Schröder, Bernd S. W. The fixed point property for closed neighborhoods of line segments in \(L^p\). (English) Zbl 1491.47049 Fixed Point Theory 20, No. 1, 299-322 (2019). Summary: We prove that, in \(L^p\)-spaces with \(p \in (1, \infty]\), 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 \(L^1\) has a fixed point. MSC: 47H10 Fixed-point theorems 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 06A07 Combinatorics of partially ordered sets 46B42 Banach lattices Keywords:dismantlable ordered set; fixed point property; line segment; closed \(L^p\)-neighborhood PDFBibTeX XMLCite \textit{B. S. W. Schröder}, Fixed Point Theory 20, No. 1, 299--322 (2019; Zbl 1491.47049) Full Text: DOI