CUBO A Mathematical Journal
Vol.15, No¯ 01, (77–96). March 2013

Existence and stability of almost periodic solutions to
impulsive stochastic differential equations

Junwei Liu and Chuanyi Zhang

Harbin Institute of Technology,

Department of Mathematics, Harbin Institute of Technology,

Harbin 150001, P.R. China.

junweiliuhit@gmail.com

ABSTRACT

This paper introduces the concept of square-mean piecewise almost periodic for im-

pulsive stochastic processes. The existence of square-mean piecewise almost periodic

solutions for linear and nonlinear impulsive stochastic differential equations is estab-

lished by using the theory of the semigroups of the operators and Schauder fixed point

theorem. The stability of the square-mean piecewise almost periodic solutions for non-

linear impulsive stochastic differential equations is investigated.

RESUMEN

Este art́ıculo introduce el concepto de periodicidad cuadrática media por tramos casi

periódica para procesos estocásticos impulsivos. La existencia de soluciones de media

cuadrática casi periódicas para ecuaciones diferenciales estocásticas impulsivas lineales

y no lineales se establece usando la teoŕıa de semigrupos de los operadores y el teo-

rema de punto fijo de Schauder. Se estudia la estabilidad de las soluciones de media

cuadrática por tramos casi periódica para ecuaciones diferenciales estocásticas impul-

sivas no lineales.

Keywords and Phrases: Square-mean piecewise almost periodic; impulsive stochastic differen-

tial equation; the semigroups of the operators; Schauder fixed point theorem; stability

2010 AMS Mathematics Subject Classification: 35B15; 35R12; 60H15; 37C75



78 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

1 Introduction

In recent years, stochastic differential systems have been extensively studied since stochastic mod-

eling plays an important role in physics, engineering, finance, social science and so on. Qual-

itative properties such as existence, uniqueness and stability for stochastic differential systems

have attracted more and more researchers’ attention. The existence of periodic, almost peri-

odic(automorphic), asymptotically almost periodic, pseudo almost periodic(automorphic) solutions

for stochastic differential equations was obtained. We refer the reader to [14, 6, 7, 17, 16, 10, 8, 1, 11]

and references therein.

On the other hand, impulsive phenomenon arises from many different real processes and

phenomena which appeared in physics, chemical technology, population dynamics, biotechnology,

medicine and economics. There has been a significant development in the theory of impulsive

differential equations. For example, the existence of almost periodic (mild) solutions of abstract

impulsive differential equations have been considered in [23, 24, 25, 4, 18, 19].

In [26], the authors combined the two directions and derived firstly some sufficient conditions

for the existence and uniqueness of almost periodic solutions for a class of impulsive stochastic

differential equations with delay. However, these above results quoted concern the case where

the activation functions satisfy Lipschitz conditions. There are few authors have considered the

problem of almost periodic solutions of impulsive stochastic differential equations without Lipschitz

activation functions. On the basis of this, this article is devoted to the discussion of this problem.

Moreover, the stability analysis on impulsive stochastic differential equations has been an

important research topic (see [20, 22, 27]). While, because the mild solutions don’t have stochastic

differentials, Ito’s formula fails to deal with the stability of mild solution to stochastic differential

equations (see [20, 9, 15]). In [9], the authors gave some properties of the stochastic convolution

which ensure the exponential stability of mild solutions.

Motivated by the above discussion, we investigate the existence and stability of almost periodic

solutions for impulsive stochastic differential equations. The paper is organized as follows, in

Section 2 we recall some definitions, the related notations and some useful lemmas. In Sections 3

and 4, we present some criteria ensuring the existence of almost periodic solutions to some linear

and nonlinear impulsive stochastic differential equations, respectively. In Section 5, we discuss the

stability of almost periodic solutions to some impulsive stochastic differential equations.

2 Preliminaries

Throughout this paper, R denotes the set of real numbers, R+ denotes the set of nonnegative real

numbers, Z denotes the set of integers, Z+ denotes the set of nonnegative integers. (H, || · ||) is

assumed to be a real and separable Hilbert space. Let (Ω,F,P) be a complete probability space

and L2(P,H) be a space of the H-valued random variables x such that E||x||2 =
∫
Ω
||x||2dP < ∞.



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Existence and stability of almost periodic solutions to impulsive ... 79

L2(P,H) is a Hilbert space equipped with the norm ||x||2 = (
∫
Ω
||x||2dP)1/2.

Definition 2.1. A stochastic process x : R+ → L2(P,H) is said to be stochastically bounded if there

exists M > 0 such that E||x(t)||2 ≤ M for all t ∈ R+.

Definition 2.2. A stochastic process x : R+ → L2(P,H) is said to be stochastically continuous in

s ∈ R+, if limt→s E||x(t) − x(s)||
2 = 0.

Let T be the set consisting of all real sequences {ti}i∈Z+ such that γ = infi∈Z+(ti+1 − ti) >

0, t0 = 0 and limi→∞ ti = ∞. x(t
+
i ) and x(t

−
i ) represent the right and left limits of x(t) at

ti, i ∈ Z
+, respectively. For {ti}i∈Z+ ∈ T, let PC(R

+,L2(P,H)) be the space consisting of all

stochastically bounded functions φ : R+ → L2(P,H) such that φ(·) is stochastically continuous at t

for any t 6∈ {ti}i∈Z+ and φ(ti) = φ(t
−
i ) for all i ∈ Z

+; let PC(R+ ×L2(P,H),L2(P,H)) be the space

formed by all stochastic processes φ : R+ × L2(P,H) → L2(P,H) such that for any x ∈ L2(P,H),

φ(·,x) is stochastically continuous at t for any t 6∈ {ti}i∈Z+ and φ(ti,x) = φ(t
−
i ,x) for all i ∈ Z

+

and for any t ∈ R+, φ(t, ·) is stochastically continuous at x ∈ L2(P,H).

Definition 2.3. For {ti}i∈Z+ ∈ T, the function φ ∈ PC(R
+,L2(P,H)) is said to be square-mean

piecewise almost periodic if the following conditions are fulfilled:

(1) {t
j
i = ti+j − ti}, j ∈ Z

+, is equipotentially almost periodic, that is, for any ǫ > 0, there

exists a relatively dense set Qǫ of R such that for each τ ∈ Qǫ there is an integer q ∈ Z such that

|ti+q − ti − τ| < ǫ for all i ∈ Z
+.

