Oscillation of neutral difference equations with positive and negative coefficients (Q2750970)

The difference equation NEWLINE\[NEWLINE\Delta\bigl( x_n-R(n)x_{n-r}\bigr) + \sum^m_{i=1} P_i(n)x_{n-\tau_i} -\sum^l_{j=1} Q_j(n)x_{n- \sigma_j}=0,\;n=0,1, \dots, \tag{1}NEWLINE\]NEWLINE is considered, where \(\{R(n)\}\), \(\{P_1(n)\},\dots,\{P_m(n)\}\), \(\{Q_1(n)\},\dots, \{Q_l(n)\}\) are nonnegative sequences, and \(r>0\), \(\tau_1, \dots, \tau_m\geq 0\) and \(\sigma_1,\dots,\sigma_l\geq 0\). This equation is an extension of NEWLINE\[NEWLINE\Delta\bigl( x_n-R(n)x_{n-r} \bigr)+ P(n)x_{n- \tau}-Q(n)x_{n-\sigma} =0,NEWLINE\]NEWLINE which has been studied by several authors. By extending their techniques in straightforward manners, several oscillation criteria are obtained for (1).