Boundedness and periodicity of Volterra systems of difference equations

We consider the systems of Volterra difference equations of nonconvolution types (1) y(n+1) = A(n)y(n)+ Sigma(s=0)(n) C(n, s)y(s) + g(n) and (2) x(n+1) = A(n)x(n) + Sigma(s=-infinity)(n) C(n, s)x(s)+g(n) (1) y(n+1) = A(n)y(n) + where A(n), C(n,s) are k x k matrices, g(n) is a k x 1 bounded vector function. The goal of this paper is to use the z-transform and Lyapunov functionals to show that all solutions y(n) of(I) are bounded. Also, we show that lim sup of y(n) of (1) is uniformly bounded. Both the bound and the uniform bound that we will obtain will not depend on the resolvent matrix solution of; the homogeneous part of (1). Furthermore, if we impose periodicity conditions on A(n), C(n, s) and g(n), then our theorems will directly imply the existence of a periodic solution of (2) by appealing to Elaydi's results in [3].