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**Existence and algorithm of solutions for generalized mixed implicit quasi-variational inequalities.**
*(English)*
Zbl 1020.49015

Summary: In this paper, by applying the auxiliary variational principle technique, some existence theorems of solutions for a class of generalized mixed implicit quasivariational inequalities are proved in Hilbert spaces. A novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed. The convergence criteria are also given. As special cases of these results, an open problem put forward by Noor is answered positively and some results of existence and an algorithm for solutions for generalized implicit quasi-complementarity problems are also obtained. Our results are new and generalize a number of known results of mixed variational inequalities, mixed quasivariational inequalities and quasi-complementary problems involving single-valued and set-valued mappings in recent literature.

### MSC:

49J40 | Variational inequalities |

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

47H10 | Fixed-point theorems |

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

47J25 | Iterative procedures involving nonlinear operators |

65K10 | Numerical optimization and variational techniques |

### Keywords:

auxiliary variational principle technique; existence theorems; generalized mixed implicit quasivariational inequalities; Hilbert spaces; iterative algorithm; implicit quasi-complementarity problems
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\textit{X. P. Ding}, Appl. Math. Comput. 113, No. 1, 67--80 (2000; Zbl 1020.49015)

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### References:

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