Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 540365, 18 pages
doi:10.1155/2010/540365
Research Article
The Existence and Exponential Stability for
Random Impulsive Integrodifferential Equations of
Neutral Type
Huabin Chen, Xiaozhi Zhang, and Yang Zhao
Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China
Correspondence should be addressed to Huabin Chen, chb 00721@126.com
Received 24 March 2010; Revised 9 July 2010; Accepted 28 July 2010
Academic Editor: Claudio Cuevas
Copyright q 2010 Huabin Chen 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.
By applying the Banach fixed point theorem and using an inequality technique, we investigate a
kind of random impulsive integrodifferential equations of neutral type. Some sufficient conditions,
which can guarantee the existence, uniqueness, and exponential stability in mean square for such
systems, are obtained. Compared with the previous works, our method is new and our results can
generalize and improve some existing ones. Finally, an illustrative example is given to show the
effectiveness of the proposed results.
1. Introduction
Since impulsive differential systems have been highly recognized and applied in a
wide spectrum of fields such as mathematical modeling of physical systems, technology,
population and biology, etc., some qualitative properties of the impulsive differential
equations have been investigated by many researchers in recent years, and a lot of valuable
results have been obtained see, e.g., 1–10 and references therein. For the general theory
of impulsive differential systems, the readers can refer to 11, 12. For an impulsive
differential equations, if its impulsive effects are random variable, their solutions are
stochastic processes. It is different from the deterministic impulsive differential equations and
stochastic differential equations. Thus, the random impulsive differential equations are more
realistic than deterministic impulsive systems. The investigation for the random impulsive
differential equations is a new area of research. Recently, the p-moment boundedness,
exponential stability and almost sure stability of random impulsive differential systems
were studied by using the Lyapunov functional method in 13–15, respectively. In 16 Wu
and Duan have investigated the oscillation, stability and boundedness in mean square of
second-order random impulsive differential systems; Wu et al. in 17 studied the existence
2
Advances in Difference Equations
and uniqueness of the solutions to random impulsive differential equations, and in 18
Zhao and Zhang discussed the exponential stability of random impulsive integro-differential
equations by employing the comparison theorem. Very recently, the existence, uniqueness
and stability results of random impulsive semilinear differential equations, the existence and
uniqueness for neutral functional differential equations with random impulses are discussed
by using the Banach fixed point theorem in 19, 20, respectively.
It is well known that the nonlinear impulsive delay differential equations of neutral
type arises widely in scientific fields, such as control theory, bioscience, physics, etc. This
class of equations play an important role in modeling phenomena of the real world. So it is
valuable to discuss the properties of the solutions of these equations. For example, Xu et al.
in 21, have considered the exponential stability of nonlinear impulsive neutral differential
equations with delays by establishing singular impulsive delay differential inequality and
transforming the n-dimensional impulsive neutral delay differential equation into a 2ndimensional singular impulsive delay differential equations; and the results about the global
exponential stability for neutral-type impulsive neural networks are obtained by using the
linear matrix inequality LMI in 9, 10, respectively.
However, most of these studies are in connection with deterministic impulses and
finite delay. And, to the best of author’s knowledge, there is no paper which investigates
the existence, uniqueness and exponential stability in mean square of random impulsive
integrodifferential equation of neutral type. One of the main reason is that the methods to
discuss the exponential stability of deterministic impulsive differential equations of neutral
type and the exponential stability for random differential equations can not be directly
adapted to the case of random impulsive differential equations of neutral type, especially,
random impulsive integrodifferential equations of neutral type. That is, the methods
proposed in 15, 16 are ineffective for the exponential stability in mean square for such
systems. Although the exponential stability of nonlinear impulsive neutral integrodifferential
equations can be derived in 22, the method used in 22 is only suitable for the deterministic
impulses. Besides, the methods introduced to deal with the exponential stability of random
impulsive integrodifferential equations in 18 and study the exponential stability in mean
square of random impulsive differential equations in 19, can not be applied to deal with
our problem since the neutral item arises. So, the technique and the method dealt with the
exponential stability in mean square of random impulsive integrodifferential equations of
neutral type are in need of being developed and explored. Thus, with these aims, we will
make the first attempt to study such problems to close this gap in this paper.
The format of this work is organized as follows. In Section 2, some necessary
definitions, notations and lemmas used in this paper will be introduced. In Section 3, The
existence and uniqueness of random impulsive integrodifferential equations of neutral type
are obtained by using the Banach fixed point theorem. Some sufficient conditions about the
exponential stability in mean square for the solution of such systems are given in Section 4.
Finally, an illustrative example is provided to show the obtained results.
2. Preliminaries
Let | · | denote the Euclidean norm in Rn . If A is a vector or a matrix, its transpose
is denoted
T
by A ; and if A is a matrix, its Frobenius norm is also represented by | · | traceAT A.
Assumed that Ω is a nonempty set and τk is a random variable defined from Ω to Dk 0, dk for all k 1, 2, . . ., where 0 < dk ≤ ∞. Moreover, assumed that τi and τj are independent
with each other as i /
j for i, j 1, 2, . . ..
