The Caristi–Kirk Fixed Point Theorem from the point of view of ball spaces

We take a fresh look at the important Caristi-Kirk Fixed Point Theorem and link it to the recently developed theory of ball spaces, which provides generic fixed point theorems for contracting functions in a number of applications including, but not limited to, metric spaces. The connection becomes clear from a proof of the Caristi-Kirk Theorem given by J.-P. Penot in 1976. We define Caristi-Kirk ball spaces and use a generic fixed point theorem to reprove the Caristi-Kirk Theorem. Further, we show that a metric space is complete if and only if all of its Caristi-Kirk ball spaces are spherically complete.