Bounded solutions for a forced bounded oscillator without friction

Under the validity of a Landesman–Lazer type condition, we prove the existence of solutions bounded on the real line, together with their first derivatives, for some second order nonlinear differential equation of the form u¨+g(u)=p(t), where the reaction term g is bounded. The proof is variational, and relies on a dual version of the Nehari method for the existence of oscillating solutions to superlinear equations.