OSCILLATION OF THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

The aim of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear delay differential equation (E) ˆ a(t) ˆ x ′′ (t) ˜ γ ˜ ′ + q(t)f (x [τ (t)]) = 0. Applying suitable comparison theorems we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of (E). Obtained results essentially improve and complement earlier ones. Various examples are considered to illustrate the main results.


Introduction
We are concerned with oscillatory behavior of the third-order functional differential equations of the form (E) a(t) x ′′ (t) γ ′ + q(t)f (x [τ (t)]) = 0 In the sequel we will assume that the following conditions are always satisfied throughout this paper: (H1) a(t), q(t) ∈ C([t 0 , ∞)), a(t), q(t) are positive, τ (t) ∈ C([t 0 , ∞)), τ (t) ≤ t, By a solution of Eq.(E) we mean a function x(t) ∈ C 2 [T x , ∞), T x ≥ t 0 , which has the property a(t)(x ′′ (t)) γ ∈ C 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.Equation (E) is said to be oscillatory if all its solutions are oscillatory.
Following Tanaka [23] we say that a nontrivial solution x(t) of (E) is strongly decreasing if it satisfies (1.1) x(t)x ′ (t) < 0 EJQTDE, 2010 No. 43, p. 1 for all sufficiently large t and it said to be strongly increasing if Recently differential equations of the form (E) and its special cases have been the subject of intensive research (see enclosed references).Grace et al. in [9] have established a useful comparison principle for studying properties of (E).They have compared Eq.(E) with a couple of the first order delay differential equations in the sense that we deduce oscillation of Eq.(E) from the oscillation of this couple of equations.
Dzurina and Baculikova in [5] improve their results for the case when Zhong et al. in [24] adapted Grace et al.'s method and extended some of their results to neutral differential equation So that again from oscillation of a suitable first order delay equation we deduce oscillation of (E 1 ) On the other hand, Saker and Dzurina in [21] studied a particular case of Eq.(E), namely the differential equation They presented conditions under which every nonoscillatory solution of (E 2 ) tends to zero as t → ∞.Those results are applicable even if the criteria presented in [9] fail.
It is useful to notice that for a very special case of (E), that is, for Hartman and Wintner in [11] have derived that (E 3 ) always has a strongly decreasing solution.Thus, the effort for obtaining criteria for all nonoscillatory solutions to be strongly decreasing appeared.Therefore, from all above mentioned results, we conclude that if the gap between t and τ (t) is small, then there exists a nonoscillatory solution of (E) and the Theorems from [9] are not applicable to deduce oscillation of (E).In this case, our goal is to prove that every nonoscillatory solution of (E) tends to zero as t → ∞.While if the difference t − τ (t) is large enough, then we can study the oscillatory character of (E).
So our aim of this article is to provide a general classification of oscillatory and asymptotic behavior of the studied equation.We present criteria for (E) to be oscillatory or for every its nonoscillatory solution to be either strictly decreasing or tend to zero as t → ∞.
At first we turn our attention to Theorem 2.2 from [9], which is the main result of the paper.Formulation of Theorem 2.2 in [9] does not match its proof and for all that we provide a corrected version of the theorem.EJQTDE, 2010 No. 43, p. 2 Theorem A. [Theorem 2.2 in [9]] Let (H1) hold and assume that there exist two functions ξ(t) and η(t and both the first order delay equations for any constant c, 0 < c < 1, and T ≥ t 0 , and are oscillatory, then every solution of Eq.(E) is oscillatory.
Remark 1.In the formulation of Theorem 2.2 in [9], there is an excess term f (g(t)) included in Eq. (E I ).Now, the reader can easily reconstruct the results from [9] pertaining to Theorem 2.2.

Main results
We start our main results with the classification of the possible nonoscillatory solutions of (E).Lemma 1.Let x(t) be a positive solution of (E).Then either (i) x ′′ (t) > 0, eventually and x(t) is either strongly increasing or strongly decreasing, or (ii) x ′′ (t) < 0, eventually and x(t) is strongly increasing.
Proof.Let x(t) be a nonoscillatory solution of Eq.(E).We may assume that x(t) > 0, eventually (if it is an eventually negative, the proof is similar).Then [a(t) [x ′′ (t)] γ ] ′ < 0, eventually.Thus, a(t) [x ′′ (t)] γ is decreasing and of one sign and it follows from hypothesis (H1) and (H2) that there exists a t 1 ≥ t 0 such that x ′′ (t) is of fixed sign for t ≥ t 1 .If we have x ′′ (t) > 0, then x ′ (t) is increasing and then either (1.1) or (1.2) hold, eventually.
On the other hand, if x ′′ (t) < 0 then x ′ (t) is decreasing, hence x ′ (t) is of fixed sign.If we have x ′ (t) < 0, then lim t→∞ x(t) = −∞.This contradicts the positivity of x(t).Whereupon x ′ (t) > 0. The proof is complete.
The following criterion eliminates case (ii) of Lemma 1.
Lemma 2. Let x(t) be a positive solution of (E).If EJQTDE, 2010 No. 43, p. 3 Proof Let x(t) be a positive solution of Eq.(E).We assume that x(t) satisfies case (ii) of Lemma 1.That is x ′′ (t) < 0 and x ′ (t) > 0, eventually.Then there exist a t 1 ≥ t 0 and a constant k, for t ≥ t 2 ≥ t 1 .Now Eq.(E), in view of (H4) and (2.2), implies An integration of this inequality yields On the other hand, since a −1/γ (s)ds.
Combining (2.5) together with (2.3), and taking into account (H3), we get Letting t → ∞ we get a contradiction to condition (2.1).Therefore, we have eliminated case (ii) of Lemma 1.Now we are prepared to provide oscillation and asymptotic criteria for solutions of Eq.(E).

