Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 473969, 13 pages
http://dx.doi.org/10.1155/2013/473969
Research Article
Almost Automorphic Solutions to Nonautonomous Stochastic
Functional Integrodifferential Equations
Li Xi-liang and Han Yu-liang
School of Mathematics and Information Sciences, Shandong Institute of Business and Technology, Yantai 264005, China
Correspondence should be addressed to Li Xi-liang; lixiliang@amss.ac.cn
Received 28 January 2013; Accepted 7 March 2013
Academic Editor: Zhiming Guo
Copyright © 2013 L. Xi-liang and H. Yu-liang. 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.
This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional
integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni”
conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost
automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic
solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.
1. Introduction
Stochastic differential equations in both finite and infinite
dimensions, which are important from the viewpoint of
applications since they incorporate natural randomness into
the mathematical description of the phenomena and hence
provide a more accurate description of it, have received
considerable attention. Based on this viewpoint, there has
been an increasing interest in extending certain classical
deterministic results to stochastic differential equations in
recent years. As a good case in point, the existence of almost
periodic or pseudo-almost periodic solutions to stochastic
evolution equations has been extensively considered in many
publications; see [1–8] and the references therein.
Integrodifferential equations are used to describe lots of
phenomena arising naturally from many fields such as fluid
dynamics, number reactor dynamics, population dynamics,
electromagnetic theory, and biological models, most of which
cannot be described by classical differential equations, and
hence they have attracted more and more attention in recent
years; see [1, 9–12] for more details.
Recently, Keck and McKibben [9, 10] proposed a
general abstract model for semilinear functional stochastic integrodifferential equations and studied the existence
and uniqueness of mild solutions to these equations.
Based on their works, the existence and uniqueness of
square-mean almost periodic solutions to some functional
integrodifferential stochastic evolution equations was carefully investigated in [1] for the autonomous case and in our
forthcoming paper for the nonautonomous case. In a very
recent paper, as a natural generalization of the notion of
square-mean almost periodicity, a new concept of squaremean almost automorphic stochastic process was introduced
by Fu and Liu [13], and the existence results of squaremean almost automorphic mild solutions to some linear
and semilinear autonomous stochastic differential equations
were formulated, while paper [14] investigated the same
issue for nonautonomous stochastic differential equations.
Under some suitable assumptions, the authors established in
a forthcoming paper the existence and uniqueness of squaremean almost automorphic solutions to a class of autonomous
functional integrodifferential stochastic evolution equations.
In this paper, we are concerned with a class of semilinear nonautonomous functional stochastic integrodifferential
equations in a real separable Hilbert space in the abstract
form:
𝑋󸀠 (𝑡) = 𝐴 (𝑡) 𝑋 (𝑡) + ∫
𝑡
−∞
+∫
𝑡
−∞
𝐶 (𝑡 − 𝑠) 𝐺 (𝑠, 𝑋 (𝑠)) 𝑑𝑊 (𝑠)
𝐵 (𝑡 − 𝑠) 𝐹2 (𝑠, 𝑋 (𝑠)) 𝑑𝑠
+ 𝐹1 (𝑡, 𝑋 (𝑡)) ,
𝑡 ∈ R,
(1)
2
Abstract and Applied Analysis
where 𝐴(𝑡) : 𝐷(𝐴(𝑡)) ⊂ L2 (P; H) → L2 (P; H) is a
family of densely defined closed (possibly unbounded) linear operator satisfying the so-called “Acquistapace-Terreni”
conditions introduced in [15], 𝐵 and 𝐶 are convolution-type
kernels in L1 (0, ∞) and L2 (0, ∞), respectively, satisfying
Assumption 3.2 in [16], 𝑊(𝑡) is a two-sided standard onedimensional Brownian motion defined on the filtered probability space (Ω, F, P, F𝑡 ), where F𝑡 = 𝜎{𝑊(𝑢)−𝑊(V); 𝑢, V ≤
𝑡}. Here 𝐹1 , 𝐹2 , 𝐺 : R × L2 (P; H) → L2 (P; H) are jointly
continuous functions satisfying some additional conditions
to be specified later in Section 3.
Motivated by the aforementioned works [1, 13, 14],
we investigate in this paper the existence and uniqueness
of square-mean almost automorphic solutions to nonautonomous equation (1). The main tools employed here are
Banach contraction principle and an estimate on the Ito
integral. The obtained results can be seen as a contribution
to this emerging field.
The paper is organized as follows. In Section 2, we review
some basic definitions and preliminary facts on square-mean
almost automorphic processes which will be used throughout
this paper. Section 3 is devoted to establish the existence,
uniqueness, and the asymptotical stability of square-mean
almost automorphic mild solution to (1). As an illustration
of our abstract result, square-mean almost automorphic
solution to a concrete nonautonomous integrodifferential
stochastic evolution equation is investigated in Section 4.
Definition 1. A standard one-dimensional Brownian motion
is a continuous, adapted real-valued stochastic process
(𝑊(𝑡), 𝑡 ≥ 0) such that
2. Preliminaries
Lemma 4. If 𝑋, 𝑋1 , and 𝑋2 are all square-mean almost
automorphic stochastic processes, then the following statements
hold true:
To begin this paper, we recall some primary definitions,
notations, lemmas, and technical results which will be used
in the sequel. For more details on almost automorphy and
stochastic differential equations, the readers are referred to
[13, 17–23] and the references therein.
Throughout this paper, we assume that (H, ‖⋅‖) is a real
separable Hilbert space, (Ω, F, P) is a probability space,
and L2 (P; H) stands for the space of all H-valued random
variables 𝑋 such that
E‖𝑋‖2 = ∫ ‖𝑋‖2 𝑑P < ∞.
Ω
(2)
(i) 𝑊(0) = 0 a.s.;
(ii) 𝑊(𝑡) − 𝑊(𝑠) is independent of F𝑠 for all 0 ≤ 𝑠 < 𝑡;
(iii) 𝑊(𝑡) − 𝑊(𝑠) is 𝑁(0, 𝑡 − 𝑠) distributed for all 0 ≤ 𝑠 ≤ 𝑡.
The following definitions and lemmas concerning squaremean almost automorphic functions can be found in [13, 14].
Definition 2. A stochastic process 𝑋 : R → L2 (P; H) is said
to be stochastically continuous if
lim E‖𝑋 (𝑡) − 𝑋 (𝑠)‖2 = 0.
(4)
𝑡→𝑠
Definition 3. A stochastically continuous stochastic process
𝑋 : R → L2 (P; H) is said to be square-mean almost
automorphic if for every sequence of real numbers {𝑠𝑛󸀠 } there
exists a subsequence {𝑠𝑛 } and a stochastic process 𝑌 : R →
L2 (P; H) such that
󵄩2
󵄩
lim E󵄩󵄩𝑋 (𝑡 + 𝑠𝑛 ) − 𝑌 (𝑡)󵄩󵄩󵄩 = 0,
𝑛→∞ 󵄩
(5)
󵄩2
󵄩
lim E󵄩󵄩󵄩𝑋 (𝑡 + 𝑠𝑛 ) − 𝑌 (𝑡)󵄩󵄩󵄩 = 0
𝑛→∞
hold for each 𝑡 ∈ R. The collection of all square-mean
almost automorphic stochastic processes is denoted by
𝐴𝐴(R; L2 (P; H)).
(i) 𝑋1 + 𝑋2 is square-mean almost automorphic;
(ii) 𝜆𝑋 is square-mean almost automorphic for every scalar
𝜆;
(iii) There exists a constant 𝑀 > 0 such that sup𝑡∈R ‖𝑋 (𝑡)‖2
≤ 𝑀. That is, 𝑋 is bounded in L2 (P; H).
Lemma 5. 𝐴𝐴(R; L2 (P; H)) is a Banach space when it is
equipped with the norm
‖𝑋‖∞ := sup‖𝑋 (𝑡)‖2 = sup(E‖𝑋 (𝑡)‖2 )
𝑡∈R
𝑡∈R
1/2
(6)
for 𝑋 ∈ 𝐴𝐴(R; L2 (P; H)).
For 𝑋 ∈ L2 (P; H), let
‖𝑋‖2 := (∫ ‖𝑋‖2 𝑑P)
Ω
1/2
.
(3)
It is routine to check that L2 (P; H) is a Banach space
equipped with the norm ‖⋅‖2 .
It is well-known that Brownian motion plays a key role
in the construction of stochastic integrals. Throughout this
paper, 𝑊(𝑡) denotes a two-sided standard one-dimensional
Brownian motion defined on the filtered probability space
(Ω, F, P, F𝑡 ), where F𝑡 = 𝜎{𝑊(𝑢) − 𝑊(V); 𝑢, V ≤ 𝑡}.
Definition 6. A jointly continuous function 𝐹 : R ×
L2 (P; H) → L2 (P; H), (𝑡, 𝑋) 󳨃→ 𝐹(𝑡, 𝑋) is said to be
square-mean almost automorphic in 𝑡 ∈ R for each 𝑋 ∈
L2 (P; H) if for every sequence of real numbers {𝑠𝑛󸀠 } there
exists a subsequence {𝑠𝑛 } and a stochastic process 𝐺 : R ×
L2 (P; H) → L2 (P; H) such that
󵄩
󵄩2
lim E󵄩󵄩𝐹 (𝑡 + 𝑠𝑛 , 𝑋) − 𝐺 (𝑡, 𝑋)󵄩󵄩󵄩 = 0,
𝑛→∞ 󵄩
(7)
󵄩
󵄩2
lim E󵄩󵄩󵄩𝐺 (𝑡 − 𝑠𝑛 , 𝑋) − 𝐹 (𝑡, 𝑋)󵄩󵄩󵄩 = 0
𝑛→∞
hold for each 𝑡 ∈ R and each 𝑋 ∈ L2 (P; H).
Abstract and Applied Analysis
3
Lemma 7. Let 𝑓 : R × L2 (P; H) → L2 (P; H), (𝑡, 𝑋) 󳨃→
𝑓(𝑡, 𝑋) be square-mean almost automorphic in 𝑡 ∈ R for
each 𝑋 ∈ L2 (P; H), and assume that 𝑓 satisfies a Lipschitz
condition in the following sense:
󵄩2
󵄩2
󵄩
󵄩
E󵄩󵄩󵄩𝑓 (𝑡, 𝜑) − 𝑓 (𝑡, 𝜓)󵄩󵄩󵄩 ≤ 𝐿E󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩
(8)
for all 𝜑, 𝜓 ∈ L2 (P; H) and each 𝑡 ∈ R, where 𝐿 > 0 is independent of 𝑡. Then for any square-mean almost automorphic
stochastic process 𝑋 : R → L2 (P; H), the stochastic process
𝐹 : R → L2 (P; H) given by 𝐹(𝑡) := 𝑓(𝑡, 𝑋(𝑡)) is square-mean
almost automorphic.
