Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 414320, 14 pages
doi:10.1155/2012/414320
Research Article
Mean Square Almost Periodic
Solutions for Impulsive Stochastic Differential
Equations with Delays
Ruojun Zhang,1 Nan Ding,2, 3 and Linshan Wang1
1
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
College of Information Engineering, Ocean University of China, Qingdao 266100, China
3
Department of Mathematics, Chongqing Three Gorges University, Chongqing 404100, China
2
Correspondence should be addressed to Ruojun Zhang, zhangru1626@sina.com
Received 28 December 2011; Revised 15 March 2012; Accepted 15 March 2012
Academic Editor: Zhiwei Gao
Copyright q 2012 Ruojun Zhang 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.
We establish a result on existence and uniqueness on mean square almost periodic solutions for a
class of impulsive stochastic differential equations with delays, which extends some earlier works
reported in the literature.
1. Introduction
Impulsive effects widely exist in many evolution processes of real-life phenomena in which
states are changed abruptly at certain moments of time, involving such areas as population
dynamics and automatic control 1–3. Because delay is ubiquitous in the dynamical system,
impulsive differential equations with delays have received much interesting in recent years,
intensively researched, some important results are obtained 4–9. And almost periodic
solutions for abstract impulsive differential equations and for impulsive neural networks
with delay have been discussed by G. T. Stamov and I. M. Stamova 10, and Stamov and
Alzabut 11.
However, besides delay and impulsive effects, stochastic effects likewise exist in real
system. A lot of dynamic systems have variable structures subject to stochastic abrupt
changes, which may result from abrupt phenomena such as stochastic failures and repairs of
components, changes in the interconnections of subsystems, sudden environment changes,
and so on 12–14. Moreover, differential descriptor systems also have abrupt changes
15, 16. Recently, a large number of stability criteria of stochastic system with delays have
2
Journal of Applied Mathematics
been reported 17–19. Almost periodic solutions to some functional integro-differential
stochastic evolution equations and to some stochastic differential equations have been
studied by Bezandry and Diagana 20, and Bezandry 21. Huang and Yang investigated
almost periodic solution for stochastic cellular neural networks with delays 22. Because
it is not easy to deal with the case of coexistence of impulsive, delay and stochastic effects
in a dynamical system, there are few results about this problems 23–25. To the best of our
knowledge, there exists no result on the existence and uniqueness of mean square almost
periodic solutions for impulsive stochastic differential equations with delays.
Motivated by the above discussions, the main aim of this paper is to study the mean
square almost periodic solutions for impulsive stochastic differential equations with delays.
By employing stochastic analysis, delay differential inequality technique and fixed points
theorem, we obtain some criteria to ensure the existence and uniqueness of mean square
almost periodic solutions.
The rest of this paper is organized as follows: in Section 2, we introduce a class of
impulsive stochastic differential equations with delays, and the relating notations, definitions
and lemmas which would be used later; in Section 3, a new sufficient condition is proposed
to ensure the existence and uniqueness of mean square almost periodic solutions; in Section
4, an example is constructed to show the effectiveness of our results. Finally, a conclusion is
given in Section 5.
2. Preliminaries
Let R −∞, ∞, N {1, 2, 3, . . .}, and B {{tk } : t0 0 < t1 < t2 < · · · < tk < tk1 <
· · · , limk → ∞ tk ∞} be the set of all sequence unbounded and strictly increasing. For x ∈ Rn
and A ∈ Rn×n , let x be any vector norm, and denote the induced matrix norm and the matrix
measure, respectively, by
Ax
,
x/
0 x
A sup
μA lim
h→0
I hA − 1
.
h
2.1
The norm and measure of vector and matrix are x maxi |xi |, A maxi nj1 |aij |, μA maxi {aii nj/ i |aij |}.
