Volume 2009, Article ID 461757, 13 pagesdoi:10.1155/2009/461757 Research Article A New Singular Impulsive Delay Differential Inequality and Its Application 1 College of Computer Science
Trang 1Volume 2009, Article ID 461757, 13 pages
doi:10.1155/2009/461757
Research Article
A New Singular Impulsive Delay Differential
Inequality and Its Application
1 College of Computer Science and Technology, Southwest University for Nationalities,
Chengdu 610041, China
2 Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China
Correspondence should be addressed to Xiaohu Wang,xiaohuwang111@163.com
Received 10 January 2009; Accepted 5 March 2009
Recommended by Wing-Sum Cheung
A new singular impulsive delay differential inequality is established Using this inequality, the invariant and attracting sets for impulsive neutral neural networks with delays are obtained Our results can extend and improve earlier publications
Copyrightq 2009 Z Ma and X Wang 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
1 Introduction
It is well known that inequality technique is an important tool for investigating dynamical behavior of differential equation The significance of differential and integral inequalities
in the qualitative investigation of various classes of functional equations has been fully illustrated during the last 40 years 1 3 Various inequalities have been established such
as the delay integral inequality in4, the differential inequalities in 5,6, the impulsive differential inequalities in 7 10, Halanay inequalities in 11–13, and generalized Halanay inequalities in14–17 By using the technique of inequality, the invariant and attracting sets for differential systems have been studied by many authors 9,18–21
However, the inequalities mentioned above are ineffective for studying the invariant and attracting sets of impulsive nonautonomous neutral neural networks with time-varying delays On the basis of this, this article is devoted to the discussion of this problem
Motivated by the above discussions, in this paper, a new singular impulsive delay differential inequality is established Applying this equality and using the meth-ods in 10, 22, some sufficient conditions ensuring the invariant set and the global attracting set for a class of neutral neural networks system with impulsive effects are obtained
Trang 22 Preliminaries
Throughout the paper, E n means n-dimensional unit matrix, Rthe set of real numbers, N the set of positive integers, andN {1, 2, , n} A ≥ B A > B means that each pair ofΔ corresponding elements of A and B satisfies the inequality “≥ > ” Especially, A is called a nonnegative matrix if A≥ 0
C X, Y denotes the space of continuous mappings from the topological space X to the topological space Y In particular, let C C−τ, 0, RΔ n , where τ > 0 is a constant.
P C a, b, R n denotes the space of piecewise continuous functions ψs : a, b → R n
with at most countable discontinuous points and at this points ψs are right continuous Especially, let P C PC−τ, 0, RΔ n Furthermore, put PCa, b , R n c ∈a,b P C a, c, R n
P C1a, b, R n {ψs : a, b → R n | ψs , ˙ψs ∈ PCa, b, R n }, where ˙ψs denotes the derivative of ψs In particular, let PC1 Δ PC1a, b, R n
H {ht : R → R | ht is a positive integrable function and satisfies
supa ≤t<bt
t −τ h s ds σ < ∞ and lim t→ ∞t
a h s ds ∞}.
For x ∈ Rn , A ∈ Rn ×n , ϕ ∈ C or ϕ ∈ PC, we define x |x1|, , |x n| T , A
|a ij| n ×n , ϕt τ ϕ1t τ , , ϕ n t τ T , ϕt τ ϕt τ , ϕ i t τ sup−τ≤θ<0 {ϕ i
θ } And we introduce the following norm, respectively,
x max
1≤i≤nx i, A max
1≤i≤n
n
j1
a ij, ϕ τ sup
−τ≤s≤0
ϕ s . 2.1
For any ϕ ∈ PC1, we define the following norm:
ϕ 1τ maxϕ τ ,ϕ
τ
For an M-matrix D defined in23, we denote
ΩM D z∈ Rn | Dz > 0, z > 0. 2.3
It is a cone without conical surface inRn We call it an “M-cone”.
