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

##### 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
Zbl 0985.39017