Asymptotic Constancy in Linear Difference Equations: Limit Formulae and Sharp Conditions

It is found that every solution of a system of linear delay difference equations has finite limit at infinity, if some conditions are satisfied. These are much weaker than the known sufficient conditions for asymptotic constancy of the solutions. When we impose some positivity assumptions on the coefficient matrices, our conditions are also necessary. The novelty of our results is illustrated by examples.