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 Approximate Proximal Point Algorithms for Variational Inequalities in Banach Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 275208 17 pages doi 2009 275208 Research Article Hybrid Approximate Proximal Point Algorithms for Variational Inequalities in Banach Spaces L. C. Ceng 1 2 S. M. Guu 3 and J. C. Yao4 1 Department of Mathematics Shanghai Normal University Shanghai 200234 China 2 Scientific Computing Key Laboratory of Shanghai Universities Shanghai 200234 China 3 Department of Business Administration College of Management Yuan-Ze University Chung-Li Taoyuan Hsien 330 Taiwan 4 Department of Applied Mathematics National Sun Yat-Sen University Kaohsiung 804 Taiwan Correspondence should be addressed to J. C. Yao yaojc@ Received 10 March 2009 Accepted 7 June 2009 Recommended by Vy Khoi Le Let C be a nonempty closed convex subset of a Banach space E with the dual E let T C E be a continuous mapping and let S C C be a relatively nonexpansive mapping. In this paper by employing the notion of generalized projection operator we study the variational inequality for short VI T - f C find x e C such that y - x Tx - f 0 for all y e C where f e E is a given element. By combining the approximate proximal point scheme both with the modified Ishikawa iteration and with the modified Halpern iteration for relatively nonexpansive mappings respectively we propose two modified versions of the approximate proximal point scheme L. C. Ceng and J. C. Yao 2008 for finding approximate solutions of the VI T - f C . Moreover it is proven that these iterative algorithms converge strongly to the same solution of the VI T - f C which is also a fixed point of S. Copyright 2009 L. C. Ceng 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 E be a real Banach space with the dual E . As usually denotes the duality pairing
