A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings.

*(English)*Zbl 1206.47077Summary: The purpose of this paper is to consider a shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings, and the set of solutions of variational inclusion problems. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng et al. (2008), Takahashi et al. (2008), S. Takahashi and W. Takahashi (2008), and many others.

##### MSC:

47J25 | Iterative procedures involving nonlinear operators |

47J22 | Variational and other types of inclusions |

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

##### Keywords:

shrinking projection method; quasi-nonexpansive mappings; strong convergence; real Hilbert space
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\textit{W. Kumam} et al., J. Inequal. Appl. 2010, Article ID 458247, 25 p. (2010; Zbl 1206.47077)

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