Multidimensional Integral Equations and Inequalities

256 Pages·2011·1.092 MB·English

by  B.G. Pachpatte

July1,2011 9:49 book˙Pachpatte
ATLANTISSTUDIESINMATHEMATICSFORENGINEERINGANDSCIENCE
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SERIESEDITOR: C.K.CHUI
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Atlantis Studies in Mathematics for Engineering
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Multidimensional Integral
Equations and Inequalities
B.G. Pachpatte
57,ShriNiketenColoney,
Aurangabad,India
AMSTERDAM–PARIS
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Dedicatedtothememoryofmyparents.

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Preface
Integralequationsthatinvolvefunctionsoftwoormoreindependentvariablesoccurfre-
quently in the study of many problems in partial differential equations which arise from
various dynamic models. The study of various multidimensional integral equations and
inequalitiesattractedtheattentionofmanyresearchersandthereexistsaveryvastlitera-
ture. Theaimofthismonographistoprovidethereaders,representativeoverviewofthe
importantrecentdevelopments,focusingonsomeselectedmultidimensionalintegralequa-
tionsandinequalities. Itisassessabletoanyonehavingareasonablebackgroundinreal
analysis,partialdifferentialequationsandacquaintancewiththeirrelatedareas.
Thematerialincludedinthemonographisrecentandhardtofindinotherbooks. Itis
self-containedandallresultsarepresentedinaneasy-to-read,informalstyleanditcould
alsoserveasatextbookforanadvancedgraduatecourse. Itwillbeaninvaluablereading
forpureandappliedmathematicians,physists,engineers,computerscientistsandwillalso
bemostvaluableasasourceofreferenceinthefield.
Itisimpossibletothankalltheindividualswhohaveinfluencedmedirectlyorindirectly
duringthewritingofthisbook, withouttheirconstantencouragementitwouldstillhave
beenremainednomorethananidea. Inparticular,Iwishtoexpressmydeepandsincere
gratitudetoProfessorCharlesChui,EditorAMESwhoofferedinvaluablesuggestionsfor
the improvement of the presentation. Also, I am grateful to Professor Jan van Mill and
Arjen Sevenster for their support and interest in the present work. It isa pleasure toac-
knowledgethefinecollaborationandassistanceprovidedbytheeditorialandproduction
staffofAtlantisPress. Last,butnotleast,Iwouldliketothanktomyfamilymembersfor
theirunderstanding,patienceandlong-lastinginspiration.
B.G.Pachpatte
vii

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Contents
Preface vii
Introduction 1
1. Integralequationsintwovariables 9
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Basicintegralinequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Volterra-typeintegralequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Volterra-Fredholm-typeintegralequation . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Integrodifferentialequationsofhyperbolic-type . . . . . . . . . . . . . . . . . . . 27
1.6 Fredholm-typeintegrodifferentialequation. . . . . . . . . . . . . . . . . . . . . . 37
1.7 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2. Integralinequalitiesandequationsintwoandthreevariables 59
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.2 Integralinequalitiesintwovariables . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.3 Integralinequalitiesinthreevariables . . . . . . . . . . . . . . . . . . . . . . . . 65
2.4 Integralequationintwovariables. . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5 Integralequationinthreevariables . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.6 Hyperbolic-typeFredholmintegrodifferentialequation . . . . . . . . . . . . . . . 83
2.7 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3. Mixedintegralequationsandinequalities 97
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 Volterra-Fredholm-typeintegralinequalitiesI . . . . . . . . . . . . . . . . . . . . 97
3.3 Volterra-Fredholm-typeintegralinequalitiesII. . . . . . . . . . . . . . . . . . . . 108
3.4 IntegralequationofVolterra-Fredholm-type . . . . . . . . . . . . . . . . . . . . . 116
3.5 Volterra-Fredholm-typeintegralequations . . . . . . . . . . . . . . . . . . . . . . 122
3.6 GeneralVolterra-Fredholm-typeintegralequations . . . . . . . . . . . . . . . . . 128
3.7 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4. Parabolic-typeintegrodifferentialequations 143
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.2 Basicintegralinequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.3 IntegrodifferentialequationofBarbashin-type . . . . . . . . . . . . . . . . . . . . 147
4.4 GeneralintegralequationofBarbashin-type . . . . . . . . . . . . . . . . . . . . . 155
ix