Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 927059, 9 pages
doi:10.1155/2010/927059
Research Article
A New Delay Vector Integral Inequality and
Its Application
Li Xiang,1 Lingying Teng,2 and Hua Wu3
1
College of Computer Science, Civil Aviation Flight University of China, Guanghan 618307, China
College of Computer Science and Technology, Southwest University for Nationalities,
Chengdu 610041, China
3
Civil Aviation Flight University of China, Guanghan 618307, China
2
Correspondence should be addressed to Li Xiang, xiangni sss@163.com
Received 22 September 2010; Accepted 14 December 2010
Academic Editor: Mohamed A. El-Gebeily
Copyright q 2010 Li Xiang et al. 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 new delay integral inequality is established. Using this inequality and the properties of
nonnegative matrix, the attracting sets for the nonlinear functional differential equations with
distributed delays are obtained. Our results can extend and improve earlier publications.
1. Introduction
The attracting set of dynamical systems have been extensively studied over the past few
decades and various results are reported. It is well known that one of the most researching
tools is inequality technique. With the establishment of various differential and integral
inequalities 1–12, the sufficient conditions on the attracting sets for different differential
systems are obtained 12–16. However, the inequalities mentioned above are ineffective
for studying the attracting sets of a class of nonlinear functional differential equations with
distributed delays.
Motivated by the above discussions, in this paper, a new delay vector integral
inequality is established. Applying this inequality and the properties of nonnegative matrix,
some sufficient conditions ensuring the global attracting set for a class of nonlinear functional
differential equations with distributed delays are obtained. The result in 16 is extended.
2. Preliminaries
In this section, we introduce some notations and recall some basic definitions.
E means unit matrix; R is the set of real numbers. A ≤ BA < B means that each pair
of corresponding elements of A and B satisfies the inequality “≤ <”. Especially, A is called
a nonnegative matrix if A ≥ 0.
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For a nonnegative matrix A ∈ Rn×n , let ρA denote the spectral radius of A. Then
ρA is an eigenvalue of A and its eigenspace is denoted by
Ωρ A z ∈ Rn | Az ρAz ,
2.1
which includes all positive eigenvectors of A provided that the nonnegative matrix A has at
least one positive eigenvector see 17.
CX, Y denotes the space of continuous mappings from the topological space X to the
topological space Y . Especially, let C C−∞, t0 , Rn , where t0 ≥ 0.
Let R
0, ∞. For x ∈ Rn , A ∈ Rn×n , ϕ ∈ C, τt ∈ CR, R
, we define
x |x1 |, |x2 |, . . . , |xn |T , A
|aij |n×n , ϕt
τt ϕ1 tτt , ϕ2 tτt , . . .,
ϕn tτt T , ϕi tτt sup0≤s≤τt |ϕi t − s|, i 1, 2, . . . , n. For τt ∞, we define
xt
τt xt
∞ .
Definition 2.1 Xu 4. ft, s ∈ UCt means that f ∈ CR
× R, R
and for any given α and
any ε > 0 there exist positive numbers B, T , and A satisfying
t
ft, sds ≤ B,
t−T
ft, sds < ε,
α
∀t ≥ A.
2.2
α
Especially, f ∈ UCt if ft, s ft − s and
∞
0
fudu < ∞.
Lemma 2.2 Lasalle 18. If M ≥ 0 and ρM < 1, then E − M−1 ≥ 0.
3. Delay Integral Inequality
Theorem 3.1. Let yt ∈ CR, Rn×1
be a solution of the delay integral inequality
yt ≤ Gt, t0 ϕt0 t
α1
Bt, s ys τs ds
t
yt ≤ ϕt,
Ψt, s
α2
s
α3
ζs, v yv τv dv ds
D,
t > t0 ,
3.1
∀t ∈ −∞, t0 ,
3.2
T
n×1
where Gt, t0 ∈ CR
× R
, Rn×n
, ϕt ∈ C−∞, t0 , R
, D d1 , . . . , dn ≥ 0, limt → ∞ t −
τt ∞, αi ∈ −∞, 0 i 1, 2, 3. Assume that the following conditions are satisfied:
H1 Gt, t0 ϕt0 → 0 as t → ∞, Bt, u bij t, un×n , Ψt, u ψij t, un×n , ζt, u ζij t, un×n , bij t, u, ψij t, u, ζij t, u ∈ UCt , and there exists a nonnegative matrix
Π πij n×n such that
t
α1
H2 ρΠ < 1.
Bt, sds t
α2
Ψt, s
s
α3
ζs, vdv ds ≤ Π,
for ∀t ≥ t0
3.3
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3
If the initial condition satisfies
ϕt < k1 z E − Π−1 D,
t ∈ −∞, t0 ,
3.4
where k1 ≥ 0, z > 0, and z ∈ Ωρ Π, then there exists a constant k ≥ k1 such that
yt < kz E − Π−1 D,
for t ≥ t0 .
