Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 971427, 10 pages
doi:10.1155/2012/971427
Research Article
On the Uniqueness and Dependence of Positive
Periodic Solutions for Delay Differential Systems
with Feedback Control
Haitao Li1 and Yansheng Liu2
1
2
School of Control Science and Engineering, Shandong University, Jinan 250061, China
Department of Mathematics, Shandong Normal University, Jinan 250014, China
Correspondence should be addressed to Haitao Li, haitaoli09@gmail.com
Received 22 March 2012; Accepted 14 July 2012
Academic Editor: S. M. Gusein-Zade
Copyright q 2012 H. Li and Y. 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.
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
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International Journal of Mathematics and Mathematical Sciences
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:
dx
−btxt λft, xt − τt, ut − δt,
dt
t ∈ R,
du
−ηtut atxt − σt,
dt
1.1
where λ > 0 is a parameter, ft, x1 , x2 ∈ CR×0, ∞×0, ∞ → 0, ∞, τt, δt, σt ∈
CR, R, and ηt, at, bt ∈ CR, 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 ft, x1 , x2 ∈ CR × 0, ∞ × 0, ∞, 0, ∞ is nondecreasing in x1 and
nonincreasing in x2 .
H2 There exists an α ∈ 0, 1 such that
f t, kx1 , k−1 x2 ≥ kα ft, x1 , x2 ,
∀k ∈ 0, 1, t ∈ R, x1 , x2 ∈ 0, ∞.
2.1
Let Cω {x ∈ CR, R : xt xt ω, t ∈ R}. Then, Cω is a Banach space with norm
x maxt∈0, ω |xt|. In this paper, we will study the system 1.1 in Cω .
Denote
s
exp t ηrdr
,
gt, s ω
exp 0 ηrdr − 1
s
exp t brdr
Gt, s .
ω
exp 0 brdr − 1
2.2
2.3
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3
Lemma 2.1 see 5. Consider p ≤ Gt, s ≤ q, where
ω
exp − 0 brdr
,
p
ω
exp 0 brdr − 1
ω
exp 0 brdr
q
.
ω
exp 0 brdr − 1
2.4
Now, we convert the system 1.1 into an operator equation. Define operators Φ and
Ψλ as follows:
Φxt Ψλ x, y t λ
tω
tω
gt, sasxs − σsds,
∀x ∈ Cω ,
t
Gt, sf s, xs − τs, Φys − δs ds,
2.5
∀x, y ∈ Cω ,
t
where gt, s and Gt, 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
⎧ 1/1−α ⎫
⎨ λq ω fs, e, Φeds 1/1−α
⎬
e
0
,
,
Cλ ≥ max
ω
⎩
⎭
e
λp 0 fs, e, Φeds
2.6
where Φ· is given in 2.5. Let
Pe λ x ∈ Cω : Cλ−1 e ≤ xt ≤ Cλ e on 0, 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
xt Ψλ x, xt,
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 ⇒ T x1 , y1 ≤ T x2 , y2 .
2.9
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International Journal of Mathematics and Mathematical Sciences
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−1 y ≥ kα T x, y ,
for x, 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−1 e ≤ xt ≤ Cλ e,
Cλ−1 e ≤ yt ≤ Cλ e,
t ∈ 0, ω.
2.11
This together with H1, H2, 2.4, and 2.6 implies that
Ψλ x, y t λ
tω
Gt, sf s, xs − τs, Φys − δs ds
t
ω
ω f s, Cλ e, Cλ−1 Φe ds ≤ λqCλα
fs, e, Φeds ≤ Cλ e,
≤ λq
0
Ψλ x, y t λ
tω
0
Gt, sf s, xs − τs, Φys − δs ds
2.12
t
ω
ω −1
−α
f s, Cλ e, Cλ Φe ds ≥ λpCλ
fs, e, Φeds ≥ Cλ−1 e.
