Ordered Non-Convex Quasi-Variational Sweeping Processes

This paper addresses the Cauchy problem for the quasi-variational sweeping process in the ordered Hilbert space $H$ $$-u^{\prime}(t) \in N_{C(t,u(t))}(u(t)) \quad \text{for a.e. } t \in (0,T), \quad u(0)=u_0,$$ where the set $\, C(t,u(t)) \subset H \,$ is non-convex and $\, N_{C(t,u(t))} \,$ denotes its normal cone. We provide an existence result based on the classical implicit time-discretization procedure and on a fixed point argument in ordered spaces.