Advances in Difference Equations
3
Let BCX, Y be the space of bounded and continuous mappings from the topological
space X into Y , and BC1 X, Y be the space of bounded and continuously differentiable
mappings from the topological space X into Y . In particular, Let BC BC−∞, 0, Rn and
BC1 BC1 −∞, 0, Rn . PCJ, Rn {φ : J → Rn |φs is bounded and almost surely
continuous for all but at most countable points s ∈ J and at these points s ∈ J, φs
and
φs− exist, φs φs
}, where J ⊂ R is an interval, φs
and φs− denote the right-hand
and left-hand limits of the function φs, respectively. Especially, let PC PC−∞, 0, Rn .
PC1 J, Rn {φ : J → Rn |φs is bounded and almost surely continuously differentiable
for all but at most countable points s ∈ J and at these points s ∈ J, φs
and φs− ,
φs φs
, φ s φ s
}, where φ s denote the derivative of φs. Especially, let
PC1 PC1 −∞, 0, Rn .
For φ ∈ PC1 , we introduce the following norm:
φ∞ max
sup φθ , sup φ θ .
−∞<θ≤0
2.1
−∞<θ≤0
In this paper, we consider the following random impulsive integrodifferential
equations of neutral type:
x t Axt Dx t − r f1 t, xt − ht 0
xξk bk τk x ξk− ,
−∞
f2 θ, xt θdθ,
k 1, 2, . . . ,
xt0 ϕ ∈ PC1 ,
t/
ξk , t ≥ 0,
2.2
2.3
2.4
where A, D are two matrices of dimension n × n; f1 : 0, ∞ × Rn → Rn and f2 : −∞, 0 ×
Rn → Rn are two appropriate functions; bk : Dk → Rn×n is a matrix valued functions for
each k 1, 2, . . .; assume that t0 ∈ 0, ∞ is an arbitrary real number, ξ0 t0 and ξk ξk−1 τk
for k 1, 2, . . .; obviously, t0 ξ0 < ξ1 < ξ2 < · · · < ξk < · · · ; xξk− limt → ξk −0 xt; h :
0, ∞ → 0, ρ ρ > 0 is a bounded and continuous function and τ max{r, ρ} r > 0.
xt : xt s xt s for all s ∈ −∞, 0. Let us denote by {Bt , t ≥ 0} the simple counting
process generated by {ξn }, that is, {Bt ≥ n} {ξn ≤ t}, and present It the σ-algebra generated
by {Bt , t ≥ 0}. Then, Ω, {It }, P is a probability space.
Firstly, define the space B consisting of PC1 −∞, T , Rn T > t0 -valued stochastic
process ϕ : −∞, T → Rn with the norm
2
ϕ E sup ϕθ2 .
−∞<θ≤T
2.5
It is easily shown that the space B, · is a completed space.
Definition 2.1. A function x ∈ B is said to be a solution of 2.2–2.4 if x satisfies 2.2 and
conditions 2.3 and 2.4.
Definition 2.2. The fundamental solution matrix {Φt expAt, t ≥ 0} of the equation
x t Axt is said to be exponentially stable if there exist two positive numbers M ≥ 1 and
a > 0 such that |Φt| ≤ Me−at , for all t ≥ 0.
4
Advances in Difference Equations
Definition 2.3. The solution of system 2.2 with conditions 2.3 and 2.4 is said to be
exponentially stable in mean square, if there exist two positive constants C1 > 0 and λ > 0
such that
E|xt|2 ≤ C1 e−λt ,
2.6
t ≥ 0.
Lemma 2.4 see 23. For any two real positive numbers a, b > 0, then
a b2 ≤ ν−1 a2 1 − ν−1 b2 ,
2.7
where ν ∈ 0, 1.
Lemma 2.5 see 23. Let u, ψ, and χ be three real continuous functions defined on a, b and
χt ≥ 0, for t ∈ a, b, and assumed that on a, b, one has the inequality
ut ≤ ψt t
2.8
χsusds.
a
If ψ is differentiable, then
t
ut ≤ ψa exp
χsds
a
t
t
exp
a
χrdr ψ sds,
2.9
s
for all t ∈ a, b.
In order to obtain our main results, we need the following hypotheses.
H1 The function f1 satisfies the Lipschitz condition: there exists a positive constant
L1 > 0 such that
f1 t, x − f1 t, y ≤ L1 x − y,
2.10
for x, y ∈ Rn , t ∈ 0, T , and f1 t, 0 0.
H2 The function f2 satisfies the following condition: there also exist a positive constant
L2 > 0 and a function k : −∞, 0 → 0, ∞ with two important properties,
0
0
ktdt 1 and −∞ kte−lt dt < ∞ l > 0, such that
−∞
f2 t, x − f2 t, y ≤ L2 ktx − y,
2.11
for x, y ∈ Rn , t ∈ 0, T , and f2 t, 0 0.
H3 Emaxi,k { kji |bj τj |2 } is uniformly bounded. That is, there exists a positive
constant L > 0 such that
⎧
⎫⎞
⎛
k ⎨
2 ⎬
bj τj ⎠ ≤ L,
2.12
E⎝max
⎭
i,k ⎩
ji
for all τj ∈ Dj and j 1, 2, . . ..
