## Abstract and Applied Analysis

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

#### Abstract

A discrete equation $\Delta y(n)=\beta (n)[y(n-j)-y(n-k)]$ with two integer delays $k$ and $j, k>j\ge 0$ 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2011, Number 1 (2011), Article ID 709427, 15 pages.

Dates
First available in Project Euclid: 12 August 2011

https://projecteuclid.org/euclid.aaa/1313171420

Digital Object Identifier
doi:10.1155/2011/709427

Mathematical Reviews number (MathSciNet)
MR2817261

Zentralblatt MATH identifier
1220.39004

#### Citation

Berezansky, L.; Diblík, J.; Růžičková, M.; Šutá, Z. Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case. Abstr. Appl. Anal. 2011 (2011), no. 1, Article ID 709427, 15 pages. doi:10.1155/2011/709427. https://projecteuclid.org/euclid.aaa/1313171420