Fractional Abstract Cauchy Problems

This paper is concerned with fractional abstract Cauchy problems with order alpha is an element of (1, 2). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP (0)) is well-posed if and only if its coefficient operator A generates an alpha-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACP(f) ).