In Dynamical Systems and Ergodic Theory by Pollicott and Yuri, there is an easy, one dimensional, fixed point theorem:
If $T$ is a continuous map on a closed interval $J$ so that $T(J)\supseteq J$,
then $T$ has a fixed point.
Here the space is mapped onto itself, in contrast to the "usual" fixed point theorems where it is mapped into itself.
Out of idle curiosity, I wonder if it's true in higher dimensions.
To be concrete, let $D=\{(x,y): x^2+y^2\leq 1\}$ be the closed unit disk in $\mathbb{R}^2$.
Suppose that $T$ is continuous on $D$ and $T(D)\supseteq D$, does $T$ have
a fixed point?
If $T$ is one-to-one, for instance, this follows by applying Kakutani's fixed point
theorem to the set valued map $x\mapsto T^{-1}(\{x\})$ on $D$.
But I'm not sure whether the result holds in general.
Best Answer
Unfortunately, this is not true, as I convinced myself using a cork coaster not unlike these. A counterexample is given by a map that maps the upper third of the disk to the lower half, the lower third to the upper half, and the middle third to a band that connects the two outside the disk.