DOI: 10.68381/jca15014 ISSN: 0944-6532
Ordered Non-Convex Quasi-Variational Sweeping Processes
Nikolai V. Chemetov, Manuel D. P. Monteiro Marques, Ulisse Stefanelli
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.