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1、NonlinearAnalysis73(2010)15151532ContentslistsavailableatScienceDirectNonlinearAnalysisjournalhomepage:www.elsevier.com/locate/naExistencetheoremsforsolutionstorandomfuzzydifferentialequationsMarekT.MalinowskiFacultyofMathematics,ComputerScienceandEconometrics,Univers
2、ityofZielonaGóra,Szafrana4a,65-516ZielonaGóra,PolandarticleinfoabstractArticlehistory:Weexaminearandomfuzzyinitialvalueproblemwithtwokindsoffuzzyderivatives.ForReceived27May2009bothcasesweestablishtheresultsofexistenceanduniquenessoflocalsolutionstorandomAccepted24Apri
3、l2010fuzzydifferentialequations.Theexistenceofglobalsolutionsisalsoobtained.'2010ElsevierLtd.Allrightsreserved.MSC:34F0534G2034A1226E50Keywords:RandomfuzzydifferentialequationsInitialvalueproblemforfuzzymappingsFuzzyderivativeFuzzystochasticprocessFuzzyrandomvariable1.
4、IntroductionFuzzydifferentialequations(FDEs)isatopicveryimportantfromthetheoreticalpointofview(seee.g.[1,2]andthereferencestherein)aswellasoftheirapplications,forexample,incivilengineering[3],inmodelinghydraulic[4]andinpopulationmodels[5].Therearemanysuggestionsonhowto
5、defineafuzzyderivativeandconsequentlyseveralwaystostudyFDEs.Varioustypesofderivativesoffuzzyfunctionswerecompared,andthesolutionsofFDEsrelatedtothemwereinvestigatedin[6].FuzzydifferentialequationswerefirstformulatedbyKaleva[7,8].HeusedtheconceptofH-differentiabilitywhi
6、chwasintroducedbyPuriandRalescu[9],andobtainedtheexistenceanduniquenesstheoremforasolutionofFDEundertheLipschitzcondition,whereasin[8]hecharacterizedthosesubsetsofthefuzzysetspaceinwhichthePeanotheoremisvalid.Sincethenthereappearedalotofpapersconcerningthetheoryandappl
7、icationsoffuzzydifferentialequations,fuzzydynamicsandfuzzydifferentialinclusions(seee.g.[10,11,5,1230]).ForasignificantcollectionofresultsfromthetheoryofFDEswerefertothemonographofLakshmikanthamandMohapatra[1],andtoDiamondandKloeden[2].Inthispaperwewillconsiderrandomf
8、uzzydifferentialequations(RFDEs)astheycanprovidegoodmodelsofdynamicsofrealphenomenawhicharesubjectedtotwokindsofuncer
