Topics in Fractional Differential Equations

402 Pages·2012·2.791 MB·English

by  Saïd Abbas, Mouffak Benchohra, Gaston M. N'Guérékata (auth.)

Developments in Mathematics
VOLUME 27
SeriesEditors:
KrishnaswamiAlladi,Universityof Florida
Hershel M.Farkas,HebrewUniversityofJerusalem
Robert Guralnick,UniversityofSouthernCalifornia
Forfurthervolumes:
http://www.springer.com/series/5834
Sa¨ıd Abbas (cid:129) Mouffak Benchohra
Gaston M. N’Gue´re´kata
Topics in Fractional
Differential Equations
123
Sa¨ıdAbbas MouffakBenchohra
LaboratoiredeMathe´matiques LaboratoiredeMathe´matiques
Universite´deSa¨ıda Universite´deSidiBel-Abbe`s
Sa¨ıda,Algeria SidiBel-Abbe`s
Algeria
GastonM.N’Gue´re´kata
DepartmentofMathematics
MorganStateUniversity
Baltimore,MD,USA
ISSN1389-2177
ISBN978-1-4614-4035-2 ISBN978-1-4614-4036-9(eBook)
DOI10.1007/978-1-4614-4036-9
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012941498
Mathematics Subject Classification (2010): 26A33, 34A60, 34A37, 39A12, 34A37, 39A12, 34A60,
34B15,26A33,34K05
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Wededicatethisbooktoourfamilymembers.
Inparticular,Sa¨ıdAbbasdedicatestothe
memoryofhisfather,tohismother,his wife
ZoubidaandhischildrenMourad,Amina,
andIlyes;MouffakBenchohramakes his
dedicationto thememoryof hisfatherYahia
BenchohraandGastonN’Gue´re´kata tothe
memoryofhisfatherJean N’Gue´re´kata.
Preface
Fractionalcalculus(FC)generalizesintegralsandderivativestonon-integerorders.
Duringthe last decade,FC was foundto play a fundamentalrole in the modeling
of a considerable number of phenomena, in particular, the modeling of memory-
dependent phenomena and complex media such as porous media. FC emerged
as an important and efficient tool for the study of dynamical systems where
classical methods reveal strong limitations. This book is devoted to the existence
anduniquenessofsolutionsforvariousclassesofDarbouxproblemforhyperbolic
differential equations or inclusions involving the Caputo fractional derivative, the
best fractional derivative of the time. Some equations present delay which may
be finite, infinite, or state-dependent. Others are subject to impulsive effect. The
toolsused includeclassical fixedpointtheoremsas wellassharp(new)onessuch
as the one by Dhage on ordered Banach algebras and the fixed point theorem
for contraction multivalued maps due to Covitz and Nadier, as well as some
generalizations of the Gronwall’s lemma. Each chapter concludes with a section
devotedtonotesandbibliographicalremarksandallabstractresultsareillustrated
byexamples.
The content of this book is new and complements the existing literature in
fractionalcalculus. It is useful for researchers and graduate students for research,
seminars, and advanced graduate courses, in pure and applied mathematics, engi-
neering,biology,andallotherappliedsciences.
We owe a great deal to R.P. Agarwal, L. Go´rniewicz, J. Henderson,J.J. Nieto,
B.A. Slimani, J.J. Trujillo, A.N. Vityuk, and Y. Zhou for their collaboration
in research related to the problems considered in this book. We express our
vii
viii Preface
appreciationto ProfessorGeorgeAnastassiouwhostronglysupportedourproject.
Finally,wethanktheeditorialassistanceofSpringer,especiallyElizabethLoewand
JacobGallay.
