Journal of Inequalities and Applications Volume 2009, Article ID 461757,13pages doi:10.1155/2009/461757
Research Article
A New Singular Impulsive Delay Differential
Inequality and Its Application
Zhixia Ma
1and Xiaohu Wang
21College of Computer Science and Technology, Southwest University for Nationalities, Chengdu 610041, China
2Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China
Correspondence should be addressed to Xiaohu Wang,[email protected]
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.
Copyrightq2009 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 in5,6, the impulsive differential inequalities in7–10, Halanay inequalities in11–13, 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 authors9,18–21.
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.
2. Preliminaries
Throughout the paper, En means n-dimensional unit matrix, Rthe set of real numbers, N
the set of positive integers, andN Δ {1,2, . . . , n}.A ≥ BA > B means that each pair of corresponding elements ofAandBsatisfies the inequality “≥> ”. Especially,Ais called a nonnegative matrix ifA≥0.
CX, Y denotes the space of continuous mappings from the topological spaceXto the topological spaceY. In particular, letCΔC−τ,0,Rn , whereτ >0 is a constant.
P Ca, b,Rn denotes the space of piecewise continuous functionsψs : a, b → Rn
with at most countable discontinuous points and at this points ψs are right continuous. Especially, letP CΔP C−τ,0,Rn . Furthermore, putP Ca, b ,Rn
c∈a,bP Ca, c,Rn .
P C1a, b,Rn {ψs : a, b → Rn | ψs ,ψ˙s ∈ P Ca, b,Rn }, where ˙ψs
denotes the derivative ofψs . In particular, letP C1ΔP C1a, b,Rn .
H {ht : R → R | ht is a positive integrable function and satisfies supa≤t<btt−τhs dsσ <∞and limt→ ∞tahs ds∞}.
For x ∈ Rn, A ∈ Rn×n,ϕ ∈ Cor ϕ ∈ P C, we define x |x
1|, . . . ,|xn| T,A |aij| n×n,ϕt τ ϕ1t τ, . . . ,ϕnt τ T,ϕt τ ϕt τ, ϕit τ sup−τ≤θ<0{ϕit
θ }.And we introduce the following norm, respectively,
xmax 1≤i≤n
xi, Amax 1≤i≤n
n
j1
aij, ϕτ sup
−τ≤s≤0
ϕs . 2.1
For anyϕ∈P C1, we define the following norm:
ϕ1τ maxϕτ,ϕτ. 2.2
For anM-matrixDdefined in23, we denote
ΩMD 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 ther-dimensional diagonal matrixKdiag{k1, . . . , kr}satisfy
ki >0, i∈S⊂ N∗Δ{1, . . . , r}, ki0, i∈S∗ΔN∗−S. 3.1
C2 LetU−PQ be anM-matrix, whereQ qij r×r ≥0 andP pij r×r satisfies
Theorem 3.1. Assume the conditions C1 and C2 hold. Let L L1, . . . , Lr and ut u1t , . . . , urt T be a solution of the following singular delay differential inequality with the initial conditionsut ∈P Ca−τ, a,Rr :
KDut ≤ht P ut Q ut τL, t∈a, b , 3.3
whereτ >0, a < b≤∞, anduit ∈Ca, b ,R, i∈S,uit ∈P Ca, b ,R , i∈S∗,ht ∈ H. Then
ut ≤dze−λtahsds−PQ −1L, t∈a, b , 3.4
provided that the initial conditions satisfy
ut ≤dze−λtahsds−PQ −1L, t∈a−τ, a, 3.5
whered≥0,z z1, . . . , zr T ∈ΩMU and the positive numberλsatisfies the following inequality:
λKPQeλσz <0, t∈a, b . 3.6
Proof. By the conditionsC2 and the definition ofM-matrix, there is a constant vectorz z1, . . . , zr Tsuch thatPQ z <0,−PQ −1exists and−PQ −1≥0.