The Acquistapace-Terreni conditions (ATCs, for short)
play an important role in the study of nonautonomous
stochastic differential equations. We state it below for the
readers’ convenience.
ATCs. There exist constants 𝜆 0 ≥ 0, 𝜃 ∈ (𝜋/2, 𝜋), 𝐿, 𝐾 ≥ 0,
and 𝛼, 𝛽 ∈ (0, 1] with 𝛼 + 𝛽 > 1 such that
󵄩󵄩
󵄩󵄩(𝐴 (𝑡) − 𝜆 0 ) 𝑅 (𝜆, 𝐴 (𝑡) − 𝜆 0 )
󵄩
󵄩󵄩
−𝛿(𝑡−𝑠)
,
󵄩󵄩𝑈 (𝑡 + 𝑠𝑛 , 𝑠 + 𝑠𝑛 ) − 𝑈 (𝑡, 𝑠)󵄩󵄩󵄩 ≤ 𝜀𝑒
󵄩
󵄩󵄩
−𝛿(𝑡−𝑠)
󵄩󵄩𝑈 (𝑡 − 𝑠𝑛 , 𝑠 − 𝑠𝑛 ) − 𝑈 (𝑡, 𝑠)󵄩󵄩󵄩 ≤ 𝜀𝑒
(9)
󵄩
× [𝑅 (𝜆 0 , 𝐴 (𝑡)) − 𝑅 (𝜆 0 , 𝐴 (𝑠))]󵄩󵄩󵄩 ≤ 𝐿|𝑡 − 𝑠|𝛼 |𝜆|𝛽
for all 𝑡 ≥ 𝑠, where 𝛿 > 0 is the constant required in
(H1).
(H3) The functions 𝐹𝑖
:
R × L2 (P; H)
→
2
L (P; H), (𝑡, 𝑋) 󳨃→ 𝐹𝑖 (𝑡, 𝑋) (𝑖 = 1, 2), and
𝐺 : R × L2 (P; H) → L2 (P; H), (𝑡, 𝑋) 󳨃→ 𝐺(𝑡, 𝑋) are
square-mean almost automorphic in 𝑡 ∈ R for each
𝑋 ∈ L2 (P; H). Moreover, 𝐹1 , 𝐹2 , and 𝐺 are Lipschitz
in 𝑋 uniformly for 𝑡 in the following sense: there
exist constants 𝐾𝑖 > 0 (𝑖 = 1, 2, 3) such that
󵄩2
󵄩
E󵄩󵄩󵄩𝐹𝑖 (𝑡, 𝑋) − 𝐹𝑖 (𝑡, 𝑌)󵄩󵄩󵄩 ≤ 𝐾𝑖 E‖𝑋 − 𝑌‖2 ,
𝑖 = 1, 2,
E‖𝐺 (𝑡, 𝑋) − 𝐺 (𝑡, 𝑌)‖2 ≤ 𝐾3 E‖𝑋 − 𝑌‖2
Definition 9. An F𝑡 progressively measurable process
(𝑋(𝑡))𝑡∈R is called a mild solution of (1) if it satisfies the
corresponding stochastic integral equation:
𝑡
𝜎
𝑎
𝑎
𝑡
𝜎
𝑎
𝑎
+ ∫ 𝑈 (𝑡, 𝜎) ∫ 𝐶 (𝜎 − 𝑠) 𝐺 (𝑠, 𝑋 (𝑠)) 𝑑𝑊 (𝑠) 𝑑𝜎
Lemma 8. Suppose that the ATCs are satisfied, and then
there exists a unique evolution family {𝑈(𝑡, 𝑠)}−∞<𝑠≤𝑡<∞ on
L2 (P; H), which governs the linear part of (1).
+ ∫ 𝑈 (𝑡, 𝜎) ∫ 𝐵 (𝜎 − 𝑠) 𝐹2 (𝑠, 𝑋 (𝑠)) 𝑑𝑠 𝑑𝜎
𝑡
+ ∫ 𝑈 (𝑡, 𝑠) 𝐹1 (𝑠, 𝑋 (𝑠)) 𝑑𝑠
𝑎
(13)
3. Main Results
In this section, we investigate the existence and uniqueness
of square-mean almost automorphic solution to the nonautonomous stochastic integrodifferential evolution equation
(1). The following assumptions are imposed on (1) which will
be assumed throughout the manuscript.
(H1) The operator 𝐴(𝑡) : 𝐷(𝐴(𝑡)) ⊂ L2 (P; H) →
L2 (P; H) is a family of densely defined closed linear
operators satisfying the ATCs, and the generated
evolution family 𝑈(𝑡, 𝑠) is uniformly exponentially
stable; that is, there exist constants 𝑀 ≥ 1 and 𝛿 > 0
such that
∀𝑡 ≥ 𝑠.
(12)
𝑋 (𝑡) = 𝑈 (𝑡, 𝑎) 𝑋 (𝑎)
for 𝑡, 𝑠 ∈ R, 𝜆 ∈ Σ𝜃 := {𝜆 ∈ C − {0} : |arg 𝜆| ≤ 𝜃}.
The following lemma can be found in [15, 24, 25].
‖𝑈 (𝑡, 𝑠)‖ ≤ 𝑀𝑒−𝛿(𝑡−𝑠) ,
(11)
for all stochastic processes 𝑋, 𝑌 ∈ L2 (P; H) and 𝑡 ∈
R.
Σ𝜃 ∪ {0} ⊂ 𝜌 (𝐴 (𝑡) − 𝜆 0 ) ,
𝐾
󵄩󵄩
󵄩
,
󵄩󵄩𝑅 (𝜆, 𝐴 (𝑡) − 𝜆 0 )󵄩󵄩󵄩 ≤
1 + |𝜆|
{𝑠𝑛 }𝑛∈N such that, for any 𝜀 > 0, there exists an 𝑁 ∈ N
such that 𝑛 > 𝑁 implies that
(10)
for all 𝑡 ≥ 𝑎 and each 𝑎 ∈ R.
Now we are in a position to show the existence and
uniqueness of square-mean almost automorphic solution to
(1).
Theorem 10. Assume that conditions (H1)–(H3) are satisfied,
then (1) has a unique square-mean almost automorphic mild
solution 𝑋(⋅) ∈ 𝐴𝐴(R; L2 (P; H)) which can be explicitly
expressed as
𝑋 (𝑡) = ∫
𝑡
−∞
+∫
𝑈 (𝑡, 𝜎) ∫
𝑡
−∞
(H2) The evolution family {𝑈(𝑡, 𝑠), 𝑡 ≥ 𝑠} generated by 𝐴(𝑡)
satisfies the following condition: from every sequence
of real numbers {𝑠𝑛󸀠 }𝑛∈N , we can extract a subsequence
+∫
𝑡
−∞
𝜎
−∞
𝑈 (𝑡, 𝜎) ∫
𝐶 (𝜎 − 𝑠) 𝐺 (𝑠, 𝑋 (𝑠)) 𝑑𝑊 (𝑠) 𝑑𝜎
𝜎
−∞
𝐵 (𝜎 − 𝑠) 𝐹2 (𝑠, 𝑋 (𝑠)) 𝑑𝑠 𝑑𝜎
𝑈 (𝑡, 𝑠) 𝐹1 (𝑠, 𝑋 (𝑠)) 𝑑𝑠
(14)
4
Abstract and Applied Analysis
provided that
Θ := 3
2
𝑀
[𝐾1 + 𝐾2 ⋅ ‖𝐵‖2L1 (0,∞) + 𝐾3 ⋅ ‖𝐶‖2L2 (0,∞) ] < 1.
𝛿2
(15)
Proof. First of all, it is not difficult to verify that the stochastic
process
𝑋 (𝑡) = ∫
𝑡
𝜎
𝑈 (𝑡, 𝜎) ∫
−∞
+∫
−∞
𝑡
𝜎
𝑈 (𝑡, 𝜎) ∫
−∞
+∫
𝐶 (𝜎 − 𝑠) 𝐺 (𝑠, 𝑋 (𝑠)) 𝑑𝑊 (𝑠) 𝑑𝜎
−∞
𝑡
𝐵 (𝜎 − 𝑠) 𝐹2 (𝑠, 𝑋 (𝑠)) 𝑑𝑠 𝑑𝜎
󵄩󵄩2
󵄩̃
lim E󵄩󵄩󵄩𝑓
󵄩󵄩 = 0
1 (𝑡 − 𝑠𝑛 ) − 𝑓1 (𝑡)󵄩
𝑛→∞ 󵄩
(16)
is well defined and satisfies
𝑋 (𝑡) = 𝑈 (𝑡, 𝑎) 𝑋 (𝑎)
𝑡
𝜎
𝑎
𝑎
𝑡
𝜎
𝑎
𝑎
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩2 = 0,
lim E󵄩󵄩󵄩󵄩𝑓1 (𝑡 + 𝑠𝑛 ) − 𝑓
1
󵄩
𝑛→∞
𝑈 (𝑡, 𝑠) 𝐹1 (𝑠, 𝑋 (𝑠)) 𝑑𝑠
−∞
We show first that S1 𝑋 is square-mean almost automorphic
whenever 𝑋 is. Indeed, assuming that 𝑋 is square-mean
almost automorphic, then Lemma 7 implies that 𝑓1 (⋅) :=
𝐹1 (⋅, 𝑋(⋅)) is also square-mean almost automorphic. Hence,
for any sequence of real numbers {𝑠𝑛󸀠 }, there exists a sub̃ : R →
sequence {𝑠𝑛 } of {𝑠𝑛󸀠 } and a stochastic process 𝑓
1
2
L (P; H) such that
+ ∫ 𝑈 (𝑡, 𝜎) ∫ 𝐶 (𝜎 − 𝑠) 𝐺 (𝑠, 𝑋 (𝑠)) 𝑑𝑊 (𝑠) 𝑑𝜎
+ ∫ 𝑈 (𝑡, 𝜎) ∫ 𝐵 (𝜎 − 𝑠) 𝐹2 (𝑠, 𝑋 (𝑠)) 𝑑𝑠 𝑑𝜎
𝑡
(20)
hold for each 𝑡 ∈ R. By assumption (H2), for any 𝜀 > 0, there
exists 𝑁1 = 𝑁1 (𝜀) ∈ N such that 𝑛 > 𝑁1 implies that
󵄩󵄩
󵄩
−𝛿(𝑡−𝑠)
∀𝑡 ≥ 𝑠,
󵄩󵄩𝑈 (𝑡 + 𝑠𝑛 , 𝑠 + 𝑠𝑛 ) − 𝑈 (𝑡, 𝑠)󵄩󵄩󵄩 ≤ 𝜀𝑒
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩2 < 𝜀,
E󵄩󵄩󵄩󵄩𝑓1 (𝑡 + 𝑠𝑛 ) − 𝑓
1
󵄩
󵄩󵄩2
󵄩󵄩 ̃
E󵄩󵄩󵄩𝑓1 (𝑡 − 𝑠𝑛 ) − 𝑓1 (𝑡)󵄩󵄩󵄩 < 𝜀 ∀𝑡 ∈ R.
