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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 theorem for an iterative algorithm of l-strict pseudocontraction | Chai and Song Fixed Point Theory and Applications 2011 2011 95 http www.fixedpointtheoryandapplications.eom content 2011 1 95 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Convergence theorem for an iterative algorithm of l-strict pseudocontraction XinKuan Chai and Yisheng Song Correspondence songyisheng123@yahoo.com.cn College of Mathematics and Information Science Henan Normal University XinXiang 453007 PR China Springer Abstract In this article we prove strong convergence of sequence generated by the following iteration sequence for a class of Lipschitzian pseudocontractive mapping T xn 1 Pnu 1 Pn anTxn 1 an xn whenever an and 3n satisfy the appropriate conditions. 2000 AMS Subject Classification 47H06 47J05 47J25 47H10 47H17. Keywords Ầ-strict pseudocontraction 2-uniformly smooth Banach space modified Mann iteration strong convergence 1. Introduction Let T be a pseudocontractive mapping defined on a real smooth Banach space E. We consider the problem of finding a solution z e E of the fixed point equation x Tx. One classical way to study pseudocontractive mappings is to use a strong pseudocontraction to approximate a pseudocontractive mapping T. More precisely take t e 0 1 and u e E define a strong pseudocontraction Tt by Ttx tu 1 - t Tx. In 1 Corollary 2 Deimling proves that Tt has a unique fixed point xt i.e. xt tu 1 t Txt. 1.1 This implicit iteration was introduced by Browder 2 for a nonexpansive mapping T in Hilbert space. Halpern 3 was the first who introduced the following explicit iteration scheme for a nonexpansive mapping T which was referred to as Halpern iteration for u x0 e K an e 0 1 xn 1 anu 1 an Txn. 1 2 Convergence of this two schemes have been studied by many researchers with various types of additional conditions. For the studies of a nonexpansive mapping T see Bruck 4 5 Reich 6 7 Song-Xu 8 Takahashi-Ueda 9 Suzuki 10 and many others. For the studies of a continuous pseudocontractive mapping T see Morales-Jung
