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Abstract and Applied Analysis
Volume 2012, Article ID 161897, 22 pages
http://dx.doi.org/10.1155/2012/161897
Research Article

Iterative Algorithms with Perturbations for Solving the Systems of Generalized Equilibrium Problems and the Fixed Point Problems of Two Quasi-Nonexpansive Mappings

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 3 September 2012; Accepted 1 November 2012

Academic Editor: Xiaolong Qin

Copyright © 2012 Rabian Wangkeeree and Uraiwan Boonkong. 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. S. Itoh and W. Takahashi, “The common fixed point theory of singlevalued mappings and multivalued mappings,” Pacific Journal of Mathematics, vol. 79, no. 2, pp. 493–508, 1978. View at Google Scholar
  2. F. Kohsaka and W. Takahashi, “Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces,” Archiv der Mathematik, vol. 91, no. 2, pp. 166–177, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,” Journal of Nonlinear and Convex Analysis, vol. 9, no. 1, pp. 37–43, 2008. View at Google Scholar · View at Zentralblatt MATH
  4. E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–146, 1994. View at Google Scholar · View at Zentralblatt MATH
  5. P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005. View at Google Scholar · View at Zentralblatt MATH
  6. S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. L. C. Ceng and J. C. Yao, “A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem,” Nonlinear Analysis, vol. 72, no. 3-4, pp. 1922–1937, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. P.-E. Mainge and A. Moudafi, “Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems,” Journal of Nonlinear and Convex Analysis, vol. 9, no. 2, pp. 283–294, 2008. View at Google Scholar · View at Zentralblatt MATH
  9. R. Wangkeeree, “An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 134148, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. R. Wangkeeree and U. Kamraksa, “An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems,” Nonlinear Analysis. Hybrid Systems, vol. 3, no. 4, pp. 615–630, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486–491, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. W. Takahashi, Convex Analysis and Approximation of Fixed Points, vol. 2, Yokohama Publishers, Yokohama, Japan, 2000.
  13. B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Y. Kurokawa and W. Takahashi, “Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces,” Nonlinear Analysis, vol. 73, no. 6, pp. 1562–1568, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. M. Hojo and W. Takahashi, “Weak and strong convergence theorems for generalized hybrid mappings in Hilbert spaces,” Scientiae Mathematicae Japonicae, vol. 73, no. 1, pp. 31–40, 2011. View at Google Scholar · View at Zentralblatt MATH
  16. Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, article 79, 2011. View at Publisher · View at Google Scholar
  17. C. S. Chuang, L. J. Lin, and W. Takahashi, “Halpern’s type iterations with perturbations in Hilbert spaces: equilibriumsolutions and fixed points,” Journal of Global Optimization. In press. View at Publisher · View at Google Scholar
  18. S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis, vol. 69, no. 3, pp. 1025–1033, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, 2009.
  20. W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
  21. P. E. Mainge, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. 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, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. W. Takahashi, “Fixed point theorems for new nonlinear mappings in a Hilbert space,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 1, pp. 79–88, 2010. View at Google Scholar · View at Zentralblatt MATH
  24. F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,” Mathematische Zeitschrift, vol. 100, pp. 201–225, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar
  26. P. Kocourek, W. Takahashi, and J. C. Yao, “Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 14, no. 6, pp. 2497–2511, 2010. View at Google Scholar · View at Zentralblatt MATH
  27. W. Takahashi, J. C. Yao, and P. Kocourek, “Weak and strong convergence theorems for generalized hybrid nonself-mappings in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 567–586, 2010. View at Google Scholar · View at Zentralblatt MATH