On the existence of mild solutions to the Cauchy problem for a class of fractional evolution equations

Abstract

We are concerned with the existence of mild solutions to the Cauchy problem for fractional evolution equations of neutral type with almost sectorial operators dq/dtq (x(t) - h(t, x(t))) = -A(x(t) - h(t, x(t))) + f (t, x(t)), t > 0, x(0) = x0, where 0 < 1, the fractional derivative is understood in the Caputo sense, A is an almost sectorial operator on a complex Banach space, and f, h are given functions. With the help of the theory of measure of noncompactness and a fixed point theorem of Darbo type, we establish a new existence theorem of mild solutions for the Cauchy problem above. By the way, the global attractive property of the solutions is also obtained. Moreover, we give two examples to illustrate our abstract results. © 2012 Liang et al;.