An intermediate value theorem 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 ordered interval $[u_{-},u_{+}].$