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Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems. (English) Zbl 1275.47120

Summary: Recently, V. Colao et al. [J. Math. Anal. Appl. 344, No. 1, 340–352 (2008; Zbl 1141.47040)] introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining the hybrid viscosity approximation method of [loc. cit.] and I. Yamada’s hybrid steepest-descent method [Stud. Comput. Math. 8, 473–504 (2001; Zbl 1013.49005)], we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the set \(\bigcap ^N_{i=1} \mathrm{Fix} (S_i)\) of fixed points of a finite family of nonexpansive mappings \(\{S_i\}^N_{i=1}\) in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element of \(\bigcap ^N_{i=1} \mathrm{Fix} (S_i) \cap \mathrm{GMEP}\), which is the unique solution of a variational inequality.

MSC:

47J25 Iterative procedures involving nonlinear operators
49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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