On oscillation and nonoscillation properties of Emden-Fowler difference equations
The discrete Emden-Fowler equation \[ \Delta(a_n|\Delta x_n|^{\alpha} \operatorname{sgn}\Delta x_n) + b_n|x_{n+1}|^{\beta} \operatorname{sgn}x_{n+1} = 0 \] with \(\alpha>0\), \(\beta>0\) and \(\{a_n\}_n\), \(\{b_n\}_n\) positive real sequences. Set \[ A_n = (a_n)^{-1/\alpha}, \quad Y_a = \sum_{n=0}^{\infty}A_n, \quad Y_b = \sum_{n=0}^{\infty}b_n \] and consider three cases: \[ \begin{aligned} Y_a &= \infty, \quad Y_b<\infty;\tag{H1}\\ Y_a&<\infty, \quad Y_b = \infty;\tag{H2} \\ Y_a&<\infty, \quad Y_b<\infty. \tag{H3}\end{aligned} \] For each of these cases qualitative behavior is analyzed and several examples are given.
scientific article; zbMATH DE number 5637483