# Existence and Stability of Square-Mean S-Asymptotically Periodic Solutions to a Fractional Stochastic Diffusion Equation with Fractional Brownian Motion

Authors: Jia Mu; Jiecuo Nan; Yong Zhou
Journal: Complexity
 (2020)

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## Abstract

In this paper, a generalized Gronwall inequality is demonstrated, playing an important role in the study of fractional differential equations. In addition, with the fixed-point theorem and the properties of Mittag–Leffler functions, some results of the existence as well as asymptotic stability of square-meanS-asymptotically periodic solutions to a fractional stochastic diffusion equation with fractional Brownian motion are obtained. In the end, an example of numerical simulation is given to illustrate the effectiveness of our theory results.


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## Body

## 1. Introduction

Originated in 1695, fractional calculus has been widely applied in physics, chemistry, economics, biology, and other fields. Recent decades have witnessed the rapid development of fractional calculus, with the emergence of many related researches [1–9]. The dynamic behavior of some complex processes in reality can be explained by fractional differential equations. For example, anomalous diffusion phenomena can be described with fractional diffusion equations. Compared with the traditional diffusion equations (first order), with fractional diffusion equations, subdiffusion or supdiffusion phenomenon can be described when its order is between 0 and 1 or between 1 and 2, respectively.In addition, the diffusion phenomenon in real life is often affected by random factors, promoting the generation of fractional stochastic diffusion equations. However, this research has not aroused much concern until recent years. In [10], a class of nonautonomous fractional stochastic reaction-diffusion equations was studied, obtaining the regularity of random attractors. The Galerkin method was applied by Wang [11] to investigate the existence of tempered pullback random attractors for nonautonomous fractional reaction-diffusion equations with multiplicative noise. Chen [12] studied the stochastic time-fractional diffusion equations with multiplicative white noise, obtaining the Hölder continuity of the solution. Peng and Huang [13] established the existence of mild solutions for a nonlocal backward problem for fractional stochastic diffusion equations.We also notice that some researches focus on the stability of solutions to fractional stochastic differential equations of orderα∈0,1. Li and Wang [14] studied the existence and asymptotic behavior of solutions to fractional stochastic delay evolution equations with integral term and Wiener process by using fractional resolvent operator theory and the Schauder fixed-point theorem. Mathiyalagan and Balachandran [15] studied the finite-time stochastic stability of fractional-order singular systems with time delay and white noise utilizing the Gronwall approach and stochastic analysis technique. In [16], applying the Laplace transform method, the authors obtained the existence, uniqueness, and Hyers–Ulam stability of solutions to a class of linear fractional differential equations involving Mittag–Leffler kernel. The resolvent operator technique and contraction mapping principle were used in [17] to study the existence and uniqueness of mild solution to fractional neutral stochastic integrodifferential equations involving impulses driven by fractional Brownian motion (FBM), and a new impulsive-integral inequality was used to obtain the exponential stability for these equations. Moreover, the existence and asymptotic stability in the p-th moment of mild solutions to a class of fractional stochastic partial differential equations with Wiener process was investigated by Zhang et al. [18]. Because the form of the equations in this paper is different from those in the above studies, the methods to prove the stability in these studies cannot be directly applied to this paper.Inspired by the above researches, in order to study the stability and periodicity of anomalous diffusion phenomena affected by random factors, we consider the fractional stochastic diffusion equation involving Dirichlet boundary conditions:(1)∂0,tαux,t+ux,t=ft,ux,t+Dt,ux,t∂BQHx,t∂t,inΩ×0,∞,u=0,on∂Ω×0,∞,ux,0=u0x,inΩ,where ∂0,tα denotes the Caputo fractional derivative, α∈1/2,3/2−H, and Ω⊂Rn is a bounded open domain, whose boundary ∂Ω is sufficiently smooth. Functions f and D satisfy some appropriate conditions, with the initial data u0 for u, ∂BQH/∂t is the fractional Brownian motion (FBM), and H∈1/2,1. In addition,(2)Aux,t=−∑i=1n∂∂xi∑j=1naijx∂∂xjux,t+bxux,t,and the functions aij satisfy(3)aij∈L∞ℝn,1≤i,j≤n,C0∑i=1nξi2≤∑i,j=1naijxξiξj,a.e.x∈ℝn,ξ∈ℝn,where C0>0 represents a constant, and bx satisfies(4)b∈L∞ℝn,bx≥B0>0,a.e.x∈ℝn.Firstly, a new generalized Gronwall inequality is given. Then, we obtain the existence and uniqueness of the S-asymptotically periodic solutions for problem (1) based on the characteristics of Mittag–Leffler functions, Hölder inequality, and the inequality for fractional stochastic integral with FBM and Banach’s fixed-point theorem. In addition, with the help of the generalized Gronwall inequality, some conditions are given to ensure the asymptotic stability of the S-asymptotically periodic solutions for problem (1). We notice that the generalized Gronwall inequality in [19] cannot be applied to Theorem 2 because the estimation obtained by that method is not stable.Compared with previous research results, the innovations of this paper include the following: (1) the equations studied contain both fractional differential operators and FBM. It is worth mentioning that the standard Brownian motion, without long memory, cannot represent all types of noise. A good long-term memory noise could be described by FBM of Hurst parameterH∈1/2,1 [20]. For instance, the continuous disturbance and long-term dependence in the financial market model can be considered as a kind of FBM [21], with the impact of nuclear waste on the environment being seen as FBM in ecological models. Other research studies on FBM can be referred to [22–30]. (2) Stability of the S-asymptotically periodic solutions is studied by means of a new generalized Gronwall inequality.The paper is organized as follows: the readers are allowed to review Section2 for the necessary basic knowledge, followed by some results of the existence and uniqueness of S-asymptotically periodic solutions in Section 3. Subsequently, the asymptotic stability of S-asymptotically periodic solutions is studied in Section 4, with a numerical simulation example in Section 5.


