We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.
Neutral differential equations are one of the most important extensions of classical ordinary differential equations and aim to give a better explanation for modeling phenomena where ordinary differential equations are insufficient. Naturally, all the questions studied in the scope of ordinary differential equations attracted the attention also for neutral differential equations. In this paper we study the oscillatory properties of second order half-linear neutral differential equations. We present oscillation criteria derived using a new approach. This approach allows us to reduce common restrictions on the deviations in arguments which are present in the currently known results of this type.
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