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: WEAK AND STRONG CONVERGENCE THEOREMS FOR NONEXPANSIVE SEMIGROUPS IN BANACH SPACES SACHIKO ATSUSHIBA AND WATARU TAKAHASHI | WEAK AND STRONG CONVERGENCE THEOREMS FOR NONEXPANSIVE SEMIGROUPS IN BANACH SPACES SACHIKO ATSUSHIBA AND WATARU TAKAHASHI Received 24 February 2005 We introduce an implicit iterative process for a nonexpansive semigroup and then we prove a weak convergence theorem for the nonexpansive semigroup in a uniformly convex Banach space which satisfies Opial s condition. Further we discuss the strong convergence of the implicit iterative process. 1. Introduction Let C be a closed convex subset of a Hilbert space and let T be a nonexpansive mapping from C into itself. For each t e 0 1 the contraction mapping Tt of C into itself defined by Ttx tu 1 - t Tx for every x e C has a unique fixed point xt where u is an element of C. Browder 4 proved that xt converges strongly to a fixed point of T as t 0 in a Hilbert space. Motivated by Browder s theorem 4 Takahahi and Ueda 20 proved the strong convergence of the following iterative process in a uniformly convex Banach space with a uniformly Gateaux differentiable norm see also 14 xk 1 x Ợ - D Txk for every k 1 2 3 . where x e C. On the other hand Xu and Ori 21 studied the following implicit iterative process for finite nonexpansive mappings T1 T2 . Tr in a Hilbert space x0 x e C and xn anxn-1 1 O-n Tnxn for every n 1 2 . where an is a sequence in 0 1 and Tn Tn r. And they proved a weak convergence of the iterative process defined by in a Hilbert space. Sun et al. 17 studied the iterations defined by and proved the strong convergence of the iterations in a uniformly convex Banach space requiring one mapping Ti in the family to be semi compact. Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 3 2005 343-354 DOI 344 Weak and strong convergence theorems In this paper using the idea of 17 21 we introduce an implicit iterative process for a nonexpansive semigroup and then prove a weak convergence theorem for the non-expansive semigroup in a uniformly convex .
