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Báo cáo hóa học: " STRONG CONVERGENCE THEOREMS FOR INFINITE FAMILIES OF NONEXPANSIVE MAPPINGS IN GENERAL BANACH SPACES"

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: STRONG CONVERGENCE THEOREMS FOR INFINITE FAMILIES OF NONEXPANSIVE MAPPINGS IN GENERAL BANACH SPACES | STRONG CONVERGENCE THEOREMS FOR INFINITE FAMILIES OF NONEXPANSIVE MAPPINGS IN GENERAL BANACH SPACES TOMONARI SUZUKI Received 2 June 2004 In 1979 Ishikawa proved a strong convergence theorem for finite families of nonexpan-sive mappings in general Banach spaces. Motivated by Ishikawa s result we prove strong convergence theorems for infinite families of nonexpansive mappings. 1. Introduction Throughout this paper we denote by N the set of positive integers and by R the set of real numbers. For an arbitrary set A we also denote by A the cardinal number of A. Let C be a closed convex subset of a Banach space E. Let T be a nonexpansive mapping on C that is Tx - TyUllx- yll 1.1 for all x y e C. We denote by F T the set of fixed points of T. We know F T 0 in the case that E is uniformly convex and C is bounded see Browder 2 Gohde 9 and Kirk 13 . Common fixed point theorems for families of nonexpansive mappings are proved in 2 4 5 and other references. Many convergence theorems for nonexpansive mappings and families of nonexpansive mappings have been studied see 1 3 6 7 10 11 12 14 15 17 18 19 20 21 and others. For example in 1979 Ishikawa proved the following theorem. Theorem 1.1 12 . Let C be a compact convex subset of a Banach space E Let T1 T2 . T- be a finite family of commuting nonexpansive mappings on C. Let fi k 1 be a finite sequence in 0 1 and put Six fiTix 1 - fi x for x e C and i 1 2 . k. Let x1 e C and define a sequence xn in C by xn 1 n n -nk-ỉ ỉ n--1 S- n - n--2 1 Sk-1 S3 n U1 1 n1 S2 n S1 no 1 J- x1 1.2 for n e N. Then xn converges strongly to a common fixed point of T1 T2 . T- . Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 1 2005 103-123 DOI 10.1155 FPTA.2005.103 104 Convergence to common fixed point The author thinks this theorem is one of the most interesting convergence theorems for families of nonexpansive mappings. In the case of k 4 this iteration scheme is as follows X2 S4S3S2S1X1 X3 S4S3S2S1S1S2 S1S3S2S1X2