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World Scientific Publishing Company, Incorporated
Topological Methods for Set-Valued Nonlinear Analysis

Topological Methods for Set-Valued Nonlinear Analysis

by Mohammad S R Chowdhury, Enayet U Tarafdar


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Topological Methods for Set-Valued Nonlinear Analysis

This book provides a comprehensive overview of the authors' pioneering contributions to nonlinear set-valued analysis by topological methods. The coverage includes fixed point theory, degree theory, the KKM principle, variational inequality theory, the Nash equilibrium point in mathematical economics, the Pareto optimum in optimization, and applications to best approximation theory, partial equations and boundary value problems.Self-contained and unified in presentation, the book considers the existence of equilibrium points of abstract economics in topological vector spaces from the viewpoint of Ky Fan minimax inequalities. It also provides the latest developments in KKM theory and degree theory for nonlinear set-valued mappings.

Product Details

ISBN-13: 9789812704672
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 04/28/2008
Pages: 628
Product dimensions: 6.60(w) x 9.80(h) x 1.60(d)

Table of Contents

Preface     vii
Introduction     1
Contraction Mappings     9
Contraction Mapping Principle in Uniform Topological Spaces and Applications     9
Banach Contraction Mapping Principle in Uniform Spaces     10
Successive Approximation     14
Further Generalization of Banach Contraction Mapping Principle     27
Fixed Point Theorems for Some Extension of Contraction Mappings on Uniform Spaces     28
An Interplay Between the Order and Pseudometric Partial Ordering in Complete Uniform Topological Space     32
Changing Norm     34
Changing the Norm     38
On the Approximate Iteration     43
The Contraction Mapping Principle Applied to the Cauchy-Kowalevsky Theorem     44
Geometric Preliminaries     45
The Linear Problem     46
The Quasilinear Problem     50
An Implicit Function Theorem for a Set of Mappings and Its Application to Nonlinear Hyperbolic Boundary Value Problem as Application of Contraction Mapping Principle     53
An Implicit Function Theorem for a Set of Mappings     55
Notations and Preliminaries     60
Results of Smiley on Linear Problem     61
Alternative Problem and Approximate Equations     66
Application to Nonlinear Wave Equations - A Theorem of Paul Rabinowitz     73
Set-Valued Contractions     83
End Points     88
Iterated Function Systems (IFS) and Attractor     91
Applications     94
Large Contractions     103
Large Contractions     104
The Transformation     105
An Existence Theorem     106
Random Fixed Point and Set-Valued Random Contraction     107
Some Fixed Point Theorems in Partially Ordered Sets     113
Fixed Point Theorems and Applications to Economics     113
Fixed Point Theorem on Partially Ordered Sets     113
Applications to Games and Economics     116
Game     117
Economy     118
Pareto Optimum     119
The Contraction Mapping Principle in Uniform Space via Kleene's Fixed Point Theorem     120
Applications on Double Ranked Sequence     124
Lattice Theoretical Fixed Point Theorems of Tarski     125
Applications of Lattice Fixed Point Theorem of Tarski to Integral Equations     131
The Tarski-Kantorovitch Principle     134
The Iterated Function Systems on (2[superscript X], [superset or implies])     136
The Iterated Function Systems on (C(X), [superset or implies])     139
The Iterated Function System on (K(X), [superset or implies])     141
Continuity of Maps on Countably Compact and Sequential Spaces     142
Solutions of Impulsive Differential Equations     146
A Comparison Result     147
Periodic Solutions     149
Topological Fixed Point Theorems     151
Brouwer Fixed Point Theorem     151
Schauder Projection     160
Fixed Point Theorems of Set Valued Mappings with Applications in Abstract Economy     162
Applications     167
Equilibrium Point of Abstract Economy     169
Fixed Point Theorems and KKM Theorems     171
Duality in Fixed Point Theory of Set Valued Mappings     174
Applications on Minimax Principles     177
Applications on Sets with Convex Sections     179
More on Sets with Convex Sections     182
More on the Extension of KKM Theorem and Ky Fan's Minimax Principle     190
A Fixed Point Theorem Equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz Theorem     195
More on Fixed Point Theorems     200
Applications of Fixed Point Theorems to Equilibrium Analysis in Mathematical Economics and Game Theory     206
Fixed Point and Equilibrium Point     207
Existence of Maximal Elements     211
Equilibrium Existence Theorems     213
Fixed Point of [psi]-Condensing Mapping, Maximal Elements and Equilibria     224
Equilibrium on Paracompact Spaces     237
Equilibria of Generalized Games     240
Applications     243
Coincidence Points and Related Results, an Analysis on H-Spaces     244
Applications to Mathematical Economics: An Analogue of Debreu's Social Equilibrium Existence Theorem     261
Variational and Quasivariational Inequalities in Topological Vector Spaces and Generalized Games     265
Simultaneous Variational Inequalities     265
Variational Inequalities for Single Valued Functions     265
Solutions of Simultaneous Nonlinear Variational Inequalities     268
Application to Nonlinear Boundary Value Problem for Quasilinear Operator of Order 2m in Generalized Divergence Form     276
Minimization Problems and Related Results     280
Extension of a Karamardian Theorem     282
Variational Inequalities for Setvalued Mappings     284
Simultaneous Variational Inequalities     287
Implicit Variational Inequalities - The Monotone Case     292
Implicit Variational Inequalities - The USC Case      296
