The Existence, Uniqueness, and Controllability of Neutral Stochastic Delay Partial Differential Equations Driven by Standard Brownian Motion and Fractional Brownian Motion.

1. Introduction

Fractional Brownian motion (fBm for short) with Hurst parameter H [member of] (0, 1) is a centered Gaussian process {[[beta].sup.H](t), t [greater than or equal to] 0} which is often used to model many complex phenomena in applications when the systems contain rough external forcing. When H = 1/2, the fBm is the standard Brownian motion which is a Markov process and martingale, and we can use the classical Ito theory to construct a stochastic integration with respect to the standard Brownian motion. However, when H [not equal to] 1/2, the fBm is neither a Markov process nor a semimartingale, and the fBm behaviors become quite different. The classical techniques based on PDE cannot apply to in this context. For more details on fBm, we refer the readers to the articles [1-5].

Neutral differential equations (NDEs for short) are a family of differential equations depending on the past as well as the present state which involve derivatives with delays as well as the function itself. This family of equations has great applications in mathematical, electrical engineering, ecology, and other fields of science. In recent years, NDEs have received more and more attention; see, for example, [6-16] and the reference therein.

The observations of stock prices processes suggest that they are not self-similar. As a result, the mixed model describes the stock prices behavior in a better way. Since the pioneering paper by Cheridito , mixed stochastic models containing both a standard Brownian motion and fBm gained a lot of attention. The main reason for this is that they allow us to model systems driven by white noise and fractional Gaussian noise. For more details on this topic see [18-21].

The controllability problem is one of the most attractive and important problems of differential equations because of their great significance and applications in physics, population dynamics, engineering, mathematical biology, and other areas of science. Controllability for stochastic models only containing standard Brownian motion has been investigated very well (see [7, 12, 16, 22]). Recently, controllability for stochastic systems only driven by fBm has gained a lot of attention; we refer to [6, 9, 11]. Up to now, there is no paper which considers the controllability for stochastic systems driven by standard Brownian motion and fBm.

In this paper, we study the existence, uniqueness, and controllability results for a class of neutral stochastic delay partial differential equations (NSDPDEs for short) driven by a standard Brownian motion and an fBm in the abstract form

[mathematical expression not reproducible] (1)

where A is the infinitesimal generator of an analytic semigroup {S(t), t [greater than or equal to] 0} of bounded linear operators in a real separable Hilbert space X. The delay functions r, [rho], [eta] : [0, +[infinity]) [right arrow] [0, [tau]] ([tau] [greater than or equal to] 0) are continuous, and f, h : [0, +[infinity]) x X [right arrow] X, g : [0, +[infinity]) x X [right arrow] [L.sup.0.sub.Q(1)]([Y.sub.1]; X), and [sigma] : [0, +[infinity]) [right arrow] [L.sup.0.sub.Q(2)]([Y.sub.2]; X) are Borel measurable satisfying appropriate conditions. The control mapping u(x) takes values in [L.sup.2]([0, T]; U), and the Hilbert space of admissible control mappings for a real separable Hilbert U, B is a bounded linear operator from U to X. {W(t), t [greater than or equal to] 0} denotes a standard Brownian motion in a real separable Hilbert space [Y.sub.1]. {[B.sup.H.sub.Q](t), t [greater than or equal to] 0} denotes a fractional Brownian motion in a real separable Hilbert space [Y.sub.2] with Hurst parameter H [member of] (1/2, 1). Let [L.sup.0.sub.Q(i)]([Y.sub.i]; X) (i = 1, 2) be the space of all [Q.sup.(i)]-Hilbert-Schmidt operators from [Y.sub.i] into X.

