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单词 BrouwerFixedPointTheorem
释义

Brouwer fixed point theorem


TheoremLet 𝐁={xn:x1} be the closed unit ballPlanetmathPlanetmath inn. Any continuous functionMathworldPlanetmathPlanetmath f:𝐁𝐁 has a fixed pointPlanetmathPlanetmath.

Notes

Shape is not important

The theorem also applies to anything homeomorphic to a closed disk, of course. In particular, we can replace B in the formulation with a square ora triangleMathworldPlanetmath.

Compactness counts (a)

The theorem is not true if we drop a point from the interior of B. For example, the map f(x)=12x has the single fixed point at 0; dropping it from the domain yields a map with no fixed points (http://planetmath.org/FixedPoint).

Compactness counts (b)

The theorem is not true for an open disk. For instance, the map f(x)=12x+(12,0,,0) has its single fixed point on the boundary of B.

TitleBrouwer fixed point theoremMathworldPlanetmath
Canonical nameBrouwerFixedPointTheorem
Date of creation2013-03-22 12:44:34
Last modified on2013-03-22 12:44:34
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id7
Authormathcam (2727)
Entry typeTheorem
Classificationmsc 55M20
Classificationmsc 54H25
Classificationmsc 47H10
Related topicFixedPoint
Related topicSchauderFixedPointTheorem
Related topicTychonoffFixedPointTheorem
Related topicKKMlemma
Related topicKKMLemma
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