## 2. Preliminaries

For the sake of convenience in writing, throughout this paper, by∞, we mean +∞. Ω,ℱ,ℱtt≥0,P denotes a complete filtered probability space, and Uandℍ are two separable Hilbert spaces. The space of bounded linear operators from U into ℍ is written as LU,ℍ. For convenience, the same notation ⋅ is applied to denote norms in U,ℍ and LU,ℍ; ⋅,⋅ is applied to denote the inner produce of U and ℍ. Moreover, L2Ω;ℍ is the space of all strongly measurable and square-integrable ℍ-valued random variables under the Banach norm E⋅21/2.A stochastic processu:0,∞⟶L2Ω;ℍ is called stochastically bounded if supt≥0Eut2<+∞, called stochastically continuous if limt⟶sEut−us2=0 for all t,s≥0 and called square-mean S-asymptotically ω-periodic if limt⟶∞Eut+ω−ut2=0, where ω>0 is a constant.Denote bySBC0,∞;L2Ω;ℍ the space of all stochastically bounded and continuous processes from 0,∞ into L2Ω;ℍ and its norm(5)u∞=supt≥0Eut21/2,where Eut2=∫Ωut2dP. Then, SBC0,∞;L2Ω;ℍ is a Banach space.We useSAPw0,∞;L2Ω;ℍ to denote the space of square-mean S-asymptotically ω-periodic stochastic process from 0,∞ into L2Ω;ℍ. Then, SAPw0,∞;L2Ω;ℍ is a Banach space with the sup norm ⋅∞ and is a linear closed subspace of SBC0,∞;L2Ω;ℍ.In the following, we introduce the definition and properties of FBM. We denote byβHtt∈ℝH∈0,1 a two-sided one-dimensional FBM [23]. Then, βH is a continuous-centered Gaussian process, whose variance function is(6)RHt,s=EβHtβHs=12t2H+s2H−t−s2H,t,s∈ℝ.In addition, ifW is a Wiener process, then(7)βHt=∫0tKHt,sdWs,KHt,s=CHs1/2−H∫stτ−sH−3/2τH−1/2dτ,for t>s, where CH=H2H−1/B2−2H,H−1/2, with B denoting the beta function.LetQ∈Lℍ,ℍ be an operator with TrQ=∑n=1∞λn<∞ and Qen=λnen for constants λn≥0n=1,2,… and a complete orthonormal basis enn=1∞ in ℍ. The infinite dimensional FBM on ℍ can be expressed by(8)BQHt=∑n=1∞βnHtQ1/2en=∑n=1∞λnβnHten,t≥0,where Q is the covariance operator and βnHtn=1∞ are two-sided one-dimensional FBMs, which are mutually independent on Ω,ℱ,ℱtt≥0,P.LetL20U,ℍ be the collection of all Q-Hilbert–Schmidt operators ξ:U⟶ℍ, where ξQ1/2 is a Hilbert–Schmidt operator, and the norm is(9)ξL202:=trξQξ∗=∑n=1∞λnξen2<∞.For convenience, setL20ℍ:=L20ℍ,ℍ. The space L20U,ℍ is a separable Hilbert space whose inner product φ,ϕL20=∑n=1∞φen,ϕen. Then, we define the stochastic integral of ϕ with regard to BQH by(10)∫0tϕsdBQHs=∑n=1∞∫0tλnϕsendβnHs=∑n=1∞∫0tϕsQ1/2endβnHs,where ϕtt∈0,T is the deterministic function with values in L20U,ℍ.Now, we recall the Mittag–Leffler function and the probability density functions which play important roles in fractional differential equations [31].Lemma 1.
The Mittag–Leffler functions(11)Eαt=∑k=0∞tkΓαk+1,Eα,αt=∑k=0∞tkΓαk+1,t∈ℝ,where Γ is the Gamma function, have the following properties:(1)
Eαt,Eα,αt>0 [32, 33].(2)
Eαt′=1/αEα,αt [34].(3)
