Existence and stability results for neutral stochastic delay differential equations driven by a fractional Brownian motion

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    In this paper we investigate the existence, uniqueness, asymptotic behavior of mild solutions to neutral stochastic differential equations with delays driven by a fractional Brownian motion in a Hilbert space. The cases of finite and infinite delays are analyzed.


  • Keywords


    Asymptotic Behaviors; Delays; Fractional Brownian Motion; Mild Solution; Wiener Integral.

  • References


      [1] L. Coutin, Z. Qian, Stochastic differential equations for fractional Brownian motions. ComptesRendus de I'Académie des Sciences-Series I-Mathematics 331 (2000) 75-80. http://dx.doi.org/10.1016/S0764-4442(00)01594-9.

      [2] Y. Hu, D. Nualart, Differential equations driven by H¨older continuous functions of order greater than 1/2, Stochastic analysis and applications 2 (2007) 399-413. http://dx.doi.org/10.1007/978-3-540-70847-6_17.

      [3] D. Nualart, Y. Ouknine, Regularization of differential equations by fractional noise, Stochastic Process and their Applications 102 (2002) 103-116. http://dx.doi.org/10.1016/S0304-4149(02)00155-2.

      [4] D. Nualart, A. R˘a¸scanu, Differential equations driven by fractional Brownian motion, CollectaneaMathematica 53 (2002) 55-81.

      [5] D. Nualart, B. Saussereau, Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stochastic Processes and their Applications119 (2009) 391-409. http://dx.doi.org/10.1016/j.spa.2008.02.016.

      [6] T. Caraballo, J. Real, T. Taniguchi, The exponential stability of neutral stochastic delay partial differential equations, Discrete and Continuous Dynamical Systems 18 (2007) 295–313. http://dx.doi.org/10.3934/dcds.2007.18.295.

      [7] W. Dai, C. C. Heyde, Ito's formula with respect to fractional Brownian motion and its application, Journal of Applied Mathematics and Stochastic Analysis 9 (1996) 439–448. http://dx.doi.org/10.1155/S104895339600038X.

      [8] D. Feyel, A. de la Pradelle, on fractional Brownian processes, Potential Analysis 10 (1999) 273–288. http://dx.doi.org/10.1023/A:1008630211913.

      [9] E. Hernandez, D. N. Keck, M. A. McKibben, On a class of measure-dependent stochastic evolution equations driven by fBm, Journal of Applied Mathematics and Stochastic Analysis 2007 (2007) 1-26. http://dx.doi.org/10.1155/2007/69747.

      [10] P. H. Bezandry, Existence of almost periodic solutions for semilinear stochastic evolution equations driven by fractional Brownian motion, Electronic Journal of Differential Equations 2012 (2012) 1–21.

      [11] T. E. Duncan, B. Maslowski, B. Pasik-Duncan, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stochastics and Dynamics 2 (2002) 225–250. http://dx.doi.org/10.1142/S0219493702000340.

      [12] B. Maslowski, D. Nualart, Evolution equations driven by a fractional Brownian motion, Journal of Functional Analysis 2 (2003) 277–305. http://dx.doi.org/10.1016/S0022-1236(02)00065-4.

      [13] T. Caraballo, M. J. Garrido-Atienza, T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis: Theory, Methods and Applications 74 (2011) 3671–3684. http://dx.doi.org/10.1016/j.na.2011.02.047.

      [14] B. Boufoussi, S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statistics & Probability Letters 82 (2012)1549–1558. http://dx.doi.org/10.1016/j.spl.2012.04.013.

      [15] T. E. Duncan, B. Maslowski, B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic process and their applications 115(8) (2005) 1357-1383.

      [16] Nguyen-Tien-Dung, Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varying time delays, Journal of the Korean Statistical society 43 (2014) 599–608. http://dx.doi.org/10.1016/j.jkss.2014.02.003.

      [17] T. A. Burton. Stability by fixed point theory for functional differential equations, Mineola, NY, Dover Publications, Inc. (2006).

      [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York. (1983). http://dx.doi.org/10.1007/978-1-4612-5561-1.

      [19] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, (1992). http://dx.doi.org/10.1017/CBO9780511666223.


