A type of minimal and maximal point theorem in locally convex product cones

Using the neighbourhoods in upper relative topologies, we introduce a quasi-metric on cones which leads to a version of Cantors's intersection theorem for upper and lower relative v-topologies. Then we prove a type of minimal and maximal point theorem for subsets of product cone topologies and obtain the corresponding Ekeland's variational principle (EVP) results for locally convex cone-valued functions.