On exact convergence rates for solutions of linear systems of Volterra difference equations†

Abstract

The asymptotic behaviour of the solution of general linear Volterra non-convolution difference equations on a finite dimensional space, is investigated. It is proved under appropriate assumptions that the solution converges to a limit, which is in general non-trivial. These results are then used to obtain the exact rate of decay of solutions of a class of convolution Volterra difference equations, which have no characteristic roots. In particular, we obtain the exact rate of convergence of the solution of equations whose kernel does not converge exponentially. A useful formula for the weighted limit of a discrete convolution is also obtained.

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.…”

“…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.…”

“…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.…”

“…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.…”

Asymptotics of a renewal-like recursion and an integral equation

“…The recursion is of the form (1) x n = n−1 j=1 w n,j x n−j + r n , w n,j = a j + bj n 201 the last m terms appear on the right-hand side for some fixed m. Now all previous terms are present, and the weights depend both on n and j.…”