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Convergence theorem for mixed equilibrium problems and variational inequality problems for relaxed cocoercive mappings. (English) Zbl 1221.47130
Summary: We introduce a new iterative scheme for finding the 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 variational inequality problems for a relaxed $$(u,v)$$-cocoercive and $$\mu$$-Lipschitz continuous mapping in Hilbert spaces. We show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions. Our results improve and extend the recent ones announced by Y. J. Cho, X. Q. Qin and M. Kang [J. Comput. Anal. Appl. 11, 294–316 (2009; Zbl 1223.47075)] and many others.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems