Fixed-point theorem: Difference between revisions

Revision as of 18:22, 28 October 2023

Revision as of 01:16, 11 November 2023

Line 18:

}}</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