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

Author

Listed:
  • F. Cianciaruso

    (Universitá della Calabria)

  • G. Marino

    (Universitá della Calabria)

  • L. Muglia

    (Universitá della Calabria)

Abstract

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).

Suggested Citation

  • F. Cianciaruso & G. Marino & L. Muglia, 2010. "Iterative Methods for Equilibrium and Fixed Point Problems for Nonexpansive Semigroups in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 491-509, August.
    • Handle: RePEc:spr:joptap:v:146:y:2010:i:2:d:10.1007_s10957-009-9628-y
    DOI: 10.1007/s10957-009-9628-y
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    References listed on IDEAS

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    1. H.K. Xu, 2003. "An Iterative Approach to Quadratic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 116(3), pages 659-678, March.
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    Cited by:

    1. Mohammad Eslamian & Ali Abkar, 2014. "Viscosity iterative scheme for generalized mixed equilibrium problems and nonexpansive semigroups," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 554-570, July.
    2. Uthai Kamraksa & Rabian Wangkeeree, 2011. "Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces," Journal of Global Optimization, Springer, vol. 51(4), pages 689-714, December.
    3. Yekini Shehu, 2013. "Strong convergence theorem for nonexpansive semigroups and systems of equilibrium problems," Journal of Global Optimization, Springer, vol. 56(4), pages 1675-1688, August.

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