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**Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems.**
*(English)*
Zbl 1198.47081

Summary: The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points \(F(S)\) of a nonexpansive mapping \(S\) and the set of solutions \(\Omega_{A }\) of the variational inequality for a monotone, Lipschitz continuous mapping \(A\). We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of \({F(S)\cap\Omega_{A}}\). As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

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

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

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\textit{L.-C. Ceng} et al., J. Glob. Optim. 46, No. 4, 635--646 (2010; Zbl 1198.47081)

### References:

[1] | Antipin A.S.: Methods for solving variational inequalities with related constraints. Comput. Math. Math. Phys. 40, 1239–1254 (2000) · Zbl 0999.65055 |

[2] | Antipin A.S., Vasiliev F.P.: Regularized prediction method for solving variational inequalities with an inexactly given set. Comput. Math. Math. Phys. 44, 750–758 (2004) |

[3] | Browder F.E.: Existence of periodic solutions for nonlinear equations of evolution. Proc. Nat. Acad. Sc. USA 55, 1100–1103 (1965) · Zbl 0135.17601 |

[4] | 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 |

[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] | Ceng L.C., Yao J.C.: On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators. J. Comput. Appl. Math. 217, 326–338 (2007) · Zbl 1144.47048 |

[7] | Geobel K., Kirk W.A.: Topics on Metric Fixed-point Theory. Cambridge University Press, Cambridge, England (1990) |

[8] | He B.-S., Yang Z.-H., Yuan X.-M.: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362–374 (2004) · Zbl 1068.65087 |

[9] | Hu S., Papageorgiou N.S.: Handbook of multivalued analysis, vol. I: theory. Kluwer Academic Publishers, Dordrecht (1997) · Zbl 0887.47001 |

[10] | Iiduka H., Takahashi W.: Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications. Adv. Nonlinear Var. Inequal. 9, 1–10 (2006) · Zbl 1090.49011 |

[11] | Korpelevich G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976) · Zbl 0342.90044 |

[12] | 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 |

[13] | 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 |

[14] | Nadezhkina N., Takahashi W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006) · Zbl 1143.47047 |

[15] | Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003) · Zbl 1035.47048 |

[16] | Opial Z.: Weak convergence of the sequence of successive approximations for nonlinear mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967) · Zbl 0179.19902 |

[17] | Solodov M.V., Svaiter B.F.: An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions. Math. Oper. Res. 25, 214–230 (2000) · Zbl 0980.90097 |

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

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

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