Metric Structures and Fixed Point Theory [1 ed.] 0367689146, 9780367689148

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Metric Structures and Fixed Point Theory

Metric Structures and Fixed Point Theory

Edited By

Dhananjay Gopal Praveen Agarwal Poom Kumam

First edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN © 2021 selection and editorial matter, Dhananjay Gopal, Praveen Agarwal and Poom Kumam; individual chapters, the contributors CRC Press is an imprint of Taylor & Francis Group, LLC The right of Dhananjay Gopal, Praveen Agarwal and Poom Kumam to be identified as the authors of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging- in- Publication Data A catalog record has been requested for this book ISBN: 978-0-367-68914-8 (hbk) ISBN: 978-1-003-13960-7 (ebk) ISBN: 978-0-367-68917-9 (pbk) Typeset in Palatino by Newgen Publishing UK

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Symmetric Spaces and Fixed Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Pradip Ramesh Patle and Deepesh Kumar Patel 2 Fixed Point Theory in b-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Nguyen Van Dung and Wutiphol Sintunavarat 3 Basics of w-Distance and Its Use in Various Types of Results . . . . . . . . . . . . . . . . . . . . . . 67 Dhananjay Gopal and Mohammad Hasan 4 G-Metric Spaces: From the Perspective of F-Contractions and Best Proximity Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Vishal Joshi and Shilpi Jain 5 Fixed Point Theory in Probabilistic Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Juan Martínez-Moreno 6 Fixed Point Theory For Fuzzy Contractive Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Dhananjay Gopal and Tatjana Došenovi´c 7 Set-Valued Maps and Inclusion Problems in Modular Metric Spaces. . . . . . . . . . . . . 245 Poom Kumam 8 Graphical Metric Spaces and Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Satish Shukla 9 Fixed Point Theory in Partial Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Dhananjay Gopal and Shilpi Jain Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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Preface One of the intuitively natural concept in the history of human beings is notion of distance. Though the notion was known but it was axiomatically formulated by Frechet in early 1900’s. The realization of the fact that Euclidean distance between two points can be given by the absolute difference, Frechet come up with a abstract formulation and generalization of the distance concept (termed as metric). It is an indisputable argument that the formulation of metric opens a new subject in mathematics called non-linear analysts after the appearance of Banach fixed point theorem. Because the underlined space of this theorem is a metric space, the theory that developed following its publications is known as the metric fixed point theory. It is well known that metric fixed point theory provides essential tools for solving problems arising in various branches of mathematics and other sciences such as split feasibility problems, variational inequality problems, nonlinear optimization problems, equilibrium problems, complementarity problems, selection and matching problems, and problems of proving an existence of solution of integral and differential equations are closely related with fixed point theory. Due to this reason over the last seventy years many people have tried to generalize the definition of metric space and corresponding fixed point theory. As a result, the notions like Hausdorff metric, symmetric, quasi-metric, semi-metric, probabilistic metric, fuzzy metric, b-metric, w-distances, G-metric, D-metric, modular metric and so on have been appeared in the literature and this trend is still going on. However, most of the materials on these topics are scattered either in the form of research papers or general articles. Also numerous books are available on fixed point theory but we felt that no book could be used systematically to follow on the above topics. Considering these aspect, the aim of this monograph is to provide a systematic survey and latest updates on the most popular generalizations of metric spaces and corresponding fixed point results which fall within the scope of metric fixed point theory. This monograph is divided into nine chapters. Each chapter contributed by different authors contains a section “Introduction” which summarizes the material needed to read the chapter independent of others and contains a necessary background, several examples, and comprehensive literature to comprehend the concepts presented therein. This could be helpful for those who want to pursue their research career in metric fixed point theory and its related areas. Chapter One presents a brief study of fixed point theorems in symmetric (semimetric) spaces. Taking into consideration the importance, we also discussed the fixed point of multivalued mappings and common fixed point theorems of two mappings with various contractive conditions. In addition, the result due to Suzuki which proves the characterization of semi- completeness by means of fixed point property is also presented which provides quite self containment to this chapter. Finally, some open problems are listed for future scope and study. Chapter Two deals with comprehensive survey and motivation of Fixed Point Theory in b-Metric Spaces. A short history of the b-metric space and its basic properties are presented in Section 1. The metrization and completion results of b-metric spaces are shown in Section 2. Section 3 is devoted to show that many fixed point results in b-metric spaces

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have been proven similar to that in metric spaces. Multivalued fixed point results in b metric spaces are presented in Section 4. Finally, Section 5 is devoted to present the relaxations of constant contractions in b-metric spaces. Chapter Three presents the basics of w-Distances and well known results like Caristi fixed point theorem, Ekeland variational principle, the nonconvex minimization theorem according to Takahashi, Ciric fixed point theorem in the setting of w-Distances. Chapter Four deals with the study of those fixed point results in G-metrics paces that cannot be obtained from the existing results in the setting of associated metric spaces. The main motive to introduce such results was to emphasize the applicative approach of fixed point theory. Chapter Five presents concise study of fixed point results concerning various classes of probabilistic contractions including altering distance functions and probabilistic nonlinear contractions. We hope that the results presented in this chapter illustrate the direction of research over the last five decades up to the most recent contributions for fixed point theory of single valued mappings in PM spaces. Chapter Six deals with a concise study of fixed point theorems for fuzzy contractive type mappings in fuzzy metric spaces. The results presented in detail were selected to illustrate the direction research in the field has taken from last four decades up to most recent contribution in the subject. Chapter Seven present recent developments of set-valued analysis based mainly on the fixed point inclusion problems. This chapter is organized as follows: The succeeding Section 2 is devoted to the mathematical analysis of fixed point inclusion problems and also the common fixed point of several classes of set-valued maps. In the Section 3, we provide the applications of fixed point results presented in the previous section. We consider here two main nonlinear problems- abstract economy and fractional integral inclusion. Chapter Eight is mainly concern to study of an approach to fixed point theory via graphical metric structure. In view of the fact that the class of contractions in graphical metric spaces is larger than that in metric spaces, we see that the fixed point results in graphical metric spaces can be applied on a larger class of existence problems which use the fixed point methods, e.g., existence of solution of an integral equation. As well as, apart from the topology of usual metric spaces, the topology generated by graphical metric spaces are non-Hausdorff which can be considered as a useful tool in theories where nonHausdorffness occurs, e.g., in algebraic geometry, in representation theory, in the theory of C?-algebras, in domain theory, in computer applications etc., this topology can be used as a substitute of existing tools. Chapter Nine is devoted to provide a systematic survey on the various aspects of partial metric spaces and concerning fixed point theory. But still there are a lot of topological as well as fixed point aspects which remain unknown about partial metric spaces.

Acknowlegments We greatly admire and are deeply indebted to our friends and colleagues working in the Fixed Point Theory for their encouragement and support, especially Professor Sompong Dhompongsa, Chiang Mai University, Thailand, Professor Yeol-Je Cho, Gyeongsang National University, Jinju, Korea, Professor Ismat Beg, Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan and Professor S. Radenovic, Faculty of Mathematics and Statistics, Ton Duc Thang University, Vietnam. In particular, we wish to express our deepest thanks to our colleagues who contributed their recent research work in the form of chapters for inclusion in this book. Our deepest gratitude and thanks are also due to our family members who always encouraged us and refreshed our energies with their sweet words while we were busy in accomplishment of this project. The editors are very thankful to Aastha Sharma, Shikha Garg and the staff at CRC Press for their unfailing support cooperation and patience in publishing this book.

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Editors Dhananjay Gopal has a doctorate in Mathematics from Guru Ghasidas University, Bilaspur, India and is currently Associate Professor of Mathematics in Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur (C.G.), India. He is author and co-author of several papers in journals, proceedings and a monograph on Background and Recent Developments of Metric Fixed Point Theory and a book on Introduction to metric spaces. He is devoted to general research on the theory of Nonlinear Analysis and Fuzzy Metric Fixed Point Theory. Dr. Gopal has active research collaborations with KMUTT, Bangkok, Thammasat University Bangkok, Jaen University Spain and in his research pursuits he has visited South Africa, Thailand, Japan and Iran.

Dr. P. Agarwal earned his Master’s degree in Mathematics from Rajasthan University in 2000. In 2006, he earned his Ph.D. (Mathematics) at the Malviya National Institute of Technology (MNIT) in Jaipur, India, one of the highest ranking universities in India. Dr. Agarwal has been actively involved in research as well as pedagogical activities for the last 20 years. His major research interests include Special Functions, Fractional Calculus, Numerical Analysis, Differential and Difference Equations, Inequalities, and Fixed Point Theorems. He is an excellent scholar, dedicated teacher, and prolific researcher. He has published 7 research monographs and edited volumes and more than 150 publications (with almost 100 mathematicians all over the world) in prestigious national and international mathematics journals. Dr. worked previously either as a regular faculty or as a visiting professor and scientist in universities in several countries, including India, Germany, Turkey, South Korea, UK, Russia, Malaysia and Thailand. He has held several positions including Visiting Professor, Visiting Scientist, and Professor at various universities in different parts of the world. Specially, he was awarded most respected International Centre for Mathematical Sciences (ICMS) Group Research Fellowship to work with Prof. Dr. Michael Ruzhansky-Imperial College London at ICMS Centre, Scotland, UK, and during 2017-18, he was awarded most respected TUBITAK Visiting Scientist Fellowship to work with Prof. Dr. Onur at Ahi Evran University, Turkey. He has been awarded by Most Outstanding Researcher-2018 (Award for contribution to Mathematics) by the Union Minister of Human Resource Development of India, Mr. Prakash Javadekar in 2018. According to Google Scholar, Dr. Agarwal is cited more than 2,553 times, and on Scopus his work is cited more than 1,191 times. Dr. Agarwal is the recipient of several notable honors and awards. Dr. Agarwal provided significant service to Anand International College of Engineering, Jaipur. Under his leadership during 2010-20, Anand-ICE consistently progressed in xi

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education and preparation of students, and in the new direction of academics, research and development. His overall impact to the institute is considerable. Many scholars from different nations, including China, Uzbekistan, Thailand and African Countries came to work under his guidance. The majority of the visiting post-doctoral scholars were sent to work under Dr. Agarwal by their employing institutions for at least one month. Dr. Agarwal regularly disseminates his research at invited talks/colloquiums (over 25 Institutions all over the world).He has been invited to give plenary/keynote lectures at international conferences in the USA, Russia, India, Turkey, China, Korea, Malaysia, Thailand, Saudi Arabia, Germany, UK, Turkey, and Japan. He has served over 40 Journals in the capacity of an Editor/Honorary Editor, or Associate Editor, and published books as an editor. Dr. Agarwal has also organized International Conferences/ workshops’/seminars/summer schools. In summary of these few inadequate paragraphs, Dr. P. Agarwal is a visionary scientist, educator, and administrator who have contributed to the world through his long service, dedication, and tireless efforts. Poom Kumam received his Ph.D. in Mathematics from Department of Mathematics, Faculty of Science, Naresuan University. Now, he is the Professor and Associate Dean for Research and Networking, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) and also the leading of Theoretical and Computational Science (TaCS) Center and director of Computational and Applied Science for Smart Innovation Cluster (CLASSIC). He served on the editorial boards of various international journals and also published more than 700 papers in Scopus and Web of Science (WoS) database. Furthermore, his research interest focuses on Fixed Point Theory and Optimization with related with optimization problems in both pure science and applied science.

List of Contributors Tatjana Došenovi´c Faculty of Technology, University of Novi Sad, Serbia Nguyen Van Dung Faculty of Mathematics Teacher Education, Dong Thap University, Cao Lanh City, Dong Thap, Province, Vietnam

Juan Martnez-Moreno Department of Mathematics, University of Jaen, Campus las Lagunillas s/n, 23071, Jaén, Spain Dosenovic Deepesh Kumar Patel Department of Mathematics, Visvesvarya National Insitute of Technology, Nagpur, India

Dhananjay Gopal Department of Mathematics Guru Ghasidas Pradip Ramesh Patle Vishwavidyalaya (A Central University), Department of Mathematics, Visvesvarya Bilaspur (C.G.) India National Insitute of Technology, Nagpur, India Mohammad Hasan Department of Mathematics, Jazan Satish Shukla University, Jazan, Kingdom of Saudi Department of Applied Mathematics and Arabia Humanities, Shri Vaishnav Institute of Technology and Science Gram Baroli, Shilpi Jain Indore, India Department of Mathematics, Poornima College of Engineering, Jaipur, India Wutiphol Sintunavarat Department of Mathematics and Statistics, Vishal Joshi Faculty of Science and Technology, Department of Applied Mathematics Thammasat University (Rangsit Center), Jabalpur Engineering College, Jabalpur, 12121 Pathumthani, Thailand India Poom Kumam Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)

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1 Symmetric Spaces and Fixed Point Theory Pradip Ramesh Patle and Deepesh Kumar Patel

CONTENTS 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Topology of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Completeness Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Fixed Points of Single-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Fixed Points of Multivalued Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Conclusion and Future Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.1 Introduction One of the intuitively natural concepts in the history of human beings is the notion of distance. The notion was known but it was axiomatically formulated by Fréchet [19] in the early 1900s. On realization of the fact that the Euclidean distance between two points can be given by the absolute difference, Fréchet come up with an abstract formulation and generalization of the distance concept (termed as metric). It is an indisputable argument that the formulation of metric opens up a new way for analysts. A large number of generalizations, improvements and extensions of the metric concept have appeared in different directions due to its fundamental role in analytic sciences and applications. As a result, notions such as Hausdorff metric, fuzzy metric, symmetric, quasi-metric, semi-metric, metric-like, partial metric, b-metric, G-metric, D-metric, 2-metric, ultra-metric, dislocated metric, modular metric, partial b-metric and so on have appeared in the literature and this trend is still going on. In an attempt to generalize the concept of metric spaces Fréchet [19], Menger [38], Chittenden [13] and Wilson [52] introduced the notion of symmetric spaces and semi-metric spaces. These spaces have been studied in great detail and have developed into a wide literature. Cicchese [15] introduced the notion of contraction mappings and proved the first fixed point theorem in semi-metric spaces. Afterwards various fixed point theorems have been extended in these spaces. In this chapter, we restrict ourselves to the merging of two interesting notions; symmetric and semi-metric. We focus on the basic properties of symmetric and semi-metric 1

Pradip Ramesh Patle and Deepesh Kumar Patel

2

spaces. Also topology on symmetric spaces and various topological properties are presented. As our main concern is the fixed point theory in symmetric and semi-metric spaces, we need to study the concept of completeness in these spaces. There are various completeness concepts in these spaces because of different notions of Cauchy-type sequences. A huge literature of fixed point theory in these spaces has developed that is impossible to summarize in a chapter so we present some of the basic results in this chapter.

1.2 Basic Concepts In this section, very basic concepts and definitions from literature of symmetric spaces are presented. Let us begin with the following definition of symmetric. Definition 1 ([52]). A function s : X × X → [0, +∞) satisfying (W1) s(x, y) = 0 if and only if x = y and s(x, y) ≥ 0, for x, y ∈ X, (W2) s(x, y) = s(y, x), for all x, y ∈ X, is called symmetric on a nonempty set X, whereas the pair (X, s) is called symmetric space (also called E-space of Fréchet). If in addition (X, s) has the property s(x, y) ≤ s(x, z) + s(z, y), for x, y, z ∈ X, then (X, s) is a metric space. This property is known as the triangular inequality. It is clear that every metric is symmetric but the converse is not true. Example 1. Let X = [0, 1] and s(x, y) = (x − y)2 . It is clear that (X, s) is a symmetric space but not a metric space. Example 2. Let X = { n1 } ∪ {0}. Let us define s : X × X → [0, ∞) as s(0, 1) = 1 = s(1, 0), s(1, n1 ) = 2 = s( n1 , 1) for n ≥ 2, s(1, 1) = 0 and s(x, y) = |x−y| for x, y ∈ X \{1}. Then (X, s) is a symmetric 3 space. Example 3. Let X = { n1 } ∪ {0} and let

⎧  ⎨ 1 ,0 = s ⎩ n 

and

⎧  1 1   ⎪ ⎪ m−n ⎪   ⎪ ⎨   1 1  1 − 1 , s = m n ⎪ m n ⎪ ⎪ ⎪ ⎩ 1

Then (x, s) is a symmetric space.

1 n

if n is odd,

1

if n is even.

if m + n is even, if m + n is odd and |m − n| = 1, if m + n is odd and |m − n| ≥ 2.

Symmetric Spaces and Fixed Point Theory

Example 4. Let X = N and s(x, y) =

|m − n| . Then (X, s) is a symmetric space. 2min{m,n}

Example 5. Let X = { n1 } ∪ {0} and let  s

1 1 , m n

3

 =

⎧  1 1 ⎨ m − n ⎩

if |m − n| ≥ 2, if |m − n| = 1,

1

s( n1 , 0) = 1 = s(1, n1 ) and s(x, x) = 0 for all x ∈ X. Then (X, s) is a symmetric space. The following families of subsets of a nonempty set X are considered in this chapter: N (X) = {P : P ⊆ X and P = 0}; / C L s (X) = {P : P ∈ N (X) and P¯ s = P}; Bs (X) = {P : P ∈ N (X) and P is bounded}; C B s (X) = {P : P ∈ C L s (X) ∩ Bs (X)} and C (X) = {P : P ∈ N (X) and P is compact}. We now recall the definition of distance between two sets. For A, B ∈ Bs (X), S and S + are the distance functions defined as

(1.1) S (A, B) = max sup s(a, B), sup s(b, A) a∈A

b∈B

 1 S + (A, B) = sup s(a, B) + sup s(b, A) , 2 a∈A b∈B

(1.2)

where s(a, B) = infs(a, b). S is known as the Pompeiu–Hausdorff distance. b∈B

Proposition 1. [41] (C B s (X), S ) is a symmetric space if (X, s) is a symmetric space. Proposition 2. [44] (C B s (X), S + ) is a symmetric space if (X, s) is a symmetric space. Remark 1. [43] S + and S are topologically equivalent, that is, k1 S (P, Q) ≤ S + (P, Q) ≤ k2 S (P, Q), where k1 = 12 and k2 = 1. It is worth mentioning here that the equivalence of two symmetric spaces does not mean that the results proved with one are equivalent to the others. This is shown by means of examples in [43] in the case of metric spaces. Many properties and concepts in symmetric spaces and metric spaces are similar (but not all, due to the absence of the triangular inequality). Let (X, s) be symmetric space. The open ball with center x ∈ X and radius r > 0 is defined by B(x, r) = {y ∈ X : s(x, y) < r}. Also if A is a subset of X, then diam(A) = sup{s(x, y) : x, y ∈ X} denotes the diameter of A. In a symmetric space (X, s), the limit point of a sequence {xn } is defined by lim s(xn , x) = 0 if and only if lim xn = x. n→∞

n→∞

Pradip Ramesh Patle and Deepesh Kumar Patel

4

1.3 Topology of Symmetric Spaces In every symmetric space (X, s) we can define the topology τs by means of a family of closed sets as follows: a set A ⊆ X is closed if and only if for each x ∈ X, d(x, A) = 0 implies x ∈ A, where d(x, A) = inf{d(x, a) : a ∈ A}. The convergence of a sequence {xn } in the topology τs need not imply s(xn , x) → 0, although the converse holds. We can ensure this by applying the following definition. Definition 2. A topological space (X, τ ) is semimetrizable if there is a symmetric function s : X × X → R such that τs = τ and that the mapping X ⊇ A → c(A) = {x ∈ X : d(x, A) = 0} is a closure operator in τs . In terms of s it can be expressed by saying that the operator c is idempotent. In this case, the space (X, s) is called a semi-metric space. It is known that the name semimetric is used by many authors for the function that is different from metric in that it need not satisfy s(x, y) = 0 implies x = y. Therefore it is recommended here not to get confused with the terminology. We now give a characterization of topology on symmetric space (X, s) as follows: Let s be a symmetric on X. For x ∈ X and ε > 0, let B(x, ε ) = {y ∈ X : s(x, y) < ε }. A topology τs on X is defined by U ∈ τs if and only if for each x ∈ U, there exists an ε > 0 such that B(x, ε ) ⊂ U. A subset N of X is a neighbourhood of x ∈ X if there exists U ∈ τs such that x ∈ U ⊂ N. Definition 3. A symmetric is called a semi-metric if for each x ∈ X and each r > 0, B(x, r) is a neighbourhood of x in the topology τs . The following two propositions are well known and they play a very important role in characterizing semi-metric spaces. Proposition 3. If (x, s) is a symmetric space, then the family {B(x, r) : r > 0} forms a local basis at x. Also if s(xn , x) → 0 then xn → x (or lim xn = x) in the topology τs . n→∞

It is worth mentioning here that this basis need not consist of open sets. Heath [23] proved this, by constructing a semimetrizable space (X, τs ) such that for any s that generates τs , there exists x ∈ X and r > 0 such that B(x, r) is not open. Proposition 4. Every symmetric space (x, s) is a semi-metric space if and only if the following conditions hold: 1. (X, τs ) is first countable; 2. For any sequence {xn } ⊆ X, s(xn , x) → 0 is equivalent to xn → x in the topology τs . As we have seen, for symmetric spaces the triangular inequality is relaxed. In order to combat various problems arising while proving results in such spaces some alternate concepts need to be satisfied, which are listed below: Definition 4. Let (X, s) be a symmetric space. We have the following properties: (W3) [52] Given {xn }, x and y in X; lim s(xn , x) = 0 and lim s(xn , y) = 0 implies x = y. n→∞

n→∞

Symmetric Spaces and Fixed Point Theory

5

(W4) [52] Given {xn }, {yn } and x in X; lim s(xn , x) = 0 and lim s(xn , yn ) = 0 implies n→∞

lim s(yn , y) = 0.

n→∞

n→∞

(CC) [11] Given {xn } and x in X such that lim s(xn , x) = 0 implies lim s(xn , y) = s(x, y) for n→∞

n→∞

some y ∈ X. (HE) [3] Given {xn }, {yn } and x in X; lim s(xn , x) = 0 and lim s(yn , x) = 0 implies n→∞

lim s(xn , yn ) = 0.

n→∞

n→∞

(W)

[39] Given {xn }, {yn } and {zn } in X; lim s(xn , yn ) = 0 and lim s(yn , zn ) = 0 implies n→∞

lim s(xn , zn ) = 0.

n→∞

n→∞

(JMS) [31] Given {xn }, {yn } and {zn } in X; lim s(xn , yn ) = 0 and lim s(yn , zn ) = 0 implies n→∞

lim s(xn , zn ) = ∞.

n→∞

n→∞

(MT) [33] there exists K ≥ 1 such that for any x, y, z ∈ X, s(x, z) ≤ K[s(x, y) + s(y, z)]. (MC) [15] there exists ε : R+ → R+ and K ≥ 1 such that for any x, y, z ∈ X, s(x, z) ≤ ε (max{s(x, y), s(y, z)}) + K min{s(x, y), s(y, z)} and lim+ ε (t) = 0. t→0

The following relations hold between the above properties. See [7, 14, 39] for more details. (i) (ii) (iii) (iv) (v)

(W) =⇒ (W4) =⇒ (W3) but (W3)  (W4). (W) =⇒ (JMS), (W) =⇒ (HE) and (MT) =⇒ (MC). (CC) =⇒ (W3) but (W3)  (CC). (MC) =⇒ (W3), (MC) =⇒ (W4), (MC) =⇒ (HE), (MC) =⇒ (W), (MC) =⇒ (JMS). The following pairs of properties are independent (in the sense that neither of which implies the other): (W) and (CC); (W4) and (CC); (W3) and (HE); (W4) and (HE); (HE) and (CC).

The following characterization of symmetric spaces with property (JMS) is given in [31]. Proposition 5. Let (X, s) be a symmetric space. Then the following conditions are equivalent. 1. (X, s) satisfies property (JMS). 2. There exist δ , ε > 0 such that for any x, y, z ∈ X, s(x, z) + s(z, y) < δ implies that s(x, y) < ε . 3. There exists r > 0 such that sup{diam(B(x, r)) : x ∈ X} < ∞. The next theorem gives a characterization of semi-metric space with open balls in terms of the new semicontinuity property of s (see [7]). Theorem 1. Let (X, s) be a symmetric space. Then the following are equivalent: (i) lim s(xn , x) = 0 implies lim sup d(xn , y) ≤ d(x, y) . . . property (SC), n→∞

n→∞

(ii) (X, s) is a semi-metric space in which all B(x, r) are open sets. Remark 2. (i) It is obvious that (CC) implies (SC). (ii) It is shown [7] that there exists a semi-metric space in which all balls B(x, r) (r > 0) are open, that does not have properties (W4), (JMS), (HE) and (CC).

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(iii) A semi-metric space in which all balls B(x, r) (r > 0) are open, is a T1 space, but it need not be a Hausdorff space. Proposition 6. Let (X, s) be a compact semi-metric space in which all balls B(x, r) (r > 0) are open and K ⊆ X a nonempty compact set. Then K is bounded. The continuity of a function is generalized with the following two notions. Definition 5. A function f : X → X is s-continuous if lim s(xn , x) = 0 implies lim s(fxn , fx) = 0. n→∞

n→∞

Definition 6. A function f : X → X is τs -continuous if lim xn = x with respect to τs implies n→∞

lim f (xn ) = f (x) with respect to τs .

n→∞

Kirk and Shahzad in [35] introduced the following concept of triangular function. Definition 7. Let (X, s) be a semi-metric space. Then X is said to be regular with a strong triangle function if there exists a function Ψ : [0, ∞)2 → [0, ∞) satisfying the following: (i) Ψ (0, 0) = 0. (ii) Ψ is nondecreasing (i.e., Ψ (s, t) ≤ Ψ (s , t ) holds for all s, t, s , t ∈ [0, ∞) with s ≤ s and t ≤ t ). (iii) Ψ is continuous at (0, 0), that is, limΨ (sn , tn ) = 0 provided {sn } and {tn } are sequences n→∞

in [0, ∞) converging to

0.  (iv) |s(x, y) − s(z, w)| ≤ Ψ s(x, z), s(y, w) . Suzuki [49] characterized regular semi-metric space with a strong triangular function in the following lemma. Lemma 1. Let (X, s) be a semi-metric space. Then the following are equivalent: (a) X is regular with a strong triangle function Ψ . (b) There exists a function ψ : [0, ∞) → [0, ∞) satisfying (i) ψ (0) = 0. (ii) ψ is nondecreasing. (iii) ψ is continuous at 0. (iv) s(x, z) ≤ s(x, y) + ψ o s(y, z). A lot of analysis of regular semi-metric space has been done. A large number of articles have been published in fixed point theory of semi-metric space using strong triangular function. For more detailed study, interested readers can refer to the recent article of Dung and Hang [17] and references therein.

1.4 Completeness Concepts We now shift our focus to the study of Cauchy sequences and completeness in symmetric and semi-metric spaces. As triangular inequality is skipped in such spaces, various ideas

Symmetric Spaces and Fixed Point Theory

7

of Cauchy sequences and completeness have been presented. The key references for this section are [7, 12, 20, 35, 50]. Definition 8. Let (X, s) be a symmetric space. A sequence {xn } in X is called an s-Cauchy sequence if for given ε > 0 there is n0 ∈ N such that s(xm , xn ) < ε , for all m, n > n0 . Definition 9. A symmetric space (X, s) is said to be s-Cauchy complete if every s-Cauchy sequence converges to some x ∈ X in τs . Definition 10. A symmetric space (X, s) is called S-complete if for every s-Cauchy sequence {xn }, there exists x ∈ X with lim s(xn , x) = 0. n→∞

Definition 11. [24] Let (X, τ ) be a topological space and s : X × X → [0, ∞) with s(x, y) = 0 ∞

if and only if x = y. Then (X, τ ) is s-complete topological space if ∑ s(xn , xn+1 ) < ∞ implies n=1

there exists x ∈ X with lim xn = x with respect to τ (s). n→∞

The idea of completeness in s-complete topological spaces generalizes the completeness in metric and quasi-metric spaces. Jachymski et al. [31] proved that an s-Cauchy complete semi-metric space (X, τ (s)) which satisfies (W4) is an s-complete topological space. For ∞

(X, s) a semi-metric space, this means that if ∑ s(xn , xn+1 ) < ∞, there exists x ∈ X such that n=1

xn → x in τ (s).

∞

Definition 12 ([24]). A symmetric space (X, s) is complete if ∑ s(xn , xn+1 ) < ∞ implies that n=1

there exists x ∈ X such that lim s(xn , x) = 0. n→∞

Recall here that, if we let s generate a topology τs , then the topological space (X, τs ) is called a symmetric space, whereas a semi-metric space means a symmetric space in which all open balls are neighbourhoods. Remark 3 ([24]). (1) In a semi-metric space, lim s(xn , x) = 0 if and only if {xn } converges to x n→∞

in τs . (2) In a symmetric space, {xn } converges to x in τs , implies lim s(xn , x) = 0; the converse is n→∞ not true. The concept of s-weakly completeness of symmetric spaces is defined by Galvin and Shore in [20]. Definition 13. A symmetric space (X, s) is said to be s-weakly complete if every decreasing sequence {Fn } of nonempty closed subsets, such that there exists a sequence {xn }, xn ∈ Fn with Fn ⊆ B(xn , 2−n ) has a nonempty intersection. Proposition 7. Let (X, s) be a semi-metric space. Then following are equivalent: (1) (X, τs ) is s-weakly complete,

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(2) every s-Cauchy sequence in X has a convergent subsequence, (3) every decreasing sequence {Fn } of nonempty closed subsets of X such that diam(Fn ) ≤ 2−n for each n has a nonempty intersection. Lemma 2 ([24]). Suppose T : X → Bs (X). Then lim s(xn , Tx) = 0 implies and is implied by there exists yn ∈ Tx satisfying lim s(xn , yn ) = 0.

n→∞

n→∞

Definition 14 ([50]). Let {xn } be a sequence in a semi-metric space (X, s) and x ∈ X. Let κ ∈ N and h : X → R be a function. (A) {xn } is said to converge to x if lim s(xn , x) = 0. n→∞

(B) {xn } is said to be Cauchy if lim sup{s(xn , xm ) : m > n} = 0. n→∞ ∞

(C) {xn } is said to be ∑-Cauchy if

∑ s(xn , xn+1 ) < ∞.

n=1

(D) {xn } is said to be (∑, =)-Cauchy if xn (n ∈ N) are all different and

∞

∑ s(xn , xn+1 ) < ∞.

n=1

(E) X is said to be Hausdorff if lim s(xn , x) = 0 and lim s(xn , y) = 0 imply x = y. n→∞

(F) X is said to be κ -Hausdorff if

n→∞

lim S(x, un(1) , . . . , un(κ ) , y) = 0

(1.3)

n→∞

implies x = y, where S(x, un(1) , . . . , un(κ ) , y) = s(x, un(1) ) + s(un(1) , un(2) ) + . . . + s(un(κ −1) , un(κ ) ) + s(un(κ ) , y).

(1.4)

X is said to be complete if every Cauchy sequence converges. X is said to be ∑-complete if every ∑-Cauchy sequence converges. X is said to be (∑, =)-complete if every (∑, =)-Cauchy sequence converges. X is said to be semicomplete if every Cauchy sequence has a convergent subsequence. (K) X is said to be ∑-semicomplete if every ∑-Cauchy sequence has a convergent subsequence. (L) X is said to be (∑, =)-semicomplete if every (∑, =)-Cauchy sequence has a convergent subsequence. (M) s is said to be sequentially lower semicontinuous if s(x, y) ≤ lim inf s(xn , yn ) provided

(G) (H) (I) (J)

n→∞

{xn } converges to x and {yn } converges to y. (N) h is said to be sequentially lower semicontinuous if h(x) ≤ lim inf h(xn ) provided {xn } n→∞ converges to x. (N) h is said to be sequentially lower semicontinuous from above if h(x) ≤ lim inf h(xn ) provided {xn } converges to x and {h(xn )} is strictly decreasing. (O) h is said to be proper if {x ∈ X : h(x) ∈ R} is nonempty. Proposition 8. Let (X, s) be a semi-metric space. Consider the following statements. (i) X is ∑-complete. (ii) X is (∑, =)-complete. (iii) X is ∑-semicomplete.

n→∞

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(iv) X is (∑, =)-semicomplete. (v) X is complete. (vi) X is semicomplete. Then the following implications hold: (a) (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) =⇒ (vi). (b) (i) =⇒ (v) =⇒ (vi). Proposition 9. A ∑-complete semi-metric space (X, s) is κ -Hausdorff for any κ ∈ N. Proposition 10. Let (X, s) be a ∑-complete semi-metric space. Then X is Hausdorff. Proposition 11. Let (X, s) be a (∑, =)-complete, Hausdorff semi-metric space. Then X is complete. For proofs of the Propositions 8–11 refer to [50]. It is not confirmed whether the Proposition 11 holds without Hausdorffness.

1.5 Fixed Points of Single-Valued Mappings This section presents extensions and generalizations of some fundamental metric fixed point theorems to a symmetric setting. In 1976, Cicchese [15] proved the first fixed point theorem for contraction mappings in semi-metric spaces. To date, a lot of fixed point theorems have been extended and generalized in the setting of symmetric and semi-metric spaces. We discuss some of the results from the ocean called the theory of fixed points in symmetric spaces. We recall that x is a fixed point of a mapping T if x = Tx is satisfied. Proposition 12. Let (X, s) be a Hausdorff semi-metric and s-Cauchy complete space and let T be a self mapping on X satisfying the following condition s(Tx, Ty) ≤ h s(x, y), for h ∈ (0, 1) and x, y ∈ X.

(1.5)

If (X, s) is bounded, that is M = sup{d(x, y) : x, y ∈ X} < ∞, then T has a unique fixed point p and for any x ∈ X, the sequence {Tn x} of iterates of T on x converges to p. Proof. Let us fix an x ∈ X. Since s(Tn x, Tn+m x) ≤ hn s(x, Tm x) ≤ hn M, for all n, m ∈ N. As hn M → 0 with n → ∞, {Tn x} is s-Cauchy. The s-Cauchy completeness of X gives rise to existence of p in X such that {Tn x} τ -converges to p. Since s is semi-metric, (1.5) implies that T is τ -continuous. Therefore, {Tn+1 x} τ -converges to Tp. Since (X, s) is Hausdorff, we may infer that p = Tp. Uniqueness of the fixed point is guaranteed using (1.5).   The boundedness of (X, s) is necessary for Proposition 12 to hold and it cannot be extended to unbounded semi-metric spaces. This is evident from the following example.

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|x − y| , for 2min{x,y} x, y ∈ X. Then clearly (X, s) is semi-metric space. Let {xn } be an s-Cauchy sequence. Then {xn } is bounded; otherwise, there is a subsequence {xnj }, such that xnj → ∞, and then for any j ∈ N, |xnj − xnk | lim s(xnj , xnk ) = lim = ∞, k→∞ k→∞ 2xnk violating the Cauchy condition. Therefore, we may infer that {xn } is constant for sufficiently large n, since it is s-Cauchy complete, but T has no fixed point though it satisfies (1.5) with h = 12 .

Example 6. Let X = N, T : X → X defined by Tx = x + 1, x ∈ X and s(x, y) =

We now consider a class of functions, sometimes termed the auxiliary functions which are used to obtain a large number of generalized forms of contractive conditions. Definition 15. Let Φ denote the set of all monotone nondecreasing functions ϕ : [0, ∞) → [0, ∞) such that lim ϕ n (t) = 0 for any t > 0, where ϕ n is the nth iterate of ϕ . n→∞

Lemma 3. If ϕ ∈ Φ then ϕ (t) < t for all t > 0 and ϕ (0) = 0. The following result is proved by Jachymski et al. [31]. Proposition 13. Let (X, s) be a Hausdorff s-Cauchy complete semi-metric space which satisfies the property (JMS). Let T be a self mapping on X and ϕ ∈ Φ satisfying s(Tx, Ty) ≤ ϕ (s(x, y)), for all x, y ∈ X; then T has a unique fixed point p ∈ X and Tn x → p, for all x ∈ X. Proposition 13 cannot be applied on metric type spaces because they are not necessarily semi-metric space. Thus it is quite important to generalize this proposition in symmetric spaces. Arandelovic and Keckic [7] obtained its symmetric space version. In order to prove the theorem we need the following lemma. Lemma 4. Let X be a nonempty set, T be a self mapping on X and n be a fixed positive integer such that the iterate Tn has a unique fixed point x∗ . Then, (1) x∗ is a unique fixed point of T. (2) If X is a topological space and any sequence {Tn x} of Picard iterates converges to x∗ , then the sequence of Picard iterates defined by T always converges to x∗ . Lemma 5. Let (X, s) be a complete symmetric space satisfying the properties (W3) and (JMS). Let T be a self mapping on X and ϕ ∈ Φ and let δ , ε be defined as in (2) of Proposition 5, satisfying s(Tx, Ty) ≤ ϕ (s(x, y)), for all x, y ∈ X and ϕ (ε ) ≤ δ /2; then T has a unique fixed point p ∈ X and Tn x → p, for all x ∈ X. Proof. From Lemma 3, for any a, b ∈ X we have s(Tp, Tq) ≤ ϕ (s(p, q)) ≤ s(p, q), which implies T is continuous.

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Let us fix an x ∈ X. Then s(Tm+n x, Tn x) ≤ ϕ n (s(Tm x, x)) for any m, n ∈ N. Taking m = 1 we get s(Tn+1 x, Tn x) ≤ ϕ n (s(Tx, x)) which implies that s(Tn+1 x, Tn x) → 0. This means there exists k ∈ N such that s(Tk+1 x, Tk x) ≤ min{δ /2, ε }. We wish to show that for all n ∈ N, s(Tk+n x, Tk x) ≤ ε .

(1.6)

It is clear from the definition of k that (1.6) is valid for n = 1. Let us assume that (1.6) is satisfied for some n ∈ N. From s(Tk+1 x, Tk x) ≤ δ /2 and

s(Tk+1 x, Tk+n+1 x) ≤ ϕ (s(Tk x, Tk+n x)) ≤ ϕ (ε ) ≤ δ /2,

it follows that

s(Tk x, Tk+1 x) + s(Tk+1 x, Tk+n+1 x) ≤ δ .

Thus Proposition 5 yields

s(Tk x, Tk+n+1 x) ≤ ε .

By induction, we get that (1.6) is satisfied for any n ≥ 1. Thus d(Tk+n x, Tk+n+m x) ≤ ϕ n (ε ), for any m, n ∈ N. Hence {Tn x} is a Cauchy sequence. Then, there exists p ∈ X such that lim Tn x = p. Since T n→∞

is continuous, we have lim Tn+1 x = Tp. By virtue of (W3), we get Tp = p. n→∞

Since s(Tn x, p) → 0, by Proposition 3, we have Tn x → p in the topology τs . To verify uniqueness, consider p∗ such that Tp∗ = p∗ , then for all n we have s(p, p∗ ) = s(Tn p, Tn p∗ ) ≤ ϕ n (s(p, p∗ )) → 0, as n → ∞. Thus p = p∗ .

 

Theorem 2. Let (X, s) be a complete symmetric space satisfying (W3) and (JMS). Let T be a self mapping on X and ϕ ∈ Φ satisfying s(Tx, Ty) ≤ ϕ (s(x, y)), for any x, y ∈ X; then T has a unique fixed point p ∈ X and the sequence of Picard iterates {Tn x} at x converges in X, and by Proposition 3, it converges to the same limit in the topology τs . Proof. Let δ , ε be defined as in (2) of Proposition 5. If ϕ (ε ) ≤ δ /2, then from Lemma 5, it follows that T has a unique fixed point y ∈ X and for each x ∈ X the sequence of Picard iterates defined by T at x converges, in the topology τs to y. Now assume that ϕ (ε ) > δ /2. Then there exists a least positive integer j > 1 such that ϕ j (ε ) ≤ δ /2. Also, we have that s(Tj x, Tj y) ≤ ϕ j (s(x, y)), for any x, y ∈ X, ϕ j ∈ Φ . By Lemma 5, Tj has a unique fixed point, say z ∈ X and for each x ∈ X the sequence of Picard iterates defined by Tj at x converges in the topology τs , to z. From Lemma 4 it follows that T has a unique fixed point z ∈ X, and for each x ∈ X the   sequence of Picard iterates defined by T at x converges in the topology τs , to z. The next result due to [6] generalizes the above theorem. This theorem has a quite different proof technique from the previous one.

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Theorem 3. Let (X, s) be an s-Cauchy complete symmetric space satisfying (W3) and (JMS). Let T : X → X be a τs -continuous map and ϕ ∈ Φ satisfying s(Tx, Ty) ≤ ϕ (s3 (x, y)), where s3 (x, y) = max{s(x, y), s(x, Tx), s(y, Ty)} for any x, y ∈ X. Then T has a unique fixed point p ∈ X and for each x ∈ X, the sequence of Picard iterates of T at x converges to p in the topology τs . Proof. Define s∗ : X × X → [0, ∞) as follows: s∗ (x, y) = 0 for x = y and s∗ (x, y) = s3 (x, y) otherwise. Then the space (X, s∗ ) is a symmetric space. Also, we have s(x, y) ≤ s∗ (x, y) for any x, y ∈ X. So, if {xn } ⊆ X is an arbitrary s∗ -Cauchy sequence in (X, s∗ ), then {xn } is an s-Cauchy sequence in (X, s). Let x, y ∈ X. From s(T2 x, Tx) ≤ ϕ (s(Tx, x)), s(T2 y, Ty) ≤ ϕ (s(Ty, y)),

(1.7)

s(Tx, Ty) ≤ ϕ (s3 (x, y)), it follows that

s∗ (Tx, Ty) ≤ ϕ (s∗ (x, y)).

(1.8)

Let δ , ε be defined as in (2) of Proposition 5. Then there exists the least positive integer j ≥ 1 such that ϕ j (ε ) ≤ δ /2. Let g = Tj . We have that T is continuous (in τs ). Then s∗ (gx, gy) = s∗ (T(Tj−1 x), T(Tj−1 y)) ≤ ϕ (s∗ (Tj−1 x, Tj−1 y))

(1.9)

∗

≤ ϕ (s (x, y)). j

Let x ∈ X and ψ = ϕ j . Then ψ ∈ Φ and s∗ (gm+n x, gn x) ≤ ψ n (s∗ (gm x, x)) for any m, n ∈ N. Therefore

(1.10)

s∗ (gn+1 x, gn x) ≤ ψ n (s∗ (gx, x)) ∗

(1.11) ∗

which implies that s (g x, g x) → 0. That means there exists k ∈ N such that s (g min{δ /2, ε }. We wish to show that for all n ∈ N, n+1

n

s∗ (gk+n x, gk x) ≤ ε .

k+1

x, gk x) ≤

(1.12)

It is clear from the definition of k that (1.12) is valid for n = 1. Let us assume that (1.12) is satisfied for some n ∈ N. From s∗ (gk+1 x, gk x) ≤ δ /2 and

s∗ (gk+1 x, gk+n+1 x) ≤ ψ (s∗ (gk x, gk+n x)) ≤ ϕ (ε ) ≤ δ /2,

it follows that Thus Proposition 5 yields

s∗ (gk x, gk+1 x) + s∗ (gk+1 x, gk+n+1 x) ≤ δ . s∗ (gk x, gk+n+1 x) ≤ ε .

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By induction, we get that (1.12) is satisfied for any n ≥ 1. Thus s∗ (gk+n x, gk+n+m x) ≤ ψ n (ε ), for any m, n ∈ N. Hence {gn x} is an s∗ -Cauchy sequence in (X, s∗ ), which implies that {gn x} is an s-Cauchy sequence in (X, s). Then there exists z ∈ X such that lim gn x = z. Since g is τs -continuous, we n→∞

have lim gn+1 x = gz. By virtue of (W3), we get gz = z. n→∞

If y is another fixed point of g, then for all n we have s∗ (y, z) = s∗ (gn y, gn z) ≤ ϕ n (s(y, z)) → 0, as n → ∞. Therefore, z is a unique fixed point of g. By Lemma 4, z is a unique fixed point of T. From s∗ (z, gn+1 x) ≤ ϕ (max{s(z, gz), s(z, gn x), s(gn x, gn+1 x)}), it follows that for each x ∈ X, the sequence of Picard iterates defined by g = Tj at x converges, in the topology τs∗ , to z, which implies their convergence in the topology τs . Therefore, by Lemma 4, we obtain that for each x ∈ X the sequence of Picard iterates   defined by T at x converges, in the topology τs , to z. Remark 4. The continuity of T in Theorem 3 is necessary and cannot be omitted; this is shown by means of the following example. Example 7. Let X = {0} ∪ { n1 : n ∈ N} and let s be defined as follows: s(0, 1) = s(1, 0) = 1; s(1, n1 ) = s( n1 , 1) = 23 for n ≥ 2; s(1, 1) = 0; otherwise s(x, y) = |x − y|. Then clearly (X, s) is a bounded s-Cauchy complete symmetric space satisfying (W3) and (JMS). Let T : X → X be defined by ⎧ x ⎨ 4 if x = 0, T(x) = ⎩ 1 if x = 0. Then T satisfies s(Tx, Ty) ≤ ϕ (s3 (x, y)) for all x, y ∈ X and ϕ (t) = 23 t. But T does not have any fixed point in X. Note that T is not continuous. We now present some fixed point results in semi-metric spaces. Let O(x) denote the orbit of T at a point x which is nothing but a sequence {xn } defined by xn = Tn x. Let O(x, y) = O(x) ∪ O(y). Theorem 4. Let (X, s) be a bounded s-Cauchy complete semi-metric space satisfying (W4). Suppose T : X → X satisfies that for x ∈ X, there exists ν (x) ∈ N such that, for any ν ≥ ν (x) and y ∈ X, s(Tν x, Tν y) ≤ ϕ (diam(O(x, y)))

(1.13)

with ϕ ∈ Φ . Then there exists z ∈ X such that lim Tn x = z in the topology τs . n→∞

We skip the proof. Corollary 1. If in addition to the hypothesis of Theorem 4, we assume that T is τs -continuous then T has a fixed point.

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Proof. Since lim Tn x = z in the topology τs , by τs -continuity of T, lim Tn+1 x = Tz in the topoln→∞

n→∞

ogy τs . Therefore, using the fact (W4) implies (W3) and (X, s) satisfies (W4), we have Tz = z. Hence z ∈ X is a fixed point.   Theorem 5. Let (X, s) be a bounded s-Cauchy complete semi-metric space with (W4), (CC) and (JMS). Suppose T is a self mapping on X satisfying s(Tx, Ty) ≤ ϕ (diam(O(x, y))) for x, y ∈ X.

(1.14)

Then T has a unique fixed point and lim Tn x = z in the topology τs , for every x ∈ X. n→∞

Proof. Following the conclusion of Theorem 4 we have that there exists a z ∈ X such that lim s(Tn x, z) = 0 for all x ∈ X. Next assume that z = Tz; that means diam(O(z)) = β > 0. Thus

n→∞

it is possible to choose two sequences {i(n)} and {j(n)} such that lim s(Ti(n) z, Tj(n) z) = β .

n→∞

Therefore one can choose δ > 0 with a corresponding ε > 0, such that ε ≤ β /2. Since there exists n0 ∈ N such that s(Tn z, z) ≤ δ /2, s(Tm z, z) ≤ δ /2, for every n, m ≥ n0 , and therefore s(Tn z, z) + s(Tm z, z) ≤ δ , by (2) of Proposition 5 we get s(Tn z, Tm z) ≤ ε ≤ β /2, for every n, m ≥ n0 . Hence i(n) = i for all infinitely many n with 0 ≤ i ≤ n0 . Thus there exists a sequence {r(n)} ⊆ {j(n)} such that lim s(Ti z, Tr(n) z) = β . Therefore either r(n) = j for infinitely many n or there n→∞

exists a sequence {t(n)} ⊆ {r(n)} with r(n) → ∞ as n → ∞ which implies s(Ti z, z) = β . In both cases, we can conclude that there exist i, j ≥ 0 such that s(Ti z, Tj z) = β . If s(Tj z, z) = β , since lim s(Tn x, z) = 0 and by using (CC), we get n→∞

β = d(Tj z, z) = lim s(Tn z, Tj z) n→∞

 ≤ lim ϕ diam O(Tn−1 z, Tj−1 z) n→∞

 ≤ lim ϕ diam O(z) = ϕ (β ), n→∞

which is a contradiction with ϕ (β ) < β , for β > 0. On the other hand, if i, j ≥ 1, by (1.14), we have

 β = d(Ti z, Tj z) ≤ ϕ diam O(Ti−1 z, Tj−1 z) ≤ ϕ (β ), which is again a contradiction. Hence β = 0; that means z is a fixed point of T.

 

Theorem 6. Let (X, s) be a bounded s-Cauchy complete semi-metric space with (W) and (CC). Suppose T is a self mapping on X satisfying (1.14); then T has a unique fixed point and lim Tn x = z in the topology τs for every x ∈ X.

n→∞

The following result is the extension of Kirk–Saliga [36] type fixed point theorem in ∑semicomplete semi-metric spaces. This result is obtained by Suzuki in [49] (see also [50]).

Symmetric Spaces and Fixed Point Theory

15

For an arbitrary set A, car(A) denotes its cardinal number. Let α be an ordinal number. We denote by α + and α − the successor and the predecessor of α , respectively. α is said to be isolated if α − exists. On the other hand, α is said to be a limit if α = 0 holds and α − does not exist. For κ ∈ N we define α + κ by

α + κ = α +...+(κ times) . Theorem 7. Let (X, s) be a ∑-semicomplete semimetric space and let T be a mapping on X. Let h : X → (−∞, +∞] be a function which is proper and bounded from below. Assume that h is sequentially lower semicontinuous from above in the sense of Definition 14. Assume also that there exists κ ∈ N satisfying the following: (i) h(Tx) ≤ h(x) for all x ∈ X. (ii) h(Tκ x) + s(x, Tx) ≤ h(x) for all x ∈ X. Then T has a fixed point. Proof. Let us define a function H from X into (−∞, +∞] by H(x) =

κ −1

∑ h(Tj x), j=0

where T0 is the identity mapping on X. Then from (ii) we have H(Tx) + s(x, Tx) ≤ H(x), for x ∈ X.

(1.15)

Arguing by contradiction, we assume Tx = x for any x ∈ X. Let Ω be the first uncountable ordinal number. Using transfinite induction, we define a net {uα : α ∈ Ω } satisfying the following: (P1 : α ) h(uα ) ≤ h(uβ ) and H(uα ) < H(uβ ) for any β ∈ Ω with β < α . (P2 : α ) h(uα ) < h(uβ ) for any β ∈ Ω with β + κ ≤ α . (P3 : α ) For any ε > 0 and for any β ∈ Ω with β < α , there exists a finite sequence (γ0 , . . . , γn ) ∈ Ω n+1 satisfying

β = γ 0 < γ 1 < . . . < γn = α , n−1

H(uα ) + ∑ s(uγj , uγj+1 ) < H(uβ ) + ε .

(1.16)

j=0

Fix u ∈ X with h(u) < ∞. It follows from (i) that H(u) < ∞ holds. Put u0 = u. Then (P1 : 0)– (P3 : 0) hold obviously. Fix α ∈ Ω with 0 < α and assume that (P1 : β )–(P3 : β ) hold for β < α . We consider the following two cases: (a) α is isolated. (b) α is a limit. Put uα = Tuα − in the first case. For any β < α , since β ≤ α − and uα − = uα hold, we have by (P1 : α − ), (i) and (1.15) h(uα ) ≤ h(uα − ) ≤ h(uβ ), H(uα ) ≤ H(uα ) + s(uα − , uα ) ≤ H(uα − ) ≤ H(uβ ).

(1.17)

Pradip Ramesh Patle and Deepesh Kumar Patel

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This means we have shown (P1 : α ). For β ∈ Ω with β + κ ≤ α , we have by (P1 : α ) and (ii) h(uα ) ≤ h(uβ +κ ) = h(Tκ uβ ), ≤ h(Tκ uβ ) + s(uβ , Tuβ ) ≤ h(uβ ).

(1.18)

This means we have shown (P2 : α ). Fix ε > 0 and β ∈ Ω with β < α . In the case where β = α − , putting γ0 = β and γ1 = α , we have by (1.15) n−1

H(uα ) + ∑ s(uγj , uγj+1 ) = H(Tuβ ) + s(uβ , Tuβ ) j=0

< H(uβ ) < H(uβ ) + ε .

(1.19)

In the other case, where β < α − , from (P3 : α − ), there exists a finite sequence (γ0 , . . . , γn ) ∈ Ω n+1 satisfying

β = γ 0 < γ 1 < · · · < γn = α − , n−1

H(uα − ) + ∑ s(uγj , uγj+1 ) < H(uβ ) + ε .

(1.20)

j=0

Putting γn+1 = α , we have by (1.15) n−1

n−1

j=0

j=0

H(uα ) + ∑ s(uγj , uγj+1 ) ≤ H(uα − ) + ∑ s(uγj , uγj+1 ) < H(uβ ) + ε .

(1.21)

Thus (P3 : α ) is proved. Therefore uα satisfying (P1 : α )–(P3 : α ) is defined for the first case. In the second case of α being a limit, let us take {βn } an increasing sequence in Ω converging to α ; that is, the following hold: βn < α for n ∈ N and for any β < α , there exists n ∈ N satisfying β < βn . For any n ∈ N, by (P3 : βn+1 ), we can choose a finite sequence (n) (n) (γ0 , . . . , γνn ) ∈ Ω νn +1 satisfying

βn = γ0(n) < γ1(n) < · · · < γν(n) = βn+1 , n νn −1

∑ s(uγ j=0

(n) j

, uγ (n) ) < H(uβn ) − H(uβn+1 ) + 2−n .

(1.22)

j+1

Since h is bounded from below, H is also bounded from below. Therefore we have ∞ νn −1

∑ ∑ s(uγ

n=1 j=0

(n) j

, uγ (n) ) < H(uβ1 ) − lim H(uβn ) + 1 < ∞.

(1.23)

n→∞

j+1

Since X is ∑-semicomplete, the sequence (1)

(2)

(2)

(3)

(3)

(β1 =)γ0 , . . . , γν(1) (= β2 = γ0 ), γ1 , . . . , γν(2) (= β3 = γ0 ), γ1 , . . . 1 2 has a subsequence {δn } such that {uδn } converges to some uα ∈ X. Note that {δn } is strictly increasing and converges to α . Without loss of generality, we may take a subsequence with the assumption that δn + κ ≤ δn+1 for n ∈ N. By (P2 : δn+1 ) we have h(uδn+1 ) < h(uδn ) for n ∈ N. Thus, {h(uδn )} is strictly decreasing. Fix ε > 0 and β ∈ Ω with β < α . Choose ν ∈ N satisfying

β < δν , s(uδν , uα ) < ε .

Symmetric Spaces and Fixed Point Theory

17

Since h is sequentially lower semicontinuous from above, from (i) we have h(uα ) ≤ lim h(uδn ) < h(uδν ) ≤ h(uβ ), n→∞

H(uα ) ≤ κ h(uα ) ≤ κ lim h(uδn ) n→∞

= κ inf{h(uγ ) : γ < α } = lim H(uδn ) n→∞

(1.24)

< H(uδν ) < H(uβ ). We have shown (P1 : α ) and (P2 : α ). We can choose a finite sequence (γ0 , . . . , γn ) ∈ Ω n+1 satisfying

β = γ 0 < γ 1 < · · · < γn = δ ν , n−1

H(uδν ) + ∑ s(uγj , uγj+1 ) < H(uβ ) + ε .

(1.25)

j=0

Putting γn+1 = α , we have by (P1 : α ) n−1

n−1

j=0

j=0

H(uα ) + ∑ s(uγj , uγj+1 ) < H(uδν ) + ∑ s(uγj , uγj+1 ) + ε < H(uβ ) + 2ε .

(1.26)

Thus uα satisfying (P1 : α )–(P3 : α ) is defined for the second case. Therefore by transfinite induction, we have defined the net {uα : α ∈ Ω } satisfying (P1 : α )–(P3 : α ) for any α ∈ Ω . Since the net {H(uα ) : α ∈ Ω } is strictly decreasing, we obtain car(Q) = car(N) < car(Ω ) < car(Q), which is a contradiction. Therefore there exists a fixed point of T.

 

The following result is the extension of the very famous and celebrated Caristi fixed point theorem in semi-metric spaces. Its proof follows from the proof of Theorem 7. Theorem 8. Let (X, s) be a ∑-semicomplete semimetric space and let T be a mapping on X. Let h : X → (−∞, +∞] be a function which is proper and bounded from below. Assume that h is sequentially lower semicontinuous from above in the sense of Definition 14. Assume also h(Tx) + s(x, Tx) ≤ h(x) for all x ∈ X.

(1.27)

Then T has a fixed point. Suzuki in [50] characterized ∑-semicompleteness of semi-metric space (X, s) via Theorem 8, which is our next result. Theorem 9. Let (X, s) be a semimetric space. Then X is ∑-semicomplete if and only if every self mapping T on X has a fixed point provided there exists a proper sequentially lower semicontinuous function h : X → [0, +∞] satisfying (1.27). Proof. The ‘if’ part follows from Theorem 8. We need to prove the ‘only if’ part. For this we assume X is not ∑-semicomplete. Then by Proposition 8, X is not (∑, =  )-semicomplete. Hence there exists a sequence {xn } in X such that all xn are different for every n ∈ N,

Pradip Ramesh Patle and Deepesh Kumar Patel

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∞ ∑n=1 s(xn , xn+1 ) < ∞ holds and it has no convergent subsequence. We define two mappings T : X → X and h : X → [0, ∞] as follows: ⎧ ⎨ xn+1 if x = xn , for some n ∈ N Tx = ⎩ if x = xn . x1 ⎧ ∞ ⎪ ⎪ ⎨ ∑s(xj , xj+1 ) if x = xn , for some n ∈ N j=n hx = ⎪ ⎪ ⎩ ∞ if x = xn .

Note that T and h are well defined because xn are all different and ∑∞n=1 s(xn , xn+1 ) < ∞ holds. Then h is proper. We will prove that h is sequentially lower semicontinuous. We will make use of a contradiction argument to prove it. Let {yn } be a sequence in X converging to y ∈ X. We assume that h(y) > lim inf h(yn ). n→∞

Then from the definition of h, there exists a subsequence {f (n)} of {n} in N such that yf (n) ∈ {xm : m > ν } for any n ∈ N, where we put ν ∈ N ∪ {0} by ⎧ ⎨ n if y = xn , for some n ∈ N ν= ⎩ 0 if y = xn . Define a function g from N to {m ∈ N : m > ν } by yf (n) = xg(n) . We consider the following two cases: (a) lim sup g(n) = ∞. n→∞

(b) μ = lim sup g(n) < ∞. n→∞

In case (a), {xn } has a subsequence converging to y. This is a contradiction. In case (b), we have ∞ = car({n ∈ N : g(n) = μ }) = car({n ∈ N : yf (n) = xμ }) ≤ car({n ∈ N : yn = xμ }), which implies that y = xμ . This is also a contradiction. Therefore we obtain that h(y) ≤ lim inf h(yn ). This means h is sequentially lower semicontinuous. n→∞

Also T satisfies (1.27) for any x ∈ X but T does not have a fixed point, which is a contradiction with the hypothesis. Thus X is ∑-semicomplete.   The following results can be obtained as corollaries to Theorems 7 and 8. Corollary 2. Let (X, s) be a (∑, =)-complete semimetric space and let T be a self mapping on X. Let h : X → (−∞, +∞] be a function which is proper and bounded from below. Assume that h is sequentially lower semicontinuous from above in the sense of Definition 14. Assume also that there exists κ ∈ N satisfying the following: (i) h(Tx) ≤ h(x) for all x ∈ X. (ii) h(Tκ x) + s(x, Tx) ≤ h(x) for all x ∈ X. Then T has a fixed point.

Symmetric Spaces and Fixed Point Theory

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Corollary 3. Let (X, s) be a (∑, =)-complete semimetric space and let T be a self mapping on X. Let h : X → (−∞, +∞] be a function which is proper and bounded from below. Assume that h is sequentially lower semicontinuous from above in the sense of Definition 14. Assume also h(Tx) + s(x, Tx) ≤ h(x) for all x ∈ X.

(1.28)

Then T has a fixed point. The next result is an extension of the Banach Contraction Principle for semi-metric spaces with ∑-completeness and ∑-semicompleteness which is a corollary to Theorem 8. Corollary 4. Let (x, s) be a semi-metric space. Assume that either of the following holds: (i) X is ∑-complete. (ii) X is ∑-semicomplete and s is sequentially lower semicontinuous. Let T be a mapping on X satisfying s(Tx, Ty) ≤ k s(x, y), for k ∈ (0, 1) and for all x, y ∈ X. Then T has a fixed point z ∈ X. Moreover, {Tn x} converges to z for all x ∈ X. We now discuss coincidence and common fixed points in symmetric spaces. For us it is impossible to discuss all the results here, so we just give one of the basic result proved by Hicks and Rhodes [25]. Recall that, x ∈ X is said to be a coincidence point of mappings f and g on a symmetric space (X, s) if fx = gx holds and y ∈ X is called a common fixed point of f and g if y = fy = gy holds. Definition 16. Let (X, s) be a symmetric space and f , g : X → X be two mappings. Then f and g are said to be (i) commuting if fgx = gfx for all x ∈ X; (ii) compatible if lim s(fgxn , gfxn ) = 0 for each sequence {xn } in X such that lim fxn = n→∞ n→∞ lim gxn ; n→∞

(iii) noncompatible if there exists a sequence {xn } in X such that lim fxn = lim gxn but lim s(fgxn , gfxn ) is either nonzero or nonexistent;

n→∞

n→∞

n→∞

(iii) weakly compatible if they commute at their coincidence points. Theorem 10. Let (X, s) be a S-complete (s-Cauchy complete) symmetric (semi-metric) space with bounded s and (W3). Suppose f , g : X → X are two s-continuous (τs -continuous) mappings satisfying (i) f and g are commuting and g(X) ⊂ f (X); (ii) s(gx, gy) ≤ ks(fx, fy) for all x, y ∈ X. Then f and g have a unique common fixed point.

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Proof. First we prove the Theorem for s a symmetric. Let M = sup{s(x, y) : x, y ∈ X}. Let us fix x0 ∈ X. We choose x1 ∈ X such that gx0 = fx1 . Again we choose x2 ∈ X such that gx1 = fx2 . In general, we choose xn such that f (xn ) = g(xn−1 ). Now, s(fxn , fxn+m ) = s(gxn−1 , gxn+m−1 ) ≤ k s(fxn−1 , fxn+m−1 ) ≤ ... ≤ kn s(fx0 , fxm ) ≤ kn M. Therefore {fxn } is an s-Cauchy sequence and the S-completeness of (X, s) gives us x ∈ X with lim s(fxn , x) = 0. Since g is s-continuous, we have lim s(gfxn , gx) = 0. Now fxn = gxn−1 n→∞

n→∞

so that lim s(gxn , x) = 0. As f is s-continuous lim s(fgxn , fx) = 0. Since fg = gf , lim s(fgxn , fx) = n→∞

n→∞

n→∞

lim s(gfxn , gx) = 0. By (W3), we get fx = gx. Also fgx = gfx. Thus f (fx) = f (gx) = g(fx) = g(gx)

n→∞

and s(gx, g(gx)) ≤ k s(fx, f (gx)) = k s(gx, g(gx)) implies g(x) = g(gx). Hence g(x) = g(gx) = f (gx) so gx is common fixed point of f and g. To prove uniqueness: If x = fx = gx and y = fy = gy, then (ii) gives s(x, y) = s(gx, gy) ≤ k s(fx, fy) = k s(x, y). Thus x = y. For s a semi-metric, s(xn , x) → 0 if and only if xn → x in the topology τs . Thus for a τs continuous f , xn → x implies fxn → fx and the above proof holds without change.   The proof of the following corollary can be obtained easily, so we skip the proof.

Corollary 5. Let (X, s) be an S-complete (s-Cauchy complete) symmetric (semi-metric) space with bounded s and (W3). Suppose f , g : X → X to be two commuting, s-continuous (τs -continuous) mappings satisfying g(X) ⊂ f (X). If there exists a positive integer m and k ∈ (0, 1) such that s(gm x, gm y) ≤ ks(fx, fy), for all x, y ∈ X, then f and g have a unique common fixed point.

1.6 Fixed Points of Multivalued Mappings We know that multivalued mappings play a major role in many areas such as studying disjunctive logic programs. We present some fixed point results for multivalued mappings in symmetric spaces. The key contribution in the fixed point theory of multivalued mappings was made by Hicks [24], Moutawakil [41] etc. A mapping T which associates with each element x of a set, a subset T(x) of set X is called a multivalued (or set-valued) mapping. A multivalued mapping T can be treated as a single-valued mapping of X into power set 2X , that is, the set of all subsets of X. Let us recall that for a multivalued mapping T, x is a fixed point of T if x ∈ Tx. Let us begin with the following theorem of Hicks. Theorem 11. Let (X, s) be a complete symmetric space with bounded s and (W4). Let T : X → (Bs (X), S ) satisfies lim s(xn , x) = 0 implies lim S(Txn , Tx) = 0. Then, there exists x ∈ X with n→∞

n→∞

x ∈ Tx if and only if there exists a sequence {xn } in X with xn+1 ∈ Txn and ∑∞n=1 s(xn , xn+1 ) < ∞. In this case, lim s(xn , x) = 0. n→∞

Symmetric Spaces and Fixed Point Theory

21

Proof. If x ∈ Tx, let xn = x for every n. Suppose the condition holds. Since (X, s) is complete, there exists x ∈ X with lim s(xn , x) = 0 . Since xn+1 ∈ Txn , n→∞

s(xn+1 , Tx) ≤ sup{s(y, Tx) : y ∈ Txn } ≤ S(Txn , Tx) → 0 as n → ∞. Now, lim s(xn , Tx) = 0, so Lemma 2 gives yn ∈ Tx such that lim s(xn , yn ) = 0. Since we have n→∞

n→∞

lim s(xn , x) = 0, (W4) gives lim s(yn , x) = 0 which in turn gives s(x, Tx) = 0 or x ∈ Tx = Tx.

n→∞

n→∞

 

The following result is the extension of Nadler’s [42] fixed point theorem in the setting of symmetric spaces. Theorem 12. Let (X, s) be a complete symmetric space with bounded s and (W4). Let T : X → B(X) satisfy S (Tx, Ty) ≤ k s(x, y), (1.29) for all x, y ∈ X and k ∈ (0, 1); then T admits at least one fixed point. Proof. Choose a fix x0 ∈ X and x1 ∈ Tx0 such that x1 ∈ / Tx1 . We choose x2 ∈ Tx1 such that s(x1 , x2 ) ≤ k + s(x1 , Tx1 ) ≤ k + S (Tx0 , Tx1 ). Following similar analogy, there exists x3 ∈ Tx2 with s(x2 , x3 ) ≤ k2 + S (Tx1 , Tx2 ). Continuing in this manner, we obtain a sequence {xn } such that xn+1 ∈ Txn , xn ∈ / Txn and s(xn , xn+1 ) < kn + S (Txn−1 , Txn ) ≤ kn + k s(xn−1 , xn ) ≤ kn + k[kn−1 + S (Txn−2 , Txn−1 )] ≤ 2kn + k2 s(xn−2 , xn−1 ) ... ≤ nkn + kn s(x0 , x1 ). Since ∑∞n=1 nkn and ∑∞n=1 kn both converge, ∑∞n=1 s(xn , xn+1 ) < ∞. Clearly, lim s(un , x) = 0 implies lim S(Tun , Tx) = 0. Therefore Theorem 11 gives x ∈ X with x ∈ Tx.

n→∞

n→∞

 

The following two results are fixed point theorems for the more general contractive condition proved in [24]. We skip the proof of these theorems as they can be easily proved with the help of the proof of Theorem 12. Theorem 13. Let (X, s) be a complete symmetric space with bounded s and (W4). Let T : X → (Bs (X), S ) be a continuous mapping and Tx be compact for each x ∈ X. For fixed x assume y → s(x, y) is continuous. Suppose S (Tx, Ty) ≤ k(s(x, y)) holds for all x, y ∈ X, and k : [0, ∞) → [0, ∞) is a nondecreasing mapping with k(0) = 0. Then T has fixed point in X if and only if there exists x0 ∞

in X with

lim s(xn , x) = 0. ∑ kn (s(x0 , Tx0 )) = ∞. In this case, we can choose xn+1 ∈ Txn with n→∞

n=1

Pradip Ramesh Patle and Deepesh Kumar Patel

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Theorem 14. Let (X, s) be a complete symmetric space with bounded s and (W4). Let T : X → (B(X), S ) be a continuous mapping and Tx be compact for each x ∈ X. For fixed x, assume y → s(x, y) is continuous. Suppose that

 S (Tx, Ty) ≤ k max{s(x, y), s(x, Tx), s(y, Ty)} holds for all x, y ∈ X, and k : [0, ∞) → [0, ∞) is a nondecreasing mapping with k(0) = 0. Then T has ∞

a fixed point in X if and only if there exists x0 in X with choose xn+1 ∈ Txn with lim s(xn , x) = 0.

∑ kn (s(x0 , Tx0 )) = ∞. In this case, we can

n=1

n→∞

Before going further we recall two definitions from [41]. Definition 17. Let (X, s) be a symmetric space and A a nonempty subset of X. Then s

s

(i) A is s-closed if and only if A = A, where A = {x ∈ X : s(x, A) = 0} and s(x, A) = inf{d(x, y) : y ∈ A}. (ii) A is s-bounded if and only if diam(A) < ∞, where diam(A) = sup{s(x, y) : x, y ∈ A}. The following result is required to prove Theorem 15. Lemma 6. Let (X, s) be an s-bounded symmetric space. Let A, B ∈ C(X) and α > 1. For each x ∈ A, there exists y ∈ B such that s(x, y) ≤ α S (A, B). Theorem 15. Let (X, s) be an s-bounded and S-complete symmetric space satisfying (W4) and T : X → C(X) be a multivalued mapping such that S (Tx, Ty) ≤ k s(x, y), k ∈ [0, 1), for every x, y ∈ X.

(1.30)

Then there exists z ∈ X such that z ∈ Tz. Proof. Let x1 ∈ X and α ∈ (k, 1). Since Tx1 is nonempty, there exists x2 ∈ Tx1 such that s(x1 , x2 ) > 0 (otherwise x1 is a fixed point of T). By (1.30), we have s(x2 , Tx2 ) ≤ S (Tx1 , Tx2 ) ≤ k s(x1 , x2 ) < α s(x1 , x2 ) < α s(x1 , x2 ). It follows that there exists x3 (say) in Tx2 such that s(x2 , x3 ) < α s(x1 , x2 ). Similarly, there exists x4 ∈ Tx3 such that s(x3 , x4 ) < α s(x2 , x3 ). Continuing in this manner, we get a sequence {xn } in X such that xn+1 ∈ Txn , s(xn , xn+1 ) > 0 and s(xn , xn+1 ) < α s(xn−1 , xn ).

Symmetric Spaces and Fixed Point Theory

23

We claim that {xn } is an s-Cauchy sequence. Indeed, we have s(xn , xn+m ) < α s(xn−1 , xn+m−1 ) < α 2 s(xn−2 , xn+m−2 ) ... < α n−1 s(x1 , xm+1 ) < α n−1 diam(X). This implies {xn } is an s-Cauchy sequence. Hence due to S-completeness of X, there exists z in X such that lim s(xn , z) = 0. Now, from Lemma 6, we have that for each n ∈ N there n→∞

exists yn ∈ Tz such that for ε > 1, s(xn+1 , yn ) ≤ ε S (Txn , Tz) ≤ ε ks(xn , z), n = 1, 2, . . . , which implies that lim s(xn+1 , yn ) = 0. In view of (W4), we have lim s(yn , z) = 0 and therefore n→∞

s

n→∞

z ∈ Tz = Tz.

 

Recently, the notion of S + -type contractions was introduced in [44] and proved fixed point theorems for multivalued mapping. The authors used the notion of α -admissibility defined by Samet et al. [46] in order to relax the requirement of satisfying the contractive condition at every pair of points in a space without altering the outcome. The idea of α -admissible mappings is interesting as it includes the case of discontinuous mappings, unlike the contraction mapping. In the rest of this chapter, the used mapping α (unless mentioned) is considered as α : X × X → [0, ∞), where X is nonempty. Definition 18. A self-mapping T : X → X is called α -admissible if for x, y ∈ X, the condition α (x, y) ≥ 1 implies that α (Tx, Ty) ≥ 1. Subsequently, multivalued α -admissibility was proposed by Mohammadi et al. [40] as follows: Definition 19. A set-valued mapping T : X → N (X) is called an α -admissible mapping if for all x ∈ X and y ∈ Tx, α (x, y) ≥ 1 implies α (y, z) ≥ 1 for each z ∈ Ty. The α -admissible pair of multivalued mapping is defined as follows. Definition 20. Let T, S : X → N (X) be two mappings. The ordered pair (T, S) is said to be α -admissible if for all x, y ∈ X, α (x, y) ≥ 1 implies α (p, q) ≥ 1, for all p ∈ Tx and q ∈ Sy. First, we extend the idea of α -completeness to the symmetric space (X, s) along the lines of [26]. Definition 21. A symmetric space (X, s) is said to be α -complete if for every sequence {xn } ∞

in X satisfying ∑ s(xn , xn+1 ) < ∞ with α (xn , xn+1 ) ≥ 1 for all n ∈ N, there exists x ∈ X such n=1

that lim s(xn , x) = 0. n→∞

Pradip Ramesh Patle and Deepesh Kumar Patel

24

Remark 5. If (X, s) is complete, then it is also α -complete. The converse need not be true as shown in the following example. Example 8. Let X = { n1 : n ∈ N} ∪ {1 + 1j : j ∈ N}. Define s : X × X → [0, ∞) by s(x, y) = |y − x| for all x, y ∈ Y. Then (X, s) is a complete symmetric space. Consider α : X × X → [0, ∞) as ⎧ ⎨ 1 if x, y ≥ 1, α (x, y) = ⎩ 0 otherwise. Here, (X, s) is also α -complete. In fact, for every sequence {xn } in Y satisfying ∞

∑ s(xn , xn+1 ) < ∞ with α (xn , xn+1 ) ≥ 1 for all n ∈ N, we have xn ∈ {1 +

j=1

1 n

: n ∈ N}. There

exists a = 1 ∈ Y such that lim s(1 + n1 , 1) = 0. n→∞

Let Φ denote the set of all monotone nondecreasing functions ϕ : [0, ∞) → [0, ∞) such that < ∞ for each t > 0, where ϕ n is the nth iterate of ϕ .

∞ ∑n=1 ϕ n (t)

Lemma 7. Given the following statements: (i) ϕ ∈ Φ ; (ii) lim ϕ n (t) = 0; n→∞

(iii) ϕ (t) < t for all t > 0. Then (i) implies (ii) implies (iii). The notion of α -S + -continuity of a mapping weakens the continuity. Definition 22. A set-valued mapping T : X → C L (X) is called α -S + -continuous on C L (X) if lim s(xn , x) = 0 and α (xn , xn+1 ) ≥ 1 for all n ∈ N implies lim S + (Txn , Tx) = 0. n→∞

n→∞

Lemma 8. [24] Let (X, s) be a symmetric space and T : X → Bs (X). Then lim s(xn , Tx) = 0 if and only if there exists yn ∈ Tx satisfying lim s(xn , yn ) = 0.

n→∞

n→∞

We now introduce the notion of α -ϕ -S + -contractive multivalued mapping. Definition 23. Let (X, s) be a symmetric space. A set-valued mapping T : X → N (X) is called α -ϕ -S + -contractive (1) if there exist two functions ϕ ∈ Φ and α such that

 α (x, y)S + (Tx, Ty) ≤ ϕ s(x, y) for all x, y ∈ X,

(1.31)

(2) for every x ∈ X, y ∈ Tx, q ≥ 1, there exists z ∈ Ty such that s(y, z) ≤ q S + (Tx, Ty).

Definition 24. In the above definition, if we put m(x, y) = max s(x, y), s(x, Tx), s(y, Ty) instead of s(x, y) in (1.31), then the mapping T is called generalized α -ϕ -S + -contractive.

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25

Remark 6. The condition (2) in the above definition holds automatically if we replace S + by H . Theorem 16. Let (X, s) be an α -complete symmetric space with (W4). Let T : X → C B s (X) be a generalized α -ϕ -S + -contraction. Assume that (i) T is α -admissible; (ii) there exist x0 in X and x1 in Tx0 such that α (x0 , x1 ) ≥ 1; (iii) T is α -S + -continuous. Then T admits a fixed point. Proof. From (ii), we have x0 ∈ X and x1 ∈ Ta0 such that α (x0 , x1 ) ≥ 1. Assume x0 = x1 , oth/ Tx1 , otherwise x1 will be a fixed point of erwise x0 is a fixed point of T. Assume also x1 ∈ T. Define a sequence {xn } in X by x1 ∈ Tx0 , x2 ∈ Tx1 , . . ., xn+1 ∈ Txn , for all n ∈ N such that / Txn . Further, using (i), we obtain α (xn , xn+1 ) ≥ 1. Because of condition (2) in Definition xn ∈ 23, we now have s(xn , xn+1 ) ≤ α (xn−1 , xn )S + (Txn−1 , Txn ) 

 ≤ ϕ max s(xn−1 , xn ), s(xn−1 , Txn−1 ), s(xn , Txn ) 

 ≤ ϕ max s(xn−1 , xn ), s(xn , xn+1 ) .

(1.32)



If max s(xn−1 , xn ), s(xn , xn+1 ) = s(xn , xn+1 ), then from (1.32) we have s(xn , xn+1 ) ≤ ϕ (s(xn , xn+1 )) and then by Lemma 7, we get s(xn , xn+1 ) < s(xn , xn+1 ), a contradiction. Therefore, s(xn , xn+1 ) ≤ ϕ (s(xn−1 , xn )). Repeating the above process, we get s(xn , xn+1 ) ≤ ϕ n (s(x0 , x1 )). Since, ∑∞n=1 ϕ n (t) < ∞ for all t > 0, so we have ∑∞n=1 s(xn , xn+1 ) < ∞. As X is an α -complete symmetric space, so there exists x ∈ X such that lim s(xn , x) = 0. Also n→∞

the α -H + -continuity of T gives us lim S + (Txn , Tx) = 0.

n→∞

Since xn+1 ∈ Txn , by using condition (2) in Definition 23 for q ≥ 1, we get s(xn+1 , Tx) ≤ qH + (Txn , Tx) → 0 as j → ∞. Thus, lim s(xn+1 , Tx) = 0. This is equivalent to lim s(xn , Tx) = 0. Therefore by Lemma 8, there n→∞

n→∞

exists yn ∈ Tx such that lim s(xn , yn ) = 0. Since lim s(xn , x) = 0, (W4) implies lim s(yn , x) = 0 n→∞

n→∞

which in turn implies s(x, Tx) = 0 and since Tx is closed, x ∈ Tx.

n→∞

 

The following result can be proved with similar lines of proof to Theorem 16. Theorem 17. Let (X, s) be an α -complete symmetric space with (W4) and T : X → C B s (X) be an α -ϕ -S + -contractive mapping. Assume that (i)–(iii) of Theorem 16 are true. Then T admits a fixed point.

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In the next result the α -H + -continuity of the mapping is relaxed. Theorem 18. Let (X, s) be an α -complete symmetric space with (CC) and T : X → C B s (X) be a generalized α -ϕ -S + -contractive mapping. Assume that (i) T is α -admissible; (ii) there exist x0 in X and x1 in Tx0 such that α (x0 , x1 ) ≥ 1; (iii)’ if {xn } is a sequence in X with lim s(xn , x) = 0 and α (xn , xn+1 ) ≥ 1 for all j ∈ N then n→∞

α (xn , x) ≥ 1. Then T admits a fixed point.

∞

Proof. Following the proof of Theorem 16, we have that ∑ s(xn , xn+1 ) < ∞ and α (xn , xn+1 ) ≥ 1 n=1

for all n ∈ N. Then by α -completeness of (X, s), there exists x ∈ X such that lim s(xn , x) = 0. n→∞

Using (iii)’, we get α (xn , x) ≥ 1 for all n ∈ N. We now claim that x ∈ Tx. Assume that x ∈ / Tx, then s(x, Tx) > 0. By using (1.31), we have s(xn+1 , Tx) ≤ α (xn , x)H + (Txn , Tx)

 ≤ ϕ max{s(xn , x), s(xn , Txn ), s(x, Tx)} . Let ε = n > n1 .

(1.33)

s(x, Tx) s(x, Tx) . Since lim s(xn , x) = 0, we can find n1 ∈ N such that s(xn , x) < for all n→∞ 2 2

Also, as lim s(xn , xn+1 ) = 0, we can find n2 ∈ N such that s(xn , Txn ) ≤ s(xn , xn+1 ) < n→∞

for all n > n2 . Thus we get

s(x, Tx) 2

max{s(xn , x), s(xn , Txn ), s(x, Tx)} = s(x, Tx) for all n ≥ n0 = max{n1 , n2 }. Therefore, (1.33) yields us s(xn+1 , Tx) ≤ ϕ (s(x, Tx))

(1.34)

for n ≥ n0 . Taking the limit as n → ∞ in (1.34) and in view of condition (CC), we get s(x, Tx) ≤ ϕ (s(x, Tx)), which is a contradiction to the consequence of Lemma 7. Thus our assumption is wrong. Hence x ∈ Tx.   Following the proof of above theorem, the next result can be proved easily. Theorem 19. Let (X, s) be an α -complete symmetric space with (CC). Let T : X → C B s (X) be an α -ϕ -S + -contraction. Then if conditions (i), (ii) and (iii)’ of Theorem 18 hold, T admits a fixed point. Now we will discuss the common fixed point theorems for multivalued mappings using S + distance in symmetric spaces. We first introduce the concept of an α -ϕ -S + -contractive pair of mappings. Definition 25. Let (X, s) be a symmetric space. Given T, S : X → N (X). (T, S) is called an α -ϕ -S + -contractive pair if

Symmetric Spaces and Fixed Point Theory

27

(1) there exist ϕ ∈ Φ and a symmetric function α : X × X → [0, ∞) such that

 α (x, y)S + (Tx, Sy) ≤ ϕ mT,S (x, y)

for all x, y ∈ X, where mT,S (x, y) = max s(x, y), s(x, Tx), s(y, Sy) , (2) for every x ∈ X, (a) y ∈ Tx, q ≥ 1, there exists z ∈ Sy such that

(1.35)

s(y, z) ≤ qS + (Tx, Sy), (b) y ∈ Sx, q ≥ 1, there exists z ∈ Ty such that s(y, z) ≤ qS + (Sx, Ty). We are now ready to prove a common fixed point result for the class of α -ϕ -S + contractive pair of multivalued mappings. Theorem 20. Let (X, s) be an α -complete symmetric space with (CC) and (T, S) of mappings T, S : X → C B s (X) be an α -ϕ -S + -contractive pair. Assume that (i) (T, S) is α -admissible; (ii) there exist x0 in X and x1 ∈ Tx0 such that α (x0 , x1 ) ≥ 1; (iii) if {xn } is any sequence in X with lim s(xn , x) = 0 and α (xn , xn+1 ) ≥ 1 for all n ∈ N, then we have α (xn , x) ≥ 1.

n→∞

Then T admits a fixed point. Proof. Let x0 ∈ X be arbitrary and x1 ∈ Tx0 . We assume x0 = x1 ; otherwise there is nothing to prove. This means s(x0 , x1 ) > 0. From (ii), we have α (x0 , x1 ) ≥ 1. Thus by virtue of 2(a) of Definition 25, we choose x2 ∈ Sx1 such that s(x1 , Sx1 ) ≤ s(x1 , x2 ) ≤ α (x0 , x1 )S + (Tx0 , Sx1 )

 ≤ ϕ mS,T (x0 , x1 )

 ≤ ϕ max{s(x0 , x1 ), s(x0 , Tx0 ), s(x1 , Sx1 )}

 = ϕ max{s(x0 , x1 ), s(x1 , Sx1 )} .

(1.36)

Clearly, from the above inequality we can conclude that max{s(x0 , x1 ), s(x1 , Sx1 )} = s(x0 , x1 ). Otherwise, the second case would lead us to a contradiction. Thus (1.36), yields us s(x1 , x2 ) ≤ ϕ (s(x0 , x1 )).

(1.37)

As x1 ∈ Tx0 and x2 ∈ Sx1 and due to α -admissibility of (T, S), we have α (x1 , x2 ) ≥ 1. Thus by virtue of 2(b) of Definition 25, we choose x3 ∈ Tx2 such that s(x2 , Tx2 ) ≤ s(x2 , x3 ) ≤ α (x1 , x2 )S + (Sx1 , Tx2 )

 ≤ ϕ mS,T (x1 , x2 )

 ≤ ϕ max{s(x1 , x2 ), s(x1 , Sx1 ), s(x2 , Tx2 )}

 = ϕ max{s(x1 , x2 ), s(x2 , Tx2 )} .

(1.38)

Pradip Ramesh Patle and Deepesh Kumar Patel

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Again, we have max{s(x1 , x2 ), s(x2 , Tx2 )} = s(x1 , x2 ). Otherwise, the second case would lead to a contradiction. Thus from (1.38), we get s(x2 , x3 ) ≤ ϕ (s(x1 , x2 )) = ϕ 2 (s(x0 , x1 )).

(1.39)

Persisting this way, a sequence {xn } in X is generated such that x2n+1 ∈ Tx2n , x2n+2 ∈ Sx2n+1 satisfying α (xn , xn+1 ) ≥ 1 and s(xn , xn+1 ) = ϕ n (s(x0 , x1 )) for all n ∈ N.

(1.40)

Since ∑∞n=1 ϕ n (t) < ∞, we have ∑∞n=1 s(xn , xn+1 ) < ∞. As the symmetric space (X, s) is α -complete, there exists x ∈ X such that lim s(xn , x) = 0.

n→∞

From (iii), we have α (xn+1 , x) ≥ 1 for all n ∈ N. We now claim that x ∈ Tx ∩ Sx. First, let us assume x ∈ / Tx; then s(x, Tx) > 0. By 2(a), we have s(x2n+2 , Tx) ≤ α (x2n+1 , x)S + (Sx2n+1 , Tx)

 ≤ ϕ max{s(x2n+1 , x), s(x, Tx), s(x2n+1 , Sx2n+1 )} . Since lim s(xn , x) = 0, we can find an integer N1 ∈ N such that s(x2n+1 , x) < ε = n→∞

(1.41) s(x,Tx) 2

for all

n > N1 . Also as {xn } is a sequence such that lim s(xn , xn+1 ) = 0, we can find an integer N2 ∈ N n→∞

such that d(x2n+1 , Sx2n+1 ) ≤ s(x2n+1 , x2n+2 ) < ε =

s(x,Tx) 2

for all n > N2 . Thus, we get

max{s(x, x2n+1 ), s(x, Tx), s(x2n+1 , Sx2n+1 )} = s(x, Tx), for all n ≥ N0 = max{N1 , N2 }. Therefore we have s(Tx, x2n+2 ) ≤ ϕ (s(x, Tx)) for all n ≥ N0 . Taking n → ∞ and in view of (CC), we get s(Tx, x) < s(x, Tx), which gives us s(x, Tx) = 0. As Tx is closed, we have x ∈ Tx. Arguing in a similar way, we can get x ∈ Sx and hence x ∈ Tx ∩ Sx.   The following result is a fixed point theorem without using H or S + distance functions in symmetric spaces. Definition 26. Let (X, s) be a symmetric space. A multivalued mapping T : X → C B s (X) is called generalized pointwise α -ϕ -contractive if there exist functions ϕ ∈ Φ and α : X × X → [0, ∞) such that for x1 , x2 ∈ X, y1 ∈ Tx1 , y2 ∈ Tx2 ,

 α (x1 , x2 )s(y1 , y2 ) ≤ ϕ M3 (x1 , x2 ) , (1.42)

where M3 (x1 , x2 ) = max s(x1 , x2 ), s(x1 , y1 ), s(x2 , y2 ) . Definition 27. A mapping T is called pointwise α -ϕ -contractive if we replace M3 (x1 , x2 ) by s(x1 , x2 ) in Definition 26. Theorem 21. Let (X, s) be an α -complete symmetric space with (W4) and mapping T : X → C B s (X) be generalized point-wise α -ϕ -contractive. Then T admits a fixed point if the following hold:

Symmetric Spaces and Fixed Point Theory

29

(i) T is α -admissible; (ii) there exist x0 in X and x1 ∈ Tx0 such that α (x0 , x1 ) ≥ 1; (iii) for every sequence {xn } in X such that lim s(xn , x) = 0 with α (xn , xn+1 ) ≥ 1, there exists a n→∞

sequence {yn } in Txn such that lim s(yn , y) = 0 for some y ∈ Tx. n→∞

Proof. Initiating with arbitrary x0 ∈ X and x1 ∈ Tx0 such that α (x0 , x1 ) ≥ 1, then following the proof of Theorem 16 we get a sequence {xn } defined by x1 ∈ Tx0 , x2 ∈ Tx1 , . . . , xn+1 ∈ Txn for all n ∈ N such that xn ∈ / Txn . Since T is α -admissible, we have α (xn , xn+1 ) ≥ 1 for all n ∈ N ∪ {0}. By (1.42), we have s(xn , xn+1 ) ≤ α (xn−1 , xn )s(xn , xn+1 ) ≤ ϕ (M3 (xn−1 , xn )) 

 ≤ ϕ max s(xn−1 , xn ), s(xn−1 , xn ), s(xn , xn+1 ) 

 ≤ ϕ max s(xn−1 , xn ), s(xn , xn+1 ) .

