Fixed Point Theory and Graph Theory

Fixed Point Theory and Graph Theory

Fixed Point Theory and Graph Theory Foundations and Integrative Approaches Edited by Monther Rashed Alfuraidan King Fahd University of Petroleum & Minerals, Department of Mathematics and Statistics, Dhahran, Saudi Arabia Qamrul Hasan Ansari Aligarh Muslim University, Department of Mathematics, Aligarh, India, and King Fahd University of Petroleum & Minerals, Department of Mathematics and Statistics, Dhahran, Saudi Arabia AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, UK 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Copyright c 2016 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-804295-3 For information on all Academic Press publications visit our website at http://www.elsevier.com Publisher: Nikki Levy Acquisition Editor: Graham Nisbet Editorial Project Manager: Susan Ikeda Production Project Manager: Poulouse Joseph Designer: Matthew Limbert

FOREWORD The present book differs in important ways from previous texts which deal largely with metric fixed point theory. Foremost, the topics collected here are not found in any single previous treatments and most are entirely new. These include detailed discussions of Caristi s theorem in a number of settings, as well as numerous approximation techniques in Banach spaces, hyperbolic spaces, and more general geodesic spaces. In recent years metric fixed point theory has increasingly moved away from traditional functional analytic settings, thus opening up new avenues of application. The present book reflects this trend by discussing in detail aspects of the theory which involve a blend of metric fixed point theory and graph theory, as well as how retractions are used to prove fixed point results in ordered sets. Many examples are included as well as potential applications. Because of the comprehensive background material this book is self-contained and easily accessible. A notable feature is the thoroughness of the bibliographic citations, many of which are not found in any previous sources. This book should be a worthwhile resource for students and researchers interested in almost any aspect of metric fixed point theory, and especially for those interested in many of the more recent trends. William A. Kirk Iowa City, IA, USA ix

ACKNOWLEDGMENTS We would like to thank our illustrious friends and colleagues in the Fixed Point Theory Research Group at the Department of Mathematics and Statistics at King Fahd University of Petroleum & Minerals who, through their encouragement and help, were instrumental in the development of this book. In particular, we are grateful to Prof. Khalid Al-Sultan, the Rector of King Fahd University of Petroleum & Minerals, and Prof. Mohammad Al-Homoud, the Vice Rector for Academic Affairs of King Fahd University of Petroleum & Minerals, for their unflinching support in the organization of the International Workshop on Fixed Point Theory and Applications during December 22 24, 2014. This Workshop provided the platform to contact and invite the delegates to contribute to this book as authors. It is heartening to note that most of them accepted our offer and generously contributed their research for inclusion in the book. The book in its present form is, in large measure, a fruit of their wholehearted cooperation. We are profoundly grateful to them. The combination of Fixed Point Theory and Graph Theory is the brainchild of our friend Prof. M. A. Khamsi, University of Texas at El Paso. He is the prime mover behind the international workshop on this topic and the subsequent plan to publish a book which combines these two subjects. We are deeply thankful and grateful to him for his encouragement, support and help. Thanks are also due to Prof. Suliman Al- Homidan, Dean, College of Science, King Fahd University of Petroleum & Minerals, for his motivating advice and help to publish this book. We would like to convey our special thanks to Mr. Graham Nisbet, Senior Acquisitions Editor, Mathematical Sciences, Elsevier, for taking a keen interest in publishing this book. Monther Rashed Alfuraidan Qamrul Hasan Ansari December 2015 xi

PREFACE Fixed point theory is without doubt one of the most important tools of modern mathematics as attested by Browder, who is considered as one of the pioneers in the development of the nonlinear functional analysis. The flourishing field of fixed point theory started in the early days of topology through the work of Poincaré, Lefschetz-Hopf and Leray-Schauder for example. Fixed point theory is widely used in different areas such as ordinary and partial differential equations, economics, logic programming, convex optimization, control theory, etc. In metric fixed point theory, successive approximations are rooted in the work of Cauchy, Fredholm, Liouville, Lipschitz, Peano and Picard. It is well accepted among experts of this subarea that Banach is responsible for laying the ground for an abstract framework well beyond the scope of elementary differential and integral equations. The fixed point theory of multivalued mappings is as important as the theory of single-valued mappings. After Nadler gave the multivalued version of the Banach fixed point theorem for contraction mappings, many authors generalized his fixed point theorem in different ways. A constructive proof of a fixed point theorem makes the conclusion more valuable in view of the fact that it yields an algorithm for computing a fixed point. The last decade has seen some excitement about monotone mappings after the publication of Ran and Reurings of the analogue to the Banach fixed point theorem in metric spaces endowed with a partial order. Their result was motivated by the problem of finding the solutions to some matrix equations, which often arise in the analysis of ladder networks, dynamic programming, control theory, stochastic filtering, statistics and many other applications. Jachymski is recognized for making the connection between classical metric fixed point theory and graph theory. This book can be treated as a foundation for both theories and presents the most up-to-date results bridging the two theories. Each chapter is written by different author(s) who attempt to render the major results understandable to a wide audience, including nonspecialists, and at the same time to provide a source for examples, references, open questions, and sometimes new approaches for those currently working in these areas of mathematics. This book should be of interest to graduate students seeking a field of interest, to mathematicians interested in learning about the subject, and to specialists. All the chapters are self-contained. Chapter 1 describes the classical Caristi fixed point theorem and its various versions, and studies Caristi-Browder operators in the setting of metric spaces, R m +-metric spaces, s(r + )-metric spaces, and Kasahara spaces. It also provides some new research directions in the Caristi-Browder operator theory. xiii