limt⟶−∞Eαt=limt⟶−∞Eα,αt=0 [32, 35].We notice that the Mittag–Leffler functionEαt is a generalization of exponential function et, which is E1t.Setutx=ux,t, ft,utx=ft,x,ux,t, and BQHutx=BQHux,t. Let ℍ=L2Ω, and we define A:DA⊂ℍ⟶ℍ by(12)DA=H2Ω∩H01Ω,Autx=Aux,t.We suppose that−A generates an exponentially stable C0-semigroup Ttt≥0 satisfying(13)Tt≤Me−δt,∀t∈0,∞,where M>0 and δ>0 are constants.Thus, (1) can be transformed into(14)D0αut+Aut=ft,ut+Dt,utdBQHtdt,t∈0,∞,u0=u0,where D0α is the Caputo-fractional derivative. Later, in the paper, u0∈L2Ω;ℍ.Remark 1.
We see that (13) is much easy to be verified. For example, let(15)Ω=0,π,ℍ=L20,π,A=−∂2∂x2,DA=u∈ℍu,u′are absolutely continuous,u″∈ℍ,u0=uπ=0.
Then,A has eigenvalues n2n∈ℕ, whose normalized eigenvectors wnt=2/πsinntn∈ℕ, −A generates an analytic, compact, and exponentially stable semigroup Ttt≥0, and(16)Ttu=∑n=1∞e−n2tu,wnwn,Tt≤e−t,t∈0,∞.Definition 1.
A stochastic continuous processu:0,∞⟶L2Ω;ℍ is said to be a mild solution of equation (14) if(17)ut=Utu0+∫0tt−sα−1Vt−sfs,usds+∫0tt−sα−1Vt−sDs,usdBQHs,where(18)Ut=∫0∞ζαθTtαθdθ,Vt=α∫0∞θζαθTtαθdθ,and the probability density function [36, 37](19)ζαθ=1αθ−1−1/αραθ−1/α,ραθ=1π∑n=0∞−1n−1θ−αn−1Γnα+1n!sinnπα,θ∈0,∞.
Later, in this paper, we need the following results.Remark 2.
(1)
ζαθ≥0, for θ∈0,∞.(2)
∫0∞θνζαθdθ=Γ1+ν/Γ1+αν, for ν∈−1,+∞ [36, 38].(3)
∫0∞e−tθθζαθdθ=1/αEα,α−t, for t∈ℝ [37].(4)
∫0∞e−tθθζαθdθ=1/αEα,α−t, for t∈ℝ [36].Lemma 2.
(1)
Ut and Vt are strongly continuous for t≥0 [38].(2)
IfTtt≥0 satisfy (14), then Ut≤MEα−δtα and Vt≤MEα,α−δtα for t≥0 [2].(3)
Vt≤Mα/δtα for t>0.Proof.
The proof of3 is as follows. In fact, for t>0, in view of e−v<1/v with v>0, we have(20)Vt≤α∫0∞θζαθTtαθdθ≤Mα∫0∞θζαθe−δtαθdθ≤Mα∫0∞θζαθδtαθdθ≤Mαδtα.
In order to get the stability of the square-mean S-asymptoticallyω-periodic solution, we need the following generalized Gronwall inequality for fractional differential equations.Lemma 3.
Letu0, λ1, λ2∈ℝ be two constants. If a continuous function u:0,+∞⟶ℝ satisfies(21)ut≤Eαλ1tαu0+λ2∫0tt−sα−1Eα,αλ1t−sαusds,then(22)ut≤Eαλ1+λ2tαu0.Proof.
We find that the solution of the equation [31](23)D0αut=λ1ut+λ2ut,t∈0,+∞,u0=u0,is given by(24)ut=Eαλ1tαu0+λ2∫0tt−sα−1Eα,αλ1t−sαusds,orut=Eαλ1+λ2tαu0.
In view of the uniqueness of solution to (23), we get (22).Remark 3.
Compared with the generalized Gronwall inequality in [19], Eαλ1tαu0 does not have to be a nondecreasing function, and λ2 is not necessarily nonnegative. This is very important to prove the stability of the solution.
Next, we give a result which is very useful for the estimations of fractional stochastic integral with FBM.Lemma 4 (see [39]).
LetH∈1/2,1 and h:0,T⟶L20U,V satisfy(25)∫abϕsL202ds<∞,for∀a,b∈0,Twithb>a,then the corresponding sum given in (10) is well defined and we obtain(26)E∫abhsdBQHs2≤2Hb−a2H−1∫abhsL202ds.


