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1、doi:10.3969/j.issn.1005-3085.2014.02.014ArticleID:1005-3085(2014)02-0286-14ExistenceofSolutiontoSingularTwo-pointBoundaryValueProblemforNonlinearFractionalDifferentialEquation∗HANRen-ji1,JIANGWei2(1-DepartmentofBasicCourses,HuishangVocationalCollege,Heifei2312
2、01;2-SchoolofMathematicalSciences,AnhuiUniversity,Heifei230039)Abstract:Inthispaper,wediscusstheexistenceofsolutiontosingularboundaryvalueforaclassofnonlinearfractionaldifferentialequation.ThedifferentialoperatoristheRiemann-Liouvillederivativeandtheinhomogeneo
3、ustermdependsonthefrac-tionalderivativeoflowerorder.OuranalysisreliesonLeray-Schauder’sfixedpointtheorem.Finally,anexampleisgiventoillustratetheeffectivenessoftheresult.Keywords:nonlinearfractionaldifferentialequation;singulartwo-pointboundaryvalueprob-lem;Schau
4、derfixedpointtheoremClassification:AMS(2000)34A08CLCnumber:O175.8Documentcode:A1IntroductionWeareinterestedinthesingularfractionalboundaryvalueproblemαµD0+u(t)=f(t,u(t),Du(t)),(1)u(0)=u(1)=u′(0)=u′′(1)=0,(2)where3<α≤4,0<µ≤α−2,f:(0,1]×R×R→R,f(t,·,·)issingulara
5、tt=00+andDαisthestandardRiemann-Liouvillefractionalderivative.Recently,fractionaldifferentialequationshavebeeninvestigatedextensivelyasvaluabletoolsinthemathematicalmodelofsystemsandprocessesinvariousfieldsofscienceandengineering.Italsoservesasanexcellenttoolf
6、orthedescriptionofhereditarypropertiesofvariousandprocesses.Fordetails,see[1–3]andtherefer-encestherein.Papers,suchas[4–9],discussfractionalboundaryvalueproblemwithnonlinearitieshavingsingularitiesinvariousspace.Received:08May2012.Accepted:06Dec2012.Biograph
7、y:HanRenji(Bornin1981),Male,Master.Researchfield:func-tionaldifferentialequationandfractionaldifferentialequation.∗Foundationitem:TheNationalNatureScienceFoundationofChina(11071001).Paper[4]investigatestheexistenceofpositivesolutiontothesingularboundaryvalueprob
8、lemfornonlinearfractionaldifferentialequationαD0+u(t)+f(t,u(t))=0,(3)u(0)=u′(1)=u′′(0)=0,(4)0+where2<α≤3isarealnumber,DαistheCaputo′sdifferential,andf:(0,1]×[0,+∞)→[0,+∞),limf(t,·)=+∞.t→0+
