Open Access

An approximation method for continuous pseudocontractive mappings

Journal of Inequalities and Applications20062006:28950

DOI: 10.1155/JIA/2006/28950

Received: 20 March 2006

Accepted: 28 May 2006

Published: 26 July 2006


Let be a closed convex subset of a real Banach space , is continuous pseudocontractive mapping, and is a fixed -Lipschitzian strongly pseudocontractive mapping. For any , let be the unique fixed point of . We prove that if has a fixed point and has uniformly Gâteaux differentiable norm, such that every nonempty closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then converges to a fixed point of as approaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004).


Authors’ Affiliations

College of Mathematics and Information Science, Henan Normal University
Department of Mathematics, Tianjin Polytechnic University


  1. Chang S-S, Cho YJ, Zhou H: Iterative Methods for Nonlinear Operator Equations in Banach Spaces. Nova Science, New York; 2002:xiv+459.MATHGoogle Scholar
  2. Gao J: Modulus of convexity in Banach spaces. Applied Mathematics Letters 2003,16(3):273–278. 10.1016/S0893-9659(03)80043-5MathSciNetView ArticleMATHGoogle Scholar
  3. Martin RH Jr.: Differential equations on closed subsets of a Banach space. Transactions of the American Mathematical Society 1973, 179: 399–414.MathSciNetView ArticleMATHGoogle Scholar
  4. Megginson RE: An Introduction to Banach Space Theory, Graduate Texts in Mathematics. Volume 183. Springer, New York; 1998:xx+596.View ArticleMATHGoogle Scholar
  5. Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000:iv+276.MATHGoogle Scholar
  6. Takahashi W, Ueda Y: On Reich's strong convergence theorems for resolvents of accretive operators. Journal of Mathematical Analysis and Applications 1984,104(2):546–553. 10.1016/0022-247X(84)90019-2MathSciNetView ArticleMATHGoogle Scholar
  7. Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar


© Song and Chen 2006

This article is published under license to BioMed Central Ltd. 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.