| We consider the problem of constructing an asymptotic representation of solutions for several classes of systems of linear difference equations with variable coefficients as well as determining the stability or instability of the trivial solution of such systems. The asymptotic representation we have in mind is related to the Levinson's Asymptotic Lemma and its difference equations analogs established by Rapoport, Benzaid and Lutz, and Elaydi. As a main tool we utilize the method of averaging developed by N. N. Bogoliubov for investigating stability of solutions of ordinary differential equations. We start with a comprehensive review of existing results in this area, particularly due to Poincare, Perron, Evgrafov, Gelfond, Levinson, Benzaid and Lutz, Elaydi, and others. We also give brief introduction into Bogoliubov's method of averaging. In Chapter 2 we investigate the stability of the trivial solutions for a certain class of linear difference equations with almost periodic, near constant coefficients. Chapters 3--5 are devoted to constructing asymptotic representations of solutions for specific classes of systems of linear difference equations. In each case we establish ways to use the ideas of the method of averaging to transform the original system to an asymptotically equivalent system for which we can construct an asymptotic representation either directly or by utilizing the difference analogs of Levinson's Asymptotic Lemma. The results in each chapter are illustrated by examples. |
