Classifications and exististence of positive solutions of second order nonlinear neutral difference equations. (English) Zbl 0894.39002

Consider the difference equation \[ \Delta (r_n \Delta (x_n - p_n x_{n-\tau})) + f(n, x_{n-\delta}) = 0, \qquad n = 0, 1, 2, \dots, \tag{1} \] where \(\tau\) is a positive integer, \(\delta\) is a nonnegative integer, \(\{ r_n \}_{n=0}^{\infty}\) is a positive sequence and \(\{ p_n \}_{n=0}^{\infty}\) is a real sequence such that \(0 \leq p_n \leq p < 1\) for \(n \geq 0.\) The function \(f: \mathbb{R} \to \mathbb{R}\) is continuous and \(x f(n, x) > 0\) for \(x \not= 0\) and \(n \geq 0\). The authors give classifications for the eventually positive solutions of (1) and find necessary and/or sufficient conditions for the existence of these solutions.


39A10 Additive difference equations