Fixed-point theorem: Difference between revisions

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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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*[[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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== 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]]