xiv Preface Chapter 2 is devoted to the fixed point theorems for self and nonself almost contraction mappings, and common fixed point theorems for almost contraction mappings. Iterative approximation of fixed points is also studied for implicit almost contraction mappings. The approximate fixed points are useful when a computational approach is involved. This subject is addressed in Chapter 3. Some known results about the concept of approximate fixed points of a mapping are presented. Most of the approximate fixed points discussed in this chapter are generated by an algorithm that allows its computational implementation. An application of these results to the case of a nonlinear semigroup of mappings is given. Chapter 4 presents the mathematical idea of viscosity methods for solving some applied nonlinear analysis problems, namely, fixed point problems, split common fixed point problems, split equilibrium problems, hierarchical fixed point problems, etc. together with examples in linear programming, semi-coercive elliptic problems and the area of finance. Some numerical experiments for an inverse heat equation are also given. Chapter 5 deals with several kinds of extragradient iterative algorithms for some nonlinear problems, namely, fixed point problems, variational inequality problems, hierarchical variational inequality problems and split feasibility problems. Several extragradient iterative algorithms for finding a common solution of a fixed point problem and a variational inequality problem are also presented and investigated. Chapter 6 concerns with the fixed point (common fixed point) problems for nonexpansive (asymptotically quasi-nonexpansive) mappings in Banach spaces and certain important classes of metric spaces. Some results about weak convergence, -convergence and strong convergence of explicit and multistep schemes of nonexpansive-type mappings to a common fixed point in the context of uniformly convex Banach spaces, CAT(0) spaces, hyperbolic spaces and convex metric spaces are presented. Chapter 7 discusses a new area that overlaps between metric fixed point theory and graph theory. This new area yields interesting generalizations of the Banach contraction principle in metric and modular spaces endowed with a graph. The bridge between both theories is mainly motivated by the applications of matrix and differential equations. The last chapter gives an overview how retractions are used to prove fixed point theorems for relation-preserving maps in continuous as well as discrete settings. After first discussing comparative retractions in the context of infinite ordered sets and analysis, the remainder of the chapter focuses on retractions that remove a single point in finite ordered sets, graphs and simplicial complexes. Monther Rashed Alfuraidan Qamrul Hasan Ansari Dhahran, Saudi Arabia; and Aligarh, India December 2015

ABOUT THE AUTHORS Monther Rashed Alfuraidan is Associate Professor of Mathematics in the Department of Mathematics & Statistics at King Fahd University of Petroleum & Minerals at Dhahran, Saudi Arabia. He obtained his Ph.D. (Mathematics) from Michigan State University. He has written more than twenty articles on graph theory, algebraic graph theory and metric fixed point theory. He has peer-reviewed many articles (among others) for: Algebraic Journal of Combinatorics, Arabian Journal of Mathematics, Fixed Point Theory and Applications and Journal of Inequality and Applications. Qamrul Hasan Ansari is Professor of Mathematics at Aligarh Muslim University, Aligarh, India. He is also joint professor at King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia. He obtained his Ph.D. (Mathematics) from Aligarh Muslim University, India. He is an associate editor of Journal of Optimization Theory and Applications, Fixed Point Theory and Applications, Fixed Point Theory and Carpathian Journal of Mathematics. He has also edited several special issues of several journals, namely, Journal of Global Optimization, Fixed Point Theory and Applications, Abstract and Applied Analysis, Journal of Inequalities and Applications, Applicable Analysis, Positivity, Filomat, etc. He has written more than 180 articles on variational inequalities, fixed point theory and applications, vector optimization, etc. in various international peer-reviewed journals. He has edited six books for Springer, Taylor & Francis and Narosa, India. He is an author of a book on metric spaces published by Narosa, India and has co-authored one book on variational inequalities and nonsmooth optimization for Taylor & Francis. Mostafa Bachar received his Ph.D. from the University of Pau (France) in Applied Mathematics; he is currently working as associate professor at King Saud University (KSA). He has previously held the position of Research Associate at the Institute of Mathematics and Scientific Computing at the University of Graz (Austria). Recently he has edited two books in Lecture Notes in Mathematics: Mathematical Biosciences Subseries, Springer. His current research is in mathematical modeling in mathematical biology and mathematical analysis. Vasile Berinde is Professor of Mathematics and Director of the Department of Mathematics and Computer Science in the Faculty of Sciences, North University Center at Baia Mare, Technical University of Cluj-Napoca, Romania. He obtained his B.Sc., M.Sc. (in Computer Science) and Ph.D. (in Mathematics) from the Babes-Bolyai University of Cluj-Napoca, under the supervision of Professor Ioan A. Rus. His main scientific interests are in nonlinear analysis and fixed point theory. He is the founder and xv

xvi About the Authors current Editor-in-Chief of the journals Carpathian Journal of Mathematics and Creative Mathematics and Informatics and editorial board member for the journals Fixed Point Theory, Fixed Point Theory and Applications, Journal of Applied Mathematics, General Mathematics, Linear and Nonlinear Analysis, Carpathian Mathematical Publications, Didactica Mathematica, Gazeta Matematică, Revista de Matematicădin Timişoara, etc. He has published more than 150 scientific articles on the iterative approximation of fixed points, fixed point theory and its applications to differential and integral equations. He has published several books, textbooks and monographs with Romanian publishers. Two of them have been undertaken by international publishers: Iterative Approximation of Fixed Points (Springer, 2007) and Exploring, Investigating and Discovering in Mathematics (Birkhäuser, 2004). Buthinah A. Bin Dehaish is Associate Professor of Mathematics at King Abdullaziz University. She obtained her Ph.D. (Mathematics) from King Abdullaziz University. She has written more than 15 articles on harmonic analysis and fixed point theory and its applications in various international peer-reviewed journals. Hafiz Fukhar-ud-din is an associate professor at the Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. He obtained his Ph.D. (Mathematics) from Tokyo Institute of Technology, Japan. He has supervised three Ph.D. and five M.Phil. students. He has written more than 40 articles on BCH-algebras and fixed point iterative methods in Banach spaces, hyperbolic spaces and convex metric spaces. He has contributed a chapter to the book Fixed Point Theory, Variational Analysis, and Optimization, Chapman & Hall/CRC, 2014. Mohamed Amine Khamsi is Professor of Mathematics at the University of Texas at El Paso. He obtained his Ph.D. (Mathematics) from the University of Paris. He has written more than 70 articles on fixed point theory and its applications in various international peer-reviewed journals. He has co-authored five books. Abdul Rahim Khan is Professor of Mathematics in the Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. He obtained a Ph.D. (Mathematics) from the University of Wales, UK. He has supervised five Ph.D. and eleven M.Phil. students. He is an editor of Carpathian Journal of Mathematics and Arabian Journal of Mathematics. He has written more than 115 articles on fixed point theory and iterative methods of nonlinear mappings on a Banach space, CAT(0) space and convex metric space, in various peer-reviewed journals. He has contributed a book chapter in Advances in Non-Archimedean Analysis, Contemporary Mathematics, American Mathematical Society, 2011 and Fixed Point Theory, Variational Analysis, and Optimization, Chapman & Hall/CRC, 2014.

