Asymptotically almost periodic mild solutions to a class of Weyl-like fractional difference equations

This paper is concentrated on a class of difference equations with a Weyl-like fractional difference in a Banach space X forms like △ α x ( n ) = A x ( n + 1 ) + F ( n , x ( n ) ) , n ∈ Z , where α ∈ ( 0 , 1 ) , the operator A generates a C 0 -semigroup on X , △ α denotes the Weyl-like fractional difference operator, F ( n , x ) : Z × X → 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.