Mathematical analysis and applications. Selected topics

750 Pages·2018·4.72 MB·English

by  Agarwal

MathematicalAnalysisandApplications
Mathematical Analysis and Applications
SelectedTopics
Editedby
MichaelRuzhansky
HemenDutta
RaviP.Agarwal
Thiseditionfirstpublished2018
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LibraryofCongressCataloging-in-PublicationData:
Names:Ruzhansky,M.(Michael),editor.|Dutta,Hemen,1981-editor.|
Agarwal,RaviP.,editor.
Title:Mathematicalanalysisandapplications:selectedtopics/editedby
MichaelRuzhansky,HemenDutta,RaviP.Agarwal.
Description:Hoboken,NJ:JohnWiley&Sons,2018.|Includes
bibliographicalreferencesandindex.|
Identifiers:LCCN2017048922(print)|LCCN2017054738(ebook)|ISBN
9781119414308(pdf)|ISBN9781119414339(epub)|ISBN9781119414346
(cloth)
Subjects:LCSH:Mathematicalanalysis.
Classification:LCCQA300(ebook)|LCCQA300.M2252018(print)|DDC
515–dc23
LCrecordavailableathttps://lccn.loc.gov/2017048922
CoverDesign:Wiley
CoverImage:©LoudRedCreative/GettyImages
Setin10/12ptWarnockProbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
v
Contents
Preface xv
AbouttheEditors xxi
ListofContributors xxiii
1 SpacesofAsymptoticallyDevelopableFunctionsand
Applications 1
SergioAlejandroCarrilloTorresandJorgeMozoFernández
1.1 IntroductionandSomeNotations 1
1.2 StrongAsymptoticExpansions 2
1.3 MonomialAsymptoticExpansions 7
1.4 MonomialSummabilityforSingularlyPerturbedDifferential
Equations 13
1.5 PfaffianSystems 15
References 19
2 DualityforGaussianProcessesfromRandomSigned
Measures 23
PalleE.T.JorgensenandFengTian
2.1 Introduction 23
2.2 ReproducingKernelHilbertSpaces(RKHSs)intheMeasurable
Category 24
2.3 ApplicationstoGaussianProcesses 30
2.4 ChoiceofProbabilitySpace 34
2.5 ADuality 37
2.A StochasticProcesses 40
2.B OverviewofApplicationsofRKHSs 45
Acknowledgments 50
References 51
vi Contents
3 Many-BodyWaveScatteringProblemsforSmallScatterers
andCreatingMaterialswithaDesiredRefraction
Coefficient 57
AlexanderG.Ramm
3.1 Introduction 57
3.2 DerivationoftheFormulasforOne-BodyWaveScattering
Problems 62
3.3 Many-BodyScatteringProblem 65
3.3.1 TheCaseofAcousticallySoftParticles 68
3.3.2 WaveScatteringbyManyImpedanceParticles 70
3.4 CreatingMaterialswithaDesiredRefractionCoefficient 71
3.5 ScatteringbySmallParticlesEmbeddedinanInhomogeneous
Medium 72
3.6 Conclusions 72
References 73
4 GeneralizedConvexFunctionsandtheirApplications 77
AdemKiliçmanandWedadSaleh
4.1 BriefIntroduction 77
4.2 GeneralizedE-ConvexFunctions 78
4.3 E𝛼-Epigraph 84
4.4 Generalizeds-ConvexFunctions 85
4.5 ApplicationstoSpecialMeans 96
References 98
5 SomePropertiesandGeneralizationsoftheCatalan,Fuss,
andFuss–CatalanNumbers 101
FengQiandBai-NiGuo
5.1 TheCatalanNumbers 101
5.1.1 ADefinitionoftheCatalanNumbers 101
5.1.2 TheHistoryoftheCatalanNumbers 101
5.1.3 AGeneratingFunctionoftheCatalanNumbers 102
5.1.4 SomeExpressionsoftheCatalanNumbers 102
5.1.5 IntegralRepresentationsoftheCatalanNumbers 103
5.1.6 AsymptoticExpansionsoftheCatalanFunction 104
5.1.7 CompleteMonotonicityoftheCatalanNumbers 105
5.1.8 InequalitiesoftheCatalanNumbersandFunction 106
5.1.9 TheBellPolynomialsoftheSecondKindandtheBessel
Polynomials 109
5.2 TheCatalan–QiFunction 111
5.2.1 TheFussNumbers 111
5.2.2 ADefinitionoftheCatalan–QiFunction 111
5.2.3 SomeIdentitiesoftheCatalan–QiFunction 112
5.2.4 IntegralRepresentationsoftheCatalan–QiFunction 114