## 3. Existence Uniqueness of Square-Mean S-Asymptoticallyω-Periodic Solutions

Lemma 5.
Letu∈SAPω0,∞,L2Ω;ℍ,(27)Γ1ut=Utu0+∫0tt−sα−1Vt−sfs,usds,and if f:0,∞×L2Ω;ℍ⟶L2Ω;ℍ satisfies.
H1 For x in every bounded subset K of L2Ω;ℍ,ft,x is S-asymptotically ω-periodic in t. Moreover, there exists a positive constant L such that(28)Eft,y−ft,z2≤LEy−z2,for t∈0,∞ and y,z∈L2Ω;ℍ.
Then,Γ1u∈SAPω0,∞,L2Ω;ℍ.Proof.
Step 1: for∀ut∈SAPω0,∞,L2Ω;ℍ, we prove that(29)limt⟶∞EΓ1ut+ω−Γ1ut2=0.
Firstly, we have(30)Γ1ut+ω−Γ1ut=Ut+ω−Utu0+∫−ω0t−sα−1Vt−sfs+ω,us+ωds+∫0tt−sα−1Vt−sfs+ω,us+ω−fs,us+ωds+∫0tt−sα−1Vt−sfs,us+ω−fs,usds.
Then,(31)EΓ1ut+ω−Γ1ut2≤4EUt+ω−Utu02+4E∫−ω0t−sα−1Vt−sfs+ω,us+ωds2+4E∫0tt−sα−1Vt−sfs+ω,us+ω−fs,us+ωds2+4E∫0tt−sα−1Vt−sfs,us+ω−fs,usds2=:I1t+I2t+I3t+I4t.
In view of Lemma2, it is obvious that(32)limt⟶∞I1t=0.
By combining Hölder inequality withH1 and using Lemma 2, we have(33)I2t≤4∫−ω0t−sα−1Vt−sds∫−ω0t−sα−1Vt−sEfs+ω,us+ω2ds≤8LEu∞2+supt≥0Eft,02∫−ω0t−sα−1Vt−sds2≤8M2LEu∞2+supt≥0Eft,02∫−ω0t−sα−1Eα,α−δt−sαds2≤8M2δ2LEu∞2+supt≥0Eft,02Eα−δt−sα−ω02=8M2δ2LEu∞2+supt≥0Eft,02Eα−δtα−Eα−δt+ωα2.
The last formula and Lemma1 yield that(34)limt⟶∞I2t=0.
SinceH1 implies limt⟶+∞Eft+ω,ut+ω−ft,ut+ω2=0, then for ε>0, there exists Tε>0 such that Eft+ω,ut+ω−ft,ut+ω2≤ε whenever u>Tε. Then, we have(35)I34≤4∫0tt−sα−1Vt−sds∫0tt−sα−1Vt−sEfs+ω,us+ω−fs,us+ω2ds≤4M2∫0tt−sα−1Eα,α−δt−sαds∫0tt−sα−1Eα,α−δt−sα⋅Efs+ω,us+ω−fs,us+ω2ds≤4M2δ1−Eα−δtα∫0Tεt−sα−1Eα,α−δt−sα⋅Efs+ω,us+ω−fs,us+ω2ds+∫Tεtt−sα−1Eα,α−δt−sαEfs+ω,us+ω−fs,us+ω2ds≤4M2δ21−Eα−δtα4LEu∞2+supt≥0Eft,02Eα−δt−Tεα−Eα−δtα+ε1−Eα−δt−Tεα.
Due to Lemma1, it is obvious that(36)limt⟶∞I3t=0.
H1 implies that(37)I44≤4L∫0tt−sα−1Vt−sds∫0tt−sα−1Vt−sEus+ω−us2ds.
Using a strategy similar to the one in the proof of (36), we get(38)limt⟶∞I4t=0.
Then,limt⟶∞EΓ1ut+ω−Γ1ut2=0.
Step 2: For∀ut∈SAPω0,∞,L2Ω;ℍ, we prove that Γ1ut is stochastically bounded and continuous.
On the one hand, for a givent0≥0, we have(39)EΓ1ut−Γ1ut02≤3EUt−Ut0u02+3E∫t0−t0t0−sα−1Vt0−sfs+t−t0,us+t−t0ds2+3E∫0t0t0−sα−1Vt0−sfs+t−t0,us+t−t0−fs,usds2=:K1t+K2t+K3t.
