if $F:X\to X$ is continuous, such that for some $\mu>0$, $\left \langle F(u), u \right \rangle\geq 0$ for all $u\in X$ with $||u||=\mu$

Question

If $(X,\left \langle \cdot , \cdot \right \rangle)$ is a Hilbert space of finite dimension and if $F:X\to X$ is continuous, such that for some $\mu>0$, $\left \langle F(u), u \right \rangle\geq 0$ for all $u\in X$ with $||u||=\mu$, then there is $u_0\in X$, $||u_0||\leq\mu$ such that $F(u_0)=0$.

I know that to solve this problem I have to use Brouwer's fixed point theorem, but I don't know how to use it. Any suggestions? Thank you very much for your help!