Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 538912, 29 pages
http://dx.doi.org/10.1155/2012/538912
Research Article

Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Received 7 October 2011; Accepted 1 November 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Tanom Chamnarnpan 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. 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
  2. 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
  3. 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
  4. 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
  5. T. Jitpeera, U. Witthayarat, and P. Kumam, “Hybrid algorithms of common solutions of generalized mixed equilibrium problems and the common variational inequality problems with applications,” Fixed Point Theory and Applications, vol. 2011, Article ID 971479, 28 pages, 2011. View at Google Scholar · View at Zentralblatt MATH
  6. T. Jitpeera and P. Kumam, “An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings,” Journal of Nonlinear Analysis and Optimization: Theory & Applications, vol. 1, no. 1, pp. 71–91, 2010. View at Google Scholar
  7. T. Jitpeera and P. Kumam, “A new hybrid algorithm for a system of equilibrium problems and variational inclusion,” Annali dell'Universita di Ferrara, vol. 57, no. 1, pp. 89–108, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. T. Jitpeera and P. Kumam, “Hybrid algorithms for minimization problems over the solutions of generalized mixed equilibrium and variational inclusion problems,” Mathematical Problems in Engineering, vol. 2011, Article ID 648617, 25 pages, 2011. View at Publisher · View at Google Scholar
  9. P. Kumam, U. Hamphries, and P. Katchang, “Common solutions of generalized mixed equilibrium problems, variational inclusions and common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings,” Journal of Applied Mathematics, vol. 2011, Article ID 953903, 27 pages, 2011. View at Google Scholar
  10. P. Sunthrayuth and P. Kumam, “A new general iterative method for solution of a new general system of variational inclusions for nonexpansive semigroups in Banach spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 187052, 29 pages, 2011. View at Google Scholar · View at Zentralblatt MATH
  11. P. Katchang and P. Kumam, “Convergence of iterative algorithm for finding common solution of fixed points and general system of variational inequalities for two accretive operators,” Thai Journal of Mathematics, vol. 9, no. 2, pp. 319–335, 2011. View at Google Scholar
  12. W. Kumam, P. Junlouchai, and P. Kumam, “Generalized systems of variational inequalities and projection methods for inverse-strongly monotone mappings,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 976505, 24 pages, 2011. View at Google Scholar
  13. P. Kumam, “A relaxed extragradient approximation method of two inverse-strongly monotone mappings for a general system of variational inequalities, fixed point and equilibrium problems,” Bulletin of the Iranian Mathematical Society, vol. 36, no. 1, pp. 227–250, 2010. View at Google Scholar
  14. P. Kumam and C. Jaiboon, “Approximation of common solutions to system of mixed equilibrium problems, variational inequality problem, and strict pseudo-contractive mappings,” Fixed Point Theory and Applications, vol. 2011, Article ID 347204, 30 pages, 2011. View at Google Scholar · View at Zentralblatt MATH
  15. C. Jaiboon and P. Kumam, “A general iterative method for addressing mixed equilibrium problems and optimization problems,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 73, no. 5, pp. 1180–1202, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Y. J. Cho, I. K. Argyros, and N. Petrot, “Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2292–2301, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Y. J. Cho, N. Petrot, and S. Suantai, “Fixed point theorems for nonexpansive mappings with applications to generalized equilibrium and system of nonlinear variational inequalities problems,” Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 45–53, 2010. View at Google Scholar
  18. Y. J. Cho and N. Petrot, “On the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 437976, 12 pages, 2010. View at Google Scholar · View at Zentralblatt MATH
  19. Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011. View at Publisher · View at Google Scholar
  20. W. A. Kirk, “Fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, pp. 1004–1006, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,” Acta Mathematica, vol. 115, pp. 271–310, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J.-C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,” in Handbook of Generalized Convexity and Monotonicity, J. P. Crouzeix, N. Haddjissas, and S. Schaible, Eds., vol. 76, pp. 501–558, Springer, New York, NY, USA, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. 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
  24. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” European Journal of Operational Research. In Press. View at Publisher · View at Google Scholar
  25. Y. Yao, Y.-C. Liou, and C.-P. Chen, “Algorithms construction for nonexpansive mappings and inverse-strongly monotone mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 5, pp. 1979–1998, 2011. View at Google Scholar
  26. Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical and Computer Modelling, vol. 55, pp. 1506–1515, 2012. View at Google Scholar
  27. Y. Yao, Y.-C. Liou, S. M. Kang, and Y. Yu, “Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 17, pp. 6024–6034, 2011. View at Publisher · View at Google Scholar
  28. Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In Press. View at Publisher · View at Google Scholar
  29. 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
  30. Y. Hao, “Some results of variational inclusion problems and fixed point problems with applications,” Applied Mathematics and Mechanics, vol. 30, no. 12, pp. 1589–1596, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. M. Liu, S. S. Chang, and P. Zuo, “An algorithm for finding a common solution for a system of mixed equilibrium problem, quasivariational inclusion problem, and fixed point problem of nonexpansive semigroup,” Journal of Inequalities and Applications, vol. 2010, Article ID 895907, 23 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. 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, vol. 29, no. 5, pp. 571–581, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 46–55, 2000. View at Google Scholar · View at Zentralblatt MATH
  34. B. Lemaire, “Which fixed point does the iteration method select?” in Recent Advances in Optimization, vol. 452 of Lecture Note in Economics and Mathematical Systems, pp. 154–167, Springer, Berlin, Germany, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. 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
  36. H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 61, no. 3, pp. 341–350, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. Y. Su, M. Shang, and X. Qin, “An iterative method of solution for equilibrium and optimization problems,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 69, no. 8, pp. 2709–2719, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. H. He, S. Liu, and Y. J. Cho, “An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4128–4139, 2011. View at Publisher · View at Google Scholar
  39. T. Jitpeera and P. Kumam, “A general iterative algorithm for generalized mixed equilibrium problems and variational inclusions approach to variational inequalities,” International Journal of Mathematics and Mathematical Sciences, Article ID 619813, 25 pages, 2011. View at Google Scholar · View at Zentralblatt MATH
  40. H. Brézis, “Opérateur maximaux monotones,” in Mathematics Studies, vol. 5, North-Holland, Amsterdam, The Netherlands, 1973. View at Publisher · View at Google Scholar
  41. 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
  42. H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. F. E. Browder, Ed., Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, vol. 18 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 1976. View at Zentralblatt MATH
  44. 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
  45. C. Klin-Eam and S. Suantai, “A new approximation method for solving variational inequalities and fixed points of nonexpansive mappings,” Journal of Inequalities and Applications, vol. 2009, Article ID 520301, 16 pages, 2009. View at Google Scholar · View at Zentralblatt MATH