H4 κ max{L, 1}|D| ∈ 0, 1.
Advances in Difference Equations
5
3. Existence and Uniqueness
In this section, to make this paper self-contained, we study the existence and uniqueness
for the solution to system 2.2 with conditions 2.3 and 2.4 by using the Picard iterative
method under conditions H1 –H4 . In order to prove our main results, we firstly need the
following auxiliary result.
Lemma 3.1. Let f1 : 0, ∞ × Rn → Rn and f2 : −∞, 0 × Rn → Rn be two continuous functions.
Then, x is the unique solution of the random impulsive integrodifferential equations of neutral type:
x t Axt Dx t − r f1 t, xt − ht xξk bk τk x ξk− ,
0
−∞
f2 θ, xt θdθ,
t
/ ξk , t ≥ 0,
3.1
k 1, 2, . . . ,
xt0 ϕ ∈ PC1 ,
if and only if x is a solution of impulsive integrodifferential equations:
i xt0 θ ϕθ, θ ∈ −∞, 0,
ii
⎡
k
k
∞ k ξi
⎣ bi τi Φt − t0 x0 xt bj τj
Φt − sDdxs − r
k0
t
Φt − sDdxs − r ξk
×
ξi−1
i1 ji
i1
k
k bj τj
i1 ji
ξi
Φt − sf1 s, xs − hsds
ξi−1
t
k
k ξi
Φt − sf1 s, xs − hsds bj τj
Φt − s
ξk
×
i1 ji
0
−∞
t
ξk
3.2
ξi−1
f2 θ, xs θdθds
Φt − s
0
−∞
f2 θ, xs θdθds Iξk ,ξk
1 t,
for all t ∈ t0 , T , where njm · 1 as m > n, kji bj τj bk τk bk−1 τk−1 · · · bi τi ,
and IΩ · denotes the index function, that is,
IΩ t ⎧
⎨1,
if t ∈ Ω ,
⎩0,
if t /
∈ Ω .
3.3
6
Advances in Difference Equations
Proof. The approach of the proof is very similar to those in 17, 19, 20. Here, we omit it.
Theorem 3.2. Provided that conditions (H1 )–(H4 ) hold, then the system 2.2 with the conditions
2.3 and 2.4 has a unique solution on B.
Proof. Define the iterative sequence {xn t} t ∈ −∞, T , n 0, 1, 2, . . . as follows:
x t 0
k
∞ k0
i1
k0
i1
bi τi Φt − t0 x0 Iξk ,ξk
1 t,
t ∈ t0 , T ,
⎡
k
k
∞ k ξi
n
⎣
x t bi τi Φt − t0 x0 bj τj
Φt − sDdxn s − r
ξi−1
i1 ji
k
k ξi
bj τj
Φt − sf1 s, xn−1 s − hs ds
i1 ji
t
ξi−1
Φt − sf1 s, xn−1 s − hs ds
3.4
ξk
0
k
k ξi
bj τj
Φt − s
f2 θ, xn−1 s θ dθds
i1 ji
t
ξi−1
Φt − sDdx s − r n
ξk
t
Φt − s
0
−∞
ξk
× Iξk ,ξk
1 t,
xtn0 θ ϕθ,
−∞
f2 θ, x
n−1
s θ dθds
t ∈ t0 , T , n 1, 2, . . . ,
θ ∈ −∞, 0, n 0, 1, 2, . . . .
Thus, due to Lemma 2.4, it follows that
2
n
1
x t − xn t
⎡
k k
!
∞ ξi
⎣
bj τj
Φt − sDd xn s − r − xn−1 s − r
k0 i1 ji
ξi−1
k
k !
ξi
bj τj
Φt − s f1 s, xn s − hs − f1 s, xn−1 s − hs ds
i1 ji
ξi−1
0
k
k ξi
bj τj
Φt − s
i1 ji
t
ξk
ξi−1
−∞
!
f2 θ, xn s θ − f2 θ, xn−1 s θ dθds
Φt − sDd xn s − r − xn−1 s − r
!
Advances in Difference Equations
t
7
!
Φt − s f1 s, xn s − hs − f1 s, xn−1 s − hs ds
ξk
2
Φt − s
f2 θ, xn s θ − f2 θ, xn−1 s θdθds Iξk ,ξk
1 t
ξk
−∞
t
0
⎧
⎧
⎫ ⎫
k
⎨
⎨
⎬ ⎬
1
2
≤ max max
|bj τj |2 , 1 |D|2 |xn
1 t − r − xn t − r|
⎩ i,k ⎩ ji
⎭ ⎭
κ
⎧
⎫
k ⎨
2 ⎬
3
bj τj , 1
max max
⎩
⎭
1−κ
ji
2
2
× |D| |A|
t
2
2
n
1
n
Φt − sx s − r − x s − r ds
t0
⎧
⎨
⎧
⎫ ⎫
k ⎨
2 ⎬ ⎬
3
b τ , 1
max max
⎩
⎩ ji j j ⎭ ⎭
1−κ
×
L21
t
2
2
n
n−1
Φt − sx s − hs − x s − hs ds
t0
⎧
⎫ ⎫
k ⎨
⎬ ⎬
3
bj τj 2 , 1
max max
⎩ ji
⎩
⎭ ⎭
1−κ
⎧
⎨
×
L22
t
Φt − s
0
−∞
t0
x s θ − x
n
n−1
!
