A periodic problem for first order differential equations with locally coercive nonlinearities

In this paper we study the periodic boundary value problem associated with a first order ODE of the form x' + g(t, x) = s where s is a real parameter and g is a continuous function, T-periodic in the variable t. We prove an Ambrosetti-Prodi type result in which the classical uniformity condition on g(t, x) at infinity is considerably relaxed. The Carathéodory case is also discussed.