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)
Integral inequalities for second-order linear oscillation. (English) Zbl 0921.34035
The author extends in various directions the classical Lyapunov inequality for solutions to the second-order differential equation $$ y''+ q(t)y=0. \tag{*} $$ This investigation attracted attention in several recent papers [see e.g. {\it B. Harris} and {\it Q. Kong}, Trans. Am. Math. Soc. 347, No. 5, 1831-1839 (1995; Zbl 0829.34025) and {\it S. Clark} and {\it D. B. Hinton}, Math. Inequal. Appl. 1, No. 2, 201-209 (1998; Zbl 0909.24033) and the reference given therein]. The principal role plays the concept of the downswing of a function, which measures (in a certain sense) how much a function can fall down in a given interval. Using this concept, several necessary conditions are obtained for the existence of conjugate/focal points of solutions to (*) in a given interval. Using these results, the following interesting nonoscillation criterion for (*) is proved. If $$ \limsup_{T\to \infty}\int_0^T tq(t) dt- \liminf_{T\to \infty}\int_0^T tq(t) dt<1, $$ then (*) is nonoscillatory.
Reviewer: O.Došlý (Brno)

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
26D10Inequalities involving derivatives, differential and integral operators