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

Authors: Haitao Li; Yansheng Liu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## 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 parameterλ is also studied. Finally, an example together with numerical simulations is worked out to illustrate the main results.


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## Body

## 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:(1.1)dxdt=-b(t)x(t)+λf(t,x(t-τ(t)),u(t-δ(t))),t∈R,dudt=-η(t)u(t)+a(t)x(t-σ(t)),
where λ>0 is a parameter, f(t,x1,x2)∈C(R×(0,+∞)×(0,+∞)→(0,+∞)),  τ(t),δ(t),σ(t)∈C(R,R), and η(t),a(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. Section2 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 anα∈(0,1) such that
(2.1)f(t,kx1,k-1x2)≥kαf(t,x1,x2),∀k∈(0,1),t∈R,x1,x2∈(0,+∞).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(2.2)g(t,s)=exp⁡{∫tsη(r)dr}exp⁡{∫0ωη(r)dr}-1,(2.3)G(t,s)=exp⁡{∫tsb(r)dr}exp⁡{∫0ωb(r)dr}-1.Lemma 2.1 (see [5]).
Considerp≤G(t,s)≤q, where
(2.4)p=exp⁡{-∫0ωb(r)dr}exp⁡{∫0ωb(r)dr}-1,q=exp⁡{∫0ωb(r)dr}exp⁡{∫0ωb(r)dr}-1.Now, we convert the system (1.1) into an operator equation. Define operators Φ and Ψλ as follows:
(2.5)Φx(t)=∫tt+ωg(t,s)a(s)x(s-σ(s))ds,∀x∈Cω,Ψλ(x,y)(t)=λ∫tt+ωG(t,s)f(s,x(s-τ(s)),Φy(s-δ(s)))ds,∀x,y∈Cω,
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 constante>0. Then, for each λ>0, we can choose a proper number Cλ>1 such that
(2.6)Cλ≥max⁡{(λq∫0ωf(s,e,Φ(e))dse)1/1-α,(eλp∫0ωf(s,e,Φ(e))ds)1/1-α},
where Φ(·) is given in (2.5). Let
(2.7)Pe(λ)={x∈Cω:Cλ-1e≤x(t)≤Cλeon⁡[0,ω]}.Lemma 2.2.
For anyλ>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
(2.8)x(t)=Ψλ(x,x)(t),
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 thatT(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
(2.9)x1≤x2,y1≥y2⟹T(x1,y1)≤T(x2,y2).Lemma 2.4 (see [12]).
Assume thatT(x,y):Pe(λ)×Pe(λ)→Pe(λ) is a mixed monotone operator and there exists α∈(0,1) such that
(2.10)T(kx,k-1y)≥kαT(x,y),forx,y∈Pe(λ),k∈(0,1).
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 anyx,y∈Pe(λ), we have
(2.11)Cλ-1e≤x(t)≤Cλe,Cλ-1e≤y(t)≤Cλe,t∈[0,ω].
This together with (H1), (H2), (2.4), and (2.6) implies that
(2.12)Ψλ(x,y)(t)=λ∫tt+ω‍G(t,s)f(s,x(s-τ(s)),Φy(s-δ(s)))ds≤λq∫0ω‍f(s,Cλe,Cλ-1Φ(e))ds≤λqCλα∫0ω‍f(s,e,Φ(e))ds≤Cλe,Ψλ(x,y)(t)=λ∫tt+ω‍G(t,s)f(s,x(s-τ(s)),Φy(s-δ(s)))ds≥λp∫0ω‍f(s,Cλ-1e,CλΦ(e))ds≥λpCλ-α∫0ω‍f(s,e,Φ(e))ds≥Cλ-1e.
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
(2.13)Ψλ(kx,k-1y)≥kαΨλ(x,y),forx,y∈Pe(λ),k∈(0,1).Proof.
