On the existence of fixed points for set-valued mappings in partially ordered sets and its application to vector equilibrium problem

Abstract

In this paper, the notions of upper order-preserving and lower
order-preserving set-valued mappings are given. By using these
notions, some fixed point theorems in the setting of partially
ordered sets equipped with the hull-kernel topology for set-valued
mappings are established. As an application of these theorems, an
existence theorem for a solution of vector equilibrium problem is
provided. The main results of this article can be viewed as the
set-valued version of the main theorems given in \cite{Tarski,
1976,1980,2024} with mild assumptions.