#### Abstract

Using Lyapunov-Krasovskii functional approach, we establish a new result to guarantee the existence of periodic solutions of a certain multidelay nonlinear functional differential equation of second order. By this work, we extend and improve some earlier result in the literature.

#### 1. Introduction

It is well known that the problem of the existence of periodic solutions of retarded functional differential equations of second order is not only very important in the background applications, but also of considerable significance in theory of differential equations. Besides, the scope of retarded functional differential equations is very general. For example, it contains ordinary differential equations, differential-difference equations, integrodifferential equations, and so on. The motivation of this paper is that in recent years the study of the existence of periodic solutions to various kinds of retarded functional differential equations of second order has become one of the most attractive topics in the literature. Especially, by using the famous continuation theorem of degree theory (see Gaines and Mawhin [1]), many authors have made a lot of interesting contributions to the topic for retarded functional differential equations of second order. Here, we would not like to give the details of these works.

On the other hand, amongst the achieved excellent results, one of them is the famous Yoshizawa’s theorem [2] for existence of periodic solutions of retarded functional differential equations, which has vital influence and has been widely used in the literature. This theorem has also been generally one of the best results in the literature from the past till now. It should be noted that in 1994, Zhao et al. [3] proved four sufficiency theorems on the existence of periodic solutions for a class of retarded functional differentials to have the existence of an -periodic solution. By this work, the authors proved that their theorems are better than Yoshizawa’s [2] periodic solutions theorem primarily by removing restrictions of the size of the constant delay . An example of application was given at the end of the paper. Namely, in the same paper, the authors applied the following Theorem A to discuss the existence of an -periodic solution of the nonlinear delay differential equation of the second order: where is an external force, is a delayed restoring force, the delay is a positive constant and the friction is proportional to the velocity, and is a positive constant. It should be noted that a feedback system with friction proportional to velocity, an external force , and a delayed restoring force , may be written as (1) (see Burton [4]).

Consider the following general nonautonomous delay differential equation: where , , is a positive constant, and we suppose that is continuous, -periodic, and takes closed bounded sets into bounded sets of ; and such that solutions of initial value problems are unique, can be either larger than , or equal to or smaller than . Here is the Banach space of continuous function with supremum norm; , is the open -ball in ; . Standard existence theory, see Burton [4], shows that if and , then there is at least one continuous solution such that on satisfying (2) for , and is a positive constant. If there is a closed subset such that the solution remains in , then . Further, the symbol will denote a convenient norm in with = . Let us assume that is continuous} and denotes the in the particular , and that . Finally, by the periodicity, we mean that that there is an such that is -periodic in the sense that if is a solution of (2) so is .

*Definition 1. *Solutions of (2) are uniform bounded at if for each there exists such that imply that (see Burton [4]).

*Definition 2. *Solutions of (2) are uniform ultimate bounded for bound at if for each there exists a such that imply that (see Burton [4]).

The first theorem given in Zhao et al. [3] is the following.

Theorem A. *If the solutions of (2) are ultimately bounded by the bound , then **(i)** equation (2) has an -periodic solution and is bounded by ,**(ii)** if (2) is autonomous, then (2) has an equilibrium solution and is bounded by (see Zhao et al. [3]).*

Regarding (1) Zhao et al. [3] proved the following theorem as example of application.

Theorem B. *Assume that the following conditions hold:**(1)** is an -periodic continuous function, is a continuous differentiable function,**(2)**, and there is a bounded set containing the origin such that on ; is the complement of the set ,**(3)**. **Then, (1) has an -periodic motion. *

Moreover, when , is a constant, under the above conditions (1) has a constant motion , and the constant satisfies . In fact, from the condition (2), there is a bounded set containing the origin such that and on ; is the complement of the set .

In this paper, we consider the following nonlinear differential equation of second order with multiple constant delays, : where are fixed constants delay with ; the primes in (3) denote differentiation with respect to , , , , , and are continuous functions in their respective arguments on , , , , and , respectively, and also depend only on the arguments displayed explicitly. The continuity of these functions is a sufficient condition for existence of the solution of (3). It is also assumed as basic that the functions , , , and satisfy a Lipschitz condition in . By this assumption, the uniqueness of solutions of (3) is guaranteed. The derivatives exist and are continuous. It should be noted that throughout the paper, sometimes, and are abbreviated as and , respectively.

We write (3) in system form as follows: where .

It is clear that (3) is a particular case of (2). It should be noted that the reason or the motivation for taking into consideration (3) comes from the following modified Liénard type equation of the form: These types of equations have great applications in theory and applications of the differential equations. Therefore, till now, the qualitative behaviors, the stability, boundedness, global existence, existence of periodic solutions, and so forth, of these type differential equations have been studied by many researchers, and the researches on these topics are still being done in the literature. For example, we refer the readers to the books of Ahmad and Rao [5], Burton [4], Gaines and Mawhin [1], and the papers of Constantin [6], Graef [7], Huang and Yu [8], Jin [9], Liu and Huang [10], Nápoles Valdés [11], Qian [12], Tunç [13–21], C. Tunç and E. Tunç [22], Zhao et al. [3], Zhou [23], and the references cited in these works.

We here give certain sufficient conditions to guarantee the existence of an -periodic solution of (3). This paper is inspired by the mentioned papers and that in the literature. Our aim is to generalize and improve the application given in [3] for (3). This paper has also a contribution to the investigation of the qualitative behaviors of retarded functional differential equations of second order, and it may be useful for researchers who work on the above mentioned topics. Finally, without using the famous continuation theorem of degree theory, which belongs to Gaines and Mawhin [1], we prove the following main result. This case makes the topic of this paper interesting.

#### 2. Main Result

Our main result is the following.

Theorem 3. *We assume that there are positive constants , , , , and such that the following conditions hold:**(i)**,
**(ii)**, and there is a bounded set containing the origin such that on ; is the complement of the set ,**(iii)**, and there is a bounded set containing the origin such that on ; is the complement of the set ,**(iv)** is an -periodic continuous function. **If , then (3) has an -periodic solution. *

Moreover, , -constant, under the above conditions (3) has a constant motion , and the constant satisfies .

*Proof. *Define the Lyapunov-Krasovskii functional :
where are some positive constants to be determined later in the proof.

Evaluating the time derivative of along system (4), we get
By noting the assumption of the theorem and the estimate , one can obtain the following estimates:
Then, it follows that
Let and . In fact, these choices imply that

In view of the continuity and periodicity of the function and the assumption , it follows that there is a bounded set with containing the origin and a positive constant such that
Therefore, we can write

From the last estimate, we can arrive that the -coordinate of the solutions of system (4) is ultimately bounded for a positive constant . On the other hand, since is a continuous periodic function, if and , then, subject to the assumptions of the theorem, it can be easily seen that there is a constant on such that
Since and as , then it can be chosen a positive constant such that
Therefore, we can conclude that there is a positive constant such that the -coordinate of the solutions of system (4) satisfies for .

In a similar manner, if and , then, subject to the assumptions of the theorem, it can be easily followed by the time derivative of the functional that there is a constant on such that

Since and as , then it can be chosen a positive constant such that

Then, we can conclude that there is a positive constant such that the -coordinate of the solutions of system (4) satisfies for . On gathering the above whole discussion, one can see that the solutions of system (4) are ultimately bounded. Therefore, (3) has an -periodic motion (solution). When , -constant, (3) has a constant motion . From (3), it can be seen that the constant is given by .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.