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**A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixed-point problems in Banach space.**
*(English)*
Zbl 1396.47004

Summary: In this paper, the common solution problem (P1) of generalized equilibrium problems for a system of inverse-strongly monotone mappings \(\{A_k\}^N_{k=1}\) and a system of bifunctions \(\{f_k\}^N_{k=1}\) satisfying certain conditions, and the common fixed-point problem (P2) for a family of uniformly quasi-\(\varphi\)-asymptotically nonexpansive and locally uniformly Lipschitz continuous or uniformly Hölder continuous mappings \(\{S_i\}^\infty_{i=1}\) are proposed. A new iterative sequence is constructed by using the generalized projection and hybrid method, and a strong convergence theorem is proved for approximating a common solution of (P1) and (P2) in Banach space.

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

47H10 | Fixed-point theorems |

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

### Keywords:

common solution; equilibrium problem; fixed-point problem; iterative sequence; strong convergence; Banach space
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\textit{D. Qu} and \textit{C. Cheng}, Fixed Point Theory Appl. 2011, Paper No. 17, 13 p. (2011; Zbl 1396.47004)

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### References:

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