Consider the following a class of Itô impulsive stochastic differential equations with
delay
dxt Axt Bft, xt Cgt, xt − h It dt σt, xtdωt, t ≥ 0, t /
tk ,
−
−
− Δxt xtk − x tk Dk x tk Vk x tk βk , t tk , k ∈ N,
xt φt,
−h ≤ t ≤ 0,
2.2
where xt x1 t, . . . , xn tT is the solution process, A, B, C, Dk ∈ Rn×n are constant matrices, ft, x f1 t, x, . . . , fn t, xT , gt, x g1 t, x, . . . , gn t, xT , It I1 t, . . . , In tT ,
σt, x σij t, xn×n is the diffusion coefficient matrix, Vk x V1k x, . . . , Vnk xT is
impulsive function, h > 0 is delay; tk ∈ B is impulsive time, βk β1k , . . . , βnk T is a constant
vector, ωt ω1 t, . . . , ωn tT is an n-dimensional Brown motion defined on a complete
Journal of Applied Mathematics
3
probability space Ω, F, P with a natural filtration {Ft }t≥0 generated by ωt, and denote
by F the associated σ-algebra generated by ωt with the probability measure P. Moreover,
Δ
the initial conditions φt φ1 t, . . . , φn tT ∈ P CBFb 0 −h, 0, Rn P CBFb 0 . Denote by
P CBFb 0 the family of all bounded F0 -measurable, P C−h, 0, Rn -valued random variable ζ,
satisfying Eζ2 Esup−h≤θ≤0 ζθ2 < ∞, where P C−h, 0, Rn {ζ : −h, 0 → Rn is
continuous}. E denotes the expectation of stochastic process.
Let H, · be a Hilbert space and Ω, F, P be a complete probability space. Define
L2 P, H to be the space of all H-value random variable Y such that
2
EY Ω
Y 2 dP < ∞.
2.3
It is then routine to check that L2 P, H is a Hilbert space when it is equipped with its
natural norm · 2 defined by
Y 2 Ω
1/2
< ∞,
Y 2 dP
2.4
for each Y ∈ L2 P, H.
Definition 2.1 see 25. For any φ ∈ P CBFb 0 , a function xt : −h, ∞ → L2 P, H is said
to be solution of system 2.2 on −h, ∞ satisfying initial value condition, if the following
conditions hold:
i xt is absolutely continuous on each interval tk , tk1 ∈ 0, ∞, k ∈ N;
ii for any tk ∈ 0, ∞, k ∈ N, xtk and xt−k exist and xtk xtk ;
iii xt satisfies 2.2 for almost everywhere in −h, ∞ and at impulsive points t tk
situated in 0, ∞, k ∈ N, may have discontinuity points of the first kind.
Obviously, the solution defined by definition 1 is piecewise continuous.
j
j
Definition 2.2 see 26. The set of sequences {tk }, tk tkj − tk , k ∈ N, j ∈ N, {tk } ∈ B is said
to be uniformly almost periodic if for any ε > 0, there exists relatively dense set of ε-almost
periods common for any sequences.
Definition 2.3. A piecewise continuous function xt : −h, ∞ → L2 P, H with discontinuity points of first kind at t tk is said to be mean square almost periodic, if
j
i the set of sequence {tk } is uniformly almost periodic;
ii for any ε > 0, there exists δ > 0, such that if the points t and t belong to one and
the same interval of continuity of xt and satisfy the inequality |t − t | < δ, then
Ext − xt 2 < ε;
iii for any ε > 0, there exists a relatively dense set T such that if τ ∈ T , then Ext τ − xt2 < ε for all t ∈ −h, ∞ satisfying the condition |t − tk | > ε, k ∈ N.
The collection of all functions xt : −h, ∞ → L2 P, H with discontinuity
points of the first kind at t tk which are mean square almost periodic is denoted by
4
Journal of Applied Mathematics
AP −h, ∞; L2 P, H, one can check that AP −h, ∞; L2 P, H is a Banach space when
it is equipped with the norm:
1/2
.
x∞ sup Ext2
2.5
t∈R
Let B1 , · 1 and B2 , · 2 be Banach space and L2 P, B1 and L2 P, B2 be their
corresponding L2 -space, respectively.
Lemma 2.4 see 20. Let f : R × L2 P, B1 → L2 P, B2 , t, x → ft, x be mean square almost
periodic in t ∈ R uniformly in x ∈ K, where K ⊂ L2 P, B1 is compact. Suppose that there exists
Lf > 0 such that
2
2
Eft, x − ft, y2 ≤ Lf Ex − y1
2.6
for all x, y ∈ L2 P, B1 and for each t ∈ R. Then for any mean square almost periodic function
ψt : R → L2 P, B1 , ft, ψt is mean square almost periodic.
In this paper, we always assume that:
0 and the sequence {Dk }, k ∈ N, is almost periodic, where I ∈ Rn×n is
A1 detI Dk /
the identity matrix;
j
A2 the set of {tk } is uniformly almost periodic and θ infk {t1k } > 0.
