Journal of Applied Mathematics and Computing

, Volume 29, Issue 1–2, pp 263–280 | Cite as

A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping

  • Poom Kumam


In this paper, we introduce an iterative scheme by a new hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under some parametric controlling conditions by the new hybrid method which is introduced by Takahashi et al. (J. Math. Anal. Appl., doi:  10.1016/j.jmaa.2007.09.062, 2007). The results are connected with Tada and Takahashi’s result [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem, J. Optim. Theory Appl. 133, 359–370, 2007]. Moreover, our result is applicable to a wide class of mappings.


Nonexpansive mapping Monotone mapping Equilibrium problem Variational inequality 

Mathematics Subject Classification (2000)

46C05 47D03 47H09 47H10 47H20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student. 63, 123–145 (1994) MATHMathSciNetGoogle Scholar
  2. 2.
    Burachik, R.S., Lopes, J.O., Svaiter, B.F.: An outer approximation method for the variational inequality problem. SIAM J. Control Optim. 43, 2071–2088 (2005) MATHCrossRefMathSciNetGoogle 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 progamming using proximal-link algorithms. Math. Program. 78, 29–41 (1997) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Genel, A., Lindenstrass, J.: An example concerning fixed points. Isr. J. Math. 22, 81–86 (1975) MATHCrossRefGoogle Scholar
  6. 6.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990) MATHGoogle Scholar
  7. 7.
    Iiduka, H., Takahashi, W.: Strong convergence theorems for nonexpansive mapping and inverse-strong monotone mappings. Nonlinear Anal. 61, 341–350 (2005) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kirk, W.A.: Fixed point theorem for mappings which do not increase distance. Am. Math. Mon. 72, 1004–1006 (1965) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953) MATHCrossRefGoogle Scholar
  10. 10.
    Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture Note in Economics and Mathematical Systems, vol. 477, pp. 187–201. Springer, New York (1999) Google Scholar
  11. 11.
    Opial, Z.: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Reich, S.: Weak convergence theorems for nonexpansive mappings. J. Math. Anal. Appl. 67, 274–276 (1979) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Rockafellar, R.T.: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Solodov, M.V., Svaiter, B.F.: A hybrid projection–proximal point algorithm. J. Convex Anal. 6, 59–70 (1999) MATHMathSciNetGoogle Scholar
  17. 17.
    Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. A 87, 189–202 (2000) MATHMathSciNetGoogle Scholar
  18. 18.
    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 (2006) Google Scholar
  19. 19.
    Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000) MATHGoogle Scholar
  21. 21.
    Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. (2007). doi: 10.1016/j.jmaa.2007.09.062 Google Scholar
  24. 24.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) MATHCrossRefGoogle Scholar
  25. 25.
    Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Yamada, I.: The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithm for Feasibility and Optimization, pp. 473–504. Elsevier, Amsterdam (2001) Google Scholar
  27. 27.
    Yao, J.-C., Chadli, O.: Pseudomonotone complementarity problems and variational inequalities. In: Crouzeix, J.P., Haddjissas, N., Schaible, S. (eds.) Handbook of Generalized Convexity and Monotonicity, pp. 501–558 (2005) Google Scholar
  28. 28.
    Zeng, L.C., Schaible, S., Yao, J.C.: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. J. Optim. Theory Appl. 124, 725–738 (2005) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2008

Authors and Affiliations

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

Personalised recommendations