Almost Automorphic Solutions of Difference Equations

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.