Note the followin theorem
Theorem : any continuous map from a unit two dimensional disk $E^2$ into itself has a fixed point.
To prove this theorem, Harper and Greenberg's book use the following argument :
If $f : E^2 \to E^2$ has no fixed point then we have $F(x) = \frac{f(x)-x}{|f(x) - x|}$.
So from the pertubation of $F$ we have a contraction $r$.
Here I cannot understand why $r$ is a contraction. If $F|_{S^1}$ has degree $1$, then it is plausible. But $F|_{S^1}$ may have degree $>1 $.