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# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## On the Converse of Caristi's Fixed Point Theorem

### Tom 52 / 2004

Bulletin Polish Acad. Sci. Math. 52 (2004), 411-416 MSC: Primary 47H10, 54H25; Secondary 03E50. DOI: 10.4064/ba52-4-7

#### Streszczenie

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.

#### Autorzy

• Szymon GłąbInstitute of Mathematics
Technical University of /L/od/x
Al. Politechniki 11
90-924 /L/od/x, Poland
and
nstitute of Mathematics