ABSTRACT
Operator equations in ordered
spaces and rst applications
Our basic method in treating implicit problems is to represent them as an
operator equation
Lu = Nu; (1.1.1)
where the dependence of the operator N on u is implicit, e.g., of the form
Nu = Q(u; Lu). For this purpose we study in this chapter the solvability of
(1.1.1) in the case when L and N are mappings from a partially ordered set
(poset) to an ordered normed space. No linearity or continuity hypotheses
are imposed on the operators L and N , since no algebraic or topological
structures are assumed for their domain. This general setting gives us tools
to treat also implicit problems involving discontinuous nonlinearities.
