zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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)].

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