  • Existence and stability results for neutral stochastic  
    delay differential equations driven by a  
    fractional Brownian motion  
    1
    2
    R. Maheswari *, S. Karunanithi  
    1
    Department of Mathematics, Sri Eshwar College of Engg., Coimbatore-641 202, Tamilnadu,  
    Department of Mathematics, Kongunadu Arts and Science College (Autonomous), Coimbatore-641 029, Tamilnadu, India  
    Corresponding author E-mail:mahesenthil12@gmail.com  
    2
    *
    Copyright © 2015R. Maheswari, S. Karunanithi. This is an open access article distributed under the Creative Commons Attribution License, which permits  
    unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.  
    Abstract  
    In this paper we investigate the existence, uniqueness, asymptotic behavior of mild solutions to neutral stochastic  
    differential equations with delays driven by a fractional Brownian motion in a Hilbert space. The cases of finite and  
    infinite delays are analyzed.  
    Keywords: Asymptotic Behaviors; Delays; Fractional Brownian Motion; Mild Solution; Wiener Integral.  
    1
    . Introduction  
    The theory of stochastic differential equations driven by a fractional Brownian motion (fBm) has been studied  
    intensively in recent years [1], [2], [3], [4], and [5]. The fBm received much attention because of its huge range of  
    potential applications in several fields like telecommunications, networks, finance markets, biology and so on [6], [7],  
    [
    8], [9]. Moreover, one of the simplest stochastic processes that is Gaussian, self-similar, and has stationary increments  
    is fBm [10]. In particular, fBm is a generalization of the classical Brownian motion, which depends on a parameter H ∈  
    (
    0, 1) called the Hurst index [9]. It should be mentioned that when H = , the stochastic process is a standard Brownian  
    2
    motion; when H 2, it behaves completely in a different way than the standard Brownian motion, in particular neither  
    is a semimartingale nor a Markov process. It is a self-similar process with stationary increments and has a long-memory  
    whenH ≠ 2. These significant properties make fBm a natural candidate as a model for noise in a wide variety of  
    physical phenomena such as mathematical finance, communication networks, hydrology and medicine. The existence  
    and uniqueness of mild solutions for a class of stochastic differential equations in a Hilbert space with a standard,  
    cylindrical fBm with the Hurst parameter in the interval , 1ꢂ has been studied [11]. Maslowski and Nualart [12] have  
    2
    studied the existence and uniqueness of a mild solution for nonlinear stochastic evolution equations in a Hilbert space  
    driven by a cylindrical fBm under some regularity and boundedness conditions on the coefficients. Recently, Caraballo  
    and et al [13] investigated the existence and uniqueness of mild solutions to stochastic delay equations driven by fBm  
    with Hurst parameter H ∈ ꢁ2 , 1ꢂ. An existence and uniqueness result of mild solutions for a class of neutral stochastic  
    differential equation with finite delay, driven by an fBm in a Hilbert space has been investigated [14] in Boufoussi and  
    Hajji. The asymptotic behavior of solutions for stochastic differential equations with fBm has only been investigated by  
    a few authors [13], [14], [15]. Moreover Nguyen [16] has studied the asymptotic behaviors of mild solutions to neutral  
    stochastic differential equations driven by an fBm. Motivated by this consideration in this paper we investigate the  
    existence and uniqueness and asymptotic behaviors of mild solutions for a neutral stochastic differential equation with  
    finite or infinite delays driven by fBm in the following form  
    d [x(t) g ꢁt, xꢃt r(t)ꢄꢂ] = [Ax(t) + f ꢁt, xꢃt ρ(t)ꢄꢂ] dt + h ꢁt, xꢃt δ(t)ꢄꢂ dW(t)  
    +
    σ(t)dB (t) ; t 0  
    Q
    x(t) = ϕ(t), t (−τ, 0ꢆ(0 <)  
    (1.1)  
    where A is an infinitesimal generator of an analytic semigroup of bounded linear operators,ꢃS(t)ꢄ in a Hilbert  
    space X with norm‖∙‖, B (t)denotes an fBm with H > on a real and separable Hilbert space Y, r, ρ, δ: ꢋ0, ∞) → ꢋ0, τ)  
    Q
    2
    are continuous, f, g: ꢋ0, ∞) × X → X , h: ꢋ0, ∞) × X → ꢌ , σ: ꢋ0, ∞) → ℒ (Y, X) , the initial data ϕ ∈ Cꢃ(−τ, 0ꢆ, Xꢄ the  
