# Ekeland's variational principle in weak and strong systems of arithmetic

## TL;DR

This paper investigates the logical strength of Ekeland's variational principle within reverse mathematics, revealing its equivalence to strong systems like $ ext{ACA}_0$ and $ ext{Pi}^1_1$-${ ext{CA}}_0$, especially in localized forms.

## Contribution

It establishes the precise logical equivalences of Ekeland's variational principle and its restrictions in the framework of reverse mathematics.

## Key findings

- Full variational principle equivalent to $ ext{Pi}^1_1$-${ ext{CA}}_0$
- Restrictions correspond to $ ext{WKL}_0$ and $ ext{ACA}_0$
- Localized version with continuous functions also equivalent to $ ext{Pi}^1_1$-${ ext{CA}}_0$

## Abstract

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