Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 259805, 28 pages
doi:10.1155/2011/259805
Research Article
Existence and Global Attractivity of Positive
Periodic Solutions for a Two-Species Competitive
System with Stage Structure and Impulse
Changjin Xu1 and Daxue Chen2
1
Guizhou Key Laboratory of Economics System Simulation, School of Mathematics
and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, China
2
Faculty of Science, Hunan Institute of Engineering, Xiangtan 411004, China
Correspondence should be addressed to Changjin Xu, xcj403@126.com
Received 10 August 2011; Accepted 17 September 2011
Academic Editor: Yuji Liu
Copyright q 2011 C. Xu and D. Chen. 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.
A class of nonautonomous two-species competitive system with stage structure and impulse
is considered. By using the continuation theorem of coincidence degree theory, we derive a
set of easily verifiable sufficient conditions that guarantee the existence of at least a positive
periodic solution, and, by constructing a suitable Lyapunov functional, the uniqueness and global
attractivity of the positive periodic solution are presented. Finally, an illustrative example is given
to demonstrate the correctness of the obtained results.
1. Introduction
In recent years, with the increasing applications of theory of differential equations in
mathematical ecology, various mathematical models have been proposed in the study of
population 1–25. But most of the previous results focused on the dynamical behaviors
including the stability, attractiveness, persistence, and periodicity of solution of the systems
which have fixed parameters and there is no impulse. Considering that harvest of many
populations are not continuous and the periodic environmental factor, it is reasonable to
investigate the systems with periodic coefficients and impulse. Impulsive differential systems
display a combination of characteristics of both the continuous-time and discrete-time
systems 26–30. In 2006, Chen 1 studied the following non-autonomous almost periodic
competitive two-species model with stage structure in one species:
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International Journal of Differential Equations
ẋ1 t −a1 tx1 t b1 tx2 t,
ẋ2 t a2 tx1 t − b2 tx2 t − ctx22 t − β1 tx2 tx2 tx3 t,
ẋ3 t x3 t dt − etx3 t − β2 tx2 t ,
1.1
where x1 t and x2 t are immature and mature population densities of one species,
respectively; x3 t represents the population density of another species; ai t, bi t, βi t i 1, 2, ct, dt, et are all continuous, almost periodic functions. The competition is
between x2 t and x3 t. Chen 1 obtained sufficient conditions for the existence of a unique,
globally attractive, strictly positive almost periodic solution for system 1.1.
Considering that the harvest is an annual harvest pulse, to describe a system more
accurately, we should consider the impulsive differential equation. Motivated by this point
of view, we revised system 1.1 into the following form:
ẋ1 t −a1 tx1 t b1 tx2 t,
t
/ tk ,
ẋ2 t a2 tx2 t − b2 tx2 t − ctx22 t − β1 tx2 tx3 t,
ẋ3 t x3 t dt − etx3 t − β2 tx2 t ,
Δxi tk xi tk − xi t−k −γik xi tk,
t /
tk ,
t
/ tk ,
1.2
i 1, 2, 3, k 1, 2, . . . , q,
where Δxi tk xi tk − xi t−k are the impulses at moments tk and t1 < t2 < · · · is a strictly
increasing sequence such that limk → ∞ tk ∞; x1 t and x2 t are immature and mature
population densities of one species, respectively, and x3 t represents the population density
of another species. The competition is between x2 t and x3 t.
Throughout the paper, we always assume the following.
H1 ai t, bi t, βi t i 1, 2, ct, dt, et are all continuous ω periodic; that is, ai t
ω ai t, bi t ω bi t, βi t ω βi t i 1, 2, ct ω ct, dt ω dt, et ω et for any t ∈ R.
H2 ai t, bi t, βi t i 1, 2, ct, dt, et are all positive.
H3 0 < γik < 1, i 1, 2, 3 for all k ∈ N, and there exists a positive integer q such that
tkq tk ω, γikq γik , i 1, 2, 3.
The principle object of this paper is by using Mawhin’s continuation theorem of
coincidence degree theory and by constructing the Lyapunov functions to investigate the
stability and existence of periodic solutions of 1.2. To the best of my knowledge, it is the
first time to deal with the existence and stability of periodic solutions of 1.2.
The organization of the paper is as follows. In Section 2, we introduce some notations
and definitions and state some preliminary results needed in later sections. We then establish,
in Section 3, some simple criteria for the existence of positive periodic solutions of system
1.2 by using the continuation theorem of coincidence degree theory proposed by Gaines
and Mawhin 31. The uniqueness and global attractivity of the positive periodic solution
are presented in Section 4. In Section 5, an illustrative example is given to demonstrate the
correctness of the obtained results.
International Journal of Differential Equations
3
2. Preliminaries
We will introduce some notations and definitions and state some preliminary results.
Consider the impulsive system
ẋt ft, x,
Δxt|ttk
t/
tk , k 1, 2, . . . ,
Ik x t−k ,
2.1
where x ∈ Rn , f : R × Rn → Rn is continuous and ft ω, x ft, x; Ik : Rn → Rn are
continuous, and there exists a positive integer q such that tkq tk ω, Ikq x Ik x with
tk ∈ R, tk1 > tk , limk → ∞ ∞, Δxt|ttk xtk − xt−k . For tk / 0 k 1, 2, . . ., 0, ω ∩ {tk } {t1 , t2 , . . . , tq }. As we know, {tk } are called points of jump.
Let us recall some definitions. For the Canchy problem,
ẋt ft, x,
t ∈ 0, ω, t /
tk ,
Δxt|ttk Ik x t−k ,
x0 x0 .
2.2
Definition 2.1. A map x : 0, ω → Rn is said to be a solution of 2.2, if it satisfied the
following conditions:
i xt is a piecewise continuous map with first-class discontinuity points in tk ∩0, ω,
and at each discontinuity point it is continuous on the left;
ii xt satisfies 2.2.
Definition 2.2. A map x : 0, ω → Rn is said to be an ω periodic solution of 2.1, if
i xt satisfies i and ii of Definition 2.1 in the interval 0, ω and
ii xt satisfies xt ω − 0 xt − 0, t ∈ R.
Obviously, if xt is a solution of 2.2 defined on 0, ω, such that x0 xω, then,
by the periodicity of 2.2 in t, the function x∗ t defined by
∗
x t ⎧ ⎨x t − jω ,
t ∈ jω, j 1 ω \ {tk },
⎩x∗ t is left continuous at t t
k
2.3
is a ω periodic solution of 2.1.
