Publication detail

# Asymptotic convergence of the solutions of a discrete system with delays

DIBLÍK, J. RŮŽIČKOVÁ, M. ŠUTÁ, Z.

Original Title

Asymptotic convergence of the solutions of a discrete system with delays

English Title

Asymptotic convergence of the solutions of a discrete system with delays

Type

journal article - other

Language

en

Original Abstract

A system of $s$ discrete equations \begin{equation*} \Delta y (n)=\beta(n)[y(n-j)-y(n-k)] \end{equation*} is considered where $k$ and $j$ are integers, $k>j\geq0$, $\beta(n)$ is a real $s\times s$ square matrix defined for $n\ge n_{0}-k$, $n_{0}\in \mathbb{Z}$ with non-negative elements $\beta _{ij}(n)$, $i,j=1,\dots,s$ such that $\sum_{j=1}^{s}\beta _{ij}(n)>0$, $y=(y_1, y_2,\dots,y_s)^T\colon {n_{0}-k,n_{0}-k+1,\dots\}\to \mathbb{R}^{s}$ and $\Delta y(n)=y(n+1)-y(n)$ for $n\ge n_{0}$. A method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically convergent under some conditions, i.e., for every solution $y(n)$ defined for all sufficiently large $n$, there exists a finite limit $\lim_{n\to\infty}y(n)$. Moreover, it is proved that the asymptotic convergence of all solutions is equivalent to the existence of one asymptotically convergent solution with increasing coordinates. Some discussion related to the so-called critical case known for scalar equations is given as well.

English abstract

A system of $s$ discrete equations \begin{equation*} \Delta y (n)=\beta(n)[y(n-j)-y(n-k)] \end{equation*} is considered where $k$ and $j$ are integers, $k>j\geq0$, $\beta(n)$ is a real $s\times s$ square matrix defined for $n\ge n_{0}-k$, $n_{0}\in \mathbb{Z}$ with non-negative elements $\beta _{ij}(n)$, $i,j=1,\dots,s$ such that $\sum_{j=1}^{s}\beta _{ij}(n)>0$, $y=(y_1, y_2,\dots,y_s)^T\colon {n_{0}-k,n_{0}-k+1,\dots\}\to \mathbb{R}^{s}$ and $\Delta y(n)=y(n+1)-y(n)$ for $n\ge n_{0}$. A method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically convergent under some conditions, i.e., for every solution $y(n)$ defined for all sufficiently large $n$, there exists a finite limit $\lim_{n\to\infty}y(n)$. Moreover, it is proved that the asymptotic convergence of all solutions is equivalent to the existence of one asymptotically convergent solution with increasing coordinates. Some discussion related to the so-called critical case known for scalar equations is given as well.

Keywords

Discrete equation, delay, asymptotic convergence, increasing solution.

RIV year

2012

Released

15.12.2012

Publisher

Elsevier Science Publishing Co

Location

USA

ISBN

0096-3003

Periodical

APPLIED MATHEMATICS AND COMPUTATION

Year of study

2012

Number

18

State

US

Pages from

4036

Pages to

4044

Pages count

9

Documents

BibTex


@article{BUT95916,
author="Josef {Diblík} and Miroslava {Růžičková} and Zuzana {Šutá}",
title="Asymptotic convergence of the solutions of a discrete system with delays",
annote="A system of $s$ discrete equations \begin{equation*} \Delta y (n)=\beta(n)[y(n-j)-y(n-k)] \end{equation*} is considered where  $k$ and $j$ are integers, $k>j\geq0$, $\beta(n)$ is a real $s\times s$ square matrix defined  for $n\ge n_{0}-k$, $n_{0}\in \mathbb{Z}$ with non-negative elements $\beta _{ij}(n)$, $i,j=1,\dots,s$ such that $\sum_{j=1}^{s}\beta _{ij}(n)>0$, $y=(y_1, y_2,\dots,y_s)^T\colon {n_{0}-k,n_{0}-k+1,\dots\}\to \mathbb{R}^{s}$ and $\Delta y(n)=y(n+1)-y(n)$ for $n\ge n_{0}$. A method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically
convergent under some conditions, i.e., for every solution $y(n)$ defined for all sufficiently large $n$, there exists a finite limit $\lim_{n\to\infty}y(n)$. Moreover, it is proved that the asymptotic convergence of all solutions  is equivalent to the existence of one asymptotically convergent solution with increasing coordinates. Some discussion related to the so-called critical case known for scalar equations is given as well.",
}