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Asymptotic behavior of a class of nonlinear delay difference equations. (English) Zbl 1010.39003
The authors study a nonlinear delay difference equation of the form \[ x_{n+1} = x_n^p f(x_{n-k_1}, x_{n-k_2},\dots , x_{n-k_r}), \quad n= 0, 1,\dots, \tag{1} \] where \(p\) is a positive constant, \(k_1,\dots , k_r\) are (fixed) nonnegative integers, and \(f \in C([0, \infty)^r\), \((0, \infty))\) and is non-increasing in each of its arguments.
They prove that if \(p< 1\) then (1) is permanent and has a unique positive fixed point. Sufficient conditions for global attractivity of all positive solutions are obtained.
In the case \(p > 1,\) (1) exhibits neither permanence nor global attractivity in general. The asymptotic behavior of solutions in the neighborhood of positive fixed points of (1) is studied.

39A11 Stability of difference equations (MSC2000)
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