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

## TL;DR

This paper introduces Caristi-Kirk and Oettli-Théra ball spaces based on ball space theory, demonstrating their properties in complete metric spaces and providing simplified proofs for key fixed point theorems and variational principles.

## Contribution

It extends ball space theory to include Caristi-Kirk and Oettli-Théra spaces, linking them to classical fixed point and variational results.

## Key findings

- In complete metric spaces, every ball contains a singleton ball.
- Provides simplified proofs for fixed point theorems and variational principles.
- Establishes strong properties of these new ball spaces.

## 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.