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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 971427, 10 pages
http://dx.doi.org/10.1155/2012/971427
Research Article

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

1School of Control Science and Engineering, Shandong University, Jinan 250061, China
2Department of Mathematics, Shandong Normal University, Jinan 250014, China

Received 22 March 2012; Accepted 14 July 2012

Academic Editor: S. M.Β Gusein-Zade

Copyright Β© 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.

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 πœ† 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: 𝑑π‘₯𝑑𝑑=βˆ’π‘(𝑑)π‘₯(𝑑)+πœ†π‘“(𝑑,π‘₯(π‘‘βˆ’πœ(𝑑)),𝑒(π‘‘βˆ’π›Ώ(𝑑))),π‘‘βˆˆπ‘,𝑑𝑒𝑑𝑑=βˆ’πœ‚(𝑑)𝑒(𝑑)+π‘Ž(𝑑)π‘₯(π‘‘βˆ’πœŽ(𝑑)),(1.1) where πœ†>0 is a parameter, 𝑓(𝑑,π‘₯1,π‘₯2)∈𝐢(𝐑×(0,+∞)Γ—(0,+∞)β†’(0,+∞)),β€‰β€‰πœ(𝑑),𝛿(𝑑),𝜎(𝑑)∈𝐢(𝐑,𝐑), and πœ‚(𝑑),π‘Ž(𝑑),𝑏(𝑑)∈𝐢(𝐑,(0,+∞)). All functions are πœ”-periodic in 𝑑 and πœ”>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 πœ† 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 πœ†. 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(𝑑,π‘₯1,π‘₯2)∈𝐢(𝐑×(0,+∞)Γ—(0,+∞),(0,+∞)) is nondecreasing in π‘₯1 and nonincreasing in π‘₯2.(H2)  There exists an π›Όβˆˆ(0,1) such that 𝑓𝑑,π‘˜π‘₯1,π‘˜βˆ’1π‘₯2ξ€Έβ‰₯π‘˜π›Όπ‘“ξ€·π‘‘,π‘₯1,π‘₯2ξ€Έ,βˆ€π‘˜βˆˆ(0,1),π‘‘βˆˆπ‘,π‘₯1,π‘₯2∈(0,+∞).(2.1)Let πΆπœ”={π‘₯∈𝐢(𝐑,𝐑)∢π‘₯(𝑑)=π‘₯(𝑑+πœ”),π‘‘βˆˆπ‘}. Then, πΆπœ” is a Banach space with norm β€–π‘₯β€–=maxπ‘‘βˆˆ[0,πœ”]|π‘₯(𝑑)|. In this paper, we will study the system (1.1) in πΆπœ”.

Denote ξ€½βˆ«π‘”(𝑑,𝑠)=expπ‘ π‘‘πœ‚ξ€Ύ(π‘Ÿ)π‘‘π‘Ÿξ€½βˆ«expπœ”0ξ€Ύ,πΊξ€½βˆ«πœ‚(π‘Ÿ)π‘‘π‘Ÿβˆ’1(2.2)(𝑑,𝑠)=exp𝑠𝑑𝑏(π‘Ÿ)π‘‘π‘Ÿξ€½βˆ«expπœ”0ξ€Ύ.𝑏(π‘Ÿ)π‘‘π‘Ÿβˆ’1(2.3)

Lemma 2.1 (see [5]). Consider 𝑝≀𝐺(𝑑,𝑠)β‰€π‘ž, where ξ€½βˆ’βˆ«π‘=expπœ”0𝑏(π‘Ÿ)π‘‘π‘Ÿξ€½βˆ«expπœ”0ξ€Ύξ€½βˆ«π‘(π‘Ÿ)π‘‘π‘Ÿβˆ’1,π‘ž=expπœ”0𝑏(π‘Ÿ)π‘‘π‘Ÿξ€½βˆ«expπœ”0𝑏(π‘Ÿ)π‘‘π‘Ÿβˆ’1.(2.4)

