Caristi's fixed point theorem, non-monotone inductive definitions, and relativized leftmost paths

This is joint work with Paul Shafer, Henry Towsner and Keita Yokoyama.
A Caristi system is a triple $(X,f,V)$, where $X$ is a complete metric space, $V$ is a lower semi-continuous function from $X$ to the positive reals, and $f: X\to X$ is an arbitrary function such that $d(x,f(x))\leqslant V(x) - V(f(x))$ always holds.
Caristi's fixed point theorem states that any Caristi system has a fixed point. This has been proven in the literature using the Ekeland variational principle, and using Caristi sequences, which are transfinite iterations of $f$.
We analyze Caristi's theorem and its known proofs in the context of reverse mathematics, where metric spaces are assumed separable and coded in the standard way. Among the results obtained, we have that, over $\mathrm{RCA}_0$:

• $\mathrm{WKL}_0$ is equivalent to Caristi's theorem restricted to compact spaces with continuous $V$.
• $\mathrm{ACA}_0$ is equivalent to Caristi's theorem restricted to compact spaces with lower semi-continuous $V$.
• Towsner's relativized leftmost path principle is equivalent to Caristi's theorem for Baire or Borel $f$.
• The arithmetical inflationary fixed point scheme is equivalent to the statement that if f is arithmetically defined, any point of $X$ can be included in a Caristi sequence containing a fixed point of $f$.

These theories are all defined over the language of second-order arithmetic and we mention them in strictly increasing order of strength.