Brouwer's Fixed Point Theorem

The Brouwer fixed-point theorem is a statement from the mathematics . It is named after the Dutch mathematician Luitzen Egbertus Jan Brouwer and says that the unit sphere has the fixed point property. With the help of this statement one can make statements about the existence of solutions of real, nonlinear systems of equations. ${\displaystyle D^{n}}$

statement

With the will -dimensional unit sphere called. Then every continuous self-mapping of itself has at least one fixed point . ${\displaystyle D^{n}=\{x\in \mathbb {R} ^{n}:\|x\|\leq 1\}}$ ${\displaystyle n}$ ${\displaystyle D^{n}}$

In quantifier notation , the statement can be passed through

${\displaystyle \forall f\in C(D^{n},D^{n}):\exists x\in D^{n}:f(x)=x}$

represent.

Proof idea

Using Stone-Weierstraß's approximation theorem , one can restrict oneself to functions. ${\displaystyle {\mathcal {C}}^{1}}$

Now it is assumed that there is no fixed point. Then is given by ${\displaystyle f}$ ${\displaystyle F\colon D^{n}\to S^{n-1}}$

${\displaystyle F(x):=x+\left({\sqrt {1-|x|^{2}+\left\langle x,{\frac {x-f(x)}{|x-f(x)|}}\right\rangle ^{2}}}-\left\langle x,{\frac {x-f(x)}{|x-f(x)|}}\right\rangle \right){\frac {x-f(x)}{|x-f(x)|}}}$,

Illustration of F in D ²

a well-defined and smooth mapping that assigns the intersection of the half-straight line from through with the sphere to every point in the full sphere. is in particular a retraction, i. h., for all true . ${\displaystyle f(x)}$ ${\displaystyle x}$ ${\displaystyle F}$ ${\displaystyle x\in S^{n-1}}$ ${\displaystyle F(x)=x}$

This can lead to a contradiction by first shows that for the following applies: . This is easy to see, since the determinant of the Jacobi matrix of F must be 0 according to the theorem of the inverse function . ${\displaystyle \omega ^{n-1}:=F^{1}\,\mathrm {d} F^{2}\wedge \cdots \wedge \mathrm {d} F^{n}}$ ${\displaystyle \mathrm {d} \omega ^{n-1}=0}$

So:

${\displaystyle 0=\int _{D^{n}}\mathrm {d} \omega ^{n-1}=\int _{S^{n-1}}\omega ^{n-1}}$

according to Stokes' theorem . But identity is on the sphere . Hence (again according to Stokes' theorem): ${\displaystyle F}$

${\displaystyle =\int _{S^{n-1}}x_{1}\mathrm {d} x^{2}\wedge \cdots \wedge \mathrm {d} x^{n}={\rm {vol}}(D^{n})\neq 0}$.

Other proofs use Sperner's lemma (see Aigner, Ziegler, The Book of Proofs , Chapter 25) or Borsuk-Ulam's theorem .

Topologically equivalent formulations

The statement of Brouwer's Fixed Point Theorem in its topological core content can be summarized as follows:

• The -dimensional sphere is never a retract of the -dimensional unit sphere . ${\displaystyle (n-1)}$ ${\displaystyle S^{n-1}}$ ${\displaystyle n}$ ${\displaystyle D^{n}}$

Or in other words:

• There is no continuous mapping of the -dimensional unit sphere onto the -dimensional sphere , which leaves the points of the fix. ${\displaystyle n}$ ${\displaystyle D^{n}}$ ${\displaystyle (n-1)}$ ${\displaystyle S^{n-1}}$ ${\displaystyle S^{n-1}}$

The following illustration is equivalent:

• A sphere is never a contractible space . ${\displaystyle S^{n-1}}$

Or in other words:

• The identical mapping of a sphere is not null homotopic . ${\displaystyle id_{S^{n-1}}}$ ${\displaystyle S^{n-1}}$

Generalizations

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

Let be a continuous mapping of a non-empty , compact, convex subset of a finite-dimensional Banach space in itself. Then has a fixed point. ${\displaystyle f}$ ${\displaystyle f}$

This statement is also sometimes referred to as Brouwer's fixed point theorem, see also his generalization on Schauder's fixed point theorem .

The fill-in phrase

The generalization of Brouwer's Fixed Point Theorem just given can in turn be drawn as a consequence of the following theorem, which is also known as the fill-in theorem:

Is a bounded open subset of and a continuous mapping and thereby ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon {\overline {\Omega }}\rightarrow \mathbb {R} ^{n}}$

${\displaystyle f(x)=x}$ for all ${\displaystyle x\in \partial {\Omega },}$

so applies . ${\displaystyle f({\overline {\Omega }})\supset \Omega }$

The connection with the fill-in theorem is obtained when one takes into account that every finite-dimensional Banach space is topologically equivalent to one and that every compact, convex subset contained therein represents a set of the kind mentioned above . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\overline {\Omega }}}$

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

literature

• Piers Bohl : About the movement of a mechanical system near a position of equilibrium . In: Journal for pure and applied mathematics , 127, 1904, pp. 179-276
• Jacques Hadamard : Note on quelques applications de l'indice de Kronecker . In: Jules Tannery : Introduction à la théorie des fonctions d'une variable (Volume 2). 2nd Edition. A. Hermann & Fils, Paris 1910, pp. 437-477 (French) Textarchiv - Internet Archive
• LEJ Brouwer : On the mapping of manifolds . (July 1910), Mathematische Annalen 71, July 25, 1911, pp. 97–115 ( correction . January 23, 1912, p. 598)
• Harro Heuser : Textbook of Analysis. Part 2. 5th revised edition. Teubner, Stuttgart a. a. 1990, ISBN 3-519-42222-0 , p. 593.
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).

Web links

Commons : Brouwer fixed point theorem  - collection of images, videos and audio files

• Eric W. Weisstein : Brouwer Fixed Point Theorem . In: MathWorld (English).
• Set theoretical topology. (PDF; 1.72 MB) Script (German)

Individual evidence

1. ↑ ^{a b} Harzheim: p. 158
2. ↑ Harzheim: pp. 157-160
3. ↑ Harzheim: p. 157