# Existence and Uniqueness of Mild Solutions to Some Neutral Stochastic Partial Functional Integrodifferential Equations with Non-Lipschitz Coefficients

Authors: Mamadou Abdoul Diop; Khalil Ezzinbi; Modou Lo
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

This paper presents the result on existence, uniqueness of mild solutions to neutral stochastic partial functional integrodifferential equations under the Carathéodory-type conditions on the coefficients. The results are obtained by using the method of successive approximation. An example is provided to illustrate the results of this work.


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## Body

## 1. Introduction

In this paper, our objective is to study the existence of mild solution for the following neutral stochastic partial functional integrodifferential equation in a real separable Hilbert space:(1.1)d[u(t)-H(t,ut)]=A[u(t)-H(t,ut)]dt+[∫0tB(t-s)[u(s)-H(s,us)]ds+F(t,ut)]dt+G(t,ut)dw(t),fort∈[0,T],u0(⋅)=φ∈CF0b([-r,0];H),r>0,
where ut(θ)=u(t+θ)forθ∈[-r,0]. The mappings H:ℝ+×Cr→ℍ, F:ℝ+×Cr→ℍ, and G:ℝ+×Cr→ℒ(𝕂,ℍ) are Borel measurable.Neutral stochastic partial functional differential equations have atracted great interest due to their applications in characterizing many problems in physics, biology, mechanics and so on.Qualitative properties such as existence, uniqueness, and stability for various stochastic differential and integrodifferential systems have been extensively studied by many researchers; see, for instance, [1–6] and the references therein. The problem of the existence and uniqueness of solution for neutral stochastic partial functional differential equations in the case where the coefficients do not satisfy the global Lipschitz condition was investigated by Taniguchi [7], Turo [8], Cao et al. [9], and recently by Jiang and Shen [10].Stimulated by the above works, we consider the existence and uniqueness of mild solutions to (1.1) under some carathéodory-type conditions to the Hilbert space with the Lipschitz condition in [11] and the non-Lipschitz condition in [12] being regarded as special cases.Our main results concerning (1.1) rely essentially on techniques using strongly continuous family of operators{R(t),t≥0}, defined on the Hilbert spaceℍand called the resolvent (precise definition will be given below).The contents of the paper are as follows. In Section2, we summarize several important working tools on the Wiener process and deterministic integrodifferential equations that will be used to develop our results. In Section 3, we study the existence of mild solutions for the neutral system (1.1) using the theory of resolvent operators and by means of successive approximation (the Picard iteration). In Section 4, we provide an example to illustrate our main approach.


## 2. Wiener Process and Deterministic Integrodifferential Equations

### 2.1. Wiener Process

Throughout this paper, let(Ω,ℱ,{ℱt}t≥0,ℙ) be a complete probability space with a normal filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is increasing and right-continuous while ℱ0 contains all ℙ-null sets). Moreover, let ℍ and 𝕂be two real separable Hilbert spaces; we denote by〈·,·〉ℍ, 〈·,·〉𝕂 their inner products and by ∥·∥ℍ, ∥·∥𝕂 their vectors norms, respectively. We denote by ℒ(ℍ,𝕂) the space of all bounded linear operator from ℍ into 𝕂, equipped with the usual operator norm ∥·∥. In this paper, we always use the same symbol ∥·∥ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let r>0 and Cr=C([-r,0];ℍ) denote the family of all continuous ℍ-valued functions ξ from [-r,0] to ℍ with norm ∥ξ∥C=supt∈[-r,0]∥ξ(t)∥ℍ. Let Cℱ0b([-r,0];ℍ) be the family of all almost surely bounded, ℱ0-measurable,C([-r,0];ℍ)-valued random variables.Let{w(t):t≥0} denote a 𝕂-valued Wiener process defined on the probability space (Ω,ℱ,{ℱt}t≥0,ℙ) with covariance operator Q; that is, E〈w(t),x〉𝕂〈w(s),y〉𝕂=(t∧s)〈Qx,y〉𝕂, for all x,y∈𝕂, where Q is a positive, self-adjoint, trace class operator on 𝕂. In particular, we denote w(t) a 𝕂-valuedQ-Wiener process with respect to {ℱt}t≥0. To define stochastic integrals with respect to the Q-Wiener processw(t), we introduce the subspace 𝕂0=Q1/2𝕂of𝕂 endowed with the inner product 〈u,v〉𝕂0=〈Q-1/2u,Q-1/2v〉𝕂 as a Hilbert space. We assume that there exists a complete orthonormal system {ei} in 𝕂, a bounded sequence of nonnegative real numbers λi such that Qei=λiei, i=1,2,…, and a sequence {βi(t)}i>1 of independent standard Brownian motions such that(2.1)w(t)=∑i=1+∞λiβi(t)ei,t≥0,
and ℱt=ℱtw, where ℱtw is the σ-algebra generated by {w(s):0≤s≤t}. Let ℒ20=ℒ2(𝕂0,ℍ) be the space of all Hilbert-Schmidt operators from 𝕂0toℍ. It turns out to be a separable Hilbert space equipped with the norm ∥v∥ℒ20=tr((vQ1/2)(vQ1/2)*) for any v∈ℒ20. Obviously, for any bounded operator v∈ℒ20, this norm reduces to ∥v∥ℒ202=tr(vQv*).


