Fixed point theorem

In mathematics, a fixed point theorem is a proposition which, under certain conditions, guarantees the existence of fixed points in a mapping . That is, the sentence guarantees the existence of a point with . ${\displaystyle f\colon X\to X}$ ${\displaystyle x\in X}$ ${\displaystyle f(x)\,=\,x}$

overview

In many areas of mathematics one looks for statements about the existence of fixed points. One of the best-known fixed point theorems is Banach's fixed point theorem . With its help the Picard-Lindelöf theorem can be proven, which ensures a unique solution of certain ordinary differential equations . In contrast to other fixed point theorems, Banach's fixed point theorem also gives the uniqueness of the fixed point.

The Schauder fixed point theorem is also important in the field of analysis. It is actually a theorem from topology and is proven using Brouwer's fixed point theorem. However, from it one can derive, for example, Peano's theorem, which also ensures the existence of a solution to an ordinary differential equation. This theorem also plays a central role in nonlinear functional analysis. In this way an application-rich version of the theorem for nonlinear compact operators can be formulated.

List of fixed point sets

Fixed point sets are listed below, broken down according to their specialist areas. This list is of course incomplete.

Analysis and functional analysis

• Fixed point theorem from Banach
• Fixed point theorem by Browder-Göhde-Kirk
• Fixed point set for entire functions
• Fixed point theorem of Kakutani
• Fixed point theorem by Krasnoselski
• Fixed point theorem by Ryll-Nardzewski
• Fixed point theorem of Schauder
• Weissinger's Fixed Point Theorem

Differential geometry

• Atiyah-Bott Fixed Point Theorem

Group theory

• Fixed point theorem (finite groups)
• Borelian Fixed Point Theorem

Association theory

• Fixed point theorem by Tarski and Knaster

logic

• Fixed point theorem by Gödel

topology

• Brouwer's Fixed Point Theorem
• Fixed point theorem by Lefschetz
• Leray shudder alternative

Computer science

• Kleene's Fixed Point Theorem

Category theory

• Lawvere's Fixed Point Theorem

See also

• Fixed point formula

literature

• Klaus Deimling: Nonlinear Functional Analysis . 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 .