Converses to fixed point theorems of Zermelo and Caristi Original Research Article

Let X be an abstract nonempty set and T be a self-map of X. Let View the MathML source and View the MathML source denote the sets of all periodic points and all fixed points of T, respectively. Our main theorem says that if View the MathML source, 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 Zermeloʹs 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 Caristiʹs theorem and its restriction to mappings satisfying Caristiʹs condition with a continuous real function ϕ.