tailieunhanh - Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 169062,

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 169062, 12 pages doi: Research Article Comparison of the Rate of Convergence among Picard, Mann, Ishikawa, and Noor Iterations Applied to Quasicontractive Maps B. E. Rhoades1 and Zhiqun Xue2 1 2 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA Department of Mathematics and Physics, Shijiazhuang Railway University, Shijiazhuang 050043, China Correspondence should be addressed to Zhiqun Xue, xuezhiqun@ Received 12 October 2010; Accepted 14 December 2010 Academic Editor: Juan J. Nieto Copyright q 2010 B. E. Rhoades and Z. Xue. This is an open access article distributed under the Creative Commons Attribution. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 169062 12 pages doi 2010 169062 Research Article Comparison of the Rate of Convergence among Picard Mann Ishikawa and Noor Iterations Applied to Quasicontractive Maps B. E. Rhoades1 and Zhiqun Xue2 1 Department of Mathematics Indiana University Bloomington IN 47405-7106 USA 2 Department of Mathematics and Physics Shijiazhuang Railway University Shijiazhuang 050043 China Correspondence should be addressed to Zhiqun Xue xuezhiqun@ Received 12 October 2010 Accepted 14 December 2010 Academic Editor Juan J. Nieto Copyright 2010 B. E. Rhoades and Z. Xue. 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 provide sufficient conditions for Picard iteration to converge faster than Krasnoselskij Mann Ishikawa or Noor iteration for quasicontractive operators. We also compare the rates of convergence between Krasnoselskij and Mann iterations for Zamfirescu operators. 1. Introduction Let X d be a complete metric space and let T be a self-map of X. If T has a unique fixed point which can be obtained as the limit of the sequence pn where pn Tnpo po any point of X then T is called a Picard operator see . 1 and the iteration defined by pn is called Picard iteration. One of the most general contractive conditions for which a map T is a Picard operator is that of Ciric 2 see also 3 . A self-map T is called quasicontractive if it satisfies d Tx Ty Ômax d x y d x Tx d y Ty d x Ty d y Tx for each x y e X where Ô is a real number satisfying 0 Ỗ 1. Not every map which has a unique fixed point enjoys the Picard property. For example let X 0 1 with the absolute value metric T X X defined by Tx 1 - x. Then T has a unique fixed point at x 1 2 but if one chooses as a starting point x0 a for any a 1 2 then successive .