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Báo cáo hóa học: " Research Article On the Existence and Convergence of Approximate Solutions for Equilibrium Problems in Banach Spaces"

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 the Existence and Convergence of Approximate Solutions for Equilibrium Problems in Banach Spaces | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 17294 14 pages doi 10.1155 2007 17294 Research Article On the Existence and Convergence of Approximate Solutions for Equilibrium Problems in Banach Spaces Nan-Jing Huang Heng-You Lan and Kok Lay Teo Received 15 June 2006 Revised 11 February 2007 Accepted 20 February 2007 Recommended by Charles Ejike Chidume We introduce and study a new class of auxiliary problems for solving the equilibrium problem in Banach spaces. Not only the existence of approximate solutions of the equilibrium problem is proven but also the strong convergence of approximate solutions to an exact solution of the equilibrium problem is shown. Furthermore we give some iterative schemes for solving some generalized mixed variational-like inequalities to illuminate our results. Copyright 2007 Nan-Jing Huang 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. 1. Introduction Let X be a real Banach space with dual X let K c X be a nonempty subset and let f K X K R -TO ro be a given bifunction. By the equilibrium problem introduced by Blum and Oettli in 1 we can formulate the following equilibrium problem of finding an x e K such that f x y 0 Vy e K 1.1 where f x x 0 for all x e K. The following is a list of special cases of problem 1.1 . 1 If f x y N x x n y x b x y - b x x for all x y e K where N K X K X n K X K X and b K X K R then the problem of finding an x e K such that N x x n y x b x y - b x x 0 Vy e K 1.2 2 Journal of Inequalities and Applications is a special case of problem 1.1 . This problem is known as the generalized mixed variational-like inequality. Problem 1.2 was considered by Huang and Deng 2 in the Hilbert space setting with set-valued mappings. 2 If X X H is a Hilbert space N x y Tx - Ay and b x y m y for all x y e K where T