Homotopy of Extremal Problems

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ISBN: 9783110189421 | 3110189429
Cover: Hardcover
Copyright: 9/30/2007

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This monograph provides a thorough treatment of parameter-dependent extremal problems with local minimum values that remain unchanged under changes of the parameter. The authors consider the theory as well the practical treatment of those problems, both in finite-dimensional as well as in infinite-dimensional spaces. Various applications are considered, e.g., variational calculus, control theory and bifurcations theory. Thorough treatment of parameter-dependent extremal problems with local minimum values. Includes many applications, e.g., variational calculus, control theory and bifurcations theory. Intended for specialists in the field of nonlinear analysis and its applications as well as for students specializing in these subjects.

Preface	p. v
Introduction	p. 1
Preliminaries	p. 5
Topological, Metric, and Normed Spaces	p. 5
Topological Spaces	p. 5
Metric Spaces	p. 7
Vector Spaces	p. 8
Normed and Banach Spaces	p. 9
Hilbert Spaces	p. 10
Concrete Spaces of Functionals	p. 11
Compactness	p. 13
Compact Sets	p. 13
Compactness Criteria in Function Spaces	p. 14
Linear Functionals and Dual Spaces	p. 15
Linear Functionals	p. 15
The Hahn-Banach Theorem	p. 16
The Dual Space	p. 16
Reflexive Spaces	p. 17
Weak Convergence	p. 17
Weak Compactness	p. 17
Linear Operators	p. 18
Definitions	p. 18
Convergence of Linear Operators	p. 19
Inverse Operators	p. 19
Unbounded Operators	p. 20
The Spectrum of a Linear Operator	p. 20
Adjoint Operators	p. 21
The Space of Linear Operators	p. 21
Completely Continuous Operators	p. 21
Embedding Operators	p. 22
Nonlinear Operators and Functionals	p. 22
Continuity	p. 22
Differentiability	p. 22
Lipschitzian and Convex Functionals	p. 25
Some Special Nonlinear Operators and their Properties	p. 29
Extensions of Mappings and the Partition of Unity Theorem	p. 30
Contraction Mappings, the Implicit Function Theorem, and Differential Equations in a Banach Space	p. 31
Contraction Mappings	p. 31
The Implicit Function Theorem	p. 32
The Cauchy Problem	p. 33
Minimizers of Nonlinear Functionals	p. 34
Critical Points	p. 34
Second-Order Necessary and Sufficient Conditions for a Minimizer	p. 35
Conditional Extrema	p. 36
Extrema of Lipschitzian and Convex Functionals	p. 37
Weierstrass Theorems	p. 39
Monotonicity	p. 41
Definitions	p. 41
Potential Operators	p. 42
Monotonicity and Convexity	p. 42
Finite-Dimensional Problems	p. 45
Nondegenerate Deformations of Smooth Functions	p. 45
Invariance of Local Minimizers	p. 45
Invariance of a Global Minimizer	p. 49
Linear Deformations	p. 54
Nondegenerate Deformations of Nonsmooth Functions	p. 56
Generalized Derivatives and Generalized Gradients	p. 56
The Deformation Theorem	p. 56
Global Deformations of Lipschitzian Functions	p. 64
Linear Deformations of Lipschitzian Functions	p. 68
Regular and Critical Points of Continuous Functions	p. 69
Deformations of Continuous Functions	p. 73
Converses of Deformation Theorems	p. 76
Real Polynomials	p. 76
Real Analytic Functions	p. 78
Smooth Functions	p. 79
Theorems of Hopf and Parusinski	p. 86
Statement of the Problem	p. 86
The Degree of a Mapping and Rotation of a Vector Field	p. 87
Hopf's Theorem	p. 91
Parusinski's Theorem	p. 97
Preparatory Lemmas	p. 98
Proof of Theorem 2.4.11	p. 99
Infinite-Dimensional Problems	p. 105
Deformations of Functionals on Hilbert Spaces	p. 105
H-Regular Functionals	p. 105
