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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 349158, 14 pages
http://dx.doi.org/10.1155/2010/349158
Research Article

A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

Received 26 January 2010; Accepted 21 April 2010

Academic Editor: Binggen Zhang

Copyright © 2010 Watcharaporn Cholamjiak and Suthep Suantai. 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. C. Shiau, K. K. Tan, and C. S. Wong, “Quasi-nonexpansive multi-valued maps and selections,” Fundamenta Mathematicae, vol. 87, pp. 109–119, 1975. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. N. Shahzad and H. Zegeye, “On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 838–844, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  3. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. H. H. Bauschke, E. Matoušková, and S. Reich, “Projection and proximal point methods: convergence results and counterexamples,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 5, pp. 715–738, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. K. P. R. Sastry and G. V. R. Babu, “Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point,” Czechoslovak Mathematical Journal, vol. 55, no. 4, pp. 817–826, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B. Panyanak, “Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 54, no. 6, pp. 872–877, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Song and H. Wang, “Erratum to Mann and Ishikawa iterative process for multivalued mappings in Banach spaces,” Comput. Math. Appl., vol. 55, no. 6, pp. 2999–3002, 2008. View at Google Scholar
  12. F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. 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 · View at MathSciNet
  14. L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. P. Cholamjiak, “A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 719360, 18 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. 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 MathSciNet
  17. W. Cholamjiak and S. Suantai, “Monotone hybrid projection algorithms for an infinitely countable family of Lipschitz generalized asymptotically quasi-nonexpansive mappings,” Abstract and Applied Analysis, vol. 2009, Article ID 297565, 16 pages, 2009. View at Google Scholar · View at MathSciNet
  18. 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 · View at MathSciNet
  19. A. Kangtunyakarn and S. Suantai, “Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 296–309, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  20. J.-W. Peng, Y.-C. Liou, and J.-C. Yao, “An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions,” Fixed Point Theory and Applications, vol. 2009, Article ID 794178, 21 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 5, pp. 1140–1152, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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 · View at MathSciNet
  24. H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 3, pp. 341–350, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. R. Khan, “Properties of fixed point set of a multivalued map,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2005, no. 3, pp. 323–331, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet