Asymptotic Stability of Fractional Stochastic Neutral Differential Equations with Infinite Delays

Abstract

We study the existence and asymptotic stability in pth moment of a mild solution to a class of nonlinear fractional neutral stochastic differential equations with infinite delays in Hilbert spaces. A set of novel sufficient conditions are derived with the help of semigroup theory and fixed point technique for achieving the required result. The uniqueness of the solution of the considered problem is also studied under suitable conditions. Finally, an example is given to illustrate the obtained theory.

1. Introduction

The stochastic differential equations have been widely applied in science, engineering, biology, mathematical finance and in almost all applied sciences. In the present literature, there are many papers on the existence and uniqueness of solutions to stochastic differential equations (see [1–4] and references therein). More recently, Chang et al. [5] investigated the existence of square-mean almost automorphic mild solutions to nonautonomous stochastic differential equations in Hilbert spaces by using semigroup theory and fixed point approach. Fu and Liu [2] discussed the existence and uniqueness of square-mean almost automorphic solutions to some linear and nonlinear stochastic differential equations and in which they studied the asymptotic stability of the unique square-mean almost automorphic solution in the square-mean sense. On the other hand, recently fractional differential equations have found numerous applications in various fields of science and engineering [6]. The existence and uniqueness results for abstract stochastic delay differential equation driven by fractional Brownian motions have been studied in [7]. In particular the stability investigation of stochastic differential equations has been investigated by several authors [8–15].

Let 𝐾 and 𝐻 be two real separable Hilbert spaces with inner products ⟨⋅,⋅⟩_𝐾 and ⟨⋅,⋅⟩_𝐻, respectively. We denote their norms by |⋅|_𝐾 and |⋅|_𝐻. To avoid confusion we just use ⟨⋅,⋅⟩ for the inner product and |⋅| for the norm. Let {𝑒_𝑖}^∞_𝑖=1 be an orthonormal basis of 𝐾. Throughout the paper, we assume that (Ω,ℱ,𝑃;𝔽)(𝔽={ℱ_𝑡}_𝑡≥0) is a complete filtered probability space satisfying that ℱ₀ contains all 𝑃-null sets of ℱ. Suppose {𝑊(𝑡)∶𝑡≥0} is cylindrical 𝐾-valued Brownian motion with a trace class operator 𝑄, denote trac𝑄=∑^∞_𝑖=1𝜆_𝑖=𝜆<∞, which satisfies that 𝑄𝑒_𝑖=𝜆_𝑖𝑒_𝑖. So, actually, 𝑊(𝑡)=∑^∞_𝑖=1𝜆_𝑖𝑊_𝑖(𝑡)𝑒_𝑖, where {𝑊_𝑖(𝑡)}^∞_𝑖=1 are mutually independent one-dimensional standard Brownian motions. Define ℒ(𝐾,𝐻) as the set of all bounded linear operators 𝐴∶𝐾→𝐻 with the following norm: |𝐴|=^∞_𝑖=1𝐴𝑒_𝑖^21/2<∞. (1) It is obvious that ℒ(𝐾,𝐻) is a Hilbert space with an inner product induced by the above norm. Let 𝐴∈ℒ(𝐾,𝐻) be called a Hilbert-Schmidt operator. We further assume that the filtration is generated by the cylindrical Brownian motion 𝑊(⋅) and augmented, that is, ℱ_𝑡=𝜎{𝑊(𝑠);0≤𝑠≤𝑡}∨𝒩, (2) where 𝒩 is the 𝑃-null sets.

The qualitative properties of stochastic fractional differential equations have been considered only in few publications. El-Borai et al. [16] studied the existence uniqueness, and continuity of the solution of a fractional stochastic integral equation. Ahmed [17] derived a set of sufficient conditions for controllability of fractional stochastic delay equations by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. Moreover, theory of neutral differential equations is of both theoretical and practical interests. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral differential equations. However, to the author's best knowledge no work has been reported in the present literature regarding the existence, uniqueness, and asymptotic stability of mild solutions for neutral stochastic fractional differential equations with infinite delay in Hilbert spaces. Motivated by this consideration, in this paper we consider the nonlinear fractional neutral stochastic differential equations with infinite delays in the following form: ^𝑐𝐷^𝛼_𝑡𝑋(𝑡)(((𝑡(𝑡)(((𝑡(((𝑡(𝑡)+𝑔𝑡,𝑋𝑡−𝜏)))=𝐴𝑋+𝑓𝑡,𝑋𝑡−𝜏)))+𝜎𝑡,𝑋𝑡−𝜈)))𝑑𝑊𝑋𝑑𝑡,𝑡≥0,₀(⋅)=𝜑∈ℬ_ℱ([𝑚(0),0],𝐻), (3)(4) where 𝐴 is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operator 𝑆(𝑡),𝑡≥0 in the Hilbert space 𝐻,𝑓∶𝑅₊×𝐻→𝐻,𝜎∶𝑅₊×𝐻→ℒ(𝐾,𝐻) are two Borel measurable mappings, and 𝑔∶𝑅₊×𝐻→𝐻 is continuous mapping. The fractional derivative ^𝑐𝐷^𝛼_𝑡,𝛼∈(0,1) is understood in the Caputo sense. In addition, let 𝜏(𝑡),𝜈(𝑡)∈𝐶(𝑅₊,𝑅₊) satisfy 𝑡−𝜏(𝑡)→∞, 𝑡−𝜈(𝑡)→∞ as 𝑡→∞. Let 𝑚(0)=max{inf_𝑠≥0(𝑠−𝜏(𝑠)),inf_𝑠≥0(𝑠−𝜈(𝑠))}. Here ℬ_ℱ0([𝑚(0),0],𝐻) denote the family of all almost surely bounded, ℱ₀-measurable, continuous random variables 𝜑(𝑡)∶[𝑚(0),0]→𝐻 with norm |𝜑|_ℬ=sup_{𝑚(0)≤𝑡≤0}𝐸|𝜑(𝑡)|_𝐻. Throughout this paper, we assume that 𝑝≥2 is an integer.

