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Existence result for fractional neutral stochastic integro-differential equations with infinite

delay

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2011 J. Phys. A: Math. Theor. 44 335201

(http://iopscience.iop.org/1751-8121/44/33/335201)

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IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL

J. Phys. A: Math. Theor. 44 (2011) 335201 (16pp) doi:10.1088/1751-8113/44/33/335201

Existence result for fractional neutral stochasticintegro-differential equations with infinite delay

Jing Cui1,2 and Litan Yan3

1 College of Information Science and Technology, Donghua University, 2999 North Renmin Rd,Songjiang, Shanghai 201620, People’s Republic of China2 Department of Mathematics, Anhui Normal University, 1 East Beijing Rd, Wuhu 241000,People’s Republic of China3 Department of Mathematics, College of Science, Donghua University, 2999 North Renmin Rd,Songjiang, Shanghai 201620, People’s Republic of China

E-mail: jcui123@126.com and litanyan@hotmail.com

Received 31 March 2011, in final form 5 June 2011Published 21 July 2011Online at stacks.iop.org/JPhysA/44/335201

AbstractThis paper is concerned with the existence of mild solutions for a class offractional neutral stochastic integro-differential equations with infinite delayin Hilbert spaces. A sufficient condition for the existence is obtained undernon–Lipschitz conditions by means of Sadovskii’s fixed point theorem. Anexample is given to illustrate the theory.

PACS numbers: 02.60.Nm, 05.10.GgMathematics Subject Classification: 34K30, 34K50, 26A33

1. Introduction

In the last two decades, fractional calculus (see, for example, Samko et al [24] andreferences therein) has attracted many physicists, mathematicians and engineers, and notablecontributions have been made to both theory and applications of fractional differentialequations. In fact, fractional differential equations are considered as an alternative modelto nonlinear differential equations. Fractional differential equations draw a great applicationin many physical phenomena such as seepage flow in porous media and in fluid dynamictraffic models. The most important advantage of using fractional differential equations inthese and other applications is their nonlocal property. This means that the next state of asystem depends not only upon its current state but also upon all of its historical states. Thisis probably the most relevant feature for making this fractional tool useful from an appliedstandpoint and interesting from a mathematical standpoint and in turn led to the sustained studyof the theory of fractional differential equations. Besides, noise or stochastic perturbation isunavoidable and omnipresent in nature as well as in man-made systems. Therefore, it is of

1751-8113/11/335201+16$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

great significance to import the stochastic effects into the investigation of fractional differentialsystems.

In this paper, we are interested in the existence of mild solutions for a class of fractionalneutral stochastic integro-differential equations with infinite delay of the form⎧⎨

⎩cDα

t [x(t) + G(t, xt )] = −Ax(t) + f (t, xt )

+∫ t

−∞ σ(t, s, xs) dW(s), t ∈ J := [0, b],x(t) = φ(t), t ∈ (−∞, 0].

(1.1)

Here, x(·) takes value in a real separable Hilbert space H with inner product (·, ·) and norm |·|H.The fractional derivative cDα , α ∈ (0, 1), is understood in the Caputo sense. −A : D(−A) ⊂H → H is the infinitesimal generator of a strongly continuous semigroup of a bounded linearoperator S(t), t � 0, on H. Let K be another separable Hilbert space with inner product (·, ·)K

and norm ‖ · ‖K. W is a given K-valued Wiener process with a finite trace nuclear covarianceoperator Q � 0 defined on a filtered complete probability space (�,F, {Ft }t�0, P ). Thehistories xt : � → Ch defined by xt = {x(t + θ), θ ∈ (−∞, 0]} belong to the phase spaceCh, which will be defined in section 2. The initial data φ = {φ(t), t ∈ (−∞, 0]} is an F0-measurable, Ch-valued random variable independent of W with finite second moments, andG : J ×Ch → H, f : J × H → H, σ : J × J × H → L0

2(K, H) are appropriate mappingsspecified later (here, L0

2(K, H) denotes the space of all Q-Hilbert–Schmidt operators from K

into H, which is going to be defined below).Recently, there has been intense interest [3, 4, 12, 13, 21, 25, 26] in the study of differential

equations of fractional order. Much of the motivation has come from mathematical physicsand various fields of science and engineering. One of the emerging branches of this study is thetheory of fractional evolution equations, say, evolution equations where the integer derivativewith respect to time is replaced by a derivative of fractional order. The increasing interest inthis class of equations is motivated both by their application to problems from viscoelasticity,heat conduction in materials with memory, electrodynamics with memory, and also becausethey can be employed to approach nonlinear conservation laws (see [8, 9, 11, 12, 23] andreferences therein).

On the other hand, neutral stochastic differential equations with infinite delay have becomeimportant in recent years as mathematical models of phenomena in both physical and socialsciences, for instance, in the theory development in Gurtin and Pipkin [7] and Nunziato [17]for the description of heat conduction in materials with fading memory. However, to the bestof our knowledge, it seems that little is known about fractional neutral stochastic differentialequations with infinite delay and the aim of this paper is to fill this gap. We refer the interestedreader to [1, 5] and references therein for fractional stochastic equations. This paper isdifferent from the previous works in which the dependence of the nonlinear map contains anintegro-differential term with infinite delay.

The paper is organized as follows. In section 2, we briefly present some basic notationsand preliminaries. In section 3, we give the mild solution and existence result of the system(1.1) by Sadovskii’s fixed point theorem. An example is given to illustrate our result. Toavoid some lengthy calculations arising from proofs of theorems, we give an appendix whichconsists of some basic estimates. These estimates are of pure analysis.

