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)
On the oscillations of mixed neutral equations. (English) Zbl 0845.34074
The author considers neutral differential equations of odd order of the form $$(x(t)+ cx(t- h)+ c^* x(t+ h^*))^{(n)}= qx(t- g)+ px(t+ g^*),\tag1$$ where $c$, $c^*$, $g$, $g^*$, $h$, $h^*$, $p$ and $q$ are real constants. It is well-known that a necessary and sufficient condition for oscillation of all solutions of (1) is that the characteristic equation $z^n(1+ ce^{- hz}+ c^* e^{h^* z})= qe^{- g z}+ pe^{g^* z}$ associated with (1) has no real roots. Since this is not easily verifiable, the author’s aim is to obtain sufficient conditions for oscillation of (1) involving the coefficients and the arguments only. A typical result is the following theorem: “Suppose that $c^*$, $g^*$, $h^*$ and $p$ are positive constants and $c$, $g$, $h$ and $q$ are nonnegative constants. Let $$\Biggl({p\over 1+ c}\Biggr)^{1/n} \Biggl({g^*\over n}\Biggr) e> 1$$ and either $$q> 0,\ \Biggl({q\over c^*}\Biggr)^{1/n} \Biggl({g+ h^*\over n}\Biggr) e> 1$$ or $$h^*> g^*,\ \Biggl({p+ q\over c^*}\Biggr)^{1/n} \Biggl({h^*- g^*\over n}\Biggr) e> 1.$$ Then the equation $(x(t)+ cx(t- h)- c^* x(t+ h^*))^{(n)}= qx(t- g)+ px(t+ g^*)$ is oscillatory.” At the end of the paper, the author notes that his results are extendable to more general neutral and nonneutral equations.

34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text: DOI