2. Preliminaries and Basic Properties

Let 𝐴 be the infinitesimal generator of an analytic semigroup 𝑆(𝑡) in 𝐻. Then, (𝐴−𝜂𝐼) is invertible and generates a bounded analytic semigroup for 𝜂>0 large enough. Therefore, we can assume that the semigroup 𝑆(𝑡) is bounded and the generator 𝐴 is invertible. It follows that (−𝐴)^𝜂,0<𝜂≤1 can be defined as a closed linear invertible operator with its domain 𝐷(−𝐴)^𝜂 being dense in 𝐻. We denote by 𝐻_𝜂 the Banach space 𝐷(−𝐴)^𝜂 endowed with the norm |𝑥|_𝜂=|(−𝐴)^𝜂𝑥|, which is equivalent to the graph norm of (−𝐴)^𝜂. For more details about semigroup theory, one can refer [18].

Lemma 1 (see [18]). Suppose that the preceding conditions are satisfied. (a)Let 0<𝜂≤1, then 𝐻_𝜂 is a Banach space.(b)If 0<𝜈≤𝜂, then the embedding 𝐻_𝜈⊂𝐻_𝜂 is compact whenever the resolvent operator of 𝐴 is compact. (c)For every 𝜂∈(0,1], there exists a positive constant 𝐶_𝜂 such that ‖𝐴^𝜂𝑆(𝑡)‖≤𝐶_𝜂/𝑡^𝜂,𝑡>0.

Definition 2 (see [19]). The fractional integral of order 𝑞 with the lower limit 0 for a function 𝑓 is defined as 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀𝑓(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠,𝑡>0,𝑞>0 (5) provided the righthand side is pointwise defined on [0,∞), where Γ(⋅) is the gamma function.

Definition 3 (see [19]). The Caputo derivative of order 𝛼 for a function 𝑓∶[0,∞)→𝑅 can be written as ^𝑐𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑓^(𝑛)(𝑠)(𝑡−𝑠)^{𝛼+1−𝑛}𝑑𝑠=𝐼^𝑛−𝛼𝑓^(𝑛)(𝑡),𝑡>0,0≤𝑛−1<𝛼<𝑛. (6) If 𝑓 is an abstract function with values in 𝐻, then integrals which appear in the above definitions are taken in Bochner's sense.
According to Definitions 2 and 3, it is suitable to rewrite the stochastic fractional equation (3) in the equivalent integral equation 𝑋(𝑡)=𝜑(0)(((𝑡+1+𝑔0,𝜑−𝑔𝑡,𝑋𝑡−𝜏)))Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1(𝑠)(((𝑠(((𝑠(𝑠)𝐴𝑋+𝑓𝑠,𝑋𝑠−𝜏)))+𝜎𝑠,𝑋𝑠−𝜈)))𝑑𝑊𝑑𝑠𝑑𝑠. (7)
In view of [18, Lemma 3.1] and by using Laplace transform, we present the following definition of mild solution of (3).

Definition 4. A stochastic process {𝑋(𝑡)∶𝑡∈[0,𝑇]},0≤𝑇<∞ is called a mild solution of (3), if (i)𝑋(𝑡) is ℱ_𝑡-adapted and is measurable, 𝑡≥0; (ii)𝑋(𝑡)∈𝐻 has càdlàg paths on 𝑡∈[0,𝑇] almost surely and for each 𝑡∈[0,𝑇], the function (𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼(𝑡−𝑠)𝑔(𝑠,𝑋(𝑠−𝜏(𝑠))) is integrable such that the following integral equation is satisfied: 𝑋(𝑡)=𝑆_𝛼(𝑡𝜑(0)(((𝑡−)+𝑔0,𝜑−𝑔𝑡,𝑋𝑡−𝜏)))_𝑡0(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼()𝑔(((𝑠+𝑡−𝑠𝑠,𝑋𝑠−𝜏)))𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝜎(𝑠,𝑋(𝑠−𝜈(𝑠)))𝑑𝑊(𝑠), (8)(iii)𝑋₀(⋅)=𝜑∈ℬ_ℱ([𝑚(0),0],𝐻), where 𝑆_𝛼(𝑡)𝑋=^∞₀𝜂_𝛼(𝜃)𝑆𝑡^𝛼𝜃𝑇𝑋𝑑𝜃,_𝛼(𝑡)𝑋=𝛼^∞₀𝜃𝜂_𝛼(𝜃)𝑆𝑡^𝛼𝜃𝑋𝑑𝜃 (9) with 𝜂_𝛼 a probability density function defined on (0,∞).

The following properties of 𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) [18] are useful.

Lemma 5. Under previous assumptions on 𝑆(𝑡),𝑡≥0 and 𝐴, (i)𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) are strongly continuous; (ii)for any 𝑋∈𝐻,𝛽∈(0,1) and 𝜃∈(0,1] one has 𝐴𝑇_𝛼(𝑡)𝑋=𝐴^1−𝛽𝑇_𝛼(𝑡)𝐴^𝛽𝐴𝑋,_𝜃𝑇_𝛼(𝑡)≤𝛼𝐶_𝜃𝑡^𝛼𝜃Γ(2−𝜃)Γ(([].1+𝛼1−𝜃)),𝑡∈0,𝑇 (10)

Definition 6. Let 𝑝≥2 be an integer. Equation (8) is said to be stable in 𝑝th moment if for arbitrarily given 𝜖>0 there exists a 𝛿>0 such that 𝐸sup_𝑡≥0|𝑋(𝑡)|^𝑝<𝜖,when𝜑_ℬ<𝛿. (11)

Definition 7. Let 𝑝≥2 be an integer. Equation (8) is said to be asymptotically stable in 𝑝th moment if it is stable in 𝑝th moment and, for any 𝜑∈ℬ_ℱ([𝑚(0,0)],𝐻), it holds lim_𝑇→∞𝐸sup_𝑡≥𝑇|𝑋(𝑡)|^𝑝=0. (12)

3. Main Result

In this section, we prove the existence, uniqueness, and stability of the solution to fractional stochastic equation (3) by using the Banach fixed point approach.

In order to obtain the existence and stability of the solution to (3), we impose the following assumptions on (3).(H1) There exist constants 𝑀≥1 and 𝑎>0 such that |𝑆(𝑡)|≤𝑀𝑒^−𝑎𝑡. (H2) There exists a positive constant 𝐿₁, for every 𝑡≥0 and 𝑥,𝑦∈𝐻, such that 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝐿₁,𝜎()𝑥−𝑦𝑡,𝑥−𝜎𝑡,𝑦≤𝐿₁𝑥−𝑦. (13)(H3) There exist 0<𝛽<1 such that 𝑔 is 𝐻_𝛽-valued, (−𝐴)^𝛽𝑔 is continuous and there exists a positive constant 𝑀_𝑔 such that (−𝐴)^𝛽𝑔(𝑡,𝑥)−(−𝐴)^𝛽𝑔𝑡,𝑦≤𝑀_𝑔𝑥−𝑦 (14) for every 𝑡≥0 and 𝑥,𝑦∈𝐻. (H4)Θ=[4^(𝑝−1)|(−𝐴)^(−𝛽)|^𝑝𝑀^𝑝_𝑔+4^𝑝−1𝑀^𝑝_𝑔𝐾(𝛼,𝛽)(𝑇^𝛼𝛽/𝛼𝛽)^𝑝+4^𝑝−1𝛼^𝑝𝑀^𝑝𝐿^𝑝₁𝐿^𝑝+ 4^𝑝−1𝐶_𝑝𝛼^𝑝𝑀^𝑝𝐿^𝑝₁𝐿^′(𝑝/2)]<1, where 𝐶_𝑃=(𝑝(𝑝−1))/2^𝑝/2, 𝑀_𝑡=∫^𝑡₀𝜃𝜂_𝛼(𝜃)𝑒^{−𝑎𝑡𝛼𝜃}𝑑𝜃, 𝐿=∫^𝑇₀𝑀_{(𝑡−𝑠)}(𝑡−𝑠)^𝛼−1𝑑𝑠 and 𝐿^′=∫^𝑇₀[𝑀_𝑡−𝑠(𝑡−𝑠)^𝛼−1]²𝑑𝑠. In addition, in order to derive the stability of the solution, we further assume that(H5)𝑔(𝑡,0)=0,𝑓(𝑡,0)=0,𝜎(𝑡,0)=0. (15) It is obvious that (3) has a trivial solution when 𝜑=0 under the assumption (H5).

Lemma 8. Let 𝑝≥2,𝑡>0 and let Φ be an ℒ(𝐾,𝐻)-valued, predictable process such that 𝐸∫^𝑡₀|Φ(𝑠)|^𝑝_ℒ𝑑𝑠<∞. Then, sup_{0≤𝑠≤𝑡}𝐸^𝑠₀Φ(𝑢)𝑑𝑊(𝑢)^𝑝≤𝑝𝑝−12^𝑝/2𝑡₀𝐸Φ(𝑠)^𝑝_ℒ^2/𝑝𝑝/2. (16)

Theorem 9. Let 𝑝≥2 be an integer. Assume that the conditions (H1)–(H4) hold, then the nonlinear fractional neutral stochastic differential equation (3) is asymptotically stable in the 𝑝th moment.

