset of X and b ∈ X. We say that b is an upper bound of Y , whenever, for all y ∈ Y , y  b. Analogously, we say that b is a lower bound of X, whenever, for all y ∈ Y , b  y. If b is an upper bound of Y such that b  b0 , for any other upper bound b0 of Y , we say that b is the least upper bound (supremum) of Y . Analogously, if b is a lower bound of Y such that b0  b, for any other lower bound b0 of Y , we say that b is the greatest lower bound of Y . The least upper bound and the greatest lower bound (infimum) of Y are denoted by sup(Y ) and inf(Y ), respectively. An element a ∈ X is called a maximal element of X, whenever the condition a  x, for some x ∈ X, implies x  a.

Topological Properties of Tvs-Metric Cone Spaces

5

A directed set (D, ) is a pre-ordered set such that, for all x, y ∈ D, the set {x, y} has an upper bound. Let X be a nonempty set. A net in X is any function from D to X, where (D, ) is a directed set. We say that a net is bounded, whenever its range is bounded and a bound of this net should be understood as a bound of its range. The following axiom of the set theory is frequently used. Zorn’s Lemma. If each chain in a partially ordered set X has an upper bound, then X contains a maximal element.

3.

O RDERED TOPOLOGICAL V ECTOR SPACES

Definitions 3.1. Let τ be a topology on a vector space E over a field K (K = R either K = C). We say that E is a topological vector space (TVS), whenever τ is Hausdorff and the vector space operations are continuous with respect to τ . We denote by B the family of all neighbourhood of θ, where θ stands for the zero element of E. Remark 3.2. Let E be a TVS. From the continuity of the sum operation, for each B ∈ B there exists A ∈ B such that A + A ⊆ B. In the sequel of this chapter, E stands for a real topological vector space with usual notations for addition and scalar product. We assume E contains a cone P , i.e., P is a nonempty closed subset of E such that P ∩ (−P ) = {θ} and for each λ ≥ 0, λP + P ⊆ P . Moreover, a partial order  is defined on E as follows: for each x, y ∈ E, x  y, if and only if, y − x ∈ P . We denote by x ≺ y whenever x  y and x 6= y. Moreover, the notations x  y means that y − x belongs to int(P ), the interior of P . As natural, the notations x  y, x  y and x  y mean y  x, y ≺ x and y  x, respectively. In what follows, P denotes a cone and  stands for the partial ordering defined by P . This cone is said to be normal (c.f. [20]), if for each B ∈ B, and x, y ∈ B, we have {z ∈ E : x  z  y} ⊆ B. The pair (E, ) is said to be a Riesz space, if given x, y ∈ E, there exists the greatest lower bound of {x, y}, which we denote by x ∧ y. This fact implies that there exists the least upper bound of {x, y}, which we denote by x ∨ y. We refer to [20] for notations and facts, about ordered vector spaces. Remark 3.3. For each a, b, c ∈ E such that a  b  c, we have a  c.

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Ra´ul Fierro

Notice that a natural uniformity could be defined on a TVS E. Indeed, let U = {UB }B∈B, where, for each B ∈ B, UB = {(x, y) ∈ E × E : x − y ∈ B}. Since, for each a ∈ E and B ∈ B, a + B = UB [a] := {x ∈ E : (x, a) ∈ UB }, we have the uniformity U is compatible with the topology τ . In particular, E is a completely regular topological space. Let C be a filter base on E. We say C is Cauchy, whenever for any B ∈ B, there exists A ∈ C such that A − A ⊆ B and we say that C converges to a point a ∈ E, whenever for any B ∈ B, there exists A ∈ C such that A ⊆ a + B. From Remark 3.2, any filter convergent is Cauchy. We assume, in what follows, that E is complete, i.e., any Cauchy filter on E is convergent. In all of this chapter, we assume that E is complete, i.e., any Cauchy filter on E is convergent. Definition 3.4. Let E be a Riesz space. We say that E is order complete (Dedekind), whenever any increasing and bounded from above net in E has a least upper bound. Theorem 3.5. Let E be an order complete Riesz space. Then, (i) every nonempty bounded from above subset of E has a least upper bound, and (ii) every nonempty bounded from below subset of E has a great lower bound. Proof. Let X be a nonempty bounded from above subset of E. Denote by b such a bound and let D be the family of all finite subset of X. Hence, (D, ) is a directed set and since E is a Riesz space, for each d = {x1 , . . . , xr } ∈ D, there exists Ud = x1 ∨ · · · ∨ xr . Accordingly, {Ud }d∈D is an increasing net in X bounded by b and since E is order complete, {Ud }d∈D has a least upper bound b∗ . Clearly, b∗ is an upper bound of X. Let b0 be another upper bound of X. Hence, b0 is an upper bound of {Ud }d∈D and consequently b∗  b0 . This, b∗ is the least upper bound of X, which proves (i). Condition (ii) follows from (i), due to for each nonempty bounded from below subset X of E, we can define −X = {x ∈ E : −x ∈ X}. We have −X is a nonempty bounded from above subset of E and by (i), −X has a least upper bound b∗ . Therefore, −b∗ is a a great lower bound of X and the proof is complete.

Topological Properties of Tvs-Metric Cone Spaces

4.

7

C ONE METRIC SPACES

Definition 4.1. A cone metric space is a pair (X, d), where X is a nonempty set and d : X × X → E is a function satisfying the following two conditions: (i) for all x, y ∈ X, d(x, y) = θ, if and only if, x = y, and (ii) for all x, y, z ∈ X, d(x, y)  d(x, z) + d(y, z). Examples 4.2. (i) Let (E1 , d1 ) and (E2 , d2 ) be two metric spaces and d : E1 × E2 → R2 defined as d(x, y) = (d1 (x, y), d2(x, y)). Then, (E1 × E2 , d) is a cone metric space. (ii) Let Q {(En , dn)}n∈N be a sequence of classical metric spaces, p ≥ 1 and ∞ d: ∞ 1} , where n=0 En → ` (C) defined as d(x, y) = {dn (x, y) ∧ Q n∈N ∞ ` (C) = {{xn }n∈N ∈ RN : supn∈N |xn | < ∞}. Then, ( ∞ E n=0 n , d) is a cone metric space. In what follows, (X, d) stands for a cone metric space. Remark 4.3. Let d : X × X → E a function. Then, (X, d) is a cone metric space, if and only if, for all x, y, z ∈ X, the following four conditions hold: a) d(x, y)  θ, b) d(x, y) = θ ⇔ x = y, c) d(x, y) = d(y, x), and d) d(x, y)  d(x, z) + d(z, y). Indeed, it is clear that these conditions are sufficient for (X, d) to be a cone metric space. Hence, let us prove that these conditions are necessary. By assuming that (X, d) is a cone metric space and making x = y in (ii), we obtain θ  2d(y, z), which proves a). Moreover, by making x = z in (ii), we have d(x, y)  d(y, x) and condition c) holds. Finally, condition b) and c) follow from (i) and, (ii) and c), respectively. In what follows, (X, d) stands for a cone metric space. Next, we define a topology on X based on its cone metric d. We denote UB BX UX

= {(x, y) ∈ X × X : d(x, y) ∈ B}, B ∈ B, = {UB : B ∈ B}, and = {U ⊂ X × X : ∃V ∈ BX , V ⊂ U }

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Ra´ul Fierro

Without loss of generality, in the sequel, we assume that every B ∈ B is balanced, i.e., for each α ∈ R such that |α| ≤ 1, we have αC ⊆ C. Notice that UX is a uniformity on X, if and only if, BX is a fundamental system of entourages. Theorem 3.5 below shows that indeed (X, UX ) is a uniform space. As usual, the diagonal of X is defined as ∆ = {(x, y) ∈ X × X : x = y}. Theorem 4.4. The family BX is a fundamental system of entourages for the uniformity UX on X. I.e., BX is a filter base satisfying the following three conditions: T (i) ∆ = C∈C UC ; (ii) for each U ∈ BX , there exists V ∈ BX , such that V ⊂ U −1 .

(iii) for each U ∈ BX , there exists W ∈ BX , such that W ◦ W ⊂ U . Proof. Conditions (i) and (ii) follow directly from the definition of cone metric. Let us prove condition (iii). Let U ∈ BX and C ∈ C satisfy U = UC . Hence, there exists B ∈ C such that B + B ⊂ C. Let W = UB and (x, y) ∈ W ◦ W . There exists z ∈ X such that (x, z) ∈ W and (z, y) ∈ W and consequently d(x, z) ∈ B and d(z, y) ∈ B. Thus d(x, y)  d(x, z) + d(z, y) ∈ B + B ⊂ C and accordingly W ◦ W ⊂ U . From (i), ∅ ∈ / BX . Let A, B ∈ C and C ∈ C such that C ⊂ A ∩ B. Hence UC ⊂ UA ∩ UB . Consequently, BX is a filter base on X × X, which concludes the proof. In the next sections, up to Section 6, we consider the space X endowed with the topology generated by UX . Hence, a base for the topology of X is given by means of the family {UB [a]; a ∈ X, B ∈ B}, where UB [a] = {x ∈ X : d(x, a) ∈ B}. Moreover, it is worth noting that X satisfies the first countability axiom, whenever E does it. In particular, when E is metrizable, X is 1◦ countable. Remark 4.5. Notice that for each   θ and a ∈ X, the ball B(a, ) = {x ∈ X : d(x, a)  } is an open set. However, in general, it is easy to give examples showing that the family of all balls is not a base for the topology of X. Consequently, the topology generated by {B(a, ) :   θ}, is, in general,

Topological Properties of Tvs-Metric Cone Spaces

9

weaker than the topology of X generated by UX . In [21], this weaker topology has been considered to prove some results. We refer to this topology as to the weak conical topology. In Section 6, we introduce some fixed point results based on this topology. S For each A ⊆ X and B ∈ B, we denote UB [A] = x∈A UB [x]. A subset B of X is said to be bounded, if there exists an (order) upper bounded B ∈ B such that B × B ⊆ UB . We denote by 2X the family of all nonempty subsets of X, by B(X) the family of all bounded subsets of X, by C(X) the family of all closed and nonempty subsets of X and CB(X) = C(X) ∩ B(X). A subfamily H = {HB }B∈B of CB(X) × CB(X) is defined as follows: HB = {(A, C) ∈ CB(X) × CB(X) : A ⊆ UB [C] and C ⊆ UB [A]}. It is well known, see [22] for instance, that H is a fundamental system of entourages for a uniformity on CB(X). The topology on CB(X) induced by H is referred to as the H-topology. Let K(X) denote the family of all nonempty compact subsets of X. Since K(X) ⊆ CB(X), we consider K(X) endowed with the induced H-topology. In [23] is proved that K(X) is H-complete, if and only if, (X, UX ) so is. In accordance with the uniformity generating the topology of X, a subset D of X is precompact, if for each B ∈ B, there exist x1 , . . . , xr ∈ X such that D ⊆ UB [x1 ] ∪ · · · ∪ UB [xr ]. Moreover, a filter F on X is a Cauchy filter (c.f. [24]), if for each B ∈ B, there exist A ∈ F such that A × A ⊆ UB . We define a noncompactness measure as follows: for D ∈ B(X), MX (D) denotes the family of all B ∈ B such that there exist x1 , . . . , xr ∈ X satisfying D ⊆ UB [x1 ] ∪ · · · ∪ UB [xr ]. Accordingly, (c.f. [24]), D ∈ B(X) is precompact, if and only if, MX (D) = B. Thus, MX (·) is a noncompactness measure. It is easy to see the following four properties hold: a’) MX (B) ⊆ MX (A) whenever A ⊆ B, b’) MX (A ∪ B) = MX (A) ∩ MX (B),

(B ∈ B(X)). (A, B ∈ B(X)).

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Ra´ul Fierro

c’) MX (A) = MX (A), (A ∈ B(X)). d’) MX (A) ⊆ B + MX (UB [A]),

(A ∈ B(X), B ∈ B),

where B + MX (UB [A]) = {B + B 0 : B 0 ∈ MX (UB [A])}. We have seen that a cone space enjoys of a Hausdorff uniform structure. The next result states that any Hausdorff uniform space has the topology induced by a cone metric. T Theorem 4.6. Let (X, U ) be a Hausdorff uniform space, i.e., U ∈U U = ∆. Then, there exists a cone metric d : X × X → E generating the topology of X. Proof. It is well-known that the uniformity U is generated by a family of separating pseudo metrics {dλ }λ∈Λ on X. Let E = RΛ be the real vector space of all functions from Λ to R, endowed with the usual operations of addition and scalar multiplication. By considering E with the product topology and defining P = {x ∈ E : ∀λ ∈ Λ, xλ ≥ 0}, we have P is a cone of E and a cone metric d : X × X → E is defined as d(x, y) = {dλ (x, y)}λ∈Λ. A local base for E is given by the family B of the all sets having the form Bλ1 ,...,λr (a, ) = {x ∈ E : max |xλi − aλi | < }, 1≤i≤r

λ1 , . . . λr ∈ Λ,

 > 0,

a ∈ E.

It is easy to see that {UB ; B ∈ B}, where UB = {(x, y) ∈ X × X : d(x, y) ∈ B}, is a fundamental system of entourages generating the uniformity U . Therefore, d is a cone metric generating the topology of X. Let D ⊆ X and C be a filterbase in X. We say that C converges to D, in the H-topology, if and only if, for each B ∈ B, there exists C ∈ C such that C ⊆ UB [D] and D ⊆ UB [C]. Theorem 4.6 below extends a classical result (Theorem 1’) by Kuratowski in [11]. T Theorem 4.7. Let C ⊆ CB(X)S be a filter base on X and D = C∈C C. Suppose (X, UX ) is complete and C∈C MX (C) = B. Then, the following two conditions hold: (i) D is compact and nonempty, and (ii) C converges to D in the H-topology.

Topological Properties of Tvs-Metric Cone Spaces

11

Proof. Since D is closed, X is complete and MX (D) = B, we have D is compact. Let F be the filter generated by C, i.e., F = {A ⊆ X : ∃C ∈ C, C ⊆ A}. In order to prove D is nonempty, we consider F ∗ , a ultrafilter such that F ⊆ F ∗ . Let B ∈ B and B 0 ∈ B such that B 0 + B 0 ⊆ B. By assumption, there exist C ∈ C and x1 , . . . , xr ∈ X such that C ⊆ UB 0 [x1 ] ∪ · · · ∪ UB 0 [xr ]. Since C ∈ F ∗ and F ∗ is a ultrafilter, there exists i ∈ {1, . . . , r} such that UB 0 [xi] ∈ F ∗ (see Corollary in Section §6.4, Chapter I in [24], for instance). Thus, UB 0 [xi ] × UB 0 [xi ] ⊆ UB and consequently, F ∗ is a Cauchy filter, which converges to some point x∗ ∈ X. But, \ \ \ {x∗ } = F ⊆ F = C F ∈F ∗

F ∈F

C∈C

and therefore D is nonempty. Next, we prove the convergence of C to E in the H-topology. Suppose there exist B ∈ B such that for any C ∈ C, C ∩ UB [D]c 6= ∅. Consequently, Ce = S c c {C ∩ U [D] : C ∈ C} is a filter base on U [D] . Moreover MX (C) ⊇ B B C∈Ce S c C∈C MX (C) = B and UB [D] is complete. These facts imply, from (i), that ∅ 6=

\

C∈Ce

C=

\

C ∩ UB [D]c = D ∩ UB [D]c,

C∈C

which is a contradiction. Therefore, for each B ∈ B, there exists C ∈ C such that C ⊆ UB [D] and the proof is complete. Example 4.8. Let f : X → X be a continuous mapping and for each B ∈ B denote ∆B = {x ∈ X : f (x) ∈ UB [x]}. Suppose S (X, UX ) is complete and C = {∆B ; B ∈ B} is a filter base satisfying B∈B MX (∆B ) = B. Then, from Theorem 4.6, the set of the all fixed points of f is compact and nonempty. Moreover, B converges to this set in the H-topology.

12

Ra´ul Fierro

Corollary 4.9. (Kuratowski [11].) Let (X, d) be a complete metric space, αX be the noncompactness measure on X, {Bn }n∈N be a decreasing sequence of T nonempty, closed and bounded subsets of X and D = n∈N Bn . Suppose limn→∞ αX (Bn ) = 0. Then, the following two conditions hold: (i) D is compact and nonempty, and (ii) B converges to D in the Hausdorff metric. Proof. Let C = {(−, ) :  > 0} and B ∈ B. Hence, there exists  > 0 such that C = (−, ). Let nS0 ∈ N such that αX (Bn0 ) < . Hence, C ∈ MX (Bn0 ) and consequently n∈N MX (Bn ) = C. Therefore, conditions (i) and (ii) follows from Theorem 4.6, which completes the proof. Remark 4.10. Note that whether C is countable, in Theorem 4.7, in order to T C is compact and nonempty, it suffices that X is sequentially complete, C∈C with respect to the uniformity UX . Let T : D ⊆ X → 2X be a set-valued mapping. We say that T is condensing, if for each A ⊆ D such that A ∈ B(X) and MX (A) 6= B, we have T (A) ∈ B(X), MX (A) ⊆ MX (T (A)) and MX (A) 6= MX (T (A)). A subfamily D of 2X is said to be stable under intersections, if for any T D0 ⊆ D, we have D∈D0 D ∈ D.

Theorem 4.11. Suppose (X, UX ) is complete. Let D be a stable under intersections subfamily of CB(X), D ∈ D and T : D → 2D be a condensing set-valued mapping. Then, there exists C ∈ D, a compact subset of D, such that T (C) ⊆ C.

Proof. Let x0 ∈ D and Σ = {K ∈ D : x0 ∈ K ⊆ D and T (K) ⊆ K}. T Due to D ∈ Σ, we T have Σ 6= ∅. Let B = K∈Σ K and C = {x0 } ∪ T (B). We have T (B) ⊆ K∈Σ T (K) ⊆ B and x0 ∈ B. Moreover, since B is closed, C ⊆ B. Thus T (C) ⊆ T (B) ⊆ C, C ∈ Σ and B = C. Since D is stable under intersections, we have C ∈ D. From properties b) and c), we obtain MX (C) = MX (T (B)) = MX (T (C)) and due to T is condensing, we have MX (C) = B. Since C is closed, we have C is compact and the proof is complete. Example 4.12. Suppose X is a complete topological vector space. Hence, under conditions and notations stated in Theorem 4.11, D can be chosen as the

Topological Properties of Tvs-Metric Cone Spaces

13

family of all nonempty convex sets and consequently, there exists a compact convex subset C of X such that T (C) ⊆ C. This fact, in case the Schauder conjecture were correct (see The Scottish Book [25], Problem 54), would imply the existence of a fixed point of T , when T is a single-valued mapping.

5.

A PPLICATIONS TO A PPROXIMATE AND F IXED P OINTS

In this section, the cone P is assumed to be normal. Recall that this means for each B ∈ B, and x, y ∈ B, we have {z ∈ E : x  z  y} ⊆ B. Let T : D ⊆ X → 2X be a set-valued mapping. A point x∗ ∈ D is said to be a fixed point of T , if x∗ ∈ T x∗ and T is said to have an approximate fixed point (see [4, 26, 27, 28, 29] for related concepts), if for any B ∈ B, there exists x ∈ X such that T x ∩ UB [x] 6= ∅. The set of all fixed points of T is denoted by Fix(T ). A concept extending the classical Hausdorff metric is stated as follows: for x ∈ X and A, B ∈ CB(X), we define s(x, B) and s(A, B) as follows: [ s(x, B) = {  θ : d(x, b)  } b∈B

and s(A, B) =

\

s(a, B) ∩

a∈A

\

s(b, A).

b∈B

Let k ≥ 0 and T : X → CB(X) be a set-valued mapping satisfying kd(x, y) ∈ s(T x, T y),

for all x, y ∈ X.

(1)

As in [5, 6, 7, 30, 31], we say T is a contraction whenever k < 1. In this work, the set-valued mapping T is said to be non-expansive, whenever (1) is satisfied with k = 1. In [18, 8, 10], other definitions of contraction for setvalued mappings are given in the context of cone metric spaces. Also, these definitions are based on extensions of the classical Hausdorff metric. The following lemma is stated and proved in [31] (see also [5]) whether E is a locally convex space and in [10] whether E is a Banach space. Since we are not assuming these conditions and the completeness condition stated here is different, we give an explicit proof, which is simple and similar to that given by Nadler in [32], for set-valued contraction with respect to the classical Hausdorff metric.

14

Ra´ul Fierro

Lemma 5.1. Suppose (X, UX ) is complete and let T : X → CB(X) be a contraction. Then, T has a fixed point. T Proof. Let x0 ∈ X and x1 ∈ T x0 . Since kd(x0 , x1 ) ∈ x∈T x0 s(x, T x1) ⊆ s(x1 , T x1), there exists x2 ∈ T x1 such that d(x1 , x2 )  kd(x0 , x1 ). It follows by induction that there exists a sequence {xn }n∈N in X such that, for each n ∈ N, xn+1 ∈ T xn and d(xn+1 , xn+2 )  kd(xn , xn+1 ). Hence, for each n, p ∈ N, d(xn , xn+p )  {kn /(1 − k)}d(x0, x1 ) and since E is normal, {xn }n∈N is a Cauchy sequence with respect to UX . Thus, there exists x∗ ∈ X such that {xn }n∈N converges to x∗ . On the other hand, kd(xn+1 , x∗ ) ∈ s(T xn , T x∗) and hence there exists yn ∈ T x∗ such that d(xn+1 , yn )  kd(xn+1 , x∗ ). Accordingly, d(x∗ , yn )  d(x∗ , xn+1 ) + kd(xn+1 , x∗ ). Let C, C 0 ∈ C and N ∈ N such that C 0 + C 0 ⊆ C and d(xn, x∗ ) ∈ C 0 for all n ≥ N . We have, d(x∗ , xn+1 ) + kd(xn+1 , x∗ ) ∈ C 0 + kC 0 ⊆ C and since E is normal, d(x∗ , yn ) ∈ C, for all n ≥ N . This proves that {yn }n∈N converges to x∗ and due to T x∗ is closed, we have x∗ ∈ T x∗ , which completes the proof. Next, we assume X is a vector space endowed with a cone norm on E. That is, there exists a function k · k : X → E satisfying the following three conditions: (a) for all x ∈ X, kxk = 0 implies x = 0, (b) for all x ∈ X and λ ∈ R, kλxk = |λ|kxk, and (c) for all x, y ∈ X, kx + yk  kxk + kyk. We denote by d the cone metric induced by k · k and, in what follows, for A, B ∈ CB(X), s(A, B) is defined in terms of this cone metric. Since E is normal, it is easy to see that the topology of X, generated by UX , makes continuous its vector space operations. Consequently, with respect to this topology, X turns out a topological vector space. Theorem 5.2. Suppose (X, UX ) is complete, D is a nonempty closed convex and bounded subset of X and T : D → 2D is a non-expansive set-valued mapping. Then, T has an approximate fixed point. Proof. For a fixed z0 ∈ D and  > 0, we define T : D → 2D by T x = z0 + (1 − )T x. Notice that T is well-defined and, for any x, y ∈ X, s(T x, Ty) = (1 − )s(T x, T y). Consequently, (1 − )d(x, y) ∈ s(T x, Ty), i.e., T is a contraction. It follows from Lemma 5.1 that there exists x ∈ D such that x ∈ T x . Let y ∈ T x such that x = z0 + (1 − )y . We have kx − y k = kz0 − y k  c, where c is an upper bound of {d(u, v) : u, v ∈ D}. Since E is normal, for each B ∈ B we can choose  > 0 such that kx − y k ∈ C. Therefore, T x ∩ UB [x ] 6= ∅ and the proof is complete.

Topological Properties of Tvs-Metric Cone Spaces

15

Given T : D ⊆ X → 2D and B ∈ B, we denote AB [T ] = {x ∈ D : T x ∩ UB [x] 6= ∅}. Theorem 5.3. Suppose (X, UX ) is complete, D is a nonempty closed convex and bounded subset of X, T : D → K(D) is a non-expansive set-valued mapS ping and B∈B MX (AB [T ]) = B. Then, Fix(T ) is compact and nonempty.

Proof. From Theorem 5.2, for each B ∈ B, AB [T ] is nonempty and hence, CT = {AB [T ]; B ∈ B} is a filter base of closed sets. On the other hand, S assumption and c’) in Section 4. C∈C MX (BC [T ]) = B holds due to the T Consequently, Theorem 4.6 implies that B∈B AB [T ] 6= ∅. Since Fix(T ) = T B∈B AB [T ] , it suffices to prove that \ \ AB [T ] = AB [T ]. (2) B∈B

B∈B

In order to obtain (2), we prove that for each B ∈ B, there exists B 0 ∈ B such that AB 0 [T ] ⊆ AB [T ]. Let B ∈ B and choose B 0 ∈ B satisfying B 0 + B 0 ⊆ B. T Let y ∈ AB 0 [T ] and suppose that T y ∩ UB [y] = ∅. Since T y = B∈B UB [T y] and T y is compact, there exists B 00 ∈ C such that UB 00 [T y] ∩ UB [y] = ∅. Notice that for each x ∈ UB 0 [y], UB 0 [x] ⊆ UB [y]. Let x ∈ UB 00 [y]. Due to d(x, y) ∈ s(T x, T y) we have, for each z ∈ T x there exists b ∈ T y such that d(z, b)  d(x, y). Since d(x, y) ∈ B 00 and E is normal, we have d(z, b) ∈ B 00 . Consequently, for each x ∈ UB 00 [y], we have T x ⊆ UB 00 [T y] and by defining Vy = UB 0 [y] ∩ UB 00 [y], we obtain that, for each x ∈ Vy , T x ∩ UB 0 [x] ⊆ UB 00 [T y] ∩ UB [y] = ∅, i.e., Vy ⊆ AB 0 [T ]c, which is a contradiction. Therefore, AB 0 [T ] ⊆ AB [T ] and the proof is complete. Corollary 5.4. Suppose D is a nonempty compact and convex subset of X and T : D → K(D) is a non-expansive set-valued mapping. Then, Fix(T ) is (compact and) nonempty. Remark 5.5. Suppose B is a base of neighborhoods of θ consisting of convex sets. Hence, {UB [0]; B ∈ B} is a base of convex neighborhoods of 0 ∈ X. Consequently, X is a locally convex space whether E so is. For each D ⊆ X, let Q(D) be the family of all nonempty compact convex subsets of D. Lemma 5.6. Suppose E is a locally convex space, C ∈ Q(X) and let T : C → Q(C) be a continuous set-valued mapping. Then, T has a fixed point.

16

Ra´ul Fierro

Proof. It directly follows from Theorem 2 by Fan in [33]. The following result extends, to locally convex spaces and for continuous multi-functions, a known theorem by Darbo in [16]. Theorem 5.7. Suppose E is a locally convex space and (X, UX ) is complete. Let D be a nonempty bounded closed and convex subset of X and T : D → C(D) be a condensing multi-function with convex images. Then, T has a fixed point. Proof. Let D be the subfamily of CB(X) consisting of all convex subsets of X. Since D is stable under intersections, Theorem 4.11 implies that there exists a compact and convex subset C of X such that T (C) ⊆ C. Since C ∈ Q(X) and for each x ∈ C, T x ∈ Q(C), it follows from Lemma 5.6 that T has a fixed point, which completes the proof.

6.

C ARISTI AND R ELATED F IXED P OINT R ESULTS

In this section, we assume X is endowed with the weak conical topology, i.e, the topology of X is generated by the sub basis Bw = {Bw (a);   θ}, where Bw (a) = {x ∈ X : d(x, a)  }. It is worth mentioning that Bw need not be a basis for some topology. Additionally, we assume that E is order complete (Dedekind), i.e., every decreasing bounded from below net in E has an infimum. In particular, since E is a Riesz space, every bounded from below subset of E has an infimum. This fact is used in this section when a kind of Hausdorff pseudo metric is defined. Let {xn }n∈N be a sequence in X and x ∈ X. It is easy to see that {xn }n∈N converges to x, if and only if, for every   θ, there exists N ∈ N such that, d(xn , x)  , for any n ≥ N. Since the topology of X need not be generated by a uniformity, we need to say what we mean by a Cauchy sequence. A sequence {xn }n∈N it said to be Cauchy, if and only if, for every   θ, there exists N ∈ N such that, for any m, n ≥ N, we have d(xm , xn)  . The cone metric space (X, d) is said to be complete, if and only if, every Cauchy sequence in X converges to some point x ∈ X. Notices that, if a subset F of X is weakly closed, then for each sequence {xn }n∈N in F converging to x ∈ E, we have x ∈ F . The reciprocal is true whenever X satisfies the first countability axiom. In what follows of this section, we assume that X satisfies the first countability

Topological Properties of Tvs-Metric Cone Spaces

17

axiom and, consequently, the topological properties can be characterized by means of sequences. Remark 6.1. If X is complete and F ⊆ X is closed, then F is complete. Let ϕ : X → E be a function. We say ϕ is lower semicontinuous, if and only if, for any α ∈ E, the set {x ∈ X : ϕ(x)  α} is closed. For this function, a Brønsted type order ϕ is defined on X as follows: x ϕ y, if and only if, d(x, y)  ϕ(x) − ϕ(y). It is easy to see that ϕ is in effect a partial ordering relation on X. In the sequel, LS(X) stands for the space of all lower semicontinuous and bounded below functions from X to E. Remark 6.2. The function ϕ defining ϕ is non-increasing. The following theorem is an extension of the well-known results by BishopPhelps lemma [34]. Theorem 6.3. Suppose X is d-complete. Then, for each ϕ ∈ LS(X) and x0 ∈ X there exists a maximal element x∗ ∈ X such that x0 ϕ x∗ . Proof. For each x ∈ X, let S(x) = {y ∈ X : x ϕ y}, x0 ∈ X and C be a chain in S(x0 ). Since S(x) = {y ∈ X : ϕ(y) + d(x, y)  ϕ(x)}, the lower semicontinuity of ϕ + d(x, ·) implies S(x) is a closed set. Let e  θ and, inductively, define an increasing sequence {xn }n∈N as xn ∈ S(xn−1 ) ∩ C

with ϕ(xn ) ≺ (1/2n)e + Ln  (1/n)e + Ln ,

where x0 is given, An = {ϕ(y) : y ∈ S(xn−1 ) ∩ C} and Ln = inf(An ). Due to ϕ is non-increasing and bounded below, An is a chain in P and consequently {xn }n∈N is well defined. Moreover, for each n, p ∈ N, xn+p ∈ An and hence d(xn , xn+p)  ϕ(xn ) − ϕ(xn+p )  (1/n)e. Thus, {xn }n∈N is a Cauchy sequence and accordingly, there exists x∗ ∈ X such that this sequence converges to x∗ . Since for each n ∈ N, S(xn ) is a closed set, we have x∗ ∈ S(xn) and thus x0  xn  x∗ . Suppose y ∈ X satisfies x∗ ϕ y. We have, for each n ∈ N, d(xn , y)  ϕ(xn ) − ϕ(y) ≺ (1/n)e and hence limn→∞ d(xn , y) = 0. This fact implies that x∗ = y and therefore x∗ ∈ X is a maximal element satisfying x0 ϕ x∗ . This concludes the proof.

18

Ra´ul Fierro

A set B ⊆ X is said to be bounded, whenever {d(x, y) : x, y ∈ B} is bounded in E. In the sequel, we denote by 2X the family of all nonempty subsets of X and by B(X) the subfamily of 2X consisting of all nonempty and bounded subsets of X. For a set-valued mapping T : X → 2X and x ∈ X, we usually denote T x instead of T (x). Theorem 6.3 enables us to state below a generalized version of Caristi’s theorem. Theorem 6.4. Suppose X is d-complete, T : X → 2X is a set-valued mapping and ϕ ∈ LS(X). The following two propositions hold: (i) If for each x ∈ X, there exists y ∈ T x such that d(x, y)  ϕ(x) − ϕ(y), then, there exists x∗ ∈ X such that x∗ ∈ T x∗ . (ii) If for each x ∈ X and y ∈ T x, d(x, y)  ϕ(x) − ϕ(y), then, there exists x∗ ∈ X such that {x∗ } = T x∗ . Proof. From Theorem 6.3, ϕ has a maximal element x∗ ∈ X. Suppose there exists y ∈ T x∗ such that d(x∗ , y)  ϕ(x∗ ) − ϕ(y). i.e., x∗ ϕ y. The maximality of x∗ implies y = x∗ and hence i) holds. Since T x∗ is nonempty, (i) implies {x∗ } ⊆ T x∗ . By applying assumption in (ii) and the maximality of x∗ , we have T x∗ ⊆ {x∗ }, which proves (ii) and the proof is complete. For single-valued mappings the following corollary holds. Corollary 6.5. Suppose X is d-complete. Let f : X → X be a mapping and ϕ ∈ LS(X) such that for each x ∈ X, d(x, f (x))  ϕ(x) − ϕ(f (x)). Then, there exists x∗ ∈ X such that x∗ = f (x∗ ). A cone metric version of the nonconvex minimization theorem according to Takahashi [35] is stated as follows. Theorem 6.6. Let ϕ ∈ LS(X) such that for any x0 ∈ X satisfying inf x∈X ϕ(x) ≺ ϕ(x0 ), the following condition holds: there exists x ∈ X \ {x0 } such that d(x0 , x)  ϕ(x0 ) − ϕ(x). Then, there exists x∗ ∈ X such that inf y∈X ϕ(y) = ϕ(x∗ ). Proof. Suppose for every z ∈ X, inf y∈X ϕ(y) ≺ ϕ(z) and let x0 ∈ X. From Theorem 6.3, ϕ has a maximal element x∗ ∈ X such that x0 ϕ x∗ . Since ϕ

Topological Properties of Tvs-Metric Cone Spaces

19

is non-increasing, ϕ(x∗ )  ϕ(x0 ) and the assumption implies that there exists x ∈ X \ {x∗ } such that x∗ ϕ x. From the maximality of x∗ we have x = x∗ , which is a contradiction. Therefore, there exists z ∈ X such that inf x∈X ϕ(x) = ϕ(z), which completes the proof. At the current context, the Hausdorff conical metric can be defined as a functions H : B(X) × B(X) → E satifying ( ) H(A, B) = sup sup d(x, B), sup d(y, A) , x∈A

(1)

y∈B

where for each x ∈ X and a nonempty subset A of X, d(x, A) = inf y∈A d(x, y). Since E is an order complete Riesz space, Theorem 3.5 (see also Theorem 1.20 in [20] assures that (1) is well-defined. Remark 6.7. When d is a standard metric on X, H is the Hausdorff metric on B(X). However, in general, (B(X), H) is not a cone metric space. An linear operator L : E → E is said to be positive, if for any x ∈ P we have Lx ∈ P . Let K+ (E) be the set of all positive, injective and continuous linear operators δ from E into itself such that, there exists 0 ≤ t < 1 satisfying 0  δx  tx, for all x ∈ P . Notice that for each δ ∈ K+ (E) and x ∈ E, |δx|  δ|x|, where |a| = a ∨ −a, for all a ∈ E. Following Berinde and Berinde in [17], a set-valued mapping T : X → B(X) is called a (δ, L)-weak contraction, if there exist an positive linear operator L : E → E and δ ∈ K+ (E) such that H(T x, T y)  δd(x, y) + Ld(y, T x),

for all x, y ∈ X.

(2)

By the symmetry of the distance, condition (2) implicitly includes the following dual inequality H(T x, T y)  δd(x, y) + Ld(x, T y),

for all x, y ∈ X.

(3)

Hence, in order to check that a set-valued mapping T : X → B(X) is a (δ, L)weak contraction, it is necessary to check both inequalities (2) and (3). Let T : X → B(X) be a set-valued mapping. We say T is H-continuous at x ∈ A, if for any sequence {xn }n∈N in A converging to x, {H(T xn , T x)}n∈N converges to θ in E. The mapping T is said to be a contraction, if there exists

20

Ra´ul Fierro

k ∈ K+ (E) such that for any x, y ∈ X, H(T x, T y)  kd(x, y). Notice that T is a contraction, if and only if, there exists 0 ≤ t < 1 such that for any x, y ∈ X, H(T x, T y)  td(x, y). When E is a Banach space, t can be chosen as the spectral ratio ρ(k) of k and hence in this case, k is a contraction, if and only if, ρ(k) < 1. Of course, any contraction is a weak contraction. A selector of T is any function f : X → X such that f (x) ∈ T x, for all x ∈ X. We say T satisfies condition (S) if, for any  > 0, there exists a selector f of T such that for each x ∈ X, d(x, f (x))  (1 + )d(x, T x). Remark 6.8. For x ∈ X and A, B ∈ B(X), it is defined s(x, B) and s(A, B) as follows: [ s(x, B) = {  θ : d(x, b)  } b∈B

and s(A, B) =

\

s(a, B) ∩

a∈A

\

s(b, A).

b∈B

Some authors such as [5, 6, 7, 30, 31] define k-contraction, for 0 ≤ k < 1, as a set-valued mapping T : X → B(X) satisfying kd(x, y) ∈ s(T x, T y),

for all x, y ∈ X.

(4)

This definition is more restrictive than our definition of contraction by making L = 0 in (2). Indeed, even though the functional H is not properly a cone metric, it is easy to see that a set-valued mapping satisfying condition (1), it also satisfies our definition of contraction. Furthermore, condition θ ∈ s(a, A) implies a ∈ A for all a ∈ X and A ⊆ X, even though A is not closed. However, it is not possible to conclude that a ∈ A, if d(a, A) = 0, even though A is closed. Consequently, condition (1) is stronger than our definition of contraction. See example below. Example 6.9. Let E = R2 and P = {(x, y) ∈ E : x, y ≥ 0}. Let X = {(0, 0), (1, 0), (0, 1), (1, 1)} and define d : X × X → E as d((a, b), (c, d)) = (|a − c|, |b − d|). Hence, (X, d) is a cone metric space. Let T : X → 2X be a set-valued mapping such that  {(0, 0), (1, 1)} if (x, y) = (0, 0) T (x, y) = {(1, 0), (0, 1)} if (x, y) 6= (0, 0).

Topological Properties of Tvs-Metric Cone Spaces

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It is easy to see that for each (a, b), (c, d) ∈ X, H(T (a, b), T (c, d)) = (0, 0) and consequently, according to our definition, T is a k-contraction, for all k ∈ K+ (E). However, s(T (0, 0), T (1, 1)) = (1, 1) = d((0, 0), (1, 1)) and therefore, T does not satisfy (1) for k < 1. Given a set-valued mapping T : X → B(X), we denote by ϕT the mapping from X to E defined as ϕT (x) = d(x, T x). Proposition 6.10. Let T : X → B(X) be a H-continuous set-valued mapping. Then, ϕT ∈ LS(X). Proof. Let u, v ∈ X and y ∈ T v. Hence, d(u, T u)  d(u, v) + d(v, y) + d(y, T u)  d(u, v) + d(v, y) + H(T v, T u). Consequently, ϕT (u)  ϕT (v) + d(u, v) + H(T u, T v) and from this, the lower semicontinuity of ϕT is obtained. Corollary 6.11. Let T : X → B(X) be a contraction. Then, ϕT ∈ LS(X). Theorem 6.12. Let L : E → E be a positive linear operator, δ ∈ K+ (E), and T : X → B(X) be a (δ, L)-weak contraction satisfying condition (S). Suppose E is d-complete and ϕT ∈ LS(X). Then, there exists x∗ ∈ X such that x∗ ∈ T x∗ . Proof. We have H(T x, T y)  δd(x, y) + Ld(y, T x),

for all x, y ∈ X.

Hence, for each y ∈ T x, we have H(T x, T y)  δd(x, y). Define ϕ : X → E as  −1 1 ϕ(x) = −δ ϕT (x), 1+ 1 where  > 0 is chosen in such a way that 1+ > δ. From assumption ϕ ∈ LS(X) and since T satisfies condition (S), there exists a selector f or T such

22

Ra´ul Fierro

that for each x ∈ X, d(x, f(x))  (1+)d(x, T x). Hence, d(f (x), T f(x))  H(T x, T f(x))  δd(x, f(x)) and thus   1 − δ d(x, f(x))  d(x, T x) − d(f (x), T f(x)). 1+ Consequently, for each x ∈ X, d(x, f(x))  ϕ(x) − ϕ(f (x)) and it follows from Corollary 6.5 that there exists x∗ ∈ X such that x∗ ∈ T x∗ , which concludes the proof. Corollary 6.13. Suppose E is d-complete and let T : X → B(X) be a contraction satisfying condition (S). Then, there exists x∗ ∈ X such that x∗ ∈ T x∗ . Proof. It follows from Corollary 6.11 and Theorem 6.12. Corollary 6.14. Suppose E is d-complete and let f : X → X be a single valued contraction. Then, there exists x∗ ∈ X such that x∗ = f (x∗ ). Remark 6.15. Since the condition d(x, T x) = 0 does not imply, even if T x is closed, that x ∈ T x, it is not possible, in the scenario of cone metric spaces, to prove existence of fixed point for weak contractions, as it was done by Berinde and Berinde in [17] for set-valued mapping defined on standard metric spaces. Consequently, Corollary 6.5 was crucial in the proof of Theorem 6.12. Some emblematic and particular cases of standard weak contractions are the Chatterjea [36] and Kannan [37] contractions. Natural extensions of these concepts are obtained for set-valued mappings defined on cone metric spaces. Corollary 6.16 below shows that, under usual conditions, the existence of fixed points is ensured. Corollary 6.16. Suppose E is d-complete and let T : X → B(X) be a setvalued mapping satisfying condition (S) and such that, ϕT ∈ LS(X) and at least one of the following two conditions holds: (i) H(T x, T y)  α[d(x, T x) + d(y, T y)] (Kannan condition) and (ii) H(T x, T y)  α[d(x, T y) + d(y, T x)] (Chatterjea condition), where α : E → E is a linear operator satisfying 2α ∈ K+ (E). Then, there exists x ∈ X such that x ∈ T (x).

Topological Properties of Tvs-Metric Cone Spaces

23

ACKNOWLEDGMENTS The research was partially supported by Chilean Council for Scientific and Technological Research, grant FONDECYT 1200525.

R EFERENCES [1] Huang L.G. and Zhang X., Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications, 332:1468–1476, 2007. [2] Agarwal R.P. and Khamsi M.A., Extension of Caristi’s fixed point theorem to vector valued metric spaces. Nonlinear Analysis. Theory Methods & Applications, 74:141–145, 2011. [3] Altun I. and Rakocevi´c V., Ordered cone metric spaces and fixed point results. Computers and Mathematics with Applications, 60(8):1145–1151, 2010. [4] Amini A., Fakhar J., and Zafarani J., Fixed point theorems for the class S-KKM mappings in abstract convex spaces. Nonlinear Analysis. Theory Methods & Applications, 66(1):14–21, 2007. [5] Azam A., Mehmood N., Ahmad J., and Radenovi´c S., Multivalued fixed point theorems in cone b-metric spaces. Journal of Inequalities and Applications, 2013:582, 2013. [6] Cho S.H. and Bae J.S., Fixed point theorems for multivalued maps in cone metric spaces. Fixed Point Theory and Applications, page 87, 2011. [7] Mehmood N., Azam A., and Ko˘cinac L.D.R., Multivalued fixed point results in cone metric spaces. Topology and its Applicatons, 179:156–170, 2015. [8] Radenovi´c S., Simi´c S., Caki´c N., and Golubovi´c Z., A note on tvs-cone metric fixed point theory. Mathematical and Computer Modelling, 54(910):2418–2422, 2011. [9] Rad G.S., Rahimi H., and Radenovi´c S., Algebraic cone b-metric spaces and its equivalence. Miskolc Mathematical Notes, 17(1):553-560, 2016.

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[10] Wardowski D., On set-valued contractions of Nadler type in cone metric spaces. Applied Mathematics Letters, 24:275–278, 2011. [11] Kuratowski C., Sur les espaces complets. Fundamenta Mathematicae, 15:301–309, 1930. [12] Himmelberg C.J., Porter J.R., and Van Vleck F.S., Fixed point theorems for condensing multifunctions. Proceedings of the American Mathematical Society, 23:635–641, 1969. [13] Reich S., Fixed points in locally convex spaces. Zeitschrift, 125(1):17–31, 1972.