By using continuity, we obtain that there must exist a positive constantλsatisfying the inequality3.6, that is,
r
j1
pijqijeλσ
zj<−λkizi, i∈ N∗. 3.7
Denote by
vt v1t , . . . , vrt T
ut PQ −1L, t∈a−τ, b . 3.8
It follows from3.3 and3.5 that
KDvt ≤ht P ut Q ut τL
≤ht P vt Q vt τ, t∈a, b ,
vt ≤dze−λtahsds, t∈a−τ, a.
3.9
In the following, we will prove that for any positive constantε,
vit ≤dε zie−λ
t
Let
℘i∈ N∗| vit > wit for some t∈a, b ,
θiinf
t∈a, b |vit > wit , i∈℘. 3.11
If inequality3.10 is not true, then℘is a nonempty set and there must exist some integer
m∈℘such thatθmmini∈℘{θi} ∈a, b .
Ifm∈S, byvmt ∈Ca, b ,R and the inequality3.5 , we can get
θm> a, vm θm wm θm
, Dvm
θm
≥w˙m
θm
, 3.12
vit ≤wit , t∈a−τ, θm , i∈ N∗, vi
θm
≤wi
θm
, i∈S. 3.13
By usingC2 ,3.3 ,3.7 ,3.12 ,3.13 , andvit τ sup−τ≤θ<0{uitθ }, i∈ N∗, we obtain that
kmDvm
θm
≤hθm r
j1
pmjvj
θm
qmj vj
θm
τ
hθm
pmmvm
θm
j /m, j∈S
pmjvj
θm
j∈S∗
pmjvj
θm
j1
qmj vj
θm
τ
≤hθm
j∈S
pmjdε zje−λ
θm a hsds
r
j1
qmjdε zje−λ
θm−τ a hsds
≤dε hθm r
j1
pmjqmjeλσ
zje−λ
θm a hsds
<−dε λkmzmh
θm
e−λθma hsds.
3.14
Sincem∈S, we havekm>0 byH1 . Then3.14 becomes
Dum
θm
<−dε λzmh
θm
e−λθma rsdsw˙mθm, 3.15
which contradicts the second inequality in3.12 .
Ifm∈S∗, thenkm0 byC1 andvmt ∈P Ca, b ,R . From the inequality3.5 , we can get
θm> a, vm
θm
≥wm
θm
, vi
θm
≤wi
θm
, i∈S,
By usingC2 ,3.3,3.7,3.16 , andvit τ sup−τ≤θ<0{vitθ }, i∈ N∗, we obtain that
0≤
r
j1
pmjvj
θm
j1
qmj vj
θm
τ
j∈S
pmjvj
θm
pmmvmθm
j /m, j∈S∗
pmjvj
θm
r j1
qmj vj
θm
τ
≤dε
j∈S
pmjzjpmmzm
j1
qmjzjeλ
θm θm−τhsds
e−λθma hsds
≤dε
r
j1
pmjzjqmjzjeλσ
e−λθma hsds
<−dε kmzmh
θm
e−λθma hsds 0.
3.17
This is a contradiction. Thus the inequality3.10 holds. Therefore, lettingε → 0 in3.10 , we have
vt ut PQ −1L≤dze−λtahsds, t∈a, b . 3.18
The proof is complete.
Remark 3.2. In order to overcome the difficulty thatut in3.3 may be discontinuous, we introduce the notation uit τ sup−τ≤s<0{uits } which is different from the notation
uit τsup−τ≤s≤0{uits }in7. However, whenuit is continuous int, we have
uit τ sup
−τ≤s<0
uits sup
−τ≤s≤0
uits , i∈ N∗. 3.19
So we can get7, Lemma 1when we chooseKEr,SN∗,ht ≡1 inTheorem 3.1.
