  We establish a coupled fixed points theorem for a meaningful class of mixed
monotone multivalued operators and then we use it to derive some results on
existence of quasisolutions and solutions to first--order functional
differential equations with state--dependent deviating arguments. Our results
are very general and can be applied to functional equations featuring
discontinuities with respect to all of their arguments, but we emphasize that
they are new even for differential equations with continuously state--dependent
delays.
