Cauchy Problems For Fractional Differential Equations With α-Caputo Fractional Derivatives

This thesis is concerned with fractional abstract Cauchy problems with order α∈(2,3). The definition of fractional integral solution operator with order α∈(2,3) is worked out based on the notion of fractional solution operator with order α∈(1,2) and the notion of fractional resolvent with order α∈[1,2], which are proposed by Professor Peng Jigen.A generation theorem for exponentially bounded fractional integral solution operator is obtained through the properties of fractional integral solution operator with order a and the method of inverse Laplace transform and so on.As the application of fractional integral solution operator with order α,this thesis studies the Cauchy problems with Caputo fractional derivative with order α. The author also proves the existence and uniqueness of the homogeneous fractional Cauchy problem (FAC4CP0) with order a and the inhomogeneous fractional Cauchy problem (FACCPf) with order a above. Finally,sufficient conditions are given to testify the existence of the mild solutions and strong solutions of the homogeneous fractional Cauchy problem(FACP0) with order a and inhomogeneous fractional Cauchy problem (FACPf) with order a.