Fixed-point theorem
A fixed point theorem in mathematics is a sentence which guarantees the existence of fixed points of a map under certain conditions. That is, the rate guarantees the existence of a point with.
Overview
In many areas of mathematics, one is looking statements about the existence of fixed points. One of the most well-known fixed point theorems is the fixed point theorem of Banach. With its help, the set of Picard - Lindelöf can be proved, which ensures a unique solution of certain ordinary differential equations. Unlike other fixed point sets of the fixed point theorem of Banach also yields the uniqueness of the fixed point.
The fixed point theorem of Schauder is also important in the area of analysis. Actually, it is a set of the topology and is proved using the fixed point theorem of Brouwer. However, one can derive from it, for example, the set of Peano, which also ensures the existence of a solution of an ordinary differential equation. Plays a central role in this set of nonlinear functional analysis. This allows an application- rich version of the theorem for nonlinear compact operators formulate.
List of fixed point sets
The following fixed point theorems are divided according to their areas of expertise listed. This list is incomplete.
Analysis
- Fixed point theorem of Banach
- Fixed point theorem of Ryll - Nardzewski
- Fixed point theorem for entire functions
Differential Geometry
- Atiyah - Bott fixed-point theorem
Lattice theory
- Fixed point theorem of Tarski and Knaster
Logic
- Fixed point theorem of Godel
- Fixed point theorem of Kakutani
Topology
- Brouwer fixed point theorem
- Lefschetz fixed point theorem
- Leray - Schauder alternative
- Fixed point theorem of Schauder
Computer science
- Fixed point theorem of Kleene
Category theory
- Fixed point set of Lawvere