2
s θ dθds| ds
2
⎧
⎧
⎫ ⎫
k
⎨
⎨
⎬ ⎬
1
2
≤ max max
|bj τj |2 , 1 |D|2 sup |xn
1 s − xn s|
⎩ i,k ⎩ ji
⎭ ⎭
κ
−∞<s≤t
⎧
⎧
⎫ ⎫
k ⎨
⎨
2 ⎬ ⎬
3
b τ , 1
max max
⎩ ji j j ⎭ ⎭
⎩
a1 − κ
2
2
× |D| |A| M
2
t
2
sup |xn
1 u − xn u| ds
t0 −∞<u≤s
⎧
⎧
⎫ ⎫
k ⎨
⎨
2 ⎬ ⎬
3
b τ , 1
max max
⎩
⎩ ji j j ⎭ ⎭
a1 − κ
×M
2
L21
L22
t
2
sup xn
1 θ − xn θ ds.
t0 −∞<θ≤s
3.5
8
Advances in Difference Equations
From condition H3 , we have
E
2
sup xn
1 s−xn s
−∞<s≤t
≤
2
n
1
n
sup x θ−x θ ds
3M2 D|2 A|2 max{1, L} t
E
a1 − κ2
t0
−∞<θ≤s
2
3M2 L21 L22 max{1, L} t
n
n−1
E sup x θ−x θ ds.
a1 − κ2
t0
−∞<θ≤s
3.6
In view of Lemma 2.5, it yields that
E
2
sup xn
1 s − xn s
−∞<s≤t
≤ Λ1
t
E
t0
2
n
n−1
sup x θ − x θ ds,
−∞<θ≤s
3.7
where Λ1 3M2 |D|2 |A|2 max{1, L}/a1 − κ2 exp3M2 L21 L22 max{1, L}/a1 − κ2 T − t0 .
Furthermore,
E
2
sup x1 s − x0 s
−∞<s≤t
≤
4κ2 M2 Eϕ2∞
t
4 max{L, 1}M2 L21 L22
E
1 − κ2 a
t0
4 max{L, 1}|D|2 |A|2 M2 t
1 2
E
sup
u
ds.
x
1 − κ2 a
t0 −∞<u≤s
1 − κ2
0 2
sup x u ds
−∞<u≤s
3.8
By 3.4, we can obtain that
E
2
sup x1 s
≤
−∞<s≤t
5LM2 Eϕ2∞ 5 max{L, 1}M2 |D|2 Eϕ2∞
1 − κ2
E
2
sup x0 s
−∞<s≤t
5 max{L, 1}M2 |D|2 |A|2
2
2
−∞<u≤s
3.9
2
E sup |x0 u| ds,
1 − κ a
t0 −∞<u≤s
2
0 2
sup ϕθ
E sup x s
−∞<θ≤0
≤ 1 LM2 φ∞
Λ2 .
2
E sup |x1 u| ds
1 − κ a
t0
t
5 max{L, 1}M2 L21 L22
≤E
t
0≤s≤t
3.10
Advances in Difference Equations
9
From the Gronwall inequality, 3.9 implies that
E
2
sup x1 t
−∞<t≤T
≤ Λ3 expΛ4 T − t0 ,
3.11
where Λ3 5LM2 Eϕ2∞ 5 max{L, 1}M2 |D|2 Eϕ2∞ /1 − κ2 5 max{L, 1}M2 L21 L22 Λ2 T − t0 /1 − κ2 a and Λ4 5 max{L, 1}M2 |D|2 |A|2 /1 − κ2 a.
From 3.8 and 3.11, we have
E
2
sup x1 s − x0 s
−∞<s≤t
≤ Λ5 ,
3.12
for all t ∈ 0, T , where
Λ5 4κ2 M2 Eϕ2∞
1 − κ2
4 max{L, 1}M2 L21 L2
1 − κ2 a
4 max{L, 1}|D|2 |A|2 M2
1 − κ2 a
Λ2 T − t0 3.13
Λ3 expΛ4 T − t0 T − t0 .
From 3.4, it follows that
2
2
x t − x1 t
⎧
⎧
⎫ ⎫
k ⎨
⎨
2
2 ⎬ ⎬ 2
1
bj τj , 1 |D| sup x2 s − x1 s
≤ max max
⎩ i,k ⎩ ji
⎭ ⎭
κ
−∞<s≤t
⎧
⎧
⎫ ⎫
k ⎨
⎨
2
2 ⎬ ⎬ 2 2 2 t
3
bj τj , 1 |D| |A| M
max max
sup x1 u − x0 u ds
⎩ ji
⎩
⎭ ⎭
a1 − κ
t0 −∞<u≤s
⎧
⎧
⎫ ⎫
k ⎨
⎨
⎬ ⎬
2
t
3
bj τj 2 , 1 M2 L2 L2
max max
sup x1 θ−x0 θ ds.