    2
    Q
    space of all continuous functions from(−τ, 0ꢆ to X and has finite second moments. Further we assume W and B are  
    independent. The main tool of this paper is the fixed point theory which was proposed by Burton [17].  
    2
    . Preliminaries  
    In this section we first recall the fBm as well as the Wiener integral with respect to it. We also establish some  
    important results which will be needed throughout the paper. Let (Ω, ℱ, P) be a complete probability space and T > 0  
    be an arbitrary fixed horizon. Let X and be two real, separable Hilbert spaces and L(Y, X) be the space of bounded  
    linear operators from Y to X.  
    Let (Y, X) be separable Hilbert space with respect to the Hilbert-Schmidt norm‖∙‖ꢍ  
    . Let (Y, X) be the space of all  
    2
    Q
    ψ ∈ ℒ(Y, X) such that ψꢐ  
    is a Hilbert Schmidt operator. The norm is given be‖ψ‖2ℒ  
    = ꢓψꢐ  
    ꢓ = tr(ψꢐψ). Then ψ  
    is called a HilbertSchmidt operator from Y to X.Let (Ω, X) denote the space of- measurable, X -valued and  
    2
    square integrable Stochastic processes. Consider a time interval ꢋ0, Tꢆ with arbitrary fixed horizon T and let  
    {
    β(t), t ∈ ꢋ0, Tꢆ} be a two sided one dimensional fBm with Hurst parameter H ∈ ꢁ , 1ꢂ with the covariance function  
    2
    R(t, s) = Eꢋβ(t)β(s)ꢆ = 2 (|t|2 + |s|2 |t − s|2), t, s.  
    It is known that β(t) with H > 2 admits the following Wiener integral representation  
    β(t) = K(t, s) dW(s),  
    where W = {W(t): t ∈ ꢋ0, Tꢆ} a Wiener is process and K(t, s) is the kernel given by  
    2  
    du, t >.  
    K(t, s) = c (u s)3⁄  
    2
    ꢂ  
    (2ꢅꢔꢀ)  
    with c =ꢘ  
    with β(∙) representing the beta function. Let K(t, s) = 0 if t ≤ s .  
    ꢁ2ꢔ2ꢅ ,ꢅꢔ ꢂ  
    We will denote bythe reproducing kernel Hilbert space of the fBm. Infactis the closure of set of indicator  
    functions ꢙ1;, t ∈ ꢋ0, Tꢆꢚ with respect to the scalar product  
    1,, 1, = R (t, s).  
    The mapping 1, β(t) can be extended to an isometry between and the first Wiener chaos and we will denote  
    by β (φ) by the previous isometry. We recall for ψ, φ ∈ ℋ their scalar product in is given by  
    ꢝ ꢝ  
    ψ, φ = H(ꢛH 1) ꢜ ψ(s)φ(t) |t s|22dsdt  
    ꢊ ꢊ  
    For the deterministic function φ L2(ꢋ0, Tꢆ), the fractional Weiner integral of φ with respect to β is defined by  
    φ(s)dβ(t) = K φ(s)dβ(s) ,  
    ∂k  
    where K φ(s) = ∫ φ(r) (r, s)dr  
    Letβn(t)ꢚnbe a sequence of two-sided one dimensional standard fBm mutually independent on (Ω, ℱ, P). When one  
    consider the following series  
    n
    β (t)en,t 0  
    nꢟꢀ  
    where {en} is a orthonormal basis in Y, this series does not necessarily converge in the space Y.Thus we consider a  
    Y - valued stochastic process B (t) given formally by the following series  
    Q
    n
    B (t) = ∑  
    β (t) ꢐ  
    en,, t 0  
    Q
    nꢟꢀ  
    If is a non-negative self-adjoint trace class operator, then this series converges in the space Y, that is, it holds that  
    2
    Q
    B (t) L (Ω, Y) then we say that the above B (t) is a Y - valued - cylindrical fBm with covariance operator. For  
    Q
    example if {λn} is a bounded sequence of non-negative real numbers such that ꢐen = λnen assuming that is a nuclear  
    operator in Y (ie.,n λn <) then the stochastic process  
    n
    en =nλnβn (t) en t 0  
    B (t) = ∑  
    β (t) ꢐ  
    Q
    nꢟꢀ  
    is well defined as a Y -valued - cylindrical fBm.  
    n
    where β (t) are real, independent fBm’s. This process is a Yvalued Gaussian, it starts from 0, zero mean and  
    covariance  
    E〈β(t), x〉〈β(s), y〉 = R(s, t)〈ꢐ(x), y〉For all x, y ∈ Y and t, s ∈ ꢋ0, Tꢆ.  
    ꢓK φ2en, (,:) < ∞  
    Definition 2.1: Let φ: ꢋ0, Tꢆ → ℒ (Y, X) such that ∑  
    (2.1)  
    Q
    nꢟꢀ  
    ꢎ  
    Then its stochastic integral with respect to the fBm B is defined, for t 0, as follows  
    Q
    n
    φ(s) dB (s) ∑  
    φ(s)ꢠλnend β (s) = ∑  
    ∫ ꢠλn (φen) (s)dβn(s)  
    Q
    nꢟꢀ  
    nꢟꢀ  
    Notice that if  
    φλnenꢍ  
    (ꢋꢊ,ꢝꢆ:ꢅ)  
    < ∞  
    (2.2)  
    nꢟꢀ  
    2
    Lemma 2.2: For any φ: ꢋ0, Tꢆ → ℒ (Y, X) satisfiesφ(s)‖  
    ds < such that (2.2) holds, and for any α, β ꢋ0, Tꢆ  
    Q
    with α > ꢥ,  
    2
    2
    α
    α
    ⁄  
