Caristi–Kirk and Oettli–Théra ball spaces and applications

Based on the theory of ball spaces introduced by Kuhlmann and Kuhlmann, we introduce and study Caristi–Kirk and Oettli–Théra 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–Théra theorem, Takahashi’s theorem and the flower petal theorem.

Remarks on Ume’s u−distance and Ekeland’s variational principle

The purpose of this paper is to present some remarks on Ume’s new concept of distance called u-distance, which generalizes w-distance and Suzuki’s t-distance. As an application of the u-distance version of Ekeland’s variational principle, we establish a generalized flower petal theorem.