Lemma2 (1) implies that limt⟶t0K1t=0. Arguing similarly as in (33), we see that(40)K2t≤6M2δ2LEu∞2+supt≥0Eft,02Eα−δt0α−Eα−δtα2,which means that limt⟶t0K2t=0.(41)K3t≤6M2∫0t0t0−sα−1Eα,α−δt0−sαds∫0t0t0−sα−1Eα,α−δt0−sα⋅Efs+t−t0,us+t−t0−fs,us+t−t02+Efs,us+t−t0−fs,us2ds≤6M2δ1−Eα−δt0α∫0t0t0−sα−1Eα,α−δt0−sαEfs+t−t0,us+t−t0−fs,us+t−t02+LEus+t−t0−us2ds.
For an arbitrary sequence of real numberstn with tn⟶t0 as n⟶∞, for ∀ut∈SAPω0,∞,L2Ω;ℍ, we have(42)Efs+tn−t0,us+tn−t0−fs,us+tn−t02+LEus+tn−t0−us2⟶0,asn⟶∞,which is due to H1. Hence,(43)Efs+tn−t0,us+tn−t0−fs,us+tn−t02+LEus+tn−t0−us2<1,for every n sufficiently large. In view of(44)∫0t0t0−sα−1Eα,α−δt0−sαds=1δ1−Eα−δt0α<1δ<∞,then it follows from Lebesgue’s dominated convergence theorem that(45)limn⟶∞∫0t0t0−sα−1Eα,α−δt0−sαEfs+tn−t0,us+tn−t0−fs,us+tn−t02+LEus+tn−t0−us2ds=0.
Additionally, according to the arbitrariness oftn, we have that(46)limt⟶t0∫0t0t0−sα−1Eα,α−δt0−sαEfs+t−t0,us+t−t0−fs,us+t−t02+LEus+t−t0−us2ds=0,which gives limt⟶t0K3t=0. Then, we know that Γ1ut is stochastically continuous.
On the other hand,(47)EΓ1ut2≤2EUtu02+2E∫0tt−sα−1Vt−sfs,usds2≤2M2Eu02+2M2∫0tt−sα−1Eα,α−δt−sαds⋅∫0tt−sα−1Eα,α−δt−sαEfs,us2ds≤2M2Eu02+4M2δ21−Eα−δtαLEu∞2+supt≥0Eft,02≤2M2Eu02+4M2δ2LEu∞2+supt≥0Eft,02<∞,which implies that Γ1ut is stochastically bounded.
By Steps 1-2, we obtainΓ1u∈SAPω0,∞,L2Ω;ℍ.Lemma 6.
Letu∈SAPω0,∞,L2Ω;ℍ and(48)Γ2ut=∫0tt−sα−1Vt−sDs,usdBQHs.
IfD:0,∞×L2Ω;ℍ⟶L20ℍ satisfies.
H2 For x in every bounded subset K of L2Ω;ℍ, Dt,x is stochastically bounded and continuous in t. Furthermore, for ∀ε>0 and K, there exists Tε>0 such that t2α+H−1Dt+ω,y−Dt,yL202<ε for t>Tε and y∈K.
H3 There exists a positive constant L1 such that(49)t2α+H−1Dt,y−Dt,zL202≤L1Ey−z2,for t∈0,∞ and y,z∈L2Ω;ℍ.
H4Dt,0=0 for t∈0,∞.
Then,Γ2u∈SAPω0,∞,L2Ω;ℍ.Proof.
Step 1: for∀ut∈SAPω0,∞,L2Ω;ℍ, we prove that(50)limt⟶∞EΓ2ut+ω−Γ2ut2=0.
