# Báo cáo hóa học: " Research Article Regularization and Iterative Methods for Monotone Variational Inequalities"

## 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 Regularization and Iterative Methods for Monotone Variational Inequalities | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 765206 11 pages doi 2010 765206 Research Article Regularization and Iterative Methods for Monotone Variational Inequalities Xiubin Xu1 and Hong-Kun Xu2 1 Department of Mathematics Zhejiang Normal University Jinhua Zhejiang 321004 China 2 Department of Applied Mathematics National Sun Yat-Sen University Kaohsiung 80424 Taiwan Correspondence should be addressed to Xiubin Xu xxu@ Received 16 September 2009 Accepted 23 November 2009 Academic Editor Mohamed A. Khamsi Copyright 2010 X. Xu and . Xu. 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 provide a general regularization method for monotone variational inequalities where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an iterative method as discretization of the regularization method. We prove that both regularization and iterative methods converge in norm. 1. Introduction Variational inequalities VIs have widely been studied see the monographs 1-3 . A monotone variational inequality problem VIP is stated as finding a point x with the following property x e C Ax x - x 0 Vx e C where C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm II II respectively and A is a monotone operator in H with domain dom A D C. Recall that A is monotone if Ax - Ay x - y 0 x y e dom A . A typical example of monotone operators is the subdifferential of a proper convex lower semicontinuous function. 2 Fixed Point Theory and Applications Variational inequality problems are equivalent to fixed point problems. As a matter of fact x solves VIP if and only if x solves the following fixed point problem FPP for any Y 0 x Pc I - yA x where Pc is the metric or nearest point .

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