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Abstract. The following aspects of oscillation theory for Delay-difference equation Mathematics subject classification (1991): 39A10.

A solution of a linear delay differential equation can have an infinite number of isolated zeros on a finite interval. As is well known, for ordinary differential equations this is impossible.As is well known, there exist upper and lower bounds for the distance between two adjacent zeros for a solution of the linear ordinary differential equationThe simplest upper bound is the following: b − a ≤ π µ , where q(t) ≥ µ 2 . Lower bounds can be obtained by Lyapunov's theorem (see Cor. 5In particular, this inequality implies the number of zeros for any solution of (1) is finite on any finite interval (see [1], Chap. IV).The situation for delay differential equations (both first and second order) is absolutely different. For these equations the phenomenon of "sticking together" for two solutions on some interval is possible.

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.

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