Advances in Difference Equations
Volume 2011 (2011), Article ID 867136, 9 pages
doi:10.1155/2011/867136
Research Article

Asymptotic Behavior of a Discrete Nonlinear Oscillator with Damping Dynamical System

Hadi Khatibzadeh1,2

1Department of Mathematics, Zanjan University, P.O. Box 45195-313, Zanjan, Iran
2School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

Received 24 December 2010; Accepted 10 February 2011

Academic Editor: Istvan Gyori

Copyright © 2011 Hadi Khatibzadeh. 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 propose a new discrete version of nonlinear oscillator with damping dynamical system governed by a general maximal monotone operator. We show the weak convergence of solutions and their weighted averages to a zero of a maximal monotone operator 𝐴 . We also prove some strong convergence theorems with additional assumptions on 𝐴 . This iterative scheme gives also an extension of the proximal point algorithm for the approximation of a zero of a maximal monotone operator. These results extend previous results by Brézis and Lions (1978), Lions (1978) as well as Djafari Rouhani and H. Khatibzadeh (2008).