Proof. Denote by 𝔹 the space of all ℱ₀-adapted process 𝜙(𝑡,𝑤)∶[𝑚(0),0]×Ω→𝑅, which is almost surely continuous in 𝑡 for fixed 𝑤∈Ω and satisfies 𝜙(𝑡,𝑤)=𝜑(𝑡) for 𝑡∈[𝑚(0),0] and 𝐸|𝜙(𝑡,𝑤)|^𝑝→0 as 𝑡→0. It is then routine to check that 𝔹 is a Banach space when it is equipped with a norm defined by |𝜑|_𝔹=sup_𝑡≥0𝐸|𝜑(𝑡)|^𝑝_𝐻. Define the nonlinear operator Ψ∶𝔹→𝔹 such that (Ψ𝑋)(𝑡)=𝜑(𝑡),𝑡∈[𝑚(0),0] and, for 𝑡≥0, (Ψ𝑋)(𝑡)=𝑆_𝛼(𝑡𝜑(0)(((𝑡−)+𝑔0,𝜑−𝑔𝑡,𝑋𝑡−𝜏)))_𝑡0(𝑡−𝑠)^𝛼−1𝐴𝑇_𝛼()𝑔(((𝑠+𝑡−𝑠𝑠,𝑋𝑠−𝜏)))𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠+^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝜎(𝑠,𝑋(𝑠−𝜈(𝑠)))𝑑𝑊(𝑠). (17) As mentioned in Luo [20], to prove the asymptotic stability it is enough to show that the operator Ψ has a fixed point in 𝐻. To prove this result, we use the contraction mapping principle. To apply the contraction mapping principle, first we verify the mean square continuity of Ψ on [0,∞). Let 𝑋∈𝔹,𝑡₁≥0 and let |ℎ| be sufficiently small, and observe that 𝐸(Ψ𝑋)𝑡₁+ℎ−(Ψ𝑋)𝑡₁^𝑝≤5^5𝑝−1_𝑖=1𝐸𝐹_𝑖𝑡₁+ℎ−𝐹_𝑖𝑡₁^𝑝. (18) Note that 𝐸𝐹₁(𝑡₁+ℎ)−𝐹₁(𝑡₁)^𝑝=𝐸𝑆_𝛼𝑡₁+ℎ−𝑆_𝛼𝑡₁𝜑(0)−𝑔0,𝜑^𝑝. (19) The strong continuity of 𝑆_𝛼(𝑡) [18] implies that the right hand of (19) goes to 0 as |ℎ|→0. In view of Lemma 5 and the Holder's inequality, the third term of (18) becomes 𝐸𝐹₃𝑡₁+ℎ−𝐹₃(𝑡₁)^𝑝=𝐸^𝑡1₀^+ℎ𝑡₁+ℎ−𝑠^𝛼−1(−𝐴)𝑇_𝛼𝑡₁𝑔×(((𝑠−+ℎ−𝑠𝑠,𝑋𝑠−𝜏)))𝑑𝑠_𝑡10𝑡₁−𝑠^𝛼−1(−𝐴)𝑇_𝛼𝑡₁𝑔×(((𝑠−𝑠𝑠,𝑋𝑠−𝜏)))𝑑𝑠^𝑝≤3^𝑝−1𝐸^𝑡1_𝑡^+ℎ₁𝑡₁+ℎ−𝑠^𝛼−1(−𝐴)𝑇_𝛼𝑡₁𝑔×(((𝑠+ℎ−𝑠𝑠,𝑋𝑠−𝜏)))𝑑𝑠^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1(−𝐴)𝑇_𝛼×𝑡₁+ℎ−𝑠𝑔(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁−𝑠^𝛼−1()×𝑇−𝐴_𝛼𝑡₁+ℎ−𝑠−𝑇_𝛼𝑡₁𝑔×(((𝑠−𝑠𝑠,𝑋𝑠−𝜏)))𝑑𝑠^𝑝≤3^𝑝−1𝐸^𝑡1_𝑡^+ℎ₁𝑡₁+ℎ−𝑠^𝛼−1(−𝐴)^1−𝛽𝑇_𝛼𝑡₁×)+ℎ−𝑠(−𝐴^𝛽𝑔(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1×(−𝐴)^1−𝛽𝑇_𝛼𝑡₁×)+ℎ−𝑠(−𝐴^𝛽𝑔(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁−𝑠^𝛼−1(−𝐴)^1−𝛽×𝑇_𝛼𝑡₁+ℎ−𝑠−𝑇_𝛼𝑡₁×)−𝑠(−𝐴^𝛽𝑔(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝≤3^𝑝−1𝐾𝛼,𝛽𝑀^𝑝_𝑔^𝑡1_𝑡^+ℎ₁𝑡₁+ℎ−𝑠^𝛼𝛽−1𝑑𝑠^𝑝−1×^𝑡1_𝑡^+ℎ₁𝑡₁+ℎ−𝑠^𝛼𝛽−1𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠+3^𝑝−1𝐾𝛼,𝛽𝑀^𝑝_𝑔×^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝑡₁+ℎ−𝑠^{𝛼(𝛽−1)}𝑑𝑠^𝑝−1×^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝑡₁+ℎ−𝑠^{𝛼(𝛽−1)}𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠+3^𝑝−1𝜖^𝑝𝑀^𝑝_𝑔(−𝐴)^{