(21)
+ ∫ 𝑈 (𝑡, 𝑠) 𝐹1 (𝑠, 𝑋 (𝑠)) 𝑑𝑠
𝑎
(17)
for all 𝑡 ≥ 𝑎 and each 𝑎 ∈ R, and hence it is a mild solution of
the original (1).
To seek the square-mean almost automorphic mild solution to (1), let us consider the nonlinear operator S acting on
the Banach space 𝐴𝐴(R; L2 (P; H)) given by
𝑡
𝑈 (𝑡, 𝜎) ∫
(S𝑋) (𝑡) := ∫
−∞
+∫
−∞
𝑡
−∞
+∫
𝜎
𝑡
−∞
𝑈 (𝑡, 𝜎) ∫
𝜎
𝐵 (𝜎 − 𝑠) 𝐹2 (𝑠, 𝑋 (𝑠)) 𝑑𝑠 𝑑𝜎
𝑈 (𝑡, 𝑠) 𝐹1 (𝑠, 𝑋 (𝑠)) 𝑑𝑠.
(18)
2
If we can show that the operator S maps 𝐴𝐴(R; L (P; H))
into itself and it is a contraction mapping, then, by Banach
contraction principle, we can conclude that there is a unique
square-mean almost automorphic mild solution to (1).
Now define three nonlinear operators acting on the
Banach space 𝐴𝐴(R; L2 (P; H)) as follows:
𝑡
−∞
(S2 𝑋) (𝑡) := ∫
𝑡
−∞
(S3 𝑋) (𝑡) := ∫
𝑡
−∞
𝑈 (𝑡, 𝑠) 𝐹1 (𝑠, 𝑋 (𝑠)) 𝑑𝑠,
𝑈 (𝑡, 𝜎) ∫
𝜎
−∞
𝜎
𝑈 (𝑡, 𝜎) ∫
−∞
𝑢 (𝑡) := ∫
𝐵 (𝜎−𝑠) 𝐹2 (𝑠, 𝑋 (𝑠)) 𝑑𝑠 𝑑𝜎,
𝐶 (𝜎−𝑠) 𝐺 (𝑠, 𝑋 (𝑠)) 𝑑𝑊 (𝑠) 𝑑𝜎.
(19)
𝑡
−∞
𝑢̃ (𝑡) := ∫
𝑡
−∞
𝐶 (𝜎 − 𝑠) 𝐺 (𝑠, 𝑋 (𝑠)) 𝑑𝑊 (𝑠) 𝑑𝜎
−∞
(S1 𝑋) (𝑡) := ∫
Now, define functions on R as follows:
𝑈 (𝑡, 𝑠) 𝑓1 (𝑠) 𝑑𝑠
(22)
̃ (𝑠) 𝑑𝑠,
𝑈 (𝑡, 𝑠) 𝑓
1
and then the above observation together with (3.1) and
Cauchy-Schwarz inequality implies that for any 𝜀 > 0 and
for the aforementioned 𝑁1 = 𝑁1 (𝜀) ∈ N if 𝑛 > 𝑁1 it yields
that
󵄩2
󵄩
E󵄩󵄩󵄩𝑢 (𝑡 + 𝑠𝑛 ) − 𝑢̃ (𝑡)󵄩󵄩󵄩
󵄩󵄩 𝑡+𝑠𝑛
𝑡
󵄩󵄩2
󵄩
̃ (𝑠) 𝑑𝑠󵄩󵄩󵄩
= E󵄩󵄩󵄩∫ 𝑈 (𝑡 + 𝑠𝑛 , 𝑠) 𝑓1 (𝑠) 𝑑𝑠 − ∫ 𝑈 (𝑡, 𝑠) 𝑓
1
󵄩󵄩
󵄩󵄩 −∞
−∞
󵄩
󵄩󵄩 𝑡
󵄩
= E 󵄩󵄩󵄩∫ 𝑈 (𝑡 + 𝑠𝑛 , 𝑠 + 𝑠𝑛 ) 𝑓1 (𝑠 + 𝑠𝑛 ) 𝑑𝑠
󵄩󵄩 −∞
󵄩󵄩2
𝑡
̃ (𝑠) 𝑑𝑠󵄩󵄩󵄩
− ∫ 𝑈 (𝑡, 𝑠) 𝑓
1
󵄩󵄩
−∞
󵄩
󵄩󵄩 𝑡
󵄩󵄩2
󵄩
󵄩
≤ 2E󵄩󵄩󵄩∫ [𝑈 (𝑡 + 𝑠𝑛 , 𝑠 + 𝑠𝑛 ) − 𝑈 (𝑡, 𝑠)] 𝑓1 (𝑠 + 𝑠𝑛 ) 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩 −∞
󵄩󵄩
󵄩󵄩2
󵄩󵄩 𝑡
󵄩
̃ (𝑠)] 𝑑𝑠󵄩󵄩󵄩
+ 2E󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝑠) [𝑓1 (𝑠 + 𝑠𝑛 ) − 𝑓
1
󵄩󵄩
󵄩󵄩 −∞
󵄩
Abstract and Applied Analysis
≤ 2𝜀2 E(∫
𝑡
󵄩
󵄩
𝑒−𝛿(𝑡−𝑠) 󵄩󵄩󵄩𝑓1 (𝑠 + 𝑠𝑛 )󵄩󵄩󵄩 𝑑𝑠)
−∞
+ 2𝑀2 E(∫
𝑡
𝑡
𝑒−𝛿(𝑡−𝑠) 𝑑𝑠) (∫
−∞
𝑡
−∞
𝑡
× (∫
−∞
2
and then, by making change of variables 𝜏 = 𝜎 − 𝑠𝑛 and 𝑟 =
𝑠 − 𝑠𝑛 , we obtain that
2
󵄩
󵄩2
𝑒−𝛿(𝑡−𝑠) E󵄩󵄩󵄩𝑓1 (𝑠 + 𝑠𝑛 )󵄩󵄩󵄩 𝑑𝑠)
𝑒−𝛿(𝑡−𝑠) 𝑑𝑠)
󵄩
̃ (𝑠)󵄩󵄩󵄩󵄩2 𝑑𝑠)
𝑒−𝛿(𝑡−𝑠) E󵄩󵄩󵄩󵄩𝑓1 (𝑠 + 𝑠𝑛 ) − 𝑓
1
󵄩
𝑡
−𝛿(𝑡−𝑠)
≤ 2𝜀 (∫
−∞
𝑒
2
+ 2𝑀 (∫
𝑡
−∞
≤
𝑡
−∞
+ 2𝑀2 (∫
2
󵄩
̃ (𝑠)󵄩󵄩󵄩󵄩 𝑑𝑠)
𝑒−𝛿(𝑡−𝑠) 󵄩󵄩󵄩󵄩𝑓1 (𝑠 + 𝑠𝑛 ) − 𝑓
1
󵄩
−∞
≤ 2𝜀2 (∫
5
󵄩
󵄩2
E󵄩󵄩󵄩V (𝑡 + 𝑠𝑛 ) − ̃V (𝑡)󵄩󵄩󵄩
𝜎
󵄩󵄩󵄩 𝑡+𝑠𝑛
= E 󵄩󵄩󵄩∫ 𝑈 (𝑡 + 𝑠𝑛 , 𝜎) ∫ 𝐵 (𝜎 − 𝑠) 𝑓2 (𝑠) 𝑑𝑠 𝑑𝜎
−∞
󵄩󵄩 −∞
󵄩󵄩2
𝑡
𝜎
̃ (𝑠) 𝑑𝑠 𝑑𝜎󵄩󵄩󵄩
− ∫ 𝑈 (𝑡, 𝜎) ∫ 𝐵 (𝜎 − 𝑠) 𝑓
2
󵄩󵄩
−∞
−∞
󵄩
󵄩󵄩 𝑡
󵄩
= E 󵄩󵄩󵄩∫ 𝑈 (𝑡 + 𝑠𝑛 , 𝜎 + 𝑠𝑛 )
󵄩󵄩 −∞
2
󵄩
󵄩2
𝑑𝑠) supE󵄩󵄩󵄩𝑓1 (𝑡 + 𝑠𝑛 )󵄩󵄩󵄩
×∫
−∞
𝑡∈R
−𝛿(𝑡−𝑠)
𝑒
2
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩2
𝑑𝑠) supE󵄩󵄩󵄩󵄩𝑓1 (𝑡 + 𝑠𝑛 ) − 𝑓
1
󵄩
𝑡∈R
2𝑀1 2 2𝑀2
⋅ 𝜀 + 2 ⋅ 𝜀,
𝛿2
𝛿
−∫
󵄩2
󵄩
lim E󵄩󵄩󵄩𝑢̃ (𝑡 − 𝑠𝑛 ) − 𝑢 (𝑡)󵄩󵄩󵄩 = 0
󵄩󵄩2
󵄩
𝐵 (𝜎 − 𝑠) 𝑓2 (𝑠 + 𝑠𝑛 ) 𝑑𝑠 𝑑𝜎󵄩󵄩󵄩
󵄩󵄩
−∞
×∫
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩2 = 0,
lim E󵄩󵄩󵄩󵄩𝑓2 (𝑡 + 𝑠𝑛 ) − 𝑓
2
󵄩
󵄩󵄩2
󵄩̃
lim E󵄩󵄩󵄩𝑓
󵄩󵄩 = 0
2 (𝑡 − 𝑠𝑛 ) − 𝑓2 (𝑡)󵄩
𝑛→∞ 󵄩
≤ 2𝜀2 E (∫
𝑡
−∞
̃V (𝑡) := ∫
𝑡
−∞
𝑈 (𝑡, 𝜎) ∫
𝜎
−∞
𝑈 (𝑡, 𝜎) ∫
𝜎
(26)
−∞
𝑡
+2𝑀2 E (∫ 𝑒−𝛿(𝑡−𝜎)
−∞
󵄩󵄩 𝜎
󵄩
× 󵄩󵄩󵄩∫ 𝐵 (𝜎−𝑠)
󵄩󵄩 −∞
2
󵄩󵄩
̃ (𝑠)]𝑑𝑠󵄩󵄩󵄩 𝑑𝜎) .
