带有多点边值的分数阶微分包含解的Filippov型存在性定理
|
摘要
很多领域的实际问题可以建立分数阶微分方程或者微分包含模型进行研究,近年来分数阶微积分受到广泛关注。2016年,文献[8]研究了一类带有多点边值条件的分数阶微分方程解的存在性。本文中,利用多值映射的不动点定理,给出了以下带有多点边值分数阶微分包含解的Filippov型存在性定理:D~αy(t)∈F(t,y(t)),t∈[0,T],T>0,y(T)=y~*+h(x),D~Py(T)=m∑i=1D~py(ηi),其中1<α≤2,0 In many research fields,some practical problems could be established to fractional differential equation models or fractional differential inclusion models for study,which have been got much attention by mathematicians in recent years. In 2016,in paper[8] the authors investigated the existence of solutions for a class of fractional differential equations. In this paper,based on fixed-point theorem for multi-value maps,we are concerned with the following fractional order differential inclusions with multi-point boundary value problems:D~αy( t) ∈ F( t,y( t)),t ∈ [0,T],T > 0,y( T) = y~*+ h( y),D~Py( T) =m∑i = 1D~py( ηi),where 1 < α≤2,0 < p < 1,D~α,D~p denote Caputo derivatives,x~*∈ R,0 < η_i< T,i = 1,2,3,...,m. h:[0,T] × R→ R is a continuous,F:[0,T]× R → P(R) in[0,T]is a multi-valued map. The Filippove theorem on the existence solutions for the problem is given. The aim of this paper is to extend known single value result[8] to multi-value framework.
引文
[1]AUBIN J P,FRANKOWSKA H.Set-valued analysis[M].Boston:Birkhauser,1990.
[2]AHMAD B,AGARWAL R P,ALSAEDI A.Fractional differential equations and inclusions with semiperiodic and three-point boundary conditions[J].Bound Value Probl,2016(28):1-20.
[3]CEMEA A.Continuous verson of Filippov’theorem for fractional differential inclusions[J].Anonlinear Anal,2010(72):204-208.
[4]CASTAING C,VALADIER M,BAI Z.Convex analysis and measurable multifunctions,lecture notes in mathematics[M].New York:Springer-Verlag,1977:580.
[5]El-Sayed A M A,IBRAHIM A G.Multivalued fractional differential equations[J].Appl Math Comput,1995(68):15-25.
[6]FRANKOWSKA H.A priori estimates for operational differential inclusions[J].J Differential Equations,1990(84):100-128.
[7]GORNIEWICZ L.Topological fixed point theory of multivalued mappings,mathematics and its applications[M].Dordrecht:Kluwer Academic Publishers,1999.
[8]HOUAS M,DAHMANI Z.On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions[J].Lobachevskii Journal of Mathematics,2016,37(2):120-127.
[9]MAHUDOV N,UNUL S.On existence of BVP’s for impulsive fractional differential equations[J].Advances in Difference Equations,2017(15):1-16.
[10]NTOUYAS S K,ETEMAD S.On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions[J].Appl Math Comput,2015(266):235-243.
[11]OUAHAB A.Filippov’s theorem for impulsive differential inclusions with fractional order[J].Electron J Qual Theory Differ Equa,2009(23):1-28.
[12]PODLUBNY I.Fractional differential equations[M].San Diego:Academic Press,1999.
[13]SAMKO S G,KILBAS A A,MARICHEV O I.Fractional integral and derivatives,theory and applications[M].Switzerland:Gordon and Breach,1993.
[14]王永庆,刘立山.Banach空间中分数阶微分方程m点边值问题的正解[J].数学物理学报,2012,32(1):246-256.
[15]杨丹丹.带有积分边值条件的分数阶微分包含解的存在性[J].浙江大学学报(理学版),2015(6):688-692.
[2]AHMAD B,AGARWAL R P,ALSAEDI A.Fractional differential equations and inclusions with semiperiodic and three-point boundary conditions[J].Bound Value Probl,2016(28):1-20.
[3]CEMEA A.Continuous verson of Filippov’theorem for fractional differential inclusions[J].Anonlinear Anal,2010(72):204-208.
[4]CASTAING C,VALADIER M,BAI Z.Convex analysis and measurable multifunctions,lecture notes in mathematics[M].New York:Springer-Verlag,1977:580.
[5]El-Sayed A M A,IBRAHIM A G.Multivalued fractional differential equations[J].Appl Math Comput,1995(68):15-25.
[6]FRANKOWSKA H.A priori estimates for operational differential inclusions[J].J Differential Equations,1990(84):100-128.
[7]GORNIEWICZ L.Topological fixed point theory of multivalued mappings,mathematics and its applications[M].Dordrecht:Kluwer Academic Publishers,1999.
[8]HOUAS M,DAHMANI Z.On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions[J].Lobachevskii Journal of Mathematics,2016,37(2):120-127.
[9]MAHUDOV N,UNUL S.On existence of BVP’s for impulsive fractional differential equations[J].Advances in Difference Equations,2017(15):1-16.
[10]NTOUYAS S K,ETEMAD S.On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions[J].Appl Math Comput,2015(266):235-243.
[11]OUAHAB A.Filippov’s theorem for impulsive differential inclusions with fractional order[J].Electron J Qual Theory Differ Equa,2009(23):1-28.
[12]PODLUBNY I.Fractional differential equations[M].San Diego:Academic Press,1999.
[13]SAMKO S G,KILBAS A A,MARICHEV O I.Fractional integral and derivatives,theory and applications[M].Switzerland:Gordon and Breach,1993.
[14]王永庆,刘立山.Banach空间中分数阶微分方程m点边值问题的正解[J].数学物理学报,2012,32(1):246-256.
[15]杨丹丹.带有积分边值条件的分数阶微分包含解的存在性[J].浙江大学学报(理学版),2015(6):688-692.