For system 1.2, seeking the periodic solutions is equivalent to seeking solutions of
the following boundary value problem:
ẋ1 t −a1 tx1 t b1 tx2 t,
t/
tk , t ∈ 0, ω, k 1, 2, . . . , q,
ẋ2 t a2 tx2 t − b2 tx2 t − ctx22 t − β1 tx2 tx3 t, t / tk , t ∈ 0, ω, k 1, 2, . . . , q,
tk , t ∈ 0, ω, k 1, 2, . . . , q,
ẋ3 t x3 t dt − etx3 t − β2 tx2 t , t /
−
Δxi tk xi tk − xi tk −γik xi tk , i 1, 2, 3, xi 0 xi ω, k 1, 2, . . . , q.
2.4
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International Journal of Differential Equations
3. Existence of Positive Periodic Solutions
In this section, based on the Mawhin’s continuation theorem, we shall study the existence of
at least one periodic solution of 1.1. To do so, we shall make some preparations.
Let X, Y be normed vector spaces; L : Dom L ⊂ X → Y is a linear mapping; N : X →
Y is a continuous mapping. The mapping L will be called a Fredholm mapping of index
zero if dimKer L codimIm L < ∞ and Im L is closed in Y . If L is a Fredholm mapping
of index zero and there exist continuous projectors P : X → X and Q : Y → Y such that
Im P Ker L, Im L Ker Q ImI −Q, it follows that L | Dom L ∩ Ker P : I −P X → Im L is
invertible. We denote the inverse of that map by KP . If Ω is an open bounded subset of X, the
mapping N will be called L-compact on Ω if QNΩ is bounded and KP I − QN : Ω → X
is compact. Since Im Q is isomorphic to Ker L, there exist isomorphisms J : Im Q → Ker L.
Now we introduce Mawhin’s continuation theorem 31 as follows.
Lemma 3.1 Continuation Theorem 31. Let L be a Fredholm mapping of index zero, and let N
be L-compact on Ω. Suppose
a for each λ ∈ 0, 1, every solution x of Lx λNx is such that x ∈
/ ∂Ω.
b QNx /
0 for each x ∈ Ker L ∂Ω, and deg{JQN, Ω ∂ Ker L, 0} /
0.
Then the equation Lx Nx has at least one solution lying in Dom L
Ω.
For convenience and simplicity in the following discussion, we always use the
notations below throughout the paper:
f
1
ω
ω
ftdt,
0
f L min ft,
t∈0,ω
f M max ft,
t∈0,ω
1
f ω
ω
ftdt,
3.1
0
where ft is a ω continuous periodic function. For any nonnegative integer p, let
Cp 0, ω; t1 , t2 , . . . , tq {x : 0, ω → Rm | xp t exist for t / t1 , . . . , tq ; xp t 0, and let
p
j
j
x t − 0 exist at t1 , t2 , . . . , tq , and x tk x tk − 0, k 1, . . . , m, j 0, 1, 2, . . . , p} with the
p
norm xp max {supt∈0,ω xj t} , where · is any norm of Rm . It is easy to see that
j1
Cp 0, ω; t1 , t2 , . . . , tq is a Banach space.
Now we are now in a position to state and prove the existence of periodic solutions of
2.4.
Theorem 3.2. In addition to H1 , H2 , H3 , assume further that the following hold:
H4 min{P1 , P2 , P3 } > 0,
H5 a1 ω >
q
ln 1 − γ1k ,
3.2
k1
H6 dω q
k1
ln 1 − γ3k > β2 ωeB4 ,
β1 β2 /
ce,
International Journal of Differential Equations
5
where
P1 |a2 − b2 |ω q
ln 1 − γ2k − cωeB6 ,
k1
P2 |a2 − b2 |ω q
ln 1 − γ2k − β1 ωeB29 ,
3.3
k1
P3 |a2 − b2 |ω q
ln 1 − γ2k − cω,
k1
and B4 , B6 , B29 are defined by 3.27, 3.32, and 3.61, respectively. Then the system 1.2 has at
least a ω periodic solution.
Proof. According to the discussion above in Section 2, we need only to prove that the
boundary value problem 2.4 has a solution. Since solutions of 2.4 remained positive for
all t ≥ 0, we let
u1 t lnx1 t,
u2 t lnx2 t,
u3 t lnx3 t,
3.4
then system 2.4 can be translated to
u̇1 t −a1 t b1 teu2 t−u1 t ,
t/
tk , t ∈ 0, ω, k 1, 2, . . . , q,
u̇2 t a2 t − b2 t − cteu2 t − β1 teu3 t ,
u̇3 t dt − ete
u3 t
− β2 te
u2 t
Δui tk ln 1 − γik ,
,
t/
tk , t ∈ 0, ω, k 1, 2, . . . , q,
3.5
t/
tk , t ∈ 0, ω, k 1, 2, . . . , q,
i 1, 2, 3,
ui 0 ui ω.
It is easy to see that if system 3.5 has one ω periodic solution u∗1 t, u∗2 t, u∗3 tT , then
∗
∗
∗
T
x1∗ t, x2∗ t, x3∗ tT eu1 t , eu2 t , eu3 t is a positive solution of system 1.2. Therefore, to
complete the proof, it suffices to show that system 3.5 has at least one ω periodic solution.
In order to use the continuation theorem of coincidence degree theory, we take
X u ∈ C 0, ω; t1 , t2 , . . . , tq ,
Y X × R3×q1 .
3.6
Then X is a Banach space with norm · 0 , and Y is also a Banch space with norm z x0 y, x ∈ X, Y ∈ R3q .
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International Journal of Differential Equations
Let the following hold:
dom L x u1 , u2 , u3 T ∈ C0, ω; t1 , t2 , . . . , tq ,
L : Dom L ⊂ X −→ Y,
q
x −→ x
, Δxtk k1 ,
N : X −→ Y,
⎞
⎛⎛
⎞ ⎛ −a1 t b1 t expu2 t − u1 t
ln 1 − γ11
⎜⎜
⎟ ⎜ ⎟
u2 t
⎜
⎟
⎜
Nu ⎜
− β1 teu3 t ⎟
⎝⎝a2 t − b2 t − cte
⎠, ⎝ln 1 − γ21 ⎠,
ln 1 − γ31
dt − eteu3 t − β2 teu2 t
⎞
⎞ ⎞
⎛ ⎛ ln 1 − γ12
ln 1 − γ1q
⎜ ⎜ ⎟
⎟ ⎟
⎜ln 1 − γ22 ⎟, . . . , ⎜ln 1 − γ2q ⎟, 0⎟
⎝
⎠
⎝
⎠ ⎠
ln 1 − γ32
ln 1 − γ3q
3.7
Obviously,
Ker L u : ut h ∈ R3 , t ∈ 0, ω ,
Im L z f, a1 , a2 , . . . , aq , d ∈ Y :
ω
q
fsds ak d 0
0
k1
3.8
X × R3×q × {0},
dimKer L 3 codimIm L.