Recall 2, consider the following linear system of system2.2
ẋt Axt, t /
tk ,
−
Δxtk Dk x tk , k ∈ N,
2.7
that if Uk t, s is the Cauchy matrix for the system
ẋt Axt,
tk−1 ≤ t < tk ,
2.8
then the Cauchy matrix for the system 2.7 is in the form
⎧
⎪
Uk t, s,
⎪
⎪
⎪
⎪
⎪
⎨U t, t I D U t , s,
k1
k
k
k k
Wt, s ⎪
i1
⎪
⎪
⎪
⎪
Uk1 t, tk I Dk Uj tj , tj1 I Di Ui ti , s,
⎪
⎩
tk−1 ≤ s ≤ t < tk ,
tk−1 ≤ s < tk ≤ t < tk1 ,
ti−1 ≤ s < ti < tk ≤ t < tk1 .
jk
2.9
As the special case of Lemma 1 in 10, we have the following lemma.
Journal of Applied Mathematics
5
Lemma 2.5. Assume that (A1), (A2) and the following condition hold. For the Cauchy matrix Wt, s
of system 2.7, there exist positive constants M and λ such that
Wt, s ≤ Me−λt−s ,
t ≥ s, t, s ∈ R.
2.10
Then for any ε > 0, t ≥ s, t, s ∈ R, |t − tk | > ε, |s − tk | > ε, k ∈ N, there must be exist a relatively
dense set T of ε-almost periodic of the matrix A and a positive constant Γ such that for τ ∈ T , it
follows:
Wt τ, s τ − Wt, s ≤ εΓe−λ/2t−s .
2.11
Lemma 2.6 see 6. Let Wt, s be the Cauchy matrix of the linear system 2.7. Given a constant
η ≥ I Dk for all k ∈ N, if η ≥ 1 and θ infk {t1k } > 0, then
Wt, s ≤ ηeμAln η/θt−s ,
t ≥ s.
2.12
Introduce the following conditions:
A3 The functions f, g : R × L2 P, H → L2 P, H are mean square almost periodic in
t ∈ R uniformly in x ∈ Θ, where Θ ⊂ L2 P, H is compact, and f0, 0 g0, 0 0.
Moreover, there exist Lf , Lg > 0 such that
2
2
Eft, x − ft, y ≤ Lf Ex − y ,
2
2
Egt, x − gt, y ≤ Lg Ex − y ,
2.13
for all stochastic processes x, y ∈ L2 P, H and t ∈ R.
A4 The function σ : R × L2 P, H → L2 P, H is mean square almost periodic in t ∈ R
uniformly in x ∈ Θ , where Θ ⊂ L2 P, H is compact, and σ0, 0 0. Moreover,
there exists Lσ > 0 such that
2
2
Eσt, x − σ t, y ≤ Lσ Ex − y ,
2.14
for all stochastic processes x, y ∈ L2 P, H and t ∈ R.
A5 The function Ii t : R → R is almost periodic in the sense of Bohr, {βk }k∈N is almost
periodic sequence and there exists a constant γ0 > 0, such that
max maxβk , supIt ≤ γ0 .
k
t
2.15
6
Journal of Applied Mathematics
A6 The sequence of functions Vk x : L2 P, H → L2 P, H is mean square almost
periodic uniformly with respect to x ∈ Θ , where Θ ⊂ L2 P, H is compact.
Moreover, there exists LV > 0 such that
2
2
EVk x − Vk y ≤ LV Ex − y
2.16
for all stochastic processes x, y ∈ L2 P, H.
Lemma 2.7 see 26. If conditions (A1)–(A6) are satisfied, then for each ε > 0, there exists ε1 , 0 <
ε1 < ε and relatively dense sets T of real numbers and Q of integral numbers, such that
i Eft τ, y − ft, y2 < ε, Egt τ, y − gt, y2 < ε, t ∈ R, τ ∈ T , |t − tk | > ε, k ∈
N, y ∈ L2 P, H;
ii Eσt τ, y − σt, y2 < ε, t ∈ R, τ ∈ T, |t − tk | > ε, k ∈ N, y ∈ L2 P, H;
iii It τ − It2 < ε, t ∈ R, τ ∈ T, |t − tk | > ε;
iv EVkq y − Vk y2 < ε, q ∈ Q, k ∈ N;
v βkq − βk 2 < ε, q ∈ Q, k ∈ N;
vi tkq − τ2 < ε1 , q ∈ Q, τ ∈ T, k ∈ N.
