Asymptotic behavior of solutions of ordinary difference equations

(1.1) y(n + 1) = Jy(n) + f(n, y(n)), where y is a d-vector, J is a constant d x d matrix and f(n, y) is a vector-valued function which is continuous in y for fixed n and becomes small in some sense as (n,y)-+(oo,0). Systems of this type have been studied by Perron, among others; see for example [10]. Perron has also investigated the analogous problem of the asymptotic behavior of solutions of the differential equation