(2) For any ǫ > 0, there exists a positive number δ = δ(ǫ) such that if the points t′ and t′′

belong to a same interval of continuity of φ and |t′ − t′′| < δ, then E||φ(t′) − φ(t′′)||2 < ǫ.

(3) For every ǫ > 0, there exists a relatively dense set Ω(ǫ) in R such that if τ ∈ Ω(ǫ), then

E||φ(t + τ) − φ(t)||2 < ǫ

for all t ∈ R+ satisfying the condition |t − ti| > ǫ, i ∈ Z
+. The number τ is called ǫ-translation

number of φ.

We denote by APT (R
+,L2(P,H)) the collection of all the square-mean piecewise almost periodic

processes, it thus is a Banach space with the norm ||x||∞ = supt∈R+ ||x(t)||2 = supt∈R+(E||x(t)||
2)

1
2

for x ∈ APT (R
+,L2(P,H)).

Lemma 2.4. Let f ∈ APT (R
+,L2(P,H)), then, R(f), the range of f is a relatively compact set of

L2(P,H).

Refer to [18] for the detailed proof of Lemma 2.4.

Definition 2.5. For {ti}i∈Z+ ∈ T, the function f(t,x) ∈ PC(R
+ × L2(P,H),L2(P,H)) is said to be

square-mean piecewise almost periodic in t ∈ R+ and uniform on compact subset of L2(P,H) if for

every ǫ > 0 and every compact subset K ⊆ L2(P,H), there exists a relatively dense subset Ω of R

such that

E||f(t + τ,x) − f(t,x)||2 < ǫ,



80 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

for all x ∈ K,τ ∈ Ω,t ∈ R+ satisfying |t − ti| > ǫ. The collection of all such processes is denoted

by APT (R
+ × L2(P,H),L2(P,H)).

Lemma 2.6. Suppose that f(t,x) ∈ APT (R
+×L2(P,H),L2(P,H)) and f(t, ·) is uniformly continuous

on each compact subset K ⊆ L2(P,H) uniformly for t ∈ R. That is, for all ǫ > 0, there exists δ > 0

such that x,y ∈ K and E||x − y||2 < δ implies that E||f(t,x) − f(t,y)||2 < ǫ for all t ∈ R+. Then

f(·,x(·)) ∈ APT (R
+,L2(P,H)) for any x ∈ APT (R

+,L2(P,H)).

Proof. Since x ∈ APT (R
+,L2(P,H)), by Lemma 2.4, R(x) is a relatively compact subset of L2(P,H).

Because f(t, ·) is uniformly continuous on each compact subset K ⊆ L2(P,H) uniformly for t ∈ R.

Then for any ǫ > 0, there exists number δ : 0 < δ ≤ ǫ
4
, such that

E||f(t,x1) − f(t,x2)||
2 <

ǫ

4
, (1)

where x1,x2 ∈ R(x) and E||x1 − x2||
2 < δ, t ∈ R. By square-mean piecewise almost periodic of f

and x, there exists a relatively set Ω of R such that the following conditions hold:

E||f(t + τ,x0) − f(t,x0)||
2 <

ǫ

4
, (2)

E||x(t + τ) − x(t)||2 <
ǫ

4
, (3)

for every x0 ∈ R(x) and t ∈ R
+, |t − ti| > ǫ, i ∈ Z

+, τ ∈ Ω. Note that (a + b)2 ≤ 2(a2 + b2) and

E||f(t + τ,x(t + τ)) − f(t,x(t))||2

≤2E||f(t + τ,x(t + τ)) − f(t + τ,x(t))||2 + 2E||f(t + τ,x(t)) − f(t,x(t))||2.

Combing (1), (2) and (3), it follows that

E||f(t + τ,x(t + τ)) − f(t,x(t))||2 < ǫ, t ∈ R+, |t − ti| > ǫ, i ∈ Z
+, τ ∈ Ω.

The proof is complete.

We obtain the following corollary as an immediate consequence of Lemma 2.6.

Corollary 2.7. Let f(t,x) ∈ APT (R
+×L2(P,H),L2(P,H)) and f is Lipschitz, i.e., there is a number

L > 0 such that

E||f(t,x) − f(t,y)||2 ≤ LE||x − y||2,

for all t ∈ R+ and x,y ∈ L2(P,H), if for any x ∈ APT (R
+,L2(P,H)), then f(·,x(·)) ∈ APT (R

+,L2(P,H)).

Definition 2.8. A sequence x : Z+ → L2(P,H) is called a square-mean almost periodic sequence if

the ǫ-translation set of x

T(x;ǫ) = {τ ∈ Z : E||x(n + τ) − x(t)||2 < ǫ, for all n ∈ Z+}

is a relatively dense set in Z for all ǫ > 0.

The collection of all square-mean almost periodic sequences x : Z+ → L2(P,H) will be denoted

by APT (Z
+,L2(P,H)).



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Existence and stability of almost periodic solutions to impulsive ... 81

Remark 2.9. If x(n) ∈ APT (Z
+,L2(P,H)), then {x(n) : n ∈ Z+} is stochastically bounded, that

is, supn∈Z+ E||x(n)||
2 < ∞.

In order to obtain our main results, we introduce the following lemmas.

Let h : R+ → R be a continuous function such that h(t) ≥ 1 for all t ∈ R+ and h(t) → ∞ as

t → ∞. We consider the space

(PC)0h(R
+,L2(P,H)) =

{

u ∈ PC(R+,L2(P,H)) : lim
t→∞

E||u(t)||2

h(t)
= 0

}

.

Endowed with the norm ||u||h = supt∈R+
E||u(t)||

2

h(t)
, it is a Banach space.

Lemma 2.10. A set B ⊆ (PC)0h(R
+,L2(P,H)) is a relatively compact set if and only if

(1) limt→∞
E||x(t)||

2

h(t)
= 0 uniformly for x ∈ B.

(2) B(t) = {x(t) : x ∈ B} is relatively compact in L2(P,H) for every t ∈ R+.

(3) The set B is equicontinuous on each interval (ti,ti+1)(i ∈ Z
+).

