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Handbook of Metric Fixed Point Theory PDF

702 Pages·2001·18.05 MB·English
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HANDBOOK 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

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