ASYMPTOTIC AND OSCILLATORY BEHAVIOR OF HIGHER ORDER QUASILINEAR DELAY DIFFERENTIAL EQUATIONS

In the paper, we offer such generalization of a lemma due to Philos (and partially Staikos), that yields many applications in the oscillation theory. We present its disposal in the comparison theory and we establish new oscillation criteria for n−th order delay differential equation (E) ` r(t) ˆ x ′(t) ̃ γ  ́(n−1) + q(t)xγ(τ (t)) = 0. The presented technique essentially simplifies the examination of the higher order differential equations.


Introduction
In this paper, we shall study the asymptotic and oscillation behavior of the solutions of the higher order delay differential equations (E) r(t) x ′ (t) γ (n−1) + q(t)x γ (τ (t)) = 0.
Whenever, it is assumed By a solution of Eq. (E) we mean a function x(t) ∈ C 1 ([T x , ∞)), with T x ≥ t 0 , which has the property r(t)(x ′ (t)) γ ∈ C n−1 ([T x , ∞)) and satisfies Eq. (E) on [T x , ∞).We consider only those solutions x(t) of (E) which satisfy sup{|x(t)| : t ≥ T } > 0 for all T ≥ T x .We assume that (E) possesses such a solution.A solution of (E) is called oscillatory if it has arbitrarily large zeros on [T x , ∞) and otherwise it is called to be nonoscillatory.An equation itself is said to be oscillatory if all its solutions are oscillatory.
The problem of the oscillation of higher order differential equations has been widely studied by many authors, who have provided many techniques for obtaining oscillatory criteria for studied equations (see e.g.[1] - [19]).
This lemma essentially simplifies the examination of n − th order differential equations of the form since it provides needed relationship between y(t) and y (n−1) (t) and this fact permit us to establish just one condition for oscillation of (1.2).This lemma is not applicable to differential equation (E).In this paper we offer a generalization of Lemma A, which works for (E) and permits to establish new oscillation criteria for it.

Main Results
The following result is a well-known lemma of Kiguradze see e.g.[6] or [14].
Lemma 2. Let z(t) be as in Lemma 1 and numbers t 1 and ℓ be assigned to z(t) by Lemma 1. Then for 2 Proof.Let ℓ be the integer assigned to function z(t) as in Lemma 1. Assume that ℓ < k − 1, then for any s, t with t ≥ s ≥ t 1 , we have Repeated integration in s from s to t yields It is easy to see that (2.4) holds also for ℓ = k − 1.
On the other hand, if ℓ ≥ 2, then for every t ≥ t 1 , we have Repeated integration from t 1 to t leads to or simply Integrating the last inequality from t 1 to t, we get (2.2).We have verified the first part of the lemma.Now assume that ℓ = 1.It follows from (2.4) that On the other hand, Proof.Note that r ′ (t) ≥ 0 implies that r −1/γ (t) is nonincreasing.Assume that ℓ is the integer associated with 2), we have It is easy to see that for any λ ∈ (0, 1) there exists a t λ ≥ t 1 such that t − t 1 ≥ λ γ/(k−2+γ) t for t ≥ t λ , which in view of (2.9) yields (2.8).
If ℓ = 0, then for any s, t Repeated integration in s from s to t yields An integration from s to t, yields Therefore, The proof is complete now.

Applications
To present usefulness of Lemma 2 and Lemma 3, we apply both to establish new oscillatory results for (E), based also on comparison principles.
Theorem 1. Assume that the first order delay differential equation is oscillatory.Moreover, for n-even the first order delay differential equation is oscillatory and for n-odd condition
If 2 ≤ ℓ ≤ n − 1, Then by Lemma 2 2) is positive and Setting to (E), we see that y(t) is a positive solution of the delay differential inequality EJQTDE, 2012 No. 89, p. 5 By Theorem 1 in [15] the corresponding equation (E 1 ) has also a positive solution.A contradiction.
If ℓ = 1, which is possible only when n is even, Lemma 2 implies and proceeding as above, we find out that (E 2 ) has a positive solution.A contradiction and the proof is finished for n even.
Assume that ℓ = 0, note that it is possible only of n is odd.Since x ′ (t) < 0, then there exists a finite lim t→∞ x(t) = c ≥ 0. We claim that c = 0.
If not, that x(τ (t)) ≥ c > 0, eventually, let us say for t ≥ t 2 .An integration of (E) from t to ∞ yields Integrating again from t 2 to ∞, we get du which contradicts (P 0 ).The proof is complete.
Employing any result (e.g.Theorem 2.1.1 in [14]) for the oscillation of (E 1 ) and (E 2 ), we immediately obtain criteria for studied properties of (E).
Moreover, for n-odd assume that (P 0 ) hold.Then (i) for n even, (E) is oscillatory; (ii) for n odd, each nonoscillatory solution of (E) satisfies lim t→∞ x(t) = 0.
The results of Theorem 1 and Corollary 1 can be simplified provided that we impose additional condition on the function r(t).
Proof.Assume that x(t) is an eventually positive solution of (E).Then (r(t) [x ′ (t)] γ ) (n−1) < 0 and there exist a t 1 ≥ t 0 and an integer ℓ with n + ℓ odd such that (2.1) holds.If n is odd suppose that lim t→∞ x(t) = 0 (for n is even this is obvious).Then it follows from Lemma 3 that .
By Theorem 1 in [15] the corresponding equation (E 3 ) has also a positive solution.A contradiction. (ii) for n odd, every nonoscillatory solution x(t) of (E) satisfies lim t→∞ x(t) = 0.