Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case

作者：Berezansky L; Diblik J^*; Ruzickova M; Suta Z

来源：Abstract and Applied Analysis, 2011, 709427.

DOI：10.1155/2011/709427

A discrete equation Delta y(n) = beta(n) [y(n-j) - y(n-k)] with two integer delays k and j, k > j >= 0 is considered for n -> infinity. We assume beta : Z(n0-k)(infinity) -> (0, infinity), where Z(n0)(infinity) = {n(0), n(0) + 1, ...}, n(0) is an element of N and n is an element of Z(n0)(infinity). Criteria for the existence of strictly monotone and asymptotically convergent solutions for n. 8 are presented in terms of inequalities for the function beta. Results are sharp in the sense that the criteria are valid even for some functions beta with a behavior near the so-called critical value, defined by the constant (k-j)(-1). Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.

• 出版日期2011