LetB˜QHτ=BQHτ+ω−BQHω for each τ∈R. Then, it is easy to find that B˜QHτ is identically distributed like BQHτ. Next, for u∈SAPω0,∞,L2Ω;ℍ, we see that(51)Γ2ut+ω−Γ2ut=∫0t+ωt+ω−sα−1Vt+ω−sDs,usdBQHs−∫0tt−sα−1Vt−sDs,usdBQHs=∫−ωtt−τα−1Vt−τDτ+ω,uτ+ωdBQHτ+ω−∫0tt−sα−1Vt−sDs,usdBQHs=∫−ωtt−τα−1Vt−τDτ+ω,uτ+ωdB˜QHτ−∫0tt−sα−1Vt−sDs,usdB˜QHs.
Then,(52)EΓ2ut+ω−Γ2ut2≤2E∫−ω0t−sα−1Vt−sDs+ω,us+ωdB˜QHs2+2E∫0tt−sα−1Vt−sDs+ω,us+ω−Ds,usdB˜QHs2=:2It+2Jt.
Moreover,H3−H4, Lemmas 2 and 4 yield that(53)It≤2Hw2H−1∫−ω0t−s2α−2Vt−s2Ds+ω,us+ωL202ds≤2Hw2H−1M2α2δ2∫−ω0t−s−2Ds+ω,us+ωL202ds=2Hw2H−1M2α2δ2t2∫0ωDs,usL202ds≤4Hw2H−1M2α2δ2t2∫0ωDs,0L202+L1s21−H−αEus2ds≤4Hw2H−1M2α2δ2t2⋅ωsupt≥0Dt,0L202+L1ω3−2H−2α3−2H−2αu∞2.
Then,(54)limt⟶∞It=0,and H2, H3, and Lemma 4 imply that(55)Jt≤4HTε2H−1∫0Tεt−s2α−2Vt−s2Ds+ω,us+ω−Ds,usL202ds+4Ht−Tε2H−1∫Tεtt−s2α−2Vt−s2Ds+ω,us+ω−Ds,usL202ds=4HTε2H−1J1t+4HJ2t,where limt⟶∞J1t=0 can be gotten similarly to (53). Moreover, without loss of generality, we may suppose for ∀ε>0, Eut+ω−ut2<ε, for t>Tε, owing to u∈SAPω0,∞,L2Ω;ℍ:(56)J2t≤2t−Tε2H−1∫Tεtt−s2α−2Vt−s2Ds+ω,us+ω−Ds,us+ωL202+Ds,us+ω−Ds,usL202ds≤2M2t−Tε2H−1∫Tεtt−s2α−2Eα,α−δt−sα2Ds+ω,us+ω−Ds,us+ωL202+Ds,us+ω−Ds,usL202ds=2M2ε1+L1Γα2t−Tε2H−1∫0t−Tετ2α−2t−τ21−α−Hdτ=2M2ε1+L1Γα2∫01μ2α−21−μ21−α−Hdμ=2M21+L1B2α−1,3−2α+HΓα2ε,where B is the beta function. Thus, limt⟶∞J2t=0. Therefore, limt⟶∞EΓ2ut+ω−Γ2ut2=0.
Step 2: For∀u∈SAPω0,∞,L2Ω;ℍ, we prove that Γ2ut is stochastically bounded and continuous.
For a given numbert0≥0 and t>t0, we get(57)EΓ2ut−Γ2ut02≤2E∫t0−t0t0−sα−1Vt0−sDs+t−t0,us+t−t0dB˜QHs2+2E∫0t0t0−sα−1Vt0−sDs+t−t0,us+t−t0−Ds,usdB˜QHs:=2N1t+2N2t.
It follows fromH3 and Lemmas 2 and 4 that(58)N1t≤2HM2Γα2t−t02H−1∫t0−t0t0−s2α−2Ds+t−t0,us+t−t0L202ds≤2HM2Γα2t−t02H−1∫0t−t0t−s2α−2Ds,usL202ds≤4HM2Γα2t−t02H−1∫0t−t0t−s2α−2dssupt≥0Dt,0L202+4HM2L1Γα2t−t02H−1∫0t−t0t−s2α−2s21−H−αdsu∞2≤4HM2supt≥0Dt,0L202Γα22α−1t−t02H−1t2α−1−t02α−1+4HM2L1u∞2t02α−2Γα23−2H−2αt−t02−2α.
This meanslimt⟶t0N1t=0.
