Oscillation theorems for a second-order nonlinear neutral difference equation (Q2719303)

This paper is concerned with the difference equation NEWLINE\[NEWLINE\Delta^2 (x_n-px^\alpha_{n- \tau})+ q_nx^\beta_{n- \sigma}=0,NEWLINE\]NEWLINE where \(\tau,\sigma\) are positive integers, \(\alpha,\beta\) are quotients of odd positive integers, \(\{q_n\}\) is a real sequence and \(p\geq 0\). Sufficient conditions in terms of \(\tau, \sigma,\alpha, \beta,p\) and \(\{q_n\}\) are derived for (1) every solution to oscillate, (2) every bounded solution to oscillate, and (3) the existence of an eventually positive solution that converges to zero.