tailieunhanh - Báo cáo hóa học: " Research Article Strong Convergence of a Modified Iterative Algorithm for Mixed-Equilibrium Problems 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 Strong Convergence of a Modified Iterative Algorithm for Mixed-Equilibrium Problems in Hilbert Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 454181 23 pages doi 2008 454181 Research Article Strong Convergence of a Modified Iterative Algorithm for Mixed-Equilibrium Problems in Hilbert Spaces Xueliang Gao and Yunrui Guo Department of Mathematics Sichuan University Chengdu Sichuan 610064 China Correspondence should be addressed to Xueliang Gao gxlmath@ Received 8 July 2008 Accepted 1 August 2008 Recommended by Ram U. Verma The purpose of this paper is to study the strong convergence of a modified iterative scheme to find a common element of the set of common fixed points of a finite family of nonexpansive mappings the set of solutions of variational inequalities for a relaxed cocoercive mapping as well as the set of solutions of a mixed-equilibrium problem. Our results extend recent results of Takahashi and Takahashi 2007 Marino and Xu 2006 Combettes and Hirstoaga 2005 Iiduka and Takahashi 2005 and many others. Copyright 2008 X. Gao and Y. Guo. 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 and preliminaries Let H be a real Hilbert space whose inner product and norm are denoted by and ll H respectively. Let C be a nonempty closed convex subset of H and let A C H be a nonlinear map. PC be the projection of H onto the convex subset C. The classical variational inequality problem denoted by VI C A is to find u e C such that Au v - u 0 Vv e C. For a given z e H u e C satisfies the inequality u - z v - u 0 Vv e C if and only if u PCz. It is known that the projection operator PC is nonexpansive. It is also known that PC satisfies x - y Pex - Pey Pex - Peyll2 for x y e H. Moreover PCx is characterized by the properties PCx e C and x-PCx PCx-y 0 Vy e C. 2 Journal of Inequalities and Applications One can see that the