### 2.2. Partial Integrodifferential Equations

In this section, we recall some fundamental results needed to establish our results. Regarding the theory of resolvent operators, we refer the reader to [13, 14]. Throughout the paper, X is a Banach space AandB(t)are closed linear operators on X. Y represents the Banach spaceD(A)equipped with the graph norm defined by
(2.2)|y|Y:=|Ay|+|y|fory∈Y.
The notationsC([0,+∞);Y), B(Y,X)stand for the space of all continuous functions from[0,+∞)intoY, the set of all bounded linear operators fromYintoX, respectively. We consider the following Cauchy problem:
(2.3)v′(t)=Av(t)+∫0tB(t-s)v(s)ds,fort≥0,v(0)=v0∈X.Definition 2.1 (see [13]).
A resolvent operator for (2.3) is a bounded linear operator-valued function R(t)∈ℒ(X) for t≥0, having the following properties. (i)
R(0)=I and |R(t)|≤Neβt for some constants N and β.(ii)
For eachx∈X, R(t)x is strongly continuous for t≥0.(iii)
R(t)∈ℒ(Y) for t≥0. For x∈Y, R(·)x∈C1([0,+∞);X)∩C([0,+∞);Y) and
(2.4)R′(t)x=AR(t)x+∫0tB(t-s)R(s)xds=R(t)Ax+∫0tR(t-s)B(s)xds,fort≥0.In what follows we suppose the following assumptions.(H1)
A is the infinitesimal generator of a strongly continuous semigroup onX.(H2)
For allt≥0, B(t) is closed linear operator from D(A)to X and B(t)∈B(Y,X). For anyy∈Y, the mapt→B(t)yis bounded and differentiable, and the derivativet→B′(t)y is bounded uniformly continuous on ℝ+.The resolvent operator plays an important role to study the existence of solutions and to give a variation of constants formula for nonlinear systems. We need to know when the linear system 2.1 has a resolvent operator. For more details on resolvent operators, we refer to [13, 14]. The following theorem gives a satisfactory answer to this problem, and it will be used in this work to develop our main results.Theorem 2.2 (see [13]).
Assume that (H1)-(H2) hold. Then there exists a unique resolvent operator of the Cauchy problem (2.3).In the following, we give some results for the existence of solutions for the following integrodifferential equation:(2.5)v′(t)=Av(t)+∫0tB(t-s)v(s)ds+q(t),fort≥0v(0)=v0∈X,
whereq:[0,+∞[→X is a continuous function.Definition 2.3 (see [13]).
A continuous functionv:[0,+∞)→X is said to be a strict solution of (2.5) if(i)
v∈C1([0,+∞);X)∩C([0,+∞);Y),(ii)
v satisfies (2.5), for t≥0.Remark 2.4.
From this definition, we deduce thatv(t)∈D(A), the functionB(t-s)v(s)is integrable, for allt≥0, ands∈[0,t].Theorem 2.5 (see [13]).
Assume that (H1)-(H2) hold. If v is a strict solution of (2.5), then
(2.6)v(t)=R(t)v0+∫0tR(t-s)q(s)ds,fort≥0.Accordingly, we make the following definition.Definition 2.6 (see [13]).
Forv0∈X, a function v:[0,+∞)→X is called a mild solution of (2.5) if v satisfies (2.6).The next theorem provides sufficient conditions for the regularity of solutions of (2.5).Theorem 2.7 (see [13]).
Letq∈C1([0,+∞);X) andv be defined by (2.6). If v0∈D(A), thenv is a strict solution of (2.5).


