  We study the periodic boundary value problem associated with the
$\phi$-Laplacian equation of the form $(\phi(u'))'+f(u)u'+g(t,u)=s$, where $s$
is a real parameter, $f$ and $g$ are continuous functions, and $g$ is
$T$-periodic in the variable $t$. The interest is in Ambrosetti-Prodi type
alternatives which provide the existence of zero, one or two solutions
depending on the choice of the parameter $s$. We investigate this problem for a
broad family of nonlinearities, under non-uniform type conditions on $g(t,u)$
as $u\to \pm\infty$. We generalize, in a unified framework, various classical
and recent results on parameter-dependent nonlinear equations.
