A Domain-Theoretic Bishop-Phelps theorem

TL;DR

This paper extends the Bishop-Phelps theorem to domain-theoretic and cone structures, introducing new support point concepts and proving existence results in semitopological cones and fixed point theorems.

Contribution

It introduces $c$-support points and $wd$-cones, and proves a Bishop-Phelps type theorem within this domain-theoretic framework.

Findings

Existence of $c$-support points in convex Scott closed sets

Bishop-Phelps type theorem for $wd$-cones

Fixed point results for mappings on $s$-cones

Abstract

In this paper, the notion of $c$-support points of a set in a semitopological cone is introduced. It is shown that any nonempty convex Scott closed bounded set has a $c$-support point in a cancellative $bd$-cone under certain condition. We also introduce the notion of $wd$-cone and then we prove a Bishop-Phelps type theorem for $wd$-cones, especially for normed cones, under appropriate conditions. Finally, using of the Bishop-Phelps technique, we obtain a result about the fixed points of a mapping on $s$-cones.