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Báo cáo hóa học: " Research Article Finite-Step Relaxed Hybrid Steepest-Descent Methods for 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 Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 598632 13 pages doi 10.1155 2008 598632 Research Article Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities Yen-Cherng Lin Department of Occupational Safety and Health General Education Center China Medical University Taichung 404 Taiwan Correspondence should be addressed to Yen-Cherng Lin yclin@mail.cmu.edu.tw Received 22 August 2007 Revised 2 January 2008 Accepted 13 March 2008 Recommended by Jong Kim The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space was studied. A new finite-step relaxed hybrid steepest-descent method for this class of variational inequalities was introduced. Strong convergence of this method was established under suitable assumptions imposed on the algorithm parameters. Copyright 2008 Yen-Cherng Lin. 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 H be a real Hilbert space with inner product and norm II II. Let C be a nonempty closed convex subset of H and let F C H be an operator. The classical variational inequality problem find u C such that VI F C F u v - u 0 Vv C 1.1 was initially studied by Kinderlehrer and Stampacchia 1 . It is also known that the VI F C is equivalent to the fixed-point equation u PcU - pF u 1.2 where Pc is the nearest point projection from H onto C that is PCx argminyeC x - y for each x H and where y 0 is an arbitrarily fixed constant. If F is strongly monotone and Lipschitzian on C and y 0 is small enough then the mapping determined by the right-hand side of this equation is a contraction. Hence the Banach contraction principle guarantees that the Picard iterates converge in norm to the unique solution of the VI F