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)
Oscillation of linear functional equations of the second order. (English) Zbl 0807.39007
The authors consider the functional equation $x(g(t)) = P(t) x(t) + Q(t) x(g(g(t)))$ where $g,P,Q$ are given and $x$ is to be found. Here $g : I \to I$ for some unbounded set $I$ of positive real numbers, $g(t) \not \equiv t$, $\lim\sb{t \to \infty,t \in I} g(t) = \infty$, and $P(t) > 0$, $Q(t) > 0$ for $t \in I$. A solution is called oscillatory if $x(t\sb n) x(t\sb{n + 1}) \le 0$ for some sequence $(t\sb n)$ satisfying ${t\sb n \in I}$ and $\lim\sb{n \to \infty} t\sb n = \infty$. The authors are interested in criteria guaranteeing that all solutions of the above equation are oscillatory. For example, they prove that this is the case if $\lim\sb{t \to \infty} \sup\sb{t \in I} Q(t) P(g(t)) > 1$ or $\lim\sb{t \to \infty} \inf\sb{t \in I} Q(t) P(g(t)) > {1 \over 4}$.

39B12Iterative and composite functional equations
39B22Functional equations for real functions