× [𝑓2 (𝑠+𝑠𝑛 )− 𝑓
2
󵄩󵄩󵄩
(28)
Let us evaluate the first term of the right-handed side by using
Cauchy-Schwarz inequality:
2
󵄩󵄩
󵄩󵄩 𝜎
󵄩
󵄩
𝑒−𝛿(𝑡−𝜎) 󵄩󵄩󵄩∫ 𝐵 (𝜎 − 𝑠) 𝑓2 (𝑠 + 𝑠𝑛 ) 𝑑𝑠󵄩󵄩󵄩 𝑑𝜎)
󵄩󵄩
󵄩󵄩 −∞
−∞
E(∫
𝑡
𝑡
≤ (∫
(27)
𝑒−𝛿(𝑡−𝜎)
2
󵄩󵄩 𝜎
󵄩󵄩
󵄩
󵄩
× 󵄩󵄩󵄩∫ 𝐵 (𝜎 − 𝑠) 𝑓2 (𝑠 + 𝑠𝑛 ) 𝑑𝑠󵄩󵄩󵄩 𝑑𝜎)
󵄩󵄩 −∞
󵄩󵄩
(25)
𝐵 (𝜎 − 𝑠) 𝑓2 (𝑠) 𝑑𝑠 𝑑𝜎
̃ (𝑠) 𝑑𝑠 𝑑𝜎,
𝐵 (𝜎 − 𝑠) 𝑓
2
𝑡
−∞
hold for each 𝑡 ∈ R. Now define functions on R as follows:
V (𝑡) := ∫
𝜎
󵄩󵄩2
̃ (𝑠)] 𝑑𝑠 𝑑𝜎󵄩󵄩󵄩
× [𝑓2 (𝑠 + 𝑠𝑛 ) − 𝑓
2
󵄩󵄩
󵄩
(24)
for each 𝑡 ∈ R. Thus, we conclude that 𝑢 = S1 𝑋 ∈
𝐴𝐴(R; L2 (P; H)).
In an analogous way, assuming that 𝑋 is square-mean
almost automorphic and using Lemma 7, one can easily see
that 𝑓2 (⋅) := 𝐹2 (⋅, 𝑋(⋅)) is also square-mean almost automorphic. Let {𝑠𝑛󸀠 } be an arbitrary sequence of real numbers, and
then there exists a subsequence {𝑠𝑛 } of {𝑠𝑛󸀠 } and a stochastic
̃ : R → L2 (P; H) such that
process 𝑓
2
𝑛→∞
󵄩󵄩2
̃ (𝑠) 𝑑𝑠 𝑑𝜎󵄩󵄩󵄩
𝐵 (𝜎 − 𝑠) 𝑓
2
󵄩󵄩󵄩
−∞
𝜎
𝑈 (𝑡, 𝜎) ∫
󵄩󵄩 𝑡
𝜎
󵄩
+ 2E 󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝜎) ∫ 𝐵 (𝜎 − 𝑠)
󵄩󵄩 −∞
−∞
for each 𝑡 ∈ R. And we can show in a similar way that
𝑛→∞
𝑡
−∞
where 𝑀1 := sup𝑡∈R E ‖ 𝑓1 (𝑡)‖2 < +∞. Hence,
󵄩
󵄩2
lim E󵄩󵄩󵄩𝑢 (𝑡 + 𝑠𝑛 ) − 𝑢̃ (𝑡)󵄩󵄩󵄩 = 0
𝐵 (𝜎 − 𝑠) 𝑓2 (𝑠 + 𝑠𝑛 ) 𝑑𝑠 𝑑𝜎
󵄩󵄩 𝑡
󵄩
≤ 2E 󵄩󵄩󵄩∫ [𝑈 (𝑡 + 𝑠𝑛 , 𝜎 + 𝑠𝑛 ) − 𝑈 (𝑡, 𝜎)]
󵄩󵄩 −∞
(23)
𝑛→∞
𝜎
−∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
󵄩󵄩 𝜎
󵄩󵄩2
󵄩
󵄩
𝑒−𝛿(𝑡−𝜎) E󵄩󵄩󵄩∫ 𝐵 (𝜎 − 𝑠) 𝑓2 (𝑠 + 𝑠𝑛 ) 𝑑𝑠󵄩󵄩󵄩 𝑑𝜎
󵄩
󵄩󵄩
−∞
󵄩 −∞
×∫
𝑡
6
Abstract and Applied Analysis
≤ (∫
𝑡
−∞
× (∫
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩2
⋅ supE󵄩󵄩󵄩󵄩𝑓2 (𝑡 + 𝑠𝑛 ) − 𝑓
2
󵄩
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
𝑡
𝑡∈R
𝜎
−∞
𝑒−𝛿(𝑡−𝜎) {∫
−∞
× {∫
𝜎
−∞
󵄩
󵄩2
‖𝐵 (𝜎 − 𝑠)‖ E󵄩󵄩󵄩𝑓2 (𝑠 + 𝑠𝑛 )󵄩󵄩󵄩 𝑑𝑠} 𝑑𝜎)
2
𝑡
≤ {∫
−∞
≤
‖𝐵 (𝜎 − 𝑠)‖ 𝑑𝑠}
+∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎} ⋅ {∫
0
‖𝐵‖2L1 (0,∞)
𝛿2
⋅ 𝜀.
(30)
2
‖𝐵 (𝑢)‖ 𝑑𝑢}
2
󵄩2
󵄩
󵄩2 ‖𝐵‖L1 (0,∞)
󵄩
⋅ supE󵄩󵄩󵄩𝑓2 (𝑠 + 𝑠𝑛 )󵄩󵄩󵄩 ≤
⋅ supE󵄩󵄩󵄩𝑓2 (𝑡)󵄩󵄩󵄩 .
𝛿2
𝑠∈R
𝑡∈R
(29)
As to the second term, in a similar manner as above, we have
the following observation:
2
󵄩󵄩
󵄩󵄩 𝜎
󵄩
̃ (𝑠)] 𝑑𝑠󵄩󵄩󵄩 𝑑𝜎)
𝑒−𝛿(𝑡−𝜎) 󵄩󵄩󵄩∫ 𝐵 (𝜎 − 𝑠) [𝑓2 (𝑠 + 𝑠𝑛 ) − 𝑓
2
󵄩󵄩
󵄩󵄩 −∞
−∞
󵄩
E(∫
≤
1
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩2
⋅ ‖𝐵‖2L1 (0,∞) ⋅ supE󵄩󵄩󵄩󵄩𝑓2 (𝑡 + 𝑠𝑛 ) − 𝑓
2
2
󵄩
𝛿
𝑡∈R
Based on the above argument, for arbitrary 𝜀 > 0, thanks to
the boundedness and square-mean almost automorphy of 𝑓2 ,
there exists certain 𝑁2 = 𝑁2 (𝜀) ∈ N such that 𝑛 > 𝑁2 implies
that
󵄩2
󵄩
E󵄩󵄩󵄩V (𝑡 + 𝑠𝑛 ) − ̃V (𝑡)󵄩󵄩󵄩
≤ 2𝜀2 E (∫
𝑡
𝑒−𝛿(𝑡−𝜎)
−∞
2
󵄩󵄩 𝜎
󵄩󵄩
󵄩
󵄩
× 󵄩󵄩󵄩∫ 𝐵 (𝜎 − 𝑠) 𝑓2 (𝑠 + 𝑠𝑛 ) 𝑑𝑠󵄩󵄩󵄩 𝑑𝜎)
󵄩󵄩 −∞
󵄩󵄩
𝑡
𝑡
𝑡
+2𝑀2 E (∫ 𝑒−𝛿(𝑡−𝜎)
−∞
≤ E [(∫ 𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
󵄩󵄩 𝜎
󵄩
× 󵄩󵄩󵄩∫ 𝐵 (𝜎−𝑠)
󵄩󵄩 −∞
−∞
𝑡
×(∫ 𝑒−𝛿(𝑡−𝜎)
−∞
󵄩󵄩 𝜎
󵄩
× 󵄩󵄩󵄩∫ 𝐵 (𝜎−𝑠)
󵄩󵄩 −∞
󵄩󵄩2
̃ (𝑠)]𝑑𝑠󵄩󵄩󵄩 𝑑𝜎)]
× [𝑓2 (𝑠+𝑠𝑛 )− 𝑓
2
󵄩󵄩
󵄩
≤ (∫
𝑡
−∞
𝑡
−∞
𝑒−𝛿(𝑡−𝜎)
󵄩󵄩 𝜎
󵄩
× E 󵄩󵄩󵄩∫ 𝐵 (𝜎 − 𝑠)
󵄩󵄩 −∞
󵄩󵄩2
̃ (𝑠)] 𝑑𝑠󵄩󵄩󵄩 𝑑𝜎)
× [𝑓2 (𝑠 + 𝑠𝑛 ) − 𝑓
2
󵄩󵄩
󵄩
≤ (∫
𝑡
−∞
× (∫
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
𝑡
−𝛿(𝑡−𝜎)
−∞
𝑒
𝜎
{∫
−∞
× {∫
𝜎
−∞
‖𝐵 (𝜎−𝑠)‖ 𝑑𝑠}
‖𝐵 (𝜎−𝑠)‖
󵄩
̃ (𝑠)󵄩󵄩󵄩󵄩2 𝑑𝑠} 𝑑𝜎)
×E 󵄩󵄩󵄩󵄩𝑓2 (𝑠+𝑠𝑛 )− 𝑓
2
󵄩
𝑡
≤ {∫
−∞
2
⋅𝜀 +
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
× (∫
2
󵄩󵄩
̃ (𝑠)]𝑑𝑠󵄩󵄩󵄩 𝑑𝜎)
× [𝑓2 (𝑠+𝑠𝑛 )− 𝑓
2
󵄩󵄩
󵄩
󵄩2
󵄩
2‖𝐵‖2L1 (0,∞) ⋅ sup𝑡∈R E󵄩󵄩󵄩𝑓2 (𝑡)󵄩󵄩󵄩
≤
𝛿2
2
+∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎} ⋅ {∫
0
2
‖𝐵 (𝑢)‖ 𝑑𝑢}
2𝑀2 ⋅ ‖𝐵‖2L1 (0,∞)
𝛿2
⋅ 𝜀,
(31)
where we use the fact that, for arbitrary 𝜀 > 0, there exists
𝑁2 = 𝑁2 (𝜀) ∈ N such that, for all 𝑛 > 𝑁2 ,
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩2 < 𝜀
(32)
E󵄩󵄩󵄩󵄩𝑓2 (𝑡 + 𝑠𝑛 ) − 𝑓
2
󵄩
holds for all 𝑡 ∈ R. This indicates that
󵄩
󵄩2
lim E󵄩󵄩V (𝑡 + 𝑠𝑛 ) − ̃V (𝑡)󵄩󵄩󵄩 = 0
(33)
𝑛→∞ 󵄩
for each 𝑡 ∈ R. In an analogous way, we can show that
󵄩2
󵄩
lim E󵄩󵄩̃V (𝑡 − 𝑠𝑛 ) − V (𝑡)󵄩󵄩󵄩 = 0
𝑛→∞ 󵄩
(34)
for each 𝑡 ∈ R. Combining (33) with (34), we obtain that V =
S2 𝑋 is square-mean almost automorphic.
