An intermediate value theorem for monotone operators in ordered Banach spaces

Abstract

We consider a monotone increasing operator in an ordered Banach space having u- and u+ as a strong super- and subsolution, respectively. In contrast with the well-studied case u+ < u -, we suppose that u- < u+. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval [u-, u+]. © 2012 Kostrykin and Oleynik; licensee Springer.