Lemma 2.8 see 26. Let condition (A2) holds. Then for each p > 0, there exists a positive integer
N such that on each interval of length p, there are no more than N elements of the sequence {tk }, that
is,
is, t ≤ Nt − s N,
2.17
where is, t is the number of points tk in the interval s, t.
3. Main Results
Theorem 3.1. Assume that (A1)–(A6) hold, then there exists a unique mean square almost periodic
solution of system 2.2 if the following conditions are satisfied: There exists a constant η ≥ 1, such
that I Dk ≤ η, k ∈ N and
μA ln η Δ
−λ < 0.
θ
3.1
Furthermore,
ρ 6η
2
L2
2
N2
L2V σ
B2 L2f C2 L2g 2
2
2λ
λ
1 − e−λ < 1.
3.2
Proof. Let D {ϕt ∈ L2 P, H : ϕt ϕ1 t, . . . , ϕn tT } ⊂ AP −h, ∞; L2 P, H satisfy2
ing the equality Eϕ2 < K, where K 2η2 γ02 1/λ N/1 − e−λ > 0.
Journal of Applied Mathematics
7
Set
xt Wt, 0φ0 t
Wt, s Bfs, xs Cgs, xs − h Is ds
0
Wt, tk Vk xtk βk t
3.3
Wt, sσs, xsdωs,
t ≥ 0.
0
0≤tk <t
where φ0 x0, it is easy to see that xt given by 3.3 is the solution of system 2.2
according to 2 and Lemma 2.2 in 27.
By Lemma 2.6 and the conditions of Theorem, we have
Wt, s ≤ ηe−λt−s ,
t ≥ s, t, s ∈ R.
3.4
For zt ∈ D, we define the operator L in the following way
Lzt t
0
Wt, s Bfs, zs Cgs, zs − h Is ds
Wt, tk Vk ztk βk t
3.5
Wt, sσs, zsdωs.
0
0≤tk <t
Define subset D∗ ⊂ D, D∗ {z ∈ D : Ez − z0 2 ≤ ρK/1 − ρ}, and z0 Isds 0≤tk <t Wt, tk βk .
We have
2
2
t
Ez0 ≤ 2E Wt, sIsds 2E
Wt, tk βk 0
0≤t <t
k
2
t
2
−λt−s
−λt−tk ≤2
ηe
supIsds 2
ηe
max βk
t
0
Wt, s
2
0
≤
2η2 γ02
s
N
1
λ 1 − e−λ
2
0≤tk <t
3.6
k
K.
Then for ∀z ∈ D∗ , from the definition of D∗ and 3.6, since a b2 ≤ 2a2 2b2 , we have
Ez2 Ez − z0 z0 2 ≤ 2E z − z0 2 z0 2
≤2
ρK
K
1−ρ
2K
.
1−ρ
3.7
8
Journal of Applied Mathematics
For ∀z ∈ D∗ , we have
t
Lz − z0 Wt, s Bfs, zs Cgs, zs − h ds
0
t
Wt, tk Vk ztk Wt, sσs, zsdωs.
0
0≤t <t
3.8
k
Since a b c2 ≤ 3a2 3b2 3c2 , it follows
2
ELz − z0 ≤ 3E
t
2
Wt, sBfs, zs Cgs, zs − hds
0
2
t
2
3E Wt, tk Vk ztk 3E
Wt, sσs, zsdωs .
0≤t <t
0
k
3.9
For first term of the right-hand side, using 3.7, A3 and Cauchy-Schwarz inequality, we
have
t
E
2
Wt, sBfs, zs Cgs, zs − hds
0
≤η
2
t
e
0
≤ η2
t
0
−λt−s
t
2
−λt−s
ds ·
e
· EBfs, zs Cgs, zs − h ds
0
t
3.10
2
2 2
2
2 2
−λt−s
−λt−s
e
ds ·
e
· 2B Lf Ezs 2C Lg Ezs − h ds
0
2K 2 2 2
2 2
.
≤ η2 ·
L
L
C
B
g
f
1 − ρ λ2
As to the second term, using 3.7, A6 and Cauchy-Schwarz inequality, we can write
2
E Wt, tk Vk ztk 0≤t <t
k
2
2
−λt−tk −λt−tk ·
≤η
e
e
EVk ztk 0≤tk <t
≤ η2
e−λt−tk 0≤tk <t
0≤tk <t
·
e−λt−tk L2V Eztk 2
0≤tk <t
2K
N2
2
.
