Fixed point theorem of Schauder

The Schauder fixed point theorem is after the mathematician Juliusz Schauder designated and indicates a sufficient condition under which an image of a fixed point has. It represents a strong generalization of Brouwer's Fixed Point Theorem, which treats continuous functions on convex, compact subsets of finite-dimensional vector spaces. Andrei Nikolajewitsch Tychonoff proved Schauder's fixed point theorem for locally convex vector spaces . Therefore this version of the theorem is also called Tychonoff's Fixed Point Theorem .

Formulations of the sentence

The Schauder Fixed Point Theorem exists in several versions.

Version for locally convex Hausdorff rooms

Let be a locally convex , Hausdorff , topological vector space and a non-empty, compact and convex subset of . Then every continuous mapping has a fixed point. Since every Banach space is a locally convex Hausdorff space, this version already includes all Banach spaces. ${\displaystyle E}$ ${\displaystyle C}$ ${\displaystyle E}$ ${\displaystyle T\colon C\to C}$

Version for all Hausdorff rooms

Let be a Hausdorff topological vector space and a non-empty, compact and convex subset of . Then every continuous mapping has a fixed point. ${\displaystyle E}$ ${\displaystyle C}$ ${\displaystyle E}$ ${\displaystyle T\colon C\to C}$

Examples

In infinite-dimensional, locally convex or normalized vector spaces, the Schauder Fixed Point Theorem does not generally apply to closed and limited sets , that is, the requirement of compactness cannot be dispensed with. Let be the closed unit sphere of the sequence space . Since is infinite-dimensional, the closed spheres are no longer compact. Also be defined by. This mapping is continuous and maps to . If it had a fixed point, it should apply. However, the only constant sequence in is the constant sequence. But it is valid and therefore has no fixed points. ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle F\colon C\to C}$
${\displaystyle x=(s_{n})_{n\in \mathbb {N} }\mapsto \left({\sqrt {1-\|x\|_{\ell ^{2}}^{2}}},s_{1},s_{2},\ldots \right)}$
${\displaystyle C}$ ${\displaystyle s_{1}=s_{2}=\ldots }$ ${\displaystyle \ell ^{2}}$ ${\displaystyle 0}$ ${\displaystyle F(0)\neq 0}$ ${\displaystyle F}$

However, if one demands that the mapping is compact , then the Schauder Fixed Point Theorem also applies to closed and restricted subsets. ${\displaystyle T}$

Remarks

Schauder proved the fixed point theorem in 1930 for standardized spaces. In the event that there is a locally convex space , the theorem was proved in 1935 by Andrei Nikolajewitsch Tichonow , while Schauder himself only had one incorrect proof. Robert Cauty was able to show in 2001 that the theorem even holds for all Hausdorff topological vector spaces. This was already suspected by Schauder, but has not yet been proven. ${\displaystyle E}$

In the known proofs, Brouwer's fixed-point theorem is used, the proof of which is by no means trivial. As an application, one can derive Peano's existence theorem from Schuder's fixed point theorem.

swell

• Dirk Werner : Functional Analysis. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .
• Klaus Deimling: Nonlinear Functional Analysis. 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 .
• Robert Cauty: Solution of the problem of the point of the shudder. In: Fundamenta Mathematicae . Vol. 170, No. 3, 2001, pp. 231-246, doi : 10.4064 / fm170-3-2

Individual evidence

1. ^ Klaus Deimling: Nonlinear equations and degrees of mapping. Springer-Verlag, Berlin / Heidelberg / New York 1974, ISBN 3-540-06888-0 , p. 130.
2. ^ Klaus Deimling: Nonlinear Functional Analysis. 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 , p. 60.
3. ^ Klaus Deimling: Nonlinear Functional Analysis. 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 , p. 90.

Web links

• Proof of the fixed point theorems of Brouwer and Schauder (the latter in the case of Banach spaces)