Open Access

Strong Convergence to Common Fixed Points of a Finite Family of Nonexpansive Mappings

Journal of Inequalities and Applications20072007:037513

DOI: 10.1155/2007/37513

Received: 24 December 2006

Accepted: 2 May 2007

Published: 7 June 2007

Abstract

We suggest and analyze an iterative algorithm for a finite family of nonexpansive mappings . Further, we prove that the proposed iterative algorithm converges strongly to a common fixed point of .

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Authors’ Affiliations

(1)
Department of Information Management, Cheng Shiu University
(2)
Department of Mathematics, Tianjin Polytechnic University
(3)
Department of Applied Mathematics, National Sun Yat-Sen University

References

  1. Atsushiba S, Takahashi W: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian Journal of Mathematics 1999,41(3):435–453.MathSciNetMATHGoogle Scholar
  2. Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1996,202(1):150–159. 10.1006/jmaa.1996.0308MathSciNetView ArticleMATHGoogle Scholar
  3. Lions P-L: Approximation de points fixes de contractions. Comptes Rendus de l'Académie des Sciences. Séries A et B 1977,284(21):A1357-A1359.Google Scholar
  4. Shimizu T, Takahashi W: Strong convergence to common fixed points of families of nonexpansive mappings. Journal of Mathematical Analysis and Applications 1997,211(1):71–83. 10.1006/jmaa.1997.5398MathSciNetView ArticleMATHGoogle Scholar
  5. Takahashi W, Tamura T, Toyoda M: Approximation of common fixed points of a family of finite nonexpansive mappings in Banach spaces. Scientiae Mathematicae Japonicae 2002,56(3):475–480.MathSciNetMATHGoogle Scholar
  6. Zeng LC, Cubiotti P, Yao J-C: Approximation of common fixed points of families of nonexpansive mappings. to appear in Taiwanese Journal of Mathematics to appear in Taiwanese Journal of MathematicsGoogle Scholar
  7. Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar
  8. Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar
  9. Kimura Y, Takahashi W, Toyoda M: Convergence to common fixed points of a finite family of nonexpansive mappings. Archiv der Mathematik 2005,84(4):350–363. 10.1007/s00013-004-1175-zMathSciNetView ArticleMATHGoogle Scholar
  10. Xu H-K: Another control condition in an iterative method for nonexpansive mappings. Bulletin of the Australian Mathematical Society 2002,65(1):109–113. 10.1017/S0004972700020116MathSciNetView ArticleMATHGoogle Scholar
  11. Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 179: 251–262.MathSciNetView ArticleMATHGoogle Scholar
  12. Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
  13. 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
  14. 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

Copyright

© Yeong-Cheng Liou et al. 2007

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.