3.5
Proof. By the condition Π ≥ 0 and Lemma 2.2, together with ρΠ < 1, this implies that
E − Π−1 exists and E − Π−1 ≥ 0.
From limt → ∞ Gt, t0 ϕt0 0, there is a T > t0 such that
Gt, t0 ϕt0 ≤
1 − ρΠ
k1 z,
2
for t ≥ T.
3.6
By the continuity of yt, together with 3.2 and 3.4, there exists a constant k ≥ k1 such that
yt < kz E − Π−1 D,
for t ∈ −∞, T .
3.7
In the following, we will prove that
yt < kz E − Π−1 D,
for t ≥ T.
3.8
If this is not true, from 3.7 and the continuity of yt, then there must be a constant t1 > T
and some integer i such that
yi t1 ei kz E − Π−1 D ,
yt ≤ kz E − Π−1 D,
where ei 0, 0, . . . , 0, 1, 0, . . . , 0.
i
for t ≤ t1 ,
3.9
3.10
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Journal of Inequalities and Applications
Using 3.1, 3.3, 3.6, 3.10, and ρΠ < 1, we obtain that
yi t1 ei yt1 ≤ ei Gt1 , t0 ϕt0 t1
α1
t1
Ψt1 , s
s
α3
α2
≤ ei
1 − ρΠ
k1 z 2
t1
α2
Ψt1 , s
Bt1 , s ys τs ds
ζs, v yv τv dv ds D
t1
Bt1 , s kz E − Π−1 D ds
α1
s
ζs, v kz E − Π−1 D dv ds D
3.11
α3
1 − ρΠ
kz Π kz E − Π−1 D E − ΠE − Π−1 D
2
1 − ρΠ
≤ ei
kz ρΠkz Π E − ΠE − Π−1 D
2
1 ρΠ
≤ ei
kz E − Π−1 D
2
< ei kz E − Π−1 D .
≤ ei
This contradicts the equality in 3.9, and so 3.8 holds. The proof is complete.
4. Applications
The delay integral inequality obtained in Section 3 can be widely applied to study the
attracting set of the nonlinear functional differential equations. To illustrate the theory, we
consider the following differential equation with distributed delays
ẋt Atxt F t, xt ,
t
xt ϕt,
−∞
gt, s, xsds ,
t ≥ t0 ,
4.1
−∞ < t ≤ t0 ,
where x x1 , . . . , xn T ∈ Rn , g ∈ CR × R × Rn , Rn , F ∈ CR × Rn × Rn , Rn , gt, s, 0 ≡ 0,
Ft, 0, 0 ≡ 0, xt xt − τt, τt ∈ CR, R
, t0 > 0, limt → ∞ t − τt ∞. We always
Journal of Inequalities and Applications
5
assume that for any ϕ ∈ C, the system 4.1 has at least one solution through t0 , ϕ denoted
by xt, t0 , ϕ or simply xt if no confusion should occur.
In order to study the attracting set, we rewrite 4.1 as
xt Φt, t0 ϕt0 t
s
Φt, sF s, xs ,
gs, v, xvdv ds,
−∞
t0
xt ϕt,
t ≥ t0 ,
4.2
−∞ < t ≤ t0 ,
where Φt, t0 be a fundamental matrix of the linear equation ẋt Atxt.
Throughout this section, we suppose the following:
t
t
A1 Ft, xt , −∞ gt, s, xsds
≤ Btxt
τt −∞ ζt − sxs
τs ds Jt, where
T
n×n
Bt ∈ CR, Rn×n
, ζt − u ζt, u ∈ CR × R, R
, Jt J1 t, . . . , Jn t ≥ 0,
Ji t ∈ CR, R
, i 1, 2, . . . , n,
A2 Φt, s
Bs wij t, sn×n , Φt, s
φij t, sn×n , ζt − s ζij t − sn×n .
∞
wij t, s, φij t, s ∈ UCt . 0 ζij sds < ∞, i, j 1, 2, . . . , n. limt → ∞ Φt, t0 0.
There are two matrices Π πij n×n ≥ 0 and D d1 , . . . , dn T ≥ 0 such that
t
Φt, s
Bsds t
t0
t
Φt, s
t0
Φt, s Jsds ≤ D,
s
−∞
ζs, vdv ds ≤ Π,
4.3
for ∀t ≥ t0 ,
t0
A3 ρΠ < 1.
Definition 4.1. The set S ⊂ C is called a global attracting set of 4.1, if for any initial value
ϕ ∈ C, the solution xt t0 , ϕ converges to S as t → ∞. That is,
distxt , S −→ 0
as t −→ ∞,
4.4
where distφ, S infψ∈S dφ, ψ, dφ, ψ is the distance of φ to ψ in Rn .
Theorem 4.2. Assume that A1 –A3 hold. Then S {ϕ ∈ C | ϕ
τ ≤ E − Π−1 D} is a global
attracting set of 4.1.
