Table Of ContentHANDBOOK OF METRIC FIXED POINT THEORY
Handbook of Metric
Fixed Point Theory
Edited by
William A. Kirk
Department ofM athematics,
The University of Iowa,
Iowa City, lA, U.S.A.
and
Brailey Sims
School ofM athematical and Physical Sciences,
The University of Newcastle,
Newcastle, Australia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5733-4 ISBN 978-94-017-1748-9 (eBook)
DOI 10.1007/978-94-017-1748-9
Printed on acid-free paper
All Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint of the hardcover 1s t edition 200 1
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording, or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface xi
Contraction mappings and extensions
W. A. Kirk
1.1 Introduction 1
1.2 The contraction mapping principle 3
1.3 Further extensions of Banach's principle 7
1.4 Caristi's theorem 14
1.5 Set-valued contractions 15
1.6 Generalized contractions 18
1. 7 Probabilistic metrics and fuzzy sets 20
1.8 Converses to the contraction principle 23
1.9 Notes and remarks 25
2
35
Examples of fixed point free mappings
B. Sims
2.1 Introduction 35
2.2 Examples on closed bounded convex sets 36
2.3 Examples on weak' compact convex sets 40
2.4 Examples on weak compact convex sets 43
2.5 Notes and remarks 47
3
49
Classical theory of nonexpansive mappings
K. Goebel and W. A. Kirk
3.1 Introduction 49
3.2 Classical existence results 50
3.3 Properties of the fixed point set 64
3.4 Approximation 69
3.5 Set-valued nonexpansive mappings 78
v
vi
3.6 Abstract theory 79
4
93
Geometrical background of metric fixed point theory
S. Pros
4.1 Introduction 93
4.2 Strict convexity and smoothness 93
4.3 Finite dimensional uniform convexity and smoothness 98
4.4 Infinite dimensional geometrical properties 108
4.5 Normal structure 118
4.6 Bibliographic notes 127
5
133
Some moduli and constants related to metric fixed point theory
E. L. Fuster
5.1 Introduction 133
5.2 Moduli and related properties 134
5.3 List of coefficients 157
6
177
Ultra-methods in metric fixed point theory
M. A. Khamsi and B. Sims
6.1 Introduction 177
6.2 Ultrapowers of Banach spaces 177
6.3 Fixed point theory 186
6.4 Maurey's fundamental theorems 193
6.5 Lin's results 195
6.6 Notes and remarks 197
7
201
Stability of the fixed point property for nonexpansive mappings
J. Garcia-Falset, A. Jimenez-Me/ado and E. Llorens-Fuster
7.1 Introduction 201
7.2 Stability of normal structure 204
7.3 Stability for weakly orthogonal Banach lattices 212
7.4 Stability of the property M(X) > 1 217
7.5 Stability for Hilbert spaces. Lin's theorem 223
7.6 Stability for the T-FPP 228
7.7 Further remarks 231
7.8 Summary 236
Contents VII
8
239
Metric fixed point results concerning measures of noncompactness
T. Dominguez, M. A. Japon and G. Lopez
8.1 Preface 239
8.2 Kuratowski and Hausdorff measures of noncompactness 240
8.3 ¢-minimal sets and the separation measure of noncompactness 244
8.4 Moduli of noncompact convexity 248
8.5 Fixed point theorems derived from normal structure 252
8.6 Fixed points in NUS spaces 257
8.7 Asymptotically regular mappings 260
8.8 Comments and further results in this chapter 264
9
269
Renormings of £1 and Co and fixed point properties
P. N. Dowling, C. J. Lennard and B. Turett
9.1 Preliminaries 269
9.2 Renormings of £1 and Co and fixed point properties 271
9.3 Notes and remarks 294
10
299
Nonexpansive mappings: boundary/inwardness conditions and local theory
w. A. Kirk and C. H. Morales
10.1 Inwardness conditions 299
10.2 Boundary conditions 301
10.3 Locally nonexpansive mappings 308
10.4 Locally pseudocontractive mappings 310
10.5 Remarks 320
11
323
Rotative mappings and mappings with constant displacement
W. Kaczor and M. Koter-Morgowska
11.1 Introduction 323
11.2 Rotative mappings 323
11.3 Firmly Iipschitzian mappings 330
11.4 Mappings with constant displacement 333
11.5 Notes and remarks 336
12
339
Geometric properties related to fixed point theory in some Banach function lattices
S. Chen, Y. Cui, H. Hudzik and B. Sims
12.1 Introduction 339
viii
12.2 Normal structure, weak normal structure, weak sum property, sum property
and uniform normal structure 343
12.3 Uniform rotundity in every direction 356
12.4 B-convexity and uniform monotonicity 358
12.5 Nearly uniform convexity and nearly uniform smoothness 362
12.6 WORTH and uniform nonsquareness 367
12.7 Opial property and uniform opial property in modular sequence spaces 368
12.8 Garcia-Falset coefficient 377
12.9 Cesaro sequence spaces 378
12.10 WCSC, uniform opial property, k-NUC and UNS for cesp 380
13
391
Introduction to hyperconvex spaces
R. Espinola and M. A. Khamsi
13.1 Preface 391
13.2 Introduction and basic definitions 393
13.3 Some basic properties of hyperconvex spaces 394
13.4 Hyperconvexity, injectivity and retraction 399
13.5 More on hyperconvex spaces 405
13.6 Fixed point property and hyperconvexity 411
13.7 Topological fixed point theorems and hyper convexity 415
13.8 Isbell's hyperconvex hull 418
13.9 Set-valued mappings in hyperconvex spaces 422
13.10 The KKM theory in hyperconvex spaces 428
13.11 Lambda-hyperconvexity 431
14
437
Fixed points of holomorphic mappings: a metric approach
T. Kuczumow, S. Reich and D. Shoikhet
14.1 Introduction 437
14.2 Preliminaries 438
14.3 The Kobayashi distance on bounded convex domains 440
14.4 The Kobayashi distance on the Hilbert ball 447
14.5 Fixed points in Banach spaces 450
