Oscillations of solutions of neutral differential equations with positive and negative coefficients (Q1600369)

The authors consider the neutral differential equation with positive and negative coefficients of the form \[ \frac{d}{dt}[y(t)-R(t)y(t-r)]+P(t)y(t-\tau)-Q(t)y(t-\sigma)=0, \] where \(r\in (0, \infty)\), and \(\tau , \sigma \in \mathbb{R}^+\) with \(\tau \geq\sigma \); \(P, Q, R\in C([t_0, \infty), \mathbb{R}^+), P(t)-Q(t-\tau+\sigma)\geq 0\) and not identically zero. Several new sufficient conditions for the oscillation of all solutions to the above equation are obtained without the following usual hypothesis: \(\int_{t_0}^{\infty}[P(s)-Q(s-\tau+\sigma)] ds=\infty\).