Now, we convert the system (1.1) into an operator equation. Define operators Ξ¦ and Ξ¨πœ† as follows: ξ€œΞ¦π‘₯(𝑑)=𝑑𝑑+πœ”π‘”(𝑑,𝑠)π‘Ž(𝑠)π‘₯(π‘ βˆ’πœŽ(𝑠))𝑑𝑠,βˆ€π‘₯βˆˆπΆπœ”,Ξ¨πœ†ξ€œ(π‘₯,𝑦)(𝑑)=πœ†π‘‘π‘‘+πœ”πΊ(𝑑,𝑠)𝑓(𝑠,π‘₯(π‘ βˆ’πœ(𝑠)),Φ𝑦(π‘ βˆ’π›Ώ(𝑠)))𝑑𝑠,βˆ€π‘₯,π‘¦βˆˆπΆπœ”,(2.5) where 𝑔(𝑑,𝑠) and 𝐺(𝑑,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 πœ†>0, we can choose a proper number πΆπœ†>1 such that πΆπœ†βŽ§βŽͺ⎨βŽͺβŽ©ξƒ©βˆ«β‰₯maxπœ†π‘žπœ”0𝑓(𝑠,𝑒,Ξ¦(𝑒))𝑑𝑠𝑒ξƒͺ1/1βˆ’π›Ό,ξƒ©π‘’βˆ«πœ†π‘πœ”0ξƒͺ𝑓(𝑠,𝑒,Ξ¦(𝑒))𝑑𝑠1/1βˆ’π›ΌβŽ«βŽͺ⎬βŽͺ⎭,(2.6) where Ξ¦(β‹…) is given in (2.5). Let 𝑃𝑒(πœ†)=π‘₯βˆˆπΆπœ”βˆΆπΆπœ†βˆ’1𝑒≀π‘₯(𝑑)β‰€πΆπœ†[]ξ€Ύ.𝑒on0,πœ”(2.7)

Lemma 2.2. For any πœ†>0, π‘₯βˆˆπ‘ƒπ‘’(πœ†) is a πœ”-periodic solution of the system (1.1) if and only if π‘₯βˆˆπ‘ƒπ‘’(πœ†) is a fixed point of the operator equation π‘₯(𝑑)=Ξ¨πœ†(π‘₯,π‘₯)(𝑑),(2.8) where Ξ¨πœ†(β‹…,β‹…) 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(π‘₯,𝑦)βˆΆπ‘ƒπ‘’(πœ†)×𝑃𝑒(πœ†)→𝑃𝑒(πœ†). Then, 𝑇 is called mixed monotone if 𝑇 is nondecreasing in π‘₯ and nonincreasing in 𝑦; that is, for π‘₯1,π‘₯2,𝑦1,𝑦2βˆˆπ‘ƒπ‘’(πœ†), we have π‘₯1≀π‘₯2,𝑦1β‰₯𝑦2ξ€·π‘₯βŸΉπ‘‡1,𝑦1ξ€Έξ€·π‘₯≀𝑇2,𝑦2ξ€Έ.(2.9)

Lemma 2.4 (see [12]). Assume that 𝑇(π‘₯,𝑦)βˆΆπ‘ƒπ‘’(πœ†)×𝑃𝑒(πœ†)→𝑃𝑒(πœ†) is a mixed monotone operator and there exists π›Όβˆˆ(0,1) such that π‘‡ξ€·π‘˜π‘₯,π‘˜βˆ’1𝑦β‰₯π‘˜π›Όπ‘‡(π‘₯,𝑦),forπ‘₯,π‘¦βˆˆπ‘ƒπ‘’(πœ†),π‘˜βˆˆ(0,1).(2.10) Then, 𝑇 has a unique fixed point in 𝑃𝑒(πœ†).

Lemma 2.5. Suppose that (H1) and (H2) hold. Then, Ξ¨πœ†βˆΆπ‘ƒπ‘’(πœ†)×𝑃𝑒(πœ†)→𝑃𝑒(πœ†), where 𝑃𝑒(πœ†) is given in (2.7).

