A short proof of the converse to the contraction principle and some related results

Jacek Jachymski

DOI: http://dx.doi.org/10.12775/TMNA.2000.014


We simplify a proof of Bessaga's theorem given in the
monograph of Deimling. Moreover, our argument let us also obtain the
following result.

Let $F$ be a selfmap of an arbitrary set $\Omega$ and $\alpha\in
(0,1)$. Then $F$ is an $\alpha$-similarity with respect to some complete
metric $d$ for $\Omega$ (that is, $d(Fx,Fy)=\alpha d(x,y)$ for all
$x,y\in\Omega$) if and only if $F$ is injective and $F$ has a unique
fixed point.

Finally we present that the converse to the Contraction Principle for
bounded spaces is independent of the Axiom of Choice.


Fixed point; periodic point; Banach's contraction; similarity; Schröder's functional equation and inequality

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