Lemma 2.11. Assume that f ∈ APT (R
+,L2(P,H)), the sequence {xi : i ∈ Z

+} is almost periodic

in L2(P,H) and {t
j
i}, j ∈ Z

+, is equipotentially almost periodic. Then for each ǫ > 0 there are

relatively dense sets Ωǫ,f,xi of R and Qǫ,f,xi of Z such that the following conditions hold:

(i) E||f(t + τ) − f(t)||2 < ǫ for all t ∈ R+, |t − ti| > ǫ, τ ∈ Ωǫ,f,xi and i ∈ Z
+.

(ii) E||xi+q − xi||
2 < ǫ for all q ∈ Qǫ,f,xi and i ∈ Z

+.

(iii) For every τ ∈ Ωǫ,f,xi, there exists at least one number q ∈ Qǫ,f,xi such that

|t
q
i − τ| < ǫ, i ∈ Z

+.

Lemma 2.10 and Lemma 2.11 are stochastic generalized versions of Lemma 4.1 in [12] and

Lemma 35 in [23], respectively, and one may refer to [23, 18, 19, 26, 2, 13, 12] for more details.

Here we omit the proofs.

Lemma 2.12. ([9]) For any r ≥ 1 and for arbitrary L2(P,H)-valued process φ(·) such that

sup
s∈[0,t]

E

∣

∣

∣

∣

∣

∣

∣

∣

∫s

0

φ(u)dw(u)

∣

∣

∣

∣

∣

∣

∣

∣

2r

≤ Cr

( ∫t

0

(E||φ(s)||2r)
1
r ds

)r

, t ≥ 0,

where Cr = (r(2r − 1))
r.

3 Almost periodic solutions for linear impulsive stochastic

differential equations

To begin, consider the following linear impulsive stochastic differential equation:
{
dx(t) = [Ax(t) + f(t)]dt + g(t)dw(t), t ≥ 0,t 6= ti, i ∈ Z

+,

△x(ti) = x(t
+
i ) − x(t

−
i ) = βi, i ∈ Z

+,
(4)



82 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

where A is an infinitesimal generator which generates a C0-semigroup {T(t) : t ≥ 0} such that

for all t ≥ 0, ||T(t)|| ≤ Me−δt with M,δ > 0 and {T(t) : t > 0} is compact. Furthermore,

f,g : R → L2(P,H) are two stochastic processes, βi is a square-mean almost periodic sequence and

w(t) is a two-sided standard one-dimensional Brownian motion, which is defined on the filtered

probability space (Ω,F,P,Fσ) with Ft = σ{w(u) − w(v) : u,v ≤ t}.

Definition 3.1. An Ft-progressive process x(t) is called a mild solution of system (4) if it satisfies

the following stochastic integral equation

x(t) = T(t)x0 +

∫t

0

T(t − s)f(s)ds +

∫t

0

T(t − s)g(s)dw(s) +
∑

0<ti<t

T(t − ti)βi.

for all t ≥ 0.

Theorem 3.2. Assume f,g ∈ APT (R
+,L2(P,H)), {βi, i ∈ Z

+} is a square-mean almost periodic

sequence, then system (4) has a square-mean piecewise almost periodic mild solution.

Proof. From semigroup theory, we know

x(t) = T(t)x0 +

∫t

0

T(t − s)f(s)ds +

∫t

0

T(t − s)g(s)dw(s),t ≥ 0,

is a mild solution to

dx(t) = [Ax(t) + f(t)]dt + g(t)dw(t),t ≥ 0.

So for system (4), if t ∈ [t0,t1),

x(t) = T(t)x0 +

∫t

0

T(t − s)f(s)ds +

∫t

0

T(t − s)g(s)dw(s),

which implies

x(t−1 ) = T(t1)x0 +

∫t1

0

T(t1 − s)f(s)ds +

∫t1

0

T(t1 − s)g(s)dw(s),

by using x(t+1 ) = x(t
−
1 ) + β1, for t ∈ (t1,t2), we get

x(t) =T(t − t1)x(t
+
1 ) +

∫t

t1

T(t − s)f(s)ds +

∫t

t1

T(t − s)g(s)dw(s)

=T(t − t1)[x(t
−
1 ) + β1] +

∫t

t1

T(t − s)f(s)ds +

∫t

t1

T(t − s)g(s)dw(s)

=T(t − t1)[T(t1)x0 +

∫t1

0

T(t1 − s)f(s)ds +

∫t1

0

T(t1 − s)g(s)dw(s) + β1]

+

∫t

t1

T(t − s)f(s)ds +

∫t

t1

T(t − s)g(s)dw(s)

=T(t)x0 +

∫t

0

T(t − s)f(s)ds +

∫t

0

T(t − s)g(s)dw(s) + T(t − t1)β1,



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Existence and stability of almost periodic solutions to impulsive ... 83

reiterating this procedure, we can prove that

x(t) = T(t)x0 +

∫t

0

T(t − s)f(s)ds +

∫t

0

T(t − s)g(s)dw(s) +
∑

0<ti<t

T(t − ti)βi, (5)

and by Definition 3.1, (5) is a mild solution of system (4), to finish the proof, we need to prove the

above process (5) is a square-mean piecewise almost periodic process.

Since f,g ∈ APT (R
+,L2(P,H)), from Lemma 31 in [23], for the two almost periodic functions

f,g, there exists a relatively dense set of their common ǫ-translation number. Moreover, {βi, i ∈

Z+} is a square-mean almost periodic sequence, then by Lemma 2.11, for each ǫ > 0, there exist

relatively dense sets Ωǫ,f,g,xi of R and Qǫ,f,g,xi of Z such that the following relations hold:

(1) E||f(t + τ) − f(t)||2 < ǫ,t ∈ R+, |t − ti| > ǫ,i ∈ Z
+,τ ∈ Ωǫ,f,g,xi.

(2) E||g(t + τ) − g(t)||2 < ǫ,t ∈ R+, |t − ti| > ǫ,i ∈ Z
+,τ ∈ Ωǫ,f,g,xi.

(3) E||xi+q − xi||
2 < ǫ,i ∈ Z+,q ∈ Qǫ,f,g,xi.

(4) For each τ ∈ Ωǫ,f,g,xi, ∃ q ∈ Qǫ,f,g,xi, s.t. |ti+q − ti − τ| < ǫ, i ∈ Z
+.

