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Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space. (English) Zbl 1223.47083
Summary: In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for monotone, Lipschitz-continuous mappings. The iterative process is based on the so-called extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parametric controlling conditions. This main theorem extends a recent result of Y.-H. Yao, Y.-C. Liou and J.-C. Yao [J. Inequal. Appl. 2007, Article ID 38752 (2007; Zbl 1137.47057)] and many others.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
49J40Variational methods including variational inequalities
91B50General equilibrium theory in economics