Caristi-Kirk and Oettli-Th\'era Ball Spaces and applications

Abstract

Based on the theory of ball spaces introduced by Kuhlmann and Kuhlmann we introduce and study Caristi-Kirk and Oettli-Th\'era ball spaces. We show that if the underlying metric space is complete, then these have a very strong property: every ball contains a singleton ball. This fact provides quick proofs for several results which are equivalent to the Caristi-Kirk Fixed Point Theorem, namely Ekeland's Variational Principles, the Oettli-Th\'era Theorem, Takahashi's Theorem and the Flower Petal Theorem.