A new method for the boundedness of semilinear Duffing equations at resonance (Q262068)

Consider the differential equation NEWLINE\[NEWLINE\ddot{x}+n^2 x+g(x)+\psi (x)=p(t),NEWLINE\]NEWLINE where \(n\geq 1\) is an integer, \(p(t)\) is \(2\pi\)-periodic, the function \(g\) has finite limits at infinity and \(\psi\) is periodic. In the linear case (\(g\equiv \psi \equiv 0\)) it is well known that all solutions are bounded whenever \(\hat{p}_n :=\frac{1}{2\pi}\int_0^{2\pi} p(t)e^{-int} dt=0\). Assuming that all functions are very smooth and the derivatives of \(g\) decay to zero fast enough, the authors prove that in the nonlinear case all solutions are bounded if NEWLINE\[NEWLINE|\hat{p}_n |<\frac{1}{\pi} |g(+\infty )-g(-\infty )|.NEWLINE\]NEWLINE This condition is sharp because it was already known that there exist unbounded solutions when the strict inequality is reversed. The proof follows along the lines of previous papers and the novelty is in the introduction of the term \(\psi\). This term leads to the appearance of some oscillatory integrals making the proof more complicated.