Operator equations in ordered spaces and first applications
S. Carl, Seppo Heikkila · 2000
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.