# Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations

Authors: Zhi-Han Zhao; Yong-Kui Chang; Juan J. Nieto
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The existence of asymptotically almost automorphic mild solutions to an abstract stochastic fractional partial integrodifferential equation is considered. The main tools are some suitable composition results for asymptotically almost automorphic processes, the theory of sectorial linear operators, and classical fixed point theorems. An example is also given to illustrate the main theorems.


---

## Body

## 1. Introduction

This paper is mainly concerned with the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to the following stochastic fractional partial integrodifferential equation in the form(1)d[x(t)-f(t,x(t))]=∫0t‍(t-s)α-2Γ(α-1)A[x(s)-f(s,x(s))]dsdt+g(t,x(t))dW(t),t≥0,x(0)=u0,
where 1<α<2, A:D(A)⊂L2(ℙ,ℍ)→L2(ℙ,ℍ) is a linear densely defined operator of sectorial type on a Hilbert space L2(ℙ,ℍ), W(t) is 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}, and u0 is an ℱ0-adapted, ℍ-valued random variable independent of the Wiener process W. Here f and g are appropriate functions to be specified later. The convolution integral in (1) is understood in the Riemann-Liouville fractional integral (see, e.g., [1, 2]). We notice that fractional order can be complex in viewpoint of pure mathematics and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proven to be valuable in various fields of science and engineering (see, e.g., [3–11] and references therein).The concept of asymptotically almost automorphic functions was firstly introduced by N’Guérékata in [12]. Since then these functions have become of great interest to several mathematicians and gained lots of developments and applications, we refer the reader to [13–16] and the references listed therein.Recently, the existence of almost automorphic and pseudo almost automorphic solutions to some stochastic differential equations has been considered in many publications such as [17–27] and the references therein. In a very recent paper [28], the authors introduced a new notation of square-mean asymptotically almost automorphic stochastic processes including a composition theorem. However, to the best of our knowledge, the existence of square-mean asymptotically almost automorphic mild solutions to the problem (1) is an untreated topic. Therefore, motivated by the works [16, 28], the main purpose of this paper is to investigate the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to the problem (1). Then, we present an example as an application of our main results.The rest of this paper is organized as follows. In Section2, we recall some basic definitions and facts which will be used throughout this paper. In Section 3, we prove some existence results of square-mean asymptotically almost automorphic mild solutions to the problem (1). Finally, we give an example as an application of our abstract results.


## 2. Preliminaries

In this section, we introduce some basic definitions, notations, and preliminary facts which will be used in the sequel. For more details on this section, we refer the reader to [28–30].Throughout the paper,(ℍ,∥·∥,〈·,·〉) stands for a real separable Hilbert space. (Ω,ℱ,ℙ) denotes a complete probability space, and L2(ℙ,ℍ) stands for the space of all ℍ-valued random variables x such that
(2)E∥x∥2=∫Ω‍∥x∥2dℙ<∞.
Note that L2(ℙ,ℍ) is a Hilbert space equipped with the norm
(3)∥x∥2:=(∫Ω‍∥x∥2dℙ)1/2,foreachx∈L2(ℙ,ℍ).
We denote by C0(ℝ+;L2(ℙ,ℍ)) the collection of all bounded continuous stochastic processes φ from ℝ+ into L2(ℙ,ℍ) such that limt→+∞E∥φ(t)∥2=0. It is then easy to check that C0(ℝ+;L2(ℙ,ℍ)) is a Banach space when it is endowed with the norm ∥φ∥C0:=supt∈ℝ+∥φ(t)∥2. Similarly, C0(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)) stands for the space of the continuous stochastic processes f:ℝ+×L2(ℙ,ℍ)→L2(ℙ,ℍ) such that
(4)limt→+∞E∥f(t,x)∥2=0
uniformly for x∈K, where K⊂L2(ℙ,ℍ) is any bounded subset. Additionally, W(t) will be 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}.


### 2.1. Sectorial Linear Operators

A closed and linear operatorA is said to be sectorial of type ϖ and angle θ if there exist 0<θ<π/2,M>0, and ϖ∈ℝ such that its resolvent exists outside the sector ϖ+Sθ:={ϖ+λ:λ∈ℂ,|arg(-λ)|<θ} and ∥(λ-A)-1∥≤M/|λ-ϖ|, λ∉ϖ+Sθ.Definition 1 (see [2]).
LetA be a closed and linear operator with domain D(A) defined on a Banach space 𝕏. We call A the generator of a solution operator if there exist ϖ∈ℝ and a strongly continuous function Sα:ℝ+→ℒ(𝕏) such that {λα:Re(λ)>ϖ}⊂ρ(A) and λα-1(λα-A)-1x=∫0∞‍e-λtSα(t)xdt, Re(λ)>ϖ,x∈𝕏. In this case, Sα(·) is called the solution operator generated by A.We note that ifA is sectorial of type ϖ with 0≤θ<π(1-α/2), then A is the generator of a solution operator given by Sα(t):=(1/2πi)∫γ‍eλtλα-1(λα-A)-1dλ,t≥0, where γ is a suitable path lying outside the sector ϖ+Sθ. Recently, Cuesta in [1] proved that if A is a sectorial operator of type ϖ<0 for some M>0 and 0≤θ<π(1-α/2), then there exists a constant C>0 such that
(5)∥Sα(t)∥≤CM1+|ϖ|tα,t≥0.Remark 2.
Note thatSα(t) is, in fact, integrable. For more details on the solution family Sα(t) and related issues, we refer the reader to [31–33].


