OSCILLATION CRITERIA IN NEUTRAL EQUATIONS OF n ORDER WITH VARIABLE COEFFICIENTS

Consider the n -order neutral delay differential equation 
 d n d t n [ y ( t ) + P ( t ) y ( t − τ ) ] + Q ( t ) y ( t − σ ) = 0 
where P , Q ∈ C [ [ t 0 , ∞ ) , ℝ ] and the delays τ and σ are nonnegative real numbers. In this paper we 
examined the oscillatory behavior of the solutions of the above equation using techniques which 
allow the relaxation of the restrictions which has been introduced previously. We illustrate new 
type of conditions which improve and extend known results, by relaxing hypotheses that P is 
constant and Q is τ -periodic.

1. INTRODUCTION AND PRELIMINARIES.A neutral delay differential equation (NDDE for short) is a differential equation in which the higher order derivative of the unknown function appears in the equation both with and without delays.
On leave from Department of Electrical Engineering, Democritus University of Thrace, Xanthi 67100, Greece.
In this paper we also study the oscillatory behavior of the solutions of Eq. (1.1).In Section 2 we establish sufficient conditions for the oscillation of all solutions of Eq. (1) with n 1, by relaxing the restrictions which have been introduced in [1], [21, [3] and [41.In Section 3 we obtain several sufficient conditions for the oscillation of all solutions of Eq. (1) with n odd.This new type of conditions improve and extend results which have been established in [6], by relaxing the hypotheses that P is constant and Q is r-periodic.
Let p max {r,a}.By a solution of Eq. (1) we mean a function y E C[t p, oo),R], for some > 0, such that y(t) + P(t)y(t-r) is n times continuously differentiable on [tl,oo) and such that Eq. (1) is satisfied for > 1.
As usual, a solution of Eq. (1.1) is called oscillatory if it has arbitrarily large zeros and non- oscillatory if it is eventually positive or eventually negative.
In the sequel, for convenience, when we write an inequality without specifying its domain of validity we will assume that it holds for all large t.
2. SUFFICIENT CONDITIONS FOR THE OSCILLATION OF THE FIRST ORDER NDDE.Throughout this section, and without any further reference to, we consider Eq. (1.1) with n 1.We will establish sufficient conditions for the oscillation of all solutions of the first order NDDE (1), which differ from the corresponding conditions in [1], [3], [4] and [61 in terms of r and a.
The following lemma extracted from [1] will be utilized in the proofs of the main results in this section.
By noting the fact that if Q(t) is r-periodic then z(t) satisfies the equation z'(t) + P(t a)z'(t r) + Q(t)z(t a) 0 (2.12) and by an argument similar to that in the proof of the Theorem 1 one can see that the following result holds.

R(t_a)[z(t-
Then by combining (2.17) and (2.18) we are led to the inequality It-" O(,), z(t-)-R(t ).z(t-I m .)d" -< 0 z(t 3r) W z(t 2r a 2rRSJW r- i_ a) t-r--,s O(s W "r) or 2rR sj +z(t-'r- 1 t-a Q(s+ ds]<O.This is a contradiction and the proof is complete. By noting the fact that z(t) satisfies (2.11) for Q(t) r-periodic and by an argument similar to that in the proof of Theorem 3 we can obtain the following result.THEOREM 4. Assume that (1.2) and (2.1) hold, P(t) <_ 1, P'(t) <_ O, O(t) is r-p,riodic and Then every solution of Eq. (1.1) oscillates.
3. SUFFICIENT CONDITIONS FOR THE OSCILLATION OF ODD ORDER NDDE.
In this section we will establish some sufficient conditions for the oscillation of all solutions of Ee I. (1.1) with n odd.Throughout this section we will assume that n is odd and Q(t) is positive in Eq. (1.1) without specifying them.
First we present a lemma which is extracted from [5] and [6] and will be utilized in the proofs of the main results in this section.
(3.11) Suppose also there exists a positive number r such that Q(t) 1 + R(t + r a) <-r and rl/"(r a)/n > .
The following result is an immediate corollary of Theorem 5 which has been established in [6].COROLLARY 1. Assume that (1.2) and (3.8) hold, P(t) p < 1 and Q(t) is r-periodic.
PROOF.Assume that one of the conditions is satisfied and that, contrary to the conclusion of the theorem, Eq. (1.1) has an eventually positive solution y(t).Consider z(t) and w(t) as they have been defined by (3.2) and (3.9), respectively.Then by Lemma 2 (ii) and (iii), (3.6ab) holds.
The following result is an immediate corollary of Theorem 6.
the hypotheses of Theorem 5. Hence every solution of this equation oscillates.