On the Converse of Caristi's Fixed Point Theorem

Szymon Głąb¹

¹ Institute of Mathematics Technical University of /L/od/x Al. Politechniki 11 90-924 /L/od/x, Poland and nstitute of Mathematics Polish Academy of Sciences /Sniadeckich 8 00-956 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 4, pp. 411-416

Résumé

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.

Détail

DOI : 10.4064/ba52-4-7

Keywords: nonempty set cardinality aleph selfmap main theorem says each periodic point fixed point under has fixed point there exist metric lower semicontinuous map phi mathbb leq phi phi separable assuming continuum hypothesis deduce compact