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 On Equivalence of Some Iterations Convergence for Quasi-Contraction Maps in Convex Metric Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 252871 10 pages doi 2010 252871 Research Article On Equivalence of Some Iterations Convergence for Quasi-Contraction Maps in Convex Metric Spaces Zhiqun Xue 1 Guiwen Lv 1 and B. E. Rhoades2 1 Department of Mathematics and Physics Shijiazhuang Railway University Shijiazhuang 050043 China 2 Department of Mathematics Indiana University Bloomington IN 47405-7106 USA Correspondence should be addressed to Zhiqun Xue xuezhiqun@ Received 23 July 2010 Accepted 9 September 2010 Academic Editor Marlene Frigon Copyright 2010 Zhiqun Xue 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. We show the equivalence of the convergence of Picard and Krasnoselskij Mann and Ishikawa iterations for the quasi-contraction mappings in convex metric spaces. 1. Introduction Let E d be a complete metric space and I 0 1 . Denote E2 E X E 12 IXI .A continuous mapping w E2 X I2 E is said to be a convex structure on E 1 if for all u z1 z2 6 E 11 12 6 I with 11 12 1 such that d u W z1 z2 11 12 11d u z1 12d u z2 W Z1 z2 1 0 Z1 W Z1 z2 0 1 z2. If E d satisfies the conditions of convex structure then E d is called convex metric space that is denoted as E d W . In the following part we will consider a few iteration sequences in convex metric space E d W . Suppose that T is a self-map of E. Picard iteration is as follows Vp0 6 E pn 1 Tpn Tn 1P0 n 0. 2 Fixed Point Theory and Applications Krasnoselskijiteration is as follows Vv0 E vn 1 W vn Tvn 1 -1 1 n 0 where 1 e 0 1 . Mann iteration is as follows Vu0 e E un 1 W un Tun 1 - an an n 0 where an e 0 1 . Ishikawa iteration is as follows Vx0 e E Xn 1 W xn Eyn 1 - an an n 0 yn W Xn TXn 1 - bn bn n 0 where an bn e 0 1 for all n 0. A mapping T E E is called contractive if