### 2.2. Square-Mean Asymptotically Almost Automorphic Processes

We recall some basic facts for a symptotically almost automorphic processes which will be used in the sequel.Definition 3 (see [22]).
A stochastic processx:ℝ→L2(ℙ,ℍ) is said to be stochastically continuous if
(6)limt→sE∥x(t)-x(s)∥2=0.Definition 4 (see [17]).
A stochastically continuous stochastic processx:ℝ→L2(ℙ,ℍ) is said to be square-mean almost automorphic if, for every sequence of real numbers {sn′}n∈ℕ, there exist a subsequence {sn}n∈ℕ and a stochastic process y:ℝ→L2(ℙ,ℍ) such that
(7)limn→∞E∥x(t+sn)-y(t)∥2=0,limn→∞E∥y(t-sn)-x(t)∥2=0
hold for each t∈ℝ. The collection of all square-mean almost automorphic stochastic processes x:ℝ→L2(ℙ,ℍ) is denoted by AA(ℝ;L2(ℙ,ℍ)).Definition 5 (see [17]).
A functionf:ℝ×L2(ℙ,ℍ)→L2(ℙ,ℍ), (t,x)→f(t,x), which is jointly continuous, is said to be square-mean almost automorphic if f(t,x) is square-mean almost automorphic in t∈ℝ uniformly for all x∈𝕂, where 𝕂 is any bounded subset of L2(ℙ,ℍ). That is to say, for every sequence of real numbers {sn′}n∈ℕ, there exists a subsequence {sn}n∈ℕ and a function f~:ℝ×L2(ℙ,ℍ)→L2(ℙ,ℍ) such that
(8)limn→∞E∥f(t+sn,x)-f~(t,x)∥2=0,limn→∞E∥f~(t-sn,x)-f(t,x)∥2=0
for each t∈ℝ and each x∈𝕂. Denote by AA(ℝ×L2(ℙ,ℍ);L2(ℙ,ℍ)) the set of all such functions.Lemma 6 (see [22]).
(
A
A
(
ℝ
;
L
2
(
ℙ
,
ℍ
)
)
,
∥
·
∥
∞
) is a Banach space equipped with the norm
(9)∥x∥∞:=supt∈ℝ∥x(t)∥2=supt∈ℝ(E∥x(t)∥2)1/2,
for x∈AA(ℝ;L2(ℙ,ℍ)).Lemma 7 (see [17]).
Letf:ℝ×L2(ℙ,ℍ)→L2(ℙ,ℍ), (t,x)→f(t,x) be square-mean almost automorphic, and assume that f(t,·) is uniformly continuous on each bounded subset 𝕂⊂L2(ℙ,ℍ) uniformly for t∈ℝ; that is, for all ɛ>0, there exists δ>0 such that x,y∈𝕂 and E∥x-y∥2<δ imply that E∥f(t,x)-f(t,y)∥2<ɛ for all t∈ℝ. Then for any square-mean almost automorphic process x:ℝ→L2(ℙ,ℍ), the stochastic process F:ℝ→L2(ℙ,ℍ) given by F(·):=f(·,x(·)) is square-mean almost automorphic.Definition 8 (see [25]).
A stochastically continuous processf:ℝ+→L2(ℙ,ℍ) is said to be square-mean asymptotically almost automorphic if it can be decomposed as f=g+h, where g∈AA(ℝ;L2(ℙ,ℍ)) and h∈C0(ℝ+;L2(ℙ,ℍ)). Denote by AAA(ℝ+;L2(ℙ,ℍ)) the collection of all the square-mean asymptotically almost automorphic processes f:ℝ+→L2(ℙ,ℍ).Definition 9 (see [28]).
A functionf:ℝ+×L2(ℙ,ℍ)→L2(ℙ,ℍ), (t,x)→f(t,x), which is jointly continuous, is said to be square-mean asymptotically almost automorphic if it can be decomposed as f=g+h, where g∈AA(ℝ×L2(ℙ,ℍ);L2(ℙ,ℍ)) and h∈C0(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)). Denote by AAA(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)) the set of all such functions.Lemma 10 (see [28]).
Iff, f1, and f2 are all square-mean asymptotically almost automorphic stochastic processes, then the following hold true: (I)
f1+f2 is square-mean asymptotically almost automorphic;(II)
λf is square-mean asymptotically almost automorphic for any scalar λ;(III)
there exists a constantM>0 such that supt∈ℝ+E∥f(t)∥2≤M.Lemma 11 (see [28]).
Suppose thatf∈AAA(ℝ+;L2(ℙ,ℍ)) admits a decomposition f=g+h, where g∈AA(ℝ;L2(ℙ,ℍ)) and h∈C0(ℝ+;L2(ℙ,ℍ)). Then {g(t):t∈ℝ}⊂{f(t):t∈ℝ+}¯.Corollary 12 (see [28]).
The decomposition of a square-mean asymptotically almost automorphic process is unique.Lemma 13 (see [28]).
A
A
A
(
ℝ
+
;
L
2
(
ℙ
,
ℍ
)
) is a Banach space when it is equipped with the norm
(10)∥f∥AAA(ℝ+;L2(ℙ,ℍ)):=supt∈ℝ∥g(t)∥2+supt∈ℝ+∥h(t)∥2,
where f=g+h∈AAA(ℝ+;L2(ℙ,ℍ)) with g∈AA(ℝ;L2(ℙ,ℍ)), h∈C0(ℝ+;L2(ℙ,ℍ)).Lemma 14 (see [28]).
A
A
A
(
ℝ
+
;
L
2
(
ℙ
,
ℍ
)
) is a Banach space with the norm
(11)∥f∥∞:=supt∈ℝ+∥f(t)∥2=supt∈ℝ+(E∥f(t)∥2)1/2.Remark 15 (see [28]).
In view of the previous lemmas it is clear that the two norms are equivalent inAAA(ℝ+;L2(ℙ,ℍ)).Lemma 16 (see [28]).
Letf∈AA(ℝ×L2(ℙ,ℍ);L2(ℙ,ℍ)) and let f(t,x) be uniformly continuous in any bounded subset 𝕂⊂L2(ℙ,ℍ) uniformly for t∈ℝ+. Then f(t,x) is uniformly continuous in any bounded subset 𝕂⊂L2(ℙ,ℍ) uniformly for t∈ℝ.Lemma 17 (see [28]).
Letf∈AAA(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)) and suppose that f(t,x) is uniformly continuous in any bounded subset 𝕂⊂L2(ℙ,ℍ) uniformly for t∈ℝ+. If u(t)∈AAA(ℝ+;L2(ℙ,ℍ)), then f(·,u(·))∈AAA(ℝ+;L2(ℙ,ℍ)).We now give the following concept of mild solution of (1).Definition 18.
LetSα(t) be an integrable solution operator on L2(ℙ,ℍ) with generator A. An ℱt-adapted stochastic process x:[0,+∞)→L2(ℙ,ℍ) is called a mild solution of the problem (1) if x(0)=u0 is ℱ0-measurable and x(t) satisfies the corresponding stochastic integral equation:
(12)x(t)=Sα(t)[u0-f(0,u0)]+f(t,x(t))+∫0t‍Sα(t-s)g(s,x(s))dW(s),t≥0.


