An Introduction

Klaus Schmitt

Department of Mathematics

University of Utah

Russell C. Thompson

Department of Mathematics and Statistics

Utah State University

Lecture Notes

I Nonlinear Analysis 1

Chapter I. Analysis In Banach Spaces 3
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Differentiability, Taylor’s Theorem . . . . . . . . . . . . . . . . . 8
4 Some Special Mappings . . . . . . . . . . . . . . . . . . . . . . . 11
5 Inverse Function Theorems . . . . . . . . . . . . . . . . . . . . . 20
6 The Dugundji Extension Theorem . . . . . . . . . . . . . . . . . 22
7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter II. The Method of Lyapunov-Schmidt 27

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Splitting Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Bifurcation at a Simple Eigenvalue . . . . . . . . . . . . . . . . . 30
Chapter III. Degree Theory 33
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Properties of the Brouwer Degree . . . . . . . . . . . . . . . . . . 38
4 Completely Continuous Perturbations . . . . . . . . . . . . . . . 42
5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter IV. Global Solution Theorems 49

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2 Continuation Principle . . . . . . . . . . . . . . . . . . . . . . . . 49
3 A Globalization of the Implicit Function Theorem . . . . . . . . 52
4 The Theorem of Krein-Rutman . . . . . . . . . . . . . . . . . . . 54
5 Global Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

II Ordinary Differential Equations 63

Chapter V. Existence and Uniqueness Theorems 65

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2 The Picard-Lindel¨of Theorem . . . . . . . . . . . . . . . . . . . . 66
3 The Cauchy-Peano Theorem . . . . . . . . . . . . . . . . . . . . . 67
4 Extension Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Dependence upon Initial Conditions . . . . . . . . . . . . . . . . 72
6 Differential Inequalities . . . . . . . . . . . . . . . . . . . . . . . 74
7 Uniqueness Theorems . . . . . . . . . . . . . . . . . . . . . . . . 78
8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Chapter VI. Linear Ordinary Differential Equations 81

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3 Constant Coefficient Systems . . . . . . . . . . . . . . . . . . . . 83
4 Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Chapter VII. Periodic Solutions 91
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3 Perturbations of Nonresonant Equations . . . . . . . . . . . . . . 92
4 Resonant Equations . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Chapter VIII. Stability Theory 103

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2 Stability Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Stability of Linear Equations . . . . . . . . . . . . . . . . . . . . 105
4 Stability of Nonlinear Equations . . . . . . . . . . . . . . . . . . 108
5 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Chapter IX. Invariant Sets 123

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2 Orbits and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4 Limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 Two Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . 129
6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Chapter X. Hopf Bifurcation 133

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2 A Hopf Bifurcation Theorem . . . . . . . . . . . . . . . . . . . . 133
TABLE OF CONTENTS vii

Chapter XI. Sturm-Liouville Boundary Value Problems 139

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2 Linear Boundary Value Problems . . . . . . . . . . . . . . . . . . 139
3 Completeness of Eigenfunctions . . . . . . . . . . . . . . . . . . . 142
4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Bibliography 147
Index 149

Preface

The subject of Differential Equations is a well established part of mathematics
and its systematic development goes back to the early days of the development
of Calculus. Many recent advances in mathematics, paralleled by
a renewed and flourishing interaction between mathematics, the sciences, and
engineering, have again shown that many phenomena in the applied sciences,
modelled by differential equations will yield some mathematical explanation of
these phenomena (at least in some approximate sense).

The intent of this set of notes is to present several of the important existence
theorems for solutions of various types of problems associated with differential
equations and provide qualitative and quantitative descriptions of solutions. At
the same time, we develop methods of analysis which may be applied to carry
out the above and which have applications in many other areas of mathematics,
as well.

As methods and theories are developed, we shall also pay particular attention
to illustrate how these findings may be used and shall throughout consider
examples from areas where the theory may be applied.

As differential equations are equations which involve functions and their
derivatives as unknowns, we shall adopt throughout the view that differential
equations are equations in spaces of functions. We therefore shall, as we
progress, develop existence theories for equations defined in various types of
function spaces, which usually will be function spaces which are in some sense
natural for the given problem.