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

Abstract

A discrete equation $\Delta y\left(n\right)=\beta \left(n\right)[y(n-j)-y(n-k)]$ with two integer delays $k$ and is considered for $n\to \infty$. We assume $\beta :{\mathbb{Z}}_{n_0-k}^{\infty }\to (0,\infty )$, where ${\mathbb{Z}}_{n_0}^{\infty }=\{n_0,n_0+1,\dots \}$, $n_0\in \mathbb{N}$ and $n\in {\mathbb{Z}}_{n_0}^{\infty }$. Criteria for the existence of strictly monotone and asymptotically convergent solutions for $n\to \infty$ 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.