On the asymptotic behaviour of Volterra difference equations

We consider the asymptotic behaviour of solutions of the difference equations of the form x(n + 1) = Ax(n) + Sigma B-n(k=0)(n - k)x(k) + y(n) in a Banach space X, where n=0,1,2,...; A,B(n) are linear bounded operators in X. Our method of study is based on the concept of spectrum of a unilateral sequence. The obtained results on asymptotic stability and almost periodicity are stated in terms of spectral properties of the equation and its solutions. To this end, a relation between the Z-transform and spectrum of a unilateral sequence is established. The main results extend previous results.