≥ λp
0
0
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−1 y ≥ kα Ψλ x, y ,
for x, 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
tω
λ
Gt, s f s, x1 s − τs, Φy1 s − δs
t
− f s, x2 s − τs, Φy2 s − δs ds ≤ 0.
Hence, Ψλ is a mixed monotone operator.
2.14
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5
In addition, for any x, y ∈ Pe λ and k ∈ 0, 1, H2 shows that
−1
Ψλ kx, k y λ
tω
t
≥ kα λ
Gt, sf s, kxs − τs, Φk−1 ys − δs ds
tω
Gt, sf s, xs − τs, Φys − δs ds
2.15
t
kα Ψλ 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 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−1 y ≥ kα Ψλ x, y ,
for x, 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 limλ → ∞ xλ ∞;
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
1/1−α
1−2α/1−α
−1
γ > 0 : γ −1 λ1 λ−1
x
λ
x
≥
x
≥
γ
λ
on
R
.
t
t
t
λ2
λ1
1 2
λ2
2
2.17
Since e > 0, we have xλ1 t > 0 and xλ2 t > 0 for t ∈ R. Thus
γ ∗ : min
λ−1
1 λ2
1−2α/1−α
min
t∈R
1/1−α
xλ1 t xλ t
min 2
, λ1 λ−1
2
t∈R xλ1 t
xλ2 t
> 0.
2.18
Obviously, for any γ satisfying 0 < γ < γ ∗ , γ ∈ D. Hence, D /
∅.
Define γ sup D. Then
1/1−α
1−2α/1−α
γ −1 λ1 λ−1
xλ2 t ≥ xλ1 t ≥ γ λ1 λ−1
xλ2 t,
2
2
t ∈ R.
2.19
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International Journal of Mathematics and Mathematical Sciences
Now let us show that γ ≥ 1. In fact, if 0 < γ < 1, then H1 and H2 imply that
λ1 ft, xλ1 t − τt, Φxλ1 t − δt
1−2α/1−α
1/1−α
−1
−1
≥ λ1 f t, γ λ1 λ−1
λ
x
γ
λ
Φx
−
τt,
−
δt
t
t
λ2
1 2
λ2
2
1/1−α
Φx
≥ λ1 f t, γxλ2 t − τt, γ −1 λ1 λ−1
−
δt
t
λ2
2
1/1−α
Φx
≥ γ α λ1 f t, xλ2 t − τt, λ1 λ−1
−
δt
t
λ2
2
−1/1−α
1/1−α
−1
x
λ
Φx
≥ γ α λ1 f t, λ1 λ−1
−
τt,
λ
−
δt
t
t
λ
1
λ
2
2
2
2
−α/1−α
≥ γ α λ1 λ1 λ−1
ft, xλ2 t − τt, Φxλ2 t − δt
2
1−2α/1−α
γ α λ1 λ−1
λ2 ft, xλ2 t − τt, Φxλ2 t − δt,
2
λ2 ft, xλ2 t − τt, Φxλ2 t − δt
−1/1−α
−1−2α/1−α
−1
−1
−1
≥ λ2 f t, γ λ1 λ2
λ1 λ2
xλ1 t − τt, γ
Φxλ1 t − δt
2.20
−1/1−α
−1
−1
xλ1 t − τt, γ Φxλ1 t − δt
≥ λ2 f t, γ λ1 λ2
−1/1−α
x
−
τt,
Φx
−
δt
≥ γ α λ2 f t, λ1 λ−1
t
t
λ1
λ1
2
−1/1−α
1/1−α
−1
x
λ
Φx
≥ γ α λ2 f t, λ1 λ−1
−
τt,
λ
−
δt
t
t
λ1
1 2
λ1
2
−α/1−α
≥ γ α λ2 λ1 λ−1
ft, xλ1 t − τt, Φxλ1 t − δt
2
−1/1−α
γ α λ1 λ−1
λ1 ft, xλ1 t − τt, Φxλ1 t − δt.
