# Mean Square Almost Periodic Solutions for Impulsive Stochastic Differential Equations with Delays

Authors: Ruojun Zhang; Nan Ding; Linshan Wang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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.


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## Body

## 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 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 Section2, 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

Letℝ=(-∞,+∞), ℕ={1,2,3,…}, and ℬ={{tk}:t0=0<t1<t2<⋯<tk<tk+1<⋯,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(2.1)‖A‖=sup⁡x≠0‖Ax‖‖x‖,μ(A)=lim⁡h→0+‖I+hA‖-1h.
The norm and measure of vector and matrix are ∥x∥=maxi|xi|, ∥A∥=maxi∑j=1n|aij|, μ(A)=maxi{aii+∑j≠in|aij|}.Consider the following a class of Itô impulsive stochastic differential equations with delay(2.2)dx(t)=[Ax(t)+Bf(t,x(t))+Cg(t,x(t-h))+I(t)]dt+σ(t,x(t))dω(t),t≥0,t≠tk,Δx(t)=x(tk)-x(tk-)=Dkx(tk-)+Vk(x(tk-))+βk,t=tk,k∈N,x(t)=ϕ(t),-h≤t≤0,
where x(t)=(x1(t),…,xn(t))T is the solution process, A,B,C,Dk∈ℝn×n are constant matrices, f(t,x)=(f1(t,x),…,fn(t,x))T, g(t,x)=(g1(t,x),…,gn(t,x))T, I(t)=(I1(t),…,In(t))T, σ(t,x)=(σij(t,x))n×n is the diffusion coefficient matrix, Vk(x)=(V1k(x),…,Vnk(x))T is impulsive function, h>0 is delay; tk∈ℬ is impulsive time, βk=(β1k,…,βnk)T is a constant vector, ω(t)=(ω1(t),…,ωn(t))T is an n-dimensional Brown motion defined on a complete probability space (Ω,ℱ,ℙ) with a natural filtration {ℱt}t≥0 generated by ω(t), and denote by ℱ the associated σ-algebra generated by ω(t) with the probability measure ℙ. Moreover, the initial conditions ϕ(t)=(ϕ1(t),…,ϕn(t))T∈PCBℱ0b([-h,0],ℝn)=ΔPCBℱ0b. Denote by PCBℱ0b the family of all bounded ℱ0-measurable, PC([-h,0],ℝn)-valued random variable ζ, satisfying E∥ζ∥2=E(sup-h≤θ≤0∥ζ(θ)∥2)<+∞, where PC([-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(2.3)E‖Y‖2=∫Ω‖Y‖2dP<∞.It is then routine to check thatL2(ℙ,ℍ) is a Hilbert space when it is equipped with its natural norm ∥·∥2 defined by(2.4)‖Y‖2=(∫Ω‖Y‖2dP)1/2<∞,
for each Y∈L2(ℙ,ℍ).Definition 2.1 (see [25]).
For anyϕ∈PCBℱ0b, a function x(t):[-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)
x(t) is absolutely continuous on each interval (tk,tk+1)∈[0,+∞),k∈ℕ;(ii)
for anytk∈[0,+∞),k∈ℕ, x(tk+) and x(tk-) exist and x(tk+)=x(tk);(iii)
x(t) 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.Definition 2.2 (see [26]).
The set of sequences{tkj}, tkj=tk+j-tk,k∈ℕ,j∈ℕ,{tk}∈ℬ is said to be uniformly almost periodic if for any ɛ>0, there exists relatively dense set of ɛ-almost periods common for any sequences.Definition 2.3.
A piecewise continuous functionx(t):[-h,+∞)→L2(ℙ,ℍ) with discontinuity points of first kind at t=tk is said to be mean square almost periodic, if(i)
the set of sequence{tkj} is uniformly almost periodic;(ii)
for anyɛ>0, there exists δ>0, such that if the points t′ and t′′ belong to one and the same interval of continuity of x(t) and satisfy the inequality |t′-t′′|<δ, then E∥x(t′)-x(t′′)∥2<ɛ;(iii)
for anyɛ>0, there exists a relatively dense set T such that if τ∈T, then E∥x(t+τ)-x(t)∥2<ɛ for all t∈[-h,+∞) satisfying the condition |t-tk|>ɛ,k∈ℕ.The collection of all functionsx(t):[-h,+∞)→L2(ℙ,ℍ) with discontinuity points of the first kind at t=tk which are mean square almost periodic is denoted by 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:(2.5)‖x‖∞=sup⁡t∈R(E‖x(t)‖2)1/2.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]).
