International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 971427 | 10 pages | https://doi.org/10.1155/2012/971427

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

Academic Editor: S. M. Gusein-Zade
Received22 Mar 2012
Accepted14 Jul 2012
Published13 Aug 2012

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 𝜆 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: 𝑑𝑥𝑑𝑡=−𝑏(𝑡)𝑥(𝑡)+𝜆𝑓(𝑡,𝑥(𝑡−𝜏(𝑡)),𝑢(𝑡−𝛿(𝑡))),𝑡∈𝐑,𝑑𝑢𝑑𝑡=−𝜂(𝑡)𝑢(𝑡)+ğ‘Ž(𝑡)𝑥(ğ‘¡âˆ’ğœŽ(𝑡)),(1.1) where 𝜆>0 is a parameter, 𝑓(𝑡,𝑥1,𝑥2)∈𝐶(𝐑×(0,+∞)×(0,+∞)→(0,+∞)),  𝜏(𝑡),𝛿(𝑡),ğœŽ(𝑡)∈𝐶(𝐑,𝐑), and 𝜂(𝑡),ğ‘Ž(𝑡),𝑏(𝑡)∈𝐶(𝐑,(0,+∞)). All functions are 𝜔-periodic in 𝑡 and 𝜔>0 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 𝜆 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 𝜆. 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)𝑓(𝑡,𝑥1,𝑥2)∈𝐶(𝐑×(0,+∞)×(0,+∞),(0,+∞)) is nondecreasing in 𝑥1 and nonincreasing in 𝑥2.(H2)  There exists an 𝛼∈(0,1) such that 𝑓𝑡,𝑘𝑥1,𝑘−1𝑥2≥𝑘𝛼𝑓𝑡,𝑥1,𝑥2,∀𝑘∈(0,1),𝑡∈𝐑,𝑥1,𝑥2∈(0,+∞).(2.1)Let 𝐶𝜔={𝑥∈𝐶(𝐑,𝐑)∶𝑥(𝑡)=𝑥(𝑡+𝜔),𝑡∈𝐑}. Then, 𝐶𝜔 is a Banach space with norm ‖𝑥‖=max𝑡∈[0,𝜔]|𝑥(𝑡)|. In this paper, we will study the system (1.1) in 𝐶𝜔.

Denote ∫𝑔(𝑡,𝑠)=exp𝑠𝑡𝜂(𝑟)𝑑𝑟∫exp𝜔0,𝐺∫𝜂(𝑟)𝑑𝑟−1(2.2)(𝑡,𝑠)=exp𝑠𝑡𝑏(𝑟)𝑑𝑟∫exp𝜔0.𝑏(𝑟)𝑑𝑟−1(2.3)

Lemma 2.1 (see [5]). Consider 𝑝≤𝐺(𝑡,𝑠)â‰¤ğ‘ž, where −∫𝑝=exp𝜔0𝑏(𝑟)𝑑𝑟∫exp𝜔0∫𝑏(𝑟)𝑑𝑟−1,ğ‘ž=exp𝜔0𝑏(𝑟)𝑑𝑟∫exp𝜔0𝑏(𝑟)𝑑𝑟−1.(2.4)

Now, we convert the system (1.1) into an operator equation. Define operators Φ and Ψ𝜆 as follows: Φ𝑥(𝑡)=𝑡𝑡+𝜔𝑔(𝑡,𝑠)ğ‘Ž(𝑠)𝑥(ğ‘ âˆ’ğœŽ(𝑠))𝑑𝑠,∀𝑥∈𝐶𝜔,Ψ𝜆(𝑥,𝑦)(𝑡)=𝜆𝑡𝑡+𝜔𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)),Φ𝑦(𝑠−𝛿(𝑠)))𝑑𝑠,∀𝑥,𝑦∈𝐶𝜔,(2.5) where 𝑔(𝑡,𝑠) and 𝐺(𝑡,𝑠) 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 𝑒>0. Then, for each 𝜆>0, we can choose a proper number 𝐶𝜆>1 such that ğ¶ğœ†âŽ§âŽªâŽ¨âŽªâŽ©îƒ©âˆ«â‰¥maxğœ†ğ‘žğœ”0𝑓(𝑠,𝑒,Φ(𝑒))𝑑𝑠𝑒1/1−𝛼,𝑒∫𝜆𝑝𝜔0𝑓(𝑠,𝑒,Φ(𝑒))𝑑𝑠1/1âˆ’ğ›¼âŽ«âŽªâŽ¬âŽªâŽ­,(2.6) where Φ(⋅) is given in (2.5). Let 𝑃𝑒(𝜆)=𝑥∈𝐶𝜔∶𝐶𝜆−1𝑒≤𝑥(𝑡)≤𝐶𝜆[].𝑒on0,𝜔(2.7)

