Two refinements of Frink's metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces (Q1731443)

A pair $(X,d)$ consisting of a nonempty set $X$ and a function $d : X \times X \to \mathbb{R}_+$ is called a quasimetric space if $d$ satisfies two first axioms of a metric and the triangle inequality in the form \[ d(x,z) \leq K \cdot \max \{d(x,y), d(y,z)\}, \quad \forall x,y,z \in X \] with $K \geq 1.$ \par Some results of [\textit{A. H. Frink}, Bull. Am. Math. Soc. 43, 133--142 (1937; Zbl 0016.08205)] concerning the construction of a metric in $X$ equivalent, in a certain sense, to a given quasimetric are improved. The authors present analogs of the Caristi fixed point theorem and the Cantor intersection theorem from which they deduce a version of the Banach fixed point theorem in quasimetric spaces which is due to \textit{I. A. Bakhtin} [Funkts. Anal. 30, 26--37 (1989; Zbl 0748.47048)].