  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 element 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. In contrast to prior results, our analysis requires neither a
Dirichlet space structure, nor restrictive assumptions on the mapping behavior
and regularity of the involved operators, nor sign conditions on the directions
that are considered in the directional derivatives. Our approach further covers
the elliptic and 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 partial differential equation.