1−𝛽𝑝𝑡1}₀𝑡₁−𝑠^𝛼−1𝑑𝑠^𝑝−1×^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠≤3^𝑝−1𝐾𝛼,𝛽𝑀^𝑝_𝑔ℎ^𝛼𝛽𝛼𝛽^𝑝−1×^𝑡1_𝑡^+ℎ₁𝑡₁+ℎ−𝑠^𝛼𝛽−1𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠+3^𝑝−1𝐾𝛼,𝛽𝑀^𝑝_𝑔×^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝑡₁+ℎ−𝑠^{𝛼(𝛽−1)}𝑑𝑠^𝑝−1×^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝑡₁+ℎ−𝑠^{𝛼(𝛽−1)}𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠+3^𝑝−1𝜖^𝑝𝑀^𝑝_𝑔(−𝐴)^1−𝛽𝑝𝑡^𝛼₁𝛼^𝑝−1×^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠, (20) where 𝐾(𝛼,𝛽)=𝛼^𝑝Γ^𝑝_(1+𝛽)𝐶^𝑝_1−𝛽/Γ^𝑝_(1+𝛼𝛽). Since 𝜖 is sufficiently small, the right hand side of the above equation tends to zero as |ℎ|→0.
Next we consider 𝐸𝐹₄𝑡₁+ℎ−𝐹₄𝑡₁^𝑝=𝐸^𝑡1₀^+ℎ𝑡₁+ℎ−𝑠^𝛼−1𝑇_𝛼𝑡₁(((𝑠−+ℎ−𝑠×𝑓𝑠,𝑋𝑠−𝜏)))𝑑𝑠_𝑡10𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₁−𝑠𝑓(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝≤3^𝑝−1𝐸^𝑡1_𝑡^+ℎ₁𝑡₁+ℎ−𝑠^𝛼−1𝑇_𝛼𝑡₁(((𝑠+ℎ−𝑠×𝑓𝑠,𝑋𝑠−𝜏)))𝑑𝑠^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1×𝑇_𝛼𝑡₁+ℎ−𝑠𝑓(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₁+ℎ−𝑠−𝑇_𝛼𝑡₁𝑓×(((𝑠−𝑠𝑠,𝑋𝑠−𝜏)))𝑑𝑠^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1×𝑇_𝛼𝑡₁+ℎ−𝑠𝑓(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₁+ℎ−𝑠−𝑇_𝛼𝑡₁×𝑓(((𝑠−𝑠𝑠,𝑋𝑠−𝜏)))𝑑𝑠^𝑝. (21) By the Holder’s inequality, we obtain ≤3^𝑝−1𝐿^𝑝₁𝑀^𝑝𝛼^𝑝𝑡1_𝑡^+ℎ₁𝑀_{𝑡1+ℎ−𝑠}𝑡₁+ℎ−𝑠^𝛼−1𝑑𝑠^𝑝−1×^𝑡1_𝑡^+ℎ₁𝑀_{𝑡1+ℎ−𝑠}𝑡₁+ℎ−𝑠^𝛼−1𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠+3^𝑝−1𝐿^𝑝₁𝑀^𝑝𝛼^𝑝×^𝑡1₀𝑀_{𝑡1+ℎ−𝑠}𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝑑𝑠^𝑝−1×^𝑡1₀𝑀_{𝑡1+ℎ−𝑠}𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝐸×|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠+3^𝑝−1𝜖^𝑝𝐿^𝑝₁𝑡^𝛼₁𝛼^𝑝−1×^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠. (22) Therefore, the right hand side of the above equation tends to zero as |ℎ|→0 and 𝜖 sufficiently small. Further, we have 𝐸𝐹₅𝑡₁+ℎ−𝐹₅𝑡₁^𝑝=𝐸^𝑡1₀^+ℎ𝑡₁+ℎ−𝑠^𝛼−1𝑇_𝛼𝑡₁(((𝑠(𝑠)−+ℎ−𝑠×𝜎𝑠,𝑋𝑠−𝜏)))𝑑𝑊_𝑡10𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₁(((𝑠(𝑠−𝑠×𝜎𝑠,𝑋𝑠−𝜏)))𝑑𝑊)^𝑝≤3^𝑝−1𝐸^𝑡1_𝑡^+ℎ₁𝑡₁+ℎ−𝑠^𝛼−1𝑇_𝛼𝑡₁(((𝑠(𝑠+ℎ−𝑠×𝜎𝑠,𝑋𝑠−𝜏)))𝑑𝑊)^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1×𝑇_𝛼𝑡₁+ℎ−𝑠𝜎(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑊(𝑠)^𝑝+3^𝑝−1𝐸^𝑡1₀𝑡₁−𝑠^𝛼−1𝑇_𝛼𝑡₁+ℎ−𝑠−𝑇_𝛼𝑡₁(((𝑠(𝑠−𝑠×𝜎𝑠,𝑋𝑠−𝜏)))𝑑𝑊)^𝑝≤3^𝑝−1𝐶_𝑝𝐸^𝑡1_𝑡^+ℎ₁𝑡₁+ℎ−𝑠^2(𝛼−1)𝑇_𝛼𝑡₁+ℎ−𝑠²×|𝜎(𝑠,𝑋(𝑠−𝜏(𝑠)