Boundedness of solutions for semilinear Duffing equations. (Q1855848)

The author studies the boundedness of all solutions of the equation \(x''+n^{2}x+f(x) = p(t)\). The main result is the following theorem: Suppose \(f(x) \in C^{\infty } ({\mathbb R}), p(t)\in C^{6} ({\mathbb R}/2 \pi {\mathbb Z})\). If \(\int^{2\pi }_{0} p(t)e^{-int}dt = 0\) and if \(f(x)\) satisfies the following conditions: the limits \(\lim \limits_{x\rightarrow \infty } f(x) = \lim \limits_{x\rightarrow -\infty } f(x)\) exist and are finite, \(\lim \limits_{| x| \rightarrow \infty } x^{5}f^{(5)}(x) = 0\), there exists an \(m\in {\mathbb N}\), \(m\geq 6\), such that the limits \(\lim \limits_{x\rightarrow \infty } x^{m}f^{(m)}(x) = c^{+}_{m}\), \(\lim \limits_{x\rightarrow -\infty } x^{m}f^{(m)}(x) = c^{-}_{m}\) exist and are finite and \(\tau _{m} = c^{+}_{m}-c^{-}_{m}\neq 0\), \(\tau _{m-j} = 0\), \(j=1,2,\ldots ,m-1\), then every solution \(x(t)\) of the considered equation is bounded: \(\sup \limits_{\mathbb{R}} (x(t)+x'(t))<\infty \). This extends partially the relevant result of \textit{A. C. Lazer} and \textit{D. E. Leach} [Ann. Mat. Pura Appl., IV. Ser. 82, 49--68 (1969; Zbl 0194.12003)] and of \textit{B. Liu} [J. Differ. Equations 145, 119--144 (1998; Zbl 0913.34032)].