So, Im L is closed in Y ; L is a Fredholm mapping of index zero. Define two projectors
1 ω
xtdt,
ω 0
ω
q
1
Qz Q f, a1 , a2 , . . . , aq , d fsds ak d, , 0, 0, . . . , 0 .
ω 0
k1
Px 3.9
It is easy to show that P and Q are continuous and satisfy Im P Ker L, Im L Ker Q ImI − Q.
Further, by direct computation, we can find that the inverse KP of L, KP : Im L →
Ker P ∩ Dom L has the following form:
t
1
KP z fsds ak −
ω
0
tk <t
ω t
0
0
fsds dt −
q
q
1
ak ak tk .
ω k1
k1
3.10
International Journal of Differential Equations
7
Moreover, it is easy to check that
⎛⎛
1 t
⎜⎜ ω F1 sds ⎜⎜
⎜⎜ 0
⎜⎜ 1 t
⎜⎜
F sds QNu ⎜⎜
⎜⎜ ω 0 2
⎜⎜ ⎜⎜
⎝⎝ 1 t
F3 sds ω 0
⎞
⎞
q
1 ln 1 − γ1k ⎟
⎟
ω k1
⎟
⎟
⎟
⎟
q
⎟
⎟
1
⎟
⎟
ln 1 − γ2k ⎟, 0, 0, . . . , 0⎟,
⎟
⎟
ω k1
⎟
⎟
⎟
⎟
q
⎠
⎠
1 ln 1 − γ3k
ω k1
⎞
ln
1
−
γ
1k
⎟
⎜
⎟
⎜ 0
t>tk
⎟
⎜
t
⎜
⎟
⎜ F sds ln 1 − γ2k ⎟
KP I − QNu ⎜
⎟
2
⎟
⎜ 0
t>tk
⎟
⎜
⎜ t
⎟
⎠
⎝
F3 sds ln 1 − γ3k
⎛
t
0
F1 sds t>tk
⎛
q
1 ω t
F
ln 1 − γ1k dt
−
sds
1
⎜ω
⎜
k1
⎜ 0 0
q
⎜1 ω t
⎜
F2 sds dt −
ln 1 − γ2k −⎜
⎜ω 0 0
k1
⎜ ⎜
q
⎝1 ω t
F3 sds dt −
ln 1 − γ3k ω 0 0
k1
⎞
q
1 ln 1 − γ1k ⎟
ω k1
⎟
⎟
q
⎟
1 ⎟
ln 1 − γ2k ⎟
⎟
ω k1
⎟
q
⎟
⎠
1 ln 1 − γ3k
ω k1
3.11
⎞
ln 1 − γ1k ⎟
⎜ F1 sds ⎟
⎜ 0
k1
⎟
⎜
ω
q
⎟
⎜
t
1 ⎜
⎟
ln 1 − γ2k ⎟,
−
−
⎜ F2 sds ⎟
ω 2 ⎜
k1
⎟
⎜
0
⎜ ω
q
⎟
⎠
⎝
F3 sds ln 1 − γ3k
⎛
ω
q
0
k1
where
F1 s −a1 s b1 seu2 s−u1 s ,
F2 s a2 s − b2 s − cseu2 s − β1 seu3 s ,
3.12
F2 s ds − eseu3 s − β2 seu2 s .
Obviously, QN and KP I − QN are continuous. Using the Ascoli-Arzela theorem, it is not
difficult to show that KP I − QNΩ is compact for any open bounded set Ω ⊂ X. Moreover,
QNΩ is bounded. Thus, N is L-compact on Ω with any open bounded set Ω ⊂ X.
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International Journal of Differential Equations
Now we are at the point to search for an appropriate open, bounded subset Ω for
the application of the continuation theorem. Corresponding to the operator equation Lu λNu, λ ∈ 0, 1, we have
"
!
u̇1 t λ −a1 t b1 teu2 t−u1 t ,
t
/ tk , t ∈ 0, ω, k 1, 2, . . . , q,
"
!
u̇2 t λ a2 t − b2 t − cteu2 t − β1 teu3 t ,
t/
tk , t ∈ 0, ω, k 1, 2, . . . , q,
"
!
u̇3 t λ d1 t − eteu3 t − β2 teu2 t ,
Δui tk λ ln 1 − γik ,
3.13
t
/ tk , t ∈ 0, ω, k 1, 2, . . . , q,
i 1, 2, 3,
ui 0 ui ω.
Suppose that ut u1 t, u2 t, u3 tT ∈ X is an arbitrary solution of system 3.13 for a
certain λ ∈ 0, 1, integrating both sides of 3.13 over the interval 0, ω with respect to t, we
obtain
ω!
b1 te
u2 t−u1 t
"
dt 0
ω!
q
ln 1 − γ1k ω
a1 tdt,
0
k1
ω
q
"
cteu2 t β1 teu3 t dt ln 1 − γ2k ,
a2 t − b2 tdt 0
0
3.14
k1
ω
ω!
q
"
eteu3 t β2 teu2 t dt dtdt ln 1 − γ3k .
0
0
k1
From 3.13 and 3.14, we obtain
ω
|u̇1 t|dt <
0
ω
a1 tdt 0
0
2a1 ω −
ω
|u̇2 t|dt <
0
ω
0
q
ω!
"
cteu2 t β1 teu3 t dt
a2 t b2 tdt 0
|u̇3 t|dt <
ω
3.15
ln 1 − γ1k ,
k1
0
2a2 ω ω
ω!
"
b1 teu2 t−u1 t dt
q
k1
dtdt 0
2dω ω!
eteu3 t β2 teu2 t
0
q
k1
3.16
ln 1 − γ2k ,
ln 1 − γ3k .
"
dt
3.17
International Journal of Differential Equations
9
Let the following hold:
ui ηi max ui t,
ui ξi min ui t,
t∈0,ω
i 1, 2, 3.
t∈0,ω
3.18
From the second and the third equations of 3.14, we can obtain
|a2 − b2 |ω q
ln 1 − γ2k >
ω
cte
dω dt ≥
ω
0
k1
q
u2 t
ω
ln 1 − γ3k >
eteu3 t dt ≥
ω
0
k1
cteu2 ξ2 dt,
0
3.19
eteu3 ξ3 dt eωeu3 ξ3 ,
0
then
u2 ξ2 < ln
#q
|a2 − b2 |ω k1
ln 1 − γ2k
cω
u3 ξ3 < ln
dω #q
3.20
,
ln 1 − γ3k
.
eω
3.21
k1
Thus
u2 t u2 ξ2 t
u̇2 tdt ≤ u2 ξ2 ξ2
< ln
ω
|u̇2 t|dt
0
|a2 − b2 |ω ln 1 − γ2k
k1
#q
cω
2a2 ω q
3.22
ln 1 − γ2k : B1 .
k1
In the following, we will consider four cases.
