Hybrid algorithms for minimization problems over the solutions of generalized mixed equilibrium and variational inclusion problems.

*(English)*Zbl 1235.65071Summary: We introduce a new general hybrid iterative algorithm for finding a common element of the set of solution of fixed point for a nonexpansive mapping, the set of solution of generalized mixed equilibrium problem, and the set of solution of the variational inclusion for a \(\beta\)-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of G. Marino and H. K. Xu [J. Math. Anal. Appl. 318, No. 1, 43–52 (2006; Zbl 1095.47038)], Y.-H. Yao and Y.-C. Liou [Abstr. Appl. Anal. 2010, Article ID 763506 (2010; Zbl 1203.49048)], J. F. Tan and S. S. Chang [Fixed Point Theory Appl. 2011, Article ID 915629, 17 p. (2011; Zbl 1214.47076)] and other authors.

##### MSC:

65K15 | Numerical methods for variational inequalities and related problems |

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

47H10 | Fixed-point theorems |

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\textit{T. Jitpeera} and \textit{P. Kumam}, Math. Probl. Eng. 2011, Article ID 648617, 25 p. (2011; Zbl 1235.65071)

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