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

Abstract

We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $$\Pi ^1_1\text{- }\mathsf {CA}_0$$ 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 ($$\mathsf {WKL}_0$$ WKL 0 ) and to arithmetical comprehension ($$\mathsf {ACA}_0$$ ACA 0 ). We also find that the localized version of Ekeland’s variational principle is equivalent to $$\Pi ^1_1\text{- }\mathsf {CA}_0$$ 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.

Metric fixed point theory and partial impredicativity

“…In this section, we study weakenings of Caristi’s theorem provable in sans-serifWsans-serifKsans-serifL0 and sans-serifAsans-serifCsans-serifA0. Caristi’s theorem follows directly from Ekeland’s variational principle, so we first recall the main results from [5] regarding the reverse mathematics of the latter. In particular, Ekeland’s variational principle is derivable in Π11false0-sans-serifCsans-serifA0, thus establishing an upper bound for Caristi's theorem.…”

“…All of these results reverse, although we won’t be needing this. We remark that [5] uses a different presentation of potentials (i.e. lower semi-continuous functions), but the two are equivalent over sans-serifRsans-serifCsans-serifA0 (see appendix A).…”

“…lower semi-continuous functions), but the two are equivalent over sans-serifRsans-serifCsans-serifA0 (see appendix A). The discontinuous cases are established by reducing to continuous cases via envelopes, given the following [5, Lemma 8.1].…”

“…Theorem 1.2 can be derived from theorem 1.3 by observing that any critical point for V will also be a fixed point of f, by the assumption that dfalse(x∗,ffalse(x∗false)false)≤Vfalse(x∗false)−Vfalse(ffalse(x∗false)false). Note however that Ekeland’s theorem is equivalent to Π11false0-sans-serifCsans-serifA0, so the transfinite methods are still hidden ‘under the hood’ [5].…”

“…The case where both f and V are continuous has also been treated by Peng & Yamazaki [10]; this case is interesting, as it can already be viewed as a generalization of the Banach fixed point theorem [2]. Using our previous work on the reverse mathematics of Ekeland’s variational principle [5], we extend this treatment to lower semi-continuous V. As we will see, weakened versions of Caristi’s theorem are equivalent to either sans-serifWsans-serifKsans-serifL0 when scriptX is compact and both functions are continuous, or sans-serifAsans-serifCsans-serifA0 when either compactness [10] or continuity is dropped (but not both).…”