Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Strong convergence theorems by a hybrid extragradient-like approximation method for asymptotically nonexpansive mappings in the intermediate sense in Hilbert spaces | Naraghirad et al. Journal of Inequalities and Applications 2011 2011 119 http content 2011 1 119 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Strong convergence theorems by a hybrid extragradient-like approximation method for asymptotically nonexpansive mappings in the intermediate sense in Hilbert spaces Eskandar Naraghirad1 Ngai-Ching Wong2 and Jen-Chih Yao3 Correspondence wong@math. department of Applied Mathematics National Sun Yat-Sen University Kaohsiung 804 Taiwan Full list of author information is available at the end of the article Springer Abstract Let C be a nonempty closed convex subset of a real Hilbert space H. Let S C C be an asymptotically nonexpansive map in the intermediate sense with the fixed point set F S . Let A C H be a Lipschitz continuous map and VI C A be the set of solutions u e C of the variational inequality Au v Ú 0 Vv e C. The purpose of this study is to introduce a hybrid extragradient-like approximation method for finding a common element in F S and VI C A . We establish some strong convergence theorems for sequences produced by our iterative method. AMS subject classifications 49J25 47H05 47H09. Keywords asymptotically nonexpansive mapping in the intermediate sense variational inequality hybrid extragradient-like approximation method monotone mapping fixed point strong convergence 1 Introduction Let H be a real Hilbert space with inner product and norm respectively. Let C be a nonempty closed convex subset of H and let PC be the metric projection from H onto C. A mapping A C H is called monotone 1-3 if Au Av u v 0 Vu v e C and A is called k-Lipschitz continuous if there exists a positive constant k such that Au Av k Hu v Vu v e C. Let S be a mapping of C into itself. Denote by F S the set of fixed points of S that is F S u e C Su u . Recall that S is nonexpansive if Su Sv Hu v Vu v e C and S is asymptotically nonexpansive 4 if there .