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

Diblík, Josef; Schmeidel, Ewa; Růžičková, Miroslava

J. Difference Equ. Appl. 15, No. 11-12, 1165-1177 (2009).

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 \(\omega\).

Reviewer: Fozi Dannan (Damascus)

MSC:

39A23	Periodic solutions of difference equations
39A10	Additive difference equations

Keywords:

Volterra difference system; asymptotically periodic solution

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