3 Singular Impulsive Delay Differential Inequality
For convenience, we introduce the following conditions
C1 Let the r-dimensional diagonal matrix K diag{k1, , k r} satisfy
k i > 0, i ∈ S ⊂ N∗ Δ {1, , r}, k i 0, i ∈ S∗ Δ N∗− S. 3.1
p ij ≥ 0, i / j, p ij 0, i / j, i ∈ N∗, j ∈ S∗. 3.2
Trang 3Theorem 3.1 Assume the conditions C1 and C2 hold Let L L1, , L r and ut
u1t , , u r t T be a solution of the following singular delay differential inequality with the initial conditions u t ∈ PCa − τ, a, R r :
KD u t ≤ ht P u
u t τ , t ∈ a, b , 3.3
where τ > 0, a < b i t ∈ Ca, b , R , i ∈ S, u i t ∈ PCa, b , R , i ∈ S∗, h t ∈ H Then
u t ≤ dze −λt a h s ds −1L, t ∈ a, b , 3.4
provided that the initial conditions satisfy
u t ≤ dze −λt a h s ds −1L, t ∈ a − τ, a, 3.5
where d ≥ 0, z z1, , z r T ∈ ΩM U and the positive number λ satisfies the following inequality:
Proof By the conditions C2 and the definition of M-matrix, there is a constant vector z
z1, , z r Tsuch that −1exists and −1≥ 0
By using continuity, we obtain that there must exist a positive constant λ satisfying the
inequality3.6 , that is,
r
j1
p ij ij e λσ z j < −λk i z i , i∈ N∗. 3.7
Denote by
v t v1t , , v r t T −1L, t ∈ a − τ, b 3.8
It follows from3.3 and 3.5 that
KD v t ≤ ht P u
u t τ
≤ ht P v
v t τ , t ∈ a, b ,
v t ≤ dze −λt a h s ds , t ∈ a − τ, a.
3.9
In the following, we will prove that for any positive constant ε,
Trang 4℘i∈ N∗| v i t > w i t for some t ∈ a, b ,
θ i inft ∈ a, b | v i t > w i t , i ∈ ℘. 3.11
If inequality3.10 is not true, then ℘ is a nonempty set and there must exist some integer
m ∈ ℘ such that θ m mini ∈℘ {θ i } ∈ a, b
If m ∈ S, by v m t ∈ Ca, b , R and the inequality 3.5 , we can get
θ m > a, v m
θ m
w m
θ m
, D v m
θ m
≥ ˙w m
θ m
v i t ≤ w i t , t ∈ a − τ, θ m , i ∈ N∗, v i
θ m
≤ w i
θ m
, i ∈ S. 3.13
By usingC2 , 3.3 , 3.7 , 3.12 , 3.13 , and v i t τ sup−τ≤θ<0 {u i ∗, we obtain that
k m D v m
θ m
≤ h θ m r
j1
p mj v j
θ m
mj
v j
θ m τ
h θ m
p mm v m
θ m
j / m, j∈S
p mj v j
θ m
j ∈S∗
p mj v j
θ m
j1
q mj
v j
θ m τ
≤ hθ m
j ∈S
p mj j e −λθm a h s ds r
j1
q mj j e −λθm−τ a h s ds
θ m r
j1
p mj mj e λσ z j e −λθm a h s ds
< m z m h
θ m
e −λθm a h s ds
3.14
Since m ∈ S, we have k m > 0 by H1 Then 3.14 becomes
D u m
θ m
< m h
θ m
e −λθm a r s ds ˙w m
θ m
which contradicts the second inequality in3.12
If m ∈ S∗, then k m 0 by C1 and v m t ∈ PCa, b , R From the inequality 3.5 , we can get
θ m > a, v m
θ m
≥ w m
θ m
, v i
θ m
≤ w i
θ m
, i ∈ S,
v i t ≤ w i t , t ∈ a − τ, θ m , i ∈ N∗. 3.16
Trang 5By usingC2 , 3.3 , 3.7 , 3.16 , and v i t τ sup−τ≤θ<0 {v i ∗, we obtain that
0≤r
j1
p mj v j
θ m
j1
q mj
v j
θ m τ
j ∈S
p mj v j
θ m
mm v m θ m
j / m, j∈S∗
p mj v j
θ m
r
j1
q mj
v j
θ m τ
j ∈S
p mj z j mm z m
j1
q mj z j e λθm θm−τ h s ds
e −λθm a h s ds
r
j1
p mj z j mj z j e λσ e −λθm a h s ds
< m z m h
θ m
e −λθm a h s ds
0.