Case 1 if u1 t > 0, u2 t > 0. From the first equation of 3.14, we have
ω
a1 ω <
0
ω
a1 ω >
0
b1 teu2 t dt q
q
ln 1 − γ1k ≤ b1 ωeu2 η2 ln 1 − γ3k ,
k1
b1 te−u1 t dt q
k1
k1
ln 1 − γ1k ≥
ω
0
b1 te−u1 η1 dt q
k1
3.23
ln 1 − γ1k ,
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International Journal of Differential Equations
that is,
#q
a1 ω − k1 ln 1 − γ1k
,
u2 η2 > ln
b1 ω
u1 η1 > ln
3.24
a1 ω −
b1 ω
.
#q
ln 1 − γ1k
k1
Then
u2 t u2 η2 −
≥ ln
η2
t
a1 ω −
u1 t u1 η1 −
#q
k1
η1
ln 1 − γ1k
b1 ω
b1 ω
a1 ω −
#q
k1
ω
|u̇2 t|dt
0
− 2a2 ω −
u̇1 tdt ≥ u1 η1 −
t
≥ ln
u̇2 tdt ≥ u2 η2 −
ω
q
3.25
ln 1 − γ2k : B2 ,
k1
|u̇1 t|dt
0
q
3.26
ln 1 − γ1k : B3 .
− 2a1 ω −
ln 1 − γ1k
k1
Thus, from 3.22 and 3.25, we obtain
|u2 t| ≤ max{|B1 |, |B2 |} : B4 .
3.27
By the first and the third equations of 3.14, we get
ω!
q
"
b1 teB4 −u1 ξ1 dt ln 1 − γ1k > a1 ω,
0
k1
ω!
q
"
eteu3 η3 β2 teB4 dt > dω ln 1 − γ3k ,
0
3.28
k1
then
b1 ωeB4
u1 ξ1 < ln
,
#q
a1 ω − k1 ln 1 − γ1k
#q
dω k1 ln 1 − γ3k − β2 ωeB4
u3 η3 > ln
.
eω
3.29
3.30
International Journal of Differential Equations
11
From 3.15, 3.17, 3.21, and 3.30, we have
u1 t u1 ξ1 t
u̇1 tdt ≤ u1 ξ1 ξ1
ω
|u̇1 t|dt
0
q
b1 ωeB4
2a
ω
ln 1 − γ1k : B5 ,
< ln
#q
1
a1 ω − k1 ln 1 − γ1k
k1
t
ω
u3 t u3 ξ3 u̇3 tdt ≤ u3 ξ3 |u̇3 t|dt
ξ3
< ln
dω eω
η3
u3 t u3 η3 −
> ln
0
ln 1 − γ3k
k1
#q
u̇3 tdt ≥ u3 η3 −
ω
3.32
ln 1 − γ3k : B6 ,
k1
|u̇3 t|dt
0
t
#q
dω 2dω q
k1
ln 1 − γ3k − β2 ωeB4
eω
3.31
− 2dω q
ln 1 − γ3k : B7 .
3.33
k1
Thus,
|u1 t| ≤ B5 ,
3.34
|u3 t| max{|B6 |, |B7 |} : B8 .
Case 2 if u1 t > 0, u2 t < 0. By the first equation of 3.14, we have
ω
a1 ω <
b1 teu2 t dt 0
q
q
ln 1 − γ1k ≤ b1 ωeu2 η2 ln 1 − γ3k ,
k1
3.35
k1
namely,
#q
a1 ω − k1 ln 1 − γ1k
.
u2 η2 > ln
b1 ω
3.36
Then
u2 t u2 η2 −
≥ ln
η2
u̇2 tdt ≥ u2 η2 −
t
a1 ω −
#q
k1
ln 1 − γ1k
b1
ω
|u̇2 t|dt
0
− 2a2 ω −
q
k1
ln 1 − γ2k : B2 .
3.37
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International Journal of Differential Equations
From 3.22 and 3.37, we obtain
3.38
|u2 t| ≤ max{|B1 |, |B2 |} : B3 .
By the first equation of 3.14, we also have
ω
0
ω
0
q
b1 teB3
dt
ln 1 − γ1k > a1 ω,
u
t
1
e
k1
q
b1 te−B3
dt
ln 1 − γ1k < a1 ω,
u
t
e 1
k1
3.39
Then
b1 ωeB3 ln 1 − γ1k > a1 ω,
u
ξ
e 1 1
k1
q
3.40
q
b1 ωe−B3 ln 1 − γ1k < a1 ω.
u
η
1
1
e
k1
that is,
b1 ωeB3
u1 ξ1 < ln
: B8 ,
#q
a1 ω − k1 ln 1 − γ1k
u1 η1 < ln
b1 ωeB3
: B9 .
#q
a1 ω − k1 ln 1 − γ1k
3.41
Thus,
u1 t u1 ξ1 t
u̇1 tdt ≤ u1 ξ1 ξ1
< B8 2a1 ω −
u1 t u1 η1 η1
|u̇1 t|dt
0
q
ln 1 − γ1k : B10 ,
k1
u̇1 tdt ≥ u1 η1 ω
3.42
|u̇1 t|dt
0
t
> B9 − 2a1 ω −
ω
q
ln 1 − γ1k : B11 .
k1
From 3.42, we have
|u1 t ≤ max{|B10 |, |B11 |} : B12 .
3.43
International Journal of Differential Equations
13
By the second equation of 3.14, we have
ω
cteB3 dt 0
ω
ω
β1 teu3 t dt > |a2 − b2 |ω q
0
cte−B3 dt 0
ω
ln 1 − γ2k ,
k1
β1 teu3 t dt < |a2 − b2 |ω 0
q
3.44
ln 1 − γ2k .
k1
Then
cωeB3 β1 ωeu3 η3 > |a2 − b2 |ω q
ln 1 − γ2k ,
k1
cωe−B3 β1 ωeu3 ξ3 < |a2 − b2 |ω q
3.45
ln 1 − γ2k : B13 ,
k1
that is,
#q
|a2 − b2 |ω k1 ln 1 − γ2k − cωeB6
u3 η3 > ln
,
β1 ω
u3 ξ3 < ln
|a2 − b2 |ω ln 1 − γ2k − cωe−B3
k1
#q
β1 ω
3.46
: B14 .