(1.43)



If max s(xn−1 , xn ), s(xn , xn+1 ) = s(xn , xn+1 ), then from (1.43) we have s(xn , xn+1 ) ≤ ϕ (s(xn , xn+1 )). Using Lemma 7, we get s(xn , xn+1 ) < s(xn , xn+1 ); that is a contradiction. Thus (1.43) gives s(xn , xn+1 ) ≤ ϕ (s(xn−1 , xn )). Repeating this process, we get s(xn , xn+1 ) ≤ ϕ n (s(x0 , x1 )). As ∑∞n=1 ϕ n (t) < ∞ for all t > 0, so we obtain ∞

∑ s(xn , xn+1 ) < ∞.

n=1

Due to α -completeness of the symmetric space X, there exists x ∈ X such that lim s(xn , x) = 0 n→∞

and by (iii), we get a sequence {yn } ∈ Txn such that lim s(yn , y) = 0 for some y ∈ Tx. n→∞

Since xn+1 ∈ Txn , we get

s(xn+1 , Tx) = inf{s(xn , Tx) : an ∈ Tan } ≤ s(yn , y) → 0 as n → ∞. Thus, we find that lim s(xn+1 , Tx) = 0. This is equivalent to lim s(xn , Tx) = 0. Therefore by n→∞

n→∞

Lemma 8, there exists zn ∈ Tx such that lim s(xn , zn ) = 0. Since lim s(xn , x) = 0, (W4) implies n→∞

n→∞

lim s(zn , x) = 0 which in turn implies s(x, Tx) = 0 and since Tx is closed, x ∈ Tx.

n→∞

 

The following results follow in a similar way to the above proof. Theorem 22. Let (X, s) be an α -complete symmetric space with (W4) and mapping T : X → C B s (X) be pointwise α -ϕ -contractive. If conditions (i)–(iii) in Theorem 21 hold, then T admits a fixed point. Remark 7. Theorem 21 and 22 also hold if condition (iii) is replaced by the α -continuity of T.

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Fixed point theorems in symmetric spaces have found applications in probabilistic metric spaces as the very general probabilistic structures admit a compatible symmetric or semi-metric such that the distribution function Fx,y is related to the symmetric s by s(x, y) < t if and only if Fx,y (t) > 1 − t where t > 0. Hicks and Rhodes [25] discussed the application of Theorems 12, 13 and 14 whereas Patle et al.[44] applied Theorem 16 in probabilistic metric spaces.

1.7 Conclusion and Future Investigations Some of the earliest results in the theory of abstract spaces and abstract distances were the contributions of Fréchet and Menger. One of the major problems in point set topology in the 19th century was the metrization problem. The study of these problems led to the development of study of semi-metrization. Early investigations in this area were made by Wilson. Starting from the work of Wilson on symmetric spaces, in this chapter we first discussed the topology of symmetric spaces. The absence of a triangular inequality in defining symmetric spaces leads to the various concepts of Cauchy-type sequences and completeness of symmetric spaces. All these concepts have been studied in detail from the point of view of their usage. The major part of this chapter dealt with the study of fixed point theorems in symmetric spaces. Taking into consideration their importance, we also discussed the fixed point of multivalued mappings and common fixed point theorems of two mappings with various contractive conditions. In addition the result due to Suzuki, which proves the characterization of semi-completeness by means of the fixed point property, is also presented, which provides some self-containment to this chapter. Although the area of fixed point theory for symmetric spaces has developed into a wide literature there is still a lot to be explored. Some of the open questions we are discussing here: • Suzuki [50] (as discussed in Theorem 9) characterized semicompleteness of semimetric spaces through the fixed point property. Can such a result be proved for completeness instead of semicompleteness? • A basic best proximity point result is discussed in a paper by Felhi [18]. Is there any possibility of establishing the new best proximity point results in symmetric spaces? • Can fixed point results for Z-contractions and F-contractions and other contractive conditions be extended to the notion of symmetric spaces? • Is it possible to prove Proposition 11 without Hausdorffness? • Does every symmetric/semimetric space have a completion? How it can be proved?

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28.

M. Imdad, J. Ali, Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A), Bull. Belg. Math. Soc. Simon Stevin 16 (2009), 421–433. M. Imdad, J. Ali, L. Khan, Coincidence and fixed points in symmetric spaces under strict contractions, J. Math. Anal. Appl. 320 (2006), 352–360. M. Imdad, A. H. Soliman, Some common fixed point theorems for a pair of tangential mappings in symmetric spaces, Appl. Math. Lett. 23(4) (2010), 351–355. J. Jachymski, J. Matkowski, T. Swiatkowski, Nonlinear contractions on semimetric spaces, J. Appl. Anal., 1 (1995), 125–134. E. Karapinar, D. K. Patel, M. Imdad, D. Gopal, Some nonunique common fixed point theorems in symmetric spaces through CLRST property, Int. J. Math. Math. Sci. Article ID 753965, 8 pages, 2013. DOI: 10.1155/2013/753965 M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems for contractive mappings, Fixed Point Theory Appl. 2010 (2010) 7. Article ID 315398. W. A. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, (2014). Kirk, W. A., Shahzad, N.: Fixed points and Cauchy sequences in semimetric spaces. J. Fixed Point Theory Appl. 17 (2015), 541–555. Kirk, W. A., Saliga, L. M., The Brezis–Browder order principle and extensions of Caristi’s theorem. Nonlinear Anal. 47 (2001), 2765–2778. H. W. Martin, Metrization of symmetric spaces and regular maps, Proc. Amer. Math. Soc. 35 (1972), 269–274. K. Menger, Untersuchungen über allgemeine, Math. Annalen, 100 (1928), 75–163. D. Mihet, A note on a paper of Hicks and Rhoades, Nonlinear Anal., 65 (2006), 1411–1413. B. Mohammadi, S. Rezapour, N Shahzad, Some results on fixed points of α -ψ -Ciric generalized multifunctions, Fixed Point Theory Appl., (2013), 2013:24, 10 DOI:10.1186/1687-1812-2013-24 D.E. Moutawakil, A fixed point theorem for multivalued maps in symmetric spaces, Applied Mathematics E-Notes, 4 (2004), 26–32. S. B. Nadler, Multivalued contraction mappings, Pacific. J. Math. 30 (1969), 475–488. H. K. Pathak, N. Shahzad, A generalization of Nadler’s fixed point theorem and its application to nonconvex integral inclusion, Topol. Methods Nonlinear Anal. 41(1) (2013), 207–227. P. R. Patle, D. Patel, H. Aydi, S. Radenovi´c, ON H+ type multivalued contractions and applications in symmetric and probabilistic spaces. Mathematics 7 (2019), 144. S. Radenovi´c, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (1) (2011), 38–50. B. Samet, C. Vetro, P. Vetro, Fixed point theorem for α -ψ Contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam (1983). Shahzad, N., Alghamdi, M., Alshehri, S., Arandelovi´c, I., Semi-metric spaces and fixed points of α -ψ -contractive maps. J. Nonlinear Sci. Appl., 9 (2016), 3147–3156. 10.22436/jnsa.009.05.104. Suzuki, T., Caristi’s fixed point theorem in semimetric spaces, J. Fixed Point Theory Appl. (2018) 20: 30. DOI: 10.1007/s11784-018-0492-y T. Suzuki, Characterization of ∑-semicompleteness via Caristi’s fixed point theorem in semimetric spaces, J. Funct. Spaces, vol. 2018, Article ID 9435470 (2018), 7 pages. DOI: 10.1155/2018/9435470 D. Turkoglu, I. Altun, A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying an implicit relation, Bol. Soc. Mat. Mexicana 13 (2007), 195–205. W. A. Wilson, On semi-metric spaces, Amer. J. Math., 53 (1931), 361–373. J. Zhu, Y.J. Cho, S.M. Kang, Equivalent contractive conditions in symmetric spaces, Comp. Math. Appl. 50 (2005), 1621–1628.

29. 30. 31. 32.

33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

51. 52. 53.

2 Fixed Point Theory in b-Metric Spaces Nguyen Van Dung and Wutiphol Sintunavarat

CONTENTS 2.1 2.2 2.3 2.4 2.5 2.6

b-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metrization and Completion of b-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Point Results on b-Metric Spaces: Similar Techniques as in Metric Spaces . . . Multivalued Fixed Point Results in b-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relaxations of Constant Contractions in b-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 38 45 51 55 60

2.1 b-Metric Spaces In 1993, Czerwik [27] introduced the notion of a b-metric with a coefficient 2. This notion was generalized later with a coefficient κ ≥ 1 in [28]. In 2010 Khamsi and Hussain [58] reintroduced the notion of a b-metric under the name metric-type. Another notion of metrictype, called s-relaxedp metric was introduced in [42, Definition 4.2], see also [57]. A b-metric is called quasi-metric in [65]. Quasi-metric spaces play an important role in the study of Gromov hyperbolic metric spaces [91, Final remarks], and in the study of optimal transport paths [100]. Definition 1 ([27], page 5). Let X be a nonempty set and D : X × X → [0, ∞) be a function such that for all x, y, z ∈ X, 1. D(x, y) = 0 if and only if x = y. 2. D(x, y) = D(y, x). 3. D(x, z) ≤ 2[D(x, y) + D(y, z)]. Then d is called a b-metric on X and (X, D) is called a b-metric space. Subsequently, in 1998, Czerwik [28] generalized this notion where the constant 2 was replaced by a constant κ ≥ 1, also with the name b-metric. In 2010, Khamsi and Hussain [58] reintroduced the notion of a b-metric under the name metric-type. 33

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Definition 2 ([58], Definition 6). Let X be a nonempty set, κ > 0 and D : X × X → [0, ∞) be a function such that for all x, y, z ∈ X, 1. D(x, y) = 0 if and only if x = y. 2. D(x, y) = D(y, x). 3. D(x, z) ≤ κ [D(x, y) + D(y, z)]. Then D is called a metric-type on X and (X, D, κ ) is called a metric-type space. From Definition 2.(3), if κ < 1, then we have D(x, z) = 0 for all x, z ∈ X. This yields that X is a singleton set from Definition 2.(1). Note that the unique fixed point of a self-map T on a singleton set always exists. Without loss of generality we assume that κ ≥ 1 is the smallest value that satisfies Definition 2.(3), and it is usually called the coefficient. Also in 2010, Khamsi [57] introduced another definition of a metric-type where the condition (3) in Definition 2 was replaced by D(x, z) ≤ κ [D(x, y1 ) + · · · + D(yn , z)] for all x, y1 , . . . , yn , z ∈ X; see [57, Definition 2.7]. In this chapter, the metric-type in the sense of Khamsi and Hussain in Definition 2, also Czerwik [28], will be called a b-metric to avoid the confusion about the metric-type in the sense of Khamsi [57]. Note that every metrictype is a b-metric and the inverse is not true; see Example 1 below. Definition 3 ([58], Definition 7). Let (X, D, κ ) be a b-metric space. 1. A sequence {xn } is called convergent to x in (X, D, κ ), written lim xn = x, if n→∞

lim D(xn , x) = 0.

n→∞

2. A sequence {xn } is called Cauchy in (X, D, κ ) if lim D(xn , xm ) = 0. n,m→∞

3. (X, D, κ ) is called complete if every Cauchy sequence in (X, D, κ ) is a convergent sequence in (X, D, κ ). The well-known examples on b-metric spaces are as follows. Example 1 ([40], Example 2.4). Let X = R, and D(x, y) = |x − y|2 for all x, y ∈ X. Then for all x, y ∈ X,   |x − z|2 ≤ 2 |x − y|2 + |y − z|2 . So (X, D, κ ) is a b-metric space with the coefficient κ = 2. Moreover, d is not a metric-type. Note that in [19] and many others, the coefficient of the b-metrics on p and Lp [0; 1] is 1 1 κ = 2 p . However, by recalculating we find that the coefficient is also κ = 2 p −1 , see [66, Example 1]. Indeed, for q ≥ 1, using the convexity of the function f (t) = tq , t ∈ R, we have for a, b > 0,  a + b q a+ b 1 1 aq bq ≤ f (a) + f (b) = + . =f 2 2 2 2 2 2 This proves that (a + b)q ≤ 2q−1 (aq + bq ). Applying this inequality with q =

1 p

the following examples with κ = 2

1 p −1

≥ 1, 0 < p ≤ 1 we can prove the triangle inequality in .

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Example 2 ( [19], Example 1.3). Let 0 < p ≤ 1, and 



∞

{xn } : xn ∈ R, n ∈ N, ∑ |xn | < ∞

 = p

p

n=1

 and D(x, y) =

∞

∑ |xn − yn |p

n=1 1 p −1

coefficient κ = 2

 1p

for all x = {xn } , y = {yn } ∈ p . Then D is a b-metric with the

.

Example 3 ([19], Example 1.4). Let 0 < p ≤ 1, and 

1 Lp [0, 1] = x : [0, 1] → R : |x(t)|p dt < ∞ 0

and D(x, y) =



coefficient κ = 2

1

0 1 p −1

|x(t) − y(t)|p

 1p

for all x, y ∈ Lp [0, 1]. Then D is a b-metric with the

.

 Example 4. Let X = 0, 1, 12 , . . . , n1 , . . . and ⎧ 0 ⎪ ⎪ ⎨ 1 D(x, y) = |x − y| ⎪ ⎪ ⎩ 4

if x = y if x = y ∈ {0, 1}  1 if x = y ∈ {0} ∪ 2n :n∈N otherwise.

Then D is a b-metric on X. Note that, in [60, Example 13] and also in [10, Example 3.9], the coefficient κ = 83 but this fact is not true since for all n,        1 1 1 4 = D 1, ≤ κ D (1, 0) + D 0, = κ 1+ . 2n 2n 2n This implies κ ≥ 4. Reconsidering the calculation in [60, Example 13], we find that D is exactly a b-metric with the coefficient κ = 4. Let (X, D, κ ) be a b-metric space. For x ∈ X and r > 0, the open ball B(a, r) is defined by B(a, r) = {y ∈ X : D(x, y) < r} and the closed ball B[a, r] is defined by B[a, r] = {y ∈ X : D(x, y) ≤ r}. Note that a subset U is open in a metric space (X, D) if and only if for each x ∈ U, there exists rx > 0 such that B(x, rx ) ⊂ U. The topology TD on the metric space (X, D) is the family consisting of all open subsets in (X, D). For a sequence {xn } ⊂ (X, D), lim xn = x if and only if lim D(xn , x) = 0. Then the convern→∞

n→∞

gence on (X, D) induces the sequential topology T on X in the sense of Franklin [43]. It is well known that the topology TD and the sequential topology T on a metric space (X, D) are coincident. Then lim xn = x in (X, D) if and only if lim xn = x in (X, T ). The family B of n→∞

n→∞

all finite intersections of the family C = {B(x, r) : x ∈ X, r > 0}

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forms a base of certain topology T D on X. Note that, on a b-metric space (X, D, κ ), there are now three topologies T , TD and T D . The relationship between them is as follows. Proposition 1 ([10], Proposition 3.3). Let (X, D, κ ) be a b-metric space. Then we have 1. T = TD . 2. TD ⊂ T D . In the literature, every b-metric space (X, D, κ ) is always understood to be a topological space with the sequential topology T . Then, (X, D, κ ) is always understood to be a sequential space. In [58, page 3127], Khamsi and Hussain asked whether the open ball B(a, r) is open and the closed ball B[a, r] is closed or not. The following example shows that there exists an open ball B(a, r) which is not open. Note that κ = 4 as mentioned in Example 4.  1 1 0, 1, , . . . , , . . . and 2 n

 Example 5 ([10], Example 3.9). Let X =

D(x, y) =

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 |x − y| ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 4

if x = y if x = y ∈ {0, 1}  if x = y ∈ {0} ∪

1 : n = 1, 2, . . . 2n



otherwise.

Then we have 1. 2. 3. 4.

D is a b-metric on X with the coefficient κ = 4. D is not a metric on X. D is not continuous in each variable. B(1, 2) is not open but it is open with respect to T D . In particular, the inverse of Proposition 1.(2) does not hold.

Note that every open ball B(a, r) is open with respect to the topology T D by the definition of T D . The following example shows that there exists a closed ball B[a, r] which is not closed and moreover, is not closed with respect to T D .  1 1 0, 1, , . . . , , . . . and 2 n

 Example 6 ([10], Example 3.10). Let X =

D(x, y) =

Then we have

⎧ 0 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨

|x − y| ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 4

if x = y if x = y ∈ {0, 1}  if x = y ∈ {0} ∪ otherwise.

1 : n = 1, 2, . . . 2n



Fixed Point Theory in b-Metric Spaces

37

1. D is a b-metric on X with the coefficient κ = 4. 2. Dis not  continuous in each variable.   1 1 3. B 1, is not closed. Moreover, B 1, is also not closed with respect to T D . 2 2 Although the open ball in a b-metric space is not open, we also have the following property which is the same as that in metric spaces, where A is the closure of a subset A. Proposition 2 ([10], Corollary 3.5). Let (X, D, κ ) be a b-metric space and A ⊂ X. Then the following statements are equivalent. 1. x ∈ A. 2. For every ε > 0, B(x, ε ) ∩ A = 0. / 3. There exists {xn } ⊂ A such that lim xn = x. n→∞

The notion of a b-metric was also generalized by modifying its axioms. In 2011, Hussain and Shah [51] introduced the notion of a cone b-metric and established some topological properties of the cone b-metric spaces and then improved some results about KKM maps in the setting of a cone b-metric space. In 2013, Alghamdi et al. [5] introduced the notion of a b-metric-like and established the existence and uniqueness of fixed points in a b-metric-like space as well as in a partially ordered b-metric-like space. In 2014, Shukla [92] introduced the notion of a partial b-metric as a generalization of a partial metric and a b-metric, and analogue to the Banach contraction principle, as well as proving a Kannan-type fixed point result in such spaces. In 2015, An et al. [9] showed that these generalizations of b-metrics may be reduced to certain b-metrics and then they are redundant. Moreover, a b-metric need not be continuous; see Example 5 and Example 6 above. This fact suggested a strengthening of the notion of a b-metric space called a strong b-metric space by Kirk and Shahzad in [59] as follows. Definition 4 ([59], Definition 12.7). Let X be a nonempty set, κ ≥ 1 and D : X × X → [0, ∞) be a function such that for all x, y, z ∈ X, 1. D(x, y) = 0 if and only if x = y. 2. D(x, y) = D(y, x). 3. D(x, z) ≤ D(x, y) + κ D(y, z). Then D is called a strong b-metric on X and (X, D, κ ) is called a strong b-metric space. Remark 1 ([59], page 122). 1. Every strong b-metric is continuous. 2. Every open ball in a strong b-metric space is open. In 2014, Bessenyei and Páles [15] introduced the notion of a triangle function and extended the Banach contraction principle in this spirit for such complete semimetric spaces that fulfil an extra regularity property.

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Definition 5 ([15], page 2). Let (X, d) be a semimetric space. A function Φ : [0, ∞] × [0, ∞] → [0, ∞] is called a triangle function for d if Φ is increasing in each of its variables, Φ (0, 0) = 0 and for all x, y, z ∈ X, d(x, y) ≤ Φ (d(x, z), d(z, y)). Lemma 1 ([15], page 2). Let (X, d) be a semimetric space and for all u, v ∈ [0, ∞],

Φd (u, v) = sup{d(x, y) : ∃p ∈ X, d(p, x) ≤ u, d(p, y) ≤ v}. Then Φd is a triangle function for d. Moreover, if Φ is a triangle function for d then Φd ≤ Φ . Definition 6 ([15], page 2). Let (X, d) be a semimetric space. Then the triangle function Φd defined as in Lemma 1 is called the basic triangle function and (X, d) is called regular if Φd is continuous at (0, 0). Remark 2 ([15], page 2). 1. Every metric space is a semimetric space with the triangle function Φ (u, v) = u + v. 2. Every ultrametric space is a semimetric space with the triangle function Φ (u, v) = max{u, v}. 3. Every b-metric space is a semimetric space with the triangle function Φ (u, v) = κ (u + v). For results on b-metric spaces and strong b-metric spaces, readers may refer to [59, Chapter 12], [32], [8], [33] and references given there.

2.2 Metrization and Completion of b-Metric Spaces It follows from Example 1, and some others, that there exists a b-metric which is not a metric. However, the class of all b-metric spaces and the class of all metric spaces are coincident in the sense that every b-metric space is a metrizable space. Recall that the metrization problem is concerned with conditions under which a topological space X is metrizable [23], where for a function D : X × X → [0, ∞) satisfying some axioms and generating a topology T on X, and for a metric D : X × X → [0, ∞), the topological space (X, T ) is called metrizable by the metric D if T and the metric topology induced by D coincide. Recall that a space X is a metric space if there exists a metric D : X × X → [0, ∞) that satisfies the following conditions for all x, y, z ∈ X. I. D(x, y) = 0 if and only if x = y. II. The symmetry: D(x, y) = D(y, x). III. The triangle inequality: D(x, z) ≤ D(x, y) + D(y, z). Some generalizations of the triangle inequality (III) were introduced such as IV. The generalized triangle inequality: If D(x, y) < ε and D(y, z) < ε then D(x, z) < 2ε . V. The uniform regular property: For every ε > 0 there exists φ (ε ) > 0 such that if D(x, y) < φ (ε ) and D(y, z) < φ (ε ) then D(x, z) ≤ ε .

Fixed Point Theory in b-Metric Spaces

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It is clear that condition (V) reduces to (IV) if φ (ε ) = ε2 , and every b-metric space (X, D, κ ) is a space with the distance function D satisfying (I), (II) and (V) with φ (ε ) = 2εκ . In 1917, Chittenden [22] showed that a space with a distance function satisfying (I), (II) and (V), that was also called a CF-metric space [14, Definition 3.2], is metrizable. Consequently, every b-metric space is metrizable [59, page 114]. Chittenden’s proof was somewhat long and complicated and, although the existence of a distance function satisfying (III) is proved, it is not defined directly in terms of the original distance function satisfying (V). In 1937 Frink [44, page 133] presented a simple and direct proof of the fact that a topological space with a distance function satisfying (I), (II) and (IV), and also (V), is metrizable without relying on Chittenden’s theorem. Frink’s metrization technique is also called the chain approach. Note that the conclusions of Theorem 1 are still true if any strict inequality in (IV) is replaced by the corresponding inequality [31, Remark 1]. Theorem 1 ([44], pages 134–135). Let (X, D) be a space satisfying (I), (II) and (IV). For any x, y ∈ X, define   d(x, y) = inf

n

∑ D(xi , xi+1 ) : x1 = x, x2 , . . . , xn+1 = y ∈ X, n ∈ N

.

(2.1)

i=1

Then the following assertions hold. 1. For all x, x1 , . . . , xn , y ∈ X, D(x, y) ≤ 2D(x, x1 ) + 4D(x1 , x2 ) + · · · + 4D(xn−1 , xn ) + 2D(xn , y). 2. d is a metric on X. 3. For all x, y ∈ X, D(x, y) ≤ d(x, y) ≤ D(x, y). 4 In particular, we get the following conclusions. (i) lim xn = x in (X, D) if and only if lim xn = x in (X, d). n→∞

n→∞

(ii) A sequence {xn } is Cauchy in (X, D) if and only if it is Cauchy in (X, d). (iii)The distance space (X, D) is metrizable by the metric d. Theorem 2 ([44], page 135). Let (X, δ ) be a space satisfying (I), (II) and (V). For all ε ≥ 0, put ψ (ε ) = min{φ (ε ), ε2 }, and put r1 = 1, . . . , rn+1 = ψ (rn ), . . . and for all x, y ∈ X, define D(x, y) =

 1 1 2n

if D(x, y) ≥ r1 if rn > D(x, y) ≥ rn+1 .

Then the following assertions hold. 1. The distance space (X, D) satisfies (I), (II) and (IV). 2. lim xn = x in (X, δ ) if and only if lim xn = x in (X, D). In particular, the distance space (X, δ ) n→∞

n→∞

is metrizable by the metric d defined as in (2.1).

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Frink’s metrization technique impacted many results. In 1998, Aimar et al. [3] improved Frink’s metrization technique to give a direct proof of a theorem of Macías and Segovia ´ and Stemin [65] on the metrization of a b-metric space (X, D, κ ). In 2009 Paluszynski pak [75] also improved Frink’s metrization technique to produce a metric d from a given b-metric space (X, D, κ ). Theorem 3 ([3], Theorem I). Let (X, D, κ ) be a b-metric space. Then there exists 0 < β ≤ 1, depending only on κ , such that   n

∑ Dβ (xi , xi+1 ) : x1 = x, x2 , . . . , xn+1 = y ∈ X, n ∈ N

d(x, y) = inf

i=1

is a metric on X satisfying 12 Dβ ≤ d ≤ Dβ . In particular, if D is a metric then d = D. Theorem 4 ([75], Proposition on page 4308). Let (X, D, κ ) be a b-metric space, 0 < p ≤ 1 satisfying (2κ )p = 2, and for all x, y ∈ X,   d(x, y) = inf

n

∑ Dp (xi , xi+1 ) : x1 = x, x2 , . . . , xn , xn+1 = y ∈ X, n ∈ N

.

(2.2)

i=1

Then d is a metric on X satisfying 14 Dp ≤ d ≤ Dp . In particular, if D is a metric then d = D. Since every b-metric space is a metrizable space, we find that all topological properties of b-metric spaces and metric spaces are coincident. However, some authors did not know this fact and did some redundant work. For example, in [10], An et al. showed a weaker result that every b-metric space with the topology induced by its convergence is a semimetrizable space and thus many properties of b-metric spaces used in the literature are obvious. Then, the authors proved the Stone-type theorem on b-metric spaces and found a sufficient condition for a b-metric space to be metrizable. In 2006, Schroeder constructed a counterexample showing that for given b-metric space (X, D, κ ) the distance function d defined by (2.1) is not a metric [91, Example 2]. The following example, that is simpler than [91, Example 2], also showed that Theorem 1.(1) and Theorem 1.(2) do not hold if a space satisfying (I), (II) and (IV) is replaced by a b-metric space. Example 7 ([31], Example 2.1). Let X = R, and D(x, y) = |x − y|2 for all x, y ∈ X. Then for all x, y ∈ X,   |x − z|2 ≤ 2 |x − y|2 + |y − z|2 . Therefore (X, D, κ ) is a b-metric space with the coefficient κ = 2. However, we find that for n large enough,         1 1 2 n−2 n−1 n−1 4 2D 0, , , ,1 ≤ + 4D + · · · + 4D + 2D n n n n n n n 0 the authors constructed a sequence {xm } where the index m depends on ε ; that is,

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m = m(ε ). However, from the fact d(xm , xs ) ≤ 4ε for m, s large enough [27, page 6] and the fact d(xt , xj ) < 2κε for t, j large enough [59, page 116] the authors asserted that {xm } is a Cauchy sequence. This is clearly a confusion since the index m depends on ε . Note that the confusion was also corrected by Kajántó and Lukács [55] recently. Miculescu and Mihail also proved a fixed point theorem for ϕ -contractions but their main result requires the continuity of the given map [70, Theorem 3.1]. In fact, a general result was proved by Bessenyei and Páles [15] in regular semimetric spaces as follows. Theorem 7 ([15], Theorem 1). Let (X, D) be a complete regular semimetric space and f : X → X be a map such that for all x, y ∈ X, D(f (x), f (y)) ≤ ϕ (D(x, y)) where ϕ : [0, ∞) → [0, ∞) is an increasing function and for each t ∈ [0, ∞) lim ϕ n (t) = 0.

n→∞

Then f has a unique fixed point x∗ ∈ X and lim f n (x) = x∗ for each x ∈ X. n→∞

Replacing regular semimetric spaces in Theorem 7 by b-metric spaces we get the following result. Theorem 8 ([59], Theorem 12.2). Let (X, D, κ ) be a complete b-metric space and f : X → X be a map such that for all x, y ∈ X, D(f (x), f (y)) ≤ ϕ (D(x, y)) where ϕ : [0, ∞) → [0, ∞) is an increasing function and for each t ∈ [0, ∞), lim ϕ n (t) = 0.

n→∞

Then f has a unique fixed point x∗ ∈ X and lim f n (x) = x∗ for each x ∈ X. n→∞

Almost certain versions of metric fixed point theorems in the setting b-metric spaces have been proved later. We recall here some such results. Theorem 9 ([54], Theorem 3.3 – Banach type fixed point in b-metric spaces).  (X, D, κ )  Let be a complete b-metric space and T : X → X be a map such that for some λ ∈ 0, κ1 and for all x, y ∈ X, D(Tx, Ty) ≤ λD(x, y). Then T has a unique fixed point x∗ and lim Tn x = x∗ for all x ∈ X. n→∞

Theorem 10 ([54], Corollary  Let (X, D, κ ) be a complete b-metric space and T : X → X be  3.12). a map such that for some λ ∈ 0, κ1 and for all x, y ∈ X, 

D(x, Ty) D(y, Tx) D(Tx, Ty) ≤ λ max D(x, y), D(x, Tx), D(y, Ty), , . 2κ 2κ Then T has a unique fixed point x∗ .

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Recall that a b-metric is not continuous in general. To overcome that fact some techniques were used in proving fixed point theorems on b-metric spaces; for example see [1, Lemma 2.1]. In 2014 Amini-Harandi [7] introduced the Fatou property to prove one of the most general fixed point theorems in b-metric spaces. Amini-Harandi’s result partially answered the question posed by Singh et al. [93]. Note that there exists a b-metric space that does not have the Fatou property [7, Remark 2.7] and the Fatou property plays a crucial role in the proof of Theorem 11. Definition 7 ([7], Definition 2.4). A b-metric space (X, D, κ ) is said to have the Fatou property if for all x, y ∈ X and lim xn = x we have n→∞

D(x, y) ≤ lim inf D(xn , y). n→∞

Theorem 11 ([7], Theorem 2.8). Let (X, D, κ ) be a complete b-metric space having the Fatou property and f : X → X be a map such that for some λ ∈ [0, κ1 ) and all x, y ∈ X, D(f (x), f (y)) ≤ λ max {D(x, y), D(x, f (x)), D(y, f (y)), D(x, f (y)), D(y, f (x))} . Then f has a unique fixed point x∗ and lim f n (x) = x∗ for all x ∈ X. n→∞

The above theorem and many other fixed point theorems in b-metric spaces were inspired by those in metric spaces, and were proved by very similar and standard techniques. Note that, for a b-metric space (X, D, κ ), the coefficient κ ≥ 1. Therefore, by using similar techniques as in metric spaces, the authors had to use [0, κ1 ) in b-metric fixed point theorems replacing [0, 1) as in metric fixed point theorems. With similar techniques as in metric spaces, almost all types of fixed point results in metric spaces have been generalized to b-metric spaces. In 2016, Radenovi´c et al. [83] obtained some equivalences between cyclic contractions and noncyclic contractions in b-metric spaces. The results improved and complemented several fixed point results for cyclic contractions in b-metric spaces in the literature with much shorter proofs. They also asked the following question. Question 3 ([83], page 162). Prove or disprove the following conjecture. p Let (X, D, κ ) be a complete b-metric space, {Ai }i=1 be nonempty closed subsets of X and T:

p 

i=1

Ai →

p 

Ai be a map satisfying the following conditions.

i=1

1. T(Ai ) ⊂ Ai+1 for all i = 1, . . . , p, where Ap+1 = A1 . 2. There exists k ∈ [0, κ1 ) such that for all x ∈ Ai , y ∈ Ai+1 , i = 1, . . . , p, D(Tx, Ty) ≤ k max{D(x, y), D(x, Tx), D(y, Ty), D(x, Ty), D(y, Tx)}. 3. (X, D, κ ) has the Fatou property. Then T has a unique fixed point. The following example shows that the statement mentioned in Question 3 is disproved in the case p = 2 and even D is a metric.

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Example 14 ([34], Example 2.1). Let X = N and d be defined as follows ⎧ ⎪ ⎨0 if x = y d(x, y) = d(y, x) = 1 if x = 2n, y = 2m − 1 for some n, m ∈ N ⎪ ⎩ 2 otherwise. Let A = {2n : n ∈ N}, B = {2m − 1 : m ∈ N} and let T : A ∪ B → A ∪ B be defined by Tx = x + 1 for all x ∈ A ∪ B. Then the following assertions hold. 1. (X, d) is a complete metric space. 2. A and B are nonempty closed subsets of X, and TA ⊂ B, TB ⊂ A. 3. For all x ∈ A and y ∈ B, d(Tx, Ty) =

1 max{d(x, Ty), d(y, Tx)}. 2

4. T is fixed point free. The following example shows that the statement mentioned in Question 3 is disproved in the case p = 3 and even D is a metric. Example 15 ([34], Example 2.2). Let X = {1, 2, 3, 4, 5, 6} and d : X × X → [0, ∞) be defined as follows ⎧ 0 if x = y ⎪ ⎪ ⎪ ⎨ 2 if (x, y) ∈ {(1, 4), (2, 5), (3, 6)} d(x, y) = d(y, x) = ⎪ 1 if (x, y) ∈ {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 1)} ⎪ ⎪ ⎩3 otherwise. 2 Let A1 = {1, 4}, A2 = {2, 5}, A3 = {3, 6} and let T : A1 ∪ A2 ∪ A3 → A1 ∪ A2 ∪ A3 be defined by T1 = 2, T2 = 3, T3 = 4, T4 = 5, T5 = 6, T6 = 1. Then the following assertions hold. 1. (X, d) is a complete metric space. 2. A1 , A2 and A3 are nonempty closed subsets of X, and TA1 ⊂ A2 , TA2 ⊂ A3 , TA3 ⊂ A1 . 3. For all x ∈ Ai and y ∈ Ai+1 , i = 1, 2, 3, where A4 = A1 , d(Tx, Ty) ≤

3 max{d(x, Ty), d(y, Tx)}. 4

4. T is fixed point free. The following example shows that the statement mentioned in Question 3 is disproved in the case p ≥ 4 and even D is a metric. Example 16 ([34], Example 2.3). For each p ≥ 4, let X = {1, 2, . . . , p} and d : X × X → [0, ∞) be defined as follows ⎧ ⎪ ⎨0 if x = y d(x, y) = d(y, x) = 1 if (x, y) ∈ {(1, 2), (2, 3), . . . , (p − 1, p), (p, 1)} ⎪ ⎩ 2 otherwise.

Fixed Point Theory in b-Metric Spaces

Let Ai = {i} for all i = 1, . . . , p and let T :

49

p  i=1

Ai →

p  i=1

Ai be defined by T(i) = i+1, i = 1, . . . , p−1,

and T(p) = 1. Then the following assertions hold. 1. (X, d) is a complete metric space. 2. Ai , i = 1, . . . , p, are nonempty closed subsets of X, and TAi ⊂ Ai+1 , where Ap+1 = A1 . 3. For all x ∈ Ai , y ∈ Ai+1 , d(Tx, Ty) =

1 max{d(x, Ty), d(y, Tx)}. 2

4. T is fixed point free. For related results on cyclic fixed point theorems in b-metric spaces, readers may refer to [74], [104], [62] and references given there for more details. In [59, Chapter 12], Kirk and Shahzad surveyed b-metric spaces, strong b-metric spaces, and related problems. An interesting work which attracted many authors to transform the results of metric spaces to the setting of b-metric spaces. It is only fair to point out that some results seem to require the full use of the triangle inequality of a metric space. In this connection, Kirk and Shahzad [59, page 127] mentioned an interesting extension of Nadler’s theorem due to Dontchev and Hager [29]. Recall that for a metric space (X, d) and A, B ⊂ X, x ∈ X, dist(x, A) = inf{d(x, a) : a ∈ A}

δ (A, B) = sup{dist(x, A) : x ∈ B} and these notations are understood similarly on b-metric spaces. Theorem 12 ([59], Theorem 12.7). Let (X, d) be a complete metric space, T be a map from X into a nonempty closed subset of X, and x0 ∈ X such that the following conditions hold: 1. dist(x0 , Tx0 ) < r(1 − k) for some r > 0 and some k ∈ [0, 1); 2. δ (Tx ∩ B(x0 , r), Ty) ≤ kd(x, y) for all x, y ∈ B(x0 , r). Then T has a fixed point in B(x0 , r). Based on the definition of δ (A, B) and the proof of Theorem [59, Theorem 12.7], the assumption (2) in the above theorem is implicitly understood as δ (Tx ∩ B(x0 , r), Ty) ≤ kd(x, y) for all x, y ∈ B(x0 , r) and Tx ∩ B(x0 , r) = 0. / The authors of [59] did not know whether Theorem 12 holds under the weaker strong b-metric assumption. The answer is negative by the following example. Example 17 ([8], Example 2.1). Let X = {1, 2, 3}, the function D : X × X → [0, ∞) be defined by D(1, 1) = D(2, 2) = D(3, 3) = 0, D(1, 2) = D(2, 1) = 2 D(2, 3) = D(3, 2) = 1, D(1, 3) = D(3, 1) = 6 and a map T : X → X be defined by T1 = 2, T2 = 3, T3 = 1. Then the following assertions hold.

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1. (X, D, κ ) is a complete strong b-metric space with the coefficient κ = 4. 2. For x0 = 1, r = 6, k = 12 , T and (X, D, κ ) satisfy the following conditions. (i) dist(x0 , Tx0 ) < r(1 − k) for some r > 0 and some k ∈ [0, 1). / (ii) δ (Tx ∩ B(x0 , r), Ty) ≤ kD(x, y) for all x, y ∈ B(x0 , r) and Tx ∩ B(x0 , r) = 0. 3. T has no fixed points. In 1999, Jachymski et al. [52] posed the following conjecture. The generalized Banach contraction conjecture in metric spaces. Let (X, d) be a complete metric space, T : X → X be a map, and J be a positive integer such that for some M ∈ (0, 1) and all x, y ∈ X,  min d(Tk x, Tk y) : 1 ≤ k ≤ J ≤ Md(x, y). Then T has a fixed point. The Banach contraction principle is simply the case J = 1, in which T is uniformly continuous. Jachymski et al. [52] proved that the generalized Banach contraction conjecture is true for J = 2 without any additional assumption on T. Moreover, it is true for J = 3 with the additional assumption that T is continuous. Also, in 1999, Jachymski and Stein [53] proved that the generalized Banach contraction conjecture is true if T is uniformly continuous. In 2000, Stein [97] proved that the generalized Banach contraction conjecture is true for arbitrarily finite J with the additional assumption that T is strongly continuous. In 2002, Merryfield et al. [68] proved that the generalized Banach contraction conjecture is true if T is continuous. Moreover, it was proved for J = 3 without any additional assumption on T. Finally, in 2002 and 2003, the generalized Banach contraction conjecture was proved completely by Arvanitakis [11], Merryfield and Stein [69]. Besides, in 2008, Reich and Zaslavski [86] generalized the conjecture to Jachymski–Schröder–Stein contraction with respect to φ . The Banach contraction principle was also extended by replacing the complete metric space by a complete b-metric space. The generalized Banach contraction conjecture in b-metric spaces. Let (X, D, κ ) be a complete b-metric space, T : X → X be a map, and J be a positive integer such that for some M ∈ (0, 1) and all x, y ∈ X,  min D(Tk x, Tk y) : 1 ≤ k ≤ J ≤ MD(x, y). Then T has a fixed point. Note that if κ = 1, then the generalized Banach contraction conjecture in b-metric spaces is again the generalized Banach contraction conjecture in metric spaces. In [39], Dung and Hang proved that the generalized Banach contraction conjecture in b-metric spaces is true if every b-metric space is a metric-type space and M ∈ (0, κ1 ) [39, Theorem 3.6]. If M ∈ (0, κ1 ) we also prove that the generalized Banach contraction conjecture in b-metric spaces is true for the case J = 2 [39, Theorem 4.4]; and for the case J = 3 if T is continuous [39, Theorem 4.5]. These results were proved by using combinatorial arguments and, in particular, the pigeonhole principle. Recently, Lu et al. [64] solved completely the generalized Banach contraction conjecture in b-metric spaces as follows. Theorem 13 ([64], Theorem 3.7). Let (X, D, κ ) be a complete b-metric space, T : X → X be a map, and J be a positive integer such that for some M ∈ (0; 1) and all x, y ∈ X,  min D(Tk x, Tk y) : 1 ≤ k ≤ J ≤ MD(x, y). Then T has a unique fixed point.

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For other generalizations and improvements of metric fixed point results to the setting of b-metric spaces, readers may refer to [105], [103], [84], [82], [73], [90], [4], [49] and references given there.

2.4 Multivalued Fixed Point Results in b-Metric Spaces For a b-metric space (X, D, κ ) the following families of subsets of X and functions were introduced in [18]. P(X) = {A : A ⊂ X} P(X) = {A ∈ P(X) : A = 0} / Pb (X) = {A ∈ P(X) : A is bounded } Pcp (X) = {A ∈ P(X) : A is compact } Pcl (X) = {A ∈ P(X) : A is closed } Pb,cl (X) = {A ∈ P(X) : A is bounded and closed }. The gap function D : P(X) × P(X) → [0, ∞] is defined by ⎧ ⎪ ⎨inf{D(a, b) : a ∈ A, b ∈ B} if A = 0/ and B = 0/ D(A, B) = 0 if A = B = 0/ ⎪ ⎩ ∞ otherwise. The Pompeiu–Hausdorff generalized function H : P(X) × P(X) → [0, ∞] is defined by ⎧ ⎪ max{sup inf D(a, b), sup inf D(b, a)} if A = 0/ and B = 0/ ⎪ ⎨ a∈A b∈B b∈B a∈A H(A, B) = 0 if A = B = 0/ ⎪ ⎪ ⎩ ∞ otherwise. The first result in b-metric multivalued fixed point results due to Czerwik [28] is as follows. Note that the b-metric D was assumed to be continuous. Theorem 14 ([28], Theorem 2). Let (X, D, κ ) be a complete b-metric space with a continuous b-metric D, and F : X → P(X) be a map such that for all x, y ∈ X, H(F(x), F(y)) ≤ ϕ (D(x, y)) where ϕ : [0, ∞) → [0, ∞) is an increasing function and for each t ∈ (0, ∞), lim ϕ n (t) = 0.

n→∞

Then F has a fixed point x∗ ∈ X, that is, x∗ ∈ F(x∗ ). After this result, many b-metric multivalued fixed point theorems have been proved. In 2008, Boriceanu [16] presented some fixed point results for multivalued contractions on a set with two b-metrics. The data dependence and the well-posedness of the fixed point problem were also discussed. Multivalued fixed point results then were studied in [17], [24].

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In 2011, Bota et al. [19] proved a version of Ekeland’s variational principle in b-metric spaces and, as a consequence, in order to obtain a Caristi type fixed point theorem. Note that Caristi’s theorem cannot be extended fully to b-metric spaces; see Example 21 in the latter. In 2012, Aydi et al. [12] gave a fixed point theorem for set-valued quasi-contraction maps in b-metric spaces. The theorem extended, unified and generalized several well-known comparable results in the literature before. The main results in [12] were then generalized in [96], [61], [63], [48], [106], [99]. However, Nadler theorem had not been fully extended in b-metric spaces. In 2014 Kirk and Shahzad [59, Chapter 12] proved Nadler type theorem in the s-relaxedp metric space which is a special case of the b-metric space as follows. Theorem 15 ([59], Theorem 12.5). Let (X, D, κ ) be a s-relaxedp metric space, and F : X → Pb,cl (X) be a map such that for some k ∈ [0, 1) and for all x, y ∈ X, H(F(x), F(y)) ≤ kD(x, y). Then F has a fixed point x∗ ∈ X, that is, x∗ ∈ F(x∗ ). In 2017, Miculescu and Mihail [71] also proved the following version of Nadler type theorem in b-metric spaces. Theorem 16 ([71], Theorem 3.3). Let (X, D, κ ) be a complete b-metric space. Then the map F : X → Pb (X) has a fixed point if and only if the following conditions hold. 1. There exist c, d ∈ [0, 1] and α ∈ [0, 1) such that for all x, y ∈ X, d H(F(x), F(y)) ≤ α max{D(x, y), cD(x, Fx), cD(y, Fy), (D(x, Fy) + D(y, Fx))}; 2 2. max{α cκ , α dκ } < 1. As a special case of multivalued fixed points, multivalued fractals play an important role in several topics of mathematics and applied sciences [13], [20]. The most common setting for the study of fractals and multivalued fractals is the case of maps on complete or compact metric spaces. In 2010, Boriceanu et al. [18] extended the study of multivalued fractals in complete b-metric spaces. Let Fi : X → P(X) be multivalued maps, i = 1, . . . , m. The system F = {F1 , . . . , Fm } is called an iterated map system. The map TF defined by TF A =

m 

Fi A

i=1

for all A ∈ Pcp (X) is called a multivalued fractal map. A fixed point of TF is called a multivalued fractal. The following important auxiliary result that was used to prove main results in [18] ensures that TF : Pcp (X) → Pcp (X) if H(Fi x, Fi y) < d(x, y) for all x = y ∈ X, i = 1, . . . , m. Lemma 3 ([18], Lemma 3.1). Let (X, D, κ ) be a b-metric space and F : X → Pcp (X) be a multivalued map satisfying H(Fx, Fy) < D(x, y) for all x = y ∈ X. Then for any A ∈ Pcp (X), we have FA ∈ Pcp (X). The next is one well-known result in the metric fixed point theory, usually called Meir– Keeler’s fixed point theorem.

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Theorem 17 ([67], page 326). Let (X, d) be a complete metric space and f : X → X be a Meir–Keeler type map, that is, for each η > 0 there exists δ > 0 such that for all x, y ∈ X,

η ≤ d(x, y) < η + δ implies d(fx, fy) < η . Then f has a unique fixed point x∗ and lim f n x = x∗ for all x ∈ X. n→∞

Relating to the above result, the following question was proposed in [18]. Question 4 ([18], Open question 1). Let (X, D, κ ) be a complete b-metric space with a continuous b-metric. Prove a fixed point theorem for a Meir–Keeler type map on (X, D, κ ). Based on multivalued Meir–Keeler type maps on metric spaces, the authors of [18] introduced the notion of a multivalued Meir–Keeler type map on a b-metric space as follows. Definition 8 ([18], Definition 3.1). Let (X, D, κ ) be a b-metric space and T : X → Pcp (X) be a map. Then T is called a multivalued Meir–Keeler type map if for each ε > 0 there exists δ > 0 such that for all x, y ∈ X, ε ≤ D(x, y) < ε + δ implies H(Tx, Ty) < ε . Relating to multivalued Meir–Keeler type maps on b-metric spaces, the following question was proposed in [18]. Question 5 ([18], Open question 2). Let (X, D, κ ) be a complete b-metric space with a continuous b-metric and Fi : X → Pcp (X), i = 1, . . . , m, be a finite family of multivalued Meir–Keeler type maps. Prove an existence and uniqueness result for the multivalued fractal map TF : Pcp (X) → Pcp (X) generated by the iterated map system F = {F1 , . . . , Fm }. Question 4 and Question 5 were studied in [46]; see [46, Theorem 2.1] and [46, Theorem 2.4] respectively. Unfortunately, there was an error in the proof of [46, Theorem 2.1] that the function φ (t) ≤ t for all t ≥ 0 does not satisfy lim φ n (t) = 0. Note that the function n→∞

φ (t) < t for all t ≥ 0 does so satisfy. [46, Theorem 2.1] was then used to prove [46, Theorem 2.4]. Therefore, Question 4 and Question 5 currently remain open. Note that by a very similar technique to the proof of [25, Main theorem] we have the following result, while Question 4 requires there is no κ in (2.5). Theorem 18. Let (X, D, κ ) be a complete b-metric space and f : X → X be a Meir–Keeler type map; that is, for each η > 0 there exists δ > 0 such that for all x, y ∈ X,

η ≤ D(x, y) < κ [η + δ ] implies d(fx, fy) < η . ∗

(2.5)

∗

Then f has a unique fixed point x and lim f x = x for all x ∈ X. n

n→∞

As generalizations of well-posed properties in metric spaces, those in b-metric spaces were introduced as follows. For corresponding notions in metric spaces, see [78, Definitions 2.1 & 2.2]. Definition 9 ([18], Definitions 3.2 & 3.3). Let (X, D, κ ) be a b-metric space, f : X → X be a map and F : X → P(X) be a multivalued map.

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1. The fixed point problem for f is called well-posed if Fixf = {x∗ } and lim xn = x∗ in n→∞

(X, D, κ ) for every sequence {xn } with lim d(xn , fxn ) = 0, where Fixf = {x ∈ X : x = fx} n→∞

is the fixed point set of f . 2. The fixed point problem for F is called well-posed with respect to D if FixF = {x∗ } and lim xn = x∗ in (X, D, κ ) for every sequence {xn } with lim D(xn , Fxn ) = 0, where n→∞

n→∞

n→∞

n→∞

FixF = {x ∈ X : x ∈ Fx} is the fixed point set of F. 3. The fixed point problem for F is called well-posed with respect to H if SFixF = {x∗ } and lim xn = x∗ in (X, D, κ ) for every sequence {xn } with lim H(xn , Fxn ) = 0, where SFixF = {x ∈ X : {x} = Fx} is the strict fixed point set of F.

Relating to the well-posedness of the fixed point problem for a multivalued fractal map on a b-metric space, the following question was proposed in [18]. Question 6 ([18], Open question 3). Let (X, D, κ ) be a b-metric space and Fi : X → Pcp (X), i = 1, . . . , m, be such that the well-posedness property of the fixed point problem for each Fi , i = 1, . . . , m, with respect to H or D takes place. Is the fixed point problem for the multivalued fractal map TF : Pcp (X) → Pcp (X) generated by the iterated map system F = {F1 , . . . , Fm } well-posed? The main results in [18] were stated for the case of multivalued maps Fi : X → Pcp (X) with i = 1, . . . , m. Therefore the following question was proposed for the case of Pcp (X) being replaced by Pb,cl (X). Question 7 ([18], Open question 4). Let (X, D, κ ) be a complete b-metric space. Prove similar results for the case of multivalued maps Fi : X → Pb,cl (X) with i = 1, . . . , m. The following example gives a negative answer to Question 6 even in the case of the b-metric being a metric. Example 18 ([46], Example 2.5). Let X = N and the function D : X × X → R and the maps f1 , f2 : X → X be defined as follows for all n, m ∈ X, D(n, n) = 0 D(0, 2n) = D(2n, 0) = D(1, 2n + 1) = D(2n + 1, 1) = 2 if n ≥ 1 D(2n, 2m + 1) = D(2m + 1, 2n) = 5 if n = m and n, m ≥ 1 D(n, n + 1) = D(n + 1, n) = 6 D(n, m) = 4 if otherwise f1 2n = 0, f1 (2n + 1) = 2n + 2, f2 2n = 2n + 1, f2 (2n + 1) = 1. Then the following assertions hold. 1. (X, D, κ ) is a complete metric space. 2. The fixed point problem for f1 and f2 are well-posed with respect to H and D. 3. The fixed point problem for multivalued fractal map TF : Pcp (X) → Pcp (X) defined by TF A = f1 A ∪ f2 A for all A ∈ Pcp (X) is not well-posed.

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Recall that if the metric space (X, d) is complete, then so is the metric space (Pb,cl (X), H); for example see [45, Theorem (4.4)]. The next is a similar result for Pb,cl (X) where (X, D, κ ) is a complete b-metric space. Note that the result for Pcp (X) where (X, D, κ ) is a complete b-metric space was proved [28, Theorem 1]. Proposition 3 ([46], Proposition 2.6). Let (X, D, κ ) be a complete b-metric space. Then (Pb,cl (X), H, κ ) is a complete b-metric space. The next example shows that the assumption F : X → Pcp (X) in Lemma 3 cannot be replaced by F : X → Pb,cl (X) even in the case of X being a metric space. Note that Lemma 3 is the important auxiliary result to state the main results in [18]. Thus, Example 19 gives an answer to Question 7, in the sense that we cannot prove fully similar results to the main results in [18] for the case of multivalued maps Fi : X → Pb,cl (X) with i = 1, . . . , m. Example 19 ([46], Example 2.7). Let X = N be endowed with discrete metric d and F : X → Pb,cl (X) be defined by Fx = X for all x ∈ X. Then 1. H(Fx, Fy) < D(x, y) for all x = y ∈ X. 2. For all A ∈ Pcp (X) we have FA ∈ Pcp (X).

2.5 Relaxations of Constant Contractions in b-Metric Spaces For the relaxation [0, κ1 ) to [0, 1), by using ϕ (t) = λt with t ≥ 0 in Theorem 8, we get a generalization of Theorem 9 where the domain of contraction constants is [0, 1). Corollary 1 ([32], Theorem 2.1 – Banach fixed point in b-metric spaces). Let (X, D, κ ) be a complete b-metric space and f : X → X be a map such that for all x, y ∈ X and some λ ∈ [0, 1), D(f (x), f (y)) ≤ λD(x, y). Then f has a unique fixed point x∗ and lim f n x = x∗ for all x ∈ X. n→∞

Some authors used various different techniques to prove fixed point results in b-metric spaces to overcome the difficulty of κ ≥ 1 in calculations. The following result is a useful technique on relaxing [0, κ1 ) to [0, 1) in b-metric fixed point theorems. It was reproved briefly in [72]. Lemma 4 ([71], Lemma 2.2; [98], Lemma 6). Let (X, D, κ ) be a b-metric space, T : X → X be a map and xn = Tn x for all n ∈ N and some x ∈ X. If there exists λ ∈ [0, 1) such that d(xn+1 , xn+2 ) ≤ λD(xn , xn+1 ) for all n ∈ N, then the sequence {xn } is Cauchy. Using Lemma 4, Aleksi´c et al. [4] and some other authors generalized, improved and complemented several fixed point results in b-metric spaces including Corollary 1; see [49, Theorem 2.6] for example. Note that Corollary 1 was also proved by using the metrization of b-metric spaces; see [31, Theorem 3.1] for example. Theorem 8 was also extended to the setting of strong b-metric spaces as follows.

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Theorem 19 ([32], Theorem 2.3). Let (X, D, κ ) be a complete strong b-metric space and T : X → X be a map such that for all x, y ∈ X, D(Tx, Ty) ≤ ϕ (max {D(x, y), D(x, Tx), D(y, Ty)}) where ϕ : [0, ∞) → [0, ∞) is increasing and satisfies lim ϕ n (t) = 0

n→∞

for each t > 0. Then T has a unique fixed point x∗ and lim Tn x = x∗ for all x ∈ X. n→∞

By using Theorem 19, many fixed point results in strong b-metric spaces have the domain of contraction constants as [0, 1) as in metric spaces; see [32, Corollaries 2.4-2.5] for example. Corollary 2 ([32], Corollary 2.4). Let (X, D, κ ) be a complete strong b-metric space and T : X → X be a map such that for some and for all x, y ∈ X, D(Tx, Ty) ≤ λ max {D(x, y), D(x, Tx), D(y, Ty)} .

(2.6)

Then T has a unique fixed point x∗ and lim Tn x = x∗ for all x ∈ X. n→∞

Corollary 3 ([32], Corollary 2.5). Let (X, D, κ ) be a complete strong b-metric space and T : X → X be a map such that for some a, b, c ≥ 0 with a + b + c < 1 and for all x, y ∈ X, D(Tx, Ty) ≤ aD(x, y) + bD(x, Tx) + cD(y, Ty).

(2.7)

Then T has a unique fixed point x∗ and lim Tn x = x∗ for all x ∈ X. n→∞

We see that Theorem 8 is generalized to Theorem 19 in the setting of strong b-metric spaces. It is natural to ask whether the conclusion of Theorem 19 holds in the setting of b-metric spaces. The following example gives a negative answer. By modifying Example 20, we may get that the upper bound of contractive constants in fixed point theorems in b-metric spaces where the contraction conditions contain D(x, y) and one more other value such as D(x, Tx), D(y, Ty), D(x, Ty), D(y, Tx) may not be relaxed to 1 as in the setting of metric spaces. For those fixed point theorems in b-metric spaces, readers may refer to [21, Theorems 2.2, 2.3, 2.4, 3.3, 3,4], [24, Corollaries 2.2, 2.3, 2.4], [54, Theorem 3.7, Corollary 3.8, Theorem 3.11] and many others. Example 20 ([32], Example 2.6). Let X = ⎧ 0 ⎪ ⎪ ⎨ 1 D(x, y) = |x − y| ⎪ ⎪ ⎩ 1

 0, 1, 12 , . . . , n1 , . . . , and

4

and let T : X → X be defined by



Tx = Then the following assertions hold.

1 1 10n

if x = y if x = y ∈ {0, 1}  1 if x = y ∈ {0} ∪ 2n : n = 1, 2, . . . otherwise,

if x = 0 if x = n1 , n = 1, 2, . . .

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1. (X, D, κ ) is a complete metric-type space with the coefficient κ = 4. Thus it is a complete b-metric space with the coefficient κ = 4. 2. D is not a strong b-metric for any κ ≥ 1. 3. For all x, y ∈ X and some a, b, c ≥ 0, a + b + c < 1, the contraction condition (2.7) holds. 4. All assumptions of Corollary 3 hold except for the assumption of a strong b-metric. However, T has no fixed points. In Example 20 we find that

1 κ

= 14 . Set

λ = a+b+c =

9 2 1 1 + + = . 5 4 4 10

  Then λ ∈ κ1 , 1 and (2.6) holds. Therefore the conclusion of Corollary 2 does not hold in the setting of metric-type and λ ∈ [0, 1). Hence the conclusion of Theorem 10 does   spaces not remain true for λ ∈ κ1 , 1 . Note that not all fixed point theorems in metric spaces can be extended fully to b-metric spaces. The following example also shows that Caristi’s theorem cannot be extended fully to b-metric spaces. Example 21 ([32], Example 2.8). Let X = [1, ∞), D(x, y) = (x − y)4 for all x, y ∈ X and T : X → X, ϕ : X → R be defined by Tx = x + 1x , ϕ (x) = 2x for all x ∈ X. Then the following assertions hold. (i) (ii) (iii) (iv) (v)

D is a continuous b-metric on X with the coefficient κ = 8. (X, D, κ ) is complete. ϕ is semi-continuous and bounded below. D(x, Tx) ≤ ϕ (x) − ϕ (Tx) for all x ∈ X. T has no fixed points.

In 2016, Petru¸sel et al. [77] proved an analogue of the fixed point theorem of Ran and Reurings [85, Theorem 2.1] to b-metric spaces as follows. Theorem 20 ([77], Theorem 3.1). Let (X, D, κ , ) be a partially ordered and complete b-metric space and f : X → X be a map satisfying the following conditions. 1. f is a continuous map. 2. f is increasing with respect to “” . 3. There exists k ∈ (0, κ1 ) such that for all x  y, D(fx, fy) ≤ kD(x, y). 4. There exists x0 ∈ X such that x0  fx0 . Then the following assertions hold. 1. Fixf = 0. / 2. For all x ∈ X which is comparable to x0 , lim f n x = x∗ ∈ Fixf . n→∞

Note that the class of metric spaces in [85, Theorem 2.1] was replaced by the larger class of b-metric spaces in Theorem 20. However, the condition k ∈ (0, 1) in [85, Theorem 2.1]

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was replaced by a weaker condition k ∈ (0, κ1 ) in Theorem 20. Petru¸sel et al. [77] used Theorem 20 to consider coupled fixed point problems in b-metric spaces for maps satisfying a symmetric contraction condition on the existence [77, Theorems 3.4 & 3.7], the uniqueness of the solution [77, Theorem 3.5], data dependence [77, Theorem 4.1], well-posedness [77, Theorem 4.3], Ulam–Hyers stability [77, Theorem 4.5], limit shadowing property [77, Theorem 4.7] of the coupled fixed point problem. Some applications to a system of integral equations and to a periodic boundary value problems were also given [77, Theorems 5.2 & 5.3]. Petru¸sel and Petru¸sel [79] presented some coupled fixed point theorems on a space endowed with two b-metrics and applied to a boundary value problem for a system of second-order differential equations. The approach in [79] was based on the application of a fixed point theorem for an appropriate map on the Cartesian product of the given spaces. An application to a boundary value problem for a system of second-order differential equations was also presented. The main results of that paper are as follows. Theorem 21 ([79], Theorem 2). Let (X, D1 , κ , 1 ) and (Y, D2 , κ , 2 ) be two partially ordered and complete b-metric spaces, T1 : X × Y → X and T2 : X × Y → Y be two maps satisfying the following conditions. 1. T1 and T2 are closed graphic. 2. T1 and T2 have the inverse mixed-monotone property, that is, (i) If x1 , x2 ∈ X and x1 1 x2 then for all y ∈ Y,  T1 (x1 , y) 1 T1 (x2 , y) T2 (x2 , y) 2 T2 (x1 , y). (ii) If y1 , y2 ∈ Y and y2 2 y1 then for all x ∈ X,  T1 (x, y1 ) 1 T1 (x, y2 ) T2 (x, y2 ) 2 T2 (x, y1 ). 3. There exists k ∈ (0, κ1 ) such that for all (x, y), (u, v) ∈ X × Y with x 1 u, v 2 y,       D1 T1 (x, y), T1 (u, v) + D2 T2 (x, y), T2 (u, v) ≤ k D1 (x, u) + D2 (y, v) . 4. There exists (x0 , y0 ) ∈ X × Y such that x0 1 T1 (x0 , y0 ) and T2 (x0 , y0 ) 2 y0 , or T1 (x0 , y0 ) 1 x0 and y0 2 T2 (x0 , y0 ). Then the following assertions hold. 1. There exists (x∗ , y∗ ) ∈ X × Y such that  x∗ = T1 (x∗ , y∗ ) y∗ = T2 (x∗ , y∗ ) and lim xn = x∗ , lim yn = y∗ , where xn+1 = T1 (xn , yn ), yn+1 = T2 (xn , yn ) for all n. Moren→∞

n→∞

over, for all (x, y) ∈ X × Y with x 1 x0 and y0   2 y, or x0 1 x and y 2 y0 , we also have lim un+1 = x∗ and lim vn+1 = y∗ , where un+1 = T1 Fn (x, y) and vn+1 = T2 Fn (x, y) for all n n→∞  n→∞  and F(x, y) = T1 (x, y), T2 (x, y) .

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2. If d and ρ are two continuous b-metrics, then D1 (xn , x∗ ) + D2 (yn , y∗ ) ≤

   skn   D1 x0 , T1 (x0 , y0 ) + D2 y0 , T2 (x0 , y0 ) . 1 − sk

Theorem 22 ([79], Theorem 10). Consider the problem ⎧   ⎪ ⎨−x (t) = f t, x(t), y (t) −y (t) = f t, y(t), x (t) ⎪ ⎩ x(a) = x(b) = y(a) = y(b) = 0

(2.8)

where x, y ∈ C2 [a, b] and t ∈ [a, b]. Let f : [a, b] × R × R → R be a continuous function satisfying the following conditions. 1. There exist α , β > 0 such that for all u1 , v1 , u2 , v2 ∈ R, |f (s, u1 , v1 ) − f (s, u2 , v2 )| ≤ α |u1 − u2 | + β |v1 − v2 | for all [a, b]. s ∈  (b−a)2 b−a   (b−a)2 b−a  < 1. 2. max α 8 + 2 , β 8 + 2 Then the following assertions hold. 1. The problem (2.8) has the unique solution (x∗ , y∗ ) ∈ C2 [a, b] × C2 [a, b], 2. For each x0 , y0 ∈ C2 [a, b] and xn+1 (t) = yn+1 (t) =  where G(t, s) =

(s−a)(b−t) b−a (t−a)(b−s) b−a



b

  G(t, s)f s, xn (s), y n (s) ds

b

  G(t, s)f s, yn (s), x n (s) ds

a

a

if s ≤ t for all s, t ∈ [a, b], we have lim xn = x∗ and lim yn = y∗ . n→∞ n→∞ if s ≥ t

Relative to the above results and comments, the authors of [79] raised the following question. Question 8 ([79], Remark 3.(2)). Can we obtain “better” results in the setting of b-metric spaces than in the classical metric spaces or not? See also [76, Remark 5.4], [77, Remark 5.4]. The following result is an extension of Theorem 20 where the condition k ∈ (0, κ1 ) is replaced by k ∈ (0, 1). In other words, [85, Theorem 2.1] is fully extended to the setting of b-metric spaces. The key technique in the proof is to use the metrization theorem on b-metric spaces of Paluszynski ´ and Stempak—see Theorem 4—and the fixed point theorem of Ran and Reurings [85, Theorem 2.1] in the setting of metric spaces. Theorem 23 ([37], Theorem 2.1). Let (X, D, κ , ) be a partially ordered and complete b-metric space and f : X → X be a map satisfying the following conditions.

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1. f is a continuous map. 2. f is increasing with respect to “”. 3. There exists k ∈ (0, 1) such that for all x  y, D(fx, fy) ≤ kD(x, y). 4. There exists x0 ∈ X such that x0  fx0 . Then the following assertions hold. 1. Fixf = 0. / 2. For all x ∈ X which is comparable to x0 , lim f n x = x∗ ∈ Fixf . n→∞

3. If every pair x, y ∈ X has a lower bound or an upper bound then f has a unique fixed point. Since Theorem 20 was used to prove [77, Theorems 3.4 & 3.7] and Theorem 21, we find that the condition k ∈ (0, κ1 ) in [77, Theorems 3.4 & 3.7] and Theorem 21 can be replaced by k ∈ (0, 1) with the same argument as in the proof of Theorem 23. Note that topological properties in the setting of b-metric spaces are the same as in metric spaces by Theorem 4. From the proof of Theorem 23 we find that Theorem 23 can be seen as a consequence of [85, Theorem 2.1] which is a fixed point theorem in metric spaces. Therefore we may conclude that fixed point results in b-metric spaces such as [77, Theorems 3.4 & 3.7], [77, Theorem 3.5], and some others in [79] and [76], and even Theorem 23 are not better than that in the classical metric spaces. This gives a partial answer to Question 8. Following Question 8, to obtain “better” results in the setting of b-metric spaces than in the classical metric spaces, we should think of quasi-normed spaces which is a special class of b-metric spaces [56, page 1102]. Note that p , 0 < p < 1, with the quasi-norm defined 1 by x = (∑∞n=1 |xn |p ) p for all x = {xn } ∈ p is a quasi-Banach space that is not normable. A similar result also holds for Lp [0, 1]. Therefore researchers may study fixed point theory in quasi-Banach spaces. For interesting ways to extend fixed point theory in quasi-Banach spaces, readers may refer to and use the ideas in [6], [81], [87], [102], [101] and references given there. In addition, b-metric fixed point theorems have been used to study the stability of functional equations in b-metric spaces and in quasi-normed spaces. Readers may refer to [35], [38], [41], [80] and references given there.

2.6 Conclusion The b-metric space is one of the interesting generalizations of the metric space. The most different properties are that a b-metric is not necessarily continuous and that the triangle inequality does not necessarily hold for more than 3 elements. The fixed point theory in b-metric spaces has been studied by many authors. However, various fixed point results in b-metric spaces have been stated and proved similarly to those in metric spaces. One of the efficient techniques to prove nice extensions of certain metric fixed point results to b-metric spaces is Frink’s metrization.

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Acknowledgement The first author would like to thank the Bualuang ASEAN Fellowship Program for financial support during the preparation of this chapter.

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3 Basics of w-Distance and Its Use in Various Types of Results Dhananjay Gopal and Mohammad Hasan

CONTENTS 3.1 3.2 3.3 3.4 3.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of w-Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results through w-Distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 68 71 73 99

3.1 Introduction Since 1922, when S. Banach proved in his PhD thesis (see [3]) the celebrated Contraction Principle for self-mappings on a complete metric space, several hundred researchers have tried to generalize or improve it. Basically, these generalizations were done in two directions; either the contractive condition was replaced by some more general one, or the environment of metric spaces was widened. In the first direction, a lot of improved results appeared, such as Kannan, Chatterjea, Zamfirescu, Hardy–Rogers, Ciri’s, Meir–Keeler, Boyd–Wong, to mention just a few. The other direction of investigation has included the introduction of semi-metric, quasi-metric, symmetric, partial metric, b-metric, and many other classes of spaces. Caristi [8], in 1976, proved a fixed point theorem in a complete metric space which generalizes the well-known result the Banach contraction principle. Ekeland [12] also obtained a nonconvex minimization theorem for a proper lower semicontinuous function, bounded from below in a complete metric space; this theorem is also called the e-variational principle. These two theorems are very useful and have many applications. Later Takahashi [55] proved the following nonconvex minimization theorem: Let X be a complete metric space and let f : X → (−∞, ∞) be a proper lower semicontinuous function, bounded from below. Suppose that, for each u ∈ X with f (p) > inf f (x), there exists v ∈ X such that u = v z∈X

and f (v) + d(u, v) ≤ f (v). Then there exists x0 ∈ X such that f (x0 ) = inf f (x). This theorem z∈X

was used to obtain Caristi’s fixed point theorem [8], Ekeland’s ε -variation principle [12] and Nadler’s fixed point theorem [36]. In the early nineties Ume [61] and Kim et al. [24] 67

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improved the nonconvex minimization theorem according to Takahashi [56] using a continuous mapping from X into itself. On the other hand Ciric [9] proved an interesting fixed point theorem which generalizes some fixed point theorems in complete metric space. Of course, not all of these attempts were useful in applications, which should be the main motive for such investigations. Some of them were not even real generalizations, since the obtained results appeared to be equivalent to those already known. In 1996, Kada et al. [20] introduced the concept of w-distance on metric space and proved a nonconvex minimization theorem which improved the result of Takahashi [55]. This theorem is also a generalization of the results of [24, 56] and [61]. Finally Kada et al. [20] proved a fixed point theorem using the concept of w-distance which generalizes the fixed point theorem of Subrahmanyam [50], Kannan [21], Ciric [9] and many more. In this chapter, we focus on recent topics on fixed point theory and its applications using the concept of w-distance. In fact, this concept has improved many well-known results such as Caristi’s fixed point theorem, Ekeland’s ε -variation principle, the nonconvex min´ c’s fixed point theorem; these results have imization theorem according to Takahashi, Ciri´ many very useful applications.

3.2 Definitions and Examples Throughout this chapter, we denote by N the set of positive integers and R the set of real numbers. We start with the definition of w-distance as follows: Definition 1. [20] Let X be a metric space with metric d. Then a function p : X × X → [0, ∞) is called a w-distance on X if the following are satisfied: (i) p(x, z) ≤ p(x, y) + p(y, z) for any x, y, z ∈ X; (ii) for any x ∈ X p(x, .) : X → [0, ∞) is lower semicontinuous; (iii) for any ε > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ε . Some examples of w-distance. Example 1. [20] Let X be a metric space with metric d. Then p = d is a w-distance on X. Proof. (i) and (ii) are obvious. We show (iii). Let ε > 0 be given and put δ = ε2 . Then if d(z, x) ≤ δ and d(z, y) ≤ δ , we have d(x, y) ≤ d(z, x) + d(z, y) ≤ δ + δ = ε .   Example 2. [20] Let X be a metric space with metric d. Then the function p : X × X → [0, ∞) defined by p(x, y) = c for every x, y ∈ X is a w-distance on X, where c is a positive real number. Proof. (i) and (ii) are obvious. We show (iii), for any ε > 0 be given and put δ = ε2 . Then if d(z, x) ≤ δ and d(z, y) ≤ δ , imply d(x, y) ≤ ε .  

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Example 3. [20] Let X be a normed linear space with norm .. Then a function p : X × X → [0, ∞) defined by p(x, y) = x + y for every x, y ∈ X, is a w-distance on X. Proof. Let x, y, z ∈ X. Then we have p(x, z) = x + z ≤ p(x, y) + p(y, z). This implies (i). (ii) is obvious. Let ε > 0 and put δ = ε2 . Then if p(z, x) ≤ δ and p(x, y) ≤ δ , we have p(x, y) = x − y ≤ x + y ≤ p(z, x) + p(z, y) ≤ δ + δ = ε . This implies (iii).

 

Example 4. [20] Let X be a normed linear space with norm .. Then a function p : X × X → [0, ∞) defined by p(x, y) = y for every x, y ∈ X, is a w-distance on X. Proof. Let x, y, z ∈ X. Then we have p(x, z) = z ≤ y + z ≤ p(x, y) + p(y, z). This implies (i). (ii) is obvious. Let ε > 0 and put δ = ε2 . Then if p(z, x) ≤ δ and p(z, y) ≤ δ , we have p(x, y) = x − y ≤ x + y ≤ p(z, x) + p(z, y) ≤ δ + δ = ε . This implies (iii).

 

Example 5. [20] Let X be a metric space and let T be a continuous mapping from X into itself. Then a function p : X × X → [0, ∞) defined by p(x, y) = max{d(Tx, y), d(Tx, Ty)} for every x, y ∈ X, is a w-distance on X. Proof. Let x, y, z ∈ X. Then if d(Tx, z) ≥ d(Tx, Tz), we have p(x, z) = d(Tx, z) ≤ d(Tx, Ty) + d(Ty, z) ≤ max{d(Tx, y), d(Tx, Ty)} + max{d(Ty, z) + d(Ty, Tz)} = p(x, y) + p(y, z) In the other case, we have p(x, z) = d(Tx, Tz) ≤ d(Tx, Ty) + d(Ty, Tz) ≤ max{d(Tx, y), d(Tx, Ty)} + max{d(Ty, z) + d(Ty, Tz)} = p(x, y) + p(y, z)

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This implies (i). Since T is continuous, it is clear that for any x ∈ X, p(x, .) : X → [0, ∞) is lower semicontinuous. Let ε > 0 and put δ = ε2 . Then if p(z, x) ≤ δ and p(z, y) ≤ δ , we have d(Tz, x) ≤ δ and d(Tz, y) ≤ δ . Therefore d(x, y) ≤ d(Tz, x) + d(Tz, y) ≤ 2δ = ε . This implies (iii).

 

Example 6. [20] Let X = R be a metric space with usual metric and let f : X → [0, ∞) be a continuous function such that  x+r inf f (u)du > 0 x∈X x

for every r > 0. Then a function p : X × X → [0, ∞) defined by  y    p(x, y) =  f (u)du x

for every x, y ∈ X, is a w-distance on X. Example 7. [52] Let X be a metric space with metric d and let F be a bounded and closed subset of X. Assume that F contains at least two points and c is a constant with c ≥ diamF, where diamF is a diameter of F. Then a function p : X × X → [0, ∞) defined by  d(x, y) if x, y ∈ F, f (x) = c if x ∈ F or y ∈ F is a w-distance on X. Example 8. [32] If X = { n1 : n ∈ N} ∪ {0}. For each x, y ∈ X, d(x, y) = x + y if x = y and d(x, y) = 0 if x = y is a metric on X. Moreover, by defining p(x, y) = y, p is a w-distance on (X, d). Clearly, every metric is a w-distance but not conversely. The following example substantiates this fact. Example 9. [18] Let (X, d) be a metric space. A function p : X × X → [0, ∞) defined by p(x, y) = k for every x, y ∈ X is a w-distance on X, where k is a positive real number. But p is not a metric since p(x, x) = k = 0 for any x ∈ X. Example 10. [27] Let X = R+ be endowed with the Euclidean metric d = |.|, k be a positive constant and p : X × X → R+ be defined by p(x, y) = yk , ∀ x, y ∈ X. Then p is a w-distance in X. Example 11. [27] Let X = R be endowed with the Euclidean metric d = |.|, k ∈ R+ , m be a positive constant and p : X × X → R+ be defined by p(x, y) = |x|k + |y|m , ∀ x, y ∈ X. Then p is a w-distance in X.

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3.3 Properties of w-Distance Next, we discuss some properties of w-distance. Lemma 1. [20] Let X be a metric space with metric d and let p be a w-distance on X. Let {xn } and {yn } be sequences in X, let {αn } and {βn } be sequences in [0, ∞) converging to 0, and let x, y, z ∈ X. Then the following hold: If p(xn , y) ≤ αn and p(xn , z) ≤ βn for any n ∈ N, then y = z. In particular, if p(x, y) = 0 and p(x, z) = 0, then y = z; (ii) if p(xn , yn ) ≤ αn and p(xn , z) ≤ βn for any n ∈ N, then {yn } converges to z; (iii) if p(xn , ym ) ≤ αn for any n, m ∈ N with m > n, then {xn } is a Cauchy sequence; (iv) if p(y, xn ) ≤ αn for any n ∈ N, then {xn } is a Cauchy sequence. (i)

Proof. We first prove (ii). Let ε > 0 be given. From the definition of w-distance, there exists δ > 0 such that p(u, v) ≤ δ and p(u, z) ≤ δ imply d(v, z) ≤ ε . Choose n0 ∈ N such that αn ≤ δ and βn ≤ δ for every n ≥ n0 . Then we have, for any n ≥ n0 , p(xn , yn ) ≤ αn ≤ δ and p(xn , z) ≤ βn ≤ δ and hence d(yn , z)ε . This implies that {yn } converges to z. It follows from (ii) that (i) holds. Let us prove (ii). Let ε > 0 be given. As in the proof of (i), choose δ > 0 and then n0 ∈ N. Then for any n, m ≥ n0 + 1 p(xn0 , xn ) ≤ αn0 ≤ δ and p(xn0 , xm ) ≤ αn0 ≤ δ and hence d(xn , xm ) ≤ ε . This implies that {xn } is a Cauchy sequence; as in the proof of (iii), we can prove (iv).   Lemma 2. [20] Let X be a metric space and let p1 , p2 be w-distances on X. Then two functions on X × X defined as follows are w-distances on X: (i) p(x, y) = max{p1 (x, y), p2 (x, y)} for every x, y ∈ X; (ii) p(x, y) = α p1 (x, y) + β p2 (x, y) for every x, y ∈ X, where α and β are nonnegative real numbers such that α = 0 or β = 0. Proof. We first prove (i). Let x, y, z ∈ X. Then we have p(x, z) = max{p1 (x, z), p2 (x, z)} ≤ max{p1 (x, y) + p1 (y, z), p2 (x, y) + p2 (y, z)} ≤ max{p(x, y) + p(y, z), p(x, y) + p(y, z)} = p(x, y) + p(y, z) It is clear that for any x, p(x, .) = max{p1 (x, .), p2 (x, .)} is lower semicontinuous. Let ε > 0 be given. Then choose ε > 0 such that p1 (z, x) ≤ δ and p1 (z, y) ≤ δ imply d(x, y) ≤ ε . If p(z, x) ≤ δ and p(z, y) ≤ δ , then p1 (z, x) ≤ δ and p1 (z, y) ≤ δ . Therefore, we have d(x, y) ≤ ε . Let us prove (ii). Without loss of generality, we may assume α > 0. Let x, y, z ∈ Z. Then we have p(x, z) = α p1 (x, z) + β p2 (x, z)

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    ≤ α p1 (x, y) + p1 (y, z) + β p2 (x, y) + p2 (y, z) = p(x, y) + p(y, z). Since for any x ∈ X, α p1 (x, .) and β p2 (x, .) are lower semicontinuous, p(x, .) = α p1 (x, .) + β p2 (x, .) is also lower semicontinuous. Let ε > o be given and then choose δ > 0 such that p1 (z, x) ≤ δ and p1 (z, y) ≤ δ imply d(x, y) ≤ ε . Put δ = αδ . Then if p(z, x) ≤ δ and p1 (z, y) ≤ δ . Therefore, we have d(x, y) ≤ ε .   Lemma 3. [20] Let X be a metric space with metric d and let p be a w-distance on X and let α be a function from X into [0, ∞). Then a function q : X × X → [0, ∞) given by; q(x, y) = max{α (x), p(x, y) for every x, y ∈ X, is a w-distance. Proof. For every x, y, z ∈ X,

q(x, z) = max{α (x), p(x, z)} ≤ max{α (x) + α (y), p(x, y) + p(y, z)} ≤ q(x, y) + q(y, z).

Therefore (i) is satisfied. (ii) is obvious. We show (iii). Let ε > 0 be fixed. Then since p is a w-distance on X, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y)δ imply d(x, y) ≤ ε . Therefore, assume q(z, x) ≤ δ and q(z, y) ≤ δ . Then p(z, x) ≤ δ and p(z, y) ≤ δ . Hence d(x, y) ≤ ε.   Lemma 4. [52] Let X be a metric space with metric d, let p be a w-distance on X, and let q be a function from X × X → [0, 1) satisfying (i), (ii) in the definition of w-distance. Suppose that q(x, y) ≥ p(x, y) for every x, y ∈ X. Then q is also a w-distance on X. In particular, if q satisfies (i), (ii) in the definition of w-distance and q(x, y) ≥ d(x, y) for every x, y ∈ X, then q is a w-distance on X. Lemma 5. [52] Let ε ∈ [0, ∞) and let X be an ε -chainable metric space with metric d. Then the function p : X × X → [0, ∞) defined by p(x, y) = inf

 k−1

∑ d(ui , ui+1 ) : {u0 , u1 , . . . , uk }

 is an ε − chain linking x and y

i=0

is a w-distance on X. Lemma 6. [53] Let X be a metric space with metric d, let p be a w-distance on X, and let {xn } be sequence in X. Suppose that lim sup min{p(xn , xm ), p(xm , xn )} = 0.

n→∞ m>n

Then {xn } is Cauchy. In particular, the following hold; (i)

If lim supm>n p(xn , xm ) = 0, then {xn } is Cauchy; n→∞

(ii) if lim supm>n p(xm , xn ) = 0 and then {xn } is Cauchy. n→∞

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3.4 Results through w-Distances In 1996, Kada et al. [20] first proved a nonconvex minimization theorem which improved the result of Takahashi [55]. This proof employs the method of Takahashi [55] and Kijima [23]. Theorem 1. [20] Let X be a complete metric space, and let f : X → (−∞, ∞) be a proper lower semicontinuous function, bounded from below. Assume that there exists a w-distance p on X such that for any u ∈ X with infx∈X f (x) < f (u), there exists v ∈ X with u = v and f (v) + p(x, y) ≤ f (u). Then there exists x0 ∈ X such that infx∈X = f (x0 ). Proof. Suppose infx∈X f (x) < f (y) for every y ∈ X and choose u ∈ X with f (u) < ∞. Then we define inductively a sequence {xn } in X, starting with u1 = u. Suppose un ∈ X is known. Then choose un+1 ∈ S(un ) such that S(un ) = {x ∈ X : f (x) + p(un , x) ≤ f (un )}, k(un ) = inf f (x) x∈S(un )

and

1 n Since f (un+1 + p(un , un+1 ) ≤ f (un ), {f (un )} is nonincreasing. In fact, if n < m, then f (un+1 ) ≤ k(un ) +

p(un , um ) ≤

m−1

∑ p(uj , uj+1 )....................................(∗)

j=n

≤

m−1

∑ f (uj ) − f (uj+1 )

j=n

= f (un ) − f (um ) ≤ f (un ) − k. From Lemma 4.1, {un } is a Cauchy sequence. Let un → v0 . Then, if m → ∞ in (∗), we have p(un , v0 ) ≤ f (un ) − k ≤ f (un ) − f (v0 ). On the other hand, by hypothesis, there exists v1 ∈ X such that v1 = v0 and f (v1 ) + p(v0 , v1 ) ≤ f (v0 ). Hence, we obtain f (v1 ) + p(un , v1 ) ≤ f (v1 ) + p(un , u0 ) + p(v0 , v1 ).................................(∗∗) ≤ f (v0 ) + p(un , v0 ) ≤ f (un ) and hence v1 ∈ S(un ). Since f (v0 ) ≤ f (un+1 ≤ k(un ) +

1 1 ≤ f (v1 ) + n n

for every n ∈ N, we have f (v0 ) ≤ f (v1 ). Then, f (v0 ) = f (v1 ). Therefore, we have p(v0 , v1 ) = 0. By the hypothesis, there exists v2 ∈ X such that v2 = v1 and f (v2 ) + p(v1 , v2 ) ≤ f (v1 ). As in

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(∗∗), we have f (v2 ) + p(un , v2 ) ≤ f (un ) and hence v2 ∈ S(un ). Therefore, we have f (v1 ) = f (v0 ) ≤ f (v2 ). This implies p(v1 , v2 ) = 0. From p(v0 , v2 ) ≤ p(v0 , v1 ) + p(v1 , v2 ) = 0, we have p(v0 , v2 ) = 0. Hence, from p(v0 , v1 ) = 0, p(v0 , v2 ) = 0 and Lemma 4.1, we have v1 = v2 . This is a contradiction.   Using Theorem 1 and Example 5, they stated the following corollary which generalizes the results of [24, 56] and [61]. Corollary 1. [20] Let X be a complete metric space with metric d, let T be a continuous mapping from X into itself and let f : x → (−∞, ∞) be a proper lower semicontinuous function, bounded from below. Assume that for any u ∈ X with infx∈X f (x) < f (u), there is v ∈ X with u = v and f (v) + max{d(Tu, v), d(Tu, Tv)} ≤ f (u). Then there exists x0 ∈ X such that infx∈X = f (x0 ). The following theorem is a generalization of Ekeland’s ε -variational principle [12]. Theorem 2. [20] Let X be a complete metric space, let p be a w-distance on X and let f : X → (−∞, ∞) be a proper lower semicontinuous function, bounded from below. Then the following hold: For every u ∈ X with f (u) < ∞, then exists v ∈ X such that f (v) ≤ f (u) and f (w) > f (v) − p(v, w) for every w ∈ X; with w = v; (ii) for any ε > 0 and u ∈ X with p(u, u) = 0 and f (u) ≤ infx∈X f (x) + ε , there exists v ∈ X such that f (v) ≤ f (u), p(u, v) ≤ 1 and f (w) > f (v)− ε p(v, w) for every w ∈ X; with w = v.

(i)

The following theorem is a generalization of Carist’s fixed point theorem [8]. Theorem 3. [20] Let X be a complete metric space, and let f : X → (−∞, ∞) be a proper lower semicontinuous function, bounded from below. Let T be a mapping from X into itself. Assume that there exists a w-distance p on X such that f (Tx) + p(x, Tx) ≤ f (x) for every x ∈ X. Then there exists x0 ∈ X such that Tx0 = x0 and p(x0 , x0 ) = 0. Proof. Since f is proper, there exists u ∈ X such that f (u) < ∞. Put Y = {x ∈ X : f (x) ≤ f (u)}. Then, since f is lower semicontinuous, Y is closed. Therefore Y is complete. Let x ∈ X. Then, since f (Tx) + p(x, Tx) ≤ f (x) ≤ f (u), we have Tx ∈ Y. Hence, Y is invariant under T. Assume that tx = x for every x ∈ Y. Then by Theorem 4.1, there exists v0 ∈ Y such that f (v0 ) = infx∈Y f (x). Since f (Tv0 ) + p(v0 , Tv0 ) ≤ f (v0 ) and f (v0 ) = infx∈Y f (x), we have f (Tv0 ) = f (v0 ) = infx∈Y f (x) and p(v0 , Tv0 ) = 0. Similarly we obtain f (T2 v0 ) = f (Tv0 ) = infx∈Y f (x) and p(Tv0 , T2 v0 ) = 0. Since p(v0 , T2 v0 ) ≤ p(v0 , Tv0 ) + p(Tv0 , T2 v0 ) = 0, we have p(v0 , T2 v0 ) = 0 and hence Tv0 = T2 v0 by Lemma 4.1. This is a contradiction. Therefore T has a fixed point x0 in Y. Since f (x0 ) < ∞ and f (x0 ) + p(x0 , x0 ) = f (Tx0 ) + p(x0 , Tx0 ) ≤ f (x0 ), we have p(x0 , x0 ) = 0.

 

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Here, Kada et al. [20] used the concept of w-distance; they first proved a fixed point theorem in a complete metric space. Then this theorem is used to prove Subrahmanyam’s ´ c’s fixed point theorem fixed point theorem [50], Kannan’s fixed point theorem [21] and Ciri´ [9]. Theorem 4. [20] Let X be a complete metric space, let p be a w-distance on X and let T be a mapping from X into itself. Suppose that there exists r ∈ [0, 1) such that p(Tx, T2 x) ≤ rp(x, Tx) for every x ∈ X and that

inf{p(x, y) + p(x, Tx); x ∈ X} > 0

for every x ∈ X with y = Ty. Then there exists z ∈ X such that z = Tz. Moreover, if v = Tv, then p(v, v) = 0. Proof. Let u ∈ X and define

un = Tn u for any

n ∈ N.

Then we have, for any n ∈ N, p(un , un+1 ) ≤ rp(un−1 , un ) ≤ · · · ≤ rn p(u, u1 ). Thus, if m > n,

p(un , um ) ≤ p(un , un+1 ) + · · · + p(um−1 , um )

≤ rn p(u, u1 ) + · · · + rm−1 p(u, u1 ) rn p(u, u1 ). ≤ 1−r By Lemma 1. {un } is a Cauchy sequence. Since X is complete, {un } converges to some point z ∈ X. Let n ∈ N be fixed. Then since {um } converges to z and p(un , .) is lower semicontinuous, we have p(un , z) ≤ lim inf p(un , um ) ≤ m→∞

rn p(u, u1 ). 1−r

Assume that z = Tz. Then, by hypothesis, we have 0 < inf{p(x, z) + p(x, Tx) : x ∈ X} ≤ inf{p(un , z) + p(un , un+1 ) : n ∈ N} ≤ inf{

rn p(u, u1 ) + rn p(u, u1 ) :∈ N} 1−r = 0.

This is a contradiction. Therefore we have z = Tz. If v = Tv, we have p(v, v) = p(Tv, T2 v) ≤ rp(v, Tv) = rp(v, v) and hence p(v, v) = 0.

 

Using the above Theorem 4, Kada et al. [20] obtained the following corollary which is a generalization of [52].