Recently, Boufoussi and Hajji in  considered the existence and uniqueness problems of a class of neutral stochastic delay differential equations; that is, B [equivalent to] 0 and g [equivalent to] 0 in (1) by means of the Banach fixed point theory. Very recently, Liu and Luo in  studied the existence and uniqueness problems of a wide class of neutral stochastic delay partial differential equations, that is, B [equivalent to] 0 in (1) by means of the Banach fixed point theory. However, they all required 1/2 < [beta] < 1 which is a stronger condition than ours (see Section 3).

In this paper, based on the above papers, we also use the Banach fixed point theory to consider a class of neutral stochastic delay partial differential equations, which is more general than ones in [8, 13]. Some conditions on the existence, uniqueness, and controllability of (1) are obtained. The related known results in Boufoussi and Hajji  and Liu and Luo in  are improved and generalized.

The contents of this paper are as follows. In Section 2, some notions, definitions, and lemmas which will be needed throughout this paper are introduced. In Section 3, the existence and uniqueness of mild solutions of (1) are proved. In Section 4, the controllability of (1) is investigated by means of the Banach fixed point theory. In Section 5, an example is given to illustrate our main results. At last, in Section 6, our conclusion is presented.

2. Preliminaries

Let ([OMEGA], F, [{[F.sub.t]}.sub.t[greater than or equal to]0], P) be a complete probability space equipped with a normal filtration [{[F.sub.t]}.sub.t[greater than or equal to]0] satisfying standard assumptions; that is, the filtration is right continuous and [F.sub.0] contains all P-null sets. Let C([-[tau], T]; [L.sup.2]([OMEGA]; X)) be the family of all continuous functions from [-[tau], T] into [L.sup.2]([OMEGA]; X). Let X and [Y.sub.i], i = 1,2, be three real separable Hilbert spaces. We denote by L(Yi; X) the space of all bounded linear operators from [Y.sub.i] to X, i = 1,2. We suppose that [mathematical expression not reproducible] is a complete orthonormal basis in [Y.sub.i]. Let [Q.sup.(i)] [member of] L([Y.sub.i]; X) be an operator defined by [Q.sup.(i)][e.sup.(i).sub.n] = [[lambda].sup.(i).sub.n] [e.sup.(i).sub.n] with finite trace [trQ.sup.(i)] = [[summation].sup.[infinity].sub.n=1][[lambda].sup.(i).sub.n] < [infinity], where [mathematical expression not reproducible] are nonnegative real numbers. Then there exists a real-valued sequence [mathematical expression not reproducible] of one-dimensional standard Brownian motions mutually independent on ([OMEGA], F, [{[F.sub.t]}.sub.t[greater than or equal to]0], P) such that

W(t) = [[infinity].summation over (n=1)] [square root of ([[lambda].sup.(1).sub.n])] [[omega].sub.n](t) [e.sup.(1).sub.n], t [greater than or equal to] 0. (2)

Consider that a [Y.sub.2]-valued stochastic process [B.sup.H.sub.Q](t) is defined by the formal infinite sum (see ).

[B.sup.H.sub.Q](t) = [[infinity].summation over (n=1)] [square root of ([[lambda].sup.(2).sub.n])] [[beta].sup.H.sub.n](t) [e.sup.(2).sub.n], t [greater than or equal to] 0, (3)

where the sequence [mathematical expression not reproducible] is mutually independent scalar fBms with Hurst parameter H [member of] (1/2, 1).

Let [L.sup.0.sub.Q(i)]([Y.sub.i]; X) be the space of all [Q.sup.(i)]-Hilbert- Schmidt operators from [Y.sub.i] to X, i = 1, 2. Now we show the following definitions.

Definition 1. Let [[xi].sub.i] [member of] L([Y.sub.i]; X) and define

[mathematical expression not reproducible]. (4)

If [mathematical expression not reproducible], then [[xi].sub.i] is called a [Q.sup.(i)]-Hilbert-Schmidt operator and the space [mathematical expression not reproducible] is a real separable Hilbert space equipped with the inner product [mathematical expression not reproducible].

