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

*

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)

ꢅ

+

σ(t)dB (t) ; t ≥ 0

Q

x(t) = ϕ(t), t ∈ (−τ, 0ꢆ(0 < ꢇ ≤ ∞)

(1.1)

where A is an infinitesimal generator of an analytic semigroup of bounded linear operators,ꢃS(t)ꢄ_ꢈ_ꢉ_ꢊ in a Hilbert

ꢀ

ꢅ

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

ꢊ

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^ꢅ 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.

ꢊ

Let ꢌ (Y, X) be separable Hilbert space with respect to the Hilbert-Schmidt norm‖∙‖_ꢍ

ꢏ

. Let ℒ (Y, X) be the space of all

2

Q

ꢎ

ꢑ

ψ ∈ ℒ(Y, X) such that ψꢐ

ꢎ

is a Hilbert Schmidt operator. The norm is given be‖ψ‖²_ℒ

ꢏ

= ꢓψꢐ

ꢎ

ꢓ = tr(ψꢐψ^∗). Then ψ

ꢒ

ꢊ

is called a ꢐ Hilbert –Schmidt operator from Y to X.Let ℒ (Ω, X) denote the space of ℱ_ꢊ- measurable, X -valued and

2

square integrable Stochastic processes. Consider a time interval ꢋ0, Tꢆ with arbitrary fixed horizon T and let

ꢀ

{

β^ꢅ(t), t ∈ ꢋ0, Tꢆ} be a two sided one dimensional fBm with Hurst parameter H ∈ ꢁ , 1ꢂ with the covariance function

2

ꢀ

R_ꢅ(t, s) = Eꢋβ^ꢅ(t)β^ꢅ(s)ꢆ =₂ (|t|²^ꢅ + |s|²^ꢅ − |t − s|²^ꢅ), t, s ∈ ℝ.

ꢀ

It is known that β^ꢅ(t) with H >₂ admits the following Wiener integral representation

ꢈ

β^ꢅ(t) = ∫ K_ꢅ(t, s) dW(s),

ꢊ

where W = {W(t): t ∈ ꢋ0, Tꢆ} a Wiener is process and K_ꢅ(t, s) is the kernel given by

ꢅ

ꢔ^ꢀ⁄₂

du, t > ꢗ.

K_ꢅ(t, s) = c_ꢅ ∫ (u − s)^ꢅ^ꢔ³^⁄

ꢈ

ꢖ

2

ꢁ_ꢕꢂ

ꢕ

ꢅ

(2ꢅꢔꢀ)

ꢑ

ꢎ

with c_ꢅ =_√_ꢘ

with β(∙) representing the beta function. Let K_ꢅ(t, s) = 0 if t ≤ s .

ꢁ2ꢔ2ꢅ ,ꢅꢔ ꢂ

We will denote by ℋthe reproducing kernel Hilbert space of the fBm. Infact ℋis the closure of set of indicator

functions ꢙ1_ꢋ_ꢊ_;_ꢈ_ꢆ, t ∈ ꢋ0, Tꢆꢚ with respect to the scalar product

〈

1_ꢋ_ꢊ_,_ꢈ_ꢆ, 1_ꢋ_ꢊ_,_ꢕ_ꢆ〉_ℋ = R_ꢅ (t, s).

The mapping 1_ꢋ_ꢊ_,_ꢈ_ꢆ → β^ꢅ(t) can be extended to an isometry between ℋ and the first Wiener chaos and we will denote

ꢅ

by β (φ) by the previous isometry. We recall for ψ, φ ∈ ℋ their scalar product in ℋ is given by

ꢝ ꢝ

ψ, φ〉_ℋ = H(ꢛH − 1) ꢜ ꢜ ψ(s)φ(t) |t − s|²^ꢅ^ꢔ²dsdt

ꢊ ꢊ

〈

For the deterministic function φ ∈ L²(ꢋ0, Tꢆ), the fractional Weiner integral of φ with respect to β^ꢅ is defined by

ꢝ

φ(s)dβ^ꢅ(t) = ∫ K φ(s)dβ^ꢅ(s) ,

∗

ꢅ

ꢊ

∫

ꢊ

ꢝ

∂k

∗

where K φ(s) = ∫ φ(r)_∂_ꢞ (r, s)dr

ꢅ

ꢕ

Letꢙβ_n^ꢅ(t)ꢚ_n_∈_ℕbe a sequence of two-sided one dimensional standard fBm mutually independent on (Ω, ℱ, P). When one

consider the following series

∞

ꢅ

n

∑

β (t)e_n_,t ≥ 0

nꢟꢀ

where {e_n} is a orthonormal basis in Y, this series does not necessarily converge in the space Y.Thus we consider a

ꢅ

Y - valued stochastic process B (t) given formally by the following series

Q

ꢑ

ꢅ

∞

ꢅ

n

B (t) = ∑

β (t) ꢐ

ꢎ

e_n_,, t ≥ 0

Q

nꢟꢀ

If ꢐ is a non-negative self-adjoint trace class operator, then this series converges in the space Y, that is, it holds that

ꢅ

2

ꢅ

Q

B (t) ∈ L (Ω, Y) then we say that the above B (t) is a Y - valued ꢐ - cylindrical fBm with covariance operator ꢐ. For

Q

example if {λ_n} is a bounded sequence of non-negative real numbers such that ꢐe_n = λ_ne_n assuming that ꢐ is a nuclear

operator in Y (ie.,∑_n^∞_ꢟ_ꢀ λ_n < ∞) then the stochastic process

ꢑ

ꢎ

ꢅ

∞

ꢅ

n

e_n= ∑^∞_n_ꢟ_ꢀ ꢠλ_nβ_n (t) e_n t ≥ 0

B (t) = ∑

β (t) ꢐ

Q

nꢟꢀ

is well defined as a Y -valued ꢐ - cylindrical fBm.

