LIPSCHITZ STABILITY AND HADAMARD DIRECTIONAL DIFFERENTIABILITY FOR ELLIPTIC AND PARABOLIC OBSTACLE-TYPE QUASI-VARIATIONAL INEQUALITIES.

This paper is concerned with the sensitivity analysis of a class of parameterized fixed-point problems that arise in the context of obstacle-type quasi-variational inequalities. We prove that if the operators in the considered fixed-point equation satisfy a positive superhomogeneity condition, then the maximal and minimal elements of the solution set of the problem depend locally Lipschitz continuously on the involved parameters. We further show that if certain concavity conditions hold, then the maximal solution mapping is Hadamard directionally differentiable and its directional derivatives are precisely the minimal solutions of suitably defined linearized fixed-point equations. Our approach covers the elliptic and the parabolic setting simultaneously and also yields Hadamard directional differentiability results in situations in which the solution set of the fixed-point equation is a continuum and a characterization of directional derivatives via linearized auxiliary problems is provably impossible. To illustrate that our results can be used to study interesting problems arising in practice, we apply them to establish the Hadamard directional differentiability of the solution operator of a nonlinear elliptic quasi-variational inequality, which emerges in impulse control and in which the obstacle mapping is obtained by taking essential infima over certain parts of the underlying domain, and of the solution mapping of a parabolic quasi-variational inequality, which involves boundary controls and in which the state-to-obstacle relationship is described by a PDE. [ABSTRACT FROM AUTHOR]