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**Modified Noor’s extragradient method for solving generalized variational inequalities in Banach spaces.**
*(English)*
Zbl 1237.65055

Summary: Motivated and inspired by Korpelevich’s and Noor’s extragradient methods, we suggest an extragradient method by using the sunny nonexpansive retraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.

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\textit{S. Fan} et al., Abstr. Appl. Anal. 2012, Article ID 493862, 11 p. (2012; Zbl 1237.65055)

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[1] | G. Stampacchia, “Formes bilineaires coercitives sur les ensembles convexes,” Comptes Rendus de l’Academie des Sciences, vol. 258, pp. 4413-4416, 1964. · Zbl 0124.06401 |

[2] | Y. Censor, A. N. Iusem, and S. A. Zenios, “An interior point method with Bregman functions for the variational inequality problem with paramonotone operators,” Mathematical Programming A, vol. 81, no. 3, pp. 373-400, 1998. · Zbl 0919.90123 |

[3] | R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, NY, USA, 1984. · Zbl 0536.65054 |

[4] | A. N. Iusem, “An iterative algorithm for the variational inequality problem,” Computational and Applied Mathematics, vol. 13, no. 2, pp. 103-114, 1994. · Zbl 0811.65049 |

[5] | P. Jaillet, D. Lamberton, and B. Lapeyre, “Variational inequalities and the pricing of American options,” Acta Applicandae Mathematicae, vol. 21, no. 3, pp. 263-289, 1990. · Zbl 0714.90004 |

[6] | M. A. Noor, On Variational Inequalities, Brunel University, London, UK, 1975. |

[7] | M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119-122, 1988. · Zbl 0655.49005 |

[8] | M. A. Noor, “Wiener-Hopf equations and variational inequalities,” Journal of Optimization Theory and Applications, vol. 79, no. 1, pp. 197-206, 1993. · Zbl 0799.49010 |

[9] | M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217-229, 2000. · Zbl 0964.49007 |

[10] | M. A. Noor, “A class of new iterative methods for general mixed variational inequalities,” Mathematical and Computer Modelling, vol. 31, no. 13, pp. 11-19, 2000. · Zbl 0953.49016 |

[11] | M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199-277, 2004. · Zbl 1134.49304 |

[12] | M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182-186, 2009. · Zbl 1163.49303 |

[13] | M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 303569, 16 pages, 2012. · Zbl 1242.49017 |

[14] | M. A. Noor and E. A. Al-Said, “Wiener-Hopf equations technique for quasimonotone variational inequalities,” Journal of Optimization Theory and Applications, vol. 103, no. 3, pp. 705-714, 1999. · Zbl 0953.65050 |

[15] | M. Aslam Noor and Z. Huang, “Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 504-510, 2007. · Zbl 1193.49009 |

[16] | M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,” Applied Mathematics and Information Sciences, vol. 6, no. 1, pp. 65-69, 2012. · Zbl 1244.65084 |

[17] | H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185-201, 2003. · Zbl 1045.49018 |

[18] | I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8, pp. 473-504, North-Holland, Amsterdam, Holland, 2001. · Zbl 1013.49005 |

[19] | J. C. Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol. 19, no. 3, pp. 691-705, 1994. · Zbl 0813.49010 |

[20] | Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In press. · Zbl 1260.47085 |

[21] | Y. Yao, M. A. Noor, and Y.-C Liou, “Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9 pages, 2012. · Zbl 1232.49011 |

[22] | Y. Yao, M. A. Noor, and Y.-C Liou, “Iterative algorithms for general multi-valued variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 768272, 10 pages, 2012. · Zbl 1232.49012 |

[23] | E. Zeidler, Nonlinear Functional Analysis and its Applications. III: Variational Methods and Applications, Springer-Verlag, New York, NY, USA, 1985. · Zbl 0583.47051 |

[24] | G. M. Korpelevich, “An extragradient method for finding saddle points and for other problems,” Ekonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747-756, 1976. · Zbl 0342.90044 |

[25] | K. Aoyama, H. Iiduka, and W. Takahashi, “Weak convergence of an iterative sequence for accretive operators in Banach spaces,” Fixed Point Theory and Applications, vol. 2006, Article ID 35390, 13 pages, 2006. · Zbl 1128.47056 |

[26] | Y. Yao and S. Maruster, “Strong convergence of an iterative algorithm for variational inequalities in Banach spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 325-329, 2011. · Zbl 1225.65067 |

[27] | Y. Yao, Y. C. Liou, C. L. Li, and H. T. Lin, “Extended extragradient methods for generalized variational inequalities,” Journal of Applied Mathematics, vol. 2012, Article ID 237083, 14 pages, 2012. · Zbl 1235.49029 |

[28] | H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1127-1138, 1991. · Zbl 0757.46033 |

[29] | R. E. Bruck Jr., “Nonexpansive retracts of Banach spaces,” Bulletin of the American Mathematical Society, vol. 76, pp. 384-386, 1970. · Zbl 0224.47034 |

[30] | F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” in Nonlinear Functional Analysis (Proceedings of Symposia in Pure Mathematics, Vol. XVIII, Part 2, Chicago, Ill., 1968), pp. 1-308, American Mathematical Society, Providence, RI, USA, 1976. · Zbl 0327.47022 |

[31] | H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240-256, 2002. · Zbl 1013.47032 |

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