An Introduction to Metric Spaces and Fixed Point Theory

304 Pages·2001·5.317 MB·English

by  Mohamed A. Khamsi, William A. Kirk

An Introduction to
Metric Spaces and
Fixed Point Theory
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
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Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX
Editors Emeriti: PETER HILTON and HARRY HOCHSTADT, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
An Introduction to
Metric Spaces and
Fixed Point Theory
MOHAMED A. KHAMSI
WILLIAM A. KIRK
A Wiley-lnterscience Publication
JOHN WILEY & SONS, INC.
New York / Chichester / Weinheim / Brisbane / Singapore / Toronto
This text is printed on acid-free paper. ©
Copyright © 2001 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging in Publication Data
Khamsi, Mohamed A.
An introduction to metric spaces and fixed point theory / M.A. Khamsi, W.A. Kirk.
p. cm. — (Pure and applied mathematics (Wiley-Interscience series of texts,
monographs, and tracts))
Includes bibliographical references and index.
ISBN 0-471-41825-0
1. Metric spaces. 2. Fixed point theory. I. Kirk, W. A. II. Title. III. Pure and applied
mathematics (John Wiley & Sons : Unnumbered)
QA611.28 K48 2001
5I4'.32—dc21 00-068491
10 9 8 7 6 5 4 3 21
Contents
Preface ix
I Metric Spaces
1 Introduction 3
1.1 The real numbers R 3
1.2 Continuous mappings in R 5
1.3 The triangle inequality in R 7
1.4 The triangle inequality in R" 8
1.5 Brouwer's Fixed Point Theorem 10
Exercises 11
2 Metric Spaces 13
2.1 The metric topology 15
2.2 Examples of metric spaces 19
2.3 Completeness 26
2.4 Separability and connectedness 33
2.5 Metric convexity and convexity structures 35
Exercises 38
3 Metric Contraction Principles 41
3.1 Banach's Contraction Principle 41
3.2 Further extensions of Banach's Principle 46
3.3 The Caristi-Ekeland Principle 55
3.4 Equivalents of the Caristi-Ekeland Principle 58
3.5 Set-valued contractions 61
3.6 Generalized contractions 64
Exercises 67
4 Hyperconvex Spaces 71
4.1 Introduction 71
4.2 Hyperconvexity 77
4.3 Properties of hyperconvex spaces 80
4.4 A fixed point theorem 84
v
vi CONTENTS
4.5 Intersections of hyperconvex spaces 87
4.6 Approximate fixed points 89
4.7 Isbell's hyperconvex hull 91
Exercises 98
5 "Normal" Structures in Metric Spaces 101
5.1 A fixed point theorem 101
5.2 Structure of the fixed point set 103
5.3 Uniform normal structure 106
5.4 Uniform relative normal structure 110
5.5 Quasi-normal structure 112
5.6 Stability and normal structure 115
5.7 Ultrametric spaces 116
5.8 Fixed point set structure—separable case 120
Exercises 123
II Banach Spaces
6 Banach Spaces: Introduction 127
6.1 The definition 127
6.2 Convexity 131
6.3 £ revisited 132
2
6.4 The modulus of convexity 136
6.5 Uniform convexity of the t spaces 138
p
6.6 The dual space: Hahn-Banach Theorem 142
6.7 The weak and weak* topologies 144
6.8 The spaces c, CQ, t\ and ^ 146
6.9 Some more general facts 148
6.10 The Schur property and £j 150
6.11 More on Schauder bases in Banach spaces 154
6.12 Uniform convexity and reflexivity 163
6.13 Banach lattices 165
Exercises 168
7 Continuous Mappings in Banach Spaces 171
7.1 Introduction 171
7.2 Brouwer's Theorem 173
7.3 Further comments on Brouwer's Theorem 176
7.4 Schauder's Theorem 179
7.5 Stability of Schauder's Theorem 180
7.6 Banach algebras: Stone Weierstrass Theorem 182
7.7 Leray-Schauder degree 183
7.8 Condensing mappings 187
7.9 Continuous mappings in hyperconvex spaces 191
Exercises 195
CONTENTS vii
8 Metric Fixed Point Theory 197
8.1 Contraction mappings 197
8.2 Basic theorems for nonexpansive mappings 199
8.3 A closer look at ί 205
λ
8.4 Stability results in arbitrary spaces 207
8.5 The Goebel-Karlovitz Lemma 211
8.6 Orthogonal convexity 213
8.7 Structure of the fixed point set 215
8.8 Asymptotically regular mappings 219
8.9 Set-valued mappings 222
8.10 Fixed point theory in Banach lattices 225
Exercises 238
9 Banach Space Ultrapowers 243
9.1 Finite representability 243
9.2 Convergence of ultranets 248
9.3 The Banach space ultrapower X 249
9.4 Some properties of X 252
9.5 Extending mappings to X 255
9.6 Some fixed point theorems 257
9.7 Asymptotically nonexpansive mappings 262
9.8 The demiclosedness principle 263
9.9 Uniformly non-creasy spaces 264
Exercises 270
