Even continuity and the Banach contraction principle

Abstract:

In 1967, Philip R. Meyers established a nice converse to the Banach Contraction Mapping Theorem. We provide a counterexample to one of his corollaries and show that if X is a metrizable topological space, f a continuous self-map on X such that: (a)f has a fixed point p which has a compact neighborhood; (b) ${f^n}(x) \to p$ as $n \to \infty$ for each x in X, then the following are equivalent: (1) f is a contraction relative to a suitable metric on X; (2) the sequence of iterates $\{ {f^n}\} _{n = 1}^\infty$ is evenly continuous.