Next, we find that(59)N2t≤2HM2t02H−1∫0t0t0−s2α−2Eα,α−δt0−sα2Ds+t−t0,us+t−t0−Ds,usL202ds≤4HM2t02H−1∫0t0t0−s2α−2Eα,α−δt0−sα2Ds+t−t0,us+t−t0−Ds,us+t−t0L202+L1s21−α−Hus+t−t0−us2ds.
Note thatN2t=0 for t0=0, and for t0>0, we have(60)∫0t0t0−s2α−2Eα,α−δt0−sα2ds=∫0t0τ2α−2Eα,α−δτα2dτ≤∫0t0/2τ2α−2Eα,α−δτα2dτ+∫t0/2t0τ2α−2Eα,α−δτα2dτ≤1Γα2∫0t0/2s2α−2ds+t022α−2∫t0/2t01δτα2dτ≤1Γα2⋅12α−1t022α−1+1δ2t022α−2∫t0/2t0τ−2αdτ=1Γα2⋅12α−1t022α−1+1δ22α−1t02−1−t022α−2⋅t01−2α=t02α−1Γα22α−122α−1+2−22−2αδ22α−1t0<∞,t02H−1∫0t0t0−s2α−2s21−α−Hds=B2α−1,3−2α−2H.
Then, by a similar argument to that used in (46), we deduce limt⟶t0N2t=0.
Moreover, we applyH3−H4 and Lemma 2 and use Lemma 4 to conclude that(61)EΓ2ut≤2Ht2H−1∫0tt−s2α−2Vt−s2Ds,usL202ds≤4M2Ht2H−1Γα2∫0tt−s2α−2Ds,0L202+L1s21−H−αEus2ds≤4L1M2Hu∞2Γα2∫011−μ2α−2μ21−H−αds=4L1M2Hu∞2B2α−1,3−2H−2αΓα2.
Applying this and the above arguments, we conclude thatΓ2ut is stochastically bounded and continuous.
By combining Steps 1 and 2, we obtainΓ2u∈SAPω0,∞,L2Ω;ℍ.
Now we state our main results.Theorem 1.
IfH1–H4 are satisfied, then equation (14) has a unique mild solution u∈SAPω0,∞,L2Ω;ℍ provided with(62)2M22L1HB2α−1,3−2α+HΓα2+Lδ2<1.Proof.
We defineΓ by(63)Γut=Utu0+∫0tt−sα−1Vt−sfs,usds+∫0tt−sα−1Vt−sDs,usdBQHs.
It follows from Lemmas5 and 6 that Γu∈SAPω0,∞,L2Ω;ℍ. For all t≥0,u,v∈SAPω0,∞,L2Ω;ℍ, we get(64)EΓut−Γvt2≤E∫0tt−sα−1Vt−sfs,us−fs,vsds+∫0tt−sα−1Vt−sDs,us−Ds,vsdBQHs2≤2E∫0tt−sα−1Vt−sfs,us−fs,vsds2+2E∫0tt−sα−1Vt−sDs,us−Ds,vsdBQHs2=:It+Jt.
On the one hand, we can see that(65)It≤2∫0tt−sα−1Vt−sds∫0tt−sα−1Vt−sEfs,us−fs,vs2ds≤2LM2supt≥0Eut−vt2∫0tt−sα−1Eα,α−δt−sαds2=2M2Lδ2u−v∞2Eα−δt−sα0t2≤2M2Lδ2u−v∞2.
On the other hand, we get(66)Jt≤4M2Ht2H−1∫0tt−s2α−2Eα,α−δt−sα2⋅Ds,us−Ds,vsL202ds≤4M2L1Ht2H−1Γα2∫0tt−s2α−2s21−α−Hdsu−v∞2=4M2L1HΓα2∫011−τ2α−2τ21−α−Hdτu−v∞2=4M2L1HB2α−1,3−2α+HΓα2u−v∞2.
Thus,(67)Γu−Γv∞2≤2M22L1HB2α−1,3−2α+HΓα2+Lδ2u−v∞.
If (62) holds, Γ is a contraction mapping. Using the Banach fixed-point theorem, we have that Γ has a unique fixed point in SAPω0,∞,L2Ω;ℍ. This completes the proof.