## 2.1. Wiener Process

Throughout this paper, let(Ω,ℱ,{ℱt}t≥0,ℙ) be a complete probability space with a normal filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is increasing and right-continuous while ℱ0 contains all ℙ-null sets). Moreover, let ℍ and 𝕂be two real separable Hilbert spaces; we denote by〈·,·〉ℍ, 〈·,·〉𝕂 their inner products and by ∥·∥ℍ, ∥·∥𝕂 their vectors norms, respectively. We denote by ℒ(ℍ,𝕂) the space of all bounded linear operator from ℍ into 𝕂, equipped with the usual operator norm ∥·∥. In this paper, we always use the same symbol ∥·∥ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let r>0 and Cr=C([-r,0];ℍ) denote the family of all continuous ℍ-valued functions ξ from [-r,0] to ℍ with norm ∥ξ∥C=supt∈[-r,0]∥ξ(t)∥ℍ. Let Cℱ0b([-r,0];ℍ) be the family of all almost surely bounded, ℱ0-measurable,C([-r,0];ℍ)-valued random variables.Let{w(t):t≥0} denote a 𝕂-valued Wiener process defined on the probability space (Ω,ℱ,{ℱt}t≥0,ℙ) with covariance operator Q; that is, E〈w(t),x〉𝕂〈w(s),y〉𝕂=(t∧s)〈Qx,y〉𝕂, for all x,y∈𝕂, where Q is a positive, self-adjoint, trace class operator on 𝕂. In particular, we denote w(t) a 𝕂-valuedQ-Wiener process with respect to {ℱt}t≥0. To define stochastic integrals with respect to the Q-Wiener processw(t), we introduce the subspace 𝕂0=Q1/2𝕂of𝕂 endowed with the inner product 〈u,v〉𝕂0=〈Q-1/2u,Q-1/2v〉𝕂 as a Hilbert space. We assume that there exists a complete orthonormal system {ei} in 𝕂, a bounded sequence of nonnegative real numbers λi such that Qei=λiei, i=1,2,…, and a sequence {βi(t)}i>1 of independent standard Brownian motions such that(2.1)w(t)=∑i=1+∞λiβi(t)ei,t≥0,
and ℱt=ℱtw, where ℱtw is the σ-algebra generated by {w(s):0≤s≤t}. Let ℒ20=ℒ2(𝕂0,ℍ) be the space of all Hilbert-Schmidt operators from 𝕂0toℍ. It turns out to be a separable Hilbert space equipped with the norm ∥v∥ℒ20=tr((vQ1/2)(vQ1/2)*) for any v∈ℒ20. Obviously, for any bounded operator v∈ℒ20, this norm reduces to ∥v∥ℒ202=tr(vQv*).