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Corollary 2. [20] Let X be a complete metric space with metric d, let T be a continuous mapping from X into itself and let f : x → (−∞, ∞) be a proper lower semicontinuous function, bounded from below. Assume that for any u ∈ X with infx∈X f (x) < f (u), there is v ∈ X with u = v and f (v) + max{d(Tu, v), d(Tu, Tv)} ≤ f (u). Then there exists x0 ∈ X such that infx∈X = f (x0 ). Using Example 1, Kada et al. [20] proved the following is Kannan’s fixed point theorem [21]. Corollary 3. [20] Let X be a complete metric space, with metric d, and let T be a mapping from X into itself. Suppose T is Kannan, that is, there exists r ∈ [0, 1) such that p(Tx, Ty) ≤ r max{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)} for every x, y ∈ X. Then T has a unique fixed point. ´ c’s fixed point theorem [9]. Using Lemma 3, Kada et al. [20] proved the following is Ciri´ Corollary 4. [20] Let X be a complete metric space, with metric d, and let T be a mapping from X into itself. Suppose T is a quasi-contraction, that is, there exists α ∈ [0, 12 ) such that p(Tx, Ty) ≤ α {d(x, Tx) + d(y, Ty)} for every x, y ∈ X. Then T has a unique fixed point. In 1996, T. Suzuki and W. Takahashi [52] gave the following definition using w-distance and proved several results. Definition 2. [52] Let X be a metric space with metric d. A set-valued mapping T from X into itself is called weakly contractive or p-contractive if there exist a w-distance p on X and r ∈ [0, 1) such that for any x1 , x2 ∈ X and y1 ∈ Tx1 there is y2 ∈ Tx2 with p(y1 , y2 ) ≤ rp(x1 , x2 ). Theorem 5. [52] Let X be a complete metric space and let T be a set-valued p-contractive mapping from X into itself such that for any x ∈ X, Tx is a nonempty closed subset of X. Then there exists x0 ∈ X such that x0 ∈ Tx0 and p(x0 , x0 ) = 0. Definition 3. [52] Let X be a metric space with metric d. A set-valued mapping T from X into itself is called weakly contractive or p-contractive if there exist a w-distance p on X and r ∈ [0, 1) such that p(Tx, Ty) ≤ rp(x, y) for every x, y ∈ X. In the case of p = d, T is called contractive. Theorem 6. [52] Let X be a complete metric space. If a mapping T from X into itself is p-contractive, then T has a unique fixed point x0 ∈ X. Further, x0 satisfies p(x0 , x0 ) = 0. Using Theorem 6, they proved a fixed point theorem which generalizes Nadler’s fixed point theorem [36] for set-valued mappings and Edelstein’s fixed point theorem [11] on an ε -chainable metric space. Before proving it, they gave some definitions and notations. Let X be a metric space with metric d. For x ∈ X and A ⊂ X, set d(x, A) = inf{d(x, y) : y ∈ A}.

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Denote by CB(X) the class of all nonempty bounded closed subsets of X. Let H be the Hausdorff metric with respect to d, that is, H(A, B) = max{sup d(u, B), sup d(v, A)} u∈A

v∈B

for every A, B ∈ CB(X). Let ε ∈ (0, ∞). A mapping T from X into CB(X) is said to be (ε , σ )uniformly locally contractive [11] if there exists σ ∈ [0, 1) such that H(Tx, Ty) ≤ σ d(x, y) for every x, y ∈ X with d(x, y) < ε . In particular, T is said to be contractive when ε = ∞. Theorem 7. [52] Let ε ∈ (0, 1] and let X be a complete and epsilon-chainable metric space with metric d. Suppose that a mapping T from X into CB(X) is (ε , σ )- uniformly locally contractive. Then there exists x0 ∈ X with x0 ∈ Tx0 . As direct consequences of the above Theorem, we obtain the following. Corollary 5. [36] Let X be a complete metric space and let T be a contractive set-valued mapping from X into CB(X). Then there exists x0 ∈ X with x0 ∈ Tx0 . Corollary 6. [11] Let ε ∈ (0, ∞) and let X be a complete and ε -chainable metric space with metric d. Suppose that a mapping T from X into itself is (ε , σ )-uniformly locally contractive. Then T has a unique fixed point. In 1997, T. Suzuki [53], generalised some well-known results in the region of the fixed point theorem using the concept of w-distance. Corollary 7. [53] Let X be a complete metric space, with metric d and p be a w-distance on X. Let T be a mapping from X into itself. Suppose there exists r ∈ [0, 1) such that either (a) or (b) holds; (a) max{p(T2 x, Tx), p(Tx, T2 x)} ≤ r max{p(Tx, x), p(x, Tx)} for every x ∈ X; (b) p(T2 x, Tx) + p(Tx, T2 x) ≤ rp(Tx, x) + rp(x, Tx) for every x ∈ X. Furthermore assume that either one of the following holds; (i) If y = Ty, then inf{p(x, Tx) + p(Tx, x) + p(x, y) :∈ X} > 0; (ii) if {xn } and {Txn } converges to y, then y = Ty; (iii) T is continuous. Then there exists x0 ∈ X such that x0 = Tx0 . Moreover, if v = Tv, then p(v, v) = 0. In general, a w-distance p on X does not satisfy that p(x, y) = p(y, x) for every x, y ∈ X. Hence p(T2 x, Tx) ≤ rp(Tx, x) differs from p(Tx, T2 x) ≤ rp(x, Tx). Therefore, the following Theorem is different from Theorem 4. Theorem 8. [53] Let X be a complete metric space, with metric d and p be a w-distance on X. Let T be a mapping from X into itself. Suppose there exists r ∈ [0, 1) such that p(T2 x, Tx) ≤ rp(Tx, x) for every x ∈ X. Assume that either one of the following holds;

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(i) If {xn } converges to y and {(Txn , xn )} converges to 0, then p(ty, y)=0; (ii) if {xn } and {Txn } converges to y, then y = Ty; (iii) T is continuous. Then there exists x0 ∈ X such that x0 = Tx0 . Moreover, if v = Tv, then p(v, v) = 0. Next, Suzuki’s [53] generalization of the Meir–Keeler fixed point theorem [31]. Theorem 9. [53] Let X be a complete metric space, let p be a w-distance on X and let T be a mapping from X into itself. Suppose that for any ε > 0, there exists δ > 0 such that for every x, y ∈ X, p(x, y) < ε + δ implies p(Tx, Ty) < ε . Then T has a fixed point in X. Here, Suzuki [53] discussed fixed point theorems for Kannan mappings with respect to a w-distance p. Let X be a metric space and let T be a mapping from X into itself. Then T is called weakly Kannan or p-Kannan if there exist a w-distance on p on X and α ∈ [0, 12 ) such that either (a) or (b) holds; (a) p(Tx, Ty) ≤ α p(Tx, x) + α p(Ty, y) for every x, y ∈ X; (b) p(Tx, Ty) ≤ α p(Tx, x) + α p(y, Ty) for every x, y ∈ X. Theorem 10. [53] Let X be a complete metric space. If a mapping T from X into itself is p-Kannan then T has a unique fixed point x0 ∈ X. Further, such that x0 satisfies p(x0 , x0 ) = 0. Definition 4. [52] A mapping T from a metric space X into itself is called weakly contractive if there exist a w-distance p on X and r ∈ [0, 1) such that p(Tx, Ty) ≤ rp(x, y) for every x, y ∈ X. Using the above Definition 3, Suzuki [53] obtained the following. Proposition 1. [53] Let X be a complete metric space with metric d and let there be a weakly contractive mapping T from X into itself. Then T is weakly Kannan. As a direct consequence of Proposition 1, Suzuki [53] obtained the following characterization of metric completeness. Corollary 8. [53] Let X be a complete metric space. Then the following are equivalent: (i) X is complete; (ii) every weakly contractive mapping from X into itself has a fixed point in X; iii every weakly Kannan mapping from X into itself has a fixed point in X. In general, w-distance p does not necessarily satisfy p(x, y) = p(y, x). So, in Suzuki’s [53] definition, a mapping T is not necessarily called weakly Kannan even if there exist a wdistance p and α ∈ [0, 12 ) such that either (c) or (d) holds: (c) p(Tx, Ty) ≤ α p(x, Tx) + α p(Ty, y) for every x, y ∈ X; (d) p(Tx, Ty) ≤ α p(x, Tx) + α p(y, Ty) for every x, y ∈ X. In favour of this statement, Suzuki [53] gives the following example.

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Example 12. [53] Let X = [0, 1] ⊂ R be a complete metric space with the usual metric. Define a w-distance p on X by ⎧ ⎪ if x = 0, ⎨9 p(x, y) = y − x if 0 < x ≤ y, ⎪ ⎩ 3x − 3y if x > y and a mapping T from X into itself by Tx =

 1 x 10

if x = 0, if x = 0.

Then (c) and (d) hold in the case of α = 13 , although T does not have a fixed point. However, Suzuki [53] gives the following. Theorem 11. [53] Let X be a complete metric space and let T be a continuous mapping from X into itself. Suppose that there exist a w-distance p on X and α ∈ [0, 12 ) such that either (c) or (d) holds. Then there exists a unique fixed point x0 ∈ X of T. Moreover, such x0 satisfies p(x0 , x0 ) = 0. In 2001, Suzuki [54] introduced the concept of τ -distance on a metric space, which is a generalized concept of both w-distance and Tataru’s distance. He also improved the generalizations of the Banach contraction principle, Carist’s fixed point theorem, Ekeland’s variational principle, and the nonconvex minimization theorem according to Takahashi. Further he discussed the relation between w-distance and Tataru’s distance ([58]) . Suzuki [54] proved two propositions, which show that the concept of τ -distance is a generalized concept of both w-distance and Tataru’s distance. Proposition 2. [54] Let p be a w-distance on a metric space X. Then p is also a τ -distance on X. Proposition 3. [54] Let {T(t)L ∈ R} be a strongly continuous semigroup of nonexpansive mappings on a subset X of a Banach space. Then Tataru’s distance p on X is also a τ distance on X. The following proposition says that Tataru’s distance [58] is a w-distance if X is compact. Proposition 4. [54] Let X be a compact metric space, let p be a τ -distance on X, and let η be a function from X × R+ into R+ satisfying the following: (i) inf{η (x, t) : t > 0} for all x ∈ X, and η is nondecreasing in the second variable; (ii) limn sup{p(xn , yn ) : m ≥ n} = 0 and limn η (xn , tn ) = 0 imply limn (yn , tn ) = 0; (iii) limn η (zn , p(zn , xn )) = 0 and limn η (zn , p(zn , yn )) = 0 imply limn d(xn , yn )) = 0. Suppose p is lower semicontinuous in its second variable and η is continuous in its first variable. Then p is a w-distance on X. The following proposition is connected with Zhong’s result [64].

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Proposition 5. [54] Let X be a metric space with metric d, let p be a w-distance on X, let h be a nondecreasing function from R+ into itself such that 0∞ (1/(1 + h(r)))dr, and let z0 ∈ X be fixed. Then a function q from X × X into R+ defined by q(x, y) =



p(z0 ,x)+p(x,y)

p(z0 ,x

dr 1 + h(r)

for all x, y ∈ X is a tau-distance on X. Shioji et al. [47] introduced the sets WC2 (X), WC0 (X), WK1 (X), WK2 (X) and WK0 (X) of mapping of X into itself as follows; T ∈ WC2 (X) if and only if there exist p ∈ W(X) and r ∈ [0, 1) such that p(Tx, Ty) ≤ p(y, x) for allx, y ∈ X; T ∈ WC0 (X) if and only if there exist p ∈ W0 (X) and r ∈ [0, 1) such that p(Tx, Ty) ≤ p(x, y) for allx, y ∈ X; T ∈ WK1 (X) if and only if there exist p ∈ W(X) and α ∈ [0, 12 ) such that p(Tx, Ty) ≤ α {p(Tx, x) + p(Ty, y)} for allx, y ∈ X T ∈ WK2 (X) if and only if there exist p ∈ W(X) and α ∈ [0, 12 ) such that p(Tx, Ty) ≤ α {p(Tx, x) + p(y, Ty)} for allx, y ∈ X T ∈ WK0 (X) if and only if there exist p ∈ W0 (X) and α ∈ [0, 12 ) such that p(Tx, Ty) ≤ α {p(Tx, x) + p(Ty, y)} for allx, y ∈ X In particular, a mapping T ∈ WK1 (X) is called P-Kannan. The following lemma was proved in [47]. Lemma 7. [47] Let X be a metric space with metric d, let p be a w-distance on X, let T be a mapping of X into itself and let u be a point in X such that lim p(Tm u, Tn u) = 0.

m,n→∞

Then for every x ∈ X, limk→∞ p(Tk u, x) and limk→∞ p(x, Tk u) exist. Moreover, let β and γ be functions from X to [0, ∞) defined by

β (x) = lim p(Tk u, x) and γ (x) = lim p(x, Tk u). k→∞

k→∞

Then the following hold: (i) β is lower semicontinuous on X; (ii) for every ε > 0, there exists δ > 0 such that β (x) ≤ δ and β (y) ≤ δ imply d(x, y) ≤ ε . In particular, the set {x ∈ X : β (x) = 0} consists of at most one point; (iii) the functions q0 and q1 from X × X to [0, ∞) defined by q0 (x, y) = β (x) + β (y) and q1 (x, y) = γ (x) + β (y) are w-distances on X.

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Shioji et al. [47] proved the theorem from the above Lemma 1. Theorem 12. [47] Let (X, d) be a metric space. Then WC1 (X) = WC0 (X) = WK1 (X) = WK0 (X) ⊂ WC2 (X) = WK2 (X) are w-distances on X. Iemoto et al. [17] also introduced the following class of mapping of X into itself. Let p be a w-distance on X. A mapping T : x → X is called p-contractively nonspreading if there exists α ∈ [0, 12 ) such that p(Tx, Ty) ≤ {p(Tx, y) + p(x, Ty)

∀ ∈ X.

In [17], they proved the following result from Lemma 4.1. Theorem 13. [17] Let (X, d) be a metric space and let p be a w-distance on X such that p(x, x) = 0 for all x ∈ X. Let T be a p-contractively nonspreading mapping of X into itself. Then T is in WC0 (X). Takahashi et al. [60] proved a fixed point theorem for mapping with w-distances in complete metric spaces. They used Lemma 1 of Section 3[20] (see also [57, 59]) to prove their following theorem. Theorem 14. [60] Let (X, d) be a complete metric space and let p ∈ W0 (X) and let {xn } be a sequence in X such that {p(xn , x)} is bounded for some x ∈ X. Let T be a mapping of X into itself. Suppose that there exist a real number r ∈ [0, 1) and a mean μ on l∞ such that

μn p(xn , Ty) ≤ rμn p(xn , y),

∀∈X

Then, the following hold: (i) T has a unique fixed point u in X; (ii) for every z ∈ X, the sequence {Tn z} converges to u. As a direct consequence of Theorem 14, Takahashi et al. [60] obtained the following theorem proved by Hasegawa et al. [15]. Theorem 15. [15] Let (X, d) be a complete metric space and let T be a mapping of X into itself. Suppose that there exist a real number r ∈ [0, 1) and an element x ∈ X such that {Tn x} is bounded and μn d(Txn , Ty) ≤ rμn d(Txn , y), ∀∈X for some mean μ on l∞ . Then, the following hold: (i) T has a unique fixed point u in X; (ii) for every z ∈ X, the sequence {Tn z} converges to u. Using Theorem 14, Takahashi et al. [60] proved new and well-known fixed point theorems in a complete metric space. First they proved a fixed point theorem for generalized hybrid mappings with w-distance in a metric space. Let (X, d) be a metric space and let p

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be a w-distance on X. A mapping T : X → X is called p-contractively generalized hybrid if there exist αβ ∈ R and r ∈ [0, 1) such that

α p(Tx, Ty) + (1 − α )p(x, Ty) ≤ r{β p(Tx, y) + (1 − β )p(x, y)} for all x, y ∈ X. They called such a mapping T a p-contractively (α , β , r) generalized hybrid mapping. We know that the class of mapping above covers well-known mappings in a metric space. For example, a p-contractively (α , β , r) generalized hybrid mapping T is pcontractive for α = 1 and β = 0, that is, there exists r ∈ [0, 1) such that

α p(Tx, Ty) + (1 − α )p(x, Ty) ≤ r{β p(Tx, y) + (1 − β )p(x, y)}

∀x, y ∈ X.

Theorem 16. [60] Let (X, d) be a complete metric space and let p be a symmetric w-distance on X. Let T : X → X be a p-contractively generalized hybrid mapping. Then T has a fixed point in X if and only if {p(Tn x, x)} is bounded and for some x ∈ X. In this case, the following hold: (i) T has a unique fixed point u in X; (ii) for every z ∈ X, the sequence {Tn z} converges to u. Using Theorem 16, Takahashi et al. [60] proved a fixed point theorem for p-contractive mappings in a complete metric space. Theorem 17. [60] Let (X, d) be a complete metric space and let p be a w-distance on X. Let T : X → X be a p-contractive mapping, that is, there exists a real number r with 0 ≤ r < 1 such that p(Tx, Ty) ≤ p(x, y) for all x, y ∈ X. Then, the following hold: (i) T has a unique fixed point u in X; (ii) for every z ∈ X, the sequence {Tn z} converges to u. The following is a fixed point theorem for p-Kannan mapping in a complete metric space. Theorem 18. [60] Let (X, d) be a complete metric space and let p be a w-distance on X. Let T ∈ WK1 (X), that is, there exists α ∈ [0 12 ) such that p(Tx, Ty) ≤ α {p(Tx, x) + p(Ty, y)} for all x, y ∈ X. Then, the following hold: (i) T has a unique fixed point u in X; (ii) for every z ∈ X, the sequence {Tn z} converges to u. Using Theorems 17 and 18, Takahashi et al. [60] proved the following fixed point theorem. Theorem 19. [60] Let (X, d) be a complete metric space and let p be a w-distance on X such that p(x, x) = 0 for all x ∈ X. Let T : X → X be p-contractively nonspreading, that is, there exists a real number γ with 0 ≤ γ < 12 such that p(Tx, Ty) ≤ γ {p(Tx, y) + p(x, Ty)} for all x, y ∈ X. Then, the following hold:

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(i) T has a unique fixed point u in X; (ii) for every z ∈ X, the sequence {Tn z} converges to u. Concerning that {p(Tn x, x)} is bounded for some x ∈ X in Theorem 16, Takahashi et al. [60] proved the following lemma. Lemma 8. [60] Let (X, d) be a complete metric space and let p be a w-distance on X such that p(x, x) = 0 for all x ∈ X. Let T : X → X be a p-contractively (α , β , r) generalized hybrid mapping, that is, there exist α , β ∈ R and r ∈ [0, 1) such that

α p(Tx, Ty) + (1 − α )p(x, Tx) ≤ r{β p(Tx, y) + (1 − β )p(x, y)} for all x, y ∈ X. Furthermore, α , β and r satisfy

β ≥ 0, α − rβ > 0 and r
0 and r
0.........(4.2) for every y ∈ X with y not a common fixed point of T1 and T2 . Then there exists z ∈ X such that z = T1 (z) = T2 (z). Moreover, if v = T1 (v) = T2 (v), then p(v, v) = 0.

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SK Mohanta [32] now supplemented Theorem 25 by examination of conditions (4.1) and (4.2) in respect of their independence. They furnished Examples 13 and 14 below to show that these two conditions are independent. Example 13. [32] Take X = {0} ∪ { 21n : n ≥ 1}, which is a complete metric space with the usual 1 for n ≥ 1. metric d of reals. Define T : X → X by T(0) = 12 and T( 21n ) = 2n+1 Example 14. [32] Take X = [2, ∞) ∪ {0, 1}, which is a complete metric space with the usual metric d of reals. Define T : X → X T(x) = 0, for x ∈ (X/{0}) = 1, for

x = 0.

Note: In the examples above we treat d as a w-distance on X in reference to Theorem 25. Theorem 26. [32] Let p be a w-distance on a complete metric space (X, d). Let T1 , T2 be mappings of X into itself. Suppose that there exists r > 1 such that max{p(T2 T1 x, T1 x), p(T1 T2 x, T2 x)} ≤ r max{p(T1 x, x), p(T2 x, x)} for every x ∈ X and that inf{p(x, y) + min{p(T1 x, x), p(T2 x, x)} : x ∈ X} > 0 for every y ∈ X with y not a common fixed point of T1 and T2 . Then there exists z ∈ X such that z = T1 (z) = T2 (z). Moreover, if v = T1 (v) = T2 (v), then p(v, v) = 0. In 2012, Imdad and Rouzkard proved the following theorem in ordered metric spaces via w-distance. Theorem 27. [18] Let (X, d, ) be a complete partially ordered metric space equipped with a w-distance p and T : X → X be a nondecreasing mapping. Suppose that (a) there exists x0 ∈ X such that (x0 , Tx0 ) ∈ X ; (b) there exist two altering distance functions ψ , φ such that

ψ (p(Tx, Ty)) ≤ ψ (Mx,y ) − φ (Mx,y ) for all x, y ∈ X , where Mx,y = max{p(x, y), min{p(x, Tx), p(y, Ty), p(Tx, x), p(Ty, y)}} (c) either T is orbitally continuous at x0 or (c’) is orbitally X -continuous, and there exists a subsequence {Tnk x0 } of {Tn x0 } converging to x∗ such that (Tnk x0 , x∗ ) ∈ X for any k ∈ N. Then FT = 0. / In what follows, Imdad and Rouzkard [18] give a sufficient condition for the uniqueness of a fixed point in Theorem 27 which runs as follows: (A): for every x, y ∈ X, there exists a lower bound or an upper bound. In [37], it is proved that condition (A) is equivalent to the following one: (B): for every x, y ∈ X, there exists z = c(x, y) ∈ X for which (x, z) ∈ X and (y, z) ∈ X .

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Theorem 28. [18] With the addition of condition (B) to the hypotheses of Theorem 4.27, the fixed point of T turns out to be unique. Moreover lim Tn (x) = x∗

n→∞

for every x ∈ X provided x∗ ∈ FT , that is, the map T : X → X is a Picard operator. In 2013, Firouz and Hosseini Ghoncheh [13] generalized Theorem 2.1 of Boujari [6]. Theorem 29. [13] Let (X, d) be a metric space and p be a w-distance on X. Suppose X is p-complete and p-bounded. Let f and g be commuting, f (X) ⊂ g(X), p-continuous and satisfying 

p(fx,fy) 0

φ (t)dt ≤ (1 − kn )



p(gx,gy)

0

φ (t)dt

for each x, y ∈ X, where real numbers {kn }∞n=1 are in [0, 1] with ∑∞n=1 kn = ∞, ϕ : R+ → R+ is a Lebesgue-integrable mapping which is summable on each compact subset of R+ , nonnegative, and for each ε > 0; 

ε

0

ϕ (t)dt > 0,

then f and g have a unique common fixed point. The correct version of Theorem 2.1 of Boujari [6] is as follows: Theorem 30. [13] Let (X, d) be a metric space and p be a w-distance on X. Suppose X is p-complete and p-bounded and f : X → X is a self-map. If there exists c ∈ (0, 1) and map g : X → X which commutes with f such that g(X) ⊂ f (X) and for each x, y ∈ X, satisfies  0

p(gx,gy)

φ (t)dt ≤ c



p(fx,fy)

0

φ (t)dt

where ϕ : R+ → R+ is a Lebesgue-integrable mapping which is summable on each compact subset of R+ , nonnegative, and for each ε > 0; 

ε 0

ϕ (t)dt > 0.

Indeed, f and g have a unique common fixed point. In 2014, Siva Ram Prashad and T. Phaneendra [41] proved the following theorem using w-distances. Theorem 31. [41] Let (X, d) be a metric space with w-distance p on it. Suppose that f , g : X → X and φ : X → [0, ∞) satisfy the inclusion g(X) ⊂ f (X) and for each u ∈ X with u = fu or u = gu {p(u, fx) + p(u, gx) + p(fgx, gfx) : x ∈ X} > 0. Also suppose that there is a base sequence xn ∞n=1 at some point x0 ∈ X it converges to some point z ∈ X {p(z, gx) ≤ rp(z, fx) + φ (fx) − φ (gx) for all x ∈ X}

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where 0 ≤ r ≤ 1 and for every sequence un ∞n=1 ⊂ X with lim p(z, fun ) = lim p(z, gun ) = 0

n→∞

we have

n→∞

lim max{p(z, fun ), p(z, gun ), p(fgun , gfun )} = 0

n→∞

Then z is a unique common fixed point of f and g. Definition 5. [2, 39] A w-distance on a quasi-metric space (X, d) is a function q : X × X → R+ satisfying the following three conditions: (W1) q(x, y) ≤ (z, x) + q(z, y) for all x, y, z ∈ X; (W2) q(x, .) : X → R+ is lower semicontinuous on (X, τd−1 ) for all x ∈ X; (W3) for each ε > 0 there exists δ > 0 such that q(x, y) ≤ δ and q(x, z) ≤ δ imply d(y, z) ≤ ε . Several examples of w-distances on quasi-metric spaces may be found in [2, 26, 30, 40]. Note that if d is a metric on X then it is a w-distance on (X, d). Unfortunately, this does not hold for quasi-metric spaces, in general. Indeed, in [[29], Lemma 2.2] the following was observed. Lemma 9. [1] If q is a w-distance on a quasi-metric space (X, d), then for each ε > 0 there exists δ > 0 such that q(x, y) ≤ δ and q(x, z) ≤ δ imply d(y, z) ≤ ε . Using w-distance in quasi-metric space, Alegre et al. [1] proved the following theorem. Theorem 32. [1] Let f be a self-map of a complete quasi-metric space (X, d). If there exist a w-distance q on (X, d) and a Jachymski function f : R+ → R+ such that φ (t) < t for all t > 0, and q(fx, fy) ≤ φ (q(x, y)) for all x, y ∈ X, then f has a unique fixed point z ∈ X. Moreover q(z, z) = 0. In 2015, Sang and Meng [44] proved the following. Theorem 33. [44] Let (X, d) be a complete partially ordered metric space equipped with a w-distance p and T : X → X be an injective, continuous and sequentially convergent mapping. Suppose that F : X → X is a nondecreasing and continuous mapping such that

ψ (p(TFx, TFy)) < ψ (p(Tx, Ty)), x = y and

ψ (p(TFx, TFy)) ≤ φ (ψ (p(Tx, Ty))), ∀ x ≥ y,

where ψ ∈ Ψ and φ : [0, +∞) → [0, +∞) is a Jachymski function. If there exists x0 ∈ X such that x0 ∈ F(x0 ), then F has a fixed point. Remark 1. [44] In Theorem 33 the monotonicity of F is not essential for the existence of a fixed point. In fact, we can replace the nondecreasing property of F with the nonincreasing property of F. In this case, the condition that x0 ≤ F(x0 ) should be replaced by x0 ≥ F(x0 ).

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Remark 2. [44] From the process of the proof of Theorem 33, the monotonicity of φ can be replaced by continuity. In fact, Sastry and Babu [45] have addressed a similar problem (see Theorem 2.1 of [45]). Theorem 34. [44] Let (X, d) be a complete partially ordered metric space equipped with a w-distance p and T : X ∈ X be an injective, continuous, and sequentially convergent mapping. Suppose that F : X → X is a nondecreasing and continuous mapping such that

ψ (p(TFx, TFy)) < ψ (MT (x, y)), x = y and where

ψ (p(TFx, TFy)) ≤ φ (ψ (MT (x, y))), ∀ x ≥ y,  p(Tx, TFy) + p(Ty, TFx)  MT (x, y) = p(Tx, Ty), p(Tx, TFx), p(Ty, TFy), 2

The following theorems were proved by Batra et al. [4] in 2015. Theorem 35. [4] Let (X, d) be a complete metric space equipped with a w-distance p and (X, d, G) satisfy the following property. For any sequence (xn )n∈N in X with xn → x and (xn , xn+1 ) ∈ E(G), there exists a subsequence (xkn )n∈N of (xn )n∈N satisfying (xkn , x) ∈ E(G) for all n ∈ N. Let T : X → X be a (p, G)-contraction and XT = {x ∈ X : (x, Tx) ∈ E(G)}. Then (i) Card FixT =Card{[x]G : x ∈ XT }; (ii) FixT =  φ if and only if XT = φ ; (iii) T has a unique fixed point if and only if there exists a point x0 ∈ XT such that XT ⊆ [x0 ]G ; (iv) for any x ∈ XT , T|[x]G is a Picard operator; (v) if XT = φ and G is weakly connected then T is a Picard operator; (vi) if X = ∪{[x]G ; x ∈ XT } then T|X is a weakly Picard operator; (vii) if T ⊆ E(G) T is a weakly Picard operator. Theorem 36. [4] Let (X, d) be a complete metric space equipped with a w-distance p and T : X → X be an orbitally G-continuous (p, G)-contraction. Let XT = {x ∈ X : (x, Tx) ∈ E(G)}. Then the following statements hold: (i) FixT = φ if and only if XT = φ ; (ii) for any x ∈ XT and and y ∈ [x]G , the sequence (Tn y)n∈N converges to a fixed point of T and limn→∞ Tn y does not depend on y; (iii) if XT = φ and G is weakly connected then T is a Picard operator; (iv) if T ⊆ E(G) T is a weakly Picard operator. In 2015, Shrivastava et al. [48] proved common fixed point theorems for three self mappings of a Polish metric space with w-distance. Theorem 37. [48] Let F be a self mapping and G and H be continuous self mappings of Polish metric space (X, d) with a w-distance p satisfying the conditions:

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(i) FX ⊂ GX ∩ HX (ii) p(Fx(w), Fy(w)) ≤α +β

p3 (Gx(w), Fx(w)) − p3 (Hy(w), Fy(w)) [p(Gx(w), Fx(w)) + p(Gx(w), Fy(w))].[p(Gx(w), Fx(w)) − p(Hy(w), Fy(w))]

p4 (Gx(w), Fx(w)) − p4 (Hy(w), Fy(w)) [p(Gx(w), Fx(w)) + p(Gx(w), Fy(w))].[p2 (Gx(w), Fx(w)) − p2 (Hy(w), Fy(w))] +γ

p2 (Hy(w), Fx(w)) − p2 (Gx(w), Fy(w)) p(Gx(w), Fx(w)) + p(Gx(w), Fy(w))

for all x, y ∈ X where α , β , γ > 0, α + β + γ < 14 and w ∈ Ω is a selector. (iii) (F, G) and (F, H) are a weakly commuting pair. Then F, G and H have a unique common fixed point in X. Theorem 38. [48] Let F be a self mapping and G and H be continuous self mappings of Polish metric space (X, d) with a w-distance p satisfying the conditions: (i)

FX ⊂ GX ∩ HX



p2 (Hy(w), Fx(w)) − p2 (Gx(w), Fy(w)) , p(Gx(w), Fx(w)) + p(Gx(w), Fy(w)) p3 (Gx(w), Fx(w)) − p3 (Hy(w), Fy(w)) , [p(Gx(w), Fx(w)) + p(Gx(w), Fy(w))].[p(Gx(w), Fx(w)) − p(Hy(w), Fy(w))]  p4 (Gx(w), Fx(w)) − p4 (Hy(w), Fy(w)) [p(Gx(w), Fx(w)) + p(Gx(w), Fy(w))].[p2 (Gx(w), Fx(w)) − p2 (Hy(w), Fy(w))]

(ii) p(Fx(w), Fy(w)) ≤ α max

for all x, y ∈ X where 0 < α < 14 and w ∈ Ω is a selector. (iii) (F, G) and (F, H) are weakly commuting pair. Then F, G and H have a unique common fixed point in X. Definition 6. [63] A symmetric w-distance on a metric space (X, d) is a w-distance p on (X, d) such that (SY) : q(x, y) = q(y, x) for all x, y ∈ X. If p is a w-distance satisfying (SY), we then say that it is a symmetric w-distance on (X, d). In what follows, Zarinfar et al. [63] assume that p is a symmetric w-distance, in the sense of the above Definition, unless the contrary is asserted. They proved the main result as follows. Theorem 39. [63] Let (X, d) be a complete metric space, let p be a w-distance on X, and let T : X → X be an Rp -contraction with respect to ρ ∈ RA . Assume that, at least, one of the following conditions holds: T is p-continuous; that is, for any sequence {xn }, if p(xn , z) → 0 then p(Txn , Tz) → 0 as n → ∞. (ii) The function ρ satisfies, if {an }, {bn } ⊂ (0, ∞) ∩ A are two sequences such that limn→∞ bn = 0 and ρ (an , bn ) > 0, for all n ∈ N, then limn→∞ an = 0.

(i)

Then T has a unique fixed point.

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Liu et al. [27] established three fixed point theorems for three classes of contractive mappings of integral type with w-distance in complete metric spaces. Theorem 40. [27] Let (X, d) be a complete metric space and let p be a w-distance in X. Assume that T : X → X satisfies that 

0

p(Tx,Ty)

ϕ (t)dt ≤ c



p(x,y)

0

ϕ (t)dt,

∀ x, y ∈ X

where c ∈ [0, 1) is a constant and ϕ ∈ Φ . Then T has a unique fixed point u ∈ X, p(u, u) = 0, limn→∞ p(Tn x0 , u) = 0 and limn→∞ Tn x0 = u for each x0 ∈ X. Theorem 41. [27] Let (X, d) be a complete metric space and let p be a w-distance in X. Assume that T : X → X satisfies 

p(Tx,Ty) 0

ϕ (t)dt ≤ a



p(Tx,x)

0

ϕ (t)dt + b



p(Ty,y)

0

ϕ (t)dt,

∀ x, y ∈ X

where ϕ ∈ Φ , a and b are nonnegative and a + b < 1. Then T has a unique fixed point u ∈ X, p(u, u) = 0, limn→∞ p(Tn x0 , u) = 0 and limn→∞ Tn x0 = u for each x0 ∈ X. Theorem 42. [27] Let (X, d) be a complete metric space and let p be a w-distance in X. Assume that T : X → X satisfies 

p(Tx,Ty) 0

ϕ (t)dt ≤ a



p(x,Tx)

0

ϕ (t)dt + b



p(y,Ty)

0

ϕ (t)dt,

∀ x, y ∈ X

where ϕ ∈ Φ , a and b are nonnegative and a + b < 1. Then T has a unique fixed point u ∈ X, p(u, u) = 0, limn→∞ p(Tn x0 , u) = 0 and limn→∞ Tn x0 = u for each x0 ∈ X. Hosseini Ghoncheh and Khalil Nabizadeh Azar [16] proved the following theorem; the proof is based on an argument similar to the one used by Branciari [7]. Theorem 43. [16] Let (X, d) be a complete metric space, let p be a w-distance on X and c ∈]0, 1[, and let f : X → X be a mapping such that for each x, y ∈ X,  0

p(fx,fy)

ϕ (t)dt ≤ c



p(x,y)

0

ϕ (t)dt,

where ϕ : R+ → R+ is a Lebesgue-integrable mapping which is summable on each compact subset of R+ , nonnegative, and for each ε > 0,  0

ε

ϕ (t)dt > 0

Then f has a unique fixed point a ∈ X such that limn→∞ f n x = a, for each x ∈ X. Razani et al. [42] proved following theorem in respect of w-distance. Theorem 44. [42] Let p be a w-distance on a complete metric space (X; d), ϕ ∈ Φ and ψ ∈ Ψ . Suppose S is a (ϕ , ψ , p)- contractive map on X [i.e., for each x, y ∈ X, ϕ p(Sx, Sy) ≤ ψϕ p(x, y)] then S has a unique fixed point in X. Moreover, limn Sn x is a fixed point of S for each x ∈ X.

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Definition 7. [33] Let X be a metric space with metric d. Then a function p : X × X → [0, ∞) is called a generalized w-distance on X if for all x, z ∈ X and for all distinct points ξ , η ∈ X, each of them different from x and z, the following conditions are satisfied: (i) p(x, z) ≤ p(x, ξ ) + p(ξ , η ) + p(η , z) for any x, y, z ∈ X; (ii) for any x ∈ X p(x, .) : X → [0, ∞) is lower semicontinuous; (iii) for any ε > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ε . From the above Definition it follows that every w-distance is a generalized w-distance. Now we consider the following example to show that a generalized w-distance may not be a w-distance. Example 15. [53] Let X = {1, 2, 3, 4} be a metric space with metric d(x, y) = |x − y| for all x, y ∈ X. Let p : X × X → [0, ∞) be defined by p(1, 2) = p(2, 1) = 3, p(1, 3) = p(3, 1) = p(2, 3) = p(3, 2) = 1, p(1, 4) = p(4, 1) = p(2, 4) = p(4, 2) = p(3, 4) = p(4, 3) = 2 and p(x, x) = 0.6 for every x ∈ X. Here, Mohanta [33] proved a fixed point theorem in a complete metric space by employing the notion of generalized w-distance. Theorem 45. [33] Let (X, d) be a complete metric space with generalized w-distance p and T : X → X be a mapping such that p(Tn x, Tn y) = an [p(x, Tx) + p(y, Ty)] for all x, y ∈ X where an > 0 is independent of x, y such that 0 < a1 < 1 and a1 + an < 1 for n ≥ 2. Suppose that inf{p(x, y) + p(x, Tx) : x ∈ X} > 0 for every y ∈ X with y = Ty. Then if the series ∑∞n=1 an is convergent, T has a fixed point in X. Moreover, if v = Tv, then p(v, v) = 0. In 2013, Rouzkard et al. [43] proved the following fixed point theorems in generalized w-distances. Theorem 46. [43] Let (X, d, ) be a complete partially ordered metric space equipped with a w-distance p and S : X → X be a nondecreasing mapping. Suppose that (a) there exists x0 ∈ X such that (x0 , Sx0 ) ∈ X ; (b) there exist ψ ∈ Ψ and φ ∈ Φ such that

ψ (p(Sx, Sy)) ≤ ψφ (p(x, y)) for all x, y ∈ X , and (c) either S is orbitally continuous at x0 or (c’) is orbitally X -continuous, and there exists a subsequence {Snk x0 } of {Sn x0 } converging to x∗ such that (Snk x0 , x∗ ) ∈ X for any k ∈ N.

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/ Moreover, if x = Sx, then p(x, x) = 0. Then FS = 0. Theorem 47. [43] Let (X, d, ) be a complete partially ordered metric space equipped with a w-distance p and S : X → X be a nondecreasing mapping. Suppose that (a) there exists x0 ∈ X such that (x0 , Sx0 ) ∈ X ; (b) there exist ψ ∈ Ψ and φ ∈ Φ such that

ψ (p(Sx, Sy)) ≤ ψφ (p(x, y)) for all x, y ∈ X , and (c") for every y ∈ X with y = Sy, inf{p(x, y) + p(x, Sx) : x ∈ X} > 0. Then FS = 0. / Moreover, if x = Sx, then p(x, x) = 0. Theorem 48. [43] Let (X, d, ) be a complete partially ordered metric space equipped with a w-distance p and S : X → X be a nondecreasing mapping. Suppose that (a) there exists x0 ∈ X such that (x0 , Sx0 ) ∈ X ; (b) there exist ψ ∈ Ψ and φ ∈ Φ such that

ψ (p(Sx, S2 x)) ≤ ψφ (p(x, Sx)) for all x, Sx ∈ X , and (c) either S is orbitally continuous at x0 or (c’) is orbitally X -continuous, and there exists a subsequence {Snk x0 } of {Sn x0 } converging to x∗ such that (Snk x0 , x∗ ) ∈ X for any k ∈ N. / Moreover, if x = Sx, then p(x, x) = 0. Then FS = 0. Theorem 49. [43] Let (X, d, ) be a complete partially ordered metric space equipped with a w-distance p and S : X → X be a nondecreasing mapping. Suppose that (a) there exists x0 ∈ X such that (x0 , Sx0 ) ∈ X ; (b) there exist ψ ∈ Ψ and γ ∈ Γ such that

ψ (p(Sx, S2 x)) ≤ ψγ (p(x, Sx)) for all x, Sx ∈ X , and (c) either S is orbitally continuous at x0 or (c’) is orbitally X -continuous, and there exists a subsequence {Snk x0 } of {Sn x0 } converging to x∗ such that (Snk x0 , x∗ ) ∈ X for any k ∈ N. / Moreover, if x = Sx, then p(x, x) = 0. Then FS = 0.

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Theorem 50. [43] Let (X, d, ) be a complete partially ordered metric space equipped with a w-distance p and S : X → X be a nondecreasing mapping. Suppose that (a) there exists x0 ∈ X such that (x0 , Sx0 ) ∈ X ; (b) there exist γ ∈ Γ and k ∈ [0, 12 ) such that

γ (p(Sx, Sy)) ≤ k{γ p(x, Sx) + γ p(y, Sy)} for all x, y ∈ X , and (c") for every y ∈ X with y = Sy, inf{p(x, y) + p(x, Sx) : x ∈ X} > 0. Then FS = 0. / Moreover, if x = Sx, then p(x, x) = 0. Shobkolaei et al. [49] introduced the concept of fuzzy w-distance and proved many fixed point theorem in fuzzy metric spaces with fuzzy w-distance which are a nice generalization of the known results in metric and ultra fuzzy metric spaces. Definition 8. [49] Let (X, M, ∗) be a fuzzy metric space. Then a function S : X × X × [0, ∞) → [0, 1] is called a fuzzy w-distance on X if the following conditions are satisfied: (i) S(x, y, t + s) ≥ S(x, z, t) ∗ S(z, y, s) + S(η , z) for any x, y, z ∈ X and t, s > 0; (ii) for each x ∈ X and S(x, ., t) : X → [0, ∞) is upper semicontinuous. That is, if there exists a sequence {yn } of X such that yn → y, then lim sup S(xn , yn , t) ≤ S(x, y, t);

n→∞

(iii) for any 0 < ε < 1, there exists 0 < δ < 1 such that S(z, x, t) ≥ 1 − δ and S(z, y, s) ≥ 1 − δ for all t, s > 0 imply M(x, y, t + s) ≥ 1 − ε . Let us give some examples of fuzzy w-distance. Example 16. [49] Every fuzzy metric is a fuzzy w-distance. Example 17. [49] Let (X, .) be a normed linear space and (X, M, ∗) be a fuzzy metric space t with M(x, y, t) = t+x−y and a ∗ b = a.b for every a, b ∈ [0, 1]. Then the function S : X × X × t for every x, y ∈ X, t, s > 0 is a fuzzy w-distance [0, ∞) → [0, 1] defined by S(x, y, t) = t+x+y on X. Example 18. [49] Let (X, .) be a normed linear space and (X, M, ∗) be a fuzzy metric space with  1 if 0 < t < 1, M(x, y, t) = 1+x−y t if t ≥ 1 t+x−y and a ∗ b = a.b for every a, b ∈ [0, 1]. Then the function S : X × X × [0, ∞) → [0, 1] defined by 1 S(x, y, t) = 1+x+y for every x, y ∈ X, t > 0 is a fuzzy w-distance on X.

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Example 19. [49] Let (X, M, ∗) be a fuzzy metric space. Let α be a function from X into [0, 1]. Define S : X × X × [0, ∞) → [0, 1] as follows: S(x, y, t) = α (x) ∗ M(x, y, t) for every x, y ∈ X, t > 0. Then S is a fuzzy w-distance on X. Lemma 10. [49] Let (X, M, ∗) be a fuzzy metric space and let S be a fuzzy w-distance on X. Let {xn } and {yn } be sequences in X, let {αn (t)} and {βn (t)} be sequences in [0, 1] converging to 1 for t > 0, and let x, y, z ∈ X, t > 0. Then the following hold: If S(xn , y, t) ≥ αn (t) and S(xn , z, t) ≥ βn (t) for any n ∈ N, then y = z. In particular, if S(x, y, t) = 1 and S(x, z, t) = 1, then y = z; (ii) if S(xn , yn , t) ≥ αn (t) and S(xn , z, t) ≥ βn (t) for any n ∈ N, then {yn } converges to z; (iii) if S(xn , ym , t) ≥ αn (t) for any n, m ∈ N with m > n, then {xn } is a Cauchy sequence; (iv) if S(y, xn , t) ≥ αn (t) for any n ∈ N, then {xn } is a Cauchy sequence. (i)

Theorem 51. [49] Let (X, M, ∗) be a complete fuzzy metric space and S be a fuzzy w-distance. Let f , g be self-mappings on X satisfying the following conditions: (i) g(X) ⊆ f (X) and f (X) is a closed subset of X; (ii) the pair (f , g) are weakly compatible; (iii)  S(gx,gy,t)   S(fx,fy,t) ϕ (s)ds ≥ ψ ϕ (s)ds, 0

0

∀ x, y ∈ X

and t > 0, where ϕ ∈ Φ and ψ ∈ Ψ . If d(t) = inf{S(x, y, t)|x, y ∈ X} > 0 for all t > 0, then f , g have a unique common fixed point in X. The following theorems were proved by Lakzian and Arabyani [28]. Theorem 52. [28] Let (X, d) be a cone metric space with w-distance p on X and T : X → X. Suppose that there exists r ∈ [0, 1) such that p(Tx, T2 x) ≤ rp(x, Tx), for every x ∈ X and inf{p(x, y) + p(x, Tx) : x ∈ X} > 0 for every y ∈ X with y = Ty. Then there is a z ∈ X such that z = Tz. Moreover, if P is a normal cone with normal constant M and v = T(v), then, p(v, v) = 0. Theorem 53. [28] Let (X, d) be a cone metric space with w-distance p. Let P be a normal cone on X. Suppose a mapping T : X → X satisfies the contractive condition p(Tx, Ty) ≤ kp(x, y), for all x, y ∈ X, where k ∈ [0, 1) is a constant. Then, T has a unique fixed point in X. For each x ∈ X, the iterative sequence {Tn (x)}n≥1 converges to the fixed point. The following theorem was proved by Sastry et al. [45].

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Theorem 54. [45] Let (X, d) be a cone metric space with w-distance p on X and T : X → X. Suppose that there exists r ∈ [0, 1) such that p(Tx, T2 x) ≤ rp(x, Tx), for every x ∈ X and y = Ty. Then there exists sy > 0 such that 0 < sy ≤ {p(x, y) + p(x, Tx) : x ∈ X} > 0 Then there is a z ∈ X such that z = Tz. Further if v = T(v), then, p(v, v) = 0. In 2012, M. Sharma [46] proved the following theorem in respect of cone metric spaces via w-distance. Theorem 55. [46] Let (X, d) be a cone metric space with w-distance p. Let P be a normal cone on X. Suppose a mapping T : X → X satisfies the contractive condition p(Tx, T2 x) ≤ α p(x, y) + β [p(x, Tx) + p(y, Ty)] + γ [p(x, Ty) + p(y, Tx)] for all x, y ∈ X. α , β , γ are nonnegative reals such that α + 2β + 2γ < 1. Then, T has a unique fixed point in X. For each x ∈ X, the iterative sequence {Tn (x)}n≥1 converges to the fixed point. The following lemma([34]) will be needed. Lemma 11. [46] Let X be a cone metric space and let p be a w-distance on X. Let {xn } and {yn } be sequences in X. Let {αn } and {βn } be sequences in P converging to θ , and let x, y, z ∈ X. Then the following hold: (i) if p(xn , yn ) ≤ αn and p(xn , z) ≤ βn for any n ∈ N, then {yn } converges to z; (ii) If p(xn , y) ≤ αn and p(xn , z) ≤ βn for any n ∈ N, then y = z. In particular, if p(x, y) = θ and p(x, z) = θ , then y = z; (iii) if p(xn , ym ) ≤ αn for any n, m ∈ N with m > n, then {xn } is a Cauchy sequence. The following theorems were proved by Mohanta and Maitra [34]. Theorem 56. [34] Let (X, d) be a cone metric space with w-distance p and ≤ be a complete ordering on E with respect to P. L1 , L2 be mappings from X into itself. Suppose that there exists r ∈ [0, 1) such that max{p(T1 x, T2 T1 x), p(T2 x, T1 T2 x)} ≤ r min{p(x, T1 x), p(x, T2 x)} for every x ∈ X and that inf{p(x, y) + min{p(x, T1 x), p(x, T2 x)} : x ∈ X} > θ for every y ∈ X with y not a common fixed point of T1 and T2 . Then there exists z ∈ X such that z = T1 z = T2 z. Moreover, if v = T1 v = T2 v, then, p(v, v) = θ . Theorem 57. [34] Let (X, d) be a complete cone metric space with w-distance on X and let T : X → X be continuous. Suppose that there exists r ∈ [0, 1) such that p(Tx, T2 x) ≤ rp(x, Tx) for every y ∈ X. Then there exists z ∈ X such that z = Tz. Moreover, if v = Tv, then, p(v, v) = θ .

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Theorem 58. [34] Let (X, d) be a complete cone metric space and P a normal cone with normal constant M. Suppose the mapping T : X → X satisfies the contractive condition to be continuous. p(Tx, Ty) ≤ k(p(Tx, x) + p(Ty, y)) for every x, y ∈ X where k ∈ [0, 12 ) is constant. Then T has a unique fixed point in X. Theorem 59. [34] Let (X, d) be a cone metric space with w-distance p and ≤ be a complete ordering on E with respect P. L1 , L2 be mappings from X into itself. Suppose that there exists r > 1 such that min{p(T2 T1 x, T1 x), p(T1 T2 x, T2 x)} ≥ r max{p(T1 x, x), p(T2 x, x)} for every x ∈ X and that inf{p(x, y) + min{p(T1 x, x), p(T2 x, x)} : x ∈ X} > θ for every y ∈ X with y not a common fixed point of T1 and T2 . Then there exists z such that z = T1 z = T2 z. Moreover, if v = T1 v = T2 v, then, p(v, v) = θ . Theorem 60. [34] Let (X, d) be a complete cone metric space and P a normal cone with normal constant M. Suppose the mapping T : X → X, if there exists r > 1 such that satisfying p(Tx, Ty) ≥ r min{p(Tx, x) + p(Ty, y), d(x, y)} for any x, y ∈ X and T is onto continuous, then T has a fixed point in X. Theorem 61. [34] Let (X, d) be a complete cone metric space P a normal cone and the mapping T : X → X is continuous, onto satisfies the condition p(Tx, Ty) ≥ k[p(Tx, x) + p(Ty, y)] for any x, y ∈ X, where

1 2

< k < 1 is a constant. Then T has a fixed point in X.

Theorem 62. [34] Let p be a w-distance in a complete cone metric space (X, d), and P be a regular cone. Let the continuous mapping T : X → X. Suppose that there exists a mapping Q : X → P such that p(x, Tx) ≤ Q(x) − Q(T(x)) for all x ∈ X. Then T has a fixed point in X. Moreover, if v = Tv, then, p(v, v) = θ . Khan and Bano [22] proved some common fixed point theorems for a pair of weakly compatible mappings in cone metric space using w-distance on X. Theorem 63. [22] Let (X, d) be a complete cone metric space with w-distance p. Let P be a normal cone with normal constant K on X. Suppose that the mappings S, T : X → X satisfy the contractive condition, p(Sx, Sy) ≥ r[p(Sx, Ty) + p(Sy, Tx) + p(Sx, Tx) + p(Sy, Ty)] where, r ∈ [0, 14 ) is a constant. If the range of T contains the range of S and T(X) is a complete subspace of X, then S and T have a unique coincidence point in X. Moreover, if S and T are weakly compatible, then S and T have a unique common fixed point.

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Theorem 64. [22] Let (X, d) be a complete cone metric space with w-distance p. Let P be a normal cone with normal constant K on X. Suppose that the mappings S, T : X → X satisfy the contractive condition, p(Sx, Sy) ≥ r[p(Tx, Ty) + p(Sx, Tx) + p(Sy, Ty) + p(Tx, Sy) + p(Sx, Ty)],

∀ x, y ∈ X,

where, r ∈ [0, 15 ) is a constant. If the range of T contains the range of S and T(X) is complete subspace of X, then S and T have a unique coincidence point in X. Moreover, if S and T are weakly compatible, then S and T have a unique common fixed point. ´ c [9] and Fisher Recently, Darko et al. [10] generalized the given fixed point results in Ciri´ [14] via w-distance in the context of complete metric spaces. They also extended some results of Kada et al. [20], and Suzuki [53]. Theorem 65. [10] Let (X, d) be a complete metric space, T : X → X a mapping and let p be a w-distance on (X, d). If for every x ∈ X there exists n = n(x) ∈ N such that   p(Tn x, Tn y) ≤ q. max p(x, y), p(x, Ty), p(x, T2 y), . . . , p(x, Tn y), p(x, Tn x) holds for every y ∈ X and some q ∈ [0, 1), and if for every y ∈ X such that y = Tn(y) y we have   inf p(x, y) + p(x, Tx) + p(x, T2 x) + · · · + p(x, Tn(y) x) : x ∈ X > 0, where 0 ≤ q < 1, then T has a unique fixed point z ∈ X such that p(z, z) = 0. Moreover, for every x ∈ X, z = limm Tm x. Theorem 66. [10] Let (X, d) be a complete metric space, p be a w-distance on X and let T : (X, d) → (X, d) be a continuous mapping which satisfies the following condition: for each x ∈ X there is a natural number n = n(x) such that for all y ∈ X  p(Tn x, Tn y) ≤ q. max p(x, y), p(x, Ty), p(x, T2 y), . . . , p(x, Tn y),  p(x, Tx), p(x, T2 x), . . . , p(x, Tn x) where 0 ≤ q < 1. Then T has a unique fixed point z ∈ X such that p(z, z) = 0. Moreover, for every x ∈ X, z = limm Tm x. Darko et al. [10] also proved a theorem in respect of generalized quasi-contraction mapping of Fisher type via w-distance. Theorem 67. [10] Let (X, d) be a complete metric space, T : (X, d) → (X, d) be a continuous mapping and let p be a w-distance. If for some fixed positive integers l and q and some λ ∈ (0, 1)   p(Tl x, Tq y) ≤ λ. max p(xTr , Ts y), p(Ts y, Tr x), p(Tr x, Tr y), p(Ts y, Ts x) where x, y ∈ X, then T has a unique fixed point u ∈ X such that p(u, u) = 0. Most Recently, Mongkolkeha and Gopal [35] established some new existence theorems ´ c type generalized F-contraction mapping in metric of common fixed points for the Ciri´ spaces with the w-distance.

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Definition 9. [35] Let (X, d) be a metric space equipped with a w-distance p. A mapping ´ c type generalized F-contraction (for short, the CF-contraction) f : X → X is called the Ciri´ if, for all x, y ∈ X, there exist F ∈ F or F ∈ F and t > 0 such that p(fx, fy) > 0 implies τ + F(p(fx, fy)) ≤ F(λMp (x, y)) for all x, y ∈ X, where 0 ≤ λ < 1 and λMp (x, y) = max{p(x, y), p(x, fx), p(y, fy), p(x, fy), p(y, fx)}. Definition 10. [35] Let (X, d) be a metric space equipped with a w-distance p. A mapping ´ c type generalized F-contraction with respect to g (for short, the f : X → X is called the Ciri´ CFg -contraction), where g : X → X is a mapping if there exist F ∈ F or F ∈ F and t > 0 such that g p(fx, fy) > 0 implies τ + F(p(fx, fy)) ≤ F(λMp (x, y)) for all x, y ∈ X, where 0 ≤ λ < 1 and g

λMp (x, y) = max{p(gx, gy), p(gx, fx), p(gy, fy), p(gx, fy), p(gy, fx)}. Lemma 12. [35] Let (X, d) be a metric space equipped with the w-distance p. Let F ∈ F or F ∈ F and f , g : X → X be two mappings such that f (X) ⊆ g(X) and g commutes with f . Assume that f and g satisfy Definition 4.9. For any x0 ∈ X, define a sequence {x} in X by f (xn ) = g(xn+1 ) for each n ≥ 0. Then, we have the following: (i)

for each x0 ∈ X, n ∈ N and i, j ∈ N ∪ {0} with i, j ≤ n, g

τ + F(p(fxi , fxj )) ≤ F(λδp (Op (x0 , n))). (ii) for each x0 ∈ X and n ∈ N, there exist i, j ∈ N with i, j ≤ n such that g

Op (x0 , n)) = max{p(gx1 , gx1 )p(gx1 , gxi )p(gxi , gx1 )} (iii) for each x0 ∈ X, g

Op (x0 , n)) ≤

1 .α (x0 ), 1−λ

where α (x0 ) := p(gx1 , gx1 ). (iv) For each n ∈ N,

τ + F(p(fxn−1 , fxn )) ≤ F( Furthermore,

λn−1 .α (x0 )). 1−λ

lim p(fxn−1 , fxn ) = 0

n→∞

Theorem 68. [35] Let (X, d) be a complete metric space equipped with the w-distance p. Let F ∈ F and f , g : X → X be two mappings such that f (X) ⊆ g(X) and g is commuted with f . Assume that the following hold: (i) f and g satisfy Definition 4.9; (ii) for all y ∈ X with gy = fy, inf{p(gx, y) + p(gx, fx)} > 0.

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Then, f and g have a unique common fixed point u in X and p(u∗ , u∗ ) = 0. Furthermore, if {gxn } converges to u∗ ∈ X, then lim p(gfxn , fu∗ ) = 0 = lim p(fgxn , gu∗ ).

n→∞

n→∞

3.5 Conclusion Fixed point theory in metric spaces has many applications. It is natural that there have been several attempts to extend it to a more general setting. One of these generalizations was introduced by Kada et al. [20] in 1996, where they gave a distance which has a nonsymmetric property. Hence, in some of the first papers that followed, the authors implicitly used some of these additional conditions. We show in this article that, nevertheless, most of these results are valid, since their proofs can be corrected, using some easy observations due to [20, 51, 52, 53] and other authors. Moreover, all the results of authors who, later on, assumed some additional requirements can be made more general, by omitting these assumptions. Some open questions and suggestions for further work are also noted.

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4 G-Metric Spaces: From the Perspective of F-Contractions and Best Proximity Points Vishal Joshi and Shilpi Jain

CONTENTS 4.1 4.2

4.3 4.4

4.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 F-Contraction in G-Metric Spaces: Application Aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2.1 F-Contraction on a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2.2 F-Contraction in G-Metric Spaces and Associated Fixed Point Results . . . . 109 4.2.3 Results with Integral Inequalities under Generalized F-Contraction . . . . . . 122 4.2.4 G − l Cyclic F-Contraction in G-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Applications in Real World Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Best Proximity Point in G-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4.2 Best Proximity Point in G-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Conclusions and Future Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.1 Introduction The first modification of a 2-metric was a D-metric which was introduced in 1984 by Dhage [23],[24],[25]. While for a 2-metric d, d(x, y, z) can be thought as a generalization of the area of a triangle with vertices at x, y, z in R2 , then, for a D-metric D, D(x, y, z) can be treated as a generalization of the perimeter of this triangle. However, in 2003, Mustafa and Sims [48] stated some remarks concerning D-metric spaces and presented some examples which showed that many of the basic claims concerning the topological structure of D-spaces were incorrect, thus nullifying many of the results claimed for D-spaces. After that, in 2005, various concepts of open balls in D-metric spaces were studied in the case of certain D-metric spaces and many results in the literature on such balls were shown to be false by Naidu et al. [56]. In 2006, to overcome the flaws of D-metric spaces which were pointed out, Mustafa and Sims [49] introduced the notion of a G-metric space by modifying axioms of a D-metric. Mustafa provided many examples of G-metric spaces in [50] and developed some of their properties. For example, he proved that G-metric spaces are provided with a Hausdorff topology which allows us to consider, among other topological notions, convergent sequences, limits, Cauchy sequences, continuous mappings, completeness and compactness. Further, he also improved various notions in G-metric spaces such as the 103

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properties of ordinary metrics derived from a G-metric, and the properties of G-metrics derived from ordinary metrics. Definition 1. [49] Let X be a nonempty set and let G : X × X × X → [0, ∞) be a function satisfying the following properties: (G-1) (G-2) (G-3) (G-4) (G-5)

G(x, y, z) = 0 if x = y = z, G(x, x, y) > 0, for all x, y ∈ X with x = y, G(x, x, y) ≤ G(x, y, z), for all x, y, z ∈ X with y = z, G(x, y, z) = G(x, z, y) = G(y, z, x) = · · · , (symmetry in all three variables), G(x, y, z) ≤ G(x, a, a) + G(a, y, z), for all x, y, z, a ∈ X.

The function G is called a generalized or a G-metric on X and the pair (X, G) is called a G-metric space. The previous properties may be easily interpreted in the setting of metric spaces. Let (X, d) be a metric space and define G : X × X × X → [0, ∞) by G(x, y, z) = d(x, y) + d(x, z) + d(y, z), for all x, y, z ∈ X. Then (X, G) is a G-metric space. In this case, G(x, y, z) can be interpreted as the perimeter of the triangle of vertices x, y and z. For example, (G − 1) means that with one point we cannot have a positive perimeter, and (G − 2) is equivalent to the fact that the distance between two different points cannot be zero. Furthermore, as the perimeter of a triangle cannot depend on the order in which we consider its vertices, we have (G−4) and (G−5) is an extension of the triangle inequality using a fourth vertex. Maybe the most controversial axiom is (G − 3) which has an obvious geometric interpretation: the length of an edge of a triangle is less than or equal to its semiperimeter, that is, d(x, y) ≤

d(x, y) + d(y, z) + d(z, x) 2

Example 1. If X is a nonempty subset of R, then the function G : X × X × X → [0, ∞), given by G(x, y, z) = |x − y| + |y − z| + |z − x|, for all x, y, z ∈ X, is a G-metric on X. Example 2. Every nonempty set X can be provided with the discrete G-metric, which is defined, for all x, y, z ∈ X, by  0, if x = y = z; G(x, y, z) = 1, otherwise. Example 3. Let X ∈ [0, ∞) be the interval of nonnegative real numbers and let G be defined by: G(x, y, z) = max{|x − y|, |y − z|, |z − x|}, for all x, y, z ∈ X, Then G is a complete G-metric on X.

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Example 4. If G is a G-metric on X, then G : X × X × X → [0, ∞), given by G (x, y, z) =

G(x, y, z) , for all x, y, z ∈ X, 1 + G(x, y, z)

is another G-metric on X. Basic properties of G-metric spaces The following lemma can be obtained easily from the definition of a G-metric space. Lemma 1. [49] . Let (X, G) be a G-metric space. Then, for any x, y, z, a ∈ X, the following properties hold. 1. G(x, y, y) ≤ 2G(y, x, x) 2. G(x, y, z) ≤ G(x, x, y) + G(x, x, z) 3. G(x, y, z) ≤ G(x, a, a) + G(y, a, a) + G(z, a, a) 4. |G(x, y, z) − G(x, a, a)| ≤ max{G(y, a, a), G(z, a, a) 5. If G(x, y, z) = 0, then x = y = z 6. G(x, y, z) ≤ G(x, a, z) + G(a, y, z) 7. G(x, y, z) ≤ 23 [G(x, y, a) + G(x, a, z) + G(a, y, z)] (1) By the property (G − 5) along with property (G − 4), we have G(x, y, y) = G(y, y, x) ≤ G(y, x, x) + G(x, y, x) = 2G(y, x, x) (2)Utilizing (G − 4) and (G − 5), taking a = x, we have G(x, y, z) = G(y, x, z) ≤ G(y, x, x) + G(x, x, z) = G(x, x, y) + G(x, x, z) (3) Applying (G − 4) and (G − 5) again, we get G(x, y, z) ≤ G(y, a, a) + G(a, y, z) = G(x, a, a) + G(y, a, z) ≤ G(x, a, a) + G(y, a, a) + G(a, a, z) (4) Again using (G − 4) and (G − 5), we have G(x, y, z) = G(z, y, x) ≤ G(z, a, a) + G(a, y, x), G(a, y, x) ≤ G(a, z, z) + G(y, z, x). Thus we have, G(x, y, z) − G(a, y, x) ≤ G(z, a, a) and G(a, y, x) − G(x, y, z) ≤ G(a, z, z). Hence, |G(x, y, z) − G(x, y, a)| ≤ max{G(a, z, z), G(z, a, a)}. (5) consider that G(x, y, z) = 0. Now, we claim that if y = z, then we must have x = y. In fact, by the property of G-metric spaces, 0 ≤ G(x, x, y) ≤ G(x, y, z) = 0

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and therefore G(x, x, y) = 0. If x = y, then G(x, x, y) > 0, thus G(x, x, y) = 0 implies x = y. Again by the symmetry, we can prove that if z = y, then x = z. Hence, we obtain that y = x = z, a contradiction with the assumption y = z; then all the arguments must be equal. Bonding between metrics and G-metrics Every metric on X generates a G-metric on X in different ways. Lemma 2. [49]. If (X, d) is a metric space, then the functions Gdm , Gds : X3 → [0, ∞) defined by Gdm (x, y, z) = max{d(x, y), d(y, z), d(z, x)}, and Gds (x, y, z) = d(x, y) + d(y, z) + d(z, x) for all x, y, z ∈ X, are G-metrics on X. Furthermore, Gdm (x, y, z) ≤ Gds (x, y, z) ≤ 3Gdm (x, y, z) for all x, y, z ∈ X. Conversely, a G-metric on X also induces some metrics on X. Lemma 3. If (X, G) is a G-metric space, then the functions dGm , dGs : X2 → [0, ∞) defined by dGm (x, y) = max{G(x, y, y), G(y, x, x)} and dGs (x; y) = G(x, y, y), G(y, x, x) for all x, y ∈ X, are metrics on X. Symmetric G-metric spaces: A G-metric space (X, G) is called symmetric if G(x, y, y) = G(y, x, x) for all x, y ∈ X. The mappings given in Examples 1, 2 and 3 are symmetric G-metrics. Moreover Gdm and Gds are symmetric G-metrics on X. In fact Gds (x, y, y) = 2Gdm (x, y, y) = 2d(x, y),

for all x, y ∈ X

Topology of a G-metric space: Here we state the canonical Hausdorff topology of a G-metric space. Definition 2. The open ball of center x ∈ X and radius r > 0 in a G-metric space (X, G) is the subset BG (x, r) = {y ∈ X : G(x, y, y) < r}. Similarly, the closed ball of center x ∈ X and radius r > 0 in a G-metric space (X, G) is the subset B¯ G (x, r) = {y ∈ X : G(x, y, y) ≤ r}. We immediately conclude that x ∈ BG (x, r) ⊆ B¯ G (x, r).

Convergent and Cauchy sequences

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Definition 3. [49] Let (X, G) be a G-metric space and let {xn } be a sequence of points of X. We say that the sequence {xn } is G-convergent to x ∈ X if lim G(x, xn , xm ) = 0;

n,m→+∞

that is, for any ε > 0, there exists N ∈ N such that G(x, xn , xm ) < ε , for all m, n > N. We call x the limit of the sequence and write xn → x or lim xn = x. n,m→+∞

Proposition 1. [49] Let (X, G) be a G-metric space. Then the following statements are equivalent: (1) (2) (3) (4)

{xn } is G-convergent to x; G(xn , xn , x) → 0 as n → +∞; G(xn , x, x) → 0 as n → +∞; G(xn , xm , x) → 0 as n, m → +∞.

Definition 4. [49] Let (X, G) be a G-metric space. A sequence {xn } is called G-Cauchy if for every ε > 0, there is N ∈ N such that G(xn , xm , xl ) < ε , for all n, m, l ≥ N; that is G(xn , xm , xl ) → 0 as n, m, l → +∞. Proposition 2. [49] Let (X, G) be a G-metric space. Then the following statements are equivalent: (1) (2)

{xn } is G-Cauchy; For every ε > 0, there is N ∈ N such that G(xn , xn , xm ) < ε for all n, m ≥ N.

Definition 5. [49] A G-metric space (X, G) is called G-complete if every G-Cauchy sequence is G-convergent in (X, G). After coining the concept of G-metric spaces, Mustafa et al. [49] finally moved to fixed point results and proved a G-metric version of the Banach Contraction principle, given as Theorem 1. [49] Let (X, G) be a complete G-metric space and let T : X → X be a mapping such that there exists α ∈ [0, 1) satisfying G(Tx, Ty, Tz) ≤ α G(x, y, z),

(4.1)

for all x, y, z ∈ X. Then T has a unique fixed point. After this milestone initiation from Mustafa et al. many researchers attempted G- metric ´ c’s fixed point result, Hardy– versions of several celebrated fixed point results like Ciri´ Rogers, fixed point results, Alber and Guerre-Delabriere’s result etc and also fixed point results in the framework of G-metric spaces in all aspects such as for admissible mappings, cyclic mappings, and expansive mappings; for the detailed progress of G- metric spaces,

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one can refer [1], [2],[3], [4], [6], [10], [19], [27], [28]. [34], [39], [45], [51], [52], [53], [54], [65] [66]. Samet et al.[68] and Jleli et al. [34] explored how some theorems in the framework of G-metric spaces in the existing literature can be obtained directly by some existing result in the setting of usual metric spaces. Further, E. Karapinar et al. [39] answered the approach of [68] and [34] with the remark that techniques used in these papers are inapplicable unless the contraction condition in the statement of the theorem can be reduced into two variables.The technique pointed out in [68] is exploited in this chapter to obtain such types of fixed point results in G-metric spaces that cannot be obtained from the existing results in the setting of associated metric spaces.

4.2 F-Contraction in G-Metric Spaces: Application Aspect The origination of fixed point theory on complete metric space is coupled with Banach contraction principle due to Banach [12], announced in 1922. The Banach contraction principle enunciates that any contractive self-mappings on a complete metric space have a unique fixed point. This principle is one of a very ascendant test for the existence and uniqueness of the solution of substantial problems arising in mathematics. Because of its implication in mathematical theory, the Banach Contraction Principle has been extended and generalized in many directions (see [36], [42], [35], [11], [13], [14], [64], [21], [61], [55], [5], [17], [41]). Recently, one of its most interesting generalizations was stated by Wardowski [79]. He introduced a new contraction called the F-contraction and established a fixed point result as a generalization of the Banach contraction principle in a dissimilar way than in the other acknowledged results from the literature. Afterward, Secelean [69] altered the condition (F2) of [79] with an equivalent and more simple one. Most recently Piri et al. [59] revealed a large class of functions by replacing condition (F3) by the condition (F3 ) in the definition of the F-contraction due to Wardowski [79]. Utilizing this new idea, they established a fixed point theorem as a generalization of results of Wardowski [79]. 4.2.1 F-Contraction on a Metric Space Wardowski [79] initiated and considered a new contraction called the F-contraction to prove a fixed point result as a generalization of the Banach contraction principle. Definition 6. [79] Let F : R+ → R be a mapping satisfying the following conditions: (F1) F is strictly increasing; (F2) for all sequence αn ⊆ R+ , lim αn = 0 if and only if lim F(αn ) = −∞; n→∞

(F3) there exists 0 < k < 1 such that lim+ α k F(α ) = 0.

n→∞

α →0

Wordowski [79], defined the class of all functions F : R+ → R by  and introduced the notion of F-contraction as follows. Definition 7. [79] Let (X, d) be a metric space. A self-mapping T on X is called an F−contraction if there exists τ > 0 such that for x, y ∈ X d(Tx, Ty) > 0 ⇒ τ + F(d(Tx, Ty)) ≤ F(d(x, y)), where F ∈ .

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Afterward Secelean [69] established the following lemma and utilized an equivalent but a more simple condition (F2 ) instead of condition (F2). Lemma 4. [69] Let F : R+ → R be an increasing map and αn be a sequence of positive real numbers. Then the following assertions hold: (a) if lim F(αn ) = −∞ then lim αn = 0; n→∞

n→∞

(b) if infF = −∞ and lim αn = 0; then lim F(αn ) = −∞. n→∞

n→∞

He forwarded the following conditions. (F2 ) infF = −∞ or (F2 ) there exists a sequence αn be a sequence of positive real numbers such that lim F(αn ) = n→∞ −∞. Very recently Piri et al. [59] replaced the condition (F3) by (F3 ) in Definition (6) due to Wardowski as follows. (F3 ) F is continuous on (0, ∞). Thus Piri and Kumam [59] established the generalization of result of Wordowski [79] using the conditions F1, F2 and F3 . Example 5. [59] Let F1 (α ) = − α1 , Then F1 , F2 , F3 , F4 ∈ Δ .

F2 (α ) = − α1 + α ,

F3 (α ) =

1 1−eα

,

F4 (α ) =

1 eα −e−α

.

4.2.2 F-Contraction in G-Metric Spaces and Associated Fixed Point Results In a paper [73] the authors attempted the G-metric version of F-contraction and named it the Generalized F-contraction, which is defined as follows: Definition 8. A mapping T : X → X is said to be a generalized F- contraction in G-metric spaces if F ∈ Δ and there exists τ > 0 such that for x, y ∈ X, [G(Tx, T2 x, Ty) > 0 ⇒ τ + F(G(Tx, T2 x, Ty)) ≤ F(G(x, Tx, y))].

(4.2)

Example 6. Let (X, G) be a G-metric space. Define F : R+ → R by F(α ) = − α1 for α > 0. Then clearly F ∈ Δ . Each mapping T : X → X satisfying (4.2) is a generalized F-contraction such that G(x, Tx, y) , G(Tx, T2 x, Ty) ≤ 1 + τ G(x, Tx, y) for all x, y ∈ X and G(Tx, T2 x, Ty) > 0. Example 7. Let (X, G) be a G-metric space. Consider F(α ) = − α1 + α for α > 0. Then obviously F ∈ Δ . In this case each mapping T : X → X satisfying (4.2) is a generalized F-contraction such that

τ+ or

−1 −1 + G(Tx, T2 x, Ty) ≤ + G(x, Tx, y) 2 G(Tx, T x, Ty) G(x, Tx, y)

τ G(Tx, T2 x, Ty) + [G(Tx, T2 x, Ty)]2 − 1 [G(x, Tx, y)]2 − 1 ≤ , G(Tx, T2 x, Ty) G(x, Tx, y)

for all x, y ∈ X and G(Tx, T2 x, Ty) > 0.

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Fixed point result: The very first result concerning generalized F-contraction in G-metric space runs as follows. Theorem 2. Let (X, G) be a G-complete metric space and T : X → X be a generalized F-contraction. Then T has a unique fixed point x∗ ∈ X and for every x0 ∈ X, the sequence {Tn x0 }∞n=1 converges to x. Proof. Take an arbitrary x0 ∈ X and construct a sequence {xn }∞n=1 of points in X in the following manner. x1 = Tx0 , x2 = Tx1 = T2 x0 , . . . , xn+1 = Txn = Tn+1 x0 , for all n ∈ N. Notice that if xn0 = xn0 +1 , for some n0 ∈ N. Then obviously T has a fixed point. Thus, we assume that xn = xn+1 , for every n ∈ N. Employing inequality (4.2) repeatedly with x = xn−1 , y = xn , one acquires F(G(Txn−1 , T2 xn−1 , Txn )) ≤ F(G(xn−1 , Txn−1 , xn )) − τ = F(G(Txn−2 , T2 xn−2 , Txn−1 )) − τ ≤ F(G(xn−2 , Txn−2 , xn−1 )) − τ − τ = F(G(Txn−3 , T2 xn−3 , Txn−2 )) − 2τ ≤ F(G(xn−3 , Txn−3 , xn−2 )) − 3τ .. . Frequent application yields F(G(Txn−1 , T2 xn−1 , Txn )) ≤ F(G(x0 , Tx0 , x1 )) − nτ . Which on making n → ∞, reduces to lim F(G(Txn−1 , T2 xn−1 , Txn )) = −∞

n→∞ 

which together with (F 2) and Lemma (4), gives lim F(G(Txn−1 , T2 xn−1 , Txn )) = 0

n→∞

or lim G(xn , xn+1 , xn+1 ) = 0.

n→∞

(4.3)

Next, we assert that {xn }∞n=1 is a G-Cauchy sequence. To the contrary, assume that there exists ε > 0, and sequences {xn(k) } and {xm(k) } of natural numbers, such that G(xm(k) , Txm(k) , xn(k) ) ≥ ε

(4.4)

with n(k) ≥ m(k) > k. Now corresponding to m(k), n(k) can be selected in such a manner that it is the smallest integer with n(k) > m(k) satisfying (4.4). Consequently, we have G(xm(k) , Txm(k) , xn(k)−1 ) < ε . (4.5)

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On using Lemma (1) and property (G5), one gets

ε ≤G(xm(k) , Txm(k) , xn(k) ) ≤G(xn(k) , xn(k)−1 , xn(k)−1 ) + G(xn(k)−1 , Txm(k) , xm(k) ) ≤G(xn(k)−1 , Txm(k) , xm(k) ) + 2G(xn(k)−1 , xn(k) , xn(k) ) ≤G(xm(k) , Txm(k) , xn(k)−1 ) + 2Sn(k)−1 . Thus we obtain

ε ≤ ε + 2Sn(k)−1 .

(4.6)

Where Sn(k)−1 = G(xn(k)−1 , xn(k) , xn(k) ). Letting k → ∞ in 4.6, and utilizing (4.3), we arrive at lim G(xm(k) , Txm(k) , xn(k) ) = ε .

n→∞

(4.7)

Consider the following, and on utilizing Lemma(1) and (G5) G(xm(k) , Txm(k) , xn(k) ) ≤G(xm(k) , xm(k)−1 , xm(k)−1 ) + G(xm(k)−1 , Txm(k) , xn(k) ) =G(xm(k) , xm(k)−1 , xm(k)−1 ) + G(xn(k) , xm(k)−1 , Txm(k) ) ≤G(xm(k) , xm(k)−1 , xm(k)−1 ) + G(xn(k) , xn(k)−1 , xn(k)−1 )+ G(xn(k)−1 , xm(k)−1 , Txm(k) ) ≤2G(xm(k)−1 , xm(k) , xm(k) ) + 2G(xn(k)−1 , xn(k) , xn(k) )+ G(xn(k)−1 , xm(k)−1 , Txm(k) ) ≤ 2Sm(k)−1 + 2Sn(k)−1 + G(xn(k)−1 , xm(k)−1 , Txm(k) ).

(4.8)

Which on making k → ∞, reduces to

ε ≤ lim G(xn(k)−1 , xm(k)−1 , Txm(k) ). n→∞

(4.9)

With a similar approach to the above, one can get G(xn(k)−1 , xm(k)−1 , Txm(k) ) ≤ Sn(k)−1 + Sm(k)−1 + G(xm(k) , Txm(k) , xm(k) ).

(4.10)

Which on letting k → ∞, gives rise to lim G(xn(k)−1 , xm(k)−1 , Txm(k) ≤ ε .

n→∞

(4.11)

Combining (4.9) and (4.11), one obtains lim G(xn(k)−1 , xm(k)−1 , Txm(k) ) = ε .

n→∞

(4.12)

Arguing as above, this leads to

and

G(xn(k)−1 , Txm(k)−1 , Txm(k) ) ≤ 2Sm(k) + G(xm(k)−1 , Txm(k)−1 , xn(k)−1 )

(4.13)

G(xm(k)−1 , Txm(k)−1 , xn(k)−1 ) ≤ Sm(k)−1 + Sm(k) + G(xm(k) , Txm(k) , xn(k)−1 ).

(4.14)

Taking limit n → ∞ in (4.13) and (4.14) and utilizing (4.12), one gets lim G(xm(k)−1 , Txm(k)−1 , xn(k)−1 ) = ε .

n→∞

(4.15)

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Employing (4.2) with x = xm(k)−1 and y = xn(k)−1 , we have

τ + F(G(Txm(k)−1 , T2 xm(k)−1 , Txn(k)−1 )) ≤ F(G(xm(k)−1 , Txm(k)−1 , xn(k)−1 )) ⇒ τ + F(G(xm(k) , Txm(k) , xn(k) )) ≤ F(G(xm(k)−1 , Txm(k)−1 , xn(k)−1 )). 

Letting k → ∞ and using (F 3), (4.7) and (4.15), we obtain

τ + F(ε ) ≤ F(ε ). This contradiction demonstrates that {xn }∞n=1 is a Cauchy sequence. Now by the completeness of G-metric space (X, G), {xn }∞n=1 converges to some point x∗ in X. Application of continuity of T gives G(x∗ , Tx∗ , x∗ ) = lim G(xn , Txn , xn+1 ) n→∞

= lim G(xn , xn+1 , xn+1 ) n→∞

= G(x∗ , x∗ , x∗ ) = 0. This amounts to say that

Tx∗ = x∗ .

Therefore x∗ is a fixed point of T. Now to show that T has a unique fixed point. In fact if x∗ , y∗ ∈ X are two distinct fixed points of T, that is, Tx∗ = x∗ = y∗ = Ty∗ , then

τ + F(G(Tx∗ , T2 x∗ , Ty∗ )) ≤ F(G(x∗ , Tx∗ , y∗ )) ⇒τ + F(G(x∗ , x∗ , y∗ )) ≤ F(G(x∗ , x∗ , y∗ )).

(4.16)

This is a contradiction. Then we must have x∗ = y∗ . Hence T has a unique fixed point. The following example substantiates the validity of the hypothesis of Theorem 2. Example 8. Consider the sequence S1 = 1 × 2, S2 = 1 × 2 + 3 × 4, S3 = 1 × 2 + 3 × 4 + 5 × 6, .. .

= Sn = 1 × 2 + 3 × 4+, · · · , +(2n − 1)2n 

n(n+1)(4n−1) . 3

0, if and only if x = y = z max{x, y, z}, otherwise. Then (X, G) is complete G-metric space. Define the mapping T : X → X by

Let X = {Sn : n ∈ N} and G(x, y, z) =

T(S1 ) = S1 , T(Sn ) = Sn−1 , for all n > 1. Now consider the mapping F(α ) = −1 + α . Clearly F ∈ Δ . α We claim that T is an F-contraction with τ = 30 > 0. First, we observe that G(TSn , T2 Sn , TSm ) > 0 ⇔ ((1 = n < 2 < m) ∨ (1 = m < 2 < n) ∨ (1 < n < m) ∨ (1 < m < n)).



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Case I for 1 = n ∧ m > 2. We have G(TS1 , T2 S1 , TSm ) = G(S1 , S1 , Sm−1 ) = Sm−1 , since S1 = Sm−1 and G(S1 , TS1 , Sm ) = G(S1 , S1 , Sm ) = Sm . Since m > 2, so we have −1 −1 < . 1 × 2 + 3 × 4 + . . . + (2m − 3)(2m − 2) 1 × 2 + 3 × 4 + . . . + (2m − 1)(2m) Now consider 30 −

1 Sm−1

+ Sm−1 = 30 −

1 + 1 × 2 + 3 × 4 + . . . + (2m − 3)(2m − 2)

1 × 2 + 3 × 4 + . . . + (2m − 3)(2m − 2) n > 1, we have G(TSn , T2 Sn , TSm ) =G(Sn−1 , TSn−2 , Sm−1 ) = Sm−1 and G(Sn , TSn , Sm ) =G(Sn , Sn−1 , Sm ) = Sm , Sn = Sm .

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similarly to as in Case I, we have

τ−

1 1 + Sm−1 ≤ − + Sm , for τ = 30. Sm−1 Sm

As desired. Case IV For n > m > 1, we have G(TSn , T2 Sn , TSm ) =G(Sn−1 , TSn−2 , Sm−1 ) = Sn−1 and G(Sn , TSn , Sm ) =G(Sn , Sn−1 , Sm ) = Sn . Thus we easily obtain

τ− As required. Therefore

1 Sn−1

+ Sn−1 ≤

−1 + Sn , for τ = 30. Sn

τ + F(G(TSn , T2 Sn , TSm )) ≤ F(G(Sn , TSn , Sm )),

for every n, m ∈ N. Hence T is an F-contractive mapping. Thus all the conditions of Theorem 2 are satisfied and S1 is a fixed point of T and is indeed unique. Another version of F-contraction is also provide by the authors in the same paper [73] named Hardy–Rogers type generalized F-contraction, defined as Definition 9. A mapping T : X → X is said to be a generalized Hardy–Rogers type F-contractive mapping in G-metric space if F ∈ Δ and there exists τ > 0, such that G(Tx, Ty, T2 y) > 0 ⇒ τ + F(G(Tx, Ty, T2 y)) ≤ F(α G(x, y, Ty) + β G(x, Tx, Ty)+

γ G(y, Ty, T2 y) + δ G(y, T2 x, T2 y) + η G(x, Tx, T2 x)).

(4.17)

For all x, y, ∈ X and α , β , γ , δ , η ≥ 0 with α + β + γ + δ + η < 1. Example 9. Let (X, G) be a G-metric space. Define F : R+ → R by F(α ) = − α1 for α > 0. Then clearly F ∈ Δ . Each mapping T : X → X satisfying (4.17) is a Hardy–Rogers type generalized F-contraction such that G(Tx,T2 x, Ty) ≤ F(α G(x, y, Ty) + β G(x, Tx, Ty) + γ G(y, Ty, T2 y) + δ G(y, T2 x, T2 y) + η G(x, Tx, T2 x)) , 1 + F(α G(x, y, Ty) + β G(x, Tx, Ty) + γ G(y, Ty, T2 y) + δ G(y, T2 x, T2 y) + η G(x, Tx, T2 x)) for all x, y ∈ X and G(Tx, T2 x, Ty) > 0. Example 10. Let (X, G) be a G-metric space. Consider F(α ) = ln(α ) + α . Then obviously F ∈ Δ . In this case each mapping T : X → X satisfying (4.17) is a generalized F-contraction such that G(Tx, Ty, T2 y)eG(Tx,Ty,T y)−{α G(x,y,Ty)+β G(x,Tx,Ty)+γ G(y,Ty,T y)+δ G(y,T x,T y)+η G(x,Tx,T x)} ≤ e− τ α G(x, y, Ty) + β G(x, Tx, Ty) + γ G(y, Ty, T2 y) + δ G(y, T2 x, T2 y) + η G(x, Tx, T2 x) 2

for all x, y ∈ X and G(Tx, T2 x, Ty) > 0.

2

2

2

2

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The next theorem is demonstrated for the Hardy-Rogers type generalized F-contractive mappings in G-metric spaces. Theorem 3. Let (X, G) be a G-complete metric space and T : X → X be a Hardy–Rogers type generalized F-contractive mapping, that is, if F ∈ Δ and there exists τ > 0, such that

τ + F(G(Tx, Ty, T2 y)) ≤F(α G(x, y, Ty) + β G(x, Tx, Ty)+ γ G(y, Ty, T2 y) + δ G(y, T2 x, T2 y) + η G(x, Tx, T2 x)).

(4.18)

For all x, y, ∈ X, G(Tx, Ty, T2 y) > 0, where α , β , γ , δ , η ≥ 0 with α + β + γ + δ + η < 1. Then T has a fixed point in X. Furthermore, if α + 2β + δ ≤ 1, then the fixed point of T is unique. Next, we furnish an illustrative example which demonstrates the validity of the hypotheses and degree of generality of Theorem 3. Example 11. Consider the subsequent sequence {Sn }n∈N as S1 = 2.1, S2 = 2.1 + 22 .2, S3 = 2.1 + 22 .2 + 23 .3, .. .

.n = (n − 1)2n+1 + 2. Sn = 2.1 + 22 .2 + 23 .3 + 24 .4 + · · · + 2n 0, if and only if x = y = z Let X = {Sn : n ∈ X} and G(x, y, z) = max{x, y, z}, otherwise. Then (X, G) is a complete G-metric space. Define the mapping T : X → X by T(S1 ) = S1 and T(Sn ) = Sn−1 for every n > 1. Now we claim that T is an F-contraction with F(α ) = ln α + α . Clearly F ∈ δ . First of all, we observe that G(TSn , TSm , T2 Sm ) > 0 ⇔ (n = 1 ∧ m > 2) ∨ (m = 1 ∧ n > 2) ∨ (1 < n < m) ∨ (1 < m < n). In all the possible cases we will show that G(Tx, Ty, T2 y)eG(Tx,Ty,T y)−{α G(x,y,Ty)+β G(x,Tx,Ty)+γ G(y,Ty,T y)+δ G(y,T x,T y)+η G(x,Tx,T x)} ≤ e−τ (4.19) α G(x, y, Ty) + β G(x, Tx, Ty) + γ G(y, Ty, T2 y) + δ G(y, T2 x, T2 y) + η G(x, Tx, T2 x) 2

2

2

2

2

for x = Sn , y = Sm , n, m ∈ N and for τ = 6 > 0 with α = 12 , β , γ , δ , η ≥ 0 such that α + β + γ + δ + η < 1 and α + 2β + δ ≤ 1. Nnotice also that according to the structure of {Sn }, it is clear that Sn = (n − 1)2n + 1 2 = (n − 2)2n + 2 + (2n − 1) = Sn−1 + 2n − 1. Consequently we conclude that Sn−1
2. Then G(Tx, Ty, T2 y) = G(TSn , TSm , T2 Sm ) = Sm−1 , G(x, y, Ty) = G(S1 , Sm , TSm ) = Sm , G(x, Tx, Ty) = G(S1 , TS1 , TSm ) = Sm−1 , G(y, Ty, T2 y) = G(Sm , TSm , T2 Sm ) = Sm , G(y, T2 x, T2 y) = G(Sm , T2 S1 , T2 Sm ) = Sm , G(x, Tx, T2 x) = G(S1 , TS1 , T2 S1 ) = 0. Then by utilizing (4.19) and also keeping in mind that Sn−1 < α Sn , for all n ∈ N, we have Sm−1 .eSm−1 −(α Sm +β Sm−1 +γ Sm +δ Sm ) ≤ eSm−1 −(α Sm +β Sm−1 +γ Sm +δ Sm ) α Sm + β Sm−1 + γ Sm + δ Sm < eSm−1 −α Sm , for m > 2 = e−(α Sm −Sm−1 ) < e−6 , since (0.5)Sm − Sm−1 > 6 ( for m > 2). Thus in this case T is a Hardy–Rogers type generalized F contractive mapping with τ = 6. Case II When m = 1, n > 2. G(Tx, Ty, T2 y) = G(TSn , TS1 , T2 S1 ) = Sn−1 , G(x, y, Ty) = G(Sn , S1 , TS1 ) = Sn , G(x, Tx, Ty) = G(Sn , TSn , TS1 ) = Sn , G(y, Ty, T2 y) = G(S1 , TS1 , T2 S1 ) = 0, G(y, T2 x, T2 y) = G(S1 , T2 Sn , T2 S1 ) = 0 or Sn−2 , G(x, Tx, T2 x) = G(Sn , TSn , T2 Sn ) = Sn . Thus from (4.19), we have Sn−1 .eSn−1 −(α Sn +β Sn +δ Sn−2 +η Sn ) < e−(α Sn −Sn−1 ) < e−6 . α Sn + β Sn + δ Sn−2 + η Sn Case III When 1 < n < m. Then consider G(Tx, Ty, T2 y) = G(TSn , TSm , T2 Sm ) = Sm−1 , G(x, y, Ty) = G(Sn , Sm , TSm ) = Sm , G(x, Tx, Ty) = G(Sn , TSn , TSm ) = Sm−1 , G(y, Ty, T2 y) = G(Sm , TSm , T2 Sm ) = Sm , G(y, T2 x, T2 y) = G(Sm , T2 Sn , T2 Sm ) = Sm , G(x, Tx, T2 x) = G(Sn , TSn , T2 Sn ) = Sn .

