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Generalized nonlinear mixed quasi-variational inequalities. (English) Zbl 0960.47036

The problem is studied to find in a Hilbert space $u$, $x\in Su$, $y\in Tu$, and $z\in Gu$ such that the inclusion $0\in N\left(x,y\right)+M\left(p\left(u\right),z\right)$ holds. Here, $M\left(·,z\right)$ is assumed to be maximal monotone where the resolvent ${\left(I+\rho M\left(·,z\right)\right)}^{-1}$ is Lipschitz with respect to $z$; $S,T,G$ are Lipschitz with respect to the Hausdorff distance, and for $p$ and $N$ Lipschitz and monotonicity conditions are assumed (always with appropriate constants).

An iterative algorithm is suggested whose convergence to a solution is proved. Moreover, for single-valued $S,T$ and $G=I$ also stability of a perturbed algorithm is proved.

##### MSC:
 47J20 Inequalities involving nonlinear operators 47J25 Iterative procedures (nonlinear operator equations) 47H05 Monotone operators (with respect to duality) and generalizations 49J40 Variational methods including variational inequalities