Editing Banach fixed-point theorem (section)

==Generalizations==
There are a number of generalizations (some of which are immediate [[Corollary|corollaries]]).<ref name=Latif2014>{{cite book |first=Abdul |last=Latif |title=Topics in Fixed Point Theory |pages=33–64 |chapter=Banach Contraction Principle and its Generalizations |publisher=Springer |year=2014 |doi=10.1007/978-3-319-01586-6_2 |isbn=978-3-319-01585-9 }}</ref>

Let ''T'' : ''X'' → ''X'' be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are:
*Assume that some iterate ''T<sup>n</sup>'' of ''T'' is a contraction. Then ''T'' has a unique fixed point.
*Assume that for each ''n'', there exist ''c<sub>n</sub>'' such that ''d''(''T''<sup>''n''</sup>(''x''), ''T''<sup>''n''</sup>(''y'')) ≤ ''c''<sub>''n''</sub>''d''(''x'', ''y'') for all ''x'' and ''y'', and that
::<math>\sum\nolimits_n c_n <\infty.</math>
:Then ''T'' has a unique fixed point.
In applications, the existence and uniqueness of a fixed point often can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map ''T'' a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric. See also the article on [[fixed point theorems in infinite-dimensional spaces]] for generalizations.

In a non-empty [[compact metric space]], any function <math>T</math> satisfying <math>d(T(x),T(y))<d(x,y)</math> for all distinct <math>x,y</math>, has a unique fixed point. The proof is simpler than the Banach theorem, because the function <math>d(T(x),x)</math> is continuous, and therefore assumes a minimum, which is easily shown to be zero.

A different class of generalizations arise from suitable generalizations of the notion of [[metric space]], e.g. by weakening the defining axioms for the notion of metric.<ref>{{cite book |first1=Pascal |last1=Hitzler  | author-link1=Pascal Hitzler|first2=Anthony |last2=Seda |title=Mathematical Aspects of Logic Programming Semantics |publisher=Chapman and Hall/CRC |year=2010 |isbn=978-1-4398-2961-5 }}</ref> Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.<ref>{{cite journal |first1=Anthony K. |last1=Seda |first2=Pascal |last2=Hitzler | author-link2=Pascal Hitzler|title=Generalized Distance Functions in the Theory of Computation |journal=The Computer Journal |volume=53 |issue=4 |pages=443–464 |year=2010 |doi=10.1093/comjnl/bxm108 }}</ref>