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**Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method.**
*(English)*
Zbl 1197.49008

Summary: In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters.

### MSC:

49J40 | Variational inequalities |

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

47H10 | Fixed-point theorems |

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

65J15 | Numerical solutions to equations with nonlinear operators |

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

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

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\textit{Y. Yao} et al., Comput. Math. Appl. 59, No. 11, 3472--3480 (2010; Zbl 1197.49008)

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### References:

[1] | Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20, 197-228 (1967) · Zbl 0153.45701 |

[2] | Liu, F.; Nashed, M. Z., Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6, 313-344 (1998) · Zbl 0924.49009 |

[3] | Yao, J. C., Variational inequalities with generalized monotone operators, Math. Oper. Res., 19, 691-705 (1994) · Zbl 0813.49010 |

[4] | Zeng, L. C.; Schaible, S.; Yao, J. C., Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl., 124, 725-738 (2005) · Zbl 1067.49007 |

[5] | Ceng, L. C.; Yao, J. C., An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput., 190, 205-215 (2007) · Zbl 1124.65056 |

[6] | Noor, M. Aslam, Some developments in general variational inequalities, Appl. Math. Comput., 191-277 (2004) · Zbl 1134.49304 |

[7] | Censor, Y.; Iusem, A. N.; Zenios, S. A., An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Program., 81, 373-400 (1998) · Zbl 0919.90123 |

[8] | Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118, 417-428 (2003) · Zbl 1055.47052 |

[9] | Nadezhkina, N.; Takahashi, W., Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128, 191-201 (2006) · Zbl 1130.90055 |

[10] | Zeng, L. C.; Yao, J. C., Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math., 10, 1293-1303 (2006) · Zbl 1110.49013 |

[11] | Korpelevich, G. M., An extragradient method for finding saddle points and for other problems, Ekonom. i Mat. Metody, 12, 747-756 (1976) · Zbl 0342.90044 |

[12] | Yao, Y.; Yao, J. C., On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., 186, 1551-1558 (2007) · Zbl 1121.65064 |

[13] | Verma, R. U., On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line, 3, 65-68 (1999) · Zbl 0970.49011 |

[14] | Verma, R. U., Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal., 4, 117-124 (2001) · Zbl 1014.47050 |

[15] | Ceng, L. C.; Wang, C.; Yao, J. C., Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res., 67, 375-390 (2008) · Zbl 1147.49007 |

[16] | Acedo, G. L.; Xu, H. K., Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67, 2258-2271 (2007) · Zbl 1133.47050 |

[17] | Suzuki, T., Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305, 227-239 (2005) · Zbl 1068.47085 |

[18] | Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060 |

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