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Zeitschrift für Analysis und ihre Anwendungen


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Volume 33, Issue 3, 2014, pp. 335–346
DOI: 10.4171/ZAA/1515

Published online: 2014-07-02

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

Gerd Herzog[1] and Peer Christian Kunstmann[2]

(1) Karlsruher Institut für Technologie (KIT), Germany
(2) Karlsruher Institut für Technologie (KIT), Germany

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 xed point theorem is valid in $(K°; d)$. We show that the properties of $\tau$ partly diff er to the classical case. Among others $\tau$ is nicely compatible with the multiplication in ordered Banach algebras.

Keywords: Ordered Banach spaces, Thompson metric, measure of noncompactness, fixed points

Herzog Gerd, Kunstmann Peer Christian: Kuratowski's Measure of Noncompactness with Respect to Thompson's Metric. Z. Anal. Anwend. 33 (2014), 335-346. doi: 10.4171/ZAA/1515