Asymptotic Stability of Fractional Stochastic Neutral Differential Equations with Infinite Delays

We study the existence and asymptotic stability in p th moment of a mild solution to a class of nonlinear fractional neutral stochastic differentialequationswithinfinitedelaysinHilbertspaces.Asetofnovelsufficientconditionsarederivedwiththehelpofsemigrouptheoryandfixedpointtechniqueforachievingtherequiredresult.Theuniquenessofthesolutionoftheconsideredproblemisalsostudiedundersuitableconditions.Finally,anexampleisgiventoillustratetheobtainedtheory.


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][2][3][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 squaremean 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][9][10][11][12][13][14][15].
Let and be two real separable Hilbert spaces with inner products ⟨⋅, ⋅⟩ and ⟨⋅, ⋅⟩ , respectively. We denote their norms by | ⋅ | and | ⋅ | . To avoid confusion we just use ⟨⋅, ⋅⟩ for the inner product and | ⋅ | for the norm. Let { } ∞ =1 be an orthonormal basis of . Throughout the paper, we assume that (Ω, F, ; F) (F = {F } ≥0 ) is a complete filtered probability space satisfying that F 0 contains allnull sets of F. Suppose { ( ) : ≥ 0} is cylindricalvalued Brownian motion with a trace class operator , denote are mutually independent one-dimensional standard Brownian motions. Define L( , ) as the set of all bounded linear operators : → with the following norm: It is obvious that L( , ) is a Hilbert space with an inner product induced by the above norm. Let ∈ L( , ) be called a Hilbert-Schmidt operator. We further assume that the filtration is generated by the cylindrical Brownian motion (⋅) and augmented, that is, where N is the -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: where is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operator ( ), ≥ 0 in the Hilbert space , : + × → , : + × → L( , ) are two Borel measurable mappings, and : + × → is continuous mapping. The fractional derivative , ∈ (0, 1) is understood in the Caputo sense. In addition, let ( ), ( ) ∈ ( + , + ) satisfy − ( ) → ∞, − ( ) → ∞ as → ∞. Let (0) = max{inf ≥0 ( − ( )), inf ≥0 ( − ( ))}. Here B F 0 ([ (0), 0], ) denote the family of all almost surely bounded, F 0 -measurable, continuous random variables ( ) : [ (0), 0] → with norm | | B = sup (0)≤ ≤0 | ( )| . Throughout this paper, we assume that ≥ 2 is an integer.

Preliminaries and Basic Properties
Let be the infinitesimal generator of an analytic semigroup ( ) in . Then, ( − ) is invertible and generates a bounded analytic semigroup for > 0 large enough. Therefore, we can assume that the semigroup ( ) is bounded and the generator is invertible. It follows that (− ) , 0 < ≤ 1 can be defined as a closed linear invertible operator with its domain (− ) being dense in . We denote by the Banach space (− ) endowed with the norm | | = |(− ) |, which is equivalent to the graph norm of (− ) . For more details about semigroup theory, one can refer [18].
Definition 2 (see [19]). The fractional integral of order with the lower limit 0 for a function is defined as Definition 3 (see [19]). The Caputo derivative of order for a function : [0, ∞) → can be written as If is an abstract function with values in , 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 In view of [18, Lemma 3.1] and by using Laplace transform, we present the following definition of mild solution of (3).
The following properties of ( ) and ( ) [18] are useful. (ii) for any ∈ , ∈ (0, 1) and ∈ (0, 1] one has Definition 6. Let ≥ 2 be an integer. Equation (8) is said to be stable in th moment if for arbitrarily given > 0 there exists a > 0 such that Definition 7. Let ≥ 2 be an integer. Equation (8) is said to be asymptotically stable in th moment if it is stable in th moment and, for any

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).
In addition, in order to derive the stability of the solution, we further assume that (H5) It is obvious that (3) has a trivial solution when = 0 under the assumption (H5).

Lemma 8.
Let ≥ 2, > 0 and let Φ be an L( , )-valued, predictable process such that ∫ 0 |Φ( )| L < ∞. Then, As mentioned in Luo [20], to prove the asymptotic stability it is enough to show that the operator Ψ has a fixed point in . 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 4 Abstract and Applied Analysis ∈ B, 1 ≥ 0 and let |ℎ| be sufficiently small, and observe that Note that The strong continuity of ( ) [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 Abstract and Applied Analysis 5 where ( , ) = Γ (1+ ) 1− /Γ (1+ ) . Since is sufficiently small, the right hand side of the above equation tends to zero as |ℎ| → 0. Next we consider × ( , ( − ( ))) By the Holder's inequality, we obtain Therefore, the right hand side of the above equation tends to zero as |ℎ| → 0 and sufficiently small. Further, we have × ( , ( − ( ))) ( ) × ( 1 + ℎ − ) ( , ( − ( ))) ( ) Abstract and Applied Analysis As above, the right hand side of the above inequality tends to zero. Similarly, we have 2 → 0 as ℎ → 0. Thus Ψ is continuous in th moment on [0, ∞).
Now, we estimate the terms on the right hand side of (24) by using the assumptions (H1), (H3), and (H4). Now, we have For ( ) ∈ B and for any > 0 there exists a 1 > 0 such that | ( − ( ))| ≤ for ≥ 1 . Therefore, For the fourth term of (24), we have Abstract and Applied Analysis 7 Also, we have By the same discussion as above, we have that (28) tends to zero as → ∞. Thus |Ψ ( )| → 0 as → ∞. We conclude that Ψ(B) ∈ B. Finally, we prove that Ψ has a unique fixed point. Indeed, for any , ∈ B, we have Therefore, Ψ is a contradiction mapping and hence there exists a unique fixed point, which is a mild solution of (3) with ( ) = ( ) on [ (0), 0] and | ( )| → 0 as → ∞.
To show the asymptotic stability of the mild solution of (3), as the first step, we have to prove the stability in th moment. Let > 0 be given and choose > 0 such that < which contradicts the definition of . Therefore, the mild solution of (3) is asymptotically stable in th moment.
In particular, when = 2 from Theorem 9 we have the following.
From Theorems 9 and 10, we can easily get the following result.