On evolution quasi-variational inequalities and implicit state-dependent sweeping processes

In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint \begin{document}$ {C(\cdot, u)} $\end{document} depends upon the unknown state \begin{document}$ u $\end{document} , which causes one of the main difficulties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a fixed point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set \begin{document}$ C $\end{document} does not depend explicitly on the state \begin{document}$ u $\end{document} (i.e. \begin{document}$ C: = C(t) $\end{document} ) given in [ 4 ], we prove a new existence result of solutions of the quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.