Therefore, we get
u3 t u3 ξ3 t
u̇3 tdt ≤ u3 ξ3 ξ3
< B14 2d1 ω −
ω
|u̇3 t|dt
0
q
ln 1 − γ3k : B15 ,
k1
u3 t u3 η3 −
η3
u̇3 tdt ≥ u3 η3 −
3.47
|u̇3 t|dt
0
t
> B14 − 2d1 ω −
ω
q
ln 1 − γ3k : B16 .
k1
Hence, we have
|u3 t| ≤ max{|B15 |, |B16 |} : B17 .
3.48
14
International Journal of Differential Equations
Case 3 if u1 t < 0, u2 t > 0. By the first equation of 3.14, we have
ω
b1 te
−u1 t
dt q
0
ln 1 − γ1k <
ω
a1 tdt,
0
k1
ω!
ω
q
"
b1 teu2 η2 −u1 t dt ln 1 − γ1k >
a1 tdt.
0
3.49
0
k1
Then
b1 ωe−u1 η1 q
ln 1 − γ1k < a1 ω,
k1
b1 ωe−B19 eu2 η2 q
3.50
ln 1 − γ1k > a1 ω.
k1
namely,
u1 η1 > ln
b1 ω
: B18 ,
#q
a1 − k1 ln 1 − γ1k
#q
a1 ω − k1 ln 1 − γ1k
: B20 .
u2 η2 > ln
b1 ωe−B19
3.51
Therefore,
u1 t u1 η1 −
η1
u̇1 tdt ≥ u1 η1 −
ω
|u̇1 t|dt
0
t
> B18 − 2a1 ω −
q
ln 1 − γ1k : B19 ,
k1
u2 t u2 ξ2 t
u̇2 tdt ≤ u2 ξ2 ξ2
< ln
|u̇2 t|dt
0
|a2 − b2 |ω u2 t u2 η2 −
ω
#q
k1
ln 1 − γ2k
cω
η2
u̇2 tdt ≥ u2 η1 −
ω
0
t
> B20 − 2a2 ω −
2a2 ω q
k1
ln 1 − γ2k : B22 .
q
k1
|u̇2 t|dt
ln 1 − γ2k : B21 ,
3.52
International Journal of Differential Equations
15
So
B19 < u1 t < 0, |u2 t| ≤ max{|B21 |, |B22 |} : B23 .
3.53
By the third equation of 3.14, we obtain
ω
ω
q
ln 1 − γ2k ,
3.54
#q
|a2 − b2 |ω k1 ln 1 − γ2k − β2 ωeB23
: B24 .
u3 η3 > ln
eω
3.55
|u3 t| ≤ max{|B23 |, |B24 |} : B25 .
3.56
ete
u3 η3 dt 0
β2 teB23 dt > |a2 − b2 |ω 0
k1
that is,
Thus,
Case 4 if u1 t < 0, u2 t < 0. By the second equation of 3.14, we have
ω
ctdt ω
0
β1 teu3 t dt > |a2 − b2 |ω 0
q
ln 1 − γ2k .
3.57
k1
Then
cω β1 ωeu3 η3 > |a2 − b2 |ω q
ln 1 − γ2k ,
3.58
k1
that is,
#q
|a2 − b2 |ω k1 ln 1 − γ2k − cω
: B26 .
u3 η3 ≥ ln
β1 ω
Therefore,
u3 t u3 ξ3 t
u̇3 tdt ≤ u3 ξ3 ξ3
< ln
dω ω
|u̇3 t|dt
0
#q
k1
ln 1 − γ3k
eω
2dω q
k1
ln 1 − γ3k : B27 ,
3.59
16
International Journal of Differential Equations
η3
ω
u̇3 tdt ≥ u3 η3 −
u3 t u3 η3 −
|u̇3 t|dt
0
t
q
> B26 − 2dω −
ln 1 − γ3k : B28 .
k1
3.60
Thus,
3.61
|u3 t| ≤ max{|B27 |, |B28 |} : B29 .
By the second equation of 3.14, we obtain
ω
cte
u2 η2 dt ω
0
β1 teB29 dt > |a2 − b2 |ω 0
q
ln 1 − γ2k ,
3.62
k1
that is,
ceu2 η2 β1 ωeB29 > |a2 − b2 |ω q
ln 1 − γ2k .
3.63
k1
Thus,
#q
|a2 − b2 |ω k1 ln 1 − γ2k − β1 ωeB29
u2 η2 > ln
.
cω
3.64
Then, from 3.16 and 3.20, we get
u2 t u2 ξ2 t
u̇2 tdt ≤ u2 ξ2 ξ2
< ln
|u̇2 t|dt
0
|a2 − b2 |ω u2 t u2 η2 −
ω
#q
k1
ln 1 − γ2k
cω
η2
u̇2 tdt ≥ u2 η2 −
q
ln 1 − γ2k : B31 ,
k1
ω
3.65
|u̇2 t|dt
0
t
> B30 − 2a2 ωω −
2a2 ω q
ln 1 − γ2k : B32 .
k1
Thus,
|u2 t| ≤ max{|B31 |, |B32 |} : B33 .
3.66
International Journal of Differential Equations
17
By the first equation of 3.14, we have
ω
b1 teu2 t−u1 η1 dt q
0
ln 1 − γ1k <
ω
a1 tdt.
3.67
0
k1
Then
b1 ωe−B33 e−u1 η1 q
ln 1 − γ1k < a1 ω.
3.68
k1
Thus,
u1 η1 > ln
b1 ωe−B32
: B34 .
#q
a1 ω − k1 ln 1 − γ1k
3.69
Hence, we have
B34 < u1 t < 0.
3.70
Based on the discussion above, we can easily obtain
u1 t ≤ max{B5 , B12 , |B19 |, |B34 |},
u2 t ≤ max{B3 , B4 , B23 , B33 },
3.71
u3 t ≤ max{B8 , B17 , B25 , B29 }.
Obviously, Bi i 1, 2, . . . , 34 are independent of λ ∈ 0, 1. Similar to the proof of Theorem
2.1 of 17, we can easily find a sufficiently large M > 0 so that we denote the set
Ω ut u1 t, u2 t, u3 tT ∈ x : u < M, u tk ∈ Ω, k 1, 2, . . . , q .
3.72
It is clear that Ω satisfies the requirement a in Lemma 3.1.
