= Solution
The <Brouwer fixed-point theorem> says that every <continuous map> $T:\overline B_R(0)\to\overline B_R(0)$ has a <fixed point>. To use the supplied <outward vector field zero theorem>, first extend the construction to all of $\mathbb R^n$. Let
$$
P_R(x)=\begin{cases}x,&|x|\leq R,\\Rx/|x|,&|x|>R,\end{cases}\qquad g(x)=x-T(P_R(x)).
$$
The <radial retraction onto a closed ball> $P_R$ is continuous, so $g$ is a continuous vector field on the required whole space. On $|x|=R$, the <Cauchy-Schwarz inequality> gives
$$
g(x)\cdot x=R^2-T(x)\cdot x\geq R^2-|T(x)|R\geq0.
$$
The supplied theorem gives $g(x_0)=0$ with $|x_0|\leq R$. There $P_R(x_0)=x_0$, hence $\boxed{T(x_0)=x_0}$. This explicit extension avoids presuming that the original map $T$ was defined outside the ball.
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