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

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 [8, 12, 17, 18] with mild assumptions.