    E ꢓ∫β φ(s)dB (s)ꢓ cH(ꢛH − 1)(α β)2nφ(s)ꢐ 2en ds  
    Q
    β
    ⁄  
    where c = c(H). If in addition,nφ(t)ꢐ 2en is uniformly convergent for t ∈ ꢋ0, Tꢆ. Then  
    2
    α
    α
    E ꢓ∫β φ(s)dB (s)ꢓ cH(ꢛH 1)(α β)2φ(s)‖  
    2
    ds  
    Q
    β
    We now suppose that 0ρ(A), where ρ(A) is the resolvent set of A, and the semigroup S(t) is uniformly bounded that  
    is to say‖S(t)‖ M for some constant M 1 and every t 0. Then, for 0 < 1, it is possible to define the  
    α
    α
    α
    fractional power operator (−A) as a closed linear operator on its domain D(−A) . Furthermore the subspace D(−A) is  
    α
    α
    α
    dense in X and the expression ‖X‖α = ‖(−A) x‖ , x ∈ D(−A) defines a norm on Xα = D(−A) . The following  
    properties are well known in [18].  
    Lemma 2.3:[18] Under the above conditions the following properties hold.  
    i)  
    Xα is a Banach space for 0 < ꢦ 1  
    ii)  
    iii) For every 0 < 1, there exists Mα such that‖(−A) S(t)‖ Mαt  
    If the resolvent operator of A is compact, then embedding Xβ Xα is continuous and compact for 0 <.  
    e , λ > 0, 0.  
    α
    α λꢈ  
    Lemma 2.4:[19] For any r1 and for arbitrary L valued predictable process φ(∙),  
    2
    ꢂ dsꢪ  
    2
    ꢕꢖp  
    2ꢞ  
    ꢎ  
    Eꢣ∫φ(u) dw(u)ꢣ ꢃr(ꢛr 1)ꢄ ꢨ∫ ꢁE‖φ(s)‖  
    ∈ꢋꢊ,ꢈ ꢆ  
    3
    . Existence and uniqueness of a solution  
    In this section we study the existence and uniqueness of mild solutions for equation (1.1). For this equation we assume  
    that the following conditions hold.  
    (
    H)A is the infinitesimal generator of an analytic semigroup,ꢃS(t)ꢄ, of bounded linear  
    operators on X. Further we suppose that 0 ρ(A) and that ‖S(t)‖ M and  
    ꢑꢬβ  
    (−A)βS(t)ꢣ ≤β  
    .
    For some constants M, Mβ and every t ꢋ0, Tꢆ  
    (
    H2) There exists a positive constants C, C2 and L ,L2 > 0 such that, for all t ≥ 0, x, y X  
    ‖f(t, x) f(t, y)‖2 L ‖x y‖2  
    2 2  
    ‖f(t, x)‖ C(1 + ‖x‖ )  
    2
    i)  
    ii)  
    iii) ‖h(t, x) h(t, y)‖ L2 ‖x y‖2  
    2
    2
    iv) ‖h(t, x)‖ C2(1 + ‖x‖ )  
    (
    H3)There exists a constant 0 < < 1 , L3 > 0 such that the function g is Xβ - valued and  
    satisfies for all t 0,x, y ∈ X such that  
    2
    ꢣ(−A)βg(t, x) − (−A)βg(t, y)ꢣ L3 ‖x y‖2  
    i)  
    2
    ꢣ(−A)βg(t, x)ꢣ C3(1 + ‖x‖2)  
    iii) The constants L3 and ꢥ, ꢣ(−A)βꢣL3 < 1  
    ii)  
    H4) The function (−A)βg is continuous in the quadratic mean sense square. For all functions X  
    (
    2
    limꢣ(−A)βgꢃt, x(t)ꢄ − (−A)βgꢃs, x(s)ꢄꢣ = 0  
    2
    (
    H5) The function σ: ꢋ0, ) → ℒ (Y, X) satisfiesσ(s)‖  
    ds <,for all T > 0.  
    Q
    Moreover we assume that ϕ ∈ ꢭꢃꢋ−τ, 0 ꢆ, ꢌ2(Ω, X)ꢄ.  
    Definition 3.1: A X valued stochastic process {x(t), t (−τ, T)} is called a mild solution of Eqn. (1.1) if  
    x(∙) ∈ ꢭꢃꢋ−τ, 0 ꢆ, ꢌ2(Ω, X)ꢄ  
    x(t) = ϕ(t) forτ t 0  
    i)  
    ii)  
    iii) For arbitrary t ꢋ0, Tꢆ we have  
    x(t) = S(t)φ(0) + g ꢁ0, φꢃ−r(0)ꢄꢂꢯ − g ꢁt, xꢃt r(t)ꢄꢂ  
    ꢈ ꢈ  
    ∫ AS(t s)g ꢁs, xꢃs r(s)ꢄꢂ ds + S(t s) f ꢁs, xꢃs − ρ(s)ꢄꢂ ds  
    ꢊ ꢊ  
    ꢈ ꢈ  
    ∫ S(t s) h ꢁs, xꢃs δ(s)ꢄꢂ dW(s) + ∫ S(t s) σ(s)dB(s)  
    ꢊ ꢊ  
    +
    Theorem 3.1: Suppose(H) to (H5) hold. Then for all T > 0, then Eqn. (1.1) has a unique mild solution on ꢋ−τ, Tꢆ.  
    Proof: Fix T > 0 and let ꢭꢋ−τ, Tꢆ,2(Ω, X),be the Banach space of all continuous functions from ꢰ– τ, Tꢱ into  
    2  
    x ∈: x(s) = ϕ(s) for s ∈ ꢋ−τ, 0ꢆ}.  
    Sis a closed subset of provided with the norm ‖∙‖ . Define the operator ψ on S by ψ(x)(t) = ϕ(t) for t ∈  
    τ, 0ꢆ and for t ꢋ0, Tꢆ  
    (Ω, X) , equipped with the supremum normξ = suꢳτ,(E‖ξ(u)‖2)2 and let us consider the set Sꢝ  
    =
    {
    ψ(x)(t) = S(t) ꢮϕ(0) + g ꢁ0, ϕꢃ−r(0)ꢄꢂꢯ − g ꢁt, xꢃt r(t)ꢄꢂ  
    ꢈ ꢈ  
    ∫ AS(t s)g ꢁs, xꢃs r(s)ꢄꢂ ds + S(t s) f ꢁs, xꢃs ρ(s)ꢄꢂ ds  
    ꢊ ꢊ  
    +
    ∫ S(t s) h ꢁs, xꢃs δ(s)ꢄꢂ dW(s) + ∫ S(t s) σ(s)dB(s)  
    Then it is clear to prove the existence of mild solutions to Eqn. (1.1) is equivalent to find a fixed point for the operator ψ.  
    Next we will show by Banach fixed point theorem that ψ has a unique fixed point. We divide the proof into two steps.  
    2
    Step 1. For arbitrary x ∈ S , let us prove that t ψ(x)(t) is continuous on the interval ꢋ0, Tꢆ in the (Ω, X) sense .Let  
    0