## 2.1. Sectorial Linear Operators

A closed and linear operatorA is said to be sectorial of type ϖ and angle θ if there exist 0<θ<π/2,M>0, and ϖ∈ℝ such that its resolvent exists outside the sector ϖ+Sθ:={ϖ+λ:λ∈ℂ,|arg(-λ)|<θ} and ∥(λ-A)-1∥≤M/|λ-ϖ|, λ∉ϖ+Sθ.Definition 1 (see [2]).
LetA be a closed and linear operator with domain D(A) defined on a Banach space 𝕏. We call A the generator of a solution operator if there exist ϖ∈ℝ and a strongly continuous function Sα:ℝ+→ℒ(𝕏) such that {λα:Re(λ)>ϖ}⊂ρ(A) and λα-1(λα-A)-1x=∫0∞‍e-λtSα(t)xdt, Re(λ)>ϖ,x∈𝕏. In this case, Sα(·) is called the solution operator generated by A.We note that ifA is sectorial of type ϖ with 0≤θ<π(1-α/2), then A is the generator of a solution operator given by Sα(t):=(1/2πi)∫γ‍eλtλα-1(λα-A)-1dλ,t≥0, where γ is a suitable path lying outside the sector ϖ+Sθ. Recently, Cuesta in [1] proved that if A is a sectorial operator of type ϖ<0 for some M>0 and 0≤θ<π(1-α/2), then there exists a constant C>0 such that
(5)∥Sα(t)∥≤CM1+|ϖ|tα,t≥0.Remark 2.
Note thatSα(t) is, in fact, integrable. For more details on the solution family Sα(t) and related issues, we refer the reader to [31–33].