Letf:ℝ×L2(ℙ,B1)→L2(ℙ,B2),(t,x)↦f(t,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
(2.6)E‖f(t,x)-f(t,y)‖22≤LfE‖x-y‖12
for all x,y∈L2(ℙ,B1) and for each t∈ℝ. Then for any mean square almost periodic function ψ(t):ℝ→L2(ℙ,B1), f(t,ψ(t)) is mean square almost periodic.In this paper, we always assume that:(A1)
det(I+Dk)≠0 and the sequence {Dk},k∈ℕ, is almost periodic, where I∈Rn×n is the identity matrix;(A2)
the set of{tkj} is uniformly almost periodic and θ=infk{tk1}>0.Recall [2], consider the following linear system of system(2.2)(2.7)ẋ(t)=Ax(t),t≠tk,Δx(tk)=Dkx(tk-),k∈N,
that if Uk(t,s) is the Cauchy matrix for the system(2.8)ẋ(t)=Ax(t),tk-1≤t<tk,
then the Cauchy matrix for the system (2.7) is in the form(2.9)W(t,s)={Uk(t,s),tk-1≤s≤t<tk,Uk+1(t,tk)(I+Dk)Uk(tk,s),tk-1≤s<tk≤t<tk+1,Uk+1(t,tk)∏j=ki+1(I+Dk)Uj(tj,tj+1)(I+Di)Ui(ti,s),ti-1≤s<ti<tk≤t<tk+1.As the special case of Lemma  1 in [10], we have the following lemma.Lemma 2.5.
Assume that (A1), (A2) and the following condition hold. For the Cauchy matrixW(t,s) of system (2.7), there exist positive constants M and λ such that
(2.10)‖W(t,s)‖≤Me-λ(t-s),t≥s,t,s∈R.
Then for any ɛ>0,t≥s,t,s∈ℝ,|t-tk|>ɛ,|s-tk|>ɛ,k∈ℕ, there must be exist a relatively dense set T of ɛ-almost periodic of the matrix A and a positive constant Γ such that for τ∈T, it follows:
(2.11)‖W(t+τ,s+τ)-W(t,s)‖≤ɛΓe(-λ/2)(t-s).Lemma 2.6 (see [6]).
LetW(t,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{tk1}>0, then
(2.12)‖W(t,s)‖≤ηe(μ(A)+(ln⁡η/θ))(t-s),t≥s.Introduce the following conditions:(A3)
The functionsf,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 f(0,0)=g(0,0)=0. Moreover, there exist Lf,Lg>0 such that(2.13)E‖f(t,x)-f(t,y)‖2≤LfE‖x-y‖2,E‖g(t,x)-g(t,y)‖2≤LgE‖x-y‖2,for all stochastic processesx,y∈L2(ℙ,ℍ) and t∈ℝ.(A4)
The functionσ: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 σ(0,0)=0. Moreover, there exists Lσ>0 such that(2.14)E‖σ(t,x)-σ(t,y)‖2≤LσE‖x-y‖2,for all stochastic processesx,y∈L2(ℙ,ℍ) and t∈ℝ.(A5)
The functionIi(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(2.15)max⁡{max⁡k|βk|,sup⁡t‖I(t)‖}≤γ0.(A6)
The sequence of functionsVk(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(2.16)E‖Vk(x)-Vk(y)‖2≤LVE‖x-y‖2
for all stochastic processes x,y∈L2(ℙ,ℍ).Lemma 2.7 (see [26]).
If conditions (A1)–(A6) are satisfied, then for eachɛ>0, there exists ɛ1,0<ɛ1<ɛ and relatively dense sets T of real numbers and Q of integral numbers, such that(i)
E∥f(t+τ,y)-f(t,y)∥2<ɛ, E∥g(t+τ,y)-g(t,y)∥2<ɛ,t∈ℝ,τ∈T, |t-tk|>ɛ,k∈ℕ, y∈L2(ℙ,ℍ);(ii)
E∥σ(t+τ,y)-σ(t,y)∥2<ɛ,t∈ℝ,τ∈T,|t-tk|>ɛ,k∈ℕ,y∈L2(ℙ,ℍ);(iii)
∥I(t+τ)-I(t)∥2<ɛ,t∈ℝ,τ∈T,|t-tk|>ɛ;(iv)
E∥Vk+q(y)-Vk(y)∥2<ɛ,q∈Q,k∈ℕ;(v)
∥βk+q-βk∥2<ɛ,q∈Q,k∈ℕ;(vi)
∥tk+q-τ∥2<ɛ1,q∈Q,τ∈T,k∈ℕ.Lemma 2.8 (see [26]).