))|²𝑑𝑠^𝑝/2+3^𝑝−1𝐶_𝑝𝐸^𝑡1₀𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−12×𝑇_𝛼𝑡₁+ℎ−𝑠²×|𝜎(𝑠,𝑋(𝑠−𝜏(𝑠)))|²𝑑𝑠^𝑝/2+3^𝑝−1𝐶_𝑝𝐸^𝑡1₀𝑡₁−𝑠^2(𝛼−1)×𝑇_𝛼𝑡₁+ℎ−𝑠−𝑇_𝛼𝑡₁−𝑠²×|𝜎(𝑠,𝑋(𝑠−𝜏(𝑠)))|²𝑑𝑠^𝑝/2≤3^𝑝−1𝐶_𝑝𝐿^𝑝₁𝑀^𝑝𝛼^𝑝×^𝑡1_𝑡^+ℎ₁𝑀_{𝑡1+ℎ−𝑠}𝑡₁+ℎ−𝑠^{𝛼−1𝑝/(𝑝−2)}𝑑𝑠^(𝑝−2)/2×^𝑡1_𝑡^+ℎ₁𝑀_{𝑡1+ℎ−𝑠}𝑡₁+ℎ−𝑠^{𝛼−1𝑝/2}×𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠+3^𝑝−1𝐶_𝑝𝐿^𝑝₁𝑀^𝑝𝛼^𝑝×^𝑡1₀𝑀_{𝑡1+ℎ−𝑠}𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^{𝛼−1𝑝/(𝑝−2)}𝑑𝑠^(𝑝−2)/2×^𝑡1₀𝑀_{𝑡1+ℎ−𝑠}𝑡₁+ℎ−𝑠^𝛼−1−𝑡₁−𝑠^{𝛼−1𝑝/2}×𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠+3^𝑝−1𝜖^𝑝𝐶_𝑝𝐿^𝑝₁𝑡₁^{𝑝𝛼−2/𝑝−2}𝑝𝛼−2/𝑝−2^(𝑝−2)/2×^𝑡1₀𝑡₁−𝑠^{𝑝(𝛼−1)/2}𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠. (23) As above, the right hand side of the above inequality tends to zero. Similarly, we have 𝐹₂→0 as ℎ→0. Thus Ψ is continuous in 𝑝th moment on [0,∞).
Next we show that Ψ(𝔹)∈𝔹. Let 𝑋∈𝔹. From (18), we have𝐸|(Ψ𝑋)(𝑡)|^𝑝≤6^𝑝−1𝐸𝑆_𝛼(𝑡)𝜑(0)^𝑝+6^𝑝−1𝐸𝑆_𝛼(𝑡)𝑔0,𝜑^𝑝+6^𝑝−1𝐸𝑔(𝑡,𝑋(𝑡−𝜏(𝑡)))^𝑝+6^𝑝−1𝐸^𝑡₀(𝑡−𝑠)^𝛼−1(−𝐴)𝑇_𝛼()(((𝑠𝑡−𝑠×𝑔𝑠,𝑋𝑠−𝜏)))𝑑𝑠^𝑝+6^𝑝−1𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝+6^𝑝−1𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()(((𝑠(𝑠𝑡−𝑠×𝜎𝑠,𝑋𝑠−𝜈)))𝑑𝑊)^𝑝. (24) Now, we estimate the terms on the right hand side of (24) by using the assumptions (H1), (H3), and (H4). Now, we have 6^𝑝−1𝐸𝑆_𝛼(𝑡)𝜑(0)^𝑝≤6^𝑝−1𝑀^𝑝∞₀𝜂_𝛼(𝜃)𝑒^{−𝛼𝑡𝛼𝜃}𝑑𝜃^𝑝𝜑^𝑝_ℬ6⟶0as𝑡⟶∞,_𝑝−1𝐸𝑆_𝛼(𝑡)𝑔0,𝜑^𝑝≤6^𝑝−1𝑀^𝑝∞₀𝜂_𝛼(𝜃)𝑒^{−𝛼𝑡𝛼𝜃}𝑑𝜃^𝑝(−𝐴)^−𝛽𝑝𝑀^𝑝_𝑔𝜑^𝑝_ℬ6⟶0as𝑡⟶∞,_𝑝−1𝐸𝑔(𝑡,𝑋(𝑡−𝜏(𝑡)))^𝑝≤6^𝑝−1(−𝐴)^−𝛽𝑝𝑀^𝑝_𝑔𝐸|𝑋(𝑡−𝜏(𝑡))|^𝑝. (25) For 𝑋(𝑡)∈𝔹 and for any 𝜖>0 there exists a 𝑡₁>0 such that 𝐸|𝑋(𝑡−𝜏(𝑡))|^𝑝≤𝜖 for 𝑡≥𝑡₁.
Therefore, 6^𝑝−1𝐸𝑔(𝑡,𝑋(𝑡−𝜏(𝑡)))^𝑝⟶0as𝑡⟶∞. (26) For the fourth term of (24), we have 6^𝑝−1𝐸^𝑡₀(𝑡−𝑠)^𝛼−1(−𝐴)𝑇_𝛼(𝑡−𝑠)𝑔(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝≤6^𝑝−1𝐸^𝑡₀(𝑡−𝑠)^𝛼−1(−𝐴)^1−𝛽𝑇_𝛼(×)𝑡−𝑠)(−𝐴^𝛽𝑔(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝≤6^𝑝−1𝑀^𝑝_𝑔𝐾𝛼,𝛽^𝑡₀(𝑡−𝑠)^𝛼𝛽−1𝑑𝑠^𝑝−1×^𝑡₀(𝑡−𝑠)^𝛼𝛽−1𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠≤6^𝑝−1𝑀^𝑝_𝑔𝐾𝑇𝛼,𝛽^𝛼𝛽𝛼𝛽^𝑝−1×^𝑡₀(𝑡−𝑠)^𝛼𝛽−1𝐸|𝑋(𝑠−𝜏(𝑠))|^𝑝𝑑𝑠⟶0as𝑡⟶∞. (27) Also, we have 