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1、InformationSciences177(2007)4329–4337www.elsevier.com/locate/insExistenceanduniquenessofsolutionforfuzzyqrandomdifferentialequationswithnon-LipschitzcoefficientsWeiyinFei*DepartmentofAppliedMathematics,AnhuiUniversityofTechnologyandScience,Wuhu,Anhui241000,PRChinaCollegeofI
2、nformationScienceandTechnology,DonghuaUniversity,Shanghai200051,PRChinaReceived5September2006;receivedinrevisedform3March2007;accepted10March2007AbstractInthispaper,aclassoffuzzyrandomdifferentialequationswithnon-Lipschitzcoefficientsisstudied.Theexistenceanduniquenessofsol
3、utionsforfuzzyrandomdifferentialequationswithnon-Lipschitzcoefficientsisfirstproved.Thenthedependenceoffuzzyrandomdifferentialequationsoninitialvaluesisdiscussed.Finallythenon-confluencepropertyofthesolutionforfuzzyrandomdifferentialequationisinvestigated.OurmaintoolistheGronwa
4、lllemma.Ó2007ElsevierInc.Allrightsreserved.Keywords:Fuzzyrandomvariables;Fuzzyrandomdifferentialequation;Existenceanduniqueness;Non-Lipschitzcoefficients;Gronwalllemma1.IntroductionThetopicoffuzzydifferentialequationshasbeenrapidlygrowinginrecentyears.Roma´n-FloresandRojas-M
5、edar[32]presentedsomeresultsonexistenceofsolutionsforthefuzzyCauchyproblemundercom-pactnessconditions.Thisproblemhasalsobeeninvestigatedbyseveralauthors,includingDingetal.[3],Kal-eva[17,18],Kloeden[20],Nieto[30],Seikkala[33],Songetal.[34]andWuandSong[41].Inpractice,weare
6、oftenfacedwithrandomexperimentswhoseoutcomesarenotexactbutareexpressedininexactlinguisticvariableswhichchangewiththetimeparameter.Thisuncertaintywillbecharacterizedbyfuzziness.Wecallitfuzzystochasticprocess.Infact,fuzzyrandomvariableshavebeenintroducedbymanyauthors.Weref
7、ertosuchinterestingworksfordetailsasthoseofBa´n[1],Giletal.[10],GongandWang[11],Hop[12,13],Huang[14],KeandLiu[19],Kra¨tschmer[21],Kwakernaak[23],Lietal.[24],LiuandLiu[27],M.Lo´pez-Dı´azandRalescu[28],PuriandqFoundationitem:SupportedbytheKeyProjectofEducationMinistryofChi
8、na(No.205073),ResearchFundforDoctorialProgramofHigherEducation(No.20060255006)andAnhuiNaturalScienceFou
