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)
Global asymptotic stability of a higher order difference equation. (English) Zbl 1198.39025
The article deals with the equation $$x_{n+1} = \frac{Ax_{n-2r-1}}{B + Cx_{n-2l}x_{n-2k}}, \quad n = 0,1,2,\ldots,\tag1$$ where $A, B, C$ are nonnegative reals, $l, r, k$ nonnegative integers, $l, r \le k$. Equation (1) has a zero equilibrium point and, if $\gamma = BA^{-1} < 1$, a nonzero equilibrium point $\bar{y} = \sqrt{1 - \gamma}$. The main results are the following: If $\gamma > 1$ then the zero is a locally asymptotic stable equilibrium point; if $\gamma < 1$ then both equilibrium points are unstable. The case $r = k$ and $\gamma = 1$ is also considered; in this case there exist periodic solutions with the prime period $2(k + 1)$ and every solution of (1) converges to a periodic solution of (1) with the period $2(k + 1)$.

39A30Stability theory (difference equations)
39A23Periodic solutions (difference equations)
39A20Generalized difference equations
Full Text: Link