Boundedness of solutions for Duffing's equations with semilinear potentials (Q5955610)

The paper concerns the equation \(\ddot{x}(t)+\arctan x(t)=\varepsilon e(t)\) where \(e\in C^\infty(S^1)\) and \(\varepsilon\) is a small parameter. The main result states that every solution \(x\) to the equation is defined on \(\mathbb{R}\), and both \(x\) and \(\dot{x}\) are bounded there. The proof method indicates the same conclusion for the more general equation \(\ddot{x}(t)+g(x(t))=\varepsilon e(t)\) where \(g\in C^\infty(\mathbb{R})\), \(x g(x)>0\) for \(x\neq 0\), and further definite hypotheses are satisfied.