Sa¨ıda,Algeria S.Abbas
SidiBel-Abbe`s,Algeria M.Benchohra
Baltimore,Maryland,USA G.M.N’Gue´re´kata
Contents
1 Introduction .................................................................. 1
2 PreliminaryBackground.................................................... 11
2.1 NotationsandDefinitions ............................................. 11
2.2 PropertiesofPartialFractionalCalculus ............................. 12
2.3 PropertiesofSet-ValuedMaps........................................ 18
2.4 FixedPointTheorems................................................. 21
2.5 GronwallLemmas..................................................... 23
3 PartialHyperbolicFunctionalDifferentialEquations................... 25
3.1 Introduction............................................................ 25
3.2 PartialHyperbolicDifferentialEquations............................ 25
3.2.1 Introduction.................................................... 25
3.2.2 ExistenceofSolutions ........................................ 26
3.2.3 AnExample.................................................... 32
3.3 PerturbedPartialDifferentialEquations.............................. 33
3.3.1 Introduction.................................................... 33
3.3.2 ExistenceofSolutions ........................................ 33
3.3.3 AnExample.................................................... 39
3.4 NeutralPartialDifferentialEquations................................ 40
3.4.1 Introduction.................................................... 40
3.4.2 ExistenceofSolutions ........................................ 40
3.4.3 AnExample.................................................... 46
3.5 DiscontinuousPartialDifferentialEquationsinBanachAlgebras.. 47
3.5.1 Introduction.................................................... 47
3.5.2 ExistenceofSolutions ........................................ 47
3.5.3 ExistenceofExtremalSolutions ............................. 51
3.5.4 AnExample.................................................... 55
3.6 UpperandLowerSolutionsMethodforPartialHyperbolic
DifferentialEquations................................................. 56
3.6.1 Introduction.................................................... 56
3.6.2 MainResult.................................................... 56
ix
x Contents
3.7 PartialFunctionalDifferentialEquationswithInfiniteDelay....... 60
3.7.1 Introduction.................................................... 60
3.7.2 ThePhaseSpaceB ........................................... 60
3.7.3 MainResults................................................... 62
3.7.4 AnExample.................................................... 71
3.8 PartialHyperbolicDifferentialEquationswithState-Dependent
Delay................................................................... 73
3.8.1 Introduction.................................................... 73
3.8.2 ExistenceofSolutionsforFiniteDelay...................... 74
3.8.3 ExistenceofSolutionsforInfiniteDelay..................... 83
3.8.4 Examples....................................................... 94
3.9 GlobalUniquenessResultsforPartialHyperbolicDifferential
Equations............................................................... 98
3.9.1 Introduction.................................................... 98
3.9.2 GlobalResultforFiniteDelay................................ 100
3.9.3 GlobalResultforInfiniteDelay .............................. 105
3.9.4 Examples....................................................... 112
3.10 NotesandRemarks.................................................... 114
4 PartialHyperbolicFunctionalDifferentialInclusions................... 115
4.1 Introduction............................................................ 115
4.2 PartialHyperbolicDifferentialInclusions............................ 115
4.2.1 Introduction.................................................... 115
4.2.2 TheConvexCase.............................................. 116
4.2.3 TheNonconvexCase.......................................... 121
4.2.4 AnExample.................................................... 124
4.3 ExistenceResultsforPartialHyperbolicDifferentialInclusions.... 125
4.3.1 Introduction.................................................... 125
4.3.2 ExistenceofSolutions ........................................ 125
4.3.3 QualitativePropertiesandTopologicalStructure
oftheSolutionSet............................................. 130
4.4 UpperandLowerSolutionsMethodforPartialDifferential
Inclusions .............................................................. 143
4.4.1 Introduction.................................................... 143
4.4.2 MainResult.................................................... 144
4.5 PartialFunctionalDifferentialInclusionswithInfiniteDelay....... 150
4.5.1 Introduction.................................................... 150
4.5.2 MainResults................................................... 150
4.5.3 AnExample.................................................... 159
4.6 FractionalOrderRiemann–LiouvilleIntegralInclusions
withtwoIndependentVariablesandMultipleTimeDelay.......... 159
4.6.1 Introduction.................................................... 159
4.6.2 ExistenceofSolutions ........................................ 160
4.6.3 AnExample.................................................... 168
4.7 NotesandRemarks.................................................... 169
Contents xi
5 ImpulsivePartialHyperbolicFunctionalDifferentialEquations ...... 171
5.1 Introduction............................................................ 171
5.2 ImpulsivePartialHyperbolicFunctionalDifferentialEquations.... 171
5.2.1 Introduction.................................................... 171