ꢅ

n

where β (t) are real, independent fBm’s. This process is a Y –valued Gaussian, it starts from 0, zero mean and

covariance

E〈β^ꢅ(t), x〉〈β^ꢅ(s), y〉 = R(s, t)〈ꢐ(x), y〉For all x, y ∈ Y and t, s ∈ ꢋ0, Tꢆ.

ꢓK φꢐ^ꢀ^⁄2e_n_,ꢓ₍_ꢋ_ꢊ_,_ꢝ_ꢆ_:_ꢡ₎ < ∞

ꢊ

∞

∗

Definition 2.1: Let φ: ꢋ0, Tꢆ → ℒ (Y, X) such that ∑

(2.1)

Q

nꢟꢀ

ꢅ

ꢍ^ꢎ

ꢅ

Then its stochastic integral with respect to the fBm B is defined, for t ≥ 0, as follows

Q

ꢈ

ꢅ

n

ꢈ

ꢅ

∞

∗

ꢅ

∫

φ(s) dB (s) ≔ ∑

∫ φ(s)ꢠλ_ne_nd β (s) = ∑

∫ ꢠλ_nꢢ (φe_n) (s)dβ_n(s)

Q

nꢟꢀ

ꢊ

Notice that if

∞

∑

ꢣφꢠλ_ne_nꢣ_ꢍ

ꢑ

ꢤ (ꢋꢊ,ꢝꢆ:ꢅ)

< ∞

(2.2)

nꢟꢀ

ꢈ

ꢊ

2

ℒ

Lemma 2.2: For any φ: ꢋ0, Tꢆ → ℒ (Y, X) satisfies ∫ ‖φ(s)‖

ꢏ

ds < ∞ such that (2.2) holds, and for any α, β ∈ ꢋ0, Tꢆ

Q

ꢊ

ꢒ

with α > ꢥ,

2

α

ꢀ_⁄

E ꢓ∫_β φ(s)dB (s)ꢓ ≤ cH(ꢛH − 1)(α − β)²^ꢅ^ꢔ^ꢀ ∑_n^∞_ꢟ_ꢀ ∫ ꢓφ(s)ꢐ 2e_nꢓ ds

ꢅ

Q

β

ꢀ_⁄

where c = c(H). If in addition, ∑_n^∞_ꢟ_ꢀ ꢓφ(t)ꢐ 2e_nꢓ is uniformly convergent for t ∈ ꢋ0, Tꢆ. Then

2

α

E ꢓ∫_β φ(s)dB (s)ꢓ ≤ cH(ꢛH − 1)(α − β)²^ꢅ^ꢔ^ꢀ ∫ ‖φ(s)‖

ꢅ

2

ℒ

ꢏ

ds

Q

β

ꢒ

We now suppose that 0 ∈ ρ(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 < ꢦ ≤ 1, it is possible to define the

α

fractional power operator (−A) as a closed linear operator on its domain D(−A) . Furthermore the subspace D(−A) is

α

dense in X and the expression ‖X‖_α = ‖(−A) x‖ , x ∈ D(−A) defines a norm on X_α= D(−A) . The following

properties are well known in [18].

Lemma 2.3:[18] Under the above conditions the following properties hold.

i)

X_αis a Banach space for 0 < ꢦ ≤ 1

ii)

iii) For every 0 < ꢦ ≤ 1, there exists M_α such that‖(−A) S(t)‖ ≤ M_αt

If the resolvent operator of A is compact, then embedding X_β ⊂ X_α is continuous and compact for 0 < ꢦ ≤ ꢥ.

e , λ > 0, ꢧ ≥ 0.

α

ꢔα ꢔλꢈ

ꢊ

Lemma 2.4:[19] For any r ≥1 and for arbitrary L valued predictable process φ(∙),

2

ꢑ

ꢩ

ꢂ dsꢪ

ꢞ

2

ꢞ

ꢈ

ꢕꢖp

ꢕ

2ꢞ

ꢍꢎ

Eꢣ∫_ꢊφ(u) dw(u)ꢣ ≤ ꢃr(ꢛr − 1)ꢄ ꢨ∫_ꢊ ꢁE‖φ(s)‖

ꢏ

ꢕ

∈ꢋꢊ,ꢈ ꢆ

ꢡ

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_ꢈ_ꢉ_ꢊ, of bounded linear

operators on X. Further we suppose that 0 ∈ ρ(A) and that ‖S(t)‖ ≤ M and

^ꢫꢑꢬβ

(−A)^ꢀ^ꢔ^βS(t)ꢣ ≤_ꢈ_ꢑ_ꢬ_β

ꢣ

.

For some constants M, M_ꢀ_ꢔ_β 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

iv) ‖h(t, x)‖ ≤ C₂(1 + ‖x‖ )

(

H₃)There exists a constant 0 < ꢥ < 1 , L₃ > 0 such that the function g is X_β - 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)^ꢔ^βꢣL₃ < 1

ii)

H₄) The function (−A)^βg is continuous in the quadratic mean sense square. For all functions X

(

2

lim_ꢈ_→_∞ꢣ(−A)^βgꢃt, x(t)ꢄ − (−A)^βgꢃs, x(s)ꢄꢣ = 0

ꢝ

ꢊ

2

ℒ

(

H₅) The function σ: ꢋ0, ∞) → ℒ (Y, X) satisfies∫_ꢊ ‖σ(s)‖

ꢏ

ds < ∞,for all T > 0.

Q

ꢒ

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) for −τ ≤ t ≤ 0

i)

ii)

iii) For arbitrary t ∈ ꢋ0, Tꢆ we have

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

−

ꢈ ꢈ

∫ S(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s) + ∫ S(t − s) σ(s)dB^ꢅ(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 ℬ ≔ ꢭꢋ−τ, Tꢆ,ꢌ²(Ω, X),be the Banach space of all continuous functions from ꢰ– τ, Tꢱ into

ꢀ

ꢌ²

x ∈ ℬ_ꢝ: x(s) = ϕ(s) for s ∈ ꢋ−τ, 0ꢆ}.