## 2.2. Partial Integrodifferential Equations

In this section, we recall some fundamental results needed to establish our results. Regarding the theory of resolvent operators, we refer the reader to [13, 14]. Throughout the paper, X is a Banach space AandB(t)are closed linear operators on X. Y represents the Banach spaceD(A)equipped with the graph norm defined by
(2.2)|y|Y:=|Ay|+|y|fory∈Y.
The notationsC([0,+∞);Y), B(Y,X)stand for the space of all continuous functions from[0,+∞)intoY, the set of all bounded linear operators fromYintoX, respectively. We consider the following Cauchy problem:
(2.3)v′(t)=Av(t)+∫0tB(t-s)v(s)ds,fort≥0,v(0)=v0∈X.Definition 2.1 (see [13]).
A resolvent operator for (2.3) is a bounded linear operator-valued function R(t)∈ℒ(X) for t≥0, having the following properties. (i)
R(0)=I and |R(t)|≤Neβt for some constants N and β.(ii)
For eachx∈X, R(t)x is strongly continuous for t≥0.(iii)
R(t)∈ℒ(Y) for t≥0. For x∈Y, R(·)x∈C1([0,+∞);X)∩C([0,+∞);Y) and
(2.4)R′(t)x=AR(t)x+∫0tB(t-s)R(s)xds=R(t)Ax+∫0tR(t-s)B(s)xds,fort≥0.In what follows we suppose the following assumptions.(H1)
A is the infinitesimal generator of a strongly continuous semigroup onX.(H2)
For allt≥0, B(t) is closed linear operator from D(A)to X and B(t)∈B(Y,X). For anyy∈Y, the mapt→B(t)yis bounded and differentiable, and the derivativet→B′(t)y is bounded uniformly continuous on ℝ+.The resolvent operator plays an important role to study the existence of solutions and to give a variation of constants formula for nonlinear systems. We need to know when the linear system 2.1 has a resolvent operator. For more details on resolvent operators, we refer to [13, 14]. The following theorem gives a satisfactory answer to this problem, and it will be used in this work to develop our main results.Theorem 2.2 (see [13]).
Assume that (H1)-(H2) hold. Then there exists a unique resolvent operator of the Cauchy problem (2.3).In the following, we give some results for the existence of solutions for the following integrodifferential equation:(2.5)v′(t)=Av(t)+∫0tB(t-s)v(s)ds+q(t),fort≥0v(0)=v0∈X,
whereq:[0,+∞[→X is a continuous function.Definition 2.3 (see [13]).
A continuous functionv:[0,+∞)→X is said to be a strict solution of (2.5) if(i)
v∈C1([0,+∞);X)∩C([0,+∞);Y),(ii)
v satisfies (2.5), for t≥0.Remark 2.4.
From this definition, we deduce thatv(t)∈D(A), the functionB(t-s)v(s)is integrable, for allt≥0, ands∈[0,t].Theorem 2.5 (see [13]).
Assume that (H1)-(H2) hold. If v is a strict solution of (2.5), then
(2.6)v(t)=R(t)v0+∫0tR(t-s)q(s)ds,fort≥0.Accordingly, we make the following definition.Definition 2.6 (see [13]).
Forv0∈X, a function v:[0,+∞)→X is called a mild solution of (2.5) if v satisfies (2.6).The next theorem provides sufficient conditions for the regularity of solutions of (2.5).Theorem 2.7 (see [13]).
Letq∈C1([0,+∞);X) andv be defined by (2.6). If v0∈D(A), thenv is a strict solution of (2.5).