2
1
⎩
⎩ ji
⎭ ⎭
a1 − κ
t0 −∞<θ≤s
3.14
By virtue of condition H3 and Lemma 2.5,
E
2
sup x2 t − x1 t
−∞<s≤t
≤ Λ1 Λ5 t − t0 .
3.15
Now, for all n ≥ 0 and t ∈ 0, T , we claim that
E
2
sup xn
1 s − xn s
−∞<s≤t
≤ Λ5
Λ1 t − t0 n
.
n!
3.16
10
Advances in Difference Equations
We will show 3.16 by mathematical induction. From 3.12, it is easily seen that 3.16 holds
as n 0. Under the inductive assumption that 3.16 holds for some n ≥ 1. We will prove that
3.16 still holds when n 1. Notice that
⎧
⎧
⎫ ⎫
k ⎨
⎨
⎬ ⎬
2 1
2
2
n
2
bj τj , 1 |D|2 sup xn
2 s − xn
1 s
x t − xn
1 t ≤ max max
⎩ i,k ⎩ ji
⎭ ⎭
κ
−∞<s≤t
⎧
⎫ ⎫
k ⎨
⎬ ⎬
3
bj τj 2 , 1 |D|2 |A|2 M2
max max
⎩ ji
⎩
⎭ ⎭
a1 − κ
⎧
⎨
×
t
2
sup xn
2 θ − xn
1 θ ds
t0 −∞<θ≤s
⎧
⎫ ⎫
k ⎨
⎬ ⎬
3
bj τj 2 , 1 M2 L2 L2
max max
2
1
⎩ ji
⎩
⎭ ⎭
a1 − κ
×
t
⎧
⎨
2
sup xn
2 θ − xn
1 θ ds.
t0 −∞<θ≤s
3.17
From condition H3 , we have
E
2
sup xn
2 s − xn
1 s
−∞<s≤t
≤
3M2 |D|2 |A|2 max{1, L}
t
E
a1 − κ2
t0
3M2 L21 L22 max{1, L} t
a1 − κ2
2
n
2
n
1
sup x θ − x θ ds
−∞<θ≤s
2
n
1
n
sup x θ − x θ ds.
E
3.18
−∞<θ≤s
t0
In view of Lemma 2.5 and 3.16, it yields that
E
2
sup xn
2 s − xn
1 s
−∞<s≤t
≤ Λ1
t
E
t0
≤
Λ1 Λ5
n!
≤ Λ5
t
2
n
1
n
sup x θ − x θ ds
−∞<θ≤s
Λ1 s − t0 n ds
t0
Λ1 t − t0 n
1
,
n 1!
t ∈ t0 , T .
That is, 3.16 holds for n 1. Hence, by induction, 3.16 holds for all n ≥ 0.
3.19
Advances in Difference Equations
11
For any m > n ≥ 1, it follows that
1/2
x − x E
m
sup |x t − x t|
n
m
n
2
−∞<t≤T
1/2
∞
2
k
1
k
E sup x t − x t
≤
3.20
−∞<t≤T
kn
1/2
∞
Λ1 T − t0 k
Λ5
≤
−→ 0,
k!
kn
as n → ∞. Thus, {xn t}n≥0 t ∈ −∞, T is a Cauchy sequence in Banach space B. Denote
the limit by x ∈ B. Now, letting n → ∞ in both sides of 3.4, we obtain the existence for
the solution of system 2.2 with conditions 2.3 and 2.4.
Uniqueness. Let x, y ∈ B be two solutions of system 2.2 with conditions 2.3 and
2.4. By the same ways as above, we can yield that
E
2
sup xs − ys
−∞<t≤T
E
2
sup xt − yt
t0 ≤t≤T
T 2
dt.
E sup xs − ys
≤ Λ1
t0
3.21
−∞<s≤t
Applying the Gronwall inequality into 3.21, it follows that
E
2
sup xt − yt
−∞<t≤T
0.
3.22
That is, x − y 0. So, the uniqueness is also proved. The proof of this theorem is completed.
4. Exponential Stability
In this section, the exponential stability in mean square for system 2.2 with initial conditions
2.3 and 2.4 is shown by using an integral inequality.
Theorem 4.1. Supposed that the conditions of Theorem 3.2 holds, then the solution of the system
2.2 with conditions 2.3 and 2.4 is exponential stable in mean square if the inequality
4M2 max{1, L} |D|2 |A|2 L21 L22
1 − κ2 a
holds.