Whenu1 t, u2 t, u3 tT ∈ ∂Ω ∩ Ker L ∂Ω ∩ R3 and u {u1 , u2 , u3 T } is a constant
vector in R3 with u u1 t, u2 t, u3 tT M, then we have
⎞
⎞
q
1 ln 1 − γ1k
⎟
⎟
⎜⎜ a1 b1 e
⎟
⎟
⎜⎜
ω k1
⎟
⎟
⎜⎜
q
⎟
⎟
⎜⎜
⎟
1 ⎟
⎜⎜
u2
u3
ln 1 − γ2k ⎟, 0, . . . , 0⎟ /
0.
QNu ⎜⎜a2 − b2 − ce β1 e ⎟
⎟
⎜⎜
ω k1
⎟
⎟
⎜⎜
q
⎟
⎜⎜
⎟
1 ⎠
⎠
⎝⎝
d − eeu3 − β2 eu2 ln 1 − γ3k
ω k1
⎛⎛
u2 −u1
3.73
18
International Journal of Differential Equations
Letting J : Im Q → Ker L, r, 0, . . . , 0, 0 → r and, by direct calculation, we get
$
deg JQNu1 , u2 , u3 T ; ∂Ω ker L; 0
⎛
−b1 eu2 −u1 b1 eu2 −u1
⎞
0
⎟
⎟
−β1 eu3 ⎟
⎠
u2
u3
0
−β2 e
−ee
sign b1 β1 β2 − b1 ce e2u2 −u1 u3 /
0.
⎜
⎜
signdet⎜
⎝
3.74
−ceu2
0
This proves that condition b in Lemma 3.1 is satisfied. By now, we have proved that Ω
verifies all requirements of Lemma 3.1, then it follows that Lu Nu has at least one solution
u1 t, u2 t, u3 tT in Dom L ∩ Ω; that is, to say, 3.5 has at least one ω periodic solution in
T
Dom L ∩ Ω. Then we know that x1 t, x2 t, x3 tT eu1 t , exu2 t , eu3 t is an ω periodic
solution of system 2.4 with strictly positive components. This completes the proof.
4. Uniqueness and Global Attractivity of Periodic Solutions
Under the hypotheses H1 , H2 , H3 , we consider the following ordinary differential
equation without impulsive:
1 − γ2k z2 t
,
ż1 t z1 t −a1 t b1 t %
0<tk <t 1 − γ1k z1 t
&
&
ż2 t z2 t a2 t − b2 t − ct
1 − γ2k z2 t − β1 t
1 − γ3k z3 t ,
%
0<tk <t
0<tk <t
ż3 t z3 t dt − et
&
0<tk <t
1 − γ3k z3 t − β2 t
0<tk <t
&
4.1
1 − γ2k z2 t ,
0<tk <t
with the initial conditions zi 0 > 0, i 1, 2, 3.
The following lemmas will be helpful in the proofs of our results. The proof of the
following Lemma 4.1 is similar to that of Theorem 1 in 18, and it will be omitted.
Lemma 4.1. Assume that H1 , H2 , H3 hold, then one has the following.
i If zt z1 t, z2 t, z3 tT is a solution of 4.1 on 0, ∞, then xi t γik zi t i 1, 2, 3 is a solution of 2.4 on 0, ∞.
%
0<tk <t
1 −
ii If xt x1 t, x2 t, x3 tT is a solution of 2.4 on 0, ∞, then zi t %
−1
0<tk <t 1 − γik xi t i 1, 2, 3 is a solution of 4.1 on 0, ∞.
Lemma 4.2. Let zt z1 t, z2 t, z3 tT denote any positive solution of system 4.1 with initial
conditions zi 0 > 0, i 1, 2, 3. Assume that the following condition holds,
L
H7 aM
2 > b2 ,
dM > eL .
4.2
International Journal of Differential Equations
19
Then there exists a T3 > 0 such that
i 1, 2, 3, for t ≥ T3 ,
0 < zi t ≤ Mi ,
4.3
where
%
M1 >
M1∗
M2 > M2∗ M3 > M3∗ 0<tk <t 1 − γ2k M2
,
L%
a1 0<tk <t 1 − γ1k
c
L
aM
2 − b2
,
0<tk <t 1 − γ2k
4.4
%
L
eL
dM − eL
.
0<tk <t 1 − γ3k
%
Proof. From the second equation of 4.1, we can obtain
&
ż2 t ≤ z2 t a2 t − b2 t − ct
1 − γ2k z2 t
0<tk <t
L
L
≤ z2 t aM
2 − b2 − c
&
4.5
1 − γ2k z2 t .
0<tk <t
By 4.5, we can derive the following.
A1 If z2 0 ≤ M2 , then z2 t ≤ M2 , t ≥ 0.
%
L
L
A2 If z2 0 > M2 , let −α1 M2 aM
0<tk <t 1 − γ2k M2 , α1 > 0. Then there
2 − b2 − c
exists ε1 > 0 such that t ∈ 0, ε1 , then z2 t > M2 , and also we have
4.6
ż2 t < −α1 < 0.
From what has been discussed above, we can easily conclude that, if z2 0 > M2 , then z2 t is
strictly monotone decreasing with speed at least α1 . Therefore, there exists a T1 > 0 such that
t > T1 , then z2 t ≤ M2 .
From the third equation of 4.1, we can obtain
ż3 t ≤ z3 t dt − et
&
1 − γ3k z3 t
0<tk <t
&
≤ z3 t dM − eL
1 − γ3k z3 t .
0<tk <t
By 4.7, we can derive the following
B1 If z3 0 ≤ M3 , then z3 t ≤ M3 , t ≥ 0.
4.7
20
International Journal of Differential Equations
%
B2 If z3 0 > M3 , let −α2 M3 dM − eL 0<tk <t 1 − γ3k M3 , α2 > 0. Then there exists
ε2 > 0 such that t ∈ 0, ε2 , then z3 t > M3 , and also we have
4.8
ż3 t < −α2 < 0.
From what has been discussed above, we can easily conclude that, if z3 0 > M3 , then z3 t is
strictly monotone decreasing with speed at least α2 . Therefore, there exists a T2 > 0 such that
t > T2 , then z3 t ≤ M3 .
From the first equation of 4.1, we can obtain
1 − γ2k M2
ż1 t ≤ z1 t −a1 t b1 t %
0<tk <t 1 − γ1k z1 t
%
0<tk <t 1 − γ2k M2
−a1 tz1 t %
0<tk <t 1 − γ1k
%
0<tk <t 1 − γ2k M2
L
≤ −a1 z1 t %
.
0<tk <t 1 − γ1k
%
0<tk <t
4.9
Then we have
z1 t ≤ M1 ,
for t ≥ T1 .
