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Boundedness in nonlinear oscillations at resonance. (English) Zbl 0926.34028

The main theorem establishes the boundedness of all solutions to the semilinear Duffing equation

x '' +n 2 x+f(x)=p(t)in(1)

under the conditions n, fC 6 (), f(0)=0, f is bounded in , x 6 f (6) (x)0 as |x|, p is a 2π-periodic function of class C 7 (/2π), and p 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)].

MSC:
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C11Qualitative theory of solutions of ODE: growth, boundedness
37J05Relations of dynamical systems with symplectic geometry and topology
37K05Hamiltonian structures, symmetries, variational principles, conservation laws