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 (H₁) is the ratio of two positive odd integers, (H₂), , are positive, (H₃), , , , , (H₄), , , and for , (H₅) for and for .

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 [1–17]) 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 Baculíková 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 [1–17].

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 (H₂), (H₃), and (H₄), 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)–(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, Theorem  2.4.1], 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).