Nonlinear Analysis and Differential Equations. An Introduction

Title page
Preface
I Nonlinear Analysis
Chapter I. Analysis in Banach Spaces
1 Introduction
2 Banach Spaces
3 Differentiability, Taylor's Theorem
4 Some Special Mappings
5 Inverse Function Theorems
6 The Dugundji Extension Theorem
7 Exercises
Chapter II. The Method of Lyapunov-Schmidt
1 Introduction
2 Splitting Equations
3 Bifurcation at a Simple Eigenvalue
Chapter III. Degree Theory
1 Introduction
2 Definition
3 Properties of the Brouwer Degree
4 Completely Continuous Perturbations
5 Exercises
Chapter IV. Global Solution Theorems
1 Introduction
2 Continuation Principle
3 A Globalization of the Implicit Function Theorem
4 The Theorem of Krein-Rutman
5 Global Bifurcation
6 Exercises
II Ordinary Differential Equations
Chapter V. Existence and Uniqueness Theorems
1 Introduction
2 The Picard-Lindelöf Theorem
3 The Cauchy-Peano Theorem
4 Extension Theorems
5 Dependence upon Initial Conditions
6 Differential Inequalities
7 Uniqueness Theorems
8 Exercises
Chapter VI. Linear Ordinary Differential Equations
1 Introduction
2 Preliminaries
3 Constant Coefficient Systems
4 Floquet Theory
5 Exercises
Chapter VII. Periodic Solutions
1 Introduction
2 Preliminaries
3 Perturbations of Nonresonant Equations
4 Resonant Equations
5 Exercises
Chapter VIII. Stability Theory
1 Introduction
2 Stability Concepts
3 Stability of Linear Equations
4 Stability of Nonlinear Equations
5 Lyapunov Stability
6 Exercises
Chapter IX. Invariant Sets
1 Introduction
2 Orbits and Flows
3 Invariant Sets
4 Limit Sets
5 Two Dimensional Systems
6 Exercises
Chapter X. Hopf Bifurcation
1 Introduction
2 A Hopf Bifurcation Theorem
Chapter XI. Sturm-Liouville Boundary Value Problems
1 Introduction
2 Linear Boundary Value Problems
3 Completeness of Eigenfunctions
4 Exercises
Bibliography
Index