Brouwer fixed-point theorem: Difference between revisions

Revision as of 04:29, 23 January 2024

Revision as of 16:00, 11 June 2024

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If there is no fixed point of the boundary of <math>K</math>, then the function

:<math>g(x)=\frac{x-f(x)}{\sup_{xy\in K}\left|xy-f(xy)\right|}</math>

is well-defined, and

<math>H(t,x) = \frac{x-tf(x)}{\sup_{xy\in K}\left|xy-tf(xy)\right|}</math>

defines a homotopy from the identity function to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so <math>g</math> also has degree one at the origin. As a consequence, the preimage <math>g^{-1}(0)</math> is not empty. The elements of <math>g^{-1}(0)</math> are precisely the fixed points of the original function ''f''.