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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 Strong Convergence Theorems for Lipschitzian Demicontraction Semigroups in Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 583423 10 pages doi 10.1155 2011 583423 Research Article Strong Convergence Theorems for Lipschitzian Demicontraction Semigroups in Banach Spaces Shih-sen Chang 1 Yeol Je Cho 2 H. W. Joseph Lee 3 and Chi Kin Chan3 1 Department of Mathematics Yibin University Yibin Sichuan 644007 China 2 Department of Mathematics Education and RINS Gyeongsang National University Chinju 660-701 Republic of Korea 3 Department of Applied Mathematics The Hong Kong Polytechnic University Kowloon Hong Kong Correspondence should be addressed to Yeol Je Cho yjcho@gsnu.ac.kr Received 24 November 2010 Accepted 9 February 2011 Academic Editor Jong Kyu Kim Copyright 2011 Shih-sen Chang et al. 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. The purpose of this paper is to introduce and study the strong convergence problem of the explicit iteration process for a Lipschitzian and demicontraction semigroups in arbitrary Banach spaces. The main results presented in this paper not only extend and improve some recent results announced by many authors but also give an affirmative answer for the open questions raised by Suzuki 2003 and Xu 2005 . 1. Introduction and Preliminaries Throughout this paper we assume that E is a real Banach space E is the dual space of E C is a nonempty closed convex subset of E R is the set of nonnegative real numbers and J E 2E is the normalized duality mapping defined by J x f e E x f x Ilf II x Ilf II 1.1 for all x e E. Let T C C be a mapping. We use F T to denote the set of fixed points of T. We also use to stand for strong convergence and for weak convergence. Definition 1.1. 1 The one-parameter family T T f t 0 of mappings from C into itself is called a nonexpansive semigroup if the following conditions are satisfied 2 .