For anyx1,y1,x2,y2∈Pe(λ) with x1≤x2,y1≥y2, it is easy to see from (H1) that
(2.14)Ψλ(x1,y1)(t)-Ψλ(x2,y2)(t)=λ∫tt+ω‍G(t,s)[f(s,x1(s-τ(s)),Φy1(s-δ(s)))λ∫tt+ω‍G(t,s)λ-f(s,x2(s-τ(s)),Φy2(s-δ(s)))]ds≤0.
Hence, Ψλ is a mixed monotone operator.
In addition, for anyx,y∈Pe(λ) and k∈(0,1), (H2) shows that
(2.15)Ψλ(kx,k-1y)=λ∫tt+ω‍G(t,s)f(s,kx(s-τ(s)),Φk-1y(s-δ(s)))ds≥kαλ∫tt+ω‍G(t,s)f(s,x(s-τ(s)),Φy(s-δ(s)))ds=kαΨλ(x,y).
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 anyλ>0, the system (1.1) has a unique positive ω-periodic solution xλ(t)∈Pe(λ).Proof.
It is easy to see from Lemmas2.5 and 2.6 that for any λ>0, Ψλ:Pe(λ)×Pe(λ)→Pe(λ) is a mixed monotone operator and
(2.16)Ψλ(kx,k-1y)≥kαΨλ(x,y),forx,y∈Pe(λ),k∈(0,1).
Consequently, Lemmas2.2 and 2.4 imply that the conclusion holds true.Theorem 2.8.
Assume that (H1) and (H2) hold. In addition, suppose thatα∈(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 limλ→∞∥xλ∥=∞;(iii)
xλ(t) is continuous in λ; that is, if λ→λ0>0, then ∥xλ-xλ0∥→0.Proof.
Suppose thatλ1>λ2>0. Let
(2.17)D={γ>0:γ-1(λ1λ2-1)1/1-αxλ2(t)≥xλ1(t)≥γ(λ1λ2-1)1-2α/1-αxλ2(t)on⁡R}.
Sincee>0, we have xλ1(t)>0 and xλ2(t)>0 for t∈R. Thus
(2.18)γ*:=min⁡{(λ1-1λ2)1-2α/1-αmin⁡t∈Rxλ1(t)xλ2(t),(λ1λ2-1)1/1-αmin⁡t∈Rxλ2(t)xλ1(t)}>0.
Obviously, for any γ satisfying 0<γ<γ*, γ∈D. Hence, D≠∅.
Defineγ-=supD. Then
(2.19)γ--1(λ1λ2-1)1/1-αxλ2(t)≥xλ1(t)≥γ-(λ1λ2-1)1-2α/1-αxλ2(t),t∈R.
Now let us show that γ-≥1. In fact, if 0<γ-<1, then (H1) and (H2) imply that
(2.20)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,γ-xλ2(t-τ(t)),γ--1(λ1λ2-1)1/1-αΦxλ2(t-δ(t)))≥γ-αλ1f(t,xλ2(t-τ(t)),(λ1λ2-1)1/1-αΦxλ2(t-δ(t)))≥γ-αλ1f(t,(λ1λ2-1)-1/1-αxλ2(t-τ(t)),(λ1λ2-1)1/1-αΦxλ2(t-δ(t)))≥γ-αλ1(λ1λ2-1)-α/1-αf(t,xλ2(t-τ(t)),Φxλ2(t-δ(t)))=γ-α(λ1λ2-1)1-2α/1-αλ2f(t,xλ2(t-τ(t)),Φxλ2(t-δ(t))),λ2f(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)))≥γ-αλ2f(t,(λ1λ2-1)-1/1-αxλ1(t-τ(t)),Φxλ1(t-δ(t)))≥γ-αλ2f(t,(λ1λ2-1)-1/1-αxλ1(t-τ(t)),(λ1λ2-1)1/1-αΦxλ1(t-δ(t)))≥γ-αλ2(λ1λ2-1)-α/1-αf(t,xλ1(t-τ(t)),Φxλ1(t-δ(t)))=γ-α(λ1λ2-1)-1/1-αλ1f(t,xλ1(t-τ(t)),Φxλ1(t-δ(t))).
