Almost automorphic solutions of non-autonomous difference equations

In the present paper, we study the non-autonomous difference equations given by u(k + 1) = A(k)u(k) + f (k) and u(k + 1) = A(k)u(k)+ g(k, u(k)) for k is an element of Z, where A(k) is a given non-singular n x n matrix with elements a(ij)(k), 1 <= i, j <= n, f : Z -> E-n is a given n x 1 vector function, g : Z x E-n -> E-n and u(k) is an unknown n x 1 vector with components u(i)(k), 1 <= i <= n. We obtain the existence of a discrete almost automorphic solution for both the equations, assuming that A(k) and f (k) are discrete almost automorphic functions and the associated homogeneous system admits an exponential dichotomy. Also, assuming the function g satisfies a global Lipschitz type condition, we prove the existence and uniqueness of an almost automorphic solution of the nonlinear difference equation. (C) 2013 Elsevier Inc. All rights reserved.