In this work, we give a characterization of the existence of minimal elements in partially ordered sets in terms of fixed point type statement. This characterization shows that the assumptions in Caristi’s fixed point theorem can, a priori, be weakened. Finally, we give a negative answer to Kirk’s problem on an extension of Caristi’s result.

In this work, we give a characterization of the existence of minimal elements in partially ordered sets in terms of fixed point of multivalued maps. This characterization shows that the assumptions in Caristi’s fixed point theorem can, a priori, be weakened. Finally, we discuss Kirk’s problem on an extension of Caristi’s theorem and prove a new positive result which illustrates the weakening me...