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Existence and uniqueness of mild solutions to impulsive fractional differential equations. (English) Zbl 1187.34108

Summary: Our aim in this paper is to study the existence and the uniqueness of the solution for the fractional semilinear differential equation:

D t α x(t)=Ax(t)+f(t,x(t)),tI=[0,T],tt k ,x(0)=x 0 X,Δx| t=t k =l k (x(t k - )),k=1,,m,(1)

where 0<α<1, the operator A:D(A)XX is a generator of 𝒞 0 -semigroup (T(t)) t0 on a Banach space 𝕏, D t α is the Caputo fractional derivative, f:I×𝕏𝕏 is a given continuous function I k :𝕏𝕏, 0=t 0 <t 1 <<t m <t m+1 =T. Δx| t=t k =x(t k + )-x(t k - ), x(t k + )=lim h0 + x(t k +h) and x(t k - )=lim h0 -x(t k +h) represent respectively the right and left limits of x(t) at t=t k .

34K30Functional-differential equations in abstract spaces
34K05General theory of functional-differential equations
34K37Functional-differential equations with fractional derivatives
34K45Functional-differential equations with impulses