Oscillation criteria for first-order neutral differential equations (Q1861843)

The author studies the first-order neutral differential equation \[ \frac{d}{dt}\left[x(t)-\sum_{i=1}^{n}p_i(t)x(t-\tau_i(t))\right] +\sum_{j=1}^{m}q_j(t)x(t-\sigma_j(t))=0,\quad t\geq t_0, \tag{1} \] with \(p_i, q_j, \tau_i, \sigma_j\in C([t_0,\infty), [0,\infty))\), \(\lim_{t\to\infty}(t-\tau_i(t))=\infty\), \(\lim_{t\to\infty}(t-\sigma_j(t))=\infty\) for \(i=1,2,\dots ,n\), \(j=1,2,\dots ,m\). A series of sufficient conditions are established for all solutions to equation (1) to be oscillatory. Some of the conditions are also necessary in the case when the coefficients \(p_i(t),\;q_j(t)\) and the deviations \(\tau_i(t),\;\sigma_j(t)\) are constant.