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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Convergence of the modified Mann’s iterative method for asymptotically -strictly pseudocontractive mappings | Zhang and Xie Fixed Point Theory and Applications 2011 2011 100 http www.fixedpointtheoryandapplications.eom content 2011 1 100 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Convergence of the modified Mann s iterative method for asymptotically -strictly pseudocontractive mappings Ying Zhang1 2 and Zhiwei Xie3 Correspondence spzhangying@126.com 1School of Mathematics and Physics North China Electric Power University Baoding Hebei 071003 P.R. China Full list of author information is available at the end of the article Springer Abstract Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm and K a nonempty closed and convex subset of E. Let T K K be an asymptotically K-strictly pseudocontractive mapping with a nonempty fixed point set. We prove that I - T is demiclosed at 0 and obtain a weak convergence theorem of the modified Mann s algorithm for T under suitable control conditions. Moreover we also elicit a necessary and sufficient condition that guarantees strong convergence of the modified Mann s iterative sequence to a fixed point of T in a real Banach spaces with the Fréchet differentiable norm. 2000 AMS Subject Classification 47H09 47H10. Keywords asymptotically K-strictly pseudocontractive mappings demiclosedness principle the modified Mann s algorithm fixed points 1 Introduction Let E and E be a real Banach space and the dual space of E respectively. Let K be a nonempty subset of E. Let J denote the normalized duality mapping from E into 2E given by J x f e E x f x 2 f 2 for all x e E where denotes the duality pairing between E and E . In the sequel we will denote the set of fixed points of a mapping T K K by F T x e K Tx x . A mapping T K K is called asymptotically K-strictly pseudocontractive with sequence Kn n 1 c 1 to such that limn K 1 see e.g. 1-3 if for all x y e K there exist a constant K e 0 1 and j x - y e J x - y such that Tx - Ty j x - y Kn II x - yll2 - K II x - y - Tnx - Tny 2
