tailieunhanh - Báo cáo hóa học: " CONVERGENCE THEOREMS FOR A COMMON FIXED POINT OF A FINITE FAMILY OF NONSELF NONEXPANSIVE MAPPINGS"

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. Halpern 5 was the first to study the convergence of the algorithm 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 234 Convergence theorems for a common fixed point Reich 9 extended

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