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As in Case I, we get the desired result. Case IV When 1 < m < n. Then G(Tx, Ty, T2 y) = G(TSn , TSm , T2 Sm ) = Sn−1 , G(x, y, Ty) = G(Sn , Sm , TSm ) = Sn , G(x, Tx, Ty) = G(Sn , TSn , TSm ) = Sn , G(y, Ty, T2 y) = G(Sm , TSm , T2 Sm ) = Sm , G(y, T2 x, T2 y) = G(Sm , T2 Sn , T2 Sm ) = Sm or Sn−2 , G(x, Tx, T2 x) = G(Sn , TSn , T2 Sn ) = Sn . Calculating as in Case II, we get the required result. Thus T is a Hardy–Rogers type F−contraction mapping and S1 is a fixed point which is indeed unique. Corollary 1. Let (X, G) be a G-complete metric space and T be a self mapping on X. Assume that there exist F ∈ Δ and τ > 0 such that

τ + F(G(Tx, Ty, T2 y)) ≤ F(δ G(y, T2 x, T2 y) + η G(x, Tx, T2 x)). For all x, y, ∈ X, G(Tx, Ty, T2 y) > 0, where δ , η ≥ 0 with δ + η < 1. Then T has a unique fixed point in X. Taking β = δ = 0 in Theorem 3, then another corollary is obtained. Corollary 2. Let (X, G) be a G-complete metric space and T be a self mapping on X. Assume that there exists F ∈ Δ and τ > 0 such that

τ + F(G(Tx, Ty, T2 y)) ≤ F(α G(x, y, Ty) + γ G(y, Ty, T2 y) + η G(x, Tx, T2 x)). For all x, y, ∈ X, G(Tx, Ty, T2 y) > 0, where α , γ , η ≥ 0 with α + γ + η < 1. Then T has a unique fixed point in X. In another paper [74] a different version of the F-contraction is established and the allied fixed point result can be utilized in several real world problems in science and engineering. ´ c type generalized F-contraction in G-metric spaces Ciri´ ´ c type generalized F-contractive Definition 10. A mapping T : X → X is said to be a Ciri´ mapping in G-metric space if F ∈ Δ and there exists τ > 0, such that G(Tx, Ty, T2 y) > 0 ⇒ τ + F(G(Tx, Ty, T2 y)) ≤ F(α M(x, y)).

(4.20)

For all x, y, ∈ X, where 0 < α < 1 and 1 M(x, y) = max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), [G(y, Ty, T2 y) + G(y, T2 x, T2 y)]}. 2 Example 12. Let (X, G) be a G-metric space. Define F : R+ → R by F(α ) = − α1 for α > 0. ´ c type generalized Then clearly F ∈ Δ . Each mapping T : X → X satisfying (4.20) is a Ciri´

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F-contraction such that G(Tx, T2 x, Ty) > 0 ⇒ G(Tx,T2 x, Ty) ≤

α max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), 12 [G(y, Ty, T2 y) + G(y, T2 x, T2 y)]} , 1 + τα max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), 12 [G(y, Ty, T2 y) + G(y, T2 x, T2 y)]} for all x, y ∈ X. Example 13. Let (X, G) be a G-metric space. Consider F(α ) = ln(α ) + α . Then obviously ´ c type generalized F ∈ Δ . In this case each mapping T : X → X satisfying (4.20) is a Ciri´ F-contraction such that G(Tx, T2 x, Ty) > 0 ⇒ G(Tx, Ty, T2 y)eG(Tx,Ty,T y)−{α max{G(x,y,Ty),G(x,Tx,Ty),G(x,Tx,T x), 2 [G(y,Ty,T y)+G(y,T x,T y)]}} ≤ e− τ , α max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), 12 [G(y, Ty, T2 y) + G(y, T2 x, T2 y)]} 2

2

1

2

2

2

for all x, y ∈ X. ´ c type generalized F-contractive mappings in The next theorem is proved for the Ciri´ G-metric spaces. ´ c type generalized Theorem 4. Let (X, G) be a G-complete metric space and T : X → X be a Ciri´ F-contractive mapping that is, if F ∈ Δ and there exists τ > 0, such that G(Tx, Ty, T2 y) > 0 ⇒ τ + F(G(Tx, Ty, T2 y)) ≤ F(α M(x, y)), for all x, y ∈ X.

(4.21)

where M(x, y) = max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), 12 [G(y, Ty, T2 y) + G(y, T2 x, T2 y)]} and α ∈ (0, 1). Then T has a fixed point in X. Moreover, if 2α ≤ 1 then the fixed point of T is unique. Next, an illustrative example is furnished, which demonstrates the validity of the hypotheses and degree of generality of Theorem 4. Example 14. Consider the sequence {Sn }n∈N as S1 = 3.1, S2 = 3.1 + 32 .3, S3 = 3.1 + 32 .3 + 33 .5, .. .

Sn = 3.1 + 32 .3 + 33 .5 + · · · + 3n .(2n − 1) = (n − 1)3n+1 + 3. Let X = {Sn : n ∈ X} and  0, if and only if x=y=z G(x, y, z) = max{x, y, z}, otherwise. Then (X, G) is a complete G-metric space. Define the mapping T : X → X by T(S1 ) = S1 and T(Sn ) = Sn−1 for every n > 1. ´ c type generalized F-contraction in the framework of G-metric Now we claim that T is a Ciri´

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spaces with F(α ) = ln α + α , then clearly F ∈ Δ . We notice the following: G(TSn , TSm , T2 Sm ) > 0 ⇔ (n = 1 ∧ m > 2) ∨ (m = 1 ∧ n > 2) ∨ (1 < n < m) ∨ (1 < m < n). For all the possible cases, we will show that G(Tx, Ty, T2 y)eG(Tx,Ty,T

α

2 y)−α

max{G(x,y,Ty),G(x,Tx,Ty),G(x,Tx,T2 x),

G(y,Ty,T2 y)+G(y,T2 x,T2 y) } 2

2 2 x,T 2 y) max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), G(y,Ty,T y)+G(y,T } 2

≤ e− τ ,

for x = Sn , y = Sm , n, m ∈ N and for τ = 24 > 0 with α = 13 . Clearly 2α ≤ 1. In view of the structure of {Sn }, one can easily notice that Sn = (n − 1)3n + 1 3 = (n − 2)3n + 3 + (3n − 2) = Sn−1 + 3n − 2. This yields Sn−1
2. Then G(Tx, Ty, T2 y) = G(TSn , TSm , T2 Sm ) = Sm−1 , G(x, y, Ty) = G(S1 , Sm , TSm ) = Sm , G(x, Tx, Ty) = G(S1 , TS1 , TSm ) = Sm−1 , G(x, Tx, T2 x) = G(S1 , TS1 , T2 S1 ) = 0, G(y, Ty, T2 y) = G(Sm , TSm , T2 Sm ) = Sm , G(y, T2 x, T2 y) = G(Sm , T2 S1 , T2 Sm ) = Sm . Utilizing (4.22) and the fact that Sn−1 < α Sn , for every n > 1 with α = 13 , we have Sm−1 .eSm−1 −α Sm 1 ≤ eSm−1 − 3 Sm α Sm 1

= e−( 3 Sm −Sm−1 ) < e−24 , since

1 Sm − Sm−1 > 24 ( for m > 2). 3

´ c type generalized F-contractive mapping with τ = 24. Thus in this case T is a Ciri´

(4.22)

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Case II When m = 1, n > 2. G(Tx, Ty, T2 y) = G(TSn , TS1 , T2 S1 ) = Sn−1 , G(x, y, Ty) = G(Sn , S1 , TS1 ) = Sn , G(x, Tx, Ty) = G(Sn , TSn , TS1 ) = Sn , G(x, Tx, T2 x) = G(Sn , TSn , T2 Sn ) = Sn , G(y, Ty, T2 y) = G(S1 , TS1 , T2 S1 ) = 0, G(y, T2 x, T2 y) = G(S1 , T2 Sn , T2 S1 ) = 0 or Sn−2 . Thus from (4.22), we have Sn−1 .eSn−1 −α Sn 1 ≤ eSn−1 − 3 Sn α Sn 1

= e−( 3 Sn −Sn−1 ) < e−24 , since

1 Sn − Sn−1 > 24 ( for n > 2). 3

Thus we get the desired result. Case III When 1 < n < m. Then consider G(Tx, Ty, T2 y) = G(TSn , TSm , T2 Sm ) = Sm−1 , G(x, y, Ty) = G(Sn , Sm , TSm ) = Sm , G(x, Tx, Ty) = G(Sn , TSn , TSm ) = Sm−1 , G(x, Tx, T2 x) = G(Sn , TSn , T2 Sn ) = Sn , G(y, Ty, T2 y) = G(Sm , TSm , T2 Sm ) = Sm , G(y, T2 x, T2 y) = G(Sm , T2 Sn , T2 Sm ) = Sm . Using (4.22), we have Sm−1 .eSm−1 −α Sm 1 ≤ eSm−1 − 3 Sm α Sm 1

= e−( 3 Sm −Sm−1 ) < e−24 , since

1 Sm − Sm−1 > 24 ( for m > 2). 3

´ c type generalized F contractive mapping with τ = 24. It follows that T is a Ciri´ Case IV When 1 < m < n. Then G(Tx, Ty, T2 y) = G(TSn , TSm , T2 Sm ) = Sn−1 , G(x, y, Ty) = G(Sn , Sm , TSm ) = Sn , G(x, Tx, Ty) = G(Sn , TSn , TSm ) = Sn , G(x, Tx, T2 x) = G(Sn , TSn , T2 Sn ) = Sn G(y, Ty, T2 y) = G(Sm , TSm , T2 Sm ) = Sm , G(y, T2 x, T2 y) = G(Sm , T2 Sn , T2 Sm ) = Sm or Sn−2 .

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Employing (4.22), one gets Sn−1 .eSn−1 −α Sn 1 ≤ eSn−1 − 3 Sn α Sn 1

= e−( 3 Sn −Sn−1 ) < e−24 , since

1 Sn − Sn−1 > 24 ( for n > 2). 3

As required. ´ c type F-contraction mapping and S1 is a fixed point which is indeed unique. Thus T is a Ciri´ Utilizing (F1) gives the resulting subsequent corollary, which is new in case of G-metric spaces. Corollary 3. Let (X, G) be a G-complete metric space and T be a self mapping on X such that G(Tx, Ty, T2 y) ≤ α max{G(x, y, Ty), G(x, Tx, Ty), 1 G(x, Tx, T2 x), [G(y, Ty, T2 y) + G(y, T2 x, T2 y)])}. 2

(4.23)

For all x, y, ∈ X and α ∈ (0, 1). Then T has a fixed point. Moreover if 2α ≤ 1 then the fixed point is unique. With a view to demonstrating the validity of Corollary (3), the following example is adopted. Example 15. Let X = [0, ∞) and

 0, if and only if G(x, y, z) = max{x, y, z},

x=y=z otherwise.

Then (X, G) is a complete G-metric space. Define a self mapping T : X → X by Tx = x3 . In view of verification of condition (4.23), subsequent terms are evaluated and accordingly various possible cases are discussed with α = 25 . 1 max{x, y}, 3 G(x, y, Ty) = max{x, y}, y G(x, Tx, Ty) = max{x, }, 3 G(x, Tx, T2 x) = x,

G(Tx, Ty, T2 y) =

x 1 1 [G(y, Ty, T2 y) + G(y, T2 x, T2 y)] = [y + max{y, }]. 2 2 9 Case I When y3 ≤ x ≤ y, immediately we have G(Tx, Ty, T2 y) = y3 and max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), G(y,Ty,T Clearly

2 y)+G(y,T 2 x,T 2 y)

2

} = y.

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1 3

y ≤ α y with α = 25 .

Case II When x ≤ y3 , then one can obtain G(Tx, Ty, T2 y) =

y 3

and

max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), G(y,Ty,T

2 y)+G(y,T 2 x,T 2 y)

2

} = y.

Therefore 1 3

y ≤ α y with α = 25 .

Case III When

x 9

≤ y ≤ x, then one can obtain

G(Tx, Ty, T2 y) =

x 3

and

max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), G(y,Ty,T

2 y)+G(y,T 2 x,T 2 y)

2

} = x.

Therefore 1 3

x ≤ α x with α = 25 .

Arguing the same with y ≤ x9 , condition (4.23) remains true. Thus all the conditions of corollary 3 are satisfied and x = 0 is the unique fixed point of T. Remark 1. On choosing suitable values of constants a, b, c, and d in Theorem 3, a multitude of the corollaries can be obtained which comprise new versions of Chatterjea type result [18], Kannan type theorem [36], Reich type result [62] and Hardy–Rogers type theorem [30] in the context of G-metric spaces, e.g., if we set a = b = c = 0 then a Chatterjea[18] type fixed point result in the context of G metric space is obtained. 4.2.3 Results with Integral Inequalities under Generalized F-Contraction Subsequent fixed point results for integral inequalities are inferred in the setting of Gmetric spaces. Theorem 5. Let (X, G) be a G-complete metric space and T : X → X be a continuous self mapping such that for x, y ∈ X with 

 ⇒ τ +F

G(Tx,Ty,T2 y) 0 G(Tx,Ty,T2 y) 0

ϕ (t)dt > 0    ϕ (t)dt ≤ F α

0

M(x,y)

 ϕ (t)dt ,

where F ∈ Δ , α ∈ (0, 1) and ϕ : [0, ∞] → [0, ∞] is a Lebesgue-integrable mapping satisfying  0

ε

ϕ (t)dt > 0

for ε > 0 and M(x, y) = max{G(x, y, Ty), G(x, Tx, Ty), G(x, Tx, T2 x), 12 [G(y, Ty, T2 y) + G(y, T2 x, T2 y)]}. Then T has a fixed point. Furthermore if 2α ≤ 1 then the fixed point is unique. Proof. Proof of the theorem is an easy consequence of Theorem 4. The following example demonstrates the validity of hypothesis of Theorem (5).



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Example 16. Consider the following sequence {Sn }n∈N as S1 = 4, S2 = 4 + 42 , S3 = 4 + 42 + 43 , .. . Sn = 4 + 42 + 43 + · · · + 4n = 43 (4n − 1). Let X = {Sn : n ∈ X} and

 0, if and only if G(x, y, z) = max{x, y, z},

x=y=z otherwise.

Then (X, G) is a complete G-metric space. Define the mapping T : X → X by T(S1 ) = S1 and T(Sn ) = Sn−1 for every n > 1. ´ c type generalized F-contraction for integral inequality of Now we assert that T is a Ciri´ Theorem 5 in the framework of G-metric spaces with F(α ) = ln α + α , clearly F ∈ Δ and for Lebesgue-integrable function ϕ (t) = 2t. First of all, we examine that 

G(Tx,Ty,T2 y)

0

ϕ (t)dt >0 ⇔ (n = 1 ∧ m > 2) ∨ (m = 1 ∧ n > 2) ∨ (1 < n < m) ∨ (1 < m < n).

For all the possible cases, we claim that  G(Tx,Ty,T2 y) 0

 G(Tx,Ty,T2 y)

ϕ (t)dte 0 α



ϕ (t)dt−α

2 2 x,T2 y)  max{G(x,y,Ty),G(x,Tx,Ty),G(x,Tx,T2 x), G(y,Ty,T y)+G(y,T } 2 0

G(y,Ty,T2 y)+G(y,T2 x,T2 y) max{G(x,y,Ty),G(x,Tx,Ty),G(x,Tx,T2 x), } 2

0

ϕ (t)dt

ϕ (t)dt

for x = Sn , y = Sm , n, m ∈ N and for τ = 50 > 0 with α = 14 . Clearly 2α ≤ 1. In view of the structure of {Sn }, one can conclude that Sn−1
2. Then G(Tx, Ty, T2 y) =G(TSn , TSm , T2 Sm ) = Sm−1 , G(x, y, Ty) =G(S1 , Sm , TSm ) = Sm , G(x, Tx, Ty) =G(S1 , TS1 , TSm ) = Sm−1 , G(x, Tx, T2 x) =G(S1 , TS1 , T2 S1 ) = 0, G(y, Ty, T2 y) =G(Sm , TSm , T2 Sm ) = Sm , G(y, T2 x, T2 y) =G(Sm , T2 S1 , T2 Sm ) = Sm .

≤ e− τ ,

(4.24)

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Then on utilizing (4.24) with α = 14 , one gets {Sm−1 }2 .e{Sm−1 } α Sm 2

2 −α {S }2 m

≤eSm−1

2

− 14 Sm 2

1

=e−( 4 Sm

2

−Sm−1 2 )

50 ( for m > 2). 4

´ c type generalized F-contractive mapping under integral Thus in this case T is a Ciri´ inequality with τ = 50. Applying the routine calculations as done in Example 14, one can easily verify that T satisfies all the conditions of Theorem 5. Whereas S1 is the unique fixed point of T. Theorem 6. Let (X, G) be a G-complete metric space and T : X → X be a continuous self mapping such that for x, y ∈ X with  0

G(Tx,Ty,T2 y)

ϕ (t)dt >0  ⇒τ + F   ≤F a 

G(Tx,Ty,T2 y)

0

G(x,y,Ty) 0

ϕ (t)dt + b

G(y,Ty,T2 y)+G(y,T2 x,T2 y) 2

d

ϕ (t)dt

0





G(x,Tx,Ty)

0

ϕ (t)dt + c

 0

G(x,Tx,T2 x)

ϕ (t)dt+

 ϕ (t)dt ,

for all x, y ∈ X. Where F ∈ Δ , a, b, c, d ≥ 0 with a + b + c + 2d < 1 and ϕ : [0, ∞] → [0, ∞] is a Lebesgue-integrable mapping satisfying 

0

ε

ϕ (t)dt > 0

for ε > 0. Then T has a fixed point. 4.2.4 G − l Cyclic F-Contraction in G-Metric Spaces After the employment of F-contraction in G-metric spaces, we present some advancement of this context named G − l cyclic F-contraction which is utilized for the existence of fixed point of such mappings. Let Φ be the set of functions φ : [0, ∞) → [0, ∞) such that (i)

φ is upper semi-continuous (i.e., for any sequence {tn } in [0, ∞) such that tn → t as n → ∞, we have lim sup φ (tn ) ≤ φ (t)); n→∞

(ii) φ (t) < t for each t > 0. Let Ψ denote the set of all continuous functions ϕ : (0, ∞) → (0, ∞). Definition 11. [75] Let (X, G) be a G-metric space. Let m be a positive integer, A1 , A2 , . . . , Am be non empty closed subsets of X and Y = ∪m i=1 Ai . An operator T : Y → Y is called a G − l cyclic F-contraction in G-metric spaces if

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(i) ∪m i=1 Ai is cyclic representation of Y with respect to T. (ii) There exist ϕ ∈ Ψ , φ ∈ Φ and F ∈ Δ , such that G(Tx, Tl+1 x, Ty) > 0 ⇒ ϕ (G(x, Tl x, y)) + F(G(Tx, Tl+1 x, Ty)) ≤ F(φ (G(x, Tl x, y))),

(4.25)

for all x ∈ Ai , y ∈ Ai+l , i = 1, 2, . . . , m, l ∈ N (with Am+1 = A1 , Am+2 = A2 Am+3 = A3 · · · ). Remark 2. If we set ϕ (G(x, Tx, y)) = τ > 0, l = 1 and utilizing the fact that φ (t) < t then Condition (4.25) reduces to a cyclic version of Wardowski’s F- contraction [79] in the context of G-metric spaces, as follows G(Tx, T2 x, Ty) > 0 ⇒ τ + F(G(Tx, T2 x, Ty)) ≤ F(G(x, Tx, y)).

(4.26)

From F1 and (4.26), it is easy to conclude that F-contraction T is a cyclic contractive mapping in the setting of G-metric spaces. The following fixed point result is presented for G − l cyclic F-contractive mapping. Theorem 7. [75] Let (X, G) be a G-complete metric space. Let T : Y → Y be a continuous G − l cyclic F-contraction on X. Then T has a unique fixed point. Moreover, the fixed point of T belongs to ∩m i=1 Ai . Proof. Let x0 ∈ A1 then construct the sequence {xn } by Txn = xn+1 for n ∈ N ∪ {0}. Since T is cyclic, x0 ∈ A1 , x1 = Tx0 ∈ A2 , x2 = Tx1 ∈ A3 , . . . , and so on. Notice that if xn0 = xn0 +l for some n0 ∈ N ∪ {0} and l ∈ N, then obviously T has a fixed point and the proof is completed. Thus we assume that xn = xn+l for every n ∈ N ∪ {0}, l ∈ N. Therefore we have G(xn , xn+l , xn+l ) > 0. Utilizing the fact that Y = ∪m i=1 Ai then for each n ∈ N ∪ {0}, there exist in ∈ {1, 2, . . . , m} such that xn ∈ Ain , xn+1 = Txn ∈ T(Ain ) ⊆ Ain +1 , . . . and with a similar approach one can find that xn ∈ Ain +l . Then employing condition (4.25) with x = xn−1 , and y = xn+l−1 , one acquires F(G(xn , xn+l , xn+l )) = F(G(Txn−1 , Tl+1 xn−1 , Txn+l−1 )) ≤F(φ (G(xn−1 , Tl xn−1 , xn+l−1 ))) − ϕ ((xn−1 , Tl xn−1 , xn+l−1 )) 0 for which we can find two sub-sequences {xn(k) } and {xm(k) } of {xn } such that G(xm(k) , xm(k)+l , xn(k)+l ) = G(xm(k) , Tl xm(k) , xn(k)+l ) ≥ ε with n(k) ≥ m(k) > k.

(4.30)

Now corresponding to m(k), we can select n(k) in such a manner that it is the smallest integer with n(k) > m(k) satisfying (4.30). Consequently, we have (4.31) G(xm(k) , Tl xm(k) , xn(k)+l−1 ) < ε . On using Lemma (1) and property (G5), one gets

ε ≤G(xm(k) , Tl xm(k) , xn(k)+l ) ≤G(xn(k)+l , xn(k)+l−1 , xn(k)+l−1 ) + G(xn(k)+l−1 , Tl xm(k) , xm(k) ) ≤G(xn(k)+l , xn(k) , xn(k) ) + G(xn(k) , xn(k)+l−1 , xn(k)+l−1 ) + G(xn(k)+l−1 , Tl xm(k) , xm(k) ) ≤G(xn(k)+l−1 , Tl xm(k) , xm(k) ) + 2G(xn(k) , xn(k)+l , xn(k)+l ) + G(xn(k) , xn(k)+l−1 , xn(k)+l−1 ) ≤G(xm(k) , Tl xm(k) , xn(k)+l−1 ) + Sn(k)+l−1 + 2Sn(k)+l . Thus we obtain

ε ≤ G(xm(k) , Tl xm(k) , xn(k)+l ) ≤ ε + Sn(k)+l−1 + 2Sn(k)+l ,

(4.32)

where Sn(k)+l = G(xn(k) , xn(k)+l , xn(k)+l ). Letting k → ∞ in (4.32) and utilizing (4.29), we arrive at lim G(xm(k) , Tl xm(k) , xn(k)+l ) = ε .

n→∞

(4.33)

Consider the following and on utilizing Lemma (1) and (G5) G(xm(k) , Tl xm(k) , xn(k)+l ) ≤G(xm(k) , xm(k)−1 , xm(k)−1 ) + G(xm(k)−1 , Tl xm(k) , xn(k)+l ) =G(xm(k) , xm(k)−1 , xm(k)−1 ) + G(xn(k)+l , xm(k)−1 , Tl xm(k) ) ≤2G(xm(k)−1 , xm(k) , xm(k) ) + G(xn(k)+l , xn(k) , xn(k) )+ G(xn(k) , xn(k)+l−1 , xn(k)+l−1 ) + G(xn(k)+l−1 , xm(k)−1 , Tl xm(k) ) ≤2G(xm(k)−1 , xm(k) , xm(k) ) + 2G(xn(k) , xn(k)+l , xn(k)+l )+ G(xn(k) , xn(k)+l−1 , xn(k)+l−1 ) + G(xn(k)+l−1 , xm(k)−1 , Tl xm(k) ) ≤ 2Sm(k) + 2Sn(k)+l + Sn(k)+l−1 + G(xn(k)+l−1 , xm(k)−1 , Tl xm(k) ). Which on making k → ∞, reduces to

ε ≤ lim G(xn(k)+l−1 , xm(k)−1 , Tl xm(k) ). n→∞

(4.34)

With a similar approach to the above, one can get G(xn(k)+l−1 , xm(k)−1 , Tl xm(k) ) ≤ 2Sn(k)+l−1 + Sn(k)+l + Sm(k)−1 + G(xm(k) , Tl xm(k) , xn(k)+l ).

(4.35)

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Which on letting k → ∞, gives rise to lim G(xn(k)+l−1 , xm(k)−1 , Tl xm(k) ) ≤ ε .

n→∞

(4.36)

Combining (4.34) and (4.36), one obtains lim G(xn(k)+l−1 , xm(k)−1 , Tl xm(k) ) = ε .

n→∞

(4.37)

Arguing as above, it follows that G(xn(k)+l−1 , xm(k)−1 , Tl xm(k) ) ≤ 2Sm(k)+l + Sm(k)+l−1 + G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 ). Letting n → ∞ and utilizing (4.37), we have

ε ≤ lim G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 ) n→∞

(4.38)

and also considering the same approach, we have G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 ) ≤ Sm(k)+l+1 + 2Sm(k)+l−1 + G(xm(k)−1 , Tl xm(k) , xn(k)+l−1 ). Which on making n → ∞, gives lim G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 ) ≤ ε .

n→∞

(4.39)

Combining (4.38) and (4.39), one gets lim G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 ) = ε .

n→∞

(4.40)

Employing (4.25) with x = xm(k)−1 and y = xn(k)+l−1 , we have

ψ (G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 )) + F(G(Txm(k)−1 , Tl+1 xm(k)−1 , Txn(k)+l−1 )) ≤F(φ (G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 ))). This implies that F(G(xm(k) ,Tl xm(k) , xn(k)+l )) ≤F(φ (G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 ))) − ϕ (G(xm(k)−1 , Tl xm(k)−1 , xn(k)+l−1 )). Which on making k → ∞, with F3 , (4.33), (4.40) and semi-continuity of φ , reduces to F(ε ) ≤ F(φ (ε )) − ϕ (ε ) < F(ε ) − ϕ (ε ), a contradiction, which demonstrates that {xn } is a G-Cauchy sequence. Since (X, G) is Gcomplete then it is G-convergent to a limit, say w ∈ X. Since x0 ∈ A1 , then the sub-sequence {xm(n−1) }∞n=1 ∈ A1 , the sub-sequence {xm(n−1)+1 }∞n=1 ∈ A2 and continuing in this manner we find that {xmn−1 }∞n=1 ∈ Am . All the m sub-sequences are G-convergent in the G-closed sets Ai and consequently, they all converge to the same limit w ∈ ∩m i=1 Ai . Application of continuity of T gives G(w, Tw, w) = lim G(xn , Txn , xn+1 ) n→∞

= lim G(xn , xn+1 , xn+1 ) n→∞

=G(w, w, w) =0.

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Which amounts to say that

Tw = w.

Therefore w is a fixed point of T. In order to prove the uniqueness of the fixed point suppose u, w ∈ ∩m i=1 Ai be two fixed points of T such that u = w and so G(Tu, Tl+1 u, Tw) >0 ⇒ F(G(u, u, w)) =F(G(Tu, Tl+1 u, Tw)) ≤F(φ G(u, Tl u, w)) − ϕ (G(u, Tl u, w)) =F(φ G(u, u, w)) − ϕ (G(u, u, w)) 0, in Theorem (7), this results in the following corollary which can be demonstrated as a G − l cyclic version of Wardowski’s Theorem [79], in the context of G-metric spaces. Corollary 4. Let (X, G) be a complete G-metric space and {Ai }m i=1 be a family of nonempty G-closed subsets of X with Y = ∪m A . Let T : Y → Y be a map satisfying i i=1 T(Ai ) ⊆ Ai+1 , i = 1, 2, . . . , m, where Am+1 = A1 , Am+2 = A2 . . . . There exists φ ∈ Φ and F ∈ Δ such that G(Tx, Tl+1 x, Ty) > 0 ⇒ τ + F(G(Tx, Tl+1 x, Ty)) ≤ F(φ (G(x, Tl x, y))) for all x ∈ Ai and y ∈ Ai+l . Then T has a unique fixed point in ∩m i=1 Ai . If we set l = 1 in the above corollary, then a cyclic Boyd–Wong type fixed point result through F-contractive mapping in the setting of G-metric spaces is obtained. Corollary 5. Let (X, G) be a complete G-metric space and {Ai }m i=1 be a family of nonempty G-closed subsets of X with Y = ∪m i=1 Ai . Let T : Y → Y be a map satisfying T(Ai ) ⊆ Ai+1 , i = 1, 2, . . . , m, where Am+1 = A1 . There exists φ ∈ Φ and F ∈ Δ such that G(Tx, T2 x, Ty) > 0 ⇒ τ + F(G(Tx, T2 x, Ty)) ≤ F(φ (G(x, Tx, y))) for all x ∈ Ai and y ∈ Ai+1 . Then T has a unique fixed point in ∩m i=1 Ai . Utilizing the fact that φ (t) < t in Corollary 5, a Wardowski type cyclic fixed point theorem in the framework of G-metric spaces is attained. Corollary 6. Let (X, G) be a complete G-metric space and {Ai }m i=1 be a family of nonempty G-closed subsets of X with Y = ∪m i=1 Ai . Let T : Y → Y be a map satisfying T(Ai ) ⊆ Ai+1 , i = 1, 2, . . . , m, where Am+1 = A1 . There exists φ ∈ Φ and F ∈ Δ such that G(Tx, T2 x, Ty) > 0 ⇒ τ + F(G(Tx, T2 x, Ty)) ≤ F((G(x, Tx, y))) for all x ∈ Ai and y ∈ Ai+1 . Then T has a unique fixed point in ∩m i=1 Ai .

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4.3 Applications in Real World Problems Fixed point theory is a fascinating emerging field of the twenty-first century. Its scope of enquiries not only covers the geometric theory of infinite dimensional function spaces and operator-theoretic real-world problems but also widens the range of interdisciplinary fields ranging from engineering to space science, hydromechanics to astrophysics, chemistry to biology, theoretical mechanics to biomechanics and economics to stochastic game theory. The deep-rooted concepts and techniques provide the tools for developing more realistic and accurate models for a variety of phenomena encountered in various applied fields. This gives fixed point theory a rather interdisciplinary character. Today, the more theoretically inclined non-mathematician (engineer, economist, geologist, pharmacologist, biologist or chemist) needs a working knowledge of fixed point theory in order to be able to conduct a complete qualitative analysis of his models. For previous works, one can confer with Cronin [20] and Miranda [47] for general references, and for recent work, refer to Browder [15] and Martin Jr. [46]. For applications to nonlinear integral equations, refer to Krasnoselskii [43] who has dealt with this work in detail in his book. For an exhaustive discussion on positive solutions of operator equations, see Krasnoselskii [44] and Amann [8]. For a different approach to the existence problems in differential equations, see Cesari [16]. Nussbaum [58] and Walther [80] dealt with applications to functional differential equations, and Gustafson [29] has dealt with nodal problems in differential equations. These references have been cited to specify the number of applications of fixed point theory. As far as is concerned with G-metric spaces, acknowledging the aforesaid remarkable initiations, many researchers have established applications of fixed point theorems in the setting of G-metric spaces; some of the noteworthy contributions are mentioned in [40], [57], [72]. In this section, we aim to offer some applications of fixed point theorems to obtain existence theorems for differential, integral and functional equations. Our treatment includes some standard well-known results as well as some recent ones. We have stayed away from widespread discussion of these areas; instead we concentrate on a few important problems arising in real world situation. As usual, in most cases, the differential equations are transformed into equivalent operator equations involving integral operators, and then, appropriate fixed point theorems or degree theoretic methods are invoked to prove the existence of desired solutions by recasting the operator equations into fixed point equations. (1) Application to spring mass system Considering the motion of a spring that is subject to a frictional force (in the case of a horizontal spring) or a damping force (in the case of a vertical spring moving through a fluid; an example is the damping force supplied by a shock absorber in a car or a bicycle). In addition to these, the motion of the spring is affected by an external force. Then such a type of system for critical damped motion is represented by  2 d u + mc du = K(t, u(t)); dt2 dt (4.41) u(0) = 0, u (0) = 0, where K : [0, I] × R+ → R is a continuous function and I > 0. The above problem is equivalent to the integral equation u(t) =

 0

t

G(t, s)K(s, u(s))ds, t ∈ [0, I],

(4.42)

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where G(t, s) is the Green’s function, given by  G(t, s) =

1−eτ (t−s) , τ

0,

0 ≤ s ≤ t ≤ I; 0 ≤ t ≤ s ≤ I.

(4.43)

Where τ > 0 is a constant, calculated in terms of c and m, mentioned in (4.41). Let X = C([0, I], R+ ) be the set of all nonnegative continuous real functions defined on [0, I]. For an arbitrary u ∈ X, we define uτ = sup {|u(t)|}.

(4.44)

t∈[0,I]

Define G : X × X × X → R+ by G(u, v, w) = max{u − vτ , v − wτ , w − uτ },

(4.45)

where uτ is defined by (4.44). Then clearly (X, G) is a complete G-metric space. Consider the self map T : X × X, defined by Tu(t) =



t 0

G(t, s)K(s, u(s))ds, t ∈ [0, I].

(4.46)

Then clearly u∗ is a solution of (4.42), if and only if u∗ is a fixed point of T. Now we prove the following theorem to guarantee the existence of a fixed point of T. Theorem 8. Suppose the following hypotheses hold: (i) K is an increasing function; (ii) there exists τ > 0 such that 

 |K(s, u) − K(s, v)| ≤ τ 2 e−τ a|u − v| + c|u − Tu| + d |v − Tv| + |v − T2 u| , for all s ∈ [0, I], u, v ∈ R+ . Where a ≥ 0, c ≥ 0 and d ≥ 0 such that a + c + 2d < 1. Then the integral equation (4.42) has a solution. Proof. It is already noted that (C([0, I], R+ ), G) is a complete G-metric space, where G(u, v, w) is given by (4.45). From assumption (i), T is increasing. Next, for all u, v ∈ X such that Tu(t) = Tv(t), we have |Tu(t) − Tv(t)| ≤ ≤

=



t

0



t



t

G(t, s)|K(s, u(s)) − K(s, v(s))|ds

G(t, s)τ 2 e−τ a|u(s) − v(s)| + c|u(s) − Tu(s)| 0

+ d |v(s) − Tv(s)| + |v(s) − T2 u(s)| ds τ 2 e−τ a|u(s) − v(s)| + c|u(s) − Tu(s)|+ 0

d |v(s) − Tv(s)| + |v(s) − T2 u(s)| G(t, s)ds

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≤ τ 2 e−τ au − vτ + cu − Tuτ + d v − Tvτ + v − T2 uτ 

t

G(t, s)ds

sup t∈[0,I] 0



= τ 2 e−τ au − vτ + cu − Tuτ + d v − Tvτ + v − T2 uτ   1 + τ t − eτ t sup τ2 t∈[0,I]

≤ e−τ au − vτ + cu − Tuτ + d v − Tvτ + v − T2 uτ . Clearly, supt∈[0,I] (1 + τ t − eτ t ) ≤ 1. This implies that

sup |Tu(t) − Tv(t)| ≤ e−τ au − vτ + cu − Tuτ + d v − Tvτ + v − T2 uτ , t∈[0,I]

or equivalently,

Tu − Tvτ ≤ e−τ au − vτ + cu − Tuτ + d v − Tvτ + v − T2 uτ . Similarly, one can derive that

Tv − T2 vτ ≤ e−τ av − Tvτ + cTu − T2 uτ + d Tv − T2 vτ + T2 v − T2 uτ and



T2 v − Tuτ ≤ e−τ aTv − uτ + cT2 u − uτ + d v − T2 vτ + v − T2 vτ .

(4.47)

(4.48)

(4.49)

Utilizing (4.47), (4.48) and (4.49), one can get max{Tu − Tvτ ,Tv − T2 vτ , T2 − Tuτ } ≤ e−τ a max{u − vτ , v − Tvτ , Tv − uτ }+ c max{u − Tuτ , Tu − T2 uτ , T2 u − uτ }+ d max{v − Tvτ , Tv − T2 vτ , T2 v − vτ }+ d max{v − T2 uτ , T2 u − T2 vτ , T2 v − vτ }.

This leads us to say that

G(Tu, Tv, T2 v) ≤ e−τ aG(u, v, Tv) + cG(u, Tu, T2 u) + d G(v, Tv, T2 v) + G(v, T2 u, T2 v) . Consequently, by passing to logarithms, one can obtain 

 ln(G(Tu, Tv, T2 v)) ≤ ln e−τ aG(u, v, Tv)+cG(u, Tu, T2 u)+d G(v, Tv, T2 v) + G(v, T2 u, T2 v) . Or



τ + ln(G(Tu, Tv, T2 v)) ≤ ln aG(u, v, Tv) + cG(u, Tu, T2 u) + d G(v, Tv, T2 v) + G(v, T2 u, T2 v) .

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Here, we notice that the function F : R+ → R defined by F(α ) = ln(α ), for each α ∈ C([0, I], R+ ) and for τ > 0, is in Δ . Consequently all the conditions of Theorem 3 are satisfied by operator T with a ≥ 0, c ≥ 0, d ≥ 0 and b = 0 such that a + c + 2d < 1. Consequently T has a fixed point which is the solution of integral equation (4.42) and hence the spring mass system has a solution.  Remark 3. : Moreover our Theorem 3 can be utilized to find the solution of the following real time problems: (i) Solution of electrical circuit equations. (ii) Solution of equations generating by the motion of pendulum. (iii) Problems related to simple harmonic motion, etc. (2) Existence and uniqueness of bounded solutions of functional equations in dynamic programming In this section, the existence of solutions for a class of functional equations through generalized F-contraction in G-metric spaces is established. Let U and V be Banach spaces, and R is the field of real numbers. Let X = B(W) denote the set of all bounded real valued function on W. We define G : X × X × X → R+ by G(x, y, z) = max{d(x, y), d(y, z), d(z, x)}, where d(x, y) = supt∈W {|h1 (t) − h2 (t)|}. Then (X, G) is a G-complete metric space. Consider the following functional equation q(x) = sup{g(x, y) + H(x, y, q(ρ (x, y)))}, x ∈ W.

(4.50)

y∈D

Where g : W × D → R and H : W × D × R → R are bounded functions. We consider W and D as the state and the decision spaces, respectively, ρ : W × D → W represents transformation of the process and q(x) represents the optimal return function with initial state x. We also define T : B(W) → B(W) by T(h(x)) = sup{g(x, y) + H(x, y, h(ρ (x, y)))}, for all x ∈ W and h ∈ X.

(4.51)

y∈D

Naturally, if functions g and H are bounded then T is well-defined. Let 1 M(h, k) = max{G(h, k, Tk), G(h, Th, Tk), G(h, Th, T2 h), [G(k, Tk, T2 k) + G(k, T2 h, T2 k)]}. 2 Now, we prove the existence and uniqueness of the solution of the functional equation (4.50). Theorem 9. Let T : X → X be an upper-semi-continuous operator defined by (4.51) and assume that the following conditions are satisfied. (i) H : W × D × R → R and g : W × D × R → R are continuous and bounded, (ii) There exists τ ∈ R+ such that |H(x, y, h(x)) − H(x, y, k(x))| ≤ e−3λ M(h, k), for all h, k, ∈ B(W), where x ∈ W and y ∈ D.

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Then the functional equation 4.50 has a bounded solution. Proof. Let λ be any arbitrary positive number, x ∈ W and h, k ∈ B(W), and we select y1 , y2 ∈ D so that (4.52) T(h(x)) < g(x, y1 ) + H(x, y1 , h(ρ (x, y1 ))) + λ, T(k(x)) < g(x, y2 ) + H(x, y2 , k(ρ (x, y2 ))) + λ.

(4.53)

On the other hand, by the definition of T, we have T(h(x)) ≥ g(x, y2 ) + H(x, y2 , h(ρ (x, y2 ))),

(4.54)

T(k(x)) ≥ g(x, y1 ) + H(x, y1 , k(ρ (x, y1 ))).

(4.55)

Utilizing (4.52) and (4.55) with T(h(x)) = T(k(x)), one can get T(h(x)) − T(k(x)) 0 is taken arbitrarily, then we conclude that G(Th, Tk, T2 k) ≤ e−3τ M(h, k), for all x ∈ W. By passing to logarithms, we have ln(G(Th, Tk, T2 k)) ≤ ln(e−3τ M(h, k)) or

τ + ln((Th, Tk, T2 k)) ≤ ln(e−2τ M(h, k)).

We notice that the function F : R+ → R defined byF(x) = ln(x), for each x ∈ W, is in Δ . This ´ c type generalized F-contraction. Thus, in view amounts to say that the operator T is a Ciri´

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of continuity of T, Theorem 4 applies to the operator T with α = e−2τ < 1, τ > 0. Indeed, T has a fixed point h∗ ∈ B(W); that is, h∗ is a bounded solution of the functional equation  (4.50). Moreover, if τ ≥ 0.35 then this solution is unique. (3) An application to Bessel type boundary value problems In this section, we present the application of fixed point theorem discussed earlier, to establish the existence of a solution of the following Bessel type boundary value problem.  2 t2 ddt2u + t du = K(t, u(t)); dt (4.61) u(0) = u(1) = 0, where k : [0, 1] × R+ → R is a continuous function. The above problem is equivalent to the integral equation u(t) =



1

0

G(t, s)K(s, u(s))ds, t ∈ [0, 1],

where G(t, s) is the Green’s function  G(t, s) =

s (1 − t2 ), 2t t (1 − s2 ), 2s

0 ≤ s < t ≤ 1; 0 ≤ t < s ≤ 1.

(4.62)

(4.63)

Let X = C([0, 1], R+ ) be the set of all nonnegative continuous real functions defined on [0, 1]. For an arbitrary u ∈ X, we define u = sup {|u(t)|}.

(4.64)

t∈[0,1]

Define G : X × X × X → R+ by G(u, v, w) = max{u − v, v − w, w − u},

(4.65)

where u is defined by (4.64). Then clearly (X, G) is a complete G-metric space. Consider the self map T : X × X, defined by Tu(t) =



1 0

G(t, s)K(s, u(s))ds, t ∈ [0, 1].

(4.66)

Then clearly u∗ is a solution of (4.62), if and only if u∗ is a fixed point of T. Now we prove the next theorem to guarantee the existence of a fixed point of T. Theorem 10. Suppose the following hypotheses hold: (i) K is a nonincreasing function, (ii) there exists τ > 0 such that |K(s, u) − K(s, v)| ≤ e−τ |u − v|, for all s ∈ [0, 1] and u, v ∈ R+ . (iii) there exist ξ , η ∈ X such that ξ (t) ≤ η (t) for t ∈ [0, 1] and that Tξ (t) ≤ η (t) and Tη (t) ≥ ξ (t) for t ∈ [0, 1]. Then the integral equation (4.62) has a solution u∗ ∈ X and it belongs to A = {u ∈ X : ξ (t) ≤ u(t) ≤ η (t), .

t ∈ [0, 1]}

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Proof. In order to prove the existence of a fixed point of T, we construct closed subsets A1 and A2 of X as follows.

and

A1 = {u ∈ X : u(t) ≤ η (t),

t ∈ [0, 1]}

A2 = {u ∈ X : ξ (t) ≤ u(t),

t ∈ [0, 1]}.

Let u ∈ A1 , that is

u(s) ≤ η (s),

for all s ∈ [0, 1].

Since G(t, s) ≥ 0 for all t, s ∈ [0, 1], now from hypotheses (i) and (iii), it is deduced that 

1 0

G(t, s)K(s, u(s))ds ≥



1

0

G(t, s)K(s, η (s))ds ≥ ξ (t),

t ∈ [0, 1].

Then we have Tu ∈ A2 . This implies that T(A1 ) ⊆ A2 . With a similar approach to the above, the other inclusion is proved. Hence Y = A1 ∪ A2 is a cyclic representation of Y with respect to T. For u ∈ A1 and v ∈ A2 then clearly Tu(t) = Tv(t), now we have |Tu(t) − Tv(t)| ≤ ≤



1

0



1

0

G(t, s)|K(s, u(s)) − K(s, v(s))|ds G(t, s)e−τ |u(s) − v(s)|ds

−τ

≤ e u − v



1

G(t, s)ds. 0

This implies that |Tu(t) − Tv(t)| ≤ e−τ u − v or equivalently, Tu(t) − Tv(t) ≤ e−τ u − v



1

G(t, s)ds 0



1

G(t, s)ds.

(4.67)

0

Similarly, one can derive that T2 u(t) − Tv(t) ≤ e−τ Tu − v and −τ

Tu(t) − T u(t) ≤ e u − Tu 2



1

G(t, s)ds

(4.68)

G(t, s)ds.

(4.69)

0



1 0

Utilizing (4.67), (4.68) and (4.69), one can get max{Tu(t) − Tv(t),T2 u(t) − Tv(t), Tu(t) − T2 u(t)} ≤ e−τ max{u − v, Tu − v, u − Tu} This leads us to say that G(Tu, T2 u, Tv) ≤ e−τ G(u, Tu, v)



1

G(t, s)ds. 0



1

G(t, s)ds. 0

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It is clear that

 0

1

√ 1 G(t, s)ds = −t log t < , for all t ∈ [0, 1]. 10

Thus, we have G(Tu, T2 u, Tv) ≤ e−τ

G(u, Tu, v) 10

Using logarithms, one can obtain ln(G(Tu, T2 u, Tv)) ≤ ln(e−τ

G(u, Tu, v) ). 10

Or

G(u, Tu, v) ). 10 Here, we notice that the function F : R+ → R defined by F(α ) = ln(α ), for each α ∈ C([0, 1], R) and for τ > 0, is in Δ . Consequently all the conditions of Corollary 5 are satisfied by operator T with φ (t) = 10t . Consequently T has a unique fixed point u∗ ∈ A1 ∪ A2 , that is u∗ ∈ A is the unique solution to integral equation (4.62) and hence the Bessel type boundary value problem has a solution. 

τ + ln(G(Tu, T2 u, Tv)) ≤ ln(

4.4 Best Proximity Point in G-Metric Spaces 4.4.1 Introduction Fixed point theory is an important tool for solving the various equations of the form Tx = x for self-mappings T defined on subsets of metric spaces. On the other hand, consider a nonself mapping T from A to B, where A and B are two nonempty subsets of a metric space. Since T is not a self-mapping, it is improbable that the equation Tx = x has a solution. In this case, therefore, it is of key significance to search for an element x that in some sense is neighbouring to Tx. That is, when the equation Tx = x has no solution, one tries to determine an approximate value of x subject to the condition that the distance between x and Tx is minimum. Best approximation theorems and best proximity point theorems are relevant in this perspective. One of the most interesting theorems is due to Fan [26], called the best approximation theorem. There have been many succeeding generalizations of Fan’s Theorem (see [60], [63], [70], [71], [77], [78] and references therein). Through best approximation theorems, we can guarantee the existence of approximate solutions but such solutions need not acquiesce to the optimal solutions. However, best proximity point theorems provide sufficient conditions that assure the existence of approximate solutions which are optimal as well. Indeed, if there is no exact solution to the fixed point equation Tx = x for a non-self mapping T : A → B, then a best proximity theorem presents sufficient conditions for the existence of an optimal approximate solution x, called a best proximity point of the mapping T, satisfying the condition that d(x, Tx) = d(A, B). A best proximity point theorem for non-self proximal contractions has been investigated in [67]. In the case of cyclic contractive mapping T : A ∪ B → A ∪ B, a point x ∈ A ∪ B is called the best proximity point if d(x, Tx) = d(A, B). Notice that a best proximity point x is a fixed point of T whenever A ∩ B = 0. / Thus it generalizes the notion of a fixed point in the case

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when A ∩ B = 0. / Further [7], [22], [37], [33], [32], and [9] examined several variants of contractions for the existence of a best proximity point. 4.4.2 Best Proximity Point in G-Metric Spaces Recently, Hussain et al. [31] introduced the best proximity point concept in the framework of G-metric spaces and established best proximity point theorems for a new class of proximal contraction mappings. Let (X, G) be a G-metric space. Suppose that A and B are nonempty subsets of a G-metric space (X, G). Then   A0 = x ∈ A : dG (x, y) = dG (A, B) for some y ∈ B , (4.70)   B0 = y ∈ B : dG (x, y) = dG (A, B) for some x ∈ A ,   where dG (A, B) = inf dG (x, y) : x ∈ A, y ∈ B . Definition 12. If T : A∪B → A∪B is a cyclic contractive mapping in a G-metric space (X, G), a point x ∈ A∪B is called the best proximity point of T if dG (x, Tx) = dG (A, B) or equivalently we can say G(x, Tx, Tx) = G(A, B, B) and G(x, x, Tx) = G(A, A, B), where   G(A, B, B) = inf G(a, b, b) : a ∈ A, b ∈ B and

  G(A, A, B) = inf G(a, a, b) : a ∈ A, b ∈ B .

Definition 13. [31] Let (X, G) be a G-metric space and let A and B be two nonempty subsets of X. Then B is said to be approximatively compact with respect to A if every sequence {yn } in B, satisfying the condition dG (x, yn ) → dG (x, B) for some x in A, has a G-convergent subsequence. Utilizing the aforementioned notion, finally Hussain et al. introduced the first best proximity point theorem in the context of G-metric spaces. Definition 14. [31] Let A and B be two nonempty subsets of a G-metric space (X, G). Let T : A → B be a non-self-mapping. We say T is a G − φ − ψ proximal contractive mapping if, for x, y, u, u∗ , v ∈ A, dG (u, Tx) = dG (A, B) dG (u∗ , Tu) = dG (A, B) dG (v, Ty) = dG (A, B) ⇓ ∗

ψ (G(u, u , v)) ≤ ψ (G(x, u, y)) − φ (G(x, u, y)) holds where ψ ∈ Ψ and φ ∈ Φ . Theorem 11. [31] Let A , B be two nonempty subsets of a G-metric space (X, G) such that (A, G) is a complete G-metric space, A0 is nonempty, and B is approximatively compact with respect to A. Assume that T : A → B is a G − φ − ψ proximal contractive mapping such that T(A0 ) ⊆ B0 . Then T has a unique best proximity point; that is, there exists a unique z ∈ A such that dG (z, TzTz) = dG (A, B) .

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After the initiation of this new concept, D. Singh et al. [76] introduced a new class of proximal contraction for getting the best proximity point in the context of G-metric spaces. First of all, generalized proximal cyclic weak φ -contractive mapping is defined in the framework of G-metric spaces. Definition 15. Let {Ai }li=1 be a family of nonempty subsets of G-metric space (X, G) such  that Y = li=1 Ai . Let T : Y → Y be a mapping satisfying T(Ai ) ⊆ Ai+1 , i = 1, 2, . . . , l, where Al+1 = A1 . The mapping T is said to be generalized proximal cyclic weak φ -contractive, if for x, u, u∗ ∈ Ai and y, v ∈ Ai+1 , dG (u∗ , Tx) = dG (Ai , Ai+1 ), dG (u, Tu∗ ) = dG (Ai , Ai+1 ), and dG (v, Ty) = dG (Ai , Ai+1 ), then we have

G(u∗ , u, v) ≤ M(x, v, y) − φ (M(x, v, y)),

(4.71)

where M(x, v, y) = min{G(x, v, y), G(x, Tx, Tx), G(y, Ty, Ty)} and φ ∈ Φ , the set of continuous function φ : [0, ∞) → [0, ∞) with φ (0) = 0 and φ (t) > 0 for t > 0 and i = 1, 2, . . . , l. Example 17. For X = R, let G : X × X × X → R+ be defined by G(x, y, z) =

  1 max |x − y| , |y − z| , |z − x| . 2

Then (X, G) is a G-metric space and dG (x, y) = |x − y|. For A = {1, 2, 3}, B = {0, 4, 5}, let T : A ∪ B → A ∪ B be defined by T(1) = 4, T(2) = 5, T(3) = 4, T(0) = 3, T(4) = T(5) = 1. Then, clearly T(A) ⊆ B and T(B) ⊆ A, dG (A, B) = 1 and T is a generalized proximal cyclic weak φ -contraction for u = 3, u∗ = 3, x = 1 ∈ A and v = 4, y = 0 ∈ B. In fact, if dG (u∗ , Tx) = dG (A, B), dG (u, Tu∗ ) = dG (A, B), and dG (v, Ty) = dG (A, B), then we have

G(u∗ , u, v) ≤ M(x, v, y) − φ (M(x, v, y)),

where M(x, v, y) = min{G(x, v, y), G(x, Tx, Tx), G(y, Ty, Ty)} and φ (t) = 2t . The following theorem is proved for generalized proximal cyclic weak φ -contractive mappings. Theorem 12. Let (X, G) be a G-metric space and {Ai }m i=1 be a family of disjoint nonempty subsets  of X with Y = li=1 Ai such that Y is a G-complete subspace of X. Let T : Y → Y be a generalized proximal cyclic weak φ -contractive mapping, that is, T : Y → Y be a mapping satisfying T(Ai ) ⊆ Ai+1 , i = 1, 2, . . . , l, where Al+1 = A1 and if for x, u, u∗ ∈ Ai and y, v ∈ Ai+1 , dG (u∗ , Tx) = dG (Ai , Ai+1 ), dG (u, Tu∗ ) = dG (Ai , Ai+1 ), and dG (v, Ty) = dG (Ai , Ai+1 ),

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then

G(u∗ , u, v) ≤ M(x, v, y) − φ (M(x, v, y)),

(4.72)

where M(x, v, y) = min{G(x, v, y), G(x, Tx, Tx), G(y, Ty, Ty)}, φ ∈ Φ and i = 1, 2, . . . , l. Suppose that Ai0 is nonempty such that T(Ai0 ) ⊆ A(i+1)0 and Ai+1 is approximatively compact with respect to Ai , for i = 1, 2, . . . , l. Then T has a best proximity point in Y. Proof. First of all, we construct a sequence of Picard iteration as usual. Define the sequence {xn } as xn = Txn−1 , n= 1,2,3,. . . . It is given that each Ai0 is nonempty so then we pick an arbitrary point x0 ∈ A10 ⊆ A1 . Since T is cyclic, x1 = Txo ∈ T(A10 ) ⊆ A20 ⊆ A2 . Then, we have

dG (x0 , Tx0 ) = dG (A1 , A2 )

or

dG (x0 , x1 ) = dG (A1 , A2 ),

where x1 ∈ A20 . Hence we must have x2 = Tx1 ∈ T(A20 ) ⊆ A30 ⊆ A3 , then

dG (x1 , Tx1 ) = dG (A2 , A3 )

or

dG (x1 , x2 ) = dG (A2 , A3 ).

Recursively, we obtain a sequence {xn } in

l

i=1 Ai0

⊆

l

i=1 Ai

satisfying

dG (xn , xn+1 ) = dG (Ai , Ai+1 ) for n ∈ N and i = 1, 2, 3, . . . , l. Also from the above setting we conclude that xn = xn+1 since each Ai is disjoint, where i = 1, 2, 3, . . . , l. Now clearly dG (u∗ , Tx) = dG (Ai , Ai+1 ), dG (u, Tu∗ ) = dG (Ai , Ai+1 ), dG (v, Ty) = dG (Ai , Ai+1 ), ∗

with u = xn+1 , x = xn−1 , u = xn+1 , y = xn , v = xn and i = 1, 2, 3, . . . , l. Therefore from (4.72), we obtain (4.73) G(xn+1 , xn+1 , xn ) ≤ M(xn−1 , xn , xn ) − φ (M(xn−1 , xn , xn )), where M(xn−1 , xn , xn ) = min{G(xn−1 , xn , xn ), G(xn−1 , Txn , Txn ), G(xn , Txn , Txn )} = min{G(xn−1 , xn , xn ), G(xn−1 , xn , xn ), G(xn , xn+1 , xn+1 )} = min{G(xn−1 , xn , xn ), G(xn , xn+1 , xn+1 )}. Now, if

M(xn−1 , xn , xn ) = G(xn , xn+1 , xn+1 ),

then from (4.73), we have G(xn , xn+1 , xn+1 ) = G(xn , xn+1 , xn+1 ) − φ (G(xn , xn+1 , xn+1 )),

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and, then

φ (G(xn , xn+1 , xn+1 )) = 0. Hence we have G(xn , xn+1 , xn+1 )) = 0; this implies that xn = xn+1 , which is a contradiction, in our setting. Then we have M(xn−1 , xn , xn ) = G(xn−1 , xn , xn ) Therefore G(xn , xn+1 , xn+1 ) ≤ G(xn−1 , xn , xn ) − φ (G(xn−1 , xn , xn )) ≤ G(xn−1 , xn , xn ).

(4.74)

Thus the sequence G(xn , xn+1 , xn+1 ) is a nonnegative, nonincreasing sequence which converges to L ≥ 0. Letting n → ∞ in (4.74), we obtain L ≤ L − φ (L); this implies that φ (L) = 0, and so, L = 0,, that is, lim G(xn , xn+1 , xn+1 ) = 0.

n→∞

(4.75)

Now we claim that {xn } is a G-Cauchy sequence in Y. On the contrary, we assume that {xn } is not G-Cauchy. Then there exist an ε > 0 and corresponding subsequences {n(k)} and {m(k)} of N satisfying n(k) > m(k) > k such that G(xm(k) , xn(k) , xn(k) ) ≥ ε ,

(4.76)

where n(k) is chosen as the smallest integer satisfying (4.76), that is, G(xm(k) , xn(k)−1 , xn(k)−1 ) < ε .

(4.77)

It is easy to conclude from (4.76) and (4.77) and the rectangle inequality, that

ε ≤ G(xm(k) , xn(k) , xn(k) ) ≤ G(xm(k) , xn(k)−1 , xn(k)−1 ) + G(xn(k)−1 , xn(k) , xn(k) )

(4.78)

< ε + G(xn(k)−1 , xn(k) , xn(k) ). Taking the limit k → ∞ in (4.78) and utilizing (4.75), we obtain lim G(xm(k) , xn(k) , xn(k) ) = ε .

k→∞

(4.79)

Observe that for every k ∈ N, there exist s(k) satisfying 0 ≤ s(k) ≤ l such that n(k) − m(k) + s(k) ≡ 1(l).

(4.80)

Therefore, for sufficiently large values of k, we have r(k) = m(k) − s(k) > 0 and xr(k) and xn(k) lie in the consecutive sets Ai and Ai+1 respectively, where 0 ≤ i ≤ l. Next using (4.72) with u∗ = xr(k) , u = xn(k)+1 , v = xn(k) , x = xr(k) , y = xn(k) , we obtain G(xr(k) , xn(k)+1 , xn(k) ) ≤ M(xr(k) , xn(k) , xn(k) ) − φ (M(xr(k) , xn(k) , xn(k) )),

(4.81)

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where M(xr(k) , xn(k) , xn(k) ) = min{G(xr(k) , xn(k) , xn(k) ), G(xr(k) , Txr(k) , Txr(k) ), G(xn(k) , Txn(k) , Txr(k )} = min{G(xr(k) , xn(k) , xn(k) ), G(xr(k) , xr(k)+1 , xr(k)+1 ), G(xn(k) , xn(k)+1 , Txr(k)+1 }. Employing the rectangle inequality repeatedly, we observe that G(xr(k) , xn(k) , xn(k) ) ≤ G(xr(k) , xr(k)+1 , xr(k)+1 ) + G(xr(k)+1 , xn(k) , xn(k) ) ≤ G(xr(k) , xr(k)+1 , xr(k)+1 ) + G(xr(k)+1 , xr(k)+2 , xr(k)+2 ) + G(xr(k)+2 , xn(k) , xn(k) ) .. . ≤

m−1

∑ [G(xi(k) , xi(k)+1 , xi(k)+1 )] + G(xm(k) , xn(k) , xn(k) ) i=r

or equivalently 0 ≤ G(xr(k) , xn(k) , xn(k) ) − G(xm(k) , xn(k) , xn(k) ) ≤

m−1

∑ G(xi(k) , xi(k)+1 , xi(k)+1 ).

(4.82)

i=r

Note that the sum on the right hand side of (4.82) contains s − 1 ≤ l (finite) number of terms and due to (4.75) each term of this sum tends to 0 as k → ∞. Therefore, lim G(xr(k) , xn(k) , xn(k) ) = lim G(xm(k) , xn(k) , xn(k) ) = ε .

n→∞

n→∞

(4.83)

Using the rectangle inequality again, we have 0 ≤ G(xr(k) , xn(k)+1 , xn(k) ) ≤ G(xr(k) , xn(k)+1 , xn(k)+1 ) + G(xn(k)+1 , xn(k)+1 , xn(k) ).

(4.84)

On letting k → ∞ and using (4.83), we deduce that lim G(xr(k) , xn(k)+1 , xn(k) ) = ε .

k→∞

(4.85)

Now passing to the limit as k → ∞ in (4.81) and using (4.75), (4.83), (4.85), we get

ε ≤ max{ε , 0, 0} − φ (max{ε , 0, 0}) = ε − φ (ε ); this implies that φ (ε ) = 0 and so ε = 0, which contradicts the assumption that {xn } is not G-Cauchy. Thus {xn } is a G-Cauchy sequence in Y. Since Y is complete, there exists z ∈ Y such that xn → z as n → ∞. Without loss of generality, we assume that z ∈ Ai for some i. On the other hand, for all n ∈ N, we can write, for each i = 1, 2, . . . , l dG (z, Ai+1 ) ≤ dG (z, Txn ) = dG (z, xn+1 ) ≤ dG (z, xn ) + dG (xn , xn+1 ) ≤ dG (z, xn ) + dG (Ai , Ai+1 ).

(4.86)

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Taking the limit as n → ∞ in (4.86), we get dG (z, Ai+1 ) ≤ dG (Ai , Ai+1 ), but

dG (Ai , Ai+1 ) ≤ dG (z, Ai+1 ), for z ∈ Ai .

Therefore, we have

lim dG (z, Txn ) = dG (z, Ai+1 ) = dG (Ai , Ai+1 ).

n→∞

(4.87)

Since Ai+1 is approximatively compact with respect to Ai , the sequence {Txn } has a subsequence {Txn(K) } that converges to some p ∈ Ai+1 . Hence dG (z, p) = lim dG (xn(k) , Txn(k) ) n→∞

= lim dG (xn(k) , xn(k)+1 ) n→∞

(4.88)

= dG (Ai , Ai+1 ) Therefore z ∈ Ai0 . Now, since T(z) ∈ T(Ai0 ) ⊆ A(i+1)0 , there exists a w ∈ Ai0 such that dG (w, Tz) = dG (Ai , Ai+1 ). Now we claim that w = z. For our assertion, utilizing (4.72) with x = xn−1 , y = xn , v = xn , u = w, u∗ = z, we have G(z, w, xn ) ≤ M(xn−1 , xn , xn ) − φ (M(xn−1 , xn , xn )) ≤ min{G(xn−1 , xn , xn ), G(xn−1 , xn , xn ), G(xn , xn+1 , xn+1 )} − φ (min{G(xn−1 , xn , xn ), G(xn−1 , xn , xn ), G(xn , xn+1 , xn+1 )}). Letting n → ∞, we have

G(z, z, w) ≤ G(z, z, z) − φ (G(z, z, z)),

and so (G(z, z, w)) = 0, which implies that z = w. Hence we have, dG (z, Tz) = dG (Ai , Ai+1 ). Therefore T has a best proximity point in Y.

(4.89) 

The following example substantiates the hypothesis of Theorem 12. Example 18. Let X = R and G : X × X × X → R+ be defined by G(x, y, z) =

1 max{|x − y| , |y − z| , |z − x|}. 2

Then clearly (X, G) is a G-metric space. Now dG (x, y) = |x − y|. Let A = {0, −1, −2, −3, −4} and B = {1, 2, 3, 4}. Define T : A ∪ B → A ∪ B by ⎧ if x = −4, ⎨ 1 0 if x = 4, T(x) = ⎩ −x + 1 otherwise. Then clearly T(A) ⊆ B and T(B) ⊆ A. Also taking φ (t) = 2t , clearly dG (A, B) = 1 and A0 = {0}. Now if we choose u = 0, u∗ = 0, x = −4 ∈ A and v = 1, y = 4 ∈ B, then dG (u∗ , Tx) = dG (A, B),

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dG (u, Tu∗ ) = dG (A, B) and

dG (v, Ty) = dG (A, B). ∗

Now with u = u = 0, x = −4, v = 1, y = 4, we verify Condition (4.72). 1 G(u∗ , u, v) = G(0, 0, 1) = . 2 Now M(x, v, y) = min{G(x, v, y), G(x, Tx, Tx), G(y, Ty, Ty)} = min{G(−4, 1, 4), G(−4, 1, 1), G(4, 0, 0)} =2 and

1 M(x, v, y) − φ (M(x, v, y)) = 2 − (2) = 1. 2

Hence G(u∗ , u, v) = That is,

1 ≤ 1 = M(x, v, y) − φ (M(x, v, y)). 2 dG (u∗ , Tx) = dG (A, B), dG (u, Tu∗ ) = dG (A, B)

and This implies that where

dG (v, Ty) = dG (A, B). G(u∗ , u, v) ≤ M(x, v, y) − φ (M(x, v, y)), M(x, v, y) = min{G(x, v, y), G(x, Tx, Tx), G(y, Ty, Ty)}.

Thus T is a generalized weak φ -proximal cyclic mapping. All the conditions of Theorem 12 are satisfied and T has a best proximity point z = 0 ∈ A ∪ B, since dG (0, T0) = dG (A, B). Now we obtain following corollaries (inspired by [38]). Corollary 7. Let (X, G) be a G-metric space and {Ai }m i=1 be a family of disjoint nonempty subsets  of X with Y = li=1 Ai such that Y is a G-complete subspace of X. Let T : Y → Y be a generalized proximal cyclic weak φ -contractive mapping, that is, T : Y → Y be a mapping satisfying T(Ai ) ⊆ Ai+1 , i = 1, 2, . . . , l, where Al+1 = A1 and if for x, u, u∗ ∈ Ai and y, v ∈ Ai+1 , dG (u∗ , Tx) = dG (Ai , Ai+1 ), dG (u, Tu∗ ) = dG (Ai , Ai+1 ) and then

dG (v, Ty) = dG (Ai , Ai+1 ), G(u∗ , u, v) ≤ kM(x, v, y),

(4.90)

where M(x, v, y) = min{G(x, v, y), G(x, Tx, Tx), G(y, Ty, Ty)}, φ ∈ Φ and k ∈ (0, 1). Suppose that Ai0 is nonempty such that T(Ai0 ) ⊆ A(i+1)0 and Ai+1 is approximatively compact with respect to Ai , for i = 1, 2, . . . , l. Then T has a best proximity point in Y.

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Proof. The proof is obvious by taking φ (t) = (1 − k)t, k ∈ (0, 1) in Theorem 12.



Corollary 8. Let (X, G) be a G-metric space and {Ai }m i=1 be a family of disjoint nonempty subsets  of X with Y = li=1 Ai such that Y is a G-complete subspace of X. Let T : Y → Y be a generalized proximal cyclic weak φ -contractive mapping, that is, T : Y → Y be a mapping satisfying T(Ai ) ⊆ Ai+1 , for i = 1, 2, . . . , l, such that Al+1 = A1 and if for x, u, u∗ ∈ Ai and y, v ∈ Ai+1 , dG (u∗ , Tx) = dG (Ai , Ai+1 ), dG (u, Tu∗ ) = dG (Ai , Ai+1 ) and dG (v, Ty) = dG (Ai , Ai+1 ), then G(u∗ , u, v) ≤ α G(x, v, y) + β G(x, Tx, Tx) + γ G(y, Ty, Ty),

(4.91)

where M(x, v, y) = min{G(x, v, y), G(x, Tx, Tx), G(y, Ty, Ty)}, φ ∈ Φ , α , β , γ > 0 with α + β + γ < 1. Suppose that Ai0 is nonempty such that T(Ai0 ) ⊆ A(i+1)0 and Ai+1 is approximatively compact with respect to Ai for i = 1, 2, . . . , l. Then T has a best proximity point in Y. Clearly, we have

α G(x, v, y) + β G(x, Tx, Tx) + γ G(y, Ty, Ty) ≤ (α + β + γ )M(x, v, y). Thus G(u∗ , u, v) ≤ (α + β + γ )M(x, v, y). Using Corollary 7, with k = (α + β + γ ) ∈ (0, 1), we obtain that T has a best proximity point.

4.5 Conclusions and Future Investigations In summary, we introduced and investigated variants of F-contraction and the notion of a best proximity point in the context of G-metric spaces. We established G-metric versions of several celebrated fixed point results by invoking the aforementioned concept. The main motive to introduce such results was to emphasize the applicative approach of fixed point theory. Justifying this, numerous applications of results obtained in real life problems occurring in science and engineering were established. For application perspective these findings give rise to a new path to researchers who are working closely in related areas. We present some problems for further investigation. • Can Meir–Keeler type contractive conditions be obtained by amalgamating the concepts of G-metric spaces and F-contraction? • Can ideas provided in this chapter be applied in generalization of G-metric spaces such as Gb -metric spaces, G∗ -metric spaces, complex valued G-metric spaces, etc? • Can the ideas of α -admissible mappings and weakly α -admissible mappings be applied to the fixed point results for G − l type F contractive mappings?

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• In electrical and electronics circuits analysis, the following integro-differential equation appears: I(t) = p(t) +



a

t

f (t, s, I(s), I (s))ds.

It is an open question as to whether the existence of a solution of the aforementioned integro-differential equation can be established from our results, proved in this chapter. • Consider the functional differential equation of the form 

x (t) = −

t

p(t, v)G(v, x(v))dv, t−r(t)

where r(t) : [0, ∞) → [0, ∞) and p(t, v) : [0, ∞)×[r0 , ∞) → R are continuous functions. Moreover, there exists an l > 0 such that G satisfies a Lipschitz condition with respect to x on [r0 , ∞) × [0, l]; that is, there exists a constant L > 0, such that |G(v, x) − G(v, y)| ≤ L|x − y| for v ≥ r0 and x, y ∈ [0, l]. Can the existence of a solution of the aforementioned functional differential equation be established from our results. • Can one establish the existence of a solution to the storekeeper’s control problem, which is stated as: For the use of optimal storage space, a storekeeper wants to keep the stores stock of goods constant. This can be mathematically modelled as a Volterra integral equation of the first kind as follows:

ζ h(ν ) +



0

ν

h(ν − ϑ )g(ϑ )dϑ = m0 ,

where ζ = number of products in stock at time ν = 0, h(ν ) = remainder of products in stock (in percent) at the time ν , g(ν ) = the velocity (products/time unit) with which new products are purchased, g(ϑ )Δ ϑ = the amount of purchased products during the time interval ϑ .

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74.

75.

76.

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5 Fixed Point Theory in Probabilistic Metric Spaces Juan Martínez-Moreno

CONTENTS 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Probabilistic Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Probabilistic B-Contraction and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Hicks Contraction and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Comparison of Hicks Contraction and Bharucha-Reid Contraction . . . . . . . . . . . . . . . 161 Generalization of Hicks Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Non-linear Probabilistic Contractions and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Altering Distance Functions in Probabilistic Metric Spaces and Fixed Points . . . . . 173 Fixed Point Theorems for Generalized β -Type Contractive Mappings . . . . . . . . . . . . 184 Motivating Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.1 Introduction The theory of probabilistic metric spaces (in short PM spaces) was introduced in 1942 by Menger[34]. The idea was to use distribution functions instead of nonnegative real numbers as values of a metric. Thus probabilistic metric spaces have notions of uncertainty built within the structure of the space and hence provide a natural framework for the study of quantum mechanical phenomena [41]. The notion of probabilistic contractive mappings was introduced by Sehgal and BaruchaReid[53] in order to obtain extensions of the Banach contraction principle, by considering PM space that can be deduced from a metric space. Therefore some classical fixed point results can be viewed as particular cases of these in PM spaces. However, there are a variety of specific results for probabilistic contractive mappings, depending on the t-norms the spaces are endowed with and other properties such as growth conditions for F [55]. Thus, these results have advantage over the results with the usual metric. In fact finding a new class of contractive mappings in PM spaces and the process of proving fixed point theorems for such mappings involves many mathematical components such as the conditions on t-norms associated with given PM spaces and other conditions on space itself. Such 149

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properties attract fixed point theorists to work in this direction. In recent years, definitive results on this line have been given by many authors [7, 8, 9, 15, 16, 22, 24, 38, 39, 40, 51]. In this chapter, we are devoted to a concise study of fixed point results concerning various classes of probabilistic contractions including probabilistic nonlinear contractions. We hope that the results presented in this chapter illustrate the direction of research over the last five decades up to the most recent contributions for fixed point theory of single valued mappings in PM spaces.

5.2 Probabilistic Metric Spaces Probabilistic metric spaces depend on the notion of a distribution function. Although there is a great agreement about the global properties that a distribution function must satisfy, in practice, we can find several papers involving different notions. For instance, Schweizer and Sklar [51] established that a distribution function is a nondecreasing function F : [−∞, ∞] → [0, 1] such that F(−∞) = 0 and F(∞) = 1. They also denoted by Δ the family of distribution functions that are left-continuous on R. Definition 1. A distance distribution function F : [−∞, ∞] → [0, 1] is a distribution function with support contained in [0, ∞]. The family of all distance distribution functions will be denoted by Δ + . We denote D+ = {F | F ∈ Δ + , lim F(x) = 1}. x→∞

Since any function from Δ + is equal zero on [−∞, 0] we can consider the set Δ + consisting of nondecreasing functions F defined on [0, ∞] left-continuous on (0, ∞) that satisfy F(0) = 0 and F(∞) = 1. Moreover, D+ consists of nondecreasing functions F defined on [0, ∞) that satisfy F(0) = 0 and lim F(x) = 1. The class D+ plays the important role in the probabilistic x→∞

fixed point theorems. In the following a specific distribution function H that is called a Heaviside function, is defined by ⎧ ⎨ 0 iff t ≤ 0, H(t) = ⎩ 1 iff t > 0. Definition 2. [51] A generalized probabilistic metric space (generalized PM-space, for short) is an ordered pair (X, F) where X is a nonempty set and F is a mapping from X × X → D+ , whose value Fx, y denoted by Fxy , satisfies the following conditions for all x, y, z ∈ X (PM1) Fxy (t) = 1 for all t > 0 if and only if x = y (PM2) Fxy (t) = Fyx (t) for all ...., (PM3) Fxy (t) = 1 and ..... The intended meaning of Fx,y (t) = u is that u is the “probability of the distance from x to y is less than t”. If F(X × X) ⊂ D+ , a distinguished particular case of generalized PM-space just called PM-space is obtained.

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Condition (PM1) is equivalent to the statement x = y if and only if Fxy = H. Conditions (PM1)–(PM3) are generalizations of the corresponding well-known conditions satisfied by a classical metric. If only (PM1) and (PM2) hold, the ordered pair (X, F) is known as a probabilistic semimetric space. Every metric space (X, d) may be regarded as a PM-space. One only has to set Fxy (t) = H(t − d(x, y)) for each x, y ∈ X. Definition 3. A binary operation Δ : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (abbreviated, t-norm) if for all a, b, c, d ∈ [0, 1] the following conditions are satisfied: (i) (ii) (iii) (iv)

Δ (a, 1) = a, Δ (a, b) ≤ Δ (c, d) whenever a ≤ c and b ≤ d, Δ (a, b) = Δ (b, a), Δ (Δ (a, b), c) = Δ (a, Δ (b, c)).

If, in addition, Δ is continuous, then Δ is called a continuous t-norm. The following are the four basic t-norms: (i) (ii) (iii) (iv)

The minimum t-norm, ΔM , is defined by ΔM (a, b) = min{a, b}. The product t-norm, Δp , is defined by Δp (a, b) = a · b. The Lukasiewicz t-norm, ΔL , is defined by ΔL (a, b) = max{a + b − 1, 0}. The weakest t-norm, the drastic product, ΔD , is defined by ⎧ ⎨ min{a, b} if max{a, b} = 1, ΔD (a, b) = ⎩ 0 otherwise.

These four basic t-norms are remarkable for several reasons. The drastic product ΔD and the minimum ΔM are the smallest and the largest t-norm, respectively (with respect to the pointwise order), that is, ΔD ≤ Δ ≤ ΔM for every t-norm Δ . The minimum ΔM is the only t-norm where each x ∈ [0, 1] is an idempotent element. The t-norm ΔM is the strongest t-norm, that is, Δ ≤ ΔM for every t-norm. Let Δ be a given t-norm. Then (by associativity) a family of mappings Δ (n) : [0, 1] → [0, 1] ; n ∈ N, is defined as follows:

Δ 1 (t) = Δ (t, t), Δ 2 (t) = Δ (t, Δ (t)), . . . , Δ n (t) = Δ (t, Δ n−1 (t)), . . . ; t ∈ [0, 1] (n)

(n)

(n)

For three important t-norms ΔM (t) = t, Δp (t) = tn , ΔL (t) = max{(n + 1)t − n, 0} for each n ∈ N. Recall that a t-norm Δ is continuous if for all convergent sequences {xn }, {yn } in [0, 1], we have Δ (lim xn , lim yn ) = lim Δ (xn , yn ). n→∞

n→∞

n→∞

It is well known that every continuous t-norm can be written as an ordinal sum of three basic t-norms, that is, the minimum, the product, and the Lukasiewicz t-norm. Definition 4. A continuous t-norm Δ is called an Archimedean t-norm if Δ (a, a) < a for each a ∈ (0, 1).

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The following property is essentially utilized to prove a fixed point result; see ([21]): If Δ is a t-norm for which there is a sequence {xn } ⊂ [0, 1] such that for every t-norm Δ , ∞ xn+i = 1 then supa 1 − δ =⇒ Δ n (t) > 1 − ε (n ≥ 1). The class of H-type triangular norms was introduced for the first time by Hadzic in 1978. By virtue of this concept, many interesting results on the existence of fixed points and the solutions of equations for nonlinear mappings in probabilistic metrics are obtained. It is evident that if Δ is of H-type, then Δ satisfies sup0 0. Further, if Δ : [0, 1] × [0, 1] → [0, 1] is defined by Δ (a, b) = min{a, b}, then (X, F, Δ ) is a Menger space. It is complete if d is complete. If (X, F, Δ ) is a Menger space with continuous t-norm then the topology induced by the family {Nx (ε , λ) : ε > 0, λ ∈ (0, 1)} is called the (ε , λ)-topology, where Nx (ε , λ) = {y ∈ X : Fxy (ε ) > 1 − λ}

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is called the (ε , λ)-neighbourhood of x (see [51]). The (ε , λ)-topology is a Hausdorff topology. Definition 7. Let (X, F, Δ ) be a Menger PM-space. Then A sequence {xn }n in X is said to be convergent to x in X in the (ε , λ) topology if for every ε > 0 and λ > 0 there exists a positive integer Nε ,λ such that Fxn x (ε ) > 1 − λ for all n ≥ Nε ,λ . (ii) A sequence {xn }n in X is said to be a Cauchy sequence if given ε > 0 and λ > 0 there exists a positive integer Nε ,λ such that Fxn xm (ε ) > 1 − λ for all n, m ≥ Nε ,λ . (iii) A Menger PM-space is said to be complete if every Cauchy sequence in X is convergent to a point in X. (i)

5.3 Probabilistic B-Contraction and Fixed Points The probabilistic version of the classical Banach Contraction Principle was first studied in 1972 by Sehgal and Bharucha-Reid [53]. Definition 8. Let (X, F) be a probabilistic metric space and f : X → X. The mapping f is called a B-contraction if there exists a k ∈ (0, 1) such that for all points p, q ∈ X and for all t>0 Ffp,fq (kt) ≥ Fp,q (t)

(5.1)

Theorem 2. Let (X, F, Δ ) be a complete Menger space, where Δ is a continuous function satisfying Δ (x, x) ≥ x for each x ∈ [0, 1]. If T is any contraction mapping of X into itself, then there is a unique p ∈ X such that Tp = p. Moreover, Tn q → p for each q ∈ X. Proof. We first prove uniqueness. Suppose p = q and fp = p, fq = q. Then by (PM-1), there exists an x > 0 and an a, with 0 ≤ a < 1, such that Fp,q (x) = a. However, for each positive integer n, we have by (5.1) a = fp,q (x) = Ff n p,f n q (x) ≥ Fp,q (

x ) kn

Since Fp,q ( kxn ) → 1 as n → ∞, it follows that a = 1. This contradicts the selection of a, and therefore, the fixed point is unique. To prove the existence of the fixed point, consider an arbitrary q ∈ X, and define pn , = f n (q), n = 1, 2, . . .. We show that the sequence {pn } is fundamental in X. Let ε , λ be positive reals. Then for m > n, we have Fpn ,pm (ε ) ≥ Δ (Fpn ,pn+1 (ε − kε ), Fpn+1 ,pm (kε )), ≥ Δ (Fq,p1 ((ε − kε )k−n , Fpn+1 ,pm (kε )). Set d = (ε − kε )k−n . it follows by (PM-3’) and the associativity property of Δ that Fpn ,pm (ε ) ≥ Δ (Fq,p1 (d), Δ (Fpn+1 ,pn+2 (kε − k2 ε )), Fpn+2 ,pm (k2 ε )), ≥ Δ (Fq,p1 (d), Fpn+2 ,pm (k2 ε )).

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By the associativity of Δ , and the hypothesis Δ (x, x) ≥ x, we have Fpn , pm (ε ) ≥ Δ (Fq,p1 (d), Fpn+2 ,pm (k2 ε )).

(5.2)

Using the same argument repeatedly, we obtain from (5.2) Fpn ,pm (ε ) ≥ Δ (Fq,p1 (d), Fpm−1 ,pm (km−n−1 ε )) ≥ Δ (Fq,p1 (d), Fq,p1 (k−n ε ) ≥ Δ (Fq,p1 (d), Fq,p1 (d) ≥ Fq,p1 ((ε − kε )k−n ) Therefore, if we choose N such that Fq,p1 ((ε − kε )k−N ) > 1 − λ, it follows that Fpn ,pm > 1 − λ for all n ≥ N. Hence, {pn } is a fundamental sequence. Since (X, F, Δ ) is a complete PMspace, there is a p ∈ X such that pn → p, that is, f n q → p. We prove that f n q → fp also. Let Ufp (ε , λ) be any neighbourhood of fp. Then p→ p implies the existence of an integer N such that pn ∈ U(ε , λ) for all n ≥ N. However,

ε Ffpn , fp(ε ) ≥ Fpn ,p ( ) ≥ Fpn ,p > 1 − λ k for all n ≥ N. Therefore, fp ∈ U(ε , λ) for all n ≥ N, that is f n q → fp. We conclude therefore fp = p. This proves the existence part of the theorem.  However, the hypothesis Δ (x, x) ≥ x is a very restrictive one, because there is an unique example of t-norm (not necessarily continuous) verifying this property. Lemma 1. The only t-norm Δ verifying Δ (t, t) ≥ t, for all t ∈ [0, 1] is the minimum t-norm Δ = ΔM . Proof. Let t, s ∈ [0, 1] such that 0 ≤ s ≤ t ≤ 1. Since Δ in nondecreasing on each argument, Δ (s, s) ≤ Δ (s, t) ≤ Δ (s, 1) = s. Combining these inequalities with the assumption s ≤ Δ (s, s), we derive that s = Δ (s, t) = min{s, t}.  As a consequence of Theorem 2, we derive the classical Banach Contraction principle as follows [Note that Δ (a, a) ≥ a, ∀a ∈ [0, b] =⇒ Δ (a, b) = min{a, b}]. Theorem 3. Let (X, d) be a complete metric space and let f : X → X satisfy the condition: there exists a constant k, 0 < k < 1, such that d(fp, fq) ≤ kd(p, q) for all p, q ∈ X. Then f has a unique fixed point p ∈ X and f n q → p, for each q ∈ X. If F : X × X → L is the mapping induced by the metric d, then it follows by Theorem 6.1 that (X, F, Δ ) is a complete Menger space, where Δ (a, b) = min{a, b}. Since for each x > 0, Ffp,fq (kx) = H(kx − d(fp, fq)), > H(kx − kd(p, q)), = H(x − d(p, q)), = Fp,q (x), it follows that f is a contraction of X into itself. The conclusion now follows by Theorem 2.

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Sherwood [54] improved Theorem 2 by introducing the notion of Probabilistic boundedness. For every x0 ∈ X, let O(x0 , f ) = {f n x0 | n ∈ N ∪ {0}}. The set O(x0 , f ) is called the orbit of f : X → X at x0 and the function DO(xo ,f ) : R → [0, 1], defined by DO(xo ,f ) (x) = sup inf ∈ O(xo , f )Fu,v (s) s 1 − λ. Hence (xn )n∈N is a Cauchy sequence and since X is complete, there follows the existence of x ∈ X such that x = lim xn . By continuity of f and xn+1 = fxn for every n ∈ N, we obtain x = fx. n→∞

It is easy to prove that if O(xo , f ) is probabilistic bounded for some x0 ∈ X, then O(p, f ) is probabilistic bounded for any p ∈ X.  Corollary 1. Let (X, F, Δ ) be a complete Menger space, Δ a t-norm of H-type and f : X → X a probabilistic B-contraction. Then there exists a unique fixed point x ∈ X of the mapping f and lim f n p = x for every p ∈ X.

n→∞

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Proof. It is easy to prove that for a t-norm Δ , which is of H-type, the orbit O(xo , f ) is probabilistic bounded for every x0 ∈ X. Indeed, let x0 ∈ X, m ∈ N and u ∈ R. Then Ff m x0 ,xn (u) = Δ (Ff m x0 ,fx0 (ku), Ffx0 ,x0 (u − ku)) ≥ Δ (Ff m−1 x0 ,x0 (u), Ffx0 ,x0 (u − ku)) .. . (m)

≥ (Ffx0 ,x0 (u − ku))Δ

(m)

and since lim Ffx0 ,x0 (u − qu) = 1, from the equicontinuity of the family (xΔ )( m ∈ N) at the u→∞

point x = 1, it follows that DO(xo ,f ) ∈ D+ .



Since the t-norm ΔM is essentially of H-type, Theorem 4 follows from the above Corollary 1. Sherwood in [54] used the functions of the form Δ ∞ Fi to obtain fixed point results. Let Δ be a triangle function and let {Fi }∞i=1 be a sequence of function in D+ . Let

ΔΔ1 (Fi ) = ΔΔ (F1 , F2 ), (ΔΔn (Fi ) = ΔΔ (ΔΔn−1 (Fi ), Fn+1 )) (n > 2). Since ΔΔ is nondecreasing in each place, the weak limit of ΔΔn (Fi ), when n → ∞, exists and will be denoted by ΔΔ∞ (Fi ). Theorem 5. Let (X, F, Δ ) be a complete Menger space, Δ a t-norm such that supa sn for every n ∈ N and (bn )sΔn < λ for every n ∈ N. let where

m+1 G(2n+i−1 t), X = N, Fn,n = H0 , Fn,n+m (t) = Δi=1

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎨ b1 G(t) = ⎪ ⎪ ⎪ ⎪ ⎩ bn+1

if t ≥ 1, if t ∈ (1, 22+s1 ], if t ∈ (22n+sn , 22n+2+sn+1 ], n ≥ 1.

Then (X, F, Δ ) is a Menger space and f : X → X be defined by f (n) = n + 1, n ∈ N. Then f is a probabilistic 1/2-contraction, but (f n (1))n∈N is not a Cauchy sequence since Fn,n+sn (1) ≤ (bn )sΔn < λ. Thus, no Archimedean t-norm has the fixed point property.



