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 Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 852030 12 pages doi 2010 852030 Research Article Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces Shuang Wang1 and Changsong Hu2 1 School of Mathematical Sciences Yancheng Teachers University Yancheng Jiangsu 224051 China 2 Department of Mathematics Hubei Normal University Huangshi 435002 China Correspondence should be addressed to Shuang Wang wangshuang19841119@ Received 6 August 2010 Accepted 5 October 2010 Academic Editor Tomonari Suzuki Copyright 2010 S. Wang and C. Hu. 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 consider two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces. We proved that the proposed algorithms strongly converge to a common fixed point of a countable family of nonexpansive mappings which solves the corresponding variational inequality. Our results improve and extend the corresponding ones announced by many others. 1. Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Recall that a mapping T C C is said to be nonexpansive if Tx-Ty x-yịị for all x y e C. We use F T to denote the set of fixed points of T. A mapping F H H is called fc-Lipschitzian if there exists a positive constant k such that Fx - Fy fc x - y Yx y e H. F is said to be n-strongly monotone if there exists a positive constant n such that Fx - Fy x - y n x - y 2 x y e H. Let A be a strongly positive bounded linear operator on H that is there exists a constant Ỹ 0 such that Ax x ỹ x 2 Vx e H. 2 Fixed Point Theory and Applications A typical problem is that of minimizing a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H min .
