CONTENTS |
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Preface to the English Edition |
v | |
Preface to the Russian Edition |
vii | |
Introduction |
1 | |
1 Metric Spaces |
7 | |
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1.1 Preliminaries |
7 |
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1.2 Some Metric Spaces of Functions |
12 |
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1.3 Energy Spaces |
14 |
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1.4 Sets in a Metric Space |
18 |
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1.5 Convergence in a Metric Space |
18 |
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1.6 Completeness |
19 |
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1.7 The Completion Theorem |
21 |
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1.8 The Lebesgue Integral and the Space Lp |
23 |
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1.9 Banach and Hubert Spaces |
27 |
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1.10 Some Energy Spaces |
32 |
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1.11 Sobolev Spaces |
47 |
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1.12 Introduction to Operators |
50 |
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1.13 Contraction Mapping Principle |
52 |
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1.14 Generalized Solutions in Mechanics |
57 |
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1.15 Separability 62 |
62 |
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1.16 Compactness, Hausdorff Criterion. |
67 |
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1.17 Arzela's Theorem and Its Applications |
70 |
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1.18 Approximation Theory |
76 |
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1.19 Decomposition Theorem, Riesz Representation |
79 |
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1.20 Existence of Energy Solutions |
83 |
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1.21 The Problem of Elastico-Plasticity |
87 |
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1.22 Bases and Complete Systems |
94 |
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1.23 Weak Convergence in a Hubert Space |
99 |
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1.24 Ritz and Bubnov-Galerkin Methods |
109 |
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1.25 Curvilinear Coordinates, Non-Homogeneous Boundary Conditions |
111 |
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1.26 The Bramble-Hilbert Lemma and Its Applications |
114 |
2 Elements of the Theory of Operators |
121 | |
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2.1 Spaces of Linear Operators |
121 |
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2.2 Banach-Steinhaus Principle |
124 |
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2.3 The Inverse Operator |
126 |
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2.4 Closed Operators |
129 |
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2.5 The Notion of Adjoint Operator |
132 |
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2.6 Compact Operators |
139 |
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2.7 Compact Operators in Hilbert Space |
144 |
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2.8 Functions Taking Values in a Banach Space |
146 |
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2.9 Spectrum of Linear Operators |
149 |
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2.10 Resolvent Set of a Closed Linear Operator |
152 |
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2.11 Spectrum of Compact Operators in Hilbert Space |
154 |
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2.12Analytic Nature of the Resolvent of a Compact Linear Operator |
162 |
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2.13 Spectrum of Holomorphic Compact Operator Function |
164 |
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2.14 Spectrum of Self-Adjoint Compact Linear Operator in Hilbert Space |
166 |
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2.15 Some Applications of Spectral Theory |
171 |
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2.16 Courant's Mmimax Principle |
175 |
3 Elements of Nonlinear Functional Analysis |
177 | |
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3.1 Frechet and Gateaux Derivatives |
177 |
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3.2 Liapunov-Schmidt Method |
182 |
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3.3 Critical Points of a Functional |
184 |
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3.4 Von Karman Equations of a Plate |
189 |
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3.5 Buckling of a Thin Elastic Shell |
195 |
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3.6 Equilibrium of Elastic Shallow Shells |
204 |
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3.7 Degree Theory |
209 |
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3.8 Steady-State Flow of Viscous Liquid |
211 |
Appendix: Hints for Selected Problems |
219 | |
References |
231 | |
Index | 235 |