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We study oscillatory properties of solutions of the Emden-Fowler type differential equation $$u^{\left(n\right)}\left(t\right)+p\left(t\right){\left|u\left(\sigma \left(t\right)\right)\right|}^{\lambda }signu\left(\sigma \left(t\right)\right)=0,$$ where $0<\lambda <1$, $p\in L_{\mathrm{loc}}({ℝ}_{+};ℝ)$, $\sigma \in C({ℝ}_{+};{ℝ}_{+})$ and $\sigma \left(t\right)\ge t$ for $t\in {ℝ}_{+}$. Sufficient (necessary and sufficient) conditions of new type for oscillation of solutions of the above equation are established. Some results given in this paper generalize the results obtained in the paper by Kiguradze and Stavroulakis (1998).

Sufficient conditions are established for the oscillation of proper solutions of the system $$\begin{array}{cc}\hfill u_1^{\text{'}}\left(t\right)& =p\left(t\right)u_2\left(\sigma \left(t\right)\right),\hfill \\ \hfill u_2^{\text{'}}\left(t\right)& =-q\left(t\right)u_1\left(\tau \left(t\right)\right),\hfill \end{array}$$ where $p,q:R_{+}\to R_{+}$ are locally summable functions, while $\tau$ and $\sigma :R_{+}\to R_{+}$ are continuous and continuously differentiable functions, respectively, and $\underset{t\to +\infty }{lim}\tau \left(t\right)=+\infty$, $\underset{t\to +\infty }{lim}\sigma \left(t\right)=+\infty$.