Therefore,(2.21)xλ1(t)=Ψλ1(xλ1,xλ1)(t)≥γ-α(λ1λ2-1)1-2α/1-αxλ2(t),xλ2(t)=Ψλ2(xλ2,xλ2)(t)≥γ-α(λ1λ2-1)-1/1-αxλ1(t).
From (2.21), we have
(2.22)γ--α(λ1λ2-1)1/1-αxλ2(t)≥xλ1(t)≥γ-α(λ1λ2-1)1-2α/1-αxλ2(t),t∈R.
Noticing that 0<γ-<1 and α∈(0,1), one can see γ-α>γ-, a contradiction with the definition of γ-. Thus, γ-≥1 and
(2.23)xλ1(t)≥γ-(λ1λ2-1)1-2α/1-αxλ2(t)≥(λ1λ2-1)1-2α/1-αxλ2(t)>xλ2(t),t∈R.
Thus, Conclusion (i) holds.
Next, let us prove Conclusion (ii).
In (2.23), let λ1 be fixed and λ=λ2; then
(2.24)xλ(t)≤(λλ1-1)1-2α/1-αxλ1(t),t∈R.
Thus ∥xλ∥≤(λλ1-1)1-2α/1-α∥xλ1∥, which means ∥xλ∥→0 as λ→0.
Similarly, letλ2 be fixed and λ=λ1; then
(2.25)xλ(t)≥(λλ2-1)1-2α/1-αxλ2(t),t∈R.
Therefore, ∥xλ∥≥(λλ2-1)1-2α/1-α∥xλ2∥, which implies that ∥xλ∥→∞ as λ→∞.
Finally, we prove Conclusion (iii).
For any fixedλ0>0, let λ>λ0. Set λ1=λ0 in (2.24); then
(2.26)xλ(t)≤(λλ0-1)1-2α/1-αxλ0(t),t∈R,
which means
(2.27)‖xλ-xλ0‖≤((λλ0-1)1-2α/1-α-1)‖xλ0‖.
As a result, ∥xλ-xλ0∥→0 as λ→λ0+. 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:(3.1)dxdt=-(2+cos⁡t)x(t)+λf(t,x(t-τ(t)),u(t-δ(t))),t∈R,dudt=-(3+sin⁡t)u(t)+3x(t-σ(t)),
where λ>0 is a parameter, τ(t),δ(t),σ(t)∈C(R,R) are 2π-periodic in t, and
(3.2)f(t,x1,x2)=(2+sin⁡t)x13+1x23.It is easy to see thatf(t,x1,x2)∈C(R×(0,+∞)×(0,+∞)→(0,+∞)) is 2π-periodic in t. η(t)=3+sint,a(t)=3,b(t)=2+cost∈C(R,(0,+∞)) are 2π-periodic in t.Since(3.3)∂f(t,x1,x2)∂x1=2+sin⁡t3x12/3>0,∀t∈R,x1,x2∈(0,+∞),∂f(t,x1,x2)∂x2=-13x24/3<0,∀t∈R,x1,x2∈(0,+∞),
we conclude that (H1) is satisfied.Now, we check (H2). As a matter of fact,forallt∈R,x1,x2∈(0,+∞), we have
(3.4)f(t,kx1,k-1x2)=k3[(2+sin⁡t)x13+1x23]≥k3f(t,x1,x2),
therefore, (H2) holds.Hence, Theorem2.7 shows that for any λ>0, the system (3.1) has a unique positive 2π-periodic solution.Let us setλ=2, τ(t)=1,δ(t)=0.1,σ(t)=2; then, the unique positive 2π-periodic solution of the system (3.1) can be shown in Figure 1.t
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x
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t
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u and x-u graphs of Example 3.1 with λ=2.
(a)
(b)Next, to illustrate Theorem2.8, 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,σ(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.Figure 2
The graphs ofxλ with different λ.



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