6^𝑝−1𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝑓(𝑠,𝑋(𝑠−𝜏(𝑠)))𝑑𝑠^𝑝≤6^𝑝−1𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(×𝑓(((𝑠𝑡−𝑠)𝑠,𝑋𝑠−𝜏)))𝑑𝑠^𝑝≤6^𝑝−1𝛼^𝑝𝑀^𝑝𝐿^𝑝₁^𝑡₀𝑀_𝑡−𝑠(𝑡−𝑠)^𝛼−1𝑑𝑠^𝑝−1×^𝑡₀𝑀_𝑡−𝑠(𝑡−𝑠)^𝛼−1𝐸|𝑋(𝑠−𝜏(𝑠)))|^𝑝6𝑑𝑠,_𝑝−1𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(𝑡−𝑠)𝜎(𝑠,𝑋(𝑠−𝜈(𝑠)))𝑑𝑊(𝑠)^𝑝≤6^𝑝−1𝐶_𝑝𝐸^𝑡₀(𝑡−𝑠)^2(𝛼−1)𝑇_𝛼(𝑡−𝑠)²×|𝜎(𝑠,𝑋(𝑠−𝜈(𝑠)))|²𝑑𝑠^𝑝/2≤6^𝑝−1𝐶_𝑝𝐿^𝑝₁𝛼^𝑝𝑀^𝑝𝑡₀𝑀_𝑡−𝑠(𝑡−𝑠)^{𝛼−1𝑝/(𝑝−2)}𝑑𝑠^{𝑝−2/𝑝}×^𝑡₀𝑀_𝑡−𝑠(𝑡−𝑠)^𝛼−1𝐸|𝑋(𝑠−𝜈(𝑠))|^𝑝𝑑𝑠. (28) By the same discussion as above, we have that (28) tends to zero as 𝑡→∞. Thus 𝐸|Ψ𝑋(𝑡)|^𝑝→0 as 𝑡→∞. We conclude that Ψ(𝔹)∈𝔹.
Finally, we prove that Ψ has a unique fixed point. Indeed, for any 𝑋,𝑌∈𝔹, we have sup_{[]𝑡∈0,𝑇}𝐸|(Ψ𝑋)(𝑡)−(Ψ𝑌)(𝑡)|^𝑝≤4^𝑝−1sup_{[]𝑡∈0,𝑇}𝐸𝑔(𝑡,𝑋(𝑡−𝜏(𝑡)))−𝑔(𝑡,𝑌(𝑡−𝜏(𝑡)))^𝑝+4^𝑝−1sup_{[]𝑡∈0,𝑇}𝐸^𝑡₀(𝑡−𝑠)^𝛼−1(−𝐴)𝑇_𝛼()×𝑔(((𝑠(((𝑠𝑡−𝑠𝑠,𝑋𝑠−𝜏)))−𝑔𝑠,𝑌𝑠−𝜏)))𝑑𝑠^𝑝+4^𝑝−1sup_{𝑡∈[0,𝑇]}𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×𝑓(((𝑠(((𝑠𝑡−𝑠𝑠,𝑋𝑠−𝜏)))−𝑓𝑠,𝑌𝑠−𝜏)))𝑑𝑠^𝑝+4^𝑝−1sup_{𝑡∈[0,𝑇]}𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼()×[𝜎(((𝑠(((𝑠(𝑠𝑡−𝑠𝑠,𝑋𝑠−𝜈)))−𝜎𝑠,𝑌𝑠−𝜈)))]𝑑𝑊)^𝑝≤4^𝑝−1(−𝐴)^−𝛽𝑝𝑀^𝑝_𝑔sup_{[]𝑡∈0,𝑇}𝐸|𝑋(𝑡)−𝑌(𝑡)|^𝑝+4^𝑝−1sup_{[]𝑡∈0,𝑇}𝐸^𝑡₀(𝑡−𝑠)^𝛼−1(−𝐴)^1−𝛽𝑇_𝛼(×)𝑡−𝑠)(−𝐴^𝛽𝑔(((𝑠(((𝑠𝑠,𝑋𝑠−𝜏)))−𝑔𝑠,𝑌𝑠−𝜏)))𝑑𝑠^𝑝+4^𝑝−1sup_{[]𝑡∈0,𝑇}𝐸^𝑡₀(𝑡−𝑠)^𝛼−1𝑇_𝛼(×𝑓(((𝑠(((𝑠𝑡−𝑠)𝑠,𝑋𝑠−𝜏)))−𝑓𝑠,𝑌𝑠−𝜏)))𝑑𝑠^𝑝+4^𝑝−1𝐶_𝑝sup_{𝑡∈[0,𝑇]}𝐸^𝑡₀(𝑡−𝑠)^2(𝛼−1)𝑇_𝛼(𝑡−𝑠)²×|𝜎(((𝑠(((𝑠𝑠,𝑋𝑠−𝜈)))−𝜎𝑠,𝑌𝑠−𝜈)))|²𝑑𝑠^𝑝/2≤4^𝑝−1(−𝐴)^−𝛽𝑝𝑀^𝑝_𝑔+4^𝑝−1𝑀^𝑝_𝑔𝐾𝑇𝛼,𝛽^𝛼𝛽𝛼𝛽^𝑝+4^𝑝−1𝛼^𝑝𝑀^𝑝𝐿^𝑝₁𝐿^𝑝+4^𝑝−1𝐶_𝑝𝛼^𝑝𝑀^𝑝𝐿^𝑝₁𝐿^′𝑝/2sup_{[]𝑡∈0,𝑇}𝐸|𝑋(𝑡)−𝑌(𝑡)|^𝑝. (29) Therefore, Ψ is a contradiction mapping and hence there exists a unique fixed point, which is a mild solution of (3) with 𝑋(𝑠)=𝜑(𝑠) on [𝑚(0),0] and 𝐸|𝑋(𝑡)|^𝑝→0 as 𝑡→∞.
To show the asymptotic stability of the mild solution of (3), as the first step, we have to prove the stability in 𝑝th moment. Let 𝜖>0 be given and choose 𝛿>0 such that 𝛿<𝜖 satisfies 6^𝑝−1[𝑀^𝑝+𝑀^𝑝|(−𝐴)^−𝛽|^𝑝𝑀^𝑝_𝑔]𝛿+6^𝑝−1[|(−𝐴)^−𝛽|^𝑝𝑀^𝑝_𝑔+𝑀^𝑝_𝑔𝐾(𝛼,𝛽)(𝑇^𝛼𝛽/𝛼𝛽)^𝑝+ 𝛼^𝑝𝑀^𝑝𝐿^𝑝₁𝐿^𝑝+𝐶_𝑝𝛼^𝑝𝑀^𝑝𝐿^𝑝₁𝐿^′𝑝/2]𝜖<𝜖.
If 𝑋(𝑡)=𝑋(𝑡,𝜑) is mild solution of (3), with |𝜑|^𝑝_ℬ<𝛿, then (Ψ𝑋)(𝑡)=𝑋(𝑡) satisfies 𝐸|𝑋(𝑡)|^𝑝<𝜖 for every 𝑡≥0. Notice that 𝐸|𝑋(𝑡)|^𝑝<𝜖 on 𝑡∈[𝑚(0),0]. If there exists 𝑡 such that 𝐸|𝑋(𝑡)|^𝑝=𝜖 and 𝐸|𝑋(𝑠)|^𝑝<𝜖 for 𝑠∈[𝑚(0),𝑡]. Then (24) show that 𝐸𝑋𝑡^𝑝≤6^𝑝−1𝑀^𝑝𝜂_𝛼(𝜃)𝑒^{−𝛼𝑡𝛼𝜃𝑝}+𝑀^𝑝𝜂_𝛼(𝜃)𝑒^{−𝛼𝑡𝛼𝜃𝑝}(−𝐴)^−𝛽𝑝𝑀^𝑝_𝑔𝛿+6^𝑝−1(−𝐴)^−𝛽𝑝𝑀^𝑝_𝑔+𝑀^𝑝_𝑔𝐾𝑇𝛼,𝛽^𝛼𝛽𝛼𝛽^𝑝+𝛼^𝑝𝑀^𝑝𝐿^𝑝₁𝐿^𝑝+𝐶_𝑝𝛼^𝑝𝑀^𝑝𝐿^𝑝₁𝐿^′𝑝/2𝜖<𝜖 (30) which contradicts the definition of 𝑡. Therefore, the mild solution of (3) is asymptotically stable in 𝑝th moment.

