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 An Extragradient Method and Proximal Point Algorithm for Inverse Strongly Monotone | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 591874 16 pages doi 2009 591874 Research Article An Extragradient Method and Proximal Point Algorithm for Inverse Strongly Monotone Operators and Maximal Monotone Operators in Banach Spaces Somyot Plubtieng and Wanna Sriprad Department of Mathematics Faculty of Science Naresuan University Phitsanulok 65000 Thailand Correspondence should be addressed to Somyot Plubtieng somyotp@ Received 6 January 2009 Accepted 22 April 2009 Recommended by Nanjing Jing Huang We introduce an iterative scheme for finding a common element of the solution set of a maximal monotone operator and the solution set of the variational inequality problem for an inverse strongly-monotone operator in a uniformly smooth and uniformly convex Banach space and then we prove weak and strong convergence theorems by using the notion of generalized projection. The result presented in this paper extend and improve the corresponding results of Kamimura et al. 2004 and Iiduka and Takahashi 2008 . Finally we apply our convergence theorem to the convex minimization problem the problem of finding a zero point of a maximal monotone operator and the complementary problem. Copyright 2009 S. Plubtieng and W. Sriprad. 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 Let E be a Banach space with norm II II let E denote the dual of E and let x f denote the value of f e E at x e E. Let T E E be an operator. The problem of finding v e E satisfying 0 e Tv is connected with the convex minimization problems and variational inequalities. When T is maximal monotone a well-known method for solving the equation 0 e Tv in Hilbert space H is the proximal point algorithm see 1 x1 x e H and xn 1 Jrnxn n 1 2 where rn c 0 to and Jr I rT 1 for all r