Mathematische

[14] Eisenfeld J. and Lakshmikantham V., Fixed point theorems through abstract cones. Journal of Mathematical Analysis and Applications, 52:25– 35, 1975. [15] Goebel K. and Kirk W.A., Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge, United Kingdom, 1990. [16] Darbo G., Punti uniti in transformazioni a codominio non compacto. Rendiconti del Seminario Matematico della Universit`a di Padova, 24:84–92, 1955. [17] Berinde M. and Berinde V., On a general class of multi-valued weakly Picard mappings. Journal of Mathematical Analysis and Applications, 326(2):772–782, 2007. [18] Fierro R., Fixed point theorems for set-valued mappings on tvs-cone metric spaces. Fixed Point Theory and Applications, 2015:221, 2015. [19] Fierro R., A noncompactness measure for tvs-metric cone spaces and some applications. Journal of Nonlinear Science and Applications, 9(5):2680– 2687, 2016. [20] Aliprantis C.D. and Tourky R., Cones and Duality. American Mathematical Society, Providence, Rhode Island, 2007. [21] C¸evik C. and Altun I., Vector metric spaces and some properties. Topological Methods in Nonlinear Analysis, 34(2):375–382, 2009.

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[22] Castaing C. and Valadier M., Convex Analysis and Measurable Multifunctions. Springer-Verlag, Berlin, 1977. [23] Saint-Raymond J., Topologie sur l’ensemble des compacts non vides d’un espace topologique s´epar´e. S´eminaire Choquet, 9 (2, exp No. 21):1–6, 1969/70. [24] Bourbaki N., Elements of Mathematics, General Topology. Part 1. Hermann, Paris, 1966. [25] Mauldin R.D., The Scottish Book: mathematics from the Scottish Caf´e. Birkh¨auser, Basel, 1981. [26] Dhompongsa S., Inthakon W., and Kaewkhao A., Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and Applications, 350(1):12–17, 2009. [27] Esp´ınola R. and Kirk W.A., Fixed points and approximate fixed points in product spaces. Taiwanese Journal of Mathematics, 5(2):405–416, 2001. [28] Reich S. and Zaslavski A.J., Approximate fixed points of nonexpansive mappings in unbounded sets. Journal of Fixed Point Theory and Applications, 13(2):627–632, 2013. [29] Suzuki T., Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and Applications, 340(2):1088–1095, 2008. [30] Azam A. and Mehmood N., Multivalued fixed point theorems in cone tvs-cone metric spaces. Fixed Point Theory and Applications, 2013:184, 2013. ´ [31] Shatanawi W., Cojbaˇ si´c V., Radenovi´c S., and Al-Rawashdeh A., Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces. Fixed Point Theory and Applications, 2012:106, 2012. [32] Nadler S.B., Multivalued contraction mappings. Pacific Journal of Mathematics, 30(2):475–487, 1969. [33] Fan K., A generalization of Tychonoff’s fixed point theorem. Mathematische Annalen, 142:305–310, 1961.

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[34] Bishop E. and Phelps R.R., The support functionals of a convex set. Proceedings of Symposia on Pure Mathematics,7:27-35 , 1963. [35] Takahashi W., Existence theorems generalizing fixed point theorems for multivalued mappings. In M.A. Th´era and J.B. Baillon, editors, Fixed Point Theory and Applications, in: Pitman Research Notes in Mathematics Series, 252:397–406. Longman Sci. Tech., Harlow, 1991. [36] Chatterjea S.K., Fixed-point theorems. C. R. Acad. Bulgare Sci., 25:727– 730, 1972. [37] Kannan R., Some results on fixed points II. The American Mathematical Monthly, 76(4):405–408, 1969.

In: An In-Depth Guide to Fixed-Point Theorems ISBN: 978-1-53619-565-1 c 2021 Nova Science Publishers, Inc. Editors: R. Sharma and V. Gupta

Chapter 2

F IXED P OINTS OF S OME M IXED I TERATED F UNCTION S YSTEMS Bhagwati Prasad∗ and Ritu Sahni† Department of Mathematics, Jaypee Institute of Information Technology, Noida, India Department of Physical Sciences, Institute of Advanced Research, Koba Institutional Area, Gandhinagar, Gujarat, India

Abstract The theory of iterated function system has been extensively studied since the last three decades and it has been very fruitfully showing its potential applications in the areas of image compression, computer graphics, chaos and fractal theory, simulation and modeling and other domains of sciences and engineering. An iterated function system (IFS) consists of a finite set of contraction maps defined on a complete space. Mihail and Miculescu [1] introduced the notion of generalized iterated function system (GIFS) and showed GIFS to be a natural generalization of the notion of iterated function system introduced by Hutchinson [2]. Llorens-Fuster et al. [3] studied the mixed iterated function systems and obtained existence and uniqueness theorems in general settings. In this chapter, we ∗ †

Corresponding Author’s Email: [email protected]. Corresponding Author’s Email: [email protected].

28

Bhagwati Prasad and Ritu Sahni intend to establish fixed point theorems for some generalized mixed type of iterated function systems defined on b-metric spaces. Our results include certain results already reported in the literature as special cases.

Keywords: Iterated function system, Meir-Keeler contraction, φ - contraction, b-metric space AMS Subject Classification: 47H10, 54H25

1.

INTRODUCTION

Fixed point theory is one of the principal tools for establishing the existence of solutions for various operator equations and thus has far reaching applications in various branches of nonlinear analysis. Fixed point theorems are of vital importance in the theory of iterated function system (IFS). Michael Barnsley and Steven Demko [4] popularized the IFS theory after Hutchinson [2] gave a formal definition of it in 1981. This notion of IFS has been extended and enriched to more general settings by various authors, see for instance, ([3-33]). Mate [16] and Rus and Triff [34] replaced contraction constant by a comparison function to obtain their results. Petrusal [25] has established more general results by taking Meir-Keeler type operators in place of contraction maps. Mihail and Miculescu [1] introduced the notion of generalized iterated function system (GIFS) and showed GIFS to be a natural generalization of the notion of IFS. Llorens-Fuster et al. [3] defined mixed iterated function system by taking more general conditions and obtained a mixed iterated function system theory for contraction and MeirKeeler contraction maps. In this chapter, we study generalized mixed iterated function system in the settings of b-metric spaces and obtain some existence and uniqueness results. In this respect, our results extend some of the well known previous results reported in [10], [23], [25], [27], and [34]-[37].

2.

P RELIMINARIES

We consider the following families of subsets of a b-metric space (X,d) P(X)= {Y|Y⊂ X,Y6= φ};K(X)={ Y∈P(X)|Y is compact.} The following generalized functionals on b-metric space are given by Boriceanu et al. [10]:

Fixed Points of Some Mixed Iterated Function Systems S (a) The gap functional: D = P (X) × P (X) → R+ {∞} is defined as

29

  inf { d(a, b)|a ∈ A, b ∈ B}, A 6= φ 6= B D(A, B) = 0, A = φ = B  ∞, otherwise

In particular, if x0 ∈ X, then D(x0 , B) = D({x0 }, B).

(b) The excess generalized functional: ρ : P (X)×P (X) toR+ ∪{∞}is defined as   sup{D(a, B)|a ∈ A}, A 6= φ 6= B 0, A = φ ρ(A, B) =  ∞, B = φ 6= A

(c) S Pompeiu-Hausdorff generalized functional: h : P (X) × P (X) → R+ {∞} is defined as   max{ρ(A, B), ρ(B, A)}, A 6= φ 6= B h(A, B) = 0, A = φ = B  ∞, otherwise

It is known that (K(X), h) is a complete b-metric space provided (X, d) is a complete b -metric space (see Czerwik [38]). The following classical theorem in metric spaces is due to Hutchinson and Barnsley (see also [39]) which however was known to Moran [40] in some different form: Theorem 2.1. [5] (see also [2], [6], [41]). metric space and

Let (X, d) be a complete

{fi : X → X, i = 1, 2, ..., n; n ∈ N }

(2.1)

a system of contractions, i.e., d(fi (x), fi(y)) ≤ Li d(x, y) for all x, y ∈ X and some Li ∈ [0, 1), i = 1, 2, ..., n. Then, there exists exactly one invariant element A∗ ∈ P (X), where P (X) is the collection of all nonempty subsets of X, of S the Hutchinson- Barnsley operator or HB operator F (X) = ni=1 fi (x), x ∈ X called the attractor (fractal) of (2.1) or equivalently, exactly one fixed point A∗ ∈ S S P (X) of the induced HB operator F ∗ (A) = F (x)x∈A(= x∈A F (x)), A ∈

30

Bhagwati Prasad and Ritu Sahni

P (x) in the space (P (x), h) where h is the Hausdorff metric. Moreover, lim h(F ∗m (A), A∗) = 0

m→∞

for every A ∈ P (x) and h(A, A∗ ) ≤

1 ∗ 1−L h(A, F (A)),

where L = maxi=1,2....,n(Li ).

Definition 2.1. [38] Let (X, d) be a b-metric space. Then a sequence {xn }n ∈ N in X is called: (i) convergent if and only if there exists x ∈ X such that d(xn , x) → 0 as n → ∞ In this case, we write lim xn = x;

n→∞

(ii) Cauchy if and only if as d(xn, xm ) → 0 as m, n → ∞ . Remark 2.1. [38] In a b-metric space, 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 2.2. [38] Let (X, d) 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., Y¯ = {x ∈ X : there exists a sequence{xn }such that lim xn = x}. n→∞

Definition 2.3. [38] Let (X, d) be a b-metric space. Then a subset Y ⊂ X is called: (i) closed if and only if for each sequence {xn }n ∈ N in Y which converges to an element x, we have x ∈ Y (i.e. Y = Y¯ ); (ii) compact if and only if for every sequence of elements of Y there exists a subsequence that converges to an element of Y; (iii) bounded if and only if diameter of Y, i.e., δ(Y ) = sup{d(a, b) : a, b, ∈ Y } < ∞.

Fixed Points of Some Mixed Iterated Function Systems

31

Definition 2.4. [38] The b-metric space (X, d) is complete if every Cauchy sequence in X converges. We need following important concepts and results of [38] (see also [10]) for our work. Lemma 2.1. [10] Let (X, d) be a b-metric space and A, B ∈ P (X). Suppose that there exists ε > 0 such that: (i) for each a ∈ A there is b ∈ B such that d(a, b) ≤ ε; (ii)for each b ∈ B there is a ∈ A such that d(a, b) ≤ ε. Then h(A, B) ≤ ε. Lemma 2.2. [10] Let (X, d) be a b-metric space. b[D(x, B) + h(A, B)], for all x ∈ X and A, B ∈ P (X) .

Then D(x, A) ≤

Lemma 2.3. [10] Let (X, d) be a b-metric space. Then for all A, B, C ∈ P (X), we have: h(A, C) ≤ b[h(A, B) + h(B, C)]. Lemma 2.4. [10] Let (X, d) be a b-metric space. Then for all A, B ∈ P (X), then (1) for each x ∈ A there exists Y ∈ B such that : d(x, y) ≤ bh(A, B); (2) Let d : X × X → R+ a continuous b-metric and A, B ∈ P (X). Then for each x ∈ A there exists y ∈ B such that d(x, y) ≤ h(A, B) . Definition 2.5. ([27], [42]) A mapping φ : R+ → R is called a comparison function if it is increasing and φn (t) → 0, n → ∞ for any t ∈ R+ . Lemma 2.5. ([9], [27]) If φ : R+ → R is a comparison function, then: (i) each iterate φk of φ , k ≥ 1, is also a comparison function; (ii) φ is continuous at zero; (iii) φ(t) < t, for any t > 0. The following concept of (c)-comparison function is introduced by Berinde [8]. Definition 2.6. [27] A function φ : R+ → R is called a (c)-comparison function if: (i) φ is increasing; (ii) there exist k0 ∈ N, A ∈ (0, 1) and a

32

Bhagwati Prasad and Ritu Sahni

convergent series of nonnegative terms ∞ X

vk

k=1

such that φk+1 (t) ≤ aφk (t) + vk

(2.2)

for k > k0 and any t ∈ R+ . Berinde [8] extended the concept of (c)-comparison to b-comparison functions in the framework of b-metric space in the following manner. Definition 2.7. [8] Let b ≥ 1 be a real number. A mapping φ : R+ → R is called a b-comparison function if: (i) φ is monotone increasing; (ii) there exist k0 ∈ N, a ∈ (0, 1) and a convergent series of nonnegative terms ∞ X

vk

k=1

such that bk+1 φk+1 (t) ≤ abk φk (t) + vk

(2.3)

Remark 2.2. It is easy to notice that, for b=1 the concept of b-comparison function reduces to that of (c)-comparison function. We need the following useful result: Lemma 2.6. [7]. If φ : R+ → R+ is a b-comparison function, then: (i) the series ∞ X bk φk (t) k=0

converges for any t ∈ R+ ; (ii) the function sb : R+ → R+ defined by sb (t) =

∞ X k=0

bk φk (t), t ∈ R+

33

Fixed Points of Some Mixed Iterated Function Systems is increasing and continuous at 0. Lemma 2.7. [24] Any b-comparison function is a comparison function.

Definition 2.8. [43] Let us consider a map f : X → X, we say that f is a (i)φ-contraction if d(f (x), f (y)) ≤ φ(d(x, y)), for any x, y ∈ X, where φ : [0, ∞) → (0, ∞] is an increasing map and φp (t) → 0 as p → ∞, for every t ≥ 0, where φp = φ ◦ φp−1 means the p-times composition of φ ( φ is named a comparison function); (ii) Banach contraction with the contraction factor a ∈ [0, 1), if it is φcontraction, where φ(t) = at ; (iii) Contractive if d(f (x), f (y)) < d(x, y), for all x, y ∈ X, x 6= y ; (iv) Meir-Keeler type mapping if, for each ε > 0 there exists δ > 0 such that x, y ∈ X, ε ≤ d(x, y) < ε + δ ⇒ d(f (x), f (y)) < ε. Obviously, any generalized Banach contraction satisfies the generalized MeirKeeler type condition and any generalized MeirKeeler type operator is generalized contractive, but the converse need not be true, see examples below (also see [37] and [44]). 1 Example 2.1. Let f : [0, 1] → [0, 1] defined by f (x) = 1+x . To show that f (x) is a Meir-Keeler contraction, choose ε=1 and for x = 0, y = 1 and for δ > 0, we have

ε ≤ |x − y| = 1 < ε + δ and |f (x) − f (y)| = |f (0) − f (1)| = Thus f (x) is a Meir-Keeler contraction. To show that f (x) is not a contraction, consider |f (x) − f (y)| =

|x−y| |(1+x)(1+y)|

When x 6= y we have |f (x)−f (y)| |x−y|

=

1 |(1+x)(1+y)|

Then for x = 0 and y = 1 + 2δ , 0, be any arbitrary number. 2 d(f (x), f (y)) = 0 − (1 + 2δ ) + 1 δ = (1 + δ2 ) + 2+δ > 1 = ε. 1+ 2 Thus f (x) is not a Meir-Keeler contraction. Definition 2.9. [10] Let f (x) be any b-metric space. An operator f : X → X is a Picard operator if: (i) Fix f = {x∗ }, where Fix f = {x ∈ X|x = f (x)} (ii)f n (x) → x∗ , as n → ∞ for all x ∈ X. Now we define generalized mixed iterated functions on the patterns of LlorensFuster et al. [3] and Mihail and Miculescu [20]. Definition 2.10. A function f : X m → X , is said to be a: (i) generalized φ-contraction, if φ- is a comparison function and for each x1 , x2 , ..., xm; y1 , y2 , ..., ym ∈ X such that xi 6= yi for some i ∈ {1, 2, ..., m}, we have, d(f (x1, x2 , ..., xm ), f (y1, y2 , ..., ym )) ≤ φ(max{d(x1 , y1 ), d(x2 , y2 ), ..., d(xm , ym )});

(ii) generalized a-contraction [1], if a ∈ [0, 1] and for each x1 , x2 , ..., xm; y1 , y2 , ..., ym ∈ X such that xi 6= yi for some i ∈ {1, 2, ..., m}, we have, d(f(x1 , x2 , ..., xm), f(y1 , y2 , ..., ym)) ≤ amax{d(x1 , y1 ), d(x2 , y2 ), ..., d(xm, ym )}

Fixed Points of Some Mixed Iterated Function Systems

35

where, contractivity factor is defined as follows: a=

d(f(x1 , ..., xm), f(y1 , ..., ym)) ; sup max{ d(x1 , y1 ), ..., d(xm, ym )} x1 , ..., xm; y1 , ...ym max{ d(x1 , y1 ), ..., d(xm, ym )} > 0

(iii) generalized Meir-Keeler contraction if for every ε > 0, there exists δ > 0 such that, for each x1 , x2 , ..., xm; y1 , y2 , ..., ym ∈ X, we have max{d(x1 , y1 ), d(x2, y2 ), ..., d(xm, ym )} < ε + δ ⇒ d(f (x1 , x2 , ..., xm), f (y1 , y2 , ..., ym)) < ε; (iv) generalized contractive if for each x1 , x2, ..., xm; y1 , y2 , ..., ym ∈ X such that xi 6= yi for some i ∈ {1, 2, ..., m}, we have, d(f(x1 , x2 , ..., xm), f(y1 , y2 , ..., ym)) < max{d(x1 , y1 ), d(x2 , y2 ), ..., d(xm, ym )}.

Now we define generalized mixed iterated function system or GMIFS from X m = ×m k=1 X → X, rather than contractions from X to itself. Definition 2.11. Let (X, d) be a complete b-metric space and m ∈ N . A generalized mixed iterated function system or GMIFS on X of order m is defined by S = (X, (fk )k=1,n ) consists of a finite family of functions ((fk )k=1,n ), fk : X m → X such that f1 , f2 , ..., fn are generalized φ- contraction or generalized Banach contraction or generalized Meir-Keeler contraction or generalized contractive. Given a metric space (X, d), K(X) denotes the set of compact subsets of X. Definition 2.12. [20] Let f : X m → X be a function. The function Ff : K(X)m = ×m k=1 K(X) → K(X) is defined by Ff (K1 , K2 , ..., Km) = f (K1 × K2 × ... × Km ) = {f (x1 , x2 , ..., xm) : xj ∈ Kj , ∀j ∈ {1, ..., m}}, ∀ K1 , K2 , ..., Km ∈ K(X), is called the set function associated to function f . Definition 2.13. Let S = (X, (fk )k=1,n ) be a GMIFS. The funcm tion → K(X) is defined by FS (K1 , K2, ..., Km) = Sn FS : K(X) F (K , K , ..., K f 1 2 m) for all K1 , K2 , ..., Km ∈ K(X) is called the k k=1 set function associated to the GMIFS S.

36

Bhagwati Prasad and Ritu Sahni

Now, we are in a position to give some results concerning the theory of generalized mixed iterated function system in connection with some of the results of [1], [3], [10], [24], [35] and [45].

3.

MAIN R ESULTS

Lemma 3.1. Let S = (X, (fk )k=1,n ) be a GMIFS satisfying generalized φ - contraction or generalized Meir-Keeler contraction. Then the set function FS : K(X)m → K(X) associated with S is also a generalized φ - contraction or generalized Meir-Keeler contraction. Proof. We have, h(FS (K1 , ..., Km ), FS (H1 , ..., Hm )) = h(

n [

fk (K1 × ... × Km ),

k=1

= h(

n [

Ffk (K1 , ..., Km ),

n [

fk (H1 × ... × Hm ))

k=1 n [

Ffk (H1 , ..., Hm ))

k=1

k=1

≤ maxk=1,n h(fk (K1 , ..., Km ), fk (H1 , ..., Hm )) ≤ φ(max( h(K1 , H1 ), ..., h(Km , Hm )))

Similarly, for Banach contraction, h(FS (K1 , ..., Km), FS (H1 , ..., Hm)) ≤ aS max( h(K1 , H1), ..., h(Km, Hm)), where contraction factor, aS = max( a1 , ..., ak). Now for Meir-Keeler contraction, fix ε > 0 and put δ = (1/aS − 1)ε, then if h(Ki , Hi) < ε + δ, i = 1, ..., m, we have h(FS (K1 , ..., Km), FS (H1 , ..., Hm)) ≤ aS max( h(K1 , H1 ), ..., h(Km, Hm)) < aS ε + aS δ = ε. The following result is proved in [46] (see also [33]). Theorem 3.1. [46] Let (X, d) be a complete b-metric space (with constant b ≥ 1) such that the b-metric is a continuous functional and f : X m → X

Fixed Points of Some Mixed Iterated Function Systems

37

be a generalized φ -contraction with φ a b-comparison function or generalized Meir-Keeler contraction. Moreover, for any x0 , x1 , ..., xm−1 ∈ X, the sequence (xn )n≥1 defined by xn+m = f (xn , xn+1 , ..., xn+m−1 ), ∀n ∈ N has the property that lim xn = α n→∞

Then there exists a unique α ∈ X such that f (α, α, ..., α) = α. The following theorem provides the conditions under which the mixed generalized iterated function system possesses an attractor. Theorem 3.2. Let (X, d) be a complete b-metric space such that the b-metric is a continuous functional and fi : X m → X, for i ∈ {1, 2, ..., m}are operators satisfying the generalized φ−contraction with φ a b-comparison function or generalized Meir-Keeler type condition. Then the fractal operatorSTf : (H(X), h) → (H(X), h) defined by the relation m Tf (Y ) = i=1 fi (Y, Y, ..., Y ) is a generalized φ−contraction or generalized Meir-Keeler type operator. Further, FixTf = {A∗ }and (Tfn (A))n∈N converges to A∗ , for each A1 , A2 , ..., Am ∈ H(X). Proof. (i) Let us consider A1 , ..., Am, B1 , ..., Bm ∈ H(X). If u ∈ Tf (A1 , ..., Am), then there exists j ∈ {1, ..., m}and x1 ∈ A1 , x2 ∈ A2 , ..., xm ∈ Am such that u ∈ fj (x1 , ..., xm). Further, for x1 ∈ A1 , x2 ∈ A2 , ..., xm ∈ Am , we can choose y1 ∈ B1 , y2 ∈ B2 , ..., ym ∈ Bm such that max{ d(x1 , y1 ), ..., d(xm, ym )} ≤ max{ h(A1 , B1 ), ..., h(Am, Bm )}. (3.1) Then, for u ∈ fj (x1 , ..., xm) and by Lemma 2.4, there exists v ∈ fj (y1 , ..., ym) such that d(u, v) ≤ h(fj (x1 , ..., xm), fj (y1 , ..., ym)) (3.2) Thus, by (3.1) and (3.2) we find for each u ∈ Tf (A1 , ..., Am), there exists v ∈ Tf (B1 , ..., Bm) such that d(u, v) ≤ h(fj (x1 , ..., xm), fj (y1 , ..., ym)) ≤ φ(max( d(x1 , y1 ), ..., d(xm, ym ))) ≤ φ(max( h(A1 , B1 ), ..., h(Am, Bm )))

(3.3)

By a similar procedure, we obtain for each v ∈ Tf (B1 , ..., Bm), there exists u ∈ Tf (A1 , ..., Am) such that d(u, v) ≤ φ(max( h(A1 , B1 ), ..., h(Am, Bm )))

(3.4)

38

Bhagwati Prasad and Ritu Sahni

Lemma 3.1, (3.3) and (3.4) together imply h(Tf (A1 , ..., Am), Tf (B1 , ..., Bm)) ≤ φ(max( h(A1 , B1 ), ..., h(Am, Bm )). Thus Tf : (H(X), h) → (H(X), h) is a generalized φ−contraction and from Theorem 3.1, we obtain that there exists a unique A∗ ∈ H(X) such that Tf (A∗ ) = A∗ and (Tfn (A))n∈N converges to A∗ , for each A ∈ H(X). (ii) We shall prove that for each ε > 0, there exists δ > 0 such that max{ h(A1 , B1 ), ..., h(Am, Bm )} < ε + δ implies h(Tf (A1 , ..., Am), Tf (B1 , ..., Bm)) < ε. Let us consider A1 , ..., Am, B1 , ..., Bm ∈ H(X) such that max{ h(A1 , B1 ), ..., h(Am, Bm )} < ε + δ. If u ∈ Tf (A1 , ..., Am), then there exists j ∈ {1, ..., m}and x1 ∈ A1 , x2 ∈ A2 , ..., xm ∈ Am such that u = fj (x1 , ..., xm). For x1 ∈ A1 , x2 ∈ A2 , ..., xm ∈ Am , we can choose y1 ∈ B1 , y2 ∈ B2 , ..., ym ∈ Bm such that max{ d(x1 , y1 ), ..., d(xm, ym )} ≤ max{ h(A1 , B1 ), ..., h(Am, Bm )} < ε + δ. We have the following alternatives: If max{ d(x1 , y1 ), ..., d(xm, ym )} ≥ ε, then ε ≤ max{ d(x1 , y1 ), ..., d(xm, ym )} < ε + δ implies d(fj (x1 , ..., xm), fj (y1 , ..., ym)) < ε. Hence D(u, Tf (B1 , ..., Bm)) ≤ d(u, fj (y1 , ..., ym)) < ε. On the other hand, if max{ d(x1 , y1 ), ..., d(xm, ym)} < ε, then we have d(fj (x1 , ..., xm), fj (y1 , ..., ym)) < max{ d(x1 , y1 ), ..., d(xm, ym )} < ε and again the conclusion D(u, Tf (B1 , ..., Bm)) < ε. Because Tf (A1 , ..., Am) is compact, we have d(Tf (A1 , ..., Am), Tf (B1 , ..., Bm)) < ε.

Fixed Points of Some Mixed Iterated Function Systems

39

Interchanging the roles of Tf (A1 , ..., Am) and Tf (B1 , ..., Bm), we obtain d(Tf (B1 , ..., Bm), Tf (A1 , ..., Am)) < ε and hence h(Tf (A1 , ..., Am), Tf (B1 , ..., Bm)) < ε, showing the fact that Tf is a generalized Meir-Keeler type operator. From Meir –Keeler fixed point result, i.e., from Theorem 3.1, we obtain that there exists a unique A∗ ∈ H(X) such that Tf (A∗ ) = A∗ and (Tfn (A))n∈N converges to A∗ , for each A ∈ H(X). The following result follows from Theorem 3.2 (i), if we put m=1. Corollary 3.1. Let (X, d) be a complete b-metric space such that the b-metric is a continuous functional and fi : X → X, for i ∈ {1, 2, ..., m} are operators satisfying φ−contraction with φ a b-comparison function. Then the fractal S operator Tf : (H(X), h) → (H(X), h) defined by the relation Tf (Y ) = m i=1 fi (Y ) is a φ−contraction. If we put m = 1, b = 1, then the following result of [26] is obtained from part (ii) of Theorem 3.2 above. Corollary 3.2. [26] Let (X, d) be a complete metric space and fi : X → X, for i ∈ {1, 2, ..., m}are operators satisfying Meir-Keeler type condition. Then the fractal operator Tf : (H(X), h) → (H(X), h) defined by the Sm relation Tf (Y ) = i=1 fi (Y ) is a Meir-Keeler type condition. Further,Fix Tf = {A∗ }and (Tfn (A))n∈N converges to A∗ , for each A ∈ H(X). The following example ensures the existence of attractor using generalized Meir-Keeler contraction and Banach contraction: Example 3.1. Let us consider f0m , f1m : Rm → R, where R = [0, 1] ∪ {2, 3, 4, 5, ..., 2n, 2n + 1, ...} is given by  Pm 1 Pm  k=1 3 xk , k=1 xk ∈ [0, 1]     P 0, m f0 (x1 , x2, ..., xm) = k=1 xk = 2n, n ∈ N, n ≥ 2     P  1 1 − n+2 , m k=1 xk = 2n + 1, n ∈ N f1 (x1 , x2 , ..., xm) =

  

3−m 3

+

Pm

0,

Pm 1 k=1 3 xk , k=1

Pm

k=1

xk ∈ [0, 1]

xk ∈ N, n ≥ 3

40

Bhagwati Prasad and Ritu Sahni

If m < 3, then f0 is Meir-Keeler contraction and f1 is a 13 −contraction. Thus, we consider mixed generalized iterated function system S m = (R, (f0m, f1m )), where m ∈ {1, 2}. Indeed,  P [0, m/3], m  k=1 xk ∈ [0, 1]     P 0, m f0m (x1 , x2 , ..., xm) = , xi ∈ R k=1 xk = 2n, n ∈ N, n ≥ 2     P  1 1 − n+2 , m k=1 xk = 2n + 1, n ∈ N and

f1m (x1 , x2 , ..., xm) =

 Pm  [1 − m/3, 1], k=1 xk ∈ [0, 1] 0,



Pm

k=1

, xi ∈ R.

xk ∈ N, n ≥ 3

If m = 2, then we have P  [0, 2/3], m  k=1 xk ∈ [0, 1]     P 0, m f02 (x1 , x2 ) = , x1 , x2 ∈ R k=1 xk = 2n, n ∈ N, n ≥ 2     P  1 1 − n+2 , m k=1 xk = 2n + 1, n ∈ N and

f12 (x1 , x2 ) =

 Pm  [1/3, 1], k=1 xk ∈ [0, 1] 

Then,

0,

Pm

k=1

, x1 , x2 ∈ R.

xk ∈ N, n ≥ 3

FS 2 (x1 , x2 ) = f0 2 (x1 , x2) ∪ f1 2 (x1 , x2 ) =           

[0, 2/3] ∪ [1/3, 1], 0,

Pm

1−

Pm

k=1

xk ∈ [0, 1]

k=1

xk = 2n, n ∈ N, n ≥ 2 x1 , x2 ∈ R ,

1 n+2

∪ {0},

which is a well known Cantor set. If m = 1, then we have

Pm

k=1

xk = 2n + 1, n ∈ N

Fixed Points of Some Mixed Iterated Function Systems P  [0, 1/3], m  k=1 xk ∈ [0, 1]     P 0, m , x1 ∈ R f01 (x1 ) = k=1 xk = 2n, n ∈ N, n ≥ 2     P  1 , m 1 − n+2 k=1 xk = 2n + 1, n ∈ N

and

f11 (x1 ) =

 Pm  [2/3, 1], k=1 xk ∈ [0, 1] 

Then,

0,

Pm

41

, x1 ∈ R.

k=1 xk ∈ N, n ≥ 3

FS 2 (x1 , x2 ) = f0 2 (x1 , x2 ) ∪ f1 2 (x1 , x2 ) =           

[0, 1/3] ∪ [2/3, 1], 0,

Pm

1−

k=1 1 n+2

Pm

k=1

xk ∈ [0, 1]

xk = 2n, n ∈ N, n ≥ 2 x1 , x2 ∈ R , ∪ {0},

Pm

k=1

xk = 2n + 1, n ∈ N

which is a well-known Cantor set. The multi-fractal operator is defined in Petrusel and Rus [27] in the following manner: Let Fi : X m → H(X), for i ∈ {1, 2, ..., m} be a finite family of upper semicontinuous multi-valued operators and Tf : (H(X), h) → (H(X), h) be defined as follows: m [ Fi (Y, Y, ..., Y ). Tf (Y ) = i=1

A number of authors have defined contractive type conditions for multivalued case (see for example, [23], [25], [27], [47]. We generalize some of them as follows: Definition 3.2. A function F : Xm → P (X) for each x1 , x2 , ..., xm, y1 , y2 , ..., ym ∈ X, such that xi 6= yi for some i ∈ {1, 2, ..., n}, is said to be:

42

Bhagwati Prasad and Ritu Sahni

(i) Generalized φ−contraction, ifφ is a comparison function and we have h(F (x1 , x2 , ..., xm ), F (y1 , y2 , ..., ym )) ≤ φ(max{ d(x1 , y1 ), d(x2 , y2 ), ..., d(xm , ym )});

(ii) Generalized Banach contraction, if we have h(F (x1 , x2 , ..., xm ), F (y1 , y2 , ..., ym )) ≤ amax{ d(x1 , y1 ), d(x2 , y2 ), ..., d(xm , ym )}

where, contractivity factor is given as follows: a=

h(f (x1 , ..., xm), f (y1 , ..., ym)) sup ; max{ d(x1 , y1 ), ..., d(xm, ym )} x1 , ..., xm; y1 , ...ym max{ d(x1 , y1 ), ..., d(xm, ym )} > 0

(iii) Generalized Meir-Keeler contraction if for every ε > 0, there exists δ > 0 such that we have max{ d(x1 , y1 ), ..., d(xm , ym )} < ε + δ ⇒ h(F (x1, ..., xm ), F (y1 , ..., ym )) < ε;

(iv) Generalized contractive if we have h(F (x1 , x2 , ..., xm ), F (y1 , y2 , ..., ym )) < max{ d(x1, y1 ), d(x2 , y2 ), ..., d(xm , ym )}.

An existence and uniqueness result for mixed generalized multi iterated function system is given as follows: Theorem 3.3. Let (X, d) be a complete b-metric space such that the b-metric is a continuous functional and Fi : X m → H(X), for i ∈ {1, 2, ..., m}are operators satisfying the generalized φ−contraction with φ a b-comparison function or generalized Meir-Keeler type condition. Then the multivalued fractal operator Tf : (H(X), h) → (H(X), h) defined by the relation Tf (Y ) =

m [

Fi (Y, Y, ..., Y ).

i=1

is a generalized φ−contraction or generalized Meir-Keeler type operator. Further, F ixTf = {A∗ } and (Tfn (A))n∈N converges to A∗ , for each A1 , A2 , ..., Am ∈ H(X). Proof. (i) Let us consider A1 , ..., Am, B1 , ..., Bm ∈ H(X). If u ∈ TF (A1 , ..., Am), then there exists j ∈ {1, ..., m}and x1 ∈ A1 , x2 ∈

Fixed Points of Some Mixed Iterated Function Systems

43

A2 , ..., xm ∈ Am such that u ∈ Fj (x1 , ..., xm). Further, for x1 ∈ A1 , x2 ∈ A2 , ..., xm ∈ Am , we can choose y1 ∈ B1 , y2 ∈ B2 , ..., ym ∈ Bm such that max{ d(x1 , y1 ), ..., d(xm, ym )} ≤ max{ h(A1 , B1 ), ..., h(Am, Bm )} (3.5) Then, for u ∈ Fj (x1 , ..., xm). by Lemma 2.4, there exists v ∈ Fj (y1 , ..., ym) such that d(u, v) ≤ h(Fj (x1 , ..., xm), Fj (y1 , ..., ym)) (3.6) Thus, by (3.5) and (3.6) we get that for each u ∈ Tf (A1 , ..., Am), there exists v ∈ Tf (B1 , ..., Bm) such that d(u, v) ≤ h(Fj (x1 , ..., xm), Fj (y1 , ..., ym)) ≤ φ(max( d(x1 , y1 ), ..., d(xm, ym ))) ≤ φ(max( h(A1 , B1 ), ..., h(Am, Bm )))

(3.7)

By a similar procedure, we obtain that for each v ∈ Tf (B1 , ..., Bm), there exists u ∈ Tf (A1 , ..., Am) such that d(u, v) ≤ φ(max( h(A1 , B1 ), ..., h(Am, Bm )))

(3.8)

Lemma 2.1 along with (3.7) and (3.8) imply H(Tf (A1 , ..., Am), Tf (B1 , ..., Bm)) ≤ φ(max( h(A1 , B1 ), ..., h(Am, Bm )). Hence Tf : (H(X), h) → (H(X), h) is a generalized φ -contraction and from Theorem 3.1, we obtain that there exists a unique A∗ ∈ H(X) such that TF (A∗ ) = A∗ and (TFn (A))n∈N converges to A∗ , for each A ∈ H(X). (ii) We shall prove that for each ε > 0, there exists δ > 0such that max{ h(A1 , B1 ), ..., h(Am, Bm )} < ε + δ implies h(TF (A1 , ..., Am), TF (B1 , ..., Bm)) < ε. Let us consider A1 , ..., Am, B1 , ..., Bm ∈ H(X) such that max{ h(A1 , B1 ), ..., h(Am, Bm )} < ε + δ. If u ∈ TF (A1 , ..., Am), then there exists k ∈ {1, ..., m}and x1 ∈ A1 , x2 ∈ A2 , ..., xm ∈ Am such that u = Fk (x1 , ..., xm).

44

Bhagwati Prasad and Ritu Sahni

For x1 ∈ A1 , x2 ∈ A2 , ..., xm ∈ Am , we can choose y1 ∈ B1 , y2 ∈ B2 , ..., ym ∈ Bm such that max{ d(x1 , y1 ), ..., d(xm, ym )} ≤ max{ h(A1 , B1 ), ..., h(Am, Bm )} < ε + δ. We have the following alternatives: If max{ d(x1 , y1 ), ..., d(xm, ym )} ≥ ε, then ε ≤ max{ d(x1 , y1 ), ..., d(xm, ym )} < ε + δ implies h(Fk (x1 , ..., xm), Fk (y1 , ..., ym)) < ε. Hence D(u, TF (B1 , ..., Bm)) ≤ d(u, Fk (y1 , ..., ym)) < ε. On the other hand if max{ d(x1 , y1 ), ..., d(xm, ym )} < ε, then we have h(Fk (x1 , ..., xm), Fk (y1 , ..., ym)) < max{ d(x1 , y1 ), ..., d(xm, ym )} < εand again the conclusionD(u, TF (B1 , ..., Bm)) < ε. Because TF (A1 , ..., Am) is compact, we have d(TF (A1 , ..., Am), TF (B1 , ..., Bm)) < ε. Interchanging the roles of TF (A1 , ..., Am) and TF (B1 , ..., Bm), we obtain d(TF (B1 , ..., Bm), TF (A1 , ..., Am)) < ε and hence h(TF (A1 , ..., Am), TF (B1 , ..., Bm)) < ε, showing the fact that TF is a generalized Meir-Keeler type operator. From Meir –Keeler fixed point result, i.e., from Theorem 3.1, we obtain that there exists a unique A∗ ∈ H(X) such that TF (A∗ ) = A∗ and (TFn (A))n∈N converges to A∗ , for each A ∈ H(X).

If we put m = 1 in Theorem 3.3(i) above, the following result of M. Boriceanu et al. [10] is obtained. Corollary 3.3. [10] Let (X, d) be a complete b-metric space such that the bmetric is a continuous functional and Fi : X → H(X), for i ∈ {1, 2, ..., m}are operators satisfying multivaluedφ -contraction with φ a b-comparison function. Then the multivalued fractal operator TF : (H(X), h) → (H(X), h) is a φ -contraction having a unique fixed point A∗ ∈ H(X) which is a multivalued

Fixed Points of Some Mixed Iterated Function Systems

45

fractal and attractor of iterated multifunction system F = (F1 , F2 , ..., Fm). The following result of Llorens-Fuster et al. (Theorem 2.4, part (B), [3]) is obtained from Theorem 3.3 when we put m = 1, b = 1 in part (i) of Theorem 3.3 above. Corollary 3.4. [3] Let (X, d) be a complete metric space and Fi : X → H(X), for i ∈ {1, 2, ..., m}are operators satisfying φi −contraction. Then the multivalued fractal operator TF : (H(X), h) → (H(X), h) with respect to the iterated multifunction system F = (F1 , F2 , ..., Fm), converges to A∗ . In case of metric space, we obtain following result of [35] when we substituteFi = F, m = 1, p = 1 in Theorem 3.3 (i) above. Corollary 3.5. [35] Let (X, d) be a complete metric space and F : X → H(X), is a multivalued φ−contraction. Then T is upper semicontinuous and TF : (H(X), h) → (H(X), h) defined by the relation TF (Y ) =

[

F (x)

x∈Y

is a set to set φ -contraction and F ixTf = {A∗ }and (TFn (A))n∈N converges to A∗ , for each x ∈ X. Further, we obtain Theorem 2.4, part (A) of [3] and Theorem 3.5 of [26] when we substitute m = 1, b = 1 in part (ii) of Theorem 3.3 above. Corollary 3.6. [3, 26] Let (X, d) be a complete metric space and Fi : X → H(X), for i ∈ {1, 2, ..., m} are operators satisfying Meir-Keeler type condition. Then the multivalued fractal operator TF : (H(X), h) → (H(X), h) with respect to the iterated multifunction system F = (F1 , F2 , ..., Fm), converges to A∗ .

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[40] Moran PAP, Additive functions of intervals and Hausdorff measures, Math. Proc. Cambridge Philos. Soc., 42(1946), pp.15-23. [41] Mandelbrot B. B., The Fractal Geometry of Nature, W.H. Freeman and Company, 1982. [42] Browder F. E., On the convergence of successive approximations for nonlinear functional equations, Indag. Math., 30, (1968), pp. 27-35. [43] Rus I. A., Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001. [44] Edelstein M., On fixed and periodic points under contractive mappings, J. London Math, Soc, 37 (1962), pp. 74-79. [45] Suzuki T., Some notes on Meir-Keeler contractions and L-functions, Bull. Kyushu Inst. Tech., Pure Appl. Math. No. 53, (2006), pp. 1-13. [46] Prasad Bhagwati and Sahni Ritu, An existence theorem for mixed iterated function systems in B-metric spaces, Proc. Inter. Con. Mathematics Education & Mathematics in Engineering & Technology (ICMET’13), (2013) pp. 140-148. [47] Covitz H., Nadler jr. S. B., Multi-valued contraction mapping in generalized metric spaces, Israel J. Math. 8(1970), pp. 5-11.

In: An In-Depth Guide to Fixed-Point Theorems ISBN: 978-1-53619-565-1 c 2021 Nova Science Publishers, Inc. Editors: R. Sharma and V. Gupta

Chapter 3

R ANDOM I TERATION S CHEME L EADING TO A R ANDOM F IXED P OINT T HEOREM AND I TS A PPLICATION Debashis Dey∗ Sudpur High School, Sudpur, Purba Bardhaman, West Bengal, India

Abstract In the present chapter, we study the strong convergence of a random iterative algorithm and thereby we obtain random fixed point for a general class of random operators in a separable Banach space. We apply the random fixed point theorem to analyze the existence of a solution of a nonlinear stochastic integral equation of the Hammerstein type. Also the well-posedness (almost surely) of a random fixed point problem is being discussed in this chapter for single and multivalued operators.

1.

INTRODUCTION

The growing importance of Random fixed point theory lies in its vast application in probabilistic functional analysis and numerous probabilistic models. The introduction of randomness however leads to several new questions of measurability of solutions, probabilistic and statistical aspects of random solutions. It ∗

Corresponding Author’s Email: [email protected].

52

Debashis Dey

is well known that random fixed point theorems are stochastic generalization of classical fixed point theorems what we call as determinstic results. The study of random fixed point theorems was initiated by the Prague school of probabilists in 1950’s. Random fixed point theorems for random contraction mappings on ˇ cek [1] and Hanˇs (see separable complete metric spaces were first proved by Spaˇ [2]-[3]). The survey article by Bharucha-Reid [4] in 1976 attracted the attention of several mathematicians and gave wings to this theory. Itoh [5] extended ˇ cek’s and Hanˇs’s theorems to multivalued contraction mappings. MukherSpaˇ jee [6] gave a random version of Schaduer’s fixed point Theorem on an atomic probability measure space. Random fixed point theorems with an application to Random differential equations in Banach spaces are obtained by Itoh [5]. Sehgal and Waters [7] had obtained several random fixed point theorems including random analogue of the classical results due to Rothe [8]. Several fixed point theorems including Kannan type [9], Chatterjeea [10] and Zamfirescu type [11] have been generalized in stochastic version (see for detail in Joshi and Bose [12], Saha et al. [13], Choudhury [14]-[15]). Recently Papageorgiou [16], Xu [17], Beg ([18]-[19]), Beg and Shahazad [20], Xu and Beg [21], Liu [22], Saha and Dey [23], Saha and Debnath [24], Saha [25] and Saha and Ganguly [26] have studied the fixed points of random maps over separable Hilbert space and Banach space. Existence of random fixed point using various iterative techniques for approximating fixed points of random nonexpansive mappings have been studied by Beg and Abbas [27], Choudhury [28], Choudhury and Ray [15], Saha and Dey [13]. A significant application of random fixed point theory was introduced by Achari [29] and subsequently developed Dey and Saha [30]. Motivated by these literature review, the author seems that this is a potential field to explore a lot of new results as well as to study the subject extensively. The aim of this chapter is to provide some aspects of deterministic results in the perspective of stochastic verse. An application of random fixed point theorem has been shown and wellposedness of random fixed point problem has been set in congruence with this study. The entire chapter is being organised into the following sections. In Section 2, we present some basic ideas and preliminaries. The Section 3 deals with the strong convergence of a random iterative algorithm namely Xu-Mann random iteration scheme leading to a random fixed point theorem. In Section 4, we study the application of random fixed point theorem to analyze the existence of a solution of a nonlinear stochastic integral equation of the Hammerstein type. Finally, well-posedness (almost surely) of a random fixed point problem for single and multivalued operators have been discussed in

Random Iteration Scheme Leading to ...