Let T > 0, [beta] = {[beta](t); t [member of] [0, T]} be a Wiener process and {[[beta].sup.H](t), t [member of] [0, T]} be the one-dimensional fBm with Hurst parameter H [member of] (1/2, 1). [[beta].sup.H] has the following stochastic integral representation:

[[beta].sup.H](t) = [[integral].sup.t.sub.0] [K.sub.H](t, s) d[beta](s), (5)

[K.sub.H](t, s) is the kernel defined as

[K.sub.H](t, s) = [c.sub.H] [s.sup.1/2-H] [[integral].sup.t.sub.s][(u - s).sup.H-3/2] [u.sup.H-1/2] du (6)

for t > s, where [c.sub.H] = (H(2H - 1)/B[(2 - 2H, H - 1/2)).sup.1/2], and B(x, x) denotes the Beta function. Set [K.sub.H](t, s) = 0 for t [less than or equal to] s. Let H denote the reproducing kernel Hilbert space of the fBm. In fact H is the closure of the set of indicator functions {[I.sub.[0;t]], t [member of] [0,T]} with respect to the inner product.

[<[I.sub.[0,t]], [I.sub.[0,s]]>.sub.H] = [R.sub.H](t, s). (7)

Then the mapping [I.sub.[0,t]] [right arrow] [[beta].sup.H](t) can be extended to an isometry between H and the first Wiener chaos of the fBm [mathematical expression not reproducible]. The image of an element [phi] [member of] H by this isometry is called the Wiener integral of [phi] with respect to [[beta].sup.H] (see ).

We recall that for [psi], [phi] [member of] H their inner product in H is given by

[<[psi], [phi]>.sub.H] = H(2H - 1)[[integral].sup.T.sub.0] [[integral].sup.T.sub.0] [psi](s) [phi](t) [[absolute value of t - s].sup.2H-2] ds dt. (8)

[K.sup.*.sub.H] is an operator from H to [L.sup.2]([0, T]) defined as

([K.sup.*.sub.H][phi])(s) = [[integral].sup.t.sub.s] [phi](r) [[partial derivative][K.sub.H](r, s)/[partial derivative]r] dr. (9)

Then

([K.sup.*.sub.H][I.sub.[0,t]])(s) = [K.sub.H](t, s) [I.sub.[0,t]](s) (10)

and [K.sup.*.sub.H] is an isometry between the set of indicator functions {[I.sub.[0;t]], t [member of] [0,T]} and [L.sup.2]([0, T]) that can be extended to H (see ).

Let [psi](s) for s [member of] [0,T] be a mapping with values in [L.sup.0.sub.2]([Y.sub.2]; X). The stochastic integral of [psi] with respect to [B.sup.H.sub.Q] is defined by

[mathematical expression not reproducible], (11)

where [[beta].sub.n] is the standard Brownian motion which was introduced in (5).

In order to set existence, uniqueness, and controllability problems, we need the following lemmas which are Lemma 2 in  and Lemma 7.7 in , respectively.

Lemma 2 (see ). For any [mathematical expression not reproducible] and 0 [less than or equal to] a [less than or equal to] b [less than or equal to] T, if [[SIGMA].sup.[infinity].sub.n=1] [square root of ([[lambda].sup.(2).sub.n])] < [infinity] and the series [[SIGMA].sup.[infinity].sub.n=1] [square root of ([[lambda].sup.(2).sub.n] [phi](t)[e.sup.(2).sub.n] is uniformly convergent for t [member of] [0,T], then one has

[mathematical expression not reproducible], (12)

where H [member of] (1/2, 1) and [c.sub.H] = [square root of (H(2H - 1)/B(2 - 2H, H - 1/2))].