Assuming that 𝑋 ∈ 𝐴𝐴(R; L2 (P; H)), then similar
argument as above ensures that 𝑔(⋅) := 𝐺(⋅, 𝑋(⋅)) ∈
𝐴𝐴(R; L2 (P; H)). As a consequence, for every sequence of
real numbers {𝑠𝑛󸀠 } there exist a subsequence {𝑠𝑛 } ⊂ {𝑠𝑛󸀠 } and a
stochastic process 𝑔̃ such that
󵄩2
󵄩
lim E󵄩󵄩𝑔 (𝑡 + 𝑠𝑛 ) − 𝑔̃ (𝑡)󵄩󵄩󵄩 = 0,
𝑛→∞ 󵄩
(35)
󵄩2
󵄩
lim E󵄩󵄩󵄩𝑔̃ (𝑡 − 𝑠𝑛 ) − 𝑔 (𝑡)󵄩󵄩󵄩 = 0
𝑛→∞
Abstract and Applied Analysis
7
󵄩󵄩 𝑡
󵄩
+ 2E 󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝜎)
󵄩󵄩 −∞
hold for each 𝑡 ∈ R. Hence, for arbitrary 𝜀 > 0, there exists
certain 𝑁3 = 𝑁3 (𝜀) ∈ N such that, for all 𝑛 > 𝑁3 , there holds
󵄩2
󵄩
E󵄩󵄩󵄩𝑔 (𝑡 + 𝑠𝑛 ) − 𝑔̃ (𝑡)󵄩󵄩󵄩 < 𝜀
×∫
(36)
−∞
𝑡
≤ 2𝜀2 E (∫ 𝑒−𝛿(𝑡−𝜎)
−∞
2
󵄩󵄩 𝜎
󵄩󵄩
󵄩
̃ (𝑠)󵄩󵄩󵄩 𝑑𝜎)
× 󵄩󵄩󵄩∫ 𝐶 (𝜎−𝑠) 𝑔 (𝑠+𝑠𝑛 ) 𝑑𝑊
󵄩󵄩
󵄩󵄩 −∞
󵄩
(37)
+ 2𝑀2 E (∫
𝑡
−∞
𝑡
̃ (𝑡) := ∫
𝜔
−∞
𝑈 (𝑡, 𝜎) ∫
𝜎
−∞
𝑈 (𝑡, 𝜎) ∫
𝜎
−∞
𝑡
−∞
for each 𝑠 ∈ R, which has the same distribution as 𝑊. Define
two functions on R as below:
𝜔 (𝑡) := ∫
𝐶 (𝜎 − 𝑠)
󵄩󵄩2
̃ (𝑠) 𝑑𝜎󵄩󵄩󵄩
× [𝑔 (𝑠 + 𝑠𝑛 ) − 𝑔̃ (𝑠)] 𝑑𝑊
󵄩󵄩󵄩
for all 𝑡 ∈ R.
The next step aims to prove the square-mean almost
automorphy of S3 𝑋. This is more complicated because
the involvement of the Brownian motion 𝑊. Consider the
̃ defined by
Brownian motion 𝑊
̃ (𝑠) = 𝑊 (𝑠 + 𝑠𝑛 ) − 𝑊 (𝑠𝑛 )
𝑊
𝜎
𝑒−𝛿(𝑡−𝜎)
󵄩󵄩 𝜎
󵄩
× 󵄩󵄩󵄩∫ 𝐶 (𝜎 − 𝑠)
󵄩󵄩 −∞
× [𝑔 (𝑠 + 𝑠𝑛 )
𝐶 (𝜎 − 𝑠) 𝑔 (𝑠) 𝑑𝑊 (𝑠) 𝑑𝜎,
2
󵄩󵄩
̃ (𝑠) 󵄩󵄩󵄩 𝑑𝜎)
−𝑔̃ (𝑠)] 𝑑𝑊
󵄩󵄩
󵄩
(38)
𝐶 (𝜎 − 𝑠) 𝑔̃ (𝑠) 𝑑𝑊 (𝑠) 𝑑𝜎.
≤ 2𝜀2 (∫
By making change of variables 𝜏 = 𝜎 − 𝑠𝑛 and 𝑟 = 𝑠 − 𝑠𝑛 , and
then using Cauchy-Schwarz inequality, we obtain that
𝑡
−∞
× (∫
𝑡
−∞
2
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩
E󵄩󵄩󵄩𝜔 (𝑡 + 𝑠𝑛 ) − 𝜔
󵄩󵄩 𝑡+𝑠𝑛
𝜎
󵄩
= E 󵄩󵄩󵄩∫ 𝑈 (𝑡 + 𝑠𝑛 , 𝜎) ∫ 𝐶 (𝜎 − 𝑠) 𝑔 (𝑠) 𝑑𝑊 (𝑠) 𝑑𝜎
󵄩󵄩 −∞
−∞
󵄩󵄩2
𝑡
𝜎
󵄩
− ∫ 𝑈 (𝑡, 𝜎) ∫ 𝐶 (𝜎 − 𝑠) 𝑔̃ (𝑠) 𝑑𝑊 (𝑠) 𝑑𝜎󵄩󵄩󵄩
󵄩󵄩
−∞
−∞
󵄩󵄩 𝑡
󵄩
= E 󵄩󵄩󵄩∫ 𝑈 (𝑡+𝑠𝑛 , 𝜎+𝑠𝑛 )
󵄩󵄩 −∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
𝑒−𝛿(𝑡−𝜎)
󵄩󵄩 𝜎
󵄩󵄩2
󵄩
̃ (𝑠)󵄩󵄩󵄩 𝑑𝜎)
× E 󵄩󵄩󵄩∫ 𝐶 (𝜎 − 𝑠) 𝑔 (𝑠+𝑠𝑛 ) 𝑑𝑊
󵄩󵄩
󵄩󵄩 −∞
󵄩
+ 2𝑀2 (∫
𝑡
−∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
𝑡
× (∫ 𝑒−𝛿(𝑡−𝜎)
−∞
󵄩󵄩 𝜎
󵄩
× E 󵄩󵄩󵄩∫ 𝐶 (𝜎 − 𝑠)
󵄩󵄩 −∞
𝜎
×∫ 𝐶 (𝜎−𝑠) 𝑔 (𝑠+𝑠𝑛 ) 𝑑𝑊 (𝑠+𝑠𝑛 ) 𝑑𝜎
−∞
󵄩󵄩2
̃ (𝑠) 󵄩󵄩󵄩 𝑑𝜎) .
