Asymptotic convergence of the solutions of a dynamic equation on discrete time scales
type: Item
instance of: scholarly article
title: Asymptotic convergence of the solutions of a dynamic equation on discrete time scales
zbMATH Open document ID: 1232.39006
DOI: 10.1155/2012/580750
published in: Abstract and Applied Analysis
publication date: 2012-01-05T00:00:00Z
review text: Summary: The authors investigate the dynamic equation \[ \Delta y(t_n) = \beta(t_n)[y(t_{n-j}) - y(t_{n-k})] \] for \(n \rightarrow \infty\), where \(k\) and \(j\) are integers such that \(k > j \geq 0\), on an arbitrary discrete time scale \(\mathbb T : = \{t_n\}\) with \(t_n \in \mathbb R, n \in \mathbb Z^{\infty}_{n_{0}-k} = \{n_0 - k, n_0 - k + 1, \dots\}, n_0 \in \mathbb N, t_n < t_{n+1}, \Delta y(t_n) = y(t_{n+1}) - y(t_n)\), and \(\lim_{n \rightarrow \infty} t_n = \infty\). We assume \(\beta : \mathbb T \rightarrow (0, \infty)\). It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent for \(n \rightarrow \infty\). The results are presented as inequalities for the function \(\beta\). Examples demonstrate that the criteria obtained are sharp in a sense.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34N05
Mathematics Subject Classification ID: 39A22
Mathematics Subject Classification ID: 39A06
zbMATH DE Number: 5994554
zbMATH Keywords: discrete time scales
zbMATH Keywords: dynamic equation
zbMATH Keywords: asymptotic convergence
zbMATH Keywords: increasing solution
Wikidata QID: Q58695593
author: Josef Diblík
author: Miroslava Růžičková
author: Zdeněk Šmarda
author: Z. Šutá
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/580750
OpenAlex ID: W2041772676
cites work: Analysis on measure chains - a unified approach to continuous and discrete calculus
cites work: Q3995705
cites work: First and second order linear dynamic equations on time scales
cites work: Convergence in asymptotically autonomous functional differential equations
cites work: Q3527108
cites work: Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays
cites work: Asymptotic convergence criteria of solutions of delayed functional differential equations
cites work: Convergence of the solutions of the equation \(\dot y(t) = \beta(t)[y(t-\delta)-y(t-\tau)]\) in the critical case
cites work: Convergence of solutions for two delays Volterra integral equations in the critical case
cites work: On oscillation of a food-limited population model with time delay
cites work: Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case
cites work: Convergence of solutions of nonhomogeneous linear difference systems with delays
cites work: Q3634194
cites work: Asymptotic constancy in linear difference equations: limit formulae and sharp conditions
cites work: Global stability and asymptotics of some classes of rational difference equations
cites work: On the difference equation \(x_{n+1}=x_nx_{n-2}-1\)
cites work: Q5487039
cites work: Nonoscillatory solutions of a second-order difference equation of Poincaré type
description: scientific article; zbMATH DE number 5994554