Brouwer fixed-point theorem

The Brouwer fixed point theorem is a statement from mathematics. It is named after the Dutch mathematician Brouwer Luitzen Egbertus January and states that the unit sphere has the fixed point property. Using this statement one can make statements about the existence of real solutions, systems of nonlinear equations.

Statement

With the -dimensional unit sphere is called. Then every continuous self-map of yourself in at least one fixed point.

In Quantorenschreibweise can the statement by

Represent.

Idea of ​​proof

Using the approximation theorem of Stone - Weierstrass can be restricted to functions.

Now it is believed, have no fixed point. Then, is given by

A well-defined and smooth image, which assigns to each point in the solid ball of the intersection of the mid - line of the through sphere. particular is a retraction, ie, valid for all.

This one leads to a contradiction by first showing that it covers'. This one is easy to see since the determinant of the Jacobian matrix from F to the set of the inverse function must be 0.

So the following applies:

By the theorem of Stokes. On the sphere, however, the identity. So This is true ( again according to the Stokes' theorem ):

Topologically equivalent formulations

The statement of Brouwer's fixed point theorem in topological its core content can thus be summarized as follows:

• The - dimensional sphere is never a retract of the -dimensional unit sphere.

Or in other words:

• There is no continuous mapping the -dimensional unit sphere in the - dimensional sphere, which makes the fix points.

This is equivalent to the following representation:

• A sphere is never a contractible space.

Or in other words:

• The identity map of a sphere is not null-homotopic.

Generalizations

By means of a continuous transformation on the simplex, which is homeomorphic to the unit sphere, can the statement of the theorem applied to arbitrary compact, convex sets in a finite -dimensional Banach space:

Also this statement is sometimes called a fixed point theorem of Brouwer, see also its generalization to the fixed point theorem of Schauder.

The filling rate

The just- mentioned generalization of Brouwer's fixed point theorem in turn can be considered as a consequence of the following theorem, which is also called filling rate:

The connection with the filling rate is obtained if one takes into account that every finite dimensional Banach space is a topologically equivalent and that each contained in a compact, convex subset represents a lot of the kind of the above.

The filling rate itself results from a direct application of the properties of the mapping degree.