nonlinear analysis & differential equations, an introduction - schmitt & thompson

As di erential equations are equations which involve functions and theirderivatives as unknowns, we shall adopt throughout the view that di eren-tial equations are equations in spaces of

An Introduction

Klaus Schmitt Department of Mathematics University of Utah Russell C Thompson Department of Mathematics and Statistics

Utah State University August 14, 2000

Copyright c1998 by K Schmitt and R Thompson

Preface

The subject of Dierential Equations is a well established part of matics and its systematic development goes back to the early days of the de-velopment of Calculus Many recent advances in mathematics, paralleled byengineering, have again shown that many phenomena in the applied sciences,modelled by dierential equations will yield some mathematical explanation ofthese phenomena (at least in some approximate sense)

mathe-The intent of this set of notes is to present several of the important existencetheorems for solutions of various types of problems associated with dierentialequations and provide qualitative and quantitative descriptions of solutions Atthe same time, we develop methods of analysis which may be applied to carryout 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 ndings may be used and shall throughout considerexamples from areas where the theory may be applied

As dierential equations are equations which involve functions and theirderivatives as unknowns, we shall adopt throughout the view that dieren-tial equations are equations in spaces of functions We therefore shall, as weprogress, develop existence theories for equations de ned in various types offunction spaces, which usually will be function spaces which are in some sensenatural for the given problem

Table of Contents

1 Introduction 3

2 Banach Spaces 3

3 Dierentiability, 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 De nition 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

Chapter V Existence and Uniqueness Theorems 65

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1 Introduction 65

2 The Picard-Lindelof Theorem 66

3 The Cauchy-Peano Theorem 67

4 Extension Theorems 69

5 Dependence upon Initial Conditions 72

6 Dierential Inequalities 74

7 Uniqueness Theorems 78

8 Exercises 79

Chapter VI Linear Ordinary Dierential Equations 81 1 Introduction 81

2 Preliminaries 81

3 Constant Coecient 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 121 1 Introduction 121

2 Orbits and Flows 122

3 Invariant Sets 123

4 Limit Sets 125

5 Two Dimensional Systems 126

6 Exercises 128

Chapter X Hopf Bifurcation 129 1 Introduction 129

2 A Hopf Bifurcation Theorem 129

TABLE OF CONTENTS vii

Chapter XI Sturm-Liouville Boundary Value Problems 135

1 Introduction 135

2 Linear Boundary Value Problems 135

3 Completeness of Eigenfunctions 138

4 Exercises 141

Nonlinear Analysis

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in Banach spaces, a version of this important result, suitable for our purposes

is stated and proved As a consequence we derive the Inverse Function theorem

in Banach spaces and close this chapter with an extension theorem for tions de ned on proper subsets of the domain space (the Dugundji extensiontheorem)

func-In this chapter we shall mainly be concerned with results for not necessarilylinear functions; results about linear operators which are needed in these noteswill be quoted as needed

iii) kuk=jjkuk, for every scalarand everyu2E,

iv) ku+vk  kuk+kvk, for allu;v;2E (triangle inequality)

A norm k
k de nes a metric d : E E ! R

+ by d(u;v) = ku?vk and(E;k
k) or simplyE (if it is understood which norm is being used) is called a

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Banach space if the metric space (E;d),dde ned as above, is complete (i.e allCauchy sequences have limits inE).

IfEis a real (or complex) vector space which is equipped with an inner product,i.e a mapping

h
;
i:EE! R (orC (the complex numbers))

satisfying

i) hu;vi=hv;ui; u;v2E

ii) hu+v;wi=hu;wi+hv;wi; u;v;w2E

iii) hu;vi=hu;vi; 2 C; u;v;2E

iv) hu;ui  0,u2E, andhu;ui= 0 if and only if u= 0,

thenE is a normed space with the norm de ned by

kuk=p

hu;ui; u2E:

IfE is complete with respect to this norm, thenE is called a Hilbert space

An inner product is a special case of what is known as a conjugate linearform, i.e a mappingb:EE! C having the properties (i){(iv) above (with

h
;
ireplaced byb(
;
)); in caseEis a real vector space, thenbis called a bilinearform

The following collection of spaces are examples of Banach spaces They willfrequently be employed in the applications presented later The veri cation thatthe spaces de ned are Banach spaces may be found in the standard literature

0<+1 HenceC0 ;R m) is a Banach space

2 BANACH SPACES 5

2.2 Spaces of dierentiable functions

R n Let ...

Nonlinear Analysis< /h2>

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