2
Therefore,
1−2α/1−α
xλ1 t Ψλ1 xλ1 , xλ1 t ≥ γ α λ1 λ−1
xλ2 t,
2
xλ2 t Ψλ2 xλ2 , xλ2 t ≥ γ
α
λ1 λ−1
2
−1/1−α
2.21
xλ1 t.
From 2.21, we have
1/1−α
1−2α/1−α
γ −α λ1 λ−1
xλ2 t ≥ xλ1 t ≥ γ α λ1 λ−1
xλ2 t,
2
2
t ∈ R.
2.22
International Journal of Mathematics and Mathematical Sciences
7
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
1−2α/1−α
1−2α/1−α
xλ1 t ≥ γ λ1 λ−1
xλ2 t ≥ λ1 λ−1
xλ2 t > xλ2 t,
2
2
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
1−2α/1−α
xλ1 t,
xλ t ≤ λλ−1
1
2.24
t ∈ R.
1−2α/1−α
xλ1 , which means xλ → 0 as λ → 0.
Thus xλ ≤ λλ−1
1 Similarly, let λ2 be fixed and λ λ1 ; then
1−2α/1−α
xλ t ≥ λλ−1
xλ2 t,
2
2.25
t ∈ R.
1−2α/1−α
xλ2 , which implies that xλ → ∞ as λ → ∞.
Therefore, xλ ≥ λλ−1
2 Finally, we prove Conclusion iii.
For any fixed λ0 > 0, let λ > λ0 . Set λ1 λ0 in 2.24; then
1−2α/1−α
xλ t ≤ λλ−1
xλ0 t,
0
2.26
t ∈ R,
which means
xλ − xλ0 ≤
λλ−1
0
1−2α/1−α
− 1 xλ0 .
2.27
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:
dx
−2 cos txt λft, xt − τt, ut − δt,
dt
du
−3 sin tut 3xt − σt,
dt
t ∈ R,
3.1
International Journal of Mathematics and Mathematical Sciences
10
10
9
9
8
8
7
7
6
6
u
x/u
8
5
5
4
4
3
3
2
2
1
0
5
10
15
20
1
2
3
4
5
6
7
8
x
t
x
u
u(x)
a
b
Figure 1: t − x, t − u and x − u graphs of Example 3.1 with λ 2.
where λ > 0 is a parameter, τt, δt, σt ∈ CR, R are 2π-periodic in t, and
√
1
ft, x1 , x2 2 sin t 3 x1 √
.
3
x2
3.2
It is easy to see that ft, x1 , x2 ∈ CR × 0, ∞ × 0, ∞ → 0, ∞ is 2π-periodic in
t. ηt 3 sin t, at 3, bt 2 cos t ∈ CR, 0, ∞ are 2π-periodic in t.
Since
∂ft, x1 , x2 2 sin t
> 0,
∂x1
3x12/3
∀t ∈ R, x1 , x2 ∈ 0, ∞,
∂ft, x1 , x2 1
− 4/3 < 0,
∂x2
3x
∀t ∈ R, x1 , x2 ∈ 0, ∞,
3.3
2
we conclude that H1 is satisfied.
Now, we check H2. As a matter of fact, for all t ∈ R, x1 , x2 ∈ 0, ∞, we have
−1
f t, kx1 , k x2
√
1
3
3
3
k 2 sin t x1 √
≥ kft, x1 , x2 ,
3
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, τt 1, δt 0.1, σt 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 τt 1, δt 0.1, σt 2; then the unique positive 2π-periodic solutions of
International Journal of Mathematics and Mathematical Sciences
9
10
9
8
x
7
6
5
4
3
2
0
5
10
15
20
t
x2
x2.1
x2.2
x2.3
x2.4
x2.5
Figure 2: The graphs of xλ with different λ.
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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