Almost Periodic Mild Solutions to a Class of Fractional Delayed Differential Equations

Abstract

In this paper, one studies the existence and uniqueness of almost periodic mild solutions to fractional delayed differential equations of the form D_t^\alpha x(t)=Ax(t)+D_t^{\alpha-1} f(t,x_t) where 1 < α < 2, A : D(A) \subset X -- X is a linear densely defined operator of sectional type on a complex Banach space X and f : R \times X -- X is jointly continuous. Let f(t; x) be almost periodic in t \in R uniformly for x. Under some additional assumptions on A and f, the existence and uniqueness of a almost periodic mild solution to above equation is obtained by using the Banach fixed-point principle. The obtaining results extent corresponding results in time delay with respect to almost periodic mild solutions for fractional differential equations.