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 On Two Iterative Methods for Mixed Monotone Variational Inequalities | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 291851 10 pages doi 2010 291851 Research Article On Two Iterative Methods for Mixed Monotone Variational Inequalities Xiwen Lu 1 Hong-Kun Xu 2 and Ximing Yin1 1 Department of Mathematics East China University of Science and Technology Shanghai 200237 China 2 Department of Applied Mathematics National Sun Yat-Sen University Kaohsiung 80424 Taiwan Correspondence should be addressed to Hong-Kun Xu xuhk@ Received 22 September 2009 Accepted 23 November 2009 Academic Editor Tomonari Suzuki Copyright 2010 Xiwen Lu et al. 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. A mixed monotone variational inequality MMVI problem in a Hilbert space H is formulated to find a point u e H such that Tu v - uf y y - y u 0 for all v e H where T is a monotone operator and y is a proper convex and lower semicontinuous function on H. Iterative algorithms are usually applied to find a solution of an MMVI problem. We show that the iterative algorithm introduced in the work of Wang et al. 2001 has in general weak convergence in an infinitedimensional space and the algorithm introduced in the paper of Noor 2001 fails in general to converge to a solution. 1. Introduction Let H be a real Hilbert space with inner product and norm II II and let T be an operator with domain D T and range R T in H .Recall that T is monotone if its graph G T x y e H X H x e D T y e Tx is a monotone set in H X H. This means that T is monotone if and only if x y x y e G T x - x y - yf 0. A monotone operator T is maximal monotone if its graph G T is not properly contained in the graph of any other monotone operator on H. Let y H R R u to TO be a proper convex and lower semicontinuous functional. The subdifferential of y dy is defined by dy x z