    < ꢧ < and |h| be sufficiently small. Then for any x S, we have  
    ψ(x)(t + h)ψ(x)(t)‖ ꢓꢃS(t + h) − S(t)ꢄ ꢮϕ(0) + g ꢁ0, ϕꢃ−r(0)ꢄꢂꢯꢓ  
    +
    +
    +
    +
    +
    ꢓg ꢁt + h, xꢃt + h − r(t + h)ꢄꢂ − g ꢁt, xꢃt − r(t)ꢄꢂꢓ  
    ꢵꢶ  
    ꢓ∫AS(t + h s)g ꢁs, xꢃs r(s)ꢄꢂ ds AS(t − s)g ꢁs, xꢃs r(s)ꢄꢂ dsꢓ  
    ꢵꢶ  
    ꢓ∫S(t + h s) f ꢁs, xꢃs ρ(s)ꢄꢂ ds S(t − s) f ꢁs, xꢃs ρ(s)ꢄꢂ dsꢓ  
    ꢵꢶ  
    ꢓ∫S(t + h s) h ꢁs, xꢃs δ(s)ꢄꢂ dw(s) S(t − s) h ꢁs, xꢃsδ(s)ꢄꢂ dw(s)ꢓ  
    ꢵꢶ  
    ꢓ∫S(t + h s) σ(s)dB(s) S(t s) σ(s)dB(s)ꢓ  
    =
    6 I(h)  
    By the strong continuity of S(t), we have  
    limꢓꢃS(t + h) S(t)ꢄ ꢮϕ(0) + g ꢁ0, ϕꢃ−r(0)ꢄꢂꢯꢓ = 0  
    The condition (H) assure that  
    ꢃS(t + h) S(t)ꢄ ꢮϕ(0) + g ꢁ0, ϕꢃ−r(0)ꢄꢂꢯꢓ ꢛM ꢓϕ(0) + g ꢁ0, ϕꢃ−r(0)ꢄꢂꢓ2(Ω)  
    Then we conclude by the Lebesgue dominated theorem that lim ꢹ|I(h)| = 0  
    β
    By using the fact that the operator (−A) is bounded, we obtained that  
    2
    2
    |I2(h)|2 ꢣ(−A)β ꢓ(−A)βg ꢁt + h, xꢃt + h r(t + h)ꢄꢂ − (−A)βg ꢁt, xꢃt − r(t)ꢄꢂꢓ  
    Then we conclude by condition (H4) that lim ꢹ|I2(h)| = 0  
    For I3(h), we suppose that h > 0, then we have  
    I3(h) ≤ ꢓ∫ Sꢃ(h) Iꢄ(−A)βS(t − s)(−A)βg ꢁs, xꢃs r(s)ꢄꢂ dsꢓ  
    ꢈꢵꢶ  
    ꢓ∫(−A)βS(t + h s)(−A)βg ꢁs, xꢃs r(s)ꢄꢂ dsꢓ  
    +
    By Holder’s inequality  
    2
    ꢓ∫Sꢃ(h) Iꢄ(−A)βS(t s)(−A)βg ꢁs, xꢃs r(s)ꢄꢂ dsꢓ  
    2
    t ꢹ ꢜ ꢓSꢃ(h) Iꢄ(−A)βS(t s)(−A)βg ꢁs, xꢃs − r(s)ꢄꢂꢓ ds  
    By using the strong continuity of S(t), we have for each s ∈ ꢋ0, tꢆ,  
    limSꢃ(h) − Iꢄ(−A)βS(t − s)(−A)βg ꢁs, xꢃs r(s)ꢄꢂ = 0  
    By using condition (H), condition (ii) in (H3) and the fact that 0 < < 1 , we obtain  
    Sꢃ(h) − Iꢄ(−A)βS(t − s)(−A)βg ꢁs, xꢃs − r(s)ꢄꢂꢓ  
    (
    )β  
    β
    ꢓꢃ– Aꢄ g ꢁs, xꢃs r(s)ꢄꢂꢓ2(ꢋ0, tꢆ × Ω)  
    (
    ꢈꢔꢕ)ꢬ  
    β
    Then we conclude by the Lebesgue dominated theorem that  
    2
    lim ꢓ∫ Sꢃ(h) Iꢄ(−A)βS(t s)(−A)βg ꢁs, xꢃs r(s)ꢄꢂ dsꢓ = 0  
    By conditions (H), condition (ii)in (H3) and Holder’s inequality we get  
    2
    ꢈꢵꢶ  
    ꢓ∫(−A)βS(t + h s)(−A)βg ꢁs, xꢃs r(s)ꢄꢂ dsꢓ  
    β  
    2βꢀ  
    h2β C (ꢹ‖x(s) r(s)‖2 + 1) ds  
    2
    3
    2
    ꢵꢶ  
    Then lim ꢓ∫ (−A)βS(t + h s)(−A)βg ꢁs, xꢃs r(s)ꢄꢂ dsꢓ = 0  
    Hence that lim ꢹ|I3(h)| = 0  
    I4(h) ≤ ꢓ∫ Sꢃ(h) IꢄS(t − s) f ꢁs, xꢃs ρ(s)ꢄꢂ dsꢓ  
    ꢵꢶ  
    +
    ꢓ∫S(t + h s) f ꢁs, xꢃs ρ(s)ꢄꢂ dsꢓ  
    2
    Now ꢹ ꢓ∫ Sꢃ(h) IꢄS(t − s) f ꢁs, xꢃs ρ(s)ꢄꢂ dsꢓ  
    2
    2
    ∫ Sꢃ(h) IꢄS(t s) C ꢁꢹꢣxꢃs ρ(s)ꢄꢣ + 1ꢂ ds  
    2
    2
    M ∫ Sꢃ(h) Iꢄ C ꢁꢣxꢃs ρ(s)ꢄꢣ + 1ꢂ ds  
    2
    Since lim ꢓ∫ Sꢃ(h) − IꢄS(t − s) f ꢁs, xꢃsρ(s)ꢄꢂ dsꢓ = 0  
    2
    ꢵꢶ  
    and ꢓ∫ S(t + h − s) f ꢁs, xꢃs ρ(s)ꢄꢂ dsꢓ  
    2
    ꢈꢵꢶ  
    M2ꢹ ꢓ∫ f ꢁs, xꢃs ρ(s)ꢄꢂ dsꢓ  
    ꢈꢵꢶ  
    2
    M2 C ꢁꢣxꢃs ρ(s)ꢄꢣ + 1ꢂ ds  
    2
    2
    Then lim ꢓ∫ S(t + h − s) f ꢁs, xꢃsρ(s)ꢄꢂ dsꢓ = 0  
    ꢵꢶ  
    Hence lim ꢹ|I4(h)| = 0  
    Now I5(h) ≤ ꢓ∫ Sꢃ(h) IꢄS(t − s) h ꢁs, xꢃs δ(s)ꢄꢂ dW(s)ꢓ  
    ꢵꢶ  
    +
    ꢓ∫S(t + h s) h ꢁs, xꢃs δ(s)ꢄꢂ dW(s)ꢓ  
    2
    Now ꢹ ꢓ∫ Sꢃ(h) IꢄS(t − s) h ꢁs, xꢃs δ(s)ꢄꢂ dW(s)ꢓ  
    2
    2
    ∫ Sꢃ(h) IꢄS(t s) C ꢁꢹꢣxꢃs δ(s)ꢄꢣ + 1ꢂ ds  
    2
    2
    M2 Sꢃ(h) Iꢄ C ꢁꢹꢣxꢃs δ(s)ꢄꢣ + 1ꢂ ds  
    2
    2
    2
    Therefore lim ꢓ∫ Sꢃ(h) IꢄS(t − s) h ꢁs, xꢃsδ(s)ꢄꢂ dW(s)ꢓ = 0  
    2
    and ꢹ ꢓ∫ S(t + h s) h ꢁs, xꢃs δ(s)ꢄꢂ dW(s)ꢓ  
    ꢵꢶ  
    ꢵꢶ  
    2
    M2 C ꢁꢣxꢃs δ(s)ꢄꢣ + 1ꢂ ds  
    2
    2
    2
    Then lim ꢓ∫ S(t + h − s) h ꢁs, xꢃsδ(s)ꢄꢂ dW(s)ꢓ = 0  
    ꢵꢶ  
    Hence lim ꢹ|I5(h)| = 0  
    For the term I6(h), we have  
    ꢈꢵꢶ  
    I6(h) ≤ ꢓ∫ Sꢃ(h) IꢄS(t − s) σ(s)dB(s)ꢓ + ꢓ∫ S(t + h s) σ(s)dB(s)ꢓ  
    By condition (H) and lemma 2.2 we get  
    2
    ꢓ∫Sꢃ(h) IꢄS(t s) σ(s)dB(s)ꢓ  
    2
    cH(ꢛH − 1)(t)2 ꢣSꢃ(h) IꢄS(t s)σ(s)ꢣ  
    ds  
    2
    cH(ꢛH − 1)(t)2M ∫ ꢣSꢃ(h) Iꢄσ(s)ꢣ  
    ds  
    2
    Since limꢣSꢃ(h) − Iꢄσ(s)ꢣ = 0 and  
    ꢏ  
    2
    Sꢃ(h) − Iꢄσ(s)ꢣℒ  
    ≤ ꢺM2σ(s)‖2  
    ∈ ℒ(ꢋ0, Tꢆ, ds)  
    ꢒ  
    we conclude by the dominated convergence theorem that  
    2
    limꢹ ꢓ∫ Sꢃ(h) − IꢄS(t − s) σ(s)dB(s)ꢓ = 0  
    Applying lemma 2.2, we get  
    ꢵꢶ  
    ꢈꢵꢶ  
    ꢓ∫S(t + h − s) σ(s)dB(s)ꢓ cH(h)2M2 σ(s)‖2  
    ds → 0  
    Hence that lim ꢹ|I6(h)| = 0  
    The above arguments show thatlimꢹ‖ψ(x)(t + h) ψ(x)(t)‖2 = 0. Hence we conclude that the function t ψ(x)(t)  
    2
    is continuous on ꢋ0, Tꢆ in the - sense.  
