Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 857860, 15 pages

http://dx.doi.org/10.1155/2011/857860

## Properties of Third-Order Nonlinear Functional Differential Equations with Mixed Arguments

Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of KoΕ‘ice, LetnΓ‘ 9, 042 00 KoΕ‘ice, Slovakia

Received 14 December 2010; Accepted 20 January 2011

Academic Editor: JosefΒ DiblΓk

Copyright Β© 2011 B. BaculΓkovΓ‘. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this paper is to offer sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments . Both cases and are considered. 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 certain asymptotic behavior of all solutions of the third-order functional differential equations
Throughout the paper, it is assumed that , , , and (*H _{1}*) is the ratio of two positive odd integers, (

*H*), , are positive, (

_{2}*H*), , , , , (

_{3}*H*), , , and for , (

_{4}*H*) for and for .

_{5}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 nonoscillatory. Equation () is said to be oscillatory if all its solutions are oscillatory.

Recently, () and its particular cases (see [1β17]) have been intensively studied. The effort has been oriented to provide sufficient conditions for every () to satisfy or to eliminate all nonoscillatory solutions. Following [6, 8, 13, 15], we say that () has property (B) if each of its nonoscillatory solutions satisfies (1.1).

We will discuss both cases

We will establish suitable comparison theorems that enable us to study properties of () regardless of the fact that (1.3) or (1.2) holds. We will compare () with the first-order advanced/delay equations, in the sense that the oscillation of these first-order equations yields property (B) or the oscillation of ().

In the paper, we are motivated by an interesting result of Grace et al. [10], where the oscillation criteria for () are discussed. This result has been complemented by BaculΓkovΓ‘ et al. [5]. When studying properties of (), the authors usually reduce () onto the corresponding differential inequalities and further study only properties of these inequalities. Therefore, the criteria obtained withhold information either from delay argument and the corresponding functions and or from advanced argument and the corresponding functions and . In the paper, we offer a technique for obtaining new criteria for property (B) and the oscillation of () that involve both arguments and . Consequently, our results are new even for the linear case of () and properly complement and extend earlier ones presented in [1β17].

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

#### 2. Main Results

The following results are elementary but useful in what comes next.

Lemma 2.1. *Assume that , , . Then,
*

*Proof. *If or , then (2.1) holds. For , setting , condition (2.1) takes the form , which is for evidently true.

Lemma 2.2. *Assume that , , . Then,
*

*Proof. *We may assume that . Consider a function . Since for , function is concave down; that is,
which implies (2.2).

The following result presents a useful relationship between an existence of positive solutions of the advanced differential inequality and the corresponding advanced differential equation.

Lemma 2.3. *Suppose that , , and satisfy (H _{2}), (H_{3}), and (H_{4}), respectively. If the first-order advanced differential inequality
*

*has an eventually positive solution, so does the advanced differential equation*

*Proof. *Let be a positive solution of (2.4) on . Then, satisfies the inequality
Let
It follows from the definition of and () that the sequence has the property
Hence, converges pointwise to a function , where . Let , , then . Since is integrable on and , it follows by Lebesgue's dominated convergence theorem that
Thus, satisfies (2.5).

We start our main results with the classification of the possible nonoscillatory solutions of ().

Lemma 2.4. *Let be a nonoscillatory solution of (). Then, satisfies, eventually, one of the following conditions *(I)*(II)**and if (1.2) holds, then also *(III)

*Proof. *Let be a nonoscillatory solution of (), say for . It follows from () that , eventually. Thus, and are of fixed sign for , large enough. At first, we assume that . Then, either or , eventually. But together with imply that . A contradiction, that is, Case (II) holds.

Now, we suppose that , then either Case (I) or Case (III) holds. On the other hand, if (1.3) holds, then Case (III) implies that , . Integrating from to , we have
which implies that as , and we deduce that Case (III) may occur only if (1.2) is satisfied. The proof is complete.

*Remark 2.5. *It follows from Lemma 2.4 that if (1.3) holds, then only Cases (I) and (II) may occur.

In the following results, we provide criteria for the elimination of Cases (I)β(III) of Lemma 2.4 to obtain property (B)/oscillation of ().

