The oscillatory property of solutions to the equation \[ {d\over dt} x(t)+ r[1+ x(t)] \prod^m_{j= 1} x^{\alpha_j}(t- \tau_j)= 0\tag{1} \] with the initial condition \[ x(t)= \phi(t)\geq - 1,\quad t\in [t_0- \tau, t_0],\tag{2} \]
\[ \phi\in C([t_0- \tau, t_0],[- 1,\infty))\quad \text{and} \quad \phi(t_0)>-1 \] is considered, where \(\tau= \max\{\tau_1,\dots, \tau_m\}\); \(r, \tau_j\in (0, \infty)\), \(\alpha_j= p_j/q_j\) are rational numbers with \(q_j\) odd, \(p_j\), \(q_j\) co-prime for \(1\leq j\leq m\).
It is shown that: 1) All solutions are oscillatory when \(\sum^m_{j= 1} \alpha_j< 1\). 2) There exists at least one non-oscillatory solution when \(\sum^m_{j= 1} \alpha_j> 1\). 3) When \(\sum^m_{j= 1} \alpha_j= 1\) every solution of (1) with (2) oscillates if and only if every solution of the equation \[ {d\over dt} y(t)+ r \prod^m_{j= 1} y^{\alpha_j}(t- \tau_j)= 0 \] oscillates.