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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: CONVERGENCE THEOREMS FOR FIXED POINTS OF DEMICONTINUOUS PSEUDOCONTRACTIVE MAPPINGS | CONVERGENCE THEOREMS FOR FIXED POINTS OF DEMICONTINUOUS PSEUDOCONTRACTIVE MAPPINGS C. E. CHIDUME AND H. ZEGEYE Received 26 August 2004 Let D be an open subset of a real uniformly smooth Banach space E. Suppose T D E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition where D denotes the closure of D. Then it is proved that i D Q I r I - T for every r 0 ii for a given y0 G D there exists a unique path t yt G D t G 0 1 satisfying yt tTyt 1 - t y0. Moreover if F T 0 or there exists y0 G D such that the set K y G D Ty Ay 1 - A y0 for A 1 is bounded then it is proved that as t 1- the path yt converges strongly to a fixed point of T. Furthermore explicit iteration procedures with bounded error terms are proved to converge strongly to a fixed point of T. 1. Introduction Let D be a nonempty subset of a real linear space E. A mapping T D E is called a contraction mapping if there exists L G 0 1 such that II Tx - Ty II L x - y II for all x y G D. If L 1 then T is called nonexpansive. T is called pseudocontractive if there exists j x - y G J x - y such that Tx - Ty j x - y x- yll2 Vx y G K 1.1 where J is the normalized duality mapping from E to 2E defined by Jx f G E x f x 2 IIf 2 . 1.2 T is called strongly pseudocontractive if there exists k G 0 1 such that Tx - Ty j x - y k x- yll2 Vx y G K. 1.3 Clearly the class of nonexpansive mappings is a subset of class of pseudocontractive mappings. T is said to be demicontinuous if xn Q D and xn x G D together imply that Txn Tx where and denote the strong and weak convergences respectively. We denote by F T the set of fixed points of T. Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 1 2005 67-77 DOI 10.1155 FPTA.2005.67 68 Fixed points of demicontinuous pseudocontractive maps Closely related to the class of pseudocontractive mappings is the class of accretive mappings. A mapping A D A Q E E is called accretive if T I - A is pseudocontractive. If E is a Hilbert .