In particular, when 𝑝=2 from Theorem 9 we have the following.

Theorem 10. Suppose that the conditions (H1)–(H3) hold. Then, the stochastic fractional differential equations (3) are mean square asymptotically stable if 4𝑀²_𝑔[|(−𝐴)^−𝛽|²+𝑉(𝛼,𝛽)(𝑇^𝛼𝛽/𝛼𝛽)²]+4𝛼²𝑀²𝐿²₁[𝐿²+𝐿^′]<1, where 𝑉(𝛼,𝛽)=𝛼²Γ²_(1+𝛽)𝐶²_1−𝛽/Γ²_(1+𝛼𝛽).

When 𝑔≡0,𝑝=2, (3) reduces to ^𝑐𝐷^𝛼_𝑡𝑋(𝑡)(𝑡)(((𝑡(((𝑡(𝑡)=𝐴𝑋+𝑓𝑡,𝑋𝑡−𝜏)))+𝜎𝑡,𝑋𝑡−𝜈)))𝑑𝑊𝑋𝑑𝑡,𝑡≥0,₀(⋅)=𝜑∈ℬ_ℱ([𝑚(0),0],𝐻). (31) From Theorems 9 and 10, we can easily get the following result.

Corollary 11. Suppose the assumptions (H1) and (H2) hold. Then, the stochastic equations (8) are mean square asymptotically stable if 2𝛼²𝑀²𝐿²₁[𝐿²+𝐿^′]<1.

Example 12. Consider the following stochastic nonlinear fractional partial differential equation with infinite delay in the following form ^𝑐𝐷^𝛼_𝑡𝑢+=𝜕𝑡,𝑦𝑔(𝑡,𝑢𝑡−𝜏,𝑦₂𝑢𝑡,𝑦𝜕𝑦²+𝑓+𝜎(𝑡)𝑡,𝑢𝑡−𝜏,𝑦𝑡,𝑢𝑡−𝜏,𝑦𝑑𝑊,𝑢()()𝑢[]𝑑𝑡𝑡,0=𝑢𝑡,𝜋=0,𝑡,𝑦=𝜙𝑡,𝑦,𝑦∈0,𝜋,𝑡≤0, (32) where 𝑊(𝑡) denotes a standard cylindrical Wiener process and a standard one-dimensional Brownian motion. To write the system (32) into the abstract form of (3), we consider the space 𝐻=𝐾=𝐿²[0,𝜋] and define the operator 𝐴∶𝐷(𝐴)⊂𝐻→𝐻 by 𝐴𝑤=𝑤^′′ with domain 𝐷(𝐴)=𝑤∈𝑋;𝑤,𝑤^′𝑤areabsolutelycontinuous,_′′∈𝑋,𝑤(0)=𝑤(𝜋)=0,𝐴𝑤=^∞_𝑛=1𝑛²𝑤,𝑤_𝑛𝑤_𝑛,𝑤∈𝐷(𝐴), (33)(34) where 𝑤_𝑛(𝑠)=√2sin(𝑛𝑠),𝑛=1,2,… is the orthogonal set of eigenvectors in 𝐴. It is well known that 𝐴 generates a compact, analytic semigroup {𝑆(𝑡),𝑡≥0} in 𝑋 and 𝑆(𝑡)𝑤=^∞_𝑛=1𝑒^−𝑛2𝑡𝑤,𝑤_𝑛𝑤_𝑛. (35) It is well known that |𝑆(𝑡)|≤𝑒^−𝜋2𝑡. Take 𝑝=2. Since 𝑀=1, we can get the inequality 4[|(−𝐴)^−𝛽|²+𝑉(𝛼,𝛽)(𝜋^𝛼𝛽/𝛼𝛽)²+𝛼²(𝐿²+𝐿^′)]<1. Further, if we impose suitable conditions on 𝑔, 𝑓, and 𝜎 to verify assumptions of Theorem 10, then we can conclude that the mild solution of (32) is mean square asymptotically stable.