# Fixed point theorems in infinite-dimensional spaces

In

mathematics , a number of**fixed point theorems in infinite-dimensional spaces**generalise theBrouwer fixed point theorem . They have applications, for example, to the proof ofexistence theorem s forpartial differential equation s.The first result in the field was the

**Schauder fixed point theorem**, proved in 1930 byJuliusz 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 ofalgebraic topology , first proved for finitesimplicial complex es, to spaces of infinite dimension. For example, the research ofJean Leray who foundedsheaf theory came out of efforts to extend Schauder's work.The

Schauder fixed point theorem states, in one version, that if "C" is anonempty closed convex subset of aBanach 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 anylocally 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."

**ee also***

Topological degree theory **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.

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