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

F. Cianciaruso, G. Marino () and L. Muglia
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F. Cianciaruso: Universitá della Calabria
G. Marino: Universitá della Calabria
L. Muglia: Universitá della Calabria

Journal of Optimization Theory and Applications, 2010, vol. 146, issue 2, No 14, 509 pages

Abstract: 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).

Keywords: Equilibrium problem; Fixed points; Semigroup of nonexpansive mappings; Variational inequalities; Iterative algorithms (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-009-9628-y

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