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**Generalized variational inequalities and associated nonlinear equations.**
*(English)*
Zbl 0953.49011

A generalized variational inequality involves two Lipschitz continuous operators. The first one is assumed to be also strongly monotone and the second one is opposite to a monotone operator. The inequality is shown to be equivalent with a nonlinear equation, which can be solved by a natural iterative algorithm of successive approximations. Some sufficient conditions are derived for the convergence of the algorithm to a solution of the variational inequality. The problem is treated on an abstract level in a real Hilbert space.

Reviewer: I.Hlaváček (Praha)

### MSC:

49J40 | Variational inequalities |

65J15 | Numerical solutions to equations with nonlinear operators |

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

### References:

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[2] | Z. Naniewicz and P. D. Panagiotopoulos: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York, 1995. · Zbl 0968.49008 |

[3] | T. L. Saaty: Modern Nonlinear Equations. Dover Publications, New York, 1981. · Zbl 0546.00003 |

[4] | R. U. Verma: An Iterative Procedure for a Random Fixed Point Theorem Involving the Theory of a Numerical Range. PanAmer. Math. J. 5(3) (1995), 71-75. · Zbl 0837.47045 |

[5] | R. U. Verma: Demiregular Convergence and the Theory of Numerical Ranges. J. Math. Anal. Appl. 193 (1995), 484-489. · Zbl 0829.47053 · doi:10.1006/jmaa.1995.1248 |

[6] | E. Zeidler: Nonlinear Functional Analysis and its Applications II/B. Springer-Verlag, New York, 1990. · Zbl 0684.47029 |

[7] | J.-C. Yao: Applications of Variational Inequalities to Nonlinear Analysis. Appl. Math. Lett. 4(4) (1991), 89-92. · Zbl 0734.49003 · doi:10.1016/0893-9659(91)90062-Z |

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