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Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. (English) Zbl 1193.34056

From the introduction: We consider the following impulsive hybrid boundary value problem for nonlinear fractional differential equations

c D q x(t)=f(t,x(t)),1<q2,tJ 1 =[0,1]{t 1 ,t 2 ,,t p },Δx(t k )=I k (x(t k - )),Δx ' (t k )=J k (x(t k - )),t k (0,1),k=1,2,,p,x(0)+x ' (0)=0,x(1)+x ' (1)=0·(1)

where c D is the Caputo fractional derivative f:[0,1]× is a continuous function, I k ,J k :, Δx(t k )=x(t k + )-x(t k - ) with x(t k + )=lim h0 + x(t k +h),x(t k - )=lim h0 - x(t k +h), k=1,2,,p for 0=t 0 <t 1 <t 2 <<t p <t p+1 =1.

We prove some existence results by means of the contraction mapping principle and Krasnoselskii’s fixed point theorem.

34B37Boundary value problems for ODE with impulses
34A08Fractional differential equations
47N20Applications of operator theory to differential and integral equations