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Let (''X'', d) be a non-empty [[complete metric space]]. Let ''T'' : ''X'' <tt>-></tt> ''X'' be a ''[[contraction mapping]]'' on ''X'', i.e: there is a [[real number]] ''q'' < 1 such that
:<math>d(Tx,Ty) \le q\cdot d(x,y)</math>
for all ''x'', ''y'' in ''X''. Then the map ''T'' admits one and only one fixed point ''x''<sup>*</sup> in ''X'' (this means ''Tx''<sup>*</sup> = ''x''<sup>*</sup>). Furthermore, this fixed point can be found as follows: start with an arbitrary element ''x''<sub>0</sub> in ''X'' and define
:<math>d(x^*, x_n) \leq \frac{q^n}{1-q} d(x_1,x_0)</math>.
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When using the theorem in practice, the most difficult part is typically to define ''X'' properly so that ''T'' actually maps elements from ''X'' to ''X'', i.e. that ''Tx'' is always an element of ''X''.
==Applications==
A standard application is the proof of the [[Picard-Lindelöf theorem]] about the existence and uniqueness of solutions to certain [[ordinary differential equation]]s. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions. The Banach fixed point theorem is then used to show that this integral operator has a unique fixed point.
==Converses==
Several converses of the Banach contraction principle exist. The following is due to [[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''<sup>n</sup> has a unique fixed point. Let ''q'' be a real number, 0 < q < 1. Then there exists a complete metric on ''X'' such that ''f'' is contractive, and ''q'' is the contraction constant.
==References==
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