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)
Dynamics of a higher order rational difference equation. (English) Zbl 1158.39301
Summary: We will investigate a nonlinear rational difference equation of higher order. Our concentration is on invariant intervals, periodic character, the character of semicycles and global asymptotic stability of all positive solutions of $$x_{n+1}=\frac{\beta x_n+\gamma x_{n-k}}{Bx_n+Cx_{n-k}}\,,\quad n=0,1,\dots\,.$$ It is worth to mention that our results solve the open problem proposed by {\it H. L. S. Kulenvić} and {\it G. Ladas} in their monograph [Dynamics of second order rational difference equations: with open problems and conjectures, Chapman & Hall/CRC, Boca Raton (2002; Zbl 0981.39011)].

39A20Generalized difference equations
Full Text: DOI
[1] Abu-Saris, R. M.; Devault, R.: Global stability of xn+1=A+xnxn-k. Appl. math. Lett. 16, 173-178 (2003) · Zbl 1049.39002
[2] Amleh, A.; Grove, E.; Ladas, G.; Georgiou, G.: On the recursive sequence xn+1=A+xn-1xn. J. math. Anal. appl. 233, 790-798 (1999) · Zbl 0962.39004
[3] Cunningham, K.; Kulenovic, M. R. S.; Ladas, G.; Valicenti, S. V.: On the recursive sequence $xn+1=\alpha +\beta $xnBxn+Cxn-1. Nonlinear anal. 47, 4603-4614 (2001) · Zbl 1042.39522
[4] Dehghan, M.; Douraki, M.; Douraki, M. J.: Dynamics of a rational difference equation using both theoretical and computational approaches. Appl. math. Comp., 1-20 (2004) · Zbl 1085.39006
[5] Devault, R.; Kosmala, W.; Ladas, G.; Schultz, S. W.: Global behavior of yn+1=p+yn-kqyn+xn-k. Nonlinear anal. 47, 4743-4751 (2001) · Zbl 1042.39523
[6] El-Owaidy, H.; Ahmed, A.; Mousa, M.: On asymptotic behaviour of the difference equation xn+1=A+xn-kxn. Appl. math. Comput. 147, 163-167 (2004) · Zbl 1042.39001
[7] Kosmala, W.; Kulenovic, M. R. S.; Ladas, G.; Teixeira, C. T.: On the recursive sequence yn+1=p+yn-1qyn+xn-1. J. math. Anal. appl. 251, 571-586 (2000) · Zbl 0967.39004
[8] Kulenovic, M. R. S.; Ladas, G.: Dynamics of second order rational difference equations with open problems and conjectures. (2002)
[9] Kulenovic, M. R. S.; Ladas, G.; Prokup, N. R.: A rational difference equation. Appl. math. Comput. 41, 671-678 (2001) · Zbl 0985.39017
[10] Elaydi, Saber N.: An introduction to difference equations. (1996) · Zbl 0840.39002
[11] Saleh, M.; Aloqeili, M.: On the rational difference equation xn+1=A+xnxn-k. Appl. math. Comput. 171, 862-869 (2005) · Zbl 1092.39019
[12] Li, W. -T.; Sun, H. -R.: Dynamics a rational difference equations. Appl. math. Comput. 157, 713-727 (2004)