2. Preliminaries

We start with some elements of fractional calculus, the stochastic integral and semigroups oflinear operators in Hilbert spaces. Some surveys and complete literatures could be found inSamko et al [24], Da Prato et al [20] and Pazy [19]. Throughout this paper, (H, | · |H) and

2

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

(K, ‖ · ‖K) denote two real separable Hilbert spaces. We denote by L(K, H) the set of alllinear bounded operators from K into H, equipped with the usual operator norm ‖ · ‖. In thispaper, we use the symbol ‖ ·‖ to denote norms of operators regardless of the spaces potentiallyinvolved when no confusion possibly arises.

Let (�,F, {Ft }t�0, P ) be a filtered complete probability space satisfying the usualcondition, which means that the filtration is a right continuous increasing family and F0

contains all P-null sets. W = (Wt)t�0 be a Q-Wiener process defined on (�,F, {Ft }t�0, P )

with the covariance operator Q such that tr Q < ∞. We assume that there exists a completeorthonormal system {ek}k�1 in K, a bounded sequence of nonnegative real numbers λk suchthat Qek = λkek , k = 1, 2, . . ., and a sequence of independent Brownian motions {βk}k�1

such that

(W(t), e)K =∞∑

k=1

√λk(ek, e)Kβk(t), e ∈ K, t � 0.

Let L02 = L2(Q

12 K, H) be the space of all Hilbert–Schmidt operators from Q

12 K to H with

the inner product 〈ϕ, φ〉L02= tr[ϕQφ∗].

We suppose that 0 ∈ ρ(−A), the resolvent set of −A, and the semigroup S(·) is uniformlybounded. That is to say, ‖S(t)‖ � M for some constant M � 1 and every t � 0. Then, forα ∈ (0, 1], it is possible to define the fractional power operator Aα as a closed linear operatoron its domain D(Aα). Furthermore, the subspace D(Aα) is dense in H and the expression

‖x‖α = ‖Aαx‖, x ∈ D(Aα),

defines a norm on Hα := D(Aα). The following properties are well known.

Lemma 2.1 (Pazy [19]). Suppose that the preceding conditions are satisfied.

(a) If 0 < β < α � 1, then Hα ⊂ Hβ and the embedding is compact whenever the resolventoperator of A is compact.

(b) For every α ∈ (0, 1], there exists a positive constant Cα such that

‖AαS(t)‖ � Cα

tα, t > 0.

Assume that h : (−∞, 0] → (0, +∞) with l = ∫ 0−∞ h(t) dt < +∞ a continuous function.

Recall that the abstract phase space Ch is defined by

Ch ={ϕ : (−∞, 0] → H, for any a > 0, (E|ϕ(θ)|2) 1

2 is a bounded and

measurable function on [−a, 0] and∫ 0

−∞h(s) sup

s�θ�0(E|ϕ(θ)|2) 1

2 ds < +∞}.

If Ch is endowed with the norm

‖ϕ‖Ch=

∫ 0

−∞h(s) sup

s�θ�0(E|ϕ(θ)|2) 1

2 ds, ϕ ∈ Ch,

then (Ch, ‖ · ‖Ch) is a Banach space (see Li–Liu [14]).

Let us now recall some basic definitions and results of fractional calculus.

Definition 2.1. The fractional integral of order α with the lower limit 0 for a function f isdefined as

Iαf = 1

�(α)

∫ t

0

f (s)

(t − s)1−αds, t > 0, α > 0,

provided the right-hand side is pointwise defined on [0,∞), where �(·) is the gamma function.

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J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

Definition 2.2. The Caputo derivative of order α with the lower limit 0 for a function f canbe written as

cDαf (t) = 1

�(n − α)

∫ t

0

f (n)(s)

(t − s)α+1−nds = I n−αf (n)(t), t > 0, 0 � n − 1 < α < n.

If f is an abstract function with values in H, then the integrals appearing in the abovedefinitions are taken in Bochner’s sense (see [18]).

At the end of this section, we recall the fixed point theorem of Sadovskii [22] which isused to establish the existence of the mild solution to the system (1.1).

Lemma 2.2. Let � be a condensing operator on a Banach space H, that is, � is continuousand takes bounded sets into bounded sets, and μ(�(B)) � μ(B) for every bounded set B ofH with μ(B) > 0. If �(N) ⊂ N for a convex, closed and bounded set N of H, then � has afixed point in H (where μ(·) denotes Kuratowski’s measure of noncompactness).

3. The mild solution and existence

In this section, we consider the system (1.1). We first present the definition of mild solutionsfor the system [6, 16].

Definition 3.1. An H-valued stochastic process {x(t), t ∈ (−∞, b]} is said to be a mildsolution of the system (1.1) if

• x(t) is Ft -adapted and measurable, t � 0;• x(t) is continuous on [0, b] almost surely and for each s ∈ [0, t), the function

(t − s)α−1ATα(t − s)G(s, xs) is integrable such that the following stochastic integralequation is verified:

x(t) = Sα(t)[φ(0) + G(0, φ)] − G(t, xt ) −∫ t

0(t − s)α−1ATα(t − s)G(s, xs) ds

+∫ t

0(t − s)α−1Tα(t − s)f (s, xs) ds

+∫ t

0(t − s)α−1Tα(t − s)

[∫ s

−∞σ(s, τ, xτ ) dW(τ)

]ds.

