Ambrosetti-Prodi type result to a Neumann problem via a topological approach

We prove an Ambrosetti-Prodi type result for a Neumann problem associated to the equation \begin{document}$u''+f(x, u(x))=μ$\end{document} when the nonlinearity has the following form: \begin{document}$f(x, u):=a(x)g(u)-p(x)$\end{document} . The assumptions considered generalize the classical one, \begin{document}$f(x, u)\to+∞$\end{document} as \begin{document}$|u|\to+∞$\end{document} , without requiring any uniformity condition in \begin{document}$x$\end{document} . The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method.