53

Section 5.

2.

P RELIMINARIES

Let (X, βX ) be a separable Banach space where βX is a σ-algebra of Borel subsets of X, and let (Ω, β, µ) denote a complete probability measure space with measure µ, and β be a σ-algebra of subsets of Ω. For more details one can see Joshi and Bose [12]. Definition 2.1. A mapping x : Ω → X is said to be an X-valued random variable, if the inverse image under the mapping x of every Borel set B of X belongs to β; that is, x−1 (B) ∈ β for all B ∈ βX . Definition 2.2. A mapping x : Ω → X is said to be a finitely valued random variable, if it is constant on each ! of finite number of disjoint sets Ai ∈ β and n [ is equal to 0 on Ω − Ai . X is called a simple random variable if it is i=1

finitely valued and µ {ω : kx (ω)k > 0} < ∞.

Definition 2.3. A mapping x : Ω → X is said to be a strong random variable, if there exists a sequence {xn (ω)} of simple random variables which converges to x (ω) almost surely, i.e., there exists a set A0 ∈ β with µ (A0 ) = 0 such that lim xn (ω) = x (ω), ω ∈ Ω − A0 . n→∞

Definition 2.4. A mapping x : Ω → X is said to be a weak random variable, if the function x∗ (x (ω)) is a real valued random variables for each x∗ ∈ X ∗ , the space X ∗ denoting the first normed dual space of X. In a separable Banach space X, the notions of strong and weak random variables x : Ω → X [see Joshi and Bose [12], Corollary 1] coincide and in respect of such a space X, x is called as a random variable. We recall the following results. Theorem 2.5. (Joshi and Bose [12], Theorem 6.1.2 (a)) Let x, y : Ω → X be a strong random variables and α, β be constants. Then the following statement holds: (a) α x (ω) + β y (ω) is a strong random variable.

54

Debashis Dey

(b) If f (ω) is a real valued random variable and x (ω) is a strong random variable, then f (ω) x (ω) is a strong random variable. (c) If {xn (ω)} is a sequence of strong random variables converging strongly to x (ω) almost surely, i.e., if there exists a set A0 ∈ β with µ (A0 ) = 0 such that lim kxn (ω) − x (ω)k = 0 for every ω ∈ / A0 , then x (ω) is a n→∞ strong random variable. Remark 2.6. If X is a separable Banach space, then the σ-algebra generated by the class of all spherical neighourhoods of X is equal to the σ-algebra of all Borel sets of X and hence every strong and also weak random variable is measurable in the sense of Definition 2.1. Let Y be another Banach space. We also need the following definitions from Joshi and Bose [12]. Definition 2.7. A mapping F : Ω × X → Y is said to be a random mapping if F (ω, x) = Y (ω) is a Y -valued random variable for every x ∈ X. Definition 2.8. A mapping F : Ω × X → Y is said to be a continuous random mapping if the set of all ω ∈ Ω for which F (ω, x) is a continuous function of x has measure one. Definition 2.9. A random mapping F : Ω × X → Y is said to be demicontinuous at the x ∈ X if kxn − xk → 0 implies F (ω, xn )

weakly → F (ω, x)

almost surely.

Theorem 2.10. (Joshi and Bose [12], Theorem 6.2.2.) Let F : Ω × X → Y be a demi-continuous random mapping where Banach space Y is separable. Then for any X-valued random variable x, the function F (ω, x (ω)) is a Y -valued random variable. Remark 2.11. (see [25]) Since a continuous random mapping is a demicontinuous random mapping, Theorem 2.5 is also true for a continuous random mapping.

55

Random Iteration Scheme Leading to ...

The following definitions are also given in their book of Joshi and Bose [12]. Definition 2.12. An equation of the type F (ω, x (ω)) = x (ω) where F : Ω × X → X is a random mapping is called a random fixed point equation. Definition 2.13. Any mapping x : Ω → X which satisfies random fixed point equation F (ω, x (ω)) = x (ω) almost surely is said to be a wide sense solution of the fixed point equation. Definition 2.14. Any X-valued random variable x (ω) which satisfies µ {ω : F (ω, x (ω)) = x (ω)} = 1 is said to be a random solution of the fixed point equation or a random fixed point of F . Remark 2.15. A random solution is a wide sense solution of the fixed point equation. But the converse is not necessarily true. This is evident from following example as found under Remark 1, Joshi and Bose [12]. Example 2.16. Let X be the set of all real numbers and let E be a non measurable subset of X. Let F : Ω × X → Y be a random mapping defined as F (ω, x) = x2 + x − 1 for all ω ∈ Ω. In this case, the real valued function x (ω), defined as x (ω) = 1 for all ω ∈ Ω is a random fixed point of F . However the real valued function y (ω) defined as  ω∈ /E  −1, y (ω) =  1, ω∈E

is a wide sense solution of the fixed point equation F (ω, x (ω)) = x (ω), without being a random fixed point of F . Theorem 2.17. (Joshi and Bose [12]) Let X be a separable Banach space and (Ω, β, µ) be a complete probability measure space. Let T : Ω × X → X be a continuous random operator satisfying kT (ω, x1) − T (ω, x2 )k ≤ k1 (ω) [kx1 − T (ω, x1 ) k + kx2 − T (ω, x2 ) k] +k2 (ω) [kx1 − T (ω, x2) k + kx2 − T (ω, x1) k] +k3 (ω) kx1 − x2 k

(2.1)

56

Debashis Dey

for all ω ∈ Ω and x1 , x2 ∈ X, ki (ω) ≥ 0; 1 ≤ i ≤ 3. are real valued random variables with 2k1 (ω) + 2k2 (ω) + k3 (ω) < 1 almost surely. Then there exists a unique random fixed point of T . Remark 2.18. (I): In the above theorem, setting k2 (ω) = k3 (ω) = 0, and k1 (ω) = λ(ω) one can find random analogue of kannan fixed point theorem [9] and in that case the operator T : Ω × X → X takes the form: kT (ω, x1 ) − T (ω, x2 )k ≤

λ(ω) [kx1 − T (ω, x1 ) k + kx2 − T (ω, x2 ) k] (2.2)

for all ω ∈ Ω and x1 , x2 ∈ X, λ(ω) ≥ 0 is real valued random variables with λ(ω) < 21 almost surely. (II): Setting k1 (ω) = k3 (ω) = 0, and k2 (ω) = λ(ω) one can find random analogue of Chatterjea fixed point theorem [10] and in that case the operator T : Ω × X → X takes the form: kT (ω, x1 ) − T (ω, x2 )k ≤

λ(ω) [kx1 − T (ω, x2 ) k + kx2 − T (ω, x1 ) k] (2.3)

for all ω ∈ Ω and x1 , x2 ∈ X, λ(ω) ≥ 0 is real valued random variables with λ(ω) < 21 almost surely. (III): Setting k1 (ω) = k2 (ω) = 0, and k3 (ω) = α(ω) one can find random analogue of famous Banach contraction principle [31] and in that case the operator T : Ω × X → X takes the form: kT (ω, x1) − T (ω, x2)k ≤ α(ω) (kx1 − x2 k)

(2.4)

for all ω ∈ Ω and x1 , x2 ∈ X, α(ω) ≥ 0 is real valued random variables with α(ω) < 1 almost surely. Remark 2.19. Note that neither Kannan operator nor Chatterjea operator is continuous in general. So random fixed point theorems for these two operators are slightly different from their deterministic approach.

3.

C ONVERGENCE OF A R ANDOM ITERATION SCHEME (X U -M ANN ITERATION ) TO A R ANDOM F IXED P OINT

The Mann iteration scheme [32] is a very old one and was obtained in 1953. Random analogue of this iteration was investigated by B.S. Choudhury in 2003 [14].

57

Random Iteration Scheme Leading to ...

Random Iteration Scheme (Mann Iteration). Let T : Ω × C → C, where C is a nonempty convex subset of a separable Banach space X, be a random operator. Let g0 : Ω → C be any measurable function. Then the random iteration scheme is defined as the following sequences of functions: gn+1 (ω) = (1 − bn ) gn (ω) + bn T (ω, gn (ω)) , n ≥ 0, ω ∈ Ω

(3.1)

where {bn } is a sequence in [0, 1]. The consideration of error terms is an important part of any theory of iteration. For this reason we need random analogue of Xu-Mann [33] iteration scheme in separable Banach space. Random Iteration Scheme (Xu-Mann Iteration). Let T : Ω × C → C, where C is a nonempty convex subset of a separable Banach space X, be a random operator. Let g0 : Ω → C be any measurable function. Then the random iteation scheme is defined as the following sequences of functions: gn+1 (ω) = an gn (ω) + bn T (ω, gn (ω)) + cn un , n ≥ 0, ω ∈ Ω

(3.2)

where {an },{bn } and {cn } are sequences in [0, 1] such that an +bn +cn = 1 and {un } is a bounded sequence in C. Clearly, this iteration process contains the first one (Mann Iteration) as it’s special case. Now we investigate the strong convergence of Xu-Mann random iteration which leads to the random fixed point for a general class of contraction mappings. For this we recall some contraction mappings in a metric space(X, d). ´ c Operator. Quasi-Contraction Mappings and Ciri´ A mapping T : X → X is said to be an a-contraction if d (T x, T y) ≤ ad (x, y) for all x, y ∈ X

(3.3)

where a ∈ (0, 1).
A mapping T is called a Kannan mapping [9] if there exists b ∈ 0, 21 such that d (T x, T y) ≤ b [d (x, T x) + d (y, T y)] for all x, y ∈ X


1

(3.4)

A similar definition is due to Chatterjea [10]: There exists c ∈ 0, 2 such that d (T x, T y) ≤ c [d (x, T y) + d (y, T x)] for all x, y ∈ X

(3.5)

58

Debashis Dey

Combining (3.3), (3.4) and (3.5), Zamfirescu [11] proved the following result in 1972. Theorem 3.1. [11] Let (X, d) be a complete metric space and T : X → X a mapping
for which there exists real numbers a, b, c satisfying a ∈ (0, 1), b, c ∈ 1 0, 2 such that for any pair x, y ∈ X, at least any of the following conditions hold: (z1 ) d (T x, T y) ≤ ad (x, y) (z2 ) d (T x, T y) ≤ b [d (x, T x) + d (y, T y)] (z3 ) d (T x, T y) ≤ c [d (x, T y) + d (y, T x)] Then T has a unique fixed point p and the Picard iteration {xn }∞ n=0 defined by xn+1 = T xn ; n = 0, 1, 2, ... converges to p for any arbitrary but fixed x0 ∈ X. Remark 3.2. The conditions (z1 )-(z3 ) can be written in the following equivalent form:   d (x, T x) + d (y, T y) d (x, T y) + d (y, T x) d (T x, T y) ≤ h max d (x, y) , , 2 2 (3.6)

for all x, y ∈ X, 0 < h < 1. A class of mappings satisfying the contractive conditions (z1 )-(z3 ) and as well as (3.6) is a subclass of mappings satisfying the following condition   d (x, T x) + d (y, T y) d (T x, T y) ≤ h max d (x, y) , d (x, T y) , d (y, T x) , (3.7) 2

for all x, y ∈ X, 0 < h < 1. ´ c The class of mappings satisfying (3.7) was introduced and investigated by Ciri´ ´ [34] in 1971. A mapping satisfying (3.7) is commonly known as Ciri´c generalised contraction which is also a subclass of mappings satisfying the following conditions d (T x, T y) ≤ h max {d (x, y) , d (x, T x) , d (y, T y) , d (x, T y) , d (y, T x)} (3.8) for all x, y ∈ X, 0 < h < 1. The class of mappings satisfying (3.8) was ´ c [35] in 1974 and a mapping satisfying (3.8) is called Ciri´ ´ c investigated by Ciri´ quasi contraction.

Random Iteration Scheme Leading to ...

59

Now we are in a state to investigate the convergence of random itera´ c quasition scheme due to Xu-Mann leading to a random fixed point of Ciri´ contractive type operator in the following theorem. In the sequel, let C be a nonempty subset of X. A function f : Ω → C is said to be measurable if f −1 (B ∩ C) ∈ β for every Borel subset B of βX . A function F : Ω × C → C is said to be a random operator if F (., x) : Ω → C is measurable for every x ∈ C. A measurable function g : Ω × C → C is said to be a random fixed point of the random operator F : Ω × C → C if F (ω, g(ω)) = g(ω) for all ω ∈ Ω. Before proving the theorem, we state a well known lemma without proof. Lemma 3.3. Let {rn }, {sn }, {tn } and {kn } be sequences of nonnegative numbers satisfying rn+1 ≤ (1 − sn ) rn + sn tn + kn for all n ≥ 1 If

∞ X

sn = ∞, lim tn = 0 and n→∞

n=1

∞ X

kn < ∞ hold, then lim rn = 0.

n=1

n→∞

Theorem 3.4. Let X be a separable Banach space and C be a nonempty convex subset of X. Let T be a random operator defined on C such that for ω ∈ Ω, T (ω, .) : C → C satisfies kT (ω, x) − T (ω, y)k ≤

h(ω) max {kx − yk , kx − T (ω, y)k , ky − T (ω, x)k ,  kx − T (ω, x)k + ky − T (ω, y)k (3.9) 2

for all x, y ∈ C, where h(ω) is a real valued random variable such that 0 < h(ω) < 1. Further it is assumed that T has a random fixed point. Also let g0 : Ω → C ∞ X be any measurable function. If bn = ∞ and cn = O(bn), then the sequence n=1

of function defined by Xu-Mann random iteration scheme (3.2) converges to a random fixed point of T .

Proof. Let p : Ω → C be a random fixed point of T . Since T satisfies (3.9), then for all x, y ∈ C and ω ∈ Ω, following cases arise. Case I. Let max =



kx − yk , kx − T (ω, y)k , ky − T (ω, x)k ,

kx − T (ω, x)k + ky − T (ω, y)k 2

kx − T (ω, x)k + ky − T (ω, y)k 2

ff

60

Debashis Dey

Then by (3.9) kT (ω, x) − T (ω, y)k ≤

h(ω) [kx − T (ω, x)k + ky − T (ω, y)k] 2

implies h(ω) [kx − T (ω, x)k + ky − xk + kx − T (ω, x)k 2 + kT (ω, x) − T (ω, y)k] ,

kT (ω, x) − T (ω, y)k ≤

which gives   h(ω) h(ω) 1− kT (ω, x) − T (ω, y)k ≤ [kx − yk + 2 kx − T (ω, x)k] 2 2

and so kT (ω, x) − T (ω, y)k ≤

1

h(ω) 2 − h(ω) 2

kx − yk +

h(ω) 1−

h(ω) 2

kx − T (ω, x)k

(3.10)

Case II. Let max =



kx − yk , kx − T (ω, y)k , ky − T (ω, x)k ,

kx − T (ω, x)k + ky − T (ω, y)k 2

ff

kx − T (ω, y)k

Then again by (3.9) kT (ω, x) − T (ω, y)k ≤ h(ω) kx − T (ω, y)k ≤ h(ω) [kx − T (ω, x)k + kT (ω, x) − T (ω, y)k] implies kT (ω, x) − T (ω, y)k ≤

h(ω) kx − T (ω, x)k 1 − h(ω)

(3.11)

Case III. Again let max =



kx − yk , kx − T (ω, y)k , ky − T (ω, x)k ,

kx − T (ω, x)k + ky − T (ω, y)k 2

ff

ky − T (ω, x)k

Then by (3.9) kT (ω, x) − T (ω, y)k ≤ h(ω) ky − T (ω, x)k ≤ h(ω) kx − yk + h(ω) kx − T (ω, x)k (3.12)

61

Random Iteration Scheme Leading to ... Case IV. Finally, let max =



kx − yk , kx − T (ω, y)k , ky − T (ω, x)k ,

kx − T (ω, x)k + ky − T (ω, y)k 2

ff

kx − yk

Then by (3.9) kT (ω, x) − T (ω, y)k ≤ h(ω) kx − yk

(3.13)

Now this is obvious that (

h(ω) 2 − h(ω) 2

)

= h(ω)

h(ω) max h(ω), , h(ω) 1 − h(ω) 1− 2

)

=

max h(ω), (

1

h(ω)

h(ω) 1 − h(ω)

Thus for all the cases (I), (II), (III) and (IV) and by (3.10), (3.11), (3.12) and (3.13) we get kT (ω, x) − T (ω, y)k ≤ h(ω) kx − yk +

h(ω) kx − T (ω, x)k (3.14) 1 − h(ω)

holds for all x, y ∈ C. Also we have assumed that p : Ω → C is a random fixed point of T . So T (ω, p(ω)) = p(ω) and taking y = p(ω) = T (ω, p(ω)) in (3.14) and (3.9) respectively we get kT (ω, x) − p (ω)k ≤ h(ω) kx − p(ω)k +

h(ω) kx − T (ω, x)k (3.15) 1 − h(ω)

and kT (ω, x) − p (ω)k ≤ h(ω) max {kx − p(ω)k , kp(ω) − T (ω, x)k ,  kx − T (ω, x)k 2 ≤ h(ω) max {kx − p(ω)k , kp(ω) − T (ω, x)k ,  kx − p(ω)k + kp(ω) − T (ω, x)k 2 ≤ h(ω) max {kx − p(ω)k , kp(ω) − T (ω, x)k}

62

Debashis Dey

Now if we consider kp(ω) − T (ω, x)k, then

max {kx − p(ω)k , kp(ω) − T (ω, x)k}

=

kT (ω, x) − p (ω)k ≤ h(ω) kp(ω) − T (ω, x)k . Which is a contradiction to the fact that p(ω) is a random fixed point of T . Hence kT (ω, x) − p (ω)k ≤ h(ω) kx − p(ω)k .

(3.16)

Now using (3.2) (Xu-Mann iteration scheme) for ω ∈ Ω and x ∈ C, kgn+1 (ω) − p(ω)k

= =

kan gn (ω) + bn T (ω, gn (ω)) + cn un − p(ω)k kan gn (ω) + bn T (ω, gn (ω)) + cn un −(an + bn + cn )p(ω)k

=

kan (gn (ω) − p(ω)) + bn (T (ω, gn (ω)) − p(ω)) +cn (un − p(ω))k an kgn (ω) − p(ω)k + bn kT (ω, gn (ω)) − p(ω)k +cn kun − p(ω)k

≤ ≤

(1 − bn − cn ) kgn (ω) − p(ω)k +bn kT (ω, gn (ω)) − p(ω)k + cn kun − p(ω)k (3.17)

Assume that sup kun − p(ω)k = M , then by (3.17) we have n≥0

kgn+1 (ω) − p(ω)k ≤ (1 − bn ) kgn (ω) − p(ω)k +bn kT (ω, gn(ω)) − p(ω)k + M cn

(3.18)

So finally (3.15) and (3.16) gives together kT (ω, x) − p (ω)k ≤ and thereby, kT (ω, gn (ω)) − p (ω)k ≤

h(ω) kx − p(ω)k h(ω) kgn (ω) − p(ω)k .

(3.19)

So from (3.18) and (3.19) kgn+1 (ω) − p (ω)k ≤

[1 − (1 − h(ω))bn ] kgn (ω) − p(ω)k + M cn . (3.20)

Then using Lemma (3.3), we get for all ω ∈ Ω, lim kgn+1 (ω) − p(ω)k = 0 and consequently gn (ω) → p(ω) and thus completes the proof. Similarly, the Theorem 3.4 can be proved using Mann iteration scheme. In that case the theorem can be stated as

Random Iteration Scheme Leading to ...

63

Theorem 3.5. Let X be a separable Banach space and C be a nonempty convex subset of X. Let T be a random operator defined on C such that for ω ∈ Ω, T (ω, .) : C → C satisfies (3.9) for all x, y ∈ C, where h(ω) is a real valued random variable such that 0 < h(ω) < 1. Further it is assumed that T has a random fixed point. Also let g0 : Ω → C be ∞ X any measurable function. If bn = ∞ , then the sequence of function defined n=1

by Mann random iteration scheme (3.1) converges to a random fixed point of T .

4.

A PPLICATION TO A R ANDOM N ONLINEAR INTEGRAL E QUATION For the sake of simplicity, we choose Kannan fixed point theorem which was provided in stochastic verse in Remark 2.18 (I). We apply the theorem in solving nonlinear stochastic integral equation of the Hammerstein type of the form: Z x(t; ω) = h(t; ω) + k(t, s; ω)f (s, x(s; ω))dµ0(s) (4.1) S

where (i)S is a locally compact metric space with metric d on S × S, µ0 is a complete σ-finite measure defined on the collection of Borel subsets of S; (ii) ω ∈ Ω, where ω is a supporting set of probability measure space (Ω, β, µ); (iii) x(t; ω) is the unknown vector-valued random variables for each t ∈ S. (iv) h(t; ω) is the stochastic free term defined for t ∈ S; (v) k(t, s; ω) is the stochastic kernel defuned for t and s in S and (vi) f (t, x) is vector-valued function of t ∈ S and x and the integral in equation (4.1) is a Bochner integral. We will further assume that S is the union of a countable family of compact sets {Cn } having the properties that C1 ⊂ C2 ⊂ ... and that for any other compact set S there is a Ci which contains it (see [36]). Definition 4.1. We define the space C(S, L2 (Ω, β, µ)) to be the space of all continuous functions from S into L2 (Ω, β, µ) with the topology of uniform convergence on compacta i.e. for each fixed t ∈ S, x(t; ω) is a vector valued random variable such that Z 2 kx(t; ω)kL2 (Ω,β,µ) = |x(t; ω)|2 dµ(ω) < ∞ Ω

64

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It may be noted that C(S, L2 (Ω, β, µ)) is locally convex space (see [37]) whose topology is defined by a countable family of seminorms given by kx(t; ω)kn = sup kx(t; ω)kL2 (Ω,β,µ) , n = 1, 2, ... t∈Cn

Moreover C(S, L2 (Ω, β, µ)) is complete relative to this topology since L2 (Ω, β, µ) is complete. We further define BC = BC(S, L2 (Ω, β, µ)) to be the Banach space of all bounded continuous functions from S into L2 (Ω, β, µ) with norm kx(t; ω)kBC = sup kx(t; ω)kL2 (Ω,β,µ) t∈S

The space BC ⊂ C is the space of all second order vector-valued stochastic process defined on S which are bounded and continuous in mean square. We will consider the function h(t; ω) and f (t, x(t; ω)) to be in the space C(S, L2 (Ω, β, µ)) with respect to the stochastic kernel. We assume that for each pair (t, s), k(t, s; ω) ∈ L∞ (Ω, β, µ) and denote the norm by kk(t, s; ω)k = kk(t, s; ω)kL∞ (Ω,β,µ) = µ − ess sup |k(t, s; ω)| . ω∈Ω

Also we will suppose that k(t, s; ω) is such that |||k(t, s; ω)||| . kx(s; ω)kL2 (Ω,β,µ) is µ0 -integrable with respect to s for each t ∈ S and x(s; ω) in C(S, L2 (Ω, β, µ)) and there exists a real valued function G defined µ0 -a.e. on S, so that G(S) kx(s; ω)kL2 (Ω,β,µ) is µ0 -integarable and for each pair (t, s) ∈ S × S, |||k(t, u; ω) − k(s, u; ω)||| . kx(u, ω)kL2 (Ω,β,µ) ≤ G(u) kx(u, ω)kL2 (Ω,β,µ) µ0 -a.e. Further, for allmost all s ∈ S, k(t, s; ω) will be continuous in t from S into L∞ (Ω, β, µ). We now define the random integral operator T on C(S, L2 (Ω, β, µ)) by Z (T x)(t; ω) = k(t, s; ω)x(s; ω)dµ0(s) (4.2) S

where the integral is a Bochner integral. Moreover, we have that for each t ∈ S, (T x)(t; ω) ∈ L2 (Ω, β, µ) and that (T x)(t; ω) is continuous in mean square by Lebesgue’s dominated convergence theorem. So (T x)(t; ω) ∈ C(S, L2 (Ω, β, µ)).

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Definition 4.2. (see [29], [38]) Let B and D be Banach spaces. The pair (B, D) is said to be admissible with respect to a random operator T (ω) if T (ω)(B) ⊂ D. Lemma 4.3. (see [39]) The linear operator T defined by (4.2) is continuous from C(S, L2 (Ω, β, µ)) into itself. Lemma 4.4. (see [39], [38]) If T is a continuous linear operator from C(S, L2 (Ω, β, µ)) into itself and B, D ⊂ C(S, L2 (Ω, β, µ)) are Banach spaces stronger than C(S, L2 (Ω, β, µ)) such that (B, D) is admissible with respect to T , then T is continuous from B into D. Remark 4.5. (see [39]) The operator T defined by (4.2) is a bounded linear operator from B into D. It is to be noted that by a random solution of the equation (4.1) we will mean a function x(t; ω) in C(S, L2 (Ω, β, µ)) which satisfies the equation (4.1) µ-a.e. We are now in a state to prove the following theorem. Theorem 4.6. We consider the stochastic integral equation (4.1) subject to the following conditions: (a) B and D are Banach spaces stronger than C(S, L2 (Ω, β, µ)) such that (B, D) is admissible with respect to the integral operator defined by (4.2); (b) x(t; ω) → f (t, x(t; ω)) is an operator from the set Q(ρ) = {x(t; ω) : x(t; ω) ∈ D, kx(t; ω)kD ≤ ρ} into the space B satisfying kf(t, x1 (t; ω)) − f(t, x2 (t; ω))kB

≤

λ(ω) [kx1 (t; ω) − f (t, x1(t; ω))kD + kx2 (t; ω) − f(t, x2 (t; ω))kD ]

(4.3)

for x1 (t; ω), x2(t; ω) ∈ Q(ρ), where 0 ≤ λ(ω) < 21 is a real valued random variable almost surely, (c) h(t; ω) ∈ D. Then there exists a unique random solution of (4.1) in Q(ρ), provided λ(ω) (1 + c(ω)) < 21 and   1 + λ(ω) c(ω)λ(ω) kh(t; ω)kD + c(ω) kf (t; 0)kB ≤ ρ 1 − 1 − λ(ω) 1 − λ(ω) where c(ω) is the norm of T (ω).

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Proof. Define the operator U (ω) from Q(ρ) into D by Z (U x)(t; ω) = h(t; ω) + k(t, s; ω)f (s, x(s; ω))dµ0(s) S

Now

k(U x)(t; ω)kD

≤

kh(t; ω)kD + c(ω) kf (t, x(t; ω))kB

≤

kh(t; ω)kD + c(ω) kf (t; 0)kB + c(ω) kf (t, x(t; ω)) − f (t; 0)kB

Then from the condition (4.3) of this theorem kf (t, x(t; ω)) − f (t; 0)kB

≤ ≤ ≤

λ(ω)[kx(t; ω) − f (t, x(t; ω))kD + kf (t; 0)kD ] ˆ ˜ λ(ω) kx(t; ω)kD + kf (t, x(t; ω))kD + kf (t; 0)kD ˜ ˆ λ(ω) kx(t; ω)kD + kf (t, x(t; ω)) − f (t; 0)kD + 2 kf (t; 0)kD

implies kf (t, x(t; ω)) − f (t; 0)kB ≤

λ(ω) 2λ(ω) ρ+ kf (t; 0)kB 1 − λ(ω) 1 − λ(ω)

(4.4)

Therefore by (4.4), we have k(U x)(t; ω)kD ≤ kh(t; ω)kD + c(ω) kf (t; 0)kB   λ(ω)ρ 2λ(ω) +c(ω) + kf (t; 0)kB 1 − λ(ω) 1 − λ(ω) c(ω)λ(ω) 1 + λ(ω) = kh(t; ω)kD + ρ+ c(ω) kf (t; 0)kB 1 − λ(ω) 1 − λ(ω) < ρ Hence (U x)(t; ω) ∈ Q(ρ). Then for x1 (t; ω), x2(t; ω) ∈ Q(ρ), we have by condition (b) k(U x1)(t; ω) − (U x2)(t; ω)kD

=

‚Z ‚ ‚ ‚ ‚ k(t, s; ω)[f (s, x1 (s; ω)) − f (s, x2 (s; ω))]dµ0 (s)‚ ‚ ‚ S

≤ ≤

D

c(ω) kf (t, x1 (t; ω)) − f (t, x2 (t; ω))kB ˆ c(ω)λ(ω) kx1 (t; ω) − f (t, x1 (t; ω))kD ˜ + kx2 (t; ω) − f (t, x2 (t; ω))kD

since c(ω)λ(ω) < 21 , U (ω) is a Kannan contraction on Q(ρ). Hence, by Theorem 2.17 and Remark 2.18(I), there exists a unique x∗ (t, ω) ∈ Q(ρ), which is a fixed point of U , that is x∗ (t, ω) is the unique random solution of the equation (4.1).

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A similar theorem can be obtained using random analogue of Chatterjea fixed point theorem as mentioned in Remark 2.18 (II). Theorem 4.7. Assume that the stochastic integral equation (4.1) subject to the following conditions: 0 (a ) Same as (a) of Theorem 4.6; 0 (b ) x(t; ω) → f (t, x(t; ω)) is an operator from the set Q(ρ) = {x(t; ω) : x(t; ω) ∈ D, kx(t; ω)kD ≤ ρ} into the space B satisfying kf(t, x1 (t; ω)) − f(t, x2 (t; ω))kB

≤

λ(ω) [kx1 (t; ω) − f (t, x2(t; ω))kD + kx2 (t; ω) − f(t, x1 (t; ω))kD ] (4.5)

for x1 (t; ω), x2(t; ω) ∈ Q(ρ), where 0 ≤ λ(ω) < 21 is a real valued random variable almost surely, 0 (c ) h(t; ω) ∈ D. Then there exists a unique random solution of (4.1) in Q(ρ), provided λ(ω) (1 + c(ω)) < 21 and   1 + λ(ω) c(ω)λ(ω) kh(t; ω)kD + c(ω) kf (t; 0)kB ≤ ρ 1 − 1 − λ(ω) 1 − λ(ω) where c(ω) is the norm of T (ω). Proof. The proof is similar to that of Theorem 4.6. The following example illustrates the Theorem 4.6. Example 4.8. Consider the following nonlinear stochastic integral equation: Z ∞ e−t−s x(t; ω) = ds 8(1 + |x(s; ω)|) 0 Comparing with (4.1), we see that h(t, ω) = 0, k(t, s; ω) =

1 −t−s 1 e , f (s, x(s; ω)) = 2 4(1 + |x(s; ω)|)

Then one can check that equation (4.3) is satisfied with λ(ω) = 31 . Comparing with integral operator equation (4.2), we see that the norm of T (ω) is c(ω) = 41 satisfying λ(ω)(1 + c(ω)) < 21 . So, all the conditions of Theorem 4.6 are satisfied and there exists a random fixed point of the integral operator T satisfying (4.2).

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5. W ELL -P OSEDNESS (A LMOST SURELY ) OF A R ANDOM F IXED P OINT P ROBLEM In this section, we first propose well-posedness almost surely of a random fixed point problem for a single valued mapping and then extend to to the multivalued mapping. Let (Ω, β, µ) be a complete probability measure space and E be a nonempty subset of a separable Banach space X. Let T : Ω × E → E be a random operator with a random fixed point x∗ (ω) ∈ E. Definition 5.1. The random fixed point problem of T : Ω × E → E is said to be almost sure well-posed if: 0 (a ) T : Ω × E → E has a unique random fixed point x∗ (ω) ∈ E; 0 (b ) for any given random variable x0 (ω) ∈ E, sequence of random variables {xn (ω)}∞ n=0 ⊂ E be such that lim kT (ω, xn (ω)) − xn (ω)k = 0, then n→∞

we have xn (ω) → x∗ (ω) ∈ E almost surely as n → ∞. We use this notion of almost sure well-posedness of random fixed point problem for Banach contraction operator(Remark 2.18 (III)) . Theorem 5.2. Let (Ω, β, µ) be a complete probability measure space and E be a nonempty subset of a separable Banach space X. Let T : Ω × E → E be a random operator satisfying kT (ω, x1) − T (ω, x2)k ≤ α(ω) kx1 − x2 k

(5.1)

almost surely for all x1 , x2 ∈ X, where α(ω) is a non-negative real random variable such that α(ω) < 1 almost surely. Then the random fixed point problem of T : Ω × E → E is almost sure well-posed. Proof. It is immediate to conclude that T : Ω × E → E is a continuous random operator and T : Ω × E → E has a unique random fixed point x∗ (ω) ∈ E (by 0 Remark 2.18 (III)). So the condition (a ) of Definition 5.1 is satisfied. Next let {xn (ω)}∞ n=0 ⊂ E be an arbitrary sequence of random variables such that lim kT (ω, xn(ω)) − xn (ω)k = 0. n→∞

Now kx∗ (ω) − xn (ω)k

≤ ≤ ≤

kx∗ (ω) − T (ω, xn (ω))k + kT (ω, xn (ω)) − xn (ω)k kT (ω, x∗ (ω)) − T (ω, xn (ω))k + kT (ω, xn (ω)) − xn (ω)k α(ω) kx∗(ω) − xn (ω)k + kT (ω, xn (ω)) − xn (ω)k

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implies that kx∗ (ω) − xn (ω)k ≤

1 kT (ω, xn(ω)) − xn (ω)k → 0 as n → ∞. 1 − α(ω)

Subsequently, xn (ω) → x∗ (ω) ∈ E almost surely. Measurability of x∗ (ω) is also clear. This proves the theorem. Theorem 5.3. Let (Ω, β, µ) be a complete probability measure space and E be a nonempty subset of a separable Banach space X. Let T : Ω × E → E be a random operator satisfying kT (ω, x1 ) − T (ω, x2 )k ≤ λ(ω) [kx1 (ω) − T (ω, x1 (ω))k + kx2 (ω) − T (ω, x2 (ω))k] (5.2)

almost surely for all x1 , x2 ∈ X, where λ(ω) is a non-negative real random variable such that λ(ω) < 21 almost surely. Then the random fixed point problem of T : Ω × E → E is almost sure well-posed. Proof. First part of the proof is immediate i.e T : Ω × E → E has a unique random fixed point x∗ (ω) ∈ E (by Remark 2.18 (I)). Next let {xn (ω)}∞ n=0 ⊂ E be an arbitrary sequence of random variables such that lim kT (ω, xn(ω)) − xn (ω)k = 0. n→∞

Now kx∗ (ω) − xn (ω)k

≤

kx∗ (ω) − T (ω, xn (ω))k + kT (ω, xn (ω)) − xn (ω)k

≤ ≤

kT (ω, x∗ (ω)) − T (ω, xn (ω))k + kT (ω, xn (ω)) − xn (ω)k λ(ω) [kx∗(ω) − T (ω, x∗ )k + kxn (ω) − T (ω, xn )k] + kT (ω, xn (ω)) − xn (ω)k

implies that kx∗ (ω) − xn (ω)k ≤ (1 + λ(ω)) kT (ω, xn (ω)) − xn (ω)k → 0 as n → ∞.

Subsequently, xn (ω) → x∗ (ω) ∈ E almost surely. Measurability of x∗ (ω) is also clear. This proves the theorem. Theorem 5.4. Let (Ω, β, µ) be a complete probability measure space and E be a nonempty subset of a separable Banach space X. Let T : Ω × E → E be a random operator satisfying kT (ω, x1 ) − T (ω, x2 )k ≤ λ(ω) [kx1 (ω) − T (ω, x2 (ω))k + kx2 (ω) − T (ω, x1 (ω))k] (5.3)

almost surely for all x1 , x2 ∈ X, where λ(ω) is a non-negative real random variable such that λ(ω) < 21 almost surely. Then the random fixed point problem of T : Ω × E → E is almost sure well-posed.

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Proof. First part of the proof is immediate i.e T : Ω × E → E has a unique random fixed point x∗ (ω) ∈ E ( by Remark 2.18 (II)). Next let {xn (ω)}∞ n=0 ⊂ E be an arbitrary sequence of random variables such that lim kT (ω, xn(ω)) − xn (ω)k = 0. n→∞

Now kx∗ (ω) − xn (ω)k

≤ ≤

kx∗ (ω) − T (ω, xn (ω))k + kT (ω, xn (ω)) − xn (ω)k kT (ω, x∗ (ω)) − T (ω, xn (ω))k + kT (ω, xn (ω)) − xn (ω)k

≤

λ(ω) [kx∗(ω) − T (ω, xn (ω))k + kxn (ω) − T (ω, x∗(ω))k] + kT (ω, xn (ω)) − xn (ω)k λ(ω) [kx∗(ω) − xn (ω)k + kxn (ω) − T (ω, xn (ω))k + kxn (ω) − T (ω, x∗ (ω))k] + kT (ω, xn (ω)) − xn (ω)k

≤

implies that kx∗ (ω) − xn (ω)k ≤

1 + λ(ω) kT (ω, xn (ω)) − xn (ω)k → 0 as n → ∞. 1 − 2λ(ω)

Subsequently, xn (ω) → x∗ (ω) ∈ E almost surely. Measurability of x∗ (ω) is also clear. This proves the theorem.

5.1.

The Multi Valued Deterministic Case

For notational advantages, we assume the underlying space as a Polish space i.e., a complete and separable metric space (E, d). Let CB(E) be the family of all closed, bounded and nonempty subsets of E and MO(E) be the families of all multivalued operators on E. For any set A, we denote by Card(A) the cardinality of A. Also let T ∈ MO(E). In the sequel, MFix(T ) stands for the set of all fixed points for T , i.e. MFix(T )={x ∈ E : x ∈ T x}. As usual, for x ∈ E and nonempty subset A of E, we denote d(x, A) = inf{d(x, y) : y ∈ A}. Definition 5.5. Let T ⊆ MO(E). The multi valued fixed point problem of T is said to be well-posed, if the following two conditions hold: T 1. For each T ∈ T , Card(M F ix(T)) = Card( T ∈T M F ix(T) = 1.

∞ 2. For any sequence {xn }∞ n=0 in E, such that {d(xn , T (xn))}n=0 con∞ verges to zero, we have {xn }n=0 converges to the unique element in T M F ix(T). T ∈T

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Theorem 5.6. Let T ⊆ MO(E) and suppose the following three conditions hold: 1. For each T ∈ T , M F ix(T) 6= ∅. 2. For each T ∈ T and x ∈ M F ix(T), there exists C(x, T ) > 0 such that d(x, y) ≤ C(x, T )d(y, T y), for all y ∈ E. 3. For each S, T ∈ T , M F ix(S) ⊆ M F ix(T). Then, the multi valued fixed point problem of T is well-posed. Proof. Conditions (1) and (2) imply there exists xT ∈ E such that {xT } = M F ix(T). But condition (3) implies for any S ∈ T , M F ix(S) = M F ix(T) and hence there exists x∗ ∈ E such that for each T ∈ T , {x∗ } = M F ix(T) = T T ∈T M F ix(T). Let {xn }∞ n=0 be a sequence in E such that for each T ∈ T , d(xn , T xn ) → 0. By condition (2), we have d(xn, x∗ ) ≤ C(x∗ , T )d(xn, T xn ) and hence ∗ {xn }∞ n=0 converges to x , which concludes the proof. We consider CB(E) endowed with the Hausdorff metric H induced by d. I.e. ( ) H(A, B) = max sup d(y, A), sup d(x, B) . y∈B

x∈A

Definition 5.7. A multivalued operator on E is any mapping T from E to CB(E). For this operator, we define recursively the sequence {T n }∞ n=0 , of mul1 n+1 n tivalued operators on E, as T = T and T (x) = T (T (x)), for all x ∈ E. We say that T is a Picard operator, whenever there existe a unique x∗ ∈ E such that x∗ ∈ T (x∗ ) and, for each x ∈ E, {d(x∗ , T n(x))}∞ n=0 converges to zero, as n goes to ∞. A single operator T : E → E is said to be Picard, whenever {T } is a Picard operator in the sense just defined. Definition 5.8. A set T ⊆ MO(E) is called 1. a Kannan multi valued contraction family with constant α ∈ [0, 1/2) if H(Sx, T y) ≤ α[d(x, Sx) + d(y, T y)], for all x, y ∈ E and S, T ∈ T , and

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Remark 5.9. Note that a Kannan or a Chatterjea multi valued contraction need not be well-posed. Indeed, the multi valued function T : R → CB(R) defined as T (x) = [−1, 1] is Kannan and Chatterjea multi valued contraction, however MFix(T )=[−1, 1].

Proposition 5.10. Let T ⊆ MO(E) be a Kannan or Chatterjea multi valued contraction family with constant α ∈ [0, 1/2) and satisfying, for each T ∈ T , the following two conditions: 1. the function hT : E → R, such that hT (x) = d(x, T (x)), is lower semicontinuous, and 2. M F ix(T) = {x ∈ E : T (x) = {x}}. Then, the multi valued fixed point problem of T is well-posed. Proof. First we verify T satisfies (2) in Theorem 5.6. Let T ∈ T , x ∈ M F ix(T) and y ∈ E. Choose u ∈ T (y) such that d(x, u) ≤ 2d(y, T (y)). Hence, whether T is a Kannan multi valued contraction family, we have d(y, x) ≤ ≤ ≤ =

d(y, u) + d(u, T (x)) 2d(y, T (y)) + H(T (x), T (y)) 2d(y, T (y)) + α[d(x, T (x)) + d(y, T (y))] (2 + α)d(y, T (y)).

In case T is a Chatterjea multi valued contraction family, we have d(y, x) ≤ 2d(y, T (y)) + α[d(x, T (y)) + d(y, T (x))] ≤ 2αd(x, y) + (2 + α)d(y, T (y)) and thus d(y, x) ≤



2+α 1 − 2α



d(y, T (y)).

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Accordingly, in any case, condition (2) in Theorem 5.6 holds with C(x, T ) = (2 + α)/(1 − 2α). Let S ∈ T . In order to prove that condition (3) in Theorem 5.6 holds, it suffices to prove that d(x, S(x)) = 0. We have d(x, S(x)) ≤ H(T (x), S(x)) ≤ αd(x, S(x)) and since α < 1, we obtain d(x, S(x)) = 0. Hence it only remains to prove T satisfies condition (1) in Theorem 5.6. Let α 1 β = β() = − . 1−α 1+ Since 0 ≤ α < 1/2, we can choose  > 0 such that β > 0. For each x ∈ E, let u(x) ∈ T (x) such that d(x, u(x)) ≤ (1 + )d(x, T (x)). In case T is a Kannan multi valued contraction family, we have d(u(x), T (u(x))) ≤ ≤ ≤ =

H(T (x), T (u(x))) α[d(x, T (x)) + d(u(x), T (u(x))] α[d(x, u(x)) + d(u(x), T (x)) + d(u(x), T (u(x))] α[d(x, u(x)) + d(u(x), T (u(x))]

and hence



 α d(u(x), T (u(x))) ≤ d(x, u(x)). 1−α In case T is a Chatterjea multi valued contraction family, we have d(u(x), T (u(x))) ≤ αd(x, T (u(x))) ≤ αd(x, u(x)) + αd(u(x), T (u(x)))

and also d(u(x), T (u(x))) ≤



α 1−α



d(x, u(x)).

Consequently, in any case, βd(x, u(x)) ≤ d(x, T (x)) − d(u(x), T (u(x))) or in a equivalent form d(x, u(x)) ≤ ϕ(x) − ϕ(u(x)), where ϕ : E → R is defined as ϕ(x) = d(x, T (x))/β. Condition (1) implies ϕ is lower semicontinuous and hence, by Theorem (2.1)´ in [40], u has a fixed point x∗ and hence x∗ ∈ M F ix(T). Therefore M F ix(T) 6= ∅ and consequently Theorem 5.6 implies that the fixed point problem of T is well-posed. This concludes the proof.

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5.2.