× [𝑔 (𝑠 + 𝑠𝑛 )− 𝑔̃ (𝑠)] 𝑑𝑊
󵄩󵄩󵄩
(39)
󵄩󵄩2
󵄩
−∫ 𝑈 (𝑡, 𝜎) ∫ 𝐶 (𝜎−𝑠) 𝑔̃ (𝑠) 𝑑𝑊 (𝑠) 𝑑𝜎󵄩󵄩󵄩
󵄩󵄩
−∞
−∞
𝑡
𝜎
󵄩󵄩 𝑡
󵄩
= E 󵄩󵄩󵄩∫ 𝑈 (𝑡+𝑠𝑛 , 𝜎+𝑠𝑛 )
󵄩󵄩 −∞
𝜎
̃ (𝑠) 𝑑𝜎
×∫ 𝐶 (𝜎 − 𝑠) 𝑔 (𝑠+𝑠𝑛 ) 𝑑𝑊
For the above-mentioned 𝜀 > 0, by using an estimate on Ito
integral established in [26], it follows that once 𝑛 > 𝑁3 , then
the first term of the above inequality can be evaluated as
−∞
󵄩󵄩2
̃ (𝑠) 𝑑𝜎󵄩󵄩󵄩
−∫ 𝑈 (𝑡, 𝜎) ∫ 𝐶 (𝜎−𝑠) 𝑔̃ (𝑠) 𝑑𝑊
󵄩󵄩
−∞
−∞
󵄩
𝑡
𝜎
󵄩󵄩 𝑡
󵄩
≤ 2E 󵄩󵄩󵄩∫ [𝑈 (𝑡 + 𝑠𝑛 , 𝜎 + 𝑠𝑛 ) − 𝑈 (𝑡, 𝜎)]
󵄩󵄩 −∞
󵄩󵄩2
̃ (𝑠) 𝑑𝜎󵄩󵄩󵄩
× ∫ 𝐶 (𝜎 − 𝑠) 𝑔 (𝑠 + 𝑠𝑛 ) 𝑑𝑊
󵄩󵄩
−∞
󵄩
𝜎
(∫
𝑡
−∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
× (∫
𝑡
−∞
𝑒−𝛿(𝑡−𝜎)
󵄩󵄩 𝜎
󵄩󵄩2
󵄩
̃ (𝑠)󵄩󵄩󵄩 𝑑𝜎)
× E󵄩󵄩󵄩∫ 𝐶 (𝜎−𝑠) 𝑔 (𝑠+𝑠𝑛 ) 𝑑𝑊
󵄩󵄩
󵄩󵄩 −∞
󵄩
8
Abstract and Applied Analysis
≤ (∫
𝑡
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
−∞
× (∫
𝑡
−∞
≤ (∫
𝜎
−∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
× (∫
𝑡
−∞
󵄩2
󵄩
⋅ sup󵄩󵄩󵄩𝑔 (𝑠 + 𝑠𝑛 ) − 𝑔̃ (𝑠)󵄩󵄩󵄩
−∞
× (∫
−∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
󵄩
󵄩2
‖𝐶 (𝜎 − 𝑠)‖2 E󵄩󵄩󵄩𝑔 (𝑠 + 𝑠𝑛 )󵄩󵄩󵄩 𝑑𝑠} 𝑑𝜎)
≤
𝑡∈R
‖𝐶‖2L2 (0,∞)
𝛿2
The above argument yields that
𝑒−𝛿(𝑡−𝜎) {∫
𝜎
−∞
2
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩
E󵄩󵄩󵄩𝜔 (𝑡 + 𝑠𝑛 ) − 𝜔
‖𝐶 (𝜎 − 𝑠)‖2 𝑑𝑠} 𝑑𝜎)
≤ 2𝜀2 (∫
≤ (∫
−∞
𝑒
2
∞
× (∫
2
𝑑𝜎) ⋅ (∫ ‖𝐶 (𝑢)‖ 𝑑𝑢)
𝑡
−∞
0
𝑡∈R
≤
𝛿2
󵄩2
󵄩
⋅ supE󵄩󵄩󵄩𝑔(𝑡)󵄩󵄩󵄩 ,
(∫
−∞
𝑒
× (∫
−∞
𝑑𝜎)
−𝛿(𝑡−𝜎)
×(∫ 𝑒
−∞
≤ (∫
𝑡
−∞
𝑒
𝑡
× (∫
−∞
𝛿2
2𝑀2 ⋅ ‖𝐶‖2L2 (0,∞)
𝛿2
⋅ 𝜀2
⋅ 𝜀,
𝜎
{∫
(42)
which implies that
2
󵄩
̃ (𝑡)󵄩󵄩󵄩󵄩 = 0
lim E󵄩󵄩󵄩𝜔 (𝑡 + 𝑠𝑛 ) − 𝜔
𝑛→∞
𝑑𝜎)
−𝛿(𝑡−𝜎)
𝑒
≤
󵄩2
󵄩
2‖𝐶‖2L2 (0,∞) ⋅ sup𝑡∈R E󵄩󵄩󵄩𝑔 (𝑡)󵄩󵄩󵄩
+
󵄩
󵄩2
{∫ E󵄩󵄩󵄩𝐶 (𝜎−𝑠)[𝑔 (𝑠+𝑠𝑛 )− 𝑔̃ (𝑠)]󵄩󵄩󵄩 𝑑𝑠} 𝑑𝜎)
−∞
−𝛿(𝑡−𝜎)
𝑒−𝛿(𝑡−𝜎)
󵄩󵄩2
̃ (𝑠) 󵄩󵄩󵄩 𝑑𝜎)
× [𝑔 (𝑠 + 𝑠𝑛 ) − 𝑔̃ (𝑠)] 𝑑𝑊
󵄩󵄩
󵄩
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
𝑡
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
󵄩󵄩 𝜎
󵄩
× E 󵄩󵄩󵄩∫ 𝐶 (𝜎 − 𝑠)
󵄩󵄩 −∞
󵄩󵄩2
̃ (𝑠) 󵄩󵄩󵄩 𝑑𝜎)
×[𝑔 (𝑠+𝑠𝑛 )− 𝑔̃ (𝑠)] 𝑑𝑊
󵄩󵄩
󵄩
𝑡
𝑡
−∞
󵄩󵄩 𝜎
𝑡
󵄩
×(∫ 𝑒−𝛿(𝑡−𝜎) E 󵄩󵄩󵄩∫ 𝐶 (𝜎−𝑠)
󵄩󵄩 −∞
−∞
≤ (∫
𝑡
−∞
𝑡∈R
and the second term can be evaluated analogously as
−𝛿(𝑡−𝜎)
𝑒−𝛿(𝑡−𝜎)
+ 2𝑀2 (∫
(40)
𝑡
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
󵄩󵄩 𝜎
󵄩󵄩2
󵄩
̃ (𝑠)󵄩󵄩󵄩 𝑑𝜎)
× E󵄩󵄩󵄩∫ 𝐶 (𝜎 − 𝑠) 𝑔 (𝑠 + 𝑠𝑛 ) 𝑑𝑊
󵄩󵄩
󵄩󵄩 −∞
󵄩
󵄩2
󵄩
⋅ supE󵄩󵄩󵄩𝑔 (𝑡)󵄩󵄩󵄩
‖𝐶‖2L2 (0,∞)
𝑡
−∞
𝑠∈R
−𝛿(𝑡−𝜎)
⋅ 𝜀.
(41)
󵄩
󵄩2
⋅ supE󵄩󵄩󵄩𝑔 (𝑠 + 𝑠𝑛 )󵄩󵄩󵄩
𝑡
2
0
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
𝑡
𝑡
∞
−∞
−∞
≤ (∫
󵄩2
󵄩
⋅ (∫ ‖𝐶 (𝑢)‖2 𝑑𝑢) ⋅ sup󵄩󵄩󵄩𝑔 (𝑠 + 𝑠𝑛 ) − 𝑔̃ (𝑠)󵄩󵄩󵄩
× {∫
𝑡
𝑡∈R
𝑒−𝛿(𝑡−𝜎)
𝜎
≤ (∫
−∞
× {∫ ‖𝐶 (𝜎−𝑠)‖2 𝑑𝑠} 𝑑𝜎)
󵄩
󵄩2
E󵄩󵄩󵄩𝐶 (𝜎 − 𝑠) 𝑔 (𝑠 + 𝑠𝑛 )󵄩󵄩󵄩 𝑑𝑠} 𝑑𝜎)
−∞
−∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎) (∫ 𝑒−𝛿(𝑡−𝜎)
𝑒−𝛿(𝑡−𝜎)
× {∫
𝑡
𝑡
−∞
𝜎
≤ (∫
𝑡
(43)
for each 𝑡 ∈ R. Analogously, we can show that
𝜎
2
−∞
‖𝐶 (𝜎 − 𝑠)‖ E
󵄩
󵄩2
× 󵄩󵄩󵄩𝑔 (𝑠 + 𝑠𝑛 ) − 𝑔̃ (𝑠)󵄩󵄩󵄩 𝑑𝑠} 𝑑𝜎)
󵄩̃
󵄩2
lim E󵄩󵄩󵄩𝜔
(𝑡 − 𝑠𝑛 ) − 𝜔 (𝑡)󵄩󵄩󵄩 = 0
𝑛→∞
(44)
for each 𝑡 ∈ R. Combining (43) with (44), we obtain that
𝜔 = S3 𝑋 is square-mean almost automorphic.
Abstract and Applied Analysis
9
In view of the above arguments, it follows from (18)
that the nonlinear operator S = S1 + S2 + S3 maps
𝐴𝐴(R; L2 (P; H)) into itself. To complete the proof, it suffices
to show that S is a contraction mapping on 𝐴𝐴(R; L2 (P; H)).
Indeed, for each 𝑋, 𝑌 ∈ 𝐴𝐴(R; L2 (P; H)), thanks to the fact
that (𝑎 + 𝑏 + 𝑐)2 ≤ 3𝑎2 + 3𝑏2 + 3𝑐2 , we have the following
observation:
≤ 3𝑀2 (∫
𝑡
−∞
× (∫
𝑡
−∞
≤ 3𝑀2 (∫
󵄩2
󵄩
𝑒−𝛿(𝑡−𝑠) E󵄩󵄩󵄩𝐹1 (𝑠, 𝑋 (𝑠)) − 𝐹1 (𝑠, 𝑌 (𝑠))󵄩󵄩󵄩 𝑑𝑠)
2
𝑡
−∞
E‖(S𝑋) (𝑡) − (S𝑌) (𝑡)‖2
𝑒−𝛿(𝑡−𝑠) 𝑑𝑠)
𝑒−𝛿(𝑡−𝑠) 𝑑𝑠)
󵄩2
󵄩
× supE󵄩󵄩󵄩𝐹1 (𝑠, 𝑋 (𝑠)) − 𝐹1 (𝑠, 𝑌 (𝑠))󵄩󵄩󵄩
󵄩2
󵄩󵄩 𝑡
󵄩󵄩
󵄩
≤ 3E󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝑠) [𝐹1 (𝑠, 𝑋 (𝑠)) − 𝐹1 (𝑠, 𝑌 (𝑠))] 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 −∞
󵄩󵄩󵄩 𝑡
+ 3E 󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝜎)
󵄩󵄩 −∞
𝑠∈R
≤ 3𝐾1 𝑀2 (∫
𝑡
−∞
𝑡∈R
3𝐾1
⋅ 𝑀2 ⋅ supE‖𝑋 (𝑡) − 𝑌 (𝑡)‖2 .
