  In this paper, we introduce for the first time a class of state-dependent
maximal monotone differential inclusions. Then the existence and uniqueness of
solutions are obtained by using an implicit discretization scheme and a kind of
hypo-monotonicity assumption respectively. In addition, a characterization for
nonsmooth Lyapunov pairs associated with such systems is provided. Our result
can be applied to study state-dependent sweeping processes and Lur'e dynamical
systems. It is new even the involved maximal monotone operators depend only on
the time.