## 2.2. Square-Mean Asymptotically Almost Automorphic Processes

We recall some basic facts for a symptotically almost automorphic processes which will be used in the sequel.Definition 3 (see [22]).
A stochastic processx:ℝ→L2(ℙ,ℍ) is said to be stochastically continuous if
(6)limt→sE∥x(t)-x(s)∥2=0.Definition 4 (see [17]).
A stochastically continuous stochastic processx:ℝ→L2(ℙ,ℍ) is said to be square-mean almost automorphic if, for every sequence of real numbers {sn′}n∈ℕ, there exist a subsequence {sn}n∈ℕ and a stochastic process y:ℝ→L2(ℙ,ℍ) such that
(7)limn→∞E∥x(t+sn)-y(t)∥2=0,limn→∞E∥y(t-sn)-x(t)∥2=0
hold for each t∈ℝ. The collection of all square-mean almost automorphic stochastic processes x:ℝ→L2(ℙ,ℍ) is denoted by AA(ℝ;L2(ℙ,ℍ)).Definition 5 (see [17]).
A functionf:ℝ×L2(ℙ,ℍ)→L2(ℙ,ℍ), (t,x)→f(t,x), which is jointly continuous, is said to be square-mean almost automorphic if f(t,x) is square-mean almost automorphic in t∈ℝ uniformly for all x∈𝕂, where 𝕂 is any bounded subset of L2(ℙ,ℍ). That is to say, for every sequence of real numbers {sn′}n∈ℕ, there exists a subsequence {sn}n∈ℕ and a function f~:ℝ×L2(ℙ,ℍ)→L2(ℙ,ℍ) such that
(8)limn→∞E∥f(t+sn,x)-f~(t,x)∥2=0,limn→∞E∥f~(t-sn,x)-f(t,x)∥2=0
for each t∈ℝ and each x∈𝕂. Denote by AA(ℝ×L2(ℙ,ℍ);L2(ℙ,ℍ)) the set of all such functions.Lemma 6 (see [22]).
(
A
A
(
ℝ
;
L
2
(
ℙ
,
ℍ
)
)
,
∥
·
∥
∞
) is a Banach space equipped with the norm
(9)∥x∥∞:=supt∈ℝ∥x(t)∥2=supt∈ℝ(E∥x(t)∥2)1/2,
for x∈AA(ℝ;L2(ℙ,ℍ)).Lemma 7 (see [17]).
Letf:ℝ×L2(ℙ,ℍ)→L2(ℙ,ℍ), (t,x)→f(t,x) be square-mean almost automorphic, and assume that f(t,·) is uniformly continuous on each bounded subset 𝕂⊂L2(ℙ,ℍ) uniformly for t∈ℝ; that is, for all ɛ>0, there exists δ>0 such that x,y∈𝕂 and E∥x-y∥2<δ imply that E∥f(t,x)-f(t,y)∥2<ɛ for all t∈ℝ. Then for any square-mean almost automorphic process x:ℝ→L2(ℙ,ℍ), the stochastic process F:ℝ→L2(ℙ,ℍ) given by F(·):=f(·,x(·)) is square-mean almost automorphic.Definition 8 (see [25]).
A stochastically continuous processf:ℝ+→L2(ℙ,ℍ) is said to be square-mean asymptotically almost automorphic if it can be decomposed as f=g+h, where g∈AA(ℝ;L2(ℙ,ℍ)) and h∈C0(ℝ+;L2(ℙ,ℍ)). Denote by AAA(ℝ+;L2(ℙ,ℍ)) the collection of all the square-mean asymptotically almost automorphic processes f:ℝ+→L2(ℙ,ℍ).Definition 9 (see [28]).
A functionf:ℝ+×L2(ℙ,ℍ)→L2(ℙ,ℍ), (t,x)→f(t,x), which is jointly continuous, is said to be square-mean asymptotically almost automorphic if it can be decomposed as f=g+h, where g∈AA(ℝ×L2(ℙ,ℍ);L2(ℙ,ℍ)) and h∈C0(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)). Denote by AAA(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)) the set of all such functions.Lemma 10 (see [28]).
Iff, f1, and f2 are all square-mean asymptotically almost automorphic stochastic processes, then the following hold true: (I)
f1+f2 is square-mean asymptotically almost automorphic;(II)
λf is square-mean asymptotically almost automorphic for any scalar λ;(III)
there exists a constantM>0 such that supt∈ℝ+E∥f(t)∥2≤M.Lemma 11 (see [28]).
Suppose thatf∈AAA(ℝ+;L2(ℙ,ℍ)) admits a decomposition f=g+h, where g∈AA(ℝ;L2(ℙ,ℍ)) and h∈C0(ℝ+;L2(ℙ,ℍ)). Then {g(t):t∈ℝ}⊂{f(t):t∈ℝ+}¯.Corollary 12 (see [28]).
The decomposition of a square-mean asymptotically almost automorphic process is unique.Lemma 13 (see [28]).
A
A
A
(
ℝ
+
;
L
2
(
ℙ
,
ℍ
)
) is a Banach space when it is equipped with the norm
(10)∥f∥AAA(ℝ+;L2(ℙ,ℍ)):=supt∈ℝ∥g(t)∥2+supt∈ℝ+∥h(t)∥2,
where f=g+h∈AAA(ℝ+;L2(ℙ,ℍ)) with g∈AA(ℝ;L2(ℙ,ℍ)), h∈C0(ℝ+;L2(ℙ,ℍ)).Lemma 14 (see [28]).
A
A
A
(
ℝ
+
;
L
2
(
ℙ
,
ℍ
)
) is a Banach space with the norm
(11)∥f∥∞:=supt∈ℝ+∥f(t)∥2=supt∈ℝ+(E∥f(t)∥2)1/2.Remark 15 (see [28]).
In view of the previous lemmas it is clear that the two norms are equivalent inAAA(ℝ+;L2(ℙ,ℍ)).Lemma 16 (see [28]).
Letf∈AA(ℝ×L2(ℙ,ℍ);L2(ℙ,ℍ)) and let f(t,x) be uniformly continuous in any bounded subset 𝕂⊂L2(ℙ,ℍ) uniformly for t∈ℝ+. Then f(t,x) is uniformly continuous in any bounded subset 𝕂⊂L2(ℙ,ℍ) uniformly for t∈ℝ.Lemma 17 (see [28]).
Letf∈AAA(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)) and suppose that f(t,x) is uniformly continuous in any bounded subset 𝕂⊂L2(ℙ,ℍ) uniformly for t∈ℝ+. If u(t)∈AAA(ℝ+;L2(ℙ,ℍ)), then f(·,u(·))∈AAA(ℝ+;L2(ℙ,ℍ)).We now give the following concept of mild solution of (1).Definition 18.
LetSα(t) be an integrable solution operator on L2(ℙ,ℍ) with generator A. An ℱt-adapted stochastic process x:[0,+∞)→L2(ℙ,ℍ) is called a mild solution of the problem (1) if x(0)=u0 is ℱ0-measurable and x(t) satisfies the corresponding stochastic integral equation:
(12)x(t)=Sα(t)[u0-f(0,u0)]+f(t,x(t))+∫0t‍Sα(t-s)g(s,x(s))dW(s),t≥0.


