On the existence of almost automorphic solutions of Volterra difference equations

Given a complex summable sequence and a discrete almost automorphic function with values in a complex Banach space X, we give criteria for the existence of discrete almost automorphic solutions of the linear Volterra difference equation , , for λ in a distinguished subset of the complex plane, determined by . We also prove the existence of a discrete almost automorphic solution of the nonlinear Volterra difference equation , , where is a discrete almost automorphic function in n for each that satisfies a global Lipschitz condition and takes values in X. Finally, we treat the same type of problem for Volterra functional difference equations.