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

. In this paper we study an analytic resolvent family for abstract fractional integro- differential system using the perturbation theory of sectorial operators. We apply this resolvent family on the existence of mild solutions for abstract semilinear Cauchy problem where D α t u represents the Caputo derivative of u for α ∈ ( 0 , 1 ) , A , ( B ( t )) t (cid:2) 0 are closed linear operators de fi ned on a common domain which is dense in a Banach space X and f satis fi es appropriated conditions. In the end, we applain the ours abstract results in the existence of mild solution of two partial integro-differential systems.