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