Periodic Solution for a Kind of Third-Order Neutral-Type Differential Equation

Shu, Axiu; Du, Bo

doi:https://doi.org/10.1155/2023/5569141

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5569141 | https://doi.org/10.1155/2023/5569141

Periodic Solution for a Kind of Third-Order Neutral-Type Differential Equation

Axiu Shu¹and Bo Du²

Academic Editor: Calogero Vetro

Received20 Apr 2023

Revised07 Jun 2023

Accepted23 Jun 2023

Published07 Jul 2023

Abstract

In this paper, we investigate a class of a third-order neutral-type differential equation with time-varying delays. Some sufficient conditions on the existence of a periodic solution are established for the considered system. Different from the previously reported research results, by utilizing the properties of neutral operators and a special variable substitution, we transform a high-order neutral equation into a first-order three-dimensional nonneutral system. The existence of a periodic solution such as the high-order neutral equation has not been given much attention in past papers due to the difficulty of estimation of prior bounds of solutions. This paper is devoted to the use of properties of neutral-type operators with a variable parameter and Mawhin’s continuation theorem for overcoming the above difficulties. The neutral term in the third-order neutral differential equation in this paper contains a variable parameter which is different from third-order neutral-type equations that have been studied. The third-order neutral-type equation studied in this paper is more general, and similar equations studied in the past are special cases of the equations studied in this paper. Finally, an example is given to elucidate the effectiveness and values of the present results. Our results are new and complement the related results of third-order functional differential equations.

1. Introduction

The third-order functional differential equations are more complex than the corresponding second-order and first-order equations. The research methods of second-order differential equations and first-order differential equations are often difficult to use to study third-order differential equations. In recent years, many authors developed some new methods for studying different types of third-order differential equations. Remili and Oudjedi [1] established some new sufficient conditions which guarantee the stability and boundedness of solutions of certain nonlinear and nonautonomous third-order differential equations with delay by using the Lyapunov function. Mahmoud [2] considered the existence and uniqueness of periodic solutions for a kind of third-order functional differential equation with a time delay. Tung [3] studied the stability and boundedness of solutions to a third-order nonlinear differential equation with retarded argument by the use of the Lyapunov function.

In the present paper, we are concerned with periodic solutions of neutral-type third-order differential equations. A neutral functional differential equation is a differential equation with a time-delay-containing derivative. It is widely used in many aspects, such as physical chemistry, mathematical biology, electrical control, and engineering; see [4–8]. Graef et al. [9] studied the stability, boundedness, and square integrability of solutions of third-order neutral-delay differential equations. In 2022, Taie and Alwaleedy [10] dealt with the existence problems of a periodic solution for the third-order neutral functional differential equation: where are continuous periodic functions on with and and are continuous functions on with . After that, using Mawhin’s continuation theorem of coincidence degree and analysis techniques, Taie and Bakhi [11] further studied the following third-order neutral functional differential equation:

Mahmoud and Farghaly [12] established some sufficient conditions for the existence of a periodic solution to the following third-order neutral functional differential equation:

For more results about neutral third-order functional differential equations, see, e.g., [13–17] and related references. For the recent advance in the theory and application of Mawhin’s continuation theorem and periodic solutions, see [18–20]. However, we find that the existing results of the neutral third-order functional differential equations mainly focus on the existence and uniqueness of periodic solutions, and there are also many studies on the stability (including asymptotic and exponential stability) of periodic solutions; see [10–12]. In this paper, we will study the existence of periodic solutions of the following third-order neutral functional differential equation: where is a constant; are continuous -periodic functions on with , and ; and , and are continuous functions on . Using Mawhin’s continuation theorem, we will establish existence theorems of periodic solution for equation (4).

We list the main contributions of this paper as follows: (1)We convert a third-order equation into a first-order three-dimensional differential system via suitable variable substitution, so that we can study the periodic solution problem of the above system conveniently(2)It should be pointed out that the first-order three-dimensional differential system studied in this paper is a nonneutral-type system, but the system in [11] is a neutral-type system. Therefore, this paper can easily use some mathematical analysis methods to study the above nonneutral-type system(3)The neutral-type equation in this paper is described by the -operator form (see [21]). Using a property we obtained earlier on neutral-type operators (see [22]), we can easily study the periodic solutions of third-order neutral-type equations. Our main results are also valid for the case of nonneutral third-order neutral-type equations(4)The neutral term in equation (4) contains variable parameter which is different from the similar equations in [10–12] that are special cases of equation (4)

The remainder of this paper are organized as follows: Section 2 gives some preparatory work for this article, including necessary lemmas, required conditions, and so on. In Section 3, we obtain some sufficient conditions for the existence of the periodic solution of equation (4). In Section 4, an example is given to show the effectiveness of the proposed approaches. Finally, some conclusions and discussions are given.

