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 Iterative Algorithm for Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 32870 8 pages doi 2007 32870 Research Article Iterative Algorithm for Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces Yonghong Yao and Rudong Chen Received 11 October 2006 Revised 8 December 2006 Accepted 11 December 2006 Recommended by Nan-Jing Huang We first introduce and analyze an algorithm of approximating solutions of maximal monotone operators in Hilbert spaces. Using this result we consider the convex minimization problem of finding a minimizer of a proper lower-semicontinuous convex function and the variational problem of finding a solution of a variational inequality. Copyright 2007 Y. Yao and R. Chen. 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. 1. Introduction Throughout this paper we assume that H is a real Hilbert space and T H 2H is a maximal monotone operator. A well-known method for solving the equation 0 e Tv in a Hilbert space H is the proximal point algorithm x1 X e H and Xn 1 JrnXn n 1 2 . where rn c 0 to and Jr I rT 1 for all r 0. This algorithm was first introduced by Martinet 1 . Rockafellar 2 proved that if liminfn TO rn 0 and T-10 0 then the sequence xn defined by converges weakly to an element of T-10. Later many researchers have studied the convergence of the sequence defined by in a Hilbert space see for instance 3-6 and the references mentioned therein. In particular Kamimura and Takahashi 7 proved the following result. Theorem . Let T H 2H be a maximal monotone operator. Let xn be a sequence defined as follows X1 u e H and Xn 1 anU 1 - Xn Jr Xn n 1 2 . 2 Fixed Point Theory and Applications where an c 0 1 and rn c 0 to satisfylimn TO an 0 TO 1 an TO andlimn TO rn TO. If T 0 0 then the sequence xn defined by converges strongly to .