• x(t) = φ(t) on (−∞, 0] satisfying ‖φ‖2Ch

< ∞,

where

Sα(t)x =∫ ∞

0ηα(θ)S(tαθ)x dθ, Tα(t)x = α

∫ ∞

0θηα(θ)S(tαθ)x dθ

with ηα a probability density function defined on (0,∞).

The following properties of Sα(t) and Tα(t) appeared in [26] are useful.

Lemma 3.1. Under previous assumptions on S(t), t � 0, and A,

• (i) for any fixed t � 0, Sα(t) and Tα(t) are linear and bounded operators such that forany x ∈ H, |Sα(t)x|H � M|x|H, |Tα(t)x|H � Mα

�(1+α)|x|H;

• (ii) Sα(t) and Tα(t) are strongly continuous;• (iii) for any x ∈ H, β ∈ (0, 1) and θ ∈ (0, 1], we have

ATα(t)x = A1−βTα(t)Aβx and ‖AθTα(t)‖ � αCθ

tαθ

�(2 − θ)

�(1 + α(1 − θ)), t ∈ [0, b].

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J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

In order to explain our theorem, we need the following assumptions.

(A0) −A is the infinitesimal generator of an analytic semigroup of bounded linear operatorsS(t) in H, 0 ∈ ρ(−A), S(t) is compact for t > 0, and there exists a positive constant Msuch that ‖S(t)‖ � M .

(A1) The function G : J × Ch → H is continuous and there exist some constants MG > 0,β ∈ (0, 1), such that G is Hβ-valued and

E|AβG(t, x) − AβG(t, y)|2H

� MG‖x − y‖2Ch

, x, y ∈ Ch, t ∈ J,

E|AβG(t, x)|2H

� MG

(‖x‖2Ch

+ 1).

(A2) For each ϕ ∈ Ch,

K(t) = lima→∞

∫ 0

−a

σ (t, s, ϕ) dW(s)

exists and is continuous. Further, there exists a positive constant Mk such that

E|K(t)|2H

� Mk.

(A3) σ : J × J × Ch → L(K, H) satisfies the following:

(3a) for each (t, s) ∈ D := J × J , σ(t, s, ·) : Ch → L(K, H) is continuous and for eachx ∈ Ch, σ(·, ·, x) : D → L(K, H) is strongly measurable;

(3b) there is a positive integrable function m ∈ L1([0, b]) and a continuous nondecreasingfunction �σ : [0,∞) → (0,∞) such that for every (t, s, x) ∈ J × J × Ch, we have∫ t

0E|σ(t, s, x)|2L0

2ds � m(t)�σ

(‖x‖2Ch

), lim inf

r→∞�(r)

rds = � < ∞.

(A4) f : J × Ch → H satisfies the following:

(4a) f (t, ·) : Ch → H is continuous for each t ∈ J and for each x ∈ Ch, f (·, x) : J → H

is strongly measurable;(4b) there is a positive integrable function n ∈ L1([0, b]) and a continuous nondecreasing

function �f : [0,∞) → (0,∞) such that for every (t, x) ∈ J × Ch, we have

E|f (t, x)|2H

� n(t)�f

(‖x‖2Ch

), lim inf

r→∞�f (r)

rds = ϒ < ∞.

(A5) Assume that the following relationship holds:

L := 20MG‖A−β‖2l2 +20l2MGC2

1−β�2(1 + β)b2αβ

β2�2(1 + αβ)

+20l2ϒM2b2α

�2(1 + α)sups∈J

n(s) +40l2�M2TrQb2α

�2(1 + α)sups∈J

m(s) < 1.

Denote by C((−∞, b], H) the space of all continuous H-valued stochastic processes{ξ(t), t ∈ (−∞, b]}. Let

Cb = {x : x ∈ C((−∞, b], H), x0 = φ ∈ Ch}.Set ‖ · ‖b be a seminorm defined by

‖x‖b = ‖x0‖Ch+ sup

s∈[0,b](E|x(s)|2) 1

2 , x ∈ Cb.

We have the following useful lemma appeared in [14].

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J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

Lemma 3.2. Assume that x ∈ Cb; then for all t ∈ J , xt ∈ Ch. Moreover,

l(E|x(t)|2) 12 � ‖xt‖Ch

� l sups∈[0,t]

(E|x(s)|2) 12 + ‖x0‖Ch

,

where l = ∫ 0−∞ h(s) ds < ∞.

The main object of this paper is to explain and prove the following theorem.

Theorem 3.1. Assume that assumptions (A0)–(A5) hold. Then there exists a mild solutionto the system (1.1).

Proof. We consider the operator Q : Cb → Cb defined by

(Qx)(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

φ(t), t ∈ (−∞, 0];Sα(t)[φ(0) + G(0, φ)] − G(t, xt ) − ∫ t

0 (t − s)α−1ATα(t − s)G(s, xs) ds

+∫ t

0 (t − s)α−1Tα(t − s)f (s, xs) ds

+∫ t

0 (t − s)α−1Tα(t − s)[ ∫ s

−∞ σ(s, τ, xτ ) dW(τ)]

ds, t ∈ J.