4.10
Set T3 max{T1 , T2 }, then we have
0 < zi t ≤ Mi ,
i 1, 2, 3,
for t ≥ T3 .
4.11
The proof is complete.
Lemma 4.3. Let H1 , H2 , H3 hold. Assume that the following condition holds.
H8 aL2 > b2M β1M
&
1 − γ3k ,
dL > eM − β2M
0<tk <t
&
1 − γ2k .
0<tk <t
4.12
Then there exists positive constants T > 0 and mi i 1, 2, 3 such that, for all t > T ,
i 1, 2, 3,
mi < zi t,
for t ≥ T,
in which
m1 <
m∗1
1 − γ2k m2
M%
,
a1
0<tk <t 1 − γ1k
b1L
%
0<tk <t
4.13
International Journal of Differential Equations
21
m2 <
m∗2
%
aL2 − b2M − β1M 0<tk <t 1 − γ3k
,
%
cM 0<tk <t 1 − γ2k
m3 <
m∗3
%
dL − eM − β2M 0<tk <t 1 − γ2k M2
.
%
eM 0<tk <t 1 − γ3k
4.14
Proof. By the second equation of 4.1, It is easy to obtain that, for t ≥ T3 ,
ż2 t ≥ z2 t
aL2
−
b2M
−c
&
1 − γ2k z2 t −
M
0<tk <t
&
1 − γ3k M3 ,
β1M
4.15
0<tk <t
where T3 is defined in Lemma 4.1.
C1 If z2 T3 ≥ m2 , then z2 t ≥ m2 , t ≥ T3 .
C2 If z2 T3 < m2 and let
μ1 z2 T3 aL2
−
b2M
−c
M
&
1 − γ2k m2 ,
4.16
0<tk <t
then there exists ε3 > 0 such that t ∈ T3 , T3 ε3 , then z2 t > m2 , and also we have
4.17
ż2 t > μ1 > 0.
Then we know that if z2 T3 < m2 , z2 t will strictly monotonically increase with speed μ2 .
Thus, there exists T4 > T3 such that if t ≥ T4 , then z2 t ≥ m2 .
By the third equation of 4.1, It is easy to obtain that for t ≥ T3 ,
ż3 t ≥ z3 t d − e
L
&
M
1 − γ3k z3 t − β2
1 − γ2k M2 ,
&
M
0<tk <t
4.18
0<tk <t
where T3 is defined in Lemma 4.2.
D1 If z2 T3 ≥ m3 , then z3 t ≥ m3 , t ≥ T3 .
D2 If z2 T3 < m3 , and let
μ2 z3 T3 d − e
L
M
&
1 − γ3k m3 −
0<tk <t
β2M
&
1 − γ2k M2 ,
4.19
0<tk <t
then there exists ε4 > 0 such that t ∈ T3 , T3 ε4 , then z3 t > m3 , and also we have
ż3 t > μ2 > 0.
4.20
22
International Journal of Differential Equations
Then we know that if z3 T3 < m3 , z3 t will strictly monotonically increase with speed μ2 .
Thus, there exists T5 > T3 such that, if t ≥ T5 , then z3 t ≥ m3 .
Finally, by the third equation of 4.1, we obtain
ż1 t ≥ z1 t
%
−aM
1
0<t <t 1 − γ2k m2
b1L % k 0<tk <t 1 − γ1k z1 t
0<tk <t 1 − γ2k m2
M
L
−a1 z1 t b1 %
.
0<tk <t 1 − γ1k z1 t
%
4.21
Thus, we have
4.22
z1 t ≥ m1 ,
for t ≥ T4 . Set T max{T4 , T5 }, then we have
zi t > mi ,
i 1, 2, 3,
for t ≥ T.
4.23
In the sequel, we formulate the uniqueness and global attractivity of the ω periodic solution
x∗ t in Theorem 4.4. It is immediate that if x∗ t is global attractivity, then x∗ t is in fact
unique.
Theorem 4.4. In addition to H1 − H8 , assume further H9 limt → ∞ inf Bi t > 0, where
B1 t ct − β2 t −
m1
%
bM
0<tk <t
1 − γ1k
&
1 − γ2k ,
0<tk <t
&
B2 t et − β1 t
1 − γ3k .
4.24
0<tk <t
Then system 2.4 has a unique positive ω periodic solution x∗ t x1∗ t, x2∗ t, x3∗ tT which is
global attractivity.
Proof. According to the conclusion of Theorem 3.2, we only need to show that the positive
periodic solution of 2.4 is global asymptotical stable. Let x∗ t x1∗ t, x2∗ t, x3∗ tT
be a positive ω periodic solution of system 2.4 let xt x1 t, x2 t, x3 tT be any
%
positive solution of system 2.4. Then z∗ t z∗1 t, z∗2 t, z∗3 tT , z∗1 t 0<tk <t 1 −
%
%
γ1k x1∗ t, z∗2 t 0<tk <t 1 − γ2k x2∗ t, z∗3 t 0<tk <t 1 − γ3k x3∗ t is the positive ω periodic
solution of 4.1, and zt is the positive solution of 4.1. It follows from Lemma 4.2 and 4.3
that there exists positive constants T > 0, Mi and mi defined by Lemmas 4.2 and 4.3, resp.
such that, for all t > T ,
mi < z∗i t ≤ Mi ,
mi < zi t ≤ Mi , i 1, 2, 3.
4.25
International Journal of Differential Equations
23
Define
V t ln z∗1 t − ln z1 t ln z∗2 t − ln z2 t ln z∗3 t − ln z3 t.
4.26
Calculating the upper-right derivative of V t along the solution of 4.1, it follows for t ≥
T that
∗
3
zi t zi t
D V t −
sgn z∗i t − zi t
∗
zi t
zi t
i1
∗
%
z2 t z2 t
∗
0<tk <t 1 − γ2k
sgn z1 t − z1 t b1 t %
−
z∗1 t z1 t
0<tk <t 1 − γ1k
&
∗
sgn z2 t − z2 t −ct
1 − γ2k z∗2 t − z2 t
0<tk <t
&
−β1 t
1 − γ3k
z∗3 t
− z3 t
0<tk <t
&
sgn z∗3 t − z3 t −et
1 − γ3k
z∗3 t − z3 t
4.27
0<tk <t
&
−β2 t
1 − γ2k
z∗2 t
− z2 t
0<tk <t
≤ −ct
&
&
1 − γ2k z∗2 t − z2 t β1 t
1 − γ3k z∗3 t − z3 t
0<tk <t
− et
0<tk <t
&
&
1 − γ3k z∗3 t − z3 t β2 t
1 − γ2k z∗2 t − z2 t D1 t,
0<tk <t
0<tk <t
where
⎧
∗
%
z2 t z2 t
⎪
0<tk <t 1 − γ2k
⎪
⎪
−
,
⎪b1 t %
⎪
z∗1 t z1 t
⎨
0<tk <t 1 − γ1k
D1 t ∗
%
⎪
⎪
z2 t z2 t
⎪
0<tk <t 1 − γ2k
⎪
⎪
−
,
⎩b1 t %
z∗1 t z1 t
0<tk <t 1 − γ1k
z∗1 t > z1 t,
4.28
z∗1 t
< z1 t.
