The main theorem establishes the boundedness of all solutions to the semilinear Duffing equation
under the conditions , , , is bounded in , as , is a -periodic function of class , and satisfies a Lazer-Leach boundedness condition [A. C. Lazer and 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 R. Ortega [Proc. Lond. Math. Soc. (in press)]. Related results have been obtained by G. R. Morris [Bull. Aust. Math. Soc. 14, 71-93 (1976; Zbl 0324.34030)], R. Dieckerhoff and E. Zehnder [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 1, 79-95 (1987; Zbl 0656.34027)], J. You [Sci. China, Ser. A 35, No. 4, 399-412 (1992; Zbl 0763.34022)], 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)].