具因果算子的分数泛函微分方程解的存在性

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1、ExistenceofSolutionsofFractionalFunctionalDifferentialEquationswithCausalOperatorsCandidateSupervisorCollegeProgramLuoXianNanProfessorZhouYongMathematicsandComputationalScienceAppliedMathematicsSpecialityTheoryandApplicationofFunctionalDifferentialEquationsDegreeUniversityDatePh.D

2、.degreeofScienceXiangTanUniversityApril,2011BanachKrasnoselskiiIRiemann-LiouvilleCaputoAbstractDuringthepastdecades,withthedevelopmentoffractionalcalculus,frac-tionaldifferentialequationshaveddevelopedrapidlyduetoitsapplicationinvarioussciences.Althoughalotofresultshavebeenobtain

3、edfortheexistenceofsolutionsforfunctionaldifferentialequationswithcausaloperators,theresultsfortheinitialvalueproblemsoffractionalfunctionaldifferentialequationswithcausaloperatorsisscarce.Thisdissertationfocusesontheexistenceofsolutionsoffractionalfunctionaldifferentialequationswith

4、causaloperators.Thisthesisconsistsoffourchapters.Maincontentsareorganizedasfollows:Inchapter1,wefirstintroducethebasicconceptsforthefractionalcalculusandfractionaldifferentialequations,thedefinitionandpropertiesofthecausaloperator.Next,wealsointroducesomeapplicationbackgroundforfract

5、ionaldifferentialequationsandsomeimportantfixedtheorems.Finally,weintroducethemainworkofthisthesis.Inchapter2,wediscussinitialvalueproblemforfractionalfunctionaldiffer-entialequationswhichcontainscausaloperatorinBanachspace.Wefirstestab-lishafractionalintegralequationswhichisequivalen

6、ttothefractionalfunctionaldifferentialequations.Byusingthetheoryoffractionalcalculusandmeasureofnoncompactness,weobtaintheexistenceresultsofsolutionsforfractionaldif-ferentialsystems.Wealsodiscussthecontinuationofsolutionsforfractionalfunctionaldifferentialsystems.Inchapter3,westudy

7、initialvalueproblemforneutralfractionalfunctionaldifferentialequationwithcausaloperator.Basedonthetheoryoffractionalcalculus,weestablishafractionalintegralequationwhichisequivalenttothefractionalfunctionaldifferentialequations.ByusingKrasnoselskii’sfixedpointtheorem,weobtaintheexiste

8、nceanduniquenessresultsofsolution