Kuhlmann, FranzViktor; Kuhlmann, Katarzyna; Paulsen, Matthias
(2018):
The CaristiKirk 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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Abstract
We take a fresh look at the important CaristiKirk 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 CaristiKirk Theorem given by J.P. Penot in 1976. We define CaristiKirk ball spaces and use a generic fixed point theorem to reprove the CaristiKirk Theorem. Further, we show that a metric space is complete if and only if all of its CaristiKirk ball spaces are spherically complete.