Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Fixed-point theorems in infinite-dimensional spaces
Summary
In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations.
The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder (a previous result in a different vein, the Banach fixed-point theorem for contraction mappings in complete metric spaces was proved in 1922). Quite a number of further results followed. One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray who founded sheaf theory came out of efforts to extend Schauder's work.
Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V. If f : C → C is continuous with a compact image, then f has a fixed point.
Tikhonov (Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X → X has a fixed point.
Browder fixed-point theorem: Let K be a nonempty closed bounded convex set in a uniformly convex Banach space. Then any non-expansive function f : K → K has a fixed point. (A function is called non-expansive if for each and .)
Other results include the Markov–Kakutani fixed-point theorem (1936-1938) and the Ryll-Nardzewski fixed-point theorem (1967) for continuous affine self-mappings of compact convex sets, as well as the Earle–Hamilton fixed-point theorem (1968) for holomorphic self-mappings of open domains.
Kakutani fixed-point theorem: Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.