## 3. Main Results

In this section, we establish the existence of square-mean asymptotically almost automorphic mild solutions to the problem (1). For that, we need the following technical results.First, we list the following basic assumptions.(H1)
The operatorA is a sectorial operator of type ϖ<0 for some M>0 and 0≤θ<π(1-α/2), and then there exists  C>0 such that
(13)∥Sα(t)∥≤CM1+|ϖ|tα,t≥0,whereSα(t) is the solution operator generated by A.(H2)
The functionf∈AAA(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)) and there exists a continuous and nondecreasing function Lf:[0,+∞)→[0,+∞) such that for each r≥0 and for all E∥x∥2≤r, E∥y∥2≤r,
(14)E∥f(t,x)-f(t,y)∥2≤Lf(r)E∥x-y∥2for allt∈ℝ+.(H3)
The functiong∈AAA(ℝ+×L2(ℙ,ℍ);L2(ℙ,ℍ)) and there exists a continuous and nondecreasing function Lg:[0,+∞)→[0,+∞) such that for each r≥0 and for all E∥x∥2≤r, E∥y∥2≤r,
(15)E∥g(t,x)-g(t,y)∥2≤Lg(r)E∥x-y∥2for allt∈ℝ+.(H4)
We have(16)supr>0[|ϖ|-1/α(1-1/α)πLg(r)rαsin(π/α)r6(CM)2-Lf(r)r(CM)2-|ϖ|-1/α(1-1/α)πLg(r)rαsin(π/α)-λr]>(1+1(CM)2)Mf+|ϖ|-1/α(1-1/α)παsin(π/α)Mg,whereMf=supt∈ℝ+E∥f(t,x(t))∥2 and Mg=supt∈ℝ+E∥g(t,x(t))∥2.(H5)
The operatorA is a sectorial operator of type ϖ with 0≤θ<π(1-α/2), and there exists ϕ(·)∈L1(ℝ+) such that
(17)∥Sα(t)∥2≤ϕ(t)∀t≥0,limt→+∞ϕ(t)=0,whereSα(t) is the solution operator generated by A.Lemma 19.
Suppose that assumption (H1) holds and letf∈AAA(ℝ+;L2(ℙ,ℍ)). If F is the function defined by
(18)F(t):=∫0t‍Sα(t-s)f(s)dW(s),t≥0,
then F∈AAA(ℝ+;L2(ℙ,ℍ)).Proof.
Sincef∈AAA(ℝ+;L2(ℙ,ℍ)), we have by definition that f=g+h, where g∈AA(R;L2(P,H)) and h∈C0(ℝ+;L2(ℙ,ℍ)). Then
(19)F(t)=∫0t‍Sα(t-s)g(s)dW(s)+∫0t‍Sα(t-s)h(s)dW(s)=∫-∞t‍Sα(t-s)g(s)dW(s)-∫-∞0‍Sα(t-s)g(s)dW(s)+∫0t‍Sα(t-s)h(s)dW(s)=G(t)+H(t),
where G(t)=∫-∞t‍Sα(t-s)g(s)dW(s) and H(t)=-∫-∞0‍Sα(t-s)g(s)dW(s)+∫0t‍Sα(t-s)h(s)dW(s).
First we prove thatG(t)∈AA(ℝ;L2(ℙ,ℍ)). Let {sn′}n∈ℕ be an arbitrary sequence of real numbers. Since g∈AA(ℝ;L2(ℙ,ℍ)), there exists a subsequence {sn}n∈ℕ of {sn′}n∈ℕ such that for a certain stochastic process g~(20)limn→∞E∥g(t+sn)-g~(t)∥2=0,limn→∞E∥g~(t-sn)-g(t)∥2=0
hold for each t∈ℝ. Now, let W~(σ):=W(σ+sn)-W(sn) for each σ∈ℝ. Note that W~ is also a Brownian motion and has the same distribution as W. Moreover, if we let G~(t)=∫-∞t‍Sα(t-s)g~(s)dW(s), then by making a change of variables σ=s-sn to get (see the equation (10.6.6) in [34])
(21)E∥G(t+sn)-G~(t)∥2=E∥∫-∞t+sn‍Sα(t+sn-s)g(s)dW(s)-∫-∞t‍Sα(t-s)g~(s)dW(s)∥2=E∥∫-∞t‍Sα(t-σ)[g(σ+sn)-g~(σ)]dW~(σ)∥2
and, hence, using the Ito’s isometry property of stochastic integral, we have the following estimations
(22)E∥G(t+sn)-G~(t)∥2≤E(∫-∞t‍∥Sα(t-σ)[g(σ+sn)-g~(σ)]∥2dσ)≤∫-∞t‍∥Sα(t-σ)∥2E∥g(σ+sn)-g~(σ)∥2dσ≤supt∈ℝE∥g(t+sn)-g~(t)∥2∫-∞t‍(CM1+|ϖ|(t-σ)α)2dσ≤(CM)2supt∈ℝE∥g(t+sn)-g~(t)∥2∫0∞‍1(1+|ϖ|sα)2ds≤(CM)2|ϖ|-1/α(1-1/α)παsin(π/α)supt∈ℝE∥g(t+sn)-g~(t)∥2.
Then by (20), we obtain that limn→∞E∥G(t+sn)-G~(t)∥2=0 for each t∈ℝ. In a similar way, we can show that limn→∞E∥G~(t-sn)-G(t)∥2=0 for each t∈ℝ. Thus we conclude that (·)∈AA(ℝ;L2(ℙ,ℍ)).