<a≤l
4.1
12
Advances in Difference Equations
Proof. From 3.2 and Lemma 2.4, we derive that
bj τj M2 ϕ2 e−at
|xt| ≤ max
∞
2
i,k
i,k
⎧
⎨
⎫ ∞ t
2 ⎬
−at−s
b τ
max max
,1
Me
|D|dxs − r
⎩ i,k ji j j
⎭ k0 t0
⎧
⎫ ⎫
k ⎨
2 ⎬ ⎬
b τ , 1
× Iξk ,ξk
1 t max max
⎩ i,k ⎩ ji j j ⎭ ⎭
⎧
⎨
×
∞ t
Me
f1 s, xs − hs ds Iξk ,ξk
1 t
−at−s t0
k0
⎧
⎫ ⎫
k ⎨
2 ⎬ ⎬
b τ , 1
max max
⎩ i,k ⎩ ji j j ⎭ ⎭
⎧
⎨
2
0
∞ t
−at−s
f2 θ, xs θdθds Iξ ,ξ t
×
Me
k k
1
−∞
t0
k0
⎡
# $ $
#
k 2 , 1 M2 |D|2 ϕ2
∞ 8 max maxi,k
ji bj τj
∞
⎢
≤
⎣
1−κ
k0
8maxi,k
⎤
# $
k 2 M2 ϕ2
b
τ
j
j
j1
∞⎥
⎦
1−κ
× Iξk ,ξk
1 te−at−t0 ×
#
$ $
#
2
k
,1
|b
τ
|
max maxi,k
j
j
ji
κ
|D|2
#
$ $
#
2
2
2
k
2
∞ t
4 max maxi,k
ji |bj τj | , 1 M |D| |A| 1 − κa
k0
#
$ $
#
2
k
2 2 ∞ t
4 max maxi,k
ji |bj τj | , 1 M L1 1 − κa
k0
∞
|xt − r|2 Iξk ,ξk
1 t
k0
e
t0
−at−s
2
|xs − r| ds Iξk ,ξk
1 t
e
−at−s
t0
2
|xs − hs| ds Iξk ,ξk
1 t
#
$ $
#
2
k
2 2
4 max maxi,k
ji |bj τj | , 1 M L2
1 − κa
∞ t
k0
t0
e
−at−s
0
−∞
2
kθ|xs θ| dθds Iξk ,ξk
1 t.
4.2
Advances in Difference Equations
13
Thus, it follows that
⎤
8 max{L, 1}M2 |D|2 L Eϕ2∞
⎥ −at−t0 ⎢
E|xt|2 ≤ ⎣
κE|xt − r|2
⎦e
1−κ
⎡
4 max{L, 1}M2 |D|2 |A|2
1 − κa
4 max{L, 1}M2 L21
1 − κa
t
t
e−at−s E|xs − r|2 ds
t0
e−at−s E|xs − hs|2 ds
t0
t
0
4 max{L, 1}M2 L22
−at−s
e
kθE|xs θ|2 dθds
1 − κa
t0
−∞
8 max{L, 1}M2 |D|2 Eϕ2∞ 8LM2 Eϕ2∞ −at−t0 ≤
e
κ sup E|xt θ|2
1−κ
1−κ
θ∈−τ,0
4 max{L, 1}M2 |D|2 |A|2 L21 t
e−at−s sup E|xs θ|2 ds
1 − κa
t0
θ∈−τ,0
4 max{L, 1}M2 L22
1 − κa
t
e
−at−s
t0
0
−∞
4.3
kθE|xs θ|2 dθds,
and it is easily seen that there exists a positive number M1 > 0 such that E|xt|2 ≤ M1 e−at−t0 ,
for all t ∈ −∞, t0 .
For the convenience, setting λ1 8 max{L, 1}M2 |D|2 LEϕ2∞ /1 − κ, λ2 4 max{L, 1}M2 |D|2 |A|2 L21 /1 − κa, and λ3 4 max{L, 1}M2 L22 /1 − κa, it implies from
4.3 that
E|xt|2 ≤
⎧
t
⎪
2
⎪
−at−t0 ⎪
e
κ
sup
E|xt
θ|
λ
e−at−s sup E|xs θ|2 ds
λ
2
⎪ 1
⎪
⎪
t0
θ∈−τ,0
θ∈−τ,0
⎨
t
⎪
λ3 e−at−s
⎪
⎪
⎪
t0
⎪
⎪
⎩
λ1 e−at−t0 ,
0
−∞
kθE|xs θ|2 dθds,
t ≥ t0
t ∈ −∞, t0 .
4.4
0
From 4.4, letting Fλ κeλr λ2 /a − λeλτ λ3 /a − λ −∞ kθe−λθ dθ − 1,
F0Fa− < 0 holds. That is, there exists a positive constant μ ∈ 0, a such that Fμ 0.
For any ε > 0 and letting
⎧
⎨
Mε max λ1 ε
⎩
λ2 eμτ
⎫
⎬
a−μ
ε
> 0.
,
λ
1
0
⎭
λ3 −∞ kθe−μθ dθ
4.5
14
Advances in Difference Equations
Now, in order to show our main result, we only claim that 4.4 implies
E|xt|2 ≤ Mε e−μt−t0 ,
4.6
t ∈ −∞, ∞.