Lemma 2.2. For any 𝜆>0, 𝑥∈𝑃𝑒(𝜆) is a 𝜔-periodic solution of the system (1.1) if and only if 𝑥∈𝑃𝑒(𝜆) is a fixed point of the operator equation 𝑥(𝑡)=Ψ𝜆(𝑥,𝑥)(𝑡),(2.8) where Ψ𝜆(⋅,⋅) 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 𝑇(𝑥,𝑦)∶𝑃𝑒(𝜆)×𝑃𝑒(𝜆)→𝑃𝑒(𝜆). Then, 𝑇 is called mixed monotone if 𝑇 is nondecreasing in 𝑥 and nonincreasing in 𝑦; that is, for 𝑥1,𝑥2,𝑦1,𝑦2∈𝑃𝑒(𝜆), we have 𝑥1≤𝑥2,𝑦1≥𝑦2𝑥⟹𝑇1,𝑦1𝑥≤𝑇2,𝑦2.(2.9)

Lemma 2.4 (see [12]). Assume that 𝑇(𝑥,𝑦)∶𝑃𝑒(𝜆)×𝑃𝑒(𝜆)→𝑃𝑒(𝜆) is a mixed monotone operator and there exists 𝛼∈(0,1) such that 𝑇𝑘𝑥,𝑘−1𝑦≥𝑘𝛼𝑇(𝑥,𝑦),for𝑥,𝑦∈𝑃𝑒(𝜆),𝑘∈(0,1).(2.10) Then, 𝑇 has a unique fixed point in 𝑃𝑒(𝜆).

Lemma 2.5. Suppose that (H1) and (H2) hold. Then, Ψ𝜆∶𝑃𝑒(𝜆)×𝑃𝑒(𝜆)→𝑃𝑒(𝜆), where 𝑃𝑒(𝜆) is given in (2.7).

Proof. For any 𝑥,𝑦∈𝑃𝑒(𝜆), we have 𝐶𝜆−1𝑒≤𝑥(𝑡)≤𝐶𝜆𝑒,𝐶𝜆−1𝑒≤𝑦(𝑡)≤𝐶𝜆[].𝑒,𝑡∈0,𝜔(2.11) This together with (H1), (H2), (2.4), and (2.6) implies that Ψ𝜆(𝑥,𝑦)(𝑡)=𝜆𝑡𝑡+𝜔𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)),Φ𝑦(𝑠−𝛿(𝑠)))ğ‘‘ğ‘ â‰¤ğœ†ğ‘žğœ”0𝑓𝑠,𝐶𝜆𝑒,𝐶𝜆−1Φ(𝑒)ğ‘‘ğ‘ â‰¤ğœ†ğ‘žğ¶ğ›¼ğœ†î€œğœ”0𝑓(𝑠,𝑒,Φ(𝑒))𝑑𝑠≤𝐶𝜆Ψ𝑒,𝜆(𝑥,𝑦)(𝑡)=𝜆𝑡𝑡+𝜔𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)),Φ𝑦(𝑠−𝛿(𝑠)))𝑑𝑠≥𝜆𝑝𝜔0𝑓𝑠,𝐶𝜆−1𝑒,𝐶𝜆Φ(𝑒)𝑑𝑠≥𝜆𝑝𝐶𝜆−𝛼𝜔0𝑓(𝑠,𝑒,Φ(𝑒))𝑑𝑠≥𝐶𝜆−1𝑒.(2.12)
Therefore, Ψ𝜆∶𝑃𝑒(𝜆)×𝑃𝑒(𝜆)→𝑃𝑒(𝜆).

Lemma 2.6. Assume that (H1) and (H2) hold. Then, Ψ𝜆 is a mixed monotone operator and Ψ𝜆𝑘𝑥,𝑘−1𝑦≥𝑘𝛼Ψ𝜆(𝑥,𝑦),for𝑥,𝑦∈𝑃𝑒(𝜆),𝑘∈(0,1).(2.13)

