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.",
address="Elsevier Science Publishing Co",
chapter="95916",
institution="Elsevier Science Publishing Co",
number="18",
volume="2012",
year="2012",
month="december",
pages="4036--4044",
publisher="Elsevier Science Publishing Co",
type="journal article - other"
}