Abstract
This paper is concentrated on a class of difference equations with a Weyl-like fractional difference in a Banach space X forms like
Delta(alpha)x(n) = Ax(n + 1) + F(n, x(n)), n is an element of Z,
where alpha is an element of (0, 1), the operator A generates a C-0-semigroup on X, Delta(alpha) denotes the Weyl-like fractional difference operator, F(n, x) : ZxX -> X is a nonlinear function. Some existence theorems for asymptotically almost periodic mild solutions to this system are obtained with the nonlinear perturbation F being of Lipschitz type or non-Lipschitz type. The results are a consequence of applications of the Banach contraction mapping theory, the Leray-Schauder alternative theorem, and Matkowski's fixed point theorem. As an application, an example is provided to show the feasibility of the theoretical results.