Proof. For any π‘₯,π‘¦βˆˆπ‘ƒπ‘’(πœ†), we have πΆπœ†βˆ’1𝑒≀π‘₯(𝑑)β‰€πΆπœ†π‘’,πΆπœ†βˆ’1𝑒≀𝑦(𝑑)β‰€πΆπœ†[].𝑒,π‘‘βˆˆ0,πœ”(2.11) This together with (H1), (H2), (2.4), and (2.6) implies that Ξ¨πœ†(ξ€œπ‘₯,𝑦)(𝑑)=πœ†π‘‘π‘‘+πœ”ξ€œπΊ(𝑑,𝑠)𝑓(𝑠,π‘₯(π‘ βˆ’πœ(𝑠)),Φ𝑦(π‘ βˆ’π›Ώ(𝑠)))π‘‘π‘ β‰€πœ†π‘žπœ”0𝑓𝑠,πΆπœ†π‘’,πΆπœ†βˆ’1ξ€ΈΞ¦(𝑒)π‘‘π‘ β‰€πœ†π‘žπΆπ›Όπœ†ξ€œπœ”0𝑓(𝑠,𝑒,Ξ¦(𝑒))π‘‘π‘ β‰€πΆπœ†Ξ¨π‘’,πœ†ξ€œ(π‘₯,𝑦)(𝑑)=πœ†π‘‘π‘‘+πœ”ξ€œπΊ(𝑑,𝑠)𝑓(𝑠,π‘₯(π‘ βˆ’πœ(𝑠)),Φ𝑦(π‘ βˆ’π›Ώ(𝑠)))𝑑𝑠β‰₯πœ†π‘πœ”0𝑓𝑠,πΆπœ†βˆ’1𝑒,πΆπœ†ξ€ΈΞ¦(𝑒)𝑑𝑠β‰₯πœ†π‘πΆπœ†βˆ’π›Όξ€œπœ”0𝑓(𝑠,𝑒,Ξ¦(𝑒))𝑑𝑠β‰₯πΆπœ†βˆ’1𝑒.(2.12)
Therefore, Ξ¨πœ†βˆΆπ‘ƒπ‘’(πœ†)×𝑃𝑒(πœ†)→𝑃𝑒(πœ†).

Lemma 2.6. Assume that (H1) and (H2) hold. Then, Ξ¨πœ† is a mixed monotone operator and Ξ¨πœ†ξ€·π‘˜π‘₯,π‘˜βˆ’1𝑦β‰₯π‘˜π›ΌΞ¨πœ†(π‘₯,𝑦),forπ‘₯,π‘¦βˆˆπ‘ƒπ‘’(πœ†),π‘˜βˆˆ(0,1).(2.13)

Proof. For any π‘₯1,𝑦1,π‘₯2,𝑦2βˆˆπ‘ƒπ‘’(πœ†) with π‘₯1≀π‘₯2,𝑦1β‰₯𝑦2, it is easy to see from (H1) that Ξ¨πœ†ξ€·π‘₯1,𝑦1ξ€Έ(𝑑)βˆ’Ξ¨πœ†ξ€·π‘₯2,𝑦2ξ€Έξ€œ(𝑑)=πœ†π‘‘π‘‘+πœ”πΊξ€Ίπ‘“ξ€·(𝑑,𝑠)𝑠,π‘₯1(π‘ βˆ’πœ(𝑠)),Φ𝑦1ξ€Έξ€·(π‘ βˆ’π›Ώ(𝑠))βˆ’π‘“π‘ ,π‘₯2(π‘ βˆ’πœ(𝑠)),Φ𝑦2(π‘ βˆ’π›Ώ(𝑠))𝑑𝑠≀0.(2.14) Hence, Ξ¨πœ† is a mixed monotone operator.
In addition, for any π‘₯,π‘¦βˆˆπ‘ƒπ‘’(πœ†) and π‘˜βˆˆ(0,1), (H2) shows that Ξ¨πœ†ξ€·π‘˜π‘₯,π‘˜βˆ’1π‘¦ξ€Έξ€œ=πœ†π‘‘π‘‘+πœ”ξ€·πΊ(𝑑,𝑠)𝑓𝑠,π‘˜π‘₯(π‘ βˆ’πœ(𝑠)),Ξ¦π‘˜βˆ’1𝑦(π‘ βˆ’π›Ώ(𝑠))𝑑𝑠β‰₯π‘˜π›Όπœ†ξ€œπ‘‘π‘‘+πœ”πΊ(𝑑,𝑠)𝑓(𝑠,π‘₯(π‘ βˆ’πœ(𝑠)),Φ𝑦(π‘ βˆ’π›Ώ(𝑠)))𝑑𝑠=π‘˜π›ΌΞ¨πœ†(π‘₯,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 πœ†>0, the system (1.1) has a unique positive πœ”-periodic solution π‘₯πœ†(𝑑)βˆˆπ‘ƒπ‘’(πœ†).

