Abstract
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.
Highlights
Based on the theory of ball spaces introduced by Kuhlmann and Kuhlmann, we introduce and study Caristi–Kirk and Oettli–Thera ball spaces
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–Thera theorem, Takahashi’s theorem and the flower petal theorem
General setting The literature on complete metric spaces contains remarkable results such as the Theorem of Caristi and Kirk ([2] and [7]), Ekeland’s principle ([4]), Takahashi’s theorem ([17]) and the flower petal theorem ([16])
Summary
General setting The literature on complete metric spaces contains remarkable results such as the Theorem of Caristi and Kirk ([2] and [7]), Ekeland’s principle ([4]), Takahashi’s theorem ([17]) and the flower petal theorem ([16]). These theorems are known to be equivalent (see, e.g., [15,16] and Remark 3 below). Their statements can be found in Sect. The authors would like to thank the referees for their numerous suggestions that helped improve the paper
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