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Báo cáo hóa học: " Research Article Convergence Comparison of Several Iteration Algorithms for the Common Fixed Point Problems"

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 Comparison of Several Iteration Algorithms for the Common Fixed Point Problems | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 824374 13 pages doi 10.1155 2009 824374 Research Article Convergence Comparison of Several Iteration Algorithms for the Common Fixed Point Problems Yisheng Song and Xiao Liu College of Mathematics and Information Science Henan Normal University 453007 China Correspondence should be addressed to Yisheng Song songyisheng123@yahoo.com.cn Received 20 January 2009 Accepted 2 May 2009 Recommended by Naseer Shahzad We discuss the following viscosity approximations with the weak contraction A for a non-expansive mapping sequence Tn yn anAyn 1 - an Tnyn xn 1 anAxn 1 - an Tnxn. We prove that Browder s and Halpern s type convergence theorems imply Moudafi s viscosity approximations with the weak contraction and give the estimate of convergence rate between Halpern s type iteration and Mouda s viscosity approximations with the weak contraction. Copyright 2009 Y. Song and X. Liu. 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 The following famous theorem is referred to as the Banach Contraction Principle. Theorem 1.1 Banach 1 . Let fE d be a complete metric space and let A be a contraction on X that is there exists p e 0 1 such that d Ax Ay pd x yfi Nx y e E. 1.1 Then A has a unique fixed point. In 2001 Rhoades 2 proved the following very interesting fixed point theorem which is one of generalizations of Theorem 1.1 because the weakly contractions contains contractions as the special cases y fi 1 - p t . Theorem 1.2 Rhoades 2 Theorem 2 . Let fE d be a complete metric space and let A be a weak contraction on E that is d Ax Ay d x y - y d x y Vx y e E 1.2 2 Fixed Point Theory and Applications for some p 0 x 0 x is a continuous and nondecreasing function such that p is positive on 0 x and p 0 0. Then A has a unique