Periodic solutions to parameter-dependent equations with a φ-Laplacian type operator

We study the periodic boundary value problem associated with the φ-Laplacian equation of the form (φ(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→± ∞. We generalize, in a unified framework, various classical and recent results on parameter-dependent nonlinear equations.