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Zbl 1180.39022
Dibl{\'\i}k, Josef; Schmeidel, Ewa; Ruužičková, Miroslava
Existence of asymptotically periodic solutions of system of Volterra difference equations.
(English)
[J] J. Difference Equ. Appl. 15, No. 11-12, 1165-1177 (2009). ISSN 1023-6198

Consider a Volterra system of two difference equations of the form $$x_{s}(n+1)=a_{s}(n)+b_{s}(n)x_{s}(n)+\sum_{i=0}^{n}K_{s1}(n,i)x_{1}(i)+\sum_{i=0}^{n}K_{s2}(n,i)x_{2}(i),\tag*$$ where $n\in \Bbb{N}:=\{0,1,2,\dots\}$, $a_{s},b_{s},x_{s}:\Bbb{N}\rightarrow \Bbb{R}$, $s=1,2,$ and $\Bbb{R}$ denotes the set of real numbers. The authors obtained sufficient conditions in terms of $K_{sp}~\{s,p=1,2\}$ for which the system ($*$) has asymptotically $\omega$-periodic solution $x$ such that $x(n)=u(n)+v(n),~n\in \Bbb{N}$ with $u_{s}(n):=c_{s}\prod_{k=0}^{n^{\ast }}b_{s}(k)$ and $\lim_{n\rightarrow \infty}v(n)=0,$ where $s=1,2$ and $n^{\ast }$ is the remainder of dividing $n-1$ by $\omega$.
[Fozi Dannan (Damascus)]
MSC 2000:
*39A23
39A10 Difference equations

Keywords: Volterra difference system; asymptotically periodic solution

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