On asymptotic behavior of solutions of discrete systems

This contribution concerns the asymptotic behavior of solutions of systems of difference equations. It reassumes paper [J. Dibl´ık: Anti-Lyapunovmethod for systems of discrete equations, Nonlinear Anal. 57 (2004), 1043-1057], where conditions are described which ensure that the graph of at least one solution of the studied system stays in a prescribed domain. Those conditions include the existence of a so called connecting function for the sets forming the given domain. The connecting function must have certain properties which enable the use of the retract method for the proof of the main result of the cited work. We show that the conditions can be made simpler, without the need to introduce the connecting function.