The Multi Valued Random Case

In the sequel, (Ω, β, µ) stands for a complete probability measure space. We say that ξ : Ω → E is a random variable, if for each open set G in E, ξ −1 (G) ∈ β. Since E is separable, in order to verify measurability of ξ, it suffices to assume that G is a ball in the metric space (E, d). Let T : Ω × E → E be a mapping. We say T is a random operator on E, if T is measurable, where Ω × E is considered as the product σ-algebra β ⊗ B(E). Given a mapping ξ : Ω → E, T (ξ) : Ω → E is defined as T (ξ)(ω) = T (ω, ξ(ω)). A multivalued operator on E is any mapping T from E to CB(E). Let T : Ω × E → CB(E) be a mapping. We say that T is a multivalued random operator on E, if T is measurable where Ω × E is considered as the product σ-algebra β ⊗ B(E). That is, for each open subset G of E, T −1 (G) ∈ β ⊗ B(E). The multivalued random operator T is said to be lower or upper semicontinuous whether for each ω ∈ Ω, T (ω, ·) is lower or upper semicontinuous, respectively. Analogously, T is a continuous multivalued random operator if T is lower and upper semicontinuous multivalued random operator. Let MRO(E) be the families of all multivalued random operators on E. Let T : Ω × E → CB(E) be a mapping. We say ξ : Ω → E is a deterministic fixed point for T , if ξ ∈ T (ξ). A random fixed point ξ for T is a deterministic fixed point for T such that T ∈ MRO(E) and ξ is measurable. We denote by MRFix(T ) the set of all random fixed point of T . Proposition 5.11. Let T : Ω × E → CB(E) be a mapping such that for each ω ∈ Ω, T (ω, ·) is lower semicontinuous. Then, the following two conditions are equivalent 1. T ∈ MRO(E) 2. For each x ∈ E, T (·, x) is measurable. Proof. For each x ∈ E, let hx : Ω → Ω × E be the function defined as hx (ω) = (ω, x). Suppose (1) holds. Hence, for any A ∈ β and B ∈ B(E), we have  A if x ∈ B −1 hx (A × B) = ∅ if x ∈ / B.

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and T (·, x) = T ◦hx . Consequently, T (·, x) is measurable. Hence, for any open subset G of E, we have −1 T (·, x)−1 (G) = {ω ∈ Ω : T (hx (ω)) ∩ G 6= ∅} = h−1 (G)) x (T

and thus, (1) implies (2). Next assume condition (2) and let D be a dense and countable subset of E and G an open subset of E. Since T (ω, ·) is lower semicontinuous, (ω, x) ∈ T −1 (G), if and only y ∈ D such that T (ω, y) ∈ G. S if, there exists −1 −1 Consequently, T (G) = y∈D T (·, y) (G) and therefore T −1 (G) ∈ β, which concludes the proof. Definition 5.12. Let T ⊆ MRO(E). The multi valued random fixed point problem of T is said to be well-posed almost surely (respectively, in probability), if the following two conditions hold: T 1. For each T ∈ T , Card(M RF ix(T)) = Card( T ∈T M RF ix(T)) = 1. 2. For any sequence {ξn }∞ n=0 of random variables such that ∞ {d(ξn, T (ξn))}n=0 converges to zero, almost surely (respectively, in probability), we have {ξn }∞ n=0 converges to the unique element in T M RF ix(T), almost surely (respectively, in probability). T ∈T

Proposition 5.13. Let T ⊆ MRO(E) and suppose the multi valued random fixed point problem of T is well-posed almost surely. Then, this multi valued random fixed point problem is well posed in probability.

Proof. Let ξ : Ω → E be the unique measurable function such that for any T ∈ T T , {ξ} = M RF ix(T) = T ∈T M RF ix(T) and {ξn }∞ n=0 be a sequence of ∞ random functions such that {d(ξn , T (ξn))}n=0 converges to zero, in probability. ∞ Let {ξnk }∞ k=0 be a subsequence of {ξn }n=0 . We need to find a further subse∞ quence of {ξnk }k=0 converging to ξ, almost surely. Since {d(ξnk , T (ξnk ))}∞ k=0 converges in probability to zero, there exists {d(ξnkj , T (ξnkj ))}∞ subsej=0 quence of {d(ξnk , T (ξnk ))}∞ , which converges to zero, almost surely. From k=0 assumption, {ξnkj }∞ converges to ξ, almost surely, and therefore, Proposition j=0 5.18 in [41] implies {ξn }∞ n=0 converges in probability to ξ, which completes the proof. Theorem 5.14. Let T ⊆ MRO(E) such that for each ω ∈ Ω, T (ω) satisfies conditions (1)-(3) in Theorem 5.6. Moreover, suppose there exists S ∈ T

76

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such that, for each ω ∈ Ω, S(ω, ·) is a Picard operator. Then, the multivalued random fixed point problem of T is well posed, almost surely. Proof. From Theorem 5.6, for each ω ∈ Ω, the multivalued fixed point problem of T (ω) is well-posed. Let ξ : Ω → E be such that {ξ(ω)} = M F ixS(ω, ·) and B = B(a, r) be the ball with centre in a ∈ E and radius r > 0. Since E is separable, in order to prove that ξ is measurable, it suffices to verify that ξ −1 (B) ∈ β. For each (ω, x) ∈ Ω × E, we have |d(a, ξ(ω)) − d(a, S n(ω, x))| ≤ d(ξ(ω), S n(ω, x)) and hence limn→∞ d(a, S n(ω, x)) = d(a, ξ(ω)). But {S n (·, x)}∞ n=0 is a sequence of multivalued measurable functions and consequently d(a, ξ) is measurable. In addition, ξ −1 (B) = {ω ∈ Ω : d(ξ(ω), a) < r} ∈ β, which concludes the proof. Example 5.15. Let T be a subfamily of MRO(E) such that for each ω ∈ Ω, T (ω) is a quasi contraction multivalued family with constant α(ω) ∈ [0, 1/2) and for each T ∈ T , M F ix(T (ω, ·)) = {x ∈ E : T (ω, x) = {x}}. Let ξ : Ω → E satisfy {ξ(ω)} = M F ix(T (ω, ·)). Hence, for each x ∈ E, we have  n α(ω) d(ξ(ω), T n(ω, x)) ≤ d(ξ(ω), x) 1 − α(ω) and, due to α(ω)/(1 − α(ω)) < 1, T (ω, ·) is a Picard operator for each ω ∈ Ω. Therefore, it follows, from Proposition 5.10 and Theorem 5.14, that the multivalued random fixed point problem of T is well posed.

ACKNOWLEDGMENT The author thanks the anonymous referee for his valuable suggestion to improve the chapter.

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77

R EFERENCES ˇ cek A., Zuf¨allige Gleichungen, Czechoslovak Mathematical Jour[1] Spaˇ nal5(80) (1955), 462-466 (German), with Russian summary. [2] Hanˇs O., Reduzierende zuf¨allige transformationen, Czechoslovak Math. Journal 7(82) (1957), 154-158 (German), with Russian summary. [3] Hanˇs O., Random operator equations, Proceedings of 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, University of California Press, California, part I, (1961), 185-202. [4] Bharucha-Reid A.T., Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., 82(5) (1976), 641-657. [5] Itoh S., Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl.67(2) (1979), 261-273. [6] Mukherjee A., Transformation aleatoives separables theorem all point fixed aleatoire, C. R. Acad. Sci. Paris Ser. A - B, 263, (1966), 393-395. [7] Sehgal V.M. and Waters C., Some random fixed point theorems for condensing operators, Proc. Amer. Math. Soc., 90 (1) (1984) 425-429. [8] Rothe E., Zu¨r Theorie der topologische ordnung und der vektorfelder in Banachschen Raumen, Composito Math., 5(1937), 177-197. [9] Kannan R., Some results on fixed points, Bull. Cal. Math. Soc., 60(1968), 71-76. [10] Chatterjea S.K., Fixed point theorems, C. R. Acad. Bulgare Sci.25 (1972), 727-730. [11] Zamfirescu T., Fixed point theorems in metric spaces, Arch. Math.(Basel) 23(1972), 292-298. [12] Joshi M.C. and Bose R.K., Some topics in non linear functional analysis, Wiley Eastern Ltd. (1984).

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[13] Saha M. and Dey D., Investigation on convergence of a random iteration ´ c quasi-contractive operator, Italian scheme to a random fixed point of Ciri´ J. Pure. Appl. Math., (accepted and to appear). [14] Choudhury B. S., Random Mann iteration scheme, Appl. Math. Lett., 16(1) (2003), 93-96. [15] Choudhury B. S. and Ray M., Convergence of an iteration leading to a solution of a random operator equation, J. Appl. Math. Stochastic Anal., 12(2), (1999), 161-168. [16] Papageorgiou N.S., Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc., 97, (1986), 507–514. [17] Xu H.K., Some random fixed point theorems for condensing and nonexpansive operators, Proc. Amer. Math. Soc. 110, (1990), 103-123. [18] Beg I., Random fixed points of random operators satisfying semicontractivity conditions, Math. Japan. 46,(1997), 151-155. [19] Beg I., Minimal displacement of random variables under Lipschitz random maps, Topol. Methods Nonlinear Anal. 19, (2002), 391-397. [20] Beg I. and Shahzad N., Random approximations and random fixed point theorems, J. Appl. Math. Stochastic Anal., 7(2), (1994), 145-150. [21] Xu H.K., Beg I., Measurability of fixed point sets of multivalued random operators, J. Math. Anal. Appl. 225, (1998), 62-72. [22] Liu Q., Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 256, (2001), 1–7. [23] Saha M. and Dey D., Some Random fixed point theorems for (θ, L)weak contractions, Hacettepe Journal of Mathematics and Statistics, 41 (6) (2012), 795-812. [24] Saha M. and Debnath L., Random fixed point of mappings over a Hilbert space with a probability measure, Adv. Stud. Contemp. Math. 1(2007), 7984.

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[25] Saha M., On some random fixed point of mappings over a Banach space with a probability measure, Proc. Nat. Acad. Sci. India, 76(A)III, (2006), 219-224. [26] Saha M. and Ganguly A., Random fixed point theorem on Ciric type contractive mapping and its consequence, Fixed Point Theory and Applications, Springer, 2012:209 doi:10.1186/1687-1812-2012-209. [27] Beg I., Abbas M., Iterative procedures for solutions of random operator equations in Banach spaces J. Math. Anal. Appl. 315, (2006), 181-201. [28] Choudhury B. S., Convergence of a random iteration scheme to a random fixed point, J. Appl. Math. Stochastic Anal., 8(2), (1995), 139-142. [29] Achari J., On a pair of random generalized non-linear contractions, Int. J. Math. Math. Sci. 6(3), (1983), 467-475. [30] Dey D. and Saha M., Application of random fixed point theorems in solving nonlinear stochastic integral equation of the Hammerstein type, Malaya Journal of Matematik, 2(1),(2013) 54-59. [31] Banach S., Sur les op´erations dans les ensembles abstraits et leur application aux e´ quations int´egrales, Fund. Math. 3, (1922)133-181 (French). [32] Mann W. R., Mean value methods in iterations, Proc. Amer. Math. Soc., 4 (1953), 506-510. [33] Xu Y., Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equation, J. Math. Anal. Appl., 224 (1998), 91101. ´ c Lj. B., Generalised contractions and fixed point theorem, Publ. Inst. [34] Ciri´ Math., 12(26), (1971), 19-26. ´ c Lj. B., A generalization of Banach’s contraction principle, Proc. [35] Ciri´ Amer. Math. Soc., 45 (1974), 727-730. [36] Arens R.F., A topology for spaces of transformations, Annals of Math. 47(2), (1946), 480-495. [37] Yosida K., Functional analysis, Academic press, New york, SpringerVerlag, Berlin, (1965).

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[38] Lee A.C.H. and Padgett W.J., On random nonlinear contraction, Math. Systems Theory ii (1977), 77-84. [39] Padgett W.J., On a nonlinear stochastic integral equation of the Hammerstein type, Proc. Amer. Math. Soc., 38, (1), 1973. [40] Caristi J., Fixed point theorem for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215, (1976), 241-251. [41] Karr A. F., Probability. Springer, New York, (1993)

In: An In-Depth Guide to Fixed-Point Theorems ISBN: 978-1-53619-565-1 c 2021 Nova Science Publishers, Inc. Editors: R. Sharma and V. Gupta

Chapter 4

S OME C OMMON F IXED P OINT T HEOREMS FOR S ELF -M APPINGS S ATISFYING R ATIONAL I NEQUALITIES C ONTRACTION IN C OMPLEX VALUED M ETRIC S PACES AND A PPLICATIONS Khaled Berrah1,∗, Oussaeif Taki Eddine2,† and Abdelkrim Aliouche2,‡ 1 Laboratory of Mathematics, Informatics and Systems (LAMIS) Larbi Tebessi University, Tebessa, Algeria 2 Department of Mathematics and Computer Sciences, Larbi Ben M’hidi University, Oum El Bouaghi, Algeria

Abstract In this chapter, we highlight some common fixed point theorems for self-mappings under different proprieties as compatible and weakly compatible mappings, satisfying rational inequalities contraction in a complex valued b-metric space. The principal results were been published in AIMS Mathematics entitled Applications and theorem on common fixed point in complex valued b-metric space. In addition, some examples ∗

Corresponding Author’s Email: [email protected]. Email: taki [email protected]. ‡ Email: [email protected].

†

82

Khaled Berrah, Oussaeif Taki Eddine and Abdelkrim Aliouche which were provided to verify our main results are given. Finally, we validate our results by proving both existence and uniqueness of a common solution of the system of Urysohn integral equations and the existence of a unique solution for linear equations system.

1.

INTRODUCTION

Banach contraction principle was an authority and a reference for many researchers during the last decades in the field of nonlinear analysis, it was used to establish the existence of a unique solution for a nonlinear integral equation in metric space [1]. The notion of metric space was firstly introduced by M. Fr´echet in 1906. Nevertheless, the term metric space was coined by Hausdorff a bit later. The utility of metric spaces in the augmentation of functional analysis is enormous. Inspired from the effect of this notion, several researchers give it a whirl many generalizations of this notion in the recent past such as: rectangular metric spaces, semimetric spaces, quasimetric spaces, quasi-semimetric spaces, pseudo metric spaces, probabilistic metric spaces, 2-metric spaces, D-metric spaces, Gmetric spaces, K-metric spaces, cone metric spaces etc. Now, there exists a considerable literature on all these generalizations of metric spaces. For more details, one can see [2, 3, 4, 5, 6, 7, 8, 9]. In 1989, Bakthtin [10] initiated the motif of b-metric space. After that, Czerwik in [11, 12] defined it such as the topical structure considered as a generalization of metric spaces. The complex valued b-metric spaces concept was introduced in 2013 by Rao et al. [13], which was more general than the wellknown complex valued metric spaces that were introduced in 2011 by Azam et al. [14]. We note that Azam et al. proved some common fixed point theorems for self-mappings satisfying rational inequalities in complex valued metric spaces which are not worthwhile in cone metric spaces[15, 16, 17, 18]. Sundry authors have studied and proved the fixed point results for mappings satisfying different contraction conditions in the framework of complex valued metric (b-metric) spaces(see [19, 20, 21, 22, 23, 13, 24]). The main purpose of this chapter, is to prove some common fixed point theorems satisfying certain partial rational expressions in complex valued bmetric spaces considered as a generalization of fixed point theorems of Azam et al. [14] and we prove a common fixed point theorem under new contraction [25].

Some Common Fixed Point Theorms for Self-Mappings ...

83

Some related results are also derived besides furnishing illustrative examples to highlight the realized improvements. Eventually, we establish the existence and the uniqueness of a common solution for the system of Urysohn integral equations. Furthermore, we prove the existence and the uniqueness of a solution for linear system in complete complex valued b-metric space.

2.

P RELIMINARIES

In this section, we give some notations, definitions and basic results that will be utilized in our subsequent discussion. Let us recall a natural relation ≤ on C, the set of complex numbers as follows: let z1 , z2 in C z1 ≤ z2 ⇔ Re(z1 ) ≤ Re(z2 ) and Im(z1 ) ≤ Im(z2 ) z1 < z2 ⇔ Re(z1 ) < Re(z2 ) and Im(z1 ) < Im(z2 ) In [14], the authors defined a partial order relation z1 - z2 on C as follows: z1 - z2 if and only if Re(z1 ) ≤ Re(z2 ) and Im(z1 ) ≤ Im(z2 ). As a result, one can infer that z1 - z2 if one of the following conditions is satisfied: (i) Re(z1 ) = Re(z2 ) , Im(z1 ) < Im(z2 ), (ii) Re(z1 ) < Re(z2 ) , Im(z1 ) = Im(z2 ), (iii) Re(z1 ) < Re(z2 ) , Im(z1 ) < Im(z2 ), (iv) Re(z1 ) = Re(z2 ) , Im(z1 ) = Im(z2 ). In (i), (ii) and (iii) we have |z1 | < |z2 |. In (iv) we have |z1 | = |z2 |, so that, |z1 | ≤ |z2 | In particular, z1 - z2 if z1 6= z2 and one of (i), (ii) and (iii) is satisfied. In this case |z1 | < |z2 |. We will write z1 ≺ z2 if only (iii) is satisfied. Further, 0 - z1  z2 ⇒ | z1 | 1. For all z, w ∈ X, define d : X × X → C by Z Z i z(u) w(u) 2 d(z(u), w(u)) = du − du , 2π u u Γ

Γ

a complex valued b-metric where Γ is a closed path in X containing a zero. We prove that d is a complex valued b-metric with s = 2 d(z(u), w(u))

= = -

-

d(z(u), w(u))

-

˛ ˛Z Z i ˛˛ w(u) ˛˛2 z(u) − du du ˛ 2π ˛ Γ u u Γ ˛ ˛Z Z Z Z ˛ i ˛ x(u) x(u) w(u) ˛˛2 z(u) − + − du du du du ˛ 2π ˛ Γ u u Γ u Γ u Γ ˛2 ˛ ˛Z ˛Z Z Z ˛ ˛ ˛ i ˛ x(u) ˛ i ˛ w(u) ˛˛2 z(u) x(u) − + − du du du du ˛ ˛ + 2π ˛ Γ u 2π ˛ Γ u u Γ u Γ ˛ ˛Z ˛ ˛Z Z Z i ˛˛ x(u) ˛˛ ˛˛ x(u) w(u) ˛˛ z(u) 2 du − du˛ ˛ du − du˛ 2π ˛ Γ u u u u Γ Γ Γ ˛ ˛Z ˛2 ˛Z Z Z i ˛˛ w(u) ˛˛2 x(u) ˛˛ z(u) i ˛˛ x(u) − − du du + ˛ + ˛ 2π ˛ Γ u 2π ˛ Γ u u Γ Γ u ˛2 ˛ ˛Z ˛Z Z Z i ˛˛ x(u) ˛˛ i ˛˛ w(u) ˛˛2 z(u) x(u) − + − du du du du ˛ ˛ 2π ˛ Γ u 2π ˛ Γ u u Γ u Γ ( ˛2 ˛ ) ˛Z ˛Z Z Z i ˛˛ x(u) ˛˛ i ˛˛ w(u) ˛˛2 z(u) x(u) 2 du − du˛ + du − du˛ 2π ˛ Γ u 2π ˛ Γ u u Γ u Γ 2 {d(z(u), x(u)) + d(x(u), w(u))} .

Now we define the mappings S, T, P, Q : X → X by u 1 Sz(u) = u, T z(u) = e 2 , P z(u) = eu − 1 and Qz(u) = u2 + u. 2

Using the Cauchy formula when the mappings S, T, P and Q are analytics we get

Some Common Fixed Point Theorms for Self-Mappings ...

d(Sz(u), T w(u)) = d(P z(u), Qw(u)) =

d(P z(u), Sz(u)) =

Z i 2π Γ Z i 2π Γ Z i 2π Γ

d(Qw(u), T w(u)) =

i 2π

d(Qw(u), Sz(u)) =

i 2π

d(P z(u), T w(u)) =

i 2π

99

eu − 1 2 du = 0, u Γ 2 u Z 2 1 u + 2 u (2π)2i e2 du − du = , u u 2π Γ 2 u Z u e2 du − du = 0, u Γ u u du − u

Z Γ Z Γ Z Γ

Z

2 eu − 1 du = 0, u Γ 2 Z u2 + 21 u u du − du = 0, u Γ u 2 u Z u e − 1 (2π)2i e2 du − du = , u u 2π Γ u2 + 21 u du − u

Z

R(z(u), w(u)) = max{2πi, 0} = 2πi. Further, 0 = d(Sz(u), T w(u)) -

πλi . 2

As all the conditions of Theorem 3 are satisfied, the mappings S, T, P and Q have a unique common fixed point in X.

4.

A PPLICATIONS

In this section we present some applications upon a partial rational inequality contraction condition in complex valued b-metric spaces. Firstly, we apply Theorem 3 to a system of Urysohn integral equations, then, we apply Corollary 3 to linear system.

4.1.

Application to Urysohn Integral Equations

Our first new result is the following

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Khaled Berrah, Oussaeif Taki Eddine and Abdelkrim Aliouche

Theorem 4. Let X = C([a, b], Rn), a > 0 and d : X × X → C defined as follows. p −1 d(z, w) = max kz(u) − w(u)k∞ 1 + a2 eitan a . u∈[a,b]

Consider the Urysohn integral equations z(u) =

z(u) =

Z Z

b

K1 (t, s, z(u))ds + g(u),

(1)

K2 (t, s, z(u))ds + h(u),

(2)

a b

a

where, u ∈ [a, b] ⊂ R and z, g, h ∈ X. Assume that K1 , K2 : [a, b] × [a, b] × Rn → Rn such that Fz , Gz ∈ X, for each z ∈ X, where Z b Z b Fz (u) = K1 (t, s, z(u))ds, Gz(u) = K2 (t, s, z(u))ds for all u ∈ [a, b]. a

a

If there exist s ≥ 1, λ ∈ (0, 1) such that the inequality A(z, w)(u) -

λ R(z, w)(u), s2

(4.1)

holds for all z, w ∈ X where R(z, w) = max {D(z, w)(u), B(z, w)(u), C(z, w)(u)  1 B(z, w)(u)C(z, w)(u) , [B(z, w)(u) + C(z, w)(u)], 2 1 + D(z, w)(u) and p −1 A(z, w)(u) = kFz (u) − Gw (u) + g(u) − h(u)k 1 + a2 eitan a, p −1 B(z, w)(u) = kz(u) − Fz (u) − g(u)k 1 + a2 eitan a, p −1 C(z, w)(u) = kw(u) − Gw (u) − h(u)k 1 + a2 eitan a , p −1 D(z, w)(u) = kz(u) − w(u)k 1 + a2 eitan a ,

Then, the system of Urysohn integral equations has a unique common solution in X.

Some Common Fixed Point Theorms for Self-Mappings ...

101

Proof. Define S, T : X → X by Sz = Fz + g, T z = Gz + h. Then, p −1 max kFz (u) − Gw (u) + g(u) − h(u)k∞ 1 + a2 eitan a , u∈[a,b] p −1 d(z, Sz) = max kz(u) − Fz (u) − g(u)k∞ 1 + a2 eitan a , u∈[a,b] p −1 d(w, T w) = max kw(u) − Gw (u) − h(u)k∞ 1 + a2 eitan a , u∈[a,b] p −1 d(z, w) = max kz(u) − w(u)k∞ 1 + a2 eitan a .

d(Sz, T w) =

u∈[a,b]

From assumption 0.9, for each u ∈ [a, b] we have

λ R(z, w)(u) s2 λ max {D(z, w)(u), B(z, w)(u), C(z, w)(u) s2  1 B(z, w)(u)C(z, w)(u) , [B(z, w)(u) + C(z, w)(u)] , , 2 1 + D(z, w)(u)

A(z, w)(u) -

which implies that max A(z, w)(u)

u∈[a,b]

-

λ max max {D(z, w)(u), B(z, w)(u), C(z, w)(u) s2 u∈[a,b] ff 1 B(z, w)(u)C(z, w)(u) [B(z, w)(u) + C(z, w)(u)] , 2 1 + D(z, w)(u)  λ max max D(z, w)(u), max B(z, w)(u) u∈[a,b] u∈[a,b] s2 1 , max C(z, w)(u), [ max B(z, w)(u) + max C(z, w)(u)] u∈[a,b] 2 u∈[a,b] u∈[a,b] ff maxu∈[a,b] B(z, w)(u) maxu∈[a,b] C(z, w)(u) , . 1 + maxu∈[a,b] D(z, w)(u) ,

-

Therefore, d(Sz, T w) -

λ max{d(z, w), d(z, Sz), d(w, T w), s2 1 d(z, Sz)d(w, T w) [d(w, Sz) + d(z, T w)] , }. 2 1 + d(z, w)

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Khaled Berrah, Oussaeif Taki Eddine and Abdelkrim Aliouche

Thus all the conditions of Theorem 3 with P = Q = IX are satisfied. Therefore, the system of Urysohn integral equations has a unique common solution in X.

4.2.

Application to Linear System

We give an application in linear system using the Corollary 3 in complete complex valued b-metric space (X = Cn , d2) where, d2 (z, w) =

" n X

#1 2

2

|zi − wi| + i|zi − wi |

i=1

2




.

Theorem 5. Let (X = Cn , d2 ) where, z = (z1 , . . ., zn )t ∈ X and w = (w1 , . . . , wn)t ∈ X, if β < n1 where, ( aij if, i 6= j and β = max {βij } , ∀1 ≤ i, j ≤ n. βij = aii + 1 if, i = j So, the following linear system of n equations and n unknowns AZ = B has a unique solution. 8 > >a11 z1 + a12 z2 + . . . + > >

> . > > : an1 z1 + an2 z2 + . . . +

a1n zn = b1 a2n zn = b2

ann zn = bn

0 B B ⇔B @

a11 a21 . .. an1

a12 a22

... ...

an2

...

10 a1n z1 a2n C B z2 CB CB . A @ .. ann zn

1

0

C B C B C=B A @

b1 b2 . .. bn

1 C C C. A

Where, z = (z1 , . . . , zn)t ∈ X and aij ∈ C, where, 1 ≤ i, j ≤ n and b1 , b2 , bn ∈ C Proof. Define T : X → X by T z = (A+I)Z −B. to prove that a linear system AZ = B has a unique solution, its enough to prove that T is a contraction. Since " n #1 X
2 2 2 |(T z)i − (T w)i| + i|(T z)i − (T w)i| d2 (T z, T w) = i=1

2 2  12   n n n X X X   = βij (zj − wj ) + i βij (zj − wj )  , j=1 j=1 i=1

Some Common Fixed Point Theorms for Self-Mappings ...

103

where βij =

(

aij if i 6= j aii + 1 if i = j

and β = max {βij } , ∀1 ≤ i, j ≤ n.

Then, d2 (T z, T w)

-

-

-

" `

n X i=1

nβ 2

max

1≤i,j≤n

´ 21

˛2 ˛ ˛2 !# 12 ! ˛ n n ˛X ˛ ˛X ˛ ˛ ˛ ˛ ˛ (zj − wj )˛ + i ˛ (zj − wj )˛ ˛ ˛ ˛ ˛ ˛ j=1

j=1

˛ n ˛2 ˛ n ˛2 !# 12 ˛X ˛ ˛X ˛ ˛ ˛ ˛ ˛ n ˛ (zj − wj )˛ + i ˛ (zj − wj )˛ ˛ ˛ ˛ ˛

"

j=1

j=1

˛2 ˛ n ˛2 !# 12 " ˛ n ˛X ˛ ˛X ˛ ˛ ˛ ˛ ˛ nβ ˛ (zj − wj )˛ + i ˛ (zj − wj )˛ ˛ ˛ ˛ ˛ j=1

=

2 βij

j=1

nβd2 (z, w).

So, we get finally d2 (T z, T w) - nβd2 (z, w) where β = max{|aij |, |aii + 1| ∀ 1 ≤ i, j ≤ n}. We conclude that T is a contraction mapping. By applying Corollary 3, the linear system has a unique solution.

ACKNOWLEDGMENTS The authors are thankful to the learned referees for their valuable comments which helped in bringing this paper to its present form.

R EFERENCES [1] Banach S., Sur les op´erations dans les ensembles abstraits et leur application aux e´ quations int´egrales, Fund. Math., 3 (1922), 133–181. [2] Branciari A., A fixed point theorem of BanachCaccippoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (12) (2000), pp. 31-37. [3] Dhage B.C., Generalized metric spaces with fixed point, Bull. Calcutta Math. Soc., 84 (1992), pp. 329-336.

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[4] Ghaler S., 2-metrische Raume und ihre topologische strukture, Math. Nachr., 26 (1963), pp. 115-148. [5] George A., Veeramani P., On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), pp. 395-399. [6] Hadzic O., Pap E., Fixed Point Theory in PM-Spaces, Kluwer Academic, Dordrecht, The Netherlands, (2001). [7] Huang L.G., Zhang X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), pp. 1468-1476. [8] Kelley J.L., General Topology, Van Nostrand Reinhold, New York., (1955) [9] Kunzi H.P.A., A note on sequentially compact quasi-pseudo-metric spaces, Monatsh. Math., 95 (1983), pp. 219-220. [10] Bakhtin I., The contraction mapping principle in quasimetric spaces, Func. An., Gos. Ped. Inst. Unianowsk., 30 (1989), 26–37. [11] Czerwik S., Contraction mappings in b-metric spaces, Acta Mat. Inf. Uni. Ostraviensis, 1 (1993), 5–11. [12] Czerwik S., Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263–276. [13] Rao K. P. R., Swamy P. R. and Prasad J. R., A common fixed point theorem in complex valued b-metric spaces, Bull. Maths. Stat. res., 1 (2013), 1–8. [14] Azam A., Fisher B. and Khan M., Common fixed point theorems in complex valued metric spaces, Numer. Func. Anal. Opt., 32 (2011), 243–253. [15] Aleksic S., Kadelburg Z., Mitrovic Z. D., et al. A new survey: Cone metric spaces, ArXiv Preprint, ArXiv: 1805.04795. [16] George R., Nabwey H. A., Rajagopalan R., et al. Rectangular cone bmetric spaces over Banach algebra and contraction principle, Fixed Point Theory Appl., 2017 (2017), 14. [17] Huang H., Deng G. and Radenovi S., Some topological properties and fixed point results in cone metric spaces over Banach algebras, Positivity, 23 (2019), 21–34.

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[18] Vetro F. and Radenovi S., Some results of Perov type in rectangular cone metric spaces, J. Fix. Point Theory A., 20 (2018), 41. [19] Berrah K., Aliouche A. and Oussaeif T., Common fixed point theorems under Patas contraction in complex valued metric spaces and an application to integral equations, Bol. Soc. Mat. Mex., (2019), 2296–4495. [20] Bhatt S., Chaukiyal S. and Dimri R. C., Common fixed point of mappings satisfying rational inequality in complex valued metric space, Int. J. Pure. Appl. Maths., 73 (2011), 159–164. [21] Dubey A. K. and Tripathi M., Common Fixed Point Theorem in Complex Valued b-Metric Space for Rational Contractions, J. Inf. Math. Sci., 7 (2015), 149–161. [22] Manro S., Some Common Fixed Point Theorems in Complex-Valued Metric Spaces Using Implicit Relation, Inter. J. Anal. Appl., 2 (2013), 62–70. [23] Manro S. and Krishnapada D., Some common fixed point theorems using CLRg property in Complex valued metric spaces, Gazi U. J. Sci., 27 (2014), 735–738 [24] Sintunavarat W. and Kumam P., Generalized common fixed point theorems in complex valued metric spaces and applications, J. Inequal. Appl., 2012 (2012), 84. [25] Berrah K., Aliouche A. and Oussaeif T., Applications and theorem on common fixed point in complex valued b-metric space, AIMS. Maths., 4(3) (2019). [26] Verma R. K. and Pathak H. K., Common fixed point theorems for a pair of mappings in complex-Valued metric spaces, J. Maths. Comput. Sci., 6 (2013), 18–26. [27] Jungck G., Common fixed points for noncontinuous nonself maps on nonmetric spaces, 4 (1996), 199–215. [28] Rouzkard F. and Imdad M., Some common fixed point theorems on complex valued metric spaces, Comput. Math. Appl., 64 (2012), 1866–1874.

In: An In-Depth Guide to Fixed-Point Theorems ISBN: 978-1-53619-565-1 c 2021 Nova Science Publishers, Inc. Editors: R. Sharma and V. Gupta

Chapter 5

B EST P ROXIMITY P OINT T HEOREMS U SING S IMULATION F UNCTIONS 1

Sanjay Mishra1,∗, Rashmi Sharma1,† and Manoj Kumar2,‡ Department of Mathematics, Lovely Professional University, Phagwara, Punjab, India 2 Department of Mathematics, Baba Mastnath University, Asthal Bohar, Rohtak, Haryana, India

Abstract The purpose of this chapter is to introduce the proximal contraction of first kind and second kind with respect to ζ, called simulation function which generalize several known types of contractions. Secondly, we prove certain fixed point theorems using simulation functions in complete Metric spaces. There is also an illustration provided for supporting our results.

1.

INTRODUCTION

Since 1922, the famous Banach fixed point theorem has intrigued many researchers with fixed point theory. The literature includes several interesting variations and versions of the above observation. However, the mappings used ∗

Email address: [email protected] (Corresponding author). Email address: [email protected]. ‡ Email address: [email protected]. †

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with both of these observations are by themselves mappings. The findings set out in this article guarantee that optimal approximate answers are possible for these fixed-point problems where no answer is required. Fixed point theory is an invaluable method to solve a T x = x equation for a T mapping specified on a metric space sub-set, a normal linear space, or a vector topology space. Several types of contractions can be evaluated for the presence of the best point of similarity in ([1], [2], [3], [4], [5], [6]). Many of the strongest proximity point theorems have been established for set weighted mappings in ([1], [7], [8], [9], [10], [11], [12]). The strongest point theorems of similarity were obtained for fairly non-expansive mappings by Anthony Eldred et al. ([13]). A classic best Fan approximation theorem ([14]) asserts that if A is a non-empty convex subset of a locally convex Hausdroff topology vector space X with a semi-norm p. Several versions of the Fan’s Theorem include Prolla([15]), Reich([16]), Sehgal and Singh([17], [18]. In comparison, Vetrivel et al.([19]) provided a coherent framework to furnish such interesting observations. In ([20]) a possible proximity point theorem was investigated for contractive non-self-mappings. The aim of this paper is to expand and generalize the class of first and second forms of proximal contraction with simulation function that is distinct in the literature from another form. We also provide an example to explain our key findings. The conclusions of this paper are generalizations of Basha’s findings in [20] and the best point theorems of proximity in the literature.

2.

P RELIMINARIES

Certain common theorems for optimal proximity points were studied in ([20], [21]). Khojasteh, Shukla and Radenovic ([22]) have recently implemented a new mapping form, named simulation functions. Later, the concept of simulation functions was marginally changed by Argoubi, Samet, and Vetro([23]). Definition 1. ([23]) A simulation function is a mapping ζ : [0, ∞) × [0, ∞) → R satisfying the following conditions: (ζ1 ) ζ(t, s) < s − t for all t, s > 0; (ζ2 ) if {tn } and {sn } are sequences in (0, ∞) such that limn→∞ tn = limn→∞ sn = l ∈ (0, ∞), then limn→∞ ζ(tn , sn ) < 0. Note that the classes of all simulation functions ζ : [0, ∞) × [0, ∞) → R denote by Z.

Best Proximity Point Theorems Using Simulation Functions

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Definition 2. ([20]) A mapping T : A → B is said to be a proximal contraction of first kind if there exists α ∈ [0, 1) such that, for all a, b, x, y ∈ A, ( d(a, T x) = d(A, B) d(b, T y) = d(A, B), implies that, d(a, b) ≤ αd(x, y). Definition 3. ([20]) A mapping T : A → B is said to be a strong proximal contraction of first kind if there exists α ∈ [0, 1) such that, for all a, b, x, y ∈ A, ( d(a, T x) = d(A, B) d(b, T y) = d(A, B), implies that, d(a, b) ≤ αd(x, y) + (β − 1)d(A, B)). Definition 4. ([20]) A mapping T : A → B is said to be a proximal contraction of second kind if there exists α ∈ [0, 1) such that, for all a, b, x, y ∈ A, ( d(a, T x) = d(A, B) d(b, T y) = d(A, B), implies that, d(a, b) ≤ αd(T x, T y). The necessary condition for a self-mapping T to be a proximal contraction of the second kind is that d(T 2 x, T 2 y) ≤ αd(T x, T y) for all x, y in the domain of T . Definition 5. ([20]) Given T : A → B and an isometry g : A → A, the mapping T is said to preserve isometric distance with respect to g if d(T gx, T gy) = (T x, T y) for all x, y ∈ A.

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MAIN R ESULTS

In this section, we prove common fixed point theorems for self-mappings in a complete complex valued b-metric spaces using partial rational contraction condition and we give some examples. In this section, we present the notions of the first and second forms of generalized proximal contraction mappings with simulation method that vary from another type in the literature. Assume that M and N are two non empty sets of a complete metric space X, d. The following notations followed : d∗ (M, N ) = inf {d∗ (a, b) : a ∈ M and b ∈ N } M0 = {a ∈ M : d∗ (a, b) = d∗ (M, N ) for some b ∈ N } N0 = {b ∈ N : d∗ (a, b) = d∗ (M, N ) for some a ∈ M } Definition 6. A mapping of F : M → N is considered to be a first-class proximal contraction if a non-negative integer is α < 1 such that for all w1 , w2 , a1 , a2 in M with respect to ζ and ζ ∈ Z if ( d∗ (w1 , F a1 ) = d∗ (M, N ) d∗ (w2 , F a2 ) = d∗ (M, N ), implies that, 0 ≤ ζ(d∗ (w1 , w2), αd∗ (a1 , a2 )). Definition 7. A mapping of F : M → N is said to be a strong proximal contraction of first class if there exists a non-negative integer α < 1 and β < 1 such that for all w1 , w2 , a1 , a2 in M with respect to ζ and ζ ∈ Z if ( d∗ (w1 , F a1 ) ≤ βd∗ (M, N ) d(w2 , F a2 ) ≤ βd∗ (M, N ), implies that, 0 ≤ ζ(d∗ (w1 , u2 ), (αd∗(a1 , a2 ) + (β − 1)d∗ (M, N )).

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Definition 8. A mapping F : M → N is said to be a second class proximal contraction if a non-negative integer occurs α < 1 such that for all w1 , w2, a1 , a2 in M with respect to ζ and ζ ∈ Z if ( d∗ (w1 , F a1 ) = d∗ (M, N ) d∗ (w2 , F a2 ) = d∗ (M, N ), implies that, 0 ≤ ζ(d∗ (u1 , u2 ), αd∗(a1 , a2 )). Each time a1 , a2 , w1 and w2 are elements in M which satisfy the requirement that d∗ (w1 , F a1 ) = d∗ (M, N ) and d∗ (w2 , F a2 ) = d∗ (M, N ). The precondition for a F self-map to be a proximal second-class contraction is 0 ≤ ζ(d∗ (F 2 a1 , F 2 a2 ), αd∗ (F a1 , F a2 )), for all a1 and a2 in the domain of F . We present and prove certain results: Theorem 1. Let X be a complete metric space with respect to ζ and ζ ∈ Z. Let M and N be non-empty, with X closed subsets so M is equally compact to N . Suppose M0 and N0 are non-empty, instead. Suppose F : M → N and g0 : M → M satisfied this: 1. F is the second type of persistent proximal contraction. 2. g0 reflects an isometry. 3. F M0 is contained in N0 . 4. M0 is contained in g0 N0 . 5. F retains isometric variance in addition to g0 . Therefore an item a exists in M such that d∗ (g0 a, F a) = d∗ (M, N ). In addition, if a∗ is another variable that holds the preceding assumption for, then F a and F a∗ are similar.

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Proof. Let a0 be a fixed point in M0 . Because F M0 is in N0 and M0 is in g0 M0 , there is an item a1 in M0 that exists d∗ (g0 a1 , F a0 ) = d∗ (M, N ). Again, because F a1 is an item of F M0 that is contained in N0 and M0 in g0 M0 , it follows that a2 is contained in M0 d∗ (g0 a2 , F a1 ) = d∗ (M, N ). Will start this phase. Having selected an in M0 , it is possible to find am+1 in M0 such that d∗ (g0 am+1 , F am ) = d∗ (M, N ). For any positive integer m then F M0 is in M0 and N0 is in g0 M0 . As F is a second type of proximal contraction, 0 ≤ζ(d∗ (g0 am+1 , F am ), αd∗(F am , F am−1 )) 0, then 𝑞 𝑝 = 𝜉 satisfies for 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝜀 > 0, 𝜖

𝜖

𝜉(𝑥, 𝑧) < 2𝑠 and 𝜉(𝑧, 𝑦) < 2𝑠, then 𝜉(𝑥, 𝑦) < 𝜖.

(1)

Proof. See Lemma 1 and Remark 2. Let 𝜉 satisfy (i), (ii) and (vi). Then, as in Frink [6], one can define 𝑑 satisfying (vii) and 𝑑 ~ 𝜉 (‘‘~’’ - means the equivalence relation). Also as in [6], p.134, one can prove that 𝑑(𝑎, 𝑏) ≤ 2𝑑(𝑎, 𝑥1 ) + 4𝑑( 𝑥1 , 𝑥2 )+ . . . + 4𝑑( 𝑥𝑛−1 , 𝑥𝑛 ) + 2𝑑(𝑥𝑛 , 𝑏),

(2)

for all chains 𝑎 = 𝑥0 , 𝑥1 , . . . , 𝑥𝑛 = 𝑏, in X. Let 𝐷(𝑎, 𝑏): = 𝑖𝑛𝑓{∑𝑛 𝑖=1 𝑑( 𝑥𝑖−1 , 𝑥𝑖 ); 𝑎 = 𝑥0 , 𝑥1 , … , 𝑥𝑛 = 𝑏}

(3)

Then, as in [6], one has, by (2) 1 𝑑(𝑎, 𝑏) 4

≤ 𝐷(𝑎, 𝑏) ≤ 𝑑(𝑎, 𝑏)

for a, b ∈ X. Lemma 2. We have 𝐷~𝑑~𝜉,

(4)

On B - Metric Spaces and Their Completion

125

𝐷 is a metric in 𝑋. Proof. See (4), (3) and (2). See also [6]. Remark 4. The formula (3) gives the method of producing many metrics from 𝑏 - metric 𝑞 𝑝 , 𝑝 > 0. In [1] there is only one metric build from a given 𝑏-metric (𝑝 is only one). Lemma 3. Let 𝜉 satisfy (i), (ii) and (vi) and let 𝑠 > 1 be fixed. Then 𝜉(𝑥, 𝑦) ≤ 𝑠[𝜉(𝑥, 𝑧) + 𝜉(𝑧, 𝑦)] for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Proof. If 𝑥 = 𝑦, then the thesis is obvious. Assume that 𝑥 ≠ 𝑦, 𝑥, 𝑦 ∈ 𝑋. Take (𝑧 ∈ 𝑋) 𝜖1 : = 𝑠𝜉(𝑥, 𝑧), 𝜖2 : = 𝑠𝜉(𝑧, 𝑦), 𝜖 ∶= 𝑚𝑎𝑥[𝜖1 , 𝜖2 ], so 𝜖 > 0. By (vi) one has 𝜉(𝑥, 𝑦) < 𝜖 ≤ ϵ1 + ϵ2 ≤ 𝑠[𝜉(𝑥, 𝑧) + 𝜉(𝑧, 𝑦)], i.e., the thesis. Remark 5. The distance function 𝜉 is in fact a 𝑏 −metric with 𝑠 > 1. Remark 6. For 𝑠 = 1 Lemma 3 is not true.