Remark 3.3. Suppose thatL L1, . . . , Lr T 0 andht ≡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
˙
xt −Dt xt At Fxt Bt Gxt−τt
Ct Hx˙t−rt Jt , t /tk,
xt Ik
t, xt−, ttk,
where x x1, . . . , xn T ∈ Rn is the neural state vector; Dt diag{d1t , . . . , dnt } > 0, At aijt n×n, Bt bijt n×n, Ct cijt n×n are the interconnection matrices
representing the weighting coefficients of the neurons;Fx f1x1 , . . . , fnxn T, Gx g1x1 , . . . , gnxn T, Hx h1x1 , . . . , hnxn T are activation functions; τt τijt n×n, rt rijt n×n are transmission delays; Jt J1t , . . . , Jnt T denotes the external inputs at time t. Ikt, y I1kt, y , . . . , Inkt, y T represents impulsive perturbations; the fixed moments of timetksatisfytk< tk1, limk→∞tk ∞, k∈N.
The initial condition for4.1 is given by
xt0s
ϕs ∈P C1, t0∈R, −τ ≤s≤0. 4.2
We always assume that for anyϕ∈P C1,4.1 has at least one solution throught 0, ϕ , denoted byxt, t0, ϕ orxtt0, ϕ simplyxt orxtif no confusion should occur .
Definition 4.1. The setS⊂P C1is called a positive invariant set of4.1, if for any initial value
ϕ∈S, we have the solutionxtt0, ϕ ∈Sfort≥t0.
Definition 4.2. The setS⊂P C1is called a global attracting set of4.1 , if for any initial value
ϕ∈P C1, the solutionxtt
0, ϕ converges toSast → ∞. That is,
distxt, S −→0 ast−→∞, 4.3
where distφ, S infψ∈Sdistφ, ψ , distφ, ψ sups∈−τ,0|φs −ψs |,forφ∈P C1.
Throughout this section, we suppose the following.
H1 Dt ∈P CR,Rn , At , Bt , Ct , τt , rt are continuous. Moreover, 0≤τijt ≤τ and 0< rijt ≤τ i, j∈ N .
H2 There exist nonnegative matricesD1 diag{d11, . . . ,d1n},D2 diag{d21, . . . ,d2n},
J J1, . . . ,Jn T,hs ∈ Hand a constantδ >0 such that
D1ht ≤Dt ≤D2ht , 0< ht ≤ 1
δ, Jt
≤Jh t . 4.4
H3 There exist nonnegative matricesA aij n×n,B bij n×n,C cij n×nsuch that
At ≤Ah t , Bt ≤Bh t , Ct ≤Ch t . 4.5
H4 There exist nonnegative matricesF diag{α1, . . . , αn},G diag{β1, . . . , βn},H diag{γ1, . . . , γn}such that for allu∈Rnthe activating functionsF· , G· andH· satisfy
H5 There exists nonnegative matrix Ik Iijk n×n, such that for allu ∈ Rn,i ∈ Nand
k∈N
Ikt, u ≤Iku. 4.7
H6 Denote by
UAF, V BG, W CH,
K
En 0
0 0
diagk1, . . . ,k2n
, L J,JT,
P
−D1U 0
D2U −δEn
Δ
pijt 2n×2n, Q
V W
V W
Δ
qijt 2n×2n,
4.8
and letD−PQ be anM-matrix, and−PQ −1L 1, 2 T ≥0, 1, 2∈
Rn.
H7 There exists a constantνsuch that
lnηk≤ν tk
tk−1
hs ds, k∈N, μ ∞
k1
lnμk<∞, 4.9
whereν < λ, and the scalarλ >0 is determined by the inequality
λKPQeλσz∗<0, 4.10
wherez∗ z1, . . . , z2n T ∈ΩMD , and
ηk, μk≥1, ηkz∗x ≥Ikz∗x, μk1≥Ik1, k∈N, z∗x
z1, . . . , zn T
. 4.11
Theorem 4.3. Assume that H1 –H7 hold. Then S {φ ∈ P C1 | φτ ≤ eμ1} is a global attracting set of 4.1.
