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

Abstract

Let (X, ) 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 (X) = 2 X be the power set of X and (X) = 2 X \\{}. In this paper, we consider the so called upward preordering on (X), which has been used by many authors (see [47, 9, 1114, 1819]). We first prove some properties of this ordering relation Let (X) denote the collection of all -closed subsets of X and (X) = (X)\\{}, which is equipped with the Fell topology Let C be a nonempty subset of X and let F: C (X) be a closed set-valued mapping. By applying the Fell topology on (X) and the Fan-KKM Theorem, we prove some existence theorems for some -minimization and -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