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 research of Jean Leray that proved influential for algebraic topology and sheaf theory was motivated by the need to go beyond the Schauder fixed point theorem, proved in 1930 by Julius Schauder.

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 countably compact, then f has a fixed point.

The Tikhonov (Tychohoff) fixed point theorem is now applied to any locally convex topological vector space V. For any non-empty compact convex set X in V, and continuous function

f:X → X,

there is a fixed point for f.

Other results are the Kakutani and Markov fixed point theorems, now subsumed in the Ryll-Nardzewski fixed point theorem (1967).