Proof. Denote ˙xt yt . Let sgn· be the sign function. For x x1, . . . , xn T, define Sgnx diag{sgnx1 , . . . ,sgnxn }.
Calculating the upper right derivativeDxt along system4.1. From4.1,H2 andH3 we have
D xt ≤ht −D1U xt
V G xt τ
W yt τJ, t∈ tk−1, tk
, t≥σ, k∈N.
On the other hand, from4.1 andH2 –H4 , we have
0≤ht
− 1
ht yt
D
2U xt
V xt
τW yt
τJ
≤ht −δ yt D2U xt V xt
τW yt
τJ
, t∈tk−1, tk , k∈N. 4.13
Let
ut xt , yt T ∈R2n, 4.14
then from4.12 –4.14 andH6 , we have
KD ut ≤ht P ut Q ut τ L, t∈tk−1, tk , k∈N. 4.15
By the conditionsH6 and the definition ofM-matrix, we may choose a vectorz∗ z1, . . . , z2n T ∈ΩMD such that
Dz∗−PQ z∗>0. 4.16
By using continuity, we obtain that there must be a positive constantλsatisfying the inequality4.10 . Letz∗x z1, . . . , zn T and z∗y zn1, . . . , z2n T, thenz∗ z∗x, z∗y T. Since
z∗>0, denote
d 1
min1≤i≤2n
zi
, 4.17
thendz∗≥e2n 1, . . . ,1 T ∈R2n. From the property ofM-cone, we have,dz∗∈ΩMD .
For the initial conditionsxt0s ϕs ,s∈ −τ,0, whereϕ ∈P C1 andt0 ∈ Rno loss of generality, we assumet0≤t1 , andt∈t0−τ, t0, we can get
xt ≤ϕt τe−λ
t
t0hsdsdz∗ x,
yt ≤ϕt τe−λ
t
t0hsdsdz∗ y,
4.18
Then4.18 yield
ut ≤dz∗ϕ1τe−λ
t
t0hsds, t0−τ≤t≤t0. 4.19
LetN∗ {1, . . . ,2n},S {1, . . . , n} NandS∗ {n1, . . . ,2n} N∗−S. Thus, all conditions ofTheorem 3.1are satisfied. ByTheorem 3.1, we have
ut ≤dz∗ϕ1τe−λ
t
t0hsds, t
Suppose that for allm1, . . . , k, the inequalities
ut ≤
m−1
j0
ηjdz∗ϕ1τe−λ
t t0hsds
m−1
j0
μj, tm−1≤t < tm, t≥t0, 4.21
hold, whereη0μ01.
From4.21 ,H5 , andH7 , we can get
xtk
≤Ik x
t−k
≤k j0
ηjdz∗xϕt 1τe
−tk t0hsds
k
j0
μj1.
4.22
Since−PQ −1L, we have
D2U V
1W 2Jδ2. 4.23
On the other hand, it follows fromH7 that
D2U
z∗x V z∗xWz ∗yeλσ< δz∗y. 4.24
Then from4.21 –4.24 , we have
ytk
≤k
j0
ηjdz∗yϕ1τe
−λtkt 0hsds
k
j0
μj2, 4.25
which together with4.22 yields that
utk
≤k
j0
ηjdz∗ϕ1τe−λ
tk t0hsds
k
j0
μj. 4.26
Then, it follows from4.21 and4.26 that
ut ≤
k
j0
ηjdz∗ϕ1τe−λ
t t0hsds
k
j0
μj
k j0
ηjdz∗ϕ1τe−λ
tk
t0hsdse−λ
t tkhsds
k
j0
μj, ∀t∈ tk−τ, tk
.
UsingTheorem 3.1again, we have
ut ≤
k
j0
ηjdz∗ϕ1τe−λ
tk
t0hsdse−λ
t tkhsds
k
j0
μj
k j0
ηjdz∗ϕ1τe−λ
t t0hsds
k
j0
μj, tk≤t < tk1.