L ·
≤η ·
1 − ρ V 1 − e−λ 2
2
3.11
Journal of Applied Mathematics
9
As far as last term is concerned, using 3.7, A4, and the Itô isometry theorem, we obtain
t
E
2
Wt, sσs, zsdωs
≤
t
0
0
≤η
2
Wt, s2 Eσs, zs2 ds
t
0
e−2λt−s L2σ Ezs2 ds
2K L2σ
·
.
≤η ·
1 − ρ 2λ
3.12
2
Thus, by combining 3.9–3.12, it follows that
ρK
L2σ
N2
2K 2 2 2
2 2
2
ELz − z0 ≤ 3η ·
.
LV B Lf C Lg 2
2
1−ρ λ
2λ
1−ρ
1 − e−λ 2
2
3.13
By Lemmas 2.5 and 2.6, one can obtain
Wt τ, s τ − Wt, s ≤ εΓe−λ/2t−s .
3.14
Let τ ∈ T, q ∈ Q, where the sets T and Q are determined in Lemma 2.7, and we assume that
0 < ε < 1, then
Lzt τ − Lzt
t
Wt τ, s τ − Wt, s Bfs τ, zs τ Cgs τ, zs τ − h Is τ ds
0
t
Wt, s Bfs τ, zs τ Cgs τ, zs τ − h Is τ
0
− Bfs, zs Cgs, zs − h Is ds
W t τ, tkq − Wt, tk Vkq z tkq βkq
0≤tk <t
Wt, tk Vkq z tkq − Vk ztk βkq − βk
0≤tk <t
t
Wt τ, s τ − Wt, sσs τ, zs τdωs
Wt, sσs τ, zs τ − σs, zsdωs.
0
0
t
3.15
10
Journal of Applied Mathematics
Therefore, we have
ELzt τ − Lzt2
t
≤ 3E Wt τ, s τ−Wt, s Bfsτ, zs τCgs τ, zsτ −h Is τ ds
0
t
Wt, s Bfs τ, zs τCgs τ, zs τ − hIs τ
0
2
− Bfs, zs Cgs, zs −h Is ds 3E
W t τ, tkq − Wt, tk Vkq z tkq βkq
0≤t <t
k
2
Wt, tk Vkq z tkq − Vk ztk βkq − βk <t
0≤tk
t
3E Wt τ, s τ − Wt, sσs τ, zs τdωs
0
2
t
Wt, sσs τ, zs τ − σs, zsdωs .
0
3.16
We first evaluate the first term of the right hand side
t
E Wt τ, s τ − Wt, s Bfs τ, zs τ Cgs τ, zs τ − h Is τ ds
0
t
Wt, s Bfs τ, zs τ Cgs τ, zs τ − h Is τ
0
≤ 2E
t
0
2
− Bfs, zs Cgs, zs −h Is ds Wt τ, s τ − Wt, s
2
× Bfs τ, zs τ Cgs τ, zs τ − h Is τ
ds
2E
t
Wt, sB fs τ, zs τ − fs, zs
0
2
C gs τ, zs τ − h − gs, zs − h Is τ − Is ds
≤ c1 ε,
3.17
where c1 96η2 /λ2 B2 L2f · K/1 − ρ 1 C2 L2g · K/1 − ρ 1 γ02 1.
Journal of Applied Mathematics
11
For the second term, we can estimate that
E
W t τ, tkq − Wt, tk Vkq z tkq βkq
0≤t <t
k
2
Wt, tk Vkq z tkq − Vk ztk βkq − βk 0≤tk <t
2
Wt τ, tkq − Wt, tk Vkq ztkq βkq ≤ 2E
0≤t <t
k
2
2E
Wt, tk Vkq ztkq − Vk ztk βkq − βk 0≤t <t
3.18
k
≤ c2 ε,
where c2 8η2 N 2 /1 − e−λ L2V · K/1 − ρ 1 γ02 1.