Let us denote for our further references that

Theorem 2.6. *Let . Assume that is a nonoscillatory solution of (). If the first-order advanced differential equation
**
is oscillatory, then Case (II) cannot hold.*

*Proof. *Let be a nonoscillatory solution of (), satisfying Case (II) of Lemma 2.4. We may assume that for . Integrating () from to , one gets
On the other hand, the substitution gives
Using (2.17) in (2.16), we find
Taking into account the monotonicity of , it follows from Lemma 2.1 that
where we have used () and (). An integration from to yields
Regarding (), it follows that is a positive solution of the differential inequality
Applying the transformation
we can easily verify that is a positive solution of the advanced differential inequality
By Lemma 2.3, we conclude that the corresponding differential equation () has also a positive solution. A contradiction. Therefore, cannot satisfy Case (II).

*Remark 2.7. *It follows from the proof of Theorem 2.8 that if at least one of the following conditions is satisfied:
then any nonoscillatory solution of () cannot satisfy Case (II). Therefore, we may assume that the corresponding integrals in (2.14)-(2.15) are convergent.

Now, we are prepared to provide new criteria for property (B) of () and also the rate of divergence of all nonoscillatory solutions.

Theorem 2.8. *Let (1.3) hold and . Assume that () is oscillatory. Then, () has property (B) and, what is more, the following rate of divergence for each of its nonoscillatory solutions holds:
*

*Proof. *Let be a positive solution of (). It follows from Lemma 2.4 and Remark 2.5 that satisfies either Case (I) or (II). But Theorem 2.6 implies that the Case (II) cannot hold. Therefore, satisfies Case (I), which implies (1.1); that is, () has property (B). On the other hand, there is a constant such that
Integrating twice from to , we have
which is the desired estimate.

Employing an additional condition on the function , we get easily verifiable criterion for property (B) of ().

Corollary 2.9. *Let and (1.3) hold. Assume that
**
Then, () has property (B).*

*Proof. *First note that (2.29) implies
By Theorem 2.8, it is sufficient to show that () is oscillatory. Assume the converse, let () have an eventually positive solution . Then, and so . Integrating () from to , we have in view of (2.28)
Using (2.30) in the previous inequalities, we get as . Therefore, , eventually. Now, using (2.28) in (), one can verify that is a positive solution of the differential inequality
But, by [14, Theoremββ2.4.1], condition (2.29) ensures that (2.32) has no positive solutions. This is a contradiction, and we conclude that () has property (B).

*Example 2.10. *Consider the third-order nonlinear differential equation with mixed arguments
where , , and is a ratio of two positive odd integers. Since
Corollary 2.9 implies that () has property (B) provided that
Moreover, by Theorem 2.8, the rate of divergence of every nonoscillatory solution of () is
For and satisfying , one such solution is .

Now, we turn our attention to the case when .

Theorem 2.11. *Let . Assume that is a nonoscillatory solution of (). If the first-order advanced differential equation
**
is oscillatory, then Case (II) cannot hold.*

*Proof. *Let be an eventually positive solution of (), satisfying Case (II) of Lemma 2.4. Then, (2.18) holds. Lemma 2.2, in view of the monotonicity of , (), and (), implies
An integration from to yields
Noting (), we see that is a positive solution of the differential inequality
Setting
one can see that is a positive solution of the advanced differential inequality
By Lemma 2.3, we deduce that the corresponding differential equation () has also a positive solution. A contradiction. Therefore, cannot satisfy Case (II).

The following result is obvious.

Theorem 2.12. *Let (1.3) hold and . Assume that () is oscillatory. Then, () has property (B) and, what is more, each of its nonoscillatory solutions satisfies (2.25).*

Now, we present easily verifiable criterion for property (B) of ().

Corollary 2.13. *Let (1.3) and (2.28) hold and . If
**
then () has property (B).*

*Proof. *The proof is similar to the proof of Corollary 2.9 and so it can be omitted.

*Remark 2.14. *Theorems 2.6, 2.8, 2.11, and 2.12 and Corollaries 2.9 and 2.13 provide criteria for property (B) that include both delay and advanced arguments and all coefficients and functions of (). Our results are new even for the linear case of ().

*Remark 2.15. *It is useful to notice that if we apply the traditional approach to (), that is, if we replace () by the corresponding differential inequality (), then conditions (2.29) of Corollary 2.9 and (2.41) of Corollary 2.13 would take the forms
respectively, which are evidently second to (2.29) and (2.41).