Contents vii
5.2.5 AsymptoticExpansionsoftheCatalan–QiFunction 115
5.2.6 CompleteMonotonicityoftheCatalan–QiFunction 116
5.2.7 Schur-ConvexityoftheCatalan–QiFunction 118
5.2.8 GeneratingFunctionsoftheCatalan–QiNumbers 118
5.2.9 ADoubleInequalityoftheCatalan–QiFunction 118
5.2.10 Theq-Catalan–QiNumbersandProperties 119
5.2.11 TheCatalanNumbersandthek-Gammaandk-BetaFunctions 119
5.2.12 SeriesIdentitiesInvolvingtheCatalanNumbers 119
5.3 TheFuss–CatalanNumbers 119
5.3.1 ADefinitionoftheFuss–CatalanNumbers 119
5.3.2 AProduct-RatioExpressionoftheFuss–CatalanNumbers 120
5.3.3 CompleteMonotonicityoftheFuss–CatalanNumbers 120
5.3.4 ADoubleInequalityfortheFuss–CatalanNumbers 121
5.4 TheFuss–Catalan–QiFunction 121
5.4.1 ADefinitionoftheFuss–Catalan–QiFunction 121
5.4.2 AProduct-RatioExpressionoftheFuss–Catalan–QiFunction 122
5.4.3 IntegralRepresentationsoftheFuss–Catalan–QiFunction 123
5.4.4 CompleteMonotonicityoftheFuss–Catalan–QiFunction 124
5.5 SomePropertiesforRatiosofTwoGammaFunctions 124
5.5.1 AnIntegralRepresentationandCompleteMonotonicity 125
5.5.2 AnExponentialExpansionfortheRatioofTwoGamma
Functions 125
5.5.3 ADoubleInequalityfortheRatioofTwoGammaFunctions 125
5.6 SomeNewResultsontheCatalanNumbers 126
5.7 OpenProblems 126
Acknowledgments 127
References 127
6 TraceInequalitiesofJensenTypeforSelf-adjointOperators
inHilbertSpaces:ASurveyofRecentResults 135
SilvestruSeverDragomir
6.1 Introduction 135
6.1.1 Jensen’sInequality 135
6.1.2 TracesforOperatorsinHilbertSpaces 138
6.2 Jensen’sTypeTraceInequalities 141
6.2.1 SomeTraceInequalitiesforConvexFunctions 141
6.2.2 SomeFunctionalProperties 145
6.2.3 SomeExamples 151
6.2.4 MoreInequalitiesforConvexFunctions 154
6.3 ReversesofJensen’sTraceInequality 157
6.3.1 AReverseofJensen’sInequality 157
6.3.2 SomeExamples 163
6.3.3 FurtherReverseInequalitiesforConvexFunctions 165
6.3.4 SomeExamples 169
viii Contents
6.3.5 ReversesofHölder’sInequality 174
6.4 Slater’sTypeTraceInequalities 177
6.4.1 Slater’sTypeInequalities 177
6.4.2 FurtherReverses 180
References 188
7 SpectralSynthesisandItsApplications 193
LászlóSzékelyhidi
7.1 Introduction 193
7.2 BasicConceptsandFunctionClasses 195
7.3 DiscreteSpectralSynthesis 203
7.4 NondiscreteSpectralSynthesis 217
7.5 SphericalSpectralSynthesis 219
7.6 SpectralSynthesisonHypergroups 238
7.7 Applications 248
Acknowledgments 252
References 252
8 VariousUlam–HyersStabilitiesofEuler–Lagrange–Jensen
General(a,b;k=a+b)-SexticFunctionalEquations 255
JohnMichaelRassiasandNarasimmanPasupathi
8.1 BriefIntroduction 255
8.2 GeneralSolutionofEuler–Lagrange–JensenGeneral
(a,b;k =a+b)-SexticFunctionalEquation 257
8.3 StabilityResultsinBanachSpace 258
8.3.1 BanachSpace:DirectMethod 258
8.3.2 BanachSpace:FixedPointMethod 261
8.4 StabilityResultsinFelbin’sTypeSpaces 267
8.4.1 Felbin’sTypeSpaces:DirectMethod 268
8.4.2 Felbin’sTypeSpaces:FixedPointMethod 269
8.5 IntuitionisticFuzzyNormedSpace:StabilityResults 270
8.5.1 IFNS:DirectMethod 272
8.5.2 IFNS:FixedPointMethod 279
References 281
9 ANoteontheSplitCommonFixedPointProblemandits
VariantForms 283
AdemKiliçmanandL.B.Mohammed
9.1 Introduction 283
9.2 BasicConceptsandDefinitions 284
9.2.1 Introduction 284
9.2.2 VectorSpace 284
9.2.3 HilbertSpaceanditsProperties 286
9.2.4 BoundedLinearMapanditsProperties 288
Contents ix
9.2.5 SomeNonlinearOperators 289
9.2.6 ProblemFormulation 294
