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Abstract and Applied Analysis
Volume 2012, Article ID 316276, 39 pages
http://dx.doi.org/10.1155/2012/316276
Research Article

A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

1Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 30 August 2011; Accepted 1 November 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Pongsakorn Sunthrayuth 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. J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401–1432, 2008. View at Google Scholar
  2. 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
  3. S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008. View at Publisher · View at Google Scholar
  4. 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
  5. C. Jaiboon and P. Kumam, “A general iterative method for addressing mixed equilibrium problems and optimization problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 5, pp. 1180–1202, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. L.-C. Ceng, A. R. Khan, Q. H. Ansari, and J.-C. Yao, “Viscosity approximation methods for strongly positive and monotone operators,” Fixed Point Theory, vol. 10, no. 1, pp. 35–72, 2009. View at Google Scholar · View at Zentralblatt MATH
  7. L. C. Ceng, H.-Y. Hu, and M. M. Wong, “Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed pointed problem of infinitely many nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 1341–1367, 2011. View at Google Scholar
  8. H.-Y. Hu and L.-C. Ceng, “A general system of generalized nonlinear mixed composite-type equilibria in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 927–959, 2011. View at Google Scholar
  9. A. Moudafi and M. Théra, “Proximal and dynamical approaches to equilibrium problems,” in Ill-Posed Variational Problems and Regularization Techniques, vol. 477 of Lecture Notes in Economics and Mathematical Systems, pp. 187–201, Springer, Berlin, Germany, 1999. View at Google Scholar · View at Zentralblatt MATH
  10. N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. A. Noor, “General variational inequalities and nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 810–822, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. 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 Zentralblatt MATH
  13. S. Plubtieng and P. Kumam, “Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 614–621, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. Saewan and P. Kumam, “A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems,” Abstract and Applied Analysis, vol. 2010, Article ID 123027, 31 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. S. Saewan and P. Kumam, “Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi-ø-asymptotically nonexpansive mappings,” Abstract and Applied Analysis, vol. 2010, Article ID 357120, 22 pages, 2010. View at Publisher · View at Google Scholar
  17. S. Saewan and P. Kumam, “A new modified block iterative algorithm for uniformly quasi-ø-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems,” Fixed Point Theory and Applications, vol. 2011, 35 pages, 2011. View at Publisher · View at Google Scholar
  18. S. Saewan and P. Kumam, “A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces,” Computers and Mathematics with Applications, vol. 62, pp. 1723–1735, 2011. View at Google Scholar
  19. S. Saewan and P. Kumam, “Strong convergence theorems for countable families of uniformly quasi-ø-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems,” Abstract and Applied Analysis, vol. 2011, Article ID 701675, 2011. View at Publisher · View at Google Scholar
  20. S. Saewan and P. Kumam, “The shrinking projection method for solving generalized equilibrium problem and common fixed points for asymptotically quasi-ø-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2011, 9 pages, 2011. View at Publisher · View at Google Scholar
  21. 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
  22. 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
  23. P. Kumam, W. Kumam, and P. Junlouchai, “Generalized systems of variational inequalities and projection methods for inverse-strongly monotone mappings,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 976505, 2011. View at Publisher · View at Google Scholar
  24. 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
  25. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. 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. 343–360, 2011. View at Google Scholar
  27. P. Katchang and P. Kumam, “A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems,” Journal of Inequalities and Applications, vol. 2010, Article ID 370197, 25 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. P. Katchang, T. Jitpeera, and P. Kumam, “Strong convergence theorems for solving generalized mixed equilibrium problems and general system of variational inequalities by the hybrid method,” Nonlinear Analysis. Hybrid Systems, vol. 4, no. 4, pp. 838–852, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. 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
  30. S. Saewan and P. Kumam, “A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems,” Abstract and Applied Analysis, vol. 2010, Article ID 123027, 31 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. Y. Yao and J.-C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. S. Saewan, P. Kumam, and K. Wattanawitoon, “Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 734126, 25 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. J. Zhao and S. He, “A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 670–680, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. 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
  35. Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, no. 79, 2011. View at Publisher · View at Google Scholar
  36. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters, vol. 4, pp. 635–641, 2010. View at Publisher · View at Google Scholar
  37. Y. Yao, Y.-C. Liou, and C.-P. Chen, “Algorithms construction for nonexpansive mappings and inversestrongly monotone mappings,” Taiwanese Journal of Mathematics, vol. 15, pp. 1979–1998, 2011. View at Google Scholar
  38. Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical & Computer Modelling. In press. View at Publisher · View at Google Scholar
  39. P. Sunthrayuth and P. Kumam, “A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces,” Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 139–150, 2010. View at Google Scholar
  40. 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, vol. 2011, Article ID 619813, 25 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  41. 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, vol. 1, no. 1, pp. 71–91, 2010. View at Google Scholar
  42. T. Jitpeera and P. Kumam, “A composite iterative method for generalized mixed equilibrium problems and variational inequality problems,” Journal of Computational Analysis and Applications, vol. 13, no. 2, pp. 345–361, 2011. View at Google Scholar
  43. 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, Article ID 217407, 27 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  44. W. Chantarangsi, C. Jaiboon, and P. Kumam, “A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 390972, 39 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. L. Yu and M. Liang, “Convergence of iterative sequences for fixed point and variational inclusion problems,” Fixed Point Theory and Applications, vol. 2011, Article ID 368137, 15 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  46. H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  47. P. L. Combettes, “Hilbertian convex feasibility problem: convergence of projection methods,” Applied Mathematics and Optimization, vol. 35, no. 3, pp. 311–330, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  48. F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  49. 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
  50. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH
  51. 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, 26 pages, 2011. View at Publisher · View at Google Scholar
  52. R. Wangkeeree, N. Petrot, P. Kumam, and C. Jaiboon, “Convergence theorem for mixed equilibrium problems and variational inequality problems for relaxed cocoercive mappings,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 425–449, 2011. View at Google Scholar · View at Zentralblatt MATH
  53. A. Kangtunyakarn, “A new iterative algorithm for the set of fixed-point problems of nonexpansive mappings and the set of equilibrium problem and variational inequality problem,” Abstract and Applied Analysis, vol. 2011, Article ID 562689, 24 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  54. 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
  55. W. Takahashi, Nonlinear Functional analysis, Yokohama Publishers, Yokohama, Japan, 2000.
  56. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “On a strongly nonexpansive sequence in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 471–489, 2007. View at Google Scholar · View at Zentralblatt MATH
  57. 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
  58. L. C. Ceng, Q. H. Ansari, and S. Schaible, “Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems,” Journal of Global Optimization. In press. View at Publisher · View at Google Scholar
  59. 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. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1922–1937, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  60. J. T. Oden, Qualitative Methods on Nonlinear Mechanics, Prentice Hall, Englewood Cliffs, NJ, USA, 1986.
  61. 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 Zentralblatt MATH
  62. J. S. Jung, “Iterative algorithms with some control conditions for quadratic optimizations,” Panamerican Mathematical Journal, vol. 16, no. 4, pp. 13–25, 2006. View at Google Scholar · View at Zentralblatt MATH
  63. 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
  64. L. Yu and M. Liang, “Convergence of iterative sequences for fixed point and variational inclusion problems,” Fixed Point Theory and Applications, Article ID 368137, 15 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  65. R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970. View at Google Scholar · View at Zentralblatt MATH