Diblík, Josef; Schmeidel, Ewa; Růžičková, Miroslava Existence of asymptotically periodic solutions of system of Volterra difference equations. (English) Zbl 1180.39022 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) Cited in 14 Documents MSC: 39A23 Periodic solutions of difference equations 39A10 Additive difference equations Keywords:Volterra difference system; asymptotically periodic solution PDF BibTeX XML Cite \textit{J. Diblík} et al., J. Difference Equ. Appl. 15, No. 11--12, 1165--1177 (2009; Zbl 1180.39022) Full Text: DOI OpenURL References: [1] Agarwal R.P., Monographs and Textbooks in Pure and Applied Mathematics, in: Difference Equations and Inequalities. Theory, Methods, and Applications, 2. ed. (2000) [2] DOI: 10.1016/0895-7177(95)00096-K · Zbl 0871.39002 [3] DOI: 10.1080/10236190600986594 · Zbl 1119.39003 [4] Diblík J., Communications of the Laufen colloquium on science (2007) [5] Elaydi S.N., Undergraduate Texts in Mathematics, in: An Introduction to Difference Equations, 3. ed. (2005) [6] DOI: 10.1080/10236199808808097 · Zbl 0891.39013 [7] DOI: 10.1016/S0362-546X(01)00520-X · Zbl 1042.39500 [8] Furumochi T., Vietnam J. Math. 30 pp 537– (2002) [9] Kocić V.L., Mathematics and its Applications, in: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) [10] DOI: 10.1016/j.na.2005.02.007 · Zbl 1224.39022 [11] Musielak J., PWN, Warszawa (1976) [12] Popenda J., Arch. Math. 35 pp 13– (1999) [13] Popenda J., Facta Univ. Ser. Math. Inform. 14 pp 31– (1999) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.