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 Juliusz 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.
Another variant of this theorem states that if U is an open subset of C containing the origin (zero), then any bounded, contractive map f on the closure of U has one, or both of the following properties: (1) f has a unique fixed point, or (2) there is a point x on the boundary of U such that f(x) = a x for some 0 < a < 1.
The Tikhonov (Tychonoff) 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).
References
- Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, Holland (1981). ISBN 90-277-1224-7
- Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5
- William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London ISBN 0-7923-7073-2