Appendix: Set Theory 273
A.l Mappings 273
A.2 Order relations and Zermelo's Theorem 274
A.3 Zorn's Lemma and the Axiom Of Choice 275
A.4 Nets and subnets 277
A.5 Tychonoff's Theorem 278
A.6 Cardinal numbers 280
A. 7 Ordinal numbers and transfinite induction 281
A.8 Zermelo's Fixed Point Theorem 284
A.9 A remark about constructive mathematics 286
Exercises 287
Bibliography 289
Index 301
Preface
This text is primarily an introduction to metric spaces and fixed point theory.
It is intended to be especially useful to those who might not have ready access to
other sources, or to groups of people with diverse mathematical backgrounds.
Because of this the text is self-contained. Introductory properties of metric
spaces and Banach spaces are included, and an appendix contains a summary
of the concepts of set theory (Zorn's Lemma, Tychonoff's Theorem, transfinite
induction, etc.) that might be encountered elsewhere in the text. Most of the
text should be accessible to reasonably mature students who have had very
little training in mathematics beyond calculus. In particular a very elementary
treatment of Brouwer's Theorem is given in Chapter 7. At the same time later
chapters of the book contain a large amount of material that might be of interest
to more advanced students and even to serious scholars.
Readers with a good background in elementary real analysis should skip
Chapters 1 and 2, and those who have had a course in functional analysis should
also skip Chapter 6. Those who have had a course in set theory will have little
use for the Appendix. Most readers will find something new in the remaining
chapters and they might find the inclusion of this other material helpful as well.
Although a number of exercises are included, only rarely are important de-
tails of the major developments left to the reader. However in order to focus
on the main development some peripheral material is included without proof,
especially in later chapters.
Despite the fact that the text is largely self-contained, extensive bibliographic
references are included.
In terms of content this text overlaps in places with three recent books on
fixed point theory: Nonstandard Methods in Fixed Point Theory by A. Aksoy
and M. A. Khamsi (Springer-Verlag, New York, Berlin, 1990, 139 pp.), Topics
in Metric Fixed Point Theory by K. Goebel and W. A. Kirk (Cambridge Univ.
Press, Cambridge, 1990, 244 pp.), and Nonlinear Functional Analysis and its
Applications I: Fixed-Point Theorems by E. Zeidler (Springer-Verlag, New York,
Berlin, Heidelberg, Tokyo, 1986, 897 pp.). However, in addition to the inclusion
of excercises, the level of presentation and the comprehensive development of
what is known in a purely metric context (especially in hyperconvex spaces) is
unique to this treatment. Among other things, it has been our hope especially to
illustrate the richness and depth of the abstract metric theory. Also a number of
Banach space results are included here which appear in none of the above books,
IX
X PREFACE
some by choice and some because they describe more recent developments in
the subject.
Material on the general theory of Banach space geometry is drawn from
many sources but one is worth special mention: Introduction to Banach Spaces
and their Geometry, Second revised edition, by B. Beauzamy (North-Holland,
Amsterdam, New York, Oxford, 1985).
This book could easily serve as a text for an introductory course in metric and
Banach spaces. In this case material should be drawn selectively from Chapters
1 through 4 along with Chapters 6 and 7, and the Appendix as needed. A
number of exercises have been included at the end of each of these chapters.
The second author lectured on portions of the material covered in the text
to students of the I. C. T. P.- Trieste Diploma Program in Mathematics during
May 1998. He wishes to thank them for providing an attentive and critical
audience. Both authors express their deep gratitude to Rafael Espfnola for
calling attention to a number of oversights in the penultimate draft of this text.
M. A. KHAMSI
W. A. KIRK
Iowa City
December 2000
An Introduction to Metric Spaces and Fixed Point Theory
by Mohamed A. Khamsi and William A. Kirk
Copyright © 2001 John Wiley & Sons, Inc.
Part I
Metric Spaces