Generalized nonlinear mixed quasi-variational inequalities.

*(English)*Zbl 0960.47036The problem is studied to find in a Hilbert space $u$, $x\in Su$, $y\in Tu$, and $z\in Gu$ such that the inclusion $0\in N(x,y)+M\left(p\right(u),z)$ holds. Here, $M(\xb7,z)$ is assumed to be maximal monotone where the resolvent ${(I+\rho M(\xb7,z))}^{-1}$ 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.

Reviewer: Martin Väth (Würzburg)

##### 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 |