From the above discussion it is clear that in order to obtain fixed theorems in Menger spaces (X, F, Δ ), where Δ is an Archimedean t-norm, one has to impose additional conditions on the distribution function F. This approach has been pursued by Tardiff [55]. To state and prove major results of Tardiff; first we need the following theorems and lemmas: Theorem 7. Let (Fi )i∈N be a sequence in D+ and Δ an Archimedean t-norm. Then either ΔΔ∞ (Fi ) is identically 0 or is in D+ . Theorem 8. Let (Fi )i∈N be a sequence in D+ . Then ΔΔ∞p (Fi ) is identically 0 if and only if ΔΔ∞L (Fi ) is identically 0. Lemma 2. Let F ∈ D+ and α ∈ (0, 1). Then G = ΔΔ∞p (Fojα i−1 ) ∈ D+ ⇔



∞ 1

ln(u)dF(u) < ∞.

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Proof. Since Δp is an Archimedean t-norm, we can apply Theorem 8. In order to prove that TT∞p (Fojα i−1 ) is in D+ , it is enough to prove that for some x ∈ R, G(x) > 0. The function G is a nondecreasing and left continuous function and therefore G has at most a countable number of discontinuous points. Let x ∈ R be a point of the continuity of G. Then n→∞

∑ni=1 i

i=1

∞ ∑i=1 (1 − δ )δ i−1

Let δ ∈ (α , 1). Since F is nondecreasing and n

βi x

n

∏ F( α i−1 ). β =1

G(x) = lim sup

n

x

∏ F( α i−1 ) ≥ G(x) ≥ ∏ F( i=1

i=1

= 1, it follows that

(1 − δ )x )). ( αδ i−1

+

Since F ∈ D , there exists y ∈ R such that F(y) > 0.We shall prove that for and σ ∈ (0, 1) n

∏ F( i=1



y

σ

)>0⇔ i−1

∞

1

ln(u)dF(u) < ∞.

Let F(y) > 0, σ ∈ (0, 1) and g be defined by ⎧ ⎨ 0 g(t) = ⎩ yσ 1−t

if t ≤ 1, if t > 1.

Since 0 < σ < 1 and F ∈ D+ , it follows that Fog ∈ D+ . Further by [27] ∞

n

∏ Fog(i) > 0 ⇔ ∑(1 − Fog(i)) < ∞ i=1

⇔ and by [14]



∞ 0

i=1 ∞



0

(1 − Fog(t))dt < ∞ ⇔

Hence we conclude that ∞

∏ Fog(i) > 0 ⇔

 0

i=1

If u = g(t) then the relation 

0

∞

(1 − Fog(t))dt < ∞

tdF(g(t)) = F(y) +

 y

∞

∞



∞ 0

tdF(g(t)) < ∞.

tdF(g(t)) < ∞.

(1 −

ln(u) − ln(y) )dF(u) ln(σ )

implies that the right side of the above inequality is finite if and only if

∞ 1

ln(u)dF(u) < ∞. 

Corollary 3. Let F ∈ D+ and α ∈ (0, 1). Then ΔΔ∞L (Fojα i−1 ) ∈ D+ if and only if 

1

∞

ln(u)dF(u) < ∞.

(5.3)

Theorem 9. Let (X, F, Δ ) be a complete Menger space and Δ a t-norm such that Δ ≥ ΔL . If for every u, v ∈ X (Theorem 8) holds, with F replaced by Fu,v , then any probabilistic k-contraction f : X → X has a unique fixed point x and x = lim f n p for every p ∈ X. n→∞

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Proof. From Lemma 2 and Theorem 8 it follows that Δ ΔL

and since Δ ≥ ΔL

∞

(Fp,f (p) ojqi−1 ) ∈ D+

ΔΔ∞ (Fp,f (p) ojqi−1 ) ∈ D+ . 

This completes the proof.

Corollary 4. Let (X, F, Δ ) be a complete Menger space and Δ a t-norm such that Δ ≥ ΔL and f : X → X a probabilistic k-contraction. If there exists p ∈ X such that 

∞

1

ln(u)dFp,fp (u) < ∞

then there exists a unique fixed point x of the mapping f and x = lim f n p. n→∞

Proof. The proof is identical to the proof of Theorem 9.



Corollary 5. Let (X, F, Δ ) be a complete Menger space and Δ a t-norm such that Δ ≥ ΔL . If for all u, v ∈ X the first moment is finite, then any probabilistic k-contraction f : X → X has a unique fixed point and x = lim f n p for every p ∈ X. n→∞

Proof. Let g(x) =

⎧ ⎨ x−1 ⎩

if x ≤ 1, if x > 1.

ln x

Using the Jensen inequality, since g is concave, we have that for u, v ∈ X 

∞

0

g(x)dFu,v (x) ≤ g(



0

∞

xdFu,v (x)) < ∞, 

and therefore (5.3) holds.

5.4 Hicks Contraction and Fixed Points Schweizer in [52] has pointed out that the result of Sehgal and Bharucha-Reid is the exception rather than the rule specifically that, for any Archimedean t-norm Δ , there exist a complete Menger space (X, F, Δ ) and a B-contraction f on (X, F) which has no fixed point. This fact might motivate Hicks to consider contraction on PM spaces differently from those of Sehgal and Bharucha-Reid. Definition 10. Let (X, F) be a probabilistic metric space and f : X → X. The mapping f is called an H-contraction if the following implication holds for every p, q ∈ X and t ∈ (0, ∞); Fp,q (t) > 1 − t =⇒ Ffp,fq (kt) > 1 − kt where k ∈ (0, 1) is given.

(5.4)

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Remark 1. Hicks observed that the metric d is related to the distribution functions given by d(p, q) < t iff Fp,q (t) > 1 − t.

(5.5)

This allows an exact translation of the contraction condition in metric spaces to PM spaces. More especially the following theorem was proved by Hicks: Theorem 10. Let (X, F) be a probabilistic semimetric space such that Fx,z (r + s) ≥ min{Fx,z (r), Fx,z (s)}. Let

d(x, y) = sup{t ≥ 0 : t ≤ 1 − Fpq (t)}.

Then (1) (2) (3) (4)

d(x, y) < t if and only if Fx,y (t) > 1 − t. d is a compatible metric for t(F). If f : X → X and 0 < k < 1, (5.4) holds if and only if d(fx, fy) ≤ kd(x, y). (X, F) is complete if and only if (X, d) is complete.

Proof. Observe that if t < r, Nx (t, t) ⊂ Nx (r, r). Also ∩{Nx (ε , ε ) : 0 < ε < 1} = {x}. For, if x = y, Fx,y = H. Thus there exists ε > 0 such that Fx,y (ε ) = δ where 0 < δ < 1. Set δ = 1 − δ1 and let ε1 = min{ε , δ1 }. Then Fx,y (ε1 ) ≤ Fx,y (ε ) = δ = 1 − δ1 ≤ 1 − ε1 gives y ∈ / Nx (ε1 , ε1 ). (1) If 1 < t, d(x, y) ≤ 1 < t and also Fx,y (t) ≥ 0 > 1 − t. Suppose d(x, y) ≤ 1 < t. Choose δ such that d(x, y) < δ < t ≤ 1. Then y ∈ Nx (δ , δ ) and Fx,y (t) ≥ Fx,y (δ ) > 1− δ > 1−t. For if we assume y∈ / Nx (δ , δ ), then d(x, y) ≥ δ , a contradiction. Conversely, suppose Fx,y (t) > 1 − t where / Nx (ε , ε ) for all ε < t, Fx,y (t) = lim− Fx,y (ε ) ≤ lim− (1 − ε ) = 1 − t, 0 < t ≤ 1. Then y ∈ Nx (t, t). If y ∈ ε →t

ε →t

a contradiction. Thus there exists 0 < ε < t such that y ∈ Nx (ε , ε ). Hence d(x, y) ≤ ε < t. (2) If d satisfies the triangular inequality, it is a metric. Also, (1) shows it is compatible with t(F). We observe that d(x, y) < ε1 and d(y, z) < ε2 implies that d(x, z) < ε1 + ε2 . For, suppose Fx,y (ε1 ) > 1 − ε1 and Fy,z (ε2 ) > 1 − ε2 . If Fx,y (ε1 ) is the minimum, Fx,z (ε1 + ε2 ) ≥ min{Fx,y (ε) , Fy,z (ε2 } > 1 − ε1 > 1 − (ε1 + ε2 ) gives d(x, z) < ε1 + ε2 . The triangular inequality follows. (3) Suppose d(fx, fy) ≤ kd(x, y) and Fx,y (t) > 1 − t. Then d(x, y) < t and d(fx, fy) < kt. Thus Ffx,fy (kt) > 1 − kt. If (5.4) holds, let ε > 0 be given. Set t = d(x, y) + ε .d(x, y) = t − ε < t gives Fx,y (t) > 1 − t and Ffx,fy (kt) > 1 − kt follows from (5.4). Thus d(fx, fy) < kt = k(d(x, y) + ε ) = kd(x, y) + kε ). Since ε > 0 was arbitrary, d(fx, fy) ≤ kd(x, y).  Remark 2. Assuming the condition in Theorem 10, we have Fx,z (r + s) ≥ Δ (Fx,y (r), Fy,z (s)) = min{Fx,y (r), Fy,z (s)}. The inequality in Theorem 10 does not require the existence of a t-norm. Condition (5.4) and the earlier definition of contraction seem to be independent for 0 < k < 1.

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The main result of Hicks [22] reads as follows: Corollary 6. Every H-contraction on a complete Menger space (X, F, ΔM ) has a unique fixed point. Proof. The Theorem 10 gives a compatible metric d such that d(fx, fy) ≤ kd(x, y). From Banach contraction Theorem f has a unique fixed point. 

5.5 Comparison of Hicks Contraction and Bharucha-Reid Contraction In 1988, B. Schweizer et al. [52] showed that the Hicks conjecture (see Remarks 6.2) is correct, that is, in general the Hicks contraction and the Bharucha-Reid contraction are independent. To present illustrative examples for this fact, we first state the following results: Lemma 3. For any distribution function F in D+ and any t > 0, F(t) > 1 − t ⇔ dL (F, H) < t

(5.6)

where dL (F, H) = inf{h : F(h+) > 1 − h}. Remark 3. It is apparent from the definition of β , (5.4) and (5.6) that f is an H-contraction if and only if, for all x > 0 and all p, p ∈ X,

β (fp, fq) < γ x =⇒ β (p, q) < x.

(5.7)

(X, F, Δ ) with Δ ≥ ΔL is actually a Banach contraction in the metric space (X, K), where β (p, q) = sup{t ≥ 0 : t ≤ 1 − Fpq (t)} = dL (Fp,q , H). This allows us to transpose some results from probabilistic metric spaces to metric spaces. Theorem 11. If (X, F, Δ ) is a PM-space such that β (p, q) = dL (Fp,q , H) with Δ ≥ δL , then β is a metric on X. Proof. It is easy to see that β (p, q) = 0 iff p = q and that β (p, q) = β (q, p). Now, we prove a triangle inequality. Suppose β (p, q) = h1 and β (q, r) = h2 . Let η1 , η2 > 0 be given. Then β (p, q) < h1 + η1 and β (q, r) < h2 + η2 . Therefore we have

τ (Fpq , Fqr )(h1 + η1 + h2 + η2 ) ≥ τW (Fpq , Fqr )(h1 + η1 + h2 + η2 ) =

sup x+y=(h1 +η1 +h2 +η2 )

max(Fpq (x) + Fqr (y) − 1, 0)

≥ Fpq ((h1 + η1 ) + Fqr (h2 + η2 ) − 1 > 1 − h1 − η1 + 1 − h2 − η2 − 1 = 1 − (h1 + η1 − h2 + η2 ).

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Since Fpr ≥ τ (Fpq , Fqr ) and dL (G, ε0 ) ≤ dL (F, ε0 ) whenever F ≤ G, we have

β (p, r) = dL (Fpr , ε0 ) ≤ dL (τ (Fpq , Fqr ), ε0 ) < h1 + η1 + h2 + η2 = β (p, q) + β (q, r) + η1 + η2 . Letting η1 , η2 → 0 completes the proof.



It is important to note that the condition Δ ≥ ΔL in the above theorem is necessary. It should be noted also that in the proofs of the above-mentioned theorems the condition F(X × X) ⊂ D+ on the probabilistic metric is not necessary. Therefore results are also valid for generalized PM-spaces. Example 2. Let Δ be a continuous t-norm and suppose it is not true that Δ ≥ ΔL . Then there exist a, b ∈ (0, 1) such that 0 < a ≤ b < 1 and 0 ≤ Δ (a, b) < ΔL (a, b) = a + b − 1. Let X = {p, q, r} and define F : X × X → D+ by ⎧ 0, if t ≤ 0, ⎪ ⎪ ⎪ ⎪ ⎨ a if 0 < t ≤ 2, Fpq (t) = ⎪ ⎪ ⎪ ⎪ ⎩ 1 if 2 < t. ⎧ 0, if t ≤ 0, ⎪ ⎪ ⎪ ⎪ ⎨ b if 0 < t ≤ 2, Fqr (t) = ⎪ ⎪ ⎪ ⎪ ⎩ 1 if 2 < t. and

Fpr (t) = Δ (Fpq , Fqr )(t) =

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Δ (a, b) ⎪ ⎪ b ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1

if t ≤ 0, if 0 < t ≤ 2, if 2 < t ≤ 4, if 4 < t.

Then one can verify that (X, F, Δ ) is a PM space. But

β (p, q) + β (q, r) = dL (Fpq , ε0 ) + dL (Fqr , ε0 ) = 1 − a + 1 − b = 1 − ΔL (a, b) < 1 − Δ (a, b) = dL (Fpr , ε0 ) = β (p, r.) Thus β is not a metric on X. Radu [43] proved a more general result: Theorem 12. Every H-contraction on a complete Menger space (X, F, Δ ) such that Δ ≥ ΔL has a unique fixed point. The following example shows that an H-contraction need not be a B-contraction.

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Example 3. Let X = {0, 1, 2, . . .} and, for x = y, define F : X × X → D+ as follows ⎧ 0, if t ≤ 2− min(x,y) , ⎪ ⎪ ⎪ ⎪ ⎨ Fx,y (t) = Fy,x (t) = 1 − 2− min(x,y) if 2− min(x,y) < t ≤ 1, ⎪ ⎪ ⎪ ⎪ ⎩ 1 if 1 < t. Then one can verify that (X, F, ΔM ) is a PM-space. Define f : X × X by f (x) = x + 1. Then f is an H contraction. Next, let γ be any number in (0, 1) and choose t ∈ (1, 1γ . Then γ t < 1 so that 1 F(γ t)f (0),f (1) = F1,2 (γ t) ≤ < 1 = F0,1 (t), 2 that is, f is not a B-contraction. Next we show that a B-contraction need not be an H-contraction. Example 4. Let X = {p, q, r} and let F : X × X → D+ be defined by ⎧ 0, if t ≤ 0, ⎪ ⎪ ⎪ ⎪ ⎨ 1 if 0 < t ≤ 2, Fpr (t) = Frp (t) = Frq (t) = Fqr (t) = 2 ⎪ ⎪ ⎪ ⎪ ⎩ 1 if t > 2. and Fpq (t) = Fqp (t) =

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

if t ≤ 0,

1 2

if 0 < t ≤ 32 ,

1

if t > 32 .

Then one can verify that (X, F, ΔM ) is a PM-space. Define f : X → X as follows f (p) = f (q) = p and f (r) = q. Since Fpq ( 3t4 ) = Fpr (t) for all t, it follows at once that f is a B-contraction on (X, F). However, Remark 3 shows that f is not an H-contraction because β (fp, fr) = β (p, q) = 12 = β (q, r). In [52] some sufficient conditions for F are given such that every B-contraction on (X, F) is an H-contraction. Theorem 13. Let (X, F) be a probabilistic semimetric space such that Ran(F) is finite and Ran(F) \ {H} is strictly increasing on [0, 1]. Then every B-contraction on (X, F) is an H-contraction.

5.6 Generalization of Hicks Contraction In 2005, Mihet[38] weakening the Hicks contraction, proposed the following:

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Definition 11. Let X be a nonempty set and F be a probabilistic distance on X. A mapping f : X → X is said to be a weak-Hicks contraction (w-H contraction for short) if there exists k ∈ (0, 1) such that for all p, q ∈ X, t ∈ (0, 1) Fp,q (t) > 1 − t =⇒ Ffp,fq (kt) > 1 − kt. Example 5. Let X be a set containing at least two elements. Consider the discrete Menger space under ΔM determined (for x = y) by the probabilistic metric ⎧ ⎨ 0, if t ≤ 1, Fxy (t) = ⎩ 1 if t > 1. Then any mapping f : X → X is a weak-Hicks contraction, while the only Hickscontractions on (X, F, ΔM ) are the constant mappings. Because in the case of weak Hicks-contraction the contraction condition takes place only for t ∈ (0, 1), that can be counterbalanced by a more convenient condition on the probabilistic metric, namely Fx,f (x) ((1)) > 0 to ensure the existence of a fixed point for a weak Hicks contraction (Theorem (given below)). min(x,y) Example 6. Let X = [0, ∞) and Fxy (t) = max(x,y) , ∀t ∈ (0, ∞), ∀x, y ∈ X. It is well known that (X, F, Δp ) is a complete Menger space under the triangular norm Δ = Δp ≥ ΔL . Also it can be easily seen that the mapping f : X → X, ⎧ ⎨ 0, if x = 0, f (x) = ⎩ 1 if x > 0.

is a weak Hicks contraction ∀k ∈ (0, 1). Indeed if Ffxfy (kt) ≤ 1 − kt then (exactly) one of the numbers x and y is 0, which implies Fxy (t) = 0 < 1−t. Note that this mapping f has two fixed points: x = 0 and x = 1. Therefore, unlike the Hicks contraction, a weak Hicks contraction is not necessarily a Picard operator (i.e., the mapping f : X → X is a weakly Picard operator and the set of its fixed points contains a unique element) on a metric space (X, k) where k(p, q) = sup{t : t ≤ 1 − Fpq (t)}. Example 7. Let X = {a, b, c} and define Fab (t) = 0 for all a, b ∈ X, a = b and t > 0. It is easy to see that (X, F, ΔL ) is a complete Menger space. Consider the mapping f : X → X defined by f (a) = a, f (b) = c, f (c) = b. Since t ∈ (0, 1), Fab (t) > 1 − t =⇒ a = b =⇒ Ffa,fb (kt) = 1, f is a weak-Hicks contraction. Note that f is not a weakly Picard operator in (X, F), for the sequence (f n (b)) does not converge. Thus the class of weak-Hicks contractions strictly contains the class of Hicks-contraction. The following results are needed to prove the existence of fixed point for weak Hicks contraction mapping. Lemma 4. Every weak-Hicks contraction is (uniformly) continuous.

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Proof. Indeed, if ε > 0 and λ ∈ (0, 1) are given, we choose δ ∈ (0, 1) such that 0 < kδ < min{ε , λ}. Then (p, q) ∈ Vδ =⇒ Fpq (δ ) > 1 − δ =⇒ Ffpfq (kδ ) > 1 − kδ =⇒ Ffpfq (kδ ) > 1 − λ =⇒ (fp, fq) ∈ Uε ,λ , where Uε ,λ = {(x, y) ∈ X × X : Fx,y (ε ) > 1 − λ} and Vδ = Uδ ,δ .



Definition 12. We say that the t-norm Δ is geometrically convergent (g-convergent) if for ∞ (1 − qi ) = 1. all q ∈ (0, 1) lim Δi=n n→∞

ΔL and the t-norm of Hadzic type are examples of g-convergent t-norms. It is proved in ∞ (1 − qi0 ) = 1 for some q0 ∈ (0, 1), then Δ is g-convergent. Also note that if Δ is [21], if lim Δi=n n→∞

g-convergent then supt 0. Proof. The direct implication is obvious, for Fuu (t) = 1, ∀t > 0. For the converse implication, first we show that there exists δ ∈ (0, 1) such that Fxf (x) (δ ) > 1 − δ . Indeed, if we suppose that Fxf (x) (δ ) ≤ 1 − δ for all δ ∈ (0, 1), then Fxf (x) (1) ≤ 0 (for Fxf (x) is left continuous), which is a contradiction. Now, using (w − H), we can prove by induction that Ff n (x)f n+1 (x) (kn δ ) > 1 − kn δ , ∀n ∈ N. Next we will show that the sequence (f n (x))n∈N is a Cauchy sequence, that is, ∀ε > 0, ∀λ ∈ (0, 1)∃n0 (= n0 (ε , λ)) : Ff n (x)f n+m (x) (ε ) > 1 − λ, ∀n ≥ n0 , ∀m ∈ N. Let ε > 0 and λ ∈ (0, 1) be given. Since the series ∑∞n=1 kn δ is convergent, there exists n1 (= n1 (ε )) such that ∑∞n=n1 kn δ < ε . Then, for all n ≥ n1 and m ∈ N we have ∞

Ff n (x)f n+m (x) (ε ) ≥ Ff n (x)f n+m (x) ( ∑ ki δ ) ≥ i=n1

Ff n (x)f n+m (x) (

n+m−1

∑

m ki δ ) ≥ Δi=1 xn+i−1

i=n ∞ where xj := Ff j (x)f j+1 (x) (k δ ) satisfies xj ≥ 1 − kj δ , ∀j ≥ n. Let n2 be such that Δi=1 (1 − kn2 +i−1 ) > 1−λ (such a number n2 does exist, for Δ is g-convergent). Then, for all n ≥ n0 := Max{n1 , n2 } and m ∈ N, j

m m (1 − kn+i−1 δ ) ≥ Δi=1 (1 − kn+i−1 ) ≥ Ff n (x)f n+m (x) (ε ) ≥ Δi=1 m ∞ Δi=1 (1 − kn2 +i−1 ) ≥ Δi=1 (1 − kn2 +i−1 ) > 1 − λ.

Thus, (f n (x))n∈N is a Cauchy sequence. By the completeness of (X, F, Δ ) and the continuity of f it follows that (f n (x))n∈N converges to a fixed point of f . This completes the proof.  Corollary 7. ([35]) Let (S, F, Δ ) be a complete Menger space where either Δ ≥ ΔL or Δ is of Hadzic type and f : X → X be a weak-Hicks contraction. If Fpf (p) (1) > 0 for some p ∈ X, then f has a fixed point.

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5.7 Non-linear Probabilistic Contractions and Fixed Points A natural generalization of the notion of probabilistic B-contraction is a notion of a ϕ probabilistic contraction or a nonlinear probabilistic contraction. Definition 13. Let f be a self map of a metric space (X, d) and ϕ : [0, ∞) → [0, ∞) such that ϕ (t) < t for all t > 0 . We say that f is a ϕ -contraction if d(fx, fy) ≤ ϕ (d(x, y)) , for all x, y ∈ X

(5.8)

Definition 14. A self map f of a metric space (X, d) is called a Matkowski contraction if f is a ϕ -contraction with a nondecreasing function ϕ such that limn→∞ ϕ n (t) = 0 for any t > 0. Notice that, in general, there is no relationship between the condition limn→∞ ϕ n (t) = 0 for any t > 0 and the condition ϕ (t) < t for all t > 0. However, Matkowski [32] showed that when ϕ is nondecreasing the first condition implies the second one. Definition 15. A nonlinear probabilistic contraction or ϕ -contraction on a Menger PM space (X, F, Δ ) is a mapping f : X → X satisfying Ffx ,fy (ϕ (t)) ≥ Fx,y (t), ∀ x, y ∈ X, t > 0,

(5.9)

for some given function ϕ : [0, ∞) → [0, ∞) satisfies ϕ (t) < t for all t > 0. Clearly, if ϕ (t) = kt, where 0 < k < 1, then a ϕ -probabilistic contraction is a B-contraction. The study of fixed point theorems for ϕ -probabilistic contraction is scattered mainly in ([12, 46]) in which the function ϕ is assumed to satisfy ∞

∑ ϕ n (t) < ∞

(Phi)

n=1

for all t > 0 and some other conditions. It has been observed that the hypothesis Φ ∈ ϕ is very strong and difficult for testing in practice, but it is necessary if one wishes to use the methods of proofs which are closely related to the corresponding methods for probabilistic B-contraction. This hypothesis and others which the authors supposed on ϕ are such that, practically only the linear function ϕ (t) = kt, 0 < k < 1 satisfies all of them. However, the proving fixed point technique for probabilistic B-contraction are no longer usable for ϕ -probabilistic contraction, if ϕ does not satisfy the condition Φ . ´ c [7] proposed a systematic study of fixed points for ϕ -probabilistic conIn 2010, Ciri´ traction by introducing new approach, substantially different from previous techniques for cases where a mapping satisfies the probabilistic linear contraction condition. He established the following result. Theorem 15. Let (X, F, Δ ) be a complete Menger probabilistic metric space with a continuous tnorm Δ of H-type and let ϕ : [0, ∞) → [0, ∞) be a function satisfying the condition (CBW):

ϕ (0) = 0, ϕ (t) < t and lim+ inf ϕ (t) < t for all t > 0 r→t

If f : X → X is a probabilistic ϕ -contraction, then f has a unique fixed point x∗ ∈ X and {f n (x0 )} converges to x∗ for each x0 ∈ X.

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However, Jachymski[24] observed some error in the proof of the above Theorem 15 which has been corrected by himself. Further, he gave a characterization of a function ϕ having the property that there exists a probabilistic ϕ -contraction (probabilistic nonlinear contraction) which is not a probabilistic B-contraction and supported the main idea of ´ c [7]. Ciri´ To establish a general fixed point theorem for probabilistic ϕ -contraction we need the following. Lemma 5. For n ∈ N, let gn : (0, ∞) → (0, ∞) and Fn , F : R → [0, 1]. Assume that sup{F(t) : t > 0} = 1 and for any t > 0, lim gn (t) = 0 and Fn (gn (t)) ≥ F(t). n→∞

If each Fn is nondecreasing, then limn→∞ Fn (t) = 1 for any t > 0. Proof. Fix t > 0 and ε > 0. By hypothesis, there is t0 > 0 such that F(t0 ) > 1 − ε . Since gn (t0 ) → 0, there is k ∈ N such that gn (t0 ) < t for all n ≥ k. By monotonicity, Fn (t) ≥ Fn (gn (t0 )) ≥ F(t0 ) > 1 − ε for n ≥ k. Hence we infer that limn→∞ Fn (t) = 1, since Fn (t) ≤ 1.



Theorem 16. Let (X, F, Δ ) be a complete probabilistic metric space such that Δ is a continuous triangular norm of H-type. Let a function ϕ : [0, ∞) → [0, ∞) be such that, for any t > 0, 0 < ϕ (t) < t and lim ϕ n (t) = 0. n→∞

If f : X → X is a probabilistic ϕ -contraction, then f has a unique fixed point x∗ ,and, for any x0 ∈ X, limn→∞ f n (x0 ) = x∗ . Proof. Let x0 ∈ X and xn = fxn−1 for n ∈ N. Observe that, for any t > 0, the sequence (Fxn ,xn+1 (ϕ n (t)))∞n=0 is nondecreasing. Indeed, given n ∈ N, ϕ n−1 (t) > 0; so, by (5.13) we get Fxn ,xn+1 (ϕ n (t)) = Ffxn−1 ,fxn (ϕ (ϕ n−1 (t))) ≥ Fxn−1 ,xn (ϕ n−1 (t)) Hence, we infer that Fxn ,xn+1 (ϕ n (t)) ≥ Fx0 ,x1 (t) ; so, by Lemma 5, we have lim Fxn ,xn+1 (t) = 1 for any t > 0

n→∞

(5.10)

Now let n ∈ N and t > 0. We show by induction that, for any k ∈ N ∪ {0} Fxn ,xn+k (t) ≥ Δ k (Fxn ,xn+1 (t − ϕ (t))).

(5.11)

This is obvious for k = 0, since Fxn ,xn = 1. Assume that (5.10) holds for some k. Hence, by (5.11) and the monotonicity of Δ , we have Fxn ,xn+k+1 (t) = Fxn ,xn+k+1 ((t − ϕ (t)) + ϕ (t)) ≥ Δ (Fxn ,xn+1 (t − ϕ (t)), Fxn+1 ,xn+k+1 (ϕ (t))) ≥ Δ (Fxn ,xn+1 (t − ϕ (t)), Fxn ,xn+k (t)) ≥ Δ (Fxn ,xn+1 (t − ϕ (t)), Δ k (Fxn ,xn+1 (t − ϕ (t)))) = Δ k+1 (Fxn ,xn+1 (t − ϕ (t))) which completes the induction. We show that xn is a Cauchy sequence, that is, limm,n→∞ Fxn ,xm (t) = 1 for any t > 0. Let t > 0

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and ε > 0. By hypothesis, {Δ n : n ∈ N} is equicontinuous at 1 and Δ n (1) = 1, so there is δ > 0 such that (5.12) if s ∈ (1 − δ , 1], then Δ n (s) > 1 − ε for all n ∈ N. Since, by , limn→∞ Fxn ,xn+1 (t − ϕ (t)) = 1, there is n0 ∈ N such that, for any n ≥ n0 , Fxn ,xn+1 (t − ϕ (t)) ∈ (1 − δ , 1]. Hence, by (5.10) and (5.12), we get Fxn ,xn+k (t) > 1 − ε for any k ∈ N ∪ {0}. This proves the Cauchy condition for xn . By completeness, xn converges to some x∗ ∈ X, that is, limn→∞ Fxn ,x∗ (t) = 1 for any t > 0. We show that x∗ is a fixed point of f . By monotonicity and continuity of Δ , we get Fx∗ ,fx∗ (t) ≥ Δ (Fx∗ ,xn+1 (t − ϕ (t)), Ffxn ,fx∗ (ϕ (t))) ≥ Δ (Fx∗ ,xn+1 (t − ϕ (t)), Fxn ,x∗ (t)) → Δ (1, 1) = 1. This yields Fx∗ ,fx∗ (t) = 1 for any t > 0, and hence x∗ = fx∗ . Finally, we show the uniqueness of a fixed point. Assume that y∗ = fy∗ . Fix t > 0. Then the sequence Fx∗ ,y∗ (ϕ n (t))∞n=0 is non-decreasing. Indeed, given n ∈ N, we have Fx∗ ,y∗ (ϕ n (t)) = Ffx∗ ,fy∗ (ϕ (ϕ n−1 (t))) ≥ Fx∗ ,y∗ (ϕ n−1 (t)), which yields Fx∗ ,y∗ (ϕ n (t)) ≥ Fx∗ ,y∗ (t). By Lemma 5 (with Fn := Fx∗ ,y∗ ), we get Fx∗ ,y∗ (t) = 1. hence  we infer that x∗ = y∗ Actually, Jachymski established Theorem 16 by assuming that the triangular norm Δ is continuous and that ϕ (t) > 0 for all t > 0. However, Alegre and Romaguera [1] remarked that the proof only uses continuity of Δ at (1, 1) which is satisfied for every triangular norm of H-type, and, on the other hand, condition ϕ (t) > 0 for all t > 0 is automatically satisfied for any probabilistic ϕ -contraction, as Jachymski observed in the first lines of Section 2 of [24]. Consequences, Scopes and Limitations of Theorem 16 1.

Theorem 16 yields Browder’s [5] fixed point theorem. We say that a self mapping f on a metric space (X, d) is Browder contraction if f is a ϕ -contraction with a nondecreasing and right continuous function ϕ .

Theorem 17. Let f be a self map of a metric space (X, d). The following statements are equivalent: (i) f is a Browder contraction. (ii) f satisfies the assumption of Theorem 16 with the probabilistic metric (X, d) induced by (X, d), and an upper semicontinuous function ϕ . Proof. It is easily seen that if functions Fx,y are defined as Fx,y (t) = H(t − d(x, y)),

(5.13)

then f is a probabilistic ϕ -contraction iff for any x, y ∈ X and t > 0, d(x, y) < t ⇒ d(fx, fy) < ϕ (t).

(5.14)

(i) ⇒ (ii): Let f be a Browder contraction. By [[23],Th.1], there exists a strictly increasing and continuous function ϕ such that f is a ϕ -contraction. It suffices to verify (5.14). Let x, y ∈ X, t > 0 and d(x, y) < t. Then, by strict monotonicity of ϕ , we have d(fx, fy) ≤ ϕ (d(x, y)) < ϕ (t),

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so (ii) holds. (ii) ⇒ (i): Let f be a probabilistic ϕ -contraction with an upper semicontinuous function ϕ and Fx,y = H(t − d(x, y). We show that f is a ϕ -contraction. Suppose, on the contrary, that d(fx0 , fy0 ) > ϕ (d(x0 , y0 )) for some x0 , y0 ∈ X. By upper semicontinuity of ϕ , there is δ > 0 such that ϕ (t) < d(fx0 , fy0 ) for any t ∈ [t0 − δ , t0 + δ ] where t0 := d(x0 , y0 ). Then we have d(x0 , y0 ) < t0 + δ and d(fx0 , fy0 ) > ϕ (t0 + δ ) which contradicts (5.14). Therefore f is a ϕ -contraction, and by [[23],Th.1], (i) holds.



Remark 4. The part of (i) ⇒ (ii) shows that if f is a ϕ -contraction with a strictly increasing function ϕ , then f is a probabilistic ϕ -contraction on the Menger space induced by (X, d). On the other hand, the part of (ii) ⇒ (i) shows that, if f is a probabilistic ϕ -contraction with an upper semicontinuous function ϕ , then f is a ϕ -contraction. Weakness of Theorem 16: It is not clear whether Theorem 16 yields the result of Boyd and Wong [4]. On the other hand, it is surprising that Theorem 16 does not imply the fixed point theorem of Matkowski [31]. This fact can be understood from the following results. Theorem 18. Let a function ϕ : [0, ∞) → [0, ∞) be nondecreasing and such that limn→∞ ϕ n (t) = 0 for any t > 0. The following statements are equivalent: (i) for any t > 0, lims→t+ ϕ (t) < t; (ii) given a metric space (X, d) and a ϕ -contraction f : X → X, there exists a function ψ : [0, ∞) → [0, ∞) such that 0 < ψ (t) < t and ψ n (t) → 0 for all t > 0, and f is a probabilistic ψ -contraction on the Menger space induced by (X, d). Proof. (i)⇒ (ii): Let f be a ϕ -contraction on (X, d) with ϕ as in (i). By [[23],Th.1 and Th.5], there is a continuous and strictly increasing function ψ : [0, ∞) → [0, ∞) such that ψ (t) < t for t > 0, and f is a ψ -contraction. Then ψ (t) > 0 and lims→t+0 ψ (t) < t for t > 0 (see [5]). By Remark 4, f is a probabilistic ψ -contraction. (ii)⇒ (i):By hypothesis, we infer that ϕ (t) < t for t > 0 (see [31]). Hence lims→t+ ϕ (s) ≤ t. Suppose, on the contrary, that lims→t+0 ϕ (s) ≤ t0 for some t0 > 0. Set X := [0, ∞) and, for x, y ∈ X, d(x, y) := max{x, y} if x = y, and d(x, y) := 0 if x = y. Define f := ϕ . By the monotonicity of ϕ , f is a ϕ -contraction on (X, d) so, by (ii), f is a probabilistic ψ -contraction with ψ as in (ii). By([23],Th. 7), there is δ > 0 such that ϕ (t) = t0 for any t ∈ (t0 , t0 + δ ). Fix t ∈ (t0 , t0 + δ ), and set x :=

(t0 + t) and y := 0. 2

Then d(x, y) < t, so, by 5.14, d(fx, fy) < ψ (t). On the other hand, d(fx, fy) = ϕ (x) = t0 . Thus, for any t ∈ (t0 , t0 + δ ), ψ (t) > t0 and ψ (t) < t < t0 + δ , which means that (t0 , t0 + δ ) is ψ -invariant. Hence ψ n (t0 + δ2 ) ∈ (t0 , t0 + δ ), so ψ n (t0 + δ2 )  0 , a contradiction.  Example 8. Consider the function ϕ : [0, ∞) → [0, ∞):

ϕ (0) := 0, ϕ (t) := 1 for t > 1, ϕ (t) :=

1 1 1 if t ∈ ( , ] for some n ∈ N. 1+n 1+n n

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It is easily seen that ϕ is nondecreasing and limn→∞ ϕ n (t) = 0 for any t > 0. Moreover, lims→t+ ϕ (s) = t for any t ∈ { n1 : n ∈ N}. Hence, by Theorem 18, there exist a metric space (X, d) and a ϕ -contraction f : X → X for which Theorem 16 (with the Menger space induced by (X, d)) cannot be applied. Significance of Theorem 16: The following theorem provides a natural characterization of a function ϕ for the existence of a probabilistic ϕ -contraction (probabilistic nonlinear contraction) which is not a probabilistic B-contraction. Theorem 19. Let a function ϕ : [0, ∞) → [0, ∞) be strictly increasing, right continuous and such that ϕ (t) < t for t > 0. The following statements are equivalent: : t > 0} = 1; (i) sup{ ϕ (t) t (ii) there exists a probabilistic ϕ -contraction on a complete Menger space, which is not a probabilistic k-contraction for any k ∈ (0, 1). Proof. (i)⇒ (ii): Let X := [0, ∞) and d be the ultrametric in X defined in the proof of (ii)⇒ (i) of Theorem 18. Since (X, d) is complete, so is the Menger space induced by (X, d). Let f := ϕ . Then f is a ϕ -contraction, so, by Remark 4, f is a probabilistic ϕ -contraction. Suppose, on the contrary, that f is a probabilistic k-contraction for some k ∈ (0, 1). By Remark 4, f is a Banach k-contraction, so, in particular, d(fx, f 0) ≤ kd(x, 0) for x > 0. : x > 0} < 1 , a contradiction. Hence ϕ (x) ≤ kx, which yields sup{ ϕ (x) x (ii)⇒ (i): Suppose, on the contrary, that ϕ (t) ≤ kt for any t ∈ [0, ∞) and some k ∈ (0, 1). Let f be a mapping on a Menger space (X, F, Δ ) as in (ii). Let x, y ∈ X and t > 0. Since Ffx,fy is nondecreasing and 5.9 holds, we get that Ffx,fy (kt) ≥ Ffx,fy (ϕ (t)) ≥ Fx,y (t), which yields that f is a probabilistic k-contraction. This contradicts (ii).



In 2015, Fang [13] attempted to improve Theorem 16 (Theorem of Jachymski [24]) by introducing a new class of function ϕw as follows: The class of function ϕ : [0, ∞) → [0, ∞) satisfying the condition: for each t > 0 there exist r ≥ t such that limn→∞ ϕ n (r) = 0 will be denoted by ϕw . Theorem 20. Let (X, F, Δ ) be a complete Menger space with Δ a t-norm of H-type. If f : X → X is a probabilistic ϕ -contraction with ϕ ∈ ϕw , then f has a unique fixed point. However, very recently Gregory et al. [16] showed that the fixed point theorems of Fang and Jachymski are equivalent by proving equivalence between the classes of gauge functions ϕ and ϕw considered in ([13, 24]). To this end, we will consider here three classes of gauge function: (i) Ψ will denote the class of gauge functions ψ satisfying: 0 < ψ (t) < t and lim ψ n (t) = 0 for all t > 0. n→∞

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(ii) Φ will denote the class of gauge functions ϕ satisfying: lim ϕ n (t) = 0 for all t > 0.

n→∞

(iii) Φw will denote the class of gauge functions ϕw satisfying: for each t > 0 there exists r ≥ t such that limn→∞ ϕwn (r) = 0. Obviously, Ψ ⊂ Φ ⊂ Φw and it is easy to verify that these inclusions are strict. Theorem 21. Let (X, F, Δ ) be a Menger space and let f : X → X be a mapping. Then, f is a probabilistic ϕw -contraction for some ϕw ∈ Φw , if and only if f is a probabilistic ϕ -contraction, for some ϕ ∈ Φ. Proof. Suppose that f : X → X is a probabilistic ϕw -contraction, for some ϕw ∈ φw . Then, Ffx,fy (ϕw (t)) ≥ Fx,y (t) for each x, y ∈ X and t > 0. Denote A := {t > 0 : limn→∞ ϕwn (t) = 0}. For each t ∈ A we consider the set Bt = {r > t : limn→∞ ϕwn (r) = 0}, which is a nonempty set since ϕw ∈ Φw . Then, by the Axiom of Choice, we can take an element rt ∈ Bt , for each t ∈ A. Obviously, rt ∈ / A, for each t ∈ A. Now, we will construct a gauge function ϕ ∈ Φ , such that f is a probabilistic ϕ -contraction, for ϕ , as follows. Define  ϕw (t), if t ∈ /A ϕ (t) = ϕw (rt ), if x ∈ A Obviously, ϕ is well-defined. We will prove that limn ϕ n (t) = 0 for all t > 0 in three steps. / A for all i ∈ N. Indeed, let t ∈ / A and i ∈ N. (I) First we prove that if t ∈ / A then ϕwi (t) ∈ n i Then, limn ϕw (ϕw (t)) = limn ϕwn+i (t) = 0, and so ϕwi (t) ∈ / A. (II) Suppose that t ∈ / A. We will prove, by induction on n, that ϕ n (t) = ϕwn (t) for all n ∈ N. Indeed, ϕ (t) = ϕw (t) since t ∈ / A. Suppose the assertion is true for n. By induction hypothesis and by (I) we have that

ϕ n+1 (t) = ϕ (ϕ n (t)) = ϕ (ϕwn (t)) = ϕw (ϕwn (t)) = ϕwn+1 (t) Therefore, limn ϕ n (t) = limn ϕwn (t) = 0. (III) Suppose that t ∈ A. We will prove, by induction on n, that ϕ n (t) = ϕwn (rt ) for all n ∈ N. The assertion is true for n = 1 by definition of ϕ . Suppose the assertion is true for n. By induction hypothesis and by (I) we have that

ϕ n+1 (t) = ϕ (ϕ n (t)) = ϕ (ϕwn (rt )) = ϕw (ϕwn (t)) = ϕwn+1 (t) Therefore, limn ϕ n (t) = limn ϕwn (rt ) = 0.

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Thus, limn→∞ ϕ n (t) = 0 for each t > 0, and then ϕ ∈ Φ . Finally, we will see that f is a probabilistic ϕ -contraction for this ϕ ∈ Φ . Let x, y ∈ X and t > 0. We distinguish two cases: 1. 2.

If t ∈ / A, then Ffx,fy (ϕ (t)) = Ffx,fy (ϕw (t)) ≥ Fx,y (t). If t ∈ A, then Ffx,fy (ϕ (t)) = Ffx,fy (ϕw (rt )) ≥ Fx,y (rt ) ≥ Fx,y (t), since rt > t and Fx,y is nondecreasing.

Therefore, f is a probabilistic ϕ -contraction, for ϕ ∈ Φ . The converse is obvious since Φ ∈ Φw .



Theorem 22. Let (X, F, Δ ) be a Menger space and let f : X → X be a mapping. Then, f is a probabilistic ϕ -contraction for some ϕ ∈ Φ , if and only if f is a probabilistic ψ -contraction, for some ψ ∈Ψ. Proof. Let (X, F, Δ ) be a Menger space and suppose that f : X → X is a probabilistic ϕ contraction, for some ϕ ∈ Φ . Then, Ffx,fy (ϕ (t)) ≥ Fx,y (t) for each x, y ∈ X and t > 0. We claim that for all t > 0 we can find r > t such that 0 < ϕ (r) < t. Indeed, suppose there exists t0 > 0 such that ϕ (r) ≥ t0 for each r ≥ t0 . Then, for a fixed r0 ≥ t0 we have that ϕ (r0 ) ≥ t0 and so ϕ 2 (r0 ) ≥ t0 . Proceeding inductively on n we obtain ϕ n (r0 ) ≥ t0 for each n ∈ N and so, limn→∞ ϕ n (r0 ) ≥ t0 , a contradiction. Besides, ϕ (r) > 0. Indeed, if ϕ (r) = 0 we have that 0 = Ffx,fx (ϕ (r)) ≥ Fx,x (r) = 1, since f is a probabilistic ϕ -contraction, a contradiction. Therefore, 0 < ϕ (r) < t. Now, for each t > 0 we consider the set Rt = r ≥ t : 0 < ϕ (r) < t, which is a nonempty set, as we have just seen. For each t > 0, we define t + inf{ϕ (r) : r ∈ Rt } . ψ (t) = 2 The function ψ is well-defined, since for each r ∈ Rt we have that ϕ (r) > 0 and clearly ψ (t) > 0, for each t > 0. Next, we will see that ψ ∈ Ψ in six steps. (I) First, note that for each t > 0 we have that 0 < ψ (t) < t. Indeed, by definition, for each t > 0 we have that t+t t = t. 0 < ≤ ψ (t) < 2 2 (II) The function ψ is nondecreasing. Indeed, if we suppose that for some 0 < s < t we have that ψ (s) > ψ (t), then inf{ϕ (r) : r ∈ Rs } > inf{ϕ (r) : r ∈ Rt }. Therefore, we can find rt ∈ Rt such that 0 < ϕ (rt ) < inf{ϕ (r) : r ∈ Rs }. Taking into account that inf{ϕ (r) : r ∈ Rs } < s, then rt ∈ Rs , since rt ≥ t > s and 0 < ϕ (rt ) < s, and hence ϕ (rt ) > inf{ϕ (r) : r ∈ Rs }, a contradiction. (III) The sequence {ψ n (t)}n is strictly decreasing for each t > 0. It is obvious, since for each t > 0 we have that 0 < ψ (t) < t. Consequently: (IV) {ψ n (t)}n converges to the infimum of the sequence in [0, ∞) with respect to the usual topology of R. (V) It is fulfilled that limn ψ n (s) ≤ limn ψ n (t) whenever 0 < s < t. This assertion is obvious since both limits exist and ψ is nondecreasing.

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(VI) We will see that limn ψ n (t) = 0 for each t > 0. Suppose that limn ψ n (t0 ) = a > 0 for some t0 > 0. We assert that limn ψ n (s) ≥ a for each s > a. Indeed, if s > a then we can find ns ∈ N such that ψ i (t0 ) ∈ (a, s) for all i ≥ ns . In particular a < ψ ns (t0 ) < s. Then, by (V ) we have that a = limn ψ n (ψ ns (t0 )) ≤ limn ψ n (s). We claim that there exists t > a such that ψ (t) < a. Indeed, given a > 0 we can find, as we have seen above, ra > a such that 0 < ϕ (ra ) < a. Now, note that for each t ∈ (a, ra ) we have that ra ∈ Rt , since ra ≥ t and 0 < ϕ (ra ) < a < t. Take δ > 0 such that δ < min{a − ϕ (ra ), ra − a}. Then, for each t ∈ (a, a + δ ) we have

ψ (t) =

t + ϕ (ra ) a + δ + a − δ < =a 2 2

Choose t > a such that ψ (t) < a. Then, limn ψ n (t) ≥ a, since t > a. On the other hand, {ψ n (t)}n is a strictly decreasing sequence with ψ (t) < a, a contradiction. Therefore, limn ψ n (t) = 0 for each t > 0, and so ψ ∈ Ψ . Finally we will see that f is a probabilistic ψ -contraction for ψ ∈ Ψ . First, note that for all t > 0 we can find rt ∈ Rt such that 0 < ϕ (rt ) < ψ (t). Contrarily, suppose that there exists t0 > 0 such that for all r ∈ Rt0 we have that ϕ (r) > ψ (t0 ). Then, inf{ϕ (r) : r ∈ Rt0 } ≥ ψ (t0 ) =

t0 + inf{ϕ (r) : r ∈ Rt0 } 2

and so inf{ϕ (r) : r ∈ Rt0 } ≥ t0 . Now, by definition t0 > inf{ϕ (r) : r ∈ Rt0 }, a contradiction. Let x, y ∈ X and t > 0. By the last paragraph we can take rt ∈ Rt such that 0 < ϕ (rt ) < ψ (t). Then, Ffx,fy (ψ (t)) ≥ Ffx,fy (ϕ (rt )) ≥ Fx,y (rt ) ≥ Fx,y (t). Therefore, f is a probabilistic ψ -contraction, for ψ ∈ Ψ . The converse is obvious since Ψ ⊂ Φ .



As a consequence of these two last theorems we obtain the next corollary. Corollary 8. Let (X, F, Δ ) be a Menger space and let f : X → X be a mapping. They are equivalent: (I) f is a probabilistic ψ -contraction, for some ψ ∈ Ψ . (II) f is a probabilistic ϕw -contraction, for some ϕw ∈ Φw . Consequently, Theorem 16 and Theorem 18 are equivalent.

5.8 Altering Distance Functions in Probabilistic Metric Spaces and Fixed Points In metric fixed point theory a new direction of generalizing the Banach contraction principle was opened by Khan et al. [28] by introducing a new contraction principle through a control function which they called an altering distance function. Definition 16. A function ψ : [0, ∞) → [0, ∞) is called an altering distance function if it has the following properties:

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(i) ψ is monotone increasing and continuous (ii) ψ (t) = 0 iff t = 0. Theorem 23. Let (X, d) be a complete metric space, ψ be an altering distance function and f : X → X be a self map satisfying

ψ (d(fx, fy)) ≤ cψ (d(x, y))

(5.15)

for all x, y ∈ X and for some 0 < c < 1. Then f has a unique fixed point. In 2008, Choudhury and Das [8] extended the notion of altering distance functions to probabilistic metric space. This extension of altering distance functions has been called Φ function, and the class is denoted by Φ . Definition 17. A function φ : [0, ∞) → [0, ∞) is said to be a Φ function if it satisfy the following conditions: (i) (ii) (iii) (iv)

φ (t) = 0 iff t = 0 φ is strictly increasing and φ (t) → ∞ as t → ∞ φ is left continuous in (0, ∞) φ is continuous at 0.

Definition 18. Let (X, F, Δ ) be a Menger space. A self map f : X → X is said to be a φ contractive if t



(5.16) Ffx,fy φ (t) ≥ Fx,y φ c where 0 < c < 1, x, y ∈ X and t > 0 and φ ∈ Φ . Theorem 24. Let (X, F, Δ ) be a complete Menger space with continuous t-norm and f : X → X be a φ -contractive and for some x0 ∈ X the sequence {xn } in X be constructed as xn = fxn−1 , n = 1, 2, . . .. If the sequence {xn } converges, then it converges to a unique fixed point of f . Proof. In view of the conditions (i) and (iv) in Definition 17, for s > 0 we can find a positive number r such that s > φ (r). Then for s > 0, we have Fxn ,xn+1 (s) ≥ Ffxn−1 ,fxn (φ (r)) r

≥ Fxn ,xn+1 φ c r

= Ffxn−1 ,fxn φ c r

≥ Fxn−1 ,xn φ 2 c ··· r

≥ Fx0 ,x1 φ n . c Therefore,

r

Fxn ,xn+1 (s) ≥ Fx0 ,x1 φ n c

(5.17)

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Taking n → ∞, we have Fxn ,xn+1 (s) → 1 as n → ∞. By the hypothesis of the theorem, {xn } is convergent. Let xn → z as n → ∞.

(5.18)

We next prove that z is a fixed point of f . From the property of φ , it follows that given ε > 0 we can find ε1 > 0 such that

ε > φ (ε1 ) > 0. Then for all n = 0, 1, 2, 3, . . . , Ffz,z (ε ) ≥ Δ (Ffz,xn (φ (ε1 )), Fxn ,z (ε − φ (ε1 ))) = Δ (Ffz,fxn−1 (φ (ε1 )), Fxn ,z (ε − φ (ε1 ))) ε



1 , Fxn ,z (ε − φ (ε1 )) . ≥ Δ Fz,xn−1 φ c  Making n → ∞ in the above inequality, by virtue of (5.18) and the fact that Δ is continuous t-norm, we have for all ε > 0, Ffz,z (ε ) = 1, that is, fz = z. We next show that the fixed point is unique. If possible, let u and v be two fixed points. As in the above corresponding to ε > 0, we make a choice of ε1 > 0 such that ε > ϕ (ε1 ). Then Fu,v ≥ Ffu,fv (ε ) ≥ Ffu,fv (φ (ε )) ε

1 ≥ Fu,v φ c ε

1 = Ffu,fv φ c ε

1 ≥ Fu,v φ 2 . c Proceeding as above we obtain for any ε > 0, Fu,v (ε ) ≥ Fu,v (φ ( cεn1 )) → 1 as n → ∞. Therefore u = v. This proves the uniqueness of the fixed point. Theorem 25. Let (X, F, ΔM ) be a complete Menger space with continuous t-norm ΔM given by ΔM (a, b) = min{a, b} and f : X → X be φ -contractive. Then f has a unique fixed point. Proof. Let x0 ∈ X. Now we construct a sequence {xn } in X as follows: xn = fxn−1 , n = 1, 2, 3, . . . In view of Theorem 24, the proof of the theorem is complete if we can prove that {xn } is a Cauchy sequence. If {xn } is not a Cauchy sequence, then there exist ε > 0, λ > 0 such that for every positive integer k there exist positive integers m(k), n(k) ≥ k such that Fxm(k) ,xn(k) (ε ) < 1 − λ.

(5.19)

We can choose m(k) < n(k) and n(k) to be the smallest integer corresponding to m(k) satisfying the above inequality. From the above statement it follows that there exist ε > 0,

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λ > 0 for which we can construct increasing sequences of integers n(k) and m(k) satisfying the following: m(k) < n(k) Fxm(k) ,xn(k)−1 (ε ) ≥ 1 − λ

(5.20)

Fxm(k) ,xn(k) (ε ) < 1 − λ

(5.21)

and Equivalently, the construction is finding a point xn(k) in the sequence with n(k) > m(k) which will fall outside the set {z : Fxm(k) ,z (ε ) ≥ 1 − λ} but the point in the sequence preceding the point xn(k) , that is, xn(k−1) , will fall inside the set. This is guaranteed by the fact the sequence is assumed not to be a Cauchy sequence. Since {x : Fx,p (ε1 ) ≥ 1 − λ} ⊆ {x : Fx,p (ε2 ) ≥ 1 − λ} for all p ∈ X, λ > 0 and 0 < ε1 < ε2 , it follows that whenever the above construction is possible for ε > 0, λ > 0, it is also possible to construct {xm(k) } and {xn(k) } satisfying (5.20) and (5.21) corresponding to ε  > 0, λ > 0 whenever ε  < ε . Again from the properties of ϕ it is immediate that given ε > 0 we can find ε  > 0 such that ε > φ (ε  ). In view of the above paragraph we make a choice of ε in (5.20) and (5.21) such that ε = φ (ε1 ) for some ε1 > 0 such that φ ( εc1 ) > ϕ (ε1 ). Such a choice is possible by virtue of conditions (i) and (ii) of Definition 17 . We have, from (5.20) and (5.21), Fxm(k) ,xn(k)−1 (φ (ε1 )) ≥ 1 − λ,

(5.22)

Fxm(k) ,xn(k) (φ (ε1 )) < 1 − λ

(5.23)

but

Then 1 − λ > Fxm(k) ,xn(k) (φ (ε1 )) = Fxm(k)−1 ,xn(k)−1 (φ (ε1 )) ε

1 ≥ Fxm(k)−1 ,xn(k)−1 φ . c Since φ ( εc1 ) > φ (ε1 ) we make a choice of the positive number η such that η < φ ( εc1 ) − φ (ε1 ), that is, φ ( εc1 ) − η > φ (ε1 ). In view of (5.17) in Theorem 24, we may choose k large enough so that Fxm(k ) , xm(k)−1 (η ) > 1 − λ1 for given 0 < λ1 < λ. With the above choice of η and k, we obtain by virtue of (5.20) and (5.23) and from the above inequality that ε

1 1 − λ > Fxm(k)−1 ,xn(k)−1 φ c ε



1 − η , Fxm(k)−1 ,xm(k) (η ) ≥ Δm Fxm(k) ,xn(k)−1 φ c ≥ Δm (Fxm(k) ,xn(k)−1 (φ (ε1 )), Fxm(k)−1 ,xm(k) (η )) ≥ Δm (1 − λ, 1 − λ1 ) = 1 − λ (since λ1 < λ implies 1 − λ < 1 − λ1 ), which is a contradiction. Hence {xn } is a Cauchy sequence. The proof of the theorem is then completed by an application of Theorem 24. 

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In the following we show that Theorem 23 can be derived from Theorem 25. Let ψ be the altering distance function and now define φ as

φ (t) = sup{a : ψ (a) < t} Let t > 0 and x, y ∈ X. If φ (t) − d(x, y) > 0 then we have d(x, y) < φ (t) = sup{a : ψ (a) < t} that is, ψ (d(x, y)) < t (by the continuity of ψ ), t − ψ (d(x, y)) > 0. Conversely, if t − ψ (d(x, y)) > 0 then we have

ψ (d(x, y)) < t that is, d(x, y) < sup{a : ψ (a) < t} = φ (t) (by continuity of ψ ) Therefore,

φ (t) − d(x, y) > 0 iff t − ψ (d(x, y)) > 0

(5.24)

Similarly, we can prove that

φ (t) − d(x, y) ≤ 0 iff t − ψ (d(x, y)) ≤ 0

(5.25)

and from 5.24, 5.25 we have H(φ (t) − d(x, y)) = H(t − ψ (d(x, y)).

(5.26)

Let us assume inequality 5.15 in a complete metric space (X, d). Then we have for all x, y ∈ X, t > 0 and 0 < c < 1, Ffx,fy (φ (t)) = H(φ (t) − d(fx, fy)) = H(t − ψ (d(fx, fy))) t ≥ H( − ψ (d(x, y))) (by inequality 5.15) c t = H(φ ( ) − d(x, y)) c t ≥ Fx,y (φ ( )) c Thus, we see that the inequality 5.15 in a complete metric space (X, d) implies the inequality 5.16 in the corresponding Menger space. By an application of Theorem 25, we can prove Theorem 23. Remark 5. In Theorem 25, a specific form of the t-norm, that is, Δ = Δm . This leads to the following question: Question (Choudhury and Das) : Does Theorem 25 remains true when Δ is an arbitrary continuous t-norm ?

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In 2009, Mihet [39], resolved the above problem by establishing the following theorems: Theorem 26. Let (X, FΔ ) be a complete Menger PM space with Δ continuous and f : X → X be a probabilistic φ -contraction mapping. If there exists x ∈ X such that the orbit of f in x, O(f , x) = {f m x : x ∈ N ∪ {0}}, is probabilistically bounded, then f has a unique fixed point. Proof. Let us define the sequence {xn }n∈N∪{0} by x0 = x and xn = fxn−1 ; n = 1, 2, . . . . In view of the conditions (i) and (iv) in the definition 17, for given s > 0 we can find a positive number r such that s > φ (r). Then, for n, m ∈ N, Fxn ,xn+m (s) ≥ Fxn ,xn+m (φ (r)) = Ffxn−1 ,fxn+m−1 (φ (r)) r r ≥ Fxn−1 ,xn+m+1 (φ ( )) = Ffxn−2 ,fxn+m−2 (φ ( )) c c r r r ≥ Fxn−2 ,xn+m−2 (φ ( 2 )) · · · ≥ Fx,xm (φ ( n )) ≥ DO(f ,x) (φ ( n )) → 1. c c c Thus, the sequence {f n (x)}n∈N is Cauchy. Denote by z its limit and let ε > 0 be given. Using the properties of a Φ -function, we can find a t > 0 such that 0 < φ (t) < ε2 . Then ε

t

≥ Fxn+1 ,fz (φ (t)) ≥ Fxn ,z φ ( ) Ffz,xn+1 2 c Taking the limit as n → ∞ we obtain Ffz,xn+1 ( ε2 ) → 1. Now, taking n → ∞ in ε

ε

, Fxn+1 ,z Ffz,z (ε ) ≥ Δ Ffz,xn+1 2 2 we deduce that Ffz,z (ε ) = 1 for every ε > 0, and hence z = fz. Next we prove the uniqueness of the fixed point. Suppose that there exists w ∈ X such that fw = w and w = z. For any given ε > 0 we can find ε1 > 0 such that ε > φ (ε1 ). Then Fw,z (ε ) = Ffw,fz (ε ) ≥ Ffw,fz (φ (ε1 )) ε

 ε

1 1 = Ffw,fz φ ≥ Fw,z φ c c ε

ε

1 1 ≥ Fw,z φ 2 · · · ≥ Fw,z φ n . c c Letting n → ∞ the above inequality we obtain z = w. This completes the proof.



Corollary 9. Let (X, FΔ ) be a complete Menger PM space under a continuous t-norm H and f : X → X be a φ -contraction such that lim(φ (r) − φ (cr)) = ∞.

r→∞

Then f has a unique fixed point. Proof. We will show that for every p ∈ X the orbit O(f , p) is probabilistically bounded. Let p ∈ X, m ∈ N and r > 0 be given. Then Ff m (p),p (φ (r)) ≥ Δ (Ff m (p),f (p) (φ (cr)), Ff (p),p (φ (r) − φ (cr))) ≥ Δ (Ff m−1 (p),f (p) (φ (cr)), Ff (p),p (φ (r)) − φ (r))) ≥ Δ (Ff m−2 (p),p (φ (r)), Δ 2 (Ff (p),p (φ (r) − φ (r)))) ≥ · · · ≥ Δ m (Ff (p),p (φ (r) − φ (r))). Since Ff (p),p ∈ D+ , it follows that limr→∞ Ff (p),p (φ (r) − φ (r)) = 1. From the equicontinuity of  {Δ m } at 1 we obtain that DO(f ,p) ∈ D+ .

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Theorem 27. Let (X, F, Δ ) be a compact Menger PM space under a t-norm Δ satisfying supa 0

n→∞

We have seen that

lim Fxn ,xn+1 (s) = 1 ∀ s > 0

n→∞

Let s > 0 and ε > 0 be given. Since supa 1 − ε . We can find k0 ∈ N such that, for every j ≥ k0 , Fxnj ,x (s) > 1 − δ and

Fxnj ,xnj +1 (s) > 1 − δ

Therefore we obtain Fxnj +1 ,x (2s) ≥ Δ (Fxnj +1 ,xnj (s)) ≥ Δ (1 − δ , 1 − δ ) > 1 − ε , ∀ j ≥ k0 Hence

xnj +1 → x

Let again s > 0 and ε > 0 be given and δ ∈ (0, 1) be such that Δ (1 − δ , 1 − δ ) > 1 − ε . We can find a t > 0, such that 0 < φ (t) < 2s . Then s t Ffx,xnj +1 ( ) ≥ Fxnj +1 ,fx (φ (t)) ≥ Fxnj ,x (φ ( )). 2 c Thus, there is an n0 ∈ N such that Ffx,xnj+1 ( 2s ) > 1 − δ and Fxnj+1 ,x ( 2s ) > 1 − δ , ∀ j > n0 . Now, from s s Ffx,x (s) ≥ Δ (Ffx,xnj +1 ( ), Fxnj+1 ,x ( )) 2 2 one obtains Ffx,x (s) > 1 − ε and thus fx = x.  Theorem 6.24 has further been generalized and extended by Choudhary and Dass in [9], whereas Babacev in [3] extended and improved the results of Choudhary and Das for a nonlinear generalized contraction wherein she used the associated t-norm as min norm as follows: Theorem 28. Let (X, F, Δ ) be a complete Menger PM-space with a continuous t-norm Δ which satisfies Δ (a, a) ≥ a for every a ∈ [0, 1]. Let c ∈ (0, 1) be fixed. If for a φ -function φ and a self mapping f on X,





t t t Ffx,fy (φ (t)) ≥ min{Fx,y φ , Fx,fx φ , Fy,fy φ , c c c



t t Fx,fy φ , Fy,fx φ } c c holds for every x, y ∈ X and for all t > 0, then f has a unique fixed point in X.

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Another different generalization is presented in [26], where the authors give a common fixed point theorem for two R-weakly commuting self-mappings satisfying nonlinear contractive type condition defined using a Φ -function. They used a proof very similar to ones used here. Very recently, Gopal et al. [15], introduced two new classes of contractive mappings (following the idea of Samet et al. [50]) and established corresponding fixed point theorems for such contractive mappings which extend, generalize and improve the results of Babacev [3] and Dutta et al. [10] Here we denote by Ψ the class of all nondecreasing functions ψ : R+ → R such that ψ is continuous at point 0, ψ (0) = 0 and ψ n (an ) → 0 whenever an → 0 as n → ∞. Definition 19. Let (X, F, Δ ) be a Menger PM-space and f : X → X be a given mapping. We say that f is an α -ψ -type contractive mapping if there exist two functions α : X × X × (0, ∞) → R+ and ψ ∈ Ψ satisfying the following inequality

1 1 −1 ≤ ψ −1 , (5.27) α (x, y, t) Ffx,fy (φ (ct)) Fx,y (φ (t)) for all x, y ∈ X and for all t > 0 such that Fx,y (φ (t)) > 0, where c ∈ (0, 1) and φ ∈ Φ . Remark 6. If α (x, y, t) = 1 for all x, y ∈ X and for all t > 0, then condition 5.29 reduces to the contractive condition given in [10]. This implies that a mapping satisfying the contractive condition of [10] is an α -ψ -type contractive mapping but the converse is not necessarily true (see Example 9 given below). Definition 20. Let (X, F, Δ ) be a PM-space, f : X → X be a given mapping and α : X × X × (0, ∞) → R+ be a function, we say that f is α -admissible if x, y ∈ X, for all t > 0, α (x, y, t) ≥ 1 =⇒ α (fx, fy, t) ≥ 1. Theorem 29. Let (X, F, Δ ) be a G-complete Menger PM-space and f : X → X be an α -ψ -type contractive mapping satisfying the following conditions: (i) f is α -admissible, (ii) there exists x0 ∈ X such that α (x0 , fx0 , t) ≥ 1, for all t > 0, (iii) if {xn } is a sequence in X such that α (xn , xn+1 , t) ≥ 1 for all n ∈ N and for all t > 0, and xn → x as n → ∞, then α (xn , x, t) ≥ 1 for all n ∈ N and for all t > 0. Then f has a fixed point, that is, there exists a point u ∈ X such that fu = u. Proof. Let x0 ∈ X be such that α (x0 , fx0 , t) ≥ 1 for all t > 0. Define a sequence {xn } in X so that xn+1 = fxn , for all n ∈ N. Clearly, we suppose xn+1 = xn for all n ∈ N, otherwise f has trivially a fixed point. Then, by using the fact that f is α -admissible, we write

α (x0 , fx0 , t) = α (x0 , x1 , t) ≥ 1 =⇒ α (fx0 , fx1 , t) = α (x1 , x2 , t) ≥ 1 and, by induction, we get

α (xn , xn+1 , t) ≥ 1, for all n ∈ N and for all t > 0.

(5.28)

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From the properties of function φ , we can find t > 0 such that Fx0 ,x1 (φ (t)) > 0. Then by an application of 5.27, we have 1 1 −1 = −1 Fx1 ,x2 (φ (ct)) Ffx0 ,fx1 (φ (ct))

1 ≤ α (x0 , x1 , t) −1 Ffx0 ,fx1 (φ (ct))

1 ≤ψ −1 . (5.29) Fx0 ,x1 (φ (t))    Obviously, Fx0 ,x1 (φ (t)) > 0 implies Fx0 ,x1 φ ct > 0 and so, again by applying 5.27, we get 1 1 −1 = −1 Fx1 ,x2 (φ (t)) Ffx0 ,fx1 (φ (t))

1 −1 ≤ α (x0 , x1 , t) Ffx0 ,fx1 (φ (t))   1    − 1 . ≤ψ Fx0 ,x1 φ ct Repeating the above procedure successively n times, we obtain   1 1    − 1 . − 1 ≤ ψn Fxn ,xn+1 (φ (t)) Fx0 ,x1 φ ctn Also, 5.29 implies that Fx1 ,x2 (φ (ct)) > 0. Then, following the above procedure, we have   1 1 n−1   ct  − 1 . −1 ≤ ψ Fxn ,xn+1 (φ (ct)) Fx1 ,x2 φ cn−1 In general, if we repeat the above step r times with r < n, we get   1 1 n−r   r  − 1 . −1 ≤ ψ Fxn ,xn+1 (φ (cr t)) Fxr ,xr+1 φ ccn−rt

(5.30)

(5.31)

(5.32)

Since ψ n (an ) → 0 whenever an → 0, then from 5.32, for all r > 0 we deduce that Fxn ,xn+1 (φ (cr t)) → 1, as n → ∞.

(5.33)

Now, let ε > 0 be given, then by using the properties of function φ we can find r > 0 such that φ (cr t) < ε . Therefore, from (5.33) we get Fxn ,xn+1 (ε ) → 1, as n → ∞, for every ε > 0. On the other hand, we know that      Fxn ,xn+p (ε ) ≥ Δ Fxn ,xn+1 (ε /p), Δ Fxn+1 ,xn+2 (ε /p), . . . . . . , Fxn+p−1 ,xn+p (ε /p) . . . .    p−times

Thus, letting n → ∞ and using 5.34, for any integer p, we have Fxn ,xn+p (ε ) → 1, as n → ∞, for every ε > 0.

(5.34)

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It follows that {xn } is a G-Cauchy sequence. Since (X, F, Δ ) is G-complete, {xn } is convergent and hence xn → u as n → ∞ for some u ∈ X. Moreover, we get   Ffu,u (ε ) ≥ Δ Ffu,xn+1 (ε /2), Fxn+1 ,u (ε /2) . (5.35) Next, using the properties of function φ , we can find t2 > 0 such that φ (t2 ) < ε2 . Again xn → u as n → ∞ and hence there exists n0 ∈ N such that, for all n > n0 , we have Fxn ,u (φ (t2 )) > 0. Then, for n > n0 , we write 1 Fxn+1 ,fu ( ε2 )

1 −1 Ffxn ,fu (φ (t2 ))

1 ≤ α (xn , u, φ (t2 )) −1 Ffxn ,fu (φ (t2 ))   1 ≤ψ −1 . Fxn ,u (φ ( tc2 ))

−1 ≤

(from (iii))

Now, letting n → ∞, since ψ (0) = 0 and the continuity of function ψ , we obtain Fxn+1 ,fu (ε /2) → 1, as n → ∞.

(5.36)

Finally, passing to the limit for n → ∞ in 5.35, from 5.36, continuity of Δ and the fact that xn → u as n → ∞, we conclude that Ffu,u (ε ) = 1, for every ε > 0 and thus fu = u. This completes the proof.



The following example shows the usefulness of Definition 19. t for all x, y ∈ X Example 9. Let X = R, Δ (a, b) = min{a, b} for all a, b ∈ [0, 1] and Fx,y (t) = t+|x−y| and for all t > 0. Clearly, (X, F, Δ ) is a complete Menger PM-space. Define the mapping f : X → X by ⎧ 2 x ⎪ ⎨ if x ∈ [0, 1], 4 fx = ⎪ ⎩ 4 otherwise,

and the function α : X × X × (0, ∞) → R+ by ⎧ ⎨ 1 α (x, y, t) = ⎩ 0

if x, y ∈ [0, 1], otherwise,

for all t > 0. If we define φ , ψ : R+ → R+ by φ (t) = ψ (t) = t, then the mapping f satisfies the hypotheses of Theorem 29. To view this, suppose at least one of x and y is in R/[0, 1], then α (x, y, t) = 0 and so inequality 5.27 trivially. If both x and y are in [0, 1] then α (x, y, t) = 1 and so   holds 5.27 holds for all c ∈ 12 , 1 . Now, let x, y ∈ X be such that α (x, y, t) ≥ 1 for all t > 0; this implies that x, y ∈ [0, 1] and by the definitions of f and α , we have fx =

y2 x2 ∈ [0, 1], fy = ∈ [0, 1]and α (fx, fy, t) = 1 for all t > 0, 4 4

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that is, f is α -admissible. Further, there exists x0 ∈ X such that α (x0 , fx0 , t) ≥ 1 for all t > 0; indeed for x0 = 1 we have α (1, f (1), t) = 1. Next, let {xn } ⊂ X be such that α (xn , xn+1 , t) ≥ 1 for all n ∈ N and for all t > 0, and xn → x as n → ∞; this implies that xn , x ∈ [0, 1] and so α (xn , x, t) ≥ 1 for all n ∈ N and for all t > 0. Thus, Theorem 29 is applicable. Moreover, f has two fixed points, x = 0 and x = 4. Finally, we show that the main Theorem of [10] is not applicable in this case, since f does not satisfy the contractive condition given in [10] . To see this, consider x ∈ [0, 1] and y = 4, then, by applying the contractive condition given in [10], we get x + 4 ≤ 4c, which gives a contradiction to the fact that c ∈ (0, 1). We prove, with next theorem, uniqueness of the fixed point. To this aim, we consider the following condition: (H): For all x, y ∈ X and for all t > 0, there exists z ∈ X such that α (x, z, t) ≥ 1 and α (y, z, t) ≥ 1. Theorem 30. Adding condition (H) to the hypotheses of Theorem 29, we obtain that f has a unique fixed point. Proof. Let u, v ∈ X be such that fu = u and fv = v. First, note that if α (u, v, t) ≥ 1, then by 5.27, it follows easily that u = v. Therefore, suppose that α (u, v, t) < 1. Now, from condition (H) there exists z ∈ X such that

α (u, z, t) ≥ 1 and α (v, z, t) ≥ 1.

(5.37)

Since f is α -admissible, from (5.37) we get

α (u, f n z, t) ≥ 1 and α (v, f n z, t) ≥ 1, for all n ∈ N and for all t > 0.

(5.38)

Then, using 5.27 and 5.38, we have



1 1 −1 = −1 Fu,f n z (φ (ct)) Ffu,f (f n−1 z) (φ (ct))

1 ≤ α (u, f n−1 z, t) −1 Ffu,f (f n−1 z) (φ (ct))

1 ≤ψ −1 . Fu,f n−1 z (φ (t)) This implies that



1 1 n −1 ≤ ψ − 1 , for all n ∈ N. Fu,f n z (φ (ct)) Fu,z (φ (t))

Finally, letting n → ∞, we obtain f n z → u. A similar argument shows that f n z → v as n → ∞. Now, uniqueness of the limit gives us u = v and hence the proof is complete.  Remark 7. In view of Remark 6, the main Theorem of [10] is a particular case of Theorem 30. Taking α (x, y, t) = 1 and φ (t) = t a proof of Theorem 1.24 is obtained. Inspired by [2, 50], we show that our theorems are also useful to deduce easily some fixed point results in ordered Menger PM-spaces. We begin by giving the following definition.

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Definition 21. Let  be an order relation on X. We say that f : X → X is a nondecreasing mapping with respect to , if x  y implies fx  fy. Theorem 31. Let (X, ) be a partially ordered set, (X, F, Δ ) be a complete Menger PM-space and f : X → X be a mapping satisfying the following inequality x, y ∈ X, x  y =⇒

1 1 −1 ≤ ψ −1 , Ffx,fy (φ (ct)) Fx,y (φ (t))

for all t > 0 such that Fx,y (φ (t)) > 0, where c ∈ (0, 1), φ ∈ Φ and ψ ∈ Ψ . Also, suppose that the following conditions hold: (i) f is a nondecreasing mapping with respect to , (ii) there exists x0 ∈ X such that x0  fx0 , Fx0 ,fx0 (φ (t)) > 0 for all t > 0, (iii) if f is G-continuous or {xn } is a nondecreasing sequence in X such that xn → x ∈ X as n → ∞, then xn  x for all n ∈ N. Then f has a fixed point. Proof. Define the function α : X × X × (0, ∞) → R+ by ⎧ ⎨ 1 if x  y, α (x, y, t) = ⎩ 0 otherwise, for all t > 0. The reader can show easily that f is α -ψ -type contractive and α -admissible. Now, let {xn } be a sequence in X such that α (xn , xn+1 , t) ≥ 1 for all n ∈ N and xn → x ∈ X as n → ∞. By the definition of α , we have xn  xn+1 for all n ∈ N. From (iii), this implies that xn  x for all n ∈ N, which gives us that α (xn , x, t) = 1 for all n ∈ N and t > 0. Thus all the hypotheses of Theorem 29 are satisfied and f has a fixed point. 

5.9 Fixed Point Theorems for Generalized β-Type Contractive Mappings In this section we introduce the notions of generalized β -type contractive and β -admissible mappings in Menger PM-spaces. Before giving our results, we quote the following: Lemma 6 ([3]). Let (X, F, Δ ) be a Menger PM-space and φ : R+ → R+ be a Φ -function. Then the following statement holds: If for x, y ∈ X, c ∈ (0, 1), we have Fx,y (φ (t)) ≥ Fx,y (φ (t/c)) for all t > 0, then x = y. Introducing the generalized contraction of nonlinear type on Menger PM-spaces, Babaˇcev [3] proved the following result, as an improvement of the results contained in [6]. Theorem 32 ([3]). Let (X, F, Δ ) be a complete Menger PM-space with continuous t-norm Δ which satisfies Δ (a, a) ≥ a for each a ∈ [0, 1]. Let c ∈ (0, 1) be fixed. If for a Φ -function φ and a self-mapping f on X, we have

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185

 Ffx,fy (φ (t)) ≥ min Fx,y (φ (t/c)) , Fx,fx (φ (t/c)) , Fy,fy (φ (t/c)) ,  Fx,fy (2φ (t/c)) , Fy,fx (2φ (t/c)) ,

(5.39)

for all x, y ∈ X and for all t > 0, then f has a unique fixed point in X. Now, we introduce the following definitions: Definition 22. Let (X, F, Δ ) be a Menger PM-space and f : X → X be a given mapping. We say that f is a generalized β -type contractive mapping if there exists a function β : X × X × (0, ∞) → (0, ∞) such that  β (x, y, t)Ffx,fy (φ (t)) ≥ min Fx,y (φ (t/c)) , Fx,fx (φ (t/c)) , Fy,fy (φ (t/c)) ,  Fx,fy (2φ (t/c)) , Fy,fx (2φ (t/c)) ,

(5.40)

for all x, y ∈ X and for all t > 0, where φ ∈ Φ and c ∈ (0, 1). Remark 8. If β (x, y, t) = 1 for all x, y ∈ X and for all t > 0, then condition 5.40 reduces to condition (5.39), but the converse is not always true (see Example 10 given below). Definition 23. Let (X, F, Δ ) be a Menger PM-space, f : X → X be a given mapping and β : X × X × (0, ∞) → (0, ∞) be a function. We say that f is β -admissible if x, y ∈ X, for all t > 0, β (x, y, t) ≤ 1 =⇒ β (fx, fy, t) ≤ 1. Theorem 33. Let (X, F, Δ ) be a complete Menger PM-space with continuous t-norm Δ which satisfies Δ (a, a) ≥ a with a ∈ [0, 1]. Let f : X → X be a generalized β -type contractive mapping satisfying the following conditions: (i) f is β -admissible, (ii) there exists x0 ∈ X such that β (x0 , fx0 , t) ≤ 1 for all t > 0, (iii) if {xn } is a sequence in X such that β (xn , xn+1 , t) ≤ 1 for all n ∈ N and for all t > 0, and xn → x as n → ∞, then β (xn , x, t) ≤ 1 for all n ∈ N and for all t > 0. Then f has a fixed point. Proof. Since T is continuous and Δ (a, a) ≥ a, for all a ∈ [0, 1], then we have

Δ (a, b) ≥ min{a, b}, and, by (PM3), we write

  Fx,y (2t) ≥ min Fx,z (t), Fz,y (t) , for all x, y, z ∈ X.

(5.41)

Now, let x0 ∈ X be such that (ii) holds and define a sequence {xn } in X so that xn+1 = fxn , for all n ∈ N. First, we suppose xn+1 = xn for all n ∈ N, otherwise f has trivially a fixed point. Now, since f is β -admissible, we have

β (x0 , fx0 , t) = β (x0 , x1 , t) ≤ 1 =⇒ β (fx0 , fx1 , t) = β (x1 , x2 , t) ≤ 1.

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Consequently, by induction, we get

β (xn , xn+1 , t) ≤ 1, for all t > 0.

(5.42)

From the properties of function φ , we can find r > 0 such that t > φ (r) and therefore we have Fxn ,xn+1 (t) ≥ Ffxn−1 ,fxn (φ (r)) ≥ β (xn−1 , xn , φ (r))Ffxn−1 ,fxn (φ (r))  ≥ min Fxn−1 ,xn (φ (r/c)) , Fxn−1 ,xn (φ (r/c)) , Fxn ,xn+1 (φ (r/c)) ,  (using 5.40) Fxn−1 ,xn+1 (2φ (r/c)) , Fxn ,xn (2φ (r/c))   = min Fxn−1 ,xn (φ (r/c)) , Fxn ,xn+1 (φ (r/c)) , Fxn−1 ,xn+1 (2φ (r/c))  ≥ min Fxn−1 ,xn (φ (r/c)) , Fxn ,xn+1 (φ (r/c)) ,   min Fxn−1 ,xn (φ (r/c)) , Fxn ,xn+1 (φ (r/c))   = min Fxn−1 ,xn (φ (r/c)) , Fxn ,xn+1 (φ (r/c)) . We shall prove that Fxn ,xn+1 (φ (r)) ≥ Fxn−1 ,xn (φ (r/c)) .

(5.43)

If we assume that Fxn ,xn+1 (φ (r/c)) is the minimum, then from Lemma 6, we get that xn = xn+1 , which leads to contradiction with the assumption xn = xn+1 and so Fxn−1 ,xn (φ (r/c)) is the minimum, that is, inequality 5.43 holds true. Since φ is strictly increasing, we have Fxn ,xn+1 (t) ≥ Fxn ,xn+1 (φ (r)) ≥ Fxn−1 ,xn (φ (r/c)) ≥ · · · ≥ Fx0 ,x1 (φ (r/cn )) , that is, Fxn ,xn+1 (t) ≥ Fx0 ,x1 (φ (r/cn )) , for arbitrary n ∈ N. Next, let m, n ∈ N with m > n, then by (PM3) we have   Fxn ,xm ((m − n)t) ≥ min Fxn ,xn+1 (t), . . . , Fxm−1 ,xm (t)    ≥ min Fx0 ,x1 (φ (r/cn )) , . . . , Fx0 ,x1 φ (r/cm−1 . Since φ is strictly increasing and φ (t) → ∞ as t → ∞, then fixed ε ∈ (0, 1) there exists n0 ∈ N such that Fx0 ,x1 (φ (r/cn )) > 1 − ε , whenever n ≥ n0 . This implies that, for every m > n ≥ n0 , we get Fxn ,xm ((m − n)t) ≥ 1 − ε . Since t > 0 and ε ∈ (0, 1) is arbitrary, we deduce that {xn } is a Cauchy sequence in the complete Menger PM-space (X, F, Δ ). Then, xn → u as n → ∞ for some u ∈ X. We will show that u is a fixed point of f . By (PM3), we have   Ffu,u (t) ≥ Δ Ffu,xn (φ (r)) , Fxn ,u (t − φ (r))   ≥ min Ffu,xn (φ (r)) , Fxn ,u (t − φ (r)) .

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Notice that, if xn = fu for infinitely many values of n, then u = fu and hence the proof finishes. Therefore, we assume that xn = fu for all n ∈ N. Thus, since lim xn = u, for any n→∞

arbitrary ε ∈ (0, 1) and n large enough, we get Fxn ,u (t − φ (r)) > 1 − ε and hence, we have

  Ffu,u (t) ≥ min Ffu,xn (φ (r)) , 1 − ε .

Since ε > 0 is arbitrary, we can write Ffu,u (t) ≥ Ffu,xn (φ (r)). Next, we get Fu,fu (t) ≥ Fxn ,fu (φ (r)) = Ffxn−1 ,fu (φ (r)) (using (iii)) ≥ β (xn−1 , u, φ (r))Ffxn−1 ,fu (φ (r))  ≥ min Fxn−1 ,u (φ (r/c)) , Fxn−1 ,xn (φ (r/c)) , Fu,fu (φ (r/c)) ,  Fxn−1 ,fu (2φ (r/c)) , Fu,xn (2φ (r/c))   ≥ min Fxn−1 ,u (φ (r/c)) , Fu,fu (φ (r/c)) , Fxn−1 ,xn (φ (r/c)) . It follows that Fu,fu (t) ≥ lim inf Fxn ,fu (φ (r)) n→∞   ≥ lim inf min Fxn−1 ,u (φ (r/c)) , Fu,fu (φ (r/c)) , Fxn−1 ,xn (φ (r/c)) n→∞   ≥ min 1 − ε , Fu,fu (φ (r/c)) , 1 − ε . Finally, since ε ∈ (0, 1) is arbitrary, we have Ffu,u (φ (r)) ≥ Fu,fu (φ (r/c)) , and so, by Lemma 6, we deduce that u = fu. This completes the proof.



The following example shows the usefulness of Definition 22. t Example 10. Let X = [ 41 , ∞), Δ (a, b) = min{a, b} for all a, b ∈ [0, 1] and Fx,y (t) = t+|x−y| for all x, y ∈ X and for all t > 0. Clearly, (X, F, Δ ) is a complete Menger PM-space. Define the mapping f : X → X by ⎧   if x ∈ 14 , 1 , ⎨ 1 fx = ⎩ 2 otherwise,

and the function β : X × X × (0, ∞) → (0, ∞) by ⎧ 1 ⎪ ⎪ ⎨ β (x, y, t) = 2(t + 1) ⎪ ⎪ ⎩ 2t + |x − y| for all t > 0.

if x, y ∈

1  ,1 , 4

otherwise,

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Now, consider φ : R+ → R+ defined by φ (t) = t and let c = 12 . To show that f is a generalized β -type contractive mapping, we have to check the following condition:   β (x, y, t)Ffx,fy (t) ≥ min Fx,y (2t), Fx,fx (2t), Fy,fy (2t), Fx,fy (4t), Fy,fx (4t) . Then, we distinguish three cases:   Case I: If x, y ∈ 14 , 1 , then the left-hand side of above inequality is equal to 1 and hence the inequality   is obviously true. Case II: If x, y ∈ / 14 , 1 , then the left-hand side of above inequality is greater than Fx,y (2t) and hence  is again true.      the inequality / 14 , 1 or y ∈ 14 , 1 and x ∈ / 14 , 1 , then we have the same Case III: If x ∈ 14 , 1 and y ∈ conclusion as Case II. On the other hand, f does not satisfy inequality (5.39). Indeed, for x = 1 and y = 2, we get   t t t t t ≥ min , 1, 1, , , = t+1 t+c t + c/2 t + c/2 t+c which gives c ≥ 1, a contradiction. We denote by Fix(f ) the set of fixed points of f . Then, to obtain uniqueness of the fixed point in Theorem 33, we introduce the following condition: (J): For all u, v ∈ Fix(f ) and for all t > 0 there exists z ∈ X such that β (z, fz, t) ≤ 1 with β (u, z, t) ≤ 1 and β (v, z, t) ≤ 1. Theorem 34. Adding condition (J) to the hypotheses of Theorem 33, we obtain that f has a unique fixed point. Proof. Let u, v ∈ X be such that fu = u and fv = v. From condition (J), there exists z ∈ X such that β (z, fz, t) ≤ 1, with β (u, z, t) ≤ 1 and β (v, z, t) ≤ 1. Since f is β -admissible, then we have

β (fz, f 2 z, t) ≤ 1, β (u, fz, t) ≤ 1, β (v, fz, t) ≤ 1 and, by induction, we get

β (zn , zn+1 , t) ≤ 1, β (u, zn , t) ≤ 1, β (v, zn , t) ≤ 1, for all t > 0

(5.44)

where zn = f n z. From the properties of function φ , we can find r > 0 such that t > φ (r) and therefore we have Fu,zn+1 (t) ≥ Fu,fzn (φ (r)) = Ffu,fzn (φ (r)) ≥ β (u, zn , φ (r))Ffu,fzn (φ (r))  ≥ min Fu,zn (φ (r/c)) , Fu,fu (φ (r/c)) , Fzn ,zn+1 (φ (r/c)) ,

 Fu,zn+1 (2φ (r/c)) , Fzn ,fu (2φ (r/c)) ,

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which implies

189

  Fu,zn+1 (t) ≥ min Fu,zn (φ (r/c)) , Fzn ,zn+1 (φ (r/c)) .

(5.45)

Now, we distinguish two cases: Case I. We assume that Fzn ,zn+1 (φ (r/c)) is the minimum. Then, by applying 5.38, we can write Fu,zn+1 (φ (r)) ≥ Fzn ,zn+1 (φ (r/c))      ≥ min Fzn−1 ,zn φ (r/c2 ) , Fzn ,zn+1 φ (r/c2 ) .   Now, if Fzn ,zn+1 φ (r/c2 ) is the minimum for some n ∈ N, by Lemma 6, we deduce that zn = zn+1 = u. Consequently, by 5.43 we deduce that β (v, u, t) ≤ 1 and so by 5.38 we have Fv,u (φ (t)) ≥ min {Fv,u (φ (t/c)) , Fv,v (φ (t/c)) , Fu,u (φ (t/c)) , Fv,u (2φ (t/c)) , Fu,v (2φ (t/c))} = Fv,u (φ (t/c)) . Again, by Lemma 6 we conclude  that u = v. On the other hand, if Fzn−1 ,zn φ (r/c2 ) is the minimum, then   Fzn ,zn+1 (φ (r/c)) ≥ Fzn−1 ,zn φ (r/c2 ) ≥ · · · ≥ Fz0 ,z1 (φ (r/cn )) and, letting n → ∞, we get

Fzn ,zn+1 (φ (r/c)) → 1.

Therefore Fu,zn+1 (t) → 1 as n → ∞, which implies zn+1 → u as n → ∞. Case II. Suppose that Fu,zn (φ (r/c)) is the minimum, then we have Fu,zn+1 (φ (r)) ≥ Fu,zn (φ (r/c))   ≥ Fu,zn−1 φ (r/c2 ) ≥ ··· ≥ Fu,z0 (φ (r/cn )) . Letting n → ∞, we obtain Fu,zn+1 (φ (r)) → 1 as n → ∞, that is, zn+1 → u as n → ∞. A similar argument shows that zn+1 → v, for n → ∞. Now, uniqueness of the limit gives us u = v and the proof is complete.



