Topics in nonlinear analysis & applications

710 Pages·1997·256.477 MB·English

by  Donald H Hyers, George Isac, Themistocles M Rassias

Topics in
NONLINEAR
ANALYSIS
APPLICATIONS
This page is intentionally left blank
Topics in
NONPAR
ANAlksiS
APPLIOJIONS
Donald H Hyers
Department of Mathematics
University of Southern California, USA
George Isac
Department of Mathematics and Computer Science
Royal Military College of Canada
Themistocles M Rassias
Department of Mathematics
National Technical University of Athens, Greece
World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS
Copyright © 1997 by World Scientific Publishing Co. Pte. Ltd.
A11 rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic
or mechanical, including photocopying, recording or any information storage and retrieval system now
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required from the publisher.
ISBN 981-02-2534-2
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Printed in Singapore by UtoPrint
CONTENTS
Preface IX
CHAPTER 1 1
Cones and Complementarity Problems
1. INTRODUCTION 1
2. CONVEX CONES 2
• Normal cones 4
• Regular and completely regular cones 8
• Well based cones 10
• Isotone projection cones 16
• Galerkin cones 23
3. COMPLEMENTARITY PROBLEMS 27
• The explicit complementarity problem 30
• The implicit complementarity problem 39
• The generalized order complementarity problem 42
4. EXISTENCE THEOREMS 47
• Galerkin cones and the generalized Karamardian condition 47
• Galerkin cones and conically coercive functions 54
• Variational inequalities and explicit complementarity pro
blems 59
• Isotone projection cones and complementarity problems 69
• Comment 81
• Complementarity problems and condition (S) 82
• S-variational inequalities and the implicit complementarity
problem 95
• Heterotonic operators and the generalized order complemen
tarity problem 104
• Topological degree and complementarity 117
CONTENTS
5. SOME SPECIAL PROBLEMS IN COMPLEMENTARITY
THEORY 132
• Boundedness of the solution-set 132
• Solution which is the least element of the feasible set 135
• The cardinality of the solution-set 139
• Nonexistence of solution 139
• Sensitivity analysis 139
• Nonlinear complementarity and quasi-equilibria 139
6. COMPLEMENTARITY AND FIXED POINTS 140
7. REFERENCES 148
CHAPTER 2 167
Metrics on Convex Cones
1. INTRODUCTION 167
2. HILBERT'S PROJECTIVE METRIC 170
3. THOMPSON'S METRIC 208
4. WORKING WITH TWO CONES 220
5. MONOTONE SEMIGROUPS AND METRICS ON CONES 228
6. REFERENCES 233
CHAPTER 3 241
Zero-Epi Mappings
1. INTRODUCTION 241
2. ZERO-EPI MAPPINGS ON BOUNDED SETS 243
3. ZERO-EPI MAPPINGS ON THE WHOLE SPACE 255
4. ZERO-EPI MAPPINGS ON CONES 271
5. ZERO-EPI FAMILIES OF MAPPINGS AND OPTIMIZATION 284
6. ZERO-EPI MAPPINGS AND COMPLEMENTARITY PROB
LEMS 304
7. ZERO-EPI MAPPINGS AND k-SET CONTRACTION 315
8. REFERENCES 318
VI
CONTENTS
CHAPTER 4 325
Variational Principles
1. INTRODUCTION 325
2. PRELIMINARIES 328
3. CRITICAL POINTS FOR DYNAMICAL SYSTEMS 343
4. VARIANTS OF EKJELAND'S VARIATIONAL PRINCIPLE 355
5. THE DROP THEOREM 365
6. STRONG FORMS AND GENERALIZATIONS OF EKE-
LAND'S PRINCIPLE 379
7. EQUIVALENCIES 426
8. EKELAND'S VARIATIONAL PRINCIPLE FOR VECTOR
VALUED FUNCTIONS 433
9. APPLICATIONS 444
• Existence of solutions for minimizing problems 444
• Coercivity condition 452
• A global variational principle on cones 454
• Density result 463
• Mountain pass lemma 464
• The Bishop-Phelps theorem 477
• Clarke's fixed point theorem 480
• Borwein's c-principle 484
• <D-accretive operators and surjectivity 488
• Zabreiko-Krasnoselskii's theorem 494
• The drop property and the geometry of Banach spaces 497
• The drop property for arbitrary sets 519
10.REFERENCES 539
CHAPTER 5 555
Maximal Element Principles
1. INTRODUCTION 555
VII
CONTENTS
2. PRELIMINARIES 556
3. VARIATION ON ZORN'S LEMMA 562
• Applications 567
• Comments 575
4. NEW MAXIMAL ELEMENT PRINCIPLES 576
• Applications 608
• A fixed point theorem for ordered sets 608
• Maximality and solvability 609
• Variable drops and general solvability 618
• A drop theorem 624
