In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form \[ y^{\prime }_i(t)-p_i(t)y_{i+1}(t)=0, \quad i=1,2,\dots , n-2, y^{\prime }_{n-1}(t)-p_{n-1}(t)|y_n(h_n(t))|^\alpha \mathop {\mathrm sgn}[y_n(h_n(t))]=0, y^{\prime }_n(t) \mathop {\mathrm sgn}[y_1(h_1(t))]+p_n(t)|y_1(h_1(t))|^\beta \, \le 0, \] where $ n\ge 3 $ is odd, $ \alpha >0$, $ \beta >0$.
The purpose of this paper is to obtain oscillation criteria for the differential system \[ \begin{aligned}{[y_1(t)-a(t)y_1(g(t))]}^{\prime}&=p_1(t)f_1(y_2(h_2(t))) \\ y_2^{\prime }(t)&=p_2(t)f_2(y_3(h_3(t))) \\ y_3^{\prime }(t)&= - p_3(t)f_3(y_1(h_1(t))), \quad t\in \mathbb R_+=[0,\infty ).\end{aligned} \].