Open Access

Remarks of Equivalence among Picard, Mann, and Ishikawa Iterations in Normed Spaces

Fixed Point Theory and Applications20072007:061434

Received: 1 April 2007

Accepted: 21 June 2007

Published: 9 August 2007


We show that the convergence of Picard iteration is equivalent to the convergence of Mann iteration schemes for various Zamfirescu operators. Our result extends of Soltuz (2005).


Authors’ Affiliations

Department of Mathematics and Physics, Shijiazhuang Railway Institute


  1. Şoltuz ŞM: The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators. Mathematical Communications 2005,10(1):81–88.MATHMathSciNetGoogle Scholar
  2. Vasilev FP: Numerical Methods for Solving Extremal Problems. 2nd edition. Nauka, Moscow, Russia; 1988:550.Google Scholar
  3. Weng X: Fixed point iteration for local strictly pseudo-contractive mapping. Proceedings of the American Mathematical Society 1991,113(3):727–731. 10.1090/S0002-9939-1991-1086345-8MATHMathSciNetView ArticleGoogle Scholar


© Xue Zhiqun. 2007

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