Next, let us show thatH(·)∈C0(ℝ+;L2(ℙ,ℍ)). Since h∈C0(ℝ+;L2(ℙ,ℍ)) and 1/(1+|ϖ|sα)2 is integrable in [0,+∞), for any sufficiently small ɛ>0, there exists a constant T>0 such that E∥h(s)∥2≤ɛ and ∫T∞‍(1/(1+|ϖ|sα)2)ds≤ɛ for all s≥T. Then, for all t≥2T, we obtain
(23)E∥H(t)∥2=E∥∫0t/2‍Sα(t-s)h(s)dW(s)+∫t/2t‍Sα(t-s)h(s)dW(s)-∫-∞0‍Sα(t-s)g(s)dW(s)∥2≤3E(∫0t/2‍∥Sα(t-s)h(s)∥2ds)+3E(∫t/2t‍∥Sα(t-s)h(s)∥2ds)+3E(∫-∞0‍∥Sα(t-s)g(s)∥2ds)≤3∫0t/2‍(CM1+|ϖ|(t-s)α)2E∥h(s)∥2ds+3∫t/2t‍(CM1+|ϖ|(t-s)α)2E∥h(s)∥2ds+3∫-∞0‍(CM1+|ϖ|(t-s)α)2E∥g(s)∥2ds≤3(CM)2supt∈ℝ+E∥h(t)∥2∫t/2t‍1(1+|ϖ|sα)2ds+3(CM)2ɛ∫0t/2‍1(1+|ϖ|sα)2ds+3(CM)2supt∈ℝE∥g(t)∥2∫t∞‍1(1+|ϖ|sα)2ds≤3(CM)2supt∈ℝ+E∥h(t)∥2∫T∞‍1(1+|ϖ|sα)2ds+3(CM)2ɛ∫0∞‍1(1+|ϖ|sα)2ds+3(CM)2supt∈ℝE∥g(t)∥2∫T∞‍1(1+|ϖ|sα)2ds≤3(CM)2ɛ(supt∈ℝ+E∥h(t)∥2+|ϖ|-1/α(1-1/α)παsin(π/α)+supt∈ℝE∥g(t)∥2|ϖ|-1/α(1-1/α)παsin(π/α)).
This inequality proves the assertion since ɛ is arbitrary. Recalling that F(t)=G(t)+H(t) for all t≥0, we get F(t)∈AAA(ℝ+;L2(ℙ,ℍ)). The proof is completed.It is easy to see that, by arguments similar to those in the proof of Lemma19, we have the following result.Lemma 20.
Suppose that assumption (H5) holds and letf∈AAA(ℝ+;L2(ℙ,ℍ)). If F is the function defined by
(24)F(t):=∫0t‍Sα(t-s)f(s)dW(s),t≥0,
then F∈AAA(ℝ+;L2(ℙ,ℍ)).Now, we are ready to establish our main results.Theorem 21.
Assume that (H1)–(H4) hold. Then there existsɛ>0 such that for each u0∈Bɛ(0,L2(ℙ,ℍ)) there exists a unique square-mean asymptotically almost automorphic mild solution x(·,u0) of the problem (1) on [0,∞) such that x(0,u0)=u0.Proof.
We define a nonlinear operatorΥ by
(25)(Υx)(t)=Sα(t)[u0-f(0,u0)]+f(t,x(t))+∫0t‍Sα(t-s)g(s,x(s))dW(s)t≥0.
First we prove thatΥ(AAA(ℝ+;L2(ℙ,ℍ)))⊆AAA(ℝ+;L2(ℙ,ℍ)). Given x∈AAA(ℝ+;L2(ℙ,ℍ)), from the properties of {Sα(t)}t≥0, f, and g, we infer that Υx is well defined and continuous. Since x(t) is bounded, we can choose a bounded subset 𝕂 of L2(ℙ,ℍ) such that x(t)∈𝕂 for all t∈ℝ+. It follows from (H2) and (H3) that both f(t,x) and g(t,x) are uniformly continuous on the bounded subset 𝕂 uniformly for t∈ℝ+. Moreover, from Lemmas 17 and 19 and taking into account (H1), it follows that Υx∈AAA(ℝ+;L2(ℙ,ℍ)).
Now, by (H4), there exists a constantr>0 such that
(26)r6(CM)2-Lf(r)r(CM)2-|ϖ|-1/α(1-1/α)πLg(r)rαsin(π/α)-λr>(1+1(CM)2)Mf+|ϖ|-1/α(1-1/α)παsin(π/α)Mg.
Let 0<λ<1. We affirm that the assertion holds for ɛ=λr. In fact, let u0∈Bɛ(0,L2(ℙ,ℍ)). Define the space 𝔻={x∈AAA(ℝ+;L2(ℙ,ℍ)):x(0)=u0,E∥x(t)∥2≤r,t≥0}. Then 𝔻 is a closed subspace of AAA(ℝ+;L2(ℙ,ℍ)). We claim that Υ𝔻⊆𝔻. If x∈𝔻 and t∈ℝ+, we get
(27)E∥(Υx)(t)∥2≤3E∥Sα(t)[u0-f(0,u0)]∥2+3E∥f(t,x(t))∥2+3E∥∫0t‍Sα(t-s)g(s,x(s))dW(s)∥2≤6(CM)2[E∥u0∥2+E∥f(0,u0)∥2]+3E∥f(t,x(t))-f(t,0)+f(t,0)∥2+3E(∫0t‍∥Sα(t-s)g(s,x(s))∥2ds)≤6(CM)2λr+6(CM)2E∥f(0,u0)∥2+6Lf(r)r+6supt∈ℝ+E∥f(t,0)∥2+6(CM)2[∫0t‍(11+|ϖ|(t-s)α)2Lg(r)rds+∫0t‍(11+|ϖ|(t-s)α)2E∥g(s,0)∥2ds]≤6(CM)2λr+6(CM)2Mf+6Lf(r)r+6Mf+6(CM)2|ϖ|-1/α(1-1/α)παsin(π/α)[Lg(r)r+Mg],
which from (26) implies that E∥(Υx)(t)∥2≤r for all t≥0, and so that Υ𝔻⊆𝔻.
Next, to complete the proof, we need to show thatΥ(·) is a contraction from 𝔻 into 𝔻. By (26), we know that
