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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 A COMMON FIXED POINT OF A FINITE FAMILY OF NONSELF NONEXPANSIVE MAPPINGS | CONVERGENCE THEOREMS FOR A COMMON FIXED POINT OF A FINITE FAMILY OF NONSELF NONEXPANSIVE MAPPINGS C. E. CHIDUME HABTUZEGEYE AND NASEER SHAHZAD Received 10 September 2003 and in revised form 6 July 2004 Let K be a nonempty closed convex subset of a reflexive real Banach space E which has a uniformly Gateaux differentiable norm. Assume that K is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. Let Ti K E i 1 . r be a family of nonexpansive mappings which are weakly inward. Assume that every nonempty closed bounded convex subset of K has the fixed point property for nonexpansive mappings. A strong convergence theorem is proved for a common fixed point of a family of nonexpansive mappings provided that Tị i 1 2 . r satisfy some mild conditions. 1. Introduction Let K be a nonempty closed convex subset of a real Banach space E. A mapping T K E is called nonexpansive if II Tx - Ty II x - y II for all x y e K. Let T K K be a non-expansive self-mapping. For a sequence an of real numbers in 0 1 and an arbitrary u e K let the sequence xn in K be iteratively defined by x0 e K Xn 1 an 1U 1 - an 1 Txn n 0. 1.1 Halpern 5 was the first to study the convergence of the algorithm 1.1 in the framework of Hilbert spaces. Lions 6 improved the result of Halpern still in Hilbert spaces by proving strong convergence of xn to a fixed point of T if the real sequence an satisfies the following conditions i limn--TO an 0 ii Z 1 an to iii limn--to. an - an-1 af 0. It was observed that both Halpern s and Lions conditions on the real sequence an excluded the natural choice an n 1 -1. This was overcome by Wittmann 12 who proved still in Hilbert spaces the strong convergence of xn if an satisfies the following conditions i limn-TO an 0 ii STO 1 an to iii sn 0 I an 1 - an ị TO. Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 2 2005 233-241 DOI 10.1155 FPTA.2005.233 234 Convergence theorems for a common fixed point Reich 9 extended