𝛿2
𝑡∈R
=
𝜎
2
𝑒−𝛿(𝑡−𝑠) 𝑑𝑠) supE‖𝑋 (𝑡) − 𝑌 (𝑡)‖2
× ∫ 𝐵 (𝜎−𝑠)
(46)
−∞
󵄩󵄩2
󵄩
× [𝐹2 (𝑠, 𝑋 (𝑠))−𝐹2 (𝑠, 𝑌 (𝑠))]𝑑𝑠 𝑑𝜎󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 𝑡
󵄩
+ 3E 󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝜎)
󵄩󵄩 −∞
As to the second term, we can proceed in the same manner
as above and obtain
𝜎
× ∫ 𝐶 (𝜎−𝑠)
−∞
3E (∫
󵄩󵄩2
󵄩
×[𝐺 (𝑠, 𝑋 (𝑠))−𝐺 (𝑠, 𝑌 (𝑠))]𝑑𝑊 (𝑠)𝑑𝜎󵄩󵄩󵄩
󵄩󵄩
≤ 3E(∫
𝑡
󵄩
󵄩
‖𝑈 (𝑡, 𝑠)‖ 󵄩󵄩󵄩𝐹1 (𝑠, 𝑋 (𝑠)) − 𝐹1 (𝑠, 𝑌 (𝑠))󵄩󵄩󵄩 𝑑𝑠)
−∞
+ 3E (∫
𝑡
−∞
−∞
𝜎
󵄩
󵄩󵄩
󵄩󵄩𝐵 (𝜎−𝑠) [𝐹2 (𝑠, 𝑋 (𝑠))−𝐹2 (𝑠, 𝑌 (𝑠))]󵄩󵄩󵄩 𝑑𝑠 𝑑𝜎)
−∞
×∫
2
≤
𝑡
‖𝑈 (𝑡, 𝜎)‖
‖𝑈 (𝑡, 𝜎)‖
2
3𝐾2
⋅ 𝑀2 ⋅ ‖𝐵‖2L1 (0,∞) ⋅ supE‖𝑋 (𝑡) − 𝑌 (𝑡)‖2 .
𝛿2
𝑡∈R
(47)
𝜎
󵄩
×∫ 󵄩󵄩󵄩𝐵 (𝜎−𝑠)
−∞
2
󵄩
× [𝐹2 (𝑠, 𝑋(𝑠))−𝐹2 (𝑠, 𝑌(𝑠))]󵄩󵄩󵄩𝑑𝑠 𝑑𝜎)
As far as the last term of the right-hand side is concerned, we
use again the estimate on the Ito integral to obtain
𝑡
+ 3E (∫ ‖𝑈 (𝑡, 𝜎)‖
−∞
3E (∫
󵄩󵄩 𝜎
󵄩
× 󵄩󵄩󵄩∫ 𝐶 (𝜎−𝑠) [𝐺 (𝑠, 𝑋 (𝑠))
󵄩󵄩 −∞
2
󵄩󵄩
󵄩
−𝐺 (𝑠, 𝑌 (𝑠))]𝑑𝑊 (𝑠) 󵄩󵄩󵄩𝑑𝜎) .
󵄩󵄩
−∞
𝑡
−∞
󵄩
󵄩
‖𝑈 (𝑡, 𝑠)‖ 󵄩󵄩󵄩𝐹1 (𝑠, 𝑋 (𝑠)) − 𝐹1 (𝑠, 𝑌 (𝑠))󵄩󵄩󵄩 𝑑𝑠)
𝑡
≤ 3𝑀2 E(∫
−∞
‖𝑈 (𝑡, 𝜎)‖
󵄩󵄩 𝜎
󵄩
× 󵄩󵄩󵄩∫ 𝐶 (𝜎 − 𝑠)
󵄩󵄩 −∞
(45)
Now, we evaluate the first term of the right-hand side with the
help of Cauchy-Schwarz inequality as follows:
3E(∫
𝑡
2
󵄩󵄩
󵄩
× [𝐺 (𝑠, 𝑋 (𝑠)) − 𝐺 (𝑠, 𝑌 (𝑠))] 𝑑𝑊 (𝑠) 󵄩󵄩󵄩 𝑑𝜎)
󵄩󵄩
≤ 3𝑀2 E (∫
𝑡
−∞
𝑒−𝛿(𝑡−𝜎)
󵄩󵄩 𝜎
󵄩
× 󵄩󵄩󵄩∫ 𝐶 (𝜎−𝑠)
󵄩󵄩 −∞
2
2
󵄩
󵄩
𝑒−𝛿(𝑡−𝑠) 󵄩󵄩󵄩𝐹1 (𝑠, 𝑋 (𝑠))−𝐹1 (𝑠, 𝑌 (𝑠))󵄩󵄩󵄩 𝑑𝑠)
2
󵄩󵄩
󵄩
× [𝐺 (𝑠, 𝑋 (𝑠))−𝐺 (𝑠, 𝑌 (𝑠))]𝑑𝑊󵄩󵄩󵄩 𝑑𝜎)
󵄩󵄩
10
Abstract and Applied Analysis
≤ 3𝑀2 (∫
𝑡
−∞
×(∫
𝑡
As a result, (50) together with (51) gives that, for each 𝑡 ∈ R,
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
‖(S𝑋) (𝑡) − (S𝑌) (𝑡)‖2 ≤ √Θ sup‖𝑋 (𝑡) − 𝑌 (𝑡)‖2
𝑡∈R
𝑒−𝛿(𝑡−𝜎)
−∞
= √Θ ‖𝑋 − 𝑌‖∞ .
󵄩󵄩 𝜎
󵄩
×E 󵄩󵄩󵄩∫ 𝐶 (𝜎−𝑠)
󵄩󵄩 −∞
It follows that
󵄩󵄩2
󵄩
×[𝐺 (𝑠, 𝑋 (𝑠))−𝐺 (𝑠, 𝑌 (𝑠))]𝑑𝑊 (𝑠)󵄩󵄩󵄩 𝑑𝜎)
󵄩󵄩
≤ 3𝑀2 (∫
𝑡
−∞
× (∫
𝑡
×∫
𝜎
‖𝐶 (𝜎 − 𝑠)‖2
−∞
×E‖𝐺 (𝑠, 𝑋 (𝑠)) − 𝐺 (𝑠, 𝑌 (𝑠))‖2 𝑑𝑠 𝑑𝜎)
≤ 3𝑀2 (∫
2
𝑡
−∞
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎) ⋅ ‖𝐶‖2L2 (0,∞)
⋅ supE‖𝐺 (𝑠, 𝑋 (𝑠)) − 𝐺 (𝑠, 𝑌 (𝑠))‖
𝑡
−∞
2
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎) ⋅ ‖𝐶‖2L2 (0,∞)
⋅ supE‖𝑋 (𝑡) − 𝑌 (𝑡)‖2
𝑡∈R
3𝐾
= 23 ⋅ 𝑀2 ⋅ ‖𝐶‖2L2 (0,∞) ⋅ supE‖𝑋 (𝑡) − 𝑌 (𝑡)‖2 .
𝛿
𝑡∈R
(48)
Thus, by combining the three inequalities together, we obtain
that, for each 𝑡 ∈ R,
E‖(S𝑋) (𝑡) − (S𝑌) (𝑡)‖
≤ {3
2
𝑀2
[𝐾1 + 𝐾2 ⋅ ‖𝐵‖2L1 (0,∞) + 𝐾3 ⋅ ‖𝐶‖2L2 (0,∞) ]}
𝛿2
Remark 11. If the functions 𝐹1 , 𝐹2 , 𝐺 in (1) are square-mean
almost periodic in 𝑡, then the unique square-mean almost
automorphic solution obtained in Theorem 10 is actually
square-mean almost periodic; see paper [27].
whenever E ‖ 𝑋(0) − 𝑋∗ (0)‖2 < 𝛿, where 𝑋(𝑡) stands for a
solution of (1) with initial value 𝑋(0). The solution 𝑋∗ (𝑡) is
said to be asymptotically stable in square-mean sense if it is
stable in square-mean sense and
󵄩2
󵄩
lim E󵄩󵄩󵄩𝑋 (𝑡) − 𝑋∗ (𝑡)󵄩󵄩󵄩 = 0.
The following Gronwall-type inequality is proved to be
useful in our asymptotic stability analysis.
Lemma 12. Let 𝑢(𝑡), 𝑏(𝑡) be nonnegative continuous functions
for 𝑡 ≥ 𝑎, and 𝛼, 𝛾 be some positive constants. If
𝑎
(49)
(55)
𝑡→∞
𝑡
𝑡∈R
𝑡 ≥ 𝑎,
(56)
then
𝑡
That is,
(S𝑌) (𝑡)‖22
≤ Θ sup‖𝑋 (𝑡) −
𝑡∈R
𝑌 (𝑡)‖22 ,
𝑢 (𝑡) ≤ 𝛼 exp {−𝛽 (𝑡 − 𝑎) + ∫ 𝑏 (𝑠) 𝑑𝑠} .
(50)
where Θ := 3(𝑀2 /𝛿2 )[𝐾1 +𝐾2 ⋅ ‖ 𝐵‖2L1 (0,∞) +𝐾3 ⋅ ‖ 𝐶‖2L2 (0,∞) ].
Notice that
2
sup‖𝑋 (𝑡) − 𝑌 (𝑡)‖22 ≤ (sup‖𝑋 (𝑡) − 𝑌 (𝑡)‖2 ) .
𝑡∈R
(53)
𝑡∈R
𝑢 (𝑡) ≤ 𝛼𝑒−𝛽(𝑡−𝑎) + ∫ 𝑒−𝛽(𝑡−𝑠) 𝑏 (𝑠) 𝑢 (𝑠) 𝑑𝑠,
⋅ supE‖𝑋 (𝑡) − 𝑌 (𝑡)‖2 .
‖(S𝑋) (𝑡) −
= sup‖(S𝑋) (𝑡) − (S𝑌) (𝑡)‖2 ≤ √Θ ‖𝑋 − 𝑌‖∞ ,
Now we are in a position to show the asymptotically stable
property of the unique square-mean almost automorphic
solution to (1). Recall that the unique square-mean almost
automorphic solution 𝑋∗ (𝑡) of (1) is said to be stable in
square-mean sense if, for arbitrary 𝜖 > 0, there exists 𝛿 > 0
such that
󵄩2
󵄩
(54)
E󵄩󵄩󵄩𝑋 (𝑡) − 𝑋∗ (𝑡)󵄩󵄩󵄩 < 𝜖, 𝑡 ≥ 0
2
𝑠∈R
≤ 3𝐾3 𝑀2 (∫
‖S𝑋 − S𝑌‖∞
which implies that S is a contraction mapping by the assumption (15) imposed on Θ. Therefore, by the Banach contraction
principle, we conclude that there exists a unique fixed point
𝑋 for S in 𝐴𝐴(R; L2 (P; H)), which is the unique squaremean almost automorphic mild solution to the functional
integrodifferential semilinear stochastic evolution equation
(1) as we have claimed. The proof is complete.
𝑒−𝛿(𝑡−𝜎) 𝑑𝜎)
𝑒−𝛿(𝑡−𝜎)
−∞
(52)
𝑡∈R
(51)
𝑎
(57)
Theorem 13. Let all the assumptions in Theorem 10 hold and
assume that
1
𝑀2
[𝐾1 + 𝐾2 ⋅ ‖𝐵‖2L1 (0,∞) + 𝐾3 ⋅ ‖𝐶‖2L2 (0,∞) ] < .
