### 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 |

### Fingerprint

### Keywords

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

### Cite this

}

*Optimization*, vol. 64, no. 2, pp. 429-451. https://doi.org/10.1080/02331934.2012.759327

**On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space.** / Vuong, Phan Tu; Strodiot, Jean Jacques; Nguyen, Van Hien.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

AU - Vuong, Phan Tu

AU - Strodiot, Jean Jacques

AU - Nguyen, Van Hien

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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.

AB - 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.

KW - Armijo-backtracking linesearch

KW - demicontractive mapping

KW - equilibrium problem

KW - extragradient method

KW - fixed point problem

KW - Lipschitz continuity

KW - viscosity approximation method

UR - http://www.scopus.com/inward/record.url?scp=84921357988&partnerID=8YFLogxK

U2 - 10.1080/02331934.2012.759327

DO - 10.1080/02331934.2012.759327

M3 - Article

VL - 64

SP - 429

EP - 451

JO - Optimization

JF - Optimization

SN - 0233-1934

IS - 2

ER -