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 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 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 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 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