In the sequel, we will estimate D1 t under the following two cases.
i If z∗1 t ≥ z1 t, then
1 − γ2k ∗
D1 t ≤ ∗ %
z2 t − z2 t
z1 t 0<tk <t 1 − γ1k
%
bM 0<tk <t 1 − γ2k ∗
≤
z2 t − z2 t.
%
m1 0<tk <t 1 − γ1k
b1 t
%
0<tk <t
4.29
24
International Journal of Differential Equations
ii If z∗1 t < z1 t, then
1 − γ2k D1 t ≤
z2 t − z∗2 t
%
z1 t 0<tk <t 1 − γ1k
b1 t
%
0<tk <t
0<tk <t 1 − γ2k ≤
z∗2 t − z2 t.
%
m1 0<tk <t 1 − γ1k
bM
%
4.30
Combining the conclusions of 4.29 and 4.30, we obtain
D1 t ≤
bM
m1
%
%
0<tk <t
0<tk <t
1 − γ2k ∗
z2 t − z2 t.
1 − γ1k
4.31
It follows from 4.31 that
&
&
1 − γ2k z∗2 t − z2 t β1 t
1 − γ3k z∗3 t − z3 t
D V t ≤ −ct
0<tk <t
− et
0<tk <t
&
&
1 − γ3k z∗3 t − z3 t β2 t
1 − γ2k z∗2 t − z2 t
0<tk <t
b
m1
%
m1
0<tk <t
%
b
1 − γ2k ∗
z2 t − z2 t
M
0<tk <t
0<tk <t
1 − γ1k
0<tk <t
%
M
1 − γ1k
− ct β2 t
4.32
&
1 − γ2k z∗2 t − z2 t
0<tk <t
&
β1 t − et
1 − γ3k z∗3 t − z3 t
0<tk <t
≤ − B1 tz∗2 t − z2 t B2 tz∗3 t − z3 t ,
where B1 t and B2 t are defined in Theorem 4.4.
By hypothesis H8 , there exist constants αi , i 2, 3 and T ∗ > T such that
Bi t ≥ αi > 0,
i 2, 3, for t ≥ T ∗ .
4.33
Integrating both sides of 4.32 on interval T ∗ , t yields
V t 3 t
i2
T∗
Bi tz∗i t − zi tds ≤ V T ∗ .
4.34
International Journal of Differential Equations
25
It follows from 4.33 and 4.34 that
3 t
i2
T∗
Bi tz∗i t − zi tds ≤ V T ∗ < ∞,
for t ≥ T ∗ .
4.35
Since z∗i t and zi t i 2, 3 are bounded for t ≥ T ∗ , so |z∗i t − zi t| i 2, 3 are uniformly
continuous on T ∗ , ∞. By Barbalat’s Lemma 32, we have
lim z∗i t − zi t lim
t→∞
−1 ∗
xi t − xi t 0,
1 − γik
t→∞
&
i 2, 3.
4.36
0<tk <t
Thus,
lim xi∗ t − xi t 0,
t→∞
i 2, 3.
4.37
By 4.37 and the first equation of 2.4, one can easily obtain that
lim x1∗ t − x1 t 0.
4.38
t→∞
By Theorems 7.4 and 8.2 in 33, we know that the positive periodic solution x∗ t x1∗ t, x2∗ t, x3∗ tT of 2.4 is uniformly asymptotically stable. The proof of Theorem 4.4 is
complete.
5. An Example
As an application of our main results, we consider the following system:
ẋ1 t −2x1 t x2 t,
t/
tk ,
ẋ2 t 4 cos tx2 t − 2 cos tx2 t − 1 − sin tx22 t
−
1
2e200π1
sin tx2 tx3 t,
t/
tk ,
(
)
49
200π
ẋ3 t x3 t 50 sin t − 50e
− cos t x2 t ,
1 sin t x3 t −
e50 3π
1
Δx1 tk x1 tk − x1 t−k − x1 tk ,
2
k 1, 2, . . . ,
1
Δx2 tk x2 tk − x2 t−k − x2 tk ,
3
k 1, 2, . . . ,
1
Δx3 tk x3 tk − x3 t−k − x3 tk ,
4
k 1, 2, . . . ,
t/
tk ,
5.1
26
International Journal of Differential Equations
in which tk2 tk 2π, 0, 2π ∩ {tk } {t1 , t2 }, a1 t 2, b1 t 1, a2 t 4 cos t, b2 t 2 cos t, β1 t 1/2e200π1 sin t, β2 t 49/e50 3π − cos t, ct 1 − sin t, dt 50 sin t,
and et 50e200π 1 sin t. By direct computation, we can obtain
a1 2,
a2 4,
β2 c 1,
|a2 − b2 | 2,
49
e 50e200π 1,
e50 3π
B1 ln
B2 ln
B6 B27 ln
,
d 50,
β1 1
,
2e200π1
b1 1,
2 .
4π 2 ln2/3
16π 2 ln 50.07,
2π
3
2 .
4π − 2 ln1/2
− 16π − 2 ln −47.07,
2π
3
5.2
3 .
100π − 2 ln3/4
200π 2 ln −0.5754,
4
2π50e200π B26 ln
2π 2 ln2/3 .
629.55,
π/e200π1
B28 B26 − 200π − 2 ln
3 .
2.9362.
4
.
.
Then B4 50.07, B29 2.9362. It is easy to check that 5.1 satisfies all the conditions
of Theorems 3.2 and 4.4; hence, 5.1 has a positive 2π periodic solution which is global
attractivity.
Acknowledgments
The authors would like to thank the referees for their helpful comments and valuable
suggestions regarding this paper. This work is supported by the National Natural Science
Foundation of China no. 10771215, the Scientific Research Fund of Hunan Provincial
Education Department no. 10C0560, the Doctoral Foundation of Guizhou College of
Finance and Economics 2010 and the Science and technology Program of Hunan Province
no. 2010FJ6021.
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