We write x(t) of (5) as

x(t) = T(t)x0 + x1(t) + x2(t) + x3(t)

where

x1(t) =

∫t

0

T(t − s)f(s)ds, x2(t) =

∫t

0

T(t − s)g(s)dw(s), x3(t) =
∑

0<ti<t

T(t − ti)βi.

(i) x1 ∈ APT (R
+,L2(P,H)). By (1), for τ ∈ Ωǫ,f,g,xi,t ∈ R

+, |t − ti| > ǫ,i ∈ Z
+, one obtains

E||x1(t + τ) − x1(t)||
2 =E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

0

T(t − s)[f(s + τ) − f(s)]ds

∣

∣

∣

∣

∣

∣

∣

∣

2

≤E

[ ∫t

0

Me−δ(t−s)||f(s + τ) − f(s)||ds

]2

≤E

[ ∫t

0

M2e−δ(t−s)ds

∫t

0

e−δ(t−s)||f(s + τ) − f(s)||2ds

]

≤
M2

δ

∫t

0

e−δ(t−s)E||f(s + τ) − f(s)||2ds

≤
M2

δ

∫t

0

e−δ(t−s)ǫds ≤
M2

δ2
ǫ.

(ii) x2 ∈ APT (R
+,L2(P,H)). Let w̃(s) = w(s + τ) −w(τ) for each s ∈ R+. Note that w̃ is also



84 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

a Brownian motion and has the same distribution as w. By Lemma 2.12 and (2), we have

E||x2(t + τ) − x2(t)||
2 =E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

0

T(t − s)g(s + τ)dw(s + τ) −

∫t

−∞

T(t − s)g(s)dw(s)

∣

∣

∣

∣

∣

∣

∣

∣

2

=E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

0

T(t − s)[g(s + τ) − g(s)]dw̃(s)

∣

∣

∣

∣

∣

∣

∣

∣

2

≤

∫t

0

E||T(t − s)[g(s + τ) − g(s)]||2ds

≤

∫t

0

M2e−2δ(t−s)E||g(s + τ) − g(s)||2ds

≤

∫t

0

M2e−2δ(t−s)ǫds =
M2

2δ
ǫ.

(iii) x3 ∈ APT (R
+,L2(P,H)). Define

r(t) = T(t − ti)βi, ti < t ≤ ti+1, i ∈ Z
+.

For ti < t ≤ ti+1, |t − ti| > ǫ, |t − ti+1| > ǫ,i ∈ Z
+, by (4), we can get

t + τ > ti + ǫ + τ > ti+q,

and

ti+q+1 > ti+1 + τ − ǫ > t + τ,

that is, ti+q+1 > t + τ > ti+q. Since (a + b)
2 ≤ 2(a2 + b2), one has

E||r(t + τ) − r(t)||2

=E||T(t + τ − ti+q)βi+q − T(t − ti)βi||
2

=E||[T(t + τ − ti+q) − T(t − ti)]βi+q + T(t − ti)[βi+q − βi]||
2

≤2E||[T(t + τ − ti+q) − T(t − ti)]βi+q||
2 + 2E||T(t − ti)[βi+q − βi]||

2

≤2||T(t + τ − ti+q) − T(t − ti)||
2E||βi+q||

2 + 2||T(t − ti)||E||βi+q − βi||
2

≤2||T(t + τ − ti+q) − T(t − ti)||
2E||βi+q||

2 + 2M2ǫ,

since {T(t) : t ≥ 0} is a C0-semigroup (see [21, 3]), for the above ǫ, there exists 0 < µ < ǫ < 1 such

that 0 < s < µ implies ||T(t − ti + s) − T(t − ti)|| < ǫ. Note that M0 = supi∈Z E||βi||
2 < ∞, so

E||r(t + τ) − r(t)||2 ≤ 2M0ǫ
2 + 2M2ǫ.

Next we will prove that r is uniformly continuous on each interval (ti,ti+1)(i ∈ Z
+). Let

t,h ∈ R+ such that ti < t,t + h < ti+1, then

E||r(t + h) − r(t)||2 ≤ ||T(t + h − ti) − T(t − ti)||
2E||βi||

2.



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Existence and stability of almost periodic solutions to impulsive ... 85

Since {T(t) : t ≥ 0} is a C0-semigroup and M0 = supi∈Z+ E||βi||
2 < ∞, we conclude that E||r(t +

h) − r(t)||2 → 0 as h → 0 independent of t and i.

Finally, by Cauchy-Schwarz inequality and (3),

E

∣

∣

∣

∣

∣

∣

∣

∣

∑

0<ti<t+τ

T(t + τ − ti)βi −
∑

0<ti<t

T(t − ti)βi

∣

∣

∣

∣

∣

∣

∣

∣

2

≤E

[

∑

0<ti<t

||T(t − ti)[βi+q − βi]||

]2

≤E

[

∑

0<ti<t

Me−δ(t−ti)||βi+q − βi||

]2

≤E

[

( ∑

0<ti<t

M2e−δ(t−ti)
) ∑

0<ti<t

e−δ(t−ti)||βi+q − βi||
2

]

≤
M2

1 − e−δγ

∑

0<ti<t

e−δ(t−ti)E||βi+q − βi||
2

≤
M2

1 − e−δγ

∑

0<ti<t

e−δ(t−ti)ǫ

≤
M2

(1 − e−δγ)2
ǫ.

In view of the above, it is clear that x3 ∈ APT (R
+,L2(P,H)).

Furthermore, since {T(t) : t ≥ 0} is a bounded C0-semigroup and {T(t) : t > 0} is compact,

by Theorem 2.1 in [5], T(·)x0 ∈ APT (R
+,L2(P,H)). By combing (i), (ii) and (iii), it follows that

(5) is a square-mean piecewise almost periodic process, so system (4) has a square-mean piecewise

almost periodic solution. The proof is complete.

4 Almost periodic solutions for nonlinear impulsive stochas-

tic differential equations

Consider the following nonlinear impulsive stochastic differential equation

{
dx(t) = [Ax(t) + f(t,x(t))]dt + g(t,x(t))dw(t), t ≥ 0,t 6= ti, i ∈ Z

+,

△x(ti) = x(t
+
i ) − x(t

−
i ) = Ii(x(ti)), i ∈ Z

+,
(6)

where f,g : R+ × L2(P,H) → L2(P,H), Ii : L
2(P,H) → L2(P,H), i ∈ Z+ and w(t) is a two-sided

standard one dimensional Brownian motion defined on the filtered probability space (Ω,F,P,Fσ)

with Ft = σ{w(u) − w(v) : u,v ≤ t}.

