| Abstract: | Our purpose is to asymptotically represent solutions of linear difference equations x(k + 1) = [A0 + 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. |
