Oscillation and nonoscillation criteria for the self-adjoint linear differential equation $${\left(t^{\alpha }{y}^{\text{'}\text{'}}\right)}^{\text{'}\text{'}}-\frac{{\gamma }_{2,\alpha }}{t^{4-\alpha }}y=q\left(t\right)y,\alpha \notin \{1,3\},$$ where $${\gamma }_{2,\alpha }=\frac{{(\alpha -1)}^2{(\alpha -3)}^2}{16}$$ and $q$ is a real and continuous function, are established. It is proved, using these criteria, that the equation $${\left(t^{\alpha }{y}^{\text{'}\text{'}}\right)}^{\text{'}\text{'}}-\left(\frac{{\gamma }_{2,\alpha }}{t^{4-\alpha }}+\frac{\gamma }{t^{4-\alpha }{ln}^{2}t}\right)y=0$$ is nonoscillatory if and only if $\gamma \le \frac{{\alpha }^2-4\alpha +5}{8}$.

This paper deals with the second order nonlinear neutral differential inequalities $\left(A_{\nu }\right)$: ${(-1)}^{\nu }x\left(t\right)\{z^{\text{'}\text{'}}\left(t\right)+{(-1)}^{\nu }q\left(t\right)f\left(x\left(h\left(t\right)\right)\right)\}\le 0,$$t\ge t_0\ge 0...$

The purpose of this paper is to obtain oscillation criteria for the differential system $$\begin{array}{cc}\hfill {[y_1\left(t\right)-a\left(t\right)y_1\left(g\left(t\right)\right)]}^{\text{'}}& =p_1\left(t\right)f_1\left(y_2\left(h_2\left(t\right)\right)\right)\hfill \\ \hfill y_2^{\text{'}}\left(t\right)& =p_2\left(t\right)f_2\left(y_3\left(h_3\left(t\right)\right)\right)\hfill \\ \hfill y_3^{\text{'}}\left(t\right)& =-p_3\left(t\right)f_3\left(y_1\left(h_1\left(t\right)\right)\right),t\in {ℝ}_{+}=[0,\infty ).\hfill \end{array}$$

A sufficient condition for the nonoscillation of nonlinear systems of differential equations whose left-hand sides are given by $n$-th order differential operators which are composed of special nonlinear differential operators of the first order is established. Sufficient conditions for the oscillation of systems of two nonlinear second order differential equations are also presented.

The purpose of this paper is to obtain oscillation criterions for the differential system of neutral type.