Initial Value Problems For Fractional Evolution Equations

In this paper,we study the existence,uniqueness and continuous dependence of the mild solutions for a class of fractional evolution equations.First,using the Krasnoselskii's fixed point theorem and the theory of resolvent operators for integral equations,we show the existence,uniqueness and continuous dependence of the mild solutions of the nonlocal Cauchy problem for a class of fractional evolution equations with finite delay:where 0<?<1.Second,we convert the limited delay to infinite delay.Then we consider the fractional evolution equation:where 0<?<1,and show its existence of a mild solution by using the Krasnosel-skii's fixed point theorem.By using the Banach fixed point theorem,we obtain the uniqueness of a mild solution to this equation.Here using the theory of resolvent operators for integral equations.we give the definition of mild solution for the fractional evolution equation,without taking the method that using by most of the authors who define the mild solutions for fractional order just.like for integer order,so the results in this paper can provide deeper insights and further explored.