Copyright © 2009 Daniela Araya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study discrete almost automorphic functions (sequences) defined on the set of integers with values in a Banach space X. Given a bounded linear operator T defined on X and a discrete almost automorphic function f(n), we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation Δu(n)=Tu(n)+f(n). We also prove the existence of a discrete almost automorphic solution of the nonlinear difference equation Δu(n)=Tu(n)+g(n,u(n)) assuming that g(n,x) is discrete almost automorphic in n for each x∈X, satisfies a global Lipschitz type condition, and takes values on X.