The Existence And Uniqueness Of Two Classes Of Fractional Implusive Differential Equation

In this paper,first of all,by using the Banach’s fixed-point theo-rem and Schauder’s fixed-point theorem,we study the existence and uniqueness of solutions for the following fractional implusive integral-differential equation:Where J:=[0,T],t ∈ J’:=J\{t1,t2,…,tm},CDq is the Caputo fractional derivative of order q ∈(0,1).f:J × Rn × Rn →Rn,Ω={(t,s):0≤s≤t≤T},h:Ω×Rn→ Rn.A(t)=(aij(t))(t ∈ J),A1,A2 are given matrices.Then,we study the existence and uniqueness of solutions for the following fractional implusive differential equation by using the Ba-nach’s fixed-point theorem,Krasnoselskii’s fixed-point theorem and the Leray-Schauder Nonlinear Alternative theorem.Where CDα is the Caputo fractional derivative of order α ∈(0,1),f ∈ C([0,1]× R→R),Iβk,j and Ip are the R-L fractional integrals oforder βk,j>0 and p ∈(0,1)respectively.β∈R 且β≠Γ(p+1)/ηp.