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Fixed Point Theory in Distance Spaces
William Kirk
•
Naseer Shahzad
Fixed Point Theory in Distance Spaces
123
William Kirk Department of Mathematics University of Iowa Iowa City, IA, USA
Naseer Shahzad Department of Mathematics King Abdulaziz University Jeddah, Saudi Arabia
ISBN 978-3-319-10926-8 ISBN 978-3-319-10927-5 (eBook) DOI 10.1007/978-3-319-10927-5 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014948344 Mathematics Subject Classification (2000): 54H25, 51K10, 54C05, 47H09 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Abstract. Traditionally, a large body of metric fixed point theory has been couched in a functional analytic framework. This aspect of the theory has been written about extensively. This survey treats the purely metric aspects of the theory—specifically results that do not depend on any algebraic structure of the underlying space. The focus is on (I) metric spaces satisfying additional geometric conditions, (II) metric spaces with geodesic structures, and (III) semimetric spaces satisfying relaxed versions of the triangle inequality.
Preface Mathematicians interested in topology typically give an abstract set a “topological structure” consisting of a collection of subsets of the given set to determine when points are “near” each other. People interested in geometry need a more rigid notion of nearness. This usually begins with assigning a symmetric “distance” to each two points of a set, resulting in the notion of a semimetric. With the addition of the triangle inequality, one passes to a metric space. This will be our point of departure. There are four classical fixed point theorems against which metric extensions are usually checked. These are, respectively, the Banach contraction mapping principle, Nadler’s well-known set-valued extension of that theorem, the extension of Banach’s theorem to nonexpansive mappings, and Caristi’s theorem. These comparisons form a significant component of this survey. This exposition is divided into three parts. In Part I we discuss some aspects of the purely metric theory, especially Caristi’s theorem and its relatives. Among other things, we discuss these theorems in the context of their logical foundations. We omit a discussion of the well-known Banach Contraction Principle and its many generalizations in Part I because this topic is well known and has been reviewed extensively elsewhere (see, e.g., [117]). In Part II we discuss classes of spaces which, in addition to having a metric structure, also have geometric structure. These specifically include the geodesic spaces, length spaces, and CAT(0) spaces. In Part III we turn to distance spaces that are not necessarily metric. These include certain distance spaces which lie strictly between the class of semimetric spaces and the class of metric spaces, as well as other spaces whose distance properties do not fully satisfy the metric axioms. We make no attempt to explain all aspects of the topics we cover nor to present a compendium of all known facts, especially since the theory continues to expand at a rapid rate. Any attempt to provide the latest tweak on the various theorems we discuss would surely be outdated before reaching print. Our objective rather is to present a concise accessible document which can be used as an introduction to the subject and its central themes. We include proofs selectively, and from time to time we mention open problems. The material in this exposition is collected together here for the first time. Those VII
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PREFACE
wishing to investigate these topics deeper are referred to the original sources. We have attempted to include details in those instances where the sources are not readily available. This might be the case, for example, when the source is in a conference proceedings. Also some results appear here for the first time. Many of the concepts introduced here have found interesting applications. Indeed some were motivated by attempts to address both mathematical and applied problems. Other concepts we discuss are more formal in nature and have yet to find any serious application; indeed some may never. However our hope is that this discussion will suggest directions for those interested in further research in this area. The first author lectured on portions of the material covered in this monograph to students and faculty at King Abdulaziz University. He wishes to thank them for providing an attentive and critical audience. Both authors express their gratitude to Rafa Espínola for calling attention to a number of oversights in an earlier draft of this manuscript. Iowa City, IA, USA Jeddah, Saudi Arabia
William Kirk Naseer Shahzad
Contents Preface
VII
Contents
IX
Part I. Metric Spaces
1
Chapter 1.
3
Introduction
Chapter 2. Caristi’s Theorem and Extensions 2.1. Introduction 2.2. A Proof of Caristi’s Theorem 2.3. Suzuki’s Extension 2.4. Khamsi’s Extension 2.5. Results of Z. Li 2.6. A Theorem of Zhang and Jiang
7 7 9 11 11 16 18
Chapter 3. Nonexpansive Mappings and Zermelo’s Theorem 3.1. Introduction 3.2. Convexity Structures
19 19 19
Chapter 4.
23
Hyperconvex Metric Spaces
Chapter 5. Ultrametric Spaces 5.1. Introduction 5.2. Hyperconvex Ultrametric Spaces 5.3. Nonexpansive Mappings in Ultrametric Spaces 5.4. Structure of the “Fixed Point Set” of Nonexpansive Mappings 5.5. A Strong Fixed Point Theorem 5.6. Best Approximation
25 25 27 28 30 31 35
Part II. Length Spaces and Geodesic Spaces
37
Chapter 6. Busemann Spaces and Hyperbolic Spaces 6.1. Convex Combinations in a Busemann Space
39 42
Chapter 7. Length Spaces and Local Contractions 7.1. Local Contractions and Metric Transforms
47 54
IX
X
CONTENTS
Chapter 8. The G-Spaces of Busemann 8.1. A Fundamental Problem in G-Spaces
61 63
Chapter 9. CAT(0) Spaces 9.1. Introduction 9.2. CAT(κ) Spaces 9.3. Fixed Point Theory 9.4. A Concept of “Weak” Convergence 9.5. Δ-Convergence of Nets 9.6. A Four Point Condition 9.7. Multimaps and Invariant Approximations 9.8. Quasilinearization
65 65 66 70 81 83 86 89 93
Chapter 10. Ptolemaic Spaces 10.1. Some Properties of Ptolemaic Geodesic Spaces 10.2. Another Four Point Condition
95 96 98
11. R-Trees (Metric Trees) The Fixed Point Property for R-Trees The Lifšic Character of R-Trees Gated Sets Best Approximation in R-Trees Applications to Graph Theory
99 100 102 105 106 109
Chapter 11.1. 11.2. 11.3. 11.4. 11.5.
Part III. Beyond Metric Spaces
111
Chapter 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12.7. 12.8.
12. b-Metric Spaces Introduction Banach’s Theorem in a b-Metric Space b-Metric Spaces Endowed with a Graph Strong b-Metric Spaces Banach’s Theorem in a Relaxedp Metric Space Nadler’s Theorem Caristi’s Theorem in sb-Metric Spaces The Metric Boundedness Property
113 113 115 116 121 124 125 128 129
Chapter 13.1. 13.2. 13.3.
13. Generalized Metric Spaces Introduction Caristi’s Theorem in Generalized Metric Spaces Multivalued Mappings in Generalized Metric Spaces
133 133 136 139
Chapter 14.1. 14.2. 14.3. 14.4.
14. Partial Metric Spaces Introduction Some Examples The Partial Metric Contraction Mapping Theorem Caristi’s Theorem in Partial Metric Spaces
141 141 143 143 144
CONTENTS
14.5. 14.6. Chapter 15.1. 15.2. 15.3.
Nadler’s Theorem in Partial Metric Spaces Further Remarks 15. Diversities Introduction Hyperconvex Diversities Fixed Point Theory
XI
148 152 153 153 155 155
Bibliography
159
Index
173
