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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.

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