## 4. Asymptotic Stability of Square-Mean S-Asymptoticallyω-Periodic Solutions

Theorem 2.
Assume thatH1 and H2′D:0,∞⟶L20ℍ is stochastically and continuous. Furthermore, for ∀ε>0, there exists Tε>0 such that t2α+H−1Dt+ω−DtL202<ε and DtL202≤t21−α−H for t>T hold. If δ2>2M2L, there exists a unique asymptotically stable S-asymptotically ω-periodic solution u∗ in square-mean sense to equation (14) with Dt,⋅=Dt.Proof.
From the proof process of Theorem1, we get the existence and uniqueness of the S-asymptotically ω-periodic solution u∗t similarly. In addition, for ∀u1∈L2Ω;ℍ, equation (14) has a unique mild solution ut with the new initial value u0=u1. And then from (17) and Lemmas 2 and 4, we have(68)Eut−u∗t2≤2EUtu1−Utu02+2E∫0tt−sα−1Vt−sfs,us−fs,u∗sds2≤2M2Eα−δtαEu1−u02+2∫0tt−sα−1Vt−sds⋅∫0tt−sα−1Vt−sEfs,us−fs,u∗s2ds≤2M2Eα−δtαEu1−u02+2M2L∫0tt−sα−1Eα,α−δt−sαds⋅∫0tt−sα−1Eα,α−δt−sαEus−u∗s2ds=Eα−δtα2M2Eu1−u02+2M2Lδ∫0tt−sα−1Eα,α−δt−sα⋅Eus−u∗s2ds=Eα2M2Lδ−δtα2M2Eu1−u02.
Ifδ2>2M2L, by Lemma 1, we obtain that the square-mean S-asymptotically ω-periodic solution u∗ to equation (14) with Dt,⋅=Dt is asymptotically stable in square-mean sense.
Now the proof could be finished.


## 5. Numerical Simulation

Example 1.
We consider the following fractional stochastic equation with FBM:(69)D00.85ut+4ut=sinut+sint+sin1tdBQ0.6tdt,t∈0,+∞,u0=0.22,where D00.85 is the Caputo fractional derivative and BQ0.6t is one-dimensional FBM with H=0.6 (Figure 1).
We notice that−4 generates an exponentially stable semigroup e−4tt≥0, δ=4, and M=1. Set ft,ut=sinut+sint and Dt=sin1/t. Then, by the Lagrange differential mean value theorems, we obtain that H1 and H2′ are satisfied with L=1. It is easy to see that δ2>2M2L. From Theorem 2, we conclude that equation (69) has a unique square-mean S-asymptotically ω-periodic solution (Figure 2), which is asymptotically stable in square-mean sense.
From the above example, we find that although there is no periodic solution for the fractional-order differential equation with finite lower limit [2], the S-asymptotically periodic solution can be found and is stable even for fractional stochastic differential equation.Figure 1
Fractional Brownian motion (FBM) withH=0.6.Figure 2
Time-series graph of asymptotically stableS-asymptotically ω-periodic solution of (69).



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