Ekeland Variational Principle in asymmetric locally convex spaces

In this paper we prove two versions of Ekeland Variational Principle in asymmetric locally convex spaces. The first one is based on a version of Ekeland Variational Principle in asymmetric normed spaces proved in S. Cobzaş, Topology Appl. 158 (8) (2011) 1073–1084. For the proof we need to study the completeness with respect to the asymmetric norm p A (the Minkowski functional) of the subspace X A of an asymmetric locally convex space X generated by a convex subset A of X (the analog of Banach disk). The second one is based on the existence of minimal elements (with respect to an appropriate order) in quasi-uniform spaces satisfying some completeness conditions, obtained as a consequence of Brezis–Browder maximality principle.