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# Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space

- Yong-Kui Chang
^{1}Email author, - Zhi-Han Zhao
^{1}and - Gaston Mandata N'Guérékata
^{2}

**2011**:9

https://doi.org/10.1186/1687-1847-2011-9

© Chang et al; licensee Springer. 2011

**Received:**20 September 2010**Accepted:**14 June 2011**Published:**14 June 2011

## Abstract

This article deals primarily with the existence and uniqueness of square-mean almost automorphic mild solutions for a class of stochastic differential equations in a real separable Hilbert space. We study also some properties of square-mean almost automorphic functions including a compostion theorem. To establish our main results, we use the Banach contraction mapping principle and the techniques of fractional powers of an operator.

**Mathematics Subject Classification (2000)**

34K14, 60H10, 35B15, 34F05.

## Keywords

- Stochastic differential equations
- Square-mean almost automorphic processes
- Mild solutions

## 1 Introduction

where
is the infinitesimal generator of an analytic semigroup of linear operators {*T*(*t*)}_{
t≥0}on
, *B*
_{
i
} , *i* = 1, 2, 3, are bounded linear operators that can be viewed as control terms, and *W*(*t*) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space
, where
. Here, *f*, *g*, and *h* are appropriate functions to be specified later.

The concept of almost automorphy is an important generalization of the classical almost periodicity. They were introduced by Bochner [1, 2]; for more details about this topic, we refer the reader to [3, 4]. In recent years, the existence of almost periodic and almost automorphic solutions on different kinds of deterministic differential equations have been considerably investigated in lots of publications [5–15] because of its significance and applications in physics, mechanics, and mathematical biology.

in a Hilbert space
, where A is an infinitesimal generator of a *C*
_{0}-semigroup {*T*(*t*)}_{
t ≥ 0}, and *W*(*t*) is a two-sided standard one-dimensional Brown motion defined on the filtered probability space
, where
.

Motivated by the above mentioned studies [18, 23], the main purpose of this article is to investigate the existence and uniqueness of square-mean almost automorphic solutions to the problem (1.1). Note that (1.1) is more general than the problem studied in [23]. We first use a sharper definition (Definition 2.1) of square-mean almost automorphic process than the Definition 2.5 in [23]. We then present some additional properties of square-mean almost automorphic processes (see Lemmas 2.4-2.5). Our main result is established by using fractional powers of linear operators and Banach contraction principle. The obtained result can be seen as a contribution to this emerging field since it improves and generalizes the results in [23].

The rest of this article is organized as follows. In section 2, we recall and prove some basic definitions, lemmas, and preliminary facts which will be used throughout this article. We also prove some additional properties of square-mean almost automorphic functions. In Section 3, we prove the existence and uniqueness of square-mean almost automorphic mild solutions to (1.1).

## 2 Preliminaries

In this section, we introduce some basic definitions, notations, lemmas, and technical results which are used in the sequel. For more details on this section, we refer the reader to [23, 24].

*x*such that

Then, it is routine to check that
is a Hilbert space equipped with the norm ||·||_{2}. We let
denote the space of all the linear-bounded operators from
into
, equipped with the usual operator norm
. In addition, *W*(*t*) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space
, where
.

*ρ*(

*A*) where

*ρ*(

*A*) is the resolvent set of

*A*; then, it is possible to define the fractional power (-

*A*)

^{ α }, for 0 <

*α*≤ 1, as a closed linear invertible operator on its domain

*D*((-

*A*)

^{ α }). Furthermore, the subspace

*D*((-

*A*)

^{ α }) is dense in and the expression

defines a norm on *D*((-*A*) ^{
α
} ). Hereafter, we denote by
the Banach space *D*((-*A*) ^{
α
} ) with norm ||*x*||_{
α
}.

The following properties hold by Pazy [25].

**Lemma 2.1** *Let* 0 < *γ* ≤ *μ* ≤ 1. *Then, the following properties hold:*

*(i)*
*is a Banach space and*
*is continuous*.