Variational Inequalities and Applications     301
Application to Minimization Problems     304
Duality in Variational Inequalities     306
Some Auxiliary Results     309
A Variational Inequality in Non-Compact Sets with Some Applications     312
Browder-Hartman-Stampacchia Variational Inequalities for Set-Valued Monotone Operators     321
A Minimax Inequality     321
An Existence Theorem of Variational Inequalities     322
Some Generalized Variational Inequalities with Their Applications     325
Some Generalized Variational Inequalities     325
Applications to Minimization Problems     333
Some Results of Tarafdar and Yuan on Generalized Variational Inequalities in Locally Convex Topological Vector Spaces     335
Some Generalized Variational Inequalities     337
Generalized Variational Inequalities for Quasi-Monotone and Quasi-Semi-Monotone Operators     340
Generalization of Ky Fan's Minimax Inequality     346
Generalized Variational Inequalities     348
Fixed Point Theorems     358
Generalization of Ky Fan's Minimax Inequality with Applications to Generalized Variational Inequalities for Pseudo-Monotone Type I Operators and Fixed Point Theorems     363
Generalization of Ky Fan's Minimax Inequality     365
Generalized Variational Inequalities     372
Applications to Fixed Point Theorems     377
Generalized Variational-Like Inequalities for Pseudo-Monotone Type I Operators     379
Existence Theorems for GV LI(T, [eta], h, X, F)     383
Generalized Quasi-Variational Inequalities     388
Generalized Quasi-Variational Inequalities for Monotone and Lower Semi-Continuous Mappings     388
Generalized Quasi-Variational Inequalities for Upper Semi-Continuous Mappings Without Monotonicity     393
Generalized Quasi-Variational Inequalities for Lower and Upper Hemi-Continuous Operators on Non-Compact Sets     397
Generalized Quasi-Variational Inequalities for Lower Hemi-Continuous Operators     398
Generalized Quasi-Variational Inequalities for Upper Hemi-Continuous Operators     404
Generalized Quasi-Variational Inequalities for Upper Semi-Continuous Operators on Non-Compact Sets     409
Non-Compact Generalized Quasi-Variational Inequalities     410
Generalized Quasi-Variational Inequalities for Pseudo-Monotone Set-Valued Mappings     415
Generalized Quasi-Variational Inequalities for Strong Pseudo-Monotone Operators     415
Generalized Quasi-Variational Inequalities for Pseudo-Monotone Set-Valued Mappings     421
Non-Linear Variational Inequalities and the Existence of Equilibrium in Economics with a Reiesz Space of Commodities     426
Existence of Equilibrium Lemma     428
Equilibria of Non-compact Generalized Games with L* Majorized Preference Correspondences     430
Existence of Maximal Elements     430
Existence of Equilibrium for Non-Compact Abstract Economies     434
Equilibria of Non-Compact Generalized Games     438
Equilibria of Generalized Games     442
Tarafdar and Yuan's Application on Existence Theorem of Equilibria for Constrained Games     445
Best Approximation and Fixed Point Theorems for Set-Valued Mappings in Topological Vector Spaces     447
Single-Valued Case     448
Set-Valued Case     452
Some Lemmas and Relevant Results     454
Degree Theories for Set-Valued Mappings     463
Degree Theory for Set-Valued Ultimately Compact Vector Fields     463
Properties of the Degree of Ultimately Compact Vector Fields     465
k-[phi]-Contractive Set Valued Mappings     467
Coincidence Degree for Non-Linear Single-Valued Perturbations of Linear Fredholm Mappings     471
An Equivalence Theorem     473
Definition of Coincidence Degree     474
Properties of the Coincidence Degree     475
On the Existence of Solutions of the Equation Lx [set membership] Nx and a Coincidence Degree Theory     478
Coincidence Degree for Set-Valued k - [phi]-Contractive Perturbations of Linear Fredholm Mappings     479
Coincidence Degree for Multi-Valued Mappings with Non-Negative Index     497
Basic Assumptions and Main Results in Akashi (1988)     497
Akashi's Basic Properties of Coincidence Degree     502
Application to Multitivalued Boundary Value Problem for Elliptic Partial Differential Equation     503
Applications of Equivalence Theorems with Single-Valued Mappings: An Approach to Non-Linear Elliptic Boundary Value Problems     507
Tarafdar's Application to Elliptic Boundary Value Problems     521
Further Results in Coincidence Degree Theory     525
Tarafdar and Thompson's Theory of Bifurcation for the Solutions of Equations Involving Set-Valued Mapping     528
Characteristic Value and Multiplicity     532
Tarafdar and Thompson's Results on the Theory of Bifurcation     532
Tarafdar and Thompson's Application on the Theory of Bifurcation     539
Tarafdar and Thompson's Results on the Solvability of Non-Linear and Non-Compact Operator Equations     542
Measure of Noncompactness and Set Contraction     542
Epi Mappings     546
Tarafdar and Thompson's (p, k)-Epi Mappings on the Whole Space     555
Tarafdar and Thompson's Applications of (p, k)-Epi Mappings in Differential Equations     556
Nonexpansive Types of Mappings and Fixed Point Theorems in Locally Convex Topological Vector Spaces     563
Nonexpansive Types of Mappings in Locally Convex Topological Vector Spaces     563
Nonexpansive Mappings     563
Set-Valued Mappings of Nonexpansive Type     571
Normal Structure and Fixed Point Theorems     572
Another Definition of Nonexpansive Set-Valued Mapping and Corresponding Results on Fixed Point Theorems     575
Fixed Point Theorems for Condensing Set-Valued Mappings on Locally Convex Topological Vector Spaces     576
Measure of Precompactness and Non-Precompactness     577
Condensing Mappings     578
Fixed Point Theorems     580
Bibliography     583
Index     605

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