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A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping. (English) Zbl 1220.47102
Summary: In this paper, we introduce an iterative scheme by a new hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality for $$\alpha$$-inverse-strongly monotone mappings in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under some parametric controlling conditions by the new hybrid method introduced by W. Takahashi, Y. Takeuchi and R. Kubota [J. Math. Anal. Appl. 341, No. 1, 276–286 (2008; Zbl 1134.47052)]. The results are connected with Tada and Takahashi’s result [A. Tada and W. Takahashi, J. Optim. Theory Appl. 133, No. 3, 359–370 (2007; Zbl 1147.47052)]. Moreover, our result is applicable to a wide class of mappings.

MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 47H20 Semigroups of nonlinear operators
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