A classification scheme for nonoscillatory solutions to a higher order neutral nonlinear difference equation. (English) Zbl 0942.39007

Consider nonlinear neutral difference equations of the form \[ \Delta^m (x_n+ px_{n-\tau}) +f(n,x_{n-\partial}) =0,\quad n=0,1,2, \dots \tag{*} \] where \(m,\tau\) and \(\partial\) and integers such that \(m\geq 2\), \(\tau>0 \), \(\partial\geq 0\) and \(p\) is a nonnegative real number different from \(1\) and \(f\): \((0,1, \dots) \times\mathbb{R} \to\mathbb{R}\) is continuous in the second variable and \(x\cdot f(n,x)>0\) for \(x\neq 0\). The authors classify all nonoscillatory solutions to (*) by means of their asymptotic behavior. Existence criteria are then provided for justification of such classification.


39A11 Stability of difference equations (MSC2000)