Let $B$ be a closed ball in $n$-dimensional space, let $f$ be a "sufficiently smooth" map from $B$ to $B$, and let $p$ be a point on the boundary of $B$.
It seems to me that something like the Picard–Lindelöf theorem should ensure that there will be a unique maximally defined solution to the differential equation $x'(t) = f(x(t)) - x(t)$ with $x(0) = p$. Is that correct? Must this solution have a limiting value which is a fixed point of $f$? And is there any way to argue for Brouwer's fixed point theorem in general along these lines?