Definition 4.1. An Ft-progressive process x(t) is called a mild solution of system (6) if it satisfies



86 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

the corresponding stochastic integral equation

x(t) = T(t)x0 +

∫t

0

T(t−s)f(s,x(s))ds+

∫t

0

T(t−s)g(s,x(s))dw(s) +
∑

0<ti<t

T(t−ti)Ii(x(ti)). (7)

for all t ≥ 0.

In order to obtain the existence of square-mean piecewise almost periodic solution to system

(6), we introduce the following assumptions:

(A1) The operator A : D(A) ⊆ L2(P,H) → L2(P,H) is the infinitesimal generator of an

exponentially stable C0-semigroup {T(t) : t ≥ 0} on L
2(P,H), i.e., ||T(t)|| ≤ Me−δt, t ≥ 0, M,δ > 0.

Moreover, T(t) is compact for t > 0.

(A2) f,g ∈ APT (R
+ × L2(P,H),L2(P,H)), for each compact set K ⊆ L2(P,H), g(t, ·),f(t, ·)

are uniformly continuous in each compact set K ⊆ L2(P,H) uniformly for t ∈ R+. Ii(x) is almost

periodic in i ∈ Z+ uniformly in x ∈ K and is a uniformly continuous function defined on the set

K ⊆ L2(P,H) for all i ∈ Z+.

(A3) FL = sup{t∈R+, E||x||2≤L} E||f(t,x)||
2 < ∞, GL = sup{t∈R+, E||x||2≤L} E||g(t,x)||

2 < ∞, IL =

sup{i∈Z+, E||x||2≤L} E||Ii(x(ti))||
2 < ∞, where L is an arbitrary positive number. Moreover, there

exist a number L0 > 0 such that 4M
2L0 +

4M
2

δ2
FL0 +

2M
2

δ
GL0 +

4M
2

(1−e−δγ)2
IL0 ≤ L0.

Theorem 4.2. Assume that the conditions (A1)-(A3) are satisfied, then the impulsive stochastic

differential equation (6) admits at least one square-mean piecewise almost periodic solution.

Proof. Let

B = {x ∈ APT (R
+,L2(P,H)) : E||x||2 ≤ L0}.

Obviously, B is a closed set of APT (R
+,L2(P,H)). Define Γ on (PC)0h(R

+,L2(P,H)),

Γx(t) = T(t)x0 +

∫t

0

T(t − s)f(s,x(s))ds +

∫t

0

T(t − s)g(s,x(s))dw(s) +
∑

0<ti<t

T(t − ti)Ii(x(ti)).

In order to show that the impulsive stochastic differential equation (6) has a square-mean piecewise

almost periodic solution, we only need to prove the operator Γ has a fixed point in B.

First we show Γx ∈ B,x ∈ B. For x ∈ B, by Lemma 2.6 and (A2), we have f(·,x(·)),g(·,x(·)) ∈

APT (R
+,L2(P,H)), by (A2) and Lemma 37 in [23], Ii(x(ti)) is a square-mean almost periodic

sequence, analogous to the proof of Theorem 3.2, we can show Γx ∈ APT (R
+,L2(P,H)).

Since (a+b+c+d)2 ≤ 4(a2 +b2 +c2 +d2), by using Cauchy-Schwarz inequality and Lemma



CUBO
15, 1 (2013)

Existence and stability of almost periodic solutions to impulsive ... 87

2.12, we obtain

E||Γx(t)||2

≤4E||T(t)x0||
2 + 4E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

0

T(t − s)f(s,x(s))ds

∣

∣

∣

∣

∣

∣

∣

∣

2

+ 4E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

0

T(t − s)g(s,x(s))dw(s)

∣

∣

∣

∣

∣

∣

∣

∣

2

+ 4E

∣

∣

∣

∣

∣

∣

∣

∣

∑

0<ti<t

T(t − ti)Ii(x(ti))

∣

∣

∣

∣

∣

∣

∣

∣

2

≤4M2E||x0||
2 + 4E

[ ∫t

0

Me−δ(t−s)||f(s,x(s))||ds

]2

+ 4

∫t

0

E||T(t − s)g(s,x(s))||2ds

+ 4E

[

∑

0<ti<t

Me−δ(t−s)||Ii(x(ti))||

]2

≤4M2L0 + 4E

[ ∫t

0

M2e−δ(t−s)ds

∫t

0

e−δ(t−s)||f(s,x(s))||2ds

]

+ 4

∫t

0

M2e−2δ(t−s)E||g(s,x(s))||2ds + 4E

[

( ∑

0<ti<t

M2e−δ(t−ti)
) ∑

0<ti<t

e−δ(t−ti)||Ii(x(ti))||
2

]

≤4M2L0 +
4M2

δ

∫t

0

e−δ(t−s)E||f(s,x(s))||2ds + 4

∫t

0

M2e−2δ(t−s)E||g(s,x(s))||2ds

+
4M2

1 − e−δγ

∑

0<ti<t

e−δ(t−ti)E||Ii(x(ti))||
2

≤4M2L0 +
4M2

δ

∫t

0

e−δ(t−s)FL0ds + 4

∫t

0

M2e−2δ(t−s)GL0ds +
4M2

1 − e−δγ

∑

0<ti<t

e−δ(t−ti)IL0

≤4M2L0 +
4M2

δ2
FL0 +

2M2

δ
GL0 +

4M2

(1 − e−δγ)2
IL0,

since 4M2L0 +
4M

2

δ2
FL0 +

2M
2

δ
GL0 +

4M
2

(1−e−δγ)2
IL0 ≤ L0, then E||Γx||

2 ≤ L0, that is, Γx ∈ B,x ∈ B.

Next we show B(t) = {Γx(t) : x ∈ B} is a relatively compact subset of L2(P,H) for each t ∈ R+.

