Research Article | Open Access

# On the Uniqueness and Dependence of Positive Periodic Solutions for Delay Differential Systems with Feedback Control

**Academic Editor:**S. M. Gusein-Zade

#### Abstract

This paper investigates a class of delay differential systems with feedback control. Sufficient conditions are obtained for the existence and uniqueness of the positive periodic solution by utilizing some results from the mixed monotone operator theory. Meanwhile, the dependence of the positive periodic solution on the parameter is also studied. Finally, an example together with numerical simulations is worked out to illustrate the main results.

#### 1. Introduction

As is known to all, the periodic environment changes and the unpredictable forces play an important role in many biological and ecological systems. Therefore, several different periodic models with feedback control have been studied by many authors (see [1–10] and references therein). For instance, Gopalsamy and Weng [2] introduced a feedback control variable into the delayed logistic model and discussed the asymptotic behavior of solutions in logistic models with feedback control. Li and Wang [5] investigated the existence and global attractivity of positive periodic solutions for a delay differential system with feedback control. The method they used involved Krasnoselskii's fixed point theorem and estimates of uniform upper and lower bounds of solutions. In a recent work [3], Guo considered the existence of nontrivial periodic solutions for a kind of nonlinear functional differential system with feedback control. By using Leray-Schauder nonlinear alternative, the author obtained several sufficient conditions for the existence of nontrivial solutions. A class of impulsive functional equations with feedback control was studied by Guo and Liu [4], and they presented the existence results of three positive periodic solutions by using Leggett-Williams fixed point theorem.

However, as we know, there are few results on the uniqueness and parameter dependence of the positive periodic solution for delay differential systems with feedback control. Motivated by this fact, this paper is devoted to investigating the uniqueness and parameter dependence of the positive periodic solution for the following nonlinear nonautonomous delay differential system with feedback control: where is a parameter, , and . All functions are -periodic in and is a constant.

The main features here are as follows. On one hand, by utilizing the mixed monotone operator theory, the existence and uniqueness of the positive periodic solution of the delay differential system (1.1) are studied in this work. As is known to us, there are few papers to investigate this topic. On the other hand, the dependence of the positive periodic solution on the parameter is studied, and some interesting results are obtained.

The rest of this paper is organized as follows. Section 2 presents the existence and uniqueness result of the system (1.1) together with the dependence of the positive periodic solution on the parameter . In Section 3, an illustrative example is worked out to support the main results of this work.

#### 2. Main Results

For convenience, let us first list some conditions.(H1) is nondecreasing in and nonincreasing in .(H2) There exists an such that Let . Then, is a Banach space with norm . In this paper, we will study the system (1.1) in .

Denote

Lemma 2.1 (see [5]). *Consider , where
*

Now, we convert the system (1.1) into an operator equation. Define operators and as follows: where and are given in (2.2) and (2.3), respectively.

For the sake of using a fixed point theorem on mixed monotone operators, choose a fixed constant . Then, for each , we can choose a proper number such that where is given in (2.5). Let

Lemma 2.2. * For any , is a -periodic solution of the system (1.1) if and only if is a fixed point of the operator equation
**
where is given in (2.5).*

* Proof. * The proof of this lemma is similar to Theorem 2.1 in [5], and thus we omit it.

Next, we recall some results from the monotone operator theory. The following results are well known (see [11–13], for details).

*Definition 2.3 (see [12]). *Assume that . Then, is called mixed monotone if is nondecreasing in and nonincreasing in ; that is, for , we have

Lemma 2.4 (see [12]). * Assume that is a mixed monotone operator and there exists such that
**
Then, has a unique fixed point in .*

Lemma 2.5. * Suppose that (H1) and (H2) hold. Then, , where is given in (2.7).*

* Proof. * For any , we have
This together with (H1), (H2), (2.4), and (2.6) implies that

Therefore, .

Lemma 2.6. *Assume that (H1) and (H2) hold. Then, is a mixed monotone operator and
*

*Proof. *For any with , it is easy to see from (H1) that
Hence, is a mixed monotone operator.

In addition, for any and , (H2) shows that

To sum up, the proof of this lemma is completed.

Finally, we present the main results of this paper.

Theorem 2.7. *Suppose that (H1) and (H2) hold. Then, for any , the system (1.1) has a unique positive -periodic solution .*

* Proof. * It is easy to see from Lemmas 2.5 and 2.6 that for any , is a mixed monotone operator and

Consequently, Lemmas 2.2 and 2.4 imply that the conclusion holds true.

Theorem 2.8. *Assume that (H1) and (H2) hold. In addition, suppose that . Then, the unique positive -periodic solution of the system (1.1), denoted by , satisfies the following properties:*(i)* is strictly increasing in ; that is, if , then ;*(ii)*, and ;*(iii)* is continuous in ; that is, if , then .*

* Proof. * Suppose that . Let

Since , we have and for . Thus
Obviously, for any satisfying , . Hence, .

Define . Then
Now let us show that . In fact, if , then (H1) and (H2) imply that

Therefore,

From (2.21), we have
Noticing that and , one can see , a contradiction with the definition of . Thus, and

Thus, Conclusion (i) holds.

Next, let us prove Conclusion (ii).

In (2.23), let be fixed and ; then
Thus , which means as .

Similarly, let be fixed and ; then
Therefore, , which implies that as .

Finally, we prove Conclusion (iii).

For any fixed , let . Set in (2.24); then
which means
As a result, as . Similarly, we can show that as .

To sum up, the proof of this theorem is completed.

#### 3. An Illustrative Example

In this section, we give an illustrative example to show how to use our new results.

*Example 3.1. * Consider the following nonlinear nonautonomous delay differential system with feedback control:
where is a parameter, are -periodic in , and

It is easy to see that is -periodic in . are -periodic in .

Since we conclude that (H1) is satisfied.

Now, we check (H2). As a matter of fact, , we have therefore, (H2) holds.

Hence, Theorem 2.7 shows that for any , the system (3.1) has a unique positive -periodic solution.

Let us set , ; then, the unique positive -periodic solution of the system (3.1) can be shown in Figure 1.

**(a)**

**(b)**

Next, to illustrate Theorem 2.8, we set , and , respectively, and let ; then the unique positive -periodic solutions of the system (3.1) with these different can be shown in Figure 2. From this figure, one can easily see that is strictly increasing in .

#### Acknowledgments

The paper is supported by NSF of Shandong (ZR2009AM006), the Key Project of Chinese Ministry of Education (no: 209072), the Science & Technology Development Funds of Shandong Education Committee (J08LI10), and Graduate Independent Innovation Foundation of Shandong University (yzc10064).

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#### Copyright

Copyright © 2012 Haitao Li and Yansheng Liu. 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.