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

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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).

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Correspondence to G. Marino.

Additional information

Communicated by F. Giannessi.

The authors thank the anonymous referees for their comments and suggestions which improved the presentation of this paper.

Research supported by Ministero dell’Universitá e della Ricerca of Italy.

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Cianciaruso, F., Marino, G. & Muglia, L. Iterative Methods for Equilibrium and Fixed Point Problems for Nonexpansive Semigroups in Hilbert Spaces. J Optim Theory Appl 146, 491–509 (2010). https://doi.org/10.1007/s10957-009-9628-y

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