S_ꢝis a closed subset of ℬ_ꢝ provided with the norm ‖∙‖_ℬ_ꢲ . Define the operator ψ on S_ꢝ by ψ(x)(t) = ϕ(t) for t ∈

−τ, 0ꢆ and for t ∈ ꢋ0, Tꢆ

(Ω, X) , equipped with the supremum norm ‖ξ‖_ℬ_ꢲ = suꢳ_ꢖ_∈_ꢋ_ꢔ_τ_,_ꢝ_ꢆ(E‖ξ(u)‖²)^⁄2 and let us consider the set S_ꢝ

=

{

ꢋ

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

−

+

ꢈ

∫ S(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s) + ∫ S(t − s) σ(s)dB^ꢅ(s)

ꢊ

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_ꢝ , let us prove that t → ψ(x)(t) is continuous on the interval ꢋ0, Tꢆ in the ꢌ (Ω, X) sense .Let

0

< ꢧ < ꢴ and |h| be sufficiently small. Then for any x ∈ S_ꢝ, 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

ꢈ

ꢵꢶ

ꢈ

ꢓ∫_ꢊAS(t + h − s)g ꢁs, xꢃs − r(s)ꢄꢂ ds − ∫ AS(t − s)g ꢁs, xꢃs − r(s)ꢄꢂ dsꢓ

ꢊ

ꢈ

ꢵꢶ

ꢈ

ꢓ∫_ꢊS(t + h − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ ds − ∫ S(t − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ

ꢊ

ꢈ

ꢵꢶ

ꢈ

ꢓ∫_ꢊS(t + h − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dw(s) − ∫ S(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dw(s)ꢓ

ꢊ

ꢈ

ꢵꢶ

ꢈ

ꢓ∫_ꢊS(t + h − s) σ(s)dB^ꢅ(s) − ∫ S(t − s) σ(s)dB^ꢅ(s)ꢓ

ꢊ

=

∑_ꢀ_ꢸ_ꢷ_ꢸ₆I_ꢷ(h)

By the strong continuity of S(t), we have

lim_ꢶ_→_ꢊꢓꢃS(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)ꢄꢂꢓ ∈ ꢌ²(Ω)

ꢓ

Then we conclude by the Lebesgue dominated theorem that lim_ꢶ_→_ꢊ ꢹ|I_ꢀ(h)| = 0

ꢔ

β

By using the fact that the operator (−A) is bounded, we obtained that

2

|I₂(h)|²≤ ꢣ(−A)^ꢔ^βꢣ ꢹ 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_ꢶ_→_ꢊ ꢹ|I₂(h)| = 0

For I₃(h), we suppose that h > 0, then we have

ꢈ

I₃(h) ≤ ꢓ∫_ꢊ Sꢃ(h) − Iꢄ(−A)^ꢀ^ꢔ^βS(t − s)(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂ dsꢓ

ꢈꢵꢶ

ꢓ∫_ꢈ(−A)^ꢀ^ꢔ^βS(t + h − s)(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂ dsꢓ

+

By Holder’s inequality

2

ꢈ

ꢓ∫_ꢊSꢃ(h) − Iꢄ(−A)^ꢀ^ꢔ^βS(t − s)(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂ dsꢓ

ꢹ

ꢈ

2

t ꢹ ꢜ ꢓSꢃ(h) − Iꢄ(−A)^ꢀ^ꢔ^βS(t − s)(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂꢓ ds

≤

ꢊ

By using the strong continuity of S(t), we have for each s ∈ ꢋ0, tꢆ,

_ꢶli_→m_ꢊSꢃ(h) − Iꢄ(−A)^ꢀ^ꢔ^βS(t − s)(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂ = 0

By using condition (H_ꢀ), condition (ii) in (H₃) and the fact that 0 < ꢥ < 1 , we obtain

Sꢃ(h) − Iꢄ(−A)^ꢀ^ꢔ^βS(t − s)(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂꢓ

ꢓ

(

^ꢫ^ꢵ^ꢀ⁾^ꢫ^ꢑꢬβ

β

ꢓꢃ– Aꢄ g ꢁs, xꢃs − r(s)ꢄꢂꢓ ∈ ꢌ²(ꢋ0, tꢆ × Ω)

≤

(

ꢈꢔꢕ)^ꢑ^ꢬ

β

Then we conclude by the Lebesgue dominated theorem that

2

ꢈ

lim_ꢶ_→_ꢊꢹ ꢓ∫_ꢊ Sꢃ(h) − Iꢄ(−A)^ꢀ^ꢔ^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∫_ꢈ(−A)^ꢀ^ꢔ^βS(t + h − s)(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂ dsꢓ

ꢹ

^ꢫ^ꢎ^ꢑꢬβ

≤ ₂_β_ꢔ_ꢀ

ꢝ

h²^β^ꢔ^ꢀ∫ C (ꢹ‖x(s) − r(s)‖² + 1) ds

2

3

ꢊ

2

ꢈ

ꢵꢶ

Then lim_ꢶ_→_ꢊ ꢓ∫_ꢈ (−A)^ꢀ^ꢔ^βS(t + h − s)(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂ dsꢓ = 0

Hence that lim_ꢶ_→_ꢊ ꢹ|I₃(h)| = 0

ꢈ

I₄(h) ≤ ꢓ∫_ꢊ Sꢃ(h) − IꢄS(t − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ

ꢈ

ꢵꢶ

+

ꢓ∫_ꢈS(t + h − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ

2

Now ꢹ ꢓ∫_ꢊ Sꢃ(h) − IꢄS(t − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ

ꢈ

2

≤

∫ Sꢃ(h) − IꢄS(t − s) C ꢁꢹꢣxꢃs − ρ(s)ꢄꢣ + 1ꢂ ds

ꢀ

ꢊ

ꢝ

2

M ∫ Sꢃ(h) − Iꢄ C ꢹ ꢁꢣxꢃs − ρ(s)ꢄꢣ + 1ꢂ ds

ꢀ

ꢊ

≤

2

Since _ꢶli_→m_ꢊꢹ ꢓ∫_ꢊ Sꢃ(h) − IꢄS(t − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ = 0