14.6 Fixed points in the Hilbert ball 454
14.7 Fixed points in finite powers of the Hilbert ball 460
14.8 Isometries on the Hilbert ball and its finite powers 465
14.9 The extension problem 469
14.10 Approximating sequences in the Hilbert ball 472
14.11 Fixed points in infinite powers of the Hilbert ball 481
14.12 The Denjoy-Wolff theorem in the Hilbert ball and its powers 483
14.13 The Denjoy-Wolff theorem in Banach spaces 490
Contents ix
14.14 Retractions onto fixed point sets 496
14.15 Fixed points of continuous semigroups 502
14.16 Final notes and remarks 507
15
517
Fixed point and non-linear ergodic theorems for semigroups of non-linear mappings
A. Lau and W. Takahashi
15.1 Introduction 517
15.2 Some preliminaries 518
15.3 Submean and reversibility 519
15.4 Submean and normal structure 523
15.5 Fixed point theorem 527
15.6 Fixed point sets and left ideal orbits 532
15.7 Ergodic theorems 538
15.8 Related results 545
16
557
Generic aspects of metric fixed point theory
S. Reich and A. J. Zaslavski
16.1 Introduction 557
16.2 Hyperbolic spaces 557
16.3 Successive approximations 558
16.4 Contractive mappings 561
16.5 Infinite products 564
16.6 (F}-attracting mappings 567
16.7 Contractive set-valued mappings 568
16.8 Nonexpansive set-valued mappings 569
16.9 Porosity 570
17
577
Metric environment of the topological fixed point theorems
K. Goebel
17.1 Introduction 577
17.2 Schauder's theorem 579
17.3 Minimal displacement problem 586
17.4 Optimal retraction problem 597
17.5 The case of Hilbert space 604
17.6 Notes and remarks 608
18
613
Order-theoretic aspects of metric fixed point theory
x
J. Jachymski
18.1 Introduction 613
18.2 The Knaster-Tarski theorem 614
18.3 Zermelo's fixed point theorem 623
18.4 The Tarski-Kantorovitch theorem 630
19
643
Fixed point and related theorems for set-valued mappings
G. Yuan
19.1 Introduction 643
19.2 Knaster-Kuratowski-Mazurkiewicz principle 644
19.3 Ky Fan minimax principle 651
19.4 Ky Fan minimax inequality-I 653
19.5 Ky Fan minimax inequality-II 659
19.6 Fan-Glicksberg fixed points in G-convex spaces 662
19.7 Nonlinear analysis of hyperconvex metric spaces 666
Index 691
Preface
The presence or absence of a fixed point is an intrinsic property of a map. However,
many necessary or sufficient conditions for the existence of such points involve a mixture
of algebraic, order theoretic, or topological properties of the mapping or its domain.
Metric fixed point theory is a rather loose knit branch of fixed point theory concerning
methods and results that involve properties of an essentially isometric nature. That
is, the class of mappings and domains satisfying the properties need not be preserved
under the move to an equivalent metric. It is this fragility that singles metric fixed
point theory out from the more general topological theory, although, as many of the
entries in this Handbook serve to illustrate, the divide between the two is often a vague
one.
The origins of the theory, which date to the latter part of the nineteenth century, rest
in the use of successive approximations to establish the existence and uniqueness of
solutions, particularly to differential equations. This method is associated with the
names of such celebrated mathematicians as Cauchy, Liouville, Lipschitz, Peano, Fred
holm and, especially, Picard. In fact the precursors of a fixed point theoretic approach
are explicit in the work of Picard. However, it is the Polish mathematician Stefan
Banach who is credited with placing the underlying ideas into an abstract framework
suitable for broad applications well beyond the scope of elementary differential and
integral equations. Around 1922, Banach recognized the fundamental role of 'metric
completeness'; a property shared by all of the spaces commonly exploited in analysis.
For many years, activity in metric fixed point theory was limited to minor extensions
of Banach's contraction mapping principal and its manifold applications. The theory
gained new impetus largely as a result of the pioneering work of Felix Browder in the
mid-nineteen sixties and the development of nonlinear functional analysis as an active
and vital branch of mathematics. Pivotal in this development were the 1965 existence
theorems of Browder, Gohde, and Kirk and the early metric results of Edelstein. By
the end of the decade, a rich fixed point theory for nonexpansive mappings was clearly
emerging and it was equally clear that such mappings played a fundamental role in
many aspects of nonlinear functional analysis with links to variational inequalities and
the theory of monotone and accretive operators.
Nonexpansive mappings represent the limiting case in the theory of contractions, where
the Lipschitz constant is allowed to become one, and it was clear from the outset
that the study of such mappings required techniques going far beyond purely metric
arguments. The theory of nonexpansive mappings has involved an intertwining of
geometrical and topological arguments. The original theorems of Browder and Giihde
exploited special convexity properties of the norm in certain Banach spaces, while Kirk
identified the underlying property of 'normal structure' and the role played by weak
compactness. The early phases of the development centred around the identification
of spaces whose bounded convex sets possessed normal structure, and it was soon
xi
