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Kuhlmann, Franz-Viktor; Kuhlmann, Katarzyna and Paulsen, Matthias (2018): The Caristi-Kirk Fixed Point Theorem from the point of view of ball spaces. In: Journal of Fixed Point Theory and Applications, Vol. 20, No. 3

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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.

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