Oscillation criteria for first order neutral differential equations with positive and neagative coefficients (Q5927561)

The authors deal with the first-order neutral differential equation NEWLINE\[NEWLINE \frac{d}{dt}[x(t)-R(t)x(t-r)]+P(t)x(t-\tau)-Q(t)x(t-\delta)=0, \tag{1} NEWLINE\]NEWLINE with \(P,Q,R\in C([t_0,\infty),\mathbb{R}^+)\), \(r\in (0,\infty)\), \(\tau,\delta\in \mathbb{R}^+\), \(\tau\geq\delta\), \(\overline P\equiv P(t)-Q(t-\tau+\delta)\geq 0\) and not identically zero. They derive, by some new technique, sufficient conditions for the oscillation of equation (1). Some examples are given to illustrate the advantage of the obtained results.