Editing Fixed-point theorem (section)

== In mathematical analysis ==
The [[Banach fixed-point theorem]] (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of [[iteration|iterating]] a function yields a fixed point.<ref>{{cite book
 | author    = Giles, John R.
 | title     = Introduction to the Analysis of Metric Spaces
 | year      = 1987
 | publisher = Cambridge University Press
 | isbn      = 978-0-521-35928-3
}}</ref>

By contrast, the [[Brouwer fixed-point theorem]] (1911) is a non-[[Constructivism (mathematics)|constructive result]]: it says that any [[continuous function]] from the closed [[unit ball]] in ''n''-dimensional [[Euclidean space]] to itself must have a fixed point,<ref>Eberhard Zeidler, ''Applied Functional Analysis: main principles and their applications'', Springer, 1995.</ref> but it doesn't describe how to find the fixed point (see also [[Sperner's lemma]]).

For example, the [[cosine]] function is continuous in [−1,&nbsp;1] and maps it into [−1,&nbsp;1], and thus must have a fixed point. This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve ''y'' = cos(''x'') intersects the line ''y'' = ''x''. Numerically, the fixed point (known as the [[Dottie number]]) is approximately ''x'' = 0.73908513321516 (thus ''x'' = cos(''x'') for this value of ''x'').

The [[Lefschetz fixed-point theorem]]<ref>{{cite journal |author=Solomon Lefschetz |title=On the fixed point formula |journal=[[Annals of Mathematics|Ann. of Math.]] |year=1937 |volume=38 |pages=819–822 |doi=10.2307/1968838 |issue=4}}</ref> (and the [[Nielsen theory|Nielsen fixed-point theorem]])<ref>{{cite book
 | last1=Fenchel | first1=Werner | author1link=Werner Fenchel
 | last2=Nielsen | first2=Jakob | author2link=Jakob Nielsen (mathematician)
 | editor-last=Schmidt | editor-first=Asmus L.
 | title=Discontinuous groups of isometries in the hyperbolic plane
 | series=De Gruyter Studies in mathematics
 | volume=29
 | publisher=Walter de Gruyter & Co.
 | location=Berlin
 | year=2003
}}</ref> from [[algebraic topology]] is notable because it gives, in some sense, a way to count fixed points.

There are a number of generalisations to [[Banach fixed-point theorem]] and further; these are applied in [[Partial differential equation|PDE]] theory. See [[fixed-point theorems in infinite-dimensional spaces]].

The [[collage theorem]] in [[fractal compression]] proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image.<ref>{{cite book
 | author     = Barnsley, Michael.
 | title      = Fractals Everywhere
 | url        = https://archive.org/details/fractalseverywhe0000barn
 | url-access = registration
 | year       = 1988
 | publisher  = Academic Press, Inc.
 | isbn       = 0-12-079062-9
}}</ref>