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1、NonlinearAnalysis69(2008)2158–2166www.elsevier.com/locate/naStepanov-likealmostautomorphicsolutionsofnonautonomoussemilinearevolutionequationswithdelayHaewonLee∗,HadiAlkahbyDivisionofNaturalSciences,DillardUniversity,NewOrleans,LA70122,USAReceived5July2007;accepted27July2007AbstractWeconsiderthe
2、existenceanduniquenessofaStepanov-likealmostautomorphicsolutiontothenonautonomoussemilinearevolutionequationswithaconstantdelay:u0(t)=A(t)u(t)+f(t,u(t−h)),t∈RwhereA(t),t∈R,generatesanexponentiallystableevolutionfamily{U(t,s)}andf:R×X→XsatisfiesaLipschitzconditionwithrespecttothesecondargument.Pub
3、lishedbyElsevierLtdMSC:43A60;34G20;47DxxKeywords:Almostautomorphic;Nonautonomousequations;Constantdelay;Regularsolutions1.IntroductionInthispaperwestudytheexistenceanduniquenessofStepanov-likealmostautomorphicsolutionstothefollowingnonautonomoussemilinearevolutionequationswithaconstantdelay:u0(t
4、)=A(t)u(t)+f(t,u(t−h)),t∈R(1.1)whereh≥0isafixedconstant,and{A(t)}t∈Rsatisfiesthe‘Acquistapace–Terreni’conditionin[1],andU(t,s)generatedbyA(t),isexponentiallystable,andf∈ASp(X)∩C(R,X)forp>1,andf:R×X→XsatisfiesaLipschitzconditionwithrespecttothesecondargument.Intheearliersixties,Bochnerintroducedthec
5、onceptofalmostautomorphicfunctionsinhispapers[2–4]inrelationtosomeaspectsofdifferentialgeometry.Thisconceptbecameageneralizationofalmostperiodicitywhichisoneofthemostattractivetopicsinthequalitativetheoryofdifferentialequationsbecauseoftheirsignificanceandapplicationsinphysics,mathematicalbiology
6、,controltheory,andotherrelatedfields.Recently,theconceptofalmostautomorphicfunctionshaswidelybeenusedintheinvestigationoftheexistenceofalmostautomorphicsolutionsofvariouskindsofevolutionequationsbyN’Guer´ekataandothers.´∗Correspondingauthor.Tel.:+15048164318;fax:+15048164885.E-mailaddresses:hlee@
7、dillard.edu(H.Lee),halkahby@dillard.edu(H.Alkahby).0362-546X/$-seefrontmatterPublishedbyElsevierLtddoi:10.1016/j.na.2007.07.053H.Lee,H.Alkahby/NonlinearAnalysis69(2008)2158–21662159Recently,N’Guer´ekataandPankovintroducedthe
