Lecture 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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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