## Abstract

In this paper, new numerical algorithms are introduced for finding the solution of a variational inequality problem whose constraint set is the common elements of the set of fixed points of a demicontractive mapping and the set of solutions of an equilibrium problem for a monotone mapping in a real Hilbert space. The strong convergence of the iterates generated by these algorithms is obtained by combining a viscosity approximation method with an extragradient method. First, this is done when the basic iteration comes directly from the extragradient method, under a Lipschitz-type condition on the equilibrium function. Then, it is shown that this rather strong condition can be omitted when an Armijo-backtracking linesearch is incorporated into the extragradient iteration. The particular case of variational inequality problems is also examined.

Original language | English |
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Pages (from-to) | 429-451 |

Number of pages | 23 |

Journal | Optimization |

Volume | 64 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

## Keywords

- Armijo-backtracking linesearch
- demicontractive mapping
- equilibrium problem
- extragradient method
- fixed point problem
- Lipschitz continuity
- viscosity approximation method