Existence and stability for Hadamard p-type fractional functional differential equations

《Existence and stability for Hadamard p-type fractional functional differential equations》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、J.Appl.Math.Comput.DOI10.1007/s12190-016-1049-0ORIGINALRESEARCHExistenceandstabilityforHadamardp-typefractionalfunctionaldifferentialequationsHaihuaWang1·YanqiongLiu1·HuanranZhu2Received:3July2016©KoreanSocietyforComputationalandAppliedMathematics2016AbstractInthispaper,westudythe

2、existenceandHyersUlamstabilityofsolutionsforaHadamardp-typefractionalorderfunctionalandneutralfunctionaldifferentialequationsinvolvinginitialvalueproblems.SufficientconditionswhichguaranteetheexistenceandHyersUlamstabilityarenewandobtained.KeywordsHadamardfractionalderivative·p-fun

3、ction·Existence·HyersUlamstabilityMathematicsSubjectClassification26A33·34K101IntroductionLetJ⊂R.DenoteC(J,R)astheBanachspaceofallcontinuousfunctionsfromJintoRwiththenormx=supt∈J

4、x(t)

5、.Givenanyr>0,letC=C([−r,0],R)denotethespaceofcontinuousfunctionson[−r,0].Foranyelementφ∈C,definet

6、henormφ∗=supθ∈[−r,0]

7、φ(θ)

8、.ThispaperisconcernedwiththeexistenceofsolutionsandstabilityforinitialvalueproblemsofHadamardp-typefractionalorderfunctionaldifferentialequations.InSect.3,weareinterestedintheexistenceandstabilityofthefollowinginitialvalueproblems:BHaihuaWangwanghoiwan@

9、163.com1DepartmentofMathematics,GuangxiScienceandTechnologyNormalUniversity,Laibin546100,Guangxi,PeoplesRepublicofChina2DepartmentofMathematics,HunanUniversityofScienceandTechnology,Xiangtan411201,Hunan,PeoplesRepublicofChina123H.Wangetal.qDa+x(t)=f(t,xt),(1.1)xa=ϕ,(a,ϕ)∈,qwhere

10、Da+istheHadamardfractionalderivativeoforder00,isanopensubsetof(0,+∞)×C.f:→Risagivenfunctionsatisfiessomeassumptionsthatwillbespecifiedlater,ϕ∈Candϕ(0)=0.xt∈Cisdefinedbyxt(θ)=x(p(t,θ)),where−r≤θ≤0,p(t,θ)isap-function.Section4isdevotedtoHadamardfractionalneutralfunctionaldif

11、ferentialequa-tions:qDa+[x(t)−g(t,xt)]=f(t,xt),(1.2)xa=ϕ,(a,ϕ)∈,wherea,f,ϕandareasinProblem(1.1),andg:C→Risagivenfunctionsuchthatg(a,ϕ)=0.Fractionaldifferentialequationshavegainedconsiderableimportanceduetotheirapplicationsinvariousscience,suchasphysics,chemistry,engineering,po

12、lymerrhe-ology,etc.Inrecentyears,