By virtue of lemma 3.1, it follows that

E|∫ t

0(t − s)α−1ATα(t − s)G(s, xs) ds|2

H

� E

[∫ t

0|(t − s)α−1A1−βTα(t − s)AβG(s, xs)|H ds

]2

�α2C2

1−β�2(1 + β)

�2(1 + αβ)E

[∫ t

0|(t − s)αβ−1AβG(s, xs)|H ds

]2

;

applying the Holder inequality and assumption (A1), we further derive that

E|∫ t

0(t − s)α−1ATα(t − s)G(s, xs) ds|2

H

�α2C2

1−β�2(1 + β)

�2(1 + αβ)

∫ t

0(t − s)αβ−1ds

∫ t

0(t − s)αβ−1E|AβG(s, xs)|2H ds

�α2C2

1−β�2(1 + β)

�2(1 + αβ)

bαβ

αβ

∫ t

0(t − s)αβ−1E|AβG(s, xs)|2H ds

�α2C2

1−β�2(1 + β)

�2(1 + αβ)

MGbαβ

αβ

∫ t

0(t − s)αβ−1

(1 + ‖xs‖2

Ch

)ds, (3.1)

which deduces that (t − s)α−1ATα(t − s)G(s, xs) is integrable on J by Bochner’s theorem (see[18]) and lemma 3.2

We shall show that Q has a fixed point, which is then a mild solution for the system (1.1).For φ ∈ Ch, define

φ(t) ={φ(t), t ∈ (−∞, 0];Sα(t)φ(0), t ∈ J.

Then φ ∈ Cb. Let x(t) = φ(t) + z(t), t ∈ (−∞, b]. It is easy to check that x satisfies (1.1) ifand only if z0 = 0 and

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J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

z(t) = Sα(t)G(0, φ) − G(t, φt + zt ) −∫ t

0(t − s)α−1ATα(t − s)G(s, φs + zs) ds

+∫ t

0(t − s)α−1Tα(t − s)f (s, φs + zs) ds

+∫ t

0(t − s)α−1Tα(t − s)

[∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds.

Set

C0b = {z ∈ Cb, z0 = 0 ∈ Ch}.

For any z ∈ C0b , we have

‖z‖b = ‖z0‖Ch+ sup

s∈[0,b](E|z(s)|2) 1

2 = sups∈[0,b]

(E|z(s)|2) 12 .

Thus,(C0

b , ‖ · ‖b

)is a Banach space. For each positive number q, set

Bq = {y ∈ C0

b , ‖y‖2b � q

};then, for each q, Bq is clearly a bounded closed convex set in C0

b . For z ∈ Bq , from lemma 3.2,we see that

‖zt + φt‖2Ch

� 2(‖zt‖2

Ch+ ‖φt‖2

Ch

)� 4

(l2 sup

s∈[0,t]E|z(s)|2 + ‖z0‖2

Ch+ l2 sup

s∈[0,t]E|φ(s)|2 + ‖φ0‖2

Ch

)

� 4l2(q + M2E|φ(0)|2

H

)+ 4‖φ‖2

Ch. (3.2)

Consider the map � on C0b defined by

(�z)(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, t ∈ (−∞, 0];Sα(t)G(0, φ) − G(t, φt + zt ) − ∫ t

0 (t − s)α−1ATα(t − s)G(s, φs + zs) ds

+∫ t

0 (t − s)α−1Tα(t − s)f (s, φs + zs) ds

+∫ t

0 (t − s)α−1Tα(t − s)[ ∫ s

−∞ σ(s, τ, φτ + zτ ) dW(τ)]

ds, t ∈ J.

From (3.1), it follows that � is well defined on Bq for each q > 0.Moreover, it is obvious that the operator Q has a fixed point if and only if � has a fixed

point. To this end, we decompose � as � = �1 + �2, where the operators �1 and �2 aredefined on Bq, respectively, by

(�1z)(t) = Sα(t)G(0, φ) − G(t, φt + zt ) −∫ t

0(t − s)α−1ATα(t − s)G(s, φs + zs) ds,

(�2z)(t) =∫ t

0(t − s)α−1Tα(t − s)f (s, φs + zs) ds

+∫ t

0(t − s)α−1Tα(t − s)

[∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds.

Thus, the theorem follows from the next theorem. �

Theorem 3.2. Assume that assumptions (A0)–(A5) hold. Then, �1 is a contractive mapping,while �2 is compact.

Proof. The theorem follows from lemmas in the appendix and Arzela–Ascoli theorem. �

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J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

Remark 1.(I). We obtain the existence of the mild solution to the system (1.1) under non-Lipschitz

conditions, which makes it more feasible that the conditions of solution can be satisfied.(II). When G = 0, that is, MG = 0, it is obvious that assumption A5 can be satisfied if b

is sufficiently small. Furthermore, if σ = 0, the system (1.1) reduces tocDα

t x(t) = −Ax(t) + f (t, xt ), t ∈ J := [0, b], α ∈ (0, 1),

x(t) = φ(t), t ∈ (−∞, 0],(3.3)

which was recently studied in [16]. In other words, in this special case, some of the results in[16] are improved.

At last, an example is provided to illustrate our results.

Example 3.1. Consider the following fractional neutral stochastic partial differential equationwith infinite delays of the form:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

cDαt [u(t, x) − G(t, u(t − h, x))] = ∂2

∂x2u(t, x) + f (t, u(t − h, x))

+∫ t

−∞ σ(s, u(s − h, x)) dW(s), 0 � x � π, h > 0, t ∈ J := [0, b];u(t, 0) = u(t, π) = 0, t ∈ J.

u(t, x) = φ(t, x), t ∈ (−∞, 0],

(3.4)

where α ∈ (0, 1), andW(t) is a standard cylindrical Wiener process defined on a stochasticbasis (�,F, P , {Ft }).

