Existence of positive solutions for neutral difference equations with positive and negative coefficients (Q1847616)

The authors give sufficient conditions for the existence of eventually positive solutions to the neutral delay difference equation of the form \[ \Delta (x_n - R_n x_{n-r}) + P_n x_{n-r} - Q_n x_{n-\sigma} = 0,\quad n = 0, 1,\dots, \] where \(\Delta x_n = x_{n+1} - x_n,\) \(P_n \geq 0\), \(Q_n \geq 0\), \(R_n \geq 0,\) \(P_n - Q_{n-r+\sigma} \geq 0\) \((\not\equiv 0)\) \((n = 0, 1,\dots)\), \(\tau\), \(\sigma\) and \(r\) are integers with \(0 \leq \sigma < \tau\), \(r > 0.\)