For the last term, using A4 and Itô isometry identity, we have
t
E Wt τ, s τ − Wt, sσs τ, zs τdωs
0
2
t
Wt, sσs τ, zs τ − σs, zsdωs
0
2
t
≤ 2E Wt τ, s τ − Wt, s, σs τ, zs τdωs
0
2
t
2E Wt, sσs τ, zs τ − σs, zsdωs
0
t
≤ 2E Wt τ, s τ − Wt, s2 σs τ, zs τ2 ds
0
2E
t
3.19
Wt, s2 σs τ, zs τ − σs, zs2 ds
0
≤ c3 ε,
where c3 2/λΓ2 L2σ K/1 − ρ 1.
Combining 3.17, 3.18 and 3.19, it follows that
ELzt τ − Lzt2 ≤ c0 ε,
where c0 3c1 c2 c3 .
So, Lz ∈ D∗ , that is L is self-mapping from D∗ to D∗ by 3.13 and 3.20.
Secondly, we will show L is contracting operator in D∗ .
3.20
12
Journal of Applied Mathematics
For ∀x, y ∈ D∗ ,
t
Lx − Ly Wt, sB fs, xs − f s, ys
0
C gs, xs − h − g s, ys − h ds
Wt, tk Vk xtk − Vk ytk 3.21
0≤tk <t
Wt, s σs, xs − σ s, ys dωs.
0
t
By a minor modification of the proof of 3.13, we can obtain
2
ELx − Ly
2
L2σ
N2
2 2
2 2
2 2
2
sup Ext − yt
≤ 6η
LV B Lf C Lg 2
2
2λ
λ
t
1 − e−λ 2
ρ x − y ∞,
3.22
and therefore, Lx − Ly∞ ≤ ρx − y∞ , it follows that L is contracting operator in D∗ , so there
exists a unique mean square almost periodic solution of 2.2 by the fixed points theorem.
4. Example
Consider the following impulsive stochastic differential equation with delay
⎡
dxi t ⎣ai xi t 2
⎤
2
bij fj xj t cij gj xj t − 0.1 Ii t⎦dt
j1
j1
0.5xi tdωi t, t ≥ 0, t /
tk ,
−
−
Δxt xtk − x tk Dk x tk Vk x t−k βk ,
xt φt,
4.1
t tk , k ∈ N,
−h ≤ t ≤ 0,
where tk k, k ∈ N, fxt sin x1 t, sin x2 tT , gxt−0.1 cos x1 t−0.1, cos x2 t−
0.1T , Vik 0.01 sin x1 t, 0.01 cos x2 tT , βk 0.1, It 0.1, 0.1T , γ0 0.1, for
convenience, we can choose
−2 0
A
,
0 −3
−0.1 0
B
,
0 −0.1
0.2 0
C
,
0 −0.2
−0.5 0
.
Dk 0 −0.5
4.2
Then μA −2, I Dk 1/2, B 0.1, C 0.2, Lf Lg 1, LV 0.01, Lσ 0.5.
Choose θ infk {t1k } 0.01, η 1, N 6. By simple calculation, we have λ −μA .
.
.
ln η/θ 2, ρ 0.8139 < 1, K 1.107, ρK/1 − ρ 4.841.
Journal of Applied Mathematics
13
Let D∗ {z ∈ D : Ez − z0 2 ≤ 4.841}, so, by Theorem 3.1, system 4.1 has a unique
mean square almost periodic solution in D∗ .
Remark 4.1. Since there exist no results for almost periodic solutions for impulsive stochastic
differential equations with delays, one can easily see that all the results in 10, 11, 20–22, 28
and the references therein cannot be applicable to system 4.1. This implies that the results
of this paper are essentially new.
5. Conclusion
In this paper, a class of Itô impulsive stochastic differential equations with delays has been
investigated. We conquer the difficulty of coexistence of impulsive, delay and stochastic
factors in a dynamic system, and give a result for the existence and uniqueness of mean
square almost periodic solutions. The results in this paper extend some earlier works reported
in the literature. Moreover, our results have important applications in almost periodic
oscillatory stochastic delayed neural networks with impulsive control.
Acknowledgment
This work is supported by the National Science Foundation of China no. 10771199.
References
1 A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore,
1995.
2 D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations: Periodic Solutions and
Applications, Longman, Harlow, UK, 1993.
3 V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World
Scientific, Singapore, 1989.
4 X. Liu, “Stability of impulsive control systems with time delay,” Mathematical and Computer Modelling,
vol. 39, no. 4-5, pp. 511–519, 2004.