About the Authors xvii Paul-Emile Maingé is an assistant professor at the Université des Antilles, Campus de Schoelcher, Martinique, France. He holds a Ph.D. degree from Bordeaux University, France (1996) in the field of Applied Mathematics and an accreditation to supervise research in Mathematics (Habilitation à Diriger des Recherches) from French West Indies University, France (2008). He has vast experience of teaching and research at university level in various institutions including Bordeaux, Paris, Guadeloupe and Martinique. His field covers several areas of Applied Mathematics such as optimization, variational analysis, convex analysis, dynamical systems, variational inequalities, fixed-point problems and partial differential equations (slow and fast diffusion problems). He has obtained the Grant of Scientific Excellence for the periods 2006/2010 and 2011/2015. He already has more than 30 research papers to his credit which were published in leading world class journals and acts as a referee for many international journals. Abdellatif Moudafi is a Full Professor, first class. He obtained a Ph.D. degree from Montpellier University, France (1991) in the field of Applied Mathematics (Nonlinear Analysis and Optimization), a doctorate (Doctorat d état) from Casablanca University, Morocco (1995) and an accreditation to supervise research in Mathematics (Habilitation à Diriger des Recherches) from Montpellier University, France (1997). He has vast experience of teaching and research at university level in various institutions including Montpellier, Clermont-Ferrand, Limoges, Marrakech, Guadeloupe, Montreal, Martinique, New York and Aix-Marseille. His field covers many areas of Applied Mathematics such as optimization, applied nonlinear analysis, variational analysis, convex analysis, asymptotic analysis, discretization of gradient dynamical systems, proximal algorithms, variational inequalities, approximation and regularization of fixed-point problems and equilibria. He has supervised successfully several Ph.D. and M.S./M.Phil. students. He has obtained the Grant of Scientific Excellence for the periods 2001/2005, 2006/2010, and 2011/2015. He is currently member of the editorial board of several reputed international journals of mathematics. He has more than 100 research papers to his credit which were published in leading world class journals and acts as a referee for over 50 international journals. Mădălina Păcurar is Associate Professor of Mathematics, Department of Statistics, Forecast and Mathematics, Faculty of Economics and Business Administration, Babes-Bolyai University of Cluj-Napoca, Romania. She obtained her Ph.D. in Mathematics from the Babes-Bolyai University of Cluj-Napoca, under the supervision of Professor Ioan A. Rus. Her scientific interest is in fixed point theory. She has published more than 40 scientific articles in the field of fixed point theory on product spaces. She has co-authored some textbooks (Mathematics for Economics) and had a monograph published (Iterative methods for fixed point approximation, Editura Risoprint, Cluj-Napoca, 2009) on her main research interest area.

xviii About the Authors Ioan A. Rus is Emeritus Professor, Faculty of Mathematics and Computer Science, Babes-Bolyai University of Cluj-Napoca, Romania. He is the founder (1969) of the fixed point research group in Cluj-Napoca, one of the oldest and most important research groups still active in this area. He supervised 26 Ph.D. theses at Babes-Bolyai University of Cluj-Napoca with topics in fixed point theory and its applications to differential and integral functional equations. He is the author of about 200 scientific papers. He has also had several books and monographs published, amongst which we mention the impressive monograph fixed point theory (Cluj University Press, Cluj- Napoca, 2008, xx+509 pp.), co-authors A. Petruşel and G. Petruşel. He is the founder of the first journal devoted entirely to fixed point theory: Seminar on Fixed Point Theory (1983 2002), now known as Fixed Point Theory: An International Journal on Fixed Point Theory, Computation and Applications since vol. 3 (2003). He is currently its Editor-in-Chief and also an editorial board member of Carpathian Journal of Mathematics, andstudia Universitatis Babeş-Bolyai. Mathematica. D. R. Sahu is professor at the Department of Mathematics at Banaras Hindu University, Varanasi, India. He obtained his Ph.D. from Pt. Ravi Shankar Shukla University, Raipur, India. He is the co-author of a book on fixed point theory published by Springer. He has written more than 100 articles on existence, approximation and applications of fixed point problems of nonlinear operators in various peer-reviewed publications. He has contributed chapters for various edited books. BerndS.W.Schröder has served as Professor and Chair of the Department of Mathematics at the University of Southern Mississippi since 2014. He obtained his Ph.D. in Mathematics (Probability Theory and Harmonic Analysis) from Kansas State University in 1992. He held a position at Hampton University from 1992 to 1997, first as a research associate from 1992 to 1993 and then as an assistant professor from 1993 to 1997. Subsequently, he moved to Louisiana Tech University, where he was an associate professor from 1997 to 2003 and a professor from 2003 to 2014. At Louisiana Tech, he served as Program Chair for Mathematics and Statistics from 2003 to 2014 and as Academic Director for Mathematics and Statistics from 2008 to 2014. He has written 42 refereed papers on ordered sets, graphs, analysis and engineering education and he has written six textbooks, ranging from first year calculus to a monograph on ordered sets.