The Deformation Principle for Minimizers	p. 105
Preparatory Lemmas	p. 106
Proof of the Main Theorem	p. 107
The Property of H-Regularity	p. 111
Deformations of Functionals on Banach Spaces	p. 112
E-Regular Functionals	p. 112
Preparatory Lemmas	p. 113
Proof of the Deformation Theorem	p. 115
Global Deformations of Functionals	p. 116
Growing Functionals	p. 116
Global Deformations	p. 119
Generalizations	p. 121
Deformations of Lipschitzian Functionals	p. 121
(P,S)-Regular Functionals	p. 121
The Deformation Theorem	p. 122
Preparatory Lemmas	p. 122
Proof of the Deformation Theorem	p. 125
Deformations of Nonsmooth Problems with Constraints	p. 127
Statement of the Problem	p. 127
The Deformation Theorem	p. 129
Linear Deformations and Conditions for Global Minimizers	p. 132
Conley Index	p. 135
Conley Index in Finite-Dimensional Problems	p. 135
Flows in Finite-Dimensional Spaces	p. 135
Index Pairs for Invariant Sets	p. 139
Conley Index	p. 142
Homotopy Invariance of the Conley Index	p. 154
The Conley Index of Nondegenerate Critical Points	p. 158
Conley Index in Infinite-Dimensional Problems	p. 163
(E,H)-Regular Functionals	p. 163
The Conley Index of Critical Points of (E,H)-Regular Functionals	p. 164
Homotopy Invariance of the Conley Index	p. 170
Applications	p. 173
Problems of Classical Analysis	p. 173
Proving Inequalities (General Principles)	p. 173
Sylvester's Criterion	p. 174
Young's Inequality	p. 177
Minkowski's Inequality	p. 178
Jensen's Inequality	p. 179
Cauchy's Inequality (Inequality of the Arithmetic and Geometric Means)	p. 180
Improvements and Extensions	p. 182
Nonlinear Programming Problems	p. 183
Extremals of Lipschitz Nonlinear Programming Problems	p. 183
Extremals of Classical Nonlinear Programming Problems	p. 184
The Deformation Theorem	p. 185
Linear Deformations of Nonlinear Programming Problems and Invariance of Global Minimizers	p. 191
Sufficient Conditions for Minimizers in Nonlinear Programming Problems	p. 193
Inequalities with Constraints	p. 199
Bernstein's Inequality	p. 204
Multicriteria Problems	p. 207
Definitions	p. 207
The Deformation Theorem	p. 208
Multicriteria Problems with Constraints	p. 210
Problems in the Calculus of Variations	p. 216
One-Dimensional Problems	p. 216
Multidimensional Integral Functionals	p. 218
The Deformation Theorem	p. 219
Deformations of Integral Functionals in the Problem of Weak Minimizers	p. 221
Functional Inequalities	p. 229
Solvability of Boundary Value Problems and Criteria for Minimizers of Integral Functionals	p. 231
Investigation of Critical Points for a Minimizer	p. 235
Stability of Solutions of Ordinary Differential Equations	p. 236
Stability of Gradient Systems	p. 236
Stability of Hamiltonian Systems	p. 238
Stability of Gradient Systems in the Large	p. 239
Conley Index and the Stability of Dynamical Systems	p. 240
Optimal Control Problems	p. 244
Statement of the Problem	p. 244
Auxiliary Results	p. 245
Deformation Theorems	p. 252
Bifurcation of Critical Points in Variational Problems	p. 255
Statement of the Problem and a Necessary Condition for Bifurcation	p. 255
Deformation Principle for Minimizers in the Study of Bifurcation Points	p. 256
Conley Index and Bifurcation Points	p. 259
Bifurcation Points of Critical Points in One-Dimensional Variational Problems	p. 265
Analysis of Bifurcation Values of a Parameter	p. 267
Bifurcations of Solutions of the Ginzburg-Landau Equations	p. 269
Additional Remarks and Bibliographic Comments	p. 273
References	p. 281
Notation	p. 293
Name Index	p. 297
Subject Index	p. 299
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