Let condition (A2) holds. Then for eachp>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,
(2.17)i(s,t)≤N(t-s)+N,
where i(s,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
(3.1)μ(A)+ln⁡ηθ=Δ-λ<0.
Furthermore,
(3.2)ρ=6η2[2λ2(‖B‖2Lf2+‖C‖2Lg2)+N2(1-e-λ)2LV2+Lσ22λ]<1.Proof.
LetD={φ(t)∈L2(ℙ,ℍ):φ(t)=(φ1(t),…,φn(t))T}⊂AP([-h,+∞);L2(ℙ,ℍ)) satisfying the equality E∥φ∥2<K¯, where K¯=2η2γ02((1/λ)+(N/(1-e-λ)))2>0.
Set(3.3)x(t)=W(t,0)ϕ0+∫0tW(t,s)[Bf(s,x(s))+Cg(s,x(s-h))+I(s)]ds+∑0≤tk<tW(t,tk)[Vk(x(tk))+βk]+∫0tW(t,s)σ(s,x(s))dω(s),t≥0.
where ϕ0=x(0), it is easy to see that x(t) given by (3.3) is the solution of system (2.2) according to [2] and Lemma  2.2 in [27].
By Lemma2.6 and the conditions of Theorem, we have
(3.4)‖W(t,s)‖≤ηe-λ(t-s),t≥s,t,s∈R.
Forz(t)∈D, we define the operator L in the following way
(3.5)(Lz)(t)=∫0tW(t,s)[Bf(s,z(s))+Cg(s,z(s-h))+I(s)]ds+∑0≤tk<tW(t,tk)[Vk(z(tk))+βk]+∫0tW(t,s)σ(s,z(s))dω(s).
Define subsetD*⊂D, D*={z∈D:E∥z-z0∥2≤ρK¯/(1-ρ)}, and z0=∫0tW(t,s)I(s)ds+∑0≤tk<tW(t,tk)βk.
We have(3.6)E‖z0‖2≤2E‖∫0tW(t,s)I(s)ds‖2+2E‖∑0≤tk<tW(t,tk)βk‖2≤2[∫0tηe-λ(t-s)sup⁡s‖I(s)‖ds]2+2[∑0≤tk<tηe-λ(t-tk)max⁡k|βk|]2≤2η2γ02(1λ+N1-e-λ)2=K¯.
Then for ∀z∈D*, from the definition of D* and (3.6), since (a+b)2≤2a2+2b2, we have
(3.7)E‖z‖2=E‖(z-z0)+z0‖2≤2E(‖z-z0‖2+‖z0‖2)≤2(ρK¯1-ρ+K¯)=2K¯1-ρ.
For∀z∈D*, we have
(3.8)‖Lz-z0‖=‖∫0tW(t,s)[Bf(s,z(s))+Cg(s,z(s-h))]ds+∑0≤tk<tW(t,tk)Vk(z(tk))+∫0tW(t,s)σ(s,z(s))dω(s)‖.
Since(a+b+c)2≤3a2+3b2+3c2, it follows
(3.9)E‖Lz-z0‖2≤3E(∫0t‖W(t,s)‖‖Bf(s,z(s))+Cg(s,z(s-h))‖ds)2+3E(‖∑0≤tk<tW(t,tk)Vk(z(tk))‖)2+3E(∫0t‖W(t,s)σ(s,z(s))dω(s)‖)2.
For first term of the right-hand side, using (3.7), (A3) and Cauchy-Schwarz inequality, we have
(3.10)E(∫0t‖W(t,s)‖‖Bf(s,z(s))+Cg(s,z(s-h))‖ds)2≤η2(∫0te-λ(t-s)ds)⋅(∫0te-λ(t-s)⋅E‖Bf(s,z(s))+Cg(s,z(s-h))‖2ds)≤η2(∫0te-λ(t-s)ds)⋅[∫0te-λ(t-s)⋅(2‖B‖2Lf2E‖z(s)‖2+2‖C‖2Lg2E‖z(s-h)‖2)ds]≤η2⋅2K¯1-ρ[2λ2(‖B‖2Lf2+‖C‖2Lg2)].
