Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces

Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces

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Article ID: iaor201110507
Volume: 51
Issue: 4
Start Page Number: 689
End Page Number: 714
Publication Date: Dec 2011
Journal: Journal of Global Optimization
Authors: ,
Keywords: programming: multiple criteria
Abstract:

In this paper, we introduce two iterative schemes (one implicit and one explicit) 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 Li et al. (2009), Cianciaruso et al. (2010) and many others.

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