For each t ∈ R+, 0 < ǫ < 1, x ∈ B, define

Γǫx(t) =T(t)x0 +

∫t−ǫ

0

T(t − s)f(s,x(s))ds +

∫t−ǫ

0

T(t − s)g(s,x(s))dw(s)

+
∑

0<ti<t−ǫ

T(t − ti)Ii(x(ti))

=T(ǫ)

[

T(t − ǫ)x0 +

∫t−ǫ

0

T(t − ǫ − s)f(s,x(s))ds

+

∫t−ǫ

0

T(t − ǫ − s)g(s,x(s))dw(s) +
∑

0<ti<t−ǫ

T(t − ǫ − ti)Ii(x(ti))

]

=T(ǫ)Γx(t − ǫ).

Since {Γx(t − ǫ) : x ∈ B} is bounded and T(ǫ) is compact, {Γǫx(t) : x ∈ B} is a relatively compact

subset of L2(P,H). Moreover, for ǫ is small enough and the points t and t − ǫ belong to the same



88 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

interval of continuity of x, then

Γx(t) − Γǫx(t) =

∫t

t−ǫ

T(t − s)f(s,x(s))ds +

∫t

t−ǫ

T(t − s)g(s,x(s))dw(s),

since (a + b)2 ≤ 2(a2 + b2), by using Cauchy-Schwarz inequality and Lemma 2.12, one has

E||Γx(t) − Γǫx(t)||2

≤2

[

E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

t−ǫ

T(t − s)f(s,x(s))ds

∣

∣

∣

∣

∣

∣

∣

∣

2

+ E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

t−ǫ

T(t − s)g(s,x(s))dw(s)

∣

∣

∣

∣

∣

∣

∣

∣

2
]

≤2E

[ ∫t

t−ǫ

Me−δ(t−s)||f(s,x(s))||ds

]2

+ 2

∫t

t−ǫ

E||T(t − s)g(s,x(s))||2ds

≤2E

[ ∫t

t−ǫ

M2e−δ(t−s)ds

∫t

t−ǫ

e−δ(t−s)||f(s,x(s))||2ds

]

+ 2

∫t

t−ǫ

M2e−2δ(t−s)E||g(s,x(s))||2ds

≤2M2ǫ

∫t

t−ǫ

E||f(s,x(s))||2ds + 2M2
∫t

t−ǫ

E||g(s,x(s))||2ds

≤2M2ǫ2FL0 + 2M
2ǫGL0,

so B(t) = {Γx(t) : x ∈ B} is a relatively compact subset of L2(P,H) for each t ∈ R+.

Finally we show {Γx : x ∈ B} is equicontinuous at each interval (ti,ti+1)(i ∈ Z
+). Let

x ∈ B, ti < t
′′ < t′ < ti+1, i ∈ Z

+, and ρ < min
{

ǫ
36M2FL0

, ǫ
36M2GL0

,1
}
,

Γx(t′) − Γx(t′′)

=T(t′)x0 +

∫t′

0

T(t′ − s)f(s,x(s))ds +

∫t′

0

T(t′ − s)g(s,x(s))dw(s)

+
∑

0<ti<t
′

T(t′ − ti)Ii(x(ti)) − T(t
′′)x0 −

∫t′′

0

T(t′′ − s)f(s,x(s))ds

−

∫t′′

0

T(t′′ − s)g(s,x(s))dw(s) −
∑

0<ti<t
′′

T(t′′ − ti)Ii(x(ti))

=[T(t′) − T(t′′)]x0 +

∫t′

t′′
T(t′ − s)f(s,x(s))ds +

∫t′

t′′
T(t′ − s)g(s,x(s))dw(s)

+

∫t′′

0

[T(t′ − s) − T(t′′ − s)]f(s,x(s))ds +

∫t′′

0

[T(t′ − s) − T(t′′ − s)]g(s,x(s))dw(s)

+
∑

0<ti<t
′′

[T(t′ − ti) − T(t
′′ − ti)]Ii(x(ti)).

Since {T(t) : t ≥ 0} is a C0-semigroup, there exists µ < ρ such that t
′ − t′′ < µ implies that

||T(t) − I||2 ≤ min

{
ǫ

36M2L0
,

ǫδ2

36M2FL0
,

ǫδ

18M2GL0
,
ǫ(1 − e−δγ)2

36M2IL0

}

.



CUBO
15, 1 (2013)

Existence and stability of almost periodic solutions to impulsive ... 89

By using Cauchy-Schwarz inequality and Lemma 2.12, we have

E||[T(t′) − T(t′′)]x0||
2 = E||[T(t′ − t′′) − I]T(t′′)x0||

2

≤ ||T(t′ − t′′) − I||2M2E||x0||
2

≤
ǫ

36M2L0
M2L0 =

ǫ

36
,

and,

E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′

t′′
T(t′ − s)f(s,x(s))ds

∣

∣

∣

∣

∣

∣

∣

∣

2

≤E

[ ∫t′

t′′
Me−δ(t

′
−s)

||f(s,x(s))||ds

]2

≤E

[ ∫t′

t′′
M2e−δ(t

′
−s)ds

∫t′

t′′
e−δ(t

′
−s)

||f(s,x(s))||2ds

]

≤M2(t′ − t′′)

∫t′

t′′
e−δ(t

′
−s)E||f(s,x(s))||2ds

≤M2FL0(t
′ − t′′)2

≤M2FL0
ǫ

36M2FL0
=
ǫ

36
,

and

E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′

t′′
T(t′ − s)g(s,x(s))dw(s)

∣

∣

∣

∣

∣

∣

∣

∣

2

≤

∫t′

t′′
E||T(t′ − s)g(s,x(s))||2ds

≤

∫t′

t′′
M2e−2δ(t

′
−s)E||g(s,x(s))||2ds

≤M2GL0(t
′ − t′′)

≤M2GL0
ǫ

36M2GL0
=
ǫ

36
,

and

E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′′

0

[T(t′ − s) − T(t′′ − s)]f(s,x(s))ds

∣

∣

∣

∣

∣

∣

∣

∣

2

=E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′′

0

[T(t′ − t′′) − I]T(t′′ − s)f(s,x(s))ds

∣

∣

∣

∣

∣

∣

∣

∣

2

≤E

[ ∫t′′

0

||T(t′ − t′′) − I||Me−δ(t
′′
−s)

||f(s,x(s))||ds

]2

≤E

[ ∫t′′

0

||T(t′ − t′′) − I||2M2e−δ(t
′′
−s)ds

∫t′′

0

e−δ(t
′′
−s)

||f(s,x(s))||2ds

]

