= Solution
Let $h_0=f_{1,0}-f_{2,0}\geq0$. The <damped free-transport evolution> preserves nonnegativity, and a nonnegative kernel makes $\tau$ preserve it as well. Every term in $\sum_{n\geq0}\tau^nF(h_0)$ is therefore nonnegative. By <linearity>, the difference $f_1-f_2$ solves the <fixed point> equation with initial value $h_0$. The assumed uniqueness identifies it with this nonnegative sum on every finite interval. Hence the <order preservation for the linear Boltzmann equation> gives
$$
\boxed{f_1(t,x,v)\geq f_2(t,x,v)\quad\text{for all }t\geq0}.
$$
Continuity upgrades pointwise limits in the series to the stated continuous solution; no pointwise differentiability or minimum principle is needed.
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