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A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings. (English) Zbl 1221.49011
Summary: We investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for $k$-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of P. Kumam [Turk. J. Math. 33, No. 1, 85–98 (2009; Zbl 1223.47083)], R. Wangkeeree [Fixed Point Theory Appl. 2008, Article ID 134148 (2008; Zbl 1170.47051], Y. Yao, Y. C. Liou andR. Chen [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 5–6, A, 1644–1654 (2008; Zbl 1223.47105)], X. Qin, M. Shang adn Y. Su [Nonlinear Anal., Theory Methods Appl. 69, No. 11, A, 3897–3909 (2008; Zbl 1170.47044)], and many others.

##### MSC:
 49J40 Variational methods including variational inequalities 47J20 Inequalities involving nonlinear operators 47H09 Mappings defined by “shrinking” properties 47J25 Iterative procedures (nonlinear operator equations) 49L25 Viscosity solutions (infinite-dimensional problems)