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Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces. (English) Zbl 1258.90083
Summary: We introduce two iterative schemes (one implicit and one explicit one) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of {\it S. Li, L. Li} and {\it Y. Su} [Nonlinear Anal., Theory Methods Appl. 70, No. 9, A, 3065--3071 (2009; Zbl 1177.47075)] and {\it F. Cianciaruso, G. Marino} and {\it L. Muglia} [J. Optim. Theory Appl. 146, No. 2, 491--509 (2010; Zbl 1210.47080)] and many others.

90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
Full Text: DOI
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