Proof. It is easy to see from Lemmas 2.5 and 2.6 that for any πœ†>0, Ξ¨πœ†βˆΆπ‘ƒπ‘’(πœ†)×𝑃𝑒(πœ†)→𝑃𝑒(πœ†) is a mixed monotone operator and Ξ¨πœ†ξ€·π‘˜π‘₯,π‘˜βˆ’1𝑦β‰₯π‘˜π›ΌΞ¨πœ†(π‘₯,𝑦),forπ‘₯,π‘¦βˆˆπ‘ƒπ‘’(πœ†),π‘˜βˆˆ(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 π›Όβˆˆ(0,1/2). Then, the unique positive πœ”-periodic solution of the system (1.1), denoted by π‘₯πœ†(𝑑), satisfies the following properties:(i)π‘₯πœ†(𝑑) is strictly increasing in πœ†; that is, if πœ†1>πœ†2>0, then π‘₯πœ†1(𝑑)>π‘₯πœ†2(𝑑),π‘‘βˆˆπ‘;(ii)limπœ†β†’0+β€–π‘₯πœ†β€–=0, and limπœ†β†’βˆžβ€–π‘₯πœ†β€–=∞;(iii)π‘₯πœ†(𝑑) is continuous in πœ†; that is, if πœ†β†’πœ†0>0, then β€–π‘₯πœ†βˆ’π‘₯πœ†0β€–β†’0.