CHAPTER 1 Caristi-Browder Operator Theory in Distance Spaces Vasile Berinde and Ioan A. Rus North University Center at Baia Mare, Department of Mathematics and Computer Science, Victoriei nr. 76, 430122 Baia Mare, Romania King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran, Saudi Arabia Babeş-Bolyai University of Cluj-Napoca, Department of Mathematics, Kogălniceanu nr. 1, Cluj-Napoca, Romania Corresponding: vasile.berinde@gmail.com Contents 1.1. From the Caristi Fixed Point Theorems to Caristi, Caristi-Kirk and Caristi-Browder Operators 1 1.2. List of Notations 4 1.3. Weakly Picard Operators on L-Spaces 4 1.4. Caristi-Browder Operators on Metric Spaces 6 1.5. Caristi-Browder Operators on R m + -Metric Spaces 12 1.6. Caristi-Browder Operators on s(r + )-Metric Spaces 16 1.7. Caristi-Browder Operators on Kasahara Spaces 19 1.8. Research Directions in the Caristi-Browder Operator Theory 21 References 22 Abstract Starting from the classical Caristi fixed point theorem and its various versions, the aim of this chapter is to study Caristi-Browder operators in the following settings: (1) metric spaces; (2) R m + -metric spaces; (3) s(r +)-metric spaces; (4) Kasahara spaces. Some new research directions in the Caristi-Browder operator theory are also indicated. 1.1. From the Caristi Fixed Point Theorems to Caristi, Caristi-Kirk and Caristi-Browder Operators The following results are well known (see Refs. [35, 37, 98, 156]). Theorem 1.1.1 (Caristi-Kirk Theorem). Let (X,d) be a complete metric space and let f : X X be such that there exists a lower semicontinuous (l.s.c.) functional http://dx.doi.org/10.1016/b978-0-12-804295-3.50001-2 c 2016 Elsevier Inc. All rights reserved. 1

2 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches ϕ : X R + with Then d(x, f (x)) ϕ(x) ϕ( f (x)), for all x X. F f := {x X : f (x)=x} /0. Theorem 1.1.2 (Caristi-Browder Theorem). Let f : X X be with closed graphic and there exists ϕ : X R + such that Then (a) F f /0; d(x, f (x)) ϕ(x) ϕ( f (x)), for all x X. (b) f n (x) x (x) F f as n for all x X. Theorem 1.1.3 (Caristi Theorem). Let Y X be a nonempty closed subset and f : Y X a metrically inward contraction (i.e., for all x Y, there exists u Y such that d(x,u)+d(u, f (x)) = d(x, f (x)),whereu= x x = f (x)). Then f has a fixed point. Remark 1.1.1. The first theorem (Caristi-Kirk Theorem in this chapter) is also known as: Caristi Theorem, Caristi-Kirk Theorem, Kirk-Caristi Theorem, Caristi-Ekeland Theorem, Caristi-Brøndsted Theorem, Caristi-Browder Theorem, Caristi-Kirk- Browder Theorem, etc. For some of the most relevant generalizations, further developments and related results (see Refs. [19, 33 35, 57, 63, 79, 92, 95, 98, 101, 134, 137, 161, 166, 183]). We summarize here some of the early contributions on the Caristi-Kirk Theorem, following the presentation by Caristi himself [38]: Wong [183] has simplified the original transfinite induction argument, Browder [35] has given a proof which avoids the use of transfinite induction, Siegel [161] has presented a simple constructive proof, Brézis and Browder [33] have given a general theorem on ordered sets which includes Caristi-Kirk Theorem, Kasahara [92] has proved an analog of the Caristi-Kirk Theorem in the setting of partially ordered sets. On the other hand, the Caristi-Kirk Theorem is actually implicit in Brøndsted s work [34] and is essentially equivalent to Ekeland s variational principle [63] (which is not formulated as a fixed point theorem). A generalization of Caristi-Kirk Theorem was given by Downing and Kirk [57]. Moreover, Kirk [98] has shown that the Caristi- Kirk Theorem characterizes completeness. Remark 1.1.2. An operator f : X X which satisfies (a) and (b) in Theorem 1.1.2 is, by definition, a weakly Picard operator and we denote (see Ref. [151]) f (x) := x (x),

Caristi-Browder Operator Theory in Distance Spaces 3 x X. It is clear that f (X)=F f and f : X F f is a set retraction. The above results give rise to the following notions: Definition 1.1.1. An operator f : (X,d) (X,d) is a Caristi operator if there exists a functional ϕ : X R + such that d(x, f (x)) ϕ(x) ϕ( f (x)), for all x X. (1.1.1) Definition 1.1.2. An operator f : (X,d) (X,d) is a Caristi-Kirk operator if f is a Caristi operator with ϕ a lower semicontinuous functional. Definition 1.1.3. By definition, an operator f : (X,d) (X,d) is a Caristi-Browder operator if it is a Caristi operator with closed graphic. Remark 1.1.3. If we consider on (X,d) the following partial order corresponding to ϕ : X R + (see Refs. [34, 92, 95, 101]): x d,ϕ y d(x,y) ϕ(x) ϕ(y), then, f : (X, d,ϕ ) (X, d,ϕ ) is progressive (i.e., x d,ϕ f (x) for all x X). If f is a progressive operator, then for the maximal element set, Max(X, d,ϕ ),wehavethe inclusion Max(X, d,ϕ ) F f, so the problem is to give conditions on d and ϕ such that Max(X, d,ϕ ) /0. The Caristi-Kirk Theorem and its extensions are developments on these topics. This aspect has been largely studied (see Refs. [1, 95, 101, 80]; see also Refs. [4, 19, 23, 33, 34, 57, 63, 71, 73, 100, 115, 137, 141, 161, 166, 177, 183, 187, 122, 134]. For some other extensions of Caristi Theorem, see Ref. [155] and the references therein (see also Ref. [36]). The aim of this chapter is to study the Caristi-Browder operators. The plan of the rest of the chapter is the following: List of notations / symbols Weakly Picard operators on L-spaces Caristi-Browder operators on metric spaces Caristi-Browder operators on R m + -metric spaces Caristi-Browder operators on s(r + )-metric spaces Caristi-Browder operators on Kasahara spaces Research directions in the Caristi-Browder operator theory - Caristi-Browder operator on partial metric spaces - Caristi-Browder operator on b-metric spaces