5.2.2 ExistenceofSolutions ........................................ 172
5.2.3 AnExample.................................................... 178
5.3 ImpulsivePartialHyperbolicDifferentialEquationsatVariable
Times................................................................... 179
5.3.1 Introduction.................................................... 179
5.3.2 ExistenceofSolutions ........................................ 179
5.3.3 NonlocalImpulsivePartialDifferentialEquations .......... 186
5.3.4 AnExample.................................................... 187
5.4 ImpulsiveDiscontinuousPartialHyperbolicDifferential
EquationsonBanachAlgebras........................................ 188
5.4.1 Introduction.................................................... 188
5.4.2 ExistenceofSolutions ........................................ 189
5.4.3 ExistenceofExtremalSolutions ............................. 193
5.4.4 AnExample.................................................... 198
5.5 ImpulsivePartialHyperbolicDifferentialEquations
withVariableTimesandInfiniteDelay............................... 199
5.5.1 Introduction.................................................... 199
5.5.2 MainResult.................................................... 200
5.5.3 AnExample.................................................... 209
5.6 ImpulsivePartialHyperbolicFunctionalDifferentialEquations
ofFractionalOrderwithState-DependentDelay .................... 211
5.6.1 Introduction.................................................... 211
5.6.2 ImpulsivePartialDifferentialEquationswithFinite
Delay........................................................... 212
5.6.3 ImpulsivePartialDifferentialEquationswithInfinite
Delay........................................................... 215
5.6.4 Examples....................................................... 220
5.7 ImpulsivePartialHyperbolicFunctionalDifferentialEquations
withVariableTimesandState-DependentDelay .................... 222
5.7.1 Introduction.................................................... 222
5.7.2 ImpulsivePartialDifferentialEquationswithFinite
Delay........................................................... 223
5.7.3 ImpulsivePartialDifferentialEquationswithInfinite
Delay........................................................... 231
5.7.4 Examples....................................................... 241
5.8 UpperandLowerSolutionsMethodforImpulsivePartial
HyperbolicDifferentialEquations.................................... 243
5.8.1 Introduction.................................................... 243
5.8.2 MainResult.................................................... 244
5.9 NotesandRemarks.................................................... 249
xii Contents
6 ImpulsivePartialHyperbolicFunctionalDifferentialInclusions ...... 251
6.1 Introduction............................................................ 251
6.2 ImpulsivePartialHyperbolicDifferentialInclusions................ 251
6.2.1 Introduction.................................................... 251
6.2.2 TheConvexCase.............................................. 252
6.2.3 TheNonconvexCase.......................................... 259
6.2.4 AnExample.................................................... 263
6.3 ImpulsivePartialHyperbolicDifferentialInclusions
withVariableTimes ................................................... 265
6.3.1 Introduction.................................................... 265
6.3.2 ExistenceofSolutions ........................................ 265
6.3.3 AnExample.................................................... 275
6.4 TheMethodofUpperandLowerSolutionsforPartial
HyperbolicFractionalOrderDifferentialInclusions
withImpulses.......................................................... 276
6.4.1 Introduction.................................................... 276
6.4.2 MainResult.................................................... 277
6.5 NotesandRemarks.................................................... 285
7 ImplicitPartialHyperbolicFunctionalDifferentialEquations......... 287
7.1 Introduction............................................................ 287
7.2 DarbouxProblemforImplicitDifferentialEquations ............... 287
7.2.1 Introduction.................................................... 287
7.2.2 Riemann–LiouvilleandCaputoPartialFractional
Derivatives..................................................... 288
7.2.3 ExistenceofSolutions ........................................ 289
7.2.4 AnExample.................................................... 294
7.3 AGlobalUniquenessResultforImplicitDifferentialEquations ... 295
7.3.1 Introduction.................................................... 295
7.3.2 ExistenceofSolutions ........................................ 295
7.3.3 AnExample.................................................... 298
7.4 FunctionalImplicitHyperbolicDifferentialEquations
withDelay ............................................................. 299
7.4.1 Introduction.................................................... 299
7.4.2 ExistenceResultswithFiniteDelay.......................... 300
7.4.3 ExistenceResultsforInfiniteDelay ......................... 305
7.4.4 ExistenceResultswithState-DependentDelay ............. 308
7.4.5 Examples....................................................... 312
7.5 DarbouxProblemforImplicitImpulsivePartialHyperbolic
DifferentialEquations................................................. 316
7.5.1 Introduction.................................................... 316
7.5.2 ExistenceofSolutions ........................................ 316
7.5.3 AnExample.................................................... 325