    Step 2. Now we are going to show that ψ is a contraction mapping in S with some T T.  
    3
    3
    3
    Let x, y ∈ S by using the inequality (a + b + c + d)2 a2 +  
    b2 +  
    c2 +  
    d2  
    k
    where = L3ꢣ(−A)βꢣ < 1, we obtain any fixed t ∈ ꢋ0, Tꢆ  
    ꢀꢔk  
    ꢀꢔk  
    ꢀꢔk  
    2
    ψ(x)(t) − ψ(y)(t)‖2 ꢓg ꢁt, xꢃt r(t)ꢄꢂ g ꢁt, yꢃt r(t)ꢄꢂꢓ  
    k
    2
    3
    + ꢀ  
    ꢓ∫AS(t − s) ꢮg ꢁs, xꢃs r(s)ꢄꢂ g ꢁs, yꢃs r(s)ꢄꢂꢯ dsꢓ  
    ꢔk  
    2
    ꢓ∫S(t s) ꢮf ꢁs, xꢃs ρ(s)ꢄꢂ f ꢁs, yꢃs ρ(s)ꢄꢂꢯ dsꢓ  
    3
    + ꢀ  
    ꢔk  
    2
    ꢓ∫S(t s) ꢮh ꢁs, xꢃs δ(s)ꢄꢂ h ꢁs, yꢃs δ(s)ꢄꢂꢯ dW(s)ꢓ  
    3
    + ꢀ  
    ꢔk  
    2
    2
    ꢣ(−A)βꢣ ꢓ(−A)βg ꢁt, xꢃt − r(t)ꢄꢂ − (−A)βg ꢁt, yꢃt − r(t)ꢄꢂꢓ  
    k
    2
    3
    ꢓ∫(−A)βS(t − s)(−A)β ꢮg ꢁs, xꢃs r(s)ꢄꢂ g ꢁs, yꢃs r(s)ꢄꢂꢯꢓ  
    + ꢀ  
    ꢔk  
    2
    3
    ꢓ∫S(t s)(−A)β ꢮf ꢁs, xꢃs ρ(s)ꢄꢂ − f ꢁs, yꢃs ρ(s)ꢄꢂꢯ dsꢓ  
    + ꢀ  
    + ꢀ  
    ꢔk  
    2
    3
    ꢓ∫S(t s)(−A)β ꢮh ꢁs, xꢃs δ(s)ꢄꢂ h ꢁs, yꢃs δ(s)ꢄꢂꢯ dW(s)ꢓ  
    ꢔk  
    By Lipchitz property of (−A)βg and f, h combined with Holder’s inequality, we obtain  
    ψ(x)(t) − ψ(y)(t)‖2 ꢢꢹ‖x(t r) y(t r)‖2  
    3
    βꢑ  
    2
    2
    ꢂ ∫ ꢹ‖x(s r) y(s r)‖2ds  
    + ꢀ  
    C M  
    3
    ꢀꢔβ  
    ꢔk  
    2βꢔꢀ  
    3
    2
    tM2C ∫ ꢹꢣxꢃsρ(s)ꢄ − yꢃsρ(s)ꢄꢣ ds  
    2
    + ꢀ  
    + ꢀ  
    ꢔk  
    3
    2
    tM2C ∫ ꢹꢣxꢃsδ(s)ꢄ − yꢃs δ(s)ꢄꢣ ds  
    2
    2
    ꢔk  
    Hencesuꢳτ,ꢹ‖ψ(x)(s) ψ(y)(s)‖2 γ(t)suτ,ꢹ‖x(s) y(s)‖2  
    3ꢫꢎ  
    3ꢫꢎ  
    ꢎ  
    3
    ꢑꢬβ  
    t2β+  
    +
    where γ(t) = + (  
    ꢀꢔk)(2βꢔꢀ)  
    ꢀꢔk  
    ꢀꢔk  
    By condition (iii)in (H3) we have,γ(0) = ꢢ = ꢣ(−A)βꢣL3 < 1.Then there exists 0 < T such that  
    0
    < ꢽ(T) < 1 and ψ is a contraction mapping on S and therefore has a unique fixed point, which is a mild solution of  
    Eqn.(1.1) on ꢰ– τ, T.This procedure can be repeated in order to extend the solution to the entire interval ꢰ– τ, Tꢱ in  
    finitely many steps. This completes the proof.  
    4
    . Stability analysis  
    In this section we establish the results for the case of finite delays, then the case of infinite delays can be proved. Our  
    method is based on the contraction mapping principle.  
    In order to prove the required results, we assume the following additional conditions.  
    λꢈ  
    (
    (
    H6) The semigroup S(t) satisfiesλ > 0, ∃M > 0such that ‖S(t)‖ Me , t 0  
    γꢕ  
    2
    H7) The function σ: ꢋ0, ) → ℒ satisfies eσ(s)‖ℒ  
    ds < .  
    Q
    We first consider the case of finite delays, τ <.  
    Theorem 4.1: (Finite delays) Assume that f(t, 0) = g(t, 0) = h(t, 0) = 0, t 0, the assumptions (H) - (H7)holds and  
    that  
    Γ(β)  
    λβ  
    2
    ꢁL3ꢣ(−A)βꢣ + M2βL3  
    + M2Lλ2+M2L2(ꢛλ)2 < 1  
    (4.1)  
    where Γ(∙) is the gamma function,Mβ is the corresponding constant in lemma (2.3). Then the mild solution to (1.1)  
    exists uniquely and is exponential decay to zero in mean square. i.e., there exists a pair of positive constants a > 0 and  
    M = M (φ, a) such that  
    ‖x(t)‖2 Me, ∀ t 0.  
    Proof: Denote by the space of all stochastic process x(t, ω): (−τ, ) × Ω X satisfying x(t) = ϕ(t), t (−τ, 0ꢆ and  
    there exists some constants a > 0 and M = M (φ, a) > 0 such that  
    ‖x(t)‖2 Me, ∀ t 0.  
    (4.2)  
    It is to check that is a Banach space endowed with a norm|x| = suꢳ ꢹ|x(t)|2. Without loss of generality, we  
    2
    assume that a <. We denote the operator Φ on by  
    (Φx)(t) = ϕ(t), t ∈ (−τ, 0ꢆ and  
    (Φx)(t) = S(t) ꢮϕ(0) + g ꢁ0, ϕꢃ−r(0)ꢄꢂꢯ − g ꢁt, xꢃt r(t)ꢄꢂ  
    ∫ AS(t s)g ꢁs, xꢃs r(s)ꢄꢂ ds + S(t s) f ꢁs, xꢃs ρ(s)ꢄꢂ ds  
    ∫ S(t s) h ꢁs, xꢃs δ(s)ꢄꢂ dW(s) + ∫ S(t s) σ(s)dB(s)  
    +
    To get desired results, it is enough to show that operator Φ has a unique fixed point in. For this purpose, we use the  
    contraction mapping principle.  
    Step 1. We first verify that Φ(ꢿ). For convenience of notation, we denote by M , i = 1,ꢛ, the finite positive  
    constants depending on φ, a. By assumption (H6) we have  
    2
    ‖ꢳ(t)‖2 M2ꢹ ꢓϕ(0) + g ꢁ0, ϕꢃ−r(0)ꢄꢂꢓ eλ M eλꢈ  
    (4.3)  
    To estimate(t), i = ꢛ, … ,ꣁ. we observe that for x ∈ ꢿ and u(t) = r(t) orρ(t) or δ(t)the following useful estimate  
    holds  
    2 2  
    ꢣxꢃt − u(t)ꢄꢣ ꢁMe() + ꢹꢣφꢃt − u(t)ꢄꢣ ꢂ  
    ꢃMe()+ ꢹE‖φ2e()ꢄ  
    (M+ ꢹ‖ϕ‖2)eτeꢈ  
    where ‖ϕ‖ = suꢳτφ(s)‖ <.  
