## 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 $$n\in \mathbb{N}:=\{0,1,2,\dots\}$$, $$a_{s},b_{s},x_{s}:\mathbb{N}\rightarrow \mathbb{R}$$, $$s=1,2,$$ and $$\mathbb{R}$$ 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 $$x(n)=u(n)+v(n),~n\in \mathbb{N}$$ 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 $$\omega$$.

### MSC:

 39A23 Periodic solutions of difference equations 39A10 Additive difference equations
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### References:

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