Fuzzy delay differential equations under generaliz

FuzzydelaydifferentialequationsundergeneralizeddifferentiabilityA.Khastana,⇑,J.J.Nietob,c,R.Rodríguez-LópezbaDepartmentofMathematics,InstituteforAdvancedStudiesinBasicSciences(IASBS),Zanjan45137-66731,IranbDepartamentodeAnálisisMatemático,FacultaddeMatemáticas,UniversidaddeSantiagodeCompostela15782,SantiagodeCompostela,SpaincDepartmentofMathematics,FacultyofScience,KingAbdulazizUniversity,P.O.Box80203,Jeddah21589,SaudiArabiaarticleinfoArticlehistory:Received28June2010Receivedinrevisedform17November2013Accepted9February2014Availableonline19February2014Keywords:FuzzyfunctionaldifferentialequationExistenceanduniquenesstheoremGeneralizeddifferentiabilityMalthusianmodelAscitestumormodelabstractWeinterpretafuzzydelaydifferentialequationusingtheconceptofgeneralizeddifferen-tiability.Inthissetting,weprovetheexistenceoftwofuzzysolutions,eachonecorre-spondingtoadifferenttypeofdifferentiability.Uniquenessisunderstoodinthesensethatthesolutionsconsidereddonothaveswitchingpoints.Theapplicabilityofthetheo-reticalresultsisillustratedwithsomerealworldexamples.2014ElsevierInc.Allrightsreserved.1.IntroductionInthemathematicaldescriptionofmanyphenomenarelativetoscientificresearch,functionaldifferentialequationsandsystemsariseasanimportanttooltoachieveamoreadequateexplanationandafaithfuladjustmenttothebehavioroftheparticularmagnitudeofinterest.Thisisreflectedinfieldssuchasengineering,economics,physics,biology,andothers[4,14,15,24,25].Itiswell-knownthatdeterminismisnotable,ingeneral,toprovideacompleteanddefinitemodelfortheanalysisofadynamicalsystem[26,28]duetotheimperfectionsandvaguenessofourperceptionofthesystemitself.Togiveanexample,uncertaintyispresentinthestudyofintelligentsystemsfromdifferentpointsofviewsuchassoftcomputingorgranularcomputing[34].Thus,fuzzinessconstitutesanadequatemechanismtointroducethesubjectivefactorswhichmightinflu-encethephenomenaintothesystem.Fuzzydifferentialequationshavebeenstudiedthroughlyinthelastyearsasadequatemodelstopredictthebehaviorofcontinuousprocessessusceptibletoimprecisionbasedonsubjectivechoices.However,themeaningofafuzzydifferentialequationstronglydependsontheselectionoftheconceptoffuzzyderivative[9].Foranoverviewofthedifferentapproachestofuzzydifferentialequations,wefirstcitetheextensionmadebyPuriandRalescuoftheHukuharaderivativefromthecon-textofset-valuedfunctionstofuzzy-valuedfunctions.See,fordetails,[36]andalsootherworkssuchas[18],wherethisapproachisfollowedtostudyfuzzydifferentialequations.Theinconvenientconsistingintheincreasingmonotonicityof2014ElsevierInc.Allrightsreserved.⇑Correspondingauthor.E-mailaddresses:[e-mail],[e-mail](A.Khastan),[e-mail](J.J.Nieto),[e-mail](R.Rodríguez-López).InformationSciences275(2014)145–167ContentslistsavailableatScienceDirectInformationSciencesjournalhomepage:’diameterfortheHukuhara-differentiablesolutions[12]producesalackofcorrespondencebetweentheprop-ertiesofordinaryandfuzzysolutions.AnotherapproachtosolvefuzzydifferentialequationsisthatproposedbyHüllermeier[16]whorewritestheproblemasafamilyofdifferentialinclusions.Thismethodhastheadvantageofrequiringjustordinaryderivativessothatitisnotnec-essarytodefineproperlytheconceptofderivativeofafunctionwithfuzzyvalues.Ontheotherhand,acompletelydifferentapproachispresentedin[33]bystudyingdifferentialequationswithfuzzyparametersandinitialconditions,consistingintheapplicationofZadeh’sextensionprincipletotheordinary(real)solutioninordertodefinethefuzzysolution.Seealso[11]forsomerelationsbetweensolutionscorrespondingtodifferentapproaches.Generalizeddifferentiabilityhasitsoriginsin[6]andhasbeenstudiedandapplied,forinstance,in[7,8,10,21,30].Withtheuseofthisnotionofdifferentiability,thelevelsetsofthesolutionstofuzzydifferentialequationsdonotnecessarilyhavemonotonicallyincreasingdiameter.Besides,therearefunctionswhicharenotdifferentiablefromthepointofviewofHuku-haraderivativebuttheybecomedifferentiablethankstogeneralizeddifferentiabilityapproach,whichwillbetheonecon-sideredinthepresentpaper.Combiningthetwoaspectsintroduced,fuzzymathematicsandfunctionaldifferentialequations,wegetfuzzyfunctionaldifferentialequations,whichhaveattractedtheinterestofmanyresearchers[13,26,28].Fuzzydifferentialequationswithoutfunctionaldependenceareconsidered,forinstance,in[1–3,5,11,23,27,29,31,32,35,38,39].Themajorcontributionofthispaperistoprovidesufficientconditionsfortheglobalexistenceofaunique(2)-solutiontoaninitialvalueproblemforfuzzyfunctionaldifferentialequations.Ourresults,usinggeneralizedderivative,areofbroaderapplicabilitythanthoseusingHukuharaderivative.Also,theintroductionofdelayinthefuzzymodelallowstoconsidermoregeneralsituations.Ourresultsextendandcomplementthoseofvariousauthorssuchas[26],wheretheexistenceofaunique(1)-solutiontothefunctionalproblemisconsidered.Notethattheexistenceofavarietyofsolutions(forin-stance,(1)-and(2)-differentiablesolutions)mighthelptoadjustmorefaithfullysom