Existence And Controllability Of Solutions For A Class Of Fractional Evolution Equations

In this paper,by using the fixed point theorem of increasing operator and the conditions of the measure of non-compactness,we first study the existence of mild solutions for the fractional evolution equation with integral boundary condition in ordered Banach space X (?), where CDq is the Caputo fractional derivative of order q?(0,1],A:D(?)(?)X?X is a densely defined and closed linear operator,-A is the infinitesimal generator of semigroup T(t)(t?0),H(x)is the nonlocal function defined as(?),f:I×X?X and g:I × X?X are appropriate functions to be specified laterSecondly,under some“compactness-type”conditions such as the measure non-compactness condition,by using Sadovskii fixed point theorem,we discuss the exact controllability for the nonlocal problem of the fractional evolution equation in the following form(?),where A is infinitesimal generator of semigroup T(t)(t?0),the control function u(·)is given in L2(I,U),U is a Hilbert space,B is a bounded linear operator from U to X,f:I × X?X is an appropriate function to be specified later,H(x)is defined as in(?)Finally,by applying Schauder fixed point theorem,the existence of mild so-lutions of the problem(?)is proved and its approximate controllability is also discussed.