Oscillation Criteria of Fourth-Order Differential Equations with Delay Terms

Abstract

The aim of this paper is to derive oscillation criteria of the following fourth-order differential equation with delay term , under the assumption The results are based on comparison with the oscillatory behaviour of second-order delay equations and the generalised Riccati transformation. Not only do the provided theorems provide an entirely new technique but also they vastly improve on a number of previously published conclusions. We give three examples to illustrate our findings.

1. Introduction

Higher-order neutral differential equations have recently been recognized as being sufficient to describe a variety of real applications [1–4]. As a result, many researchers have studied the qualitative behaviour of solutions of these equations (see [5–8]). The research of oscillation and oscillatory behaviour of these equations, which has been investigated using multiple approaches and techniques, has received special attention (see [9–11]). The attempt to improve the work and obtain a generalised platform that covers all special cases inspires the investigation of fourth- and higher-order equations.

In this work, we are concerned with oscillation of fourth-order delay differential equations of the form

where . Throughout this work, we suppose the following: (i) and is a quotient of odd positive integers(ii)The following condition holds:

for , , and (iii), , , (), and such that

By a solution of (1), we mean a function , , that has the property and fulfills (1) on . If a solution of (1) has arbitrarily large zeros on , then it is considered oscillatory; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

Next, we give some previous findings in the literature that are relevant to the present work. Grace [12] has studied the equation

in addition to Agarwal et al. [13] and Xu and Xia [14] who have studied the equation

subject to condition (2). Zhang et al. [15] obtained oscillatory criteria of the equation

with the condition

Baculikova et al. [16] used the comparison theory to prove that if

is oscillatory, then

is oscillatory for even . Grace et al. [7] presented oscillation criteria for fourth-order delay differential equations of the form

under the assumption

Using the Riccati transformation, an oscillation criterion for fourth-order neutral delay differential equation of the form

was obtained by Chatzarakis et al. [17]. By using the technique of the Riccati transformation and the theory of comparison with first-order delay equations, Bazighifan and Abdeljawad [18] established some new oscillation criteria for fourth-order advanced differential equations with -Laplacian-like operator of the form

Very recently, Bazighifan et al. [19] established new criteria for the oscillatory behaviour of the following fourth-order differential equations with middle term

by the comparison technique and employing the Riccati transformation under the condition

For convenience, in the present work, we denote where . The generalised Riccati transformation is defined as

We remark that in the study of the asymptotic behaviour of the positive solutions of (1), there are only two cases:

Case 1.

Case 2.

In this work, using the Riccati approach and a comparison with a second-order equation, we shall obtain oscillation criteria for (1).

2. Some Significant Auxiliary Lemmas

The following lemmas serve as a basis for our findings.

Lemma 1 (see [20]). Let be a ratio of two odd numbers; and are constants. Then,

Lemma 2 (see [17]). Let and for all . Then,

Lemma 3 (see [21]). The equation where , , and , is nonoscillatory if and only if there exist and such that for .

Lemma 4 (see [22]). Suppose that ; then, for every and .

3. Oscillation Criteria

In this section, we shall obtain some oscillation criteria for equation (1).

Lemma 5. Suppose that is a solution of (1) such that and for all . If we have the function defined in (17), where , then for all , where is large enough.

Proof. Let be a solution of (1) where and for all . Thus, from Lemma 4, we get for all and for every large . From (17), we have that for , and Using (25) and (17), we acquire Letting , , and and by using Lemma 1, we get From Lemma 2, we obtain , and hence, From (1), (27), and (28), we obtain This implies that Thus, The proof is completed.

Lemma 6. Let be a solution of (1) such that and for and . If the function is defined in (18) such that , then for all , where is large enough.

Proof. Let be a solution of (1) where and for and . From Lemma 2, we have that . Integrating this inequality from to , we obtain Hence, from (3) we have By integrating (1) from to and since , we get Now letting yields and so Integrating this from to gives From (18), we have that for and by differentiating, we get Now, using Lemma 1 with , , and yields From (1), (40), and (41), we have the following: This implies that Thus, The proof is completed.

Lemma 7. Let be a solution of (1) with . If such that for some , then does not fulfill Case 1.

Proof. Let be a solution of (1) such that . From Lemma 5, we obtain that (24) holds. Using Lemma 1 with and , we get Now, integrating from to yields which contradicts (45). So, the proof is complete.

Lemma 8. Let be a solution of (1) with and for and . If such that then does not fulfill Case 2.

Proof. Let be a solution of (1) such that . From Lemma 6, we get that (33) holds. Using Lemma 1 with we obtain Integrating from to gives which contradicts (49). This completes the proof.

Theorem 9. Let such that (45) and (49) hold for some . Then, equation (1) is oscillatory.

Proof. The proof is very similar to the proofs of Lemmas 7 and 8.
Now, by using the comparison method, we develop additional oscillation results for (1) in the following theorem:

Theorem 10. Let (2) hold and assume that are both oscillatory; then, (1) is oscillatory.

Proof. Assume the contrary that (1) has a positive solution , and by virtue of Lemma 3 and if we set in (24), then we get Hence, we have that (53) is nonoscillatory, which is a contradiction. If we set in (33), then we obtain Thus, equation (54) is nonoscillatory, which is a contradiction. The proof is now complete.

It is well known (see Řehák [23])) that if

then equation (21) with is oscillatory.

Theorem 11. Let (2) hold. Assume that and for some constant and then every solution of (1) is oscillatory.
The proof is obvious.

4. Examples

In this section, we provide some examples to prove that the results of Section 3 are valid.

Example 1. Consider where .
Let , , , and .
Hence, we have If we set , , and , then condition (45) becomes Therefore, from Lemma 7, if , then (61) has no positive solution satisfying . Also, condition (49) becomes From Lemma 8, if , then (61) has no positive solution satisfying . Thus, from Theorem 9 every solution of (61) is oscillatory if

Example 2. Consider the following differential equation representing equation (1), where and are constants.
Here, , , , and . Hence, If we set , , and , then condition (45) yields Therefore, from Lemma 7, if , then (65) has a solution satisfying . Also, from condition (49) we have Thus, from Theorem 9, every solution of (65) is oscillatory if

Example 3. Consider where .
Let , , , and . When is used, condition (59) becomes and condition (60) gives Therefore, from Theorem 11, all solutions of (70) are oscillatory if .

5. Conclusion

In this paper, we have established some new sufficient criteria which ensure that every solution of the fourth-order differential equations (1) is oscillatory. The approach we used was based on comparisons with the oscillatory behaviour of second-order delay equations and the Riccati transformation. Several illustrative examples have also been presented.

Data Availability

No data were used to support this study.

Disclosure

This work is part of UKM’s research # DIP-2021-018.

Conflicts of Interest

All authors have declared they do not have any competing interests.