Open Access Research

An intermediate value theorem for monotone operators in ordered Banach spaces

Vadim Kostrykin1* and Anna Oleynik12

Author Affiliations

1 FB 08 - Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, Mainz, D-55099, Germany

2 Department of Mathematics, University of Uppsala, P.O. Box 480, Uppsala, S-75106, Sweden

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Fixed Point Theory and Applications 2012, 2012:211  doi:10.1186/1687-1812-2012-211

Published: 22 November 2012

Abstract

We consider a monotone increasing operator in an ordered Banach space having <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M1">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M2">View MathML</a> as a strong super- and subsolution, respectively. In contrast with the well-studied case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M3">View MathML</a>, we suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M4">View MathML</a>. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5">View MathML</a>.

MSC: 47H05, 47H10, 46B40.

Keywords:
fixed point theorems in ordered Banach spaces