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Existence of asymptotically periodic solutions of system of Volterra difference equations. (English) Zbl 1180.39022
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 $\omega$.

39A23Periodic solutions (difference equations)
39A10Additive difference equations
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