Abstract
In this article, we investigate the boundedness property of the solutions of linear and nonlinear discrete Volterra equations in both convolution and non-convolution case. Strong interest in these kind of discrete equations is motivated as because they represent a discrete analogue of some integral equations. The most important result of this article is a simple new criterion, which unifies and extends several earlier results in both discrete and continuous cases. Examples are also given to illustrate our main theorem.
Highlights
1 Introduction We consider the nonlinear system of Volterra difference equations n x(n + 1) = f (n, j, x(j)) + h(n), n ≥ 0, j=0 (1:1)
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
We study the boundedness of solutions of convolution cases and we get a result parallel to the corresponding result of Lipovan [18] for integral equation
Summary
The main result on the boundedness of solutions of a linear Volterra difference system in. 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. Cuevas and Pinto [4] have shown that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear Volterra difference equation with unbounded delay and its perturbation.
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