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: Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings | Peng Fixed Point Theory and Applications 2011 2011 12 http content 2011 1 12 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings Jian-Wen Peng Correspondence jwpeng6@yahoo. School of Mathematics Chongqing Normal University Chongqing 400047 PR China SpringerOpen0 Abstract In this article we introduce some new iterative schemes based on the extragradient method and the hybrid method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone Lipschitz-continuous mapping in Hilbert spaces. We obtain some strong convergence theorems and weak convergence theorems. The results in this article generalize improve and unify some well-known convergence theorems in the literature. Keywords Generalized equilibrium problem Extragradient method Hybrid method Nonex-pansive mapping Strong convergence Weak convergence 1. Introduction Let H be a real Hilbert space with inner product . . and induced norm - . Let C be a nonempty closed convex subset of H. Let F be a bifunction from C X C to R and let B C H be a nonlinear mapping where R is the set of real numbers. Moudafi 1 Moudafi and Thera 2 Peng and Yao 3 4 Takahashi and Takahashi 5 considered the following generalized equilibrium problem Find x e C Such that F x y Bx y x 0 Vy e C. The set of solutions of is denoted by GEP F B . If B 0 the generalized equilibrium problem becomes the equilibrium problem for F C X C R which is to find x e C such that F x y 0 for all y e C. The set of solutions of is denoted by EP F . The problem is very general in the sense that it includes as special cases optimization problems variational inequalities .
