Oscillatory property of higher order nonlinear difference equations (Q1921256)

The paper is devoted to the study of oscillatory and asymptotic behaviour of solutions of higher order nonlinear difference equations of the form \[ \Delta \bigl( r_n (\Delta^{d - 1} x_n)^\delta \bigr) + F(n, x_n) = 0, \] where \(n \in N(n_0) = \{n_0, n_0 + 1, n_0 + 2, \dots,\}\) \((n_0\) is a fixed nonnegative integer), \(d\) is an integer and \(d > 1\), \(\Delta\) is the forward difference operator defined by \(\Delta x_n = x_{n + 1} - x_n\), \(\delta\) is a quotient of odd positive integers, \(x : N(n_0) \to \mathbb{R}\), \(r : N(n_0) \to (0, + \infty)\), \(F : N(n_0) \times \mathbb{R} \to \mathbb{R}\), and for any \(n \in N (n_0)\), \(F\) is continuous as a function of \(x \in \mathbb{R}\). The purpose of this paper is to give some sufficient and necessary conditions for the existence of nonoscillatory solutions of the equation in the cases \(xF(n,x) > 0\) and \(xF(n,x) < 0\) for \(x \neq 0\) and \(n \in N(n_0)\).