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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 Hybrid Steepest Descent Method with Variable Parameters for General Variational Inequalities Yanrong Yu and Rudong Chen | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 19270 14 pages doi 10.1155 2007 19270 Research Article Hybrid Steepest Descent Method with Variable Parameters for General Variational Inequalities Yanrong Yu and Rudong Chen Received 16 April 2007 Accepted 2 August 2007 Recommended by Yeol Je Cho We study the strong convergence of a hybrid steepest descent method with variable parameters for the general variational inequality GVI F g C . Consequently as an application we obtain some results concerning the constrained generalized pseudoinverse. Our results extend and improve the result of Yao and Noor 2007 and many others. Copyright 2007 Y. Yu 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 Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F H - H be an operator such that for some constants k n 0 F is k-Lipschitzian and n-strongly monotone on C that is F satisfies the following inequalities Fx - Fy II k x - y II and Fx - Fy x - y n x - yll2 for all x y E C respectively. Recall that T is nonexpansive if Tx - Tyll x - yll for all x y E H. We consider the following variational inequality problem find a point u E C such that VI F C F u v - v 0 Vv E C. 1.1 Variational inequalities were introduced and studied by Stampacchia 1 in 1964. It is now well known that a wide class of problems arising in various branches of pure and applied sciences can be studied in the general and unified framework of variational inequalities. Several numerical methods including the projection and its variant forms Wiener-Hofp equations auxiliary principle and descent type have been developed for solving the variational inequalities and related optimization problems. The reader is referred to 1-18 and the references therein. 2 .