The following example shows the crucial role of the condition Δ (a, a) ≥ a in establishing existence of the fixed point. Example 11. Let X = R+ , Δ = Δp and

Fxy (t) =

⎧ min{x, y} ⎪ ⎪ ⎨ max{x, y} ⎪ ⎪ ⎩

1

for all x = y, otherwise,

for all t > 0. Clearly, (X, F, Δp ) is a complete Menger PM-space. Define the mapping f : X → X by fx = x + 1, for all x ∈ X and the function β : X × X × (0, ∞) → (0, ∞) by β (x, y, t) = 1, for all x, y ∈ X and for all t > 0.

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Now, it can be easily shown that all the hypotheses of Theorem 33 are satisfied except that Δp (a, a) ≥ a with a ∈ (0, 1), but here the mapping f has no fixed point. Also, Theorem 33 does not hold for any t-norm Δ ≤ ΔP . Our last existence theorem is obtained modifying the generalized β -type contractive condition 5.38 as shown below. Theorem 35. Let (X, F, Δ ) be a complete Menger PM-space with continuous t-norm Δ and f : X → X. Assume that there exists β : X × X × (0, ∞) → (0, ∞) such that   β (x, y, t)Ffx,fy (φ (t)) ≥ min Fx,y (φ (t/c)) , Fx,fx (φ (t/c)) , Fy,fy (φ (t/c)) , Fy,fx (φ (t/c)) , (5.46) for all x, y ∈ X and for all t > 0, where c ∈ (0, 1) and φ ∈ Φ . Also suppose that the following conditions hold: (i) f is β -admissible, (ii) there exists x0 ∈ X such that β (x0 , fx0 , t) ≤ 1, (iii) for each sequence {xn } in X such that β (xn , xn+1 , t) ≤ 1 for all n ∈ N and for all t > 0, there exists k0 ∈ N such that β (xm−1 , xn−1 , t) ≤ 1 for all m, n ∈ N with m > n ≥ k0 and for all t > 0, (iv) if {xn } is a sequence in X such that β (xn , xn+1 , t) ≤ 1 for all n ∈ N and t > 0, and xn → x as n → ∞, then β (xn , x, t) ≤ 1 for all n ∈ N and for all t > 0. Then f has a fixed point. Proof. Let x0 ∈ X be such that β (x0 , fx0 , t) ≤ 1, for all t > 0. Define a sequence {xn } in X such that xn+1 = fxn , for all n ∈ N. Clearly, we suppose xn+1 = xn for all n ∈ N, otherwise f has trivially a fixed point. Since f is β -admissible, then we have

β (x0 , fx0 , t) = β (x0 , x1 , t) ≤ 1 =⇒ β (fx0 , fx1 , t) = β (x1 , x2 , t) ≤ 1, and, by induction, we get

β (xn , xn+1 , t) ≤ 1, for all n ∈ N.

(5.47)

From the properties of function φ , we can find r > 0 such that t > φ (r) and therefore we have Fxn ,xn+1 (t) ≥ Fxn ,xn+1 (φ (r)) ≥ β (xn−1 , xn , (φ (r)) Ffxn−1 ,fxn (φ (r))   ≥ min Fxn−1 ,xn (φ (r/c)) , Fxn−1 ,xn (φ (r/c)) , Fxn ,xn+1 (φ (r/c)) , Fxn ,xn (φ (r/c))   = min Fxn−1 ,xn (φ (r/c)) , Fxn ,xn+1 (φ (r/c)) . Next, if Fxn ,xn+1 (φ (r/c)) is the minimum, then Fxn ,xn+1 (φ (r)) ≥ Fxn ,xn+1 (φ (r/c)) and, from Lemma 6, we have xn = xn+1 , which leads to a contradiction with the assumption that xn+1 = xn for all n ∈ N.

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On the other hand, if Fxn−1 ,xn (φ (r/c)) is the minimum, then Fxn ,xn+1 (t) ≥ Fxn−1 ,xn ((φ (r/c))   ≥ Fxn−2 ,xn−1 (φ (r/c2 ) ≥ ··· ≥ Fx0 ,x1 (φ (r/cn )) , which, letting n → ∞, gives us Fxn ,xn+1 (t) → 1.

(5.48)

Now, we claim that {xn } is a Cauchy sequence. Suppose {xn } is not a Cauchy sequence, then for any ε > 0 and λ ∈ (0, 1) there are subsequences {xn(k) } and {xm(k) } of {xn } such that m(k) < n(k) with Fxm(k) ,xn(k) (ε ) ≤ 1 − λ

(5.49)

Fxm(k) ,xn(k)−1 (ε ) > 1 − λ.

(5.50)

and

Since φ is continuous at 0 and strictly monotone increasing with φ (0) = 0, then there exists ε1 > 0 such that φ (ε1 ) < ε . Thus, by the above arguments it is possible to obtain increasing sequences of integers m(k) and n(k), with m(k) < n(k), such that

and

Fxm(k) ,xn(k) (φ (ε1 )) ≤ 1 − λ

(5.51)

Fxm(k) ,xn(k)−1 (φ (ε1 )) > 1 − λ.

(5.52)

Since ε ∈ (0, 1) and φ ∈ Φ , we can choose η > 0 such that 0 < η < φ (ε1 /c) − φ (ε1 ), that is,

(5.53)

φ (ε1 /c) − η > φ (ε1 )

and so, from 5.52, we get Fxm(k) ,xn(k)−1 (φ (ε1 /c) − η ) > Fxm(k) ,xn(k)−1 (φ (ε1 )) > 1 − λ.

(5.54)

Then, for any 0 < λ1 < λ < 1, by 5.33 it is possible to find a positive integer N1 such that, for all k > N1 , we have Fxm(k)−1 ,xm(k) (φ (η )) > 1 − λ1 and Fxn(k)−1 ,xn(k) (φ (η )) > 1 − λ1 .

(5.55)

Now, by (PM3), we get

Fxm(k)−1 ,xn(k)−1 (φ (ε1 /c)) ≥ Δ Fxm(k)−1 ,xm(k) (η ) , Fxm(k) ,xn(k)−1 (φ (ε1 /c) − η ) .

(5.56)

Let λ2 be such that 0 < λ2 < λ1 < λ < 1, be arbitrary. Then by (5.55) there exists a positive integer N2 such that for all k > N2 , we have Fxm(k)−1 ,xm(k) (η ) > 1 − λ2 .

(5.57)

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Now, using 5.54, 5.56 and 5.57, for all k > max{N1 , N2 } we have Fxm(k)−1 ,xn(k)−1 (φ (ε1 /c)) > Δ (1 − λ2 , 1 − λ) . Since λ2 is arbitrary and Δ is continuous, we get Fxm(k)−1 ,xn(k)−1 (φ (ε1 /c)) > 1 − λ.

(5.58)

By using 5.45, 5.51, 5.52, 5.55 and 5.58, we have 1 − λ ≥ Fxm(k) ,xn(k) (φ (ε1 )) = Ffxm(k)−1 ,fxn(k)−1 (φ (ε1 ))   ≥ β xm(k)−1 , xn(k)−1 , φ (ε1 ) Ffxm(k)−1 ,fxn(k)−1 (φ (ε1 ))  ≥ min Fxm(k)−1 ,xn(k)−1 (φ (ε1 /c)) , Fxm(k)−1 ,xm(k) (φ (ε1 /c)) ,  Fxn(k)−1 ,xn(k) (φ (ε1 /c)) , Fxn(k)−1 ,xm(k) (φ (ε1 /c))

(by using (iii))

> min {1 − λ, 1 − λ, 1 − λ, 1 − λ} = 1 − λ, which is a contradiction, therefore {xn } is a Cauchy sequence in the complete Menger PMspace. Thus xn → u as n → ∞ for some u ∈ X. Now, we show that u is a fixed point of f . Since φ ∈ Φ , we have that for all x, y ∈ X and t > 0 there exists r > 0 such that t > φ (r) and therefore we have   Ffu,u (t) ≥ Δ Ffu,xn (φ (r)) , Fxn ,u (t − φ (r)) . (5.59) Since t > φ (r), thus (t − φ (r)) > 0. Also, since u = lim xn , for arbitrary δ ∈ (0, 1) there exists n0 ∈ N such that for all n ≥ n0 we get

n→∞

Fxn ,u (t − φ (r)) > 1 − δ .

(5.60)

Hence, from 5.59 and 5.60, we have

  Ffu,u (t) ≥ Δ Ffu,xn (φ (r)) , 1 − δ .

Notice that, if xn = fu for infinitely many values of n, then u = fu and hence the proof finishes. Therefore, we assume that xn = fu for all n ∈ N. Consequently, since δ > 0 is arbitrary and the t-norm T is continuous, we get Fu,fu (t) ≥ Fxn ,fu (φ (r)) ≥ Ffxn−1 ,fu (φ (r))   ≥ min Fxn−1 ,u (φ (r/c)) , Fxn−1 ,fxn−1 (φ (r/c)) , Fu,fu (φ (r/c)) , Fu,fxn−1 (φ (r/c)) . Letting n → ∞ in the above inequality and using the fact that T is continuous, we have Ffu,u (φ (r)) ≥ Fu,fu (φ (r/c)) and hence, by using Lemma 6, we get fu = u. This completes the proof.



Theorem 36. Adding condition (J) to the hypotheses of Theorem 35, we obtain that f has a unique fixed point.

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Proof. The proof can be completed using a similar technique as in the proof of Theorem 34. Therefore, to avoid repetition, we omit the details.  Theorems 33 and 35 are also valid if we replace condition iii) by continuity of function f in a similar way to the previous section. Inspired by [2, 50], we obtain ordered version of Theorem 33. Proof. Define the function β : X × X × (0, ∞) → R+ by ⎧ ⎨ 1 if x  y, β (x, y, t) = ⎩ ∞ otherwise, for all t > 0.



5.10 Motivating Ideas The Banach contraction principle is a powerful classical result in nonlinear analysis that has been extended in many directions. It has been extended to some larger classes of contractive mappings by replacing the strict contractive condition by weaker conditions of various types. A comparative study of some of these results has been made by Rhoades [47] or Jachymski [25]. In this sense, for example, Meir and Keeler [33] formulated their fixed point theorem for contractive mappings with a purely metric condition. They defined weakly uniformly strict contraction mappings and proved a fixed point theorem that generalized the Banach fixed point theorem and extended the principle to wider classes of maps than those covered in [47]. A mapping T on a metric space X is said to be a Meir–Keeler contraction if, for every ε > 0, there exists δ > 0 such that ∀ x, y ∈ X,

ε ≤ d(x, y) < ε + δ =⇒ d(Tx, Ty) < ε .

(5.61)

This idea was extended by numerous mathematicians. In the context of probabilistic (fuzzy) metrics, authors try to use a probabilistic (fuzzy) analogue of different contractive condition. Meir–Keeler type and others are not well studied in probabilistic cases. Manro et al. [30] modified the Meir–Keeler type contractive condition as follows: A mapping T on a PM-space (X, F, Δ ) is said to be a Meir–Keeler contraction if, for every ε > 0, there exists δ ∈ (0, ε ) such that ∀ x, y ∈ X,

ε − δ < Fx,y (t) < ε =⇒ FTx,Ty (t) ≥ ε .

However, if we identify nonnegative real numbers with their left-continuous unit step functions, then every metric space is always a Menger metric space and (5.61) in the probabilistic case can be rewritten as ∀ x, y ∈ X,

Hε +δ < Fx,y ≤ Hε =⇒ Hε < FTx,Ty ,

(5.62)

Juan Martínez-Moreno

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where Ha (t) =

⎧ ⎨ 0 ⎩

1

iff 0 ≤ t ≤ a, iff t > a.

This identification allows us to propose the following Meir–Keeler contractive type condition: A mapping T on a PM-space (X, F, Δ ) is said to be a Meir–Keeler contraction if, for all x, y ∈ X, and for every t > 0, there exists s ∈ (0, t) such that 0 < Fx,y (t) =⇒ FTx,Ty (s) = 1.

(5.63)

We show that every Meir–Keeler contraction is asymptotically regular. If T : X → X is a Meir–Keeler contraction on a PM-space (X, F, Δ ), then lim FTn x,Tn+1 x (t) = 1, for all t > 0 and n→∞

x ∈ X. Let x0 ∈ X be arbitrary and let {xn = Tn x0 } be the Picard sequence of T based on x0 . If there exists some n0 ∈ N such that xn0 +1 = xn0 , then x0 is a fixed point of T. In the contrary case, assume that xn+1 = xn for all n ≥ 0. Since, x1 = x0 , there exists t0 > 0 such that 0 < Fx0 ,x1 (t0 ) < 1. By (5.63), there exists s1 ∈ (0, t0 ) such that Fx1 ,x2 (s1 ) = 1. Repeating the procedure, there are two strictly decreasing sequences {sn }, {tn }, 0 < · · · < sn+1 < tn < sn < · · · < s2 < t1 < s1 < t0 , such that 0 < Fxn ,xn+1 (tn ) < 1 = Fxn ,xn+1 (sn ). In general, mathematicians are interested in studying and unifying different types of contraction. In the probabilistic case, relationships and definitions of classical and new contractions are not well described yet. On the other hand, inspired by the successful metric theory of Lowen’s approach spaces, Roldan et al. [49] presented a family of spaces, called fuzzy approach spaces, that are appropriate to handle, at the same time, the approach measure conception and the probabilistic metric one. Let X be a set and let Ω = P(X) be the set of subsets of X. A fuzzy approach space is a pair (X, F) where F : X × Ω → Δ + is a fuzzy set satisfying the following properties for all x ∈ X, all A, B, C ∈ Ω and all s, t ∈ [0, ∞]: (FA1) Fx,A (0) = 0. (FA2) If ∅ ∈ Ω , then Fx,∅ (t) = 0 for all t < ∞. (FA3) If {x} ∈ Ω , then Fx,{x} (t) = 1 for all t > 0. (FA4) Fx,A (·) : [0, ∞[ → [0, 1] is a left continuous function. (FA5) If A ∪ B ⊆ C, then Fx,C (t) ≥ max (Fx,A (t), Fx,B (t)). (FA6) For all r ∈ [0, ∞[, the sets A(r) = { y ∈ X : Fy,A (t) = 1, ∀t > r } and A(∞) = { y ∈ X : Fy,A (∞) = 1 } are in Ω , and Fx,A (t + s) ≥ Fx,A(r) (t) if s > r ≥ 0, and Fx,A (∞) ≥ Fx,A(∞) (t). (FA7) If x, y ∈ X are such that Fx,{y} (t) = 1 for every 0 < t < ∞, then x = y. Here we present the main result in [49]; however, in order not to enlarge the present chapter we will not include its proof.

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Theorem 37. Let (X, F) be a complete fuzzy approach space and let T : XßX be a self-mapping such that there exists λ ∈ (0, 1) satisfying FTx,TA (λt) ≥ Fx,A (t) for all x ∈ X, A ∈ Ω \ 0/ and all t > 0. If there exist x0 ∈ X and t0 > 0 such that Fx0 , {Tx0 }(t0 ) = FTx0 , {x0 }(t0 ) = 0, then T has, at least, a fixed point. Furthermore, if for all fixed points of T, x and y, we have lim Fx,y (t) = 0, then T has a t→∞

unique fixed point.

5.11 Conclusion The notion of probabilistic contractive mappings was introduced by Sehgal and BaruchaReid[53] in order to obtain extensions of the Banach contraction principle, by considering PM space that can be deduced from a metric space. Therefore some classical fixed point results can be viewed as particular cases of these in PM spaces. However, there are a variety of specific results for probabilistic contractive mappings, depending on the t-norms the spaces are endowed with and other properties like the growth condition for F [55]. Hence, these results have advantage over the results with the usual metrics. In fact finding a new class of contractive mappings in PM spaces and the process of proving fixed point theorems for such mappings involves many mathematical components, such as the conditions on t-norms associated with given PM spaces and other conditions on the space itself. Such properties attract fixed point theorists to work in this direction. The main aim of this chapter is to provide a systematic survey on fixed point results concerning various classes of probabilistic contractions including probabilistic nonlinear contractions. We hope that the results presented in this chapter illustrate the latest research and developments on the topic.

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6 Fixed Point Theory For Fuzzy Contractive Mappings Dhananjay Gopal and Tatjana Došenovi´c

CONTENTS 6.1 6.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Kramosil and Michalek Fuzzy Metric Spaces and Grabiec’s Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.3 George and Veeramani’s Fuzzy Metric Space and Fuzzy Contractive Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.4 Fuzzy Z -Contractive Mappings and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.5 Countable Extension of t-Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.6 Fixed Point Theorems in Fuzzy Metric Space of Khan Type . . . . . . . . . . . . . . . . . . . . . . . 220 6.7 Fixed Point Theorems in Fuzzy Metric Spaces of Caristi Type . . . . . . . . . . . . . . . . . . . . 230 6.8 Fixed Point Theorems in Fuzzy Metric Spaces of Nadler Type . . . . . . . . . . . . . . . . . . . . 232 6.9 Convex Fuzzy Metric Space and Fixed Point Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 6.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

6.1 Introduction In 1965, Zadeh [67] introduced the concept of the fuzzy set which transformed and stimulated almost all branches of science and engineering including mathematics. A fuzzy set on a set can be defined by assigning to each element of a set a value in [0, 1] representing its grade of membership in the fuzzy set. Mathematically, a fuzzy set A of X is a mapping A : X → [0, 1]. The concept of fuzziness found place in probabilistic metric spaces due to Menger [38]. The main reason behind this was that, in some cases, uncertainty in the distance between two points was due to fuzziness rather than randomness. With this idea, in 1975, Kramosil and Michalek [36] extended the concept of probabilistic metric spaces to the fuzzy situation. However, in order to strengthen and to obtain a Hausdorff topology (the so-called M-topology), George and Veeramani [15, 16] imposed some stronger conditions on fuzzy metrics and modified the concept of fuzzy metrics due to Kranmosil and Michalek. 199

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6.2 Kramosil and Michalek Fuzzy Metric Spaces and Grabiec’s Fixed Point Theorems Definition 1. [36] The triple (X, M, ∗) is a fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X2 × [0, ∞) satisfying the following axioms: (KM1) (KM2) (KM3) (KM4) (KM5)

M(x, y, 0) = 0; M(x, y, t) = 1, for all t > 0 if and only if x = y; M(x, y, t) = M(y, x, t); M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s); The function M(x, y, ·) : [0, ∞) → [0, 1] is left continuous, for all x, y, z ∈ X and t, s > 0.

In what follows, fuzzy metric spaces in the sense of Kramosil and Michalek [36] will be referred as KM-fuzzy metric space. In 1988, M. Grabiec [14] initiated the study of fixed point theory in fuzzy metric space and established a fuzzy Banach contraction theorem and a Fuzzy Edelstein contraction theorem. In order to obtain his theorems, Grabiec introduced the following notions: Definition 2. [14] Let (X, M, ∗) be a fuzzy metric space. Then (i)

a sequence {xn }n∈N in X is said to be convergent to x ∈ X, if lim M(xn , x, t) = 1 for all n→∞

t > 0. (ii) a sequence {xn }n∈N in X is said to be Cauchy (or G-Cauchy) if lim M(xn , xn+p , t) = 1 n→∞

for each p ∈ N and t > 0.

A fuzzy metric space in which every Cauchy (or G-Cauchy) sequence is convergent is called complete (or G-complete). It is called compact if every sequence contains a convergent subsequence. Lemma 1. M(x, y, ·) is nondecreasing for all x, y ∈ X. Proof. Suppose M(x, y, t) > M(x, y, s) for some 0 < t < s. Then M(x, y, t) ∗ M(y, y, s − t) ≤ M(x, y, s) < M(x, y, t). By (KM2) M(y, y, s − t) = 1, and thus M(x, y, t) < M(x, y, t), a contradiction.   Lemma 2. Let lim xn = x and lim yn = y. Then n→∞

(i)

n→∞

lim M(xn , yn , t) ≥ M(x, y, t) for all t > 0;

n→∞

(ii) If M(x, y, ·) is continuous, then lim M(xn , yn , t) = M(x, y, t) for all t > 0. n→∞

Theorem 1 (Fuzzy Banach contraction theorem). Let (X, M, ∗) be a complete fuzzy metric space such that (6.1) lim M(x, y, t) = 1 for all x, y ∈ X. t→∞

Let f : X → X be a mapping satisfying M(fx, fy, kt) ≥ M(x, y, t)

(6.2)

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for all x, y ∈ X where 0 < k < 1. Then f has a unique fixed point. Proof. Let x ∈ X and xn = f n x (n ∈ N). By a simple induction we get   t M(xn , xn+1 , kt) ≥ M x, x1 , n−1 k

(6.3)

for all n and t > 0. Thus for any positive integer p we have p

      t t M(xn , xn+p , t) ≥ M xn , xn+1 , ∗ · · · M xn+p−1 , xn+p , p p     t t ≥ M x, x1 , n ∗ · · · ∗ M x, x1 , n by (6.3). pk pk According to (6.1) we now have p

   lim M(xn+p , xn , t) ≥ 1 ∗ 1 ∗ 1 · · · ∗ 1 = 1,

n→∞

that is, {xn } is Cauchy, hence convergent. Call the limit y. Thus, we have     t t M(fy, y, t) ≥ M fy, fxn , ∗ M xn+1 , y, 2 2     t t ≥ M y, xn , ∗ M xn+1 , y, → 1∗1 = 1 2k 2 as n → ∞ and by (KM4). By (KM2) we get fy = y, a fixed point. To show uniqueness, assume fz = z for some z ∈ X. Then     t t 1 ≥ M(z, y, t) = M(fz, fy, t) ≥ M z, y, = M fz, fy, k k     t t ≥ M z, y, 2 ≥ . . . ≥ M z, y, n → 1 k k as n → ∞. By (KM2), z = y.

 

Theorem 2 (Fuzzy Edelstein contraction theorem). Let (X, M, ∗) be a compact fuzzy metric space with M(x, y, ·) continuous for all x, y ∈ X. Let f : X → X be a mapping satisfying M(fx, fy, t) > M(x, y, t)

(6.4)

for all x = y and t > 0. Then f has a unique fixed point. Proof. Let x ∈ X and xn = f n x(n ∈ N). Assume xn = xn+1 for each n (if not, fxn = xn ). Now assume xn = xm (n = m). For otherwise we get M(xn , xn+1 , t) = M(xm , xm+1 , t) > M(xm−1 , xm , t) > · · · > M(xn , xn+1 , t) where m > n, a contradiction. Since X is compact, {xn } has a convergent subsequence {xni }. Let y = lim xni . We also assume that y, fy ∈ / {xn : n ∈ N} (If not, choose i→∞

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a subsequence with such a property). According to the above assumption we may now write M(fxni , fy, t) > M(xni , y, t) for all i ∈ N and t > 0. Since M(x, y, ·) is continuous for all x, y in X, by (6.4) we obtain lim M(fxni , fy, t) ≥ lim M(xni , y, t) = 1 i→∞

for each t > 0, hence

i→∞

lim fxni = fy.

(6.5)

lim f 2 xni = f 2 y

(6.6)

i→∞

Similarly, we obtain

i→∞

(recall that fy = fxni for all i). Now, observe that M(xn1 , fxn1 , t) < M(fxn1 , f 2 xn1 , t) < · · · < M(xni , fxni , t) < M(fxni , f 2 xni , t) < · · · < M(xni+1 , fxni+1 , t) < M(fxni+1 , f 2 xni+1 , t) < · · · < 1 for all t > 0. Thus {M(xni , fxni , t)} and {M(fxni , f 2 xni , t)} (t > 0) are convergent to a common limit. Therefore, by (6.5), (6.6) and ((ii) of lemma 2) we get M(y, fy, t) = M(lim xni , f (lim xni ), t) i→∞

i→∞

= lim M(xni , fxni , t) i→∞

= lim M(fxni , f 2 xni , t) i→∞

= M(fy, f 2 y, t) for all t > 0. Suppose y = fy. Then, by (6.4), M(y, fy, t) < M(fy, f 2 y, t) for t > 0, a contradiction. Hence y = fy, a fixed point. Uniqueness follows at once from (6.4).   These results of Grabiec [14] have further been generalized and extended by Subrahmanyam [55] and Vasuki [59] as follows: Theorem 3. [55] Let (X, M, ∗) be a complete fuzzy metric space and let f , g : X → X be maps that satisfy the following conditions: (i) g(X) ⊆ f (X), (ii) f is continuous, (iii) M(g(x), g(y), α t) ≥ M(f (x), f (y), t) for all x, y in X and 0 < α < 1. Then, f and g have a unique common fixed point provided f and g commute. Theorem 4. [59] Let {fn } be a sequence of mappings of a complete fuzzy metric space (X, M, ∗) into itself such that for any two mappings fi , fj we have M(fim x, fjm y, αij t) ≥ M(x, y, t) for some m ∈ N and 0 < αij < k < 1, i, j = 1, 2, . . . , x, y ∈ X. Then, the sequence {fn } has a unique common fixed point.

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203

In [15, 60] it has been observed that the notion of G-completeness has a disadvantage; it is a very strong notion of completeness, in fact, if d is the Euclidean metric in R, then the induced fuzzy metric (Md , ∗) of Example 2.1 given in [60] is not G-complete. In order to strengthen and to obtain a Hausdorff topology (the so called M-topology), George and Veeramani [15, 16] imposed some stronger conditions on fuzzy metrics and modified the concept of fuzzy metrics due to Kranmosil and Michalek.

6.3 George and Veeramani’s Fuzzy Metric Space and Fuzzy Contractive Mappings Definition 3 (George and Veeramani [15]). The triple (X, M, ∗) is a fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X2 × (0, ∞) satisfying the following axioms: (GV1) (GV2) (GV3) (GV4) (GV5)

M(x, y, t) > 0; M(x, y, t) = 1 if and only if x = y; M(x, y, t) = M(y, x, t); M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s); M(x, y, .) : (0, ∞[→ [0, 1] is continuous, x, y, z ∈ X and t, s > 0.

The axiom (GV1) is justified by the authors because in the same way that a classical metric space does not take the value ∞ then M cannot take the value 0. The axiom (GV2) gives the idea that only when x = y the degree of nearness of x and y is perfect, or simply 1, and then M(x, y, t) = 1 for each x ∈ X and for each t > 0. In this manner the value 0 and ∞ in the classical theory of metric space are identified with 1 and 0, respectively, in this fuzzy theory. Finally, in (GV5) the authors only assume that the variable t behaves nicely, that is, they assume that for fixed x and y, the function t → M(x, y, t) is continuous without any imposition for M as t → ∞. In what follows, fuzzy metric spaces in the sense of (George and Veeramani [15]) will be referred as GV-fuzzy metric spaces. Example 1. Let X = R. Define a ∗ b = ab for all a, b ∈ [0, 1] and    |x − y| −1 M(x, y, t) = exp t for all x, y ∈ X and t ∈ (0, ∞). Then (X, M, ∗) is a fuzzy metric space. The next example shows that every metric space induces a fuzzy metric space. Example 2. Let (X, d) be a metric space. Define a ∗ b = ab for all a, b ∈ [0, 1] and M(x, y, t) =

ktn , k, m, n ∈ N. ktn + md(x, y)

Then (X, M, ∗) is a fuzzy metric space. In particular, taking k = m = n = 1, we get M(x, y, t) = which is called a standard fuzzy metric.

t , t + d(x, y)

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George and Veeramani proved in [15, 16] that every fuzzy metric M on X generates a topology τM on X which has as a base the family of open sets of the form {BM (x, r, t) : x ∈ X, 0 < r < 1, t > 0}, where BM (x, r, t) = {y ∈ X : M(x, y, t) > 1 − r} for all x ∈ X, r ∈ (0, 1) and t > 0. If (X, d) is a metric space then the topology generated by d coincides with the topology τMd generated by the fuzzy metric Md . Remark 1 (George and Veeramani [15]). The metric space (X, d) is complete if and only if the standard fuzzy metric space (X, Md , ∗) is complete. Definition 4 (George and Veeramani [15]). Let (X, M, ∗) be a fuzzy metric space. Then a sequence {xn }n∈N in X is said to be Cauchy (or M-Cauchy) if for each ε ∈ (0, 1) and each t > 0 there is n0 ∈ N such that M(xn , xm , t) > 1 − ε for all n, m ≥ n0 . A fuzzy metric space in which every Cauchy (or M-Cauchy) sequence is convergent is called complete (or M-complete). Definition 5. [21, 33] Let (X, M, ∗) be a fuzzy metric space. The fuzzy metric (M, ∗) (or the fuzzy metric space (X, M, ∗) ) is said to be non-Archimedean or strong if it satisfies for each x, y, z ∈ X and each t > 0 M(x, y, t) ≥ M(x, y, t) ∗ M(y, z, t). In order to obtain a fuzzy version of classical Banach contraction theorem, Gregori and Sapena [19] introduced the following concepts: Definition 6. Let (X, M, ∗) be a fuzzy metric space. A mapping f : X → X is said to be fuzzy contractive if there exists k ∈ (0, 1) such that   1 1 −1 ≤ k −1 M(f (x), f (y), t) M(x, y, t) for each x, y ∈ X and t > 0. Definition 7. Let (X, M, ∗) be a fuzzy metric space. A sequence {xn } in X is said to be fuzzy contractive if there exists k ∈ (0, 1) such that   1 1 −1 ≤ k −1 M(xn+1 , xn+2 , t) M(xn , xn+1 , t) for all t > 0, n ∈ N. Recall that a sequence {xn } in a metric space (X, d) is said to be contractive if there exists k ∈ (0, 1) such that d(xn+1 , xn+2 ) ≤ kd(xn , xn+1 ), for all n ∈ N. Proposition 1. Let (X, Md , ∗) be the standard fuzzy metric space induced by the metric d on X. The sequence {xn } in X is contractive in (X, d) iff {xn } is fuzzy contractive in (X, Md , ∗). Theorem 5. [15] A sequence {xn } in a fuzzy metric space (X, M, ∗) converges to x if and only if M(xn , x, t) → 1 as n → ∞.

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Theorem 6 (Fuzzy Banach contraction theorem). Let (X, M, ∗) be a complete fuzzy metric space (in the sense of George and Veeramani) in which fuzzy contractive sequences are Cauchy. Let f : X → X be a contractive mapping with contractive constant k. Then, f has a unique fixed point. Proof. Fix x ∈ X. Let xn = f n (x), n ∈ N. We have for t > 0   1 1 −1 ≤ k −1 M(f (x), f 2 (x), t) M(x, x1 , t) and by induction

  1 1 −1 ≤ k −1 M(xn+1 , xn+2 , t) M(xn , xn+1 , t)

for all n ∈ N. Then {xn } is a fuzzy contractive sequence, so it is a Cauchy sequence and hence, xn converges to y, for some y ∈ X. We will see y is a fixed point for f . By Theorem 5, we have   1 1 −1 ≤ k −1 → 0 M(f (y), f (xn ), t) M(y, xn , t) as n → ∞, then lim M(f (y), f (xn ), t) = 1 for each t > 0, and, therefore, lim f (xn ) = f (y), that is, n→∞

lim xn+1 = f (y) and then f (y) = y.

n→∞

n→∞

To show uniqueness, assume f (z) = z for some z ∈ X. Then for t > 0 we have 1 1 −1 = −1 M(y, z, t) M(f (y), f (z), t)   1 −1 ≤k M(y, z, t)   1 n ≤ ··· ≤ k −1 → 0 M(y, z, t)

as n → ∞. Hence, M(y, z, t) = 1 and then y = z. Now suppose (X, Md , ∗) is a complete standard fuzzy metric space induced by the metric d on X. From Remark 1 (X, d) is complete, then if {xn } is a fuzzy contractive sequence, by Proposition 1 it is contractive in (X, d), hence convergent. Therefore, from Theorem 6, we have the following corollary, which can be considered the fuzzy version of the classic Banach contraction theorem on complete metric space.   Corollary 1. Let (X, Md , ∗) be a complete standard fuzzy metric space and let f : X → X be a fuzzy contractive mapping. Then, f has a unique fixed point. Theorem 7 (Fuzzy Banach contraction theorem). Let (X, M, ∗) be a G-complete fuzzy metric space (in the sense of Kramosil and Michalek) and let f : X → X be a fuzzy contractive mapping. Then, f has a unique fixed point. Proof. Let k ∈ (0, 1) and suppose f satisfies   1 1 −1 ≤ k −1 M(f (x), f (y), t) M(x, y, t)

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t > 0. Fix x ∈ X. Let xn = f n (x), n ∈ N. We have seen in the proof of Theorem 6 that {xn } is a fuzzy contractive sequence satisfying   1 1 −1 ≤ k −1 M(xn+1 , xn+2 , t) M(xn , xn+1 , t) n ∈ N. Thus, 1 − 1 ≤ k2 M(xn+1 , xn+2 , t)



 −1

1

M(xn−1 , xn , t)   1 n ≤ ··· ≤ k −1 M(x1 , x2 , t)

→ 0 as n → ∞ and, therefore, lim M (xn , xn+1 , t) = 1, ∀t > 0. Then for a fixed p ∈ N we have n→∞

  t M(xn , xn+p , t) ≥ M xn , xn+1 , p 

t ∗ · · · ∗ M xn+p−1 , xn+p , p



(6.7) p

   → 1 ∗ ··· ∗ 1 = 1

(6.8)

and thus, {xn } is a G-Cauchy sequence. Therefore, {xn } converges to y for some y ∈ X. Now, imitating the proof of Theorem 6 one can prove that y is the unique fixed point for f .   Remark 2. In the above Theorem 7 it has been proved that each fuzzy contractive sequence is G-Cauchy whereas in the Theorem 6 it was assumed that fuzzy contractive sequences are M-Cauchy. This raises the following question: Question (Gregori and Sapena [19]). Is a fuzzy contractive sequence a Cauchy sequence in George and Veeramani’s sense? The above problem generated much interest to fuzzy fixed point theorists in working on various aspects of fuzzy contractive mapping and associated fixed points. In this direction Tirado [56, 57] introduced the following: Definition 8. We say that the mapping T is a Tirado contraction [56] (see also [41]) if the following condition is satisfied: there exists k ∈ (0, 1) such that 1 − M(Tx, Ty, t) ≤ k (1 − M(x, y, t)) for all x, y ∈ X and t > 0. The constant k is called the contractive constant of T. Tirado [56] proved the following theorem as a consequence of his study. Theorem 8. Let (X, M, ∗L ) be a complete fuzzy metric space. If T is a self-mapping of X with the property that there is k ∈ (0, 1) such that 1 − M(Tx, Ty, t) ≤ k(1 − M(x, y, t)) for all x, y ∈ X and t > 0, then T has a unique fixed point.

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On the other hand, Mihet [40] introduced the concept of point convergence and improved the result of Gregori and Sapena [19]. Definition 9. Let (X, M, ∗) be a fuzzy metric space. A sequence (xn ) in X is said to be pointwise convergent to x ∈ X (we write xn →p x) if there exists t > 0 such that lim M(xn , x, t) = 1.

n→∞

It is easy to see that, endowed with point convergence, a GV-fuzzy metric space (X, M, ∗) is a space with convergence in the sense of Fréchet, that is, one of the following holds: (1) Every sequence in X has at most one limit point. (2) Every constant sequence, xn = x, ∀n ∈ N, is convergent and lim xn = x. n→∞

(3) Every subsequence of a convergent sequence is also convergent and has the same limit as the whole sequence. Remark 3. It is worth noting that if the point convergence in a fuzzy metric space (X, M, ∗) is Fréchet, then (GV2) holds (thus, the uniqueness of the limit in the point convergence characterizes, in a sense, a fuzzy metric space in the sense of George and Veeramani). Indeed, let x, y ∈ X, x = y. If M(x, y, t) = 1 for some t > 0, then the sequence (xn )n∈N ⊂ X defined as x, y, x, y, . . . has two distinct limits, for the equality M(x, x, t) = M(y, x, t) = 1 implies xn →p x, while M(x, y, t) = M(y, y, t) = 1 implies xn →p y. In the next example we will see that there exist p-convergent but not convergent sequences. Example 3. Let (xn )n∈N ⊂ (0, ∞), xn 1 and X = (xn ) ∪ {1}. Define M(xn , xn , t) = 1∀n ∈ N, ∀t > 0, M(1, 1, t) = 1∀t > 0, M(xn , xm , t) = min{xn , xm } ∀n, m ∈ N, ∀t > 0 and

min{xn , t} if 0 < t < 1, M(xn , 1, t) = if t > 1. xn for all n ∈ N. Then (X, M, TM ), where TM (a, b) = min{a, b}, is a fuzzy metric space (see [20], Example 2]). Since lim M(xn , 1, 12 ) = 12 , (xn ) is not convergent. Nevertheless it is p-convergent n→∞

to 1, for lim M(xn , 1, 2) = 1. n→∞

Theorem 9. Let (X, M, ∗) be a GV-fuzzy metric space and f : X → X be a fuzzy contractive mapping. Suppose that, for some x ∈ X, the sequence (xn )n∈N , xn = f n (x) of its iterates has a p-convergent subsequence. Then f has a unique fixed point. It should be noted that a similar theorem does not hold in KM-fuzzy metric spaces. This is illustrated in the following: Example 4. Let X be the set N = {1, 2, . . .}. We define (for p = q) the fuzzy mapping M by ⎧ 0 if t = 0, ⎨ 1 − 2− min{p,q} if 0 < t ≤ 1, M(p, q, t) = ⎩ 1 if t > 1.

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As 1 − 1/2− min(p,r) ≥ min{1 − 1/2− min(r,q) , 1 − 1/2− min(p,q) }∀p, q, r ∈ N∗ , (X, M, TM ) is a KM-fuzzy metric space satisfying M(x, y, t) = 0 for all x, y ∈ X and t > 0. The mapping f : N → N, f (x) = x + 1 is fuzzy contractive. Indeed, if t > 1 then   1 1 1 −1 = 0 ≤ −1 M(f (p), f (q), t) 2 M(p, q, t) for every p, q ∈ N∗ , while if 0 < t ≤ 1 and p < q, then 1 1 − 1 = p+1 M(f (p), f (q), t) 2 −1 1 1 = ≤ p+1 2 −2 2



 1 −1 . M(p, q, t)

As lim M(f n (x), 1, x) = 1 for every x ∈ X and s > 1, it follows that xn →p 1. Nevertheless, 1 is n→∞

not a fixed point of f . Remark 4. (1) We note that in Example 3, as well as in Example 4, there are essentially no nonconstant convergent sequences. (2) It will be natural to continue the study of these convergence spaces, by finding some more examples and introducing a similar concept for Cauchy sequence, p-completeness, etc. Also, it would be interesting to compare different types of contraction maps in fuzzy metric spaces. On the other hand, Yun et al. [66] introduced the notion of minimal slope of a map between fuzzy metric spaces and studied various properties of fuzzy contractive mapping which support the above question proposed by Gregori and Sapena [19]. In 2008, Mihet [41], provided a partial answer to the above question proposed by Gregori and Sapena in the affirmative by introducing the notion of fuzzy Ψ -contractive mapping as follows: Definition 10. [41] Let Ψ be the class of all mappings ψ : [0, 1] → [0, 1] such that ψ is continuous, nondecreasing and ψ (t) > t, ∀ t ∈ (0, 1). Let (X, M, ∗) be a fuzzy metric space and ψ ∈ Ψ . A mapping f : X → X is called a fuzzy ψ -contractive mapping if the following implication takes place: M(x, y, t) > 0 =⇒ M(f (x), f (y), t) ≥ ψ (M(x, y, t)). A fuzzy ψ -contractive sequence in a fuzzy metric space (X, M, ∗) is any sequence {xn }n∈N in X such that M(xn+2 , xn+1 , t) ≥ ψ (M(xn+1 , xn , t)) for all n ∈ N and t > 0. Example 5. Let X = [0, ∞), a ∗ b = min{a, b} ∀a, b ∈ [0, 1] and  0 if t ≤ |x − y|, M(x, y, t) = 1 if t > |x − y|.

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It is well known that (X, M, ∗) is KM-fuzzy metric space. Let ψ be a mapping in Ψ . Since ψ (1) = 1 and M(x, y, t) > 0 =⇒ M(x, y, t) = 1 =⇒ ψ (M(x, y, t)) = 1. It follows that any fuzzy contractive mapping on (X, M, ∗) satisfying |x − y| < t =⇒ |f (x) − f (y)| < t, that is, |f (x) − f (y)| ≤ |x − y| ∀x, y ∈ X. Conversely, if f : X → X is such that |f (x) − f (y)| ≤ |x − y| ∀x, y ∈ X, then f is a fuzzy ψ contractive mapping for every ψ in Ψ such that ψ (0) = 0. Thus, the mappings f : X → X, f (x) = x + 1, g(x) = x are fuzzy ψk -contractive on (X, M, ∗). Remark 5 (Mihet [39], Example 3.4). The sequence {xn }n∈N , xn = n + 1 in the fuzzy metric space considered in the above Example 5, although fuzzy ψk -contractive, is not M-Cauchy. t is in Ψ t + k(1 − t) and a ψk -fuzzy contractive mapping is a fuzzy contractive mapping in the sense of Geogori and Sepena [19]. We note that, for every k ∈ (0, 1), the mapping ψk : [0, 1] → [0, 1], ψk (t) =

Theorem 10. Let (X, M, ∗) be an M-complete non-Archimedean fuzzy metric space and f : X → X be a fuzzy ψ - contractive mapping. If there exists x ∈ X such that M(x, f (x), t) > 0 for every t > 0, then f has a fixed point. Proof. Let x ∈ X be such that M(x, f (x), t) > 0, ∀ t > 0 and xn = f n (x), n ∈ N. We have M(x1 , x2 , t) ≥ ψ (M(x0 , x1 , t)) ≥ M(x0 , x1 , t) > 0 ∀t > 0. Hence

M(x2 , x3 , t) ≥ ψ (M(x1 , x2 , t)) ≥ M(x1 , x2 , t) > 0 ∀t > 0.

By induction, M(xn+1 , xn+2 , t) ≥ M(xn , xn+1 , t) > 0 for all t > 0. Therefore, for every t > 0, M(xn , xn+1 , t)n∈N is a nondecreasing sequence of numbers in (0, 1]. Fix a t > 0 and denote lim M(xn , xn+1 , t) by l. We have l ∈ (0, 1] (for M(x0 , x1 , t) > 0) and since M(xn , xn+1 , t) ≥

n→∞

ψ (M(xn−1 , xn , t)) and ψ is continuous, l ≥ ψ (l). This implies l = 1, and therefore lim M(xn , xn+1 , t) = 1 ∀t > 0.

n→∞

If {xn } is not M-Cauchy, then there are ε ∈ (0, 1) and t > 0 such that for each k ∈ N there exist m(k), n(k) ∈ N with m(k) > n(k) ≥ k and M(xm(k) , xn(k) , t) ≤ 1 − ε .

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Let, for each k, m(k) be the least integer exceeding n(k) satisfying the above property, that is, M(xm(k)−1 , xn(k)−1 , t) > 1 − ε and M(xm(k) , xn(k) , t) ≤ 1 − ε . Then, for each positive integer k, 1 − ε ≥ M(xm(k) , xn(k) , t) ≥ ∗(M(xm(k)−1 , xn(k) , t), M(xm(k)−1 , xm(k) , t)) ≥ ∗(1 − ε , M(xm(k)−1 , xm(k) , t)). Since lim ∗(1 − ε , M(xm(k)−1 , xm(k) , t)) = ∗(1 − ε , 1) = 1 − ε , it follows that k→∞

lim M(xm(k) , xn(k) , t) = 1 − ε .

k→∞

Let us denote M(xn(k) , xn(k)+1 , t) by zn . Then M(xm(k) , xn(k) , t) ≥ ∗2 (zn , M(xm(k)+1 , xn(k)+1 , t), zm ) ≥ ∗2 (zn , M(xm(k) , xn(k) , t), zm ). Letting k → ∞ we obtain 1 − ε ≥ ∗2 (1, ψ (1 − ε ), 1) = ψ (1 − ε ) > 1 − ε , which is a contradiction. Thus {xn } is a Cauchy sequence. If lim xn = y, then from M(f (y), f (xn ), t) ≥ ψ (M(y, xn , t)) it follows xn+1 → f (y). From here we n→∞

deduce that M(y, f (y), t) ≥ ∗2 (M(y, xn , t), M(xn , xn+1 , t), M(xn+1 , f (y), t)) −−→ 1 n→∞

for all t > 0, hence f (y) = y.

 

Theorem 11. Let (X, M, ∗) be an M-complete non-Archimedean fuzzy metric space satisfying the condition M(x, y, t) > 0, ∀ t > 0 and f : X → X be a fuzzy ψ -contractive mapping, Then f has a unique fixed point. Example 6. Let X = (0, ∞), a ∗ b = ab, ∀ a, b ∈ [0, 1] and M(x, y, t) =

min(x, y) ∀ t ∈ (0, ∞), ∀ x, y > 0. max(x, y)

√ Then, (X, M, ∗) is an M-complete non-Archimedean fuzzy metric space. Since t > t ∀ √ t∈ √ (0, 1), the mapping f : X → X, f (x) = x is a fuzzy ψ -contractive mapping with ψ (t) = t. Thus, all the conditions of Theorem 11 are satisfied, and the fixed point of f is x = 1. Some other generalizations of results of Geogori and Sepena [19] and Mihet [41] can be found in [1, 17, 18, 56, 62]. Recently, Wardowski [65] introduced the concept of H contractive mappings, as a generalization of that of fuzzy contractive mappings, and established the conditions guaranteeing the existence and uniqueness of fixed points for these types of contractions in M-complete fuzzy metric spaces in the sense of George and Veeramani. Definition 11. Let H be the family of mappings η : (0, 1] → [0, ∞) satisfying the following conditions:

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(H1) η transforms (0, 1] onto [0, ∞); (H2) η is strictly decreasing. Then the mapping f : X → X is called fuzzy H -contractive (see Wardowski [65]) with respect to η ∈ H if there exists k ∈ (0, 1) satisfying the following condition:

η (M(fx, fy, t)) ≤ kη (M(x, y, t)) for all x, y ∈ X and t > 0. Theorem 12. [65] Let (X, M, ∗) be an M-complete fuzzy metric space and f : X → X be a fuzzy H -contractive mapping with respect to η ∈ H such that: (a) ∏ki=1 M(x, fx, ti ) = 0 for all x ∈ X, k ∈ N and a sequence (ti )i∈N ⊂ (0, ∞), ti  0; (b) r ∗ s > 0 =⇒ η (r ∗ s) ≤ η (r) + η (s) for all r, s ∈ {M(x, fx, t) : x ∈ X, t > 0}; (c) {η (M(x, fx, ti )) : i ∈ N} is bounded for all x ∈ X and any sequence (ti )i∈N ⊂ (0.∞), ti  0. Then f has a unique fixed point x∗ ∈ X and for each x0 ∈ X the sequence (f n x0 )n∈N converges to x∗ . In a recent note, Gregori and Miñana [23] observed that the main idea of Wardowski [65] is correct and different from the known concepts in the literature but they also showed that the class of fuzzy H -contractive mappings are included in the class of fuzzy Ψ -contractive mappings, as well as pointing out some drawbacks of the conditions used in the above Theorem 12. Remark 6 (See Gregori and Miñana [23]). If η ∈ H then the mappings η · k : (0, 1] → [0, ∞) and η −1 : [0, ∞) → (0, 1], defined in its obvious sense, are two bijective continuous mappings which are strictly decreasing. In view of the above remark, we observe that every fuzzy H -contractive mapping is a fuzzy ψ -contraction with ψ (t) = η −1 (kη (t)) for all t ∈ (0, 1] (see [23]). In this direction of research work, a recent paper of Mihet [43] provides a larger prospective and further scope to study fixed points of fuzzy H -contractive mappings.

6.4 Fuzzy Z -Contractive Mappings and Fixed Points Most recently Shukla et al. [53] unified different classes of fuzzy contractive mappings by introducing a new class of fuzzy contractive mappings called fuzzy Z -contractive mappings. First, we define the Z -contraction in GV-fuzzy metric spaces. Denote by Z the family of all functions ζ : (0, 1] × (0, 1] → R satisfying the following condition:

ζ (t, s) > s for all t, s ∈ (0, 1). Example 7. Consider the following functions ζ defined from (0, 1] × (0, 1] into R by:

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ζ (t, s) = ψ (s), where ψ : (0, 1] → (0, 1] is a function such that s < ψ (s) for all s ∈ (0, 1); 1 + t; (ii) ζ (t, s) = s+t s (iii) ζ (t, s) = . t

(i)

Then, in all the cases ζ ∈ Z . Remark 7. By the above definition it is obvious that ζ (t, t) > t for all 0 < t < 1. Definition 12. Let (X, M, ∗) be a fuzzy metric space and T : X → X be a mapping. Suppose there exists ζ ∈ Z such that M(Tx, Ty, t) ≥ ζ (M(Tx, Ty, t), M(x, y, t))

(6.9)

for all x, y ∈ X, Tx = Ty, t > 0. Then T is called a fuzzy Z -contractive mapping with respect to the function ζ ∈ Z . Example 8. Every Tirado contraction with contractive constant k is a fuzzy Z -contraction with respect to the function ζT ∈ Z defined by ζT (t, s) = 1 + k(s − 1) for all s, t ∈ (0, 1]. Example 9. Every fuzzy contractive mapping with contractive constant k is a fuzzy Z s for all contraction with respect to the function ζGS ∈ Z defined by ζGS (t, s) = k+(1−k)s s, t ∈ (0, 1]. Example 10. In view of Remark 14, every H -contractive mapping with respect to η ∈ H is a fuzzy Z -contraction with respect to the function ζW ∈ Z defined by ζW (t, s) = η −1 (kη (s)) for all s, t ∈ (0, 1]. Example 11. Every ψ -contractive mapping is a fuzzy Z -contraction with respect to the function ζM defined by ζM (t, s) = ψ (s) for all s, t ∈ (0, 1]. Example 12. Let X = R and d be the usual metric on X. Then (X, Md , ∗m ) is a complete fuzzy t for all x, y ∈ X, t > 0, is the standard fuzzy metric metric space, where Md = t + d(x, y) induced by d (see [15]). Let T : X → X be an Edelstein’s mapping (contractive mapping) on metric space (X, d), that is, d(Tx, Ty) < d(x, y) for all x, y ∈ X; then T is a fuzzy Z -contractive mapping with respect to the function ζm ∈ Z defined by

s+t , if t > s; 2 ζm (t, s) = 1, otherwise. (i.e., the arithmetic mean of s and t) for Indeed, the above fact remains true, if instead s+t 2 t > s, one takes the geometric or harmonic mean of s and t. Remark 8. If (X, M, ∗) is an arbitrary fuzzy metric space and T : X → X is a Edelstein’s mapping on (X, M, ∗), that is, M(Tx, Ty, t) > M(x, y, t) for all x, y ∈ X, t > 0, then T is a fuzzy Z -contractive mapping with respect to the function ζm ∈ Z defined in the previous example. Therefore, we conclude that for any given fuzzy Edelstein’s mapping we always have ζ (= ζm ) ∈ Z such that the fuzzy Edelstein’s mapping is a fuzzy Z -contractive mapping,

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and so, the contractive mappings considered by Tirado [56], Gregori and Sapena [19], Wardowski [65] and Mihe¸t [41] are included in this new class. However, there are fuzzy Z -contractive mapping which do not belong to any of these considered classes (see, e.g., Example 13, Example 15 and Example 17). The following example shows that a fuzzy Z -contractive mapping may not have a fixed point even in an M-complete fuzzy metric space. Example Let X = N and define the fuzzy set M on X × X × (0, ∞) by M(n, m, t) =  n 13. m , for all n, m ∈ X, t > 0. Then (X, M, ∗p ) is an M-complete fuzzy metric space. min m n Define a mapping T : X → X by Tn = n + 1 for all n ∈ X. Then T is a fuzzy Z -contractive mapping with respect to the function ζm ∈ Z defined in Example 12. Notice that T is a fixed point free mapping on X. The above example motivates us for the consideration of a space having some additional property so that the existence of a fixed point of fuzzy Z -contractive mapping can be ensured. Definition 13. Let (X, M, ∗) be a fuzzy metric space, T : X → X a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, T, ζ ) has the property (S), if for any Picard sequence {xn } with initial value x ∈ X, that is, xn = Tn x for all n ∈ N such that inf M(xn , xm , t) ≤ m>n

inf M(xn+1 , xm+1 , t) for all n ∈ N, t > 0 implies that

m>n

lim inf ζ (M(xn+1 , xm+1 , t), M(xn , xm , t)) = 1 for all t > 0.

n→∞ m>n

The following example shows that there exists a function ζ such that the mappings introduced by Tirado [56] form a quadruple (X, M, T, ζ ) which satisfies the property (S), where (X, M, ∗) is an arbitrary fuzzy metric space. Example 14. Let (X, M, ∗) be an arbitrary fuzzy metric space and T : X → X be a fuzzy Tirado contraction. Then, the quadruple (X, M, T, ζ ) has the property (S) with ζ (t, s) = 1 + k(s − 1) for all t, s ∈ (0, 1]. Indeed, if x ∈ X and {xn } is a Picard sequence with initial value x such that inf M(xn , xm , t) ≤ inf M(xn+1 , xm+1 , t) for all n ∈ N, t > 0, then lim inf M(xn , xm , t) must exist for

m>n

m>n

n→∞ m>n

all t > 0. Suppose lim inf M(xn , xm , t) = a(t), t > 0, then a(t) ≤ 1. By definition of ζ , for every

t > 0 we have

n→∞ m>n

lim inf ζ (M(xn+1 , xm+1 , t), M(xn , xm , t)) = 1 + k(a(t) − 1).

n→∞ m>n

Also, by the contractivity condition we obtain 1 + ka(t) ≤ k + a(t), and so, 1 ≤ a(t). This shows that a(t) = 1 for all t > 0, that is, lim inf ζ (M(xn+1 , xm+1 , t), M(xn , xm , t)) = 1.

n→∞ m>n

We next state the main results of this paper. The following theorem generalizes Theorem 8 (see also Corollary 3.9 in [41]) for arbitrary t-norms. Theorem 13. Let (X, M, ∗) be an M-complete fuzzy metric space and T : X → X be a fuzzy Z contraction. If the quadruple (X, M, T, ζ ) has the property (S), then T has a unique fixed point u ∈ X.

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Proof. First, we show that if the fixed point of T exists, then it is unique. Suppose, u, v are two distinct fixed points of T, that is, Tu = u and Tv = v and there exists s > 0 such that M(u, v, s) < 1. Then by the condition (6.9) and definition of ζ we have M(u, v, s) = M(Tu, Tv, s) ≥ ζ (M(Tu, Tv, s), M(u, v, s)) > M(u, v, s). This contradiction shows that M(u, v, t) = 1 for all t > 0, and so, u = v. It proves uniqueness. We shall show the existence of a fixed point of T. Let x0 ∈ X and define the Picard sequence {xn } by xn = Txn−1 for all n ∈ N. If xn = xn−1 for any n ∈ N, then Txn−1 = xn = xn−1 is a fixed point of T. Therefore, we assume that xn = xn−1 for all n ∈ N, that is, no consecutive terms of the sequence {xn } are equal. Further, if xn = xm for some n < m, then as no consecutive terms of the sequence {xn } are equal from (6.9), we have M(xn+1 , xn+2 , t) ≥ ζ (M(xn+1 , xn+2 , t), M(xn , xn+1 , t)) > M(xn , xn+1 , t) that is, M(xn , xn+1 , t) < M(xn+1 , xn+2 , t). Similarly one can prove that M(xn , xn+1 , t) < M(xn+1 , xn+2 , t) < · · · < M(xm , xm+1 , t). Since xn = xm , we have xn+1 = Txn = Txm = xm+1 , and so, the above inequality yields a contradiction. Thus, we can assume that xn = xm for all distinct n, m ∈ N. Now, for t > 0, let an (t) = inf M(xn , xm , t) m>n

Then it follows from (6.9) and by definition of ζ that M(xn+1 , xm+1 , t) = M(Txn , Txm , t) ≥ ζ (M(Txn , Txm , t), M(xn , xm , t)) > M(xn , xm , t)

(6.10)

for each t > 0. Therefore, we have M(xn , xm , t) < M(xn+1 , xm+1 , t) for all n < m. Taking the infimum over m(> n) in the above inequality we obtain inf M(xn , xm , t) ≤ inf M(xn+1 , xm+1 , t)

m>n

m>n

that is, an (t) ≤ an+1 (t) for all n ∈ N. Thus, {an (t)} is bounded and monotonic for each t > 0. Suppose, lim an (t) = a(t), t > 0. We claim that a(t) = 1 for each t > 0. If s > 0 and a(s) < 1, n→∞

then, using the fact that the quadruple (X, M, T, ζ ) has the property (S), we obtain lim inf ζ (M(xn , xm , s), M(xn+1 , xm+1 , s)) = 1.

(6.11)

n→∞ m>n

From inequality (6.10) we have inf M(xn+1 , xm+1 , s) ≥ inf ζ (M(Txn , Txm , s), M(xn , xm , s)) ≥ inf M(xn , xm , s)

m>n

that is,

m>n

m>n

an+1 (s) ≥ inf ζ (M(Txn , Txm , s), M(xn , xm , s)) ≥ an (s). m>n

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215

Letting n → ∞ and using (6.11) in the above inequality we obtain lim inf M(xn , xm , s) = a(s) = 1.

n→∞ m>n

This contradiction verifies our claim. By definition of an we have lim M(xn , xm , t) = 1 for n,m→∞

all t > 0. Hence, {xn } is an M-Cauchy sequence and by M-completeness of X there exists u ∈ X such that (6.12) lim M(xn , u, t) = 1 for all t > 0. n→∞

We shall show that u is a fixed point of T. Suppose Tu = u. Without loss of generality we can assume that xn = u and xn = Tu for all n ∈ N, and so, there exists s > 0 such that M(u, Tu, s) < 1, M(xn , u, s) < 1 and M(xn+1 , Tu, s) = M(Txn , Tu, s) < 1 for all n ∈ N. Then, we have M(xn , u, s) < ζ (M(Txn , Tu, s), M(xn , u, s)) ≤ M(Txn , Tu, s) = M(xn+1 , Tu, s). Letting n → ∞ and using (6.12) we obtain 1 ≤ M(u, Tu, s). This contradiction shows that M(u, Tu, t) = 1 for all t > 0, and so, Tu = u. Thus, the existence of a fixed point follows.   Remark 9. Example 8 and Example 14 show that the above theorem generalizes Theorem 8 for arbitrary t-norms. The next example shows that this generalization is proper. Example 15. Let {xn } be a strictly increasing sequence of real numbers such that 0 < xn ≤ 1 for all n ∈ N and limn→∞ xn = 1. Let X = {xn : n ∈ N} ∪ {1} and define a fuzzy set M on X × X × (0, ∞) by:

1, if x = y; M(x, y, t) = for all x, y ∈ X, t ∈ (0, ∞). min{x, y}, otherwise Then, (X, M, ∗m ) is an M-complete fuzzy metric space. Define a function ζ : (0, 1]×(0, 1] → R by

t, if t > s; ζ (t, s) = √ for all s, t ∈ (0, 1] s, if t ≤ s. and a mapping T : X → X by Txn = xn+1 for all n ∈ N and T1 = 1. Then, ζ ∈ Z and the quadruple (X, M, T, ζ ) has the property (S). Furthermore, the mapping T is a fuzzy Z -contractive mapping with respect to the function ζ . Thus, all the conditions of Theorem 13 are satisfied and we can conclude the existence of a fixed point of T. Indeed, x = 1 is the unique fixed point of T. Remark 10. In view of the above example, we can conclude that the mapping T is not a 1 Tirado contraction. For instance, take the sequence {xn } defined by xn = 1 − 2 for all 2n n ∈ N; in the above example. Then, we have 1 − M(Txn , Txn+1 , t) = 1 − M(xn+1 , xn+2 , t) = 1 − xn+1 and 1 − M(xn , xn+1 , t) = 1 − xn , for all t > 0. Therefore, for sufficient large n, there exists no k such that k ∈ [0, 1) and 1 − M(Txn , Txn+1 , t) ≤ k[1 − M(xn , xn+1 , t)] for all t > 0.

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Thus, Theorem 13 is an actual generalization of the fixed point result of Tirado [56], that is, Theorem 8. We next introduce another condition (S ) which is weaker than the condition (S). Definition 14. Let (X, M, ∗) be a fuzzy metric space, T : X → X a mapping and ζ ∈ Z . Then we say that the quadruple (X, M, T, ζ ) has the property (S ) if for any Picard sequence {xn } with initial value x ∈ X, that is, xn = Tn x for all n ∈ N such that inf M(xn , xm , t) ≤ m>n

inf M(xn+1 , xm+1 , t) for all n ∈ N, t > 0 and 0 < lim inf M(xn , xm , t) < 1 for all t > 0 implies that

m>n

n→∞ m>n

lim inf ζ (M(xn+1 , xm+1 , t), M(xn , xm , t)) = 1 for all t > 0.

n→∞ m>n

The following example verifies the fact that condition (S ) is weaker than condition (S). Example 16. Let ε > 0 be fixed and X = [ε , ∞). Define a fuzzy set M on X × X × (0, ∞) by: ⎧ if x = y; ⎨ 1, 1 for all x, y ∈ X, t ∈ (0, ∞). M(x, y, t) = , otherwise ⎩ 1 + max{x, y} Then, (X, M, ∗m ) is a fuzzy metric space. Define T : X → X by Tx = 2x for all x ∈ X. Suppose ζ : (0, 1] × (0, 1] → R be defined by ζ (t, s) = ψ (s) for all t, s ∈ (0, 1], where ψ ∈ Ψ is such that ψ (0) = 0. Then, it is easy to see that the quadruple (X, M, T, ζ ) satisfies the condition (S ) trivially. On the other hand, the quadruple (X, M, T, ζ ) does not satisfy the condition (S). Indeed, for any x ∈ X, t > 0 we have inf M(Tn x, Tm x, t) = inf M(2n x, 2m x, t) = 0 < 1.

m>n

m>n

Therefore, inf M(T x, T x, t) ≤ inf M(T n

m>n

m

m>n

n+1

x, Tm+1 x, t) for all n ∈ N, t > 0. However,

lim inf ζ (M(xn+1 , xm+1 , t), M(xn , xm , t)) = lim inf ψ (M(xn , xm , t)) = 0 = 1.

n→∞ m>n

n→∞ m>n

In the next theorem, we see that the condition (S ) enables us to extend the result of Mihet [41] for fuzzy Z -contraction, but with an additional assumption to Theorem 13. Theorem 14. Let (X, M, ∗) be an M-complete fuzzy metric space, T : X → X be a fuzzy Z contraction and the quadruple (X, M, T, ζ ) has the property (S ). In addition, suppose that lim inf M(Tn x, Tm x, t) > 0 for all x ∈ X, t > 0. Then T has a unique fixed point u ∈ X.

n→∞ m>n

Proof. Because, lim inf M(Tn x, Tm x, t) > 0 for all x ∈ X, t > 0, following the lines of the proof n→∞ m>n

of Theorem 13 and using the property (S ) we obtain the required result.

 

In the next example, we show that the class of fuzzy Z -contractions is wider than that of fuzzy ψ -contractions and verify the merit of fuzzy Z -contractive mappings over fuzzy ψ -contractive mappings. For this, we use the idea of Example 15. Example 17. Let X = {xn : n ∈ N} ∪ {1}, where {xn } is an arbitrary sequence such that xn ∈ (0, 1), xn < xn+1 for all n ∈ N and limn→∞ xn = 1. Define a fuzzy set M on X × X × (0, ∞) by:

1, if x = y; M(x, y, t) = for all x, y ∈ X, t ∈ (0, ∞). min{x, y}, otherwise

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Then, (X, M, ∗m ) is an M-complete fuzzy metric space. Define a mapping T : X → X by Txn = xn+1 for all n ∈ N and T1 = 1. Then, we claim that T is not a fuzzy ψ -contraction. On contrary, suppose that T is a fuzzy ψ -contraction. Therefore, there exists ψ ∈ Ψ such that ψ (M(xn , xm , t)) ≤ M(Txn , Txm , t) for all n, m ∈ N, n < m, that is, xn < ψ (xn ) ≤ xn+1 . Since ψ ∈ Ψ we can choose the sequence {xn } as follows: let x1 ∈ (0, 1) and xn+1 = for all n ∈ N. Then, by (6.13) we obtain xn < ψ (xn ) ≤

(6.13) xn + ψ (xn ) 2

xn + ψ (xn ) . 2

The above inequalities contradict the definition of ψ . Therefore, T is not a fuzzy ψ contraction. On the other hand, we have shown in Example 15 that the mapping T is a fuzzy Z -contractive mapping as well as that the condition (S ) is satisfied. Now the existence and uniqueness of a fixed point of T is assured by Theorem 14. Indeed, 1 is the unique fixed point of T. Corollary 2. Let (X, M, ∗) be an M-complete fuzzy metric space, T : X → X be a fuzzy ψ contractive mapping and lim inf M(Tn x, Tm x, t) > 0 for all x ∈ X, t > 0. Then T has a unique n→∞ m>n

fixed point u ∈ X.

Proof. In view of Example 11 we need only to show that the quadruple (X, M, T, ζ ) has the property (S ), where ζ (t, s) = ψ (s). Suppose, x ∈ X and {xn } is a Picard sequence with initial value x such that inf M(xn , xm , t) ≤ inf M(xn+1 , xm+1 , t) and for every t > 0, m>n

m>n

0 < lim inf M(xn , xm , t) = a(t) < 1. Then, by definition of ψ we have for every t > 0 n→∞ m>n

lim inf ζ (M(xn+1 , xm+1 , t), M(xn , xm , t)) = ψ (a(t)).

n→∞ m>n

Also, by ψ -contractivity we obtain ψ (a(t)) ≤ a(t), and so, a(t) = 1, that is, lim inf ζ (M(xn+1 , xm+1 , t), M(xn , xm , t)) = 1 for all t > 0.

n→∞ m>n

Therefore, the quadruple (X, M, T, ζ ) has the property (S ).

 

Remark 11. Since the class of fuzzy ψ -contractions consists of the class of fuzzy contractive mappings [19], Tirado’s contraction [56] and Wardowski’s contraction [65], therefore, fixed point results for these contractions can be obtained by the above corollary. Remark 12. It is clear from the definition that every fuzzy Z -contractive mapping is a fuzzy Edelstein’s mapping (contractive mapping). Also, Remark 8 shows that for every fuzzy Edelstein’s mapping T there exists a function ζmean ∈ Z such that T is a fuzzy Z -contractive mapping with ζmean ∈ Z . In view of the existence of a fixed point of mapping T, notice that for a fuzzy Edelstein’s mapping the quadruple (X, M, T, ζmean ) need not to have the property (S), e.g., in Example 13, T is a fuzzy Edelstein’s mapping but the quadruple (X, M, T, ζ ) does not possess the property (S). Indeed, in this example for any Picard sequence {xn } with initial value x ∈ X we have inf M(xn , xm , t) ≤ inf M(xn+1 , xm+1 , t) for all n ∈ N, t > 0, but m>n

m>n

lim inf ζmean (M(xn+1 , xm+1 , t), M(xn , xm , t)) = 0 = 1 for all t > 0.

n→∞ m>n

Dhananjay Gopal and Tatjana Došenovi´c

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Therefore, condition (S) of Theorem 13 is not satisfied. Also, one can see that the condition: lim inf M(Tn x, Tm x, t) > 0 for all x ∈ X, t > 0 of Theorem 14 is not satisfied, while the n→∞ m>n

condition (S’) is satisfied. Thus, we conclude that: (i)

Theorem 13 and Theorem 14 are not applicable to fuzzy Edelstein mappings with ζ = ζmean . (ii) The condition lim inf M(Tn x, Tm x, t) > 0 for all x ∈ X, t > 0 of Theorem 14 is not n→∞ m>n

superfluous. Remark 13. Motivated by the results of Tirado [56] and Mihe¸t [41], we introduced the class of fuzzy Z -contractive mappings and showed that the mappings of this new class have a unique fixed point on an arbitrary M-complete fuzzy metric space having the property (S) and (S ). With suitable examples, we showed that the class of fuzzy Z -contractive mappings is weaker than the existing ones in the literature. Further, it will be interesting to apply this new approach in general settings, e.g., in fuzzy metric-like settings (see [51, 52]). It will also be interesting to generalize the class of fuzzy Z -contractive mappings for weaker contractive conditions, e.g., (ε , δ )-type contractive conditions (see [42]).

6.5 Countable Extension of t-Norms A t-norm ∗ can be extended to a countable infinite operation taking for any sequence (xn )n∈N from [0, 1] the value ∞

n

*i=1 xi = n→∞ lim *i=1 xi . ∞

n

The sequence (*i=1 xi )n∈N is nonincreasing and bounded from below, hence the limit *i=1 xi exists. In the fixed point theory (see [29, 31]) it is of interest to investigate the classes of t-norms ∗ and sequences (xn ) from the interval [0, 1] such that lim xn = 1 and n→∞

∞

∞

lim *i=n xi = lim *i=1 xn+i = 1.

n→∞

n→∞

It is obvious that ∞

∞

lim *i=n xi = 1 ⇐⇒ ∑(1 − xi ) < ∞

n→∞

i=1

for ∗ = ∗L and ∗ = ∗P . For ∗ ≥ ∗L we have the following implication ∞

∞

lim *i=n xi = 1 ⇒ ∑(1 − xi ) < ∞.

n→∞

i=1

Example 18. Some of the important classes of t-norms are the following.

(6.14)

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219

(i) The Dombi family of t-norms (∗D λ )λ∈[0, ∞] , is defined by ⎧ ∗D (x, y), λ=0 ⎪ ⎪ ⎪ ⎨ ∗M (x, y), λ=∞ 1 ∗D (x, y) = λ ⎪  λ  λ 1/λ , λ ∈ (0, ∞). ⎪ ⎪ ⎩ 1 + 1−x + 1−y x y (ii) The Aczél–Alsina family of t-norms (∗AA λ )λ∈[0, ∞] is defined by ⎧ λ=0 ⎨ ∗D (x, y), (x, y), λ=∞ ∗ (x, y) = ∗AA M λ ⎩ −((− log x)λ +(− log y)λ )1/λ , λ ∈ (0, ∞). e (iii) The family (∗SW λ )λ∈[−1, ∞] , of Sugeno–Weber t-norms is given by ⎧ ∗ (x, y), λ = −1 ⎪ ⎨ D (x, y), λ =∞ ∗ SW P ∗λ (x, y) = x + y − 1 + λxy ⎪ ⎩ max(0, ), λ ∈ (−1, ∞). 1+λ (iv) The Schweizer–Sklar family of t-norms (∗SS λ )λ∈[−∞, ∞] is defined by ⎧ ∗M (x, y), λ = −∞ ⎪ ⎪ ⎨ λ=0 ∗P (x, y), SS ∗λ (x, y) = (max(xλ + yλ − 1, 0))1/λ , λ ∈ (−∞, 0) ∪ (0, ∞) ⎪ ⎪ ⎩ λ = ∞. ∗D (x, y), SW The condition ∗ ≥ ∗L is fulfilled by the families (∗SS λ )λ∈(−∞, 1) , (∗λ )λ∈[0, ∞] . D On the other hand, there exists a member of the family (∗λ )λ∈(0, ∞) which is incomparable with ∗L , and there exists a member of the family (∗AA λ )λ∈(0, ∞) which is incomparable with ∗L . In [29] the following results are obtained:

a)

If (∗D λ )λ∈(0, ∞) is the Dombi family of t-norms and (xn )n∈N a sequence of elements from (0, 1] such that lim xn = 1 then we have the following equivalence n→∞

∞

∞ lim (∗D ∑ (1 − xn )λ < ∞ ⇐⇒ n→∞ λ )i=n xi = 1.

(6.15)

n=1

b)

If (∗SW λ )λ∈(−1, ∞] is the Sugeno–Weber family of t-norms and (xn )n∈N a sequence of elements from (0, 1] such that lim xn = 1 then we have the following equivalence n→∞

∞

∞ lim (∗SW ∑ (1 − xn ) < ∞ ⇔ n→∞ λ )i=n xi = 1.

(6.16)

n=1

c)

The equivalence (2) holds also for the family (∗AA λ )λ∈(0, ∞) , that is, ∞

∞ lim (∗AA ∑ (1 − xn )λ < ∞ ⇐⇒ n→∞ λ )i=n xi = 1.

n=1

In [29] the following proposition is obtained.

(6.17)

Dhananjay Gopal and Tatjana Došenovi´c

220

Proposition 2. Let (xn )n∈N be a sequence of numbers from [0, 1] such that lim xn = 1 and t-norm ∞

n→∞

∞

∗ is of H-type. Then lim *i=n xi = lim *i=1 xn+i = 1. n→∞

n→∞

Theorem 15. [72] Let (X, M, ∗) be a fuzzy metric space, such that lim M(x, y, t) = 1. Then if for some σ0 ∈ (0, 1) and some x0 , y0 ∈ X the following hold: ∞

1 ) = 1, σ0i

∞

1 )=1 σi

lim *i=n M(x0 , y0 ,

n→∞

then

lim *i=n M(x0 , y0 ,

n→∞

t→∞

for every σ ∈ (0, 1).

6.6 Fixed Point Theorems in Fuzzy Metric Space of Khan Type M.S. Khan et al. [34] improved the Banach fixed point theorem in metric spaces by introducing a control function called an altering distance function. As altering distance is used monotone, increasing and continuous function ϕ : [0, ∞) → [0, ∞) such that ϕ (t) = 0 if and only if t = 0. In [34] the Banach contraction principle in complete metric space (X, d) is generalized by the condition

ϕ (d(fx, fy)) ≤ aϕ (d(x, y)), for all x, y ∈ X and some 0 < a < 1, where f : X → X. This result was a motivation for further studies in metric spaces, as well as in the probabilistic metric space ([8, 9, 64]). Definition 15. [2] A function φ : R → R+ is said to be a Φ -function if it satisfies the following conditions: (i) (ii) (iii) (iv)

φ (t) = 0 if and only if t = 0, φ (t) is increasing and ϕ (t) → ∞ as t → ∞, φ is left continuous in (0, ∞), φ is continuous at 0.

Definition 16. [2] Let (X, M, ∗) be a fuzzy metric space. A self map f : X → X is said to be φ -contractive if t (6.18) M(fx, fy, φ (t)) ≥ M(x, y, φ ( )), c where 0 < c < 1, x, y ∈ X, t > 0 and φ is a Φ -function. Mihet [40] introduce the following concept of convergence (which is called pconvergence). Definition 17. [40] Let (X, M, ∗) be a fuzzy metric space. A sequence {xn } in X is said to be point convergent or p-convergent to x ∈ X if there exists t > 0 such that lim M(xn , x, t) = 1. We write xn →p x and call x the p-limit of {xn }.

n→∞

Fuzzy Contractive Mappings

221

Using the Φ function in the paper [2] the following fixed point result is obtained. Theorem 16. [2] Let (X, M, ∗) be a GV-fuzzy metric space and f : X → X be a φ -contraction. Suppose that for some x0 ∈ X the sequence {f n x0 } has a p-convergent subsequence. Then f has a unique fixed point. Without using a p-convergent subsequence, but with restrictions on t-norms the following result is obtained. This is a fuzzy version of the result presented in [70]. Theorem 17. Let (X, M, ∗) be a complete non-Archimedean fuzzy metric space, lim M(x, y, t) = 1. t→∞

Let f : X → X be continuous and a φ -contraction. If there exists x0 ∈ X and x1 = fx0 such that t−norm ∗ satisfies condition ∞ r lim *i=n M(x0 , x1 , φ ( i )) = 1, (6.19) n→∞ c for every r > 0, 0 < c < 1, then there exists a unique fixed point z of the mapping f . In [13], using the same Φ -function, a fixed point theorem for George and Veeramani contractive type mappings is proved, in probabilistic metric space. In this theorem, the authors proved that a sequence is Cauchy in the Grabiec sense. Now, we present the fuzzy version of the mentioned theorem. Theorem 18. Let (X, M, ∗) be a KM-complete fuzzy metric space, lim M(x, y, t) = 1 and f : X → X t→∞

be a self-mapping satisfying the following inequality 1 1 − 1 ≤ ψ( − 1), M(fx, fy, φ ((ct))) M(x, y, φ (t))

(6.20)

where x, y ∈ X, 0 < c < 1, φ is a Φ -function and ψ : [0, 1) → [0, 1) is such that ψ is continuous, ψ (0) = 0 and ψ n (an ) → 0, whenever an → 0 as n → ∞ and t > 0 is such that M(x, y, φ (t)) > 0. Then f has a unique fixed point. Now, with some additional conditions of the mappings φ and ψ , we prove a fixed point theorem for George and Veeramani contractive type mappings, but with the improvement that a sequence is Cauchy in George and Veeramani sense. Also, we generalize the contractive condition (6.20). Theorem 19. Let (X, M, ∗) be a GV-complete fuzzy metric space, lim M(x, y, t) = 1, t-norm ∗ is of t→∞

H-type and f : X → X be a self-mapping satisfying the following inequality: 1 1 − 1 ≤ ψ( − 1), t t M(fx, fy, φ (t)) min{M(x, y, φ ( c )), M(x, fx, φ ( c )), M(y, fy, φ ( ct )), M(fx, y, φ ( ct ))} (6.21) where x, y ∈ X, 0 < c < 1, t > 0, and φ is a nondecreasing and continuous mapping, with conditions (i) and (ii) of Φ −function and φ (a + b) ≥ φ (a) + φ (b). Let ψ : [0, 1) → [0, 1) be such that ψ is continuous, nondecreasing and ψ (t) < t. Then f has a unique fixed point.

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222

Proof. Let x0 ∈ X and x1 = fx0 . We construct a sequence {xn } as xn+1 = fxn = f n x0 , n ∈ N. Using (6.21) for x = xn−1 and y = xn we have 1 M(xn , xn+1 , φ (t)) ≤ ψ( = ψ(

−1 =

1 −1 M(fxn−1 , fxn , φ (t))

1 − 1) min{M(xn−1 , xn , φ ( ct )), M(xn−1 , xn , φ ( ct )), M(xn , xn+1 , φ ( ct )), M(xn , xn , φ ( ct ))} 1 min{M(xn−1 , xn , φ ( ct )), M(xn , xn+1 , φ ( ct ))}

− 1).

for every t > 0. If min = M(xn , xn+1 , φ ( ct )) then we have the following: 1 M(xn , xn+1 , φ (t))

− 1 ≤ ψ(

1 M(xn , xn+1 , φ ( ct ))

− 1)
M(xn , xn+1 , φ ( )), t > 0. c Since the function φ is nondecreasing we get a contradiction. Therefore min = M(xn−1 , xn , φ ( ct )), and t t M(xn , xn+1 , φ (t)) > M(xn−1 , xn , φ ( )) ≥ M(x0 , x1 , φ ( n )). c c Letting n → ∞ we have that lim M(xn , xn+1 , φ (t)) = 1, t > 0.

n→∞

∞

Let σ ∈ (c, 1), and μ = σc . As σ ∈ (0, 1), the series ∑ σ i is convergent, and there exists i=1

∞

m0 ∈ N such that ∑ σ i < 1. Therefore, for every m > m0 and s ∈ N the following is satisfied: i=m0

ε >ε

∞

m+s

i=m0

i=m

∑ σ i > ε ∑ σ i.

Since φ (0) = 0 there exists ε1 such that ε = φ (ε1 ). Let s ∈ N. Then, M(xm+s+1 , xm , ε ) = M(xm+s+1 , xm , φ (ε1 )) m+s

≥ M(xm+s+1 , xm , φ (ε1 ∑ σ i )) i=m

≥ ∗(∗(· · · ∗ (M(xm+s+1 , xm+s , φ (ε1 σ m+s )),    (s+1)−times

. . . , M(xm+1 , xm , φ (ε1 σ m ))))

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σ ≥ ∗(∗(· · · ∗ (M(xm+s+1 , xm+s , φ (ε1 ( )m+s )),    c (s+1)−times

σ . . . , M(xm+1 , xm , φ (ε1 ( )m )))) c m+s ε1 ≥ *i=m M(x1 , x0 , φ ( i )) μ ∞ ε1 ≥ *i=m M(x1 , x0 , φ ( i )) μ Using condition (ii) we have that lim M(x1 , x0 , φ ( με1i )) = 1. Let λ ∈ (0, 1) and n0 ∈ N. Then by i→∞

Proposition 2 we obtain that ∞

M(xm+s+1 , xm , ε ) ≥ *i=m M(x1 , x0 , φ (

ε1 )) > 1 − λ, for all m ≥ n0 . μi

Therefore, the sequence (xn )n∈N is Cauchy, and since the space X is complete, lim xn = x ∈ X. n→∞

Let us prove that x is a fixed point of the mapping f . Suppose contrarily, that x = fx. Then, 1 −1 ≤ M(fxn , fx, φ (t))

ψ( = ψ(

1 min{M(xn , x, φ ( ct )), M(xn , fxn , φ ( ct )), M(x, fx, φ ( ct )), M(fxn , x, φ ( ct ))}

− 1)

1 − 1). min{M(xn , x, φ ( ct )), M(xn , xn+1 , φ ( ct )), M(x, fx, φ ( ct )), M(xn+1 , x, φ ( ct ))}

Letting n → ∞ in the above we obtain that 1 1 1 − 1 ≤ ψ( − 1) < − 1. t M(x, fx, φ (t)) M(x, fx, φ ( c )) M(x, fx, φ ( ct )) Therefore, M(x, fx, φ (t)) > M(x, fx, φ ( ct )). A contradiction, and so x = fx. Let us prove that x is a unique fixed point. Suppose the contrary, that is, that there exists another fixed point u ∈ X, u = fu. Then using (6.21) we have 1 1 −1 = −1 M(x, u, φ (t)) M(fx, fu, φ (t)) ≤ ψ(

1 min{M(x, u, φ ( ct )), M(x, fx, φ ( ct )), M(u, fu, φ ( ct )), M(fx, u, φ ( ct ))}

− 1)

= ψ(

1 − 1) min{M(x, u, φ ( ct )), M(x, x, φ ( ct )), M(x, x, φ ( ct )), M(x, u, φ ( ct ))}

= ψ(

1 − 1) M(x, u, φ ( ct ))


0, f satisfies the following condition:

ϕ (M(f (x), f (y), t)) ≤ k(t) · ϕ (M(x, y, t)),

(6.22)

where x, y ∈ X and x = y, then f has a unique fixed point. Theorem 21. [50] Let (X, M, ∗) be a compact fuzzy metric space and f a continuous self-map on X and suppose that ϕ : [0, 1] → [0, 1] satisfies properties (P1) and (P2). If for any t > 0, f satisfies the following condition: ϕ (M(f (x), f (y), t)) < ϕ (M(x, y, t)), (6.23) where x, y ∈ X and x = y, then f has a unique fixed point. In the paper [12] the mentioned results were improved using a more general contraction condition. Theorem 22. Let (X, M, ∗) be a complete fuzzy metric space and f : X → X. Let the function ϕ : [0, 1] → [0, 1] satisfy conditions (P1) and (P2). If there exists function k : (0, ∞) → (0, 1) such that:

ϕ (M(f (x), f (y), t)) ≤ k(t) · max{ϕ (M(x, f (x), t)), ϕ (M(y, f (y), t)), ϕ (M(x, y, t))},

(6.24)

for any x, y ∈ X, x = y, t > 0, then f has a unique fixed point. Remark 14. (i) Obviously, in comparison to (6.22), condition (6.24) has an advantage when M(f (x), f (y), t) = M(x, y, t), for some x, y ∈ X, x = y, t > 0. (ii) With appropriate changes in the proof of Theorem 22, condition (6.24) could be replaced by another one:

ϕ (M(f (x), f (y), t)) ≤ k1 (t) · min {ϕ (M(f (x), y, t)), ϕ (M(x, f (x), t)), ϕ (M(x, f (y), t)), ϕ (M(y, f (y), t))} + k2 (t) · ϕ (M(x, y, t)),

(6.25)

x, y ∈ X, x = y, t > 0, where k1 : (0, ∞) → [0, 1), k2 : (0, ∞) → (0, 1), k1 (t) + k2 (t) < 1. Trivially, if we take k1 (t) = 0, t > 0 we have a generalization of condition (6.22). Example 19. [12] Let X = {A, B, C, D, E} be a subset of R2 , where A = (0, 0), B = (1, 0), C = (0, 1), D = (2, 0), E = (0, −2). Let f : X → X be defined by f (A) = f (C) = f (D) = A, f (B) = C, f (E) = D. √ 2d(x,y) Let ϕ (τ ) = 1 − τ , τ ∈ [0, 1] and M(x, y, t) = e− t , t > 0, where by d(x, y) is denoted Euclidean distance in R2 . Note that, by (X, M, ∗) is given a complete fuzzy metric space

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with respect to the t-norm ∗(x, y) = x · y. Also, ϕ satisfies conditions (P1) and (P2) and function k : (0, ∞) → (0, 1) defined by  5 1 − e− t , t ∈ (0, 2] k(t) = t , t ∈ (2, ∞) t+ 1 5

satisfies condition (6.24). Therefore, by Theorem 22, it follows that f has a unique fixed point. On the other hand, if we take points A and B condition (6.22) is not satisfied, that is, 1

1

1 − e− t > k(t) (1 − e− t ), t ∈ (0, 2]. The same holds for pairs (A, E), (B, D) and (B, E). Theorem 23. [12] Let (X, M, ∗) be a compact fuzzy metric space and f : X → X be a continuous function. Let function ϕ : [0, 1] → [0, 1] satisfy conditions (P1) and (P2). If

ϕ (M(f (x), f (y), t)) < max{ϕ (M(x, f (x), t)), ϕ (M(y, f (y), t)), ϕ (M(x, y, t))},

(6.26)

x, y ∈ X, x = y, t > 0, then f has a unique fixed point. Theorem 24. Let (X, M, ∗) be a complete fuzzy metric space and f : X → X. If there exist k1 , k2 : (0, ∞) → (0, 1) and an altering distance function ϕ such that the following condition

ϕ (M(fx, fy, t)) ≤ k1 (t) · min{ϕ (M(x, y, t)), ϕ (M(x, fx, t)), ϕ (M(x, fy, 2t)), ϕ (M(y, fy, t))} (6.27) +k2 (t) · ϕ (M(fx, y, 2t)), x, y ∈ X, x = y, t > 0 is satisfied, then f has a unique fixed point. Example 20. [64] Let X = {A, B, C, D} be a subset of R2 , where A = (0, 0), B = (1, 0), C = (1, 4). Let D be an arbitrary point on a circle of radius 1 with center in C. Let f : X → X be defined by f (A) = f (B) = f (D) = A, f (C) = B. t , t > 0, where by d(x, y) is denoted Euclidean Let ϕ (τ ) = 1 − τ , τ ∈ [0, 1] and M(x, y, t) = t+d(x,y) 2 distance in R . Note that, by (X, M, ∗) is defined a complete fuzzy metric space with respect to the triangular norm ∗(a, b) = min{a, b} and ϕ satisfies conditions (P1) and (P2). Functions k1 , k2 : (0, ∞) → (0, 1) are defined by  2 e , t ∈ (0, 1] t+e2 k1 (t) = k2 (t) = t , t ∈ (1, ∞) t+ 1 e2

Now, every pair of points x, y ∈ X, x = y, satisfies condition (6.27) and by Theorem 24 [64] it follows that f has a unique fixed point. On the other hand, for x = C and y = D condition (6.22) given in [50] as a classical generalization of the Banach principle of contraction is not satisfied. Nevertheless, condition (6.27) is not a full generalization of condition (6.22). Theorem 25. [64] Let (X, M, ∗) be a complete fuzzy metric space and f : X → X. If there exist k1 , k2 : (0, ∞) → [0, 1), k3 : (0, ∞) → (0, 1), ∑3i=1 ki (t) < 1, and altering distance function ϕ satisfy conditions (P1) and (P2) such that the following condition is satisfied:

ϕ (M(fx, fy, t)) ≤ k1 (t) · ϕ (M(x, fx, t)) + k2 (t) · ϕ (M(y, fy, t)) + k3 (t) · ϕ (M(x, y, t)), for all x, y ∈ X, x = y and t > 0, then f has a unique fixed point.

(6.28)

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Remark 15. If in (6.28) we take that k1 (t) = k2 (t) = 0, t > 0, we get condition (6.22) and Theorem 25 is a generalization of the result given in [50]. In the following theorems conditions (6.29) and (6.31) proposed in [65] are used to obtain fixed point results in complete and compact strong fuzzy metric spaces. Theorem 26. Let (X, M, ∗) be a complete non-Archimedean fuzzy metric space with positive tnorm ∗ and let f : X → X. If there exists an altering distance function ϕ and ai = ai (t), i = 1, 2, . . . , 5, ai > 0, a1 + a2 + 2a3 + 2a4 + a5 < 1, such that

ϕ (∗(r, s)) ≤ ϕ (r) + ϕ (s), r, s ∈ {M(x, fx, t) : x ∈ X, t > 0}

(6.29)

ϕ (M(fx, fy, t)) ≤ a1 ϕ (M(fx, x, t)) + a2 ϕ (M(fy, y, t)) + a3 ϕ (M(fx, y, t))

(6.30)

and +a4 ϕ (M(x, fy, t)) + a5 ϕ (M(x, y, t)), x, y ∈ X, t > 0 then f has a unique fixed point. Example 21. Let X = {A, B, C, D} be a subset of R2 , where A = (0, 0), B = (1, 0), C = (1, 4) and D = (0, 4). Let f : X → X be defined by f (A) = f (B) = f (D) = A, f (C) = B. t Let ϕ (τ ) = 1 − τ , τ ∈ [0, 1] and M(x, y, t) = t+|x−y| , t > 0. Note that, by (X, M, T) is defined a strong complete fuzzy metric space with respect to the triangular norm T(a, b) = a · b and ϕ satisfies conditions (AD1) and (AD2). Functions ai (t), i ∈ {1, 2, . . . , 5} are defined by √ t + 17 1 − a2 (t) a2 (t) = √ , a1 (t) = a3 (t) = a4 (t) = a5 (t) = , t > 0. 7 17(t + 1)

Now, every pair of points x, y ∈ X, x = y, satisfies condition (6.30) and by Theorem 26 it follows that f has a unique fixed point. Note that, for x = C and y = D condition (6.22) is not satisfied. Theorem 27. Let (X, M, ∗) be a compact non-Archimedean fuzzy metric space and let ϕ : [0, 1] → [0, 1] satisfy conditions (P1) and (P2) such that

ϕ (∗(r, s)) ≤ ϕ (r) + ϕ (s), r, s ∈ {M(x, fx, t) : x ∈ X, t > 0}.

