  We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in
$\mathsf{RCA}_0$. Furthermore, we show that Caristi's fixed point theorem for
both Baire and Borel functions is equivalent to the transfinite leftmost path
principle, which falls strictly between $\mathsf{ATR}_0$ and
$\Pi^1_1\mbox{-}\mathsf{CA}_0$. We also exhibit several weakenings of Caristi's
theorem that are equivalent to $\mathsf{WKL}_0$ and to $\mathsf{ACA}_0$.
