Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Asymptotic Behavior of a Discrete Nonlinear Oscillator with Damping Dynamical System | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 867136 9 pages doi 2011 867136 Research Article Asymptotic Behavior of a Discrete Nonlinear Oscillator with Damping Dynamical System Hadi Khatibzadeh1 2 1 Department of Mathematics Zanjan University . Box 45195-313 Zanjan Iran 2 School of Mathematics Institute for Research in Fundamental Sciences IPM . Box 19395-5746 Tehran Iran Correspondence should be addressed to Hadi Khatibzadeh hkhatibzadeh@ 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. 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 Brezis and Lions 1978 Lions 1978 as well as Djafari Rouhani and H. Khatibzadeh 2008 . 1. Introduction Let H be a real Hilbert space with inner product and norm . We denote weak convergence in H by and strong convergence by . Let A be a nonempty subset of H X H which we will refer to as a nonlinear possibly multivalued operator in H. A is called monotone resp. strongly monotone if y2-y1 x2 -x1 0 resp. y2 -y1 x2 -X1 a x1 -x2 2 for some a 0 for all Xi yi A i 1 2. A is maximal monotone if A is monotone and I A is surjective where I is the identity operator on H. Nonlinear oscillator with damping dynamical system u f Ỵu f Au f B 0 u u 0 u0 u 0 u1
