DOI QR코드

DOI QR Code

WEAK AND STRONG CONVERGENCE THEOREMS FOR A SYSTEM OF MIXED EQUILIBRIUM PROBLEMS AND A NONEXPANSIVE MAPPING IN HILBERT SPACES

  • Plubtieng, Somyot ;
  • Sombut, Kamonrat
  • Received : 2010.06.07
  • Published : 2013.03.31

Abstract

In this paper, we introduce an iterative sequence for finding solution of a system of mixed equilibrium problems and the set of fixed points of a nonexpansive mapping in Hilbert spaces. Then, the weak and strong convergence theorems are proved under some parameters controlling conditions. Moreover, we apply our result to fixed point problems, system of equilibrium problems, general system of variational inequalities, mixed equilibrium problem, equilibrium problem and variational inequality.

Keywords

nonexpansive mapping;system of mixed equilibrium problems;fixed point;weak convergence;strong convergence

References

  1. F. Acker and M. A. Prestel, Convergence d'un schema de minimisation alternee, Ann. Fac. Sci. Toulouse Math. (5) 2 (1980), no. 1, 1-9. https://doi.org/10.5802/afst.541
  2. R. P. Agarwal, Y. J. Cho, and N. Petrot, Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces, Fixed Point Theory Appl. 2011 (2011):31, 10 pp; doi:10.1186/1687-1812-2011-31. https://doi.org/10.1186/1687-1812-2011-31
  3. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123-145.
  4. L. C. Ceng, C. Y. Wang, and J. C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. 67 (2008), no. 3, 375-390. https://doi.org/10.1007/s00186-007-0207-4
  5. L. C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), no. 1, 186-201. https://doi.org/10.1016/j.cam.2007.02.022
  6. 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, Comput. Math. Appl. 60 (2010), no. 8, 2292-2301. https://doi.org/10.1016/j.camwa.2010.08.021
  7. Y. J. Cho and N. Petrot, An optimization problem related to generalized equilibrium and fixed point problems with applications, Fixed Point Theory 11 (2010), no. 2, 237-250.
  8. Y. J. Cho and N. Petrot, On the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces, J. Inequal. Appl. 2010 (2010), Article ID 437976, 12 pp.; doi:10.1155/2010/437976. https://doi.org/10.1155/2010/437976
  9. Y. J. Cho, J. I. Kang, and X. Qin, Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems, Nonlinear Anal. 71 (2009), no. 9, 4203-4214. https://doi.org/10.1016/j.na.2009.02.106
  10. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), no. 1, 117-136.
  11. S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming 78 (1997), no. 1, 29-41. https://doi.org/10.1016/S0025-5610(96)00071-8
  12. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., vol. 28, Cambridge Univ. Press, 1990.
  13. 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, J. Comput. Appl. Math. 235 (2011), no. 14, 4128-4139. https://doi.org/10.1016/j.cam.2011.03.003
  14. M. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  15. A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Nonlinear Convex Anal. 9 (2008), no. 1, 37-43.
  16. A. Moudafi, From alternating minimization algorithms and system of variational in-equalities to equilibrium problems, Comm. Appl. Nonlinear Anal. 16 (2009), no. 3, 31-35.
  17. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpan-sive mappings, Bull. Amer. Math. Soc. 73 (1967), 595-597.
  18. M. O. Osilike and D. I. Igbokwe, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl. 40 (2000), no. 4-5, 559-567. https://doi.org/10.1016/S0898-1221(00)00179-6
  19. 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 J. Math. 12 (2008), no. 6, 1401-1432.
  20. S. Plubtieng and K. Sombut, Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space, J. Inequal. Appl. 2010 (2010), Art. ID 246237, 12 pp.
  21. S. Plubtieng and T. Thammathiwat, A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities, J. Global Optim. 46 (2010), no. 3, 447-464. https://doi.org/10.1007/s10898-009-9448-5
  22. X. Qin, S. S. Chang, and Y. J. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2963-2972. https://doi.org/10.1016/j.nonrwa.2009.10.017
  23. R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), no. 5, 877-898. https://doi.org/10.1137/0314056
  24. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
  25. W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, 2009.
  26. W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), no. 2, 417-428. https://doi.org/10.1023/A:1025407607560
  27. S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), no. 1, 506-515. https://doi.org/10.1016/j.jmaa.2006.08.036
  28. R. U. Verma, Projection methods, algorithms, and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41 (2001), no. 7-8, 1025-1031. https://doi.org/10.1016/S0898-1221(00)00336-9
  29. H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), no. 1, 279-291. https://doi.org/10.1016/j.jmaa.2004.04.059
  30. Y. Yao, Y. C. Liou, and J. C. Yao, A new hybrid Iterative algorithm for fixed point problems, variational inequality problems, and mixed equilibrium problems, Fixed Point Theory Appl. 2008 (2008), Art. ID 417089, 15 pp.; doi:10.1155/2008/417089. https://doi.org/10.1155/2008/417089
  31. Y. Yao, Y. J. Cho, and Y. C. Liou, Iterative algorithms for variational inclusions, mixed equilibrium and fixed point problems with application to optimization problems, Cent. Eur. J. Math. 9 (2011), no. 3, 640-656. https://doi.org/10.2478/s11533-011-0021-3
  32. Y. Yao, Y. J. Cho, and Y. C. Liou, Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems, European J. Oper. Res. 212 (2011), no. 2, 242-250. https://doi.org/10.1016/j.ejor.2011.01.042