Oscillation of solutions of nonlinear difference equation with a super-linear neutral term (Q1635303)

The authors consider the nonlinear neutral difference equation of the form \[ \Delta(a_n\Delta(x_n+p_nx_{n-k}^\alpha))+q_nx_{n+1-l}^\beta=0, \] where \(a\) is a positive real sequence, \(p,q\) are nonnegative real sequences, \(p_n<1\), \(k\) is a positive integer, \(l\) is a nonnegative integer, \(\alpha\geq 1\) and \(\beta\) are ratios of odd positive integers. It is assumed that \(\sum_{n=n_0}^\infty 1/a_n<\infty\). Sufficient conditions which guarantee oscillation of all solutions of the equation are established. The proof is by contradiction with the help of a Riccati-type substitution.