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: A relaxed hybrid steepest descent method for common solutions of generalized mixed equilibrium problems and fixed point problems | Onjai-uea et al. Fixed Point Theory and Applications 2011 2011 32 http content 2011 1 32 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access A relaxed hybrid steepest descent method for common solutions of generalized mixed equilibrium problems and fixed point problems Nawitcha Onjai-uea1 3 Chaichana Jaiboon2 3 and Room Kumam1 3 Correspondence chaichana. j@ 2Department of Mathematics Faculty of Liberal Arts Rajamangala University of Technology Rattanakosin Rmutr Bangkok 10100 Thailand Full list of author information is available at the end of the article SpringerOpen0 Abstract In the setting of Hilbert spaces we introduce a relaxed hybrid steepest descent method for finding a common element of the set of fixed points of a nonexpansive mapping the set of solutions of a variational inequality for an inverse strongly monotone mapping and the set of solutions of generalized mixed equilibrium problems. We prove the strong convergence of the method to the unique solution of a suitable variational inequality. The results obtained in this article improve and extend the corresponding results. AMS 2000 Subject Classification 46C05 47H09 47H10. Keywords relaxed hybrid steepest descent method inverse strongly monotone mappings nonexpansive mappings generalized mixed equilibrium problem 1. Introduction Let H be a real Hilbert space C be a nonempty closed convex subset of H and let PC be the metric projection of H onto the closed convex subset C. Let S C C be a nonexpansive mapping that is Sx - Sy x - y for all x y e C. We denote by F S the set fixed point of S. If C c H is nonempty bounded closed and convex and S is a nonexpansive mapping of C into itself then F S is nonempty see for example 1 2 . A mapping f C C is a contraction on C if there exists a constant h e 0 1 such that fx - f y h x - y for all x y e C. In addition let D C H be a nonlinear mapping ộ C R u be a real-valued function and let F
