Nonlinearity & Functional Analysis

Lectures on Nonlinear Problems in Mathematical Analysis


  • Melvyn Berger, University of Massachusetts

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.
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Book information

  • Published: September 1977
  • ISBN: 978-0-12-090350-4

Table of Contents


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 Fredholrn Operators

2.7 Proper Mappings


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


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


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


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


Appendix A On Differentiable Manifolds

Appendix B On the Hodge-Kodaira Decomposition for Differential Forms