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Journal of Applied Mathematics and Computing

, Volume 32, Issue 1, pp 19–38 | Cite as

A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space

  • Phayap Katchang
  • Poom Kumam
Article

Abstract

In this paper, we introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of the variational inequality for β-inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in a Hilbert space. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. As applications, at the end of paper we utilize our results to study some convergence problem for finding the zeros of maximal monotone operators. Our results are generalizations and extensions of the results of Yao and Liou (Fixed Point Theory Appl. Article ID 384629, 10 p., 2008), Yao et al. (J. Nonlinear Convex Anal. 9(2):239–248, 2008) and Su and Li (Appl. Math. Comput. 181(1):332–341, 2006) and some recent results.

Keywords

Nonexpansive mapping Fixed point Equilibrium problem Variational inequality Viscosity approximation method 

Mathematics Subject Classification (2000)

46C05 47D03 47H09 47H10 47H20 

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References

  1. 1.
    Aslam Noor, M., Yao, Y., Liou, Y.C.: Extragradient method for equilibrium problems and variational inequalities. Alban. J. Math. 2, 125–138 (2008) MATHGoogle Scholar
  2. 2.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) MATHMathSciNetGoogle Scholar
  3. 3.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005) MATHMathSciNetGoogle Scholar
  4. 4.
    Flam, S.D., Antipin, A.S.: Equilibrium programming using proximal-link algorithms. Math. Program. 78, 29–41 (1997) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Kumam, P.: Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space. Turk. J. Math. 33, 1–19 (2008) MathSciNetGoogle Scholar
  6. 6.
    Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241(1), 46–55 (2000) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems. Lecture Note in Economics and Mathematical Systems, vol. 477, pp. 187–201. Springer, New York (1999) Google Scholar
  8. 8.
    Osilike, M.O., Igbokwe, D.I.: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput. Math. Appl. 40, 559–567 (2000) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Plubtieng, S., Punpaeng, R.: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Appl. Math. Comput. 197, 584–558 (2008) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rockafellar, R.T.: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305(1), 227–239 (2005) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Su, Y., Li, S.: Strong convergence theorems on two iterative method for non-expansive mappings. Appl. Math. Comput. 181(1), 332–341 (2006) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Su, Y., Shang, M., Qin, X.: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. (2007). doi: 10.1016/j.na.2007.08.045
  15. 15.
    Tada, A., Takahashi, W.: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In: Takahashi, W., Tanaka, T. (eds.) Nonlinear Analysis and Convex Analysis, pp. 609–617. Yokohama Publishers, Yokohama (2007) Google Scholar
  16. 16.
    Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 311, 506–515 (2007) CrossRefGoogle Scholar
  17. 17.
    Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58(5), 486–491 (1992) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Yao, Y., Liou, Y.-C.: Iterative algorithms for nonexpansive mapping. Fixed Point Theory Appl. Article ID 384629, 10 p. (2008). doi: 10.1155/2008/384629
  20. 20.
    Yao, Y., Liou, Y.-C., Chen, R.: Convergence theorems for fixed point problems and variational inequality problems. J. Nonlinear Convex Anal. 9(2), 239–248 (2008) MATHMathSciNetGoogle Scholar
  21. 21.
    Yao, Y., Liou, Y.-C., Yao, J.-C.: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. Article ID 64363, 12 p. (2007). doi: 10.1155/2007/64363

Copyright information

© Korean Society for Computational and Applied Mathematics 2009

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut’s University of Technology ThonburiBangkokThailand

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