  We investigate the existence, non-existence, multiplicity of positive
periodic solutions, both harmonic (i.e., $T$-periodic) and subharmonic (i.e.,
$kT$-periodic for some integer $k \geq 2$) to the equation \begin{equation*}
\Biggl{(} \dfrac{u'}{\sqrt{1-(u')^{2}}} \Biggr{)}' + \lambda a(t) g(u) = 0,
\end{equation*} where $\lambda > 0$ is a parameter, $a(t)$ is a $T$-periodic
sign-changing weight function and $g \colon \mathopen{[}0,+\infty\mathclose{[}
\to \mathopen{[}0,+\infty\mathclose{[}$ is a continuous function having
superlinear growth at zero. In particular, we prove that for both $g(u)=u^{p}$,
with $p>1$, and $g(u)= u^{p}/(1+u^{p-q})$, with $0 \leq q \leq 1 < p$, the
equation has no positive $T$-periodic solutions for $\lambda$ close to zero and
two positive $T$-periodic solutions (a 'small' one and a 'large' one) for
$\lambda$ large enough. Moreover, in both cases the 'small' $T$-periodic
solution is surrounded by a family of positive subharmonic solutions with
arbitrarily large minimal period. The proof of the existence of $T$-periodic
solutions relies on a recent extension of Mawhin's coincidence degree theory
for locally compact operators in product of Banach spaces, while subharmonic
solutions are found by an application of the Poincar\'e--Birkhoff fixed point
theorem, after a careful asymptotic analysis of the $T$-periodic solutions for
$\lambda \to +\infty$.
