# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
Oscillations of higher-order neutral equations. (English) Zbl 0566.34055

Consider the neutral delay differential equation of order n $\left(*\right)\phantom{\rule{1.em}{0ex}}\left({d}^{n}/d{t}^{n}\right)\left[y\left(t\right)+py\left(t-\tau \right)\right]+qy\left(t-\sigma \right)=0,$ $t\ge {t}_{0}$ where q is a positive constant, the delays $\tau$ and $\sigma$ are nonnegative constants and the coefficient p is a real number. Theorem 1. (a) Assume that n is odd and that $p<-1$. Then every nonoscillatory solution of (*) tends to $+\infty$ or -$\infty$ as $t\to \infty$. (b) Assume that n is odd or even and that $p>-1$. Then every nonoscillatory solution of (*) tends to zero as $t\to \infty$. Theorem 2. Assume that n is odd. Then each of the following four conditions implies that every solution of (*) oscillates: (i) $p<-1$ and ${\left(-q/\left(1+p\right)\right)}^{1/n}\left(\tau -\sigma \right)/n>1/e$; (ii) $p=-1$; (iii) $p>-1$ and ${\left(q/\left(1+p\right)\right)}^{1/n}\left(\sigma -\tau \right)/n>1/e$; (iv) $-1 and ${q}^{1/n}\sigma /n>1/e·$

Theorem 3. Assume that n is even. Then each of the following two conditions implies that all solutions of (*) oscillate: (i) $p\ge 0$; (ii) $-1\le p<0$ and ${\left(q/p\right)}^{1/n}\left(\sigma -\tau \right)/n>1/e$. Theorem 4. Assume that n is even, $p<-1$, and ${\left(q/p\right)}^{1/n}\left(\sigma -\tau \right)/n>1/e$. Then every bounded solution of (*) oscillates.

##### MSC:
 34K99 Functional-differential equations 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
##### Keywords:
neutral delay differential equation