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

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

#### Abstract

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.

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.

#### Keywords

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

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