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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 a Finite Family of Nonexpansive Mappings | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 76971 9 pages doi 10.1155 2007 76971 Research Article Strong Convergence Theorems for a Finite Family of Nonexpansive Mappings Meijuan Shang Yongfu Su and Xiaolong Qin Received 23 May 2007 Accepted 2 August 2007 Recommended by J. R. L. Webb We modified the classic Mann iterative process to have strong convergence theorem for a finite family of nonexpansive mappings in the framework of Hilbert spaces. Our results improve and extend the results announced by many others. Copyright 2007 Meijuan Shang 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. 1. Introduction and preliminaries Let H be a real Hilbert space C a nonempty closed convex subset of H and T C C a mapping. Recall that T is nonexpansive if II Tx - Ty II x - y II for all x y e C. A point x e C is called a fixed point of T provided Tx x. Denote by F T the set of fixed points of T that is F T x e C Tx x . Recall that a self-mapping f C C is a contraction on C if there exists a constant a e 0 1 such that II f x - f y II a x - y II for all x y e C. We use nC to denote the collection of all contractions on C that is nC f I f C C a contraction . An operator A is strongly positive if there exists a constant Y 0 with the property Ax x ỹ x 2 Vx e H. 1.1 Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems see e.g. 1 2 and the references therein . A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H min 7- Ax x - x b 1.2 xeC 2 2 Fixed Point Theory and Applications where C is the fixed point set of a nonexpansive mapping s and b is a given point in H. In 2 it is proved that the sequence xn defined by the iterative method .
