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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 \(n\in \mathbb{N}:=\{0,1,2,\dots\}\), \(a_{s},b_{s},x_{s}:\mathbb{N}\rightarrow \mathbb{R}\), \(s=1,2,\) 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:

[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)
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