Abstract. Oscillation criteria generalizing a series of earlier results are established for first order linear delay differential inequalities and equations.

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Sufficient conditions for the nth order linear differential equation 𝑢(𝑛) (𝑡) + 𝑝(𝑡)𝑢(τ(𝑡)) = 0, 𝑛 ≥ 2, to have Property 𝐴 or Property 𝐵 are established in both the delayed and the advanced cases. These conditions essentially improve many known results not only for differential equations with deviating arguments but for ordinary differential equations as well.

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Abstract. In this paper the Sturmian Comparison Method is developed for the first order neutral differential equation of the type(1)Using this method, a general theorem is proved on the location of zeros of (1), which is then applied to obtain two concrete results. The first one of them turns out to be the best possible in the case where P and Q are constants. The second one is concerned, for the first time, with the oscillation theory of first order neutral differential equations, in the case where the coefficient Q(t) is oscillatory.Mathematics subject classification (2000): 34K11, 34K40.