3. COMPLETION OF B-METRIC SPACES Proposition Every 𝑏 − metric space (𝑋, 𝑞) has a completion.

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Proof. Let (𝑋, 𝑑) be a metric space, with the metric 𝑑 induced by 𝑞 (see e.q. [6], [1]), equivalent to 𝑞 𝑝 , 𝑝 > 0. Let 𝐼: (𝑋, 𝑞) → (𝑋, 𝑑) be the identity mapping. Then (𝑋, 𝑑) has a completion 𝑌 (see e.q. [7]), and this complete metric space 𝑌 we consider as the completion of the 𝑏 − metric space (𝑋, 𝑞). Note that such space 𝑌 is only one, in accuracy to the isometry, in the sense, that any other such space is isometric to 𝑌 . Clearly, we have that if {𝑥𝑛 } is a Cauchy sequence in (𝑋, 𝑞), then {𝑦𝑛 } where 𝑦𝑛 = 𝐼𝑥𝑛 , 𝐼: (𝑋, 𝑞) → (𝑋, 𝑑), 𝐼 − identity map, is also a Cauchy sequence in (𝑋, 𝑑), by the equivalence relation 𝑑 ∼ 𝑞 𝑝 , 𝑝 > 0.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]

Paluszynski, M. and Stempak, K. On quasi-metric and metric spaces, Proc. Amer. Math. Soc., 137(2009), 4307 - 4312. Czerwik, S. Contraction mappings in b-metric spaces, Acta Math. Informatica Universi- tatis Ostraviensis, 1(1993), 5 - 11. Czerwik, S. Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46(1998), no. 2, 263 - 276. Cobzas, S. and Czerwik, S. The completion of generalized b-metric spaces and fixed points, Fixed Point Theory (accepted). Kirk, W. and Shahzad, N. Fixed Point Theory in Distance Spaces, Cham: Springer, 2014. Frink, A. H. Distance functions and the metrization problem, Bull. Amer. Math. Soc., 43(1937), 13 - 142. Lusternik, L. A. and Sobolew, V. J. Functional analysis (in Polish), PWN, Warsaw, 1959. An, T. V., Tuyen, L. Q. and Dung, N. V. Answers to Kirk - Shahzad’s questions on strong b-metric spaces, Taiwanese J. Math., 20(2016), 5, 1175 - 1184.

In: An In-Depth Guide to Fixed-Point Theorems ISBN: 978-1-53619-565-1 Editors: R. Sharma and V. Gupta © 2021 Nova Science Publishers, Inc.

Chapter 7

ON BANACH CONTRACTION PRINCIPLE IN GENERALIZED B-METRIC SPACES Stefan Czerwik Institute of Mathematics, Silesian University of Technology, Gliwice, Poland

ABSTRACT Motivated from the work of Nadler, Frink and Paluszyński, we present some fixed point theorems for (𝜖, 𝛾) - uniformly locally contractive mappings and 𝜖 − chainable generalized 𝑏 - metric spaces, which are some extensions of a famous Banach contraction principle for complete metric spaces.

1. INTRODUCTION The aim of this chapter is to prove some fixed point results for mappings in generalized b - metric spaces, extending the well-known fixed point principle of Banach [2]. This study is inspired from the work of Nadler ([2], [3]), Frink [4], Paluszyński [5] and Aimar [6]. 

Corresponding Author’s Email: [email protected].

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A 𝑏 - metric on a nonempty set 𝑋 is a function 𝑑: 𝑋 × 𝑋 → [0, ∞), satisfying the conditions: 𝑑(𝑥, 𝑦) = 0 ⇔ 𝑥 = 𝑦, 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥), 𝑑(𝑥, 𝑦) ≤ 𝑠[𝑑(𝑥, 𝑧 + 𝑑(𝑧, 𝑦)], for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and fixed number 𝑠 ≥ 1. The pair (𝑋, 𝑑) is called a 𝑏 −metric space (see [7], [8], [9], [10]). If d is a function with values in [0, ∞), then it is called a generalized 𝑏 − metric (see [2], [7], [10], [11]). A generalized 𝑏 − metric space (𝑋, 𝑑) is said to be complete, if and only if every Cauchy sequence in 𝑋 is 𝑑-convergent to a point in 𝑋. A sequence {𝑥𝑛 }∞ 𝑛=1 of points of X is said to be an iterative sequence of 𝑇: 𝑋 → 𝑋 at 𝑥0 ∈ 𝑋 , if and only if 𝑥𝑛 = 𝑇 (𝑥𝑛−1 ) for each 𝑛 ∈ 𝑁 (the set of all natural numbers). A function 𝑇: 𝑋 → 𝑋 is called an (𝜖, 𝛼) - uniformly locally contractive mapping, where 𝜖 > 0 and 0 ≤ 𝛼 < 1, if and only if (see [2], [12]) 𝑑[𝑇 (𝑥), 𝑇 (𝑦)] ≤ 𝛼𝑑(𝑥, 𝑦) for all 𝑥, 𝑦 ∈ 𝑋, 𝑑(𝑥, 𝑦) < 𝜖.

(1)

A generalized 𝑏 − metric space (𝑋, 𝑑) is called an 𝜖 − chainable, for 𝜖 > 0, if and only if for 𝑥, 𝑦 ∈ 𝑋 with 𝑑(𝑥, 𝑦) < ∞ there exists an 𝜖 − chain from 𝑥 to y (i.e., a finite set of points 𝑥 = 𝑧0 , … , 𝑧𝑘 = 𝑦 with d(𝑧𝑛−1, 𝑧𝑛 ) < 𝜖 for 𝑛 = 1, … , 𝑘). We shall be using the following Proposition ([4], [5], [7]) Let 𝑑 be a 𝑏-metric on a nonempty set 𝑋. If the number 𝑝 ∈ (0,1] is given by the equation (2𝑠)𝑝 = 2, then the mapping 𝑞: 𝑋 × 𝑋 → [0, ∞) defined by 𝑞(𝑥, 𝑦) ≔ 𝑖𝑛𝑓{ ∑𝑛𝑖=1 𝑑𝑝 (𝑥𝑖−1 , 𝑥𝑖 ): 𝑥 = 𝑥0 , … , 𝑥𝑛 = 𝑦}, is a metric on 𝑋, satisfying the inequalities

(2)

On Banach Contraction Principle in Generalized B-Metric Spaces 𝑞(𝑥, 𝑦) ≤ 𝑑𝑝 (𝑥, 𝑦) ≤ 4𝑞(𝑥, 𝑦),

129 (3)

for all x, y ∈ X. Finally, by ‘‘∼’’ we denote the equivalence relation in X.

2. MAIN RESULTS Theorem 1 (Banach). Let (𝑋, 𝑑) be a complete 𝑏 - metric space. If 𝑇: 𝑋 → 𝑋 satisfies 𝑑[𝑇(𝑥), 𝑇(𝑦)] ≤ 𝛼𝑑(𝑥, 𝑦), 𝑥, 𝑦 ∈ 𝑋

(4)

0 ≤ 𝛼 < 1,

(5)

and

𝑇 has exactly one fixed point 𝑢 ∈ 𝑋 𝑇 𝑛 (𝑥) → 𝑢, 𝑛 → ∞, 𝑥 ∈ 𝑋. Proof. The proof is different than the presented in [7]. Consider the cases: Case 1. 𝛼 = 0, (the proof is trivial); Case 2. 0 < 𝛼 < 1. Let ∈ 𝑋, 𝑥𝑛 = 𝑇 𝑛 (𝑥), 𝑛 ∈ 𝑁 ; then 𝑑[𝑇(𝑥), 𝑇 2 (𝑥)] ≤ 𝛼𝑑(𝑥, 𝑇(𝑥)) 𝑑[𝑇 𝑛 (𝑥), 𝑇 𝑛+1 (𝑥)] ≤αn 𝑑(𝑥, 𝑇(𝑥)), 𝑥 ∈ 𝑋, 𝑛 ∈ 𝑁.

(6)

By [7], Theorem 2.2 (see also [4], [5], [6]) for 0 < p ≤ 1 such that (2𝑠)𝑝 = 2, there exists a metric 𝑞 ∶ 𝑋 × 𝑋 → [0, ∞), 𝑞 ∼ 𝑑𝑝 , given by 𝑞(𝑥, 𝑦) ≔ 𝑖𝑛𝑓{ ∑𝑛𝑘=1 𝑑𝑝 (𝑥𝑘−1 , 𝑥𝑘 ): 𝑥 = 𝑥0 , … , 𝑥𝑛 = 𝑦} .

(7)

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clearly, 𝑞 is symmetric, satisfies the triangle inequality and 𝑞 ≤ 𝑑𝑝 . One has 𝑞(𝑥, 𝑥1 ) ≤ 𝑑𝑝 (𝑥, 𝑥1 ), 𝑞(𝑥1 , 𝑥2 ) ≤ [𝛼𝑑(𝑥, 𝑇 (𝑥)]𝑝 , and hence 𝑞[𝑇 𝑛 (𝑥), 𝑇 𝑛+1 (𝑥)] ≤ 𝜉 𝑛 𝑎, 𝑛 ∈ 𝑁, 𝜉 = 𝛼 𝑝 < 1, 𝑎 = 𝑑 𝑝 (𝑥, 𝑇(𝑥)). (8) Since 𝑞 ∼ 𝑑 p , for 𝑛, 𝑚 ∈ 𝑁 by (5) 𝑞(𝑥𝑛 , 𝑥𝑛+𝑚 ) ≤ 𝑞[𝑇 𝑛 (𝑥), 𝑇 𝑛+𝑚 (𝑥)] ≤ 𝑞(𝑥𝑛 , , 𝑥𝑛+1 )+. . . +𝑞(𝑥𝑛+𝑚−1 , 𝑥𝑛+𝑚 ) ≤ 𝜉 𝑚 𝑎 + . . . + 𝜉 𝑛+𝑚−1 𝑎 ≤ (1 − 𝜉)−1 𝜉 𝑛 𝑎 → 0, 𝑛 → ∞. 𝑝 Therefore {𝑥𝑛 }∞ 𝑛=1 , is a Cauchy sequence in (𝑋, 𝑞) and since 𝑞~𝑑 , in (𝑋, 𝑑) as well. By the completeness of (𝑋, 𝑞) and the Banach contraction principle, 𝑥𝑛 → 𝑢 ∈ (𝑋, 𝑞). We have also, by (4), that 𝑇 𝑛 (𝑥) → 𝑇(𝑢), i. e., 𝑇(𝑢) = 𝑢. The uniqueness part is trivial. This completes the proof.

Remark 1. One has the estimation 𝑞(𝑇 𝑛 (𝑥), 𝑢) ≤ (1 − 𝜉)−1 𝜉 𝑛 𝑑 𝑝 (𝑥, 𝑇(𝑥)), 𝑥 ∈ 𝑋, 𝑚 ∈ N, 0 < 𝑝 ≤ 1.

(9)

Theorem 2. Let (𝑋, 𝑑) be a generalized complete b - metric space and let 𝑥0 ∈ 𝑋. Let 𝑇: 𝑋 → 𝑋 be an (𝜖, 𝜆) - uniformly locally contractive mapping. Then the following alternative holds: either (2.1) for each iterative sequence {𝑥𝑛 }∞ 𝑛=1 of 𝑇 at 𝑥0 ∈ 𝑋 , 𝑑(𝑥𝑛−1 , 𝑥𝑛 ) ≥ 𝜖, 𝜖 > 0, 𝑛 ∈ 𝑁,

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or (2.2) the iterative sequence {𝑥𝑛 }∞ 𝑛=1 of 𝑇 at 𝑥0 ∈ 𝑋, converges to a fixed point of 𝑇 . Proof. Assume that (2.1) does not hold. So, there exists m ∈ N such that 𝑑(𝑥𝑚 , 𝑥𝑚+1 ) < 𝜖, for 𝜖 > 0, 𝜖 - fixed. One has 𝑑[𝑇(𝑥𝑚 ), 𝑇(𝑥𝑚+1 )] ≤ 𝜆𝑑(𝑥𝑚 , 𝑥𝑚+1 ) < 𝜆𝜖 < 𝜖.

(10)

Similarly, by (10), 𝑑[𝑇 (𝑥𝑚+1 ), 𝑇 (𝑥𝑚+2 )] ≤ 𝜆𝑑(𝑥𝑚+1 , 𝑥𝑚+2 ) < 𝜆2 𝑑(𝑥𝑚 , 𝑥𝑚+1 ) < 𝜆2 𝜖 < 𝜖, and consequently, by induction principle, for 𝑛 ∈ 𝑁, 𝑑[𝑇 𝑛 (𝑥𝑚 ), 𝑇 𝑛 (𝑥𝑚+1 )] < 𝜆𝑛 𝑑(𝑥𝑚 , 𝑥𝑚+1 ) < 𝜆𝑛 𝜖.

(11)

By Paluszyński, Stempak [5], there exists a generalized complete metric 𝑞: 𝑋 × 𝑋 → [0, ∞), given by 𝑞(𝑥, 𝑦): = = 𝑥0 , 𝑥1 , … , 𝑥𝑛 = 𝑦, 𝑥, 𝑦 ∈ 𝑋, 𝑑(𝑥, 𝑦) < ∞}, +∞, 𝑖𝑓 𝑑(𝑥, 𝑦) = ∞,

𝑖𝑛𝑓{∑𝑛𝑖=1 𝑑𝑝 (𝑥𝑖−1 , 𝑥𝑖 ): 𝑥

{

where 0 < 𝑝 ≤ 1 is such that (2𝑠)𝑝 = 2. One can prove that 𝑞~𝑑𝑝 (see also [5]) Clearly, q is symmetric, satisfies the triangle inequality and 𝑞 ≤ 𝑑𝑝 .

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Stefan Czerwik One has also 𝑞(𝑥𝑚+1 , 𝑥𝑚+2 ) ≤ 𝑑𝑝 (𝑥𝑚+1 , 𝑥𝑚+2 ) ≤ [𝜆𝑑(𝑥𝑚 , 𝑥𝑚+1 )]𝑝 ≤ 𝜆𝑝 𝑑𝑝 (𝑥𝑚 , 𝑥𝑚+1 ) = 𝜉𝑧,

where 𝜉 = 𝜆𝑝 < 1, 𝑧 = 𝑑 𝑝 (𝑥𝑚 , 𝑥𝑚+1 ).

Similarly, 𝑞(𝑇 2 (𝑥𝑚 ), 𝑇 2 (𝑥𝑚+1 )) ≤ 𝜉 2 𝑧,

and, by induction. 𝑞(𝑇 𝑛 (𝑥𝑚 ), 𝑇 𝑛 (𝑥𝑚+1 ) ≤ 𝜉 𝑛 𝑧, 𝑛 ∈ 𝑁.

(12)

Moreover, 𝑞(𝑥𝑛 , 𝑥𝑛+𝑘 ) = 𝑞[𝑇 𝑛 (𝑥𝑚 ), 𝑇 𝑛+𝑘−1 (𝑥𝑚+1 )] ≤ 𝑞(𝑥𝑛 , 𝑥𝑛+1 )+. . . +𝑞(𝑥𝑛+𝑘−1 , 𝑥𝑛+𝑘 ) ≤ 𝜉 𝑛 𝑧 + . . . + 𝜉 𝑛+𝑘−1 𝑧 ≤ (1 − 𝜉)𝜉 𝑛 𝑧.

Hence the sequence {𝑥𝑛 }∞ 𝑛=1 is a Cauchy sequence in a generalized complete metric space (𝑋, 𝑞), so because 𝑞 ~ 𝑑 𝑝 , also in (𝑋, 𝑑𝑝 ). Thus 𝑥𝑛 → 𝑢 ∈ (𝑋, 𝑞) and 𝑑 𝑝 (𝑥𝑛 , 𝑢) < 𝜖 for 𝑛 sufficiently large. We also have 𝑞(𝑥𝑚+1 , 𝑇(𝑢)) = 𝑞[𝑇(𝑥𝑛 ), 𝑇(𝑢)] ≤ 𝑑 𝑝 [𝑇(𝑥𝑛 ), 𝑇(𝑢)] ≤ 𝜆𝑝 𝑑 𝑝 (𝑥𝑛 , 𝑢),

i.e., 𝑥𝑛 → 𝑇(𝑢); therefore 𝑇(𝑢) = 𝑢. The proof is completed. Theorem 3. Let (X, d) be a generalized complete b - metric space and 𝑥0 ∈ 𝑋. Let T: X → X be a contraction mapping 𝑑[𝑇 (𝑥), 𝑇 (𝑦)] ≤ 𝛼𝑑(𝑥, 𝑦), 𝑥, 𝑦 ∈ 𝑋, 𝑑(𝑥, 𝑦) < ∞,

(13)

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and 0 ≤ 𝛼 < 1

(14)

Then the following alternative holds: either a) for the iterative sequence {𝑥𝑛 }∞ 𝑛=1 of 𝑇 at 𝑥0 ∈ 𝑋, 𝑑(𝑥𝑛−1 , 𝑥𝑛 ) = ∞, for each 𝑛 ∈ 𝑁 or b) the iterative sequence {𝑥𝑛 }∞ 𝑛=1 , of 𝑇 at 𝑥0 ∈ 𝑋 converges to a fixed point 𝑇. Proof. Let a) does not hold. Then the iterative sequence {𝑥𝑛 }∞ 𝑛=1 of 𝑇 at 𝑥0 ∈ 𝑋 has the property that 𝑑(𝑥𝑁−1 , 𝑥𝑁 ) < ∞ for some 𝑁 ≥ 1. Take 𝜖 > 0 such that 𝑑(𝑥𝑁−1 , 𝑥𝑁 ) < 𝜖. Therefore 𝑇 is an (𝜖, 𝛼) −uniformly locally contractive mapping. Consequently 𝑇 satisfies (2.2) of Theorem 2, which is just b) of our theorem, and the proof is completed. Theorem 4. Let (X, d) be a complete 𝜖 − chainable generalized 𝑏-metric space and let 𝑥0 ∈ 𝑋. If 𝑇: 𝑋 → 𝑋 be an (𝜖, 𝛼) − uniformly contractive mapping, then the following alternative holds: either c) for the iterative sequence {𝑥𝑛 }∞ 𝑛=1 of 𝑇 at 𝑥0 ∈ 𝑋, 𝑑(𝑥𝑛−1 , 𝑥𝑛 ) = ∞, each 𝑛 ∈ 𝑁 or d) the iterative sequence {𝑥𝑛 }∞ 𝑛=1 , of 𝑇 at 𝑥0 ∈ 𝑋 converges to a fixed point 𝑇. Proof. Let c) does not hold. Define 𝑑𝜖 : 𝑋 → [0, ∞) by (see [5]) 𝑑𝜖(𝑥, 𝑦): 𝑛

𝑖𝑛𝑓 {∑ ={ 𝑖=1

𝑑𝑝 (𝑥𝑖−1 , 𝑥𝑖 ): 𝑥 = 𝑥0 , 𝑥1 , … , 𝑥𝑛 = 𝑦, 𝑖𝑠 𝑎𝑛 𝜖 − 𝑐ℎ𝑎𝑖𝑛 }, 𝑓𝑟𝑜𝑚 𝑥 𝑡𝑜 𝑦, 𝑓𝑜𝑟 𝑑(𝑥, 𝑦) < ∞, ∞, 𝑖𝑓 𝑑(𝑥, 𝑦) = ∞,

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where (2𝑠)𝑝 = 2, 0 < 𝑝 ≤ 1. One can verify that dϵ is a generalized complete metric space, 𝑑𝜖 ~ 𝑑𝑝 . Let 𝑑(𝑥, 𝑦) < ∞. For 𝑥 = 𝑧0 , 𝑧1 , . . . , 𝑧𝑛 = 𝑦, 𝑑(𝑧𝑖−1 , 𝑧1 ) < 𝜖, 𝑖 = 1, . . . , 𝑛, 𝑑𝜖 [𝑇(𝑥), 𝑇(𝑦)] ≤ ∑𝑛𝑖=1 𝑑𝜖 [𝑇(𝑧𝑖−1 ), 𝑇(𝑧𝑖 )] ≤ 𝛼 𝑝 ∑𝑛𝑖=1 𝑑𝑝 (𝑧𝑖−1 , 𝑧𝑖 ) ≤ 𝛼 𝑝 𝑑𝜖 (𝑥, 𝑦) ≤ 𝜉𝑑𝜖 (𝑥, 𝑦), i.e., 𝑑𝜖 [𝑇(𝑥), 𝑇(𝑦)] ≤ 𝜉𝑑𝜖 (𝑥, 𝑦), 𝑥, 𝑦, ∈ 𝑋, 𝜉 = 𝛼 𝑝 < 1, 𝑑(𝑥, 𝑦) < ∞,

(15)

where (2𝑠)𝑝 = 2, 0 < 𝑝 ≤ 1. Since (𝑋, 𝑑𝜖 )is complete, our assertion follows from Theorem 3.

REFERENCES [1]

[2] [3] [4] [5] [6] [7] [8]

Banach, S. Sur les operations dans les ensembles abstroits at lear application aux equations integrals [On operations in abstract sets and application to integral equations], Fund. Math. 3(1922), 133–181. Covitz H. and Nadler, S. B. Jr., Multi - valued contraction mappings in generalized metric spaces, Israel J. Math. 8(1970), 5–11. Nadler, S. B. Jr., Multi-valued contraction mappings, Pacific J. Math., 30(1969) 415 - 487. Frink, A. H. Distance functions and the metrization problem, Bull. Amer. Math. Soc. 43(1937), 133–142. Paluszyński M. and Stempak, K. On quasi-metric spaces, Proc. Amer. Math. Soc. 137(12), (2009), 4307–4312. Aimar, H. Iaffei B. and Nitti, L. On the Macias - Segovia metrization of quasi-metric spaces, Rev. U. Mat. Argentina 41(2), (1998), 67–75. Cobzas S. and Czerwik, S. The completion of generalized b - metric spaces and fixed points, Fixed Point Theory, (to be published). Czerwik, S. Contraction mappings in b - metric spaces, Acta Math. Inform. Univ. Ostrav. 1(1993), 5–11.

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Czerwik, S. Nonlinear set-volumed contraction mappings in b - metric spaces, Atti Semin. Mat. Fis. Univ. Modena 46(1998), no.2, 263–276. [10] Luxemburg, W. A. J. On the convergence of successive approximations in the theory of ordinary differentia equations, Can. Math. Bull. 1(1958), 9–20. [11] Czerwik S. and Krl, K. Completion of generalized metric spaces, Indian J. Math. 58(2), (2016), 231–237. [12] Edelstein, M. An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12(1961), 7–10.

In: An In-Depth Guide to Fixed-Point Theorems ISBN: 978-1-53619-565-1 Editors: R. Sharma and V. Gupta © 2021 Nova Science Publishers, Inc.

Chapter 8

METRIC FIXED POINT THEORY IN CONTEXT OF CYCLIC CONTRACTIONS Ashish Kumar Department of Mathematics, Himalayan School of Science and Technology, Swami Rama Himalayan University, Dehradun, India

ABSTRACT Banach contraction principle (Bcp) has proven fruitful in the field of nonlinear analysis and continues to play a pivotal role in fixed point theory. It has been generalized and extended in several ways. Kirk, Srinivasan and Veeramani in 2003 studied cyclic contraction and presented a new approach to extend Banach contraction as well as other acclaimed nonlinear contractions of Edelstein, BoydWong, Caristi and Geraghty. The significance of cyclic contraction lies in the fact that unlike most of the contractive conditions, cyclic contraction has the distinction that it does not require the map under consideration to be continuous throughout the domain. In due course of time, it received substantial attention of mathematicians and researchers. Subsequently, a great deal of research work has been reported in this direction. The purpose of this chapter is to present a brief development of metrical fixed point theory in the context of cyclic contractions.



Corresponding Author’s Email: [email protected].

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Keywords: fixed point, best proximity point, cyclic contraction, cyclic   contraction, cyclic Meir-Keeler contraction, p-cyclic contraction, property UC, WUC property, W-WUC property, comparison function AMS Subject Classification: 47H10, 54H25

INTRODUCTION We attempt to present a brief development of metrical fixed point theory pertaining to cyclic maps. Best proximity theorems for cyclic maps have also been discussed. The whole chapter is divided into three sections. In section one cyclic fixed point theorems for single valued maps are presented. Section two is devoted for cyclic fixed point theorems concerning multivalued maps and hybrid maps which contain a pair of single valued and multivalued maps. Finally, section three is intended to discuss best proximity point theorems for single valued as well as multi valued maps.

1. SINGLE VALUED CYCLIC FIXED POINT THEOREMS Classical Banach contraction Principle (Bcp) is the fundamental result in fixed point theory and has vast range of applications in science and technology. Let (X, d) be a metric space. Bcp states that a self-map T of a complete metric space X admits a unique fixed point if T is a Banach contraction, that is, if T satisfies the condition:

d (Tx, Ty )  kd ( x, y ), x, y  X , 0  k 1.

(1.1)

Inspired by the elegant and constructive proof of Bcp a great deal of research work has been carried out by researchers and various generalizations and extensions of Bcp were obtained. For an excellent comparison of various contractive conditions one may refer to Rhoades [1].

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Throughout the chapter, let X denotes a metric space (X, d), R the set of real numbers, R  the set of positive real numbers, N the set of natural numbers and

M ( x, y )  max{ d ( x, y ) , d ( x, Tx), d ( y, Ty ) , [d ( x, Ty )  d ( y, Tx)] / 2}, for a pair of self maps of X,

M f , T ( x, y)  max{ d ( fx, fy) , d ( fx, Tx), d ( fy, Ty) }. Edelstein [2] generalized (1.1) and proved that a self map T of X satisfying the condition:

d (Tx, Ty )  d ( x, y ), x  y, x, y  X ,

(1.2)

has a unique fixed point provided X is compact. A remarkable generalization of (1.1) was obtained by replacing the constant k in (1.1) by a real valued function. To be precise, Boyd and Wong [3] used the following condition:

d (Tx, Ty )   (d ( x, y )),

(1.3)

where  :[0, )  R  is upper semi-continuous from the right and satisfies

0   (t )  t for t  0. Geraghaty [4] used the following condition:

d (Tx, Ty )   (d ( x, y )) d ( x, y ), x, y  X ,

(1.4)

where  :[0, )  [0,1) is such that  (t n ) 1 implies t n  0. Kirk et al. [5] adopted an entirely different approach to extend Bcp by formulating the concept of cyclic map and cyclic contraction. The remarkable aspect of a cyclic map is that it is not needed to be continuous to ensure the existence of a fixed point.

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In particular, let A and B be non-empty subsets of a metric space X. A map T : A  B  A  B is called cyclic if it satisfies the condition:

T ( A)  B and T ( B )  A.

(1.5)

It is called a cyclic contraction if condition (1.5) and the following condition are satisfied

d (Tx, Ty )  kd ( x, y ), 0  k  1,  x  A and y B.

(1.6)

The following result is essentially due to Kirk et al. [5]. Theorem 1.1. Let A and B be two non-empty closed subsets of a complete metric space X. Let T : X  X satisfies the conditions (1.5) and (1.6). Then T has a unique fixed point in A  B. The fundamental idea involved in the proof of Theorem 1.1 is, to generate a Cauchy sequence {T n ( x)} converging to a point z  X , then by the virtue of condition (1.6) an infinite number of terms of {T n ( x)} belong to A and an infinite number of terms belong to B, leading to z  A  B and hence A  B is non-empty. Use of conditions (1.5) and (1.6) make it easy to see that T : A  B  A  B restricted to A  B is a Banach contraction. Conclusion follows from the application of Bcp. In the same paper Kirk et al. [5] extended the concept of cyclic maps to a collection of finite non-empty sets X { Ai }ip1 , popularly known as cyclic representation of X with respect to the map T under consideration. Definition 1.1. Let X be a non-empty set, p a positive integer and T : X  X be a self-map of X. Then X  ip1 Ai is a cyclic representation of X with respect to T if the following conditions are satisfied

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141

Ai , i 1, 2, ., ., p are non-empty sets,

ii. T ( A1 )  A2      T ( Ap 1 )  Ap , T ( Ap )  A1 . Using the novel idea of cyclic representation, Kirk et al. [5] did the pioneering work and extended the celebrated results of Edelstein [2], Boyd and Wong [3], Geraghty [4], Caristi [6] including Bcp. The point of departure from the previous results again, like Theorem 1.1, lies in the fact that in Kirk et al. [5] formulation the condition of continuity of map under consideration is muted. This was a significant development in the domain of metrical fixed point theory and was qualified to have enough potential to open up the scope for a new direction of study. For the sake of completeness we cite here the results obtained by Kirk et al. [5]. In this series the following is the extension of Bcp. Theorem 1.2 [5]. Let { A}ip1 be non-empty closed subsets of a complete metric space, and suppose T :  ip1 Ai  ip1 Ai satisfy the following conditions (where Ap 1  A1 ) :

T ( Ai )  Ai 1 for 1  i  p,

(1.7)

d (Tx, Ty)  kd ( x, y ), 0  k 1,  x  Ai and y Ai 1 for 1  i  p. (1.8) Then T has a unique fixed point. The following is the extension of Edelstein’s [2] theorem. Theorem 1.3 [5]. Let { A}ip1 be non-empty closed subsets of a complete metric space, at least one of which is compact, and suppose T :  ip1 Ai  ip1 Ai satisfy the conditions (1.7) (where Ap 1  A1 ) and

d (Tx, Ty )  d ( x, y ) whenever x  Ai , y  Ai 1 and x  y, (1  i  p ). (1.9) Then T has a unique fixed point.

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Replacing the constant k in Theorem 1.2 by a real valued function Kirk et al. [5] obtained the following extension of Geraghty’s theorem [4]. Theorem 1.4 [5]. Let { A}ip1 be non-empty closed subsets of a complete metric space, and suppose T :  ip1 Ai  ip1 Ai satisfy the conditions (1.7) (where

Ap 1  A1 ) and

d (Tx, Ty )   (d ( x, y))d ( x, y),  x  Ai , y Ai 1 for 1  i  p, (1.10) where  :[0, )  [0,1) is such that  (t n ) 1 implies t n  0. Then T has a unique fixed point. The following extension of Boyd and Wong result is essentially due to Kirk et al. [5]. Theorem 1.5 [5]. Let { A}ip1 be non-empty closed subsets of a complete metric space, and suppose T :  ip1 Ai  ip1 Ai satisfy the conditions (1.7) (where

Ap 1  A1 ) and

d (Tx, Ty)  (d ( x, y)) ,  x  Ai , y  Ai 1 for 1  i  p,

(1.11)

where  : R  [0, ) is upper semi-continuous from the right and satisfies

0   (t )  t for t  0. Then T has a unique fixed point. Now we cite the following theorem which extends Caristi’s theorem [6]. Theorem 1.6 [5]. Let {A}ip1 , Ap 1  A1 be non-empty closed subsets of a complete metric space, and suppose T : X  X satisfy the conditions (1.7) and

d ( x, Tx) i ( x) i 1 (T ( x)),  x  Ai for 1  i  p,

(1.12)

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where each  i : Ai  R is lower semi-continuous and bounded below. Then T has a fixed point. The concepts of cyclic map, cyclic representation and above results received substantial attention of researchers and subsequently fixed point theory witnessed a massive growth along these lines (see, for instance, [7]-[47], [50], [64]-[76], [82], [83], [85]-[91]). Immediately after Kirk et al. [5] paper, Rus [7] used the concept of fixed point structures to generalize the results obtained by Kirk et al. [5] some important lemma were also obtained by Rus [7]. Petrusel [8] used the fixed point structures technique and a lemma of Rus [7] to obtain the periodic points for various maps such as contractive maps, nonexpansive maps, generalized contractions, Knaster-Tarski type maps and Perov type maps. For a detailed study of fixed point structures technique and its application to cyclical maps one may refer to Rus [7], Petrusel [8], Petric [12] Rus et al. [50] and references thereof. It is worth noticing that conditions (1.1), (1.2), (1.3) and (1.4) imply the continuity of map T throughout the space. Therefore the natural question that arises is, whether there exists a contractive condition which does not force map T to be continuous. R. Kannan [48] in 1968 answered this question positively and obtained a fixed point theorem using the following condition:

d (Tx, Ty )  k[d ( x, Tx)  (d ( y, Ty )], k [0,1 / 2).

(1.13)

It is important to note that it was Kannan who first considered the distance between the points x, Tx, y , Ty in condition (1.13) which actually formed the base for a new direction of study in the ambit of metrical fixed point theory. The lurking potential of Kannan condition (1.13) was soon recognized and as its outcome Chatterjea condition, Zamfirescu condition, quasi contraction of Ciric, Hardy and Rogers condition and all the other conditions involving the distance between the points x, y, Tx, Ty originated from condition (1.13) and subsequently many researchers were attracted to work along the similar lines. Consider the following conditions:

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d (Tx, Ty )  k[d ( x, Ty )  (d ( y, Tx)], x, y  X , k [0,1 / 2). (Chatterjea) (1.14) For x, y  X , and 0  a  1, 0  b, c  1 / 2 at least one of the following three are satisfied

d (Tx, Ty )  ad ( x, y ),

(1.15B)

d (Tx, Ty )  b[d ( x, Tx)  (d ( y, Ty )], (Zamfirescu)

(1.15K)

d (Tx, Ty )  c[d ( x, Ty )  (d ( y, Tx)],

(1.15C)

d (Tx, Ty )  k max{ d ( x, Tx) , d ( y, Tx)}, x, y  X , k [0,1). (Bianchini) (1.15)

d (Tx, Ty )  a d ( x, y )  bd ( x, Tx)  cd ( y, Ty ) , x, y  X , (Reich) (1.16) where a, b, c are nonnegative and satisfy a  b  c  1.

d (Tx, Ty )  a d ( x, y )  bd ( x, Tx)  c d ( y, Ty )  e d ( x, Ty )  f d ( y, Tx) , x, y  X .

d ( x, y )  bd ( x, Tx)  c d ( y, Ty )  e d ( x, Ty )  f d ( y, Tx) , x, y  X . (Hardy and Rogers)

(1.17)

where a, b, c, e, f are non-negative and satisfy a  b  c  e  f  1. For every x, y  X there exist non-negative numbers q, r, s and t which may depend on both x and y such that

sup{q  r  s  2t }  1, and

d (Tx, Ty )  q d ( x, y )  rd ( x, Tx)  s d ( y, Ty )  t [d ( x, Ty )  d ( y, Tx)]. (Ciric) (1.18)

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d (Tx, Ty )  k max{ d ( x, y ) , d ( x, Tx), d ( y, Ty ) , d ( x, Ty ), d ( y, Tx)}, x, y  X , k [0,1).

(Tx, Ty )  k max{ d ( x, y ) , d ( x, Tx), d ( y, Ty ) , d ( x, Ty ), d ( y, Tx)}, x, y  X , k [0,1). (Ciric)

(1.19)

We remark that Zamfirescu conditions (1.15B, 1.15K, 1.15C) contain Banach contraction, Kannan’s condition (1.13) and Chatterjea’s condition (1.14). Reich condition (1.16) reduces to Banach contraction if b = c = 0 in (1.16). Further conditions (1.13) and (1.14) can be obtained from condition (1.16) with the suitable choices of values for constants b and c. Notice that condition (1.19) is quasi contraction essentially due to Ciric [49] it is the most general contractive condition listed as (24) by Rhoades [1]. Rus [7] obtained the following cyclical extension of Kannan’s theorem by using the technique of fixed point structure. Theorem 1.7 [7]. Let { A}ip1 be non-empty closed subsets of a complete metric space, and suppose T :  ip1 Ai  ip1 Ai satisfy the conditions (1.7) (where

Ap 1  A1 ) and d (Tx, Ty)  k [ d ( x, Tx)  d ( y, Ty)],  x  Ai , y  Ai1 for 1  i  p, (1.20) where k [0,1 / 2). Then T has a unique fixed point x  *

p i 1

Ai .

Petric and Zlatanov [13] obtained cyclic extension of Zamfirescu’s theorem they also proved Theorem 1.7 without using fixed point structures. Further, they provided the estimates of error and rate of convergence for Theorem 1.7 as well as for Theorem 1.2. For other cyclic extensions of some fundamental results of Chatterjea, Bianchini, Reich, Hardy and Rogers, Ciric and Ciric-Reich-Rus type maps one may refer to the papers of Petric ([10], [11]). Berinde [51] introduced the notion of weak contraction and demonstrated that it contains the well-known results of Kannan’s, Chatterjea, Zamfirescu and of course Bcp as well. A self-map T : X  X of a metric space X is called a weak contraction if there exist a constant  (0,1) and some L  0 such that

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d (Tx, Ty )   d ( x, y )  Ld ( y, Tx) , x, y  X .

(1.21)

It is important to note that a weak contraction includes quasi contraction (1.19) for k 1 / 2. However, a quasi contraction with k 1 / 2 does not satisfy weak contraction. The following result is due to Berinde and Petric [16]. Theorem 1.8 [16]. Let A and B be two non-empty closed subsets of a complete metric space X, and T : A  B  A  B be a cyclic map. Let there exist a constant  (0,1) and some L  0 such that

d (Tx, Ty )   d ( x, y )  Ld ( y, Tx) ,  x  A, y  B or x  B, y  A. (1.22) Then T has a unique fixed point in A  B. Pacurar [15] defined cyclic strict Berinde operator and obtained a result similar to Theorem 1.8. Recently Babu et al. [26] defined  -cyclic contraction, almost Geraghty cyclic contraction and almost  -cyclic contraction and obtained fixed point theorems generalizing previously obtained theorems. In the sequel, we need the functions called comparison function and (c)comparison function. Definition 1.2 [53]. A function  :[0, )  [0, ) is called a comparison function if it satisfies: i.

 is increasing,

ii. { n (t )}n0 , n N , converges to 0 as t  [0,  ). Definition 1.3 [53]. A function  :[0, )  [0, ) is called a (c)-comparison function if it satisfies condition (i) of definition (1.2) and the following condition:

147

Metric Fixed Point Theory in Context of Cyclic Contractions 

i.



n

(t ) converges to 0 for all t  0.

n 0

We remark that  satisfies condition (i) of Definition 1.3 if and only if there exist 0  c  1 , k 0  N and a convergent series of positive terms,



u n 0

n

,

such that

 k 1 (t )  c  k (t )  u k , for all t  [0, ) and k  k 0 . (see, for instance, [52], [53]). Notice that a (c)-comparison function is a comparison function but the converse is not true, e.g., a function defined as  (t )  at , t [0, ), a (0,1) is a (c)-comparison function as well as a comparison function. On the other t hand,  (t )  , t [0, ) is a comparison function but not a (c)-comparison 1 t function (see also [52], [53]). Further note that some authors call a (c)comparison function as a strong comparison function also (see [21]). As defined in [52] a self-map of metric space X is said to be a  -contraction if there exists a comparison function  :[0, )  [0, ) such that

d (Tx, Ty )   ( d ( x, y )), for all x, y  X . Notice that Browder function, Boyd-Wong function, Matkowski function and Geraghty function are all the examples of some classes of comparison functions. The following definition appeared in Pacurar and Rus [9]. Definition 1.4. [9]. Let X be a metric space, p  N , A1 , ...... Ap be nonempty closed subsets of X, and Y :  ip1 Ai . A mapping T : Y  Y is called a cyclic

 -contraction if

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 ip1 Ai is a cyclic representation of Y with respect to T,

ii. there exists a comparison function  : R   R  such that

d (Tx, Ty)   ( d ( x, y)),  x  Ai , y  Ai 1 , where Ap1  A1. Pacurar and Rus [9] generalized the Theorem 1.2 [Th. 1.3, 5] by using the concept of cyclic  -contraction where  is a (c)-comparison function. Indeed, following is the result of Pacurar and Rus [9]. Theorem 1.9. [9]. Let A1 , ...... Ap , p N be non-empty closed subsets of a complete metric space X, and Y :  ip1 Ai . Let  : R   R  be a (c)comparison function and T : Y  Y a map. Assume that: i.

 ip1 Ai is a cyclic representation of Y with respect to T,

ii. T is a cyclic  -contraction. Then: (1) T has a unique fixed point x *  ip1 Ai and the Picard iteration

{x n }n0 given by xn  Txn 1 , n  1 converges to x * for any starting point x0 Y , (2) the following estimates hold: 

d ( xn , x * )   n (d ( x0 , x1 )) , n 0 

d ( xn , x* )   (d ( xn , xn1 )) , n 0

(3) for any xY , 

d ( x, x* )   n (d ( x, Tx)). n0

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149

We remark that the above theorem is not only an existence and uniqueness result it also provides the estimates of error in approximating the fixed point. Further, by considering different conditions on  various deductions can be derived from Theorem 1.9 e. g., if  (t )  at , 0  a  1, in Theorem 1.9 then we can obtain Theorem 1.2. If  (t ) is upper semi-continuous from right and satisfies  (t )  t then one obtains Theorem 1.5. Moreover, as a consequence of Theorem 1.9, Petrusel and Rus [9] demonstrated its applications to the concepts of good Picard operators, special Picard operators, data dependence, well posedness of fixed point problem and limit shadowing property etc. Notice that the ordinary version of Theorem 1.9 can be obtained if we take

Ai  X for every i  1, 2, ...... p. To be specific, the following is the ordinary counterpart of Theorem 1.9. Theorem 1.10. Let T : X  X be a self-map of a complete metric space X,

 : R   R  be a (c)-comparison function such that d (Tx, Ty )   ( d ( x, y )), for all x, y  X . Then: (1) T has a unique fixed point x *  X and the Picard iteration {x n }n0 given by xn  Txn 1 , n  1 converges to x * for any starting point

x0  X , (2) the following estimates hold: 

d ( xn , x* )   n (d ( x0 , x1 )), n0 

d ( xn , x * )   (d ( xn , xn1 )), n 0

(3) for any x X , 

d ( x, x* )    n (d ( x, Tx)). n 0

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Radenovic [32] proved the equivalence between Theorem 1.9 and Theorem 1.10 by asserting that the conclusions of Theorem 1.9 can be obtained via Theorem 1.10 applied to the map T :  ip1 Ai  ip1 Ai . Agarwal et al. [37] disproved the equivalence of these two theorems by maintaining that in Theorem 1.9 the Picard sequence converges to the fixed point of T for any

x0  ip1 Ai . However, when Theorem 1.10 is applied with X  ip1 Ai then the convergence holds only for x0   ip1 Ai , not for an arbitrary point x 0 of the space. Agarwal et al. [37] (see also [36]) further improved Theorem 1.9 by dropping the requirement of closedness of Ai for all i  1, 2, ..., p, instead they required the closedness of A1 only. The following example establishes the generality of Theorem obtained by Agarwal et al. (Th. 4.4, [37]) over Theorem 1.9. Example 1.1 [37]. Let X  R be equipped with usual metric d ( x, y)  x  y . Let A  [0, 2] and B  (1,  ) be subsets of X and T : A  B  A  B be defined as Tx  2 for x  A  B. Then for any (c)-comparison function  : R   R  the condition d (Tx, Ty )   ( d ( x, y )) for x  A, y  B is satisfied and T has a unique fixed point x  2 A  B  (1, 2]. Notice that B  (1,  ) is not closed. Radenovic [32] posed a question that: prove or disprove Theorem 1.9 when (c)-comparison function of Theorem 1.9 is replaced by a comparison function. He and Chen [34] answered positively the question and proved the following theorem. Theorem 1.11 [34]. Let A1 , ...... Ap , p N be non-empty closed subsets of a complete metric space X, and Y :  ip1 Ai . Let  : R   R  be a comparison function and T : Y  Y be a cyclic  -contraction. Then T has a unique fixed point x *  ip1 Ai and the Picard iteration {x n }n0 given by xn  Txn 1 , n  1 converges to x * for any starting point x0 

p i 1

Ai

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151

Magdas [21] defined cyclic  -contraction of Ciric type and generalized certain results of Kirk [5], Pacurar and Rus [9], Petric [10, 11], Petric and Zlatanov [13], He and Chen [34] and Agarwal et al. [37]. Theorem 1.12 [21]. Let A1 , ...... Ap , p N be non-empty closed subsets of a complete metric space X, and Y :  ip1 Ai . Let  : R   R  be a strong comparison function, and T : Y  Y a cyclic  -contraction of Ciric type. That is, following conditions are satisfied: i.

 ip1 Ai is a cyclic representation of Y with respect to T;

ii.

d (Tx, Ty)   ( M ( x, y)),  x  Ai , y  Ai1 , where Ap1  A1.

Then all the conclusions of Theorem 1.9 are true. Motivated by various classes of comparison functions López de Hierro and Shahzad [54] studied a new class of function and defined R-function and Rcontraction. Abbas et al. [44] introduced the notion of cyclic R-contraction and obtained fixed point theorem for cyclic R-contraction. They have shown that cyclic R-contraction is general than cyclic Boyd-Wong contraction, cyclic Geraghty contraction and cyclic Meir-Keeler contraction (defined below) (see example 2.3 and remark 2.4, [44]). A remarkable generalization of Bcp was obtained by A. Meir and E. Keeler [55] in 1969 they formulated a contractive condition which significantly generalizes the well-known nonlinear contractive conditions of Boyd and Wong [3], Browder [56] and Rakotch [57]. Let T be a self-map of a metric space X such that given   0 there exists   0 such that for all x, y  X ,

  d ( x, y )      d (Tx, Ty )   .