ꢈ

2

ꢈ

ꢵꢶ

and ꢹ ꢓ∫_ꢈ S(t + h − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ

2

ꢈꢵꢶ

M²ꢹ ꢓ∫_ꢈ f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ

≤

ꢈꢵꢶ

2

M²∫ C ꢹ ꢁꢣxꢃs − ρ(s)ꢄꢣ + 1ꢂ ds

2

ꢀ

ꢈ

≤

2

Then _ꢶli_→m_ꢊꢹ ꢓ∫_ꢈ S(t + h − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ = 0

ꢈ

ꢵꢶ

Hence lim_ꢶ_→_ꢊ ꢹ|I₄(h)| = 0

ꢈ

Now I₅(h) ≤ ꢓ∫_ꢊ Sꢃ(h) − IꢄS(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s)ꢓ

ꢈ

ꢵꢶ

+

ꢓ∫_ꢈS(t + h − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s)ꢓ

2

Now ꢹ ꢓ∫_ꢊ Sꢃ(h) − IꢄS(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s)ꢓ

ꢈ

2

≤

∫ Sꢃ(h) − IꢄS(t − s) C ꢁꢹꢣxꢃs − δ(s)ꢄꢣ + 1ꢂ ds

2

ꢊ

ꢈ

2

M²∫ Sꢃ(h) − Iꢄ C ꢁꢹꢣxꢃs − δ(s)ꢄꢣ + 1ꢂ ds

2

ꢊ

≤

2

Therefore _ꢶli_→m_ꢊꢹ ꢓ∫_ꢊ Sꢃ(h) − IꢄS(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s)ꢓ = 0

ꢈ

2

and ꢹ ꢓ∫_ꢈ S(t + h − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s)ꢓ

ꢈ

ꢵꢶ

ꢈ

ꢵꢶ

2

M²∫ C ꢹ ꢁꢣxꢃs − δ(s)ꢄꢣ + 1ꢂ ds

2

≤

2

ꢈ

2

Then _ꢶli_→m_ꢊꢹ ꢓ∫_ꢈ S(t + h − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s)ꢓ = 0

ꢈ

ꢵꢶ

Hence lim_ꢶ_→_ꢊ ꢹ|I₅(h)| = 0

For the term I₆(h), we have

ꢈ

ꢈꢵꢶ

I₆(h) ≤ ꢓ∫_ꢊ Sꢃ(h) − IꢄS(t − s) σ(s)dB^ꢅ(s)ꢓ + ꢓ∫_ꢈ S(t + h − s) σ(s)dB^ꢅ(s)ꢓ

By condition (H_ꢀ) and lemma 2.2 we get

2

ꢈ

ꢓ∫_ꢊSꢃ(h) − IꢄS(t − s) σ(s)dB^ꢅ(s)ꢓ

ꢹ

ꢈ

2

ℒ

cH(ꢛH − 1)(t)²^ꢅ^ꢔ^ꢀ ∫ ꢣSꢃ(h) − IꢄS(t − s)σ(s)ꢣ

≤

ꢏ

ds

ꢊ

ꢒ

ꢝ

2

ℒ

cH(ꢛH − 1)(t)²^ꢅ^ꢔ^ꢀM ∫ ꢣSꢃ(h) − Iꢄσ(s)ꢣ

≤

ꢏ

ds

ꢊ

ꢒ

2

Since _ꢶli_→m_ꢊꢣSꢃ(h) − Iꢄσ(s)ꢣ = 0 and

ℒꢒ^ꢏ

2

Sꢃ(h) − Iꢄσ(s)ꢣ_ℒ

≤ ꢺM²‖σ(s)‖²

∈ ℒ^ꢀ(ꢋ0, Tꢆ, ds)

ꢣ

ꢏ

ℒꢒ

ꢒ

we conclude by the dominated convergence theorem that

2

ꢈ

_ꢶli_→m_ꢊꢹ ꢓ∫_ꢊ Sꢃ(h) − IꢄS(t − s) σ(s)dB^ꢅ(s)ꢓ = 0

Applying lemma 2.2, we get

ꢈ

ꢵꢶ

ꢈꢵꢶ

ꢓ∫_ꢈS(t + h − s) σ(s)dB^ꢅ(s)ꢓ ≤ cH(h)²^ꢅ^ꢔ^ꢀM²∫_ꢈ ‖σ(s)‖_ℒ²

ꢹ

ꢏ

ds → 0

ꢒ

Hence that lim_ꢶ_→_ꢊ ꢹ|I₆(h)| = 0

The above arguments show that_ꢶli_→m_ꢊꢹ‖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 ꢌ - sense.

Step 2. Now we are going to show that ψ is a contraction mapping in S_ꢝ_ꢑ with some T_ꢀ ≤ T.

