Non-Conflicting Ordering Cones and Vector Optimization in Inductive Limits

Let $(E,\xi)={\rm ind}(E_n, \xi_n)$ be an inductive limit of a sequence $(E_n, \xi_n)_{n\in N}$ of locally convex spaces and let every step $(E_n, \xi_n)$ be endowed with a partial order by a pointed convex (solid) cone $S_n$. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence $(S_n)_{n\in N}$ of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.