INTEGRAL OSCILLATION CRITERIA FOR THIRD-ORDER DIFFERENTIAL EQUATIONS WITH DELAY ARGUMENT

In the present paper, some new criteria for property A and the oscillation of third order nonlinear delay differential equations of the type � a(t) h b(t)y ' (t) � ' i�' + p(t)f (y(�(t))) = 0.

We further assume that (E) is in canonical form, that is, By a solution of equation (E), we mean a function y(t) ∈ C 1 ([T y , ∞)), T y ≥ t 0 , which has the property a(t) (b(t)y ′ (t)) ′ γ ∈ C 1 ([T y , ∞)) satisfies the equation (E) on [T y , ∞).We consider only those solutions y(t) of (E) which satisfy sup{|y(t)| : t ≥ T } > 0 for all T ≥ T y .We assume that (E) possesses such a solution.A solution of (E) is called oscillatory if it has arbitrarily large zeros on [T y , ∞) and otherwise, it is called to be nonoscillatory.An equation is said to be oscillatory if all its solutions are oscillatory.
Differential equations of third order have long been considered as valuable tools in the modeling of many phenomena in different areas of applied mathematics and physics.For instance, such equations are encountered in the study of entry-flow phenomenon [4], the propagation of electrical pulses in the nerve of a squid approximated by the famous Nagumo's equation [10], the feedback nuclear reactor problem [12], the regulation of a steam turbine [8] and so on.
Hence, a great deal of work has been done in recent decades and the investigation of oscillatory and asymptotic properties for these equations has taken the shape of a well-developed theory turned mainly toward functional differential equations.In fact, the development of oscillation theory for the third order differential equations began in 1961 with the appearance of the work of Hanan [3] and Lazer [7].Since then, many authors contributed to the subject studying different classes of equations and applying various techniques.A systematic survey of the most significant efforts in this theory can be found in the excellent monographs of Swanson [11], Greguš [2] and the very recent-one of Padhi and Pati [9].
Motivated by recent oscillation results of Koplatadze [6] exploited for higherorder differential equations with deviating argument of the type we derive in the paper some useful monotonic properties of nonoscillatory solutions which permit us to achieve such new sufficient conditions for (E) to have property A or to be oscillatory that are different from most known.
As is convenient, we state here that all functional inequalities considered in this article are assumed to hold eventually, i.e., they are satisfied for all t large enough.

Preliminary Results
We start with the classification of possible nonoscillatory solutions of (E).Without loss of generality we can deal only with eventually positive solutions of (E).
Lemma 1. Assume that y(t) is an eventually positive solution of (E).Then y(t) satisfies one of the following conditions eventually.
Proof.The proof follows immediately from the canonical form of (E) and so we omit it.
The following result presents very useful monotonic properties of nonoscillatory solutions of (E).For a sake of brevity, we define the functions where t * is large enough.
Lemma 2. Let y(t) be a positive solution of (E) satisfying (N 2 ) and Proof.Assume on the contrary that (E) possesses an eventually positive solution y(t) satisfying (N 2 ) for t ≥ t * .It follows from Lemma 1 that ds. ( This yields So, we deduce On the other hand, since b(t)y ′ (t) is increasing for any t ≥ t * , it is easy to see that Therefore, there exists a t 2 > t 1 such that for any t ≥ t 2 Using this fact, we arrive at eventually and we conclude that y(t) B(t) is increasing.The proof is complete.
The following result is elementary but useful in what comes next.
To simplify our formulations of the main results, we recall the following definition.
Definition 1.We say that (E) enjoys property A if every its nonoscillatory solution satisfies (N 0 ) .Property A of third order differential equations has been widely studied in the literature, see [1,5] and references cited therein.

Criteria for Property A of (E)
Employing our lemmas, we provide in this section several limsup type criteria for (E) to have property A.
Proof.Assume on the contrary that (E) possesses an eventually positive solution y(t) satisfying (N 2 ), t ≥ t * .An integration of (E) from t to ∞ yields b(t)y ′ (t)

Taking the monotonicity properties (i) -(iii) of Lemma 2 and (H 3 ) into account, one can verify that
which in view of (H 3 ) yields Taking lim sup as t → ∞ on both sides of the previous inequality, we are led to the contradiction with assumptions of the theorem.The proof is complete.
The criterion obtained covers super-linear and half-linear case of (E).In the following corollaries, it is always assumed that δ is the ratio of odd positive integers.
Corollary 3. Let (2) hold, γ ∈ (0, 1 and then the differential equation then the differential equation We are about to provide another criterion for property A that is applicable when (E) is of sub-linear type.
Assume the the contrary, that is lim t→∞ y(t) C(t) = ℓ > 0. By the L'Hospital rule Combining ( 3), ( 4) and (11), one gets On the other hand, an integration of (E) from t * to ∞ yields which in view of (12) gives This contradicts with (9) and we conclude that y(t)/C(t) → 0 as t → ∞.

Now, setting
the condition (8) together with (H 3 ) implies Taking lim sup as t → ∞ on both sides of the previous inequality, we are led to contradiction with assumptions of our theorem.The proof is complete.
For the half-linear case f (u) = u γ , Theorem 2 reduces to Corollary 2, while in sub-linear case, we have the following result.

Oscillation of (E)
Due to the presence of the delay argument, we are also able to eliminate possible eventually positive solutions of (N 0 )-type and so attain oscillation of the equation (E).
γ (u) t u p(s) dsdu 1/γ dv > K 3 , (E x).For e.g.λ = 1/3 it occurs if k > 0.057703.And really, in the opposite cases if e.g.k = 0.0282, then (E x ) has not property A, since it possesses a positive solution y(t) = t 0,766 satisfying (N 2 ) .Moreover, taking Theorem 9 into account, we are also able to eliminate positive solutions satisfying (N 0 ) provided that