Almost Periodic Solutions of Differential Equations in Banach Spaces

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1、ALMOSTPERIODICSOLUTIONSOFDIFFERENTIALEQUATIONSINBANACHSPACESYoshiyukiHinoDepartmentofMathematicsandInformaticsChibaUniversity,Chiba,JapanToshikiNaitoDepartmentofMathematicsUniversityofElectro-Communications,Tokyo,JapanNguyenVanMinhDepartmentofMathematicsHanoiUn

2、iversityofScience,Hanoi,VietnamJongSonShinDepartmentofMathematicsKoreaUniversity,Tokyo,JapanPrefaceAlmostperiodicsolutionsofdifferentialequationshavebeenstudiedsincetheverybeginningofthiscentury.Thetheoryofalmostperiodicsolutionshasbeende-velopedinconnectionwith

3、problemsofdifferentialequations,dynamicalsystems,stabilitytheoryanditsapplicationstocontroltheoryandotherareasofmathemat-ics.TheclassicalbooksbyC.Corduneanu[50],A.M.Fink[67],T.Yoshizawa[231],L.AmerioandG.Prouse[7],B.M.LevitanandV.V.Zhikov[137]gaveaverynicepresen

4、tationofmethodsaswellasresultsinthearea.Inrecentyears,therehasbeenanincreasinginterestinextendingcertainclassicalresultstodifferentialequationsinBanachspaces.Inthisbookwewillmakeanattempttogathersystematicallycertainrecentresultsinthisdirection.Weoutlinebrieflyth

5、econtentsofourbook.Themainresultspresentedhereareconcernedwithconditionsfortheexistenceofperiodicandalmostperiodicsolutionsanditsconnectionwithstabilitytheory.Inthequalitativetheoryofdifferentialequationstherearetwoclassicalresultswhichserveasmodelsformanyworksi

6、nthearea.Namely,TheoremAAperiodicinhomogeneouslinearequationhasauniqueperiodicsolution(withthesameperiod)if1isnotaneigenvalueofitsmonodromyoperator.TheoremBAperiodicinhomogeneouslinearequationhasaperiodicsolution(withthesameperiod)ifandonlyifithasaboundedsoluti

7、on.Inourbook,amainpartwillbedevotedtodiscussthequestionashowtoex-tendtheseresultstothecaseofalmostperiodicsolutionsof(linearandnonlinear)equationsinBanachspaces.Tothisend,inthefirstchapterwepresentintroduc-tionstothetheoryofsemigroupsoflinearoperators(Section1),

8、itsapplicationstoevolutionequations(Section2)andtheharmonicanalysisofboundedfunctionsontherealline(Section3).InChapter2wepresenttheresultsconcernedwithautonomousaswellasperi