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Báo cáo hóa học: " Research Article An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert 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 An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 173430 15 pages doi 10.1155 2011 173430 Research Article An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces Ming Tian College of Science Civil Aviation University of China Tianjin 300300 China Correspondence should be addressed to Ming Tian tianming1963@126.com Received 9 December 2010 Accepted 13 February 2011 Academic Editor Shusen Ding Copyright 2011 Ming Tian. 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 use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated by our proposed schemes. 1. Introduction Let H be a real Hilbert space and C a closed convex subset of H and let ộ be a bifunction of C X C into R where R is the set of real numbers. The equilibrium problem for ộ C X C R is to find x e C such that EP ộ x y 0 Vy e C 1.1 denoted the set of solution by EP ộ . Given a mapping T C H let ộ x ỳ Tx y - x for all x y e C then z e EP ộ if and only if Tz y - z 0 for all y e C that is z is a solution of the variational inequality. Numerous problems in physics optimizations and economics reduce to find a solution of 1.1 . Some methods have been proposed to solve the equilibrium problem see for instance 1 2 . A mapping T of C into itself is nonexpansive if Tx- Ty x-y for all x y e C. The set of fixed points of T is denoted by F T . In 2007 Plubtieng and Punpaeng 3 S. Takahashi and W. Takahashi 4 and Tada and W. Takahashi 5 considered iterative methods for finding an element of EP ộ n F T . 2 Journal of .