An intermediate value theorem in ordered Banach spaces

TL;DR

This paper proves an intermediate value theorem for monotone operators in ordered Banach spaces with a normal, minihedral cone, establishing the existence of fixed points between a super- and subsolution.

Contribution

It extends the intermediate value theorem to cases where the super- and subsolutions are ordered with $u_- < u_+$, under specific cone conditions.

Findings

Existence of fixed points in ordered intervals under new conditions

Extension of intermediate value theorem to ordered Banach spaces

Applicable to monotone increasing operators with specific boundary conditions

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 $[u_-,u_+].$