Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

Abstract

A linear Volterra difference equation of the formx(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i),wherex:N0→R,a:N0→R,K:N0×N0→Randb:N0→R∖{0}isω-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on∏j=0ω-1b(j)is assumed. The results generalize some of the recent results.

Existence of bounded solution of Volterra difference equations via Darbo's fixed-point theorem

“…A Theorem 2 is related to results obtained by Diblík et al in paper [6]. By Schauder's fixed-point theorem, the authors prove sufficient conditions for the existence of weighted asymptotically periodic solutions under more restrictive assumptions.…”

Some stability conditions for scalar Volterra difference equations

“…Therefore, the qualitative theory of these types of equations is developed by many authors. For example, the boundedness of solutions of discrete Volterra equations was studied in [2,5,10] or [13]- [18], the periodicity was investigated in papers [6,8,15,18]. A survey of the fundamental results on the stability of linear Volterra difference equations, of both convolution and non-convolution type, can be found in [7], see also [3,4,11,12,17] or [19].…”

Asymptotic behavior of solutions of discrete Volterra equations

“…In particular, the boundedness of solutions was studied by, i.e., Crisci et al [2], Diblík and Schmeidel [6], Gronek and Schmeidel [7], Győri and Horvath [10], Győri and Awwad [8], Kolmanovskii and Shaikhet [12], Migda and Migda [19], Migda and Morchało [20] or Morchało [21]. Asymptotically periodic solutions were studied, for example, by Baker and Song [1], Diblík et al [4,5] or Győri and Reynolds [11]. To the best of our knowledge, there are a few papers dealing with the asymptotic behavior of solutions of higher order Volterra difference equations, for example, the second order difference equation of Volterra type was studied in Medina [13].…”