Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analyis, Melvyn S. Berger

✍ Scribed by Melvyn Stuart Berger

• Publisher: Academic Press
• Year: 1977
• Tongue: English
• Leaves: 439
• Category: Library

✦ Synopsis

Background material; Nonlinear operators; Local analysis; Analysis in the large.

✦ Table of Contents

Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis
Copyright Page
Contents
Preface
Notation and Terminology
Suggestions for The Reader
PART I: PRELIMINARIES
Chapter 1. Background Material
1.1 How Nonlinear Problems Arise
1.2 Typical Difficulties Encountered
1.3 Facts from Functional Analysis
1.4 Inequalities and Estimates
1.5 Classical and Generalized Solutions of Differential Systems
1.6 Mappings between Finite-Dimensional Spaces
Chapter 2. Nonlinear Operators
2.1 Elementary Calculus
2.2 Specific Nonlinear Operators
2.3 Analytic Operators
2.4 Compact Operators
2.5 Gradient Mappings
2.6 Nonlinear Fredholm Operators
2.7 Proper Mappings
Notes
PART II: LOCAL ANALYSIS
Chapter 3. Local Analysis of a Single Mapping
3.1 Successive Approximations
3.2 The Steepest Descent Method for Gradient Mappings
3.3 Analytic Operators and the Majorant Method
3.4 Generalized Inverse Function Theorems
Notes
Chapter 4. Parameter Dependent Perturbation Phenomena
4.1 Bifurcation Theory–A Constructive Approach
4.2 Transcendental Methods in Bifurcation Theory
4.3 Specific Bifurcation Phenomena
4.4 Asymptotic Expansions and Singular Perturbations
4.5 Some Singular Perturbation Problems of Classical Mathematical Physics
Notes
PART III: ANALYSIS IN THE LARGE
Chapter 5. Global Theories for General Nonlinear Operators
5.1 Linearization
5.2 Finite-Dimensional Approximations
5.3 Homotopy, the Degree of Mappings, and Its Generalizations
5.4 Homotopy and Mapping Properties of Nonlinear Operators
5.5 Applications to Nonlinear Boundary Value Problems
Notes
Chapter 6. Critical Point Theory for Gradient Mappings
6.1 Minimization Problems
6.2 Specific Minimization Problems from Geometry and Physics
6.3 lsoperimetric Problems
6.4 Isoperimetric Problems in Geometry and Physics
6.5 Critical Point Theory of Marston Morse in Hilbert Space
6.6 The Critical Point Theory of Ljusternik and Schnirelmann
6.7 Applications of the General Critical Point Theories
Notes
Appendix A. On Differentiable Manifolds
Appendix B. On the Hodge-Kodaira Decomposition for Differential Forms
References
Index
Pure and Applied Mathematics