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

We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to Π ^1_1- 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 ( 𝖶𝖪𝖫_0 ) and to arithmetical comprehension ( 𝖠𝖢𝖠_0 ). We also find that the localized version of Ekeland’s variational principle is equivalent to Π ^1_1- 𝖢𝖠_0 , even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.