*(ii) The function s* → (-*A*)^{
μ
}
*T*(*s*) *is continuous in the uniform operator topology on* (0, ∞), *and there exists* *M*
_{
μ
}> 0 *such that* ||(-A)^{μ}T(t)|| ≤ M_{μ}e^{-δt}t^{-μ}
*for each t* > 0.

*(iii) For each* *x* ∈ *D*((-*A*) ^{
μ
} ) *and t* ≥ 0, (-*A*) ^{
μ
}
*T*(*t*)*x* = *T*(*t*)(-*A*) ^{
μ
}
*x*.

*(iv)* (-*A*)^{-μ
}
*is a bounded linear operator in*
*with D*((-*A*) ^{
μ
} ) = *Im*((-*A*
^{)-μ
}).

**Definition 2.2**

*(compare with*[23])

*A stochastically continuous stochastic process*

*is said to be square-mean almost automorphic if for every sequence of real numbers*,

*there exist a subsequence*{

*s*

_{ n }}

_{ n∈ℕ}

*and a stochastic process*

*hold for each t* ∈ ℝ.

The collection of all square-mean almost automorphic stochastic processes is denoted by .

**Lemma 2.2** ([23]) *If x, x*
_{1}
*and x*
_{2}
*are all square-mean almost automorphic stochastic processes, then the following hold true:*

*(i) x*
_{1} + *x*
_{2}
*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* sup_{
t ∈ ℝ}||*x*(*t*)||_{2} ≤ *M*. *That is, x is bounded in*
.

*for*
.

Let be defined as and note that , are Banach spaces; then, we state the following lemmas (cf. [3, 13]):

**Lemma 2.4** *Let*
. *Then, we have*

*(I)*
.

*(II)*
.

**Lemma 2.5** *Let*
*and assume that*
. *Then*,
.

**Definition 2.3**([23])

*A function*, (

*t*,

*x*) →

*f*(

*t*,

*x*),

*which is jointly continuous, is said to be square-mean almost automorphic in t*∈ ℝ

*for each*

*if for every sequence of real numbers*,

*there exist a subsequence*{

*s*

_{ n }}

_{ n∈ℕ}

*and a stochastic process*

*such that*

*for each* *t* ∈ ℝ *and each*
.

**Theorem 2.1**([23])

*Let*, (

*t*,

*x*) →

*f*(

*t, x*)

*be square-mean almost automorphic in*

*t*∈ ℝ

*for each*,

*and assume that f satisfies Lipschitz condition in the following sense:*

*for all*
*and for each* t ∈ ℝ, *where*
*is independent of t. Then, for any square-mean almost automorphic process*
, *the stochastic process*
*given by F*(*t*) = *f* (*t*, *x*(*t*)) *is square-mean almost automorphic*.

**Definition 2.4**

*An*

*-progressively measurable stochastic process*{

*x*(

*t*)}

_{ t ∈ ℝ}

*is called a mild solution of problem (1.1) on R if the function s*→

*AT*(

*t*-

*s*)

*f*(

*s*,

*B*

_{1}

*x*(

*s*))

*is integrable on*(-∞,

*t*)

*for each*

*t*∈ ℝ,

*and x*(

*t*)

*satisfies the corresponding stochastic integral equation*

*for all* *t* ≥ *a and for each* *a* ∈ ℝ.

## 3 Main results

In this section, we investigate the existence of a square-mean almost automorphic solution for the problem (1.1). We first list the following basic assumptions:

(H1) The operator
is the infinitesimal generator of an analytic semigroup of linear operators {*T*(*t*)}_{
t≥0}on
and *M*, *δ* are positive numbers such that ||*T*(*t*)||≤ *Me*
^{-δt
}for *t* ≥ 0.

(H2) The operators
for *i* = 1, 2, 3, are bounded linear operators and
.

