Asymptotic behavior of solutions of discrete Volterra equations

Abstract

Abstract. We consider the nonlinear discrete Volterra equations of non-convolution typeWe present sufficient conditions for the existence of solutions with prescribed asymptotic behavior, especially asymptotically polynomial and asymptotically periodic solutions. We use o(n s ), for a given nonpositive real s, as a measure of approximation. We also give conditions under which all solutions are asymptotically polynomial.

Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type

“…There are relatively few works devoted to the study of equations of type (2); see, for example, [1][2][3][4]. In [5], the asymptotic behaviors of nonoscillatory solutions of the higher-order integrodynamic equation on time scales are presented.…”

On the asymptotic behavior of solution of certain systems of Volterra equations

“…For the system of difference equations given by (14), Theorem 1 of [5] cannot be applied since c 1 c 2 = 0 .…”

“…where n ∈ N := In the last years, there has been an interest among many authors to study the asymptotic behavior of solutions of Volterra difference equations. The results were published, e.g., by Appleby et al [1], by Appleby and Patterson [2], by Berezansky et al [3], by Gajda et al [7], by Gil and Medina [8], by Gronek and Schmeidel [9], by Györi and Horváth [10], by Györi and Reynolds [11], by Medina [12,13], by Migda and Migda [14,17], by Migda et al [18], and by Song and Baker [19].…”

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].…”