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Zbl 1175.49012
Wangkeeree, Rabian; Wangkeeree, Rattanaporn
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)
[J] Nonlinear Anal., Hybrid Syst. 3, No. 4, 719-733 (2009). ISSN 1751-570X

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 2000:
*49J40 Variational methods including variational inequalities
47H09 Mappings defined by "shrinking" properties
47N10 Appl. of operator theory in optimization, math. programming, etc.

Keywords: nonexpansive mapping; monotone $L$-Lipschitz continuous mapping; variational inequality; mixed equilibrium problem; fixed point; Hilbert space

Citations: Zbl 1170.47051; Zbl 1221.49011

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