Proof. For any 𝑥1,𝑦1,𝑥2,𝑦2∈𝑃𝑒(𝜆) with 𝑥1≤𝑥2,𝑦1≥𝑦2, it is easy to see from (H1) that Ψ𝜆𝑥1,𝑦1(𝑡)−Ψ𝜆𝑥2,𝑦2(𝑡)=𝜆𝑡𝑡+𝜔𝐺𝑓(𝑡,𝑠)𝑠,𝑥1(𝑠−𝜏(𝑠)),Φ𝑦1(𝑠−𝛿(𝑠))−𝑓𝑠,𝑥2(𝑠−𝜏(𝑠)),Φ𝑦2(𝑠−𝛿(𝑠))𝑑𝑠≤0.(2.14) Hence, Ψ𝜆 is a mixed monotone operator.
In addition, for any 𝑥,𝑦∈𝑃𝑒(𝜆) and 𝑘∈(0,1), (H2) shows that Ψ𝜆𝑘𝑥,𝑘−1𝑦=𝜆𝑡𝑡+𝜔𝐺(𝑡,𝑠)𝑓𝑠,𝑘𝑥(𝑠−𝜏(𝑠)),Φ𝑘−1𝑦(𝑠−𝛿(𝑠))𝑑𝑠≥𝑘𝛼𝜆𝑡𝑡+𝜔𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)),Φ𝑦(𝑠−𝛿(𝑠)))𝑑𝑠=𝑘𝛼Ψ𝜆(𝑥,𝑦).(2.15)
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 𝜆>0, the system (1.1) has a unique positive 𝜔-periodic solution 𝑥𝜆(𝑡)∈𝑃𝑒(𝜆).

Proof. It is easy to see from Lemmas 2.5 and 2.6 that for any 𝜆>0, Ψ𝜆∶𝑃𝑒(𝜆)×𝑃𝑒(𝜆)→𝑃𝑒(𝜆) is a mixed monotone operator and Ψ𝜆𝑘𝑥,𝑘−1𝑦≥𝑘𝛼Ψ𝜆(𝑥,𝑦),for𝑥,𝑦∈𝑃𝑒(𝜆),𝑘∈(0,1).(2.16)
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 𝛼∈(0,1/2). Then, the unique positive 𝜔-periodic solution of the system (1.1), denoted by 𝑥𝜆(𝑡), satisfies the following properties:(i)𝑥𝜆(𝑡) is strictly increasing in 𝜆; that is, if 𝜆1>𝜆2>0, then 𝑥𝜆1(𝑡)>𝑥𝜆2(𝑡),𝑡∈𝐑;(ii)lim𝜆→0+‖𝑥𝜆‖=0, and limğœ†â†’âˆžâ€–ğ‘¥ğœ†â€–=∞;(iii)𝑥𝜆(𝑡) is continuous in 𝜆; that is, if 𝜆→𝜆0>0, then ‖𝑥𝜆−𝑥𝜆0‖→0.

