  We deal with a planar differential system of the form \begin{equation*}
\begin{cases} \, u' = h(t,v), \\ \, v' = - \lambda a(t) g(u), \end{cases}
\end{equation*} where $h$ is $T$-periodic in the first variable and strictly
increasing in the second variable, $\lambda>0$, $a$ is a sign-changing
$T$-periodic weight function and $g$ is superlinear. Based on the coincidence
degree theory, in dependence of $\lambda$, we prove the existence of
$T$-periodic solutions $(u,v)$ such that $u(t)>0$ for all $t\in\mathbb{R}$. Our
results generalize and unify previous contributions about Butler's problem on
positive periodic solutions for second-order differential equations (involving
linear or $\phi$-Laplacian-type differential operators).