ꢀ

3

Let x, y ∈ S_ꢝ by using the inequality (a + b + c + d)² ≤ a² +

b²+

c²+

d²

k

where ꢢ = L₃ꢣ(−A)^ꢔ^βꢣ < 1, we obtain any fixed t ∈ ꢋ0, Tꢆ

ꢀꢔk

2

ꢀ

‖ψ(x)(t) − ψ(y)(t)‖² ≤ ꢓg ꢁt, xꢃt − r(t)ꢄꢂ − g ꢁt, yꢃt − r(t)ꢄꢂꢓ

ꢹ

k

2

3

ꢈ

+ _ꢀ

ꢓ∫_ꢊAS(t − s) ꢮg ꢁs, xꢃs − r(s)ꢄꢂ − g ꢁs, yꢃs − r(s)ꢄꢂꢯ dsꢓ

ꢔk

2

ꢓ∫_ꢊS(t − s) ꢮf ꢁs, xꢃs − ρ(s)ꢄꢂ − f ꢁs, yꢃs − ρ(s)ꢄꢂꢯ dsꢓ

3

ꢈ

+ _ꢀ

ꢔk

2

ꢓ∫_ꢊS(t − s) ꢮh ꢁs, xꢃs − δ(s)ꢄꢂ − h ꢁs, yꢃs − δ(s)ꢄꢂꢯ dW(s)ꢓ

3

ꢈ

+ _ꢀ

ꢔk

2

ꢣ(−A)^ꢔ^βꢣ ꢓ(−A)^βg ꢁt, xꢃt − r(t)ꢄꢂ − (−A)^βg ꢁt, yꢃt − r(t)ꢄꢂꢓ

ꢀ

≤

k

2

3

ꢈ

ꢓ∫_ꢊ(−A)^ꢀ^ꢔ^βS(t − s)(−A)^β ꢮg ꢁs, xꢃs − r(s)ꢄꢂ − g ꢁs, yꢃs − r(s)ꢄꢂꢯꢓ

+ _ꢀ

ꢔk

2

3

ꢈ

ꢓ∫_ꢊS(t − s)(−A)^β ꢮf ꢁs, xꢃs − ρ(s)ꢄꢂ − f ꢁs, yꢃs − ρ(s)ꢄꢂꢯ dsꢓ

+ _ꢀ

ꢔk

2

3

ꢈ

ꢓ∫_ꢊS(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

ꢈ^ꢎ^β^ꢬ^ꢑ

ꢈ

2

ꢂ ∫_ꢊ ꢹ‖x(s − r) − y(s − r)‖²ds

+ _ꢀ

C M

ꢁ

3

ꢀꢔβ

ꢔk

2βꢔꢀ

3

ꢈ

2

tM²C ∫ ꢹꢣxꢃs − ρ(s)ꢄ − yꢃs − ρ(s)ꢄꢣ ds

2

+ _ꢀ

ꢀ

ꢔk

ꢊ

3

ꢈ

2

tM²C ∫ ꢹꢣxꢃs − δ(s)ꢄ − yꢃs − δ(s)ꢄꢣ ds

2

ꢔk

ꢊ

Hencesuꢳ_ꢕ_∈_ꢋ_ꢔ_τ_,_ꢈ_ꢆꢹ‖ψ(x)(s) − ψ(y)(s)‖² ≤ γ(t)_ꢕ_∈s_ꢋu_ꢔꢳ_τ_,_ꢈ_ꢆꢹ‖x(s) − y(s)‖²

ꢎ

3ꢫ^ꢎ

ꢎ

ꢈ^ꢎ3ꢫ^ꢎ

ꢎ

ꢈ^ꢎ

3

ꢻ

ꢫ

ꢻ

ꢼ

ꢑꢬβ

ꢑ

ꢎ

t²^β+

+

where γ(t) = ꢢ +₍

ꢀꢔk)(2βꢔꢀ)

ꢀꢔk

By condition (iii)in (H₃) we have,γ(0) = ꢢ = ꢣ(−A)^ꢔ^βꢣL₃ < 1.Then there exists 0 < ꢴ_ꢀ ≤ T such that

0

< ꢽ(T_ꢀ) < 1 and ψ is a contraction mapping on S_ꢝ_ꢑ and therefore has a unique fixed point, which is a mild solution of

Eqn.(1.1) on ꢰ– τ, T_ꢀꢱ.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.

ꢔ

λꢈ

(

H₆) The semigroup S(t) satisfies ∃λ > 0, ∃M > 0such that ‖S(t)‖ ≤ Me , t ≥ 0

∞

ꢊ

γꢕ

2

H₇) The function σ: ꢋ0, ∞) → ℒ satisfies∫_ꢊ e ‖σ(s)‖_ℒ

ꢏ

ds < ∞ .

Q

ꢒ

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^ꢎ(β)

_λꢎβ

2

ꢁL₃ꢣ(−A)^ꢔ^βꢣ + M_ꢀ²_ꢔ_βL₃

+ M²L_ꢀλ^ꢔ²+M²L₂(ꢛλ)^ꢔ²ꢂ < 1

ꢺ

(4.1)

where Γ(∙) is the gamma function,M_ꢀ_ꢔ_β 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≤ M^∗e^ꢔ^ꢾ^ꢈ, ∀ 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≤ M^∗e^ꢔ^ꢾ^ꢈ, ∀ t ≥ 0.

ꢹ

(4.2)

It is to check that ꢿ is a Banach space endowed with a norm|x| = suꢳ_ꢈ_ꢉ_ꢊ ꢹ|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

ꢈ

−

∫ AS(t − s)g ꢁs, xꢃs − r(s)ꢄꢂ ds + ∫ S(t − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ ds

ꢊ

ꢈ

∫ S(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dW(s) + ∫ S(t − s) σ(s)dB^ꢅ(s)

+

ꢊ

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 e^ꢔ^λ^ꢈ ≤ M e^ꢔ^λ^ꢈ

∗

ꢹ

(4.3)

ꢀ

To estimate ꢳ_ꢷ(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 ≤ ꢁM^∗e^ꢔ^ꢾ^ꢃ^ꢈ^ꢔ^ꢖ⁽^ꢈ⁾^ꢄ + ꢹꢣφꢃt − u(t)ꢄꢣ ꢂ

ꢹ

ꢃM^∗e^ꢔ^ꢾ^ꢃ^ꢈ^ꢔ^ꢖ⁽^ꢈ⁾^ꢄ+ ꢹE‖φ‖_ꣂ²e^ꢔ^ꢾ^ꢃ^ꢈ^ꢔ^ꢖ⁽^ꢈ⁾^ꢄꢄ

≤

(M^∗+ ꢹ‖ϕ‖_ꣂ²)e^ꢾ^τe^ꢔ^ꢾ^ꢈ

where ‖ϕ‖_ꣂ = suꢳ_ꢔ_τ_ꣃ_꣄_ꢸ_ꢊ‖φ(s)‖ < ∞.

