Asymptotic Representation of the Solutions of Linear Volterra Difference Equations

Abstract

Recommended by Elena BravermanThis article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. We give examples showing the accuracy of our results.

On the boundedness of the solutions in nonlinear discrete Volterra difference equations

“…It is worth to note that in this case our Theorem 4.5 is applicable for any x 0 ℝ + , but the results in [2,[8][9][10][11][12]14] are not applicable in this case.…”

“…We consider the nonlinear system of Volterra difference equations In recent years, there has been an increasing interest in the study of the asymptotic behavior of the solutions of both convolution and non-convolution-type linear and nonlinear Volterra difference equations (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein). Appleby et al [2], under appropriate assumptions, have proved that the solutions of the discrete linear Volterra equation converge to a finite limit, which in general is non-trivial.…”

“…Appleby et al [2], under appropriate assumptions, have proved that the solutions of the discrete linear Volterra equation converge to a finite limit, which in general is non-trivial. The main result on the boundedness of solutions of a linear Volterra difference system in [2] was improved by Györi and Horváth [8]. In terms of the kernel of a linear system Györi and Reynolds [10] found necessary conditions for the solutions to be bounded.…”

“…In terms of the kernel of a linear system Györi and Reynolds [10] found necessary conditions for the solutions to be bounded. Also Györi and Reynolds [9] studied some connections between results obtained in [2,8]. Elaydi et al [6] have shown that under certain conditions there is a one-to-one correspondence between bounded solutions of linear Volterra difference equations with infinite delay and its perturbation.…”

Asymptotic behavior of solutions of discrete Volterra equations

“…Asymptotic behavior of solutions of first order Volterra difference equations has been studied by many authors. 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].…”