≤||T(t′ − t′′) − I||2
M2

δ

∫t′′

0

e−δ(t
′′
−s)E||f(s,x(s))||2ds

≤||T(t′ − t′′) − I||2
M2

δ

∫t′′

0

e−δ(t
′′
−s)FL0ds

≤
ǫδ2

36M2FL0

M2

δ2
FL0 =

ǫ

36
,



90 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

and

E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′′

0

[T(t′ − s) − T(t′′ − s)]g(s,x(s))dw(s)

∣

∣

∣

∣

∣

∣

∣

∣

2

≤

∫t′′

0

E||[T(t′ − t′′) − I]T(t′′ − s)g(s,x(s))||2ds

≤

∫t′′

0

||T(t′ − t′′) − I||2M2e−2δ(t
′′
−s)E||f(s,x(s))||2ds

≤||T(t′ − t′′) − I||2
∫t′′

0

M2e−2δ(t
′′
−s)GL0ds

≤
ǫδ

18M2GL0

M2

2δ
GL0 =

ǫ

36
,

and

E

∣

∣

∣

∣

∣

∣

∣

∣

∑

0<ti<t
′′

[T(t′ − ti) − T(t
′′ − ti)]Ii(x(ti))

∣

∣

∣

∣

∣

∣

∣

∣

2

=E

∣

∣

∣

∣

∣

∣

∣

∣

∑

0<ti<t
′′

[T(t′ − t′′) − I]T(t′′ − ti)Ii(x(ti))

∣

∣

∣

∣

∣

∣

∣

∣

2

≤E

[

∑

0<ti<t
′′

||T(t′ − t′′) − I||||T(t′′ − ti)||||Ii(x(ti))||

]2

≤E

[

( ∑

0<ti<t
′′

||T(t′ − t′′) − I||2e−δ(t
′′
−ti)

) ∑

0<ti<t
′′

e−δ(t
′′
−ti)||Ii(x(ti))||

2

]

≤||T(t′ − t′′) − I||2
M2

1 − e−δγ

∑

0<ti<t
′′

e−δ(t
′′
−ti)E||Ii(x(ti))||

2

≤||T(t′ − t′′) − I||2
M2

1 − e−δγ

∑

0<ti<t
′′

e−δ(t
′′
−ti)IL0

≤
ǫ(1 − e−δγ)2

36M2IL0

M2

(1 − e−δγ)2
IL0 =

ǫ

36
,



CUBO
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Existence and stability of almost periodic solutions to impulsive ... 91

so that, for x ∈ B and t′ − t′′ < µ, t′,t′′ ∈ (ti,ti+1), i ∈ Z,

E||Γx(t′) − Γx(t′′)||2

≤6E||[T(t′) − T(t′′)]x0||
2 + 6E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′

t′′
T(t′ − s)f(s,x(s))ds

∣

∣

∣

∣

∣

∣

∣

∣

2

+ 6E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′

t′′
T(t′ − s)g(s,x(s))dw(s)

∣

∣

∣

∣

∣

∣

∣

∣

2

+ 6E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′′

0

[T(t′ − s) − T(t′′ − s)]f(s,x(s))ds

∣

∣

∣

∣

∣

∣

∣

∣

2

+ 6E

∣

∣

∣

∣

∣

∣

∣

∣

∫t′′

0

[T(t′ − s) − T(t′′ − s)]g(s,x(s))dw(s)

∣

∣

∣

∣

∣

∣

∣

∣

2

+ 6E

∣

∣

∣

∣

∣

∣

∣

∣

∑

0<ti<t
′′

[T(t′ − ti) − T(t
′′ − ti)]Ii(x(ti))

∣

∣

∣

∣

∣

∣

∣

∣

2

≤ ǫ,

which shows that {Γx : x ∈ B} is equicontinuous at each interval (ti,ti+1)(i ∈ Z
+).

Since {Γx : x ∈ B} ⊆ (PC)0h(R
+,L2(P,H)) and {Γx : x ∈ B} satisfies the conditions of Lemma

2.10, {Γx : x ∈ B} is a relatively compact set, moreover, the Lebesgue dominated convergence

theorem and our assumptions on f, g and Ii imply that Γ is continuous, then Γ is a compact

operator. It follows from Schauder fixed point theorem that Γ has a fixed point in B. Thus x is a

square-mean piecewise almost periodic solution of system (6). The proof is complete.

Note that the uniformly continuous is weaker than the Lipschitz continuous, if (A2) is replaced

by the following condition:

(A2’) f,g ∈ APT (R
+,L2(P,H)), Ii(x) is almost periodic in i ∈ Z

+ uniformly for x ∈ L2(P,H),

and there exists positive numbers L1,L2,L such that

E||f(t,x) − f(t,y)||2 ≤ L1E||x − y||
2,

E||g(t,x) − g(t,y)||2 ≤ L2E||x − y||
2,

E||Ii(x) − Ii(y)||
2 ≤ LE||x − y||2,

for all x,y ∈ L2(P,H),t ∈ R+,

then by Lemma 2.7 and Theorem 3.2, we can also get the almost periodic solution of system (6).

Corollary 4.3. Suppose that the conditions (A1), (A2’) and (A3) are satisfied, then the impulsive

stochastic differential equation (6) has a square-mean piecewise almost periodic solution.

5 Stability

In this section we consider the stability of square-mean piecewise almost periodic solution to system

(6) with Lipschitz activation function. In the sequel, we will need the following lemma.



92 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

Lemma 5.1. ([23]) Let a nonnegative piecewise continuous function u(t) satisfy for t ≥ t0 the

inequality

u(t) ≤ C +

∫t

t0

v(τ)u(τ)dτ +
∑

t0<τi<t

βiu(τi),

where C ≥ 0, βi ≥ 0, v(τ) > 0, and τ
′
is are discontinuity points of first type of the function u(t).

Then the following estimate holds for the function u(t),

u(t) ≤ C
∏

t0<τi<t

(1 + βi)e
∫
t

t0
v(τ)dτ

.

