Vol. 20(2019) No. 1

 

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  The fixed point property for closed neighborhoods of line segments in Lp
 
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Fixed Point Theory, Volume 20, No. 1, 2019, 299-322, February 1st, 2019

DOI: 10.24193/fpt-ro.2019.1.20

Authors: Bernd S.W. Schröder

Abstract: We prove that, in Lp-spaces with p ∈ (1, ∞], closed neighborhoods of line segments are dismantlable and hence every monotone operator on these neighborhoods has a fixed point. We also give an example that, for p = 1, closed neighborhoods of line segments need not be dismantlable. It is an open question whether every monotone self map of a closed neighborhood of a line segment in L1 has a fixed point.

Key Words and Phrases: Dismantlable ordered set, fixed point property, line segment, closed Lp-neighborhood.

2010 Mathematics Subject Classification: 06A07, 46B42, 47H07, 47H10.

Published on-line: February 1st, 2019.

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