Abstract
We consider a kind of second-order neutral functional differential equation. On the basis of Mawhin’s coincidence degree, the existence and uniqueness of periodic solutions are proved. It is indicated that the result is related to the deviating arguments. Moreover, we present two simulations to demonstrate the validity of analytical conclusion.
1. Introduction
In this paper, we consider a kind of second-order neutral functional differential equations in the following form: where , are continuous periodic functions defined on with period , , are continuous periodic functions defined on and have the same sign, and are constants such that .
As we know, neutral differential equations are widely used in physics, biology, medicine, chemistry, economics, ecology, aerospace, and so on. The research of their theory and algorithm is greatly important. Many authors devote themselves to research such kind of NFDE and get some results; see papers [1–15]. These papers were devoted mainly to studying the following types of equations:By using the coincidence degree theory, the existence and uniqueness of -periodic solutions for (2) are obtained which are not related to delays. Since the time delay in some equations with practical application background is often very small, it is easier to miss. We know that even a small delay is also likely to have an important impact on the stability of the system. Therefore it is necessary for us to consider the influence of delays. Our work is based on such a background. In this paper, our aim is to establish some criteria to guarantee the existence and uniqueness of periodic solution for (1) by using Mawhin’s continuation theorem.
To state our main theorems, we also need some notations as follows:
Throughout this paper we assume that , and , for all , ().
2. Main Lemmas
Lemma 1 (Gaines and Mawhin [1]). Let be a Banach spaces. Suppose that is a Fredholm operator with index zero and is -compact on , where is an open bounded subset of . Moreover, assume that all the following conditions are satisfied:(a), for all and .(b), for all .(c).Then the equation has at least one solution in .
Lemma 2 (see [5]). Let . Suppose that there exists a constant such that Then
Lemma 3 (see [10]). If , then has continuous bounded inverse on , and (a), for all ;(b), for all ;(c), for all .
Let with the norm , and with the norm . Clearly, and are two Banach spaces. We also defined operators and in the following, respectively: where
Next define a nonlinear operator by setting By Hale’s terminology [9], a solution of (1) is such that and (1) are satisfied on . In general, . But, under the condition , we see from Lemma of [4] that . So a -periodic solution of (1) must be such that . Meanwhile, according to Lemma 3, we can easily get that , and . Therefore, the operator is a Fredholm operator with index zero. Define the continuous projectors and by setting Set ; then has continuous inverse defined by where Therefore, it is easy to see from (7) and (8) that is -compact on , where is an open bounded set in .
In view of (7) and (8), the operator equationis equivalent to the following equation:
Lemma 4 (see [8]). Let , with , for all . Then , where is inverse function of .
The following lemma which we obtained in [16] gives more accurate inequality about the deviating argument than Lemma in [4].
Lemma 5 (see [16]). Let be a constant. Then for for all , we have
Lemma 6. Assume that the following conditions are satisfied: One of the following conditions holds:(a), , for all , ,(b), , for all , . One of the following conditions holds:(a)There exists two positive constants such that ; (b)There exists three positive constants such that ; where and .
Then (1) has at most one -periodic solution.
Proof. Suppose that and are two -periodic solutions of (1). Then we have Set ; we obtain Integrating (18) from 0 to , we have By using the integral mean value theorem, we find that there is such that From , (20) implies that Since is a continuous function on , it follows that there exists a constant such that Let , where and is an integer. Then (22) implies that there exists a constants such that From Lemma 2, using Schwarz inequality and the following relation:we obtain Case 1. If holds, multiplying both sides of (18) by and then integrating them from 0 to , using (26), we haveSince , it follows that there exists a constant such that . From Lemma 2, using Schwarz inequality and the following relation: we obtain Substituting (30) into (27), we get Since , thus (31) implies that Hence, for all . Therefore, (1) has at most one -periodic solution.
Case 2. If holds, multiplying both sides of (18) by and then integrating them from 0 to , using (26) and (30), we have Since , thus (33) implies that Hence, for all . Therefore, (1) has at most one -periodic solution.
Lemma 7 (see [17]). Assume that holds and the following conditions are satisfied: There exists a constant such that one of the following conditions holds:(a), for all , ;(b), for all , .If is a -periodic solution of (13), then
3. Main Result
Theorem 1. Let and hold. Assume that either the condition () or the condition () is satisfied and the following inequality holds:Then (1) has a unique -periodic solution.
Proof. From Lemma 5, together with Lemma 6, it is easy to see that (1) has at most one -periodic solution. Thus, to prove Theorem 1, it suffices to show that (1) has at least one -periodic solution.
Let be a -periodic solution of (13). If holds, multiplying both sides of (13) by and then integrating them from 0 to , we have Since following from (37), we have As , using (35), from (39) we getSince , thus (40) implies that there exists a positive constant , such that If , then there exists a constant such that Thus Let , and take . It is easy to see that is -compact on . We have from (42), (43) that the conditions and in Lemma 1 hold.
Furthermore, define continuous functions and by setting If the condition holds, then Hence, using the homotopy invariance theorem, we have If the condition holds, then Hence, using the homotopic invariance theorem, we haveWhich completes the condition in Lemma 1.
Up to now, all conditions in Lemma 1 are satisfied, and hence (1) has a unique -periodic solution.
Theorem 2. Let and hold. Assume that either the condition () or the condition () is satisfied and the following inequality holds:Then (1) has a unique -periodic solution.
Proof.
Case 1. If holds, multiplying the two sides of (13) by and integrating them on , we have where .
As , there is a constant such that Case 2. If holds, multiplying the two sides of (13) by and integrating them on , using the methods similar to those used in Case , we can show that (51) holds.
The rest of the proof is similar to that of Theorem 1 and is omitted.
4. Application
Finally, we give two examples to illustrate our main results.
Example 1. Consider the following equation:In this example , , , , , , , ; then , , , , which implies that and Thus by applying Theorem 1, we have that (52) has a unique -periodic solution (see Figures 1 and 2).
Example 2. Consider the following equation:In this example , , , , , , , , ; then , , , which implies that and Thus by applying Theorem 2, we have that (54) has a unique -periodic solution.
5. Result and Discussions
Remark 1. Obviously, (1) which we study in this paper is more general. Even if for the case of , the conditions imposed on and approaches to estimate a priori bound of solutions to (1) are different from the corresponding ones of the past work [1–3, 6].
Remark 2. In the past works, many scholars studied some kinds of second-order neutral functional differential equations and obtained some good results of existence and uniqueness of periodic solutions. However, these results are not related to delays. As we know that even a small delay is also likely to have an important impact on the system. Therefore, we focus on the relationship between the existence of periodic solutions and the delays. By using Mawhin’s coincidence degree, the existence and uniqueness of periodic solutions of (1) are obtained. The interesting is that we get some existence results which are related to the delays . That is different from the past results (Theorem 4 [3], Theorem 3.1 [5], and Theorem 3.1 [17]).
Remark 3. From the above examples, we see the results are related to the deviating argument , which are different from the theorems in papers [1–6, 15, 17–19] and the references therein. The studies indicate this kind of system with time delays can exhibit periodic solutions, which shows that second-order neutral functional delayed differential equation has the potential to reproduce the complex dynamics of real applied background in physics, economics, ecology, mechanics, and so forth.
Disclosure
The author carried out the main part of this article and the main theorem.
Competing Interests
The author declares that no conflict of interests exists.
Acknowledgments
This research was sponsored by the National Science Foundation of China (Grant no. 11401385).