Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 8825413, 7 pages
https://doi.org/10.1155/2020/8825413

Research Article

Some New Oscillation Results for Fourth-Order Neutral Differential Equations with a Canonical Operator

Omar Bazighifan1,2

¹Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen
²Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512, Yemen

Correspondence should be addressed to Omar Bazighifan; o.bazighifan@gmail.com

Received 10 September 2020; Accepted 22 September 2020; Published 17 October 2020

Academic Editor: Hijaz Ahmad

Copyright © 2020 Omar Bazighifan. 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.

By this work, our aim is to study oscillatory behaviour of solutions to 4th-order differential equation of neutral type where . By using the comparison method with first-order differential inequality, we find new oscillation conditions for this equation.

1. Introduction

Neutral differential equations are getting an increasing importance among the practitioners in the recent past, since they are used in numerous applications in technology and natural science [1]. During the last one and half decade, a multidimensional application of differential equation with delay has been observed in the field of statistics, science, and engineering [2–6].

In this paper, we are concerned with the oscillation of solutions of the even-order nonlinear differential equation:

where and and are quotient of odd positive integers and

The operator is said to be in canonical form if ; otherwise, it is called noncanonical. The main results are obtained under the following conditions:

 F1. , , , F2. , , , , F3. For some the differential inequality

 is oscillatory, where . F4. For some the differential inequality

 is oscillatory. F5. For some the equation

is oscillatory.

By a solution of (1), we mean a function which has the property and satisfies (1) on .

Over the years, several studies have been carried out on oscillation of neutral differential equations [7–13].

Li et al. [14] have studied oscillation fourth-order differential equations with neutral type.

In [15], Bazighifan has considered differential equation with positive and negative coefficients and studied oscillatory behaviour of equation:

under the condition

In [16], Liu et al. have considered an even-order differential equation of the form

where and obtained some new oscillation results for this equation.

Zafer [17], Zhang and Yan [18] proved that the equation

is oscillatory if

where and is even.

Xing et al. [19] and Moaaz et al. [20] proved that the equation

is oscillatory if

where is an even and .

Now, we consider the equation

By applying conditions (10), (11), (13), and (14) to equation (15), we obtain Table 1.

Table 1: Comparison of results.

The condition
(10)
(11)
(13)
(14)

The criterion

From above, we get that [20] enriched the results in [17–19].

Our aim of this paper is complement results in [17–20].

By the comparison method with first-order differential inequality, we obtain new conditions of oscillation for this equation. Some examples are provided to show effectiveness of the obtained results.

2. Main Results

Lemma 1. Reference [21] If satisfies , , and eventually. Then, for every eventually.

Lemma 2 (see [22], Lemma 2.2.3). Let and . If , then

for every .

Lemma 3 (see [23], Lemma 1 and 2). Let , then

where is a positive real number.

Theorem 1. Assume that

If holds, then (1) is oscillatory. Proof. Assume to the contrary that (1) has a nonoscillatory solution in . Without loss of generality, we let be an eventually positive solution of (1). Then, there exists a such that , and for . Since , we have

for . From (1), we get

It follows from definition of and Lemma 3 that

From (20) and (21), we obtain

which with (18) gives

Since , we get that . Thus, from Lemma 2, we get

Combining (23) and (24), we see that

If we set

then it is easy to see that

Thus, from (25), we get that is a positive solution of

which is a contradiction. The proof is complete.

Theorem 2. Assume that (18) holds. If holds, then (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 1, we get (25). If we set , then is a positive solution of (4), which is a contradiction. The proof is complete.

Corollary 1. Let and (18) holds. If and

where or , then (1) is oscillatory.

Proof. It is well-known (see, e.g., [24], Theorem 2.1.1]) that condition (29) implies the oscillation of (3) and (5).

Theorem 3. Assume that and . If holds, then (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 1, we get (19). From definition of , we get

which with (1) gives

From Lemma 2, we obtain

Combining (31) and (32), we get

Hence, if we set , then we get that is a positive solution of the inequality

In view of ([12], Corollary 1), the associated delay differential equation (5) also has a positive solution, which is a contradiction. The proof is complete.

Corollary 2. Let , and . If

then (1) is oscillatory.

Proof. It is well-known (see, e.h., [24], Theorem 2.1.1) that condition (35) implies the oscillation of (5).

Theorem 4. Assume that and . If there exists a positive functions such that

for some and every , where

then (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 3, we find (19) and (31). From (19), we have is of one sign.

In the case where , we define a generalized Riccati substitution by

By differentiating and using (31), we obtain

From Lemma 1, we have that , and hence,

Using Lemma 2, we get

for all . Thus, by (40)–(42), we obtain

Since , there exist and a constant such that

for all . Using the inequality

with and , we find

This implies that

which contradicts (36).

For , integrating (31) from to , we obtain

From Lemma 1, we see that , and hence,

For (48), letting and using (49), we get

Integrating (50) from to , we get

for all . Now, we define

Then for . By using (44) and (51), we obtain

Thus, we find

and so

Then, we obtain

which contradicts (37). This completes the proof.

Example 1. Consider the differential equation

where . We note that , , , , , and . Hence, it is easy to see that

Using Corollary 1, the equation (57) is oscillatory if

From Corollary 2, the equation (57) is oscillatory if

Example 2. Consider the differential equation

where and . We note that , , , , and . Hence, it is easy to see that

Using Corollary 1, the equation (61) is oscillatory if

From Corollary 2, if

then (61) is ocillatory.
Finally, if we set and , then we have

Thus, from Theorem 4, equation (61) is oscillatory if

Data Availability

No data were used in the study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors declare that it has been read and approved the final manuscript.

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