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**Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces.**
*(English)*
Zbl 1195.47044

Summary: The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of the variational inequality for an \(\alpha \)-inverse-strongly monotone mapping, and the set of solutions of the generalized equilibrium problem 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. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well-known strong convergence results in the literature.

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

49J40 | Variational inequalities |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

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\textit{S. Saewan} et al., Abstr. Appl. Anal. 2010, Article ID 734126, 25 p. (2010; Zbl 1195.47044)

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