To rewrite this system into the abstract form (1.1), let H = L2([0, π ]) with the norm ‖ · ‖.Define A : H → H by AZ = z′′ with the domain

D(A) = {x ∈ H, x, x ′ are absolutely continuous, x ′′ ∈ H and x(0) = x(π) = 0};then A generates a symmetric C0-semigroup e−tA in H and there exists a complete orthonormalset {zn, n = 1, 2, . . .} of eigenvectors of A with

zn(s) =√

2

πsin(ns), n = 1, 2, . . . .

Then the operator A− 12 is given by

A− 12 ξ =

∞∑n=1

n(ξ, zn)zn

on the space D(A− 12 ) = {ξ(·) ∈ H,

∑∞n=1 n(ξ, zn)zn ∈ H} (for more details, see [19]).

Now, we present a special phase space Ch. Let h(s) = e2s , s < 0; then l = ∫ 0−∞ h(s) ds =

12 . Let

‖ϕ‖Ch=

∫ 0

−∞h(s) sup

s�θ�0(E|ϕ(θ)|2) 1

2 ds;

then it follows from [10] that (Ch, ‖ · ‖Ch) is a Banach space.

For (t, ϕ) ∈ J × Ch, where ϕ(θ)(ξ) = φ(θ, ξ), (θ, ξ) ∈ (−∞, 0] × [0, π ], letu(t)(ξ) = u(t, ξ) and define the functions G, f : J × Ch → H, σ : J × Ch → L0

2(H, H) forthe infinite delay as follows:

(−A)12 G(t, ϕ)(x) =

∫ 0

−∞μ1(θ)ϕ(θ)(x) dθ,

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J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

f (t, ϕ)(x) =∫ 0

−∞μ2(t, x, θ)G1(ϕ(θ)(x)) dθ,

σ (t, ϕ)(x) =∫ 0

−∞μ3(t, x, θ)G2(ϕ(θ)(x)) dθ;

hence, we can impose some hypotheses on μi, i = 1, 2, 3, and Gk, k = 1, 2 (see [2]), to satisfythe assumptions stated in theorem 3.1; we omit it here. Thus, there exists a mild solution forthe system (3.4).

Acknowledgments

The project was sponsored by NSFC (10871041). The authors would like to express theirsincere gratitude to the anonymous referees for the valuable suggestions and comments.

Appendix. Some basic estimates

In this appendix, our main object is to prove theorem 3.2.

Lemma A.1. Under assumptions (A0)–(A5), there exists a positive number q such that�(Bq) ⊂ Bq .

Proof. If it is not true, then for each positive number q, there exists a function zq(·) ∈ Bq ,but �(zq) /∈ Bq , that is, E|(�zq)(t)|2

H> q for some t = t (q) ∈ J . An elementary inequality

can show that

q � E|�(zq)(t)|2H

� 5E|Sα(t)G(0, φ)|2H

+ 5E∣∣G(

t, φt + zqt

)∣∣2H

+ 5E

∣∣∣∣∫ t

0(t − s)α−1ATα(t − s)G

(s, φs + zq

s

)ds

∣∣∣∣2

H

+ 5E

∣∣∣∣∫ t

0(t − s)α−1Tα(t − s)f

(s, φs + zq

s

)ds

∣∣∣∣2

H

+ 5E

∣∣∣∣∫ t

0(t − s)α−1Tα(t − s)

[∫ s

−∞σ(s, φτ + zq

τ

)dW(τ)

]ds

∣∣∣∣2

H

= 55∑

i=1

Ii . (A.1)

In what follows, K(α, β) is the number defined by

K(α, β) := α2C21−β�2(1 + β)

�2(1 + αβ). (A.2)

Let us now estimate each term above Ii, i = 1, · · · , 5. By lemma 3.2 and assumptions(A0)–(A1), we have

I1 � M2‖A−β‖2E|AβG(0, φ)|2H

� M2‖A−β‖2MG

(1 + ‖φ‖2

Ch

), (A.3)

9

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

I2 � ‖A−β‖2E∣∣AβG

(t, φt + z

qt

)∣∣2H

� MG‖A−β‖2(1 +

∥∥φt + zqt

∥∥2Ch

)� MG‖A−β‖2

(1 + 4l2

(q + M2E|φ(0)|2

H

)+ 4‖φ‖2

Ch

). (A.4)

By a standard calculation involving lemma 3.1 and the Holder inequality, we can deduce that

I3 � E

[∫ t

0

∣∣(t − s)α−1A1−βTα(t − s)AβG(s, φs + zq

s

)∣∣H

ds

]2

� K(α, β)

∫ t

0(t − s)αβ−1ds

∫ t

0(t − s)αβ−1E

∣∣AβG(s, φs + zq

s

)∣∣2H

ds

� K(α, β)bαβ

αβ

∫ t

0(t − s)αβ−1E

∣∣AβG(s, φs + zq

s

)∣∣2H

ds;in view of assumption (A1) and (3.2), we further derive that

I3 � K(α, β)bαβ

αβ

∫ t

0(t − s)αβ−1MG

(1 +

∥∥φs + zqs

∥∥2Ch

)ds

� MGK(α, β)b2αβ

(αβ)2

(1 + 4l2

(q + M2E|φ(0)|2

H

)+ 4‖φ‖2

Ch

). (A.5)