Then by assumption (i) in (H₃) we have

2

‖ꢳ₂(t)‖²≤ ꢣ(−A)^ꢔ^βꢣ ꢹ ꢓ(−A)^βg ꢁt, xꢃt − r(t)ꢄꢂ − (−A)^β g(t, 0)ꢓ

ꢹ

2

L₃ꢣ(−A)^ꢔ^βꢣ ꢹꢣxꢃt − r(t)ꢄꢣ²

≤

2

L₃ꢣ(−A)^ꢔ^βꢣ (M^∗ + ꢹ‖ϕ‖_ꣂ²)e^ꢾ^τe^ꢔ^ꢾ^ꢈ

M e

2

≤

∗

ꢔꢾꢈ

(4.4)

Using lemma 2.3, Holder’s inequality and assumption (i) in (H₃) obtain that

2

ꢈ

‖ꢳ₃(t)‖²≤ ꢹ ꢓ∫_ꢊ AS(t − s)g ꢁs, xꢃs − r(s)ꢄꢂ dsꢓ

ꢹ

ꢈ

2

ꢣ(−A)^ꢀ^ꢔ^βS(t − s)ꢣds ꢜꢣ(−A)^ꢀ^ꢔ^βS(t − s)ꢣꢹ ꢓ(−A)^βg ꢁs, xꢃs − r(s)ꢄꢂꢓ ds

≤

ꢊ

ꢈ

2

_ꢔ_βL₃ꢜ(t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ds ꢜ(t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − r(s)ꢄꢣ ds

M

ꢀ

≤

ꢊ

Γ(β)

_λβ

ꢈ

_ꢀ²_ꢔ_βL₃

∫ (t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾(M^∗ + ꢹ‖φ‖_ꣂ²)e^ꢾ^τe^ꢔ^ꢾ^ꢕds

≤

M

ꢊ

Γ(β)

ꢈ

_ꢀ²_ꢔ_βL₃_λ_β(M^∗ + ꢹ‖φ‖²_ꣂ )e^ꢾ^τe^ꢔ^ꢾ^ꢈ ∫ (t − s)^β^ꢔ^ꢀ e⁽^ꢾ^ꢔ^λ⁾⁽^ꢈ^ꢔ^ꢕ⁾ds

M

ꢊ

Γ^ꢎ(β)

(M^∗+ ꢹ‖ϕ‖_ꣂ²)e^ꢾ^τe^ꢔ^ꢾ^ꢈ

_λ_β₍_λ_ꢔ_ꢾ₎β

≤

M

_ꢀ²_ꢔ_βL₃

We therefore have ꢹ‖ꢳ₃(t)‖² ≤ M e^ꢔ^ꢾ^ꢈ

∗

(4.5)

3

Similarly, we obtain by assumption(i) iꣅ (H₂) that

2

ꢈ

‖ꢳ₄(t)‖²= ꢹ ꢓ∫_ꢊ S(t − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ

ꢹ

ꢈ ꢈ

2

M²L_ꢀ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ds ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − ρ(s)ꢄꢣ ds

ꢊ ꢊ

≤

M²L_ꢀλ^ꢔ^ꢀ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾(M^∗ + ꢹ‖ϕ‖_ꣂ²)e^ꢾ^τe^ꢔ^ꢾ^ꢕds

ꢈ

≤

ꢊ

M²L_ꢀλ^ꢔ^ꢀ(λ − a)^ꢔ^ꢀ(M^∗ + ꢹ‖ϕ‖_ꣂ²)e^ꢾ^τe^ꢔ^ꢾ^ꢈ

≤

M e^ꢔ^ꢾ^ꢈ

‖ꢳ₅(t)‖²= ꢹ ꢓ∫_ꢊ S(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dw(s)ꢓ

∗

(4.6)

4

2

ꢈ

ꢹ

ꢈ

2

M²L₂∫ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − δ(s)ꢄꢣ ds

≤

ꢊ

ꢈ

2

M²L₂∫ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾ds ∫_ꢊ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − δ(s)ꢄꢣ ds

ꢊ

ꢀ

M²L₂λ^ꢔ^ꢀ∫ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾(M^∗ + ꢹ‖ϕ‖_ꣂ²)e^ꢾ^τe^ꢔ^ꢾ^ꢕds

ꢈ

≤ ₂

ꢊ

^ꢀM²L₂λ^ꢔ^ꢀ(ꢛλ − a)^ꢔ^ꢀ(M^∗ + ꢹ‖ϕ‖_ꣂ²)e^ꢾ^τe^ꢔ^ꢾ^ꢈ

≤ ₂

M e^ꢔ^ꢾ^ꢈ

∗

(4.7)

≤

5

By using lemma 2.2 we get that

2

ꢈ

‖ꢳ₆(t)‖²= ꢹ ꢓ∫_ꢊ S(t − s) σ(s)dB^ꢅ(s)ꢓ

ꢹ

ꢈ

M²cH(ꢛH − 1)t²^ꢅ^ꢔ^ꢀ ∫ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾‖σ(s)‖_ꢍ²

≤

ꢏ

ds

ꢊ

ꢒ

From this inequality we can infer that

‖ꢳ₆(t)‖²≤ M²cH(ꢛH − 1)t²^ꢅ^ꢔ^ꢀe^ꢔ²^λ^′^ꢈ ∫ e²^γ^ꢕ‖σ(s)‖²_ℒ

ꢈ

ꢹ

ꢏ

(4.8)