9.2.7 PreliminaryResults 294
9.2.8 StrongConvergencefortheSplitCommonFixed-PointProblemsfor
TotalQuasi-AsymptoticallyNonexpansiveMappings 296
9.2.9 StrongConvergencefortheSplitCommonFixed-PointProblemsfor
DemicontractiveMappings 302
9.2.10 ApplicationtoVariationalInequalityProblems 306
9.2.11 OnSynchronalAlgorithmsforFixedandVariationalInequality
ProblemsinHilbertSpaces 307
9.2.12 Preliminaries 307
9.3 ANoteontheSplitEqualityFixed-PointProblemsinHilbert
Spaces 315
9.3.1 ProblemFormulation 315
9.3.2 Preliminaries 316
9.3.3 TheSplitFeasibilityandFixed-PointEqualityProblemsfor
Quasi-NonexpansiveMappingsinHilbertSpaces 316
9.3.4 TheSplitCommonFixed-PointEqualityProblemsfor
Quasi-NonexpansiveMappingsinHilbertSpaces 320
9.4 NumericalExample 322
9.5 TheSplitFeasibilityandFixedPointProblemsfor
Quasi-NonexpansiveMappingsinHilbertSpaces 328
9.5.1 ProblemFormulation 328
9.5.2 PreliminaryResults 328
9.6 Ishikawa-TypeExtra-GradientIterativeMethodsfor
Quasi-NonexpansiveMappingsinHilbertSpaces 329
9.6.1 ApplicationtoSplitFeasibilityProblems 334
9.7 Conclusion 336
References 337
10 StabilitiesandInstabilitiesofRationalFunctionalEquations
andEuler–Lagrange–Jensen(a,b)-SexticFunctional
Equations 341
JohnMichaelRassias,KrishnanRavi,andBeriV.SenthilKumar
10.1 Introduction 341
10.1.1 GrowthofFunctionalEquations 342
10.1.2 ImportanceofFunctionalEquations 342
10.1.3 FunctionalEquationsRelevanttoOtherFields 343
10.1.4 DefinitionofFunctionalEquationwithExamples 343
10.2 UlamStabilityProblemforFunctionalEquation 344
10.2.1 𝜖-StabilityofFunctionalEquation 344
10.2.2 StabilityInvolvingSumofPowersofNorms 345
10.2.3 StabilityInvolvingProductofPowersofNorms 346
10.2.4 StabilityInvolvingaGeneralControlFunction 347
x Contents
10.2.5 StabilityInvolvingMixedProduct–SumofPowersofNorms 347
10.2.6 ApplicationofUlamStabilityTheory 348
10.3 VariousFormsofFunctionalEquations 348
10.4 Preliminaries 353
10.5 RationalFunctionalEquations 355
10.5.1 ReciprocalTypeFunctionalEquation 355
10.5.2 SolutionofReciprocalTypeFunctionalEquation 356
10.5.3 GeneralizedHyers–UlamStabilityofReciprocalTypeFunctional
Equation 357
10.5.4 Counter-Example 360
10.5.5 GeometricalInterpretationofReciprocalTypeFunctional
Equation 362
10.5.6 AnApplicationofEquation(10.41)toElectricCircuits 364
10.5.7 Reciprocal-QuadraticFunctionalEquation 364
10.5.8 GeneralSolutionofReciprocal-QuadraticFunctional
Equation 366
10.5.9 GeneralizedHyers–UlamStabilityofReciprocal-Quadratic
FunctionalEquations 368
10.5.10 Counter-Examples 373
10.5.11 Reciprocal-CubicandReciprocal-QuarticFunctional
Equations 375
10.5.12 Hyers–UlamStabilityofReciprocal-CubicandReciprocal-Quartic
FunctionalEquations 375
10.5.13 Counter-Examples 380
10.6 Euler-Lagrange–Jensen(a,b;k =a+b)-SexticFunctional
Equations 384
10.6.1 GeneralizedUlam–HyersStabilityofEuler-Lagrange-JensenSextic
FunctionalEquationUsingFixedPointMethod 384
10.6.2 Counter-Example 387
10.6.3 GeneralizedUlam–HyersStabilityofEuler-Lagrange-JensenSextic
FunctionalEquationUsingDirectMethod 389
References 395
11 AttractoroftheGeneralizedContractiveIteratedFunction
System 401
MujahidAbbasandTalatNazir
11.1 IteratedFunctionSystem 401
11.2 GeneralizedF-contractiveIteratedFunctionSystem 407
11.3 IteratedFunctionSysteminb-MetricSpace 414
11.4 GeneralizedF-ContractiveIteratedFunctionSysteminb-Metric
Space 420
References 426