Open Access

Strong Convergence Theorems of the CQ Method for Nonexpansive Semigroups

Fixed Point Theory and Applications20072007:059735

https://doi.org/10.1155/2007/59735

Received: 25 January 2007

Accepted: 19 March 2007

Published: 30 April 2007

Abstract

Motivated by T. Suzuki, we show strong convergence theorems of the CQ method for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. The results presented extend and improve the corresponding results of Kazuhide Nakajo and Wataru Takahashi (2003).

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Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University

References

  1. Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America 1965,53(6):1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar
  2. Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. Nonlinear Analysis 1998,34(1):87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleMATHGoogle Scholar
  3. Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proceedings of the American Mathematical Society 2003,131(7):2133–2136. 10.1090/S0002-9939-02-06844-2MathSciNetView ArticleMATHGoogle Scholar
  4. Xu H-K: A strong convergence theorem for contraction semigroups in Banach spaces. Bulletin of the Australian Mathematical Society 2005,72(3):371–379. 10.1017/S000497270003519XMathSciNetView ArticleMATHGoogle Scholar
  5. Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. to appear in Applied Mathematics LettersGoogle Scholar
  6. Chen R, Zhu Z: Viscosity approximation fixed points for nonexpansive and -accretive operators. Fixed Point Theory and Applications 2006, 2006: 10 pages.MathSciNetMATHGoogle Scholar
  7. Chen R, Song Y, Zhou H: Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings. Journal of Mathematical Analysis and Applications 2006,314(2):701–709. 10.1016/j.jmaa.2005.04.018MathSciNetView ArticleMATHGoogle Scholar
  8. Yao Y, Chen R: Iterative algorithm for approximating solutions of maximal monotone operators in Hilbert spaces. Fixed Point Theory and Applications 2007, 2007: 8 pages.MathSciNetMATHGoogle Scholar
  9. Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar
  10. Browder FE: Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America 1965,54(4):1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleMATHGoogle Scholar
  11. Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleMATHGoogle Scholar
  12. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
  13. Yanagi K: On some fixed point theorems for multivalued mappings. Pacific Journal of Mathematics 1980,87(1):233–240.MathSciNetView ArticleMATHGoogle Scholar
  14. Megginson RE: An Introduction to Banach Space Theory, Graduate Texts in Mathematics. Volume 183. Springer, New York, NY, USA; 1998:xx+596.View ArticleMATHGoogle Scholar
  15. Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar

Copyright

© H. He and R. Chen. 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.