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It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and of course everyone should know Picard's Theorem in ODEs. There are also results about local and global structure OF the fixed points themselves, and quite some famous conjectures (also labeled FPT for the purpose of this question).

Many results are so far removed from my field that I am sure there are plenty of FPTs out there that I have never encountered. I know of several, and will post later if you do not beat me to them :)

Community wiki rules apply. One FPT per answer, preferably with an inspiring list of interesting applications.

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    $\begingroup$ Not a FPT but a book: "Fixed point theory" by Granas and Dugundji. $\endgroup$ – jbc Apr 10 '13 at 9:01
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    $\begingroup$ Also: Journal of Fixed Point Theory and Applications, Fixed Point Theory and Applications, Fixed Point Theory, Advances in Fixed Point Theory, and JP Journal of Fixed Point Theory and Applications. $\endgroup$ – Rodrigo A. Pérez Apr 10 '13 at 13:19
  • $\begingroup$ Not so surprising, perhaps: whenever you want to construct an object whose definition involves the object again, you want to construct some fixed point. This is a general and natural thing to want. $\endgroup$ – Qiaochu Yuan Apr 11 '13 at 1:46
  • $\begingroup$ Nash actually used the Brouwer FPT. David Gale suggested to him that he could use the Kakutani FPT. $\endgroup$ – Michael Greinecker Jun 27 '13 at 6:49
  • $\begingroup$ Some nice order-theoretical fixed point theorems (and also related interesting problems) are discussed in B. Schröder's survey dx.doi.org/10.1016/S0304-3975(98)00273-4 (see also dx.doi.org/10.1007/s40065-012-0049-7) and his book dx.doi.org/10.1007/978-1-4612-0053-6 . $\endgroup$ – Michał Kukieła Apr 19 '14 at 17:08

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There is a celebrated fixed point theorem of A. Borel with applications to algebraic geometry (Ann. of Math. (1)64(1956)).

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The Nielsen fixed point theorem gives a lower bound on the number of fixed points of any map homotopic to a fixed map in terms of the Nielsen number. For closed surfaces, pseudo-Anosov homeomorphisms realize the Nielsen number in a given mapping class.

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THEOREM:   Let $n$ be a non-negative integer. Let $X$ be a Hausdorff compact space such that $X\times I^n$ has the fixed point property. Then for every continuous $f:X\times I^n\rightarrow X$ there exists $x\in X$ such that

$$\dim\{y\in I^n : f(x\ y)=x\}\ \ge\ n-\dim X$$

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THEOREM:   Let $n$ be a non-negative integer. Let $X$ be a compact Hausdorff ANR, and $f:X\times I^n\rightarrow X$ be a continuous mapping. Assume that the Lefschetz number of the induced mapping $f_0: x\mapsto f(x\ 0)$ of $X$ into itself, is not equal to $0$. Then there exists $x\in X$ such that

$$\dim\{y\in I^n : f(x\ y) = y\}\ \ge\ n-\dim X$$

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Veblen fixed point theorem A mapping $f$ from the class of ordinals to itself is called normal if it is strictly increasing and continuous, i.e.

1- $\alpha<\beta$ implies $f(\alpha)<f(\beta)$.

2- for any limit ordinal $f(\lambda)=sup\{f(\alpha):\ \alpha<\lambda\}$.

Any normal mapping of the class of ordinals has fixed point and the set of fixed points is unbounded.

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