As to the second term, using (3.7), (A6) and Cauchy-Schwarz inequality, we can write
(3.11)E(‖∑0≤tk<tW(t,tk)Vk(z(tk))‖)2≤η2(∑0≤tk<te-λ(t-tk))⋅(∑0≤tk<te-λ(t-tk)E‖Vk(z(tk))‖2)≤η2(∑0≤tk<te-λ(t-tk))⋅(∑0≤tk<te-λ(t-tk)LV2E‖z(tk)‖2)≤η2⋅2K¯1-ρ[LV2⋅N2(1-e-λ)2].
As far as last term is concerned, using (3.7), (A4), and the Itô isometry theorem, we obtain
(3.12)E(∫0t‖W(t,s)σ(s,z(s))dω(s)‖)2≤∫0t‖W(t,s)‖2E‖σ(s,z(s))‖2ds≤η2∫0te-2λ(t-s)Lσ2E‖z(s)‖2ds≤η2⋅2K¯1-ρ⋅Lσ22λ.
Thus, by combining (3.9)–(3.12), it follows that
(3.13)E‖Lz-z0‖2≤3η2⋅2K¯1-ρ[2λ2(‖B‖2Lf2+‖C‖2Lg2)+N2(1-e-λ)2LV2+Lσ22λ]=ρK¯1-ρ.
By Lemmas 2.5 and 2.6, one can obtain
(3.14)‖W(t+τ,s+τ)-W(t,s)‖≤ɛΓe-(λ/2)(t-s).
Let τ∈T,q∈Q, where the sets T and Q are determined in Lemma 2.7, and we assume that 0<ɛ<1, then
(3.15)‖Lz(t+τ)-Lz(t)‖=‖∫0t[W(t+τ,s+τ)-W(t,s)][Bf(s+τ,z(s+τ))+Cg(s+τ,z(s+τ-h))+I(s+τ)]ds+∫0tW(t,s){[Bf(s+τ,z(s+τ))+Cg(s+τ,z(s+τ-h))+I(s+τ)]-[Bf(s,z(s))+Cg(s,z(s-h))+I(s)]ds}+∑0≤tk<t[W(t+τ,tk+q)-W(t,tk)][Vk+q(z(tk+q))+βk+q]+∑0≤tk<tW(t,tk)[Vk+q(z(tk+q))-Vk(z(tk))+βk+q-βk]+∫0t[W(t+τ,s+τ)-W(t,s)][σ(s+τ,z(s+τ))]dω(s)+∫0tW(t,s)[σ(s+τ,z(s+τ))-σ(s,z(s))]dω(s)‖.
Therefore, we have
(3.16)E‖Lz(t+τ)-Lz(t)‖2≤3E‖∫0t[W(t+τ,s+τ)-W(t,s)][Bf(s+τ,z(s+τ))+Cg(s+τ,z(s+τ-h))+I(s+τ)]ds+∫0tW(t,s){[Bf(s+τ,z(s+τ))+Cg(s+τ,z(s+τ-h))+I(s+τ)]-[Bf(s,z(s))+Cg(s,z(s+-h))+I(s)]ds}∫0t‖2+3E‖∑0≤tk<t[W(t+τ,tk+q)-W(t,tk)][Vk+q(z(tk+q))+βk+q]+∑0≤tk<tW(t,tk)[Vk+q(z(tk+q))-Vk(z(tk))+βk+q-βk]‖2+3E‖∫0t[W(t+τ,s+τ)-W(t,s)][σ(s+τ,z(s+τ))]dω(s)+∫0tW(t,s)[σ(s+τ,z(s+τ))-σ(s,z(s))]dω(s)‖2.
We first evaluate the first term of the right hand side
(3.17)E‖∫0t[W(t+τ,s+τ)-W(t,s)][Bf(s+τ,z(s+τ))+Cg(s+τ,z(s+τ-h))+I(s+τ)]ds+∫0tW(t,s){[Bf(s+τ,z(s+τ))+Cg(s+τ,z(s+τ-h))+I(s+τ)]-[Bf(s,z(s))+Cg(s,z(s+-h))+I(s)]ds}∫0t‖2≤2E[∫0t‖W(t+τ,s+τ)-W(t,s)‖×‖Bf(s+τ,z(s+τ))+Cg(s+τ,z(s+τ-h))+I(s+τ)‖∫0t]2ds+2E[∫0t‖W(t,s)‖‖B(f(s+τ,z(s+τ))-f(s,z(s)))+C(g(s+τ,z(s+τ-h))-g(s,z(s-h)))(I(s+τ)-I(s))∫0t]‖2ds≤c1ɛ,
where c1=(96η2/λ2)[∥B∥2Lf2·((K¯/(1-ρ))+1)+∥C∥2Lg2·((K¯/(1-ρ))+1)+γ02+1].
