A fixed point theorem for the topological $2$-disc

Question

If there is a continuous function from the topological $2$-disc $D$ to itself such that it is identity map on boundary, then there exists a fixed point in the $D$'s interior?

Notice that this map can be injective or not.

Answer 1

One way to do this explicitly is to pass from the interior of the disk to the upper half plane, for example using the map $f(z) = i \frac{1+z}{1-z}$ Then let $g$ be the map of the upper half plane that scales $y$ by some amount without changing $x$, so $g(x+iy) = x+icy$ for some $c > 0$. $f^{-1}gf$ will then move every point in the interior, but without changing the behavior as you limit to the boundary.