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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
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