A Caristi type fixed point theorem which characterizes metric completeness

In this paper, we improve Caristi-Jachymski-Stein Jr and Banach-Caristi type fixed point theorems by relaxing the strong continuity assumption of the mapping with some weaker continuity notions. As an application, we show that the weaker version of the Caristi-Jachymski-Stein Jr fixed point theorem characterizes the completeness of the metric space and the Cantor intersection property.