Applications of Fell Topology to Closed Set-Valued Optimizations in Partially Ordered First Countable Topological Vector Spaces. (26th April 2023)

• Record Type: Journal Article
• Authors: Li, Jinlu
• Is Part Of: Numerical functional analysis and optimization. Volume 44:Number 6(2023)
• Page Start: 461
• Page End: 489
• Publication Date: 2023-04-26
• Subjects: Closed set-valued optimization problem -- Fell topology -- first countable topological vector space -- upward preordering relation on power sets; 06A06 -- 22A26 -- 49J30 -- 49J35 -- 91B15 -- 91B64; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2023.2180753
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: Let ( X, τ ) be a topological vector space equipped with a partial order ≼, which is induced by a nonempty pointed closed and convex cone K in X . Let P ( X ) = 2 X be the power set of X and P 0 ( X ) = 2 X \{ Ø }. In this paper, we consider the so called upward preordering ≼ u on 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 ≼ u . Let C ( X ) denote the collection of all τ -closed subsets of X and C 0 ( X ) = C ( X )\{ Ø }, which is equipped with the Fell topology τ F . Let C be a nonempty subset of X and let F : C → C 0 ( X ) be a closed set-valued mapping. By applying the Fell topology τ F on C ( X ) and the Fan-KKM Theorem, we prove some existence theorems for some ≼ u -minimization and ≼ 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 R n .