Proof. Suppose that πœ†1>πœ†2>0. Let 𝐷=𝛾>0βˆΆπ›Ύβˆ’1ξ€·πœ†1πœ†2βˆ’1ξ€Έ1/1βˆ’π›Όπ‘₯πœ†2(𝑑)β‰₯π‘₯πœ†1ξ€·πœ†(𝑑)β‰₯𝛾1πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†2(𝑑)on𝐑.(2.17)
Since 𝑒>0, we have π‘₯πœ†1(𝑑)>0 and π‘₯πœ†2(𝑑)>0 for π‘‘βˆˆπ‘. Thus π›Ύβˆ—ξƒ―ξ€·πœ†βˆΆ=min1βˆ’1πœ†2ξ€Έ1βˆ’2𝛼/1βˆ’π›Όminπ‘‘βˆˆπ‘π‘₯πœ†1(𝑑)π‘₯πœ†2,ξ€·πœ†(𝑑)1πœ†2βˆ’1ξ€Έ1/1βˆ’π›Όminπ‘‘βˆˆπ‘π‘₯πœ†2(𝑑)π‘₯πœ†1ξƒ°(𝑑)>0.(2.18) Obviously, for any 𝛾 satisfying 0<𝛾<π›Ύβˆ—, π›Ύβˆˆπ·. Hence, π·β‰ βˆ….
Define 𝛾=sup𝐷. Then π›Ύβˆ’1ξ€·πœ†1πœ†2βˆ’1ξ€Έ1/1βˆ’π›Όπ‘₯πœ†2(𝑑)β‰₯π‘₯πœ†1(𝑑)β‰₯π›Ύξ€·πœ†1πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†2(𝑑),π‘‘βˆˆπ‘.(2.19) Now let us show that 𝛾β‰₯1. In fact, if 0<𝛾<1, then (H1) and (H2) imply that πœ†1𝑓𝑑,π‘₯πœ†1(π‘‘βˆ’πœ(𝑑)),Ξ¦π‘₯πœ†1ξ€Έ(π‘‘βˆ’π›Ώ(𝑑))β‰₯πœ†1𝑓𝑑,π›Ύξ€·πœ†1πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†2(π‘‘βˆ’πœ(𝑑)),π›Ύβˆ’1ξ€·πœ†1πœ†2βˆ’1ξ€Έ1/1βˆ’π›ΌΞ¦π‘₯πœ†2(π‘‘βˆ’π›Ώ(𝑑))β‰₯πœ†1𝑓𝑑,𝛾π‘₯πœ†2(π‘‘βˆ’πœ(𝑑)),π›Ύβˆ’1ξ€·πœ†1πœ†2βˆ’1ξ€Έ1/1βˆ’π›ΌΞ¦π‘₯πœ†2β‰₯(π‘‘βˆ’π›Ώ(𝑑))π›Ύπ›Όπœ†1𝑓𝑑,π‘₯πœ†2ξ€·πœ†(π‘‘βˆ’πœ(𝑑)),1πœ†2βˆ’1ξ€Έ1/1βˆ’π›ΌΞ¦π‘₯πœ†2β‰₯(π‘‘βˆ’π›Ώ(𝑑))π›Ύπ›Όπœ†1π‘“ξ‚€ξ€·πœ†π‘‘,1πœ†2βˆ’1ξ€Έβˆ’1/1βˆ’π›Όπ‘₯πœ†2ξ€·πœ†(π‘‘βˆ’πœ(𝑑)),1πœ†2βˆ’1ξ€Έ1/1βˆ’π›ΌΞ¦π‘₯πœ†2β‰₯(π‘‘βˆ’π›Ώ(𝑑))π›Ύπ›Όπœ†1ξ€·πœ†1πœ†2βˆ’1ξ€Έβˆ’π›Ό/1βˆ’π›Όπ‘“ξ€·π‘‘,π‘₯πœ†2(π‘‘βˆ’πœ(𝑑)),Ξ¦π‘₯πœ†2ξ€Έ=(π‘‘βˆ’π›Ώ(𝑑))π›Ύπ›Όξ€·πœ†1πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπœ†2𝑓𝑑,π‘₯πœ†2(π‘‘βˆ’πœ(𝑑)),Ξ¦π‘₯πœ†2ξ€Έ,πœ†(π‘‘βˆ’π›Ώ(𝑑))2𝑓𝑑,π‘₯πœ†2(π‘‘βˆ’πœ(𝑑)),Ξ¦π‘₯πœ†2(ξ€Έπ‘‘βˆ’π›Ώ(𝑑))β‰₯πœ†2𝑓𝑑,π›Ύξ€·πœ†1πœ†2βˆ’1ξ€Έβˆ’1/1βˆ’π›Όπ‘₯πœ†1(π‘‘βˆ’πœ(𝑑)),π›Ύβˆ’1ξ€·πœ†1πœ†2βˆ’1ξ€Έβˆ’1βˆ’2𝛼/1βˆ’π›ΌΞ¦π‘₯πœ†1(π‘‘βˆ’π›Ώ(𝑑))β‰₯πœ†2𝑓𝑑,π›Ύξ€·πœ†1πœ†2βˆ’1ξ€Έβˆ’1/1βˆ’π›Όπ‘₯πœ†1(π‘‘βˆ’πœ(𝑑)),π›Ύβˆ’1Ξ¦π‘₯πœ†1β‰₯(π‘‘βˆ’π›Ώ(𝑑))π›Ύπ›Όπœ†2π‘“ξ‚€ξ€·πœ†π‘‘,1πœ†2βˆ’1ξ€Έβˆ’1/1βˆ’π›Όπ‘₯πœ†1(π‘‘βˆ’πœ(𝑑)),Ξ¦π‘₯πœ†1(β‰₯π‘‘βˆ’π›Ώ(𝑑))π›Ύπ›Όπœ†2π‘“ξ‚€ξ€·πœ†π‘‘,1πœ†2βˆ’1ξ€Έβˆ’1/1βˆ’π›Όπ‘₯πœ†1ξ€·πœ†(π‘‘βˆ’πœ(𝑑)),1πœ†2βˆ’1ξ€Έ1/1βˆ’π›ΌΞ¦π‘₯πœ†1β‰₯(π‘‘βˆ’π›Ώ(𝑑))π›Ύπ›Όπœ†2ξ€·πœ†1πœ†2βˆ’1ξ€Έβˆ’π›Ό/1βˆ’π›Όπ‘“ξ€·π‘‘,π‘₯πœ†1(π‘‘βˆ’πœ(𝑑)),Ξ¦π‘₯πœ†1(ξ€Έ=π‘‘βˆ’π›Ώ(𝑑))π›Ύπ›Όξ€·πœ†1πœ†2βˆ’1ξ€Έβˆ’1/1βˆ’π›Όπœ†1𝑓𝑑,π‘₯πœ†1(π‘‘βˆ’πœ(𝑑)),Ξ¦π‘₯πœ†1ξ€Έ.(π‘‘βˆ’π›Ώ(𝑑))(2.20)
