An Introduction to Metric Spaces and Fixed Point Theory

304 Pages·2010·5.32 MB·English

by  Mohamed A. Khamsi, William A. Kirk(auth.)

An Introduction to
Metric Spaces and
Fixed Point Theory
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
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.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any
form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise,
except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without
either the prior written permission of the Publisher, or authorization through payment of the
appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA
01923, [phone], fax [phone]. Requests to the Publisher for permission should be
addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York,
NY 10158-0012, [phone], fax [phone], E-Mail:
Contents
Preface ix
I Metric Spaces
1 Introduction 3
1.1 The real numbers R 3
1.2 Continuous mapings 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 1
2 Metric Spaces 13
2.1 The metric topology 15
2.2 Examples of metric spaces 19
2.3 Completenes 26
2.4 Separability and conectednes 3
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 5
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 7
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 Aproximate fixed points 89
4.7 Isbel's hyperconvex hul 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 10
5.5 Quasi-normal structure 12
5.6 Stability and normal structure 15
5.7 Ultrametric spaces 16
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 £2 revisited 132
6.4 The modulus of convexity 136
6.5 Uniform convexity of the tp spaces 138
6.6 The dual space: Hahn-Banach Theorem 142
6.7 The weak and weak* topologies 14
6.8 The spaces c, CQ, t\ and ^ 146
6.9 Some more general facts 148
6.10 The Schur property and £j 150
6.1 More on Schauder bases in Banach spaces 154
6.12 Uniform convexity and reflexivity 163
6.13 Banach latices 165
Exercises 168
7 Continuous Mapings in Banach Spaces 171
7.1 Introduction 171
7.2 Brouwer's Theorem 173
7.3 Further coments on Brouwer's Theorem 176
7.4 Schauder's Theorem 179
7.5 Stability of Schauder's Theorem 180
7.6 Banach algebras: Stone Weierstras Theorem 182
7.7 Leray-Schauder degre 183
7.8 Condensing mapings 187
7.9 Continuous mapings in hyperconvex spaces 191
Exercises 195
CONTENTS vi
8 Metric Fixed Point Theory 197
8.1 Contraction mapings 197
8.2 Basic theorems for nonexpansive mapings 19
8.3 A closer lok at ίλ 205
8.4 Stability results in arbitrary spaces 207
8.5 The Goebel-Karlovitz Lema 21
8.6 Orthogonal convexity 213
8.7 Structure of the fixed point set 215
8.8 Asymptoticaly regular mapings 219
8.9 Set-valued mapings 2
8.10 Fixed point theory in Banach latices 25
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 mapings to X 25
9.6 Some fixed point theorems 257
9.7 Asymptoticaly nonexpansive mapings 262
9.8 The demiclosednes principle 263
9.9 Uniformly non-creasy spaces 264
Exercises 270
Apendix: Set Theory 273
A.l Mapings 273
A.2 Order relations and Zermelo's Theorem 274
A.3 Zorn's Lema and the Axiom Of Choice 275
A.4 Nets and subnets 27
A.5 Tychonof'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