*β*∈ (0, 1) such that is square-mean almost automorphic in

*t*∈ ℝ for each . Let

*L*

_{ f }> 0 be such that for each

*t*∈ ℝ for each . Moreover,

*g*and

*h*satisfy Lipschitz conditions in

*φ*uniformly for

*t*, that is, there exist positive numbers

*L*

_{ g }, and

*L*

_{ h }such that

for all *t* ∈ ℝ and each *φ*,
.

**Theorem 3.1** *Let*
*and* *α* < *β* < 1. If *the conditions (H1)-(H4) are satisfied, then the problem (1.1) has a unique square-mean almost automorphic mild solution*
*provided that*

(3.1)

*where* Γ(·) *is the gamma function*.

*x*is well defined. Indeed, let , then

*s*→

*B*

_{ i }

*x*(

*s*) is in as , ,

*i*= 1, 2, 3 in virtue of Lemma 2.5, and hence, by Theorem 2.1, the function

*s*→ (-

*A*)

^{ β }

*f*(

*s*,

*B*

_{1}

*x*(

*s*)) belongs to whenever . Thus, using Lemma 2.2 (iii), it follows that there exists a constant

*N*

_{ f }> 0 such that sup

_{ t∈ℝ}

*E*||(-

*A*)

^{ β }

*f*(

*t*,

*B*

_{1}

*x*(

*t*))||

^{2}≤

*N*

_{ f }. Moreover, from the continuity of

*s*→

*AT*(

*t*-

*s*) and

*s*→

*T*(

*t*-

*s*) in the uniform operator topology on (-∞,

*t*) for each

*t*∈ ℝ and the estimate

it follows that *s* → *AT*(*t* - *s*)*f* (*s*, *B*
_{1}
*x*(*s*)), *s* → *T*(*t* - *s*)*g*(*s*, *B*
_{2}
*x*(*s*)) and *s* → *T*(*t* - *s*)*h*(*s*, *B*
_{3}
*x*(*s*)) are integrable on (-∞, *t*) for every *t* ∈ ℝ, therefore, Λ*x* is well defined.

_{1}

*x*, Λ

_{2}

*x*, and Λ

_{3}

*x*acting on the Banach space defined by

*s*

_{ n }}

_{ n∈ℕ}of such that for certain stochastic process

for each *t* ∈ ℝ. Thus, we conclude that
.

*s*→

*g*(

*s*,

*B*

_{2}

*x*(

*s*)) belongs to whenever . Since for every sequence of real numbers , there exists a subsequence such that for certain stochastic process

for each *t* ∈ ℝ. Thus, we conclude that
.

*s*→

*h*(

*s*,

*B*

_{3}

*x*(

*s*)) is in whenever . Since , for every sequence of real numbers , there exists a subsequence such that for certain stochastic process

*t*∈ ℝ. The next step consists of showing that . Let for each σ ∈ ℝ. Note that is also a Brownian motion and has the same distribution as

*W*. Moreover, if we let , then by making a change of variables

*σ*=

*s*-

*s*

_{ n }we get

for each *t* ∈ ℝ. Thus, we conclude that
. Since
, and in view of the above, it is clear that Λ maps
into itself.

*t*∈ ℝ, , we see that

*x*(·) for Λ in , such that Λ

*x*=

*x*, that is

In conclusion, is a mild solution of equation (1.1) and . The proof is completed.

**Remark 3.1** *The results of Theorem 3.1 can be used to study the existence and uniqueness of square-mean almost automorphic mild solutions to the example in*[18].

## Declarations

### Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments and suggestions to improve this paper. This study was supported by NNSF of China (10901075), Program for New Century Excellent Talents in University (NCET-10-0022), the Key Project of Chinese Ministry of Education (210226), the Scientific Research Fund of Gansu Provincial Education Department (0804-08), and Qing Lan Talent Engineering Funds (QL-05-16A) from Lanzhou Jiaotong University.

## Authors’ Affiliations

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