Converses to fixed point theorems of Zermelo and Caristi

Abstract

Let X be an abstract nonempty set and T be a self-map of X. Let Per T and Fix T denote the sets of all periodic points and all fixed points of T, respectively. Our main theorem says that if PerT= Fix T≠∅ , then there exists a partial ordering ≼ such that every chain in ( X,≼) has a supremum and for all x∈ X, x≼ Tx. This result is a converse to Zermelo's fixed point theorem. We also show that, from a purely set-theoretical point of view, fixed point theorems of Zermelo and Caristi are equivalent. Finally, we discuss relations between Caristi's theorem and its restriction to mappings satisfying Caristi's condition with a continuous real function ϕ.