Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case
type: Item
instance of: scholarly article
title: Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case
zbMATH Open document ID: 1220.39004
DOI: 10.1155/2011/709427
author: Leonid Berezansky
author: Josef Diblík
author: Miroslava Růžičková
author: Z. Šutá
published in: Abstract and Applied Analysis
publication date: 2011-07-22T00:00:00Z
review text: Summary: A discrete equation \(\Delta y(n) = \beta(n)[y(n - j) - y(n - k)]\) with two integer delays \(k\) and \(j\), \(k > j \geq 0\) is considered for \(n \rightarrow \infty\). We assume \(\beta : \mathbb Z^{\infty}_{n_{0}-k} \rightarrow (0, \infty)\), where \(\mathbb Z^{\infty}_{n_0} = \{ n_0, n_0 + 1, \dots \}\), \(n_0 \in \mathbb N\) and \(n \in \mathbb Z^{\infty}_{n_0}\). Criteria for the existence of strictly monotone and asymptotically convergent solutions for \(n \rightarrow \infty\) 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.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A06
Mathematics Subject Classification ID: 34K06
zbMATH DE Number: 5930775
zbMATH Keywords: discrete equation with two delays
zbMATH Keywords: asymptotics
zbMATH Keywords: monotone solution
zbMATH Keywords: critical value
zbMATH Keywords: asymptotic convergence
Wikidata QID: Q58654413
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2011/709427
OpenAlex ID: W2001961465
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description: scientific article; zbMATH DE number 5930775