Proof. It follows from A1 -A2 and 4.2 that
xt ≤ Φt, t0 ϕt0 t
t0
t
t0
Φt, s
s
−∞
Φt, s
Bsxs
τs ds
ζs −
4.5
vxv
τv dv ds
D,
t ≥ t0 .
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Journal of Inequalities and Applications
For the initial conditions xt ϕt, −∞ < t ≤ t0 , where ϕ ∈ C, we have
xt
≤ k0 z,
−∞ < t ≤ t0 ,
4.6
where z ∈ Ωρ Π, z > 0, k0 ϕ/min1≤i≤n zi ≥ 0, ϕ max1≤i≤n ϕi t0 ∞ , and so
xt
≤ k0 z E − Π−1 D,
4.7
−∞ < t ≤ t0 .
By 4.5-4.7, A1 –A3 and Theorem 3.1, then there exists a constant k > k0 such that
xt
< kz E − Π−1 D,
4.8
for t ≥ t0 .
From 4.8, there exists a constant vector σ ≥ 0, such that
lim xt
σ ≤ kz E − Π−1 D.
4.9
t → ∞
Next we will show that σ ∈ S. From limt → ∞ Φt, t0 0, wij , φij ∈ UCt , and
ζij sds < ∞, for any ε > 0 and e 1, 1, . . . , 1T ∈ Rn×1
, there exist a positive number A
0
such
that
for
all
t
≥ t0 A
and a positive constant matrix R ∈ Rn×n
∞
εe
Φt, t0 ϕt0 < ,
4
t
−∞
ζt − v kz E − Π−1 D dv ≤ Re,
t
Φt, s
ds ≤ R,
t−A
Φt, s
ds ≤
t0
t−A
t0
εe
,
Φt, s
Bs kz E − Π−1 D ds ≤
4
∞
A
4.10
t0
εR−1 e
.
ζu kz E − Π−1 D du ≤
4
ε −1
R ,
4
4.11
4.12
4.13
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According to the definition of superior limit and limt → ∞ t−τt ∞, there exists sufficient
large t2 ≥ t0 2A, such that for any t ≥ t2 ,
xt
rt < σ εe,
where rt 2A sup τs.
t−2A<s<t
4.14
So, from A1 , 4.5, and 4.10-4.14, when t ≥ t2 , we obtain
xt ≤ Φt, t0 ϕt0 t
t0
t
Φt, s
s
t0
−∞
Φt, s
Bsxs
τs ds
ζs − vxv
τv dv ds D
t−A t εe
≤
Φt, s
Bsxs
τs ds
4
t0
t−A
t−A
Φt, s
t0
t
Φt, s
t−A
t
Φt, s
t−A
≤
s
−∞
ζs − vxv
τv dv ds
s−A
−∞
s
s−A
ζs − vxv
τv dv ds
ζs − vxv
τv dv ds D
t
ε
εe εe
Φt, s
Bsxs
τs ds R−1 Re
4
4
4
t−A
∞
t
ζu kz E − Π−1 D du ds
Φt, s
t−A
t
t−A
A
Φt, s
t
s
s−A
ζs − vdv dsxt
rt D
t
εR−1 e
3εe
≤
Φt, s Bsdsxtrt Φt, s
ds
4
4
t−A
t0
s
t
ζs − vdv dsσ εe D
Φt, s
t−A
≤
t
s−A
ε
3εe
Φt, s
Bsdsσ εe RR−1 e
4
4
t0
s
t
ζs − vdv dsσ εe D
Φt, s
t0
−∞
≤ εe Πσ εe D.
4.15
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Journal of Inequalities and Applications
Due to 4.9 and the definition of superior limit, there exists t3 ≥ t2 , such that xt3 > σ − εe.
So,
σ − εe < εe Πσ εe D,
t ≥ t3 .
4.16
Letting ε → 0, we have σ ≤ E − Π−1 D, that is σ ∈ S, and the proof is completed.
Corollary 4.3. If xt 0 is an equilibrium point of system 4.1, suppose that the conditions of
Theorem 4.2. hold and Jt 0, then the equilibrium xt 0 is globally asymptotically stable.
Remark 4.4. In Corollary 4.3., if Bt Bλ, ζt − u Kt − uλ, B bij n×n > 0,
λ diagλ1 , . . . , λn > 0, Kt − u Kt, u ∈ CR × R, Rn×n
, Kt − s kij t − sn×n ,
∞
k
sds
<
a
,
At
diag−α
,
.
.
.
,
−α
,
α
>
0
i
1,
2,
.
.
. , n, then we can get Theorems
ij
ij
1
n
i
0
3 and 4 in 16. In fact Φt, t0 diage−α1 t−t0 , . . . , e−αn t−t0 ∈ UCt , Theorems 3 and 4 in 16
satisfy all conditions of Corollary 4.3.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant
no. 10971147. Thanks are due to the instruction of Professor Xu.
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