Boundedness in nonlinear oscillations at resonance
The main theorem establishes the boundedness of all solutions to the semilinear Duffing equation \[ x''+ n^2x+ f(x)= p(t)\quad\text{in }\mathbb{R}\tag{1} \] under the conditions \(n\in\mathbb{N}\), \(f\in C^6(\mathbb{R})\), \(f(0)= 0\), \(f\) is bounded in \(\mathbb{R}\), \(x^6f^{(6)}(x)\to 0\) as \(| x|\to\infty\), \(p\) is a \(2\pi\)-periodic function of class \(C^7(\mathbb{R}/2\pi\mathbb{Z})\), and \(p\) satisfies a Lazer-Leach boundedness condition [\textit{A. C. Lazer} and \textit{D. E. Leach}, Ann. Mat. Pura Appl., IV. Ser. 82, 49-68 (1969; Zbl 0194.12003)]. A variant of this theorem is obtained. The proofs employ the Poincaré map of a Hamiltonian system generated from (1) by a sequence of transformations. The conclusion then follows from a modification of Moser's twist theorem found recently by \textit{R. Ortega} [Proc. Lond. Math. Soc. (in press)]. Related results have been obtained by \textit{G. R. Morris} [Bull. Aust. Math. Soc. 14, 71-93 (1976; Zbl 0324.34030)], \textit{R. Dieckerhoff} and \textit{E. Zehnder} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 1, 79-95 (1987; Zbl 0656.34027)], \textit{J. You} [Sci. China, Ser. A 35, No. 4, 399-412 (1992; Zbl 0763.34022)], \textit{R. Ortega} [J. Lond. Math. Soc., II. Ser. 53, No. 2, 325-342 (1996; Zbl 0860.34017)] and the author [J. Differ. Equations 145, 119-144 (1998; Zbl 0913.34032)].
scientific article; zbMATH DE number 1278770