Advanced Ordinary Differential Equations

232 Pages·1.257 MB·English

by  Kartsatos A.G.

Advanced Ordinary
Differential Equations
Third Edition
Athanassios G. Kartsatos
Hindawi Publishing Corporation
http://www.hindawi.com
Advanced Ordinary
Differential Equations
HindawiPublishingCorporation
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ISBN977-5945-15-1
Advanced Ordinary
Differential Equations
Third Edition
Athanassios G. Kartsatos
Hindawi Publishing Corporation
http://www.hindawi.com

DEDICATION
TothememoryofmyfatherYorgos
TomymotherAndromachi

PREFACE
This book has been designed for a two-semester course in Advanced Ordinary
Differential Equations. It is based on the author’s lectures on the subject at the
UniversityofSouthFlorida.Althoughwrittenprimarilyforgraduate,oradvanced
undergraduate,studentsofmathematics,thebookiscertainlyquiteusefultoen-
gineers,physicists,andotherscientistswhoareinterestedinvariousstability,as-
ymptotic behaviour, and boundary value problems in the theory of differential
systems.
TheonlyprerequisitesforthebookareafirstcourseinOrdinaryDifferential
EquationsandacourseinAdvancedCalculus.
The exercises at the end of each chapter are of varying degree of difficulty,
and several among them are basic theoretical results. The bibliography contains
references to most of the books and related papers which have been used in the
text.
Theauthormaintainsthatthisfunctional-analytictreatmentisasolidintro-
ductiontovariousaspectsofNonlinearAnalysis.
Barring some instances in Chapter Nine, no knowledge of Measure Theory
isrequired.TheBanachspacesofcontinuousfunctionshavesufficientstructure
forthedevelopmentofthebasicprinciplesofFunctionalAnalysisneededforthe
presenttheory.
Finally, the author is indebted to Hindawi Publishing Corporation for the
publicationofthebook.
A.G.Kartsatos
Tampa,Florida

CONTENTS
Dedication v
Preface vii
Chapter1. Banachspaces 1
1. Preliminaries 1
2. TheconceptofarealBanachspace;thespaceRn 2
3. Boundedlinearoperators 6
4. ExamplesofBanachspacesandlinearoperators 11
Exercises 15
Chapter2. Fixedpointtheorems;theinversefunctiontheorem 21
1. TheBanachcontractionprinciple 21
2. TheSchauder-Tychonovtheorem 24
3. TheLeray-Schaudertheorem 28
4. Theinversefunctiontheorem 30
Exercises 36
Chapter3. Existenceanduniqueness;continuation;
basictheoryoflinearsystems 41
1. Existenceanduniqueness 41
2. Continuation 45
3. Linearsystems 49
Exercises 54
Chapter4. Stabilityoflinearsystems;perturbedlinearsystems 61
1. Definitionsofstability 61
2. Linearsystems 62
3. Themeasureofamatrix;furtherstabilitycriteria 66
4. Perturbedlinearsystems 71
Exercises 76
Chapter5. Lyapunovfunctionsinthetheoryofdifferentialsystems;
thecomparisonprinciple 81
1. Lyapunovfunctions 82
2. Maximalandminimalsolutions;
thecomparisonprinciple 84
3. ExistenceonR 89
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