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Remarks on Caristi’s fixed point theorem. (English) Zbl 1175.54056
Summary: We give a characterization of the existence of minimal elements in partially ordered sets in terms of fixed points 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 mentioned before.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54E50Complete metric spaces
06A06Partial order
47H10Fixed point theorems for nonlinear operators on topological linear spaces