Theorem 5.2. Assume the conditions of Corollary 4.3 are fulfilled. Assume further that 1
γ
ln(1+

4M
2
L

1−e−δγ
) + 4M

2
L1

δ
+ 4M2L2 < 0. Then system (6) has an exponentially stable almost periodic

solution.

Proof. By Corollary 4.3, system (6) has a mild square-mean piecewise almost periodic solution

u(t),

u(t) = T(t)u0 +

∫t

0

T(t − s)f(s,u(s))ds +

∫t

0

T(t − s)g(s,u(s))dw(s) +
∑

0<ti<t

T(t − ti)Ii(u(ti)).

Let u(t) = u(t,0,ϕ) and v(t) = v(t,0,ψ) be two solutions of equation (6), then

u(t) = T(t)ϕ +

∫t

0

T(t − s)f(s,u(s))ds +

∫t

0

T(t − s)g(s,u(s))dw(s) +
∑

0<ti<t

T(t − ti)Ii(u(ti)),

v(t) = T(t)ψ +

∫t

0

T(t − s)f(s,v(s))ds +

∫t

0

T(t − s)g(s,v(s))dw(s) +
∑

0<ti<t

T(t − ti)Ii(v(ti)).

Since (a + b + c + d)2 ≤ 4(a2 + b2 + c2 + d2), by (A2’), Cauchy-Schwarz inequality and Lemma



CUBO
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Existence and stability of almost periodic solutions to impulsive ... 93

2.12, we have

E||u(t) − v(t)||2

=E

∣

∣

∣

∣

∣

∣

∣

∣

T(t)ϕ − T(t)ψ +

∫t

0

T(t − s)[f(s,u(s)) − f(s,v(s))]ds

+

∫t

0

T(t − s)[g(s,u(s)) − g(s,v(s))]dw(s) +
∑

0<ti<t

T(t − ti)[Ii(u(ti)) − Ii(v(ti))]

∣

∣

∣

∣

∣

∣

∣

∣

2

≤4E||T(t)[ϕ − ψ]||2 + 4E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

0

T(t − s)[f(s,u(s)) − f(s,v(s))]ds

∣

∣

∣

∣

∣

∣

∣

∣

2

+ 4E

∣

∣

∣

∣

∣

∣

∣

∣

∫t

0

T(t − s)[g(s,u(s)) − g(s,v(s))]dw(s)

∣

∣

∣

∣

∣

∣

∣

∣

2

+ 4E

∣

∣

∣

∣

∣

∣

∣

∣

∑

0<ti<t

T(t − ti)[Ii(u(ti)) − Ii(v(ti))]

∣

∣

∣

∣

∣

∣

∣

∣

2

≤4M2e−2δtE||ϕ − ψ||2 + 4E

[ ∫t

0

Me−δ(t−s)||f(s,u(s)) − f(s,v(s))||ds

]2

+ 4

∫t

0

E||T(t − s)[g(s,u(s)) − g(s,v(s))]||2ds

+ 4E

[

∑

0<ti<t

Me−δ(t−ti)||Ii(u(ti)) − Ii(v(ti))||

]2

≤4M2e−2δtE||ϕ − ψ||2 + 4E

[ ∫t

0

M2e−δ(t−s)ds

∫t

0

e−δ(t−s)||f(s,u(s)) − f(s,v(s))||2ds

]

+ 4

∫t

0

M2e−2δ(t−s)E||g(s,u(s)) − g(s,v(s))||2ds

+ 4E

[

( ∑

0<ti<t

M2e−δ(t−ti)
) ∑

0<ti<t

e−δ(t−ti)||Ii(u(ti)) − Ii(v(ti))||
2

]

≤4M2e−δtE||ϕ − ψ||2 +
4M2

δ

∫t

0

e−δ(t−s)E||f(s,u(s)) − f(s,v(s))||2ds

+ 4M2
∫t

0

e−δ(t−s)E||g(s,u(s)) − g(s,v(s))||2ds

+
4M2

1 − e−δγ

∑

0<ti<t

e−δ(t−ti)E||Ii(u(ti)) − Ii(v(ti))||
2

≤4M2e−δtE||ϕ − ψ||2 +

(

4M2L1

δ
+ 4M2L2

) ∫t

0

e−δ(t−s)E||u(s) − v(s)||2ds

+
4M2L

1 − e−δγ

∑

0<ti<t

e−δ(t−ti)E||u(ti) − v(ti)||
2.



94 Junwei Liu and Chuanyi Zhang CUBO
15, 1 (2013)

Then,

eδtE||u(t) − v(t)||2 ≤4M2E||ϕ − ψ||2 +

(

4M2L1

δ
+ 4M2L2

) ∫t

0

eδsE||u(s) − v(s)||2ds

+
4M2L

1 − e−δγ

∑

0<ti<t

eδtiE||u(ti) − v(ti)||
2.

Let Υ(t) = eδtE||u(t) − v(t)||2, then

Υ(t) ≤ 4M2Υ(0) +

(

4M2L1

δ
+ 4M2L2

) ∫t

0

Υ(s)ds +
4M2L

1 − e−δγ

∑

0<ti<t

Υ(ti).

By Lemma 5.1, we have

Υ(t) ≤4M2Υ(0)
∏

0<ti<t

(

1 +
4M2L

1 − e−δγ

)

e
∫
t

0

(

4M2L1
δ

+4M
2
L2

)

ds

=4M2Υ(0)
∏

0<ti<t

(

1 +
4M2L

1 − e−δγ

)

e

(

4M2L1
δ

+4M
2
L2

)

t

≤4M2Υ(0)

(

1 +
4M2L

1 − e−δγ

)
t
γ

e

(

4M2L1
δ

+4M
2
L2

)

t

=4M2Υ(0)e

(

1
γ

ln(1+ 4M
2L

1−e−δγ
)+

4M2L1
δ

+4M
2
L2

)

t
,

that is,

E||u(t) − v(t)||2 ≤ 4M2E||ϕ − ψ||2e

(

1
γ

ln(1+ 4M
2L

1−e−δγ
)+

4M2L1
δ

+4M
2
L2

)

t
.

Since 1
γ
ln(1+ 4M

2
L

1−e−δγ
)+ 4M

2
L1

δ
+4M2L2 < 0. The square-mean piecewise almost periodic solution

of system (6) is exponentially stable. This completes the proof.

Received: December 2012. Revised: February 2013.

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