chapter  1
14 Pages

Operator equations in ordered spaces and first applications

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.