(28)r6(CM)2-Lf(r)r(CM)2-|ϖ|-1/α(1-1/α)πLg(r)rαsin(π/α)-λr>0.
That is,
(29)r6(CM)2>Lf(r)r(CM)2+|ϖ|-1/α(1-1/α)πLg(r)rαsin(π/α)+λr.
Then, one has
(30)6Lf(r)+6(CM)2|ϖ|-1/α(1-1/α)παsin(π/α)Lg(r)+6λ(CM)2<1.
For any x,y∈𝔻 and t≥0, we have
(31)E∥(Υx)(t)-(Υy)(t)∥2≤2E∥f(t,x(t))-f(t,y(t))∥2+2E∥∫0t‍Sα(t-s)[g(s,x(s))-g(s,y(s))]dW(s)∥2≤2Lf(r)supt∈ℝ+E∥x(t)-y(t)∥2+2(CM)2∫0t‍(11+|ϖ|(t-s)α)2×E∥g(s,x(s))-g(s,y(s))∥2ds≤2Lf(r)supt∈ℝ+E∥x(t)-y(t)∥2+2(CM)2|ϖ|-1/α(1-1/α)παsin(π/α)Lg(r)×supt∈ℝ+E∥x(t)-y(t)∥2≤[2Lf(r)+2(CM)2|ϖ|-1/α(1-1/α)παsin(π/α)Lg(r)]×supt∈ℝ+E∥x(t)-y(t)∥2.
Thus, we get
(32)∥Υx-Υy∥∞=supt∈ℝ+(E∥(Υx)(t)-(Υy)(t)∥2)1/2≤2Lf(r)+2(CM)2|ϖ|-1/α(1-1/α)παsin(π/α)Lg(r)∥x-y∥∞.
It follows from (30) that Υ is a contraction mapping on 𝔻. So by the Banach contraction mapping principle, we draw a conclusion that there exists a unique fixed point x(·) for Υ in 𝔻. It is clear that x is a square-mean asymptotically almost automorphic mild solution of (1). The proof is complete.The next result is proved using the similar steps as in the proof of the previous result, so we omit the details.Theorem 22.
Assume that (H1)–(H3) hold. IfLf(r)≡Lf and Lg(r)≡Lg for all r≥0 and Lf+(CM)2|ϖ|-1/α(1-1/α)πLg/αsin(π/α)<1/2, then for every u0∈L2(ℙ,ℍ), there exists a unique square-mean asymptotically almost automorphic mild solution x(·,u0) of the problem (1) on [0,∞) such that x(0,u0)=u0.Theorem 23.
Suppose that assumptions (H2), (H3), and (H5) hold. IfLf(r)≡Lf and Lg(r)≡Lg for all r≥0 and Lf+Lg∥ϕ∥1<1/2, then for every u0∈L2(ℙ,ℍ), there exists a unique square-mean asymptotically almost automorphic mild solution x(·,u0) of the problem (1) on [0,∞) such that x(0,u0)=u0.Proof.
Consider the nonlinear operatorΥ given by
(33)(Υx)(t)=Sα(t)[u0-f(0,u0)]+f(t,x(t)),+∫0t‍Sα(t-s)g(s,x(s))dW(s),t≥0.
First we prove thatΥ maps AAA(ℝ+;L2(ℙ,ℍ)) into itself. Given x∈AAA(ℝ+;L2(ℙ,ℍ)), from the properties of {Sα(t)}t≥0, f and g, we infer that Υx is well defined and continuous. Since x(t) is bounded, we can choose a bounded subset 𝕂 of L2(ℙ,ℍ) such that x(t)∈𝕂 for all t∈ℝ+. It follows from conditions (H2) and (H3) that both f(t,x) and g(t,x) are uniformly continuous on the bounded subset 𝕂 uniformly for t∈ℝ+. Moreover, from Lemmas 17 and 20 and taking into account (H5), it follows that Υx∈AAA(ℝ+;L2(ℙ,ℍ)).
Next we prove thatΥ is a contraction mapping from AAA(ℝ+;L2(ℙ,ℍ)) into itself. Note that we have already proved Υ(AAA(ℝ+;L2(ℙ,ℍ)))⊆AAA(ℝ+;L2(ℙ,ℍ)). Moreover, for any x,y∈AAA(ℝ+;L2(ℙ,ℍ)) and t≥0, we have
(34)E∥(Υx)(t)-(Υy)(t)∥2≤2E∥f(t,x(t))-f(t,y(t))∥2+2E∥∫0t‍Sα(t-s)[g(s,x(s))-g(s,y(s))]dW(s)∥2≤2Lfsupt∈ℝ+E∥x(t)-y(t)∥2+2∫0t‍ϕ(t-s)E∥g(s,x(s))-g(s,y(s))∥2ds≤2Lfsupt∈ℝ+E∥x(t)-y(t)∥2+2Lgsupt∈ℝ+E∥x(t)-y(t)∥2∫0∞‍ϕ(s)ds≤[2Lf+2Lg∥ϕ∥1]supt∈ℝ+E∥x(t)-y(t)∥2.
Therefore
(35)∥Υx-Υy∥∞=supt∈ℝ+(E∥(Υx)(t)-(Υy)(t)∥2)1/2≤2Lf+2Lg∥ϕ∥1∥x-y∥∞.
That is, Υ is a contraction mapping on AAA(ℝ+;L2(ℙ,ℍ)). By the Banach contraction mapping principle, Υ has a unique fixed point x(t)∈AAA(ℝ+;L2(ℙ,ℍ)). It is clear that x is a square-mean asymptotically almost automorphic mild solution of (1). The proof is then complete.


