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**A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems.**
*(English)*
Zbl 1204.65062

Summary: We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an \(\alpha \)-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

47J25 | Iterative procedures involving nonlinear operators |

47H10 | Fixed-point theorems |

47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

47H05 | Monotone operators and generalizations |

### Keywords:

hybrid iterative scheme; common fixed points; quasi-nonexpansive mappings; variational inequality; inverse-strongly monotone operator; mixed equilibrium problem; maximal monotone operator; Banach space; convergence
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\textit{S. Saewan} and \textit{P. Kumam}, Abstr. Appl. Anal. 2010, Article ID 123027, 31 p. (2010; Zbl 1204.65062)

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