Fixed-point Theorems in Infinite-dimensional Spaces

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. 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.

The Schauder fixed-point theorem states, in one version, that if C is a nonempty closed convex subset of a Banach space V and f is a continuous map from C to C whose image is compact, then f has a fixed point.

The Tikhonov (Tychonoff) fixed point theorem is applied to any locally convex topological vector space V. It states that for any non-empty compact convex set X in V, any continuous function

ƒ:X → X,

has a fixed point.

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's fixed-point theorem states that:

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.