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In this paper we study an analytic resolvent family for abstract fractional integrodifferential system using the perturbation theory of sectorial operators. We apply this resolvent family on the existence of mild solutions for abstract semilinear Cauchy problem Dα t u(t) = Au(t)+ ∫ t 0 B(t− s)u(s)ds+ f (t,u(t)), t ∈ (0,τ), u(0) = u0 ∈ X , where Dα t u represents the Caputo derivative of u for α ∈ (0,1), A,(B(t))t 0 are closed linear operators defined on a common domain which is dense in a Banach space X and f satisfies appropriated conditions. In the end, we applain the ours abstract results in the existence of mild solution of two partial integro-differential systems. Mathematics subject classification (2010): 26A33, 34K30, 35R11, 45N05.

Fractional resolvent operator with α ∈ (0,1) and applications