## 4. An Example

In this section, we apply the results obtained previously to investigate the existence of square-mean asymptotically almost automorphic mild solutions for the following partial stochastic fractional differential system:(36)∂∂t[u(t,ξ)-∫-∞t‍a(t)a1(t-s)u(s,ξ)ds]=Jtα-1(∂2∂ξ2-ν)[u(t,ξ)-∫-∞ta(t)a1(t-s)u(s,ξ)ds]+∫-∞t‍a(t)a2(t-s)u(s,ξ)dsdW(t),u(t,0)=u(t,π)=0,t≥0,u(0,ξ)=u0(ξ),ξ∈I=[0,π],
where a∈AAA(ℝ+;ℝ), a1,a2:ℝ+→ℝ are continuous functions, and ν>0 is a fixed constant.Letℍ=L2([0,π]) with the norm ∥·∥ and A:D(A)⊆ℍ→ℍ be the operator defined by Ax=x′′-νx domain D(A)={x∈ℍ:x′′∈ℍ,x(0)=x(π)=0}. It is well known that Δx=x′′ is the infinitesimal generator of an analytic semigroup {S(t)}t≥0 on ℍ. Furthermore, A is sectorial of type ϖ=-ν<0.In the sequel, we assume(37)Lf=∥a∥∞(∫-∞0‍|a1(-s)|2ds)1/2<∞,Lg=∥a∥∞(∫-∞0‍|a2(-s)|2ds)1/2<∞.
Now, we can define the functions f,g:ℝ+×L2(ℙ,ℍ)→L2(ℙ,ℍ) by
(38)f(t,x)(ξ)=a(t)∫-∞0‍a1(-s)x(s,ξ)ds,g(t,x)(ξ)=a(t)∫-∞0‍a2(-s)x(s,ξ)ds,Jtα-1h(t)=∫0t‍(t-s)α-2Γ(α-1)h(s)ds,
which permits to transform the system (36) into the abstract system (1). Moreover, it is not difficult to see that f,g are continuous and Lipschitz in the second variable with Lipschitz constants Lf and Lg, respectively.The next result is a consequence of Theorem22.Theorem 24.
Under the previous assumptions, (36) has a unique mild solution x∈AAA(ℝ+;L2(ℙ,ℍ)) whenever Lf and Lg are small enough.



---

*Source: 138068-2013-12-15.xml*