Together with assumption (A4) and (3.2), we have

I4 � E

[∫ t

0

∣∣(t − s)α−1Tα(t − s)f(s, φs + zq

s

)∣∣H

ds

]2

�(

Mα

�(1 + α)

)2 ∫ t

0(t − s)α−1ds

∫ t

0(t − s)α−1E

∣∣f (s, φs + zq

s

)∣∣2H

ds

�(

Mα

�(1 + α)

)2bα

α

∫ t

0(t − s)α−1E

∣∣f (s, φs + zq

s

)∣∣2H

ds

�(

Mα

�(1 + α)

)2bα

α

∫ t

0(t − s)α−1n(s)�f

(∥∥φs + zqs

∥∥2Ch

)ds

�(

Mα

�(1 + α)

)2b2α

α2�f

(4l2

(q + M2E|φ(0)|2

H+ 4‖φ‖2

Ch

)sups∈J

n(s). (A.6)

A similar argument involves Burkholder–Davis–Gundy’s inequality and assumptions (A2)–(A3); we obtain

I5 � E

[∫ t

0

∣∣∣∣(t − s)α−1Tα(t − s)

[∫ s

−∞σ(s, τ, φτ + zq

τ

)dW(τ)

]∣∣∣∣H

ds

]2

�(

Mα

�(1 + α)

)2bα

α

∫ t

0(t − s)α−1E

∣∣∣∣∫ s

−∞σ(s, τ, φτ + zq

τ

)dW(τ)

∣∣∣∣2

H

ds

�(

Mα

�(1 + α)

)2bα

α

∫ t

0(t − s)α−1

(2Mk + 2 Tr Q

∫ s

0E

∣∣σ (s, τ, φτ + zq

τ

)∣∣2L0

2dτ

)ds

�(

Mα

�(1 + α)

)2bα

α

∫ t

0(t − s)α−1

(2Mk + 2 Tr Qm(s)�σ

(‖φs + zqs

)‖2Ch

)ds

�(

Mα

�(1 + α)

)2 2Mkb2α

α2+

(Mα

�(1 + α)

)2 2TrQb2α

α2

× �σ

(4l2

(q + M2E|φ(0)|2

H

)+ 4‖φ‖2

Ch

)sups∈J

m(s). (A.7)

10

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

Combining these estimates (A.1)–(A.7) yields

q � E|�(zq)(t)|2H

� L0 + 20MG‖A−β‖2l2q +20l2MGC2

1−β�2(1 + β)b2αβ

β2�2(1 + αβ)q

+ 5

(Mbα

�(1 + α)

)2

�f

(4l2

(q + M2E|φ(0)|2

H

)+ 4‖φ‖2

Ch

)sups∈J

n(s)

+ 10 Tr Q

(Mbα

�(1 + α)

)2

�σ

(4l2

(q + M2E|φ(0)|2

H

)+ 4‖φ‖2

Ch

)sups∈J

m(s), (A.8)

where

L0 = 5M2‖A−β‖2MG

(1 + ‖φ‖2

Ch

)+ 5MG‖A−β‖2(1 + 4l2M2E|φ(0)|2

H+ 4‖φ‖2

Ch

)

+5MGC2

1−β�2(1 + β)

�2(1 + αβ)

b2αβ

β2

(1 + 4l2M2E|φ(0)|2

H+ 4‖φ‖2

Ch

)+

10MkM2b2α

�2(1 + α).

Dividing both sides of (A.8) by q and taking q → ∞, we obtain that

20MG‖A−β‖2l2 +20l2MGC2

1−β�2(1 + β)b2αβ

β2�2(1 + αβ)

+20l2ϒM2b2α

�2(1 + α)sups∈J

n(s) +40l2�M2TrQb2α

�2(1 + α)sups∈J

m(s) � 1,

which is a contradiction by assumption (A5). Thus, for some positive number q, �(Bq) ⊂ Bq .�

Lemma A.2. Let assumptions (A0)–(A5) hold. Then �1 is contractive.

Proof. Let u, v ∈ Bq . Then

E|(�1u)(t) − (�1v)(t)|2H

� 2E|G(t, φt + ut ) − G(t, φt + vt )|2H+ 2E

∣∣∣∣∫ t

0(t − s)α−1ATα(t − s)G(s, φs + zs) ds

∣∣∣∣2

H

� 2‖A−β‖2MG‖ut − vt‖2Ch

+ 2K(α, β)E

×[ ∫ t

0(t − s)αβ−1|Aβ(G(s, φs + us) − G(s, φs + vs))|H ds

]2

� 2‖A−β‖2MG‖ut − vt‖2Ch

+2K(α, β)bαβ

αβ

∫ t

0(t − s)αβ−1

×E|Aβ(G(s, φs + us) − G(s, φs + vs))|2H ds

� 2‖A−β‖2MG‖ut − vt‖2Ch

+2MGK(α, β)bαβ

αβ

∫ t

0(t − s)αβ−1‖us − vs‖2

Chds.

Hence,

E|(�1u)(t) − (�1v)(t)|2H

� 4MGl2

(‖A−β‖2 + K(α, β)

b2αβ

(αβ)2

)sup

0�s�t

E|u(s) − v(s)|2H,

where we have used the fact that u0 = v0 = 0; K(α, β) is defined in (A.2).

11

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

Thus,

‖�1u − �1v‖2b � 4MGl2

(‖A−β‖2 + K(α, β)

b2αβ

(αβ)2

)‖u − v‖2

b,

so �1 is a contraction by assumption (A5). �

Let q > 0 and �2(Bq) ⊂ Bq .

