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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 643740, 21 pages
http://dx.doi.org/10.1155/2011/643740
Research Article

Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 4 February 2011; Accepted 22 March 2011

Academic Editor: Yonghong Yao

Copyright © 2011 Rattanaporn Wangkeeree. 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.

Linked References

  1. N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 1, pp. 87–99, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566–575, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. T. D. Benavides, G. López Acedo, and H.-K. Xu, “Construction of sunny nonexpansive retractions in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 66, no. 1, pp. 9–16, 2002. View at Publisher · View at Google Scholar
  6. H.-K. Xu, “A strong convergence theorem for contraction semigroups in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 72, no. 3, pp. 371–379, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. R. Chen and H. He, “Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space,” Applied Mathematics Letters, vol. 20, no. 7, pp. 751–757, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. R. D. Chen, H. M. He, and M. A. Noor, “Modified Mann iterations for nonexpansive semigroups in Banach space,” Acta Mathematica Sinica, vol. 26, no. 1, pp. 193–202, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization. An International Journal, vol. 19, no. 1-2, pp. 33–56, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. H.-K. Xu, “Approximations to fixed points of contraction semigroups in Hilbert spaces,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 157–163, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. Li, L. Li, and Y. Su, “General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3065–3071, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. E. Zeidler, Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization, Springer, New York, NY, USA, 1985.
  17. W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
  18. J.-P. Gossez and E. Lami Dozo, “Some geometric properties related to the fixed point theory for nonexpansive mappings,” Pacific Journal of Mathematics, vol. 40, pp. 565–573, 1972. View at Google Scholar · View at Zentralblatt MATH
  19. J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509–520, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. A. Aleyner and Y. Censor, “Best approximation to common fixed points of a semigroup of nonexpansive operators,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 137–151, 2005. View at Google Scholar · View at Zentralblatt MATH
  21. A. Aleyner and S. Reich, “An explicit construction of sunny nonexpansive retractions in Banach spaces,” Fixed Point Theory and Applications, no. 3, pp. 295–305, 2005. View at Google Scholar · View at Zentralblatt MATH
  22. R. E. Bruck Jr., “Nonexpansive retracts of Banach spaces,” Bulletin of the American Mathematical Society, vol. 76, pp. 384–386, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. S. Reich, “Asymptotic behavior of contractions in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 44, pp. 57–70, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. H. Piri and H. Vaezi, “Strong convergence of a generalized iterative method for semigroups of nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 907275, 16 pages, 2010. View at Google Scholar · View at Zentralblatt MATH