Fixed-point theorem: Difference between revisions

Revision as of 07:06, 30 March 2020

Revision as of 18:09, 16 August 2020

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By contrast, the [[Brouwer fixed-point theorem]] is a non-[[Constructivism (mathematics)|constructive result]]: it says that any [[continuous function|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 [&minus;1,1] and maps it into [&minus;1, 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 is approximately ''x''=0.73908513321516 (thus ''x''=cos(''x'') for this value of ''x'').