Journal of Optimization Theory and Applications

, Volume 146, Issue 2, pp 491–509 | Cite as

Iterative Methods for Equilibrium and Fixed Point Problems for Nonexpansive Semigroups in Hilbert Spaces

  • F. Cianciaruso
  • G. Marino
  • L. Muglia


In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup (T(s))s≥0 in Hilbert spaces. We prove that both approaches converge strongly to a common element z of the set of the equilibrium points and the set of common fixed points of (T(s))s≥0. Such common element z is the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

The results presented here belong to the area of research exemplified by Marino and Xu (J. Math. Anal. Appl. 318:43–52, 2006), Moudafi (J. Math. Anal. Appl. 241:46–55, 2000; Numer. Funct. Anal. Optim. 28(11):1347–1354, 2007), Plubtieng and Punpaeng (J. Math. Anal. Appl. 336:455–469, 2007), Takahashi and Takahashi (J. Math. Anal. Appl. 331(1):506–515, 2007), Xu (J. Optim. Theory Appl. 116:659–678, 2003).

Equilibrium problem Fixed points Semigroup of nonexpansive mappings Variational inequalities Iterative algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006) CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000) CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Moudafi, A.: On finite and strong convergence of a proximal method for equilibrium problems. Numer. Funct. Anal. Optim. 28(11), 1347–1354 (2007) CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Plubtieng, S., Punpaeng, R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455–469 (2007) CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331(1), 506–515 (2007) CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003) CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banch spaces. Arch. Ration. Mech. Anal. 24, 82–89 (1967) CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Moudafi, A., Théra, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 187–201. Springer, Berlin (1999) Google Scholar
  10. 10.
    Shimizu, T., Takahashi, W.: Strong convergence to common fixed points of families of nonexpansive mappings. J. Math. Anal. Appl. 211, 71–83 (1997) CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2, 1–17 (2002) Google Scholar
  12. 12.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997) CrossRefGoogle Scholar
  13. 13.
    Brezis, H.: Analyse Fonctionelle. Masson, Paris (1983) Google Scholar
  14. 14.
    Moudafi, A.: Krasnoselski-Mann iteration for Hierarchical fixed-point problems. Inverse Probl. 23, 1–6 (2007) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversitá della CalabriaArcavacata di RendeItaly

Personalised recommendations