Asymptotic behaviour of the solutions of a nonautonomous linear delay difference system

It is proved under appropriate assumptions that the solutions of a linear system of nonautonomous delay difference equations have finite limit at infinity. The results are based on a transformation of the delay difference system into a first-step recursion, where the companion matrices are well treatable from our point of view. Our theory is illustrated by examples, including a class of linear delay difference equations with unbounded coefficients.