Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 867623, 18 pages
doi:10.1155/2008/867623
Research Article

Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations

E. Messina,1 Y. Muroya,2 E. Russo,1 and A. Vecchio3

1Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II”, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy
2Department of Mathematics, Waseda University, 3-4-1 Ohkubo Shinjuku-ku, Tokyo 169-8555, Japan
3Istituto per le Applicazioni del Calcolo “Mauro Picone”, Sede di Napoli, Via P. Castellino 111, 80131 Napoli, Italy

Received 19 June 2007; Revised 14 January 2008; Accepted 5 May 2008

Academic Editor: A. Matsumoto

Copyright © 2008 E. Messina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider nonlinear difference equations of unbounded order of the form xi=bi−∑j=0iai,jfi−j(xj),  i=0,1,2,…, where fj(x)  (j=0,…,i) are suitable functions. We establish sufficient conditions for the boundedness and the convergence of xi as i→+∞. Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.