(6.31)

If the continuous mapping f : X → X satisfies the following condition:

ϕ (M(fx, fy, t)) ≤ a1 ϕ (M(fx, x, t)) + a2 ϕ (M(fy, y, t)) + a3 ϕ (M(fx, y, t))

(6.32)

+a4 ϕ (M(x, fy, t)) + a5 ϕ (M(x, y, t)), x, y ∈ X, x = y, t > 0, for ai = ai (x, y, t) > 0, i = 1, 2, . . . , 5, a1 + a2 + 2a3 + 2a4 + a5 < 1, then f has a unique fixed point in X. Using the same altering distance, it is possible to improve the following result.

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Theorem 28. [7] Let (X, M, ∗) be a fuzzy metric space such that M(x, y, t) → 1 as t → ∞. Let f , g : X → X be two self-mappings of X such that there exist k ∈ (0, 1) and q = q(x, y, t) ≥ 0 such that M(fx, gy, kt) + q(1 − max{M(x, gy, kt), M(y, fx, kt)}) ≥ M(x, y, t), (6.33) for each x, y ∈ X and all t > 0. If the family of t-norms {∗(p) (x)}p∈N is equicontinuous at the point x = 1 and X is complete, then f and g have a common fixed point. If in (6.33), q(x, y, t) ≤ λ(M(x, y, t) − M(x, y, kt))/(1 − M(x, y, kt)), for all x = y, λ ∈ [0, 1),

(6.34)

then the common fixed point is unique. Remark 16. A generalization of Theorem 28 is obtained if instead of condition (6.33) the following conditions hold: M(fx, gy, kt) + q min{ϕ (M(x, gy, kt)), ϕ (M(y, fx, kt))} ≥ M(x, y, t), x, y ∈ X, t > 0, ∞

lim *i=n M(x0 , fx0 ,

n→∞

1 ) = 1, μi

(6.35) (6.36)

for some x0 ∈ X, μ ∈ (k, 1) and some altering distance function ϕ . With ϕ (t) = 1 − t and t-norm ∗ of H-type (see Proposition 2) we obtain the result from [7]. If we take ϕ (t) = (1 − t)α , α > 1 we have a stronger result than in [7]. Moreover, if instead of the condition (6.36) the following conditions hold ∞

1

∑(1 − M(x0 , x1 , μ i ))λ < ∞;

(6.37)

i=1

and

∞

1

∑(1 − M(x0 , x1 , μ i )) < ∞;

(6.38)

i=1

AA SW then we obtain the same result for ∗ = ∗D λ (∗ = ∗λ ), λ > 0, and ∗ = ∗λ , λ > −1, respectively.

Theorem 29. [64] Let (X, M, ∗) and (X, M1 , ∗) be fuzzy metric spaces such that lim M1 (x, y, t) = 1 t→∞

and M(x, y, t) ≥ M1 (x, y, t), x, y ∈ X, t > 0. Let f : X → X be a continuous function such that for some x0 ∈ X sequence {f n x0 } has a convergent subsequence in (X, M, ∗). If there exist k ∈ (0, 1), μ ∈ (k, 1), q = q(x, y, t) ≥ 0, and an altering distance function ϕ such that the following conditions are satisfied: M1 (fx, fy, kt) + q min{ϕ (M1 (fx, y, kt)), ϕ (M1 (fy, x, kt))} ≥ M1 (x, y, t), x, y ∈ X, t > 0. and

∞

lim *i=n M1 (x0 , fx0 ,

n→∞

(6.39)

1 ) = 1, μi

then f has a fixed point. If, in addition, the following is satisfied q(x, y, t) ≤ λ(M1 (x, y, t) − M1 (x, y, kt))/ϕ (M1 (x, y, kt)), x, y ∈ X, x = y, λ ∈ [0, 1),

(6.40)

then f has a unique fixed point. Multivalued generalization of the Banach contraction principle in metric space (X, d) is done by Nadler [44] in the following way:

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Let X be an arbitrary nonempty set, CB(X) the family of all nonempty, closed and bounded subsets of X, and C(X) the family of all nonempty and closed subsets of X. S.B. Nadler [44] proved on a metric space (X, d) generalization of the Banach contraction principle for multi-valued mappings f : X → CB(X) of the form D(fx, fy) ≤ qd(x, y),

(6.41)

where D is the Hausdorff metric defined below and q ∈ (0, 1). D(A, B) = max{sup inf d(x, y), sup inf d(x, y)}. x∈X y∈Y

y∈Y x∈X

Later on, the probabilistic versions of condition (6.41) are given in [27, 28, 29, 30] where notions of weakly demicompact mapping, f -strongly demicompact and weakly commuting mapping are introduced. Further, Hausdorff distance between sets in fuzzy metric spaces is introduced and used in [39] for the study of the existence of a coincidence point using two multivalued and one single valued mappings. A generalization of the Nadler contraction principle in a fuzzy metric space (X, M, ∗) follows. Definition 19 (26). Let (X, M, ∗) be a fuzzy metric space, A a nonempty subset of X and F : A → CB(A). The mapping F is the fuzzy Nadler contraction, where q ∈ (0, 1), if the following condition is satisfied: for all u, v ∈ A, all x ∈ Fu and all δ > 0 there exists y ∈ Fv such that for all t > δ M(x, y, t) ≥ M(x, y,

t−δ ). q

(6.42)

Remark 17. Since the function M(x, y, ·) is continuous, for f being a single valued mapping the fuzzy Nadler contraction coincides with the notion of the fuzzy Banach contraction. Let A and B be two nonempty subsets of X, and define the Hausdorff–Pompeiu fuzzy metric as (see [37])

 ˜ M(A, B, t) = min inf E(x, B, t), inf E(y, A, t) , t > 0, x∈A

y∈B

where E(x, B, t) = sup M(x, y, t). y∈B

The following definitions are necessary for the next theorems. Definition 20. [27, 29] Let (X, M, ∗) be a fuzzy metric space, A a nonempty subset of X and f : A → 2X \ {0}. / The mapping f is weakly demicompact if for every sequence {xn }n∈N from A such that xn+1 ∈ fxn , n ∈ N and lim M(xn+1 , xn , t) = 1, t > 0, there exists a convergent subsequence {xnk }k∈N .

n→∞

Definition 21. [26] Let (X, M, ∗) be a fuzzy metric space, A a nonempty subset of X, f : A → A and F : A → C(A). The mapping F is f -strongly demicompact if for every sequence {xn }n∈N from A such that lim M(fxn , yn , t) = 1, t > 0, for some sequence {yn }n∈N , yn ∈ Fxn , n ∈ N, there n→∞

exists a convergent subsequence {fxnk }k∈N .

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t where d is the usual metric Example 22. Let X = D = R, M(x, y, t) = Md (x, y, t) = t+d(x,y) on R and ∗ = ∗P . Then (X, M, ∗) is a fuzzy metric space. Let f (x) = arctg x and F(x) = [arctg x − 1, arctg x + 1]. Then, for every sequence (xn )n∈N , we can find a sequence (yn )n∈N such that yn ∈ F(xn ), that is, yn = arctg x + n1 , and Md (f (xn ), yn , t) = t+d(f (xt n ),yn ) → 1, n → ∞. Since the sequence (arctg xn )n∈N is bounded and has a convergent subsequence, there exists a convergent subsequence, that is, for xn = n(−1)n , the sequence f (xn ) = arctg(n(−1)n ) has a convergent subsequence obtained by selecting even members.

Definition 22. [25] A mapping F : X → C(X) is weakly commuting with f : X → X if for every x ∈ X the following holds: f (Fx) ⊆ F(fx). In [10] a multivalued generalization of the result in Shen et al. [50] was obtained. Theorem 30. Let (X, M, ∗) be a complete fuzzy metric space and ∗ be a t-norm of H-type. Let f : X → X be a continuous mapping and F, G : X → CB(X) be weakly commuting with f . If there exist k : (0, ∞) → (0, 1) and an altering distance function ϕ such that the following condition is satisfied: ˜ ϕ (M(Fx, Gy, t)) ≤ k(t) · ϕ (M(fx, fy, t)), x, y ∈ X, x = y, t > 0, (6.43) then there exist x ∈ X such that fx ∈ Fx ∩ Gx. If in Theorem 30 we take that F = G and that f is an identical mapping we get the following corollary. Corollary 3. Let (X, M, ∗) be a complete fuzzy metric space, F : X → C(X), and one of the following conditions is satisfied: (a) F is weakly demicompact mapping, or (b) (X, M, ∗) is strong fuzzy metric space and ∗ is a t-norm of H-type. If there exist k : (0, ∞) → (0, 1) and an altering distance function ϕ such that: ˜ ϕ (M(Fx, Fy, t)) ≤ k(t) · ϕ (M(x, y, t)), x, y ∈ X, t > 0,

(6.44)

then there exist x ∈ X such that x ∈ Fx. Moreover, if the mapping F in Corollary 3 is single-valued we get the result in [50]. t Example 23. (a) Let X = [0, 2], ∗ = ∗P , M(x, y, t) = t+d(x,y) , where d is Euclidian metric. Then (X, M, ∗) is a fuzzy metric space. Let F(x) = {1, 2}, x ∈ X. Since F is weakly demicompact and condition (6.44) is satisfied, by Corollary 3(a) it follows that there exist x ∈ X such that x ∈ Fx.

(b) Let X = [0, 2], ∗ = ∗M , M∗ (x, y, t) = t+d∗t(x,y) , where d∗ is ultrametric. Ultrametric space is metric space where instead of a triangle inequality condition the following is satisfied: d∗ (x, z) ≤ max{d∗ (x, y), d∗ (y, z)}. Then (X, M∗ , T) is a non-Archimedean fuzzy metric space ([20]). For F(x) = {1, 2}, x ∈ X, condition (6.44) is satisfied and by Corollary 3(b) it follows that there exist x ∈ X such that x ∈ Fx.

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6.7 Fixed Point Theorems in Fuzzy Metric Spaces of Caristi Type One of the most important results in in fixed point theory and nonlinear analysis is Caristi fixed point theorem in metric spaces [6]. In this section we analyze whether it is possible to translate the results of Caristi in the context of fuzzy metric spaces. The fixed point theorem of Caristi represents a generalization of the famous Banach contraction principle and has a great significance in control theory, optimal theory, economic theory, theory of global analysis, geometric theory of Banach space, theory differential equations and in the problems of minimization. Theorem 31. [6] Let (X, d) be a complete metric space and φ : M → R a lower-semicontinuous function with a finite lower bound. Let f : X → X be any (not necessarily continuous) function such that d(x, f (x)) ≤ φ (x) − φ (f (x)) for every x ∈ X. Then f has a fixed point. In [29] the following theorem is proved. Theorem 32. Let ∗ be a t-norm. Then the following are satisfied: (i)

Suppose that there exists a strictly increasing sequence (bn )n∈N from the interval [0, 1) such that lim bn = 1 and ∗(bn , bn ) = bn . Then the t-norm ∗ is of H-type. n→∞

(ii) If t-norm ∗ is continuous and of H-type, then there exists a sequence (bn )n∈N as in (i). A theorem which we have proved is a fuzzy generalization of a theorem given in [25]. Theorem 33. Let (X, M, ∗) be a complete GV fuzzy metric space in the sense that lim M(x, y, t) = 1, t→∞

and t-norm ∗ be continuous and of H-type, f : X → X a continuous mapping, Φn : S → R+ (n ∈ N), and μ a mapping of R+ onto R+ such that μ is nondecreasing and

μ (a + b) ≤ μ (a) + μ (b) for all a, b ∈ R+ . If for every x ∈ X, every s > 0 and every n ∈ N

μ (s) > Φn (x) − Φn (f (x)) ⇒ M(x, f (x), s) > bn where (bn )n∈N is a monotone increasing sequence from (0, 1) such that lim bn = 1 and ∗(bn , bn ) = bn n→∞

for every n ∈ N, then there exists a fixed point x∗ ∈ X of the mapping f and x∗ = lim f n (x0 ) for arbitrary x0 ∈ X.

n→∞

In the next theorem (bn )n∈N is a monotone increasing sequence from (0, 1) such that lim bn = 1 but the members of sequence {bn }n∈N are not idempotent elements of the map-

n→∞

ping ∗ in a general case. This theorem is a fuzzy version of the theorem proved in [69]. Theorem 34. Let (X, M, ∗) be a complete fuzzy metric space in the sense of George and Veeramani, t-norm ∗ is of H-type, f : X → X a continuous mapping, Φn : S → R+ (n ∈ N) and μ a nondecreasing

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m

that is, M(v1 , vm , u1 + u2 + . . . um ) ≥ *i=1 bs(n) > bn , that is, dn (v1 , vm ) ≤ u1 + u2 + · · · + um . Therefore, (6.48) is satisfied. Let x0 ∈ X and xm = f m (x0 ), (m ∈ N). Then for every (m ∈ N)

μ [dn (xm+1 , xm )] = μ [dn (f (xm ), xm )] ≤ Φn (xm ) − Φn (f (xm )), and for every k ∈ N we have the following k

∑ μ [dn (xi+1 , xi )] ≤ Φn (x0 ) − Φn (xk+1 ) i=0

≤ Φn (x0 ). Using that μ is sub-additive we conclude the following: k

μ [∑ dn (xi+1 , xi )] ≤ Φn (x0 ) i=0

and therefore

k

∑ dn (xi+1 , xi ) ≤ sup{u | u > 0, μ (u) = Φn (x0 )} = Mn , i=0

that is, the series

∞

∑ dn (xi+1 , xi )

(6.50)

i=0

is convergent. Condition (6.50) is valid for every n ∈ N, so it is satisfied for s(n) and then we have m+p−1

dn (xm , xm+p ) ≤

∑

ds(n) (xi , xi+1 )

i=m ∞

≤ ∑ ds(n) (xi , xi+1 ). i=m

Using (6.48) it follows that the sequence (xn )n∈N is a Cauchy sequence. Let x∗ = lim xn = lim f n (x0 ). Using continuity of mapping f it follows that x∗ = f (x∗ ).

n→∞

n→∞

 

6.8 Fixed Point Theorems in Fuzzy Metric Spaces of Nadler Type Although it has already been mentioned, we will repeat that S.B. Nadler [21] has proved a generalization of the well-known Banach contraction principle for multi-valued mappings f : X → CB(X), where (X, d) is a classical metric space and CB(X) is the family of all nonempty, closed and bounded subsets of X. There are various extensions of the Banach contraction mappings for single-valued and multi-valued mappings performed in fuzzy metric spaces. Theorem 35. Let (X, M, ∗) be a complete fuzzy metric space, lim M(x, y, t) = 1, A non-empty and t→∞

closed subset of X. Let f : A → A be a continuous mapping and F, G : A → CB(A) such that for every q ∈ (0, 1) the following holds:

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233

for every u, v ∈ A, x ∈ Fu and δ > 0, there exists y ∈ Gv such that for every ε > δ M(x, y, ε ) ≥ M(fu, fv,

ε −δ ). q

(6.51)

If F and G are weakly commuting with f and: (i) F or G are f -strongly demicompact (ii) there exist x0 , x1 ∈ A, fx1 ∈ Fx0 and μ ∈ (q, 1) such that t-norm ∗ satisfies the following: ∞

lim *i=n M(fx0 , fx1 ,

n→∞

1 ) = 1. μi

Then there exists x ∈ A such that fx ∈ Fx ∩ Gx. Proof. Let x0 , x1 ∈ A such that fx1 ∈ Fx0 . Using (6.56) for x = fx1 , u = x0 , v = x1 and δ = q there exists x2 ∈ A such that fx2 ∈ Gx1 and M(fx1 , fx2 , ε ) ≥ M(fx0 , fx1 ,

ε −q ). q

Continuing in this way we can construct a sequence (xn )n∈N from A such that the following conditions are satisfied: (a) fx2n+1 ∈ Fx2n and fx2n+2 ∈ Gx2n+1 n (b) M(fxn , fxn+1 , ε ) ≥ M(fxn−1 , fxn , ε −q ). q Using (b) we have M(fxn , fxn+1 , ε ) ≥ M(fx1 , fx0 ,

ε − nqn ). qn

) = 1 (lim ( ε −nq ) = ∞), we obtain Since ε > 0, lim M(fx1 , fx0 , ε −nq qn qn n

n→∞

n

n→∞

lim M(fxn , fxn+1 , ε ) = 1.

n→∞

(6.52)

If we suppose that F is f -strongly demicompact, that is, condition (i) is satisfied, using lim M(fx2n , fx2n+1 , ε ) = 1

n→∞

and

fx2n+1 ∈ Fx2n

we conclude that there exists a convergent subsequence (fx2nk )k∈N of a sequence (fx2n )n∈N . We will show that a sequence (fxn )n∈N is convergent if t-norm ∗ satisfies condition (ii). ∞

Let σ = μq . As σ ∈ (0, 1),the series ∑ σ i is convergent, and there exists m0 ∈ N such that ∞

i=1

∑ σ i < 1. Therefore, for every m > m0 and s ∈ N the following is satisfied:

i=m0

ε >ε

∞

m+s

i=m0

i=m

∑ σ i > ε ∑ σ i.

Dhananjay Gopal and Tatjana Došenovi´c

234

Then, M(fxm+s+1 , fxm , ε ) ≥ M(fxm+s+1 , fxm , ε

m+s

∑ σ i)

i=m

≥ ∗(∗(· · · ∗(M(fxm+s+1 , fxm+s , εσ m+s ),    (s+1)−times

M(fxm+s , fxm+s−1 , εσ m+s−1 )), . . . , M(fxm+1 , fxm , εσ m ))

εσ m+s − (m + s)qm+s ), qm+s

≥ ∗(∗(· · · ∗(M(fx1 , fx0 ,    (s+1)−times

M(fx1 , fx0 ,

εσ m+s−1 − (m + s − 1)qm+s−1 ), . . . , qm+s−1

εσ m − mqm )) qm ε = ∗(∗(· · · ∗(M(fx1 , fx0 , q m+s − (m + s)),    (σ ) M(fx1 , fx0 ,

(s+1)−times

M(fx1 , fx0 ,

ε

− (m + s − 1)), . . .

( σq )m+s−1

ε − m)) ( σq )m m+s ε = *i=m M(fx1 , fx0 , i − i). μ M(fx1 , fx0 ,

Since μ ∈ (q, 1), there exist m1 (ε ) > m0 such that for all s ∈ N we have

ε μm

−m >

ε , 2μ m

for every m > m1 (ε ). Now,

ε ) 2μ i ∞ ε ≥ *i=m M(fx1 , fx0 , i ) 2μ m+s

M(fxm+s+1 , fxm , ε ) ≥ *i=m M(fx1 , fx0 ,

∞

∞

Using lim *i=m M(fx1 , fx0 , μ1i ) = 1, we conclude that lim *i=m M(fx1 , fx0 , 2εμ i ) = 1, for every ε > m→∞

m→∞

0. It was inequality that for every ε > 0, λ ∈ (0, 1), there exists m2 (ε , λ) > m1 (ε ) such that M(fxm+s+1 , fxm , ε ) > 1 − λ, for all m > m2 (ε , λ) and all s ∈ N. Therefore, the sequence (fxn )n∈N is Cauchy, and since the space X is complete, lim fxn exists. n→∞

Therefore, in both cases (i) and (ii) there exists a subsequence (fxnk )k∈N such that x = lim fx2nk ∈ A. k→∞

Also, using (6.52) it follows that x = lim f2nk+1 . k→∞

It remains to be proved that fx ∈ Fx ∩ Gx. Since Fx and Gx are closed, it is enough to show that fx ∈ Fx ∩ Gx, that is, that for every ε > 0 and λ ∈ (0, 1) there exists r1 (ε , λ) ∈ Fx and r2 (ε , λ) ∈ Gx such that r1 (ε , λ) ∈ U(fx, ε , λ)

and

r2 (ε , λ) ∈ U(fx, ε , λ).

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235

Since in fuzzy metric spaces t-norm ∗ is always continuous, we have sup ∗(x, x) = 1 and x 1 − λ.

Using the continuity of f and x = lim fx2nk it follows that there exists k1 ∈ N such that k→∞

ε M(fx, ffx2nk , ) > 1 − δ (λ), 3

for all k ≥ k1 .

Furthermore, condition (6.52) ensures the existence of k2 ∈ N such that

ε M(ffx2nk , ffx2nk +1 , ) > 1 − δ (λ), 3 Let δ =

ε . 12

for all k ≥ k2 .

Using that F is weakly commuting with f we conclude that ffx2nk +1 ∈ f (Fx2nk ) ⊆ F(fx2nk ),

and using (6.56), there exists r2 (ε , λ) ∈ Gx such that

ε ε M(ffx2nk+1 , r2 (ε , λ), ) > M(ffx2nk , fx, ) 3 4q > 1 − δ (λ). Then,

ε ε M(fx, r2 (ε , λ), ε ) ≥ (M(fx, ffx2nk , ) ∗ (M(ffx2nk , ffx2nk +1 , ), 3 3 ε ∗ M(ffx2nk +1 , r2 (ε , λ), ))) > 1 − λ. 3 Therefore, r2 (ε , λ) ∈ U(fx, ε , λ). Analogously, we can prove that r1 (ε , λ) ∈ Fx ∩ U(fx, ε , λ).

 

Corollary 4. Let (X, M, (∗D λ )λ∈(0, ∞) ) be a complete fuzzy metric space, lim M(x, y, t) = 1, A nont→∞

empty and closed subset of X. Let f : A → A be a continuous mapping and F, G : A → CB(A) such that for every q ∈ (0, 1) the following holds: for every u, v ∈ A, x ∈ Fu and δ > 0, there exists y ∈ Gv such that for every ε > δ M(x, y, ε ) ≥ M(fu, fv,

ε −δ ). q

(6.53)

If F and G are weakly commuting with f and: (i) F or G are f -strongly demicompact; (ii) there exist x0 , x1 ∈ A, fx1 ∈ Fx0 and μ ∈ (q, 1) such that t-norm ∗ satisfies the following: ∞

1

∑ (1 − M(fx0 , fx1 , μ i ))λ < ∞

n=1

Then there exists x ∈ A such that fx ∈ Fx ∩ Gx.

Dhananjay Gopal and Tatjana Došenovi´c

236

Corollary 5. Let (X, M, (∗AA λ )λ∈(0, ∞) ) be a complete fuzzy metric space, lim M(x, y, t) = 1, A nont→∞

empty and a closed subset of X. Let f : A → A be a continuous mapping and F, G : A → CB(A) such that for every q ∈ (0, 1) the following holds: for every u, v ∈ A, x ∈ Fu and δ > 0, there exists y ∈ Gv such that for every ε > δ M(x, y, ε ) ≥ M(fu, fv,

ε −δ ). q

(6.54)

If F and G are weakly commuting with f and: (i) F or G are f -strongly demicompact; (ii) there exist x0 , x1 ∈ A, fx1 ∈ Fx0 and μ ∈ (q, 1) such that t-norm ∗ satisfies the following: ∞

1

∑ (1 − M(fx0 , fx1 , μ i ))λ < ∞

n=1

Then there exists x ∈ A such that fx ∈ Fx ∩ Gx. Corollary 6. Let (X, M, (∗SW λ )λ∈(−1, ∞) ) be a complete fuzzy metric space, lim M(x, y, t) = 1, A be t→∞

nonempty and a closed subset of X. Let f : A → A be a continuous mapping and F, G : A → CB(A) such that for every q ∈ (0, 1) the following holds: for every u, v ∈ A, x ∈ Fu and δ > 0, there exists y ∈ Gv such that for every ε > δ M(x, y, ε ) ≥ M(fu, fv,

ε −δ ). q

(6.55)

If F and G are weakly commuting with f and: (i) F or G are f -strongly demicompact; (ii) there exist x0 , x1 ∈ A, fx1 ∈ Fx0 and μ ∈ (q, 1) such that t-norm ∗ satisfies the following: ∞

1

∑ (1 − M(fx0 , fx1 , μ i )) < ∞

n=1

Then there exists x ∈ A such that fx ∈ Fx ∩ Gx. Theorem 36. Let (X, M, ∗) be a complete fuzzy metric space, lim M(x, y, t) = 1, and A be a t→∞

nonempty and closed subset of X. Let f : A → A be a continuous mapping and F, G : A → CC(A) (where CC(A) is the set of all nonempty and compact subsets of X) such that for every q ∈ (0, 1) the following holds: for every u, v ∈ A, x ∈ Fu, there exists y ∈ Gv such that for every ε > δ

ε M(x, y, ε ) ≥ M(fu, fv, ). q

(6.56)

If F and G are weakly commuting with f and; (i) F or G are f -strongly demicompact; (ii) there exist x0 , x1 ∈ A, fx1 ∈ Fx0 and μ ∈ (q, 1) such that t-norm ∗ satisfies the following: ∞

lim *i=n M(fx0 , fx1 ,

n→∞

1 ) = 1. μi

Fuzzy Contractive Mappings

237

Then there exists x ∈ A such that fx ∈ Fx ∩ Gx. Let Φ be the set of all continuous functions φ : [0, 1]6 → R such that (φ1 ) φ (t1 , t2 , t3 , t4 , t5 , t6 ) is nondecreasing in respect with t6 . (φ2 ) φ (u, v, v, u, 1, T(u, v)) ≥ 0 implies u ≥ v. (φ3 ) φ (u, 1, 1, u, 1, u) ≥ 0 implies u ≥ 1. Example 24. Let T = TP and φ (t1 , t2 , t3 , t4 , t5 , t6 ) = t1 t2 t4 − t3 t5 t6 . Theorem 37. Let (X, M, ∗) be a complete fuzzy metric space, lim M(x, y, t) = 1, and A be a t→∞

nonempty and closed subset of X. Let f : A → A be a continuous function and F, G : X → CB(A) such that for some φ ∈ Φ there exists k ∈ (0, 1) such that for every u, v ∈ A and every t > 0, x ∈ Fu there exists y ∈ Gv such that

φ (M(x, y, kt), M(fu, fv, t), M(x, fu, t), M(y, fv, kt), M(x, fv, t), M(y, fu, (k + 1)t)) ≥ 0

(6.57)

Let F and G be weakly commuting with f and (i)

F or G are f -strongly demicompact, or (ii) there exist x0 , x1 ∈ A such that for every fx1 ∈ Fx0 and μ ∈ (k, 1): ∞

lim *i=n M(fx0 , fx1 ,

n→∞

1 ) = 1. μi

Then, there exists x ∈ A such that fx ∈ Fx ∩ Gx.

6.9 Convex Fuzzy Metric Space and Fixed Point Results W. Takahashi [58] introduced the notion of metric space with a convex structure. This class of metric space includes normed linear space and metric space of the hyperbolic type. Definition 23. Metric space (X, d) has a convex structure in the sense of Takahashi if there exists a mapping W : X × X × [0, 1] → X such that for every (u, x, y, δ ) ∈ X × X × X × [0, 1] d(u, W(x, y, δ )) ≤ δ d(u, x) + (1 − δ )d(u, y). The previous definition is generalized in probabilistic metric spaces in the paper [25]. Using results present in ([25], [71]) we define a convex structure in the sense of fuzzy metric spaces. Definition 24. The mapping W : X × X × [0, 1] → X is a convex structure on X if for every (x, y) ∈ X × X, W(x, y, 0) = y, W(x, y, 1) = x, and for every δ ∈ (0, 1), u ∈ X, ε > 0

ε ε M(u, W(x, y, δ ), 2ε )) ≥ M(u, x, ) ∗ M(u, y, ). δ 1−δ

(6.58)

Dhananjay Gopal and Tatjana Došenovi´c

238

In the next theorem we will suppose that a convex structure W on (X, M, ∗) satisfies the following condition: (6.59) M(W(x, z, δ ), W(y, z, δ ), εδ ) ≥ M(x, y, ε ) for every (x, y, z) ∈ X × X × X, every ε > 0 and δ ∈ (0, 1). Definition 25. Let (X, M, ∗) be a fuzzy metric space with a convex structure W. Subset M of X is W-starshaped if there exists x0 ∈ X such that the set {W(x, x0 , λ) : x ∈ M, λ ∈ (0, 1)} ⊂ X. The point x0 is called a star-center of X. Definition 26. Let (X, M, ∗) be a fuzzy metric space with a convex structure W and S a nonempty subset of X. A mapping f : S → X is (W, x0 )-convex if for every (x, λ) ∈ S × [0, 1] W(fx, x0 , λ) = f (W(x, x0 , λ)).

(6.60)

Lemma 3. Let (X, M, ∗) be a fuzzy metric space, S a nonempty subset of X which is W-starshaped with a star center in x0 , f : S → X is the mapping which is (W, x0 )-convex. Then f (S) is (W, x0 )convex with a star-center in x0 . Proof. Let u ∈ f (S). Then there exists x ∈ S such that u = f (x). Let us prove that for every λ ∈ [0, 1], z = W(u, x0 , λ) ∈ f (S). As u = f (x) it follows that

z = W(f (x), x0 , λ).

Since the mapping f is (W, x0 )-convex it follows that z = f (W(x, x0 , λ)). Using the condition that the subset S is W-starshaped it follows that W(x, x0 , λ) ⊂ S, that is, z ∈ f (S).   Lemma 4. Let (X, M, ∗) be a fuzzy metric space, S a nonempty subset of X, f : S → X a continuous mapping, L : S → C(X) (where C(X) is the family of all closed and nonempty subsets of X) and the following condition is satisfied: For every u, v ∈ S, every x ∈ Lu and every δ > 0, there exists y ∈ Lv such that M(x, y, ε ) ≥ M(fu, fv,

ε −δ ) q

(6.61)

for every ε > 0. Then the mapping L is closed. Proof. Let (xn )n∈N be a sequence from S such that lim xn = x, and let yn ∈ Lxn , for every n ∈ N n→∞

such that lim yn = y. Let us prove that y ∈ Lx. As Lx = Lx it is enough to show that y ∈ Lx. n→∞

Let ε > 0 and λ ∈ (0, 1) be given. It remains to be proved that b ∈ Lx such that b ∈ Uy (ε , λ), that is, M(b, y, ε ) > 1 − λ. Using condition (6.61), for u = xn , v = x and δ = ε4 from yn ∈ Lxn it follows that there exists bn ∈ Lx such that M(yn , bn , ε ) ≥ M(fxn , fx,

ε ). 4q

Fuzzy Contractive Mappings

239

Therefore,

ε ε M(y, bn , ε ) ≥ M(y, yn , ) ∗ M(yn , bn , ) 2 2 ε ε ≥ M(y, yn , ) ∗ M(fxn , fx, ). 2 4q Using continuity of the mapping f it follows that lim fxn = fx, that is, lim Ffxn ,fx (ε ) = 1. Also, n→∞

n→∞

lim M(y, yn , ε ) = 1, for every ε > 0. Using continuity of t-norm ∗ we have that sup ∗(x, x) = 1,

n→∞

x 1 − λ, and there exists n0 (η ) such that for every n ≥ n0 (η ) the following is satisfied:

ε ε M(y, yn , ) > η and M(fxn , fx, ) > η . 2 4q Now, we have that M(y, bn , ε ) ≥ η ∗ η > 1 − λ, that is, bn0 ∈ Uy (ε , λ) ∩ Lx.

 

The next theorem is a fuzzy generalization of a theorem presented in [71]. Theorem 38. Let (X, M, ∗) be a complete fuzzy metric space, lim M(x, y, t) = 1, with a convex t→∞

structure W, S a nonempty, closed and W-starshaped subset of X with a star center in x0 . Let f : M → M be a continuous function, (W, x0 )-convex mapping and L : S → CB(f (S)) such that L(S) is a compact set and the following condition is satisfied: for every u, v ∈ S, x ∈ Lu and δ > 0 there exists y ∈ Lv such that M(x, y, ε ) ≥ M(fu, fv, ε − δ ), for every ε > 0.

(6.62)

If L is weakly commuting with f , then there exists x ∈ S such that fx ∈ Lx. Proof. Let (kn )n∈N be a sequence from (0, 1) such that lim kn = 1 and let Ln x = W(Lx, x0 , kn ), n→∞

that is, Ln x =



W(z, x0 , kn ), n ∈ N, x ∈ M.

z∈Lx

Using Lemma 3 it follows that f (S) is (W, x0 )-convex. We will show that Ln x ⊂ f (S), that is, that for every z ∈ Ln x it follows z ∈ f (M). Since z ∈ Ln x = W(Lx, x0 , kn ) there exists u ∈ Lx such that z = W(u, x0 , kn ). Using the assumption that Lx ⊂ f (S) it follows that u ∈ f (S), and so W(u, x0 , kn ) ⊂ f (S), that is, z ∈ f (S). Therefore Ln x ⊂ f (S). Using (6.59) it follows that a mapping W is continuous with respect to the first variable and since Lx is closed it follows that Lx is compact (as a subset of L(S)), and therefore W(Lx, x0 , kn ) is closed for every n ∈ N. Now we have that Ln x is closed for every n ∈ N and x ∈ S. Now, we will show that for every u, v ∈ S and every x ∈ Ln u and δ > 0 there exists y ∈ Ln v such that ε −δ M(x, y, ε ) ≥ M(fu, fv, ), ε > 0. kn

Dhananjay Gopal and Tatjana Došenovi´c

240

Let u, v ∈ S, δ > 0 and x ∈ Ln u = W(Lu, x0 , kn ). Then, there exists z ∈ Lu such that x = W(z, x0 , kn ), and using (6.62) there exists y ∈ Lv such that M(z, y , ε ) ≥ M(fu, fv, ε −

δ ). kn

Let y = W(y , x0 , kn ) ∈ Ln v. Then M(x, y, ε ) = M(W(z, x0 , kn ), W(y , x0 , kn ), ≥ M(z, y ,

ε ) kn

≥ M(fu, fv,

ε kn ) kn

ε −δ ). kn

It remains to be proved that Lm is f -strongly demicompact. Let (xn )n∈N be a sequence from S such that for every ε > 0 the following is satisfied: lim M(fxn , yn , ε ) = 1

n→∞

for some sequence (yn )n∈N , yn ∈ Lm xn . Since Ln (M) = W(L(M), x0 , kn ), n ∈ N is relatively compact, using yn ∈ Lm xn we have that (yn )n∈N has a convergent subsequence (ynk )k∈N , and let lim ynk = z. Then k→∞

ε ε M(fxnk , z, ε ) ≥ M(fxnk , ynk , ) ∗ M(ynk , z, ), 2 2 and therefore lim fxnk = z. Now we have that Lm is a f -strongly demicompact. We will show k→∞

that a mapping Ln is weakly commuting with respect to f , that is, that for every x ∈ S Ln (fx) ⊂ f (Ln x) = W(L(fx), x0 , kn ). Let u ∈ Ln x = W(Lx, x0 , kn ). Then there exists z ∈ Lx such that u = W(z, x0 , kn ), and so f (u) = f (W(z, x0 , kn )) = W(fz, x0 , kn ) ⊂ W(f (Lx), x0 , kn ). Using the condition that a mapping L is weakly commuting with respect to f it follows that W(f (Lx), x0 , kn ) = W(L(fx, x0 , kn )), that is, f (u) ∈ Ln (fx). Now, all the conditions of Theorem 35 are satisfied, and so for every n ∈ N there exists xn ∈ S such that fxn ∈ Ln xn . Since Ln xn = n∈N

 z∈Lxn

W(z, x0 , kn ) there exists zn ∈ Lxn such that fxn = W(zn , x0 , kn ). Then for every

M(fxn , zn , ε ) = M(zn , W(zn , x0 , kn ), ε ) ε ε ) ) ∗ M(zn , x0 , ≥ M(zn , zn , 2kn 2(1 − kn ) ε ) = 1 ∗ M(zn , x0 , 2(1 − kn ) ε ). = M(zn , x0 , 2(1 − kn )

Fuzzy Contractive Mappings

231

mapping of R+ onto R+ such that μ (a + b) ≤ μ (a) + μ (b), for every a, b ∈ R+ . If for every x ∈ X, every s > 0 and every n ∈ N the following is satisfied

μ (s) > Φn (x) − Φn (f (x)) ⇒ M(x, f (x), s) > bn ∗

∗

∗

∗

(6.45)

then there exists x ∈ X such that x = f (x ) and x = lim f (x0 ), for arbitrary x0 ∈ X. n

n→∞

Proof. To start, it is an idea to show that (6.45) implies

μ [dn (x, f (x))] ≤ Φn (x) − Φn (f (x)),

(6.46)

for every n ∈ N and every x ∈ X, where dn (x, y) = sup{u | u ∈ R, M(x, y, u) ≤ bn }. In order to prove (6.46), we prove that the following implication is valid: s > Φn (x) − Φn (f (x)) ⇒ μ [dn (x, f (x))] ≤ s.

(6.47)

Let s > Φn (x) − Φn (f (x)). As μ : R+ → R+ , there exists s1 > 0 such that

μ (s1 ) = s > Φn (x) − Φn (f (x)). Using (6.45) we have that M(x, fx, s1 ) > bn , that is, dn (x, f (x)) < s1 , and so

μ [dn (x, f (x))] ≤ μ (s1 ) = s. Therefore, (6.47) is true. Since the t-norm ∗ is of H-type, it follows that for every α ∈ (0, 1), there exists β ∈ (0, 1) such that ∗(∗(· · · ∗(1 − β , 1 − β ), . . . , 1 − β ) > 1 − α ,    n−times

for every n ∈ N. Let 1 − α = bn , n ∈ N. Then there exists s(n) ∈ N such that 1 − β ≤ bs(n) , and therefore ∗(∗(· · · ∗(bs(n) , bs(n) ), . . . , bs(n) ) > bn ,    n−times

for every n ∈ N. We will prove that for every final set {v1 , v2 , . . . vm } ⊂ X, is dn (v1 , vm ) ≤

m−1

∑ ds(n) (vi , vi+1 )

(6.48)

i=1

Let u1 , u2 , . . . um−1 ∈ R be such that ds(n) (v1 , v2 ) < u1 , ds(n) (v2 , v3 ) < u2 , . . . , ds(n) (vm−1 , vm ) < um−1 . Then, using the definition of dn M(v1 , v2 , u1 ) > bs(n) M(v2 , v3 , u2 ) > bs(n) ... M(vm−1 , vm , um−1 ) > bs(n)

(6.49)

Fuzzy Contractive Mappings

ε n→∞ 2(1−kn )

Using lim

241

ε = ∞, we have that lim M(zn , x0 , 2(1−k ) = 1 because zn ∈ L(S) which is n) n→∞

bounded. Then, lim M(fxn , zn , ε ) = 1. Since zn ∈ Lxn and L(S) is compact it follows that n→∞

there exists a convergent subsequence (znk )k∈N of a sequence (zn )n∈N . Using lim fxnk = z it n→∞

follows that lim znk = z. It remains to be proved that fz ∈ Lz. Using lim znk = z and continuity n→∞

n→∞

of the mapping f it follows that lim fznk = fz. Since znk ∈ Lxnk it follows that n→∞

fznk ∈ f (Lxnk ) ⊂ L(fxnk ). Using Lemma 4 we have that a mapping L is closed, that is, fz ∈ Lz.  

6.10 Conclusion The notion of fuzzy metric spaces was introduced for the first time by I. Kramosil and J. Michalek in 1975, thus providing axioms to the fuzzy metric spaces; this requires that a function of the distance has supremum 1, in relation to the axiomatics of the probability of metric spaces. The modified definition of the fuzzy metric spaces was introduced by A. George and P. Veeramani in 1994; this relaxes the axiomatics of fuzzy metric spaces and it is desired that the infimum of the function of the distance is 0, in relation to the probability approximation space. Today, they are studying fuzzy metric spaces in terms of both definitions. The triangular norm (t-norm) first appeared in the framework of probabilistic metric spaces in the work of Karl Menger. This is a crucial operation in several fields (fuzzy sets, fuzzy logic. . . ) and their applications. Recently, many authors have observed that the various contraction mappings in metric spaces may be exactly translated into probabilistic or fuzzy metric spaces endowed with special t-norms, such as a minimum t-norm. Starting with the famous Banach contraction principle, a huge number of mathematicians started to formulate better contractive conditions for which a fixed point exists. In this chapter, we have identified some of the first but no less important contraction conditions that have been formulated by well-known mathematicians Grabiec, Khan, Mihet, Nadler, Caristi, and Takahashi and made into the framework of fuzzy metric space. It is our hope that the material presented in this chapter will be enough to stimulate scientists and students to investigate further this challenging field.

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72.

7 Set-Valued Maps and Inclusion Problems in Modular Metric Spaces Poom Kumam

CONTENTS 7.1 7.2

7.3

7.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Fixed Point Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.2.1 Fixed Point Inclusion in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.2.2 Fixed Point Inclusion in Modular Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.2.3 Common Fixed Points in Modular Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.3.1 Abstract Economies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.3.2 Fractional Integral Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

7.1 Introduction Solving an operator equation is a natural problem in mathematics, and we have informally or formally understood the intuitive concept of this for a very long time. In conventional mathematical expression, we are talking about solving Ax = y. Contrary to the above equation, we are sometimes unable to deliver a suitable map A that has the prescribed properties we need to made Ax = y solvable. Set-valued maps may be introduced into the situation in such a case. Suppose that X and Y are nonempty sets. When each point x ∈ X is associated to a subset F(x) ⊂ Y, we say that F(·) is a set-valued map from X into Y. We denote this by F : X ⇒ Y. If F(x) = {f (x)} is singleton for all x ∈ X, then we just view F as a single-valued map. Moreover, if f : X → Y is a single-valued map that is extracted from F, that is, f (x) ∈ F(x) for all x ∈ X, we say that f is a selection of F. To some extent, the selection of desired properties is of special interest. As we have evolved into the realm of set-valued maps, the operator equation is also generalized. The inclusion problem Ax  y is then brought into attention. This form of problem arises extensively in optimal control as in the differential inclusion problem, and is central in game theory, where the Kakutani fixed point theorem is central. Amongst so many inclusion problems, we would like to mention two of them explicitly. The first is the fixed point inclusion problem, and the second is the zero point problem. 245

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Suppose that X is a nonempty set, Z is a linear space, and F : X ⇒ X and T : Z ⇒ Z are two set-valued maps. Then, we define Fixed Point Inclusion by: Find x¯ ∈ X such that x¯ ∈ F¯x. Here, we call x¯ the fixed point of F. Fixed point inclusion is extensively studied, as it gives opportunity to several applications. The first important investigation was done by [19] in 1969, in an attempt to extend the Banach contraction principle [7]. Several investigations were done later on in various different directions. In 2010, Chistyakov [14] introduced the concept of modular metric space. This space is regarded as a modular space without linear structure. It is well known that modular space due to [21] is widely studied and plays an important role in function space theory. In Chistyakov’s work, he used the similar relationship to that which metric and Banach spaces have. This ended up with modular metric space being an extension to both modular and metric spaces. Fixed point theory in modular metric spaces is a relatively new area, measured by the few existing research papers. More interestingly, fixed point theory in this space has so many approaches and each of them are, on their own, useful in different ways. We shall be sticking with the approach used in [12, 13, 18]. Studies with this approach can also be found in [8, 9, 10, 11]. A different approach is due to Abdou and Khamsi [2], where they studied fixed point theory in modular metric space by relying on a special assumption about regularity. For more researches in fixed point theory in modular metric spaces, see [1, 3, 4, 5, 15, 16, 17, 22]. In this chapter, we shall present recent developments of set-valued analysis based mainly on fixed point inclusion problems. We investigate problems related to KKM property and contractions. Whenever it is suitable, we shall deliver some examples to support and illustrate the theorem. This chapter will be organized as follows: The next Section 7.2 is devoted to the mathematical analysis of fixed point inclusion problems and also the common fixed point of several classes of set-valued maps. In Section 7.3, we provide applications of fixed point results presented in the previous section. We consider here two main nonlinear problems – abstract economy and fractional integral inclusion.

7.2 Fixed Point Inclusion As we have mentioned in Section 7.1, fixed point inclusion is one of the most important classes of inclusion problems. Fixed point inclusion has a long history and it is diverted into many aspects and approaches. Based on so many mathematical structures, we can differentiate fixed point inclusions into ones that rely mainly on topology and geometry of the underlying space. Examples of topological fixed point inclusion theorems are Kakutani’s and Schauder’s principle, while famous geometric examples are mostly generalizations of Nadler’s theorem. There are often different benefits in applying the topological and geometric natures to each problem we are using. While topological assumptions are more intuitive, geometric theorems usually give a simple construction of the solution as a bonus.

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7.2.1 Fixed Point Inclusion in Metric Spaces It is essential to introduce the distance between two subsets of a metric space. Let (X, d) be a metric space. Suppose that A, B ⊂ X are nonempty closed bounded sets, and x ∈ X, a ∈ A and b ∈ B. Then, we adopt the following notions: ⎧ d(x, A) := infa∈A d(x, a), ⎪ ⎪ ⎨ d(A, B) := supa∈A d(a, B), ⎪ ⎪ ⎩ H(A, B) := max{d(A, B), d(B, A)}. Among the three, H(·, ·) is the most important. It is the so-called Hausdorff–Pompieu distance, and is a metric on the space CB(X) of nonempty closed bounded subsets of X. Moreover, the space (CB(X), H) is complete if and only if (X, d) is complete. Let us now mention Nadler’s theorem, which is one of the first results on fixed point inclusion in metric space. The map involved in this theorem is simply the Lipschitz continuity with the metrics d and H, respectively. Theorem 1 ([20]). Let (X, d) be a complete metric space with the corresponding Hausdorff– Pompieu metric H, and let T : X → CB(X) be a set-valued map. Suppose that there exists a constant 0 ≤ k < 1 such that the following condition is satisfied for all x, y ∈ X: H(Tx, Ty) ≤ kd(x, y). Then, T has a fixed point. This theorem is considered fundamental to every fixed point theorist who is interested in the set-valued counterpart. This very result has been extensively studied and extended into several maps in various spaces. It is worth mentioning that almost every such generalization relies on the key proof line – showing that the map is Picard or a weakly Picard map. The following theorem is our main consideration for fixed point inclusion in general metric spaces. This result is very general and yet has a simple form. After the statement and brief proof of the theorem, we shall provide some discussion and useful ideas about deducing its corollaries. Definition 1. Let X be a topological space. The set-valued map T : X ⇒ X is said to be weakly Picard if for all x ∈ X and x0 ∈ Tx, there exists a convergent sequence (xn ) ⊂ X such that xn ∈ Txn−1 and its limit is a fixed point of T. 7.2.2 Fixed Point Inclusion in Modular Metric Spaces In this subsection, we shall consider fixed point inclusions that are studied within a modular metric space. With certain conditions, we can extend Nadler’s theorem to the context of modular metric spaces successfully. A modular metric space is a relatively new concept. It generalizes and unifies both modular and metric spaces. It is therefore not necessarily equipped with a linear structure. Before we go further, let us first give basic definitions and related properties of a modular metric space.

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Definition 2 ([14]). Let X be a nonempty set. A function w : (0, ∞) × X × X → [0, +∞] is said to be a metric modular on X if the following conditions are satisfied for any s, t > 0 and x, y, z ∈ X: 1. x = y if and only if wt (x, y) = 0 for all t > 0. 2. wt (x, y) = wt (y, x). 3. ws+t (x, y) ≤ ws (x, z) + wt (z, y). Here, we use wt (·, ·) := w(t, ·, ·). In this case, we say that (X, w) is a modular metric space. Notice that the value of a metric modular can be infinite. Since we are focusing on the generalized metric space approach, we shall not be discussing modular space theory here. Suppose that (X, d) is a metric space; then wt (·, ·) := d(·, ·) is a metric modular on X. Now, we turn to the basic definitions we need in this particular space. We start by giving the topology of the space. Let (X, w) be a modular metric space. By defining an open ball with Bw (x; r) := {z ∈ X ; supt>0 wt (x, z) < r}, we can define a Hausdorff topology on X having the collection of all such open balls as a base. The convergence in this topology can therefore be written by: (xn ) → x¯ iff sup wt (xn , x¯ ) → 0, t>0

where (xn ) ⊂ X and x¯ ∈ X. With this characterization, we now have a good hint to define Cauchy sequences. A sequence (xn ) ⊂ X is said to be Cauchy if for any given ε > 0, there exists n0 ∈ N such that sup wt (xm , xn ) < ε t>0

whenever m, n > n0 . Naturally, X is said to be complete if the Cauchy sequence in X converges. We shall first discuss fixed point inclusion theorems that rely mainly on topological properties. To be precise, we shall study fixed point inclusion that involves the class of maps with the KKM property. Definition 3. Suppose that D is a bounded subset of a modular metric space X. We define 

1. co(D) := {B ⊂ X : B is a closed ball in X such that D ⊂ B}; 2. A(X) := {D ⊂ X ; D = co(D)}. If D ∈ A(X), we say that D is admissible; 3. D is said to be subadmissible if A ∈ D implies that co(A) ⊂ D, where D denotes the family of all nonempty finite subsets of D. Definition 4. Let X and Y be two Hausdorff spaces and F : X ⇒ Y be a set-valued operator with nonempty values. F is said to be: 1. upper semicontinuous if for each nonempty closed set B ⊂ Y, the set F− (B) := {x ∈ X : F(x) ∩ B = 0} / is closed in X;

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2. lower semicontinuous if for each nonempty open set B ⊂ Y, the set F− (B) := {x ∈ X : F(x) ∩ B = 0} / is open in X; 3. continuous if it is both upper and lower semicontinuous; 4. closed if its graph ΓF := {(x, y) ∈ X × Y : y ∈ F(x)} is closed in the product topology; 5. firmly compact if for each nonempty bounded set A ⊂ X, the set F(A) :=



{F(x) ; x ∈ A}

is bounded and relatively compact. Remark 1. It is well known that if a set-valued operator F : X ⇒ Y is upper semicontinuous, then F is closed. The converse holds when the space Y is compact. Definition 5. Let M be a modular metric space, X a subadmissible subset of M and Y a topological space. Let F, G : X ⇒ Y be two set-valued operators. If for each A ∈ X, we have F( co(A)) ⊂ G(A), then G is said to be a generalized KKM operator with respect to F. Definition 6. Let M be a modular metric space, X a subadmissible subset of M and Y a topological space. A set-valued operator F : X ⇒ Y is said to satisfy the KKM property if for any generalized KKM operator G : X ⇒ Y with respect to F, the family {G(x) : x ∈ X} has the finite intersection property. In general, we write KKM(X, Y) for the class of F : X ⇒ Y that satisfies the KKM property. Definition 7. Let M be a modular metric space and X be a nonempty subset of M. A multivalued mapping F : X ⇒ M is said to have the approximate fixed point property if for any ε > 0, there exists xε ∈ X such that F(xε ) ∩ Bw (xε ; ε ) = 0. / In other words, there exists y ∈ F(xε ) such that supt>0 wt (xε , y) < ε . The first important result regarding the class KKM is the approximate fixed point property defined above. The subsequent result is the fixed point inclusion theorem, in which the result is obtained via adopting the approximate fixed point property. Theorem 2. Let (M, w) be a modular metric space, X a nonempty subadmissible subset of M and F ∈ KKM(X, X). If F(X) is totally bounded, then F has the approximate fixed point property. Proof. Let Y := F(X). By thetotal boundedness of Y, for each ε > 0, there exists a bounded set A ∈ X such that Y ⊂ x∈A Bw (x; ε ). Now, define a set-valued operator G : X ⇒ X by / G(x) = Y \ Bw (x; ε ) for all x ∈ X. Thus, G(x) is closed for each x ∈ X and x∈A G(x) = 0. Therefore, G is not a generalized KKM operator with respect to F. This implies that there  exists a finite subset C := {x0 , x1 , . . . , xm } ofX such that F( co(C)) ⊂ G(C) = m i=0 G(xi ). Thus, m / i=0 G(xi ). By the definition of G, it follows that thereexists xε ∈ F( co(C)) such that xε ∈ xε ∈ m i=0 Bw (xi ; ε ). Therefore, xi ∈ Bw w(xε ; ε ) for each xi ∈ C. Hence, C ⊂ Bw (xε ; ε ) ⊂ Bw (xε ; ε ) so that co(C) ⊂ Bw (xε ; ε ). Suppose that xε ∈ F(xε ) for some xε ∈ co(C); then we have xε ∈ Bw (xε ; ε ). This further implies that xε ∈ Bw (xε ; ε ). Therefore, xε ∈ F(xε ) ∩ Bw (xε ; ε ). That is, /

F(xε ) ∩ Bw (xε ; ε ) = 0.

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Theorem 3. Let (M, w) be a modular metric space, X be a nonempty subadmissible subset of M and F ∈ KKM(X, X). If F is closed and firmly compact, then F has a fixed point. Proof. By the firm compactness of F, we can see that F(X) is bounded and compact. According to Theorem 2, F has the approximate fixed point property. Therefore, for each n ∈ N,

there exist xn , xn ∈ X such that xn ∈ F(xn ) ∩ Bw xn ; n1 . Now, since Y := F(X) is compact, we may assume that (xn ) converges to some x0 ∈ Y. Therefore, for any ε > 0, there exists ε  N ∈ N such supt>0 wt (x0 , xn ) < 2 whenever n ∈ N and n ≥ N. Observe that for n ∈ N with  that 2 n > max N, ε , we have sup wt (x0 , xn ) ≤ sup[w 2t (x0 , xn ) + w 2t (xn , xn )] t>0

t>0

≤ sup wt (x0 , xn ) + sup wt (xn , xn ) t>0

t>0

ε 1 < + 2 n < ε. Thus, (xn ) converges also to x0 . Since xn ∈ F(xn ), we have (xn , xn ) ∈ ΓF . Since (xn , xn ) is a sequence in ΓF which converges to (x0 , x0 ) in product topology and ΓF is closed, we may

conclude that (x0 , x0 ) ∈ ΓF . Therefore, x0 ∈ Fx0 . We next show that we can apply this theorem to study a single-valued fixed point of a very general map. Lemma 1. Let M be a modular metric space, Y a topological space and X a nonempty subadmissible subset of M. Suppose that f : Y → X is continuous. If F ∈ KKM(X, Y), then (f ◦ F) ∈ KKM(X, X). Proof. Let G : X ⇒ Y be a generalized KKM operator with respect to (f ◦ F) and let A :=  {x1 , x2 , . . . , xn } ∈ X. Therefore, (f ◦ F)( co(A) ∩ X) ⊂ ni=1 G(xi ). Note also that F( co(A) ∩ X) ⊂ f −1 (f ◦F)( co(A)∩X) ⊂ f −1 (G(xi )). Hence, we have that f −1 ◦G is a generalized KKM operator with respect to F. Now, since F ∈ KKM(X, Y), the family {(f −1 ◦ G)(x) : x ∈ X} has the finite intersection property, so does the family {G(x) : x ∈ X}. This shows that (f ◦F) ∈ KKM(X, X).

Corollary 1. Let Mw be a modular metric space, Y a topological space and X a nonempty subadmissible subset of Mw . Suppose that f : X → X is continuous and f (A) is bounded and compact for every nonempty bounded set A ⊂ X. If I ∈ KKM(X, X), then f has a fixed point. Proof. According to Lemma 1, we obtain f = (f ◦ I) ∈ KKM(X, X). Since f is continuous, it follows that f is closed. Therefore, we can apply Theorem 3 to obtain the desired result.

We next give another route of investigation of fixed point inclusion in modular metric spaces. This time, we shall rely more on analytical assumptions. Briefly stated, we shall use the contractivity assumptions. Before we can press on into the main exploration, we need the following knowledge of metric modular of sets. We write Cl(X) to denote the set of all nonempty closed subsets of X. For any subset A ⊂ Xw and point x ∈ X, we denote wt (x, A) := infy∈A wt (x, y). Given two subsets A, B ∈ Cl(X), define wt (A, B) := supx∈A wt (x, B). Most importantly, the Hausdorff–Pompieu metric modular Wt (A, B) := max{wt (A, B), wt (B, A)}.

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Lemma 2. Let (X, w) be a modular metric space, A ∈ Cl(X), and x ∈ X. Then, wt (x, A) = 0 for all t > 0 ⇐⇒ x ∈ A. Definition 8. Given a modular metric space (X, w) and an arbitrary point x ∈ X, a subset Y ⊂ X is said to be reachable from x if inf sup wt (x, y) = sup wt (x, Y) < ∞.

y∈Y t>0

t>0

This lemma gives a simple criterion of when the reachability holds. Lemma 3. Let (X, w) be a modular metric space with w being (l.s.c.), and Y ⊂ X a nonempty compact subset. For a point x ∈ X, if either infy∈Y supt>0 wt (x, y) < ∞ or supt>0 wt (x, Y) < ∞, then Y is reachable from x. The following lemma is essential in showing the solvability of fixed point inclusion for the contractivity condition. Lemma 4. Suppose that Y, Z ∈ Cl(X) are nonempty and z ∈ Z. If Y is reachable from z, then for each ε > 0, there exists a point yε ∈ Y such that supt>0 wt (z, yε ) ≤ supt>0 Wt (X, Y) + ε . Now, we state the notions of contraction and of Kannan’s contraction. Note that these two concepts are not generalizations of one another. Definition 9. Let (X, w) be a modular metric space. A set-valued operator F : X ⇒ X is said to be a contraction if there exists a constant k ∈ [0, 1) such that Wt (Fx, Fy) ≤ kwt (x, y),

(7.1)

for all t > 0 and x, y ∈ X. If k is restricted in [0, 12 ) and (7.1) is replaced with the following inequality: Wt (F(x), F(y)) ≤ k[wt (x, F(x)) + wt (y, F(y))]. Then, we call F a Kannan’s contraction. Now, we present the main existence theorems. Theorem 4. Let (X, w) be a complete modular metric space with w being l.s.c., and F a contraction on X having compact values with contraction constant k. Suppose that there exists a pair of points x0 ∈ X and x1 ∈ F(x0 ) with the following properties: 1. the set {x0 , x1 } is bounded, 2. F(x1 ) is reachable from x1 . Then, F has at least one fixed point. Proof. Since F(x1 ) is reachable from x1 , by using Lemma 4, we may choose x2 ∈ F(x1 ) such that sup wt (x1 , x2 ) ≤ sup wt (F(x0 ), F(x1 )) + k. t>0

t>0

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From the above evidence and the hypothesis that {x0 , x1 } is bounded, we reach the following inequalities: sup wt (x2 , F(x2 )) ≤ sup wt (F(x1 ), F(x2 )) w>0

t>0

≤ k sup wt (x1 , x2 ) t>0

≤ k[sup Wt (F(x0 ), F(x1 )) + k] t>0

≤ k sup wt (x0 , x1 ) + k2 2

t>0

< ∞. By the assumptions, we apply Lemma 3 to guarantee that F(x2 ) is actually reachable from x2 . Inductively, by this procedure, we define a sequence (xn ) in X, with the supplement from Lemma 4, satisfying the following properties for all n ∈ N: ⎧ xn ∈ F(xn−1 ), ⎪ ⎪ ⎨ supt>0 wt (xn , xn+1 ) ≤ supt>0 Wt (F(xn−1 ), F(xn )) + kn , ⎪ ⎪ ⎩ F(xn ) is reachable from xn . Hence, by the contractivity of F, we have sup wt (xn , xn+1 ) ≤ sup Wt (F(xn−1 ), F(xn )) + kn t>0

t>0

≤ k sup wt (xn−1 , xn ) + kn t>0

≤ k[k sup wt (xn−2 , xn−1 ) + kn−1 ] + kn t>0

≤ k2 sup wt (xn−2 , xn−1 ) + 2kn . t>0

Thus, by induction, we have sup wt (xn , xn+1 ) ≤ kn sup wt (x0 , x1 ) + nkn . t>0

t>0

Moreover, it follows that sup ∑ wt (xn , xn+1 ) ≤ sup wt (x0 , x1 ) ∑ kn + ∑ nkn < ∞. t>0 n∈N

t>0

n∈N

n∈N

Without loss of generality, suppose m, n ∈ N and m > n. Observe that sup wt (xn , xm ) ≤ sup[w t>0

t>0

t m−n

(xn , xn+1 ) + · · · + w

t m−n

(xm−1 , xm )]

≤ sup wt (xn , xn+1 ) + · · · + sup wt (xm−1 , xm ) t>0

≤

t>0

∞

∑ sup wt (xn , xn+1 )

n=n∗ t>0

< ε,

for all m > n ≥ n∗ for some n∗ ∈ N. Hence, (xn ) is a Cauchy sequence so that the completeness of Xw implies that (xn ) converges to some point x ∈ Xw . Consequently, we may

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conclude from the contractivity of F that the sequence (F(xn )) converges to F(x). Since xn ∈ F(xn−1 ), we have for any t > 0, 0 ≤ wt (x, F(x)) ≤ w 2t (x, xn ) + W 2t (F(xn−1 ), F(x)), which implies that wt (x, F(x)) = 0 for all t > 0. Since F(x) is closed, it then follows from Lemma 2 that x ∈ F(x).

Example 1. Suppose that X = [0, 1] and w : (0, +∞) × X × X → [0, +∞] is defined by wt (x, y) =

1 |x − y|. (1+t)

Clearly, X, and w is an l.s.c. metric modular. Notice that any two-point subset is bounded. Now, we define a set-valued operator F : X ⇒ X by

F(x) :=

 x+1 ,1 , 2

for every x ∈ X. Observe that F has compact values on X. Note that for each t > 0 and x, y ∈ X, we have Wt (Fx, Fy) =

1 |x − y| 2(1+t)

≤ 12 wt (x, y).

Therefore, F is a contraction with contraction constant k = 12 . Moreover, it is easy to see that conditions 1 and 2 hold. Finally, we have that 1 is a fixed point of F (and it is unique). Next, we shall show that the fixed point in the above theorem need not be unique, as we shall see in the following example: Example 2. Suppose that X is defined as in the previous example. Consider the operator G : X ⇒ X given by

 x+1 G(x) := 0, , 2 for each x ∈ X. Note that this operator G is also a contraction with constant k = 12 and takes compact values on X. Also, conditions 1 and 2 hold. However, every point in X is a fixed point of G. This shows the nonuniqueness of fixed points for a set-valued contraction. Theorem 5. Replacing F in Theorem 4 with a Kannan’s contraction yields the same existence result. Proof. Since F(x1 ) is reachable from x1 , by using Lemma 4, we may choose x2 ∈ F(x1 ) such that sup wt (x1 , x2 ) ≤ sup Wt (F(x0 ), F(x1 )) + k. t>0

t>0

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Now, observe that sup wt (x2 , F(x2 )) t>0

≤ sup Wt (F(x1 ), F(x2 )) t>0

≤ k sup wt (x1 , F(x1 )) + k sup wt (x2 , F(x2 )) t>0

t>0

≤ k sup Wt (F(x0 ), F(x1 )) + k sup wt (x2 , F(x2 )) t>0

t>0

≤ k sup wt (x0 , F(x0 )) + k sup wt (x1 , F(x1 )) + k sup wt (x2 , F(x2 )) t>0

t>0

t>0

≤ k sup wt (x0 , x1 ) + k sup wt (x1 , F(x1 )) + k sup wt (x2 , F(x2 )). t>0

t>0

t>0

k Writing ξ := 1−k < 1, we obtain, from the boundedness of {x0 , x1 } and the reachability of F(x1 ) from x1 , that

sup wt (x2 , F(x2 )) ≤ ξ sup wt (x0 , x1 ) + ξ sup wt (x1 , F(x1 )) < ∞. t>0

t>0

t>0

Thus, from the assumptions and Lemma 4, we may see that F(x2 ) is reachable from x2 . Inductively, we can construct a sequence (xn ) in X with exactly the same properties appearing in the proof of Theorem 4. Now, consider further that sup wt (xn , xn+1 ) t>0

≤ sup Wt (F(xn−1 ), F(xn )) + kn t>0

≤ k sup wt (xn−1 , F(xn−1 )) + k sup wt (xn , F(xn )) + kn t>0

t>0

≤ k sup wt (xn−1 , F(xn−1 )) + k sup wt (xn , xn+1 ) + kn . t>0

t>0

Moreover, we get sup wt (xn , xn+1 ) ≤ ξ sup wt (xn−1 , xn ) + t>0

t>0

kn 1−k

≤ ξ 2 sup wt (xn−2 , xn−1 ) + t>0

kn kn + (1 − k)2 (1 − k)

≤ ξ 2 sup wt (xn−2 , xn−1 ) + 2 · t>0

kn (1 − k)2

.. . ≤ ξ n sup wt (x0 , x1 ) + nξ n . t>0

As in the proof of Theorem 4, the sequence (xn ) converges to some x ∈ X. Observe now that sup wt (x, F(x)) t>0

= sup wt ({x}, F(x)) t>0

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≤ sup wt ({x}, F(xn )) + sup wt (F(xn ), F(x)) t>0

t>0

= sup wt (x, F(xn )) + sup wt (F(xn ), F(x)) t>0

t>0

≤ sup wt (x, xn+1 ) + sup Wt (F(xn ), F(x)) t>0

t>0

≤ sup wt (x, xn+1 ) + k sup wt (xn , F(xn )) + k sup wt (x, F(x)) t>0

t>0

t>0

= (1 + k) sup wt (x, xn+1 ) + k sup wt (x, F(x)). t>0

t>0

Thus, we have sup wt (x, F(x)) ≤ t>0

1+k 1−k

sup wt (x, xn+1 ). t>0

Letting n → ∞ to conclude the theorem.



7.2.3 Common Fixed Points in Modular Metric Spaces It is also an interesting question to ask about common fixed points between two or more maps. To simplify the problem, we may consider the case of a common fixed point between a single-valued map and a set-valued map. That is, if f : X → X and F : X ⇒ X, their common fixed point is the point x¯ ∈ X such that x¯ = f x¯ ∈ F¯x. This approach is relatively simple compared to common fixed points between set-valued maps, as it can be extended from common fixed points of single-valued maps with typical twists in the commutativity assumptions. In the case where we have a family of set-valued maps {Ti }i∈I from a set X into itself, the  common fixed point of the Ti s is an element x¯ ∈ X such that x¯ ∈ i∈I Ti x¯ . This problem can be viewed as a strong form of intersection property. Since we are focusing on set-valued maps particularly, we shall be dealing with this kind of problem rather that the hybrid of single- and set-valued maps. Our approach to study this inclusion problem is not based on any commutativity, but rather on the analytical structures of the space and also on the relationship between each map. We use here analytic assumptions on each t > 0, rather than topological assumptions, since some topological properties in this space are difficult to check and are rarely satisfied. Definition 10. Let (X, w) be a modular metric space. A subset G ⊂ X is said to be tproximinal at a fix t > 0 if for any x ∈ X \ G, we can find y ∈ G such that wt (x, y) = infg∈G wt (x, g). If G is t-proximinal at each t > 0, then G is said to be pointwisely proximinal. Before going further, we need to mention the following trivial property. Proposition 1. Let (X, w) be a modular metric space and let A, B be two nonempty pointwisely proximinal subsets of X. Suppose that a ∈ A and b ∈ B. Then, at each t > 0, we have wt (a, b) ≤ Ws (A, B) + diamt−s (B) for any s ∈ (0, t). Now, we state the common fixed point theorem.

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Theorem 6. Let (X, w) be a complete modular metric space and let F := {Ft }t>0 be a family of setvalued operators in which Ft : X ⇒ X has bounded, closed, and pointwisely proximal values for each t > 0. Suppose that the following conditions are satisfied: 1. For each t > 0, there exists a constant kt ∈ (0, 1) such that Wt (Ft x, Ft y) ≤ kt w4t (x, y), for all x, y ∈ X. 2. If 0 < s ≤ t, then Fs x ⊂ Ft x for each x ∈ X. 3. Let t, ε > 0, there exists 0 < s < t such that diams (Fs x) ≤ 12 (1 − ks )ε , for each x ∈ X. 4. There exists x0 ∈ X such that wt (x0 , y) < ∞ for all y ∈ Fx0 and t > 0. Then, the following hold: 1. 2. 3. 4.

F has a unique common selection f : X → X. The sequence {f n x0 } pointwisely converges to a fixed point x∗ ∈ X of f . If there exists another fixed point y∗ ∈ X of f , then wσ (x∗ , y∗ ) = ∞ for some σ > 0. x is a common fixed point of the class F if and only if it is a fixed point of f .

Proof. 1 Let t > 0 and note that {Ft x}t>0 is a family of nonempty closed bounded subsets  of X. Together with hypotheses 2 and 3, it is easy to see that t>0 Ft x = 0. / Suppose y, z ∈  F x. Let ε > 0, and choose 0 < s < t satisfying hypothesis 3; we have t t>0 wt (y, z) ≤ ws (y, z) ≤ diams (Fs x) < ε . 

Thus, we conclude that y = z. Therefore, t>0 Ft x = {fx}. Here, we obtain the mapping f : X → X which is the unique common selection of F. 2 Let x0 ∈ X satisfying condition 4 and define a sequence {xn } by xn := fxn−1 for each n ∈ N. Consider the case where xn = xn−1 only; otherwise the result is obtained. We first show that the orbit O(x0 ; ∞) is pointwisely bounded. Let t > 0 and without loss of generality, assume that w4t (x0 , x1 ) > 0. Note that xn = fxn−1 ∈ Ft xn−1 for all t > 0 and n ∈ N. Therefore, w4t (x0 , x1 ) < ∞. Setting ε := w4t (x0 , x1 ), we may find 0 < s < 2t such that diam 2s (F 2s x) ≤ 12 (1 − k 2s )ε for each x ∈ X. Moreover, observe that ε = w4t (x0 , x1 ) ≤ w2s (x0 , x1 ). Let n ∈ N and i, j ∈ {1, 2, . . . , n}, we may deduce that ws (xi , xj ) ≤ Δ 2s (F 2s xi−1 , F 2s xj−1 ) + diam 2s (F 2s xj−1 ) ≤ k 2s w2s (xi−1 , xj−1 ) + 12 (1 − k 2s )ε ≤ 12 (1 + k 2s ) diam2s (O(x0 ; n)) < diam2s (O(x0 ; n)) ≤ diams (O(x0 ; n)).

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This implies that diams (O(x0 ; n)) = ws (x0 , xk ) for some k ∈ {1, 2, . . . , n}. Moreover, since w2s (xi , xj ) ≤ ws (xi , xj ), the above inequalities imply that w2s (xi , xj ) < diam2s (O(x0 ; n)). Thus, we obtain the following inequalities: diam2s (O(x0 ; n)) = w2s (x0 , xk ) ≤ ws (x0 , x1 ) + ws (x1 , xk ) ≤ ws (x0 , x1 ) + 12 (1 + k 2s ) diam2s (O(x0 ; n)). This implies that 12 (1 − k 2s ) diam2s (O(x0 ; n)) ≤ ws (x0 , x1 ) for all n ∈ N. Thus, we have diam4t (O(x0 ; ∞)) ≤ diam2s (O(x0 ; ∞))



= sup diam2s (O(x0 ; n)) ≤ n∈N

 ws (x0 , x1 ).

2 1−k s

2

This yields that O(x0 ; ∞) is pointwisely bounded. Next, we show that the sequence {xn } is pointwisely Cauchy. For each n ∈ N, set Qn := {xn , xn+1 , . . .}. It is easy to see that f (Qn ) = Qn+1 . Since the orbit O(x0 ; ∞) is pointwisely bounded, then the sequence { diamt (Qn )} converges to some wt ≥ 0 at each t > 0. It is worth noting that diamt (Qn ) ≥ wt for all n ∈ N. Assume that {xn } is not t-Cauchy, then wt > 0. Choose 0 < η < t for which diam η2 (F η2 x) ≤ 12 (1 − k η2 )wt for each x ∈ X. Let n ∈ N and xp , xq ∈ Qn . Observe that wη (fxp , fxq ) ≤ Δ η2 (F η2 xp , F η2 xq ) + diam η2 (F η2 xq ) ≤ k η2 w2η (xp , xq ) + 12 (1 − k η2 )wt ≤ 12 (1 + k η2 ) diamη (Qn ). Moreover, we may derive that diamη (Qn+1 ) = diamη (f (Qn )) = sup wη (fxp , fxq ) ≤ 12 (1 + k η2 ) diamη (Qn ). p,q≥n

Taking n → ∞, we have wη ≤ 12 (1 + k η2 )wη . Therefore, wη = 0. Since diamη (Qn ) ≥ diamt (Qn ), we have 0 = wη ≥ wt > 0, which is a contradiction. Therefore, wt = 0 and {xn } is a pointwisely Cauchy sequence. By the pointwise completeness of X, {xn } pointwisely converges to some x∗ ∈ X. Now, we show that x∗ is actually a fixed point of f . Assume the contrary, that x∗ = fx∗ . Therefore, there exists some t0 > 0 such that wt0 (x∗ , fx∗ ) = 0. Consider first the case when wt0 (x∗ , fx∗ ) = ∞. Observe the following: wt0 (x∗ fx∗ ) ≤ w t0 (x∗ , fxn ) + w t0 (fxn , fx∗ ) 2

2

≤ w t0 (x∗ , fxn ) + Δ t0 (F t0 xn , F t0 x∗ ) + diam t0 (F t0 x∗ ) 2

4

4

4

4

4

≤ w t0 (x∗ , fxn ) + k t0 wt0 (xn , x∗ ) + diam t0 (F t0 x∗ ). 2

4

4

4

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Since F t0 x∗ is pointwisely bounded, letting n → ∞ in the above inequality, we obtain 4

wt0 (x∗ fx∗ ) ≤ diam t0 xn (F t0 x∗ ) < ∞, 4

4

which is a contradiction. This shows that wt0 (x∗ fx∗ ) = ξ < ∞. Now, choose 0 < ν < t0 such that diam ν2 (F ν2 x) ≤ 12 (1 − k ν2 )ξ for all x ∈ X. Thus, observe that wt0 (fxn , fx∗ ) ≤ wν (fxn , fx∗ ) ≤ Δ ν2 (F ν2 xn , F ν2 x∗ ) + diam ν2 (F ν2 x∗ ) ≤ k ν2 w2ν (xn , x∗ ) + 12 (1 − k ν2 )ξ . Letting n → ∞, we have

ξ ≤ 12 (1 − k ν2 )ξ .

This yields a contradiction. Therefore, x∗ is a fixed point of f . 3 Let y∗ ∈ X be another fixed point of f and assume that wt (x∗ , y∗ ) < ∞ for all t > 0. Hence, we may find t1 > 0 such that wt1 (x∗ , y∗ ) = ζ > 0. Again, choose 0 < υ < t1 satisfying diam υ2 (F υ2 x) ≤ 12 (1 − k υ2 )ζ for all x ∈ X. Thus, we have wυ (x∗ , y∗ ) = wυ (fx∗ , fy∗ ) ≤ Δ υ2 (Fx∗ , Fy∗ ) + diam υ2 (Fy∗ ) ≤ k υ2 w2υ (x∗ , y∗ ) + 12 (1 − k υ2 )wt1 (x∗ , y∗ ) ≤ 12 (1 + k υ2 )wυ (x∗ , y∗ ), which is impossible. Therefore, wσ (x∗ , y∗ ) = ∞for some σ > 0. 4 It is obvious from the facts that x∗ = fx∗ ∈ t>0 Ft x∗ and f is the only selection of F.



1 Example 3. Let X = [0, 1] and let w : (0, ∞) × X × X → [0, ∞] given by wt (x, y) := 1+t |x − y|. It is clear that (X, w) is a complete modular metric space. For each t > 0, we define a set-valued operator Ft : X ⇒ X by ⎧   t3 ⎪ x if t < 1, ⎪ 0, 1+4t ⎪ ⎪ ⎨ {0} if t = 1, Ft x := ⎪ ⎪   ⎪ ⎪ 1 ⎩ 0, t3 (1+4t) x if t > 1,

for all x ∈ X. For each t > 0, Ft has bounded closed and pointwisely proximal values. It is easy to see that for each t > 0, we have Wt (Ft x, Ft y) ≤ kt w4t (x, y), where the constants kt are given by

⎧ t ⎪ ⎪ ⎨ k kt := ⎪ ⎪ ⎩ 1 t

if t < 1, if t = 1, if t > 1,

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for any fixed k ∈ (0, 1). Thus, we have that condition 1 of Theorem 6 holds. t3 1 and B(t) := t3 (1+4t) are nondecreasing, we have Since the two functions A(t) := 1+4t immediately condition 2 of Theorem 6. For condition 3 of Theorem 6, that t, ε > 0 are chosen arbitrarily. Now, we  we suppose choose μ > 0 such that μ < min t, ε , 12 . Observe that diamμ (Fμ x) =

μ3 x 1+4μ

≤ μ3 < 12 μ (1 − μ ) < 12 (1 − μ )ε = 12 (1 − kμ )ε . Therefore, condition 3 of Theorem 6 is satisfied. Finally, since every x ∈ X satisfies condition 4 of Theorem 6, we have all the required conditions of Theorem 6. We now have the following conclusions: 1. The function f : X → X given by fx = 0 for all x ∈ X is the unique common selection of F. 2. f n x pointwisely converges to 0 and f 0 = 0. 3. 0 is the only fixed point of f . 4. 0 is a the only common fixed point of the class F.

7.3 Applications In this section, we shall illustrate some uses of the theorems presented in the previous section. We focus on applications to abstract economy and integral inclusion. Abstract economy is a general problem considered in several research areas, explicitly or implicitly. It is widely used in economics itself, but also arises in computer sciences such as network allocation problems. Abstract economy is actually a noncooperative game with feasibility constrained to the action. For another application, we investigate the solvability of the fractional inclusion problem in which we also include delayed behaviours. Fractional integral is used in describing various natural phenomena, and also in electric transmission, travelling waves, fluid dynamics, and in control engineering. 7.3.1 Abstract Economies We shall begin with a game. A game is, generally speaking, a situation with players as its main component, each having partial control over some outcome. In general, these players will have conflicting preference over the outcomes. Formally stated, the set of choices under player i’s control is denoted by Xi . Members in Xi are called strategies. Letting N := {1, 2, . . . , n} denote the set of players, X := ∏i∈N Xi is the set of strategy vectors. Each of the vectors determines the outcome. Players have preference over the outcomes, and so this induces preferences over the strategy vectors.

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 i : X → 2X in which We shall first describe player i’s preference by a set-valued operator U  i (x) represents the set of all strategy vectors preferred to x. However, recall that each U player has control only over their strategies, that is, the ith component of x. Therefore, it would be better to represent the preference in terms of the good reply sets introduced in the following. Given a strategy vector x ≡ (x1 , x2 , . . . , xn ) ∈ X and a strategy yi ∈ Xi , we shall write x|yi to represent the vector (x1 , x2 , . . . , xi−1 , yi , xi+1 , . . . , xn ). We shall say that x|yi is a good  i (x). By this relation, we may define, for each i ∈ N, reply for player i to the vector x if x|yi ∈ U Xi  i (x)}. This operator is called a set-valued operator Ui : X → 2 by Ui (x) := {yi ∈ Xi ; x|yi ∈ U a good reply correspondence. Now, we define rigorously a game to be the tuple (N, (Xi ), (Ui )), where N := {1, 2, . . . , n} is the set of players and for each i ∈ N, Xi is the set of strategies chosen by player i while Ui is the good reply correspondence from X := ∏i∈N Xi into Xi . In reality, a choice of strategies, most of the time, cannot be made independently to other players’ choices or without constraints. Suppose that N := {1, 2, . . . , n} is the set of players. We shall take theses constraints into account by introducing, for each i ∈ N, a set-valued operator Fi : X → 2Xi . This operator tells which strategies are actually feasible for player i to take when other players’ choices are kept fixed. Note that the operator Fi is generally independent of the choice made by player i himself. The jointly feasible vectors are thus the fixed points of the operators F := ∏i∈N Fi : X → 2X . A game with these constraints is called a generalized game or an abstract economy. It is written explicitly in tuple form as (N, (Xi ), (Fi ), (Ui )). A Nash’s equilibrium for a game (both an ordinary and a generalized one) is the situation where no player has a good reply. For a game, it is the case that Ui (x) = 0/ for each i ∈ N. / For an abstract economy, it is the case that x ∈ F(x) but for each i ∈ N, Ui (x) ∩ Fi (x) = 0. Theorem 7. Let (N, (Xi ), (Fi ), (Ui )) be an abstract economy satisfying, for each i ∈ N, the following conditions: 1. 2. 3. 4.

there exists a metric modular wi on each Xi ; / is nonempty and subadmissible in Xi ; the set Yi := {x ∈ Xi ; Fi (x) ∩ Ui (x) = 0} the operator Gi := Fi |Yi is invariant, closed and firmly compact; G := ∏i∈N Gi ∈ KKM(Y, Y), where Y := ∏i∈N Yi .

Then, this abstract economy has a Nash’s equilibrium. Proof. Suppose that N = {1, 2, . . . , n}. Let t > 0, then the function wt := maxi∈N wit is also a metric modular on X := ∏i∈N Xi . We now claim the following facts: 1. Y is subadmissible in Xw ; 2. G is closed and firmly compact on Xw . To prove 1, let A ∈ Y be nonempty. For i ∈ N, let Ai denote the set of all components in the ith coordinate of elements in A. Then, we set

ε0i := inf{r > 0 ; there exists z ∈ [Xi ]w such that Bwi (z; r) ⊃ Ai } ≥ 0. Now we define ε0 := maxi∈N ε0i . For ε > ε0 ≥ 0, we define Biε := Bwi (zi ; ε ),

(7.2)

where zi ∈ [Xi ]w is a point in which Biε ⊃ Ai . Then, the product ∏i∈N Biε is actually the ball Bw (z; ε ), where z := (z1 , z2 , . . . , zn ) ∈ Xw . Moreover, we have Bw (z; ε ) ⊃ A. Let Ziε ⊂ [Xi ]w be the

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261

set of all points zi satisfying (7.2). Consider that the intersection Thus, we can finally conclude the result that co(A) ⊂





∏ Biε = ∏

ε >ε0 zi ∈(∏i∈N Ziε ) i∈N

 

i∈N ε >ε0 z ∈Zi i



ε > ε0

 zi ∈Ziε

Biε = co(Ai ) ⊂ Yi .

Biε ⊂ Y.

ε

This proves 1, and 2 is clear. The conclusion then follows from Theorem 3.



7.3.2 Fractional Integral Inclusion It is natural to raise the situation of set-valued integrals, which has proved its importance in practical applications especially in engineering. In 1965, Aumann [6] introduced the concept of definite set-valued integrals on real line and Euclidean spaces. Suppose that Ψ is an interval [0, T], where T > 0. Let F : Ψ → 2R be a set-valued operator. A selection of F is the function f : Ψ → R ∪ {±∞} such that f (t) ∈ F(t) almost everywhere t ∈ Ψ . We write F to denote the set containing all integrable selections of F. According to Aumann [6], the set-valued integral is determined by the operator J in the following:   f (t)dt ; f ∈ F , JΨ F(t)dt := Ψ

that is, the set of the integrals of integrable selections of F. On the other hand, in elementary calculus, one deals with derivatives and integrals, including the higher-integer-order iterations. Here, in fractional integral, one looks at a broader concept where the real-order iteration is taken into account. There are many approaches to study this kind of extension. In our context, we shall use the classical notion introduced by Riemann and Liouville; the latter was the first to point out the possibility of fractional calculus, in 1832. Given a function f ∈ L1 (Ψ , μ ), the fractional integral of order α > 0 is given by  1 (t − τ )α −1 f (τ )dτ . iΨα f (t)dt := Γ (α ) Ψ Naturally, we may further consider the following fractional integral: JΨα F(t)dt := {iΨα f (t)dt ; f ∈ F } . In this particular subsection, we shall use notations a bit differently from those of earlier sections. This is due to the conventional uses of variables and functions that are common to integral and differential equations. Suppose that Ψ is the interval mentioned in the previous section. Let us assume throughout the section that the real line R is equipped with the metric modular

ωλR (x, y) :=

1 |x − y|, 1+λ

for λ > 0 and x, y ∈ R. Thus, for the space C(Ψ ) of all continuous (in ω R -topology) realvalued functions on Ψ , we shall use the metric modular

ωλC(Ψ ) (ϕ , ψ ) := sup ωλR (ϕ (t), ψ (t)), t∈Ψ

for λ > 0 and ϕ , ψ ∈ C(Ψ ). Note that both ω R and ω C(Ψ ) satisfy Fatou’s property. Also note that the set R is second countable, that is, it has a countable base, with respect to ω R topology. Moreover, it is clear that the set {ϕ , ψ } is bounded w.r.t. ω C(Ψ ) , for any ϕ , ψ ∈

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C(Ψ ). Suppose that F : Ψ × R → 2R is a set-valued operator with nonempty compact values, and u ∈ C(Ψ ). We shall use the following notation to explain the collection of integrable selections:  SF (u) := f ∈ L1 (Ψ , μ ) ; f (t) ∈ F(t, u(t)) a.e. t ∈ Ψ . It is clear that SF (u) is closed. Next, for each i ∈ {0, 1, . . . , N}, N ∈ N, assume that βi : Ψ → R is continuous, and τi : Ψ → R+ is a function with τi (t) ≤ t. We write B := max0≤i≤N supt∈Ψ βi (t). The main aim of this section is to consider the fractional integral inclusion: N

u(t) − ∑ βi (t)u(t − τi (t)) ∈ JΨα F(t, u(t))dt,

α ∈ (0, 1].

(FII)

i=0

In the above inclusion, the summation here is interpreted to be the delay term. We shall define a set-valued operator Λ : C(Ψ ) → 2C(Ψ ) by 

N

Λ (u) := w ∈ C(Ψ ) ; w(t) = ∑ βi (t)u(t − τi (t)) + iΨα f (t, u(t))dt,

 f ∈ SF (u) .

i=0

Note here that for any ϕ ∈ C(Ψ ), we have Λ (ϕ ) is reachable from ϕ w.r.t. ω C(Ψ ) . To restrict the operator Λ with a nice property, we assume that SF (u) is nonempty. Lemma 5. The operator Λ given above is compact valued if SF (u) is nonempty. Proof. For the proof, we shall show the compactness by its sequential characterization. Suppose that u ∈ C(Ψ ), and (wn ) is an arbitrary sequence in Λ (u). By definition, there corresponds a convergent sequence (fn ) in SF (u) ⊂ F(·, u(·)) satisfying N

wn (t) = ∑ βi (t)u(t − τi (t)) + iΨα fn (t, u(t))dt. i=0

The conclusion then follows.



Now, we shall state the solvability result for problem (FII). It is clear that u ∈ C(Ψ ) solves (FII) if and only if u is a fixed point of Λ . Theorem 8. Suppose that F defined above is compact-valued, and SF (u) is nonempty. Assume further that 1. for any given u, v ∈ C(Ψ ) and a selection f ∈ SF (u) of F, there corresponds a function f  ∈ SF (v) such that  R ωλ (f (t, u(t)), f  (t, v(t))) = ωλR (f1 (t, u(t)), F(t, v(t))),

ωλR (f (t, u(t)), f  (t, v(t))) ≤ LωλC(Ψ ) (u, v), for all t ∈ Ψ ; 2.

(N+1)BΓ (α )+LTα Γ (α )

< 1.

Then, Λ has a fixed point.

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Proof. For each u, v ∈ C(Ψ ), we may choose, from the assumption, functions f1 , f2 such that ⎧ f1 ∈ SF (u), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ f2 ∈ SF (v), ⎪ ωλR (f1 (t, u(t)), f2 (t, v(t))) = ωλR (f1 (t, u(t)), F(t, v(t))), ⎪ ⎪ ⎪ ⎪ ⎩ R ωλ (f1 (t, u(t)), f2 (t, v(t))) ≤ LωλC(Ψ ) (u, v), for each t ∈ Ψ . Consider the two functions w1 ∈ Λ (u) and w2 ∈ Λ (v), respectively as follows: 

α w1 (t) := ∑N i=0 βi (t)u(t − τi (t)) + iΨ f1 (t, u(t))dt, α w2 (t) := ∑N i=0 βi (t)v(t − τi (t)) + iΨ f2 (t, v(t))dt.

Now, consider the following computation:

ωλR (w1 (t), w2 (t)) N

≤ ∑ βi (t)ωλR (u(t − τi (t)), v(t − τi (t)) i=0

C(Ψ )

+ωλ

(iΨα f1 (t, u(t))dt, iΨα f2 (t, u(t))dt) C(Ψ )

≤ (N + 1)Bωλ

(u, v) + iΨα ωλR (f1 (t, u(t)), f2 (t, v(t)))

LTα C(Ψ ) ω (u, v) Γ (α ) λ

 (N + 1)BΓ (α ) + LTα = ωλC(Ψ ) (u, v). Γ (α ) C(Ψ )

≤ (N + 1)Bωλ

(u, v) +

It follows that

ΩλC(Ψ ) (Λ (u), Λ (v))

 (N + 1)BΓ (α ) + LTα ≤ ωλC(Ψ ) (u, v). Γ (α )

The proof ends here by applying Theorem 4.