2. Preliminaries

Let with the norm . For , then is a Banach space.

Lemma 1 (see [22]). Let where is a -periodic continuous function and is a constant. If , then operator has continuous inverse on , satisfying where .

Remark 2. In 2009, we obtained Lemma 1 which has important properties for the neutral-type operator . When is a constant , Zhang [23] obtained a similar lemma which is a special case of Lemma 1. Hence, our results generalize Zhang’s work.
Let and be two Banach spaces, and let be a linear operator, which is a Fredholm operator with index zero (meaning that is closed in and ). If is a Fredholm operator with index zero, then there exist continuous projectors such that , and is invertible. Denote by the inverse of .

Let be an open-bounded subset of ; a map is said to be -compact in if is bounded and the operator is relatively compact. We first give the famous Mawhin’s continuation theorem.

Lemma 3 (see [6]). Suppose that X and Y are two Banach spaces, and is a Fredholm operator with index zero. Furthermore, is an open-bounded set, and is -compact on . If all the following conditions hold: (1)(2)(3)then equation has a solution on .

Let . By Lemma 1, then , and equation (4) can be rewritten by the following equation:

Furthermore, let Then, equation (7) can be rewritten by the following system:

Obviously, system (8) is equivalent to equation (4). Thus, if is a -periodic solution of system (8), then must be a -periodic solution of equation ((4)). In the following, we mainly study the existence of a periodic solution of system (8).

Throughout this paper, we assume the following:

() There exist positive constants , and such that

() There exists a positive constant such that

() There exist positive constants , and such that

3. Existence of Periodic Solution

Give the following notations: where is a continuous function on . Let with the norm . Then, is a Banach space. Let . Define a linear operator and a nonlinear operator

It is easy to see that system (8) can be converted to the operator equation . By the definition of , . Hence, is a Fredholm operator with index zero. Let projectors and be defined by, respectively,

Let

Then, has continuous inverse defined by where

Theorem 4. Suppose that assumptions () and () hold. Then, equation (4) has at least one -periodic solution.

Proof. Consider the following operator equation: where and are defined by (14) and (15), respectively. By (19), we have By (20) and (21), we get and substitute into (22); then, Integrate both sides of (23) over , by , then In view of the mean value theorem of integral and (24), there exists a such that From assumption () and (25), we have where . Due to the periodicity of and , there exists a point such that If , by Lemma 1 and (26), we have If , by Lemma 1 and (26), we have By (27), we have Thus, Similar to the above proof, if , by (28), we have From , there exists a point such that . Then, Similar to the proof of (32), we have From assumption () and (23), we have where From (33) and (35), we have From (32) and (36), we have From (30) and (37), we have From (31) and (37), we have Integrate both sides of (20) over , then . There exists a point such that . Hence, we have Similar to the proof of (40), by (21), we have From assumption () and (22), we have It follows by (41) and (43) that In view of (21) and (44), we have In view of (40) and (45), we have If , let Let . It is easy to see that condition (4) of Lemma 3 holds. For each , then If , then (or ). For , by , we get . By assuption (), we get which yields a contradiction. If , we have similiar results. Hence, condition (7) of Lemma 3 holds. Let the isomorphism For and we have

By assumption (), for and we have Hence, and condition (8) of Lemma 3 holds. Using Lemma 3, we obtain that system (8) has at least one -periodic solution ; i.e., equation (4) has at least one -periodic solution .

If , in view of (39), (44), and (46), let

Similar to the above proof, we also obtain that equation (4) has at least one -periodic solution .

Remark 5. In [11], the authors used the following lemma for obtaining the existence and uniqueness of a periodic solution.

