Fixed point theorems in infinite-dimensional spaces
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 Leraywho founded sheaf theorycame out of efforts to extend Schauder's work.
Schauder fixed point theoremstates, in one version, that if "C" is a nonemptyclosed 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", and continuous function
:"f":"X" → "X",
there is a fixed point for "f".
Other results are the Kakutani and Markov fixed point theorems, as well as the
Ryll-Nardzewski fixed point theorem(1967).
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."
Topological degree theory
* 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.
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