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Zbl 1185.37025
Dehghan, Mehdi; Douraki, Majid Jaberi
Global attractivity and convergence of a difference equation. (English)
[J] Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 16, No. 3, 347-361 (2009). ISSN 1201-3390; ISSN 1918-2538/e

The authors study the global behavior of the difference equation $$x_{n+1}= {\beta x_{n-k+1}+\gamma x_{n-2k+1}\over A+ Cx_{n-2k+1}},\qquad n\ge 0,$$ where $\beta$, $\gamma$, $A$, $C$ are positive constants and the initial conditions $x_{2k+1},\dots, x_1,x_0$, $k\ge 1$, are nonnegative. The case where $k= 1$ was studied by {\it M. R. S. Kulenović}, {\it G. Ladas} and {\it N. R. Prokup} [Comput. Math. Appl. 41, No. 5--6, 671--678 (2001; Zbl 0985.39017)].\par A certain change of variable is given to simplify the equations. It is shown that zero is always an equilibrium point which satisfies a necessary and suffient condition for its local asymptotic stability. With a specific assumption on the parameters, there is a unique positive equilibrium point whose global stability is discussed. The authors examine the nature of semicycles of solutions and discuss invariant intervals.
[Geoffrey R. Goodson (Towson)]

MSC 2000:: *37B40 Topological entropy
37B20 Notions of recurrence
37E99 Low-dimensional dynamical systems
39A30

Keywords: difference equation; global attractor; invariant intervals; semicycles

Citations: Zbl 0985.39017

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