## 3. Existence of Mild Solutions of (1.1)

Definition 3.1.
A process{u(t),t∈[0,T]}, 0≤T<+∞, is called a mild solution of (1.1) if(i)
u(t) is ℱt-adapted,t≥0with∫0T∥u∥ℍ2dt<+∞ a.s.;(ii)
u(t)∈ℍhas continuous paths on t∈[0,T] a.s., and, for each t∈[0,T], u(t) satisfies the integral equation
(3.1)u(t)-H(t,ut)=R(t)[φ(0)-H(0,φ)]+∫0tR(t-s)F(s,us)ds+∫0tR(t-s)G(s,us)dw(s),
for anyu0(·)=φ∈Cℱ0b([-r,0];ℍ).In the rest of this paper we replaceX by ℍ in (H1) and (H2).To guarantee the existence and uniqueness of a mild solution to (1.1), the following much weaker conditions, instead of the non-Lipschitz condition, are used.(H3)
(a)
There exists a functionL(t,y):ℝ+×ℝ+→ℝ+ such that L(t,y) is locally integrable in t≥0 and is continuous monotone nondecreasing and concave in y for any fixedt∈[0,T]. Moreover, for any fixed t∈[0,T] and ζ∈ℍ, the following inequality is satisfied:
(3.2)‖F(t,ζ)‖H2+‖G(t,ζ)‖L202≤L(t,‖ζ‖C2),t∈[0,T].(b)
For any constantK*>0, the differential equation
(3.3)dxdt=K*L(t,x),t∈[0,T],
has a global solution for any initial valuex0.(H4)
(a)
There exists a functionλ(t,y):ℝ+×ℝ+→ℝ+ such that λ(t,y) is locally integrable in t≥0 for any fixed y≥0 and is continuous monotone nondecreasing and concave in y for any fixed t∈[0,T]. G(t,0)=0 for any fixedt∈[0,T]. Moreover, for any fixed t∈[0,T] and ζ,η∈ℍ, the following inequality is satisfied:
(3.4)‖F(t,ζ)-F(t,η)‖H2+‖G(t,ζ)-G(t,η)‖L202≤λ(t,‖ζ-η‖C2),t∈[0,T].(b)
For any constantK¯>0, if a nonnegative function z(t) satisfies that
(3.5)z(t)≤K¯∫0tλ(s,z(s))ds,t∈[0,T],
then z(t)=0 holds for any t∈[0,T].(H5)
The mappingH(t,x) satisfies that there exists a positiveK such that, for any ζ,η∈ℍ and t≥0,
(3.6)‖H(t,ζ)-H(t,η)‖H≤K‖ζ-η‖C.Remark 3.2.
Letλ(t,u)=β(t)λ¯(u), t∈[0,T], whereβ(t)≥0 is locally integrable andλ¯(u)is a concave nondecreasing function fromℝ+toℝ+such thatλ¯(0)=0,λ¯(u)>0 for u>0and∫0+(1/λ¯(u))du=∞. Then, by the comparison theorem of differential equations we know that assumption (H4-(b)) holds.
Now let us give some concrete examples of the functionλ¯(·). Letξ>0and letδ∈(0,1)be sufficient small. Define
(3.7)λ¯1(u)=ξu,u≥0,λ¯2(u)={ulog⁡⁡(u-1),0≤u≤δ,δlog⁡⁡(δ-1)+λ¯2′(δ-)(u-δ),u>δ,
whereλ¯2′denotes the derivative of functionλ¯2. They are all concave nondecreasing functions satisfying∫0+(1/λ¯i(u))du=∞(i=1,2).In the following, we establish the existence and uniqueness theorem of the mild solution.Theorem 3.3.
If (H1)–(H5) hold, then there exists a unique mild solution to (1.1), provided that
(3.8)K<1.Proof.
To obtain the existence of the solution to (1.1), we consider the Picard iteration which is defined by
(3.9)u0(t)=R(t)φ(0),t∈[0,T],u0(t)=φ(t),t∈[-r,0],
andunforn≥1is defined by
(3.10)un(t)=φ(t)fort∈[-r,0],(3.11)un(t)-H(t,utn)=R(t)[φ(0)-H(0,φ)]+∫0tR(t-s)F(s,usn-1)ds+∫0tR(t-s)G(s,usn-1)dw(s),t∈[0,T].
The proof is divided into the following three steps.Step 1.
