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October  2014, 19(8): 2681-2690. doi: 10.3934/dcdsb.2014.19.2681

On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type

 1 University of Bialystok, ul. Akademicka 2, 15-267 Białystok 2 Poznan University of Technology, ul. Piotrowo 3A, 60-965 Poznań, Poland, Poland

Received  November 2013 Revised  May 2014 Published  August 2014

A Volterra difference equation of the form $$x(n+2)=a(n)+b(n)x(n+1)+c(n)x(n)+\sum\limits^{n+1}_{i=1}K(n,i)x(i)$$ where $a, b, c, x \colon\mathbb{N} \to\mathbb{R}$ and $K \colon \mathbb{N}\times\mathbb{N}\to \mathbb{R}$ is studied. For every admissible constant $C \in \mathbb{R}$, sufficient conditions for the existence of a solution $x \colon\mathbb{N} \to\mathbb{R}$ of the above equation such that $x(n)\sim C \, n \, \beta(n),$ where $\beta(n)= \frac{1}{2^n}\prod\limits_{j=1}^{n-1}b(j)$, are presented. As a corollary of the main result, sufficient conditions for the existence of an eventually positive, oscillatory, and quickly oscillatory solution $x$ of this equation are obtained. Finally, a conditions under which considered equation possesses an asymptotically periodic solution are given.
Citation: Ewa Schmeidel, Karol Gajda, Tomasz Gronek. On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681
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