The Existence And Lower Semi-continuity For The Solution To The Vector Equilibrium Problem

In the Hausdorff topological vector space, in the case of the topological interior of the ordering cone is empty, we use the Browder's fixed point theorem, FKKM theorem and Park's fixed point theorem to prove the existence of efficient solution and strong solution for the vector equilibrium problem instead of the scalarization; and in the normed linear space, we introduce the concept of global efficient solution and Henig efficient solution for the parametric set-valued vector equilibrium problem and obtain the scalarization results of the sets of the global efficient solution and Henig efficient solution for the parametric set-valued vector equilibrium problem. Basing on the scalarization results, we study the lower semi-continuity to the global efficient solution mapping and Henig efficient solution mapping for the parametric set-valued vector equilibrium problem.