Asymptotic convergence of the solutions of a discrete equation with several delays

A discrete equation Delta y(n) = (i=1)Sigma(s)beta(i)(n)[y(n - j(i)) - y(n - k(i))] with integer delays k(i) and j(i), k(i) > j(i) >= 0 is considered. It is assumed that beta(i) : Z(n0-k)(infinity) -> vertical bar 0, infinity), n(0) is an element of Z; k = max{k(1), k(2),..., ks}, Z(p)(infinity) := {p, p + 1,...}, p is an element of Z, Sigma(s)(i-1) beta(i)(n) > 0; s is an element of Z(1)(infinity) is a fixed integer and n is an element of Z(n0)(infinity). Criteria of asymptotic convergence of solutions are expressed in terms of inequalities for the functions beta(i), i = 1,..., s. Some general properties of solutions are derived as well. It is, e. g., proved that, for the asymptotical convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient. A crucial role in the analysis of convergence is played by an auxiliary inequality derived from the form of a given discrete equation. (C) 2011 Elsevier Inc. All rights reserved.