Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 816529, 26 pages
http://dx.doi.org/10.1155/2012/816529
Research Article

Viscosity Approximation Method for System of Variational Inclusions Problems and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

1Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin (RMUTR), Bangkok 10100, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

Received 22 January 2012; Accepted 2 March 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Chaichana Jaiboon and Poom Kumam. 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. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Mass, USA, 1990. View at Publisher · View at Google Scholar
  2. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
  3. W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. S.-S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,” Applied Mathematics and Mechanics English Edition, vol. 29, no. 5, pp. 571–581, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994. View at Google Scholar · View at Zentralblatt MATH
  6. S. D. Flåm and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. P. Cholamjiak and S. Suantai, “A new hybrid algorithm for variational inclusions, generalized equilibrium problems, and a finite family of quasi-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2009, Article ID 350979, 20 pages, 2009. View at Google Scholar · View at Zentralblatt MATH
  8. P. Cholamjiak and S. Suantai, “Convergence analysis for a system of equilibrium problems and a countable family of relatively quasi-nonexpansive mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 141376, 17 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. P. Cholamjiak and S. Suantai, “Existence and iteration for a mixed equilibrium problem and a countable family of nonexpansive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2725–2733, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. P. Cholamjiak and S. Suantai, “An iterative method for equilibrium problems and a finite family of relatively nonexpansive mappings in a Banach space,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 3825–3831, 2010. View at Publisher · View at Google Scholar
  11. A. N. Iusem and M. Nasri, “Korpelevich's method for variational inequality problems in Banach spaces,” Journal of Global Optimization, vol. 50, no. 1, pp. 59–76, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. Y. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9 pages, 2012. View at Google Scholar
  13. Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical & Computer Modelling, vol. 55, pp. 1506–1515, 2012. View at Google Scholar
  14. Y. Yao, M. A. Noor, Y. C. Liou, and S. M. Kang, “Iterative algorithms for general multi-valued variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 768272, 10 pages, 2012. View at Google Scholar
  15. Y. Su, M. Shang, and X. Qin, “An iterative method of solution for equilibrium and optimization problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2709–2719, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. H. Brézis, “Opérateur maximaux monotones,” in Mathematics Studies, vol. 5, North-Holland, Amsterdam, The Netherlands, 1973. View at Google Scholar
  17. B. Lemaire, “Which fixed point does the iteration method select?” in Recent Advances in Optimization (Trier, 1996), vol. 452 of Lecture Notes in Economics and Mathematical Systems, pp. 154–167, Springer, Berlin, Germany, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. 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
  22. S. Takahashi, W. Takahashi, and M. Toyoda, “Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 147, no. 1, pp. 27–41, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH