A generalization of the Caristi fixed point theorem in metric spaces

In the present paper we ensure in metric spaces the existence of fixed points for functions not forced to be continuous. In order to obtain our objective we introduce the definition of function with partially complete graph. For functions verifying our new property we first establish two necessary and sufficient conditions on the existence of a fixed point and of a contractive fixed point respectively. These results allow us to remove the completeness on the metric space in the Caristi fixed point theorem and in its more recent generalizations. Moreover, we also prove a viable version of our Caristi type theorem.