Variational principles using a non-symmetric non-triangular distance

We consider Borwein-Preiss and Ekeland variational principles using a distance function that neither is symmetric nor enjoy the triangular inequality. All the given results rely exclusively on the convergence and continuity behaviors induced synthetically by the distance function itself without any topological implications. At the end of the paper, we also present two applications; the Caristi fixed point theorem and an existence theorem for equilibrium problems.