Oscillation Results for Third-Order Semi-Canonical Quasi-Linear Delay Differential Equations

: The main purpose of this paper is to study the oscillatory properties of solutions of the third-order quasi-linear delay differential equation where L y ( t ) = ( b ( t )( a ( t )( y ′ ( t )) α ) ′ ) ′ is a semi-canonical differential operator. The main idea is to transform the semi-canonical operator into canonical form and then obtain new oscillation results for the studied equation. Examples are provided to illustrate the importance of the main results.

From the review of literature, it is clear that most of the papers are dealing with the oscillatory and asymptotic behavior of solutions of (1.1) under the canonical type condition see, for example [1, 2, 5-7, 9, 13-16, 18, 21, 29, 30, 32, 34] and the references cited therein. This is due to the fact that the examination of canonical type equation is much simpler than non-canonical equations. Therefore in this paper we rst introduce a technique that connect the semi-canonical equation (1.1) with that of canonical type equation and then we examine the behavior of its solutions. While considering nonoscillatory solutions of (1.1), we can restrict our attention to positive ones. From the well-known results in [17,23,27], it follows that the set of positive solutions of (1.1) has the following structure: Lemma 1.1. Assume that y(t) is an eventually positive solution of (1.1), then y(t) satis es one of the following three options: eventually for all su ciently large t.
So, if we want to obtain oscillation criteria for semi-canonical equation (1.1), we have to eliminate three above mentioned cases, which may lead to three conditions. To overcome this, we assume a simple condition that yields to a canonical form and this essentially simpli es the examination of (1.1).

Main Results
Throughout the paper, we employ the following notations:

1)
then the semi-canonical operator Ly has the following unique canonical representation Proof. Direct calculation shows that This shows that (2.2) is in the canonical form, that is, . However Trench proved in [32] that there exists the only one canonical representation of L (up to multiplicative constants with product 1) and so our canonical form is unique. This completes the proof. Now it follows from Theorem 2.1 that (1.1) can be written in the canonical form as and in this case we say y ∈ N or and in this case we denote that y ∈ N .

4)
and Proof. The proof is similar to that of [13,Lemma 2] and [8, Lemma 3.1] and therefore skipped.
Our next lemma is a particular case (n = , m = ) of Corollary 1 of Philos [28].

lim t→∞ σ(t) = ∞ and σ(t) < t for every t ≥ t . If y is a positive bounded solution of the di erential inequality
then there exists a positive solution x of the di erential equation such that x(t) ≤ y(t) for all large t, and lim t→∞ x(t) = monotonically.
In the remaining part of the paper, we always assume condition (2.1) holds without further mention.
then every positive solution y(t) of (1.1) does not satisfy N .
Proof. Assume that y(t) is an eventually positive solution of (1.1). By Corollary 2.3, the function y(t) is a positive solution of (2.3). Now assume to the contrary that y(t) ∈ N . Integrating (2.3) from s to t yields Again integrating twice from s to t, one gets By setting s = σ(t) and α = β, we obtain a contradiction to (2.6). This completes the proof. Proof. Let y(t) be an eventually positive solution of (1.1). Then by Corollary 2.3, the function y(t) is also a positive solution of (2.3). Now assume to the contrary that y(t) ∈ N . Since d(t)(c(t)(y ′ (t)) α ) ′ is positive and decreasing, we can verify that Integrating again from t to t yields Using this in (2.3), we see that w(t) = d(t)(c(t)(y ′ (t)) α ) ′ is a positive solution of the di erential inequality This is a contradiction since by Theorem 2.
Since w(t) is a positive bounded solution of the last inequality then by Lemma 2.5, we see that the corresponding equation has also a positive solution. But by Theorem 3.9.3 of [17], the condition (2.10) implies that w(t) is oscillatory. This contradiction completes the proof of the theorem.
Theorem 2.10. Assume that there exists a function ζ (t) ∈ C ′ ([t , ∞)) such that If for all su ciently large t ≥ t , the rst order delay di erential equations and Proof. Let y(t) be an eventually positive solution of (1.1). It follows from Corollary 2.3 that y(t) is also a positive solution of (2.3) and either y(t) ∈ N or y(t) ∈ N . If y(t) ∈ N , then by using the fact that Integrating from t to t, we are led to Hence Combining the last inequality together with (2.3), we obtain Therefore, it is clear that w(t) is a positive solution of di erential inequality w ′ (t) + Q (t)w β/α (σ(t)) ≤ for t ≥ t . Since w(t) is a positive bounded solution of the last inequality and therefore by Lemma 2.5, we conclude that there exists a positive solution w(t) of equation (2.12) with lim t→∞ w(t) = , which contradicts the fact that equation (2.12) is oscillatory. Next, we assume that y(t) ∈ N . An integration of (2.3) from t to ζ (t) yields Then Integrating the last inequality from t to ζ (t), we have Dividing the last inequality by c(t) and then integrating it from t to ∞, one gets Set the right-hand side of the last inequality by w(t). Then y(t) ≥ w(t) > and it is easy to verify that Since w(t) is a positive bounded solution of the last inequality, then by Lemma 2.5 we see that the corresponding di erential equation (2.13) has also a positive solution with lim t→∞ w(t) = . This contradicts the assumption that (2.13) is oscillatory, and hence we conclude that (1.1) oscillates. This completes the proof of the theorem.
Using (2.4) in the above inequality yields η α (σ(t)) ∞ t F(s)ds ≤ which contradicts (2.19) as t → ∞. The proof for the case y(t) ∈ N is similar to that of Theorem 2.6. The proof is now completed.

Examples
In this section, we present two examples to illustrate our main results.