ꢊ

ꢒ

′

where λ = λ⋀γ.Indeed, if λ < ꢽ, then λ = λ and we have

∞

‖ꢳ₆(t)‖²≤ M²cH(ꢛH − 1)t²^ꢅ^ꢔ^ꢀe^ꢔ²^λ^ꢈ ∫ e²^λ^ꢕ‖σ(s)‖_ꢍ²

ꢹ

ꢏ

ds

ꢊ

ꢒ

M²cH(ꢛH − 1)t²^ꢅ^ꢔ^ꢀe^ꢔ²^λ^′^ꢈ ∫ e²^γ^ꢕ‖σ(s)‖_ℒ²

∞

≤

ꢏ

ds

ꢊ

ꢒ

′

If γ < ꣀ then λ = γ and we have

‖ꢳ₆(t)‖²≤ M²cH(ꢛH − 1)t²^ꢅ^ꢔ^ꢀe^ꢔ²^λ^′^ꢈ ∫ e²^γ^ꢕ‖σ(s)‖²_ℒ

ꢈ

ꢹ

ꢏ

ds

ꢊ

ꢒ

since suꢳ_ꢈ_ꢉ_ꢊ ꢁt²^ꢅ^ꢔ^ꢀ

e

^ꢔ^λ^′^ꢈꢂ < ∞, this together with (4.8) we have

∗

‖ꢳ₆(t)‖²≤ M e^ꢔ^ꢾ^ꢈ

ꢹ

(4.9)

6

Combining (4.3)-(4.7), (4.9) we see that there exists M^̅^∗ > 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)‖² ≤ ꢺ ∑ⁿ_ꢷ_ꢟ_ꢀ I_ꢷ

ꢹ

Since x(t) = y(t) = φ(t), t∈ (−τ, 0ꢆ, this implies that

2

ꢣxꢃt − r(t)ꢄ − yꢃt − r(t)ꢄꢣ ≤ s_ꢈu_ꢉ_ꢊꢳꢹ‖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

L₃ꢣ(−A)^ꢔ^βꢣ ꢹꢣxꢃt − r(t)ꢄ − yꢃt − r(t)ꢄꢣ

≤

2

L₃ꢣ(−A)^ꢔ^βꢣ s_ꢈu_ꢉ_ꢊꢳꢹ‖x(t) − y(t)‖²

2

I₂= ꢹ ꢓ∫_ꢊ AS(t − s) [g ꢁs, xꢃs − r(s)ꢄꢂ − g ꢁs, yꢃs − r(s)ꢄꢂ] dsꢓ

ꢈ

ꢈ ꢈ

2

_ꢔ_βL₃∫ (t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ds ∫_ꢊ(t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − r(s)ꢄ − yꢃs − r(s)ꢄꢣ ds

M

ꢀ

ꢊ

≤

Γ(β)

_λβ

ꢈ

2

M

_ꢀ²_ꢔ_βL₃

∫ (t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − r(s)ꢄ − yꢃs − r(s)ꢄꢣ ds

≤

ꢊ

Γ^ꢎ(β)

_λꢎβ

M

_ꢀ²_ꢔ_βL₃

s_ꢈu_ꢉ_ꢊꢳꢹ‖x(t) − y(t)‖²

≤

By assumption (i) iꣅ (H₂)

ꢈ

I₃= ꢹ ꢓ∫_ꢊ S(t − s) [f ꢁs, xꢃs − ρ(s)ꢄꢂ − f ꢁs, yꢃs − ρ(s)ꢄꢂ] dsꢓ

ꢈ

2

M²L_ꢀ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ds ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − ρ(s)ꢄ − yꢃs − ρ(s)ꢄꢣ ds

≤

ꢊ

2

M²L_ꢀλ^ꢔ^ꢀ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − ρ(s)ꢄ − yꢃs − ρ(s)ꢄꢣ ds

ꢈ

≤

ꢊ

M²L_ꢀλ^ꢔ²s_ꢈu_ꢉ_ꢊꢳꢹ‖x(t) − y(t)‖²

By assumption(iii) iꣅ (H₂) , we have

ꢈ

I₄= ꢹ ꢓ∫_ꢊ S(t − s) [h ꢁs, xꢃs − δ(s)ꢄꢂ − h ꢁs, xꢃs − δ(s)ꢄꢂ] dw(s)ꢓ

ꢈ

2

M²L₂∫ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾ds ∫ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − δ(s)ꢄ − yꢃs − δ(s)ꢄꢣ ds

≤

ꢊ

ꢈ

2

M²L₂(ꢛλ)^ꢔ^ꢀ∫ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − δ(s)ꢄ − yꢃs − δ(s)ꢄꢣ ds

≤

ꢊ

M²L₂(ꢛλ)^ꢔ²s_ꢈu_ꢉ_ꢊꢳꢹ‖x(t) − y(t)‖²

≤

Thusꢹ‖(Φx)(t) − (Φy)(t)‖²

_L₃Γ²(β)

2

ꢺ ꢨL₃ꢣ(−A)^ꢔ^βꢣ + M_ꢀ²_ꢔ_β

+ M²L_ꢀλ^ꢔ²+M²L₂(ꢛλ)^ꢔ²ꢪ s_ꢈu_ꢉ_ꢊꢳꢹ‖x(t) − y(t)‖²

≤

λ²^β

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_ꢈ_→_∞ ꢹ‖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

lim_ꢈ_→_∞ꢹ‖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

ꢈ

−

∫ AS(t − s)g ꢁs, xꢃs − r(s)ꢄꢂ ds + ∫ S(t − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ ds

ꢊ

ꢈ

∫ S(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dw(s) + ∫ S(t − s) σ(s)dB^ꢅ(s)

+

ꢊ

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 thatlim_ꢈ_→_∞ꢹ‖Ψx(t)‖ = 0 for all

′

x ∈ ꢿ .

