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 trichotomy character of $x_{n+1}=(\alpha+\beta x_n+\gamma x_{n-1})/(A+x_n)$. (English) Zbl 1005.39017
An analysis of the periodicity, convergence and boundedness of the solutions of the second order difference equation in the title is presented. The parameters $\alpha$, $\beta$, $\gamma$, $A$ and the initial conditions $x_{-1}$ and $x_{0}$ are nonnegative and the denominator is always positive. The main result is the following. Theorem 1. (a) Assume that $\gamma =\beta +A.$ Then every solution of the given equation converges to a period two solution. (b) If $\gamma >\beta +A$, then the equation possesses unbounded solutions. (c) If $\gamma <\beta +A,$ then every solution of the equation has a finite limit. In the case (b), the difference equation has a positive equilibrium which is a saddle point. In case (c), the equation may possess a unique equilibrium (zero or positive) which is globally asymptotically stable, or two equilibrium points of which one is zero (and this is unstable) and the other is positive (locally asymptotically stable).

39A11Stability of difference equations (MSC2000)
39B05General theory of functional equations
Full Text: DOI