    Then by assumption (i) in (H3) we have  
    2
    2
    ‖ꢳ2(t)‖2 ꢣ(−A)β ꢓ(−A)βg ꢁt, xꢃt − r(t)ꢄꢂ − (−A)β g(t, 0)ꢓ  
    2
    L3ꢣ(−A)βꢣ ꢹꢣxꢃt r(t)ꢄꢣ2  
    2
    L3ꢣ(−A)βꢣ (M + ꢹ‖ϕ‖2)eτeꢈ  
    M e  
    2
    ꢔꢾꢈ  
    (4.4)  
    Using lemma 2.3, Holder’s inequality and assumption (i) in (H3) obtain that  
    2
    ‖ꢳ3(t)‖2 ꢓ∫ AS(t − s)g ꢁs, xꢃs r(s)ꢄꢂ dsꢓ  
    2
    ꢣ(−A)βS(t − s)ꢣds ꢜꢣ(−A)βS(t s)ꢣꢹ ꢓ(−A)βg ꢁs, xꢃs − r(s)ꢄꢂꢓ ds  
    2
    2
    βL3 ꢜ(t − s)β eλ()ds ꢜ(t − s)β eλ()ꢹꢣxꢃs − r(s)ꢄꢣ ds  
    M
    Γ(β)  
    λβ  
    2βL3  
    ∫ (t − s)β eλ()(M + ꢹ‖φ2)eτeds  
    M
    Γ(β)  
    2βL3 λβ (M + ꢹ‖φ2 )eτe ∫ (t − s)β e(λ)()ds  
    M
    Γ(β)  
    (M+ ꢹ‖ϕ‖2)eτeꢈ  
    λβ(λ)β  
    M
    2βL3  
    We therefore have ꢹ‖ꢳ3(t)‖2 M eꢈ  
    (4.5)  
    3
    Similarly, we obtain by assumption(i) iꣅ (H2) that  
    2
    ‖ꢳ4(t)‖2 = ꢓ∫ S(t s) f ꢁs, xꢃs ρ(s)ꢄꢂ dsꢓ  
    ꢈ ꢈ  
    2
    M2L eλ()ds eλ()ꢹꢣxꢃsρ(s)ꢄꢣ ds  
    ꢊ ꢊ  
    M2Lλ eλ()(M + ꢹ‖ϕ‖2)eτeds  
    M2Lλ(λ a)(M + ꢹ‖ϕ‖2)eτeꢈ  
    M eꢈ  
    ‖ꢳ5(t)‖2 = ꢓ∫ S(t s) h ꢁs, xꢃs δ(s)ꢄꢂ dw(s)ꢓ  
    (4.6)  
    4
    2
    2
    M2L2 e2λ()ꢹꢣxꢃsδ(s)ꢄꢣ ds  
    2
    M2L2 e2λ()ds e2λ()ꢹꢣxꢃs δ(s)ꢄꢣ ds  
    M2L2λ e2λ()(M + ꢹ‖ϕ‖2)eτeds  
    2  
    M2L2λ(ꢛλ a)(M + ꢹ‖ϕ‖2)eτeꢈ  
    2  
    M eꢈ  
    (4.7)  
    5
    By using lemma 2.2 we get that  
    2
    ‖ꢳ6(t)‖2 = ꢓ∫ S(t s) σ(s)dB(s)ꢓ  
    M2cH(ꢛH − 1)t2 e2λ()σ(s)‖2  
    ds  
    ds  
    From this inequality we can infer that  
    ‖ꢳ6(t)‖2 M2cH(ꢛH − 1)t2e2λ e2γσ(s)‖2ℒ  
    (4.8)  
    where λ = λγ.Indeed, if λ <, then λ = λ and we have  
    ‖ꢳ6(t)‖2 M2cH(ꢛH − 1)t2e2λ e2λσ(s)‖2  
    ds  
    M2cH(ꢛH − 1)t2e2λ e2γσ(s)‖2  
    ds  
    If γ < then λ = γ and we have  
    ‖ꢳ6(t)‖2 M2cH(ꢛH − 1)t2e2λ e2γσ(s)‖2ℒ  
    ds  
    since suꢳ ꢁt2ꢀ  
    e
    λꢂ <, this together with (4.8) we have  
    ‖ꢳ6(t)‖2 M eꢈ  
    (4.9)  
    6
    Combining (4.3)-(4.7), (4.9) we see that there exists M̅ > 0 and a > 0 such that  
    2
    ∗ ꢔꢾꢈ  
    ‖(Φx)(t)‖ ≤ M e , t 0.  
    ̅
    Hence we conclude that Φ (ꢿ).  
    Step 2. We now show that Φ is a contraction mapping. For any x, y, we have  
    ‖(Φx)(t) − (Φy)(t)‖2n Iꢷ  
    Since x(t) = y(t) = φ(t), t (−τ, 0ꢆ, this implies that  
    2
    ꢣxꢃt − r(t)ꢄ − yꢃt − r(t)ꢄꢣ suꢳꢹ‖x(t) y(t)‖2  
    Then we have by assumption(i) in (H3)  
    2
    I= ꢓg ꢁt, xꢃt r(t)ꢄꢂ − g ꢁt, yꢃt r(t)ꢄꢂꢓ  
    2
    2
    L3ꢣ(−A)βꢣ ꢹꢣxꢃt − r(t)ꢄ − yꢃt − r(t)ꢄꢣ  
    2
    L3ꢣ(−A)βꢣ suꢳꢹ‖x(t) y(t)‖2  
    2
    I2 = ꢓ∫ AS(t − s) [g ꢁs, xꢃs r(s)ꢄꢂ g ꢁs, yꢃs − r(s)ꢄꢂ] dsꢓ  
    ꢈ ꢈ  
    2
    2
    βL3 ∫ (t s)β eλ()ds(t s)β eλ()ꢹꢣxꢃs r(s)ꢄ yꢃs r(s)ꢄꢣ ds  
    M
    Γ(β)  
    λβ  
    2
    M
    2βL3  
    ∫ (t s)β eλ()ꢹꢣxꢃs − r(s)ꢄ − yꢃs − r(s)ꢄꢣ ds  
    Γ(β)  
    λβ  
    M
    2βL3  
    suꢳꢹ‖x(t) − y(t)‖2  
    By assumption (i) iꣅ (H2)  
    I3 = ꢓ∫ S(t s) [f ꢁs, xꢃs ρ(s)ꢄꢂ f ꢁs, yꢃs ρ(s)ꢄꢂ] dsꢓ  
    2
    M2L eλ()ds eλ()ꢹꢣxꢃsρ(s)ꢄ − yꢃsρ(s)ꢄꢣ ds  
    2
    M2Lλ eλ()ꢹꢣxꢃsρ(s)ꢄ − yꢃs − ρ(s)ꢄꢣ ds  
    M2Lλ2suꢳꢹ‖x(t) − y(t)‖2  
    By assumption(iii) iꣅ (H2) , we have  
    I4 = ꢓ∫ S(t − s) [h ꢁs, xꢃs δ(s)ꢄꢂ h ꢁs, xꢃs δ(s)ꢄꢂ] dw(s)ꢓ  
    2
    M2L2 e2λ()ds e2λ()ꢹꢣxꢃsδ(s)ꢄ − yꢃs − δ(s)ꢄꢣ ds  
    2
    M2L2(ꢛλ) e2λ()ꢹꢣxꢃsδ(s)ꢄ − yꢃs − δ(s)ꢄꢣ ds  
    M2L2(ꢛλ)2suꢳꢹ‖x(t) − y(t)‖2  
    Thusꢹ‖(Φx)(t) − (Φy)(t)‖2  
    L3 Γ2(β)  
    2
    ꢺ ꢨL3ꢣ(−A)β + M2β  
    + M2Lλ2+M2L2(ꢛλ)2 suꢳꢹ‖x(t) y(t)‖2  
    λ2β  
    By the condition (4.1), we claim that Φ is a contractive. So, applying the Banach fixed point principle, the proof is  
    complete.  
    Now we consider the case of infinite delays. We assume that t r(t), t ρ(t), t δ(t) as t.  
    Theorem 4.2:[Infinite delays]Under the conditions of theorem 4.1, the mild solution to (1.1) exists uniquely and  
    2
    converges to zero in mean square i.e., lim ꢹ‖x(t)‖ = 0  
    Proof: Denote by the space of all stochastic processes x(t, ω): (−∞,) × Ω X satisfying  
    x(t) = ϕ(t), t ∈ (−, oꢆ and  
    limꢹ‖x(t)‖2 = 0  
    (4.10)  
    We define the operator Ψ on by (Ψx)(t) = ϕ(t), t (−, 0ꢆand  
    (Ψx)(t) = S(t) ꢮϕ(0) + g ꢁ0, ϕꢃ−r(0)ꢄꢂꢯ − g ꢁt, xꢃt r(t)ꢄꢂ  
    ∫ AS(t s)g ꢁs, xꢃs r(s)ꢄꢂ ds + S(t s) f ꢁs, xꢃs − ρ(s)ꢄꢂ ds  
    ∫ S(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dw(s) + S(t − s) σ(s)dB(s)  
    +
