Font Size:
Asymptotic Estimates of Solutions of a System of Delay Difference Equations with Continuous Time
Last modified: 2014-02-13
Abstract
In this paper we study the asymptotic behaviour of solutions of the system of difference equations with continuous time
\[x(t)=A(t)x(t-1)+B(t)x(p(t)),\]
where $x(t)$ is an $n$-dimensional column vector, $A(t)=(a_{ij}(t))$, $B(t)=(b_{ij}(t))$ are $n\times n$ real matrix functions and the lag function $p(t)$ is a real function such that $p(t)<t$ and $\lim_{t\to\infty}p(t)=\infty$.
We obtain asymptotic estimates of solutions of the considered system for the special cases when the lag function is between two known functions such as $p_1t\le p(t)\le p_2t$ for real numbers $0<p_1\le p_2<1$, $\sqrt[p_2]{t}\le p(t)\le \sqrt[p_1]{t}$ for natural numbers $1<p_1\le p_2$ and $p(t)=t-\delta(t)$, where $p_1\le \delta(t)\le p_2$ for positive integers $1\le p_1<p_2$.
\[x(t)=A(t)x(t-1)+B(t)x(p(t)),\]
where $x(t)$ is an $n$-dimensional column vector, $A(t)=(a_{ij}(t))$, $B(t)=(b_{ij}(t))$ are $n\times n$ real matrix functions and the lag function $p(t)$ is a real function such that $p(t)<t$ and $\lim_{t\to\infty}p(t)=\infty$.
We obtain asymptotic estimates of solutions of the considered system for the special cases when the lag function is between two known functions such as $p_1t\le p(t)\le p_2t$ for real numbers $0<p_1\le p_2<1$, $\sqrt[p_2]{t}\le p(t)\le \sqrt[p_1]{t}$ for natural numbers $1<p_1\le p_2$ and $p(t)=t-\delta(t)$, where $p_1\le \delta(t)\le p_2$ for positive integers $1\le p_1<p_2$.
Keywords
Functional Equations, Difference Equations with Continuous Time, Asymptotic Behaviour