Table Of ContentLecture Notes in Mathematics 2076
Valeri Obukhovskii
Pietro Zecca
Nguyen Van Loi
Sergei Kornev
Method of
Guiding Functions
in Problems of
Nonlinear Analysis
Lecture Notes in Mathematics 2076
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
Valeri Obukhovskii Pietro Zecca
(cid:2)
Nguyen Van Loi Sergei Kornev
(cid:2)
Method of Guiding
Functions in Problems
of Nonlinear Analysis
123
ValeriObukhovskii PietroZecca
SergeiKornev DipartimentodiMatematicaeInformatica
DepartmentofPhysicsandMathematics “UDini”
VoronezhStatePedagogicalUniversity Universita`diFirenze
Voronezh,Russia Firenze,Italy
NguyenVanLoi
FacultyofFundamentalScience
PetroVietNamUniversity
BaRia,Vietnam
ISBN978-3-642-37069-4 ISBN978-3-642-37070-0(eBook)
DOI10.1007/978-3-642-37070-0
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013937327
MathematicsSubjectClassification(2010):34C25,34C23,47J15,58E07,34C15,34B15,34A60,
34G25,34H05,47H04,47H08,47H09,47H11,47J05,
49J52,34K09,34K13,34K18,34K30,34K35,54H25,
91A23,93C10,47N50
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Acknowledgements
The work of V. Obukhovskii and S. Kornev was supported by the Russian FBR
Grants 11-01-00328and 12-01-00392.Professor V. Obukhovskiiis much obliged
totheUniversityofFlorencewhichsupportedhisworkoverthebook.Theworkof
P.ZeccawassupportedbyaUniversityofFirenzeResearchGrant.
v
Contents
1 Background................................................................... 1
1.1 Multimaps............................................................... 1
1.1.1 GeneralProperties.............................................. 1
1.1.2 MeasurableMultifunctionsandSuperposition
Multioperator ................................................... 5
1.1.3 Single-ValuedApproximations ................................ 8
1.2 TopologicalDegree ..................................................... 12
1.3 CoincidenceDegree..................................................... 18
1.4 PhaseSpaces ............................................................ 22
1.5 Notation ................................................................. 24
2 MethodofGuidingFunctionsinFinite-DimensionalSpaces ........... 25
2.1 PeriodicProblemforaDifferentialInclusion ......................... 25
2.2 Non-smoothGuidingFunctions........................................ 36
2.3 IntegralGuidingFunctions ............................................ 39
2.4 GeneralizedPeriodicProblems......................................... 43
2.4.1 Preliminaries.................................................... 43
2.4.2 TheSettingoftheProblem..................................... 44
2.4.3 ApplicationtoDifferentialGames............................. 45
2.4.4 ExistenceTheorem,CorollariesandExample ................ 46
2.5 GlobalBifurcationProblems........................................... 50
2.5.1 AbstractResult.................................................. 51
2.5.2 GlobalBifurcationofPeriodicSolutions...................... 52
2.5.3 Application 1: Differential Inclusion
withaBoundedNonlinearity................................... 63
2.5.4 Application2:GlobalBifurcationforFunctional
DifferentialInclusions.......................................... 64
2.5.5 Application3:FeedbackControlSystem...................... 65
vii
viii Contents
3 MethodofGuidingFunctionsinHilbertSpaces ......................... 69
3.1 IntegralGuidingFunctionsforDifferentialInclusions
inHilbertSpaces........................................................ 69
3.1.1 TheSettingoftheProblem..................................... 69
3.1.2 ExistenceofPeriodicSolutions................................ 72
3.1.3 ApproximationConditions..................................... 77
3.1.4 Application1:ControlProblemof a Partial
DifferentialEquation ........................................... 80
3.2 Non-smooth Guiding Functions for Functional
DifferentialInclusionswithInfiniteDelayinHilbertSpaces......... 83
3.2.1 SettingoftheProblem.......................................... 83
3.2.2 ExistenceTheorem ............................................. 86
3.2.3 Application:ExistenceofPeriodicSolutions
foraGradientFunctionalDifferentialInclusion.............. 90
3.3 BifurcationProblem .................................................... 93
3.3.1 TheSettingoftheProblem..................................... 93
3.3.2 GlobalBifurcationTheorem ................................... 96
3.3.3 Application 3: Ordinary FeedbackControl
SystemsinaHilbertSpace..................................... 102
4 Second-OrderDifferentialInclusions...................................... 105
4.1 ExistenceTheoreminanOne-DimensionalSpace.................... 105
4.2 Applications............................................................. 110
4.2.1 EquationswithDiscontinuousNonlinearities................. 110
4.2.2 BoundaryValueProblem....................................... 113
4.2.3 ASecond-OrderDifferentialEquation ........................ 114
4.2.4 FeedbackControlSystems..................................... 114
4.2.5 A Model of a Motion of a Particle in a
One-DimensionalPotential..................................... 118
4.3 ExistenceTheoreminHilbertSpaces.................................. 120
4.3.1 ApplicationtoaSecond-OrderFeedbackControl
SysteminHilbertSpace........................................ 122
4.3.2 Example......................................................... 127
5 NonlinearFredholmInclusionsandApplications........................ 131
5.1 Preliminaries ............................................................ 131
5.2 OrientedCoincidenceIndex............................................ 133
5.2.1 TheCaseofaFiniteDimensionalTriplet ..................... 134
5.2.2 TheCaseofaCompactTriplet................................. 138
5.2.3 OrientedCoincidenceIndexforCondensingTriplets......... 139
5.3 CalculationoftheOrientedCoincidenceIndexbytheMGF......... 145
5.3.1 TheMainResult ................................................ 145
5.3.2 Example......................................................... 153
Contents ix
5.4 GlobalBifurcationProblem............................................ 155
5.4.1 AbstractResult.................................................. 155
5.4.2 GlobalBifurcationforFamiliesofPeriodicTrajectories..... 158
5.4.3 Example......................................................... 163
References......................................................................... 167
Index............................................................................... 175