    Since (Ψx)(t) = (Φx)(t) on ꢋ0, ), this implies that Ψ is contractive. Hence it remains to check Φ(ꢿ) ⊂ ꢿ. In order to  
    2
    obtain this claim, we need to show thatlimꢹ‖Ψx(t)‖ = 0 for all  
    x ∈ .  
    By the definition of, assumption (H7)and the fact t r(t), t. We get  
    lim ꢹ‖P(t)‖2 = limꢹ‖P2(t)‖2 = limꢹ‖P6(t)‖ = 0  
    2
    We further have ꢹ‖ꢳ3(t)‖2 ꢓ∫ AS(t − s)g ꢁs, xꢃs r(s)ꢄꢂ dsꢓ  
    2
    2
    βL3 ∫ (t − s)β eλ()ds ∫ (t − s)β eλ()ꢹꢣxꢃs r(s)ꢄꢣ ds  
    M
    M
    βL3Γ(β)λβ (t − s)β eλ()ꢹꢣxꢃs − r(s)ꢄꢣ ds  
    2
    2
    2
    For any x ∈ ꢿand ε > 0 it follows from (4.10) that there exists s > 0 such that ꢹꢣxꢃs r(s)ꢄꢣ <for all  
    s ≥ s.Thus we obtain  
    ‖ꢳ3(t)‖2 M L3Γ(β)λβ (t s)β eλ()ꢹꢣxꢃs − r(s)ꢄꢣ ds + M L3Γ2(β)λ2β  
    2
    2 2  
    ϵ
    ꢀꢔβ ꢀꢔβ  
    which proves that  
    limꢹ‖P3(t)‖2 M L3Γ2(β)λ2β,∀ϵ > 0 and hence, lim ꢹ‖P3(t)‖2 = 0  
    2
    ꢀꢔβ  
    2
    ‖ꢳ4(t)‖2 = ꢓ∫ S(t − s) f ꢁs, xꢃs ρ(s)ꢄꢂ dsꢓ  
    ꢈ ꢈ  
    2
    M2L eλ()ds eλ()ꢹꢣxꢃsρ(s)ꢄꢣ ds  
    ꢊ ꢊ  
    M2Lλ eλ()ꢹꢣxꢃsρ(s)ꢄꢣ ds  
    2
    2
    For any x ∈ ꢿand ε > 0 there exists a s > 0 such that ꢹꢣxꢃs r(s)ꢄꢣ < for all s s.  
    2
    Thus we obtain ꢹ‖ꢳ4(t)‖2 M2Lλꢀ  
    λ(ꢈꢔꢕ)  
    ꢹꢣxꢃs − ρ(s)ꢄꢣ ds + M2Lλ2  
    e
    ϵ
    which proves that lim E‖P4(t)‖2 M2Lλ2ϵ,∀ϵ > 0 and hence,im E‖P4(t)‖2 = 0  
    2
    ‖ꢳ5(t)‖2 = ꢓ∫ S(t s) h ꢁs, xꢃs δ(s)ꢄꢂ dw(s)ꢓ  
    2
    M2L2 e2λ()ꢹꢣxꢃsδ(s)ꢄꢣ ds  
    2
    For any x ∈ ꢿand ε > 0 there exists as > 0 such that ꢹꢣxꢃs r(s)ꢄꢣ <for all s s.  
    2
    Thus we obtain ꢹ‖ꢳ5(t)‖2 M2L2 ∫  
    ꢔ2λ(ꢈꢔꢕ)  
    ꢹꢣxꢃs − δ(s)ꢄꢣ ds + M2L2(ꢛλ)ϵ  
    e
    2
    which proves that lim ꢹ‖P5(t)‖ M2L2(ꢛλ)ϵ, ∀ϵ > 0 and hence, lim ꢹ‖P5(t)‖2 = 0  
    Once again, by applying the Banach fixed point principle, we complete the proof of the theorem.  
    To illustrate the obtained theory, let us end this section with an example.  
    5
    . Example  
    Consider the following neutral stochastic partial differential equations with delays driven by a fBm in the following  
    form  
    2  
    t ꢰ  
    z(t, ξ) + αGꢃt, z(t τ, ξ)ꢄꢱ =ξ2 z(t, ξ) + α2Fꢃt, z(t τ, ξ)ꢄ ꣈t + α3φꢃt, z(t τ, ξ)ꢄdw(t) + Θ(t)dB (t)  
    Q
    z(t, 0) = z(t, π) = 0  
    (5.1)  
    z(t, ξ) = ϕ(t, ξ), t ∈ (−∞, 0ꢆ, 0 ξ π  
    where α, α2, α3 > 0 are constants.  
    Let X = L (0, π) with the norm ‖∙‖ and inner product 〈∙,∙〉 .Define A: X → X by Ax = x" with domain  
    2
    "
    D(A) = ꢙx X: x, x are absolutely coꣅtiꣅuous x X, x(0) = x(π) = 0ꢚ. Then Eqn. (5.1) can be written in the form  
    of equation (1.1) with the co-efficient g(t, x) = αGꢃt, z(t τ, ξ)ꢄ, f(t, x) = α2Fꢃt, z(t τ, ξ)ꢄ and  
    h(t, x) = α3φꢃt, z(t τ, ξ)ꢄ, σ(t) = Θ(t).  
    For the operator A, it is known from Pazy [18] that the following properties hold.  
    2
    Ax =n A2〈x, en〉 en, x ∈ D(A) , where en(t) = √π siꣅ(ꣅt) , = 1,ꢛ,꣉, is the orthogonal set of eigenvectors.  
    A is the infinitesimal generator of an analytic semigroup ꢃS(t)ꢄin X.  
    ꢈ  
    S(t)x =n en 〈x, en〉 en, For all x ∈ X and for every t > 0 .  
    π
    Furthermore‖S(t)‖ ≤ e  
    , t 0.  
    The bounded linear operator(−A)34 is well defined and is given by  
    (
    −A)34x = ∑n(ꣅ)32〈x, en en  
    with domain Dꢃ(−A)34 = ꢙx X:n en〈x, en〉 en Xꢚ . Furtheremore ꢣ(−A)34 = 1 and ꢣ(−A)34 ≤  
    t (4)‖S(t)‖dt < π.Thus (H) holds with M = 1, λ = π2 ,(H2) holds with L = α22 ,(H3) and (H4) holds  
    Γ(3꣊4)  
    2
    with β = ꣉꣊ꢺ,L2 = ꢣ(−A)34 α2 = α2 and (H6) holds with γ = .Consequently we conclude by theorem 4.2, that  
    2
    the stochastic partial equation (5.1) has a unique mild solution and that this solution converges to zero in mean square if  
    the parameters α, α2, α3 satisfy the following conditions.  
    αꢎ  
    ꢎ  
    4
    3
    αꢎ  
    αꢎ  
    αꢎ  
    Γ2 4 π + π + π < .  
    π+  
    4
    6
    . Conclusion  
    In this paper we derived existence, conditions ensuring the exponential decay to zero in mean square of a neutral  
    stochastic delay differential equations driven by fBm. In addition we also established the case of infinite delays which  
    has not yet been discussed in the context of neutral stochastic delay differential equations driven by fBm.  
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Article ID: 4175
 
DOI: 10.14419/ijamr.v4i2.4175




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