Asymptotic convergence of the solutions of a discrete system with delays

Asymptotic convergence of the solutions of a discrete system with delays

Peer-reviewed article not indexed in WoS or Scopus

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.

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.

Discrete equation, delay, asymptotic convergence, increasing solution.

Discrete equation, delay, asymptotic convergence, increasing solution.

DIBLÍK, J.; RŮŽIČKOVÁ, M.; ŠUTÁ, Z.

2013

15.12.2012

Elsevier Science Publishing Co

USA

0096-3003

APPLIED MATHEMATICS AND COMPUTATION

2012

18

United States of America

4036

4044

9