DOI QR코드

DOI QR Code

ITERATIVE METHODS FOR GENERALIZED EQUILIBRIUM PROBLEMS AND NONEXPANSIVE MAPPINGS

  • Received : 2010.09.28
  • Published : 2011.01.31

Abstract

In this paper, a composite iterative process is introduced for a generalized equilibrium problem and a pair of nonexpansive mappings. It is proved that the sequence generated in the purposed composite iterative process converges strongly to a common element of the solution set of a generalized equilibrium problem and of the common xed point of a pair of nonexpansive mappings.

Keywords

equilibrium problem;nonexpansive mapping;inverse-strongly monotone mapping;contractive mapping

References

  1. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123-145.
  2. F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228. https://doi.org/10.1016/0022-247X(67)90085-6
  3. L. C. Ceng and J. C. Yao, Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Appl. Math. Comput. 198 (2008), no. 2, 729-741. https://doi.org/10.1016/j.amc.2007.09.011
  4. O. Chadli, N. C. Wong and J. C. Yao, Equilibrium problems with applications to eigen-value problems, J. Optim. Theory Appl. 117 (2003), no. 2, 245-266. https://doi.org/10.1023/A:1023627606067
  5. S. S. Chang, H. W. Joseph Lee and C. K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal. 70 (2009), no. 9, 3307-3319. https://doi.org/10.1016/j.na.2008.04.035
  6. J. Chen, L. Zhang and T. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl. 334 (2007), no. 2, 1450-1461. https://doi.org/10.1016/j.jmaa.2006.12.088
  7. V. Colao, G. Marino and H. K. Xu, An iterative method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl. 344 (2008), no. 1, 340-352. https://doi.org/10.1016/j.jmaa.2008.02.041
  8. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), no. 1, 117-136.
  9. H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005), no. 3, 341-350. https://doi.org/10.1016/j.na.2003.07.023
  10. S. M. Kang, S. Y. Cho and Z. Liu, Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings, J. Inequal. Appl. 2010 (2010), Article ID 827082, 16 pages.
  11. A. Moudafi and M. Thera, Proximal and dynamical approaches to equilibrium problems, Ill-posed variational problems and regularization techniques (Trier, 1998), 187-201, Lecture Notes in Econom. and Math. Systems, 477, Springer, Berlin, 1999.
  12. X. Qin, Y. J. Cho and S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009), no. 1, 20-30. https://doi.org/10.1016/j.cam.2008.06.011
  13. Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 561-597.
  14. S. Plubtieng and R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2008), no. 2, 548-558. https://doi.org/10.1016/j.amc.2007.07.075
  15. X. Qin, M. Shang and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Modelling 48 (2008), no. 7-8, 1033-1046. https://doi.org/10.1016/j.mcm.2007.12.008
  16. T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005), no.1, 227-239. https://doi.org/10.1016/j.jmaa.2004.11.017
  17. 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
  18. S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a non-expansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008), no. 3, 1025-1033. https://doi.org/10.1016/j.na.2008.02.042
  19. H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240-256. https://doi.org/10.1112/S0024610702003332

Cited by

  1. Existence of solutions for generalized equilibrium problem in G-convex space vol.62, pp.9, 2011, https://doi.org/10.1016/j.camwa.2011.08.055