let $D$ be the unit disk, consider a continuously differentiable transformation $T$ on an open set containing D such that $T(x,y)=(u,v)$ whose Jacobian is never $0$ in $D$. Suppose that T is near the idendity map in the sense that $$|T(p)-p| \le \frac{1}{3}$$ for all $p \epsilon D$ Prove that there is a point $p_0$ with $T(p_0)= (0,0)$
Do I need to use the fixed point theorem here? OR how can I use that fact that T is invertible locally? I couldn't think of anything, any help would be appreciated.
HINT: Yes, set it up as an application of the Brouwer fixed point theorem. Can you find a function $f$ so that $f(x)=x \iff T(x)=0$? Can you find a closed ball that $f$ maps to itself?