(1.23)

Then T is called a Meir-Keeler contraction. For a fundamental comparison of various Meir-Keeler type conditions one may refer to Jachymski [58] (see also

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[59], [60] and [61]). Meir-Keeler proved that a self-map T of a metric space X satisfying condition (1.23) has a unique fixed point provided X is complete. In 2009, Karpagam and Agrawal [40] considered a Meir-Keeler contraction map T on the union of p subsets A1 , A2 , . . , Ap for all x  Ai , y  Ai 1 where

Ap 1  A1 with T ( Ai )  Ai 1 for 1  i  p, they called it p-cyclic Meir-Keeler contraction and obtained the following result. Theorem 1.13 [40]. Let A1 , A2 , . . , Ap be non-empty closed subsets of a complete metric space X and T :  ip1 Ai  ip1 Ai be a p-cyclic Meir-Keeler contraction. Then  ip1 Ai is non-empty and for any x  Ai ,1 i  p, the sequence {T pn x} converges to a unique fixed point in  ip1 Ai . Generalizing the concept of cyclic contraction Karpagam and Agrawal [41] introduced the idea of cyclic orbital contraction. Definition 1.5 [41]. Let A and B be non-empty subsets of a metric space X, T : A  B  A  B be a cyclic map such that for some x A, there exists a

k x (0,1) such that

d (T 2n x, Ty)  k x d (T 2n1 x, y), n N , y A. Then T is called a cyclic orbital contraction. Notice that cyclic contraction (cf. condition 1.6) becomes Banach contraction if A  B, however a cyclic orbital contraction need not be a Banach contraction even if A  B. Karpagam and Agrawal [41] proved that a cyclic orbital contraction map T : A  B  A  B has a unique fixed point provided that A and B are nonempty closed subsets of a complete metric space X. In the same paper they extended the idea of cyclic orbital contraction to cyclic orbital Meir-Keeler contraction which reads as.

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153

Definition 1.6 [41]. Let A and B be non-empty subsets of a metric space X. Suppose T : A  B  A  B is a cyclic map such that, for some x A, and for each   0, there exists a   0 such that

d (T 2 n1 x, y)      d (T 2 n x, Ty)   , n N , y  A. Then T is called a cyclic orbital Meir-Keeler contraction. We remark that if T : A  B  A  B is cyclic orbital Meir-Keeler contraction then

d (T 2 n x, Ty )  d (T 2 n1 x, y), n N , y  A , (see [41]). The following theorem for cyclic orbital Meir-Keeler contraction is essentially due to Karpagam and Agrawal [41]. Theorem 1.14 [41]. Let A and B be non-empty closed subsets of a complete metric space X such that d ( A, B )  0. Let T : A  B  A  B be a cyclic orbital Meir-Keeler contraction. Then there exists a fixed point x0  A  B such that for each x  A the sequence {T 2 n x} converges to x0 . Chen [45, 46] defined weaker Meir-Keeler function, used it further to define cyclic weaker (   )  contraction and obtained fixed point theorems. Later in 2015, Radenovic [33] proved that the concept of cyclic weaker (   )  contraction with additional conditions imposed on it, as in Chen [46], is superfluous. He also established that the main result of Chen [46] on cyclic weaker (   )  contraction is equivalent to its non-cyclic (ordinary) counterpart. Chen [45, 46] introduced the concept of cyclic orbital stronger Meir-Keeler  x  contraction and cyclic orbital weaker Meir-Keeler  x  contraction and obtained fixed point theorems. Further, in another paper Chen [47] generalized the above concepts by introducing the idea of generalized

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cyclic orbital stronger Meir-Keeler ( x ,  )  contraction and obtained existence and uniqueness theorems. Karapinar et al. [42] generalized the theorem of Karpagam and Agrawal [41] by introducing the notion of cyclic orbital generalized contraction, they deduced the cyclic versions of Boyd-Wong and Matkowski fixed point theorems from their main result. The following definition appears in [42]. Definition 1.7 [42]. Let A and B be non-empty subsets of a metric space X and

T : A  B  A  B be a cyclic map. If there exist x 0  A, and a function

 : R   R  with  (t )  t for all t  0 such that d (T 2 n x0 , Ty)   (M (T 2 n1 x0 , y)), n N , y A. Then T is called a cyclic orbital generalized contraction for x0 and  . The following theorem for cyclic orbital generalized contractions is essentially due to Karapinar et al. [42]. Theorem 1.15 [42]. Let A and B be non-empty closed subsets of a complete metric space X and T : A  B  A  B be a cyclic orbital generalized contraction for an x 0  A, and  : (0, )  (0, ). If  satisfies the following condition, for each   0 there exists a   0 such that for all t  (0,  ),

  t       (t )   .

(1.24)

Then T has a fixed point in x  A  B such that T n x0  x. Consider the following conditions on a metric space X:

d (Tx, Ty )  M ( x, y ), for all x, y  X , x  y. For any   0, there exists a   0 such that for all x, y  X ,

(1.25)

Metric Fixed Point Theory in Context of Cyclic Contractions

  M ( x, y )      d (Tx, Ty )   .

155 (1.26)

For any   0, there exists a   0 such that for all x, y  X ,

  M ( x, y )      d (Tx, Ty )   .

(1.27)

We remark that in a metric space condition (1.27) is weaker than condition (1.26), however in a metrically convex space both the conditions (1.27) and (1.26) are equivalent (see Jachymski [58] and Kumar et al. [60]). It is also well known that condition (1.26) implies conditions (1.25) and (1.27) however, the converse of this is not true. Further, in general the existence of a fixed point is not ensured by condition (1.27). The technique used for ensuring the existence of fixed point by condition (1.27) consists in the use of condition (1.27) along with an appropriate additional contractive condition which may be condition (1.25) or a variant of condition (1.25) (see for instance, [58] and [60]). This approach was used by Karapinar et al. [42] to prove Theorem 1.15 for cyclic orbital generalized contraction, the same technique was used earlier, among others, by Matkowski [62], Ciric [63] and Jachymiski [58]. The cyclic orbital maps were then extended to p-cyclic orbital maps such as p-cyclic orbital contraction, p-cyclic orbital Meir-Keeler maps, p-cyclic orbital Boyd-Wong maps etc. We remark that the concept of p-cyclic map has a restriction that it requires the distance between successive sets

A1 , A2 , . . , Ap , p N , be equal (see [40], [64] and [65]). In order to relax this restriction the concept of p-summing maps was introduced in 2012 by Petric and Zlatanov [66] (see also [65]). The class of p-summing maps is wider than the class of p-cyclic maps in the sense that p-summing maps do not require invariance of distance between successive sets A1 , A2 , . . , Ap , p  N . For further development and results concerning p-cyclic maps, p-cyclic orbital contractions and p-summing maps one may refer to [42], [45]-[47] and [64][73] and references thereof. Alber and Guerre-Delabriere [77] coined the concept of a new class of map called weakly contractive map. This class of map significantly generalizes the

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Banach contraction. A self-map T of X is called weakly contractive if for each

x, y  X ,

d (Tx, Ty )  d ( x, y )   (d ( x, y )),

(1.28)

where  : [0, )  [0, ) is a continuous and nondecreasing function such that

 (0)  0 and  (t )  0 for all t  0. We remark that weakly contractive map is a wider class of map and is closely related to famous nonlinear contractions of Boyd and Wong and Reich type e. g., if  (t )  (1  k )t , for t  0 and 0  k  1, in (1.28) then it reduces to (1.1). Further, let  be a lower semi-continuous map from the right then

 (t )  t   (t ) is upper semi-continuous from the right and (1.28) becomes d (Tx, Ty )   (d ( x, y )). Therefore, weakly contractive maps for which  is lower semi-continuous from the right are Boyd-Wong type. Moreover, condition (1.28) reduces to Reich type contractive condition if k (t ) 1 

 (t ) t

for t  0 and k (0)  0 (see

for instance, [78]). Alber and Guerre-Delabriere [77] obtained a fixed point theorem using weakly contractive map in the framework of Hilbert spaces. Rhoades [79] relaxed the requirement of Hilbert space and obtained an interesting generalization of Bcp in the settings of metric spaces for weakly contractive map. To be specific, Rhoades proved that a self map T of a complete metric space has a unique fixed point if T is a weakly contractive map. In 2011, Karapinar [18] studied weakly contractive maps in the context of cyclic maps and introduced the idea of cyclic weak  -contraction as follows:

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Definition 1.8 [18]. Let A1 , ...... Ap , p N be non-empty closed subsets of a metric space X, and Y :  ip1 Ai . A mapping T : Y  Y is called a cyclic weak

 -contraction if i.

 ip1 Ai is a cyclic representation of Y with respect to T, and

ii. there exists a continuous, non-decreasing function  :[0,  )  [0,  ) with  (t )  0 for t  0 and  (0)  0 such that

d (Tx, Ty)  d ( x, y)  ( d ( x, y )),  x  Ai , y  Ai 1 , where Ap1  A1. Karapinar [18] obtained the following theorem which extends [Th. 1, 79] of Rhoades. Theorem 1.16 [18]. Let A1 , ...... Ap , p N be non-empty closed subsets of a complete metric space X, and Y :  ip1 Ai . Let T : Y  Y be a cyclic weak

 -contraction. Then T has a unique fixed point x ip1 Ai . Harjani et al. [30] obtained a variant of Theorem 1.16 in a compact metric space by relaxing the condition of closedness of non-empty subsets Ai of the space; they required the function  :[0,  )  [0,  ) to be non-decreasing only. Subsequently, Kadelburg et al. [35] improved the theorem of Harjani et al. [Th. 2.1, 30] by relaxing the requirement of compactness of space. Here it is appropriate to refer to remark 1 [pp. 2, 35] and remark 3 [pp. 6, 35]. In these remarks Kadelburg et al. [35] interestingly underlined the fact that there are two types of cyclic fixed point theorems, the first type of theorems assume a map T : Y  Y , where Y  X   ip1 Ai , in this case the closedness of subsets A1 , ...... Ap , p N is not necessary. The second type of theorems consider the map T : Y  Y , where Y   ip1 Ai , Y  X , in this case it is sufficient to consider that one of the sub set in A1 , ...... Ap , p N is closed.

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Ashish Kumar Karapinar and Sadarangani [20] defined cyclic weak (   )-contraction,

which contains cyclic weak  -contraction as a special case, and obtained the following unique fixed point theorem. Theorem 1.17 [20]. Let A1 , ...... Ap , p N be non-empty subsets of a complete metric space X. Let X  ip1 Ai be a cyclic representation of X with respect to T and T : X  X be a cyclic weak (   )-contraction i.e., there exist

 :[0, )  [0, ) and  :[0, )  [0, ) with  (t )  0, (t )  0 for t  0 and  (0)  0  (0) continuous,

non-decreasing

functions

such that

 (d (Tx, Ty))  (d ( x, y))   (d ( x, y)) ,  x  Ai , y  Ai 1 , where

Ap1  A1.

(1.29)

Then T has a unique fixed point. Notice that above theorem does not require the assumption of closedness of non-empty subsets A1 , ...... Ap , p N of the metric space X. We remark that Theorem 1.17 is general than Theorem 1.9 [Th. 2.1, 9] and Theorem of Agarwal et al. [Th. 4.4, 37] in two aspects. First, in Theorem 1.17 none of the Ai , i 1, 2, ... , p, p  N is a closed subset of X whereas in Theorem 1.9 all the Ai , i 1, 2, ... , p, p  N are closed subsets of X similarly Theorem of Agarwal et al. [37] assumes A1 as closed subset of X. Second, cyclic weak (

   )-contraction used in Theorem 1.17 is wider than cyclic  -contraction where  is a c-comparison function (see [20]). In particular, consider the condition (1.29) of Theorem 1.17 with  (t ) 

t2 , for t  (0,  ) and  (t ) is 1 t

an identity map. Then both  (t ) and  (t ) are continuous and non-decreasing

Metric Fixed Point Theory in Context of Cyclic Contractions functions. Also, ( -  )(t )  t  

evidently

 (   )

n

159

t t2 t and therefore (   ) n (t )   ,  1 nt 1 t 1 t

(t ) diverges for every t  (0,  ) . This violates the

n0

definition of (c)-comparison function. Hence (   )(t ) is not a (c)-comparison function. Mishra and Pant [22] (see also Karapinar and Moradi [23]) defined generalized cyclic weak ( ,  )  contraction which is general than cyclic weak (   )-contraction. Let A1 , ...... Ap , p N be non-empty subsets of a metric space X. A cyclic p p map T :  i 1 Ai   i 1 Ai is called a generalized cyclic weak ( ,  ) 

contraction if

 (d (Tx, Ty))  ( M ( x, y))   ( M ( x, y)) ,  x  Ai , y  Ai 1 , Ap1  A1 ,

(1.30)

where  and  are continuous and non-decreasing and  (t )  0, (t )  0 iff

t  0. The following theorem is due to Mishra and Pant [22]. Theorem 1.18 [22]. Let A1 , ...... Ap , p N be non-empty closed subsets of a complete metric space X and T :  ip1 Ai  ip1 Ai be a generalized cyclic weak

( , ) -contraction on X. Then T has a unique fixed point x ip1 Ai . The following example demonstrates the generality of above theorem over Theorem 1.17. Example 1.2 [22]. Let X  {1, 2, 3, 4, 5} be endowed with the metric d defined by

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Ashish Kumar

13 3 , d (1, 4)  , d (3,4)  2. 8 2 7 15 d (1, 5)  d (2, 4)  , d (2, 3)  (4, 5)  1, d (2,5)  . 4 8

d (1, 2)  d (1, 3)  d (3, 5) 

Suppose A1  {1, 2, 3} and A2  {1, 4, 5} then A1  A2  X . Consider a map

T : X  X defined by

T 1  1, T 2  T 3  4, T 4  1, T 5  2. Define  (t )  2t and  (t ) 

t for all t  0. 20

Then it is easy to see that all the conditions of Theorem 1.18 are satisfied and 1 A1  A2 is the unique fixed point of T. However, T does not satisfy the condition (1.29) of Theorem 1.17 and condition (1.8) of Theorem 1.2 for

x  3, y  5. Nashine [25] considered the cyclic extension of following condition to obtain his result, applications pertaining to integral equations are also discussed in [25].

 (d (Tx, Ty ))   ( M ( x, y ))   ( ( M ( x, y ))),

(1.31)

where  :[0, )  [0, ) is non-decreasing, continuous and  (0)  0   (t ) for

t  0 and  :[0, )  [0, ) is non-decreasing, right continuous and  (0)  0. For the results concerning Kannan type cyclic weak contractions and Chatterjea type cyclic weak contractions one may refer to [24], [25], [27], [29], [31], [38] and papers referred therein. For the concepts related to cyclic weaker ( ,  )  contraction and generalized cyclic orbital weaker Meir-Keeler

( x ,  )  contraction, corresponding fixed point theorems and useful comments one can refer to [33], [43],[45]-[47] and references thereof.

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Metric Fixed Point Theory in Context of Cyclic Contractions

Gabeleh and Karami [39] obtained the following theorem with general control functions  and  . Theorem 1.19 [39]. Let A and B be two non-empty closed subsets of a complete metric space X and T : A  B  A  B be a cyclic map such that for all

( x, y )  A  B ,

 (d (Tx, Ty ))  ( M ( x, y ))   ( M ( x, y )),

(1.32)

where functions  , : cl (rand ) [0, ) are such that i.

 (0)  (0)  0 iff t  0,

ii. lim inf

 ( )  lim sup  t ( )  lim inf  t ( ) for all t  0,

 t

where cl(rand) denotes the closure of the value of the metric d defined on X  X . Then A  B is non-empty and T has a unique fixed point in A  B. Gabeleh and Karami [39] also obtained the cyclic extension of Wardowski and Dung [80] fixed point theorem. Replacing an identity map in Banach contraction (condition 1.1) Jungck [81] in 1976, considered two self-maps on a metric space and obtained a common fixed point theorem using following condition

d (Tx, Ty )  kd ( fx, fy ), x, y  X , 0  k  1. Jungck’s idea attracted researchers and subsequently vigorous research activity concerning existence of coincidence and common fixed point started. However, there are only a few papers devoted to the study of common fixed point theorems of cyclic maps see, for instance, Kirk et al. [5], Abbas et al. [82], Chaipunya et al. [83], Sintunavarat and Kummam [85], Sridarat and Suantai [88] and Saksirikun et al. [91].

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Kirk [5] reformulated Theorem 1.1 [Th 1.1, 1] as a common fixed point theorem for two maps defined on the closed subsets A and B of a metric space X. Later, Abbas et al. [82] defined generalized cyclic contraction for a pair of single valued maps (T, f) and obtained coincidence and fixed point theorems for such maps. Definition 1.9 [82]. Let { A}ip1 be finite, non-empty closed subsets of a metric space X, X   ip1 Ai . The pair of maps T , f :  ip1 Ai   ip1 Ai is called generalized cyclic contraction if i.

T ( X 1 )  f ( X 2 ),T ( X 2 )  f ( X 3 ),. . . ,T ( X p1 )  f ( X p ) and T ( X p )  f ( X 1 ),

ii. there exist two control functions  ,  :[0, )  [0, ) defined as  (t ) is a continuous, non-decreasing and  (t ) is lower semi-continuous with

 (t )   (t )  0 iff t  0, such that

 (d (Tx,Ty))  (M f ,T ( x, y))   (M f ,T ( x, y)),  x  Ai , y  Ai 1 , where Ap1  A1. Abbas et al. [82] called the collection Ai , i 1, . . . , p, a cyclic (T, f) cover of X. Theorem 1.20 [82]. Let Y be a non-empty subset of a complete metric space X, f and T be two self-maps on Y. Suppose that finite collection { A1 , A2 , . . . , Ap } of non-empty closed subsets of X is a cyclic (T, f) cover of Y such that f ( Ai ) is closed for each i ( 1, . . . , p ). If (T, f) is a generalized cyclic contraction pair, then f and T have a coincidence point z ip1 f ( Ai )  Z provided that

T ( Z )  f ( Z )  Z where f(Z) is closed. Moreover, if T and f are weakly compatible, then T and f have a unique common fixed point.

Metric Fixed Point Theory in Context of Cyclic Contractions

163

2. MUTLI-VALUED CYCLIC FIXED POINT THEOREMS In the rest of the chapter let CB(X) denotes the family of all non-empty closed and bounded subsets of X, C(X) the family of all non-empty compact subsets of X, P(X) the family of all non-empty subsets of X and H the Hausdorff metric induced by d, i.e.,

H ( A, B)  max {sup d ( x, B), sup d ( y, A)}, x A

yB

for all A, B  CB ( X ), where d ( x, B)  inf d ( x, y ). Let

f : X  X be a

y B

single valued map and T : X  CB ( X ) a multivalued map then a point x X is called: a fixed point of multivalued map T if xTx, a coincidence point of f and T if fxTx, and a common fixed point of f and T if x  fxTx. Let

C ( f , T ) denotes the set of all coincidence points of f and T. A multivalued map T : A  B  CB ( X ) is said to be cyclic (on A and B) if

Tx  B for all x A and Ty  A for all y B. Nadler Jr. [84] introduced the concept of multivalued contraction and proved the multivalued version of Bcp which states that; if a multivalued map T : X  CB ( X ) is a multivalued contraction i.e., for x, y  X and k (0,1), the following condition is satisfied

H (Tx, Ty )  k d ( x, y ).

(2.1)

Then T has at least one fixed point in X provided that X is complete. Papers proposing generalizations and extensions of different cyclic fixed point theorems for single valued maps in metric spaces are numerous. However, the study of fixed points of cyclic multivalued maps is a domain which requires some more attention of the researchers and mathematicians, only a few papers have been reported in this area see, for instance, Petric [12], Mishra and Pant

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Ashish Kumar

[22], Rus et al. [50], Sintunavarat and Kumam [85], Nammeatee and Kaewkhao [86], Chuensupantharat and Kumam [87], Sridarat and Suantai [88], Popa [89], Magdas [90], Saksirikunet et al. [91] and De la Sen et al. [127]. Neammanee and Kaewkhao [86] extended Theorem 1.2 [Th. 1.3, 5] to multivalued case and obtained the following theorem. Theorem 2.1 [86]. Let A1 , ...... Ap , p  N be non-empty closed subsets of a complete metric space X, and let the multivalued map T :  ip1 Ai  CB( X ) satisfy the conditions (where Ap 1  A1 ):

Tx  Ai1 , for all x  Ai ,1 i  p, and there exists a constant

k (0,1) such that, for all

(2.2)

x Ai and

y Ai 1,1 i  p , H (Tx, Ty )  k d ( x, y ).

(2.3)

Then T has at least one fixed point in  ip1 Ai . Notice that if i  1, 2 in the above theorem then it is [Th. 2.1, 86] which is the multivalued extension of Theorem 1.1 (cf. section 1). Motivated by Berinde’s weak contraction [51], Neammanee and Kaewkhao [86] further obtained the following cyclic multivalued theorem. Theorem 2.2 [86]. Let A and B be non-empty closed subsets of a complete metric space X and T : A  B  CB ( X ) be a cyclic multivalued map. If there exist two constants k  (0,1) and L  0 such that for all x A and y B,

H (Tx, Ty )  k d ( x, y )  L min{d ( y, Tx), d ( x, Ty )}. Then T has at least one fixed point in A  B.

(2.4)

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165

Mishra and Pant [22] defined multivalued cyclic weak ( ,  )  contraction and obtained the following theorem which is the multivalued version of Theorem 1.17 [Karapinor Th. 6, 20]. Theorem 2.3 [22]. Let A1 , ...... Ap , p  N be non-empty closed subsets of a complete metric space X such that X   ip1 Ai . Let T : ip1 Ai  C ( X ) be a multivalued cyclic weak ( ,  )  contraction i, e., for all x  Ai , y  Ai 1 and

Tx  Ai 1 (where Ap 1  A1 ) the following condition is satisfied

 ( H (Tx, Ty ))  (d ( x, y ))   (d ( x, y )) ,

(2.5)

where functions  and  are same as in Theorem 1.18. Then T has a fixed point in  ip1 Ai . Taking hybrid pair of maps, means a pair of maps consisting of a single valued map and a multivalued map, Sintunavarat and Kumam [85] defined the cyclic generalized multivalued contraction and obtained the existence of a coincidence and common fixed point for such maps. This concept generalize and extend the concepts of multivalued T-weak contraction and generalized multivalued T-weak contraction, studied by Kamran [92], to cyclic multivalued contraction. Definition 2.1 [85]. Let A and B be non-empty subsets of a metric space X. f : A  B  A  B be a single valued map and T : A  B  CB ( X ) be a multi valued map. The multi valued map T is said to be cyclic (on A and B) generalized multi valued contraction if T satisfies the following conditions: i.

T ( A)  f ( B ) and T ( B )  f ( A),

ii. there exist two functions  :[0, )  [0,1) and  :[0, )  [0, ) satisfying lim sup  (r ) 1 and lim sup  (r ) for every t [0,  ), r t

r t

such that for all x A and y B,

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Ashish Kumar

H (Tx, Ty )   (d ( fx, fy )) d ( fx, fy )   ( N ( x, y )) N ( x, y ), where N ( x, y )  min{ d ( fx, fy ), d ( fx, Tx), d ( fy, Ty ), d ( fx, Ty ), d ( fy, Tx)}. Theorem 2.4 [85]. Let A and B be non-empty subsets of a metric space X, f : A  B  A  B be a single valued map such that f(A) and f(B) are closed and T : A  B  P ( X ) be a cyclic (on A and B) generalized multi valued contraction. If f(X) is complete, then f and T have at least one coincidence point z  X . Moreover, if ffz = fz, then f and T have at least one common fixed point in f ( A)  f ( B ). Recently, Sridarat and Suantai [88] combined the multivalued contractive conditions of Mizoguchi and Takahashi [93] and Berinde and Berinde [94] and obtained a common fixed point theorem for a cyclic multivalued map. In line with the cyclic representation of single valued maps, Sridarat and Suantai [88] defined the cyclic representation of multivalued maps. Let A1 , ...... Ap , p  N be non-empty subsets of a metric space X such that X  ip1 Ai and

T : X  2 X be a multivalued map. Then, X  ip1 Ai is said to be cyclic representation

of

X

with

respect

to

T

if

T : Ai  CB( Ai 1 ) for

1 i  p, Ai 1  A1 . The following theorem is due to Sridarat and Suantai [88]. Theorem 2.5 [88]. Let A1 , ...... Ap , p  N be non-empty closed subsets of a complete metric space X, Y   ip1 Ai and S , T : Y  2Y . Assume that

 ip1 Ai is a cyclic representation of Y with respect to S, T and there exists a function  :[0,  )  [0,1) with lim sup  (t ) 1 for every r [0,  ) and t r

L  0 such that, for x  y,

Metric Fixed Point Theory in Context of Cyclic Contractions

167

H (Sx, Ty)   (d ( x, y)) d ( x, y)  L D( y, Tx),  x Ai , y  Ai 1 , Ap 1  A1 . (2.6) Then S and T have a common fixed point. Suzuki [95] adopted an entirely different approach to generalize Bcp by restricting the domain of contractive condition. The significance of Suzuki’s generalization can be realized by noting that it characterizes the metric completeness while Bcp cannot characterize the metric completeness. In due course of time several generalizations and extensions of Suzuki theorem have been obtained. Motivated by one such generalization given by Doric and Lazovic [96], Saksirikun et al. [91] obtained a coincidence point and common fixed point theorem for a hybrid pair of single valued and multivalued map under cyclic conditions in the following manner. Let T : X  CB ( X ) be a multivalued map. For each A  X , we put

T ( A)   Tx . xA

Theorem 2.6 [91]. Define a non-increasing function

if 0  r  1 / 2 1, .  1  r , if 1 / 2  r  1

 (r )  

Let A1 , ...... Ap , p  N be non-empty subsets of a complete metric space X,

f : X  X be a single valued map and T : X  CB ( X ) be a multivalued map such that T ( Ai )  f ( Ai 1 ) for each i  1, . . . , p  1 and T ( Ap )  f ( A1 ). Assume that there is j {1, . . . , p} such that f ( Aj ) is a closed set and if there exists k [0,1) such that for all x  Ai , y  Ai 1 ,  ( r ) d ( fx, Tx)  d ( fx, fy ) implies

H (Tx, Ty )  k max{ d ( fx, fy ), d ( fx, Tx), d ( fy, Ty ),1 / 2[d ( fx, Ty )  d ( fy, Tx)]},

H (Tx, Ty )  k max{ d ( fx, fy ), d ( fx, Tx), d ( fy, Ty ),1 / 2[d ( fx, Ty )  d ( fy, Tx)]},

(2.7)

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Ashish Kumar

where i {1, . . . , p} and Ap 1  A1 , Then there is z A j such that zC ( f , T ). In addition if ffz  fz and either f ( A j 1 ) or f ( Aj 1 ) is a closed set then fz is the common fixed point of f and T.

3. CYCLIC BEST PROXIMITY POINT THEOREMS Let us recall that, if A and B are non-empty subsets of a metric space X then T : A  B  A  B is called a cyclic map if T ( A)  B and T ( B )  A, T is called cyclic contraction [5] if d (Tx, Ty )  kd ( x, y ), 0  k  1,  x  A and

y B. A point x  A  B is a fixed point of T if Tx  x. The central part of the proof of Theorem 1.1 [5] (cf. section 1) is to show that T restricted to A B is a Banach contraction and finally application of Bcp ensures the existence of unique fixed point in A  B   (empty set). Hence it is evident that for the existence of a fixed point of map T the conditions of Theorem 1.1 (cf. section 1) necessitate that A  B be non-empty. There are numerous examples where a given map T is fixed point free; in those cases it is desirable to look for a point which is in the close proximity of Tx. Indeed, a point x  A  B is called a best proximity point of T if d ( x, Tx)  d ( A, B ) where d ( A, B )  inf{d ( x, y ) : x  A, y  B}. Eldred

and

Veeramani

[97]

considered the above situation and presented an interesting generalization of cyclic contraction [5] by modifying it so that it can include the cases when A B is empty. Notice that if A B is non-empty then a fixed point and a best proximity point are the same. Definition 3.1 [97]. Let A and B be non-empty subsets of a metric space X. A map T : A  B  A  B is called a cyclic contraction if it satisfies the conditions: (i) T ( A)  B and T ( B )  A, (ii) d (Tx, Ty )  kd ( x, y )  (1  k ) d ( A, B), 0  k  1,  x  A and y B.

Metric Fixed Point Theory in Context of Cyclic Contractions

169

Notice that cyclic contraction implies that for all x A and y B , the following condition is satisfied

d (Tx, Ty )  d ( x, y ).

(3.1)

A cyclic map satisfying condition (3.1) is called cyclic relatively nonexpansive map which contains nonexpansive maps as a subclass. In 2005, Eldred, Kirk and Veeramani [98] obtained the existence of best proximity point for a cyclic relatively nonexpansive map using geometric notion of proximal normal structure for non-empty, weakly compact and convex subsets A and B of a Banach space. In 2006, Eldred and Veeramani [97] were the first to obtain the existence of a best proximity point for a cyclic contraction in the framework of a metric space. Further, in the settings of a uniformly convex Banach space they proved existence, uniqueness and convergence theorem for a best proximity point using cyclic contraction. Subsequently, the study of best proximity point using cyclic maps became the frontier area of research activities and many interesting results were obtained see, for instance, [99], [101]-[110], [113]-[138], [140]-[147]. The following fundamental result [Prop. 3.1, 97] is instrumental in proving the existence of best proximity point in a metric space. Proposition 3.1 [97]. Let A and B be non-empty subsets of a metric space X and

T : A  B  A  B be a cyclic contraction. Then starting with any x 0 in A  B we have d ( xn , Txn )  d ( A, B), where xn1  Txn , n  0,1, 2, ... . The following best proximity point theorem obtained in metric space settings is due to Eldred and Veeramani [97]. Theorem 3.1 [97]. Let A and B be non-empty closed subsets of a complete metric space X, T : A  B  A  B be a cyclic contraction. Let x 0  A and define xn 1  Txn . Suppose {x 2n } has a convergent subsequence in A. Then there exists x in A such that d ( x, Tx)  d ( A, B ).

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Ashish Kumar A variant of Theorem 3.1 was also obtained in [97] by replacing the

condition that the sequence {x 2n } has a convergent subsequence in A by the condition of bounded compactness of one of the subsets A or B. Indeed they proved the following. Theorem 3.2 [97]. Let A and B be non-empty closed subsets of a metric space X and let T : A  B  A  B be a cyclic contraction. If either A or B is boundedly compact, then there exists x in A  B such that d ( x, Tx)  d ( A, B ). The following is the main result of Eldred and Veeramani [97] which gives the sufficient conditions for existence, uniqueness and convergence of a best proximity point in a uniformly convex Banach space. Theorem 3.3 [97]. Let A and B be non-empty closed and convex subsets of a uniformly convex Banach space. Let T : A  B  A  B be a cyclic contraction, then there exists a unique best proximity point x in A (i. e.,

x  Tx  A  B ). Further, if x0  A and xn1  Txn , then {x 2n } converges to the best proximity point. In 2008, Bari, Suzuki and Vitero [99] introduced the notion of cyclic MeirKeeler contraction as a generalization of cyclic contraction in the context of best proximity point theorems. Definition 3.2 [99]. Let A and B be non-empty subsets of a metric space X. A map T : A  B  A  B is called a cyclic Meir-Keeler contraction if it satisfies the conditions: (i) T ( A)  B and T ( B )  A, (ii) for every   0, there exists a   0 such that

d ( x, y )  d ( A, B)      d (Tx, Ty )  d ( A, B)   , for all x A, y  B.

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Lim [100] gave the characterization of Meir-Keeler contraction and defined the L-function as follows. Definition 3.3. A function  from [0, ) to itself is called an L-function if

 (0)  0,  ( s )  0 for s  (0, ), and for every s  (0,  ) there exists   0 such that  (t )  s for all t  [ s, s   ]. Here we cite an important proposition and a lemma in connection with cyclic Meir-Keeler contraction and L-function. Proposition 3.2 [99]. Let A and B be non-empty subsets of a metric space X and T : A  B  A  B be a cyclic map. Then T is cyclic Meir-Keeler contraction if and only if there exists a (non-decreasing, continuous) L-function  such that (i) d ( x, y )  d ( A, B )  0  d (Tx, Ty )  d ( A, B )   (d ( x, y )  d ( A, B )) and (ii) d ( x, y )  d ( A, B )  0  d (Tx, Ty )  d ( A, B )  0

for

all

x  A, y  B. Lemma 3.1 [99]. Let A and B be non-empty subsets of a metric space X, T : A  B  A  B be a cyclic Meir-Keeler contraction and  be an Lfunction. Then for every ( x, y ) A  B, the following hold (i) d (Tx, Ty )  (d ( x, y )) and (ii) d (Tx, Ty )  d ( A, B )   (d ( x, y )  d ( A, B )). Therefore, every cyclic Meir-Keeler contraction is a cyclic nonexpansive map. In metric space settings Bari et al. [99] obtained the following generalization of Theorem 3.1 by using cyclic Meir-Keeler contraction, they do not require the completeness of space and closedness of subsets A and B.

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Theorem 3.4 [99]. Let A and B be non-empty subsets of a metric space X,

T : A  B  A  B be a cyclic Meir-Keeler contraction. Let x 0  A and define xn 1  Txn . Suppose {x 2n } has a convergent subsequence {x 2 nk } converging to x A . Then x is a best proximity point, that is,

d ( x, Tx)  d ( A, B ) holds. Further, in [99], Theorem 3.3 was generalized by considering cyclic MeirKeeler contraction and the following existence, uniqueness and convergence result was obtained. Theorem 3.5 [99]. Let A and B be non-empty subsets of a uniformly convex Banach space. Let A be closed and convex and T : A  B  A  B be a cyclic Meir-Keeler contraction. Then there exists a unique best proximity point in A. Further, for each x A, {T 2 n x} converges to the best proximity point. Notice that the above theorem does not require subset B to be closed and convex. It can be noted that, just like Matkowski [62], Ciric [63] and Jachymski [58], Bari et al. [99] considered the following combination of a cyclic MeirKeeler contraction and a cyclic contractive condition and obtained the existence, uniqueness and convergence of best proximity point in a uniformly convex Banach space. For every   0, there exists   0 such that for all x A and y B,

x  y  d ( A, B)      Tx  Ty  d ( A, B)   ,

(3.2)

if x A and y B satisfy x  y  d ( A, B), then

Tx  Ty  x  y .

(3.3)

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All the above results are concerning to a map T on the union of two nonempty subsets A and B of metric space X, Karapagam and Agarwal [40] considered map T on the union of p subsets A1 , ... , Ap ( p  2) of X such that

T ( Ai )  Ai 1 for 1 i  p where Ap 1  A1 and called it p-cyclic map. Under such map the distance between the adjacent sets are equal (see [40], [65]). Considering non-empty subsets Ai ,1  i  p, ( p  2) a point x Ai is called a best proximity point if d ( x, Tx)  d ( Ai , Ai 1 ). Karapagam and Agarwal [40] defined p-cyclic Meir-Keeler contraction and proved a best proximity point theorem in metric space they also gave the sufficient conditions for existence and convergence of p-cyclic maps in uniformly convex Banach space. We cite here the following result proved in a metric space setting. Theorem 3.6 [40]. Let A1 , ... , Ap ( p  2) be non-empty subsets of a metric space X and T :  ip1 Ai  ip1 Ai be a p-cyclic Meir-Keeler contraction i.e., if for every   0, there exists   0 such that

d ( x, y)  d ( Ai , Ai 1 )      d (Tx, Ty)  d ( Ai , Ai 1 )   ,

(3.4)

for all x Ai , y  Ai1 , 1 i  p. If for some i and for some x Ai , the sequence

{T pn x} in Ai contains a convergent subsequence {T

pn j

x} converging to

x0  Ai , then x0 is the best proximity point in Ai . Karapagam and Agarwal [41] introduced the concept of cyclic orbital MeirKeeler contraction as a generalization of cyclic Meir-Keeler contraction and obtained the following best proximity point theorem. Theorem 3.7 [41]. Let A and B be non-empty closed and convex subsets of a uniformly convex Banach space. Let T : A  B  A  B be a cyclic orbital Meir-Keeler contraction i.e., for some x A, and for each   0, there exists

  0 such that

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d (T 2n1 x, y)  d ( A, B)      d (T 2n x, Ty)  d ( A, B)   , n N , y  A. (3.5) Then there exists a best proximity point x0  A, such that for every x A, the sequence {T 2 n x} converges to the best proximity point x0 . In 2008, Thagafi and Shahzad [101] defined cyclic  -contraction, which contains Banach contraction as a subclass, and studied the existence and convergence of best proximity point of such maps and thus generalized Theorem 3.1 and Theorem 3.3. A cyclic map on the union of two non-empty subsets A and B of metric space X is called cyclic  -contraction if  : [0, )  [0, ) is a strictly increasing map such that

d (Tx, Ty )  d ( x, y )   (d ( x, y ))   (d ( A, B )) for all x A and y B. (3.6) Eldred and Veeramani [97] posed a question that whether it is possible to ensure the existence of a best proximity point when the ambient space is a reflexive Banach space. Thagafi and Shahzad [101] answered the question affirmatively and obtained the following theorem. Theorem 3.8 [101]. Let A and B be non-empty weakly closed subsets of a reflexive Banach space and T : A  B  A  B be a cyclic contraction. Then there exists ( x, y ) ( A  B ) such that x  y  d ( A, B). Going one more step in the direction of question raised in [97], Thagafi and Shahzad [101] further raised a question about the existence of best proximity point for cyclic  -contraction in a reflexive Banach space. This question was immediately answered positively by Rezapour, Derafshpour and Shahzad [102] (see also Abkar and Gabeleh [103] for similar outcome). To be specific, they proved that the conclusion of Theorem 3.8 remains true if cyclic contraction is replaced by cyclic  -contraction.

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Vetro [104] extended the concept of cyclic  -contraction to a family of p ( p  2 ) non-empty subsets of a metric space and defined p-cyclic  contraction which generalizes the notion of p-cyclic contraction. To be precise, a p-cyclic map T on the union of p non-empty subsets of a metric space X is called p-cyclic  -contraction if there exists a strictly increasing function

 : [0, )  [0, ) such that d (Tx, Ty)  d ( x, y)   (d ( x, y))   (d ( Ai , Ai 1 )) for all x Ai and y  Ai 1 , i 1, ... , p.

(3.7)

For p-cyclic  -contraction, Vetro [104] obtained the existence of a best proximity point in a metric space as well as in a reflexive Banach space. Further, he also proved the existence, uniqueness and convergence of best proximity point in a uniformly convex Banach space. Haddadi and Moshtaghioun [105] considered the following condition on cyclic map T : A  B  A  B and obtained best proximity point theorems in metric space and uniformly convex Banach space to generalized Theorems 3.2 and 3.3 [Th. 3.4 and Th. 3.10, 97].

d (Tx, Ty )   d ( x, y )   [d ( x, Tx)  d ( y, Ty )]   d ( A, B),

(3.8)

for all x, y  A  B, where  ,  ,   0 and   2     1. Karapinar [106] in 2012 obtained unique best proximity point theorem in uniformly convex Banach space by defining generalized cyclic contraction, different variants of generalized cyclic contraction were also discussed in [106] to obtain best proximity point theorems. Let A and B be non-empty subsets of a metric space X. Then the cyclic map T : A  B  A  B is called generalized cyclic contraction if for some

k  (0,1),

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k d (Tx, Ty )  [ d ( x, y )  d ( x, Tx)  d ( y, Ty )]  (1  k ) d ( A, B), 3 for all x  A, y  B. (3.9) Mishra et al. [108] extended the concept of cyclic  -contraction [101] by introducing the notion of cyclic (   ) -weakly contraction and obtained the existence of best proximity point in a metric space as well as proved the existence, uniqueness and converges of best proximity point in uniformly convex Banach space generalizing the results of [97] and [101]. Indeed, let A and B be non-empty subsets of a metric space X. Then a cyclic map T : A  B  A  B is called cyclic (   ) -weakly contraction if for all

x  A, y  B,

 (d (Tx, Ty ))  (d ( x, y ))   (d ( x, y ))   (d ( A, B)),

(3.10)

where  ,  :[0, )  [0, ) are continuous and monotone nondecreasing functions and  (t )   (t )  0 iff t  0. Further, Gabeleh and Shahzad [110] proved the existence of best proximity point in convex metric space by using the following condition for all x A and

y B d (Tx, Ty )  k max {d ( x, y ), d ( x, Tx), d ( y, Ty )}  (1  k ) d ( A, B) , k [0,1). (3.11) Motivated by Kirk’s pointwise contraction [111] (see also Kirk and Xu [112]) Anuradha and Veeramani [113] obtained the existence of a best proximity point in a Banach space for a new class of map called proximal pointwise contraction. Kosuru and Veeramani [114] generalized the main result of [113], they observed that every proximal pointwise contraction is a pointwise cyclic contraction and have shown through example [Ex. 2.3, 114] that the converse of this is not true. Akbar and Gable [115] introduced asymptotic

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proximal pointwise contraction and obtained uniqueness and convergence theorem. Mongkolkeha and Kumam [116] further studied asymptotic proximal pointwise weaker Meir-Keeler type  -contraction for a cyclic map and obtained unique best proximity point theorem in a uniformly convex Banach space. Applying a new approach, Harandi [118] considered two cyclical contractive conditions and defined the concept of cyclic strongly quasicontraction which includes cyclic contraction as a subclass. Definition 3.4 [118]. Let A and B be non-empty subsets of a complete metric space X and T : A  B  A  B be a cyclic map. Then T is said to be cyclic strongly quasi-contraction if: d (Tx, Ty )  k max{ d ( x, y ), d ( x, Tx), d ( y, Ty ), d ( x, Ty ), d ( y, Tx)}  (1  k ) d ( A, B), (3.12)

and

d (T 2 x, T 2 y)  kd ( x, y )  (1  k )d ( A, B),

(3.13)

for all x  A, y  B where k [0,1). Notice that condition (3.12) in the above definition is the cyclic extension of quasi-contraction studied by Ciric [49] (cf. condition 1.19 section 1) which is known to be most general contractive condition. Harandi [118] obtained the following theorems, the first one of these theorems (Theorem 3.9 below, Th. 2.3, [118]) obtained in a metric space for cyclic quasi-contraction map was used to prove the second one which establishes the existence, uniqueness and convergence of best proximity point in uniformly convex Banach space for cyclic strongly quasi-contraction (Theorem 3.10 below, Th. 2.6, [118]).

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Theorem 3.9 [118]. Let A and B be non-empty subsets of a metric space and let

T : A  B  A  B be a cyclic quasi-contraction. For x0  A  B, define

xn1  Txn for each n  0. Then lim n d ( xn , xn1 )  d ( A, B). Theorem 3.10 [118]. Let A and B be non-empty closed and convex subsets of a uniformly convex Banach space and let T : A  B  A  B be a cyclic strongly quasi-contraction. For x0  A , define xn 1  Txn for each n  0. Then there exists a unique

x A such that lim n x2 n  x , T 2 x  x

and

x  Tx  d ( A, B). Harandi [118] raised the question: Does the conclusions of Theorem 3.10 remain true for cyclic quasi-contraction. Dung and Radenovic [119] pointed out an error in the proof of Theorem 3.9 (above) and provided a counterexample to show that Theorem 3.9 is not true if the map T : A  B  A  B is a cyclic quasi-contraction. Further, as the proof of Theorem 3.10 depends on Theorem 3.9 therefore the conclusions of Theorem 3.10 are also not true. Dung and Radenovic [119] obtained the revised versions of Theorems 3.9 and 3.10 respectively which are cited below. Theorem 3.11 [119]. Let A and B be non-empty subsets of a metric space and let the cyclic map T : A  B  A  B be such that for all x  A, y  B and for some k [0,1),

d (Tx, Ty )  k max{ d ( x, y ), d ( x, Tx), d ( y, Ty ), [d ( x, Ty )  d ( y, Tx)] / 2}  (1  k ) d ( A, B). d (Tx, Ty )  k max{d ( x, y ), d ( x, Tx), d ( y, Ty ),[d ( x, Ty )  d ( y, Tx)] / 2}  (1  k ) d ( A, B). For

x0  A  B,

define

lim n d ( xn , xn1 )  d ( A, B).

x n 1  Tx n

for

(3.14) each

n  0.

