Several converses of the Banach contraction principle exist. The following is due to [[Czesław Bessaga]], from 1959:
Let <math>''f\,\colon'' : ''X\to'' → ''X</math>'' be a map of an abstract [[set (mathematics)|set]] such that each [[iterated function|iterate]] ''f<mathsup>f^n</mathsup>'' has a unique fixed point. Let <math>''q\in'' ∈ (0, 1)</math>, then there exists a complete metric on <math>''X</math>'' such that <math>''f</math>'' is contractive, and <math>''q</math>'' is the contraction constant.
Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if <math>''f\,\colon'' : ''X\to'' → ''X</math>'' is a map on a [[T1 space|''T''<sub>1</sub>'' topological space]] with a unique [[fixed point (mathematics)|fixed point]] <math>''a</math>'', such that for each <math>''x\'' in ''X</math>'' we have ''f<mathsup>f^n</sup>''(''x'')\mapsto → ''a</math>'', then there already exists a metric on <math>''X</math>'' with respect to which <math>''f</math>'' satisfies the conditions of the Banach contraction principle with contraction constant <math>1/2</math>.<ref>{{cite journal |first=Pascal |last=Hitzler | authorlink1=Pascal Hitzler|first2=Anthony K. |last2=Seda |title=A 'Converse' of the Banach Contraction Mapping Theorem |journal=Journal of Electrical Engineering |volume=52 |issue=10/s |year=2001 |pages=3–6 }}</ref> In this case the metric is in fact an [[ultrametric]].
