Existence and stability for a generalized differential mixed quasi-variational inequality

In the present paper, we investigate a generalized differential mixed quasi-variational inequality consisting of a system of an ordinary differential equation and a generalized mixed quasi-variational inequality. By using an important result concerning the measurable selection, we prove the existence of Carathèodory weak solution to the generalized differential mixed quasi-variational inequality. Then, with the existence result, we establish two stability results for the generalized differential mixed quasi-variational inequality under different conditions, i.e., upper semicontinuity and lower semicontinuity of the Carathèodory weak solution with respect to the parameter, which is a perturbation of some mappings in the generalized mixed quasi-variational inequality.