Then

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Theorem 3.12 [119]. Let A and B be non-empty convex subsets of a uniformly convex Banach space and let cyclic map T : A  B  A  B be such that for all x  A, y  B and for some k [0,1) conditions (3.14) and (3.13) are satisfied. For x0  A  B , define xn 1  Txn for each n  0. Then there exists a unique

x A such that lim n x2 n  x , T 2 x  x and x  Tx  d ( A, B). After obtaining the revised version of Theorem 3.10, Dung and Radenovic [119] posed the question: Prove or disprove Theorem 3.10. Dung and Hang [120] answered the questions raised by both Hadandi [118] and Dung and Radenovic [119]. They illustrated through an example that conclusions of Theorem 3.10 do not remain true if map T : A  B  A  B in Theorem 3.10 is a cyclic quasi-contraction. Further, they answered positively to the above question of Dung and Radenovic [119] and proved Theorem 3.10. If we summarize the above discussion then it may be pointed out that in general two approaches have been used for ensuring the existence of best proximity point of map T. In first approach, the underlying space is a metric space; cyclic map on the union of subsets of metric space assumes different contractive conditions and invariably all the authors define a sequence {x 2n } which has a convergent subsequence in a subset A of metric space X, leading to a best proximity point which is not unique. The second approach pertaining to the best proximity point theorems ensure the existence, uniqueness and convergence of best proximity point with a heavy framework such as either a uniformly convex Banach space or a reflexive Banach space. Moreover, on nonempty subsets of the space some conditions e.g., closedness and convexity are also imposed. Suzuki, Kikkawa and Vetro [128] in 2009, presented a new approach by introducing a geometric like property called property UC for a pair of subsets (A, B) of a metric space X. The significance of property UC is that a metric space structure along with property UC and a suitable cyclic contractive type condition is sufficient not only for the existence of best proximity point but also for uniqueness and convergence of iterates to best proximity point. Therefore, a uniformly convex Banach space or a reflexive Banach space becomes

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redundant when one uses property UC. For study along these lines one may refer to [128]-[138] and references thereof. Definition 3.5 [128]. Let A and B be non-empty subsets of a metric space X. Then pair (A, B) is said to satisfy the property UC if {x n } and {z n } are sequences in A and { y n } is a sequence in B such that: If lim n d ( xn , y n )  d ( A, B)  lim n d ( z n , y n ), then lim n d ( xn , z n )  0 . Following important properties of property UC are worth mentioning. Property 3.1 [128]. The following pair of subsets satisfies the property UC i.

Every pair of non-empty subsets A, B of a metric space X such that

d ( A, B )  0. ii. Every pair of non-empty subsets A, B of a uniformly convex Banach space such that A is convex. iii. Every pair of non-empty subsets A, B of a uniformly convex in every direction Banach space such that A is convex and relatively compact. Suzuki et al. [128] obtained uniqueness and convergence of a best proximity point in a metric space using property UC. Theorem cited below is the generalization of Theorem 3.3. Theorem 3.13 [128]. Let A and B be non-empty subsets of a metric space such that pair (A, B) satisfies the property UC. Assume that A is complete. Let T : A  B  A  B be a cyclic map such that for all x A and y B, the condition (3.11) is satisfied. Then T has a unique best proximity point x A, 2n x is a unique fixed point of T 2 in A and for every x0  A, the sequence {T x0 }

converges to the best proximity point x . Further, T has at least one best

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proximity point in B. Moreover, if (B, A) satisfies the property UC, then Tx is a unique best proximity point in B and {T 2 n y} converges to Tx for every y B. We remark that cyclic contractive condition (3.11) used in the above theorem is general enough to include cyclic contraction. Therefore, the conclusions of Theorem 3.13 remain true if condition (3.11) is replaced by cyclic contraction (see Corollary 1, pp.6, [128]). The following theorem for cyclic Meir-Keeler contraction generalizes Theorem 3.5. Theorem 3.14 [128]. Let A and B be non-empty subsets of a metric space such that pair (A, B) satisfies the property UC. Assume that A is complete. Let T : A  B  A  B be a cyclic Meir-Keeler contraction then the conclusions of Theorem 3.13 hold. We remark that in the light of property 3.1 (ii) it may be ascertained that the property UC naturally comes into force for a pair of subsets A, B of a uniformly convex Banach space (UCED Banach space) when subset A is convex (convex and relative compact). Therefore, Theorem 3.13 contains Theorem 3.3 and Theorem 3.14 contains Theorem 3.5. Suzuki et al. [128] also obtained a generalization of Edelstein’s Theorem [2] using cyclical conditions and property UC. However, they demonstrated through examples that cyclical generalizations of Caristi Theorem [6] and Subrahmanyam Theorem [139] cannot be obtained using property UC. Fakhar el al. [129] defined set-valued cyclic Meir-Keeler contraction and obtained best proximity point theorem in a metric space using property UC. Motivated by Lim’s characterization for Meir-Keeler contraction they used Lfunction to prove the existence of best proximity point. Perhaps, they obtained the first best proximity point theorem for a cyclic set valued map. As defined in [129], a set valued map T : A  B  A  B, where A and B are non-empty subsets of a metric space X, is called a set valued (multivalued) cyclic MeirKeeler contraction if T ( A)  B, T ( B )  A and for every   0, there exists

  0 such that

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d ( x, y )  d ( A, B)      H (Tx, Ty )  d ( A, B)   for all x  A, y  B. (3.15) Following is the main result of Fakhar el al. [129] for set-valued cyclic Meir-Keeler contraction. Theorem 3.15 [129]. Let A and B be non-empty subsets of a metric space X such that pair (A, B) satisfies the property UC. Assume that A is complete and T : A  B  A  B is a set valued cyclic Meir-Keeler contraction such that T(M) is compact for any M C ( A)  C ( B ). Then T has a best proximity point x in A, i.e., d ( x, Tx)  d ( A, B) . Further, if yTx and d ( x, y )  d ( A, B ), then 2 y is the best proximity point in B and x is a fixed point of T .

Neammanee and Kaewkhao [86] in 2011 defined cyclic multivalued contraction and obtained a best proximity point theorem in a metric space using property UC. Let A and B be non-empty subsets of a metric space X. A cyclic multivalued map T : A  B  CB ( X ) is said to be cyclic contraction (on A and B) if

H (Tx, Ty )  k d ( x, y )  (1  k ) d ( A, B), k (0,1) for all x A and y  B.

(3.16)

The following theorem for a multivalued cyclic contraction in a metric space using property UC appeared in [86]. Theorem 3.16 [86]. Let A and B be non-empty subsets of a metric space such that pair (A, B) satisfies the property UC and A is complete. Let T : A  B  CB ( X ) be a cyclic (on A and B) multivalued contraction. Then T has a best proximity point x in A. Recently Magdas [132] extended above theorem by considering the following multivalued Ciric type cyclic map.

Metric Fixed Point Theory in Context of Cyclic Contractions

H (Tx, Ty )   ( M ( x, y )  d ( A, B))  d ( A, B),

183 (3.17)

where  :[ 0, )  [ 0, ) is a comparison function. To take up the discussion further, in [131] the main subject of study was to introduce two properties for a pair of subsets of a metric space namely the WUC property and W-WUC property. These properties stand weaker than the property UC. Espinola and Fernandez-Leon [131] obtained existence, uniqueness and convergence theorems using properties WUC and W-WUC. Further, they succeeded in giving a partial answer in affirmation to the question raised by Eldred and Veeramani [97] that whether a best proximity point exists when A and B are non-empty closed and convex subsets of a reflexive Banach space. For this, they used an additional hypothesis of strict convexity of subsets A and B and required the space to hold property H. We first note the definitions of properties WUC and W-WUC. Definition 3.6 [131]. Let A and B be non-empty subsets of a metric space X. Then pair (A, B) is said to satisfy the WUC property if for any sequence {x n } of points in A such that for every   0 there exists a y B satisfying that

d ( xn , y )  d ( A, B)   for all n  N , then it is the case that {xn } is convergent. Definition 3.7 [131]. Let A and B be non-empty subsets of a metric space X. Then pair (A, B) is said to satisfy the W-WUC property if for any sequence {x n } of points in A such that for every   0 there exists a y B satisfying that

d ( xn , y )  d ( A, B)   for all n  N ,

then there exists a convergent

subsequence {xnk } of {x n }. Notice that W-WUC property is even more general than WUC property. This generality is shown by the following example.

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Example 3.1 [136]. Let X = R be endowed with the usual metric and

1 1 1 A  {1  , n N }  {1  , n N }  { 1} and B  {3  , n N }  {0} be n n n subsets of metric space X. It is clear that d ( A, B )  1 and the pair (A, B) has the W-WUC property. Moreover, for the sequence { (1) n 

(1) n } and for every n

  0 we have (1) n  (1)  0  d ( A, B)   for sufficiently large n but this n

n

sequence does not converge. Therefore, the pair (A, B) does not have the WUC property. The following property establishes relation between property UC and WUC property. Property 3.2 [131]. Let A and B be non-empty subsets of a metric space X such that A is complete. Suppose the pair (A, B) has the property UC. Then pair (A, B) has the WUC property. For the detailed discussion of property UC, WUC property, W-WUC property and HW property and relation among them one may refer to [130], [131], [133], [136] and references thereof. Espinola and Fernandez-Leon [131] proved existence, uniqueness and convergence of a best proximity point for cyclic contraction in a metric space. They replaced property UC by the weaker property WUC even without adding any extra assumption and extended some previously obtained results (see also the remark after Theorem 3.13). Indeed, they obtained the following theorem. Theorem 3.17 [131]. Let A and B be non-empty subsets of a metric space such that pair (A, B) satisfies the WUC property. Assume that A is complete. Let T : A  B  A  B be a cyclic contraction. Then T has a unique best 2n

proximity point x in A, and the sequence {T x0 } converges to x for every

x0  A.

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A similar result for the pair (A, B) of subsets of a metric space satisfying WWUC property was obtained in the same paper. Further an analogous unique best proximity point theorem in strictly convex metric space using property HW was obtained in [133]. The authors in [133] also provided another partial answer to the question posed by Eldred and Veeramani [97]. Before proceeding further we require the following definition. Definition 3.8 [131]. Let A and B be two non-empty subsets of a metric space X. Then A is Chebyshev set for proximal points with respect to B if for any

x B such that d ( x, A)  d ( A, B ) we have that PA (x) is a singleton, where PA ( x)  { y  A : d ( x, y )  d ( x, A)}. Set A is said to be a Chebyshev set with respect to B if PA (x) is a singleton for any x B. We turn attention towards Theorem 3.1 and Theorem 3.2 again. These theorems establish the existence of a best proximity point for a cyclic map T : A  B  A  B in a metric space but do not guarantee the uniqueness and convergence of iterates to the best proximity point. Espinola and FernandezLeon [131] proved the existence, uniqueness and convergence of a best proximity point for cyclic contraction in a metric space with an additional assumption of Chebyshev set. Theorem 3.18 [131]. Let A and B be non-empty closed subsets of a metric space and T : A  B  A  B be a cyclic contraction. Suppose A is boundedly compact and is a Chebyshev set for proximal points with respect to B. Then T 2n

has a unique best proximity point x in A, and the sequence {T x0 } converges to x for every x0  A. Considering cyclic Meir-Keeler contraction, Piatek [130] explored that whether Theorem 3.14 can be obtained by replacing property UC by WUC property or HW property. In this pursuit, Piatek [130] interestingly demonstrated by means of examples that WUC property do not lead to a best proximity point for a cyclic Meir-Keeler contraction in a natural way as

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Theorem 3.14 naturally confirms the existence, uniqueness and convergence of a best proximity point. In order to guarantee the existence, uniqueness and convergence of a best proximity point for cyclic Meir-Keeler contraction with WUC property Piatek [130] needed some extra conditions. Before going to the main result obtained in [130] we consider the following examples in support of above discussion. Example 3.2 [130]. With the suitable choice of metric d (see, [130]), let A be the set of the odd numbers and B the set of the even numbers, and consider the mapping T : N  N given by T (i )  i  1. Then T is a cyclic Meir-Keeler contraction on A  B with no best proximity points and the pair (A, B) enjoys the WUC property. We remark that the above example typically shows two significant things. First, that the sequences of iterates {T n (i) }n1 are unbounded for each i N , this outcome is not in line with the proposition 3.3 proved in [97] which asserts that such sequences of iterates are bounded when T : A  B  A  B is a cyclic contraction. Second, that the pair (A, B) has the WUC property. However, property UC is missing for pair (A, B). Therefore, it is clear that WUC property is wider than property UC. For the next example, let the metric d : N  N  [0, ) be defined as

i j 0, 3, i  j and i  j is even  ( j  (3i 1)) / 2 1  d (i, j )  d ( j , i )  3   , j  3i  1and i  j is odd k 2 k  1  ( 3i 1 j ) / 2 1  3   2k , i  j  3i  1and i  j is odd k 1  Example 3.3 [130]. Let A be the set of the odd numbers and B the set of the even numbers, and consider the mapping T : ( N , d )  ( N , d ) given by

T (i )  i  1. Then T is a cyclic Meir-Keeler contraction on A  B with no best

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proximity points and the pair (A, B) enjoys the WUC property. Moreover, the orbits of T are bounded. The above example presents another situation where the pair (A, B) satisfies WUC property but lacks property UC. Further, it is interesting to note in above example that even after adding the condition that orbits of T are bounded the best proximity is not ensured. Therefore it becomes necessary to rely on some additional conditions to prove existence, uniqueness and convergence of best proximity point for cyclic Meir-Keeler contraction. The following theorem includes these conditions. Theorem 3.19 [130]. Let A and B be non-empty subsets of a metric space, assume A is complete. Let the pair (A, B) satisfies the WUC property and T : A  B  A  B be a cyclic Meir-Keeler contraction for which i.

there is x X with bounded orbit {T n x : n N },

ii. for

each

r  d ( A, B )

there

is

 0

such

that

d ( A, B )    r  d ( A, B )     ( ). Then T has a unique best proximity point x in A, x is a fixed point of T 2 . Moreover, the sequence {T x0 } converges to x for each x0  A. 2n

Recently Fakhar et al. [136] studied cyclic Meir-Keeler contractions in a metric space and obtained existence, uniqueness and convergence of fixed point for single valued maps using W-WUC property. They have shown that the condition that; subset A is Chebysev set with respect to subset B is necessary in their formulation for the uniqueness of fixed point when the pair (A, B) satisfies W-WUC property. They further proved the existence of a best proximity point for set valued cyclic Meir-Keeler contraction in a metric space using WUC property. Theorem 3.20 [136]. Let A and B be non-empty subsets of a complete metric space such that pair (A, B) satisfies the WUC property. Assume that A is closed and T : A  B  A  B is a set valued cyclic Meir-Keeler contraction such

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that T(M) is compact for any M C ( A)  C ( B ). If the following conditions hold:

i. there is U C ( A) with bounded orbit {T n (U ) : n  N }, ii. for

each

r  d ( A, B )

there

is

 0

such

that

d ( A, B )    r  d ( A, B )     ( ). Then T has a best proximity point x in A, i.e., d ( x, Tx)  d ( A, B) . Further, if yTx and d ( x, y )  d ( A, B ), then y is the best proximity point in B and

x is a fixed point of T 2 . Let A and B be non-empty subsets of a metric space X and T, f be the pair of self-maps on A  B. Then T and f are cyclic if f ( A)  T ( A)  B and

f ( B )  T ( B )  A. A point x  A  B is called a common best proximity point for pair (T, f) if d ( x, Tx)  d ( x, fx)  d ( A, B ). For the development and the results pertaining to common best proximity point theorems one may refer to the papers [140]-[147] and references thereof. We close our exposition regarding the development of cyclic fixed point theorems and cyclic best proximity point theorems. Owing to the diversity of the field and limitation in accessing the vast literature we do not claim it to be exhaustive. We attempted to focus mainly on the results obtained in metric spaces. Analogous results obtained in partial metric spaces, partially ordered metric spaces, dislocated metric spaces, metric like spaces, b-metric spaces etc. have not been included in this exposition.

ACKNOWLEDGMENTS Author is grateful to learned referees for their appreciations, observations and valuable suggestions to improve upon the original typescript. Author also expresses his thankfulness to Prof. Rajinder Sharma.

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[103] Abkar, A. and Gabeleh, M. (2010). “Results on the Existence and Convergence of Best Proximity Points.” Fixed Point Theory and Applications 2010: 1-10. doi:10.1155/2010/386037. [104] Vetro, C. (2010). “Best proximity points: convergence and existence theorems for p-cyclic mappings.” Nonlinear Analysis 73: 2283-2291. [105] Haddadi, M. R. and Moshtaghioun, S. M. (2011). “Some results on the best proximity pairs.” Abstract and Applied Analysis 2011: 1-9. doi:10.1155/2011/158430. [106] Karapinar, E. (2012). “Best proximity points of cyclic mappings.” Applied Mathematics Letters 25: 1761-1766. [107] Karapinar, E., and Erhan, I. M. (2011). “Best proximity point on different type contractions.” Applied Mathematics & Information Sciences 5 (3): 558-569. [108] Mishra, S. N., Pant, R and Rao, D. S. (2012). “Fixed and best proximity points for cyclic weakly contraction mappings.” Advances in Fixed Point Theory 2 (2): 135-145. [109] Petric, M. (2011). “Best proximity point theorems for weak cyclic Kannan contractions.” Filomat 25 (1): 145-154. [110] Gabeleh, M. and Shahzad, N. (2014). “Some new results on cyclic relatively nonexpansive mappings in convex metric spaces.” Journal of Inequality and Applications 2014 (350): 1-14. doi:10.1186/1029-242X2014-350. [111] Kirk, W. A. (1970). “Mappings of generalized contractive type.” Journal of Mathematical Analysis and Applications 32: 567–572. [112] Kirk, W. A. and Xu, H. K. (2008). “Asymptotic pointwise contractions.” Nonlinear Analysis 69: 4706–4712. [113] Anuradha, J and Veeramani, P. (2009). “Proximal pointwise contraction.” Topology and its Applications 156: 2942-2948. [114] Kosuru, G. S. R. and Veeramani, P. (2011). “A note on existence and convergence of best proximity points for pointwise cyclic contractions.” Numerical Functional Analysis and Optimization 32(7): 821–830. [115] Abkar, A. and Gabeleh, M. (2011). “Best proximity points for asymptotic cyclic contraction mappings.” Nonlinear Analysis 74: 7261–7268.

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[116] Mongkolkeha, C. and Kumam, P. (2013). “Best proximity points for asymptotic proximal pointwise weaker Meir-Keeler type  -contraction mappings.” Journal of the Egyptian Mathematical Society 21: 87-90. [117] Harandi, A. A. (2018). “Existence and convergence theorems for best proximity points.” Asian-European Journal of Mathematics 11 (1):1-7. doi: 10.1142/S1793557118500055. [118] Harandi, A. A. (2013). “Best proximity point theorems for cyclic strongly quasi-contraction mappings.” Journal of Global Optimization 56:1667– 1674. doi.org/10.1007/s10898-012-9953-9. [119] Dung, N. V. and Radenovic, S. (2016). “Remarks on theorems for cyclic quasi-contractions in uniformly convex Banach spaces.” Kragujevac Journal of Mathematics 40 (2): 272-279. [120] Dung, N. V. and Hang, V. T. L. (2017). “Best proximity point theorems for cyclic quasi-contraction maps in uniformly convex Banach spaces.” Bulletin of Australian Mathematical Society 95(1): 149-156. [121] Gabeleh, M. (2015). “Best proximity points and fixed point results for certain maps in Banach spaces.” Numerical Functional Analysis and Optimization, 36:1013–1028. DOI: 10.1080/01630563.2015. 1041143. [122] Chen, C. and Lin, C. (2012). “Best periodic proximity points for cyclic weaker Meir-Keeler contractions.” Journal of Applied Mathematics 2012: 1-7. doi: 10.1155/2012/782389. [123] Klanarong, C. and Chaobankoh, T. (2018). “Best proximity point theorems for a Berinde MT-cyclic contraction on a semisharp proximal pair.” International Journal of Mathematics and Mathematical Sciences 2018: 1-7. doi.org/ 10.1155/2018/9510402. [124] Du, W. S. and Lakzian, H. (2012). “Nonlinear conditions for the existence of best proximity points.” Journal of Inequalities and Applications 2012 (206): 1-7. [125] Parvaneh, V., Haddadi, M. R. and Aydi, H. (2020). “On best proximity point results for some type of mappings.” Journal of Function Spaces 2020:1-6. doi.org/10.1155/2020/6298138.

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[126] Sahmim, S. Felhi, A. and Aydi, H. (2019). “Convergence and best proximity points for generalized contraction pairs,” Mathematics 7(2): 112. [127] Sen, M. De La, Singh, S. L., Gordji, M. E., Ibeas, A. and Agarwal, R. P. (2013) “Best proximity and fixed point results for cyclic multivalued mappings under a generalized contractive condition.” Fixed Point Theory and Applications 2013 (324): 1-21. [128] Suzuki, T., Kikkawa, M. and Vetro, C. (2009). “The existence of best proximity points in metric spaces with the property UC.” Nonlinear Analysis 71(7-8): 2918-2926. [129] Fakhar, M., Soltani, Z. and Zafarani, J. (2009). “Existence of best proximity points for set-valued cyclic Meir-Keeler contractions.” Fixed Point Theory 10 (2): 1-9. [130] Piatek, B. (2011). “On cyclic Meir-Keeler contractions in metric spaces.” Nonlinear Analysis 74: 35-40. [131] Espinola, R. and Leon, A. F. (2011). “On best proximity points in metric spaces and Banach spaces.” Canadian Journal of Mathematics 63 (3): 533-550. [132] Magdas, A. (2017). “Best proximity problems for Ciric type multivalued operators satisfying a cyclic condition.” Studia Univ. Babeș -Bolyai Mathematica 62 (3): 395-405. doi: 10.24193/subbmath.2017.3.11. [133] Leon, A. F. (2010). “Existence and uniqueness of best proximity points in geodesic metric spaces.” Nonlinear Analysis 73: 915-921. [134] Veeramani, P. and Rajesh, S. (2014). “Best proximity points.” In Nonlinear Analysis, Approximation Theory, Optimization and Applications, edited by Ansari, Q. A., 1-32. India: Springer. [135] Mongkolkeha, C. and Kumam, P. (2018). “Best Proximity Point Theorems for Cyclic Contractions Mappings.” In Background and Recent Developments of Metric Fixed Point Theory, edited by Gopal, D., Kumam, P. and Abbas, M., 201-228.New York: CRC Press Taylor & Francis Group. [136] Fakhar, M., Mirdamadi, F. and Soltani, Z. (2018). “Some results on best proximity points of cyclic Meir-Keeler contraction mappings.” Filomat 32 (6): 2081-2089.

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[137] Aydi, H., Lakzian, H., Mitrovi, Z. D. and Radenovi, S. (2020). “Best proximity points of MT-cyclic contractions with property UC.” Numerical Functional Analysis and Optimization 41(7): 871–882. doi.org/10.1080/01630563.2019.1708390. [138] Gabeleh, M. and Shahzad, N. (2013). “Existence and Convergence Theorems of Best Proximity Points.” Journal of Applied Mathematics 2013:1-6. doi.org/10.1155/2013/101439. [139] Subrahmanyam, P. V. (1974). “Remarks on some fixed point theorems related to Banach’s contraction principle.” Journal of Mathematical and Physical Sciences 8: 445-457. [140] Gabeleh, M., Mary, P. J., Eldred, A. A. and Otafudu, O. O. (2017). “Cyclic pairs and common best proximity points in uniformly convex Banach spaces.” Open Mathematics 2017 (15): 711-723. doi 10.1515/math-2017-0059. [141] Baseri, A. A., Mazaheri, H. and Narang, T. D. (2016). “Common best proximity points for cyclic  -contraction maps.” International Journal of Analysis and Applications 12 (1): 1-9. [142] Gabeleh, M., Otafudu, O. O. and Shahzad, N. (2018). “Coincidence Best Proximity Points in Convex Metric Spaces.” Filomat 32 (7): 2451–2463. [143] Abkar, A. and Norouzian, M. (2020). “Coincidence-best proximity points of contraction pairs in uniformly convex metric spaces.” Journal of Mathematical Extension 14(1): 137-157. [144] Chen, C. M., and Kuo, C. C. (2019). “Best proximity point theorems of cyclic Meir-Keeler-Kannan-Chatterjea contractions.” Results in Nonlinear Analysis 2 (2): 83-91. [145] Kumam, P., Aydi, H., Karapınar, E. and Sintunavarat, W. (2013). “Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems.” Fixed Point Theory and Applications 2013 (242): 1-15. doi: 10.1186/1687-1812-2013-242.

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[146] Khammahawonga, K., Ngiamsunthornb, P. S. and Kumam, P. (2016). “On best proximity points for multivalued cyclic F-contraction mappings.” International Journal of Nonlinear Analysis and Applications 7 (2): 363-374. doi.org/10.22075/ijnaa.2017.2322. [147] Shatanawi, W. (2013). “Best proximity point on nonlinear contractive condition.” Journal of Physics: Conference Series 2013 (435): 1-9. doi: 10.1188/1742-6596/435/1/012006.

In: An In-Depth Guide to Fixed-Point Theorems ISBN: 978-1-53619-565-1 c 2021 Nova Science Publishers, Inc. Editors: R. Sharma and V. Gupta

Chapter 9

A N I NVESTIGATION OF THE F IXED P OINT A NALYSIS AND P RACTICES ¨ Ozer ¨ 1,∗ and A. Shatarah2,† O. Department of Mathematics, Faculty of Science and Arts, Kirklareli University, Kirklareli, Turkey 2 Department of Mathematics, Faculty of Sciences and Art Al Ula, Taibah University, Saudi Arabia 1

Abstract The concept of metric spaces and the related fixed point theorems are of utmost importance in the field of functional analysis. It acts as a major tool to workout the solutions on the problems related to optimization, approximation theory, operator theory, quantum physics, engineering, dynamical systems, computer sciences and other branches of applied mathematics. In 1922, Stefan Banach, a polish mathematician, constructed a result famously known as “Banach’s fixed point theorem” or “Banach contraction principle” which acts as a benchmark in an area of Fixed Point Theroy and its Applications. Thereafter, Fixed point Theory paves a way to many interesting and attractive results in different metric spaces and finds its applications to many real life situations. In this book chapter, we begin with some basic notations, definitions, propositions and theorems considering complete metric spaces such as ∗ †

Email: [email protected] (Corresponding author). Email: [email protected].

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¨ Ozer ¨ O. and A. Shatarah G− metric, s−metric, b−metric, C− metric space, cone metric, d− metric and so on. Then, we define some fixed point theorems in the such spaces. Besides, we recognize a notation called as C ∗-algebra defined recently and several types of C ∗-algebra valued metric space with theories. Finally, we present a fixed point result on the C ∗-algebra valued b− metric space. An application to differential and integral equations is also provided.

1.

INTRODUCTION

In recent years, metric spaces developed in different directions such as fuzzy metric space, generalized metric space, C− metric space, Sb -metric space, D− metric space, cone metric space, quaternion metric space, 2-metric space and so on. After the notions of cone metric space and C ∗ - algebra valued metric space, a lot of fixed point results have been demonstrated and extended. In ([1]), Tianqing extended some results on coupled fixed theorems with mappings satisfying different contractive conditions in the context of complete C ∗ -algebra valued meric spaces. In ([2, 3]), Czerwik considered contractive mappings and attempted to generalize it in a different way on a b- metric spaces. In ([4]), Cosentino et al. gave some common fixed point theorems for self mappings defined on a metric-type space, an ordered metric or a normal cone metric spaces. They also provided some numerical examples and an application to integral equations to support their main results. In ([5]), Jha in his work shed light on some applications of Banach Contraction Principle on a system of linear algebraic equations, ordinary differential equations and integral equations. In ([6]), Lin published a book on the theory of the classification of amenable ∗ C -Algebras which has a significant role in dynamical systems, quantum mechanics and operator representation. In ([7]), Gholamian et al. constructed a new kind of improved metric spaces by using the concepts of b-metric spaces and C ∗ -algebra valued metric spaces. They also investigated a solution of integral equations for its existence and uniqueness. In ([8]), Harandi demonstrated some significant results on the coupled fixed points for quasi contraction type mapping in partially metric spaces and proved tripled fixed point theorems in that metric space too. In ([9]), Murphy’s book has been contributed with significant insights re-

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lated to the fields of basic theorems of operator theory, the Kaplanski density theorem, K−theory, tensor products and representation theory of C ∗ -Algebras. In ([10]), Jiang and Sun considered the mapping with C ∗ - algebra-valued metric spaces and establised some fixed point theorems on it. They proposed a new type of metric spaces and gave some related fixed point theorems for self contractive maps and expansive conditions on such space. In ([11-13]), Ozer and Omran considered different types of C ∗ -algebra valued metric spaces and established related fixed point theorems in it. For more details, see the references provided under the references section.

2.

P RELIMINARIES

2.1.

Introduction to Functional Analysis

Definition 2.1. Assume that X is a set (X 6= ∅), d is a function defined from X × X to R+ and satisfying following conditions. 1. d(α1 , α2 ) = 0 if and only if α1 = α2 . 2. d(α1 , α2 ) = d(α2 , α1 ). 3. d(α1 , α2 ) 6 d(α1 , α3 ) + d(α3 , α2 ), for any α1 , α2 , α3 in X. Then the space (X, d) is called by metric space. Example 2.2. (Discrete metric space) Let X 6= φ be any set. Then the function d : X × X → R is defined by   0, x = y . d(x, y) = 1, x 6= y Then, (X, d) is a discrete metric space. Definition 2.3. Assume that X be a set (X 6= ∅), d be a function defined from X × X to R+ and satisfying following conditions. 1. d(α1 , α2 ) = d(α2 , α1 ). 2. d(α1 , α2 ) 6 d(α1 , α3 ) + d(α3 , α2 ), for any α1 , α2 , α3 in X, if the addition in the right side of the above inequality makes sense.

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Then, the space (X, d) is called by C- metric space. Example 2.4. Let X = {a, b, c} and d be defined as d(a, a) = d(b, b) = d(c, c) = d(a, b) = d(b, a) = 0, d(a, c) = d(c, a) = d(b, c) = d(c, b) = 2. Then, it easily seen that (X,d) is a C-metric space. Definition 2.5. Suppose that X be a non empty set. Then s−metric space on X with a function S : X 3 → R+ satisfies the following conditions: 1. S(x1 , x2 , x3 ) = 0 if and only if x1 = x2 = x3 . 2. S(x1 , x2 , x3 ) 6 S(x1 , x1 , b) + S(x2 , x2 , b) + S(x3 , x3 , b), for all x1 , x2 , x3 , b ∈ X. Also, the pair (X, S) is called by s-metric space. Example 2.6. Let X = Rn and k . k be a norm on X. Then S(δ1 , δ2 , δ3 ) = kδ2 + δ3 − 2δk + kδ2 − δ3 | is an s-metric on R, for all δ1 , δ2 , δ3 , δ ∈ Rn . Definition 2.7. Let X be a set (X 6= ∅), b ≥ 1 be a real number and d be a function defined from X × X to IR+ , satisfying the conditions as follows: 1. d(α1 , α2 ) = 0 if and only if α1 = α2 . 2. d(α1 , α2 ) = d(α2 , α1 ). 3. d(α1 , α2 ) 6 b [d(α1 , α3 ) + d(α3 , α2 )], for any α1 , α2 , α3 in X. Therefore, the space (X, d) is called by b− metric space. Example 2.8. Let X=[0,2] and d : X × X → [0, ∞) be defined by  (x − y)2 , x, y ∈ [0, 1]      | x12 − y12 |, x, y ∈ [1, 2] d(x, y) =      |x − y|, otherwise

So, we can see that (X, d) is a b− metric space.

          

.

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Definition 2.9. Assume that X be a non empty set. Then G-metric space on X is determined with a function G : X 3 → R+ which satisfies the following conditions for all x1 , x2 , x3 ∈ X. 1. G(x1 , x2 , x3 ) = 0 if and only if x1 = x2 = x3 . 2. G(x1 , x1 , x2 ) 6 G(x1 , x1 , x3 ) for all x1 , x2 , x3 ∈ X with x2 6= x3 3. 0 < G(x1 , x1 , x2 ) for all x1 , x2 , x3 ∈ X with x1 6= x2 4. G(x1 , x2 , x3 ) = G(x1 , x3 , x2 ) = G(x2 , x3 , x1 ) = ... ( symmetry in all three variables). 5. G(x1 , x2 , x3 ) ≤ G(x1 , a, a)+G(a, x2 , x3 ) for all x1 , x2 , x3 , a ∈ X. So, the pair (X, G) is called by G− metric space. Example 2.10. Let X = {a, b} and G(a, a, a) = G(b, b, b) = 0, G(a, a, b) = 1, G(a, b, b) = 2 with symmetry in variables. Thus; (X,G) is found as a Gmetric space. Definition 2.11. Let us consider X be a non empty set. Then D-metric space can be defined with a special function D : X 3 → R+ providing satisfies the following conditions for all x1 , x2 , x3 ∈ X. 1. D(x1 , x2 , x3 ) = 0 if and only if x1 = x2 = x3 . 2. D(x1 , x2 , x3 ) = D(p(x1 , x3 , x2 )), where p is a permutation of x1 , x2 , x3 ( symmetry). 3. D(x1 , x2 , x3 ) ≤ D(x1 , x2 , a) + D(x1 , a, x3 ) + D(a, x2 , x3 ) for all x1 , x2 , x3 , a ∈ X. Thus; the pair (X, D) is called as D− metric space. Example 2.12. Let X = R and D∞ (x, y, z) = max{|x − y|, |y − z|, |z − x|}. So, (R, D∞) is obtained as a D− metric spaces. Remark 2.13. Let E be a Banach space over R, P be a subset of E, with a cone in E for int P 6= φ. it is seen that we define a partial ordering ≤ with respect to P by x ≤ y if and only if y − x ∈ P.

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Definition 2.14. Given that X be a set (X 6= ∅), d be a function defined from X × X to E and ensuring conditions as follows: 1. 0 < d(α1 , α2 ) for all α1 , α2 in X and also d(α1 , α2 ) = 0 if and only if α1 = α2 ; 2. d(α1 , α2 ) = d(α2 , α1 ) for all α1 , α2 in X, 3. d(α1 , α2 ) 6 d(α1 , α3 ) + d(α3 , α2 ), for any α1 , α2 , α3 in X. Thereby, the space (X, d) is defined by cone metric space. Example 2.15. Let us consider E = R2 , P = {x, y ∈ E|x, y ≥ 0} ⊂ R2 , X = R and d : X × X → E such that d(x, y) = (|x − y|, α|x − y|), where α ≥ 0 is a constant. Here, (X, d) is a cone metric space. Definition 2.16. A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if it satisfies the following conditions. 1. ∗ is associative and commutative. 2. ∗ is continuous. 3. a ∗ 1 = a. 4. a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1]. Example 2.17. If we consider a ∗ b = ab and a ∗ b = min{a, b}, then ∗ is defined as continuous t-norm. Definition 2.18. Let U be a set and m : U → [0, 1] be a membership function. Then the pair of (U, m) is called by a fuzzy set. Definition 2.19. Let X 6= φ, ∗ be a continuous t-norm. If M is a fuzzy set on X 2 × (0, 1) and satisfies the conditions as follows. 1. M (x, y, t) > 0. 2. M (x, y, t) = 1 if and only if x = y.

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3. M (x, y, t) = M (y, x, t) 4. M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s) > 5. M (x, y, .) : (0, ∞) → [0, 1] is continuous for each x, y, z ∈ X and t, s > 0, then the 3-tuple (X, M, ∗) is called by a fuzzy metric space. Example 2.20. Let (X, d) be a metric space and ψ : R+ → (0, 1) be an increasing and continuous function such that lim ψ(t) = 1. Four different typical t→∞ examples of this function are given as follows: ψ(x) =

−1 x πx , ψ(x) = sin( ), ψ(x) = 1 − e−x , ψ(x) = e x . x+1 2x + 1

If a ∗ b ≤ ab, for all a, b ∈ [0, 1] and we define M (x, y, t) = [ψ(t)]d(x,y), for all x, y ∈ X and each t ∈ (0, ∞), then (X, M, ∗) is determined as a fuzzy metric space. Definition 2.21. Supposing that X be a non empty set and s ≥ 0 be a given real number. Then an Sb -metric space on X can be defined with a function Sb : X 3 → R+ providing the following conditions are satisfied. 1. Sb (x1 , x2 , x3 ) = 0 if and only if x1 = x2 = x3 . 2. Sb (x1 , x2 , x3 ) 6 s[Sb (x1 , x1 , b) + Sb (x2 , x2 , b) + Sb (x3 , x3 , b)], for all x1 , x2 , x3 , b ∈ X. In this case, (X, Sb) is called by Sb -metric space. Example 2.22. Let X = R2 and k . k2 be a norm on X. Then Sb (δ1 , δ2 , δ3 ) = |δ2 + δ3 − 2δ| + |δ2 − δ3 | is an Sb -metric on R, for all δ1 , δ2 , δ3 , δ ∈ R2 . Definition 2.23. Let us consider X a non empty set, k · k be a function defined from X to IR+ satisfying the following conditions:

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1. kαk = 0 if and only if α = 0. 2. kλαk = |λ| kαk. 3. kα1 + α2 ) 6 kα1 k + kα2 k, for any α1 , α2 , α in X. Hence, the space (X, k · k) is called by normed space. Definition 2.24. A normed space (X, k · k) is called as a Banach space if every Cauchy sequence in X converges to x in X. Theorem 2.25. The pair (X, k · k) is a Banach space if and only if every absolutely convergent series is convergent to x in X. Proof. It can be seen in the reference ([14]). Example 2.26. 1. Let I be a unit interval with the lebesgue measure µ. Bep sides, L (I) denotes the vector space of equivalence classes for 1 ≤ p < ∞ and Lebesgue measurable functions are defined as R f : I → R, for which I |f (t)|pdµ(t) < ∞. The set of the Lp (I) also contains f such that Z 1 kf kp = ( |f (t)|pdµ(t)) p . I

Thus, the space

(Lp(I), k ·

kp) is a Banach space.

2. Let I be a unit interval with the lebesgue measure µ and L∞ (I) denotes the vector space of equivalence classes for bounded Lebesgue measurable functions f : I → R. If the normed of f is defined kf k∞ = sup |f (t)|. Then, space (L∞ (I), k · k∞ ) is a Banach space. 3. A sequence (xn ) in a normed space (X, k.k) is said to be absolutely p∞ X summable, if kxn kp < ∞. n=1

Let us consider norm as for (xn ) ∈ `p(X). (`p(X) is the normed space of all absolutely p-summable sequences) kxn kα(p) = (

∞ X n=1

1

kxn kp ) p .

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So, we get (`p (X), k · kα(p)) as a Banach space. Definition 2.27. Let X be a vector space. An inner product on X is a mapping h., .i : X × X → C ensuring the following properties: 1. hx, xi ≥ 0 and hx, xi = 0 if and only if x = 0. 2. hx + y, zi = hx, zi + hy, zi. 3. hαx, yi = αhx, yi. 4. hx, yi = hy, xi for x, y, z ∈ X and α ∈ C. So (X, h., .i) is called normed by Inner product space. Theorem 2.28. A Hilbert space is a complete inner product space. Example. Let T be the unit circle with lebesgue measure µ. Then the space L2 (T ) is a Hilbert space with the following inner product Z hf, gi = f (z) g(z) dµ(z). T

Finally, the concept of standard metric spaces is a fundamental tool in topology, functional analysis and nonlinear analysis. This structure has attracted a considerable attention from mathematicians across the globe because of the development of the fixed point theory in standard metric spaces. In recent years, several generalizations of standard metric spaces have been appeared viz. b− metric space is a generalization of the concept metric space (when b = 1, b− metric space become metric space) and the Sb - metric space is a generalization of b-metric space and s− metric space together. Also, when we omit the first condition in metric space, we get new metric space which is called C− metric space and so on.

2.2.

Linear Operators on Banach Spaces

In this section, we review some basic facts and theorems about linear operators on Banach spaces. Definition 2.29. Let X, Y be complex normed spaces. A mapping T : X → Y is called linear operator if following conditions are satisfied.

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1. T (x + y) = T (x) + T (y) for all x, y ∈ X. 2. T (βx) = β T (x) for all β ∈ C. Definition 2.30. A linear operator T : X → Y is said to be bounded if there is a k > 0 such that kT xkY ≤ kkxkX for all x in X. Theorem 2.31. Let T : X → Y be a linear operator. T is called as continuous if xn → x in X then T xn → T x in Y . Proof. For proof, reader can look at the reference ([14]). Theorem 2.32. Let T : X → Y be a linear operator where X, Y are normed spaces. Then, T is a continuous linear operator if and only if T is bounded. Proof. Reference ([14])includes the proof of the theorem. Definition 2.33. Let L(X, Y ) be a representation of the space of all bounded linear operators from X into Y . If we define the supremum norm such as kT k = sup kT xk, for T ∈ kxk=1

L(X, Y ), then the space (L(X, Y ), k.k) become normed space. Theorem 2.34. If Y is a Banach space, then L(X, Y ) is a Banach space. Proof. Readers can see the proof in the reference ([14]). Example 2.35.

1. If T : C[0, 1] → C[0, 1] is defined as Z x T f (x) = f (x − y)g(y)dy = f ∗ g. 0

Then, T is bounded linear operator. 2. Let T : L2 [0, 1] → L2 [0, 1] be defined by T f = g.f ; such that g ∈ L∞ [0, 1]. Then, T is bounded linear operator. Theorem 2.36. ( Hahn-Banach Theorem) Let X be a normed space and W be a subspace of X and f be a linear functional defined on W . Then there is a linear functional F defined on X such that F (w) = f (w) for all w in W. Also, kF k = kf k.

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Proof. For proof, see ([14]) . Definition 2.37. A linear operator T defined from the normed space X to the normed space Y is called isometry if kT (x)kY = kxkX , ∀x ∈ X. Definition 2.38. Let X and Y be normed spaces. If there exists a bijective isometry linear operator T : X → Y, then X and Y are called isometrically isomorphic. Definition 2.39. Let X be a normed space and T : X → X be a bounded linear operator. λ ∈ C is called a point of resolvent of T if (T − λI) has a bounded inverse operator. Remark 2.40. The set of all points of resolvent of T is denoted by ρ(T ). Definition 2.41. Let X be a Banach space and T : X → X be a linear bounded operator. Then λ ∈ C is called a point of spectrum of T if λ ∈ C \ ρ(T ). Remark 2.42. Following properties are hold. 1. The set of all points of spectrum of T is denoted by σ(T ). 2. σ(T ) 6= φ. 3. σ(T ) is compact set in the C. 4. σ(T ) ∪ ρ(T ) = C and σ(T ) ∩ ρ(T ) = φ. 5. |λ| ≤ kT k, f or all λ ∈ σ(T ). Example 2.43. On the Hilbert sequence’s space X = l 2 , we can define a linear operator T : l 2 → l 2 by (x1 , x2 , , ... ) → (0, x1, x2 , ... ), where x = (xi) ∈ l 2 . The operator is called the right-shift operator. T is bounded ( and kT k = 1) since kT (x)k2 =

∞ X i=1

And also, ρ(T ) = C − {0}.

|xi |2 = kxk2 .