By the definition of ꢿ^′, assumption (H₇)and the fact t − r(t) → ∞, t → ∞. We get

lim ꢹ_ꢈ_→_∞‖P_ꢀ(t)‖² = l_ꢈi_→m_∞ꢹ‖P₂(t)‖² = l_ꢈi_→m_∞ꢹ‖P₆(t)‖ = 0

2

ꢈ

We further have ꢹ‖ꢳ₃(t)‖² ≤ ꢹ ꢓ∫_ꢊ AS(t − s)g ꢁs, xꢃs − r(s)ꢄꢂ dsꢓ

ꢈ

2

_ꢔ_βL₃∫ (t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ds ∫ (t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − r(s)ꢄꢣ ds

≤

M

ꢀ

ꢊ

_ꢔ_βL₃Γ(β)λ^ꢔ^β∫ (t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − r(s)ꢄꢣ ds

ꢈ

2

ꢀ

ꢊ

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)λ^ꢔ^β ∫ (t − s)^β^ꢔ^ꢀ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − r(s)ꢄꢣ ds + M L₃Γ²(β)λ^ꢔ²^β

ꢕ

ꢑ

2

2 2

ϵ

ꢀꢔβ ꢀꢔβ

ꢊ

ꢹ

which proves that

lim_ꢈ_→_∞ꢹ‖P₃(t)‖² ≤ M L₃Γ²(β)λ^ꢔ²^β,∀ϵ > 0 and hence, lim_ꢈ_→_∞ ꢹ‖P₃(t)‖² = 0

2

ꢀꢔβ

2

ꢈ

‖ꢳ₄(t)‖²= ꢹ ꢓ∫_ꢊ S(t − s) f ꢁs, xꢃs − ρ(s)ꢄꢂ dsꢓ

ꢹ

ꢈ ꢈ

2

M²L_ꢀ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ds ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − ρ(s)ꢄꢣ ds

ꢊ ꢊ

≤

M²L_ꢀλ^ꢔ^ꢀ∫ e^ꢔ^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − ρ(s)ꢄꢣ ds

ꢈ

2

ꢊ

≤

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 ≤ M²L_ꢀλ^ꢔ^ꢀ

ꢕ

ꢑ

ꢔλ(ꢈꢔꢕ)

ꢹꢣxꢃs − ρ(s)ꢄꢣ ds + M²L_ꢀλ^ꢔ²

∫

e

ϵ

ꢊ

which proves that lim_ꢈ_→_∞ E‖P₄(t)‖² ≤ M²L_ꢀλ^ꢔ²ϵ,∀ϵ > 0 and hence,im_ꢈ_→_∞ E‖P₄(t)‖² = 0

2

ꢈ

‖ꢳ₅(t)‖²= ꢹ ꢓ∫_ꢊ S(t − s) h ꢁs, xꢃs − δ(s)ꢄꢂ dw(s)ꢓ

ꢹ

ꢈ

2

M²L₂∫ e^ꢔ²^λ⁽^ꢈ^ꢔ^ꢕ⁾ꢹꢣxꢃs − δ(s)ꢄꢣ ds

ꢊ

≤

2

For any x ∈ ꢿ^′and ε > 0 there exists as_ꢀ > 0 such that ꢹꢣxꢃs − r(s)ꢄꢣ < ꣇for all s ≥ s_ꢀ.

ꢕ

ꢑ

2

Thus we obtain ꢹ‖ꢳ₅(t)‖² ≤ M²L₂ ∫

ꢔ2λ(ꢈꢔꢕ)

ꢹꢣxꢃs − δ(s)ꢄꢣ ds + M²L₂(ꢛλ)^ꢔ^ꢀϵ

e

ꢊ

2

which proves that lim_ꢈ_→_∞ ꢹ‖P₅(t)‖ ≤ M²L₂(ꢛλ)^ꢔ^ꢀϵ, ∀ϵ > 0 and hence, lim_ꢈ_→_∞ ꢹ‖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

_꣈^꣈_tꢰ

ꢅ

z(t, ξ) + α_ꢀGꢃt, z(t − τ, ξ)ꢄꢱ =_꣈_ξ₂z(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) = α_ꢀGꢃt, 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_ꢟ_ꢀ A²〈x, e_n〉 e_n, x ∈ D(A) , where e_n(t) = √_π siꣅ(ꣅt) , ꣅ = 1,ꢛ,꣉, … is the orthogonal set of eigenvectors.

∗

A is the infinitesimal generator of an analytic semigroup ꢃS(t)ꢄ_ꢈ_ꢉ_ꢊin X.

ꢎ_ꢈ

S(t)x = ∑^∞_n_ꢟ_ꢀ e^ꢔⁿ 〈x, e_n〉 e_n, For all x ∈ X and for every t > 0 .

ꢎ

ꢔ

π

ꢈ

Furthermore‖S(t)‖ ≤ e

, t ≥ 0.

The bounded linear operator(−A)³^꣊⁴ is well defined and is given by

(

−A)³^꣊⁴x = ∑^∞_n_ꢟ_ꢀ(ꣅ)³^꣊²〈x, e_n〉 e_n

with domain Dꢃ(−A)³^꣊⁴ꢄ = ꢙx ∈ X: ∑^∞_n_ꢟ_ꢀ e^ꢔⁿ^ꢎ^ꢈ〈x, e_n〉 e_n ∈ Xꢚ . Furtheremore ꢣ(−A)³^꣊⁴ꢣ = 1 and ꢣ(−A)³^꣊⁴ꢣ ≤

ꢀ

∞

ꢀ

t ^ꢔ⁽^ꢀ^꣊⁴⁾‖S(t)‖dt <_πꢼ_⁄_ꢎ.Thus (H_ꢀ) holds with M = 1, λ = π²,(H₂) holds with L_ꢀ = α₂² ,(H₃) and (H₄) holds

∫

Γ(3꣊4)

ꢊ

2

ꢀ

with β = ꣉꣊ꢺ,L₂ = ꢣ(−A)³^꣊⁴ꢣ α_ꢀ² = α_ꢀ² 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.

αꢑ^ꢎ

ꢫꢑ^ꢎ

4

3

αꢑ^ꢎ

αꢎ^ꢎ

αꢼ^ꢎ

ꢀ

Γ²ꢁ₄ꢂ_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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