4 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches - Nonself Caristi-Browder operators - Multivalued Caristi-Browder operators Several other important aspects related to Caristi s fixed point theorem, like its equivalence to Ekeland s variational principle, Takahashi s minimization theorem, lower semicontinuity, drop theorem, etc., are not covered in this survey, but an almost comprehensive bibliographic coverage is given in the list of references, see in particular Refs. [2, 3, 5 18, 20, 22, 24, 39 56, 58 60, 64 66, 69, 70, 72, 74, 76, 78, 81 83, 85, 86, 88 91, 93, 94, 96, 97, 99, 102 109, 111, 116, 120, 121, 123 125, 127, 130 139, 141 144, 146 149, 159 182, 184, 185, 187 193]. 1.2. List of Notations N = {0,1,2,...}; N = N \{0}; R denotes the set of real numbers: R = R \{0}; R + =[0, ); Let X be a nonempty set and let f : X X be an operator. Then - 1 X denotes the identity operator; - F f := {x X : f (x)=x} the set of fixed points of f ; - f 0 := 1 X, f 1 := f,...,f n := f f n 1 (n N ) the iterates of f ; Let (X,+,R, ) be a real normed space. Then - or denotes the strong convergence; - denotes the weak convergence; Let (X,d) be a metric space. Then - P(X) := {A : A X}; - P cb (X) := {A : A P(X),i.e., A is nonempty, closed and bounded}; - ForA,B P(X), D(A,B)=inf{d(a,b) : a A,b B} denotes the distance between A and B, - ForA,B P cb (X), H d (A,B)=max{sup{D(a,B) : a A},sup{D(b,A) : b B}} denotes the Pompeiu-Hausdorff metric on P cb (X) induced by the metric d. 1.3. Weakly Picard Operators on L-Spaces Let X be a nonempty set and s(x) := {{x n } n N : x n X, n N}. Let c(x) s(x) be a nonempty subset of s(x) and Lim : c(x) X be an operator. We consideron the triple (X, c(x), Lim) the following Fréchetaxioms: (c 1 )Ifx n = x for all n N, then{x n } n N c(x) and Lim{x n } n N = x.

Caristi-Browder Operator Theory in Distance Spaces 5 (c 2 )If{x n } n N c(x) and Lim{x n } n N = x, then all its subsequences are in c(x) and have the same limit, x. By definition, a triple (X, c(x), Lim) which satisfies the above axioms is called an L-space (see Refs. [151, 154] and the references therein). In what follows we shall use the notation (X, ), instead of (X,c(X),Lim) and instead of Lim{x n } n N = x. x n x as n, Example 1.3.1. Let (X,τ) be a topological Hausdorff space. Then (X, τ ) is an L- space. Example 1.3.2. Let (X,d) be a metric space. Then (X, d ) is an L-space. Example 1.3.3. Let (X,+,R, ) be a normed space. Then, (X, ) and (X, ) are L-spaces. Example 1.3.4. Let (X,d) be an R m +-metric space, i.e., d : X X R m +,andd satisfies the standard axioms of a metric. Then, (X, ) d is an L-space. Example 1.3.5. Let (X,d) be a b-metric space (or quasi-metric space), i.e., d : X X R + satisfies the first two standard axioms of a metric, while the triangle inequality is replaced by d(x,y) s[d(x,z)+d(z,y)], for all x,y,z X, where s > 1isgiven, (see Refs. [21, 26, 27]). Then (X, d ) is an L-space. Example 1.3.6. Let X be a nonempty set. The functional d : X X R + is called a semimetric if it satisfies the first two standard axioms of a metric, while the pair (X,d) is called a semimetric space. Ifd : X X R + is continuous, then (X, ) d is an L-space. Example 1.3.7. Let X be a nonempty set, (G,+,, ) an L-space ordered group and G + := {g G g 0}. Letd : X X G + be a G + -metric, i.e., d satisfies the standard axioms of a metric. Then (X, ) d is an L-space. Let (X, ) be an L-space and f : X X be an operator.

6 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches By definition, f is a weakly Picard operator (WPO) if the sequence { f n (x)} n N of its iterates at x convergesfor all x X and its limit (which may depend on x)is a fixed point of f. If f is a WPO, then we consider the operator f : X X defined by f (x) := Lim f n (x), for all x X. If f is a WPO with F f = {x }, then, by definition, f is called a Picard operator (PO). If (X,d) is a metric space, f : X X is an operator and ψ : R + R + is a function then, by definition, f is a ψ-wpo if (i) ψ is increasing, continuous at 0 and ψ(0)=0; (ii) d(x, f (x)) ψ(d(x, f (x)) for all x X. For more considerations on weakly Picard operator theory (see Refs. [28, 30, 151, 156, 158]). 1.4. Caristi-Browder Operators on Metric Spaces We start with some remarks an Caristi operators on a metric space (X,d) (see Refs. [35, 61, 62, 77, 154]). Remark 1.4.1. An operator f : X X is a Caristi operator if and only if ( ) (c) d f k (x), f k+1 (x) is convergent for all x X, k=0 (see Ref. [62]). Indeed, if f is a ϕ-caristi operator then n ( ) d f k (x), f k+1 (x) ϕ(x) ϕ( f n+1 (x)) ϕ(x). k=0 If we have (c) satisfied, then where and hence (1.1.1) is satisfied. d(x, f (x)) = θ(x) θ( f (x)), for all x X, θ(x)= k=0 ( ) d f k (x), f k+1 (x), Remark 1.4.1 gives rise to the following problem.

Caristi-Browder Operator Theory in Distance Spaces 7 Problem 1.4.1. Which generalized contractions are Caristi operators? Example 1.4.1. Let (X,d) be a complete metric space and f : X X a Banach contraction, i.e., a map satisfying d( f (x), f (y)) ad(x,y), for all x,y X, (1.4.2) where 0 a < 1 is a constant. Then f is a Caristi operator (see Refs. [30, 150, 156] for details). Example 1.4.2. An operator f : X X is called a graphic contraction if there exists a constant 0 a < 1 such that d( f 2 (x), f (x)) ad(x, f (x)), for all x X. Then f is a Caristi operator (see Refs. [30, 150, 156] for details). Example 1.4.3. Let (X,d) be a complete metric space and f : X X a Kannan contraction, i.e., a map for which there exists b ( 0, 1 2) such that d( f (x), f (y)) b[d(x, f (x)) + d(y, f (y))], for all x,y X. (1.4.3) Then f is a Caristi operator (see Refs. [30, 90, 150, 156] for details). Example 1.4.4. Let (X,d) be a complete metric space and f : X X a Chatterjea contraction, i.e., a map for which there exists c ( 0, 1 2) such that d( f (x), f (y)) c[d(x, f (y)) + d(y, f (x))], for all x,y X. (1.4.4) Then f is a Caristi operator (see Refs. [30, 42, 150, 156] for details). Example 1.4.5. Let (X,d) be a complete metric space and f : X X be a Zamfirescu contraction, i.e., a map having the property that, for each pair x,y X, at least one of the conditions (1.4.2), (1.4.3), and (1.4.4) is satisfied. Then f is a Caristi operator (see Refs. [30, 150, 156, 186] for details). Example 1.4.6. Let (X,d) be a complete metric space and let f : X X be a Cirić quasi-contraction, i.e., a map for which there exists 0 < h < 1suchthat d( f (x), f (y)) h max{d(x,y),d(x, f (x)),d(y, f (y)),d(x, f (y)),d(y, f (x))}, (1.4.5) for all x,y X. Thenf is a Caristi operator (see Refs. [30, 46, 150, 156] for details). Example 1.4.7. Let (X,d) be a complete metric space and f : X X a ϕ-contraction, i.e., a map for which there exists a comparison function, i.e., a function ϕ : R + R +

