Banach fixed-point theorem: Difference between revisions

Revision as of 03:40, 3 February 2005

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==Converses==

Several converses of the Banach contraction principle exist. The following is due to [[Czeslaw Bessaga]], from [[1959]]:

Let <math>f:X\rightarrow X</math> be a map of an abstract [[set (mathematics)|set]] such that each [[iterated function|iterate]] ''f''ⁿ has a unique fixed point. Let ''q'' be a real number, 0 &lt; q &lt; 1. Then there exists a complete metric on ''X'' such that ''f'' is contractive, and ''q'' is the contraction constant.