5 Z. Yang and D. Xu, “Existence and exponential stability of periodic solution for impulsive delay
differential equations and applications,” Nonlinear Analysis, vol. 64, no. 1, pp. 130–145, 2006.
6 Z. Yang and D. Xu, “Robust stability of uncertain impulsive control systems with time-varying delay,”
Computers and Mathematics with Applications, vol. 53, no. 5, pp. 760–769, 2007.
7 X. Fu and X. Li, “Global exponential stability and global attractivity of impulsive Hopfield neural
networks with time delays,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 187–
199, 2009.
8 C. Li, J. Sun, and R. Sun, “Stability analysis of a class of stochastic differential delay equations with
nonlinear impulsive effects,” Journal of the Franklin Institute, vol. 347, no. 7, pp. 1186–1198, 2010.
9 R. P. Agarwal and F. Karako, “A survey on oscillation of impulsive delay differential equations,”
Computers and Mathematics with Applications, vol. 60, no. 6, pp. 1648–1685, 2010.
10 G. T. Stamov and I. M. Stamova, “Almost periodic solutions for impulsive neural networks with
delay,” Applied Mathematical Modelling, vol. 31, no. 7, pp. 1263–1270, 2007.
11 G. T. Stamov and J. O. Alzabut, “Almost periodic solutions for abstract impulsive differential
equations,” Nonlinear Analysis, vol. 72, no. 5, pp. 2457–2464, 2010.
12 X. Mao, Exponential Stability of Stochastic Differential Equations, vol. 182, Marcel Dekker, New York, NY,
USA, 1994.
13 L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, NY, USA, 1972.
14 A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, NY, USA,
1976.
14
Journal of Applied Mathematics
15 Z. Gao and X. Shi, “Stochastic state estimation and control for stochastic descriptor systems,” in
Proceedings of the IEEE International Conference on Robotics, Automation and Mechatronics (RAM ’08),
pp. 331–336, 2008.
16 Z. Gao and S. X. Ding, “Actuator fault robust estimation and fault-tolerant control for a class of
nonlinear descriptor systems,” Automatica, vol. 43, no. 5, pp. 912–920, 2007.
17 L. Ma and F. Da, “Mean-square exponential stability of stochastic Hopfield neural networks with
time-varying discrete and distributed delays,” Physics Letters A, vol. 373, no. 25, pp. 2154–2161, 2009.
18 L. Hu and A. Yang, “Fuzzy model-based control of nonlinear stochastic systems with time-delay,”
Nonlinear Analysis, vol. 71, no. 12, pp. e2855–e2865, 2009.
19 H. Zhang, H. Yan, and Q. Chen, “Stability and dissipative analysis for a class of stochastic system
with time-delay,” Journal of the Franklin Institute, vol. 347, no. 5, pp. 882–893, 2010.
20 P. H. Bezandry and T. Diagana, “Existence of almost periodic solutions to some stochastic differential
equations,” Applicable Analysis, vol. 86, no. 7, pp. 819–827, 2007.
21 P. H. Bezandry, “Existence of almost periodic solutions to some functional integro-differential
stochastic evolution equations,” Statistics and Probability Letters, vol. 78, no. 17, pp. 2844–2849, 2008.
22 Z. Huang and Q. G. Yang, “Existence and exponential stability of almost periodic solution for
stochastic cellular neural networks with delay,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 773–
780, 2009.
23 Z. Yang, D. Xu, and L. Xiang, “Exponential -stability of impulsive stochastic differential equations
with delays,” Physics Letters A, vol. 359, no. 2, pp. 129–137, 2006.
24 L. Xu and D. Xu, “Exponential -stability of impulsive stochastic neural networks with mixed delays,”
Chaos, Solitons and Fractals, vol. 41, no. 1, pp. 263–272, 2009.
25 C. Li and J. Sun, “Stability analysis of nonlinear stochastic differential delay systems under impulsive
control,” Physics Letters A, vol. 374, no. 9, pp. 1154–1158, 2010.
26 A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, Vyshcha Shkola, Kiev,
Russia, 1987.
27 L. Xu and D. Xu, “Mean square exponential stability of impulsive control stochastic systems with
time-varying delay,” Physics Letters A, vol. 373, no. 3, pp. 328–333, 2009.
28 X. Li, “Existence and global exponential stability of periodic solution for delayed neural networks
with impulsive and stochastic effects,” Neurocomputing, vol. 73, no. 4–6, pp. 749–758, 2010.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014