Lemma 6 (see [17]). Let where is a constant and is a -periodic function. If , then the operator has a continuous inverse on satisfying where

When is a periodic function and is a constant, Lemma 1 gives the corresponding results for neutral-type operator in Lemma 6. It is easy to see that the results of Lemma 1 are more general than Lemma 6 in some aspects. Therefore, the results in this paper have a better applicability than those in [11].

Remark 7. In [11], the authors changed the third-order neutral-type differential equation into a first-order three-dimensional neutral-type system. However, in the present paper, we changed the third-order neutral-type differential equation into a first-order three-dimensional nonneutral-type system. Generally, neutral-type systems are more complex than nonneutral-type systems, and the research process is more difficult. Therefore, the research method in this article is simpler and easier to understand.

Theorem 8. Suppose that assumptions () and () hold. Then, equation (4) has at least one -periodic solution provided that or

Proof. To prove Theorem 8, we will use the relevant conclusions of Theorem 4. From assumption () and (23), if , we have From (30) and (57), we have From (32), (33), and (58), we have In view of (55) and (59), there exists constant such that From (33) and (60), we have From (32) and (61), we have From (30) and (62), we have From assumption () and (22), if , we have It follows by (41) and (65) that In view of (21) and (66), we have In view of (40) and (67), we have Hence, when , from (63), (66), and (68), we obtain the estimate of , and . On the other hand, when , similar to the above proof, by (56), we also obtain the estimate of , and . The remaining proof is similar to Theorem 4, so we omit it.

Remark 9. In [11], the authors obtained the uniqueness of a periodic solution for a third-order neutral-type differential equation. However, they imposed a strong condition on the nonlinear term as follows:
() There exists a positive constant such that , for all .
In the present paper, we particularly want to remove the above condition to obtain the uniqueness of the periodic solution. However, the equations studied in this paper have strong nonlinearity, and it is difficult to find better conditions for the uniqueness of the periodic solution. We hope to solve the above problems in future work.

Remark 10. In general, Mawhin’s continuation theorem is one of the important tools for studying boundary value problems of differential equations, especially in the study of periodic solutions of functional differential equations; see, e.g., [21–23]. However, few scholars use this theorem to study the existence of third-order neutral-type differential equations with time-varying delays. We find that the existing research methods for equation (4) are based on the fixed-point theorem. However, equation (4) is a higher-order equation, and it is difficult to obtain Green’s function. Therefore, we cannot convert equation (4) into an operator equation. Therefore, using the fixed-point theorem will be very difficult. By using Mawhin’s continuation theorem, as long as an estimate of the prior bound of equation (4) is given in a suitable function space, the existence of periodic solutions can be obtained.

4. Example

Using equation (4), we consider the following third-order neutral-type differential equation: where

Let , then , and equation (69) can be rewritten by the following equation:

Furthermore, let Then, equation (71) can be rewritten by the following system:

Obviously, there exist positive constants , and such that

Hence, assumption (H) holds. From the representations of , and , assumption () also holds. Therefore, we have verified that all the conditions of Theorem 4 are satisfied. Thus, we obtain that system (72) has at least one periodic solution ; i.e., equation (69) has at least one periodic solution . Figure 1 shows the evolution of states of system (72).

5. Conclusions and Discussions

By utilizing an appropriate variable substitution, we transform a third-order neutral-type differential equation into a first-order three-dimensional nonneutral-type system. Combining Mawhin’s continuation theorem of coincidence degree and properties of a neutral-type operator, we establish some sufficient conditions on the existence of periodic solutions for the considered equation. Finally, we give an example to verify the main results of this paper. Our results show that the existence of periodic solutions for the third-order equation can be guaranteed under the conditions that the nonlinear term is bounded and the neutral-type operator is noncritical ( in Lemma 1). For the case where a neutral-type operator is critical ( in Lemma 1), there are no results on the existence of periodic solutions for a third-order neutral-type differential equation which is a key issue for our future research. Besides, based on our paper, one can further investigate the problems of an almost-periodic solution, pseudo-almost-periodic solution, pseudo-almost-automorphic solution, and so on.

Data Availability

No data were generated or analyzed during the current study.

Conflicts of Interest

The authors declare no competing interests.

Acknowledgments

This paper is supported by the Natural Science Foundation of Huaian (HAB202231).

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Copyright © 2023 Axiu Shu and Bo Du. 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.

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