• Lipschitzianess tests 626
• Maximal element principles and general Newton-
Kantorovich processes 633
• Comments 640
5. PARETO EFFICIENCY 641
6. REFERENCES 672
Symbol Index 685
Subject Index 691
VIII
PREFACE
"Pourquoi aspirons-nous a aller de I 'avant,
toujours de I 'avant? -Parce que la vie est
seulement dans le devoilement."
Leon Tolstoi -Journaux et cornets Vol. Ill
(1905-1910) Bibliotheque de la Pleiade, pp 232
For a long period of time a great deal of mathematical activity has been
devoted to problems and theories associated with linearity. This activity
has been supported by the interest to extend several known results of linear
algebra from finite-dimensional spaces to infinite-dimensional spaces. By
combining linear structure and topological structure, the modern nonlinear
analysis arrived at its present form and structure. This, in turn, provides an
interplay of functional analysis, differential equations and geometry which
are associated with mathematical models that are defined in physics,
economics, biology, chemistry, engineering, etc. The reality and complexity
of several practical problems, imposed the necessity to study nonlinear
problems.
It is unanimously accepted that for several nonlinear problems, linearization
is not always the right approach. We believe that this fact has influenced an
impressive body of research in nonlinear analysis after 1960. The initial
development of nonlinear analysis was justified by the study of nonlinear
functional equations. In particular, topics like Fixed Point Theory,
Topological Degree, Nonlinear Fredholm Alternatives and Surjectivity
Theorems, Monotone, Pseudomonotone and Accretive Operators,
Bifurcation Theory etc. are now considered as classical subjects in
Nonlinear Analysis.
IX
New domains, as for example, Variational and Quasivariational Inequa
lities, Non Smooth Analysis, Convex Analysis, Variational Principles,
Complementarity Theory etc. are now subjects of active research interest.
The variety of fields in nonlinear analysis is now very impressive. The
present book, Topics in Nonlinear Analysis and Applications is a selec
tion of five chapters: Cones and Complementarity Problems, Metrics on
Convex Cones, Zero-Epi Mappings, Variational Principles, and Maximal
Element Principles. Every chapter is, in some sense, independent, but there
are several common elements. An attempt, however, has been made to
present the material in an integrated and a self-contained way.
The chapters presented in this book represent new directions in research. In
every chapter, we try to discuss results with the proofs that are considered
(to us at least) significant and related to some new ideas in nonlinear
analysis, as well as to some new mathematical models or practical
problems. Because of these aspects, every chapter is not exhaustive; hence
the title of the book, "Topics in Nonlinear Analysis and Applications". It
is hoped that every chapter can be considered as the subject of a stimulating
graduate course or as a source of new research in nonlinear analysis for a
self-study or seminar. Some 500 references have been cited here, including
preprints. As a rule, we have studied the original sources and in some cases
have retrieved some forgotten but useful results. Each chapter begins with
an introduction, and ends with a list of references. The book concludes
with a symbol index and a subject index.
The first chapter deals with the notion of convex cones and with the
nonlinear complementarity theory. The most important kinds of convex
cones considered in the nonlinear complementarity theory, and in the study
of other problems in nonlinear analysis, are discussed. In particular, the
concepts of normal cone, regular and completely regular cones, the well
based cones, the isotone projection cones, the Galerkin cones etc. are all
studied. The complementarity theory is related to optimization and game
theory. Generally, it is related to the study of economical equilibrium or to
the study of equilibrium as considered in some problems in elasticity or in
X