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)
Oscillatory property for second order linear delay differential equations. (English) Zbl 0626.34077
This interesting paper deals with the equation (1) $x''(t)+a(t)x(g(t))=0,$ where $a\in C[0,\infty)\to [0,\infty)$, a(t)$\not\equiv 0$ on $[t\sb 0,\infty)$ $(t\sb 0\ge 0)$; $g\in C[0,\infty)\to [0,\infty)$; $0\le g(t)\le t$, $t\ge 0$, $\lim\sb{t\to \infty}g(t)=\infty$. The function sequence $\{\alpha\sb n(t)\}$ for $n=1,2,..$. and $t\ge t\sb 0$, where $\alpha\sb 0(t)=\epsilon \int\sp{\infty}\sb{t}\frac{g(s)}{s}a(s)ds,\alpha\sb n(t)=\int\sp{\infty}\sb{t}\alpha\sp 2\sb{n-1}(s)ds+\alpha\sb 0(t),$ $n=1,2,..$. and $0<\epsilon <1$ is introduced here. Sufficient conditions for (1) to be oscillatory are formulated by the functions $\alpha\sb n(t)$. The main result of the paper extends the well-known oscillation criteria of Hille, Kneser and Opial in the ordinary differential equation case and Erbe in the delay differential equation case.
Reviewer: J.Ohriska

34K99Functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text: DOI
[1] Erbe, L.: Oscillation criteria for second order nonlinear delay equations. Canad. math. Bull. 16, 49-56 (1973) · Zbl 0272.34095
[2] Hille, E.: Nonoscillation theorems. Trans. amer. Math. soc. 64, 234-252 (1948) · Zbl 0031.35402
[3] Kneser, A.: Untersuchungen über die reelen nullstellen der integrale linearer differentialgleichungen. Math. ann. 42, 409-435 (1893) · Zbl 25.0522.01
[4] Opial, Z.: Sur LES integrales oscillantes de l’equation differentielle u” + $f(t)$ u = 0. Ann. polon. Math. 4, 308-313 (1958) · Zbl 0083.07701
[5] Wong, J. S. W: Second order oscillation with retarded arguments. Ordinary differential equation, 581-596 (1972)