Some Oscillation Criteria for a Class of Third-Order Delay Differential Equations

Abstract

In this paper, we study oscillatory properties of solutions to a class of third-order differential equation where and and are quotients of odd positive integers. By using the generalized Riccati technique, we obtain some oscillation and asymptotic criteria when and . Finally, some examples are given to show the effectiveness of the criteria obtained.

1. Introduction

Oscillation phenomena take part in different models from real-world applications (see, e.g., the papers [1, 2]), where oscillation and delay situations take part in models from mathematical biology when their formulation includes cross-diffusion terms. The oscillation theory of differential equations is an active area of research. Many scholars have studied the oscillation of solutions to various classes of differential equations and made some progress, especially for the first-order and second-order equations (see, e.g., the monographs [3, 4], the papers [5, 6], and the references cited therein). Third-order differential equations originate in many different fields of applied mathematics and physics, for example, the deflection of a buckling beam with a fixed or varying cross section, three-layer beam, electromagnetic waves, and the rising tide caused by gravitational blowing. The theory of third-order differential equations can be applied to many fields, such as to biology, population growth, engineering, generic repression, control theory, and climate model. Recently, a great deal of interest in asymptotic properties and oscillatory behavior of solutions to different classes of third-order delay differential equations has been manifested (see, e.g., the papers [7–19] and the references cited therein).

In particular, by using the Riccati transformation technique, the oscillation properties of the equationhave been considered by Liu et al. [20] under the conditions , , and . Based on the properties of comformable fractional differential and integral, Feng and Meng have considered the following fractional-order dynamic equation on time scales [21]:and obtained some oscillation criteria where for , and is a quotient of two odd positive integers. By using the integral averaging technique, the equationhas been studied by Moaaz et al. [22] under the conditions that , , and is nondecreasing. However, most of the oscillation criteria have been obtained under the condition or . Motivated by the above works, we further study the oscillation criteria of third-order equation when and .

In this paper, we mainly study the oscillation solution of the following third-order delay differential equation:where and is a quotient of odd positive integers. Throughout this paper, we assume that(A1), , , .(A2), , , and .(A3) and , and is a quotient of odd positive integers.

A function is called a solution of (4) if satisfies (4) on and if , and where . We consider solutions of (4) which satisfy . A solution of (4) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (4) is said to be oscillatory if all its solutions are oscillatory.

2. Preliminaries

In this section, we give the following lemmas which are used as tools in establishing new oscillation criteria for (4).

We introduce the following notation:where and are positive real constants.

Lemma 1. Assume that is the positive solution of (4) andThen, one of the following two cases holds.(I), , .(II), , for , where is sufficiently large.

Proof. Let be the positive solution of (4). Without loss of generality, we may assume that there exists a sufficiently large such that and .
From (4), we obtainHence, is strictly decreasing on . Because , , we can deduce that is eventually of one sign.
We can claim that for . In fact, if , according to the monotonicity of , we havefor . Integrating (9) from to , we obtainLetting , from (6), we have . Thus, there exists a sufficiently large such that for . Furthermore, we have . Dividing by and integrating on , we haveFrom (7), we have . This contradicts , which implies . We complete the proof.

Lemma 2. Assume that satisfies case (II). Supposeand then .

Proof. Let be an eventually positive solution of equation (4). Because has property (II), we obtain .
We claim . Otherwise, if we assume , then it follows that for .
From (A3), we haveIntegrating the above inequality on , we havewhich meansIntegrating inequality (15) on , we obtainFurther integrate the above inequality on to getLetting , by (12), we can deduce that , which leads to a contradiction. So, we have , and then . We complete the proof.

Lemma 3. Assume (6) and (7) hold. If is a positive solution of equation (4) with case (I) for , where is sufficiently large, then for , the following inequality holds:where , are positive real constants.

Proof. Let be a positive solution of (4). We assume that there exists a such that and for . By Lemma 1, we knowFrom (13), we can see that is decreasing on . Hence,Thus,Since is positive and increasing, we have for . Moreover, due to the monotonicity of , we havewhere for . Hence,Integrating the above inequality from to , we havewhere . Thus,Furthermore,where . Then, we havefor , where , . The proof is complete.

Lemma 4. (see [23, Theorem 41]). Assuming that and are nonnegative real numbers, then

3. Oscillation Results

In this section, we establish some new oscillation criteria for (4). For the following theorem, we introduce a class of function . Let

The function is said to belong to the class , if(i) for , for .(ii) has a continuous and nonpositive partial derivative on with respect to .

Theorem 1. Assume that (6), (7), and (12) hold. If there exists a function such that for all sufficiently large , the following condition holds:where , then every solution of equation (4) is oscillatory or tends to zero.

Proof. Let be a nonoscillatory solution of (4). Without loss of generality, we may assume that there exists a sufficiently large such that and for . According to Lemma 1, is either the case of (I) or the case of (II).
Now we assume satisfies the (I); defineThen, , andFrom (13), we knowBy Lemma 3 and (A2), it is obvious thatThus,So, we can obtainBy Lemma 3, we haveBy Lemma 4, letThen,Combining (30) and (37), we obtainIntegrating (39) from to ,Letting , we havewhich contradicts condition (30).
If satisfies (II), then from Lemma 2, it is obvious that . The proof is complete.

Theorem 2. Assume that (6), (7), and (12) hold. If there exist functions and such thatfor , where , , then every solution of equation (4) is oscillatory or tends to zero.

Proof. Let be a nonoscillatory solution of (4). Without loss of generality, we may assume that there exists a sufficiently large such that and for . According to Lemma 1, is either the case of (I) or the case of (II). Now we assume satisfies (I). Let be defined as in Theorem 1; by (37), we haveSubstituting with in the above inequality, multiplying each side by , and integrating from to , we obtainBy Lemma 4, letThen,Combining (45) and (47), we haveBecause for , we can get . Hence,Dividing , we haveLetting , we havewhich contradicts (43).
If satisfies (II), then from Lemma 2, it is obvious that . The proof is complete.

4. Examples

Example 1. Consider the equationwhere , , , , , , . Then, we haveNow we check conditions (6), (7), and (12).So, conditions (6), (7), and (12) hold. Because , there exists a sufficiently large such that for . Furthermore, , and letting , we have the following three situations.
For ,For , from the above, it is obvious that (30) holds.
For ,From above, we knowTherefore, by Theorem 1, every solution of (52) is oscillatory or tends to zero.

Example 2. Consider the equationwhere , , , , , , which satisfy conditions (A1)–(A3). Similar to the Example 1, we can easily verify that (6), (7), and (12) hold. Furthermore, we choose , . By Theorem 2, we have andThen, by Theorem 2, we know that any solution of (58) is oscillatory or tends to zero.

5. Conclusions

In this work, by using the generalized Riccati technique, we establish some new oscillation criteria of (4) and give some examples to verify the oscillation criteria. These new oscillation criteria complement some known results for third-order delay differential equations. Furthermore, in future work we will study the forced oscillatory behavior of this equation with forcing term.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Youth Project of Chongqing Three Gorges University (no. 19QN10).