Nonlinear analysis and differential equations: and introduction

158 Pages·1998·0.731 MB·English

by  Schmitt K., Thompson R.C.

Nonlinear Analysis and Differential Equations
An Introduction
Klaus Schmitt
Department of Mathematics
University of Utah
Russell C. Thompson
Department of Mathematics and Statistics
Utah State University
November 11, 2004
ii
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Copyright c1998 by K. Schmitt and R. Thompson
(cid:13)
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Preface
The subject of Differential Equations is a well established part of mathe-
matics and its systematic development goes back to the early days of the de-
velopment 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).
Theintentofthissetofnotesistopresentseveraloftheimportantexistence
theorems for solutions of various types of problems associated with differential
equationsandprovidequalitativeandquantitativedescriptionsofsolutions. At
the same time, we develop methods of analysis which may be applied to carry
outthe aboveandwhichhaveapplicationsinmanyotherareasofmathematics,
as well.
Asmethodsandtheoriesaredeveloped,weshallalsopayparticularattention
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 differen-
tial 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.
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Table of Contents
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
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1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2 The Picard-Lindel¨of Theorem . . . . . . . . . . . . . . . . . . . . 66
3 The Cauchy-PeanoTheorem. . . . . . . . . . . . . . . . . . . . . 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 NonresonantEquations . . . . . . . . . . . . . . 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
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Part I
Nonlinear Analysis
1