We claim that the sequence{un(t),n≥0} is bounded. From (3.11), for0≤t≤T,
(3.12)Esup⁡0≤s≤t‖un(s)-H(s,usn(s))‖H2≤3Esup⁡0≤s≤t‖R(t)[φ(0)-H(0,φ)]‖H2+3Esup⁡0≤s≤t‖∫0tR(t-s)F(s,usn-1)ds‖H2+3Esup⁡0≤s≤t‖∫0tR(t-s)G(s,usn-1)dw(s)‖H2=:3(I1+I2+I3).
By (H5),
(3.13)I1≤M(1+K)2E‖φ‖C2,
where M=sup0≤t≤T|R(t)|2.
Note that(3.14)E(sup⁡0≤s≤t⁡‖usn-1‖C2)≤E(sup⁡0≤s≤t⁡‖un-1(s)‖H2)+E‖φ‖C2.
By (H3) and the Jensen inequality, we obtain
(3.15)I2≤MT∫0tL(τ,E(sup⁡0≤l≤τ‖un-1(l)‖H2+‖φ‖C2))dτ.
By (H3), Liu and Hu [15, Theorem 1.2.5, page 14], and the Jensen inequality, there exists a positive constantC1such that
(3.16)I3≤C1∫0tEG(τ,‖un-1(τ)‖H2)dτ≤C1∫0tEL(τ,‖un-1(τ)‖H2)dτ≤C1∫0tL(τ,E‖un-1(τ)‖H2)dτ.
SinceL is continuous monotone nondecreasing iny, we obtain
(3.17)I3≤C1∫0tL(τ,E(sup⁡0≤l≤τ‖un-1(l)‖H2+‖φ‖C2))dτ.
Recall that, fora,b∈X, ɛ∈(0,1), ∥a-b∥X2≤1/(1-ɛ)∥a∥X2+(1/ɛ)∥b∥X2. Hence, substituting (3.13)−(3.16) into (3.12) yields
(3.18)E(sup⁡0≤s≤t‖un(s)‖H2+‖φ‖C2)≤11-KEsup⁡0≤s≤t‖un(s)-H(s,usn)‖H2+1KEsup⁡0≤s≤t‖H(s,usn)‖H2+E‖φ‖C2≤3M(1+K)21-KE‖φ‖C2+E‖φ‖C2+KE(sup⁡0≤s≤t‖un(s)‖H2+‖φ‖C2)+3(MT+C1)1-K∫0tL(τ,E(sup⁡0≤l≤τ‖un-1(l)‖H2+‖φ‖C2))dτ≤3M(1+K)2(1-K)2E‖φ‖C2+11-KE‖φ‖C2+3(MT+C1)(1-K)2∫0tL(τ,E(sup⁡0≤l≤τ‖un-1(l)‖H2+‖φ‖C2))dτ.
Assumption (H3-(b)) indicates that there is a solutionxtthat satisfies
(3.19)xt=C2E‖φ‖C2+C3∫0tL(r,xr)dr,
where C2=3M(1+K)2/(1-K)2+1/(1-K), C3=3(MT+C1)/(1-K)2.
SinceE∥φ∥C2<∞, from (3.17), we have E(sup0≤s≤t∥usn∥C2)≤xt≤xT<∞, which shows the boundedness of the {un(t),n≥0}.Step 2.
We claim that{un(t),n≥0}is a Cauchy sequence. For all n,m≥0and t∈[0,T], from (3.11), (H4), and Step 1, we have
(3.20)Esup⁡0≤s≤t‖un+1(s)-H(s,usn+1(s))-um+1(s)+H(s,usm+1(s))‖H2≤2Esup⁡0≤s≤t‖∫0sR(t-s)[F(s,uτn+1)-F(s,uτm+1)]dτ‖H2+2Esup⁡0≤s≤t‖∫0tR(t-s)[G(τ,usn)-G(τ,usm)]dw(s)‖H2≤2(M+C4)∫0sL(τ,E(sup⁡0≤l≤τ‖un(l)-um(l)‖H2))dτ,
where C4is a generic constant used by Liu and Hu [15,Theorem 1.2.6, page 14]. Therefore,
(3.21)Esup⁡0≤s≤t‖u2+1(s)-um+1(s)‖H2≤11-KEsup⁡0≤s≤t‖un+1(s)-H(s,usn+1(s))-um+1(s)+H(s,usm+1(s))‖H2+1KEsup⁡0≤s≤t‖H(s,usn+1)-H(s,usm+1)‖H2≤2M(1+C4)1-K∫0sL(τ,E(sup⁡0≤l≤τ‖un(l)-um(l)‖H2))dτ+KE(sup⁡0≤s≤t‖un+1(s)-um+1(s)‖H2).
Let
(3.22)z(t)=lim⁡n,m→+∞⁡sup⁡⁡(Esup⁡0≤s≤t⁡‖un(s)-um(s)‖H2).
By assumption (H4-(b)) and the Fatou lemma, we have
(3.23)z(t)≤C5∫0tλ(s,z(s))ds,
whereC5=2M(1+C4)/(1-K)2. By assumption (H4-(b)) we obtainz(t)=0. This shows that {un(t),n≥0}is Cauchy.Step 3.
We claim the existence and uniqueness of the solution to (1.1). The Borel-Cantelli lemma shows that, as n→∞, un(t)→u(t) holds uniformly for0≤t≤T. Hence, taking limits on both sides of (3.11), we obtain thatu(t) is a solution to (1.1). This shows the existence. The uniqueness of the solution could be obtained by the same procedure as Step 2. The proof is complete.Remark 3.4.
Ifλ(t,u)=K1ufor some constantK1, then condition (H4) implies a global Lipschitz condition.


