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

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.

### MSC:

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

### Keywords:

difference equation; global attractivity; stability; equilibrium
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