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: Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces | Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011 2011 11 http content 2011 1 11 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces Uthai Kamraksa 1 and Rabian Wangkeeree1 2 Correspondence uthaikam@ department of Mathematics Faculty of Science Naresuan University Phitsanulok 65000 Thailand Full list of author information is available at the end of the article SpringerOpen0 Abstract We first prove the existence of a solution of the generalized equilibrium problem GEP using the KKM mapping in a Banach space setting. Then by virtue of this result we construct a hybrid algorithm for finding a common element in the solution set of a GEP and the fixed point set of countable family of nonexpansive mappings in the frameworks of Banach spaces. By means of a projection technique we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solution set of GEP and common fixed point set of nonexpansive mappings. AMS Subject Classification 47H09 47H10 Keywords Banach space Fixed point Metric projection Generalized equilibrium problem Nonexpansive mapping 1. Introduction Let E be a real Banach space with the dual E and C be a nonempty closed convex subset of E. We denote by N and R the sets of positive integers and real numbers respectively. Also we denote by J the normalized duality mapping from E to 2E defined by Jx x e E x x x 2 x 2 Vx e E where - denotes the generalized duality pairing. We know that if E is smooth then J is single-valued and if E is uniformly smooth then J is uniformly norm-to-norm continuous on bounded subsets of E. We shall still denote by J the single-valued duality mapping. Let f C X C R be a bifunction and A C E be a nonlinear mapping. We consider the following generalized
