Nonlinearity and Functional Analysis
Lectures on Nonlinear Problems in Mathematical Analysis
Published on 27. October 1977
Book
Hardback
417 pages
978-0-12-090350-4 (ISBN)
Description
Nonlinearity and Functional Analysis is a collection of lectures that aim to present a systematic description of fundamental nonlinear results and their applicability to a variety of concrete problems taken from various fields of mathematical analysis. For decades, great mathematical interest has focused on problems associated with linear operators and the extension of the well-known results of linear algebra to an infinite-dimensional context. This interest has been crowned with deep insights, and the substantial theory that has been developed has had a profound influence throughout the mathematical sciences.
This volume comprises six chapters and begins by presenting some background material, such as differential-geometric sources, sources in mathematical physics, and sources from the calculus of variations, before delving into the subject of nonlinear operators. The following chapters then discuss local analysis of a single mapping and parameter dependent perturbation phenomena before going into analysis in the large. The final chapters conclude the collection with a discussion of global theories for general nonlinear operators and critical point theory for gradient mappings.
This book will be of interest to practitioners in the fields of mathematics and physics, and to those with interest in conventional linear functional analysis and ordinary and partial differential equations.
This volume comprises six chapters and begins by presenting some background material, such as differential-geometric sources, sources in mathematical physics, and sources from the calculus of variations, before delving into the subject of nonlinear operators. The following chapters then discuss local analysis of a single mapping and parameter dependent perturbation phenomena before going into analysis in the large. The final chapters conclude the collection with a discussion of global theories for general nonlinear operators and critical point theory for gradient mappings.
This book will be of interest to practitioners in the fields of mathematics and physics, and to those with interest in conventional linear functional analysis and ordinary and partial differential equations.
More details
Language
English
Place of publication
San Diego
United States
Publishing group
Elsevier Science Publishing Co Inc
Target group
Professional and scholarly
Product notice
Laminated cover
Dimensions
Height: 229 mm
Width: 152 mm
Thickness: 25 mm
Weight
756 gr
ISBN-13
978-0-12-090350-4 (9780120903504)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Melvyn S. Berger
Nonlinearity and Functional Analysis
Lectures on Nonlinear Problems in Mathematical Analysis
E-Book
10/1977
Academic Press
€54.95
Available for download
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Content
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
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