3.17
This is a contradiction Thus the inequality3.10 holds Therefore, letting ε → 0 in 3.10 ,
we have
The proof is complete
Remark 3.2 In order to overcome the di fficulty that ut in 3.3 may be discontinuous, we introduce the notation u i t τ sup−τ≤s<0 {u i
u i t τ sup−τ≤s≤0 {u i 7 However, when u i t is continuous in t, we have
u i t τ sup
−τ≤s<0
sup
−τ≤s≤0
, i∈ N∗. 3.19
So we can get7, Lemma 1 when we choose K Er , S N∗, ht ≡ 1 inTheorem 3.1
Remark 3.3 Suppose that L L1, , L r T 0 and ht ≡ 1 inTheorem 3.1, then we can get
10, Theorem 3.1
4 Applications
The singular impulsive delay differential inequality obtained in Section 3 can be widely applied to study the dynamics of impulsive neutral differential equations To illustrate the theory, we consider the following nonautonomous impulsive neutral neural networks with delays
x t x
t − τt
˙x
t − rt k ,
x t I k
t, x
t−
, t t k ,
4.1
Trang 6where x x1, , x n T ∈ Rn is the neural state vector; Dt diag{d1t , , d n t } >
0, At a ij t n ×n , Bt b ij t n ×n , Ct c ij t n ×n are the interconnection matrices representing the weighting coefficients of the neurons; Fx f1x1 , , f n x n T , G x
g1x1 , , g n x n T , H x h1x1 , , h n x n T are activation functions; τt
τ ij t n ×n , r t r ij t n ×n are transmission delays; Jt J1t , , J n t T denotes
the external inputs at time t I k t, y I 1k t, y , , I nk t, y T represents impulsive
perturbations; the fixed moments of time t k satisfy t k < t k , lim k t k
The initial condition for4.1 is given by
x
t0
ϕs ∈ PC1, t0∈ R, − τ ≤ s ≤ 0. 4.2
We always assume that for any ϕ ∈ PC1,4.1 has at least one solution through t0, ϕ ,
denoted by xt, t0, ϕ or x t t0, ϕ simply xt or x tif no confusion should occur
Definition 4.1 The set S ⊂ PC1is called a positive invariant set of4.1 , if for any initial value
ϕ ∈ S, we have the solution x t t0, ϕ ∈ S for t ≥ t0
Definition 4.2 The set S ⊂ PC1is called a global attracting set of4.1 , if for any initial value
ϕ ∈ PC1, the solution x t t0, ϕ
where distφ, S infψ ∈Sdistφ, ψ , distφ, ψ sups ∈−τ,0 |φs − ψs |, for φ ∈ PC1
Throughout this section, we suppose the following
H1 Dt ∈ PCR, R n , At , Bt , Ct , τt , rt are continuous Moreover, 0 ≤ τ ij t ≤ τ and 0 < r ij t ≤ τ i, j ∈ N
H2 There exist nonnegative matrices D1 diag{ d11, , d 1n}, D2 diag{ d21, , d 2n},
J J1, , J n T , hs ∈ H and a constant δ > 0 such that
D1h t ≤ Dt ≤ D2h t , 0 < ht ≤ 1
δ ,
J t ≤ Jht 4.4
H3 There exist nonnegative matrices A a ij n ×n , B b ij n ×n , C c ij n ×nsuch that
A t ≤ Ah t , B t ≤ Bht , C t ≤ Ch t 4.5
H4 There exist nonnegative matrices F diag{α1, , α n}, G diag{β1, , β n}, H diag{γ1, , γ n } such that for all u ∈ R n the activating functions F· , G· and H·
satisfy
F u ≤ Fu ,
G u ≤ G u ,
H u ≤ H u 4.6
Trang 7H5 There exists nonnegative matrix I k I k
ij n ×n , such that for all u ∈ Rn , i ∈ N and
k∈ N
H6 Denote by
U A F, V B G, W C H,
K
E n 0
0 0
diagk1, , k 2n
, L J, J T
,
P
− D1 U 0
D2 U −δE n
Δ
p ij t 2n×2n , Q
V W
V W
Δ
q ij t 2n×2n ,
4.8
Rn
H7 There exists a constant ν such that
ln η k ≤ ν
t k
t k−1
h s ds, k ∈ N, μ∞
k1
ln μ k < ∞, 4.9
where ν < λ, and the scalar λ > 0 is determined by the inequality
where z∗ z1, , z 2n T ∈ ΩM D , and
η k , μ k ≥ 1, η k z∗x ≥ I k z∗x , μ k 1≥ I k 1, k ∈ N, z∗
x z1, , z n T
. 4.11
Theorem 4.3 Assume that H1 –H7 hold Then S {φ ∈ PC1 | φ τ ≤ e μ 1} is a global attracting set of 4.1
Proof Denote ˙x t yt Let sgn· be the sign function For x x1, , x n T, define Sgnx diag{sgnx1 , , sgnx n }.
Calculating the upper right derivative D xt along system4.1 From 4.1 , H2 andH3 we have
D
x t ≤ ht − D1 U
x t V G
x t τ
W
y t τ , t∈ t k−1, t k
, t ≥ σ, k ∈ N. 4.12
Trang 8On the other hand, from4.1 and H2 –H4 , we have
0≤ ht
− 1
h t
y t D2 U
x t V
x t τ W
y t τ
≤ ht − δ y t D2 U xt V
x t τ W
y t τ , t ∈ t k−1, t k , k ∈ N.
