#### Abstract

The objective of this paper is to offer sufficient conditions for certain asymptotic properties of the third-order functional differential equation , where studied equation is in a canonical form, that is, . Employing Trench theory of canonical operators, we deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.

#### 1. Introduction

We are concerned with the oscillatory and asymptotic behavior of all solutions of the third-order functional differential equations: In the sequel, we will assume and is the ratio of two positive odd integers,, , .

Moreover, we assume that is in a canonical form, that is,

By a solution of we mean a function ,, which has the property and satisfies on . We consider only those solutions of which satisfy for all . We assume that possesses such a solution. A solution of is called oscillatory if it has arbitrarily large zeros on and otherwise it is called to be nonoscillatory. Equation is said to be oscillatory if all its solutions are oscillatory.

Recently, and its particular cases (see enclosed references) have been intensively studied. We establish new comparison theorems that permit to study properties of via properties of the second-order differential equations, in the sense that the oscillation of the second-order equations yields desired properties of .

Our results complement and extend earlier ones presented in [1β23].

*Remark 1.1. *All functional inequalities considered in this paper are assumed to hold eventually; that is, they are satisfied for all large enough.

*Remark 1.2. *It is sufficient to deal only with positive solutions of .

#### 2. Main Results

We begin with the classification of the possible nonoscillatory solutions of .

Lemma 2.1. *Let be a positive solution of . Then satisfies, eventually, one of the following conditions:*(I)*, , ;*(II)*, , .*

*Proof. *The proof follows immediately from the canonical form of .

To simplify formulation of our main results, we recall the following definition:

*Definition 2.2. *We say that enjoys property (A) if all its positive solutions satisfy case (I) of Lemma 2.1.

Property (A) of has been studied by various authors; see enclosed references. We offer new technique for investigation property (A) of based on comparison theorems and Trench theory of canonical operators.

*Remark 2.3. *It is known that condition
implies property (A) of . Consequently, in the sequel, we may assume that the integral on the left side of (2.1) is convergent.

Now, we offer a comparison result in which we reduce property (A) of to the absence of certain positive solution of the suitable second-order inequality.

Theorem 2.4. * If the second-order differential inequality
**
has not any solution satisfying
**
then has property (A).*

*Proof. *Assuming the contrary, let be a solution of satisfying the Case (II) of Lemma 2.1. Using the monotonicity of , we see that
Then evaluating and then integrating from to , we are lead to
Setting to , we get
Integrating to , we see that satisfies
Let us denote the right hand side of (2.5) by . Then holds and moreover,
Consequently, is a solution of the differential inequality , which contradicts our assumption.

Since is in noncanonical form, we apply Trench theory [24] to transform it to canonical form, which is more suitable for investigation.

Denote

Theorem 2.5. * If the differential equation
**
is oscillatory, then has not any solution satisfying .*

*Proof. *Let be a positive solution of , such that holds. By direct computation, we can verify Trench result that the operator
is equivalent to
Therefore the differential inequality can be written in the form
Applying the substitution , we can see that is a positive solution of the differential inequality
Moreover, since
our inequality is in canonical form, but according to Theorem 2 of [18], we get that the corresponding differential equation has also a positive solution. A contradiction. The proof is complete.

Combining Theorems 2.4 and 2.5, we get the following criterion for property (A) of .

Theorem 2.6. * If the second-order differential equation is oscillatory, then has property (A).*

*Remark 2.7. *We do not stipulate whether or not is a delayed or advanced argument. Using any oscillatory condition for , we obtain criteria for property (A) of third-order equation . We offer several such results.

Theorem 2.8. *Let and . If
**
then has property (A).*

*Proof. *By Theorem 2.6, it is sufficient to prove that is oscillatory. Assume the contrary, that is, let be a positive solution of . Then
An integration of from to leads to
where . Integrating again from to , we obtain
Let us denote
Then
Multiplying the previous inequality with and then integrating from to , we have
Letting be , we get a contradiction with (2.13).

*Example 2.9. *Consider the third-order nonlinear delay differential equation
with , and . It is easy to check that condition (2.13) is fulfilled and then Theorem 2.8 implies that enjoys property (A).

Our results are new even for . Employing a generalization of Hilleβs criterion [10] for oscillation of with , we get in view of Theorem 2.6.

Theorem 2.10. *Let and . If
**
then has property (A).*

Now, we are prepared to provide another criterion for property (A) based on the Riccati transformation.

Let us denote

Theorem 2.11. *Let and . If
**
and for some **
then has property (A).*

*Proof. *By Theorem 2.6, it is sufficient to prove that is oscillatory. Assume the contrary, that is, let be a positive solution of . Then satisfies (2.14). Since is decreasing, then there exists that
We claim that . If not, then it is easy to see that
Setting the last inequality to , we get
where is arbitrary. Integrating the previous inequality from to , one gets
Letting be , we get a contradiction with (2.22) and we conclude that
On the other hand, it follows from that for any constant , we have
We choose a positive constant , such that . It follows from (2.28) that
eventually. Integrating from to , we obtain
or equivalently
We set
Then and
which in view of (2.29) implies
It follows from (2.14) that
Therefore
where we have used (2.32). Applying the inequality , we are led to
Integrating from to , we obtain in view of (2.23)
as . A contradiction. The proof is complete now.

*Example 2.12. *Consider once more the third-order nonlinear delay differential equation
with , and . It is easy to check that condition (2.22) is fulfilled and the condition (2.22) reduces to
Choosing the condition (2.40) holds true and then Theorem 2.11 implies that enjoys property (A).

In this paper, we have presented new comparison theorems for deducing property (A) of from the oscillation of the suitable second-order delay differential equation. Our results here generalize those presented for linear differential equations [19, 25].

#### Acknowledgment

Research is in this paper supported by S.G.A. KEGA 019-025TUKE-4/2010.