Harnack type inequalities and multiple solutions in cones of nonlinear problems

The paper presents an abstract theory regarding operator equations and systems in ordered Banach spaces. We obtain existence, localization and multiplicity results of positive solutions using Krasnosel'skiï's fixed point theorem in cones, and a Harnack type inequality. Concerning systems, the localization is established by the vector version of Krasnosel'skiï's theorem, where the compression-expansion conditions are expressed on components. The approach is sufficiently general to cover and unify a large number of results on particular classes of problems. It also can guide future research in this direction.