Therefore, π‘₯πœ†1(𝑑)=Ξ¨πœ†1ξ€·π‘₯πœ†1,π‘₯πœ†1ξ€Έ(𝑑)β‰₯π›Ύπ›Όξ€·πœ†1πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†2(π‘₯𝑑),πœ†2(𝑑)=Ξ¨πœ†2ξ€·π‘₯πœ†2,π‘₯πœ†2ξ€Έ(𝑑)β‰₯π›Ύπ›Όξ€·πœ†1πœ†2βˆ’1ξ€Έβˆ’1/1βˆ’π›Όπ‘₯πœ†1(𝑑).(2.21)
From (2.21), we have π›Ύβˆ’π›Όξ€·πœ†1πœ†2βˆ’1ξ€Έ1/1βˆ’π›Όπ‘₯πœ†2(𝑑)β‰₯π‘₯πœ†1(𝑑)β‰₯π›Ύπ›Όξ€·πœ†1πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†2(𝑑),π‘‘βˆˆπ‘.(2.22) Noticing that 0<𝛾<1 and π›Όβˆˆ(0,1), one can see 𝛾𝛼>𝛾, a contradiction with the definition of 𝛾. Thus, 𝛾β‰₯1 and π‘₯πœ†1(𝑑)β‰₯π›Ύξ€·πœ†1πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†2(ξ€·πœ†π‘‘)β‰₯1πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†2(𝑑)>π‘₯πœ†2(𝑑),π‘‘βˆˆπ‘.(2.23)
Thus, Conclusion (i) holds.
Next, let us prove Conclusion (ii).
In (2.23), let πœ†1 be fixed and πœ†=πœ†2; then π‘₯πœ†(𝑑)β‰€πœ†πœ†1βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†1(𝑑),π‘‘βˆˆπ‘.(2.24) Thus β€–π‘₯πœ†β€–β‰€(πœ†πœ†1βˆ’1)1βˆ’2𝛼/1βˆ’π›Όβ€–π‘₯πœ†1β€–, which means β€–π‘₯πœ†β€–β†’0 as πœ†β†’0.
Similarly, let πœ†2 be fixed and πœ†=πœ†1; then π‘₯πœ†(𝑑)β‰₯πœ†πœ†2βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†2(𝑑),π‘‘βˆˆπ‘.(2.25) Therefore, β€–π‘₯πœ†β€–β‰₯(πœ†πœ†2βˆ’1)1βˆ’2𝛼/1βˆ’π›Όβ€–π‘₯πœ†2β€–, which implies that β€–π‘₯πœ†β€–β†’βˆž as πœ†β†’βˆž.
Finally, we prove Conclusion (iii).
For any fixed πœ†0>0, let πœ†>πœ†0. Set πœ†1=πœ†0 in (2.24); then π‘₯πœ†(𝑑)β‰€πœ†πœ†0βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όπ‘₯πœ†0(𝑑),π‘‘βˆˆπ‘,(2.26) which means β€–β€–π‘₯πœ†βˆ’π‘₯πœ†0β€–β€–β‰€ξ‚€ξ€·πœ†πœ†0βˆ’1ξ€Έ1βˆ’2𝛼/1βˆ’π›Όξ‚β€–β€–π‘₯βˆ’1πœ†0β€–β€–.(2.27) As a result, β€–π‘₯πœ†βˆ’π‘₯πœ†0β€–β†’0 as πœ†β†’πœ†+0. Similarly, we can show that β€–π‘₯πœ†βˆ’π‘₯πœ†0β€–β†’0 as πœ†β†’πœ†0.
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: 𝑑π‘₯𝑑𝑑=βˆ’(2+cos𝑑)π‘₯(𝑑)+πœ†π‘“(𝑑,π‘₯(π‘‘βˆ’πœ(𝑑)),𝑒(π‘‘βˆ’π›Ώ(𝑑))),π‘‘βˆˆπ‘,𝑑𝑒𝑑𝑑=βˆ’(3+sin𝑑)𝑒(𝑑)+3π‘₯(π‘‘βˆ’πœŽ(𝑑)),(3.1) where πœ†>0 is a parameter, 𝜏(𝑑),𝛿(𝑑),𝜎(𝑑)∈𝐢(𝐑,𝐑) are 2πœ‹-periodic in 𝑑, and 𝑓𝑑,π‘₯1,π‘₯2ξ€Έ=(2+sin𝑑)3√π‘₯1+13√π‘₯2.(3.2)

