Ordered Non-Convex Quasi-Variational Sweeping Processes
Nikolai V. Chemetov, Manuel D. P. Monteiro Marques, Ulisse Stefanelli · Journal of convex analysis · 2008
This paper addresses the Cauchy problem for the quasi-variational sweeping process in the ordered Hilbert space H 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, − u ′ ( t ) ∈ N C ( t , u ( t ) ) ( u ( t ) ) for a.e. t ∈ ( 0 , T ) , u ( 0 ) = u 0 , where the set \, C(t,u(t)) \subset H \, C ( t , u ( t ) ) ⊂ H is non-convex and \, N_{C(t,u(t))} \, 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.