Existence of solutions of the abstract Cauchy problem of fractional order

In this paper, we study the differentiability of mild solutions for a class of fractional abstract Cauchy problem. We consider the fractional derivative of order α∈(1,2) in the sense of Caputo. In the first part, we establish the existence of classical solutions for the homogeneous Cauchy problem in terms of the α-resolvent family corresponding to the problem and, in the second part, we establish the existence of classical solutions for the inhomogeneous abstract Cauchy problem in terms of regularity properties of the α-resolvent family and of the forcing function.