Schauder fixed point theorem

The fixed point theorem of Schauder is named after the mathematician Juliusz Schauder and gives a sufficient condition under which a mapping has a fixed point. It represents a strong generalization of the fixed point theorem of Brouwer, of continuous functions on convex compact subsets of finite dimensional vector spaces treated. Andrei Nikolayevich Tychonoff proved the fixed point theorem of Schauder for locally convex vector spaces. Therefore, this version of the theorem is also called the fixed point theorem of Tychonoff.

Formulations of the sentence

Schauder fixed point theorem exists in several versions.

Version for locally convex Hausdorff spaces

Be a locally convex, Hausdorff shear, topological vector space and a non-empty, compact and convex subset of. Then every continuous map a fixed point. Since every Banach space is a locally convex Hausdorff space, this version already includes so all Banach spaces.

Version for all Hausdorff spaces

Be a Hausdorff shear, topological vector space and a non-empty, compact and convex subset of. Then every continuous map a fixed point.

Examples

For infinite-dimensional locally convex or normed vector spaces of the Schauder fixed point theorem does not need for closed and bounded compact sets instead to apply. Be the closed unit ball of the sequence space. Since is infinite -dimensional, the closed balls are not compact. Be also by defined. This map is continuous and forms gradually. They possess a fixed point it would have to apply. However, the only constant is the constant sequence in sequence. However applies and thus has no fixed points. One calls, however, that the mapping is compact, so the Schauder fixed point theorem is also true for closed and bounded subsets.

Comments

Schauder fixed point theorem proved the in 1930 for normed spaces. In the event that a locally convex space, the pack has already been demonstrated in 1935 by Andrei Nikolaevich Tikhonov, while shudder even just had a faulty evidence. Robert Cauty was able to show in 2001 that the sentence is true even for all Hausdorff topological vector spaces. This was even suspected of horror, but could not be proven until now.

Thus, the Brouwer fixed point theorem is in the known proofs much used, the proof is by no means trivial. As a corollary, one can derive the existence theorem of Peano from the Schauder 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 du problème de point fixe de shudder. In: Fundamenta Mathematicae. 170 (2001), pp. 231-246.
Convex set Peano existence theorem
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