Asymptotic Representation of Solutions of Perturbed Systems of Linear Difference Equations

Our purpose is to asymptotically represent solutions of linear difference equations x ( k + 1) = [ A 0 + A ( k )] x ( k ) when k → + ∞ and A ( k ) is “small” by transforming them into so‐called L ‐diagonal form. Two properties are then responsible for the asymptotic equivalence of an L ‐diagonal form to a diagonal one: a dichotomy condition on the diagonal part, and a growth condition on the perturbation term. In this manner, we derive some known asymptotic results from a central point of view and also several new extensions and generalizations of them. Some examples are constructed which demonstrate the need for a dichotomy‐type condition, which shows incidentally that results of M. A. Evgrafov are incorrect, since they omit such a condition.