4.13 Let
u t xt , yt T ∈ R2n , 4.14 then from4.12 –4.14 and H6 , we have
KD
u t ≤ ht P
u t u t τ , t ∈ t k−1, t k , k ∈ N. 4.15
By the conditionsH6 and the definition of M-matrix, we may choose a vector z∗
z1, , z 2n T ∈ ΩM D such that
By using continuity, we obtain that there must be a positive constant λ satisfying the
inequality4.10 Let z∗
x z1, , z n T and z∗y z n , , z 2n T , then z∗ z∗
x , z∗y T Since
z∗> 0, denote
min1≤i≤2n
then dz∗≥ e 2n 1, , 1 T ∈ R2n From the property of M-cone, we have, dz∗∈ ΩM D
loss of generality, we assume t0≤ t1 , and t ∈ t0− τ, t0, we can get
x t ≤ϕ t
τ e −λ
t
t0 h s ds dz∗x ,
y t ≤ϕt
τ e −λ
t
t0 h s ds dz∗y ,
4.18
Then4.18 yield
u t ≤ dz∗ϕ
1τ e −λ
t
LetN∗ {1, , 2n}, S {1, , n} N and S∗ ∗− S Thus, all
conditions ofTheorem 3.1are satisfied ByTheorem 3.1, we have
u t ≤ dz∗ϕ
1τ e −λ
t
Trang 9Suppose that for all m 1, , k, the inequalities
u t ≤m−1
j0
η j dz∗ϕ
1τ e −λ
t t0 h s ds m−1
j0
μ j , t m−1≤ t < t m , t ≥ t0, 4.21
hold, where η0 μ0 1
From4.21 , H5 , and H7 , we can get
x
t k ≤ I k
x
t−k
≤k
j0
η j dz∗xϕ t
1τ e−
tk
t0 h s ds k
j0
μ j 1. 4.22
Since −1L, we have
D2 U V
On the other hand, it follows fromH7 that
D2 U
z∗x V z∗
x Wz∗y
e λσ < δz∗y 4.24 Then from4.21 –4.24 , we have
y
t k ≤k
j0
η j dz∗yϕ
1τ e −λ
tk
t0 h s ds k
j0
which together with4.22 yields that
u
t k ≤k
j0
η j dz∗ϕ
1τ e −λ
tk
t0 h s ds k
j0
Then, it follows from4.21 and 4.26 that
u
t ≤k
j0
η j dz∗ϕ
1τ e −λ
t t0 h s ds k
j0
μ j
k
j0
η j dz∗ϕ
1τ e −λ
tk
t0 h s ds e −λ
t
tk h s ds k
j0
μ j , ∀ t ∈ t k − τ, t k
4.27
Trang 10UsingTheorem 3.1again, we have
u t ≤k
j0
η j dz∗ϕ
1τ e −λtk t0 h s ds e −λ
t
tk h s ds k
j0
μ j
k
j0
η j dz∗ϕ
1τ e −λ
t
t0 h s ds k
j0
μ j , t k ≤ t < t k
4.28
By mathematical induction, we can conclude that
u t ≤k
j0
η j dz∗ϕ
1τ e −λ
t
t0 h s ds k
j0
μ j , t k ≤ t < t k , k ∈ N. 4.29
Noticing that η k ≤ e νtk
tk−1 h s ds, byH7 , we can use 4.29 to conclude that
u t ≤ dz∗ϕ
1τ e ν
t t0 h s ds e −λ
t t0 h s ds μ
dz∗ϕ
1τ e −λ−ν
t
This implies that the conclusion of the theorem holds
By usingTheorem 4.3with d 0, we can obtain a positive invariant set of 4.1 , and the proof is similar to that ofTheorem 4.3
Theorem 4.4 Assume that H1 –H7 with I k E n hold Then S {φ ∈ PC1 | φ τ ≤ 1} is a positive invariant set and also a global attracting set of 4.1
Remark 4.5 Suppose that c ij ≡ 0, i, j ∈ N in H5 , and ht ≡ 1, then we can get Theorems 1
and 2 in9
Remark 4.6 If I k t, xt− x ∈ R n then 4.1 becomes the nonautonomous neutral neural networks without impulses, we can get Theorem 4.1 in22
5 Illustrative Example
The following illustrative example will demonstrate the effectiveness of our results
Example 5.1 Consider nonlinear impulsive neutral neural networks:
˙x1t − 7 2t
x1
t − τ11t −1
4cos t˙x2
t − r12t − 1.5cost,
˙x2t − 6 2t
x2t − 2 cos tx2
t − τ22t
4sin t tan
˙x1
t − r21t t /
k,
5.1
... Trang 7H5 There exists nonnegative matrix I k I k
ij...
Trang 5By usingC2 , 3.3 , 3.7 , 3.16 , and v i t τ... class="text_page_counter">Trang 6
where x x1, , x n T ∈ Rn is the neural state vector;