Global attractivity in a class of higher-order nonlinear difference equation. (English) Zbl 1168.39301

Li, Wantong; Zhang, Yanhong; Su, Youhui

Acta Math. Sci., Ser. B, Engl. Ed. 25, No. 1, 59-66 (2005).

In this paper the global attractivity of the nonlinear difference equation \[ x_{n+1}=\frac{a+bx_{n}}{A+x_{n-k}},\quad n=0,1,... \] is investigated, where \(a,b,A \in (0,\infty),k\) are positive numbers and the initial conditions \(x_{-k},...,x_{-1}\) and \(x_0\) are arbitrary positive numbers. It is shown that the unique positive equilibrium of the equation is global attractive. As a corollary the result gives a positive confirmation of the conjecture presented by Kocic and Ladas.

Reviewer: Ahmed Hegazi (Mansoura)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis

Keywords:

difference equation; global attractivity; stability; equilibrium