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: Research Article Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 914702 12 pages doi 2010 914702 Research Article Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups Kriengsak Wattanawitoon and Poom Kumam Department of Mathematics Faculty of Science King Mongkut s University of Technology Thonburi KMUTT Bangmod Thrungkru Bangkok 10140 Thailand Correspondence should be addressed to Poom Kumam Received 26 September 2009 Accepted 24 November 2009 Academic Editor Tomonari Suzuki Copyright 2010 K. Wattanawitoon and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We prove convergence theorems of modified Ishikawa iterative sequence for two nonexpansive semigroups in Hilbert spaces by the two hybrid methods. Our results improve and extend the corresponding results announced by Saejung 2008 and some others. 1. Introduction Let C be a subset of real Hilbert spaces H with the inner product and the norm II . T C C is called a nonexpansive mapping if Tx - Ty x - y Nx y e C. We denote by F T the set of fixed points of T that is F T x e C x Tx . Let T t t 0 be a family of mappings from a subset C of H into itself. We call it a nonexpansive semigroup on C if the following conditions are satisfied i T 0 x x for all x e C ii T s f T s T t for all s t 0 iii for each x e C the mapping t T t x is continuous iv T F x - T t yW x - y for all x y e C and t 0. 2 Fixed Point Theory and Applications The Mann s iterative algorithm was introduced by Mann 1 in 1953. This iterative process is now known as Mann s iterative process which is defined as xn 1 anxn 1 - an Txn n 0 where the initial guess x0 is taken in C arbitrarily and the sequence an Xo is in the interval 0 1 . In 1967 Halpern 2 first introduced the following .
