## 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.

### Citations:

Zbl 1141.47040; Zbl 1013.49005
Full Text:

### References:

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