Estimates for the norms of solutions of difference systems with several delays

We derive explicit stability conditions for time-dependent difference equations with several delays in C n (the set of n complex vectors) and estimates for the size of the solutions. The growth rates obtained here are not necessarily decay rates. 1. Introduction. Stability of systems of difference equations with delays has been discussed by many authors, for example, see Gil' and Cheng [6], Zhang [11], Elaydi and Zhang [5], Pituk [10], Agarwal [1], and the references therein. In the stability literature, one can find two major trends; stability using the first approximation Lyapunov method and the direct Lyapunov functional method. For this latter trend, see Zhang and Chen [12], Crisci et al. [4], Lakshmikantham and Trigiante [7], and Agarwal and Wong [2]. By this method many very strong results are obtained. But finding Lyapunov functionals is usually a difficult task. In this note, we consider a class of perturbed difference equations with several delays and, by means of a Gronwall inequality and the recent estimates for the powers A k of a