Global attractivity and convergence of a difference equation. (English) Zbl 1185.37025

Dehghan, Mehdi; Douraki, Majid Jaberi

Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 16, No. 3, 347-361 (2009).

The authors study the global behavior of the difference equation \[ x_{n+1}= {\beta x_{n-k+1}+\gamma x_{n-2k+1}\over A+ Cx_{n-2k+1}},\qquad n\geq 0, \] where \(\beta\), \(\gamma\), \(A\), \(C\) are positive constants and the initial conditions \(x_{2k+1},\dots, x_1,x_0\), \(k\geq 1\), are nonnegative. The case where \(k= 1\) was studied by M. R. S. Kulenović, G. Ladas and N. R. Prokup [Comput. Math. Appl. 41, No. 5–6, 671–678 (2001; Zbl 0985.39017)].
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.

Reviewer: Geoffrey R. Goodson (Towson)

MSC:

37B40	Topological entropy
37B20	Notions of recurrence and recurrent behavior in topological dynamical systems
37E99	Low-dimensional dynamical systems
39A30	Stability theory for difference equations

Keywords:

difference equation; global attractor; invariant intervals; semicycles

Citations:

Zbl 0985.39017