本书发展了处理非线性常微分方程和偏微分方程的拓扑和解析方法。本书适合对泛函分析感兴趣的研究生和数学研究人员阅读参考。 Since its first appearance as a set of lecture notes published by the Courant Institute in 1974, this book has served as an introduction to various subjects in nonlinear functional analysis. The current edition is a reprint of these notes, with added bibliographic references. Topological and analytic methods are developed for treating nonlinear ordinary and partial differential equations. The first two chapters of the book introduce the notion of topological degree and develop its basic properties. These properties are used in later chapters in the discussion of bifurcation theory (the possible branching of solutions as parameters vary), including the proof of Rabinowitz's global bifurcation theorem. Stability of the branches is also studied. The book concludes with a presentation of some generalized implicit function theorems of Nash-Moser type with applications to Kolmogorov-Arnold-Moser theory and to conjugacy problems. After more than 20 years, this book continues to be an excellent graduate level textbook and a useful supplementary course text.

前辅文
Preface to New Edition
Preface
Chapter 1. Topological Approach: Finite Dimensions
  1.1. A Simple Remark
  1.2. Sard's Theorem
  1.3. Finite-Dimensional Degrec Thcory
  1.4. Properties of Degree
  1.5. Further Properties and Remarks
  1.6. Some Applications to Nonlinear Equations
  1.7. Borsuk's Theorem
  1.8. Mappings in Different Dimensions
Chapter 2. Topological Degree in Banach Space
  2.1. Schauder Fixed-Point Theorem
  2.2. An Application
  2.3. Leray-Schauder Degree
  2.4. Some Compact Operators
  2.5. Elliptic Partial Differential Equations
  2.6. Mildly Nonlinear Perturbations of Linear Operators
  2.7. Calculus in Banach Space
  2.8. The Leray-Schauder Degree for Isolated Solutions, the Index
Chapter 3. Bifurcation Theory
  3.1. The Morse Lemma
  3.2. Application of thc Morsc Lemma
  3.3. Krasnoselski's Theorem
  3.4. A Theorem of Rabinowitz
  3.5. Extension of Krasnoselski's Theorem
  3.6. Stability of Solutions
  3.7. The Number of Global Solutions of a Nonlinear Problem
Chapter 4. Further Topological Methods
  4.1. Extension of Leray-Schauder Degree
  4.2. Applications to Partial Differential Equations
  4.3. Framed Cobordism
  4.4. Stable Cohomotopy Theorem
  4.5. Cohomotopy Groups
  4.6. Stable Cohomotopy Theory
  4.7. Application to Existence of Global Solutions
Chapter 5. Monotone Operators and the Min-Max Theorem
  5.1. Monotone Operators in Hilbert Space
  5.2. Min-Max Theorem
  5.3. Dense Single-Valuedness of Monotone Operators
Chapter 6. Generalized Implicit Function Theorems
  6.1. CωSmoothing: The Analytic Case
  6.2. Analytic Smoothing on Function Spaces and Analytic Mappings
  6.3. C∞ Smoothing and Mapping of Finite Order
  6.4. A Theorem of Kolmogorov, Arnold, and Moser
  6.5.Conjugacy Problems
Bibliography