*Example 2.16. *Consider the third-order nonlinear differential equation with mixed arguments
where , is a ratio of two positive odd integers and . It is easy to see that conditions (2.14) and (2.15) for () reduce to
respectively. It follows from Corollary 2.13 that () has property (B) provided that
Moreover, (2.25) provides the following rate of divergence for every nonoscillatory solution of ():

Now, we eliminate Case (I) of Lemma 2.4, to get the oscillation of ().

Theorem 2.17. *Let be a nonoscillatory solution of (). Assume that there exists a function such that
**
If the first-order advanced differential equation
**
is oscillatory, then Case (I) cannot hold.*

*Proof. *Let be an eventually positive solution of (), satisfying Case (I). It follows from () that
Integrating from to , we have
Therefore,
An integration from to yields
Consequently, is a positive solution of the advanced differential inequality
Hence, by Lemma 2.3, we conclude that the corresponding differential equation () also has a positive solution, which contradicts the oscillation of (). Therefore, cannot satisfy Case (I).

Combining Theorem 2.17 with Theorems 2.6 and 2.11, we get two criteria for the oscillation of ().

Theorem 2.18. *Let (1.3) hold and . Assume that both of the first-order advanced equations () and () are oscillatory, then () is oscillatory.*

*Proof. *Assume that () has a nonoscillatory solution. It follows from Remark 2.5 that satisfies either Case (I) or (II). But both cases are excluded by the oscillation of () and ().

Corollary 2.19. *Let . Assume that (1.3), (2.28), (2.29), and (2.46) hold. If
**
then () is oscillatory.*

*Proof. *Conditions (2.29) and (2.52) guarantee the oscillation of () and (), respectively. The assertion now follows from Theorem 2.18.

*Example 2.20. *We consider once more the third-order differential equation () with the same restrictions as in Example 2.10. We set , where . Then condition (2.52) takes the form
which by Corollary 2.19, implies the oscillation of ().

The following results are obvious.

Theorem 2.21. *Let (1.3) hold and . Assume that both of the first-order advanced equations () and () are oscillatory, then () is oscillatory.*

Corollary 2.22. *Let . Assume that (1.3), (2.28), (2.41), (2.46), and (2.52) hold. Then () is oscillatory.*

*Example 2.23. *We recall again the differential equation () with the same assumptions as in Example 2.16. We set with . Then condition (2.52) reduces to
which, by Corollary 2.22, guarantees the oscillation of ().

The following result is intended to exclude Case (III) of Lemma 2.4.

Theorem 2.24. *Let be a nonoscillatory solution of (). Assume that (1.2) holds. If the first-order delay differential equation
**
is oscillatory, then Case (III) cannot hold.*

*Proof. *Let be a positive solution of (), satisfying Case (III) of Lemma 2.4. Using that is increasing, we find that
Integrating the inequality from to , we have
Thus,
Combining (2.57) with (2.55), we find
It follows from [16, Theorem 1] that the corresponding differential equation () also has a positive solution. A contradiction. For that reason, cannot satisfy Case (III).

The following results are immediate.

Theorem 2.25. *Let (1.2) hold and . Assume that both of the first-order advanced equations () and () are oscillatory, then () has property (B).*

Theorem 2.26. *Let (1.2) hold and . Assume that all of the three first-order advanced equations (), (), and () are oscillatory, then () is oscillatory.*

Theorem 2.27. *Let (1.2) hold and . Assume that both of the first-order advanced equations () and () are oscillatory, then () has property (B).*

Theorem 2.28. *Let (1.2) hold and . Assume that all of the three first-order advanced equations (), (), and () are oscillatory, then () is oscillatory.*

#### 3. Summary

In this paper, we have presented new comparison theorems for deducing the property (B)/oscillation of () from the oscillation of a set of the suitable first-order delay/advanced differential equation. We were able to present such criteria for studied properties that employ all coefficients and functions included in studied equations. Our method essentially simplifies the examination of the third-order equations, and, what is more, it supports backward the research on the first-order delay/advanced differential equations. Our results here extend and complement latest ones of Grace et al. [10], Agarwal et al. [1β3], Cecchi et al. [6], Parhi and Pardi [15], and the present authors [4, 8]. The suitable illustrative examples are also provided.

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