Oscillation criteria for the higher order nonlinear neutral difference equations with several variable delay arguments (Q2781731)

This paper is concerned with the difference equation NEWLINE\[NEWLINE\Delta^d \left(a_nx_n- \sum^m_{i=1} b_i(n)x_{n-k_i} \right)= \sum^s_{j=1} f_j(n,x_n, x_{ n-\tau_j(n)}) \tag{1}NEWLINE\]NEWLINE for \(n\in N=\{0,1,2, \dots\}\), where \(k_i\), \(\tau_j (n) \in N\) for \(i=1,2, \dots,m\) and \(j=1, \dots,s\). Under a certain set of conditions on the coefficient sequences, the delays and the functions \(f_j\), it is shown that (1) is oscillatory if, and only if, the following functional relation NEWLINE\[NEWLINE\Delta^d\left( a_nx_n-\sum^m_{i=1} b_i(n)x_{n-k_i}\right)\geq \sum^s_{ s=1} f_j(n,x_n, x_{n-\tau_j(n)})NEWLINE\]NEWLINE does not have any eventually negative solutions. It is also shown that under another set of conditions, (1) is oscillatory.