Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Weak and strong convergence theorems of implicit iteration process on Banach spaces | Lin et al. Fixed Point Theory and Applications 2011 2011 96 http content 2011 1 96 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Weak and strong convergence theorems of implicit iteration process on Banach spaces Lai-Jiu Lin 1 Chih-Sheng Chuang1 and Zenn-Tsun Yu2 Correspondence maljlin@. department of Mathematics National Changhua University of Education Changhua 50058 Taiwan Full list of author information is available at the end of the article Springer Abstract In this article we first consider weak convergence theorems of implicit iterative processes for two nonexpansive mappings and a mapping which satisfies condition C . Next we consider strong convergence theorem of an implicit-shrinking iterative process for two nonexpansive mappings and a relative nonexpansive mapping on Banach spaces. Note that the conditions of strong convergence theorem are different from the strong convergence theorems for the implicit iterative processes in the literatures. Finally we discuss a strong convergence theorem concerning two nonexpansive mappings and the resolvent of a maximal monotone operator in a Banach space. 1 Introduction Let E be a Banach space and let C be a nonempty closed convex subset of E. A mapping T C E is nonexpansive if Tx - Ty x - y for every x y e C. Let F T x e C x Tx denote the set of fixed points of T. Besides a mapping T C E is quasinonexpansive if F T 0 and Tx - y x - y for all x e C and y e F T . In 2008 Suzuki 1 introduced the following generalized nonexpansive mapping on Banach spaces. A mapping T C E is said to satisfy condition C if for all x y e C 1 2 x- Tx x- y Tx- Ty x- y . In fact every nonexpansive mapping satisfies condition C but the converse may be false 1 Example 1 . Besides if T C E satisfies condition C and F T 0 then T is a quasinonexpansive mapping. However the converse may be false 1 Example 2 . Construction of approximating fixed points of .