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Definition 2.44. Let X and Y be normed space and Z ⊂ X be an open subset of X. A function f : Z → Y is said to be Frechet differentiable at x ∈ Z if there exist a bounded linear operator µ : X → Y such that kf (x + h) − f (x) − µhkY . khkY = 0 khk→0 lim

Remark 2.45. If there exists a unique operator µ, then we can write Df (x) = µ and it is called as the Frechet derivative of f at x. Definition 2.46. Let X1 , X2, ... , Xn and Y be Banch spaces with f : Y be a function for a fix point x = (x1 , x2 , ... ) ∈

n Y

n Y

Xi →

i=1

Xi . It is said that f has

i=1

an i-th partial differential at the point a if the function φi : Xi → Y which is defined by φi (a) = f (x1 , ... xi−1 , a, xi+1, ..., xn) is Frechet differentiable at the point a. In this case, we define ∂i f (x) = Dφi (xi ) and we call ∂i f (x) is the i-th partial derivative of f at the point x. Example 2.47. Let f be a map, f : U ⊂ Rn → Rm with open set U. If f is a Frechet differentiable at a point a ∈ U, then its derivative is Df (a) : Rn → Rm Df (a)(v) = Jf (a)v where Jf (a) denotes the jacobian matrix of f at a. Furthermore, the partial derivatives of f at a are given by ∂f (a) = Df (a)(ei ) = Ji (a)ei , xi where ei is the orthonormal basis of Rn .

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215

Complete Metric Spaces

In this section, we will consider the complete G- metric space, complete smetric space, complete b-metric space, complete cone metric space, complete D- metric space,...etc. Definition 2.48. Suppose that (X, d) be a metric space and δn be a sequence in X. Then, we obtain followings. 1. If δn is converge to δ in X for a given  > 0, then there exist N in N such that d(δn , δ) 6 , ∀n > N. 2. If δn is a Cauchy sequence in X for a given  > 0, then there exist N in N, such that d(δn , δm) 6 , ∀n, m > N. 3. A Metric Space (X, d) is complete if every Cauchy sequence is convergent to δ ∈ X. Example 2.49. Let I be a unit interval. Then, l p(I) denotes the sequence space for 1 ≤ p < ∞ and sequences are defined as (xn ) ∈ ∞ X p |xn |p < ∞. Also, the set of the l p(I) contains (xn ) such l (I) such that n=0

that

d(x, y) = (

∞ X

1

|xn − yn |p) p ,

n=0

where x = (xn ) and y = (yn ). Thus; the space (l p(I), d) is obtained as a complete metric space. Definition 2.50. Supposing that (X, S) be an s-metric space and δn be a sequence in X. Then, we have followings. 1. If δn is converge to δ in X for a given  > 0, then there exist N in N such that S(δn , δn, δ) 6 , ∀n > N. 2. If δn is a Cauchy sequence in X for a given  > 0, then there exist N in N, such that S(δn, δn , δm ) 6 , ∀n, m > N.

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3. An s-Metric Space (X, S) is complete if every Cauchy sequence is convergent to δ ∈ X. Example 2.51. R represents the set of all real numbers. So, S(δ1 , δ2 , δ3 ) = |δ1 − δ3 | + |δ2 − δ3 | is an s-metric on R, for all δ1 , δ2 , δ3 ∈ R. Besides, (R, S) is called as the complete usual s-metric space. Definition 2.52. Supposing that (X, d) be a b-metric space and δn be a sequence in X. Hence, we get followings. 1. If δn is converge to δ in X for a given  > 0, then there exist N in N such that d(δn , δ) 6 , ∀n > N. 2. If δn is a Cauchy sequence in X for a given  > 0, then there exist N in N, such that S(δn, δm ) 6 , ∀n, m > N. 3. A b-metric space (X, d) is complete if every Cauchy sequence is convergent to δ ∈ X. Example 2.53. Let X = [0, ∞) with   x + y, x 6= y d(x, y) = . 0, x = y Then, (X,d) is a complete metric space. Definition 2.54. Suppose that (X, G) be a G− metric space and δ0 ∈ X, r > 0. Then, we give some results as follows: 1. BG (δ0 , r) = {δ ∈ X : G(δ0 , δ, δ) < r} is called G− ball with center δ0 and radius r. 2. The family of the all G− balls forms is a base of a topology τ (G) on X, and τ (G) is called a G− metric topology. 3. If δn is G− convergence to δ in X if δn → δ is in the G− metric topology τ (G).

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4. If δn is G− Cauchy if G(δn , δm , δl ) → 0 satisfies as m, n, l → ∞. 5. A G -Metric Space (X, G) is G− complete if every G− Cauchy sequence is G− convergent to δ ∈ X. Example 2.55. Let (R, d) be a usual metric space and G(x, y, z) = d(x, y) + d(y, z) + d(x, z) Therefore, (X, G) is a G− complete metric space. Definition 2.56. Suppose that (X, d) be a C− metric space and δn be a sequence in X. Then, following statements holds true. 1. If δn is d− converge to δ in X for a given  > 0, then there exist N in N such that |d(δn, δ)| < , ∀n > N. 2. If δn is a d-Cauchy sequence in X for a given  > 0, then there exist N in N, such that |d(δn , δm )| 6 , ∀n, m > N. 3. A C− Metric Space (X, d) is d-complete if every d-Cauchy sequence is d− convergent to δ ∈ X. Example 2.57. Let C[0, 1] = {f | f : [0, 1] → [0, ∞), f is continuous} and let d be defined by Z 1 d(f, g) = max{f (x), g(x)} dx, 0

for all f, g ∈ C + [0, 1]. Then (C + [0, 1],d) is a d− complete C− metric space. Definition 2.58. Suppose that (X, D) be a D-Metric Space and δn be a sequence in X. Then, following holds true. 1. If δn is D-converge to δ in X for a given  > 0, then there exist N in N such that D(δn , δm , δ) 6 , ∀n, m > N. 2. If δn is a D-Cauchy sequence in X for a given  > 0, then there exist N in N, such that D(δn , δm, δp) 6 , ∀n, m, p > N. 3. A D-Metric Space (X, D) is a D− complete metric space if every D− Cauchy sequence is D− convergent to δ ∈ X.

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Example 2.59. Let X = R and D1 (x, y, z) = |x − y| + |y − z| + |z − x| Then (R, D1) is a D− complete metric space. Definition 2.60. Suppose that (X, d) be a cone metric space and δn be a sequence in X. Then, followings conditions are satisfied. 1. If δn is converge to δ in X if for a every c ∈ E with 0 0 then there exist N in N, such that M (δn , δm , t) 6 1 − , for all n, m ¿N.

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[10] Ma, Z., Jiang, L., and Sun, H., C ∗ -algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2014, (2014) 206. ¨ ¨ Omran, O., S., Common Fixed Point Theorems in C ∗ -Algebra [11] Ozer, O., Valued b-Metric Spaces, AIP Conference Proceedings 1773, 050005 (2016) ¨ ¨ Omran, O., S., On The Generalized C ∗ - Valued Metric Spaces [12] Ozer, O., Related With Banach Fixed Point Theory, International Journal of Advanced and Applied Sciences, Vol.4, Issue.2, (2017), 35-37. ¨ ¨ Omran, O., S., A Note on C ∗ - Algebra Valued G-Metric Space [13] Ozer, O., Related with Fixed Point Theorems, Bulletin of the Karaganda University, Mathematics, No: 3(95), 44-50 (2019). [14] Kreyzig, E., Introductory Functional Analysis with Applications, New York: John Wiley and Sons (1978). [15] Sedghi, S., Dung, V., Fixed point theorems in s-Metric spaces, Journal of Mathematics Bechhk (2014), 113-124. [16] Berinde, V., Pacurar, M., A constructive approach to coupled fixed point theorems in metric space, Carpathian j.Math. Vol. 31 (2015), 277-287. [17] Aydi, H., and Karapinar, E., Tripled fixed points in ordered metric spaces, Bull. Math. Anal. Appl. 4 (2012). [18] Sedghi, S. Shobe, N., A common fixed point theorem in s- metric space, Mathematical Science, (2018), 137-143. [19] Khomdrem, B., Rohen, Y., Some common coupled fixed point theorems in Sb -metric spaces, Fasciculi Mathematic, (2018).

Additional Resources Abbas, M., Chema, I., Razani A., Existence of common fixed point for b-metric rational type contraction, Filomat 30(6) (2016), 1413-1429. Afra, J.M., Double contraction in S-metric spaces, International Journal of Mathematical Analysis, 9(3), (2015), 117-125.

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Then, the pair (S, ∗) is said to be a ∗− Algebra. Definition 2.67. A C ∗ - Algebra K is a Banach algebra over C (the field of complex numbers) with a map k → k∗ , for k ∈ K which satisfies the following properties: 1. It is an involution that k∗∗ = k, ∀ k ∈ K. 2. For all k1 , k2 in K, we have (k1 + k2 ) = k1∗ + k2∗ and (k1 k2 )∗ = k2∗ k1∗ , for all k1 , k2 in K. 3. For every complex number µ and every k1 in K, we have (µk1 )∗ = µ ¯k1∗ . 4. We have kk1 k2∗ k = kk1 kkk2∗ k, For all k1 in K (operator norm) Remark 2.68. The first three identities says that K is K ∗ -algebra. The last identity is called as C ∗ - identity and it is equivalent to kk1 k1∗ k = kk1 k2 . Definition 2.69. Let K1 and K2 be C ∗ - Algebras. Then the mapping µ : K1 → K2 is a C ∗ -Homomorphism if the following conditions are satisfied : 1. µ(k1∗ ) = (µ(k1 ))∗ 2. µ(k1 k2 ) = µ(k1 )µ(k2 ) 3. µ(k1 + k2 ) = µ(k1 ) + µ(k2 ) for all k1 , k2 in K. Remark 2.70. We can say C ∗ - identity( in the remark 2.68) if any homomorphism between C ∗ - Algebras is bounded and norm of less than or equal to 1. Example 2.71. 1. If Mn (C) is a set of all square matrices n × n over C with the involution conjugate transpose, then Mn (C) is C ∗ - Algebra. 2. If H is a Hilbert space, then a B(H) ( which is a collection of all bounded operators, with γ ∗ is the dual of the operator γ : H → H,) is C ∗ - Algebra.

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Definition 2.72. Assume that K be (unital) a C ∗ - Algebra, k in K and k is self adjoint (k∗ = k). Then k is called a positive element if σ(k)(spectrum of k) is a positive real number. Remark 2.73. Let k be a positive element. Then we can denote it by k > 0 and we can assume K+ = {k ∈ K : k > 0} which is called a positive set of K. Lemma 2.74. Let us V be a C ∗ - Algebra. Then, the following statements are true; 1. If k1 ∈ K is a normal, then k1 k1∗ > 0. 2. If k1 ∈ K is a self adjoint and kk1 k 6 1, then k1 > 0. 3. If k1 , k2 ∈ K+ , then k1 + k2 ∈ K+ . 4. K+ is closed in K. Remark 2.75. Let k1 , k2 ∈ K+ . Then it can be defined k1 6 k2 for k2 −k1 > 0. This implies that (K+ , 6) is a partially ordered relation.

2.5.

Fixed Points

Definition 2.76. Let X 6= φ be a set, x ∈ X and f : X → X be a map. Then, x is called a fixed point of f if f (x) = x. Definition 2.77. Let (X, S) be a s- metric space. Then the operator δ, which is defined from X up to X is said to be contractive on X if there exist α ∈ V with α 6 1 and satisfies S(δ(µ1 ), δ(µ2 ), δ(µ3)) 6 α S(µ1 , µ2 , µ3 ), f or all µ1 , µ2 µ3 in X. Now, we present an implicit relation to discuss some fixed point theorems on a s− metric space, ([15]). Let Mk0 be the family of all continuous operators, and Mk ∈ Mk0 such that Mk : K 5 → R. Then, we can consider the following conditions: (C1) Consider that k1 , k2 , k3 ∈ R. If k2 6 Mk (k1 , k1 , 0, k3 , k2 ) and k3 6 2k1 + k2 , then k2 6 k k1 , for some k ∈ R and k < 1. (C2) If k2 6 Mk (k2 , 0, k2 , k2 , 0), then k2 = 0 for all k2 ∈ K.

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Theorem 2.78. Let Tk : (X, S) → (X, S) be a map, where (X, S) is a complete s-metric space, and S(Tk x1 , Tk x2 , Tk x3 ) 6 Mk (S(x1 , x1 , x2), S(Tk x1 , Tk x1 , x1 ), S(Tk x1 , Tk x1 , x2 ), S(Tk x2 , Tk x2 , x1 ), S(Tk x2 , Tk x2 , x2))

for all x1 , x2 , x3 ∈ X and some Mk ∈ Mk0 . Then we have the following results: 1. If Mk satisfies the condition (C1), then Tk has a fixed point. 2. If Mk satisfies the condition (C2) and Tk has a fixed point, then the fixed point is unique. Proof. For proof, reader can look at the reference ([15]). Definition 2.79. Let X 6= φ be a set, (x, y) ∈ X × X and F : X × X → X be a map. Then (x, y) is called a coupled fixed point of F if F (x, y) = x and F (y, x) = y. Definition 2.80. Let (X, d) be a metric space. Then the operator δ : X → X is said to be contractive on X if there exist α ∈ R with α 6 1 and satisfies d(δ(µ1 ), δ(µ2 )) 6 α d(µ1 , µ2 ), f or all µ1 , µ2 in X. Theorem 2.81. Let (X, d) be a metric space. Assume that the mapping δ : X × X → X satisfying d(δ(µ, η), δ(θ, φ)) 6 αd(µ, θ) + βd(η, φ) 1 1 f or all µ, η, φ, θ ∈ X and α, β ∈ R such that α 6 , β 6 . 2 2 In this case, δ has a unique couple fixed point. Proof. For proof, reader can look at the reference ([16]). Definition 2.82. Let X 6= φ be a set, (x, y, z) ∈ X × X × X and F : X × X × X → X be a map. Then (x, y, z) is called a tripled fixed point of F if F (x, y, z) = x, F (y, z, x) = y and F (z, x, y) = z.

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Theorem 2.83. Suppose that (X, d) be a metric space. If the mapping δ : X × X × X → X satisfies following inequality d(δ(µ, η, ζ), δ(θ, φ, s)) 6 αd(µ, θ) + βd(η, φ) + γd(ζ, s), for all µ, η, ζ, θ, φ, s ∈ X and α, β, γ ∈ R such that α 6

(1)

1 1 1 , β6 , γ6 , 3 3 3

then δ has a unique tripled fixed point. Proof. Readers can see the proof of the theorem in the reference ([17]). Definition 2.84. Let X 6= φ be a set, x ∈ X, F : X × X → X be a map and δ : X → X. Then x is called as a common fixed point of F if F (x, x) = δ(x) = x. Definition 2.85. Let (X, S) be an S-metric space. A pair {f, g} is said to be compatible if and only if lim S(f gxn, f gxn, gf xn) = 0, whenever {xn } is a n→∞ sequence in X such that lim f xn = lim gxn = t, for some t ∈ X. n→∞

n→∞

Theorem 2.86. Let f, g, R, and T be self maps of a complete S-metric space (X, S), with f (X) j T (X), g(X) j R(X) and pairs {f, R} with {g, T } are compatible. If S(f x, f y, gz) ≤ q max{S(Rx, Ry, T z), S(f x, f x, Rx), S(gz, gz, T z), S(f y, f y, gz)},

for each x, y, z ∈ X, with 0 < q < 1, then f, g, R, and T have a unique common fixed point in X. Proof. We can see proof of the theorem in the reference ([18]). Definition 2.87. Let X 6= φ be a set, (x, y) ∈ X × X, F : X × X → X be a map with δ : X → X. So, (x, y) is called a common coupled fixed point of F if F (x, y) = δ(x) = x and F (y, x) = δ(y) = y. Definition 2.88. The mapping F : X × X → X and g : X → X are w-compatible if g(F (x, y)) = F (g(x), g(y)) whenever g(x) = F (x, y) and g(y) = F (y, x). Remark 2.89. Let Ψ denote the class of all function ψ : [0, ∞) → [0, ∞) such that ψ is increasing, continuous, ψ(t) < 2t , f or all t > 0 and ψ(0) = 0. It easy to see that for every ψ ∈ Ψ, we can choose k in (0, 21 ) such that ψ(t) ≤ kt.

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Theorem 2.90. Supposing that (X, Sb) be an Sb -metric space. If F : X ×X → X and g : X → X are two mappings such that Sb (F (x, y), F (u, v), F (a, b)) ≤

1 ψ(Sb (g(x), g(u), g(a)) + Sb (g(y), g(v), g(b))) s

for some ψ ∈ Ψ, for all x, y, u, v, a, b ∈ X, while F and g satisfying the following conditions. 1. F (X × X) j g(X). 2. g(X) is complete. 3. F and g are w-compatible then, F and g have a unique common coupled fixed point. Proof. In the reference ([19]), we can easily see the proof of the theorem.

3.

C ∗ - A LGEBRA VALUED b-M ETRIC SPACE

Ma et al. introduced the concept of C ∗ - Algebra valued b-metric space and studied some fixed point theorem in it. Some examples related to C ∗ - Algebra valued b-metric space is discussed in this part. Definition 3.1. Suppose that X be a set (X 6= ∅), V be a C ∗ - Algebra , b ∈ V such that kbk ≥ 1 and d be a function defined from X ×X up to V+ . If following conditions are satisfied, 1. d(α1 , α2 ) = 0 if and only if α1 = α2 . 2. d(α1 , α2 ) = d(α2 , α1 ). 3. d(α1 , α2 ) 6 b [d(α1 , α3 ) + d(α3 , α2 )] b∗ , for any α1 , α2 , α3 in X then, the space (X, V, d) is called C ∗ - Algebra valued b-metric space. Example 3.2. Let X = R and V = Mn (R). If we define metric function as follows; d(µ, θ) = Diagonal M atrix(α1 |µ − θ|p , α2 |µ − θ|p , ... , αn |µ − θ|p ), where µ, θ ∈ R and αi > 0 (i = 1, 2, 3, ...n) are constants and p > 1, then it is seen that (X, V, d) be a C ∗ - algebra valued b-metric space.

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Definition 3.3. Suppose that (X, V, d) be a C ∗ - algebra valued b-metric space, {αn } be a sequence in X. Then αn converges to α in X if there exists N in the set of natural numbers and n > N, such that kd(αn, α)kV+ 6 , f or given  > 0. Definition 3.4. Let (X, V, d) be a C ∗ - algebra valued b-metric space, {αn } be a sequence in X. Then αn is a cauchy sequence in X if there exists N in the set of natural numbers and n, m > N, such that kd(αn, αm )kV+ 6 ,for given  > 0. Definition 3.5. The tripled C ∗ - algebra valued b-metric space (X, V, d) is complete if every Cauchy sequence is convergent to x in X. Definition 3.6. Let (X, V, d) be a C ∗ - algebra valued b-metric space. The operator δ defined X → X is said to be contractive on X if there exist α ∈ V, kαk 6 1 and satisfies following inequalities. d(δ(µ1 ), δ(µ2)) 6 α∗ d(µ1 , µ2 )α, for all µ1 , µ2 in X.

4.

MAIN R ESULTS

Theorem 4.1. Suppose that (X, V, d) be a C ∗ - algebra valued b-metric space and the mapping δ : X → X such that d(δ(µ), δ(θ)) 6 α M ax {d(µ, δ(µ)), d(θ, δ(θ)), d(µ, θ)} α∗ + β {d(µ, δ(θ)) + d(θ, δ(µ))} β ∗ ,

(1)

f or µ, θ ∈ X, α, β ∈ V+ with kα I +2β b Ik 6 1, f or b ∈ V+ and kbk ≥ 1. Then, δ has a unique fixed point in C ∗ - algebra valued b-metric space. Proof. Let µo ∈ X and {µn } be a sequence in X defined by µn = δ(µn−1 ) = δ n (µo ), n = 1, 2, 3, 4, ...

(2)

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226 By (1) and (2), we get d(µn , µn+1 )

= ≤ + ≤

d(δ(µn−1 ), δ(µn )) α M ax {d(µn−1, δ(µn−1 )), d(µn , δ(µn )), d(µn−1 , µn )} α∗ β {d(µn−1, δ(µn )), d(µn , δ(µn−1))} β ∗ α M ax {d(µn−1, µn )), d(µn , µn+1 )), d(µn−1, µn )} α∗

+ ≤ +

β {d(µn−1, µn+1 ), d(µn , µn )} β ∗ α M ax {d(µn−1, µn )), d(µn , µn+1 )), d(µn−1, µn+1 )} α∗ β d(µn−1 , µn ) β , ∗

which implies that d(µn , µn+1 ) ≤ α M ax {d(µn−1 , µn )), d(µn, µn+1 )), d(µn−1 , µn )} α∗ + (b β) {d(µn−1 , µn ) + d(µn , µn+1 )} (b β)∗ ≤ α B1 α∗ + (b β) {d(µn−1 , µn ) + d(µn , µn+1 ) (b β)∗ }, where B1 = M ax {d(µn−1 , µn ), d(µn, µn+1 )}. Now, we have two cases as follows: Case 1. If B1 = d(µn , µn+1 ), then we get d(µn , µn+1 ) ≤ α d(µn , µn+1 ) α∗ + (b β) {d(µn−1 , µn ) + d(µn , µn+1 )} (b β)∗ ≤ α d(µn , µn+1 ) α∗ + (b β) d(µn−1 , µn ) (b β)∗ + (b β)∗ d(µn , µn+1 ) (b β)∗ So, we have B1

≤

α B1 α∗ + (b β) d(µn−1 µn ) (b β)∗ + (b β) B1 (b β)∗

≤

α B12 [α2 B1 + (α∗ )2 B1 + (b β)2 B1 + ((b β)∗ )2 B1 ] 2 (B1 ) 2 α∗

+

(b β) d(µn−1, µn ) (b β)∗ + (b β) (B1 ) 2 (B1 ) 2 (b β)∗ 1 1 (1 (α (B1 ) 2 ) (α (B1 )( ))∗ + (b β) d(µn−1 , µn ) (b β)∗ + ((b β) B1 )) 2 2

≤

1

1

1

1

≤ + +

1

1

((b β) B12 )∗ 1 1 1 1 1 [(α B12 )2 + (α∗ (B1∗ ) 2 )2 + ((b β) B12 )2 + ((b β)∗ (B1∗ ) 2 )2 ] 2 (b β) d(µn−1, µn ) (b β)∗ (b β) d(µn−1, µn ) (b β)∗ .

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Hence, it implies that: 1

(B1 ) 2 [α2 B1 + (α∗ )2 B1 + (b β)2 B1 + ((b β)∗ )2 B1 ] ≤ (b β) d(µn−1 , µn ) (b β)∗ Then, we obtain B1 [I −

1 2 1 [α I + (α∗ )2 I] + [(b β)2 I + ((b β)∗ )2 I]] ≤ (b β) d(µn−1, µn ) (b β)∗ 2 2

and B1 [I − α α∗ I − (b β) ((b β)∗ ) I] ≤ (b β) d(µn−1 , µn ) (b β)∗ If we consider these inequalities, we obtain B1

≤

(b β) d(µn−1 , µn ) (b β)∗ [I − α α∗ I − (b β) ((b β)∗ ) I]

≤

(b β) d(µn−1 , µn ) (b β)∗ (I − α I − (b β) I)(I − α I − (b β) I)∗

≤

(b β) I (b β)∗ I d(µn−1 , µn ) (I − α I − (b β) I) (I − α I − (b β) I)∗

≤ K d(µn−1 , µn ) K ∗ , where K=

(b β) I . (I − α I − (b β) I)

It gives us following results. d(µn , µn+1 ) ≤ K d(µn−1 , µn ) K ∗ , with kKk < 1 and kK ∗ k < 1. So, we obtain result as follows d(µn , µn+1 ) ≤ K 2 d(µn−2 , µn−1 ) (K ∗ )2

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If this process continues, we get d(µn , µn+1 ) ≤ K n d(µ0 , µ1 ) (K ∗ )n Therefore, it is seen that δ is a contractive mapping. Now, we have to show that whether or not {µn }∞ n=1 is a Cauchy sequence in X. Let m, n ∈ N, m > n, and d(µn , µm )

≤

b [d(µn , µn+1 ) + d(µn+1 , µm )] b∗ ,

≤

b d(µn , µn+1 ) b∗ + b2 [d(µn+1 , µn+2 ) + d(µn+2 , µm )] (b∗ )2 ,

≤

b d(µn , µn+1 ) + b2 d(µn+1 , µn+2 ) (b∗ )2 + b2 d(µn+2 , µm ) (b∗ )2 ,

≤

b d(µn , µn+1 ) b∗ + b2 d(µn+1 , µn+2 ) (b∗ )2 + b3 d(µn+2 , µx+3 ) (b∗ )3 + ...,

. . . ≤

1

1

1

(b K n d(µ0 , µ1 ) 2 )(bK nd(µ0 , µ1 ) 2 )∗ + (b2 K n+1 d(µ0 , µ1 ) 2 ) 1

1

1

(b2 K n+1 d(µ0 , µ1 ) 2 )∗ + (b3 K n+2 d(µ0 , µ1 ) 2 )(b3 K n+2 d(µ0 , µ1 ) 2 )∗ + ..., ≤

1

1

kb K nd(µ0 , µ1 ) 2 k2 + kb2 K n+1d(µ0 , µ1 ) 2 k2 1

+kb3 K n+2 d(µ0 , µ1 ) 2 k2 + ..., ≤

kb K nIk2 kd(µ0 , µ1 )k + kb2 K n+1 Ik2 kd(µ0 , µ1 )k + kb3 K n+2 Ik2 kd(µ0 , µ1 )k + ...,

≤

kb K n Ik2 kd(µ0 , µ1 )k[1 + kbKIk2 + kbKIk4 + kbKI|6 + ... ]

≤

kb K nIk2 kd(µ0 , µ1 )k , 1 − kbKIk2

by Geometric Series. Then, we get d(µn , µm ) → 0 as n, m → ∞, since, following result is satisfied

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kb K n Ik2 → 0 as n → ∞ ( kKk < 1 ). 1 − kbKIk2 ∗ Therefore, {µn }∞ n=1 is a Cauchy sequence in X. So, {µn } converges to µ ∈

X.

We need to prove µ∗ is the fixed point of δ. Let us consider

d(µ∗, δ(µ∗ ))

≤ ≤ ≤

b[d(µ∗, µn+1 ) + d(µn+1 , δ(µ∗ ))] b d(µ∗ , µn+1 ) b∗ + b d(δ(µn ), δ(µ∗ )) b∗ b d(µ∗ , µn+1 ) b∗ + (b α)M ax{d(µn , δ(µn )), d(µ∗ , δ(µ∗)),

≤

d(µn , µ∗ )}(b α)∗ + (b β){d(µn , δ(µ∗ )) + d(µ∗, δ(µn ))}(b β)∗ , b d(µ∗ , µn+1 ) b∗ + (b α)M ax{d(µn , µn+1 ), d(µ∗, δ(µ∗ )), d(µn , µ∗ )}(b α)∗ + (b β){d(µn , δ(µ∗ )) + d(µ∗, µn+1 )}(b β)∗ , b d(µ∗ , µn+1 ) b∗ + (b α)M ax{d(µn , µn+1 ), d(µ∗, δ(µ∗ )), d(µn , µ∗ )}(b α)∗ + (b2 β){d(µn , µ∗ )d(µ∗ , δ(µ∗ ))}(b2 β)∗ + (b β)d(µ∗ , µn+1 )(b β)∗ ,

≤

So, we have d(µ∗ , δ(µ∗ )) − (b2 β)d(µ∗ , δ(µ∗))(b2 β)∗ ≤ b d(µ∗ , µn+1 ) b∗ + (b β)d(µ∗ , µn+1 )(b β)∗ + b α)B2 (b α)∗ + (b2 β)d(µn , µ∗ )(b2 β)∗ , where B2 = M ax{d(µn , µn+1 ), d(µ∗ , δ(µ∗ )), d(µn, µ∗ )}. We should consider three different situations as follows. 1. If B2 = d(µn , µn+1 ), then we get d(µ∗ , δ(µ∗ )) − (b2 β)d(µ∗ , δ(µ∗ ))(b2 β)∗ ≤ b d(µ∗ , µn+1 ) b∗ + (b β)d(µ∗ , µn+1 ) (b β)∗ + (b2 α)d(µn , µ∗ )(b2 α)∗ + (b2 α)d(µ∗ , µn+1 )(b2 α)∗ + (b2 β)d(µn , µ∗ )(b2 β)∗,

and d(µ∗ , δ(µ∗ )) [I − (b2 β)(b2 β)∗ I] ≤ [b b∗ I + (b β)(b β)∗ I + (b2 α)(b2 α)∗ I] d(µ∗ , µn+1 ) + [(b2 β)(b2 β)∗ + (b2 α)(b2 α)∗ I]d(µn , µ∗).

It gives us; d(µ∗ , δ(µ∗ )) ≤

b b∗ I + (b β)(b β)∗ I + (b2 α)(b2 α)∗ I d(µ∗ , µn+1 )+ I − (b2 β)(b2 β)∗ I

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(b2 β)(b2 β)∗ I + (b2 α)(b2 α)∗ I d(µn , µ∗ ), I − (b2 β)(b2 β)∗ I Taking limit as n → ∞ , we obtain d(µ∗ , δ(µ∗ )) → 0 as n → ∞. it implies that µ∗ = δ(µ∗ ). So, µ∗ is the fixed point of δ. 2. If B2 = d(µn , µ∗ ), then we get d(µ∗ , δ(µ∗ )) − (b2 β)d(µ∗ , δ(µ∗ ))(b2 β)∗ ≤ b d(µ∗ , µn+1 ) b∗ + (b β)d(µ∗ , µn+1 )

(b β)∗ + (b α)d(µn , µ∗ )(b α)∗ + (b2 β)d(µn , µ∗ )(b2 β)∗ , and d(µ∗ , δ(µ∗)) [I − (b2 β)(b2 β)∗ I] ≤ [b b∗ I + (b β)(b β)∗ I] d(µ∗ , µn+1 ) + [(b2 β)(b2 β)∗ + (b α)(b α)∗ I]d(µn , µ∗ ). So, we have d(µ∗ , δ(µ∗)) ≤

b b∗ I + (b β)(b β)∗ I d(µ∗ , µn+1 )+ I − (b2 β)(b2 β)∗ I

(b2 β)(b2 β)∗ I + (b α)(b α)∗ I d(µn , µ∗ ). I − (b2 β)(b2 β)∗ I Taking limit as n → ∞ , we obtain d(µ∗ , δ(µ∗ )) → 0 as n → ∞, this proves that µ∗ = δ(µ∗ ). hence µ∗ is the fixed point of δ.

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3. If suppose that B2 = d(µ∗ , δ(µ∗ )), then we get d(µ∗ , δ(µ∗ )) − (b α)d(µ∗ , δ(µ∗ ))(b α)∗ − (b2 β)d(µ∗ , δ(µ∗ ))(b2 β)∗ ≤ b d(µ∗ , µn+1 ) b∗ + (b β)d(µ∗ , µn+1 )(b β)∗ + (b2 β)d(µn , µ∗ )(b2 β)∗ , and d(µ∗ , δ(µ∗ )) [I −(b α)(b α)∗ I −(b2 β)(b2 β)∗ I] ≤ [b b∗ I +(b β)(b β)∗ I] d(µ∗ , µn+1 ) + [(b2 β)(b2 β)∗ ]d(µn , µ∗ ). Then d(µ∗ , δ(µ∗ )) ≤

b b∗ I + (b β)(b β)∗ I d(µ∗ , µn+1 )+ I − (b α)(b α)∗ I − (b2 β)(b2 β)∗ I

(b2 β)(b2β)∗ I d(µn , µ∗ ). I − (b α)(b α)∗ I − (b2 β)(b2 β)∗ I Taking limit n → ∞ for two sides, we get d(µ∗ , δ(µ∗ )) → 0 as n → ∞, so, we obtain µ∗ = δ(µ∗ ). Therefore µ∗ is the fixed point of δ. Case 2. If B1 = d(µn−1 , µn ). In this case, we can change indices by substituting m = n − 1. So, it easily seen that we can back to case1 and get result.

4.1.

Uniqueness of Fixed Point

We need to prove that µ∗ is unique fixed point of δ. Assume that µ0 is another fixed point of δ and following equation is satisfied δ(µ0 ) = µ0 .

¨ Ozer ¨ O. and A. Shatarah

232 d(µ∗ , µ0 ) = d(δ(µ∗ ), δ(µ0 ))

≤ + ≤

α M ax{d(µ∗, δ(µ∗ )), d(µ0 , δ(µ0 )), d(µ∗ , µ0 )}α∗ β {d(µ0 , δ(µ∗ )) + d(µ∗ , δ(µ0 ))} β ∗ α M ax{d(µ∗, µ∗ ), d(µ0 , µ0 ), d(µ∗, µ0 )} α∗

+ ≤ ≤

β {d(µ0 , µ∗ ) + d(µ∗, µ0 )} β ∗ α d(µ∗ , µ0 )} α∗ + 2 (β d(µ0 , µ∗ ) β ∗ ) [α α∗I + 2 β β ∗ I] d(µ0 , µ∗ ).

This is a contradiction. So, we obtain µ∗ = µ0 . Hence µ∗ is the unique fixed point of δ. Corollary 4.2. Let (X, V, d) be a C ∗ - algebra valued b-metric space, δ be a mapping from X → X satisfying d(δ(µ), δ(θ)) 6 α {d(µ, δ(µ)) + d(θ, δ(θ))} α∗ + β {d(µ, δ(θ)) + d(θ, δ(µ))} β ∗ ,

(1)

where µ, θ ∈ X, α, β ∈ V+ such that k2α I + β (2bI + I)k < 1 where b ∈ V+ , with kbk ≥ 1. Then, δ has a unique fixed point.

5.

SOLUTION OF D IFFERENTIAL E QUATIONS AND INTEGRAL E QUATIONS U SING F IXED P OINT T HEORY

In this section, we introduce the concept of Fixed Point Theory to solve Differential Equations and integral equations.

5.1.

Lipschitz Mapping

Definition 5.1. Let (X, V, d) be a C ∗ - algebra valued b-metric space. The Map δ : X → X is said to be Lipschitzian (contractive) if there exists a constant σ(δ) > 0 such that d(δ(µ), δ(θ)) ≤ σ(δ) d(µ, θ) (σ(δ))∗, ∀ µ, θ ∈ X. Definition 5.2. A Lipschitzian mapping with Lipschitzian constant. σ(δ), is called contraction constant, where kσ(δ)k < 1.

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Definition 5.3. If (X, V, d) is a C ∗ - algebra valued b-metric space and δ : V ⊂ X → X is a map, then a solution of δ(x) = x is called a fixed point of δ. Example 5.4. Let (X, V, d) be a C ∗ - algebra valued b-metric space and f (x, y) be continuous operator on [x1 , x2 ] × [y1 , y2 ], where [x1 , x2 ] and [y1 , y2 ] denote lines segments in open sets U, V ⊂ X. Then the initial value problem is to find a continuous Frechet differentiable operator y on V satisfying the differential equation dy = f (x, y), y(x0 ) = y0 . dx If we define integral operator δ : L(U ) → L(U ) such that Z x δ(y(t)) = y0 + f (t, y(t)) dt x0

then, a solution of initial value problem corresponds with a fixed point of δ. Example 5.5. Let us consider the Fredholm integral equation for an unknown function y : [a, b] → R Z b y(x) = f (x) + λ k(x, t)y(t).dt (1) a

where, k(x, t) is continuous on [a, b] × [a, b] and f (x) is continuous on [a, b]. we define an integral operator T : C[a, b] → C[a, b] by T y(x) = f (x) + λ

Z

b

k(x, t)y(t).dt

(2)

a

Then, a solution of the Fredholm integral equation for an unknown function corresponds with a fixed point of T. Theorem 5.6. Let k(x, t) is a continuous on [a, b] × [a, b] with M = sup{|k(x, t)| : x, t ∈ [a, b]}, f is a contraction function on [a,b], and λ is a real number such that M (b − a)|λ| < 1. Then the fredholm integral equation (1) has a unique solution.

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Proof. It is suffecient to show that the mapping T defined by (2) is a contraction. For two continuous functions y1 , y2 , we have Z b kT y1 − T y2 k = sup |λ| | k(x, t)[y1 (t) − y2 (t)].dt| x∈[a,b]

a

≤ |λ| sup

Z

b

|k(x, t)||y1(t) − y2 (t)|.dt

x∈[a,b] a

Z

≤ |λ| M sup

b

|y1 (t) − y2 (t)|.dt

t∈[a,b] a

= |λ| M ky1 − y2 k

Z

b

dt

a

≤ ky1 − y2 k. So, y1 = y2 . Example 5.7. We can consider Volterra integral equation as follow Z t x(t) = f (t) + µ k(t, s)x(s)ds

(2)

a

where f (t) ∈ C[a, b] as initial function x0 (t). So, we introduce the fixed point iteration. Z t xn+1 (t) = (T (xn))(t) = f (t) + µ k(t, s)x(s)ds, t ∈ C[a, b], n ≥ 0, a

for n is a natural number. x1 (t), x2(t), ..., xn(t), ...

We can also find approximate sequence of

The approximation sequence will be converging to a function that indicates the completion of the problem above. Example 5.8. we can look at the volterra integral equation (2) and supposing that f ∈ C[a, b], µ ∈ [0, 1) and k(t, s) is a continuous function on R = {(t, s).a < s < t, a < t < b}. We can define operator as follows (T x)(t) = f (t) + µ

Z

t

k(t, s)x(s)ds

a

Obviously, the solution of equation (2) is the fixed point of operator T.

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Remark 5.9. Fredholm equations arise naturally in the theory of signal processing. The operators involved are the same as linear filters. They also commonly arise in linear forward modeling and inverse problems. In physics, the solution of such integral equations allows for experimental spectra to be related to various underlying distributions, for instance the mass distribution of polymers in a polymeric melt or the distribution of relaxation times in the system. In addition, Fredholm integral equations also arise in fluid mechanics problems involving hydrodynamic interactions near finite-sized elastic interfaces.

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ABOUT THE EDITORS Dr. Rajinder Sharma Faculty University of Technology and Applied Sciences-Sohar (Formerly College of Applied Sciences-Sohar), Oman Email: [email protected]

Dr. Rajinder Sharma is associated with Ministry of Higher Education, Oman and is currently posted as a faculty of Mathematics at University of Technology and Applied Sciences-Sohar and would like to revolve his career around teaching and research. He has more than 17 years of teaching and research experience. His research focuses on the Non Linear Analysis and Mathematical Modeling. In addition to his research work on FPT and its applications, he avidly studied Non Linear Mathematical models in Epidemiology and Population dynamics and published articles in the said area. He acted as a reviewer to many peer reviewed Scopus indexed International Mathematical Journal viz. Italian Journal of Pure and Applied Mathematics, Thai Journal of Mathematics, Annals of Fuzzy Mathematics and Informatics (Korea), Journal of Intelligent and Fuzzy Systems, Computational and Mathematical Methods, MFPT in Science, Engineering and Behavioral Sciences (Springer), Cogent Mathematics and Statistics (Taylor and Francis), Journal of Applied Research and Technology (JART), Mat lab) etc. Additionally, he is actively reviewing articles for Mathematical

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About the Editors

Reviews (An American Mathematical Society entity) and also acted as an external examiner for Master’s and PhD thesis to different universities worldwide. After joining UTAS-Sohar in 2011, he had more than 25 articles in his credit with CAS-Sohar affiliation in peer reviewed Indexed International Journals published from USA, Italy, Jordan, Malaysia, Bulgaria, Bosnia, Brazil, Spain and Herzegovina and Romania. He is a member of SIAM-USA and Ex-member Technology Transfer Office-UTAS-Sohar. He is a recipient of World Intellectual Property Organization (WIPO) scholarship for the advanced courses titled Advanced Course on Software Licensing Including Open Source, Intellectual Property, Traditional Knowledge and Traditional Cultural Expressions and Promoting Access to Medical Technologies and Innovation. He is a country representative (Oman) to IMMC and member of many international scientific research bodies. During COVID19-Pandemic, his lectures are available on Lecturers without Borders organization’s webpage to help and support the students worldwide. Most recently, he acted as a Judge for SCUDEM V 2020-A SIMIODE challenge contest using Differential Equations Modeling.

Dr. Vishal Gupta Professor Maharishi Markandeshwar (Deemed to be University), Mullana, India

Dr. Vishal Gupta, having more than 11 years of teaching experience, is currently working as professor in Department of Mathematics, Maharishi Markandeshwar (Deemed to be University), Mullana, India. He received his PhD degree in Mathematics in 2010. Also, he earned the degree of MPhil in Mathematics and MEd. He has published one research book with international publisher and his immense contribution in journals of national and international repute is more than eighty. He has presented more than fifty research papers in national and international conferences. His research

About the Editors

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interests are fixed point theory, operator theory, aggregation function, fuzzy set theory and fuzzy mappings, topology, applications of fixed point theory in medical science, in graph theory and differential and integral equations.

INDEX

A algorithm, 51, 52

B Banach spaces, 52, 65, 77, 78, 79, 198, 200, 201, 202, 211 best proximity point, vii, x, 109, 111, 113, 115, 117, 118, 119, 138, 169, 170, 171, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 186, 187, 188, 189, 191, 195, 196, 197, 198, 199, 200, 201, 202

cyclic



contraction, 138

cyclic contraction, viii, x, 137, 138, 140, 147, 153, 162, 163, 169, 170, 171, 176, 177, 178, 182, 183, 184, 186, 187, 192, 193, 197, 199, 200, 201 cyclic Meir-Keeler contraction, 138, 152, 153, 171, 172, 173, 174, 175, 182, 183, 187, 188, 189, 193, 196, 198, 201

D derivatives, 214 differential equations, 52, 193 dynamical systems, 204

C

F

Cantor set, 40, 41 comparison function, 28, 31, 32, 33, 34, 37, 39, 42, 44, 138, 147, 148, 149, 150, 151, 152, 159, 184 completion of b-metric space, 121 convergence, 11, 25, 46, 51, 52, 59, 64, 78, 118, 135, 146, 151, 170, 171, 173, 174, 175, 176, 178, 179, 181, 182, 184, 186, 187, 188, 189, 194, 195, 198, 199, 216

Fixed point, vii, viii, ix, 23, 24, 25, 28, 29, 31, 33, 35, 37, 39, 41, 43, 45, 46, 47, 48, 49, 77, 79, 80, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 104, 105, 108, 118, 119, 126, 134, 137, 138, 164, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227,

Index

252 229, 231, 232, 233, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245 fractal theory, 27 functional analysis, 51, 82, 203, 211 fuzzy set theory, ccxlix

O obstacles, xi operations, 5, 10, 14, 134, 237 optimization, 203 orbit, 188, 189

G graph, 27, 237, 238, 239, ccxlix

H Hilbert space, 52, 157, 196, 211, 220, 237

I if and only if, 5, 7, 8, 9, 10, 16, 17, 20, 30, 75, 83, 128, 147, 172, 205, 206, 207, 208, 209, 210, 211, 212, 223, 224 iteration, x, 48, 52, 56, 57, 58, 59, 62, 63, 78, 79, 149, 150, 151, 194, 234

K K+, 19, 20, 21, 22, 221

L linear function, 46, 77, 212

M metric equivalent to b - metric, 121 metric spaces, x, 2, 3, 4, 7, 13, 22, 23, 24, 25, 28, 29, 46, 47, 48, 49, 52, 77, 82, 87, 91, 99, 103, 104, 105, 110, 121, 126, 127, 134, 135, 157, 164, 190, 191, 192, 193, 194, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 207, 211, 219, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245

P p-cyclic contraction, 138, 176, 195 probability, 52, 53, 55, 63, 68, 69, 74, 75, 78, 79 property UC, 138, 181, 182, 183, 184, 185, 186, 187, 188, 200, 201

R radius, 76, 216 real numbers, 55, 58, 122, 139, 216

S set theory, ix, 5 structure, 10, 82, 119, 146, 170, 181, 198, 211 successive approximations, 46, 135, 194 symmetry, 19, 122, 207

T topology, ix, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 14, 16, 63, 64, 79, 108, 216, 244, ccxlix

V variables, 53, 54, 56, 63, 68, 69, 70, 75, 78, 207 vector, 1, 2, 3, 5, 10, 12, 14, 23, 63, 64, 86, 108, 112, 210, 211, 242

Index W WUC property, 138, 184, 185, 186, 187, 188, 189 W-WUC property, 138, 184, 185, 186, 189

253

b b - metric, 121, 122, 123, 125, 127, 129