Oscillation of higher-order neutral nonlinear difference equations (Q1808629)

The paper deals with the higher-order neutral nonlinear difference equation \[ \Delta(r_n(\Delta^d (x_n - p_n x_{n-r}))^{\delta}) + f(n, x_{n-\sigma}) = 0, \tag{1} \] where \(n \in \{ n_0+1, n_0+2,...\}, n_0\) and \(\sigma\) are nonnegative integers, \(\tau\) and \(d\) are positive integers, \(\Delta x_n = x_{n+1} - x_n,\) \(\delta\) is a quotient of odd positive integers, \(0 \leq p_n < 1, r_n > 0,\) and \(f\) is continuous in \(x\) for \(x \in \mathbb{R}.\) A solution of (1) is called nonoscillatory if it is eventually positive or eventually negative. Some necessary and sufficient conditions for the existence of nonoscillatory solutions to (1) are obtained. The author gives necessary and sufficient conditions for all bounded solutions to (1) to be oscillatory or to tend to zero.