Fractional Order Abstract Cauchy Problems

The fractional abstract Cauchy problems with order ? > 0 are discussed in this paper.Three chapters are included. The research background related to the fractional abstract Cauchy problems and preliminaries are introduced in the first chapter. The definition of the family of integral operator is given in the second chapter. Based on the property of the integral operator, the generation theorem of the index bounded integral operator T?(t)are obtained. The third chapter is divided into two sections, the order of a homogeneous Cauchy problem and the order of a inhomogeneous abstract Cauchy problem are discussed:The necessary and sufficient condition of the solution's well-posedness to the homogeneous Cauchy problem and the sufficient condition of the existence and uniqueness of mild and classical solution to the inhomogeneous Cauchy problem are discussed. The R-L fractional order derivative of absolute value function is discussed in the appendix.