Applications of Fell Topology to Closed Set-Valued Optimizations in Partially Ordered First Countable Topological Vector Spaces

Abstract

Let (X, $\tau$) be a topological vector space equipped with a partial order $\preccurlyeq ,$ which is induced by a nonempty pointed closed and convex cone K in X. Let $\mathcal{P}$(X) = 2^X be the power set of X and ${\mathcal{P}}_0$(X) = 2^X\{$\mathrm{Ø}$}. In this paper, we consider the so called upward preordering ${\preccurlyeq }^u$ on $\mathcal{P}$(X), which has been used by many authors (see [4$-$7, 9, 11$-$14, 18$-$19]). We first prove some properties of this ordering relation ${\preccurlyeq }^u.$ Let $\mathcal{C}$(X) denote the collection of all $\tau$-closed subsets of X and ${\mathcal{C}}_0$(X) = $\mathcal{C}$(X)\{$\mathrm{Ø}$}, which is equipped with the Fell topology ${\tau }_F.$ Let C be a nonempty subset of X and let F: C $\to {\mathcal{C}}_0$(X) be a closed set-valued mapping. By applying the Fell topology ${\tau }_F$ on $\mathcal{C}$(X) and the Fan-KKM Theorem, we prove some existence theorems for some ${\preccurlyeq }^u$-minimization and ${\preccurlyeq }^u$-maximization problems with respect to F subject to the subset C for first countable topological vector spaces. These results will be applied to solve some closed ball-valued optimization problems in partially ordered Hilbert spaces. Some examples will be provided in ${\mathbb{R}}^n.$

Keywords:

• Closed set-valued optimization problem
• Fell topology
• first countable topological vector space
• upward preordering relation on power sets

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 06A06
• 22A26
• 49J30
• 49J35
• 91B15
• 91B64

Acknowledgments

The author is very grateful to Professors Akhtar Khan, Robert Mendris, and Christiane Tammer for their valuable suggestions that improved the presentation of this work.