8 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches satisfying (i ϕ ) ϕ is nondecreasing; (ii ϕ ) the sequence {ϕ n (t)} converges to zero, for each t R +, such that d( f (x), f (y)) ϕ(d(x,y)), for all x,y X. (1.4.6) If ϕ is a strong comparison function, then f is a Caristi operator (see Refs. [30, 150, 156] for more details). Example 1.4.8. Let (X,d) be a complete metric space and let f : X X be an almost contraction, i.e., a map for which there exist a constant δ (0,1) and some L 0such that d( f (x), f (y)) δ d(x,y)+ld(y, f (x)), for all x,y X. (1.4.7) Then f is a Caristi operator (see Refs. [30, 29, 150, 156] for details). Remark 1.4.2. We note that the Caristi operators in Examples 1.4.1 1.4.7 are POs, while the Caristi operator in Example 1.4.8 is a WPO. Remark 1.4.3. For f : X X, let us denote F f (X,R + ) := {ϕ : X R + : d(x, f (x)) ϕ(x) ϕ( f (x)) for all x X}. We note that (a) f is Caristi operator if and only if F f (X,R + ) /0; (b) ϕ f (x) := k=0 ( ) d f k (x), f k+1 (x) is the first element of the poset (F f (X,R + ), ) if and only if F f (X,R + ) /0. Remark 1.4.4. For ϕ : X R + let us denote F ϕ (X,X) := { f : X X d(x, f (x)) ϕ(x) ϕ( f (x)) for all x X}. Let f,g F ϕ (X,X) and X = X 1 X 2 be a partition of X. Leth : X X be defined by Then h F ϕ (X,X). { f (x), for x X1, h(x) := g(x), for x X 2. Remark 1.4.5. As 1 X F ϕ (X,X), it is impossible to control the set F f \Max(X, d,ϕ

Caristi-Browder Operator Theory in Distance Spaces 9 ) in F ϕ (X,X). So, the following problem arises. Problem 1.4.2. Let (X, ) be a poset with Max(X, ) /0. Let f : X X be a progressive operator, i.e., x f (x) for all x X. The problem is to study the set F f \ Max(X, ). Remark 1.4.6. Let CO(X,X) := { f : X X f is a Caristi operator}. We remark that if f CO(X,X),then f k CO(X,X) for all k N. Theorem 1.4.1. Let (X,d) be a complete metric space and let f : X X be a Caristi- Browder (CB) operator. Then (a) fisawpo; (b) if f is a ϕ-cb operator with ϕ(x) c d(x, f (x)) for some c > 0, thenfis ψ-wpo with ψ(t)=c t, t R +. Proof. (a) This is just the Caristi-Browder Theorem. (b) We have d (x, f (x)) n k=0 From this relation we have that d (x, f (x)) ( ) d f k (a), f k+1 (x) + d ( f n+1 (x), f (x) ), for all x X. k=0 d ( ) f k (x), f k+1 (x) ϕ(x) c d(x, f (x)). Theorem 1.4.2. Let (X,d) be a complete metric space and let f i : X X, i = 1,2, be ϕ i -CB operators with ϕ i (x) c i d(x, f i (x)), i= 1,2, forsomec i > 0. In addition, we suppose that there exists η > 0 such that Then d( f 1 (x), f 2 (x)) η, for all x X. H d (F f1,f f2 ) η max{c 1,c 2 }. Proof. By Theorem 1.4.1 we have that f i is a c i -WPO, i = 1,2. This implies that d (x, f i (x)) c i d(x, f i (x)), for all x X and all i = 1,2.

10 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches Let x 2 F f2.then d (x 2, f1 (x 2)) c 1 d(x 2, f 1 (x 2 )) = c 1 d( f 2 (x 2 ), f 1 (x 2 )) c 1 η. If x 1 F f1, then in a similar way we have that d(x 1, f2 (x 1)) c 2 η. Now the proof follows by Lemma 8.1.3(e) in Ref. [150]. In order to prove Theorem 1.4.3 we shall need the following lemma [151]. Lemma 1.4.1 [151, Abstract Gronwall Lemma]. Let (X,d, ) be an ordered metric space (i.e., a set X endowed with a metric d and a partial order relation which is closed with respect to d) and let f : X X be an operator satisfying the following two conditions: (i) fisapo(f f = {x f }); (ii) f is increasing. Then (a) x f (x) x x f ; (b) x f (x) x x f. Theorem 1.4.3. Let (X,d, ) be a complete ordered metric space and let f : X X be an operator. We suppose that (i) f : (X,d) (X,d) is a Caristi-Browder operator; (ii) f : (X, ) (X, ) is an increasing operator. Then (a) x X, x f (x) x f (x); (b) x X, x f (x) x f (x). Proof. By Theorem 1.4.1, f is a WPO with respect to. d On the other hand, the operator f : (X, ) (X, ) is increasing. So, we are in the conditions of the Abstract Gronwall Lemma and the conclusion follows. Using an abstract result for WPOs [157] (see also Ref. [238, Theorem 6.3.3]), from the above result we obtain the following Gronwall type results.