It is easily seen that 4.6 holds for any t ∈ −∞, t0 . Assume, for the sake of contradiction,
that there exists a t1 > t0 and
E|xt|2 < Mε e−μt−t0 ,
E|xt1 |2 Mε e−μt1 −t0 .
t ∈ −∞, t1 ,
4.7
Then, it, from 4.4, implies that
E|xt1 |2 ≤ λ1 e−at1 −t0 κMε eμτ e−μt1 −t0 λ2 Mε
t1
e−at1 −t0 −s sup e−μs
θ ds
t0
λ3 Mε
t1
e−at1 −t0 −s
t0
0
−∞
θ∈−τ,0
kθe−μs
θ dθds
0
λ3
eμτ
−μθ
≤ λ1 − Mε λ2
kθe dθ e−at1 −t0 a − μ a − μ −∞
0
λ3
λ2 eμτ
μr
−μθ
Mε κe kθe dθ e−μt1 −t0 .
a − μ a − μ −∞
4.8
From the definitions of μ and Mε , we have
κeμr 0
λ3
λ2 eμτ
a−μ a−μ
0
−∞
kθe−μθ dθ 1,
λ3
eμτ
kθe−μθ dθ
a − μ a − μ −∞
0
a−μ
λ3
eμτ
−μθ
kθe dθ ε λ1 ≤ λ1 − λ2
0
μτ
a − μ a − μ −∞
λ2 e λ3 −∞ kθe−μθ dθ
λ1 − Mε λ2
4.9
< 0.
Thus, 4.8 yields that
E|xt1 |2 < Mε e−μt1 −t0 ,
4.10
which contradicts 4.7, that is, 4.4 holds.
As ε > 0 is arbitrarily small, in view of 4.6, it follows that
E|xt|2 ≤ Me−μt−t0 ,
t ≥ t0 ,
4.11
Advances in Difference Equations
15
where M max{λ1 a − μ/λ2 eμτ λ3
That is,
0
−∞
kθe−μθ dθ, λ1 } > 0.
E|xt|2 ≤ M e−αt−t0 ,
4.12
t ≥ t0 ,
where α ∈ 0, a and
M max
⎧
⎨ 8 max{L, 1}M2 |D|2 L Eϕ2∞ a − μ
⎩
1−κ
⎛
⎜
×⎝
4 max{L, 1}M2 |D|2 |A|2 L21 eμτ
1 − κa
4 max{L, 1}M2 L22
1 − κa
⎫
8 max{L, 1}M2 |D|2 L Eϕ2∞ a − μ ⎬
⎭
1−κ
0
−∞
⎞−1
⎟
kθe−μθ dθ⎠ ,
> 0.
4.13
The proof is completed.
In particular, when D ≡ 0, τ ≡ 0, and f2 ≡ 0, system 2.2 is turned into the following
form:
x t Axt f1 t, xt, t /
ξk , t ≥ 0,
−
xξk bk τk x ξk , k 1, 2, . . . ,
4.14
xt0 x0 .
Remark 4.2. Obviously, we can also give the existence, uniqueness, and exponential stability
in mean square for the solution of system 4.14 by employing the Picard iterative method
and a similar impulsive-integral inequality proposed in 24. So, the following corollary can
be given as follows.
Corollary 4.3. Under conditions (H1 ) and (H3 ), the existence, uniqueness, and exponential stability
in mean square for the solution of system 4.14 can be obtained only if the inequality
max{1, L}M2 L21 < a2
4.15
holds.
Proof. The proofs of this corollary are very similar to those of Theorems 3.2 and 4.1. So, we
omit them.
Remark 4.4. Recently, in 19, Anguraj and Vinodkumar have derived Corollary 4.3 by using
the fixed point theorem. Obviously, our results are more general than those obtained in 19.
Thus, we can generalize and improve the results in 19.
16
Advances in Difference Equations
5. An Illustrative Example
Let τk be a random variable defined in Dk ≡ 0, dk for all k 1, 2, . . ., where 0 < dk ≤ ∞.
j for i, j 1, 2, . . ..
Furthermore, assume that τi and τj are independent of each other as i /
Consider the random impulsive integrodifferential equations of neutral type:
1 dx2 t − r
dx1 t
1 dx1 t − r
−0.2x1 t √
− √
δ1 costx1 t − ht
dt
dt
dt
2 100
2 100
0
2
δ3
−θ−1/2 eπ θ x1 s θds,
−∞
1 dx2 t − r
dx2 t
1 dx1 t − r
−0.1x1 t − 0.1x2 t √
√
δ2 sintx2 t − ht
dt
dt
dt
3 100
4 100
0
2
δ4
−θ−1/2 eπ θ x2 s θds,
−∞
5.1
as ξk−1 ≤ t < ξk , k 1, 2, . . . , and for all k 1, 2, 3, . . .,
x1 ξk ckτk · x1 ξk− ,
x2 ξk ckτk · x2 ξk− ,
5.2
where ξ0 t0 and ξk ξk−1 τk for all k 1, 2, . . ., ck is a function of k. Denote that
c maxk {ck} and there is ρ : 0 ≤ ρ < 1 such that Ecτk2 ≤ ρ for all k 1, 2, . . .. and
|δi | ≤ L1 i 1, 2 and |δi | ≤ L2 i 3, 4. And |D| 0.0846, |A| 0.2449, κ 0.0846.
The corresponding linear homogeneous equations:
⎞
dx1 t
,
+
,+
⎜ dt ⎟
x1 t
⎟ −0.2 0
⎜
.
