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

Recommended by A. Matsumoto

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.