Proof. Suppose that 𝜆1>𝜆2>0. Let 𝐷=𝛾>0∶𝛾−1𝜆1𝜆2−11/1−𝛼𝑥𝜆2(𝑡)≥𝑥𝜆1𝜆(𝑡)≥𝛾1𝜆2−11−2𝛼/1−𝛼𝑥𝜆2(𝑡)on𝐑.(2.17)
Since 𝑒>0, we have 𝑥𝜆1(𝑡)>0 and 𝑥𝜆2(𝑡)>0 for 𝑡∈𝐑. Thus 𝛾∗𝜆∶=min1−1𝜆21−2𝛼/1−𝛼min𝑡∈𝐑𝑥𝜆1(𝑡)𝑥𝜆2,𝜆(𝑡)1𝜆2−11/1−𝛼min𝑡∈𝐑𝑥𝜆2(𝑡)𝑥𝜆1(𝑡)>0.(2.18) Obviously, for any 𝛾 satisfying 0<𝛾<𝛾∗, 𝛾∈𝐷. Hence, 𝐷≠∅.
Define 𝛾=sup𝐷. Then 𝛾−1𝜆1𝜆2−11/1−𝛼𝑥𝜆2(𝑡)≥𝑥𝜆1(𝑡)≥𝛾𝜆1𝜆2−11−2𝛼/1−𝛼𝑥𝜆2(𝑡),𝑡∈𝐑.(2.19) Now let us show that 𝛾≥1. In fact, if 0<𝛾<1, then (H1) and (H2) imply that 𝜆1𝑓𝑡,𝑥𝜆1(𝑡−𝜏(𝑡)),Φ𝑥𝜆1(𝑡−𝛿(𝑡))≥𝜆1𝑓𝑡,𝛾𝜆1𝜆2−11−2𝛼/1−𝛼𝑥𝜆2(𝑡−𝜏(𝑡)),𝛾−1𝜆1𝜆2−11/1−𝛼Φ𝑥𝜆2(𝑡−𝛿(𝑡))≥𝜆1𝑓𝑡,𝛾𝑥𝜆2(𝑡−𝜏(𝑡)),𝛾−1𝜆1𝜆2−11/1−𝛼Φ𝑥𝜆2≥(𝑡−𝛿(𝑡))𝛾𝛼𝜆1𝑓𝑡,𝑥𝜆2𝜆(𝑡−𝜏(𝑡)),1𝜆2−11/1−𝛼Φ𝑥𝜆2≥(𝑡−𝛿(𝑡))𝛾𝛼𝜆1𝑓𝜆𝑡,1𝜆2−1−1/1−𝛼𝑥𝜆2𝜆(𝑡−𝜏(𝑡)),1𝜆2−11/1−𝛼Φ𝑥𝜆2≥(𝑡−𝛿(𝑡))𝛾𝛼𝜆1𝜆1𝜆2−1−𝛼/1−𝛼𝑓𝑡,𝑥𝜆2(𝑡−𝜏(𝑡)),Φ𝑥𝜆2=(𝑡−𝛿(𝑡))𝛾𝛼𝜆1𝜆2−11−2𝛼/1−𝛼𝜆2𝑓𝑡,𝑥𝜆2(𝑡−𝜏(𝑡)),Φ𝑥𝜆2,𝜆(𝑡−𝛿(𝑡))2𝑓𝑡,𝑥𝜆2(𝑡−𝜏(𝑡)),Φ𝑥𝜆2(𝑡−𝛿(𝑡))≥𝜆2𝑓𝑡,𝛾𝜆1𝜆2−1−1/1−𝛼𝑥𝜆1(𝑡−𝜏(𝑡)),𝛾−1𝜆1𝜆2−1−1−2𝛼/1−𝛼Φ𝑥𝜆1(𝑡−𝛿(𝑡))≥𝜆2𝑓𝑡,𝛾𝜆1𝜆2−1−1/1−𝛼𝑥𝜆1(𝑡−𝜏(𝑡)),𝛾−1Φ𝑥𝜆1≥(𝑡−𝛿(𝑡))𝛾𝛼𝜆2𝑓𝜆𝑡,1𝜆2−1−1/1−𝛼𝑥𝜆1(𝑡−𝜏(𝑡)),Φ𝑥𝜆1(≥𝑡−𝛿(𝑡))𝛾𝛼𝜆2𝑓𝜆𝑡,1𝜆2−1−1/1−𝛼𝑥𝜆1𝜆(𝑡−𝜏(𝑡)),1𝜆2−11/1−𝛼Φ𝑥𝜆1≥(𝑡−𝛿(𝑡))𝛾𝛼𝜆2𝜆1𝜆2−1−𝛼/1−𝛼𝑓𝑡,𝑥𝜆1(𝑡−𝜏(𝑡)),Φ𝑥𝜆1(=𝑡−𝛿(𝑡))𝛾𝛼𝜆1𝜆2−1−1/1−𝛼𝜆1𝑓𝑡,𝑥𝜆1(𝑡−𝜏(𝑡)),Φ𝑥𝜆1.(𝑡−𝛿(𝑡))(2.20)
Therefore, 𝑥𝜆1(𝑡)=Ψ𝜆1𝑥𝜆1,𝑥𝜆1(𝑡)≥𝛾𝛼𝜆1𝜆2−11−2𝛼/1−𝛼𝑥𝜆2(𝑥𝑡),𝜆2(𝑡)=Ψ𝜆2𝑥𝜆2,𝑥𝜆2(𝑡)≥𝛾𝛼𝜆1𝜆2−1−1/1−𝛼𝑥𝜆1(𝑡).(2.21)
From (2.21), we have 𝛾−𝛼𝜆1𝜆2−11/1−𝛼𝑥𝜆2(𝑡)≥𝑥𝜆1(𝑡)≥𝛾𝛼𝜆1𝜆2−11−2𝛼/1−𝛼𝑥𝜆2(𝑡),𝑡∈𝐑.(2.22) Noticing that 0<𝛾<1 and 𝛼∈(0,1), one can see 𝛾𝛼>𝛾, a contradiction with the definition of 𝛾. Thus, 𝛾≥1 and 𝑥𝜆1(𝑡)≥𝛾𝜆1𝜆2−11−2𝛼/1−𝛼𝑥𝜆2(𝜆𝑡)≥1𝜆2−11−2𝛼/1−𝛼𝑥𝜆2(𝑡)>𝑥𝜆2(𝑡),𝑡∈𝐑.(2.23)
Thus, Conclusion (i) holds.
Next, let us prove Conclusion (ii).
In (2.23), let 𝜆1 be fixed and 𝜆=𝜆2; then 𝑥𝜆(𝑡)≤𝜆𝜆1−11−2𝛼/1−𝛼𝑥𝜆1(𝑡),𝑡∈𝐑.(2.24) Thus ‖𝑥𝜆‖≤(𝜆𝜆1−1)1−2𝛼/1−𝛼‖𝑥𝜆1‖, which means ‖𝑥𝜆‖→0 as 𝜆→0.
Similarly, let 𝜆2 be fixed and 𝜆=𝜆1; then 𝑥𝜆(𝑡)≥𝜆𝜆2−11−2𝛼/1−𝛼𝑥𝜆2(𝑡),𝑡∈𝐑.(2.25) Therefore, ‖𝑥𝜆‖≥(𝜆𝜆2−1)1−2𝛼/1−𝛼‖𝑥𝜆2‖, which implies that â€–ğ‘¥ğœ†â€–â†’âˆž as ğœ†â†’âˆž.
Finally, we prove Conclusion (iii).
For any fixed 𝜆0>0, let 𝜆>𝜆0. Set 𝜆1=𝜆0 in (2.24); then 𝑥𝜆(𝑡)≤𝜆𝜆0−11−2𝛼/1−𝛼𝑥𝜆0(𝑡),𝑡∈𝐑,(2.26) which means ‖‖𝑥𝜆−𝑥𝜆0‖‖≤𝜆𝜆0−11−2𝛼/1−𝛼‖‖𝑥−1𝜆0‖‖.(2.27) As a result, ‖𝑥𝜆−𝑥𝜆0‖→0 as 𝜆→𝜆+0. Similarly, we can show that ‖𝑥𝜆−𝑥𝜆0‖→0 as 𝜆→𝜆0.
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: 𝑑𝑥𝑑𝑡=−(2+cos𝑡)𝑥(𝑡)+𝜆𝑓(𝑡,𝑥(𝑡−𝜏(𝑡)),𝑢(𝑡−𝛿(𝑡))),𝑡∈𝐑,𝑑𝑢𝑑𝑡=−(3+sin𝑡)𝑢(𝑡)+3𝑥(ğ‘¡âˆ’ğœŽ(𝑡)),(3.1) where 𝜆>0 is a parameter, 𝜏(𝑡),𝛿(𝑡),ğœŽ(𝑡)∈𝐶(𝐑,𝐑) are 2𝜋-periodic in 𝑡, and 𝑓𝑡,𝑥1,𝑥2=(2+sin𝑡)3√𝑥1+13√𝑥2.(3.2)