⎝ dx t ⎠
x2 t
−0.1 −0.1
2
dt
⎛
5.3
And, the fundamental solution matrix of system 5.3 can be given by
+
Φt ,
e−0.2t
0
.
e−0.2t − e−0.1t e−0.1t
5.4
Then, it is easily obtain that
|Φt| ≤ Me−0.1t ,
t ≥ 0,
5.5
√
where M 2 > 1 and a 0.1 > 0, and it is easily seen that we can derive that the functions f1
and f2 satisfy conditions H1 and H2 with the Lipschitz coefficients L1 and L2 , respectively.
On the other hand, hypothesis H3 and H4 are easily verified. In view of Theorems 3.2 and
Advances in Difference Equations
17
4.1, the existence, uniqueness, and exponential stability in mean square of system 5.1 with
5.2 are obtained if the constants L1 and L2 satisfy the following inequality:
L21 L22 < 0.02771.
5.6
Acknowledgment
The authors would like to thank the referee and the editor for their careful comments and
valuable suggestions on this work.
References
1 A. Anokhin, L. Berezansky, and E. Braverman, “Exponential stability of linear delay impulsive
differential equations,” Journal of Mathematical Analysis and Applications, vol. 193, no. 3, pp. 923–941,
1995.
2 Y. Hino, S. Murakani, and T. Naito, Functional Differential Equations with Infinite Delay, vol. 1473 of
Lecture Notes in Mathematics, Springer, Berlin, Germany, 1991.
3 J. Hale and S. M. Verguyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied
Mathematical Science, Springer, New York, NY, USA, 1993.
4 J. Henderson and A. Ouadhab, “Local and global existence and uniqueness results for second and
higher order impulsive functional differential equations with infinite delay,” The Australian Journal of
Mathematical Analysis and Applications, vol. 4, pp. 1–26, 2007.
5 A. Korzeniowski and G. S. Ladde, “Modeling hybrid network dynamics under random perturbations,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 2, pp. 143–149, 2009.
6 X. Liu, X. Shen, and Y. Zhang, “A comparison principle and stability for large-scale impulsive delay
differential systems,” The ANZIAM Journal, vol. 47, no. 2, pp. 203–235, 2005.
7 X. Liu, “Stability results for impulsive differential systems with applications to population growth
models,” Dynamics & Stability of Systems. Series A, vol. 9, no. 2, pp. 163–174, 1994.
8 X. Liu and X. Liao, “Comparison method and robust stability of large-scale dynamic systems,”
Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 11, no. 2-3, pp. 413–430, 2004.
9 R. Rakkiyappan, P. Balasubramaniam, and J. Cao, “Global exponential stability results for neutraltype impulsive neural networks,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 122–
130, 2010.
10 R. Samidurai, S. M. Anthoni, and K. Balachandran, “Global exponential stability of neutral-type
impulsive neural networks with discrete and distributed delays,” Nonlinear Analysis: Hybrid Systems,
vol. 4, no. 1, pp. 103–112, 2010.
11 V. Lakshmikantham, D. D. Baı̆nov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6
of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
12 A. M. Samoı̆lenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series
on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995.
13 S. Wu and X. Meng, “Boundedness of nonlinear differential systems with impulsive effect on random
moments,” Acta Mathematicae Applicatae Sinica, vol. 20, no. 1, pp. 147–154, 2004.
14 S. Wu, X. Guo, and Y. Zhou, “p-moment stability of functional differential equations with random
impulses,” Computers & Mathematics with Applications, vol. 52, no. 12, pp. 1683–1694, 2006.
15 S. Wu, X. Guo, and R. Zhai, “Almost sure stability of functional differential equations with random
impulses,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 15, no. 3, pp. 403–415,
2008.
16 S. Wu and Y. Duan, “Oscillation, stability, and boundedness of second-order differential systems with
random impulses,” Computers & Mathematics with Applications, vol. 49, no. 9-10, pp. 1375–1386, 2005.
17 S. Wu, X. Guo, and S. Lin, “Existence and uniqueness of solutions to random impulses,” Acta
Mathematicae Applicatae Sinica, vol. 22, pp. 595–600, 2004.
18 D. Zhao and L. Zhang, “Exponential asymptotic stability of nonlinear Volterra equations with random
impulses,” Applied Mathematics and Computation, vol. 193, no. 1, pp. 18–25, 2007.
19 A. Anguraj and A. Vinodkumar, “Existence, uniqueness and stability results of random impulsive
semilinear differential systems,” Nonlinear Analysis: Hybrid Systems, vol. 4, pp. 475–483, 2010.
18
Advances in Difference Equations
20 A. Anguraj and A. Vinodkumar, “Existence and uniqueness of neutral functional differential
equations with random impulses,” International Journal of Nonlinear Science, vol. 8, no. 4, pp. 412–418,
2009.
21 D. Xu, Z. Yang, and Z. Yang, “Exponential stability of nonlinear impulsive neutral differential
equations with delays,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1426–
1439, 2007.
22 L. Xu and D. Xu, “Exponential stability of nonlinear impulsive neutral integro-differential equations,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2910–2923, 2008.
23 W. Walter, Differential and Integral Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete,
Band 55, Springer, New York, NY, USA, 1970.
24 H. Chen, “Impulsive-integral inequality and exponential stability for stochastic partial differential
equations with delays,” Statistics & Probability Letters, vol. 80, no. 1, pp. 50–56, 2010.