Resumen de Ekeland’s variational principle in weak and strong systems of arithmetic

David Fernández Duque, Paul Shafer, Keita Yokoyama

• We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to \Pi ^1_1\text{- }\mathsf {CA}_0, a strong theory of second-order arithmetic, while natural restrictions (e.g. to compact spaces or to continuous functions) yield statements equivalent to weak König’s lemma (\mathsf {WKL}_0) and to arithmetical comprehension (\mathsf {ACA}_0). We also find that the localized version of Ekeland’s variational principle is equivalent to \Pi ^1_1\text{- }\mathsf {CA}_0, even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.