Kuratowski's Measure of Noncompactness with Respect to Thompson's Metric

Abstract

It is known that the interior of a normal cone $K$ in a Banach space is a complete metric space with respect to Thompson's metric $d$. We prove that Kuratowski's measure of noncompactness $\tau$ in $(K°;d)$ has the Mazur-Darbo property and that, as a consequence, an analog of Darbo-Sadovskii's fixed point theorem is valid in $(K°;d)$. We show that the properties of $\tau$ partly differ to the classical case. Among others $\tau$ is nicely compatible with the multiplication in ordered Banach algebras.