𝛿2
4
(58)
Then the unique square-mean almost automorphic mild solution 𝑋∗ (𝑡) of (1) is asymptotically stable in square-mean sense.
Abstract and Applied Analysis
11
Proof. Let 𝑋(𝑡) be any mild solution of (1) with initial value
𝑋(0). Then, on account of (H1)-(H2) and the assumptions
imposed on 𝐵 and 𝐶, along the same line as in [9] we could
show that, for any 𝑡 ≥ 0,
Define 𝑌(𝑡) := E ‖ 𝑋(𝑡) − 𝑋∗ (𝑡)‖2 , and it yields that
𝑡
𝑌 (𝑡) ≤ 4𝑀2 𝑌 (0) 𝑒−𝛿𝑡 + 𝜅 ∫ 𝑒−𝛿(𝑡−𝑠) 𝑌 (𝑠) 𝑑𝑠.
0
Hence, it follows from Lemma 12 that
󵄩2
󵄩
E󵄩󵄩󵄩𝑋 (𝑡) − 𝑋∗ (𝑡)󵄩󵄩󵄩
󵄩󵄩
󵄩
= E 󵄩󵄩󵄩𝑈 (𝑡, 0) [𝑋 (0) − 𝑋∗ (0)]
󵄩󵄩
𝑌 (𝑡) ≤ 4𝑀2 𝑌 (0) exp {(−𝛿 + 𝜅) 𝑡} .
𝑡
0
× [𝐹1 (𝑠, 𝑋 (𝑠)) − 𝐹1 (𝑠, 𝑋∗ (𝑠))] 𝑑𝑠
𝑡
+ ∫ 𝑈 (𝑡, 𝜎)
4. Applications
0
𝜎
× ∫ 𝐵 (𝜎 − 𝑠)
0
× [𝐹2 (𝑠, 𝑋 (𝑠)) − 𝐹2 (𝑠, 𝑋∗ (𝑠))] 𝑑𝑠 𝑑𝜎
𝑡
+ ∫ 𝑈 (𝑡, 𝜎)
To illustrate the applications of our abstract results, let us
consider the following nonautonomous functional integrodifferential stochastic partial differential equation:
𝑡
𝜕𝑋 𝜕2 𝑋
+
𝐶 (𝑡 − 𝑠) 𝐺 (𝑠, 𝑋 (𝑠, 𝑥)) 𝑑𝑊 (𝑠)
=
∫
𝜕𝑡
𝜕𝑥2
−∞
0
+∫
𝜎
𝑡
−∞
×∫ 𝐶 (𝜎−𝑠)
𝐵 (𝑡 − 𝑠) 𝐹2 (𝑠, 𝑋 (𝑠, 𝑥)) 𝑑𝑠 + 𝐹1 (𝑡, 𝑋 (𝑡, 𝑥))
(62)
0
󵄩󵄩2
󵄩
× [𝐺 (𝑠, 𝑋(𝑠))−𝐺 (𝑠, 𝑋 (𝑠))]𝑑𝑊(𝑠)𝑑𝜎󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩2
≤ 4E󵄩󵄩󵄩𝑈 (𝑡, 0) [𝑋 (0) − 𝑋∗ (0)]󵄩󵄩󵄩
∗
󵄩󵄩2
󵄩󵄩 𝑡
󵄩
󵄩
+ 4E󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝑠) [𝐹1 (𝑠, 𝑋 (𝑠)) − 𝐹1 (𝑠, 𝑋∗ (𝑠))]󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 0
󵄩󵄩 𝑡
󵄩
+ 4E 󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝜎)
󵄩󵄩 0
for 𝑡 ∈ R and 𝑥 ∈ Ω, where Ω ⊂ R𝑛 is a bounded subset
whose boundary 𝜕Ω is both of class 𝐶2 and locally on one
side of Ω. Suppose further that (62) satisfies the following
boundary conditions:
𝑛
∑ 𝑛𝑖 (𝑥) 𝑎𝑖𝑗 (𝑡, 𝑥)
𝑖,𝑗=1
𝜎
𝑡 ∈ R, 𝑥 ∈ 𝜕Ω,
𝑛
𝜕
𝜕
(𝑎𝑖𝑗 (𝑡, 𝑥)
)+𝑐 (𝑡, 𝑥) ,
𝜕𝑥
𝜕𝑥
𝑖
𝑗
𝑖,𝑗=1
𝐴 (𝑡, 𝑥) = ∑
0
󵄩󵄩2
󵄩
× [𝐹2 (𝑠, 𝑋 (𝑠))−𝐹2 (𝑠, 𝑋∗ (𝑠))] 𝑑𝑠 𝑑𝜎󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 𝑡
󵄩
+4E 󵄩󵄩󵄩∫ 𝑈 (𝑡, 𝜎)
󵄩󵄩 0
(63)
𝑡 ∈ R, 𝑥 ∈ Ω,
(64)
where 𝑎𝑖𝑗 (𝑖, 𝑗 = 1, 2, . . . , 𝑛) and 𝑐 satisfy the following
conditions.
(H4) The coefficients 𝑎𝑖𝑗 (𝑖, 𝑗 = 1, 2, . . . , 𝑛) are symmetric,
that is, 𝑎𝑖𝑗 = 𝑎𝑗𝑖 for all 𝑖, 𝑗 = 1, 2, . . . , 𝑛. In addition,
𝜎
×∫ 𝐶 (𝜎−𝑠)
𝜇
𝑎𝑖𝑗 ∈ 𝐶𝑏 (R; L2 (P; 𝐶 (Ω)))
0
󵄩󵄩2
󵄩
×[𝐺 (𝑠, 𝑋(𝑠))−𝐺 (𝑠, 𝑋∗ (𝑠))]𝑑𝑊(𝑠) 𝑑𝜎󵄩󵄩󵄩
󵄩󵄩
󵄩2
󵄩
≤ 4𝑀2 𝑒−𝛿𝑡 E󵄩󵄩󵄩𝑋 (0) − 𝑋∗ (0)󵄩󵄩󵄩
∩ 𝐶𝑏 (R; L2 (P; 𝐶1 (Ω)))
0
for all 𝑖, 𝑗 = 1, 2, . . . , 𝑛 and
(59)
+ 𝐾3 ⋅ ‖
(65)
∩ 𝐴𝐴 (R; L2 (P; L2 (Ω)))
𝑡
󵄩2
󵄩
+ 𝜅 ∫ 𝑒−𝛿(𝑡−𝑠) E󵄩󵄩󵄩𝑋 (𝑠) − 𝑋∗ (𝑠)󵄩󵄩󵄩 𝑑𝑠,
𝐵‖2L1 (0,∞)
𝑑𝑋 (𝑡, 𝑥)
= 0,
𝑑𝑥𝑖
where 𝑛(𝑥) = (𝑛1 (𝑥), 𝑛2 (𝑥), . . . , 𝑛𝑛 (𝑥)) is the outer unit
normal vector. A family of operators 𝐴(𝑡, 𝑥) defined by
𝜕2 𝑋/𝜕𝑥2 = 𝐴(𝑡, 𝑥)𝑋(𝑡, 𝑥) is formally assigned to be
×∫ 𝐵 (𝜎−𝑠)
where 𝜅 := (4𝑀 (𝐾1 + 𝐾2 ⋅ ‖
(61)
Straightforwardly, we obtain that 𝑌(𝑡) converges to 0 exponentially fast if −𝛿 + 𝜅 < 0, which is equivalent to
our condition (58). Thus, we come to the conclusion that
the unique square-mean almost automorphic mild solution
𝑋∗ (𝑡) of (1) is asymptotically stable in square-mean sense.
The proof is completed.
+ ∫ 𝑈 (𝑡, 𝑠)
2
(60)
𝐶‖2L2 (0,∞) ))/𝛿.
𝑐∈
𝜇
𝐶𝑏
(R; L2 (P; L2 (Ω))) ∩ 𝐶𝑏 (R; L2 (P; 𝐶 (Ω)))
∩ 𝐴𝐴 (R; L2 (P; L1 (Ω)))
(66)
12
Abstract and Applied Analysis
for some 𝜇 ∈ (1/2, 1], where Ω means the closure of
Ω.
(H5) There exists 𝛿0 > 0 such that
𝑛
󵄨 󵄨2
∑ 𝑎𝑖𝑗 (𝑡, 𝑥) 𝜂𝑖 𝜂𝑗 ≥ 𝛿0 󵄨󵄨󵄨𝜂󵄨󵄨󵄨
𝑖,𝑗=1
(67)
for all (𝑡, 𝑥) ∈ R × Ω and 𝜂 ∈ R𝑛 .
Now, let H = L2 (Ω) and let H2 (Ω) be the Sobolev space
of order 2 on Ω. For each 𝑡 ∈ R, define an operator 𝐴(𝑡) on
L2 (P; H) by
𝐴 (𝑡) 𝑋 = 𝐴 (𝑡, 𝑥) 𝑋
∀𝑋 ∈ D (𝐴 (𝑡)) ,
(68)
where
{
D (𝐴 (𝑡)) = {𝑋 ∈ L2 (P, H2 (Ω)) :
{
𝑛
∑ 𝑛𝑖 (⋅) 𝑎𝑖𝑗 (𝑡, ⋅)
𝑖,𝑗=1
}
𝑑𝑋 (𝑡, ⋅)
= 0 on 𝜕Ω} .
𝑑𝑥𝑖
}
(69)
Under assumptions (H4)-(H5), the existence of an evolution
family 𝑈(𝑡, 𝑠) satisfying (H1) is guaranteed; see, for example,
[28].
And thus, as an immediate consequence of Theorem 10, it
yields the following.
Theorem 14. Under assumptions (H2), (H3), (H4), and (H5),
the nonautonomous integrodifferential stochastic evolution
equation (62)-(63) has a unique mild solution which is squaremean almost automorphic provided that (15) holds. If, in
addition, (58) is valid, then the unique almost automorphic
solution is asymptotically stable in square-mean sense.
Acknowledgments
This paper is jointly supported by the National Natural
Science Foundation of China under Grant nos. 11201266 and
11171191 and the Tianyuan Youth Foundation of National
Natural Science Foundation of China under Grant nos.
11026150 and 11026098.
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