Fixed-point theorem: Difference between revisions

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| isbn = 0-8218-5080-6

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</ref> Some authors claim that results of this kind are amongst the most generally useful in mathematics.<ref>{{cite book

| author1 = Dugundji, James

| author2 = Granas, Andrzej

| title = Fixed Point Theory

| year = 2003

| publisher = Springer-Verlag

| isbn = 0-387-00173-5

}}</ref>

== 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

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}}</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 (Seesee 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

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}}</ref> from [[algebraic topology]] is notable because it gives, in some sense, a way to count fixed points.

There are a number of generalizationsgeneralisations 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

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The above technique of iterating a function to find a fixed point can also be used in [[set theory]]; the [[fixed-point lemma for normal functions]] states that any continuous strictly increasing function from [[ordinal number|ordinals]] to ordinals has one (and indeed many) fixed points.

Every [[closure operator]] on a [[possetposet]] has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place.

Every [[involution (mathematics)|involution]] on a [[finite set]] with an odd number of elements has a fixed point; more generally, for every involution on a finite set of elements, the number of elements and the number of fixed points have the same [[parity (mathematics)|parity]]. [[Don Zagier]] used these observations to give a one-sentence proof of [[Fermat's theorem on sums of two squares]], by describing two involutions on the same set of triples of integers, one of which can easily be shown to have only one fixed point and the other of which has a fixed point for each representation of a given prime (congruent to 1 mod 4) as a sum of two squares. Since the first involution has an odd number of fixed points, so does the second, and therefore there always exists a representation of the desired form.<ref>{{citation

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*[[Browder fixed-point theorem]]

*[[Brouwer fixed-point theorem]]

*[[Rothe's fixed-point theorem]]

*[[Caristi fixed-point theorem]]

*[[Diagonal lemma]], also known as the fixed-point lemma, for producing self-referential sentences of [[first-order logic]]

*[[Lawvere's fixed-point theorem]]

*[[Discrete fixed-point theorem]]s

*[[Earle-Hamilton fixed-point theorem]]

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*[[Tychonoff fixed-point theorem]]

{{div col end}}

== See also ==

* [[Trace formula (disambiguation)|Trace formula]]

== Footnotes ==

{{Reflist}}

<references/>

== References ==

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==External links==

*[http://www.math-linux.com/spip.php?article60 Fixed Point Method]

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[[Category:Closure operators]]