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Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings. (English) Zbl 1175.49012
Summary: We introduce an iterative scheme based on the extragradient approximation method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of the variational inequality problem for a monotone $L$-Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Applications to optimization problems are given. The results obtained in this paper improve and extend the recent ones announced by {\it R. Wangkeeree} [Fixed Point Theory Appl. 2008, Article ID 134148, 17 p. (2008; Zbl 1170.47051)], {\it P. Kumam} and {\it P. Katchang} [Nonlinear Anal., Hybrid Syst. 3, No. 4, 475--486 (2009; Zbl 1221.49011)] and many others.

##### MSC:
 49J40 Variational methods including variational inequalities 47H09 Mappings defined by “shrinking” properties 47N10 Applications of operator theory in optimization, convex analysis, programming, economics
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