Caristi-Browder Operator Theory in Distance Spaces 11 Corollary 1.4.1. Let (X,d) be a complete ordered metric space and let f : X Xbe an increasing almost contraction. Then (a) x X, x f (x) x f (x); (b) x X, x f (x) x f (x). Proof. From Ref. [156, Theorem 6.3.3], we know that the following two statements are equivalent: (WP 1 ) There exists an L-space structure on the set X,, such that the mapping f : (X, ) (X, ) is a WPO; (WP 4 ) There exists a complete metric d on X and a number α (0,1) such that (i) f : (X,d) (X,d) has closed graphic; (ii) d ( f 2 (x), f (x) ) α d(x, f (x)),forallx,y X. On the other hand, from Ref. [30, Theorem 2.11], we know that (WP 1 ) holds, with the L-space structure. d Hence, (WP 4 ) also holds. Thus, by (ii) and Example 1.4.2, f is a Caristi operator and, by (i), f is a Caristi-Browder operator. The conclusion now follows by Theorem 1.4.3. Example 1.4.9. Let X = R with the usual metric d(x,y)= x y and let f : R R be defined by f (x)=0, if x (,3] and f (x)= 1 2,ifx (3,+ ). Then (a) f is a Kannan operator with b = 1 7 (see Ref. [129, Example 1.3.2]); hence f is an almost contraction too. (b) f is a Caristi operator (by Example 1.4.3); (c) f is not a Caristi-Browder operator (with respect to d). Indeed, f has no closed graphic, since for x n = 3 + 1 n (3,+ ) we have x n 3as n, f (x n )= 1 2 1 2 as n, but f (3)=0 1 2. Remark 1.4.7. Corollary 1.4.1 provides only sufficient conditions for an operator to have properties (a) and (b). Indeed, as shown by Example 1.4.9, f is nonincreasing and is not a Caristi-Browder operator, hence Corollary 1.4.1 cannot be applied. However, both conclusions of Corollary 1.4.1 hold since f (x)=0forallx R and {x R : x f (x)} =(,0), {x R : x f (x)} =[0, ). For more considerations on weakly Picard operator theory on metric spaces, see Refs. [19, 30, 151, 156, 158].

12 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches 1.5. Caristi-Browder Operators on R m + -Metric Spaces Let (X,d) be an R m + -metric space. By definition an operator f : X X is a Caristi operator if there exists ϕ : R m + Rm + such that d(x, f (x)) ϕ(x) ϕ( f (x)), for all x X. If, in addition, f has closed graphic, then f is called a Caristi-Browder operator. As in the case of ordinary metric spaces (i.e., d(x,y) R + ) we have the following. Remark 1.5.1. An operator f : X X is a Caristi operator if and only if ( ) d f k (x), f k+1 (x) is convergent for all x X. k=0 This remark is useful to give examples of Caristi operators in an R m +-metric space. Example 1.5.1. AmatrixS R+ m m is said to be convergent to zero if and only if S n 0as. Also, by definition, an operator f : X X is an S-contraction if there exists a matrix S R+ m m convergent to zero such that d( f (x), f (y)) Sd(x,y), for all x,y X. By the properties of matrices convergent to 0 (see Ref. [156]), and by Remark 1.5.1, it follows that any S-contraction is a Caristi operator. Example 1.5.2. An operator f : X X is called a graphic contraction if there exists amatrixs R+ m m convergent to zero such that d( f 2 (x), f (x)) Sd(x, f (x)), for all x X. By Remark 1.5.1, it follows that each graphic contraction is a Caristi operator. Example 1.5.3. Let X be a partially ordered set such that every pair x,y X has a lower and an upper bound. Furthermore, let d beametriconx such that (X,d) is a complete R m +-metric space and suppose also that f : X X is continuous, monotone (i.e., increasing or decreasing) and satisfies the following assumptions: (i) there exists a matrix A R+ m m convergent to zero such that d( f (x), f (y)) Ad(x,y), for all x y; (ii) there exists x 0 X such that x 0 f (x 0 ) or x 0 f (x 0 ). Then f is a Caristi-Browder operator (see Theorem 2.1 in Ref. [87] for details).

Caristi-Browder Operator Theory in Distance Spaces 13 Example 1.5.4. Let X be a partially ordered set such that every pair x,y X has a lower and an upper bound. Let d and ρ be two R m + -metrics on X. Letf : X X and suppose the following assumptions are satisfied: (i) d(x,y) ρ(x,y) for all x y; (ii) (X,d) is a generalized ordered complete R m + -metric space; (iii) f : (X,d) (X,d) is a continuous mapping; (iv) f is a monotone mapping; (v) there exists a matrix A R+ m m convergent to zero such that d( f (x), f (y)) Ad(x,y), for all x y; (vi) there exists x 0 X such that x 0 f (x 0 ) or x 0 f (x 0 ). Then f is a Caristi-Browder operator (see Ref. [87, Theorem 3.1] for details). Remark 1.5.2. For more details on the matrices convergent to zero and for the fixed point theory of operators on R m + -metric spaces, see Ref. [156, pp. 82 86] and the references therein (see also Ref. [25]). Theorem 1.5.1. Let (X,d) be a complete R m + -metric space and f : X X a Caristi- Browder operator. Then the following assertions hold. (a) fisawpo; (b) If f is a ϕ-cb operator with ϕ(x) Cd(x, f (x)) for some C R+ m m,then d (x, f (x)) Cd(x, f (x)), for all x X. Proof. (a) Let f be a ϕ-cb operator. Then we have n ( ) d f k (x), f k+1 (x) ϕ(x) ϕ ( f n+1 (x) ) ϕ(x), for all x X. k=0 Since R m + is regular (i.e., {x n} n N R m + increasing and bounded with respect to {x n } converges w. r. t. R m ), it follows that ( ) d f k (x), f k+1 (x) converges for all x X. k=0 This implies that f n (x) x (x) as n, for all x X.