7.4 Conclusion In this chapter, we make an attempt to provide some aspects of recent developments concerning set-valued analysis based mainly on fixed point inclusion problems. Important applications on abstract economy and fractional integral inclusion have been discussed in detail. It is our hope that the material presented in this chapter will be enough to stimulate scientists and students to investigate further this challenging field.

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Acknowledgements This project was supported by Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT. Moreover, this research work was financially supported by King Mongkut’s University of Technology Thonburi through the KMUTT 55th Anniversary Commemorative Fund.

References 1. 2. 3. 4. 5. 6. 7. 8.

9.

10.

11. 12. 13.

14. 15. 16. 17.

A. Abdou. Some fixed point theorems in modular metric spaces. Journal of Nonlinear Science and Applications, 9:4381–4387, 2016. A. A. Abdou and M. A. Khamsi. Fixed point results of pointwise contractions in modular metric spaces. Fixed Point Theory and Applications, 2013(1):1–11, 2013. A. A. Abdou and M. A. Khamsi. On the fixed points of nonexpansive mappings in modular metric spaces. Fixed Point Theory and Applications, 2013(1):1–13, 2013. M. R. Alfuraidan. The contraction principle for mappings on a modular metric space with a graph. Fixed Point Theory and Applications, 2015(1):1–10, 2015. M. R. Alfuraidan. The contraction principle for multivalued mappings on a modular metric space with a graph. Canadian Mathematical Bulletin, 59:3–12, 2016. R. J. Aumann. Integrals of set-valued functions. Journal of Mathematical Analysis and Applications, 12(1):1 – 12, 1965. S. Banach. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3:133–181, 1922. P. Chaipunya, Y. J. Cho, and P. Kumam. On circular metric spaces and common fixed points for an infinite family of set-valued operators. Vietnam Journal of Mathematics, 42(2):205–218, 2014. P. Chaipunya, Y. Je Cho, and P. Kumam. Geraghty-type theorems in modular metric spaces with an application to partial differential equation. Advances in Difference Equations, 2012(1):83, 2012. P. Chaipunya and P. Kumam. Topological aspects of circular metric spaces and some observations on the kkm property towards quasi-equilibrium problems. Journal of Inequalities and Applications, 2013(1):1–9, 2013. P. Chaipunya and P. Kumam. Common fixed points for an uncountable family of weakly contractive operators. Carpathian Journal of Mathematics, 31(3):307–312, 2015. P. Chaipunya and P. Kumam. An observation on set-valued contraction mappings in modular metric spaces. Thai Journal of Mathematics, 13(1):9–17, 2015. P. Chaipunya, C. Mongkolkeha, W. Sintunavarat, and P. Kumam. Fixed-point theorems for multivalued mappings in modular metric spaces. Abstract and Applied Analysis, 2012:14 p., 2012. V. V. Chistyakov. Modular metric spaces. I: Basic concepts. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 72(1):A, 1–14, 2010. M. E. Ege and C. Alaca. Fixed point results and an application to homotopy in modular metric spaces. Journal of Nonlinear Science and Applications, 8:900–908, 2015. D. Jain, A. Padcharoen, P. Kumam, and D. Gopal. A new approach to study fixed point of multivalued mappings in modular metric spaces and applications. Mathematics, 4(3):51, 2016. E. Kilinc and C. Alaca. A fixed point theorem in modular metric spaces. Advances in Fixed Point Theory, 4(2):199–206, 2014.

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C. Mongkolkeha, W. Sintunavarat, and P. Kumam. Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory and Applications, 2011(1):93, 2011. S. B. Nadler. Multi-valued contraction mappings. Pacific Journal of Mathematics, 30:475–488, 1969. S. B. Nadler. Multi-valued contraction mappings. Pacific Journal of Mathematics, 30(2):475–488, 1969. H. Nakano. Modulared semi-ordered linear spaces. Tokyo mathematical book series. Maruzen Co., 1950. H. Rahimpoor, A. Ebadian, M. E. Gordji, and A. Zohri. Common fixed point theorems in modular metric spaces, International Journal of Pure and Applied Mathematics, 99(3):373–383, 2015.

8 Graphical Metric Spaces and Fixed Point Theorems Satish Shukla

CONTENTS 8.1 8.2 8.3 8.4 8.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Some Basic Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Fixed Point Theorems in Graphical Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 An Application to Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

8.1 Introduction The French mathematician Maurice Fréchet [2] introduced the concept of metric spaces, which is at the center of several research activities. According to need and scope, this concept has been generalized by several mathematicians. In ordinary metric spaces, the metric function satisfies the triangular inequality for all points situated in the space. Most of the proofs of fixed point theorems use the triangular inequality, but with limited utility. Shukla et al. [8] pointed out this fact and introduced the notion of graphical metric spaces. They replaced the triangular inequality for ordinary metric functions with a weaker one, which is satisfied by only those points which are situated on some path included in the graphical structure associated with the space. Moreover, they observed that the contraction mappings in this new setting are more competent than those in usual metric spaces and can be applied to obtain the solution of differential equations. This chapter is devoted to the study of fixed point theorems in graphical metric spaces and their application to integral equations.

8.2 Some Basic Notations and Definitions Some of the following definitions and notations were introduced by Shukla et al. [8] and Jachymski [3]. 267

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Let X be a nonempty set and let Δ denote the diagonal of the cartesian product X × X. Consider a directed graph G, without parallel edges, such that the set V(G) of its vertices coincides with X, and the set E(G) of its edges contains all loops, that is, E(G) ⊇ Δ ; then we say that X is endowed with graph G = (V(G), E(G)). The conversion of the graph G is denoted by G−1 and it is defined by: V(G−1 ) = V(G) and E(G−1 ) = {(x, y) ∈ X × X : (y, x) ∈ E(G)}.  we denote the undirected graph obtained from G by including all the edges of G−1 . By G More precisely, we define  = V(G) and E(G)  = E(G) ∪ E(G−1 ). V(G) If x and y are vertices in a graph G, then a path in G from x to y of length l ∈ N is a sequence {xi }li=0 of l + 1 vertices such that x0 = x, xl = y and (xi−1 , xi ) ∈ E(G) for i = 1, 2, . . . , l. A graph G is called connected if, there is a path between any two vertices. Moreover, two vertices x and y of a directed graph are connected if there is a path from x to y and a path from y to x. G is weakly connected if, treating all of its edges as being undirected, there is  a path from every vertex to every other vertex. More precisely, G is weakly connected if G is connected. We define a relation P on X by: (xPy)G if and only if there is a directed path from x to y in G. We write z ∈ (xPy)G if z is contained in some directed path from x to y in G. For l ∈ N, we denote [x]lG = {y ∈ X : there is a directed path from x to y of length l}. A sequence {xn } in X is said to be G-termwise connected if (xn Pxn+1 )G for all n ∈ N. A connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph. Throughout this chapter, we assume that the graphs under consideration are directed, with nonempty sets of vertices and edges. Definition 1. Let X be a nonempty set endowed with a graph G and dG : X × X → R be a function satisfying the following conditions: (GM1) (GM2) (GM3) (GM4)

dG (x, y) ≥ 0 for all x, y ∈ X; dG (x, y) = 0 if and only if x = y; dG (x, y) = dG (y, x) for all x, y ∈ X; (xPy)G , z ∈ (xPy)G implies dG (x, y) ≤ dG (x, z) + dG (z, y) for all x, y, z ∈ X.

Then the mapping dG is called a graphical metric on X, and the pair (X, dG ) is called a graphical metric space. Example 1. Every metric space (X, d) is a graphical metric space with graph G, where V(G) = X and E(G) = X × X. Next, we consider an ordered metric space and show that it is always possible to construct a graphical metric space which is not a metric space. Definition 2. Let (X, ) be a partially ordered set and d : X × X → R be a mapping satisfying the following conditions:

Graphical Metric Spaces

(OM1) (OM2) (OM3) (OM4)

269

d (x, y) ≥ 0 for all x, y ∈ X; d (x, y) = 0 if and only if x = y; d (x, y) = d (y, x) for all x, y ∈ X; x  z  y implies d (x, y) ≤ d (x, z) + d (z, y) for all x, y, z ∈ X.

Then the mapping d is called an ordered metric on X and the pair (X, d ) is called an ordered metric space. Remark 1. Let (X, d ) be an ordered metric space and GI be a graph defined by V(GI ) = X and E(GI ) = {(x, y) ∈ X × X : x  y}. Then (X, dGI ) is a graphical metric space, where dGI (x, y) = d (x, y) for all x, y ∈ X. Thus, every ordered metric space is a graphical metric space with this graph GI . The graph GI is called the induced graph (induced by the partial order ). Notice that such a graphical metric need not be a metric, as shown in the following example. Example 2. Let X = N be endowed with the order  : = {(x, y) ∈ N × N : y is divisible by x}. Define a mapping d : X × X → R by

 xy, d (x, y) := 0,

if x = y, if x = y.

Then, d is an ordered metric on X and (X, d ) is an ordered metric space. Let GI be the graph induced by the partial order  . Then, the graphical metric dGI ≡ d is not a metric. Indeed, for x = 1, y = 5, z = 4 we have d (y, z) = 20 > 5 + 4 = d (y, x) + d (x, z). Remark 2. Let G be an undirected graph, V(G) = X. Let Gj , j = 1, 2, . . . , n be the connected components of G such that   ∪nj=1 E(Vj ) . V(G) = ∪nj=1 V(Gj ), E(G) = Δ For x, y ∈ X, let lxy be the length of the shortest path from x to y with lxx = 0. Now define dG : X × X → R by  lxy , if x, y ∈ Vi , i ∈ {1, 2, . . . , n}, dG (x, y) = (8.1) 1, if x ∈ Vi , y ∈ Vj , i, j ∈ {1, 2, . . . , n}, i = j. Then, dG is a graphical metric on X and (X, dG ) is a graphical metric space. The following examples show that such a graphical metric need not be a metric. Example 3. Let G be an undirected graph, where V(G) = {xi , i ∈ {1, 2, . . . , 6}}, E(G) = Δ ∪ {ei , i ∈ {1, 2, 3, 5}} and ei = {xi , xi+1 }. Let G1 , G2 , where V(G1 ) = {x1 , x2 , x3 , x4 }, E(G1 ) = {e1 , e2 , e3 } and V(G2 ) = {x5 , x6 }, E(G2 ) = {e5 }. Then, V(G) = V(G1 ) ∪ V(G2 ) and G1 , G2 are the connected components of G. Now, dG defined by (8.1) is a graphical metric on X. Note that dG is not a metric on X since dG (x1 , x4 ) > dG (x1 , x6 ) + dG (x6 , x4 ). Example 4. Let X = [0, 1] and G be the graph defined by V(G) = X and E(G) = Δ ∪ {(x, y) ∈ X × X : x, y ∈ (0, 1], x ≤ y}.

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Define a mapping dG : X × X → R by ⎧ ⎪ if x = y, ⎨0, dG (x, y) = xy, if x, y ∈ (0, 1], x = y, ⎪ ⎩ x + y, otherwise. Then, dG is a graphical metric on X and (X, dG ) is a graphical metric space. Obviously, dG is not a metric on X. Remark 3. The reachability relationship (i.e., Given a graph G and two vertices x and y, determine if there is a path from x to y) in any directed graph containing cycles gives rise to a reflexive transitive relation P, which is not a partial order. Clearly, P is a partial order if the directed graph is acyclic. Example 5. Let Q+ and I+ be the set of nonnegative rational and irrational numbers, respectively. Let X = R and G be the graph defined by V(G) = X and E(G) = Δ ∪ [Q+ × {0}] ∪ [{0} × I+ ]. Define a mapping dG : X × X → R by dG (x, y) = (x − y)2

for all x, y ∈ X.

Then, dG is a graphical metric on X and (X, dG ) is a graphical metric space. Obviously, dG is not a metric on X. Also, graph G cannot be induced by a partial order. Let (X, dG ) be a graphical metric space. An open ball BG (x, ε ) with center x ∈ X and radius ε > 0 is defined by BG (x, ε ) = {y ∈ X : (xPy)G , dG (x, y) < ε }. Since E(G) ⊇ Δ , then we have x ∈ BG (x, ε ) and so BG (x, ε ) = 0/ for all x ∈ X and ε > 0. The collection B = {BG (x, ε ) : x ∈ X, ε > 0} is a neighbourhood system for the topology τG on X induced by the graphical metric dG . Explicitly, a subset U of X is called open if for every x ∈ U there exists an ε > 0 such that BG (x, ε ) ⊂ U. Of course, a subset C of X is called closed if its complement X \ C is open. Lemma 1. Every open ball in X is an open set. Proof. Let y ∈ BG (x, ε ) for some x ∈ X and ε > 0. Then, let δ = ε − dG (x, y) > 0 and z ∈ BG (y, δ ). By definition, we have (xPy)G and (yPz)G so that (xPz)G . Now, by (GM4), we write dG (z, x) ≤ dG (z, y) + dG (y, x) < δ + dG (y, x) = ε − dG (y, x) + dG (y, x) = ε . We conclude that BG (y, δ ) ⊂ BG (x, ε ).



We next define the concepts of convergence, Cauchy sequence and completeness in a graphical metric space. Definition 3. Let (X, dG ) be a graphical metric space and {xn } be a sequence in X. Then, {xn } is said to be convergent and converges to x ∈ X if, given ε > 0, there exists n0 ∈ N such that dG (xn , x) < ε for all n > n0 . In this case, we say that x is a limit of the sequence {xn }. It is clear that the sequence {xn } is convergent and converges to x, if and only if lim dG (xn , x) = 0. n→∞

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Remark 4. The limit of a sequence in  a graphical metric space may not be unique. To see 1 this, let X = 2N ∪ {0}, where 2N = : n ∈ N and G be the graph with V(G) = X and 2n E(G) = {(x, y) ∈ X × X : y ≤ x}. Define a mapping dG : X × X → R by ⎧ ⎪ if x = y, ⎨0, dG (x, y) = xy, if x, y ∈ 2N , x = y, ⎪ ⎩ 1/2, otherwise. Then dG is a graphical metric on X and (X, dG ) is a graphical metric space. Consider the 1 sequence {xn } in X, where xn = n for all n ∈ N, then for any fixed k ∈ N we have 2

 1 1 1 dG , k = n+k → 0 as n → ∞. n 2 2 2  1 1 Therefore the sequence converges to k for every fixed k ∈ N. 2n 2 We now show that the graphical metric topology τG is T1 but not generally Hausdorff. Lemma 2. Let (X, dG ) be a graphical metric space with induced graphical metric topology τG . Then τG is T1 but not generally Hausdorff, that is, T2 . Proof. We shall show that for every x ∈ X the singleton set {x} is a closed subset of X, that is, the set X \ {x} is an open subset of X. Suppose, y ∈ X \ {x}, then y = x, that is, dG (x, y) > 0. > 0. Then we note that x ∈ / BG (y, ε ), otherwise dG (x, y) < ε = dG (x,y) , which Now, let ε = dG (x,y) 2 2 is a contradiction. This shows that BG (y, ε ) ⊂ X \ {x}, and so, X \ {x} is open. By Remark 4, we deduce that a graphical metric space is not generally Hausdorff.

1 is a limit of Remark 5. Let (X, dG ) be the graphical metric space in Remark 4, then 2 



 1 1 1 1 1 , d , the sequence , but for any k ∈ N, k > 1 we have lim = 0 = d . G G n→∞ 2n 2n 2k 2 2k Therefore, a graphical metric need not be continuous. Definition 4. Let (X, dG ) be a graphical metric space and {xn } be a sequence in X. Then, {xn } is said to be a Cauchy sequence if, given ε > 0, there exists n0 ∈ N such that dG (xn , xm ) < ε for all n, m > n0 . It is clear that the sequence {xn } is a Cauchy sequence, if and only if lim dG (xn , xm ) = 0. n,m→∞

Definition 5. A graphical metric space (X, dG ) is said to be complete if every Cauchy sequence in X converges in X. Let G be a graph such that V(G ) = X; then (X, dG ) is said to be G -complete if every G -termwise connected Cauchy sequence in X converges in X.

8.3 Fixed Point Theorems in Graphical Metric Spaces In this section, we establish some fixed point results in graphical metric spaces. Before that, we discuss some definitions and properties which will be used in the following.

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Definition 6. Let (X, dG ) be a graphical metric space, T : X → X a mapping and G be a subgraph of G such that E(G ) ⊇ . Then, T is called a (G, G )-graphical contraction on X if the following conditions hold: (GC1) T preserves edges in E(G ), that is, (x, y) ∈ E(G ) implies (Tx, Ty) ∈ E(G ); (GC2) there exists λ ∈ [0, 1) such that dG (Tx, Ty) ≤ λdG (x, y), for all x, y ∈ X with (x, y) ∈ E(G ). Note that we may treat G as a weighted graph by assigning to each edge the graphical distance between its vertices, and a graphical contraction decreases the weights of edges of G by the factor λ ∈ [0, 1). A sequence {xn } with initial value x0 ∈ X is said to be a TPicard sequence (or Picard sequence generated by T) if, xn = Txn−1 for all n ∈ N. In further discussion, we always assume that G is a subgraph of G such that E(G ) ⊇ . Example 6. Let (X, d) be a metric space and G0 be a graph such that V(G0 ) = X. Then, every G0 -contraction (in the sense of Jachymski [3]) on X, is a (G, G )-graphical contraction with E(G) = X × X and G = G0 . Remark 6. Let (X, d ) be an ordered metric space and T : X → X be a mapping and  be a suborder of  . Then, T is said to be nondecreasing with respect to  if x  y implies Tx  Ty. Assume that T is a nondecreasing mapping with respect to  and there exists λ ∈ [0, 1) such that d (Tx, Ty) ≤ λd (x, y), for all x, y ∈ X with x  y. Then, T is called an (,  )-ordered contraction on the ordered metric space (X, d ). An (,  )-ordered contraction on an ordered metric space is a (GI , GI )-graphical contraction on the graphical metric space (X, dGI ), where GI and GI are the graphs induced by  and  respectively. Note that the (,  )-ordered contractions are an extension of the mappings considered by Ran and Reurings [1]. The following theorem provides sufficient conditions for the convergence of a Picard sequence generated by a (G, G )-graphical contraction on a G -complete graphical metric space. Theorem 1. Let (X, dG ) be a G -complete graphical metric space and T : X → X be a (G, G )graphical contraction. Suppose that the following conditions hold: 1. 2.

there exists x0 ∈ X such that Tx0 ∈ [x0 ]lG for some l ∈ N; if a G -termwise connected T-Picard sequence {xn } converges in X, then there exists a limit z ∈ X of {xn } and n0 ∈ N such that (xn , z) ∈ E(G ) or (z, xn ) ∈ E(G ) for all n > n0 .

Then, there exists x∗ ∈ X such that the T-Picard sequence {xn }, with initial value x0 ∈ X, is G termwise connected and converges to both x∗ and Tx∗ . Proof. Let x0 ∈ X be such that Tx0 ∈ [x0 ]lG for some l ∈ N, and {xn } be the T-Picard sequence with initial value x0 . Then, there exists a path {yi }li=0 such that x0 = y0 , Tx0 = yl and (yi−1 , yi ) ∈ E(G ) for i = 1, 2, . . . , l. Since T is a (G, G )-graphical contraction, by (GC1), (Tyi−1 , Tyi ) ∈ E(G )

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for i = 1, 2, . . . , l. Therefore, {Tyi }li=0 is a path from Ty0 = Tx0 = x1 to Tyl = T2 x0 = x2 of length l and so x2 ∈ [x1 ]lG . Continuing this process, we obtain that {Tn yi }li=0 is a path from Tn y0 = Tn x0 = xn to Tn yl = Tn Tx0 = xn+1 of length l, and so, xn+1 ∈ [xn ]lG for all n ∈ N. Thus {xn } is a G -termwise connected sequence. Since, (Tn yi−1 , Tn yi ) ∈ E(G ) for i = 1, 2, . . . , l and n ∈ N, by (GC2) we have dG (Tn yi−1 , Tn yi ) ≤ λdG (Tn−1 yi−1 , Tn−1 yi ) ≤ · · · ≤ λn dG (yi−1 , yi ).

(8.2)

Since, G is a subgraph of G and the sequence {xn } is a G -termwise connected sequence, for any n ∈ N we obtain from (GM4) and (8.2) that dG (xn , xn+1 ) = dG (Tn x0 , Tn+1 x0 ) = dG (Tn y0 , Tn yl ) l

l

i=1

i=1

≤ ∑ dG (Tn yi−1 , Tn yi ) ≤ ∑ λn dG (yi−1 , yi ) = λ Dl , n

where Dl = ∑li=1 dG (yi−1 , yi ). Again, since the sequence {xn } is a G -termwise connected sequence, therefore for n, m ∈ N with m > n we have dG (xn , xm ) ≤

m−1

m−1

∑ d (xi , xi+1 ) ≤ ∑ λi Dl G

i=n



= λn

m−1

∑ λi−n

i=n

 Dl

i=n

≤

λn Dl . 1−λ

Since λ ∈ [0, 1), we obtain lim dG (xn , xm ) = 0. Therefore, {xn } is a Cauchy sequence in X. By n,m→∞

G -completeness of X, the sequence {xn } converges in X and by condition (B), there exists x∗ ∈ X, n0 ∈ N such that (xn , x∗ ) ∈ E(G ) or (x∗ , xn ) ∈ E(G ) for all n > n0 and lim dG (xn , x∗ ) = 0.

n→∞

Thus, the sequence {xn } converges to x∗ ∈ X. Now, if (xn , x∗ ) ∈ E(G ) for all n > n0 , using (GC2) we obtain dG (xn+1 , Tx∗ ) = dG (Txn , Tx∗ ) ≤ λdG (xn , x∗ ) for all n > n0 . Since lim dG (xn , x∗ ) = 0, we obtain n→∞

lim dG (xn+1 , Tx∗ ) = 0.

n→∞

A similar result holds if (x∗ , xn ) ∈ E(G ) and hence, the sequence {xn } converges to both x∗ and Tx∗ .

Remark 7. The above theorem ensures only the convergence of a Picard sequence generated by a (G, G )-graphical contraction on a G -complete graphical metric space. The following example shows that one should not recognize this theorem as an existence theorem in a G -complete graphical metric space.

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1 : n ∈ N and G = G be the graph defined by 2n V(G) = X and E(G) = Δ ∪ {(x, y) ∈ 2N × 2N : y ≤ x} . Define a mapping dG : X × X → R by ⎧ ⎪ if x = y, ⎨0, dG (x, y) = xy, if x, y ∈ 2N , x = y, ⎪ ⎩ 1/2, otherwise.

Example 7. Let X = 2N ∪ {0}, where 2N =

Then, dG is a graphical metric on X and (X, dG ) is a G-complete graphical metric space. Note that (X, d) is not a metric space. Define a mapping T : X → X by  x , if x ∈ 2N , Tx = 21 , if x = 0. 2 Notice that T is a graphical contraction with λ = 1/4. For each x ∈ 2N we have (x, Tx) ∈ E(G), that is, Tx ∈ [x]1G . Also, any G-termwise connected and convergent sequence in X is either a constant sequence  or a monotonic decreasing subsequence (with respect to the usual order) 1 of the sequence , and therefore has at least one limit z ∈ X such that the condition (B) 2n of Theorem 1 holds true; but T has no fixed point in X. As we know, the limit of a convergent sequence in graphical metric spaces may not be unique. Therefore we define the following property: Definition 7. Let (X, dG ) be a graphical metric space and T : X → X be a mapping. Then, the quadruple (X, dG , G , T) is said to have the property (S) if: whenever a G -termwise connected T-Picard sequence {xn } has two limits x∗ and y∗ , where x∗ ∈ X, y∗ ∈ T(X), then x∗ = y∗ .

(S)

We denote the set of all fixed points of T by Fix(T), and use the notation XT = {x ∈ X : (x, Tx) ∈ E(G )}. Remark 8. If we choose E(G) = X × X, then it is easy to see that the quadruple (X, dG , G , T) has the property (S) for arbitrary subgraph G . Example 8. Let X, G and dG as in Example 4. Consider the subgraph G of G defined by V(G ) = X and E(G ) = Δ ∪ {(x, y) ∈ X × X : x, y ∈ (0, 1), x ≤ y}. Define a mapping T : X → X by  x, if x ∈ Q ∩ [0, 1], Tx = 1, otherwise. Then, the quadruple (X, dG , G , T) has the property (S). In the next theorem, we give a sufficient condition for the existence of a fixed point of a (G, G )-graphical contraction. Theorem 2. Let (X, dG ) be a G -complete graphical metric space and T : X → X be a (G, G )graphical contraction. Suppose that the following conditions hold:

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275

there exists x0 ∈ X such that Tx0 ∈ [x0 ]lG for some l ∈ N; if a G -termwise connected T-Picard sequence {xn } converges in X, then there exists a limit z ∈ X of {xn } and n0 ∈ N such that (xn , z) ∈ E(G ) or (z, xn ) ∈ E(G ) for all n > n0 .

Then, there exists x∗ ∈ X such that the T-Picard sequence {xn } with initial value x0 ∈ X, is G termwise connected and converges to both x∗ and Tx∗ . In addition, if the quadruple (X, dG , G , T) has the property (S), then T has a fixed point in X. Proof. By Theorem 1, the T-Picard sequence {xn } with initial value x0 converges to both x∗ and Tx∗ . Since x∗ ∈ X and Tx∗ ∈ T(X), therefore by the property (S) we must have Tx∗ = x∗ . Thus, x∗ is a fixed point of T.

Remark 9. In the above theorem, we use an additional property (S) (which is not used in the usual metric cases; see, for example [3]) for the existence of a fixed point of T. Notice that, in Example 7 all the conditions of Theorem 2, except the property (S), are satisfied but Fix(T) = 0. / Therefore, in Theorem 2, the property (S) is not superfluous. The following example shows that, in Theorem 2, the fixed point of mapping T may not be unique and a (G, G )-graphical contraction in a graphical metric space, may not be a G-contraction (in the sense of Jachymski [3]) in usual metric space. Therefore the generalizations of fixed point theorems in graphical metric spaces are proper generalizations. Example 9. Let X and G as in Example 4. Assume G = G and define a mapping dG : X × X → R by ⎧ 0, if x = y, ⎪ ⎪ ⎨ 1 , if x, y ∈ (0, 1], x = y, dG (x, y) = ln ⎪ xy ⎪ ⎩ 1, otherwise. Then, dG is a graphical metric on X and (X, dG ) is a G-complete graphical metric space. Obviously, dG is not a metric on X. Define a mapping T : X → X by √ Tx = x for all x ∈ X. Then, T is a graphical contraction with λ = 1/2. Notice that, all the conditions of Theorem 2 are satisfied and T has two fixed points, namely Fix(T) = {0, 1}. On the other hand, one can see that the results from usual metric spaces are not applicable. In particular, the mapping T is not a G-contraction (in the sense of Jachymski [3]) with respect to the usual metric defined on X. In the next theorem, we give necessary conditions for uniqueness of a fixed point of a (G, G )-graphical contraction. Theorem 3. Let (X, dG ) be a G -complete graphical metric space and T : X → X be a (G, G )graphical contraction. Suppose that all the conditions of Theorem 2 are satisfied, then T has a fixed point. In addition, if for all x, y ∈ XT we have (xPy)G or (yPx)G , then the fixed point of T is unique.

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Proof. The existence of a fixed point of T follows from Theorem 2. Suppose that the given condition is satisfied in XT . Let x∗ , y∗ ∈ Fix(T) be two distinct fixed points of T. Since, E(G ) contains all the loops, therefore Fix(T) ⊆ XT and so x∗ , y∗ ∈ XT . By the assumption, we have (x∗ Py∗ )G or (y∗ Px∗ )G . Suppose that (x∗ Py∗ )G (the proof for the case (y∗ Px∗ )G is similar), that is, there exists a sequence {xi }li=0 , x0 = x∗ , xl = y∗ and (xi , xi+1 ) ∈ E(G ) for i = 0, 1, . . . , l − 1. Since T is a (G, G )-graphical contraction, by successive use of (GC1) we have (Tn xi , Tn xi+1 ) ∈ E(G ) for i = 0, 1, . . . , l − 1 and for all n ∈ N. Therefore, by (GC2) we obtain dG (Tn xi , Tn xi+1 ) ≤ λdG (Tn−1 xi , Tn−1 xi+1 ) ≤ · · · ≤ λn dG (xi , xi+1 ) for i = 0, 1, . . . , l − 1 and for all n ∈ N. Now by (GM4) we have l−1

l−1

i=0

i=0

dG (Tn x∗ , Tn y∗ ) ≤ ∑ dG (Tn xi , Tn xi+1 ) ≤ λn ∑ dG (xi , xi+1 ). Since x∗ , y∗ ∈ Fix(T), we have Tn x∗ = x∗ , Tn y∗ = y∗ ; therefore letting n → ∞, it follows from the above inequality that dG (x∗ , y∗ ) = 0, that is, x∗ = y∗ . Thus, the fixed point of T is unique.

In the next theorem, we assume that the graph G associated with the graphical metric space under consideration is weighted, and the weight of any edge is equal to the graphical distance between the vertices of that edge. If x ∈ XT , then the edge (x, Tx) is said to be the corresponding edge of x. Theorem 4. Let (X, dG ) be a graphical metric space and T : X → X be a (G, G )-graphical contraction. Suppose that XT = 0/ and there exists a point in XT such that its corresponding edge is with minimum weight in XT . Then, T has a fixed point in X. In addition, if for all x, y ∈ XT we have (xPy)G or (yPx)G , then the fixed point of T is unique. Proof. By hypothesis, XT = 0/ and x∗ ∈ XT are such that the corresponding edge of x∗ has a minimum weight in XT , that is, dG (x∗ , Tx∗ ) ≤ dG (x, Tx), for all x ∈ XT .

(8.3)

We shall show that x∗ is a fixed point of T. Since, x∗ ∈ XT we have (x∗ , Tx∗ ) ∈ E(G ), and T is a (G, G )-graphical contraction by (GC1) we have (Tx∗ , TTx∗ ) ∈ E(G ), that is, Tx∗ ∈ XT . Let wc (x) = dG (x, Tx), for all x ∈ X, be the weight of the corresponding edge of x and wc (x∗ ) = 0, then by (8.3) we get 0 < wc (x∗ ) ≤ wc (x) for all x ∈ XT . Now by (GC2) we have wc (Tx∗ ) = dG (Tx∗ , TTx∗ ) ≤ λdG (x∗ , Tx∗ ) = λwc (x∗ ) < w(x∗ ), which is a contradiction since Tx∗ ∈ XT . This contradiction shows that wc (x∗ ) = 0, that is, dG (x∗ , Tx∗ ) = 0 or Tx∗ = x∗ . Therefore, x∗ is a fixed point of T. Uniqueness of the fixed point is obtained by following similar arguments as given in Theorem 3.

We next derive several known results as consequences of our main results. Corollary 1 (see [5], Banach). Let (X, d) be a complete metric space and T : X → X be a contraction mapping. Then, T has a unique fixed point x∗ ∈ X.

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Proof. Define the graph G = G by V(G) = X and E(G) = X × X; then it is easy to see that all the conditions of Theorem 3 are satisfied and so the conclusion follows.

Let (X, ) be a partially ordered set; then A ⊆ X is called well-ordered if, for all x, y ∈ A, either x  y or y  x. The following corollary is a slight improvement of the result of Ran and Reurings [1]. Corollary 2. Let (X, ) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Suppose that T : X → X is a nondecreasing mapping with respect to  and the following conditions hold: 1.

there exists λ ∈ [0, 1) such that d(Tx, Ty) ≤ λd(x, y), for all x, y ∈ X with x  y;

2. 3.

there exists x0 ∈ X such that x0  Tx0 ; if {xn } is a sequence such that xn  xn+1 for all n ∈ N and converges to z ∈ X, then there exists n0 ∈ N such that xn  z or z  xn for all n > n0 .

Then, T has a fixed point in X. Furthermore, the fixed point of T is unique, if the set {x : x  Tx} is well-ordered. Proof. Let G = X × X and G = GI , that is, the graph induced by  . Then, it is easy to see that all the conditions of Theorem 3 are satisfied and so the conclusion follows.

Kirk et al. [9] used the cyclic contractive condition and obtained fixed point results for mappings satisfying such a type of condition. Let (X, d) be a metric space. Let A and B be two nonempty closed subsets of X such that the mapping T : A ∪ B → A ∪ B satisfies the following conditions: 1. T(A) ⊆ B and T(B) ⊆ A; 2. d(Tx, Ty) ≤ λd(x, y), for all x ∈ A and y ∈ B, where λ ∈ [0, 1). Then, T is called a cyclic contraction. Notice that, if x, y ∈ Fix(T), then by (a) x, y ∈ A ∩ B. Corollary 3 (see [9], Kirk et al.). Let A and B be two nonempty closed subsets of a complete metric space X, and suppose that T : A ∪ B → A ∪ B is a cyclic contraction. Then, T has a unique fixed point in A ∩ B. Proof. Let us define G = X × X, and the subgraph G by V(G ) = A ∪ B, E(G ) =  ∪ {(x, y) : (x, y) ∈ [A × T(A)] ∪ [B × T(B)]}. Then, by the definition, T is a (G, G )-graphical contraction on A ∪ B. Since, A and B are nonempty, therefore there exists x0 ∈ A ∪ B such that (x0 , Tx0 ) ∈ E(G ), that is, Tx0 ∈ [x0 ]1G . Notice that any G -termwise connected T-Picard sequence is a 2-cyclic sequence (see [7]), therefore, if a G -termwise connected T-Picard sequence {xn } converges to some z ∈ A ∪ B, then z ∈ A ∩ B, that is, (xn , z) ∈ E(G ) for all n ∈ N. All other conditions of Theorem 2 are satisfied, so T has a fixed point x∗ ∈ A ∩ B. Furthermore, if x∗ , y∗ ∈ Fix(T), then x∗ , y∗ ∈ A ∩ B, that is, (x∗ , y∗ ) ∈ E(G ). Thus, by (GC2) we have x∗ = y∗ .



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Let (X, d) be a metric space and ε > 0. Then, (X, d) is called ε -chainable if, given x, y ∈ X, there exists l ∈ N and a sequence {xn }li=0 such that x0 = x, xl = y and d(xi , xi−1 ) < ε for i = 1, 2, . . . , l. Next, we obtain the result of Edelstein [4] in ε -chainable metric spaces. Corollary 4 (see [4], Edelstein). Let (X, d) be a complete and ε -chainable metric space and T : X → X be such that there exists λ ∈ [0, 1) and d(Tx, Ty) ≤ λd(x, y), for all x, y ∈ X with d(x, y) < ε .

(8.4)

Then, T has a unique fixed point in X. Proof. Let us define G = X × X, and the subgraph G by V(G ) = X, E(G ) = {(x, y) ∈ X × X : d(x, y) < ε }. If (x, y) ∈ E(G ), then the definition of G and (8.4) imply that (Tx, Ty) ∈ E(G ) and T is a (G, G )-graphical contraction. Also, ε -chainability of (X, d) implies that Tx0 ∈ [x0 ]lG for some l ∈ N and for all x0 ∈ X. Also, if {xn } is a sequence converging to some z ∈ X, then we can choose n0 ∈ N such that d(xn , z) < ε for all n > n0 . All the other conditions of Theorem 3 are satisfied and hence T has a unique fixed point in X.



8.4 An Application to Integral Equations Let I > 0 and X = C([0, I], R) be the set of all real continuous functions on [0, I]. We give a typical application of fixed point theory to the study of integral equations on X. We show that, under some conditions the existence of a lower or upper solution of an integral equation ensures the solution of the integral equation.  Let B = x ∈ X : 0 < inf x(t) and x(t) ≤ 1, t ∈ [0, I] and define the graphs G, G by G = G , t∈[0,I]

V(G) = X and E(G) = Δ ∪ {(x, y) ∈ X × X : x, y ∈ B, x(t) ≤ y(t), for all t ∈ [0, I]}. Now, consider the graphical metric dG : X × X → R given by ⎧ ⎪ 0,  ⎪  if x = y, ⎪ ⎨ 1 , if x, y ∈ B, x = y, dG (x, y) = sup ln x(t)y(t) ⎪ t∈[0,I] ⎪ ⎪ ⎩ 1, otherwise, so that (X, dG ) is a G -complete graphical metric space. We consider the following integral equation: x(t) =



I 0

S(t, u)f (u, x(u))du,

(8.5)

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where f : [0, I] × R → R and S : [0, I] × [0, I] → [0, ∞) are continuous functions. A function α ∈ C([0, I], R) is said to be a lower solution of equation (8.5) if

α (t) ≤

 0

I

S(t, u)f (u, α (u))du,

t ∈ [0, I].

Obviously, an upper solution of (8.5) satisfies the reverse inequality. We shall show that the existence of a lower solution of (8.5) ensures the existence of a solution of (8.5). We consider the operator T : X → X defined by T(x)(t) =

 0

I

S(t, u)f (u, x(u))du

(8.6)

and give sufficient conditions for the existence of a fixed point of (8.6) in X. Clearly, such a fixed point is a solution of the integral equation (8.5). Theorem 5. Suppose that the following conditions are satisfied: (i)

f (u, ·) : R → R is increasing on (0, 1], for every u, t ∈ [0, I]. Moreover inf S(t, u) > 0 and S(t, u)f (u, 1) ≤ I−1 ;

t∈[0,I]

(ii) there exist λ ∈ (0, 1) and μ ∈ [1, ∞) such that, for x, y ∈ X with (x, y) ∈ E(G), we have for every u, v ∈ [0, I] f (u, x(u))f (v, y(v)) ≥ [x(u)y(v)]λ and

 I 0

I 0

S(t, u)S(t, v)dudv ≥ μ ,

for all t ∈ [0, I].

Then the existence of a lower solution of (8.5) in B, ensures the existence of solution of (8.5) in X. Proof. Notice that the operator T is well-defined. Also, by condition (ii), for x, y ∈ X with (x, y) ∈ E(G) and t ∈ [0, I], we obtain  

 1 1 ln = ln  I  I T(x)(t)T(y)(t) 0 0 S(t, u)S(t, v)f (u, x(u))f (v, y(v))dudv   1 ≤ ln  I  I λ 0 0 S(t, u)S(t, v)[x(u)y(v)] dudv ⎞ ⎛ ⎜ ≤ ln ⎜ ⎝ ⎛ ≤ ln ⎝

 inf [x(t)y(t)]λ

t∈[0,I]

1 II 0

⎞

1 ⎠ μ inf [x(t)y(t)]λ t∈[0,I]

≤ λdG (x(t), y(t)).

0

S(t, u)S(t, v)dudv

⎟ ⎟ ⎠

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Then, we have dG (T(x), T(y)) = sup t∈[0,I]

 ln

1 T(x)(t)T(y)(t)

 ≤ λdG (x(t), y(t)).

Further, for x, y ∈ X with (x, y) ∈ E(G), we have x, y ∈ B and x(t) ≤ y(t) for all t ∈ [0, I] and, by condition (i), we get inf T(x)(t) > 0 and t∈[0,I]

T(x)(t) = and T(x)(t) =

 0

I



I 0

S(t, u)f (u, x(u))du ≤

S(t, u)f (u, x(u))du ≤

 0

I

 0

I

S(t, u)f (u, 1)du ≤ 1

S(t, u)f (u, y(u))du = T(y)(t).

Consequently, the existence of a lower solution of (8.5), say α ∈ B, implies that property (A) of Theorem 2 holds true, that is, T(α ) ∈ [α ]1G . It is easy to see that the property (B) of Theorem 2 holds true and the quadruple (X, dG , G , T) has the property (S). Thus, all the conditions of Theorem 2 are satisfied and hence the operator T has a fixed point, that is a solution of the integral equation (8.5) in X.

By analogous reasoning, one can establish Theorem 5 in the case of an upper solution of integral equation (8.5); in this case one considers the following set E(G) = {(x, y) ∈ X × X : x, y ∈ B, x(t) ≥ y(t), for all t ∈ [0, I]}, retaining the rest as above. To avoid repetition, we omit the details.

8.5 Conclusion In this chapter, an approach to fixed point theory via graphical metric is discussed. In view of the fact that the class of contractions in graphical metric spaces is larger than that in metric spaces, we see that the fixed point results in graphical metric spaces can be applied to a larger class of existence problems which use the fixed point methods, e.g., existence of a solution of an integral equation. As well as, apart from the topology of usual metric spaces, the topology generated by graphical metric spaces is non-Hausdorff which can be considered a useful tool in theories where non-Hausdorffness occurs, e.g., in algebraic geometry, in representation theory, in domain theory, in computer applications etc.

References 1. 2.

A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc., 132, (2004), 1435–1443. Fréchet M., Sur quelques points du calcul fonctionnel, Rendiconti Circolo Mat. Palermo, 22 (1906), 1–74.

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3. 4. 5. 6. 7. 8. 9.

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J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359–1373. M. Edelstein, An extension of Banach’s contraction principle, Proc. Amer. Math. Soc., 12 (1961), 7–10. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181. S. Shukla, H-P A. Künzi, Some fixed point theorems for multi-valued mappings in graphical metric spaces, Mathematica Slovaca, 70(3), 2020, 719–732. S. Shukla, M. Abbas, Fixed point results of cyclic contractions in product spaces, Carpathian J. Math., 31 (2015), 119–126. S. Shukla, S. Radenovi´c, C. Vetro, Graphical metric space: a generalized setting in fixed point theory, RACSAM (2016). doi:10.1007/s13398-016-0316-0 W.A. Kirk, P.S. Srinivasan, and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4 (2003), 79–89.

9 Fixed Point Theory in Partial Metric Spaces Dhananjay Gopal and Shilpi Jain

CONTENTS 9.1 9.2 9.3 9.4 9.5 9.6 9.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Banach Contraction and Generalized Contraction on Partial Metric Spaces . . . . . . 285 Caristi Fixed Point Theorem in Partial Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Rus Problem of Metric Fixed Point Theory in the Case of Partial Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Completion of Partial Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Ekeland’s Variational Principle in Partial Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

9.1 Introduction In 1992 Matthews [11] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. The main difference comparing to the standard metric is that the self-distance of an arbitrary point need not be equal to zero. This notion was originally motivated by the experience of computer science. The authors showed how a mathematics of nonzero self-distance for metric spaces has been established, and leads to interesting research into the foundations of topology [5], [9]. Many authors studied partial metric spaces and their topological properties, as well as fixed point results We begin with the definition of a partial metric space by Matthews [11] as follows: Definition 1. [11] Let X be a nonempty set and p : X × X −→ [0, +∞) be a function such that for all x, y, z ∈ X, (i) (ii) (iii) (iv)

0 ≤ p(x, x) ≤ p(x, y). If p(x, x) = p(x, y) = p(y, y) then x = y. p(x, y) = p(y, x). p(x, z) ≤ p(x, y) + p(y, z) − p(y, y).

Then p is called a partial metric and the pair (X, p) is called a partial metric space. 283

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We mostly abbreviate a partial metric space as PMS, but sometimes we may even abbreviate ‘partial metric space’ to PMS if it is clear from the context. Apparently, every matrix on X is a partial metric on X and a partial p is a metric if and only if p(x, x) = 0 ∀ x ∈ X. To simplify the notation, let us set p˜ (x, y) = p(x, y) − p(x, x). Similarly, in the definition of a ball in a metric space, we use Bp (x, ε ) = {y ∈ X : p˜ (x, y) < ε } to denote an open p-ball for all x ∈ X and ε > 0. Analogously, let Bp [x, ε ] = {y ∈ X : p˜ (x, y) ≤ ε } be called a closed p-ball. Each partial metric p on X generates a topology Tp on X with a base consisting of open p-balls Bp (x, ε ) = {y ∈ X : p˜ (x, y) < ε }. Since in (X, p), p(x, x) is not necessarily zero, and y ∈ Bp (x, ε ) does not imply x ∈ Bp (y, ε ). Then, in general, Tp is an asymmetric topology on X. Example 1. [13] Let X = [0, ∞) and p : X × X → [0, ∞) be defined by p(x, y) = max{x, y} ∀ x, y ∈ X. Then (X, p) is a partial metric space. Example 2. Let L be the collection of nonempty closed bounded intervals in R. For all [a, b], [c, d] ∈ L , let p([a, b], [c, d]) = max{b, d} − min{a, c}. Then (L , p) is a partial metric space. Example 3. [8] Let the set of rational numbers be written as Q = {x1 , x2 , . . .}. Define, p : R × R → R+ by ⎧ ⎪ 1, if x = y ∈ R − Q ⎪ ⎪ ⎪ 3 ⎪ ⎪2, if x = y ∈ R − Q ⎨ p(x, y) = 13 , (9.1) if x = y ∈ Q ⎪ ⎪ 1 1 ⎪ 1 + m + n , if x = xm , y = xn and m = n; m, n ∈ N ⎪ ⎪ ⎪ ⎩1 + 1 , if n Then (L , p) is a partial metric space. Definition 2 ([5], Definitions 4–6 & [12], Definition 2.1). Let (X, p) be a partial metric space. (i)

A sequence {xn } is called convergent to x in (X, p), written as lim xn = x, if n→∞

lim p(xn , x) = lim p(xn , xn ) = p(x, x).

n→∞

n→∞

(ii) A sequence {xn } is called Cauchy if lim p(xn , xm ) exists and is finite. n,m→∞

(iii) (X, p) is called complete if each Cauchy sequence {xn } converges to some x and lim p(xn , xm ) = p(x, x).

m,n→∞

(iv) A sequence {xn } is called 0-Cauchy if lim p(xn , xm ) = 0. n,m→∞

(v) (X, p) is called 0-complete if each 0-Cauchy sequence converges to some x and p(x, x) = 0.

Some other pertinent facts about partial metrics:

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(i)

285

If p is partial metric on X, then the function ps : X × X → R+ given by ps (x, y) = 2p(x, y) − p(x, x) − p(y, y) x, y ∈ X

is a metric on X. (ii) A sequence {xn } is a Cauchy sequence in (X, p) iff it is a Cauchy sequence in the metric space (X, ps ). (iii) A partial metric space (X, p) is complete iff the metric space (X, ps ) is complete. Moreover, given a sequence {xm } in (X, p) and x ∈ X one has limn→∞ ps (x, xn ) iff p(x, x) = limn→∞ p(x, xn ) = limn→∞ p(xn , xm ).

9.2 Banach Contraction and Generalized Contraction on Partial Metric Spaces Definition 3. [2] A mapping f : (X, p) → (X, p) is called a Banach contraction if there exists some k ∈ [0, 1) such that p(fx, fy) ≤ k(p(x, y)) Theorem 1. [10] Let (X, p) be a complete partial metric space and suppose for some k ∈ [0, 1), f : X → X satisfies p(f (x), f (y)) ≤ kp(x, y) for all x, y ∈ X. Then there exists a unique x∗ ∈ X such that x∗ = f (x∗ ) and p(x∗ , x∗ ) = 0. Proof. Suppose u ∈ X. Then for each n, j ∈ N, p(f n+j+1 (u), f n (u)) ≤ p(f n+j+1 (u), f n+j (u)) + p(f n+j (u), f n (u)) − p(f n+j (u), f n+j (u)) ≤ kn+j p(f (u), u) + p(f n+j (u), f n (u)). Thus for each n, j ∈ N p(f n+j+1 (u), f n (u)) ≤ kn+j p(f (u), u) + kn+j−1 p(f (u), u) + p(f n+j−1 (u), f n (u)) .. . ≤ (kn+j + kn+j−1 + · · · + kn )p(f (u), u) + p(f n (u), f n (u)) ≤ (kn+j + kn+j−1 + · · · + kn )p(f (u), u) + kn p(u, u)   1 − kj+1 = kn p(f (u), u) + kn p(u, u) 1−k   1 − kj+1 n =k p(f (u), u) + p(u, u) 1−k   p(f (u), u) = kn + p(u, u) 1−k Therefore, if m > n, we say that p(f m (u), f n (u)) ≤ kn



p(f (u), u) 1−k



+ p(u, u)

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It follows that

lim p(f m (u), f n (u)) = 0

m,n→∞

Thus for all m ∈ N, ps (f m (u), f n (u)) ≤ 2p(f m (u), f n (u)) → 0 as n → ∞ Therefore {f n (u)} is Cauchy sequence in (X, ps ). Since (X, ps ) is complete, so is (X, ps ) and the sequence {f n (u)} converges to some x∗ ∈ X with respect to the metric ps . Therefore p(x∗ , x∗ ) = lim p(f n (u), x∗ ) = lim p(f m (u), f n (u)) = 0. n→∞

m,n→∞

Also p(fx∗ , x∗ ) ≤ p(fx∗ , f n+1 (u)) + p(f n+1 (u), x∗ ) − p(f n+1 (u), f n+1 (u)) ≤ kp(x∗ , f n (u)) + p(f n+1 (u), x∗ ). Letting n → ∞ we see that p(fx∗ , x∗ ) = 0. We now have p(x∗ , x∗ ) = p(fx∗ , x∗ ) = 0 and by (ii) p(fx∗ , fx∗ ) ≤ p(fx∗ , x∗ ) = 0, so by (i) x∗ = fx∗ . Now suppose there exists y∗ ∈ X such that y∗ = fy∗ . Then p(x∗ , y∗ ) = p(fx∗ , fy∗ ) ≤ kp(x∗ , y∗ ). Since k < 1 we conclude p(x∗ , y∗ ) = 0 and so x∗ = y∗ . Thus the fixed point is unique.



In [13] Romaguera established the following fixed point theorem which was an improvement over the results given in [1]. In order to simplify the notion, given a partial metric space (X, p) and f : X → X, we define 1 Pf (x, y) = max{p(x, y), p(x, fx), p(y, fy), [p(x, fy) + p(y, fx)]} ∀ x, y ∈ X 2 Definition 4. A mapping f : (X, p) → (X, p) is called a generalized contraction if there exists φ : [0, ∞) → [0, ∞) such that φ (t) < t for all t > 0, φ (0) = 0 and p(fx, fy) ≤ φ (Pf (x, y)) ∀ x, y ∈ X. In the following the letters N and ω will denote the set of all positive integer numbers and the set of all nonnegative integer numbers, respectively. Lemma 1. [13] Let (X, p) be a partial metric space and f : X → X be a map. Then, for each x ∈ X, we have Pf (x, y) = max{p(x, fx), p(fx, fx2 )}. Proof. let x ∈ X. Then max{p(x, fx), p(fx, f 2 x)} ≤ Pf (x, fx) 1 ≤ max{p(x, fx), p(fx, f 2 x), [p(x, f 2 x) + p(fx, fx)]} 2 1 ≤ max{p(x, fx), p(fx, f 2 x), [p(x, fx) + p(fx, f 2 x)]} 2 = max{p(x, fx), p(fx, f 2 x)} The proof is complete.



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Lemma 2. [13] Let (X, p) be a partial metric space and f : X → X be a map such that p(fx, fy) = φ (Pf (x, y)), for all x, y ∈ X, where φ : [0, ∞) → [0, ∞) is a function such that φ (t) < t for all t > 0. If x ∈ X satisfies that f n x = f n+1 x for all n ∈ ω , then the following hold: (a) Pf (f n x, f n+1 x) = p(f n x, f n+1 x) for all n ∈ ω (b) p(f n x, f n+1 x) ≤ φ (p(f n−1 x, f n x)) < p(f n−1 x, f n x) for all n ∈ N. Proof.

(a) Let x ∈ X be such that f n x = f n+1 x for all n ∈ ω . Then p(f n x, f n+1 x) > 0 for all n ∈ ω . By Lemma 1, Pf (f n x, f n+1 x) = max{p(f n x, f n+1 x), p(f n+1 x, f n+2 x)}. Since

p(f n+1 x, f n+2 x) ≤ φ (pf (f n x, f n+1 x)) < p(f n x, f n+1 x),

it follows that Pf (f n x, f n+1 x)) = p(f n x, f n+1 x) for all n ∈ ω . (b) Taking into account (a), we deduce that p(f n x, f n+1 x) ≤ φ (Pf (f n−1 x, f n x)) = φ (p(f n−1 x, f n x)) < p(f n−1 x, f n x), for all n ∈ N.



Let us recall that a function φ : [0, ∞) → [0, ∞) is upper semicontinuous from the right provided that for each t ≥ 0 and each sequence (tn )n∈N such that tn ≥ t and limn→∞ tn = t, it follows that lim supn→∞ φ (tn ) ≤ φ (t). Theorem 2. [13] Let (X, p) be a complete partial metric space and let f : X → X be a map such that p(fx, fy) ≤ φ (Pf (x, y)), for all x, y ∈ X, where φ : [0, ∞) → [0, ∞) is upper semicontinuous from the right function such that φ (t) < t for all t < 0. Then f has a unique fixed point z ∈ X. Moreover p(z, z) = 0. Proof. Let x ∈ X. If there is n ∈ ω such that f n x = f n+1 x, then f n x is a fixed point of f and the uniqueness of f n x follows as in the last part of the proof below. Hence, we shall assume that f n x = f n+1 x for all n ∈ ω . Put x0 = x and construct the sequence (xn )n∈ω where xn = f n x0 for all n ∈ ω . Thus xn+1 = fxn and p(xn , xn+1 ) > 0 for all n ∈ ω . By Lemma 2 (b), there is c ≥ 0 such that lim p(xn , xn+1 ) = lim φ (p(xn , xn+1 )) = c

n→∞

n→∞

If c > 0, we have c = lim sup φ (p(xn , xn+1 )) ≤ φ (c) < c, n→∞

a contradiction. Therefore limn→∞ p(xn , xn+1 ) = 0. Next we show that limn,m→∞ p(xn , xm ) = 0. This will be done by adapting a technique of Boyd and Wong [3, Theorem 1]. Indeed, assume the contrary. Then there exist ε > 0 and sequences (nk )k∈N , (mk )∈N in N, with mk >

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nk ≥ k, and such that p(xnk , xmk ) ≥ k for all k ∈ N. From the fact that limn→∞ p(xn , xn+1 ) = 0 we can suppose, without loss of generality, that p(xnk , xmk−1 ) < ε . For each k ∈ N we have

ε ≤ p(xnk , xmk ) ≤ p(xnk , xmk−1 ) + p(xmk−1 , xmk ) < ε + p(xmk−1 , xmk ), and, hence, limk→∞ p(xnk , xmk ) = ε . Now let k0 ε N be such that p(xnk+1 , xnk ) < ε and p(xmk+1 , xmk ) < ε for all k ≥ k0 . Then p(xnk , xmk ) ≤ Pf (xnk , xmk ) ≤ p(xnk , xmk ) +

 1

p(xmk , xmk+1 ) + p(xnk+1 , xnk ) , 2

for all k ≥ k0 . So limk→∞ Pf (xnk , xmk ) = ε . Since Pf (xnk , xmk ) ≥ ε for all k ∈ N, and φ is upper semicontinuous from the right, we deduce that lim sup φ (Pf (xnk , xmk )) ≤ φ (ε ). k→∞

On the other hand, for each k ∈ N we have

ε ≤ p(xnk , xmk ) ≤ p(xnk , xnk+1 ) + p(xnk+1 , xmk+1 ) + p(xmk+1 , xmk ) ≤ p(xnk , xnk+1 ) + φ (Pf (xnk , xmk ) + p(xmk+1 , xmk ), so

ε ≤ lim sup φ (Pf (xnk , xmk ) ≤ φ (ε ), k→∞

a contradiction because φ (ε ) < ε . Consequently limn,m→∞ p(xn , xm ) = 0, and, thus, (xn )n∈ω is a Cauchy sequence in the complete partial metric space (X, p). Hence, there is z ∈ X such that lim p(xn , xm ) = lim p(z, xn ) = p(z, z) = 0.

n,m→∞

n→∞

We show that z is a fixed point of f . To this end we first note that p(z, fz) = lim Pf (z, xn ), n→∞

so

lim sup φ (Pf (z, xn )) ≤ φ (p(z, fz)). n→∞

On the other hand, since for each n ∈ ω , p(z, fz) ≤ p(z, xn ) + p(xn , fz), it follows that p(z, fz) ≤ lim sup p(z, xn ) + p(xn , fz) = lim sup p(xn , fz) n→∞

n→∞

≤ lim sup φ (Pf (xn−1 , z)) ≤ φ (p(z, fz)). n→∞

Therefore p(z, fz) = 0 and thus z = fz.

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Finally, let u ∈ X be such that fu = u. Then, p(u, z) = p(fu, fz) ≤ φ (Pf (u, z)) = φ (p(u, z)). Hence p(u, z) = 0, that is, u = z. This concludes the proof.



Corollary 1. [13] Let (X, p) be a complete partial metric space and let f : X → X be a map such that p(fx, fy) ≤ φ (p(x, y)), for all x, y ∈ X, where φ : [0, ∞) → [0, ∞) is an upper semicontinuous from the right function such that φ (t) < t for all t > 0. Then f has a unique fixed point z ∈ X. Moreover p(z, z) = 0. Corollary 2. [13] (Boyd and Wong [3]) Let (X, d) be a complete metric space and let f : X → X be a map such that d(fx, fy) ≤ φ (d(x, y)), for all x, y ∈ X, where φ : [0, ∞) → [0, ∞) is an upper semicontinuous from the right function such that φ (t) < t for all t > 0. Then f has a unique fixed point.

9.3 Caristi Fixed Point Theorem in Partial Metric Spaces In 2010, Romaguara [12] extended the celebrated result of Kirk that a metric space X is complete iff every Caristi self-mapping for X has a point to partial metric space by proposing the following alternative definitions of Caristi mapping. Definition 5. [12] A self-mapping f of a partial metric space (X, p) is called a p-Caristi mapping on X if there is a function φ : X → [0, ∞) which is lower semicontinuous for (X, p) and satisfies p(x, fx) ≤ φ (x) − φ (fx), for all x ∈ X. Definition 6. [12] A self-mapping f of a partial metric space (X, p) is called a ps -Caristi mapping on X if there is a function φ : X → [0, ∞) which is lower semicontinuous for (X, ps ) and satisfies p(x, fx) ≤ φ (x) − φ (fx), for all x ∈ X. Note that every p-Caristi mapping is ps -Caristi but the converse is not true. In a first attempt to generalize Kirk’s characterization of metric completeness to the partial metric framework, one can conjecture that a partial metric space (X, p) is complete if and only if every p-Caristi mapping on X has a fixed point. However, the following example shows that this conjecture is false (see [12]). Example 4. [12] On the set N of natural numbers construct the partial metric p given by

1 1 p(n, m) = max , n m . Lemma 3. [12] Let (X, p) be a partial metric space. Then, for each x ∈ X, the function px : X → [0, ∞) given by px (y) = p(x, y) is lower semicontinuous for (X, ps ).

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Proof. Assume that limn ps (y, yn ) = 0, then px (y) ≤ px (yn ) + p(yn , y) − p(yn , yn ) = px (yn ) + ps (yn , y) − p(yn , y) + p(y, y) This yields lim infn px (yn ) ≥ px (y) because p(y, y) ≤ p(y, yn ).

(9.2) 

Theorem 3. [12] A partial metric space (X, p) is 0-complete if and only if every ps -Caristi mapping f on X has a fixed point. Proof. Suppose that (X, p) is 0-complete and let f be a ps -Caristi mapping on X; then, there is a φ : X → [0, ∞) which is a lower semicontinuous function for (X, ps ) and satisfies p(x, fx) ≤ φ (x) − φ (fx) ∀ x ∈ X

(9.3)

Ax = {y ∈ X : p(x, y) ≤ φ (x) − φ (y)}.

(9.4)

Now, for each x ∈ X define

Observe that Ax = φ because fx ∈ Ax . Moreover Ax is closed in the metric space (X, ps ) since y → p(x, y) + φ (y) is lower semicontinuous for (X, ps ). Fix x0 ∈ X. Take x1 ∈ Ax0 such that φ (x1 ) < infy∈Ax0 φ (y) + 2−1 . Clearly Ax1 ⊆ Ax0 . Hence, for each x ∈ Ax1 we have p(x1 , x) ≤ φ (x1 ) − φ (x) < inf +2−1 − φ (x) y∈Ax1

≤ φ (x) + 2−1 − φ (x) = 2−1 .

(9.5)

Following this process we construct a sequence (xn )n∈ω in X such that its associated sequence (Axn )n∈ω of closed subsets in (X, ps ) satisfies (i) Axn+1 ⊆ Axn , xn+1 ∈ Axn for all n ∈ ω , (ii) p(xn , x) < 2−n for all x ∈ Axn , n ∈ N. Since p(xn , xn ) ≤ p(xn , xn+1 ), and, by (i) and (ii), p(xn , xm ) < 2−n for all m > n, it follows that limn,m p(xn , xm ) = 0, so (xn )n∈ω is a 0-Cauchy sequence in (X, p), and by our hypothesis, there exists z ∈ X such that limn p(z, xn ) = p(z, z) = 0, and thus limn ps (z, xn ) = 0. Therefore z ∈ ∩n∈ω Axn . Finally, we show that z = fz. To this end, we first note that p(xn , fz) ≤ p(xn , z) + p(z, fz) ≤ φ (xn ) − φ (z) + φ (z) − φ (fz),

(9.6)

for all n ∈ ω . Consequently fz ∈ ∩n∈ω Axn , so by (ii), p(xn , fz) < 2−n for all n ∈ N. Since p(z, fz) ≤ p(z, xn ) + p(xn , fz), and limn p(z, xn ) = 0, it follows that p(z, fz) = 0. Hence ps (z, fz) = 0 since ps (z, fz) ≤ 2p(z, fz), so z = fz. Conversely, suppose that there is a 0-Cauchy sequence (xn )n∈ω of distinct points in (X, p) which is not convergent in (X, ps ). Construct a subsequence (yn )n∈ω of (xn )n∈ω such that p(yn , yn+1 ) < 2−(n+1) for all n ∈ ω . Put A = {yn : n ∈ ω }, and define f : X → X by fx = y0 if x ∈ X\A, and fyn = yn+1 for all n ∈ ω . Observe that A is closed in (X, ps ).

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Now define φ : X → [0, ∞) by φ (x) = p(x, y0 ) + 1 if x ∈ X\A, and φ (yn ) = 2−n for all n ∈ ω . Note that φ (yn+1 ) < φ (yn ) for all n ∈ ω and that φ (y0 ) < φ (x) for all x ∈ X\A. From this fact and the preceding lemma we deduce that φ is lower semicontinuous for (X, ps ). Moreover, for each x ∈ X\A we have p(x, fx) = p(x, y0 ) = φ (x) − φ (y0 ) = φ (x) − φ (fx),

(9.7)

and for each yn ∈ A we have p(yn , fyn ) = p(yn , yn+1 ) < 2−(n+1) = φ (yn ) − φ (yn+1 ) = φ (yn ) − φ (fyn ).

(9.8)

Therefore f is a Caristi ps -mapping on X without fixed point, a contradiction. This concludes the proof. 

9.4 Rus Problem of Metric Fixed Point Theory in the Case of Partial Metric Spaces In 2008, Rus [14] investigated the basic problem of metric fixed point theory in the case of partial metric space, existence and uniqueness, the convergence of successive approximation and well-posedness of the fixed point problem. Consequently, he presented some interesting open questions: If T : (X, p) → (X, p) is a generalized contraction, which condition satisfies T with respect to ps ? How to use metric fixed point theorems in a metric space (X, ps ) to give analogous fixed point results in a partial metric space (X, p)? Motivated and inspired by the observations of Rus [14], in 2013, Samet et al. [15], presented fixed point results which subsume many partial metric fixed point theorems (including Matthews[11]) as particular cases. Theorem 4. [15] Let (X, p) be a complete metric space, ϕ : X → [0, ∞) be a lower semicontinuous function and T : X → X be a given mapping. Suppose that for any 0 < a < b < ∞, there exists 0 < γ (a, b) < 1 such that for all x, y ∈ X a ≤ p(x, y) + ϕ (x) + ϕ (y) ≤ b ⇒ p(Tx, Ty) + ϕ (Tx) + ϕ (Ty) ≤ γ (a, b) [p(x, y) + ϕ (x) + ϕ (y)] .

(9.9)

Then T has a unique fixed point x∗ ∈ X. Moreover, we have ϕ (x∗ ) = 0. Theorem 5. [15] Let (X, p) be a complete metric space, ϕ : X → [0, ∞) be a lower semicontinuous function and T : X → X be a given mapping. Suppose that there exists a constant γ ∈ (0, 1) such that for all x, y ∈ X, p(Tx, Ty) + ϕ (Tx) + ϕ (Ty) ≤ γ [p(x, y) + ϕ (x) + ϕ (y)] . Then T has a unique fixed point x∗ ∈ X. Moreover, we have ϕ (x∗ ) = 0.

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Theorem 6. [15] Let (X, p) be a complete metric space, ϕ : X → [0, ∞) be a lower semicontinuous 1 function and T : X → X be a given mapping. Suppose that there exists a constant γ ∈ (0, ) such 2 that for all x, y ∈ X, p(Tx, Ty) + ϕ (Tx) + ϕ (Ty) ≤ γ [p(x, Tx) + p(y, Ty) + ϕ (x) + ϕ (y) + ϕ (Tx) + ϕ (Ty)] . Then T has a unique fixed point x∗ ∈ X. Moreover, we have ϕ (x∗ ) = 0. Theorem 7. [15] Let (X, p) be a complete metric space, ϕ : X → [0, ∞) be a lower semicontinuous function and T : X → X be a given mapping. Suppose that there exist α , β , γ ∈ (0, ∞) such that α + β + γ < 1, p(Tx, Ty) + ϕ (Tx) + ϕ (Ty) ≤ α [p(x, y) + ϕ (x) + ϕ (y)] + β [p(x, Tx) + ϕ (x) + ϕ (Tx)] + γ [p(y, Ty) + ϕ (y) + ϕ (Ty)] . for all x, y ∈ X. Then T has a unique fixed point x∗ ∈ X. Moreover, we have ϕ (x∗ ) = 0. Theorem 8. [15] Let (X, p) be a complete metric space, ϕ : X → [0, ∞) be a lower semicontinuous 1 function and T : X → X be a given mapping. Suppose that there exists a constant k ∈ (0, ) such 2 that for all x, y ∈ X, p(Tx, Ty) + ϕ (Tx) + ϕ (Ty) ≤ k [p(x, Ty) + p(y, Tx) + ϕ (x) + ϕ (y) + ϕ (Tx) + ϕ (Ty)] . Then T has a unique fixed point x∗ ∈ X. Moreover, we have ϕ (x∗ ) = 0. Corollary 3. [15] Let (X, p) be a complete partial metric space and T : X → X be a given mapping. Suppose that for any 0 < a < b < ∞, there exists 0 < γ (a, b) < 1 such that for all x, y ∈ X a ≤ p(x, y) ≤ b ⇒ p(Tx, Ty) ≤ γ (a, b)p(x, y).

(9.10)

Then T has a unique fixed point x∗ ∈ X. Moreover, we have p(x∗ , x∗ ) = 0. Proof. From ps (x, y) = 2p(x, y) − p(x, x) − p(y, y), for all x, y ∈ X, we have p(x, y) =

ps (x, y) + p(x, x) + p(y, y 2

Note that since (X, p) is complete this implies (X, ps ) is a complete metric space. Consider the function ϕ : X → [0, ∞) defined by

ϕ (x) = p(x, x), ∀x ∈ X. Then from (9.10), for any 0 < a < b < ∞, there exists 0 < γ (a, b) < 1 such that for all x, y ∈ X, 2a ≤ ps (x, y) + ϕ (x) + ϕ (y) ≤ 2b ⇒ ps (Tx, Ty) + ϕ (Tx) + ϕ (Ty) ≤ γ (a, b) [ps (x, y) + ϕ (x) + ϕ (y)] Now, T satisfies condition (9.9) of Theorem 4 with p = ps . Finally, to apply Theorem 4, we have to check the lower semi-continuity of the function ϕ (with respect to the topology of ps ). Let {xn } be a sequence in X such that lim n → ∞ps (xn , x) = 0, where x ∈ X. Then we get limn→∞ p(xn , xn ) = p(x, x), that is, limn→∞ ϕ (xn ) = ϕ (x). Thus, ϕ is continuous and therefore the desired result follows immediately from Theorem 4. 

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Corollary 4. [15] Let (X, p) be a complete partial metric space and T : X → X be a given mapping. Suppose that there exists 0 < γ (a, b) < 1 such that for all x, y ∈ X p(Tx, Ty) ≤ γ (a, b)p(x, y).

(9.11)

Then T has a unique fixed point x∗ ∈ X. Moreover, we have p(x∗ , x∗ ) = 0. The followings are easy consequences of Theorems 6–8. Corollary 5. [15] Let (X, p) be a complete partial metric space and T : X → X be a given mapping. 1 Suppose that there exists a constant γ ∈ (0, ) such that 2 p(Tx, Ty) ≤ γ [p(x, Tx) + p(y, Ty)] , for all x, y ∈ X. Then T has a unique fixed point x∗ ∈ X. Moreover, we have p(x∗ , x∗ ) = 0. Corollary 6. [15] Let (X, p) be a complete metric space and T : X → X be a given mapping. Suppose that there exists α , β , γ ∈ (0, ∞) such that α + β + γ < 1 , p(Tx, Ty) ≤ α p(x, y) + β p(x, Tx) + γ p(y, Ty) for all x, y ∈ X. Then T has a unique fixed point x∗ ∈ X. Moreover, we have p(x∗ , x∗ ) = 0. Corollary 7. [15] Let (X, p) be a complete metric space and T : X → X be a given mapping. Suppose 1 that there exists a constant k ∈ (0, ) such that for all x, y ∈ X, 2 p(Tx, Ty) ≤ k [p(x, Ty) + p(y, Tx)] . Then T has a unique fixed point x∗ ∈ X. Moreover, we have p(x∗ , x∗ ) = 0.

9.5 Completion of Partial Metric Spaces Ge and Lin [7] recently introduced the notions of symmetrical density and sequential density in partial metric spaces and proved the existence and uniqueness theorems in the classical sense for the completion of partial metric spaces. Definition 7 ([7], Definition 3). Let f : X −→ Y be a map from a partial metric space (X, p) to a partial metric space (Y, q). Then f is called an isometry if q(f (x), f (y)) = p(x, y) for all x, y ∈ X. Definition 8 ([7], Definition 4). Let (X, p) be a partial metric space. A complete partial metric space (X∗ , p∗ ) is called a completion of (X, p) if there is an isometry f : X −→ X∗ such that f (X) is dense in (X∗ , p∗ ). Definition 9 ([7], Definition 5). Let (X, p) be a partial metric space and Y be a subset of X.

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1. Y is called sequentially dense in X if for each x ∈ X there exists a sequence {xn } ⊂ Y such that lim p(xn , x) = lim p(xn , xn ) = p(x, x). n→∞

n→∞

2. Y is called symmetrically dense in X if for each x ∈ X and each ε > 0 there exists y ∈ Y such that y ∈ B(x, ε ) and x ∈ B(y, ε ), where B(x, ε ) = {y ∈ X : p(x, y) < p(x, x) + ε }. The main results of Ge and Lin are as follows. Theorem 9 ([7], Theorem 1). Every partial metric space has a completion. Theorem 10 ([7], Proposition 1). Let (X∗ , p) and (Y∗ , q) be two complete partial metric spaces, and X and Y be symmetrically dense subsets of X∗ and Y∗ respectively. If h : X −→ Y is an isometry then there is a unique isometric extension f : X∗ −→ Y∗ which is an extension of h. Lemma 4 ([7], Lemma 3). Let (X, p) be a partial metric space and Y be a subset of X. Then Y is sequentially dense in (X, T ) if and only if Y is symmetrically dense in (X, T ). Remark 1 ([7], Remark 3). The symmetrical density and the density are equivalent in metric spaces. Let (X, p) be a partial metric space. The proof of Theorem 9 in [7] constructed a completion of a partial metric space (X, p) in which (X, p) is not only dense but also symmetrically dense. On the other hand, Theorem 10 presented a uniqueness for the completion of a partial metric space with respect to the isometry under symmetrical densities. However, Ge and Lin [7] did not know whether there exists a completion of (X, p) in which (X, p) is dense but not symmetrically dense. In fact, they proposed the following question: Question:[[7], Question 1] For a partial metric space (X, p), can one construct a completion in which (X, p) is dense but not symmetrically dense? Very recently Dung [6] provided an example to answer affirmatively the above question. Before proceeding to the example of Dung [6], we first need the following: Lemma 5. Let (X, p) be a partial metric space. Then (i)

lim xn = x in the topological space (X, T ) if and only if lim p(xn , x) = p(x, x). In particular,

n→∞

if lim xn = x in (X, p) then lim xn = x in (X, T ). n→∞

n→∞

n→∞

(ii) A subset Y of X is dense in (X, T ) if and only if for each x ∈ X there exists a sequence {xn } ⊂ Y such that lim xn = x in (X, T ). n→∞

Example 5. [6] Let X =

Then

1 n

 : n = 1, 2, . . . ∪ {0} and ⎧ ⎨ 2 1 p(x, y) = ⎩ |x − y| + 2

if x = y = 0 if x = y = n1 otherwise.

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1. (X, p)  is a completepartial metric space. 2. Y = n1 : n = 1, 2, . . . is dense but not symmetrically dense in (X, T ). Proof. (1). For any x, y ∈ X, it is clear that 0 ≤ p(x, x) ≤ p(x, y) and p(x, y) = p(y, x) . Moreover, if p(x, x) = p(x, y) = p(y, y) then x = y. For any x, y, z ∈ X, we will show that p(x, z) ≤ p(x, y) + p(y, z) − p(y, y). Indeed, if x = z = y then p(x, z) = p(x, y) + p(y, z) − p(y, y). If x = z = 0, y = 0 then p(x, z) = 2 < |0 − y| + 2 + |y − 0| + 2 − 1 = p(x, y) + p(y, z) − p(y, y). If x = z = n1 , y =

1 n

then

     1  1 p(x, z) = 1 <  − y + 2 + y −  + 2 − 2 ≤ p(x, y) + p(y, z) − p(y, y). n n

If x = z and y = x or y = z then p(x, z) = p(x, y) + p(y, z) − p(y, y). If x = z = y then p(x, z) = |x − z| + 2 ≤ |x − y| + 2 + |y − z| + 2 − 2 ≤ p(x, y) + p(y, z) − p(y, y). By the above, p is a partial metric on X. Now, we prove that (X, p) is complete. Let {xn } be a Cauchy sequence in (X, p). Then there exists l ∈ R such that lim p(xn , xm ) = l. On the contrary, suppose that there exist n,m→∞

subsequences {xnk } and {xnl } of {xn } such that xnl = 0 and xnk = 0 for all n ∈ N. Therefore lim p(xnl , xnl ) = lim p(0, 0) = 2 and lim p(xnk , xnk ) = 1.

nl →∞

nk →∞

n→∞

This leads to a contradiction with lim p(xn , xm ) = l. Therefore there exists n0 such that n,m→∞

xn = 0 for all n ≥ n0 , or xn = 0 for all n ≥ n0 . If xn = 0 for all n ≥ n0 then lim p(xn , xn ) = p(0, 0) n→∞

and lim p(xn , 0) = p(0, 0). This shows that lim xn = 0 in (X, p). If xn = 0 for all n ≥ n0 then n→∞

n→∞

lim p(xn , xn ) = 1. Since lim p(xn , xm ) = l, it follows that

n→∞

n,m→∞

l = lim p(xn , xn+k ) = lim p(xn , xn ) = 1 n→∞

n→∞

for all k ∈ N. By definition of p there exists n1 ≥ n0 such that xn = xn1 for all n ≥ n1 . Therefore lim p(xn , xn ) = lim p(xn , xn1 ) = p(xn1 , xn1 ).

n→∞

n→∞

This shows that lim xn = xn1 in (X, p). n→∞

By the above, the Cauchy sequence {xn } is convergent in (X, p). Therefore (X, p) is complete. (2). We see that     1 1 , 0 = lim + 2 = 2 = p(0, 0). lim p n→∞ n→∞ n n By Lemma 5.(i), lim n1 = 0 in (X, T ). Then, by Lemma 5.(ii), Y is dense in (X, T ). n→∞

However, for each sequence {xn } ⊂ Y, we have lim p(xn , xn ) = lim 1 = 1 = 2 = p(0, 0).

n→∞

n→∞

This shows that lim xn = 0 in (X, p). Therefore Y is not sequentially dense in (X, T ). By n→∞

Lemma 4, Y is not symmetrically dense in (X, T ).



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Dung [6] also modified the definition of completion of partial metric space and proposed the following: Definition 10. Let (X, p) be a partial metric space. 1. A complete metric space (X∗ , d∗ ) is called a metric completion of (X, p) if there is a map f : X −→ X∗ such that f (X) is dense in (X∗ , d∗ ) and d∗ (f (x), f (y)) = p(x, y) for all x = y ∈ X. 2. A 0-complete partial metric space (X∗ , p∗ ) is called a 0-completion of (X, p) if there is an isometry f : X −→ X∗ such that f (X) is dense in (X∗ , p∗ ). Theorem 11. Let (X, p) be a partial metric space. Then 1. (X, p) has a metric completion. 2. (X, p) has a 0-completion. The following question remains unsolved: Question: Find the condition such that the density and the symmetrical density in a partial metric space (X, p) are equivalent?

9.6 Ekeland’s Variational Principle in Partial Metric Spaces In 2017, Han et al., [8] studied various properties and principles of partial metric spaces and discussed Ekeland’s variation principle: Definition 11. [8] Let(X, p) be a partial metric space. If for each x ∈ X and each sequence {xn } in X, limn→∞ p˜ (xn , x) = 0 implies limn→∞ p˜ (x, xn ) = 0, then (X, p) is sequentially symmetric. Definition 12. [8] Let (X, p) be a partial metric space. If p˜ (xn , xm ) → 0 , m > n → ∞, implies xn → x, n → ∞ for some x ∈ X, then (X, p) is called a strong complete partial metric space. Theorem 12. [8] Let (X, p) be a strong complete partial metric space and let f : X → R be a lower semi-continuous function bounded from below. Let ε > 0 be given, and y0 ∈ X be such that f (y0 ) ≤ infx∈X f (x) + ε . Then there exists a point y∗ such that f (y∗) ≤ f (y0 ), p˜ (y0 , y∗ ) < 1, and y∗ is the minimizer of fy∗ (x) = f (x) + ε p(y∗ , x). Theorem 13. [8] Let (X, p) be a complete sequentially symmetric partial metric space, and let f : X → R be a lower semi-continuous function bounded from below. Assume that a : X → R is a positive continuous function and ϕ : [0, ∞) → [0, ∞) is a bi-Lipschitz mapping (i.e., ϕ and ϕ 1 are both Lipschitz) such that ∞ ∑k=1 a(yk )ϕ (p(yk , yk+1 )) < ∞ implies yk → y for some y ∈ X. Let ε > 0 be given, and y0 ∈ X be such that f (y0 ) ≤ infx∈X f (x) + ε . Then there exists a point y∗ such that f (y∗) ≤ f (y0 ), p(y0 , y∗ ) − p(y0 , y0 ) < 1 , and y∗ is the minimizer of fy∗ (x) = f (x) + ε a(x)ϕ (p(y∗ , x)).

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9.7 Conclusion In this chapter, we make an attempt to provide a systematic survey of the various aspects of partial metric spaces and concerning fixed point theory. However, there are stilla lot of things remaining unknown about partial metric systems, for example: Is every complete partial metric on X equivalent to some bounded complete partial metric? (ii) If Y ⊂ X and p is a partial metric on Y, then there is always a partial metric P on X such that P|Y = p. The question is whether topological properties that hold in (Y, p) still hold in (X, P)?

(i)

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

I. Altun, F. Sola, and H. Simsek, Generalized contractions on partial metric spaces, Topol Appl., 157(18)(2010), 2778–2785. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181. D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Soc., 20 (1969), 458–464. M. Bukatin, R. Kopperman, S. Matthews, and H. Pajoohesh, Partial metric spaces, Amer. Math. Monthly 116 (2009), 708–718. M. Bukatin, R. Kopperman, S. Matthews, and H. Pajoohesh, Partial metric spaces, Amer. Math. Monthly 116 (2009), 708–718. N. Dung, On the completion of partial metric spaces, Quaestiones Mathematicae, 40(5) (2017), 589–597. X. Ge, S. Lin, Completions of partial metric spaces, Topology and its Applications, 182 (2015), 16–23. S. Han, J. Wu, and D. Zhang, Properties and principles on partial metric spaces, Topol Appl., 230 (2017), 77–98. R. D. Kopperman, S. G. Matthews, and H. Pajoohesh, Completions of partial metrics into value lattices, Topol Appl., 156 (2009), 1534–1544. W. Kirk, N. Shahzad, Fixed point theory in distance spaces, Cham: Springer (2014). S. G. Matthews, Partial metric topology, Papers on general topology and applications, Proc. 8th Summer Conf., Queen’s College, 1992. S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl., 2010 (2010), 1–6. S. Romaguera, Fixed point theorems for generalized contractions on partial metric spaces. Topol Appl., 159(1)(2012), 194–199. I. A. Rus, Fixed point theory in partial metric spaces, An. Univ. Vest. Timis., Ser. Mat.-Inform., 46(2) (2008), 141–160. B. Samet, C. Vetro, and F. Vetro, From metric spaces to partial metric spaces, Fixed Point Theory Appl., (2013), 5.

Index (G, G )-graphical contraction, 272 (ε , σ )-uniformly locally contractive, 77 (∑, =)-complete, 8, 9, 18 (∑, =)-semicomplete, 9 0-Cauchy sequence, 284, 290 0-complete partial metric space, 284, 290 B-contraction, 153 D-metric, 103 F-contraction, 108, 109 G-Cauchy, 107, 182 G-Cauchy sequence, 200, 206 G-closed set, 127, 128 G-closed sets, 128 G-complete, 200, 203, 205 G-convergent, 107 G-metric, 103, 104 G-metric space, 104 G − l cyclic F-contraction, 124 H-contraction, 159 H-type triangular norms, 152 M-Cauchy sequence, 204, 206, 209 M-complete, 204, 210, 211 M-complete fuzzy metric space, 213, 215 M-topology, 199 S-complete, 7, 22 T-Picard sequence, 272 W-starshaped, 238 Φ -function, 174 Φ −function, 220 α -ψ -type contractive mnap, 180 α -ϕ -S + -contractive, 24–27 α -admissible, 23, 25, 26, 180 α -complete, 23–27 β -admissible, 185 ε -chainable, 72, 76, 77 κ -Hausdorff, 9 H -contractive mapping, 210, 211 S + -distance, 3 S + -type contractions, 23 φ -contractive map, 174 φ −contractive mapping, 220 ∑-Semicomplete, 8, 14–19 ∑-complete, 8, 9, 19 τ -distance, 79

ε -chainable space, 278 ϕ -contraction, 166 b-metric, 33–35, 37 b-metric space, 33, 34, 37, 40, 42 b-metric-like, 37 p-Caristi mapping, 289 p-bounded, 86 p-complete, 86 p-completeness, 208 p-continuous, 86 p-contractive, 76, 81–83 p-convergent, 207 p−convergent sequence, 220 ps -Caristi mapping, 289, 290 s-Cauchy complete, 7, 9, 10, 12, 13 s-Cauchy sequence, 7, 8, 10, 23 s-bounded, 22 s-closed, 21 s-complete, 7 s-continuous, 6, 19, 20 s-weakly complete, 7 t-norm, 151, 200, 203 t-proximinal, 255 w-distance, 68–84, 86–89, 98 (∑, =)-Cauchy sequence, 8 (∑, =)-semicomplete, 8 0-complete partial metric, 296 0-completion, 296 Abstract economies, 259 Admissible, 248 Altering distance function, 220, 224 Altering distance functions, 173 Archimeadean t-norm, 151 Banach contraction, 285 Banach contraction conjecture, 50 Banach Contraction Principle, 19, 107, 108 Banach contraction principle, 37, 50, 67, 79, 108 Bessel type BVP, 134 Bharucha-Reid contraction, 161 Bounded, 3, 6, 9, 10, 13–21

299

Index

300

Caristi fixed point theorem, 289 Caristi type, 17, 18, 52, 74, 230 Cauchy sequence, 6, 8, 11, 34, 39, 43, 45, 71, 72, 94, 153, 248, 270 Ciric type generalized F- contraction, 117 Closed p-ball, 284 Closed ball, 35, 36, 106 Closed set, 4 Closed sets, 7, 8 Commuting, 86 Compact, 3, 6, 21, 200, 249 Complete b-metric space, 34, 43, 47 Complete partial metric space, 285, 287–289, 291–293 Completion, 293, 294, 296 Cone b-metric space, 37 Connected component, 268 Continuous, 69, 70, 74, 75, 79 Continuous t-norm, 151 Contraction, 251 Convex fuzzy metric space, 237 Convex structure, 237 Cyclic contraction, 47, 277 Definite set-valued integral, 261 Distance distribution function, 150 Dynamic programming, 132 E-space, 2 Ekeland variational principle, 67, 68, 74 Extension of t-norms, 218 Fatou property, 47 Firmly compact, 249 Fixed point inclusion, 246, 247 Fractional integral inclusion, 261 Fuzzy Z -contractive mappings, 211, 212 Fuzzy ψ -contractive mapping, 208, 210, 211 Fuzzy ψk -contractive, 209 Fuzzy ψk -contractive mapping, 209 Fuzzy w-distance, 93, 94 Fuzzy w-distance, 93, 94 Fuzzy Banach contraction, 200, 205 Fuzzy contractive mappings, 199 Fuzzy Edelstein contraction, 200, 201 Fuzzy Nadler contraction, 228 Fuzzy set, 199, 200

Generalized β -type contractive mapping, 184, 185 Generalized w-distance, 91 Generalized contraction, 285, 286, 291, 297 Generalized hybrid map, 82, 83 Generalized KKM operator, 249 Generalized PM-space, 150 Geometrically convergent, 165 Grabiec’s fixed point theorems, 200 Graphical metric, 268 Graphical metric spaces, 267, 271 GV-fuzzy metric space, 203, 207 Hardy-Rogers type F−contraction, 114 Hausdorff metric, 1, 77, 228 Hausdorff space, 8–10 Heaviside function, 150 Hick contraction, 159 Idempotent operator, 4 Integral inequality, 122 Isometry, 293, 294, 296 Kannan’s contraction, 251 KKM map, 37 KKM property, 249 KM-fuzzy metric space, 200, 207, 208 Lebesgue-integrable mapping, 86, 90 Lower semicontinuous, 67, 68, 70–74, 87, 91, 249, 289–291 Lower solution, 279 Lukasiewicz t-norm, 151 Matkowski contraction, 166 Meir-Keeler contraction, 193, 194 Meir-Keeler FPT, 52, 53 Menger PM-space, 152 Metric completion, 296 Metric modular, 248 Metric space, 2, 68–71 Metrizable, 38–40 Modular metric space, 248 Modular metric spaces, 245, 247, 255 Multivalued fractal, 52–54 Nadler types, 21, 76, 232 Nash’s equilibrium, 260 Non-Archimedean, 204, 209, 210

Index

Non-Archimedean Menger PM space, 152 Nonconvex minimization theorem, 67, 68, 73, 79 Nonspreading map, 81, 82 Normed linear space, 69, 93 Open p-ball, 284 Open ball, 36, 37, 103, 106 Orbit, 13, 155 Orbitally continuous, 83–85, 91 Ordered metric, 269 Partial b-metric, 37 Partial metric, 37, 283, 284 Partial metric space, 283, 284, 289, 293, 294 Partially ordered metric space, 85, 87, 91, 92 Path, 268 Picard operator, 86, 88 Point wise convergent, 207 Pompeiu-Hausdorff distance, 3, 51, 247 Pompeiu-Hausdorff metric modular, 250 Probabilistic k-contraction, 156 Probabilistic metric spaces, 149, 150 Proper mapping, 8, 67 Property (S ), 216 Property (S), 213 Property (W3), 4 Property (W4), 5 Property (CC), 5 Property (HE), 5 Property (JMS), 5 Property (S), 274 Property (W), 5 Reachable subset, 251 Regular, 38 Semi-metric, 4 Semi-metric space, 4–6, 38 Semicomplete, 8, 9

301

Semimetrizable, 4, 40 Sequentially dense, 294, 295 Sequentially lower semicontinuous, 8, 15, 17–19 Sequentially symmetric, 296 Set-valued maps, 245 Spring mass system, 129 Strong b-metric space, 37, 38, 56, 57 Strong complete partial metric space, 296 Strong triangle function, 6 Strongly continuous semigroup, 79 Strongly demicompact, 228 Subadmissible, 248 Symmetric G-metric spaces, 106 Symmetric spaces, 1–5, 7, 9–11 Symmetrically dense, 294, 295 Tatarus distance, 79 Tirado’s contraction, 206 Topology of a G-metric spaces, 106 Triangle function, 38 Triangular function, 6, 157 Triangular inequality, 2, 3 Ultrametric space, 38, 229 Upper semicontinuous, 168, 169, 248, 287–289 Upper solution, 279 Weak-Hicks contraction, 164 Weakest t-norm, 151 Weakly commuting, 89, 229 Weakly connected graph, 268 Weakly contractive, 76, 78 Weakly demicompact, 228 Weakly Picard, 164, 247 Well-ordered set, 277 Well-posed, 53, 54 Well-posedness, 291