## 4. Application

We conclude this work with an example of the form(4.1)∂∂t[x(t,ξ)-∫-r0h(t,x(t+θ,ξ))dθ]=∂2∂ξ2[x(t,ξ)-∫-r0h(t,x(t+θ,ξ))dθ]+∫0tb(t-s)∂2∂ξ2[x(s,ξ)-∫-r0h(t,x(t+θ,ξ))dθ]ds+∫-r0f(t,x(t+θ,ξ))dθ+g(t,x(t+θ,ξ))dw(t),fort≥0,ξ∈[0,π],x(t,0)-∫-r0h(t,x(t+θ,0))dθ=0,fort≥0,x(t,π)-∫-r0h(t,x(t+θ,π))dθ=0,fort≥0,x(θ,ξ)=x0(θ,ξ),forθ∈[-r,0],ξ∈[0,π],
wherew(t) denotes an ℝ-valued Brownian motion, h,f,g:ℝ+×ℝ→ℝare continuous functions,b:ℝ+→ℝis continuous, and x0:[-r,0]×[0,π]→ℝis a given continuous function such that x0(·)∈L2([0,π]) is ℱ0-measurable and satisfies E∥x0∥2<∞.Letℍ=L2([0,π])with the norm∥·∥ anden:=2/πsin(nx), (n=1,2,3,…) denote the completed orthonormal basis inℍ. Letw(t):=∑n=1∞λnβn(t)en(λn>0), whereβn(t)are one-dimensional standard Brownian motion mutually independent on a usual complete probability space(Ω,ℱ,{ℱt}t≥0,ℙ).DefineA:ℍ→ℍbyA=∂2/∂z2, with domainD(A)=H2([0,π])∩H01([0,π]).ThenAh=-∑n=1∞n2〈h,en〉en, h∈D(A), whereen, n=1,2,3,…, is also the orthonormal set of eigenvectors ofA. It is wellknown that A is the infinitesimal generator of a strongly continuous semigroup onℍ; thus,(H1) is true.LetB:D(A)⊂ℍ→ℍbe the operator defined by B(t)(z)=b(t)Azfort≥0and z∈D(A).We suppose that(i)
there exists a positive constantLh, 0<πLhr<1, such that
(4.2)|h(t,ζ1)-h(t,ζ2)|≤Lh|ζ1-ζ2|,fort≥0,ζ1,ζ2∈R;(ii)
there exists a constantLf, 0<πLfr<1, such that
(4.3)|f(t,ζ1)-f(t,ζ2)|2≤Lfλ(t,‖ζ1-ζ2‖C2),fort≥0,ζ1,ζ2∈R;(iii)
fort≥0andζ1,ζ2∈ℝ, |g(t,ζ1)-g(t,ζ2)|2≤λ(t,∥ζ1-ζ2∥C2).LetC=C([-r,0];ℍ) and define the operators H,F,G:ℝ+×C→ℍ by(4.4)H(t,ϕ)(ξ)=∫-r0h(t,ϕ(θ)(ξ))dθ,forξ∈[0,π],ϕ∈C,F(t,ϕ)(ξ)=∫-r0f(t,ϕ(θ)(ξ))dθ,forξ∈[0,π],ϕ∈C,G(t,ϕ)(ξ)=g(t,ϕ(θ)(ξ)).
If we put(4.5)u(t)=x(t,ξ),fort≥0,ξ∈[0,π],φ(θ)(ξ)=x0(θ,ξ),forθ∈[-r,0],ξ∈[0,π].
then (4.1) takes the following abstract form:
(4.6)d[u(t)-H(t,ut)]=A[u(t)-H(t,ut)]dt+[∫0tB(t-s)[u(s)-H(s,us)]ds+F(t,ut)]dt+G(t,ut)dw(t),fort∈[0,T]u0=φ.
Moreover, if b is bounded and C1 function such that b′ is bounded and uniformly continuous, then (H1) and (H2) are satisfied, and hence, by Theorem 2.2, (2.3) has a resolvent operator (R(t))t≥0 on ℍ. Thus, all the assumptions of Theorem 3.3 are fulfilled. Therefore, the existence of a unique mild solution of (4.1) follows.



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*Source: 748590-2012-04-05.xml*