On the Converse of Caristi's Fixed Point Theorem
Volume 52 / 2004
Bulletin Polish Acad. Sci. Math. 52 (2004), 411-416
MSC: Primary 47H10, 54H25; Secondary 03E50.
DOI: 10.4064/ba52-4-7
Abstract
Let $X$ be a nonempty set of cardinality at most $2^{\aleph _0}$ and $T$ be a selfmap of $X$. Our main theorem says that if each periodic point of $T$ is a fixed point under $T$, and $T$ has a fixed point, then there exist a metric $d$ on $X$ and a lower semicontinuous map $\phi :X\to {\mathbb R}_+$ such that $d(x,Tx)\leq \phi (x)-\phi (Tx)$ for all $x\in X$, and $(X,d)$ is separable. Assuming CH (the Continuum Hypothesis), we deduce that $(X,d)$ is compact.
