A counterexample in dimension $2$: take $K$ the unit closed disk of $\mathbb{C}$, and $f:K\to K$ the map
$$f(z)=i\frac{z}{|z|}\min(2|z|,1)$$ 
for $z\neq0$, and $f(0)=0$, which is the only fixed point of $f$. Then, starting by $u _ 0\in \partial K$ produces a sequence in $\partial K$.