It is easy to see that 𝑓(𝑑,π‘₯1,π‘₯2)∈𝐢(𝐑×(0,+∞)Γ—(0,+∞)β†’(0,+∞)) is 2πœ‹-periodic in 𝑑. πœ‚(𝑑)=3+sin𝑑,π‘Ž(𝑑)=3,𝑏(𝑑)=2+cosπ‘‘βˆˆπΆ(𝐑,(0,+∞)) are 2πœ‹-periodic in 𝑑.

Since ξ€·πœ•π‘“π‘‘,π‘₯1,π‘₯2ξ€Έπœ•π‘₯1=2+sin𝑑3π‘₯12/3>0,βˆ€π‘‘βˆˆπ‘,π‘₯1,π‘₯2ξ€·βˆˆ(0,+∞),πœ•π‘“π‘‘,π‘₯1,π‘₯2ξ€Έπœ•π‘₯21=βˆ’3π‘₯24/3<0,βˆ€π‘‘βˆˆπ‘,π‘₯1,π‘₯2∈(0,+∞),(3.3) we conclude that (H1) is satisfied.

Now, we check (H2). As a matter of fact, forallπ‘‘βˆˆπ‘,π‘₯1,π‘₯2∈(0,+∞), we have 𝑓𝑑,π‘˜π‘₯1,π‘˜βˆ’1π‘₯2ξ€Έ=3βˆšπ‘˜ξƒ¬(2+sin𝑑)3√π‘₯1+13√π‘₯2ξƒ­β‰₯3βˆšξ€·π‘˜π‘“π‘‘,π‘₯1,π‘₯2ξ€Έ,(3.4) therefore, (H2) holds.

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

Let us set πœ†=2, 𝜏(𝑑)=1,𝛿(𝑑)=0.1,𝜎(𝑑)=2; then, the unique positive 2πœ‹-periodic solution of the system (3.1) can be shown in Figure 1.

fig1
Figure 1: π‘‘βˆ’π‘₯,π‘‘βˆ’π‘’ and π‘₯βˆ’π‘’ graphs of Example 3.1 with πœ†=2.

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

971427.fig.002
Figure 2: The graphs of π‘₯πœ† with different πœ†.

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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