It is easy to see that 𝑓(𝑡,𝑥1,𝑥2)∈𝐶(𝐑×(0,+∞)×(0,+∞)→(0,+∞)) is 2𝜋-periodic in 𝑡. 𝜂(𝑡)=3+sin𝑡,ğ‘Ž(𝑡)=3,𝑏(𝑡)=2+cos𝑡∈𝐶(𝐑,(0,+∞)) are 2𝜋-periodic in 𝑡.

Since 𝜕𝑓𝑡,𝑥1,𝑥2𝜕𝑥1=2+sin𝑡3𝑥12/3>0,∀𝑡∈𝐑,𝑥1,𝑥2∈(0,+∞),𝜕𝑓𝑡,𝑥1,𝑥2𝜕𝑥21=−3𝑥24/3<0,∀𝑡∈𝐑,𝑥1,𝑥2∈(0,+∞),(3.3) we conclude that (H1) is satisfied.

Now, we check (H2). As a matter of fact, forall𝑡∈𝐑,𝑥1,𝑥2∈(0,+∞), we have 𝑓𝑡,𝑘𝑥1,𝑘−1𝑥2=3√𝑘(2+sin𝑡)3√𝑥1+13√𝑥2≥3√𝑘𝑓𝑡,𝑥1,𝑥2,(3.4) therefore, (H2) holds.

Hence, Theorem 2.7 shows that for any 𝜆>0, the system (3.1) has a unique positive 2𝜋-periodic solution.

Let us set 𝜆=2, 𝜏(𝑡)=1,𝛿(𝑡)=0.1,ğœŽ(𝑡)=2; then, the unique positive 2𝜋-periodic solution of the system (3.1) can be shown in Figure 1.

Next, to illustrate Theorem 2.8, we set 𝜆=2,2.1,2.2,2.3,2.4, and 2.5, respectively, and let 𝜏(𝑡)=1,𝛿(𝑡)=0.1,ğœŽ(𝑡)=2; then the unique positive 2𝜋-periodic solutions of the system (3.1) with these different 𝜆 can be shown in Figure 2. From this figure, one can easily see that 𝑥𝜆(𝑡) is strictly increasing in 𝜆.

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 © 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.


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