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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 $\ddot u + g(u) = p(t)$, 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.