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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
Article

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).

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

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversitá della CalabriaArcavacata di RendeItaly

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