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.
Farajzadeh et al. (Wed,) studied this question.