Fixed point of compact operator in Banach space

Question

Let $M:=\overline{B(o,1)} $ be a closed unit ball of Banach space X. Assume that $F: M \rightarrow X$ is a nonlinear compact operator and $F(\partial M) \subset M$.

Then F has a fixed point in M.

What is the intuition behind this ? Does it follows directly from Schauder Fixed Point Theorem? Its far from being obvious to me.

Thanks