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**Projection methods, algorithms, and a new system of nonlinear variational inequalities.**
*(English)*
Zbl 0995.47042

Let \(H\) be a real Hilbert space and \(T: K \to H\) a strongly monotone and Lipschitz continuous mapping from a closed convex subset \(K \subset H\) into \(H\). The paper is concerned with the problem of finding elements \(x^*, y^* \in K\) such that \(x^*=P_K [y^*-\rho T(y^*)], y^*=P_K [x^*-\gamma T(x^*)]\), where \(\rho, \gamma >0\) and \(P_K\) is the projection of \(H\) onto \(K\). To solve the problem the author proposes and studies the following iterative algorithm: \(x^{k+1}=(1-a^k)x^k+ a^k P_K [y^k-\rho T(y^k)], y^k=P_K [x^k-\gamma T(x^k)]\), where \(0 \leq a^k <1\) and \(\sum_{k=0}^{\infty} a^k =\infty\). The strong convergence of \(\{ x^k\}\) to \(x^*\) is established provided that \(\rho\) and \(\gamma\) are sufficiently small.

Reviewer: Mikhail Yu.Kokurin (Yoshkar-Ola)

### MSC:

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

47J25 | Iterative procedures involving nonlinear operators |

49J40 | Variational inequalities |

### Keywords:

system of nonlinear variational inequalities; system of complementarity problems; iterative algorithms; projection methods; strongly monotone and Lipschitz continuous mapping
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\textit{R. U. Verma}, Comput. Math. Appl. 41, No. 7--8, 1025--1031 (2001; Zbl 0995.47042)

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### References:

[1] | Noor, M.A., An implicit method for mixed variational inequalities, Appl. math. lett., 11, 4, 109-113, (1998) · Zbl 0941.49005 |

[2] | Noor, M.A., An extragradient method for general monotone variational inequalities, Adv. nonlinear var. inequal., 2, 1, 25-31, (1999) · Zbl 1007.49507 |

[3] | Verma, R.U., A class of projection-contraction methods applied to monotone variational inequalities, Appl. math. lett., 13, 8, 55-62, (2000) · Zbl 0988.47041 |

[4] | Verma, R.U., Nonlinear variational and constrained hemivariational inequalities involving relaxed operators, Zamm, 77, 5, 387-391, (1997) · Zbl 0886.49006 |

[5] | Baiocchi, C.; Capelo, A., Variational and quasivariational inequalities, (1984), Wiley & Sons New York · Zbl 0551.49007 |

[6] | Chan, D.; Pang, J.S., Iterative methods for variational and complementarity problems, Math. programming, 24, 284-313, (1982) · Zbl 0499.90074 |

[7] | Ding, X.P., A new class of generalized strongly nonlinear quasivariational inequalities and quasicomplementarity problems, Indian J. pure appl. math., 25, 11, 1115-1128, (1994) · Zbl 0821.49013 |

[8] | Dunn, J.C., Convexity, monotonicity and gradient processes in Hilbert spaces, J. math. anal. appl., 53, 145-158, (1976) · Zbl 0321.49025 |

[9] | Guo, J.S.; Yao, J.C., Extension of strongly nonlinear quasivariational inequalities, Appl. math. lett., 5, 3, 35-38, (1992) · Zbl 0778.49009 |

[10] | He, B.S., A projection and contraction method for a class of linear complementarity problems and its applications, Applied math. optim., 25, 247-262, (1992) · Zbl 0767.90086 |

[11] | He, B.S., A new method for a class of linear variational inequalities, Math. programming, 66, 137-144, (1994) · Zbl 0813.49009 |

[12] | He, B.S., Solving a class of linear projection equations, Numer. math., 68, 71-80, (1994) · Zbl 0822.65040 |

[13] | He, B.S., A class of projection and contraction methods for monotone variational inequalities, Applied math. optim., 35, 69-76, (1997) · Zbl 0865.90119 |

[14] | Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities, (1980), Academic Press New York · Zbl 0457.35001 |

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

[16] | Zeidler, E., Nonlinear functional analysis and its applications I, (1986), Springer-Verlag New York |

[17] | Marcotte, P.; Wu, J.H., On the convergence of projection methods, J. optim. theory appl., 85, 347-362, (1995) · Zbl 0831.90104 |

[18] | R.U. Verma, A class of iterative algorithms and solvability of nonlinear inequalities involving multivalued mappings, J. of Computational Analysis and Applications (to appear). · Zbl 1094.49503 |

[19] | R.U. Verma, An extension of a class of iterative procedures for nonlinear variational inequalities, J. of Computational Analysis and Applications (to appear). · Zbl 1033.65052 |

[20] | Verma, R.U., A class of quasivariational inequalities involving cocoercive mappings, Adv. nonlinear var. inequal., 2, 2, 1-12, (1999) · Zbl 1007.49512 |

[21] | Verma, R.U., An extension of a class of nonlinear quasivariational inequality problems based on a projection method, Math. sci. res. hot-line, 3, 5, 1-10, (1999) · Zbl 0954.49008 |

[22] | Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math. (basel), 58, 486-491, (1992) · Zbl 0797.47036 |

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