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)
Critical point theory and a theorem of Amaral and Pera. (English) Zbl 0603.34036

For the differential equation u '' (t)+g(t,u(t))=0 where g is 2π- periodic in t, L. Amaral and M. P. Pera [Boll. Unione Mat. Ital., V. Ser., C, Anal. Funz. Appl. 18, 107-117 (1981; Zbl 0472.34028)] showed that there is a 2π-periodic solution if there are constants H 1 and ν, with ν<1, such that for large x,

H 1 g(t,x/x)νand 0 2π lim |x| infg(t,x) xdt>0·

In the paper under review it is shown that if (sgn x) g(t,x) is bounded below, then the integral condition just stated can be replaced by

0 2π lim |x| inf(sgnx)g(t,x)dt>0·

This theorem and the theorem of Amaral and Pera are derived using the variational methods introduced by P. Rabinowitz [Nonlinear analysis, Collect. Pap. Honor H. Rothe, 161-177 (1978; Zbl 0466.58015)].

Reviewer: C.Chicone
34C25Periodic solutions of ODE
34C05Location of integral curves, singular points, limit cycles (ODE)
34C15Nonlinear oscillations, coupled oscillators (ODE)