Asymptotic Behaviour of Solutions of Systems of Discrete Equations via Liapunov Type Technique
by
J. Diblík
Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 8, 616 00 Brno, Czech Republic

The second Liapunov method serves as a powerful tool for the investigation of the stability of the trivial solution of ordinary differential equations systems and discrete equations systems. In the presented contribution, a Liapunov-type qualitative approach is used for the investigation of asymptotic behaviour of the solutions of systems of discrete equations. Conditions for the existence of continuum of solutions, the graphs of which remain within a given prescribed set, are formulated for the general systems of discrete equations u(k+1)=u(k)+F(k, u(k)). An additional advantage of the presented approach consists in the fact that no assumption concerning the existence of the trivial solution (or the existence of an equilibrium point) of systems considered is assumed. The asymptotic behaviour of solutions of some classes of linear difference systems is given by means of concrete asymptotic formulae. Illustrative examples are considered, too.

Date received: March 18, 2003