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Abstract and Applied Analysis
Volume 2014, Article ID 271208, 17 pages
http://dx.doi.org/10.1155/2014/271208
Research Article

Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces

1Computational Science and Engineering Research Cluster (CSEC), King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Khru, Bangkok 10140, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Khru, Bangkok 10140, Thailand
3Department of Mathematics, Faculty of Science and Agriculture, Rajamangala University of Technology Lanna, Phan, Chiangrai 57120, Thailand

Received 27 April 2014; Accepted 26 June 2014; Published 24 July 2014

Academic Editor: Xiaolong Qin

Copyright © 2014 Poom Kumam and Thanyarat Jitpeera. 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. L. Ceng and J. 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 · View at Scopus
  2. R. S. Burachik, J. O. Lopes, and G. J. P. Da Silva, “An inexact interior point proximal method for the variational inequality problem,” Computational & Applied Mathematics, vol. 28, no. 1, pp. 15–36, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. 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 MathSciNet
  4. 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 MathSciNet · View at Scopus
  5. P. Kumam, “Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space,” Turkish Journal of Mathematics, vol. 33, no. 1, pp. 85–98, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. P. Kumam, “A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 4, pp. 1245–1255, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  7. P. Kumam, “A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 263–280, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. P. Kumam and C. Jaiboon, “A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 510–530, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. P. Kumam and P. Katchang, “A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 475–486, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. A. Moudafi and M. Thera, “Proximal and dynamical approaches to equilibrium problems,” in Lecture note in Economics and Mathematical Systems, pp. 187–201, Springer, New York, NY, USA, 1999. View at Google Scholar
  11. Z. Wang and Y. Su, “Strong convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Application Mathematics & Informatics, vol. 28, no. 3-4, pp. 783–796, 2010. View at Google Scholar
  12. R. Wangkeeree, “Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 719–733, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Y. Yao, M. A. Noor, S. Zainab, and Y. C. Liou, “Mixed equilibrium problems and optimization problems,” Journal of Mathematical Analysis and Applications, vol. 354, no. 1, pp. 319–329, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Y. Yao, Y. J. Cho, and R. Chen, “An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3363–3373, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. J.-C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,” in Handbook of Generalized Convexity and Generalized Monotonicity, vol. 76 of Nonconvex Optimization and Its Applications, pp. 501–558, Springer, New York, NY, USA, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  16. L. C. Zeng, S. Schaible, and J. C. Yao, “Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities,” Journal of Optimization Theory and Applications, vol. 124, no. 3, pp. 725–738, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. 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
  18. H. Brezis, Opérateurs Maximaux Monotones, vol. 5 of Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1973.
  19. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000. View at MathSciNet
  20. S. Plubtieng and R. Punpaeng, “A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 455–469, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. J. Peng and J. Yao, “Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1816–1828, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. X. Qin, S. Y. Cho, and S. M. Kang, “Some results on variational inequalities and generalized equilibrium problems with applications,” Computational and Applied Mathematics, vol. 29, no. 3, pp. 393–421, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. Y.-C. Liou, “An iterative algorithm for mixed equilibrium problems and variational inclusions approach to variational inequalities,” Fixed Point Theory and Applications, vol. 2010, Article ID 564361, 15 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  24. N. Petrot, R. Wangkeeree, and P. Kumam, “A viscosity approximation method of common solutions for quasi variational inclusion and fixed point problems,” Fixed Point Theory, vol. 12, no. 1, pp. 165–178, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  25. T. Jitpeera and P. Kumam, “A new hybrid algorithm for a system of mixed equilibrium problems, fixed point problems for nonexpansive semigroup, and variational inclusion problem,” Fixed Point Theory and Applications, vol. 2011, article 217407, 27 pages, 2011. View at Google Scholar · View at MathSciNet
  26. 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 MathSciNet
  27. 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, 17 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. Y. Liou and Y. Yao, “Iterative algorithms for nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 384629, 10 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  29. Y. Yao, Y. C. Liou, and J. C. Yao, “Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2007, Article ID 64363, 12 pages, 2007. View at Google Scholar · View at MathSciNet
  30. 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 MathSciNet
  31. A. Kangtunyakarn and S. Suantai, “A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4448–4460, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. H. 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 · View at MathSciNet · View at Scopus
  34. 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 · View at MathSciNet · View at Scopus
  35. 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 MathSciNet · View at Scopus