A short proof of the converse to the contraction principle and some related results (Q1590134)

This article deals with some reasoning in connection with Bessaga's well-known theorem about the converse to the constraction principle. The author offers a simpler proof of this theorem and gives the following modification: the map \(F\) of an arbitrary set \(\Omega\) into itself is an \(\alpha\)-similarity with respect to some complete metric \(d\) for \(\Omega\) (that means \(d(Fx, Fy)= \alpha d(x,y)\), \((x,y\in\Omega)\), \(\alpha\) is a fixed number from \((0,1)\)) if and only if \(F\) is injective and \(F\) has a unique fixed point. The situations when this result can be proved without the axiom of choice is also described.