Nonlinearity and Functional Analysis - 1st Edition - ISBN: 9780120903504, 9780080570440

Nonlinearity and Functional Analysis

1st Edition

Lectures on Nonlinear Problems in Mathematical Analysis

Authors: Melvyn Berger
Hardcover ISBN: 9780120903504
eBook ISBN: 9780080570440
Imprint: Academic Press
Published Date: 28th September 1977
Page Count: 417
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Description


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

Notes

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 Fredholrn 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 Isoperimetric Problems

6.4 Isoperimetric Problems in Geometry and Physics

6.5 Critical Point Theory of Marston Morse in Hilbert Space

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

Table of 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

Notes

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 Fredholrn 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 Isoperimetric Problems

6.4 Isoperimetric Problems in Geometry and Physics

6.5 Critical Point Theory of Marston Morse in Hilbert Space

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

Details

No. of pages:
417
Language:
English
Copyright:
© Academic Press 1977
Published:
Imprint:
Academic Press
eBook ISBN:
9780080570440
Hardcover ISBN:
9780120903504

About the Author

Melvyn Berger

Affiliations and Expertise

University of Massachusetts