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

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Introduction
The third-order functional differential equations are more complex than the corresponding second-order and firstorder 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][5][6][7][8]. Graef et al. [9] studied the stability, boundedness, and square integrability of solutions of third-order neutraldelay 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 dðtÞ, δðtÞ, aðtÞ, bðtÞ, τ i ðtÞ, eðtÞ are continuous periodic functions on ℝ with jdðtÞj ≠ 1, δ ′ ðtÞ < 1 and f and g are continuous functions on ℝ × ℝ with f ð0Þ = gð0Þ = 0. 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][14][15][16][17] and related references. For the recent advance in the theory and application of Mawhin's continuation theorem and periodic solutions, see [18][19][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][11][12]. In this paper, we will study the existence of periodic solutions of the following third-order neutral functional differential equation: where γ > 0 is a constant; dðtÞ, aðtÞ, bðtÞ, τ i ðtÞ, c i ðtÞ, eðtÞ are continuous T-periodic functions on ℝ with jdðtÞj ≠ 1, Ð T 0 c i ð tÞdt ≠ 0, and Ð T 0 eðtÞdt = 0; and f , g, and h are continuous functions on ℝ. 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 threedimensional 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 D-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 xðtÞ − dðtÞxðt − γÞ in equation (4) contains variable parameter cðtÞ which is different from the similar equations in [10][11][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.

Remark 2.
In 2009, we obtained Lemma 1 which has important properties for the neutral-type operator A. When cðtÞ is Journal of Function Spaces a constant c, Zhang [23] obtained a similar lemma which is a special case of Lemma 1. Hence, our results generalize Zhang's work.
Let X and Y be two Banach spaces, and let L : DðLÞ ⊂ X ⟶ Y be a linear operator, which is a Fredholm operator with index zero (meaning that ImL is closed in Y and dimKerL = codimImL < +∞). If L is a Fredholm operator with index zero, then there exist continuous projectors P : Let Ω be an open-bounded subset of X; a map N : ΩÞ is bounded and the operator K p ðI − QÞNð ΩÞ is relatively compact. We first give the famous Mawhin's continuation theorem.
Throughout this paper, we assume the following: (H 1 ) There exist positive constants k 1 , k 2 , and k 3 such that (H 2 ) There exists a positive constant D such that (H 3 ) There exist positive constants k 1 , k 2 , k 3 , and k 4 such that

Existence of Periodic Solution
Give the following notations: where f is a continuous function on ℝ. Let with the norm kyk X = max 1≤i≤3 fjy i j 0 g. Then, X is a Banach space. Let X = Y . Define a linear operator 3 Journal of Function Spaces and a nonlinear operator It is easy to see that system (8) can be converted to the Hence, L is a Fredholm operator with index zero. Let projectors P and Q be defined by, respectively, Let Then, L P has continuous inverse L −1 P defined by where ðGzÞðtÞ = Ð t 0 zðsÞds: Theorem 4. Suppose that assumptions (H 1 ) and (H 2 ) hold. Then, equation (4) has at least one T-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 α is a constant and γ ∈ C 2 ðℝ, ℝÞ is a T-periodic function. If jαj ≠ 1, then the operator A has a continuous inverse A Journal of Function Spaces When α is a periodic function and γ is a constant, Lemma 1 gives the corresponding results for neutral-type operator A 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 threedimensional neutral-type system. However, in the present paper, we changed the third-order neutral-type differential equation into a first-order three-dimensional nonneutraltype 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.
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 f as follows: (H 4 ) There exists a positive constant k 3 such that f ðuðtÞÞ = k 3 , for all u ∈ ℝ.
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][22][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 fixedpoint 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.  Journal of Function Spaces verified that all the conditions of Theorem 4 are satisfied. Thus, we obtain that system (72) has at least one periodic solution y = ðy 1 , y 2 , y 3 Þ T ; i.e., equation (69) has at least one periodic solution x 1 ðtÞ = ðA −1 y 1 ÞðtÞ. Figure 1 shows the evolution of states of system (72).

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 thirdorder equation can be guaranteed under the conditions that the nonlinear term is bounded and the neutral-type operator is noncritical (cðtÞ ≠ ±1 in Lemma 1). For the case where a neutral-type operator is critical (cðtÞ = ±1 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.