For the second term, we can estimate that(3.18)E‖∑0≤tk<t[W(t+τ,tk+q)-W(t,tk)][Vk+q(z(tk+q))+βk+q]+∑0≤tk<tW(t,tk)[Vk+q(z(tk+q))-Vk(z(tk))+βk+q-βk]‖2≤2E‖∑0≤tk<t[W(t+τ,tk+q)-W(t,tk)][Vk+q(z(tk+q))+βk+q]‖2+2E‖∑0≤tk<tW(t,tk)[Vk+q(z(tk+q))-Vk(z(tk))+βk+q-βk]‖2≤c2ɛ,
where c2=(8η2N2/(1-e-λ))[LV2·((K¯/(1-ρ))+1)+γ02+1].
For the last term, using (A4) and Itô isometry identity, we have(3.19)E‖∫0t[W(t+τ,s+τ)-W(t,s)][σ(s+τ,z(s+τ))]dω(s)+∫0tW(t,s)[σ(s+τ,z(s+τ))-σ(s,z(s)]dω(s)‖2≤2E‖∫0t[W(t+τ,s+τ)-W(t,s),σ(s+τ,z(s+τ))]dω(s)‖2+2E‖∫0tW(t,s)[σ(s+τ,z(s+τ))-σ(s,z(s))]dω(s)‖2≤2E∫0t‖W(t+τ,s+τ)-W(t,s)‖2‖σ(s+τ,z(s+τ))‖2ds+2E∫0t‖W(t,s)‖2‖σ(s+τ,z(s+τ))-σ(s,z(s))‖2ds≤c3ɛ,
where c3=(2/λ)[Γ2Lσ2(K¯/(1-ρ))+1].
Combining (3.17), (3.18) and (3.19), it follows that(3.20)E‖Lz(t+τ)-Lz(t)‖2≤c0ɛ,
where c0=3(c1+c2+c3).
So,Lz∈D*, that is L is self-mapping from D* to D* by (3.13) and (3.20).
Secondly, we will showL is contracting operator in D*.
For∀x,y∈D*,
(3.21)‖Lx-Ly‖=‖∫0tW(t,s)B[f(s,x(s))-f(s,y(s))]+C[g(s,x(s-h))-g(s,y(s-h))]ds+∑0≤tk<tW(t,tk)[Vk(x(tk))-Vk(y(tk))]+∫0tW(t,s)[σ(s,x(s))-σ(s,y(s))]dω(s)‖.
By a minor modification of the proof of (3.13), we can obtain
(3.22)E‖Lx-Ly‖2≤6η2[2λ2(‖B‖2Lf2+‖C‖2Lg2)+N2(1-e-λ)2LV2+Lσ22λ]sup⁡tE‖x(t)-y(t)‖2=ρ‖x-y‖∞2,
and therefore, ∥Lx-Ly∥∞≤ρ∥x-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(4.1)dxi(t)=[aixi(t)+∑j=12bijfj(xj(t))+∑j=12cijgj(xj(t-0.1))+Ii(t)]dt+0.5xi(t)dωi(t),t≥0,t≠tk,Δx(t)=x(tk)-x(tk-)=Dkx(tk-)+Vk(x(tk-))+βk,t=tk,k∈N,x(t)=ϕ(t),-h≤t≤0,
where tk=k,
k∈ℕ,
f(x(t))=[sinx1(t),sinx2(t)]T,
g(x(t-0.1))=[cosx1(t-0.1),cosx2(t-0.1)]T,
Vik=[0.01sinx1(t),0.01cosx2(t)]T,
βk=0.1,I(t)=[0.1,0.1]T,
γ0=0.1, for convenience, we can choose(4.2)A=[-200-3],B=[-0.100-0.1],C=[0.200-0.2],Dk=[-0.500-0.5].
Then μ(A)=-2,∥I+Dk∥=1/2,∥B∥=0.1,∥C∥=0.2,Lf=Lg=1,LV=0.01,Lσ=0.5. Choose θ=infk{tk1}=0.01,η=1,N=6. By simple calculation, we have λ=-(μ(A)+(lnη/θ))=2,ρ≐0.8139<1,K¯≐1.107,(ρK¯/(1-ρ))≐4.841.LetD*={z∈D:E∥z-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.



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*Source: 414320-2012-05-20.xml*