Functional Differential Equations: Advances and Applications

365 Pages·2016·3.803 MB·English

by  Constantin Corduneanu, Yizeng Li, Mehran Mahdavi

FUNCTIONAL DIFFERENTIAL EQUATIONS
PUREANDAPPLIEDMATHEMATICS
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EditorsEmeriti:MYRONB.ALLENIII,DAVIDA.COX,PETERHILTON,
HARRYHOCHSTADT,PETERLAX,JOHNTOLAND
Acompletelistofthetitlesinthisseriesappearsattheendofthisvolume.
FUNCTIONAL
DIFFERENTIAL
EQUATIONS
Advances and Applications
CONSTANTINCORDUNEANU
YIZENGLI
MEHRANMAHDAVI
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LibraryofCongressCataloging-in-PublicationData:
Names:Corduneanu,C.,author.|Li,Yizeng,1949–author.|Mahdavi,Mehran,
1959–author.
Title:Functionaldifferentialequations:advancesandapplications/
ConstantinCorduneanu,YizengLi,MehranMahdavi.
Description:Hoboken,NewJersey:JohnWiley&Sons,Inc.,[2016]|Series:
Pureandappliedmathematics|Includesbibliographicalreferencesand
index.
Identifiers:LCCN2015040683|ISBN9781119189473(cloth)
Subjects:LCSH:Functionaldifferentialequations.
Classification:LCCQA372.C6672016|DDC515/.35–dc23LCrecordavailableat
http://lccn.loc.gov/2015040683
Setin10.5/13ptsTimesbySPiGlobal,Pondicherry,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
Thisbookisdedicatedtothememoryofthree
pioneersoftheTheoryofFunctional
DifferentialEquations:
A.D.MYSHKIS,theauthorofthefirstbook
onnon-classicaldifferentialequations
N.N.KRASOVSKII,forsubstantialcontributions,
particularlyforextendingLiapunov’s
methodinStabilityTheory,tothe
caseofdelayequationswithfinite
delay,thusopeningthewaytothe
methodsofFunctionalAnalysis
J.K.HALE,forcreatingavasttheoryinthe
studyoffunctionaldifferentialequations,
byusingthemoderntoolsof
FunctionalAnalysis
Eachcreatedaresearchschool,with
manydistinguishedcontributors

CONTENTS
PREFACE xi
ACKNOWLEDGMENTS xv
1 Introduction,Classification,ShortHistory,AuxiliaryResults,
andMethods 1
1.1 ClassicalandNewTypesofFEs, 2
1.2 MainDirectionsintheStudyofFDE, 4
1.3 MetricSpacesandRelatedConcepts, 11
1.4 FunctionsSpaces, 15
1.5 SomeNonlinearAuxiliaryTools, 21
1.6 FurtherTypesofFEs, 25
2 ExistenceTheoryforFunctionalEquations 37
2.1 LocalExistenceforContinuousorMeasurableSolutions, 38
2.2 GlobalExistenceforSomeClassesofFunctional
DifferentialEquations, 43
2.3 ExistenceforaSecond-OrderFunctionalDifferential
Equation, 50
2.4 TheComparisonMethodinObtainingGlobalExistence
Results, 55
2.5 AFunctionalDifferentialEquationwithBounded
SolutionsonthePositiveSemiaxis, 59
vii
viii CONTENTS
2.6 AnExistenceResultforFunctionalDifferentialEquations
withRetardedArgument, 64
2.7 ASecondOrderFunctionalDifferentialEquationwith
BoundedSolutionsonthePositiveSemiaxis, 68
2.8 AGlobalExistenceResultforaClassofFirst-Order
FunctionalDifferentialEquations, 72
2.9 AGlobalExistenceResultinaSpecialFunctionSpace
andaPositivityResult, 76
2.10 SolutionSetsforCausalFunctionalDifferentialEquations, 81
2.11 AnApplicationtoOptimalControlTheory, 87
2.12 FlowInvariance, 92
2.13 FurtherExamples/Applications/Comments, 95
2.14 BibliographicalNotes, 98
3 StabilityTheoryofFunctionalDifferentialEquations 105
3.1 SomePreliminaryConsiderationsandDefinitions, 106
3.2 ComparisonMethodinStabilityTheoryofOrdinary
DifferentialEquations, 111
3.3 StabilityunderPermanentPerturbations, 115
3.4 StabilityforSomeFunctionalDifferentialEquations, 126
3.5 PartialStability, 133
3.6 StabilityandPartialStabilityofFiniteDelaySystems, 139
3.7 StabilityofInvariantSets, 147
3.8 AnotherTypeofStability, 155
3.9 VectorandMatrixLiapunovFunctions, 160
3.10 AFunctionalDifferentialEquation, 163
3.11 BriefCommentsontheStartandEvolutionofthe
ComparisonMethodinStability, 168
3.12 BibliographicalNotes, 169
4 OscillatoryMotion,withSpecialRegardtothe
AlmostPeriodicCase 175
4.1 TrigonometricPolynomialsandAP -Spaces, 176
r
4.2 SomePropertiesoftheSpacesAP (R,C), 183
r
4.3 AP -SolutionstoOrdinaryDifferentialEquations, 190
r
4.4 AP -SolutionstoConvolutionEquations, 196
r
4.5 OscillatorySolutionsInvolvingtheSpaceB, 202
4.6 OscillatoryMotionsDescribedbyClassicalAlmost
PeriodicFunctions, 207
4.7 DynamicalSystemsandAlmostPeriodicity, 217