  We study the periodic boundary value problem associated with the second order
nonlinear differential equation $$ u" + c u' + \left(a^{+}(t) - \mu \,
a^{-}(t)\right) g(u) = 0, $$ where $g(u)$ has superlinear growth at zero and at
infinity, $a(t)$ is a periodic sign-changing weight, $c\in\mathbb{R}$ and
$\mu>0$ is a real parameter. We prove the existence of $2^{m}-1$ positive
solutions when $a(t)$ has $m$ positive humps separated by $m$ negative ones (in
a periodicity interval) and $\mu$ is sufficiently large. The proof is based on
the extension of Mawhin's coincidence degree defined in open (possibly
unbounded) sets and applies also to Neumann boundary conditions. Our method
also provides a topological approach to detect subharmonic solutions.