14 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches But f has a closed graphic, so x (x) F f. (b) We have d (x, f (x)) This implies that n k=0 d (x, f (x)) ( ) d f k (x), f k+1 (x) + d ( f n+1 (x), f (x) ), for all x X. k=0 ( ) d f k (x), f k+1 (x) ϕ(x) Cd(x, f (x)). In a similar way to the considerations in Section 1.4 we have the following. Theorem 1.5.2. Let (X,d) be a complete R m +-metric space and f i,i= 1,2, ϕ i -CB operator with ϕ i (x) C i d(x, f i (x)) for some c i R+ m m. In addition we suppose that there exists η i R m + such that Then d( f 1 (x), f 2 (x)) η, for all x X. H d (F f1,f f2 ) Cd(x, f (x)), where C R+ m m such that C i C, i = 1,2. Theorem 1.5.3. Let (X,d, ) be a complete ordered R m + -metric space and f : X X an operator. We suppose that (i) f : (X,d) (X,d) is a Caristi-Browder operator; (ii) f : (X, ) (X, ) is an increasing operator. Then (a) x X, x f (x) x f (x); (b) x X, x f (x) x f (x). By Theorem 1.5.3 and Examples 1.5.1 1.5.4 we have the following results. Corollary 1.5.1. Let (X,d) be a complete ordered R m + -metric space and let f : X X be an increasing S-contraction. Then (a) x X, x f (x) x f (x);

Caristi-Browder Operator Theory in Distance Spaces 15 (b) x X, x f (x) x f (x). Corollary 1.5.2. Let (X,d) be a complete ordered R m + -metric space and let f : X X be an increasing graphic contraction. If f has closed graph, then (a) x X, x f (x) x f (x); (b) x X, x f (x) x f (x). Corollary 1.5.3. Let X be a partially ordered set such that every pair x,y X has a lower and an upper bound. Furthermore, let d be a metric on X such that (X,d) is a complete R m +-metric space and suppose also that f : X X is continuous, increasing and satisfies the following assumptions: (i) there exists a matrix A R+ m m convergent to zero such that d( f (x), f (y)) Ad(x,y), for all x y; (ii) there exists x 0 X such that x 0 f (x 0 ) or x 0 f (x 0 ). Then (a) x X, x f (x) x f (x); (b) x X, x f (x) x f (x). Proof. By Example 1.5.3, f is a Caristi-Browder operator. The conclusion now follows by Theorem 1.5.3. Corollary 1.5.4. Let X be a partially ordered set such that every pair x,y X has a lower and an upper bound. Let d and ρ be two R m + -metrics on X. Let f : X X and suppose the following assumptions are satisfied: (i) d(x,y) ρ(x,y), for all x y; (ii) (X,d) is a generalized ordered complete R m + -metric space; (iii) f : (X,d) (X,d) is a continuous mapping; (iv) f is increasing; (v) there exists a matrix A R+ m m convergent to zero such that d( f (x), f (y)) Ad(x,y), for all x y; (vi) there exists x 0 X such that x 0 f (x 0 ) or x 0 f (x 0 ).

16 Fixed Point Theory and Graph Theory: Foundations and Integrative Approaches Then (a) x X, x f (x) x f (x); (b) x X, x f (x) x f (x). For more considerations on weakly Picard operator theory on R m + -metric spaces, see Refs. [28, 30, 151, 156, 158]. 1.6. Caristi-Browder Operators on s(r + )-Metric Spaces Let s(r) := {{x n } n N : x n R,n N}. Then (s(r),+,r, ) is a linear ordered L- space, where is the termwise convergence. Let X be a nonempty set. By definition, a functional d : X X s(r + ) is an s(r + )-metric if d satisfies the corresponding standard Fréchet s axioms of a metric. Remark 1.6.1. A functional d : X X s(r + ), (x,y) (d k (x,y)) k N is an s(r + )- metric on X if and only if (i) d k is a pseudometric, for all k N, i.e., d k satisfies: (a) d k (x,x)=0forallx,y X; (b) d k (x,y)=d k (y,x) for all x,y X; (c) d k (x,z) d k (x,y)+d k (y,z) for all x,y,z X; (ii) for all x,y X, x y, k N such that d k (x,y) 0. Example 1.6.1. X := s(r), d(x,y) :=( x 0, y 0,..., x n y n,...). Example 1.6.2. X := C([a,b],s(R)), d(x,y) :=( x 0 y 0,..., x n y n,...). Example 1.6.3. X := C(R +,R), d(x,y) :=( x(0) y(0), x [0,1] y [0,1],..., x [0,n] y [0,n],...). Definition 1.6.1 [156, Definition 6.2.2]. An s(r + )-metric space (X,d) is complete (in the Weierstrass sense) if, for any {x n } X, n N d(x n,x n+1 ) converges {x n } converges. Let (X,d) be an s(r + )-metric space. By definition, an operator f : X X is a Caristi operator if there exists ϕ : X s(r + ) such that d(x, f (x)) ϕ(x) ϕ( f (x)), for all x X. (c)

Caristi-Browder Operator Theory in Distance Spaces 17 If, in addition, f has a closed graphic, then f is called a Caristi-Browder operator. Remark 1.6.2. Let ϕ : X s(r + ), x (ϕ 0 (x),...,ϕ n (x),...). Then condition (c) take the following form: d k (x, f (x)) ϕ k (x) ϕ k ( f (x)), for all x X, k N. For a more restrictive condition (c), see for example Ref. [110]. Theorem 1.6.1. Let (X,d) be a complete s(r + )-metric space and let f : X Xbea Caristi-Browder operator. Then the following assertions hold. (a) fiswpo; (b) If f is a ϕ-cb operator with for some c k 0, k N, then ϕ k (x) c k d k (x, f (x)), for all x X,k N, d k (x, f (x)) c k d k (x, f (x)), for all x X,k N. Proof. (a) Let f be a ϕ-cb operator. Then we have n i=0 This means that d k ( f i (x), f i+1 (x) ) ϕ k (x) ϕ k ( f n+1 (x) ) ϕ k (x), for all x X. i=0 andthisimpliesthat d k ( f i (x), f i+1 (x) ) converges for all x X, for all k N, f n (x) d x (x) as n, for all x X. But f has a closed graphic (as a Caristi-Browder operator), so x (x) F f. (b) We have d k (x, f (x)) n i=0 andthisimpliesthat d k (x, f (x)) d k ( f i (x), f i+1 (x) ) +d k ( f n+1 (x), f (x) ), for all x X and all k N i=0 d k ( f i (x), f i+1 (x) ) ϕ(x) c k d k (x, f (x)), for all k N.