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A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings. (English) Zbl 1179.49011
Summary: The purpose of this paper is to consider a new hybrid relaxed extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions. Our results extend and improve the recent results of G. Cai and C. S. Hu [Nonlinear Anal., Hybrid Syst. 3, No. 4, 395–407 (2009; Zbl 1223.47071)], A. Kangtunyakarn and S. Suantai [Nonlinear Anal., Theory Methods Appl. 71, No. 10 (A), 4448–4460 (2009; Zbl 1167.47304)] and S. Thianwan [Nonlinear Anal., Hybrid Syst. 3, No. 4, 605–614 (2009; Zbl 1219.49008)] and many others.
MSC:
49J40Variational methods including variational inequalities
49M30Other numerical methods in calculus of variations
47H09Mappings defined by “shrinking” properties