4.28
By mathematical induction, we can conclude that
ut ≤
k
j0
ηjdz∗ϕ1τe−λ
t t0hsds
k
j0
μj, tk≤t < tk1, k∈N. 4.29
Noticing thatηk≤eν
tk
tk−1hsds, byH7 , we can use4.29 to conclude that
ut ≤dz∗ϕ1τeν
t
t0hsdse−λ
t
t0hsdseμ dz∗ϕ1τe−λ−ν
t
t0hsdseμ, t
k−1 ≤t < tk, k∈N.
4.30
This implies that the conclusion of the theorem holds.
By usingTheorem 4.3withd 0, we can obtain a positive invariant set of4.1, and the proof is similar to that ofTheorem 4.3.
Theorem 4.4. Assume thatH1 –H7 withIk Enhold. ThenS {φ ∈P C1 |φτ ≤ 1}is a positive invariant set and also a global attracting set of 4.1.
Remark 4.5. Suppose thatcij ≡0, i, j∈ NinH5 , andht ≡1, then we can get Theorems 1 and 2 in9.
Remark 4.6. IfIkt, xt− x ∈ Rn 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 −
7cos2tx1t sinttan
x1
t−τ11t
−1
4costx˙2
t−r12t −1.5 cost,
˙
x2t −
6sin2tx2t −2 costx2
t−τ22t 1
4sinttan
˙
x1
t−r21t
2.5 sint, t /k,
with
x1t I1
t, xt−,
tk, k∈N, x2t I2
t, xt−,
5.2
whereτijt 1/4 |cosij t | ≤ 1/4 Δ τ,rijt 1/4−1/8 |sinij t |,i, j 1,2,
Ikt, x a1kt x1b1kt x2 , a2kt x1b2kt x2 T, k∈N.
The parameters of conditionsH3 –H9 are as follows:
ht δ1, D1diag{7,6}, D2diag{8,7}, σ 1
4, J 1.5,2.5
T, U 0 0 0 0 , V 1 0 0 2
, W 1
4 0 1 1 0 , K ⎛ ⎜ ⎜ ⎝
1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
⎞ ⎟ ⎟
⎠, P
−D1U 0
D2U −δE2 ⎛ ⎜ ⎜ ⎝
−7 0 0 0
0 −6 0 0
8 0 −1 0
0 7 0 −1
⎞ ⎟ ⎟ ⎠, Q V W V W ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 0 0 1
4
0 2 1
4 0
1 0 0 1
4
0 2 1
4 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,
D−PQ
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
6 0 0 −1
4
0 4 −1
4 0
−9 0 1 −1 4 0 −9 −1
4 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . 5.3
It is easy to prove thatDis anM-matrix and
ΩMDz1, z2, z3, z4 T
>0|9z2 1
4z3< z4<24z1,9z1 1
4z4< z3<16z2
. 5.4
Let z∗ 1,1,15,20 T, then z∗ ∈ ΩMD and z∗x 1,1 T. Let λ 0.1 which satisfies the inequality
Now, we discuss the asymptotical behavior of the system5.1 as follows.
i Ifa1kt b2kt 0, b1kt 1/2 e1/5 2k
1sint , a2kt 1/2 e1/5 2k
1−cost , then
Ike1/5 2k.0 1
1 0
. 5.6
Thusηk μke1/5 2k
≥1, lnηk e1/5 2k
≤0.04,ν0.04< λ, andμ1/24. Clearly, all conditions ofTheorem 4.3are satisfied, byTheorem 4.3,S {φ ∈P C1 |φ
τ ≤
e1/24
1≈e1/241.196,1.746 T}is a global attracting set of5.1.
ii Ifa1kt cost, b2kt sint, b1kt a2kt 0, thenIk E2. ByTheorem 4.4,
S{φ∈P C1|φ
τ ≤1≈1.196,1.746 T}is a positive invariant set of5.1 .
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant no. 10671133 and the Scientific Research Fund of Sichuan Provincial Education Department
08ZA044 .
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