Lemma A.3. Let assumptions (A0)–(A5) hold. Then �2 maps bounded sets to bounded setsin Bq.

Proof. For each t ∈ J , z ∈ Bq , from (3.2), we have

‖zt + φt‖2Ch

� 4l2(q + M2E|φ(0)|2

H

)+ 4‖φ‖2

Ch:= q ′.

By the similar argument as lemma 4.1, we obtain

E|�2z(t)|2H � 2E

∣∣∣∣∫ t

0(t − s)α−1Tα(t − s)f (s, φs + zs) ds

∣∣∣∣2

H

+ 2E

∣∣∣∣∫ t

0(t − s)α−1Tα(t − s)

[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds

∣∣∣∣2

H

� 2

(Mα

�(1 + α)

)2bα

α

∫ t

0(t − s)α−1n(s)�f

(∥∥φs + zqs

∥∥2Ch

)ds

+ 2

(Mα

�(1 + α)

)2bα

α

∫ t

0(t − s)α−1

(2Mk + 2 Tr Qm(s)�σ

(∥∥φs + zqs

∥∥2Ch

))ds

� 2

(Mα

�(1 + α)

)2b2α

α2�f (q ′) sup

s∈J

n(s) + 4

(Mα

�(1 + α)

)2Mkb

2α

α2

+ 4

(Mα

�(1 + α)

)2 Tr Qb2α

α2�σ(q ′) sup

s∈J

m(s)

:= �,

which implies that for each z ∈ Bq , ‖�2z‖2b � �. �

Lemma A.4. Let assumptions (A0)–(A5) hold. Then the set {�2z, z ∈ Bq} is anequicontinuous family of functions on J.

Proof. Let 0 < ε < t < b and δ > 0 such that ‖Tα(s1) − Tα(s2)‖ < ε, for every s1, s2 ∈ J

with |s1 − s2| < δ. For z ∈ Bq , 0 < |h| < δ, t + h ∈ J, we have

E|�2z(t + h) − �2z(t)|2H= E

∣∣∣∣∫ t+h

0(t + h − s)α−1Tα(t + h − s)f (s, φs + zs) ds

+∫ t+h

0(t + h − s)α−1Tα(t + h − s)

[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds

−∫ t

0(t − s)α−1Tα(t − s)f (s, φs + zs) ds

−∫ t

0(t − s)α−1Tα(t − s)

[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds

∣∣∣∣2

H

12

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

� 6E

∣∣∣∣∫ t

0[(t + h − s)α−1 − (t − s)α−1]Tα(t + h − s)f (s, φs + zs) ds

∣∣∣∣2

H

+ 6E

∣∣∣∣∫ t+h

t

(t + h − s)α−1Tα(t + h − s)f (s, φs + zs) ds

∣∣∣∣2

H

+ 6E

∣∣∣∣∫ t

0(t − s)α−1[Tα(t + h − s) − Tα(t − s)]f (s, φs + zs) ds

∣∣∣∣2

H

+ 6E

∣∣∣∣∫ t+h

t

(t + h − s)α−1Tα(t + h − s)

[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds

∣∣∣∣2

H

+ 6E

∣∣∣∣∫ t

0[(t + h − s)α−1 − (t − s)α−1]Tα(t + h − s)

×[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds

∣∣∣∣2

H

+ 6E

∣∣∣∣∫ t

0(t − s)α−1[Tα(t + h − s) − Tα(t − s)]

×[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds

∣∣∣∣2

H

.

Applying lemma 3.1 and the Holder inequality, we obtain

E|�2z(t + h) − �2z(t)|2H� 6N(α)

∫ t

0(t + h − s)α−1 − (t − s)α−1 ds

×∫ t

0[(t + h − s)α−1 − (t − s)α−1]E|f (s, φs + zs)|2H ds

+ 6N(α)hα

α

∫ t+h

t

(t + h − s)α−1E|f (s, φs + zs)|2H ds

+ 6ε2 bα

α

∫ t

0(t − s)α−1E|f (s, φs + zs)|2H ds

+ 6N(α)hα

α

∫ t+h

t

(t + h − s)α−1E

∣∣∣∣∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

∣∣∣∣2

H

ds

+ 6N(α)

∫ t

0(t + h − s)α−1 − (t − s)α−1 ds

×∫ t

0[(t + h − s)α−1 − (t − s)α−1]E

∣∣∣∣∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

∣∣∣∣2

H

ds

+ 6ε2 bα

α

∫ t

0(t − s)α−1E

∣∣∣∣∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

∣∣∣∣2

H

ds;

by assumptions (A3)–(A4), we have

E|�2z(t + h) − �2z(t)|2H � 6N(α)�f (q ′)∫ t

0(t + h − s)α−1 − (t − s)α−1ds

×∫ t

0[(t + h − s)α−1 − (t − s)α−1]n(s) ds + 6N(α)

hα

α�f (q ′)

×∫ t+h

t

(t + h − s)α−1n(s) ds +6ε2bα

α�f (q ′)

∫ t

0(t − s)α−1n(s) ds

13

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

+ 6N(α)hα

α

∫ t+h

t

(t + h − s)α−1(2Mk + 2 Tr Qm(s)�σ (q ′)) ds

+ 6N(α)

∫ t

0(t + h − s)α−1 − (t − s)α−1 ds

×∫ t

0[(t + h − s)α−1 − (t − s)α−1](2Mk + 2 Tr Qm(s)�σ (q ′)) ds

+6ε2bα

α

∫ t

0(t − s)α−1(2Mk + 2 Tr Qm(s)�σ (q ′)) ds, (A.9)

with N(α) = ( Mα�(1+α)

)2. Therefore, for ε sufficiently small, the right-hand side of (A.9)tends to zero as h → 0. On the other hand, the compactness of Tα(t), t > 0 (lemma 3.4in [26]), implies the continuity in the uniform operator topology. Thus, the set {�2z, z ∈ Bq}is equicontinuous.

