Ekeland's variational principle in premetric spaces

We prove a variant of the Ekeland's variational principle in premetric spaces, i.e., spaces with distance function which is not symmetric and does not satisfy the triangle inequality. In the same space setting we present an extended Ekeland's variational principle and prove that it is equivalent to an Oettli-Thera type theorem. We apply the last results to get a sufficient condition for existence of a solution to the equilibrium problem.