Scalar and Asymptotic Scalar Derivatives

by Isac, George; Nemeth, Sandor Zoltan

• ISBN13: 9780387739878
• ISBN10: 0387739874
• Format: Hardcover
• Copyright: 2008-07-01
• Publisher: Springer Verlag

Summary

This book is devoted to the study of scalar and asymptotic scalar derivatives and their applications to some problems in nonlinear analysis, Riemannian geometry and applied mathematics. The theoretical results are developed in particular with respect to the study of complementarity problems, monotonicity of nonlinear mappings and the non-gradient type monotonicity on Riemannian manifolds. Scalar and Asymptotic Derivatives: Theory and Applications also presents the material in relation to Euclidean spaces, Hilbert spaces, Banach spaces, Riemannian manifolds, and Hadamard manifolds.This book is intended for researchers and graduate students working in the fields of nonlinear analysis, Riemannian geometry and applied mathematics. It fills a gap in the literature as the first book to appear on the subject.

Table of Contents

Scalar Derivatives in Euclidean Spaces	p. 1
Scalar Derivatives of Mappings in Euclidean Spaces	p. 1
Some Basic Results Concerning Skew-Adjoint Operators	p. 2
The Scalar Derivative and its Fundamental Properties	p. 3
Case n = 2. The Relation of the Scalar Derivative with the Complex Derivative	p. 7
Miscellanea Concerning Scalar Differentiability	p. 9
Characterization of Monotonicity by Scalar Derivatives	p. 12
Computational Formulae for the Scalar Derivative	p. 15
Scalar Derivatives and Directional Derivatives	p. 15
Applications	p. 20
Monotonicity, Scalar Differentiability, and Conformity	p. 24
The Coefficient of Conformity and the Conformal Derivative	p. 25
Monotone Vector Fields and Expansive Maps	p. 27
Asymptotic Derivatives and Asymptotic Scalar Derivatives	p. 31
Asymptotic Differentiability in Banach Spaces	p. 31
Hyers-Ulam Stability and Asymptotic Derivatives	p. 34
Asymptotic Differentiability Along a Convex Cone in a Banach Space	p. 45
Asymptotic Differentiability in Locally Convex Spaces	p. 49
The Asymptotic Scalar Differentiability	p. 64
Some Applications	p. 71
Scalar Derivatives in Hilbert Spaces	p. 79
Calculus	p. 79
Introduction	p. 79
Some Basic Results Concerning Skew-Adjoint Operators	p. 80
Scalar Derivatives and Scalar Differentiability	p. 81
Characterization of Monotone Mappings by Using Scalar Derivatives	p. 83
Computational Formulae for the Scalar Derivatives	p. 86
Inversions	p. 90
Fixed Point Theorems Generated by Krasnoselskii's Fixed Point Theorem	p. 93
Surjectivity Theorems	p. 94
Variational Inequalities and Complementarity Problems	p. 97
Duality in Nonlinear Complementarity Theory	p. 103
Preliminaries	p. 104
Complementarity Problem	p. 104
Exceptional Family of Elements	p. 104
Infinitesimal Exceptional Family of Elements	p. 106
A Duality and Main Results	p. 107
Duality of Implicit Complementarity Problems	p. 112
Implicit Complementarity Problem	p. 112
Exceptional Family of Elements for an Ordered Pair of Mappings	p. 113
Infinitesimal Exceptional Family of Elements for an Ordered Pair of Mappings	p. 114
A Duality and Main Results	p. 115
Duality of Multivalued Complementarity Problems	p. 119
Preliminaries	p. 120
Approachable and Approximable Mappings	p. 121
Complementarity Problem	p. 122
Inversions of Set-Valued Mappings	p. 122
Exceptional Family of Elements	p. 123
Infinitesimal Exceptional Family of Elements	p. 125
A Duality and Main results	p. 127
The Asymptotic Browder-Hartman-Stampacchia Condition and Interior Bands of ¿-Solutions for Nonlinear Complementarity Problems	p. 132
Preliminaries	p. 134
The Browder-Hartman-Stampacchia Condition	p. 135
The asymptotic Browder-Hartman-Stampacchia condition	p. 138
Infinitesimal Interior-Point-¿-Exceptional Families	p. 142
Results Related to Properties (a) and (b) of the Interior Band Mapping u	p. 143
Comments	p. 149
REFE-Acceptable Mappings and a Necessary and Sufficient Condition for the Nonexistence of Regular Exceptional Families of Elements	p. 149
REFE-Acceptable Mappings	p. 149
Mappings Without Regular Exceptional Family of Elements. A necessary and Sufficient Condition	p. 157
Scalar Derivatives in Banach Spaces	p. 161
Preliminaries	p. 161
Semi-inner Products	p. 162
Inversions	p. 163
Scalar Derivatives	p. 165
Fixed Point Theorems in Banach Spaces	p. 166
A Fixed Point Index for a-condensing Mappings	p. 166
An Altman-type Fixed Point Theorem	p. 168
Integral Equations	p. 171
Applications of Krasnoselskii-Type Fixed Point Theorems	p. 172
Applications of Altman-Type Fixed Point Theorems	p. 175
Monotone Vector Fields on Riemannian Manifolds and Scalar Derivatives	p. 179
Geodesic Monotone Vector Fields	p. 180
Geodesic Monotone Vector Fields and Convex Functionals	p. 181
Geodesic Monotone Vector Fields and the First Variation of the Length of a Geodesic	p. 182
Closed Geodesics and Geodesic Monotone Vector Fields	p. 184
The Geodesic Monotonicity of Position Vector Fields	p. 185
Geodesic Scalar Derivative	p. 189
Geodesic Monotone Vector Fields on <$>{\op S}^n<$>	p. 192
Geodesic Monotone Vector Fields on <$>{\op H}^n<$>	p. 197
Killing Monotone Vector Fields	p. 200
Expansive One-Parameter Transformation Groups	p. 200
Geodesic Scalar Derivatives and Conformity	p. 206
Projection Maps on Hadamard Manifolds	p. 207
Some Basic Consequences of the Comparison Theorems	p. 208
The Complementary Vector Field of a Map	p. 214
Projection maps generating monotone vector fields	p. 214
Nonexpansive Maps	p. 219
Some Other Consequences of the Comparison Theorems	p. 219
Nonexpansive Maps Generating Monotone Vector Fields	p. 222
Zeros of Monotone Vector Fields	p. 222
Homeomorphisms and Monotone Vector Fields	p. 223
Preliminary Results	p. 224
Homeomorphisms of Hadamard Manifolds	p. 227
References	p. 231
Index	p. 241
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