�

Lemma A.5. Let assumptions (A0)–(A5) hold. Then �2 maps Bq onto a precompact set inBq.

Proof. Let 0 < t � b be fixed and ε be a real number satisfying 0 < ε < t . For δ > 0,define an operator �

ε,δ2 on Bq by

(�

ε,δ2 z

)(t) = α

∫ t−ε

0

∫ ∞

δ

θ(t − s)α−1ηα(θ)S((t − s)αθ)f (s, φs + zs) ds

+ α

∫ t−ε

0

∫ ∞

δ

θ(t − s)α−1ηα(θ)S((t − s)αθ)

[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds

= S(εαδ)α

∫ t−ε

0

∫ ∞

δ

θ(t − s)α−1ηα(θ)S((t − s)αθ − εαδ)f (s, φs + zs) ds

+ S(εαδ)α

∫ t−ε

0

∫ ∞

δ

θ(t − s)α−1ηα(θ)S((t − s)αθ − εαδ)

×[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]ds.

Since S(t), t > 0, is a compact operator, the set{�

ε,δ2 z)(t), z ∈ Bq

}is precompact in H for

every ε ∈ (0, t), δ > 0. Moreover, for each z ∈ Bq , we have

E∣∣(�2z)(t) − (

�ε,δ2 z

)(t)

∣∣2H

� 4α2E

∣∣∣∣∫ t

0

∫ δ

0θ(t − s)α−1ηα(θ)S((t − s)αθ)f (s, φs + zs) dθ ds

∣∣∣∣2

H

+ 4α2E

∣∣∣∣∫ t

t−ε

∫ ∞

δ

θ(t − s)α−1ηα(θ)S((t − s)αθ)f (s, φs + zs) dθ ds

∣∣∣∣2

H

+ 4α2E

∣∣∣∣∫ t

0

∫ δ

0θ(t − s)α−1ηα(θ)S((t − s)αθ)

×[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]dθ ds

∣∣∣∣2

H

+ 4α2E

∣∣∣∣∫ t

t−ε

∫ ∞

δ

θ(t − s)α−1ηα(θ)S((t − s)αθ)

14

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

×[ ∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

]dθ ds

∣∣∣∣2

H

=4∑

i=1

Ji. (A.10)

A similar argument as before can show that

J1 � 4α2E

[ ∫ t

0

∫ δ

0|θ(t − s)α−1ηα(θ)S((t − s)αθ)f (s, φs + zs)|H dθ ds

]2

� 4α2M2∫ t

0(t − s)α−1 ds

∫ t

0(t − s)α−1E|f (s, φs + zs)|2H ds

(∫ δ

0θηα(θ) dθ

)2

� 4αM2bα�f (q ′)∫ t

0(t − s)α−1n(s) ds

( ∫ δ

0θηα(θ) dθ

)2

(A.11)

and

J2 � 4α2M2∫ t

t−ε

(t − s)α−1 ds

∫ t

t−ε

(t − s)α−1E|f (s, φs + zs)|2H ds

( ∫ ∞

0θηα(θ) dθ

)2

� 4αM2εα�f (q ′)∫ t

t−ε

(t − s)α−1n(s) ds

( ∫ ∞

0θηα(θ) dθ

)2

� 4αM2εα�f (q ′)�2(1 + α)

∫ t

t−ε

(t − s)α−1n(s) ds, (A.12)

where we have used the equality (see [15, 26])∫ ∞

0θξηα(θ) dθ = �(1 + ξ)

�(1 + αξ).

Similarly, employing Burkholder–Davis–Gundy’s inequality, we further derive that

J3 � 4αM2bα

∫ t

0(t − s)α−1E

∣∣∣∣∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

∣∣∣∣2

H

ds

( ∫ δ

0θηα(θ)dθ

)2

� 4αM2bα

∫ t

0(t − s)α−1(2Mk + 2 Tr Q�σ(q ′)m(s)) ds

( ∫ δ

0θηα(θ) dθ

)2

,

(A.13)

and

J4 � 4αM2εα

∫ t

t−ε

(t − s)α−1E

∣∣∣∣∫ s

−∞σ(s, τ, φτ + zτ ) dW(τ)

∣∣∣∣2

H

ds

( ∫ ∞

0θηα(θ) dθ

)2

� 4αM2εα

�2(1 + α)

∫ t

t−ε

(t − s)α−1(2Mk + 2 Tr Q�σ(q ′)m(s)) ds. (A.14)

Recalling (A.10), from (A.11)–(A.14), we see that for each z ∈ Bq ,

E∣∣(�2z)(t) − (

�ε,δ2 z

)(t)

∣∣2H

→ 0 as ε → 0+, δ → 0+.

Therefore, there are relatively compact sets arbitrary close to the set {�2(t), z ∈ Bq}; hence,the set {(�2z)(t), z ∈ Bq} is also precompact in Bq. �

15

J. Phys. A: Math. Theor. 44 (2011) 335201 J Cui and L Yan

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