Theorem 1. Let (2.1) hold. If the first order delay equation
is oscillatory, then every solution of Eq.(E) is either oscillatory or strongly decreasing.
Proof.Let x(t) be a nonoscillatory solution of Eq.(E).We may assume that x(t) > 0 for t ≥ t 0 .From Lemma 2 we see that x ′′ (t) > 0 and x(t) is either strongly increasing or strongly decreasing.
EJQTDE, 2010 No. 43, p. 4 Assume that x(t) is strongly increasing, that is x ′ (t) > 0, eventually.Using the fact that a(t) [x ′′ (t)] γ is decreasing, we are lead to Integrating (2.7) from t 1 to t, we have There exists a t 2 ≥ t 1 such that for all t ≥ t 2 , one gets where y(t) = a(t)(x ′′ (t)) γ .Combining (2.8) together with (E), we see that where we have used (H3).Thus, y(t) is a positive and decreasing solution of the differential inequality Hence, by Theorem 1 in [19] we conclude that the corresponding differential equation (E 4 ) also has a positive solution, which contradicts to oscillation of (E 4 ).Therefore x(t) is strongly decreasing.Adding an additional condition, we achieve stronger asymptotic behavior of nonoscillatory solutions of Eq.(E).

Lemma 3. Assume that x(t) is a strongly decreasing solution of Eq.(E). If
(2.9) Proof.We may assume that x(t) is positive.It is clear that there exists a finite lim t→∞ x(t) = ℓ.We shall prove that ℓ = 0. Assume that ℓ > 0.
Integrating Eq.(E) from t to ∞ and using x[τ (t)] > ℓ and (H3), we obtain , where ℓ 1 = f 1/γ (ℓ) > 0. Integrating the last inequality from t to ∞, we get Now integrating from t 1 to t, we arrive at Letting t → ∞ we have a contradiction with (2.9) and so we have verified that lim t→∞ x(t) = 0.
Combining Theorem 1 and Lemma 3 we get:

1) and (2.9) holds. If the equation (E 4 ) is oscillatory then every solution of Eq.(E) is oscillatory or tends to zero as t → ∞.
For a special case of Eq.(E), we have: Corollary 1. Assume that (2.9) holds and Assume that β is a quotient of odd positive integers.If the delay equation is oscillatory then every solution of the equation is oscillatory or tends to zero as t → ∞.
In Theorems 1 and 2 and Corollary 1 we have established new comparison principles that enable to deduce properties of the third order nonlinear differential equation (E) from oscillation of the first order nonlinear delay equation (E 4 ).Consequently, taking into account oscillation criteria for (E 4 ), we immediately obtain results for (E).
Corollary 2. Assume (2.9) and is oscillatory or tends to zero as t → ∞.
Proof.Condition (2.12) (see Theorem 2.1.1 in [16]) guarantees oscillation of (E 6 ) with β = γ.Now we eliminate the strongly decreasing solutions of (E) to get an oscillation result.We relax condition (2.9) and employ another one.Our method is new and complements the one presented in [9].Theorem 3. Let (2.1) hold and τ ′ (t) > 0. Assume that there exist a function If both first order delay equations (E 4 ) and are oscillatory, then Eq. (E) is oscillatory.
Proof.Let x(t) be a nonoscillatory solution of Eq.(E).We may assume that x(t) > 0. From Theorem 1, we see that x(t) is strongly decreasing (i.e., x ′ (t) < 0).Integration of (E) from t to ξ(t) yields Then Integrating from t to ξ(t) once more, we get Finally, integrating from t to ∞, one gets Let us denote the right hand side of (2.14) by z(t).Then z(t) > 0 and one can easily verify that z(t) is a solution of the differential inequality Then Theorem 1 in [19] shows that the corresponding differential equation (E 8 ) has also a positive solution.This contradiction finishes the proof.
When choosing ξ(t) we are very particular about two conditions ξ(t) > t and τ (ξ(ξ(t))) < t hold.Unfortunately there is no general rule how to choose function ξ(t) to obtain the best result for oscillation of (E 8 ).We suggest for function ξ(t) "to be close to" the inverse function of τ (t).In the next example the reader can see the details.

Example 1. Let us consider third order differential equation
It is easy to verify that (2.In this paper we provide a full classification of nonoscillatory solutions of (E).Our partial results guarantee described asymptotic behavior of all nonoscillatory solutions (boundedness, convergence to zero and nonexistence).Our criteria improve and properly complement known results even for simple cases of (E).Our conclusions are precedented by illustrative examples that confirm upgrading of known oscillation criteria.If we apply known/new criteria for both nonlinear first order equations (E 4 ) and (E 8 ) to be oscillatory, we obtain more general criteria for asymptotic properties of nonlinear third order equation (E).