Periodic solutions to parameter-dependent equations with a $\varphi$-Laplacian type operator

We study the periodic boundary value problem associated with the $\varphi$-Laplacian equation of the form $(\varphi (u^{\mathrm{\prime }}){)}^{\mathrm{\prime }}+f(u)u^{\mathrm{\prime }}+g(t,u)=s$, 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 $u\to \pm \mathrm{\infty }$. We generalize, in a unified framework, various classical and recent results on parameter-dependent nonlinear equations.