Lemma 3 (see ). For any r [greater than or equal to] 1 and for arbitrary predictable process [mathematical expression not reproducible], one has

[mathematical expression not reproducible]. (13)

We assume that 0 [member of] [rho](A), where [rho](A) is the resolvent set of A and the analytic semigroup {S(t), t [greater than or equal to] 0} in X is uniformly bounded; that is, [[parallel]S(t)[parallel].sub.X] [less than or equal to] M for some positive constant M [greater than or equal to] 1. Then, for [alpha] [member of] (0, 1], it is possible under some circumstance to define the fractional power [(-A).sup.[alpha]] as a closed linear operator with domain D([(-A).sup.[alpha]]). Moreover, the subspace D([(-A).sup.[alpha]]) is dense in X and the expression

[[parallel]v[parallel].sub.[alpha]] = [[parallel][(- A).sup.[alpha]]v[parallel].sub.X], v [member of] D([(- A).sup.[alpha]]) (14)

defines a norm on D([(-A).sup.[alpha]]).

Let [H.sub.[alpha]] denote the Banach space D([(-A).sup.[alpha]]) equipped with the norm [[parallel]x[parallel].sub.[alpha]], then the following properties are well known (see ).

Lemma 4 (see ). Suppose that the preceding conditions are satisfied.

(1) If 0 < [beta] [less than or equal to] [alpha], then the injection [H.sub.[alpha]] [??] [H.sub.[beta]] is continuous.

(2) For every 0 < [beta] [less than or equal to] 1, there exists [M.sub.[beta]] > 0 such that

[parallel][(-A).sup.[beta]] S(t)[parallel] [less than or equal to] [M.sub.[beta]] [t.sup.-[beta]], 0 [less than or equal to] t [less than or equal to] T. (15)

3. Existence and Uniqueness of Mild Solution

In this section, the Banach fixed point theory is used to investigate the existence and uniqueness problems of (1). Now, we introduce the definition of mild solution of (1).

Definition 5. An X-valued stochastic process x(t) is called a mild solution of (1) if x(x) [member of] C([-[tau], T]; [L.sup.2]([OMEGA]; X)) and x(t) = [phi](t) for t [member of] [-[tau], 0] and for t [member of] [0,T] satisfies

[mathematical expression not reproducible]. (16)

In order to set the existence and uniqueness problems, we assume that the following conditions hold:

(C.1) A is the infinitesimal generator of an analytic semigroup {S(t), t [greater than or equal to] 0} of bounded linear operators in X and there exists a positive constant M [greater than or equal to] 1 such that [[parallel]S(t)[parallel].sub.X] [less than or equal to] M for 0 [less than or equal to] t [less than or equal to] T.

(C.2) The mappings f : [0, +[infinity]) x X [right arrow] X and [mathematical expression not reproducible] satisfy the following conditions: for any x, y [member of] X and t [member of] [0,T], there exist nonnegative constants [L.sub.f], [C.sub.f], [L.sub.g], and [C.sub.g], such that

(i) [[parallel]f(t, x)-f(t, y)[parallel].sub.X] [less than or equal to] [L.sub.f][[parallel]x - y[parallel].sub.X], [[parallel]f(t,x)[parallel].sup.2.sub.X] [less than or equal to] [C.sub.f](1 + [[parallel]x[parallel].sup.2.sub.X]);

(ii) [mathematical expression not reproducible].

(C.3) The mapping h(t, x) [member of] D([(-A).sup.[beta]]) satisfies the following conditions: for any x, y [member of] X and t [greater than or equal to] 0, there exist nonnegative constants [L.sub.h], [C.sub.h] [greater than or equal to] 0 and [beta] [member of] (0, 1) such that

(i) [[parallel][(-A).sup.[beta]]h(t, x) - [(-A).sup.[beta]]h(t, y)[parallel].sub.X] [less than or equal to] [L.sub.h][[parallel]x - y[parallel].sub.X];

(ii) [[parallel][(-A).sup.[beta]]h(t, x)[parallel].sup.2.sub.X] [less than or equal to] [C.sub.h](1 + [[parallel]x[parallel].sub.2.sub.X]);

(iii) the constants [L.sub.h] and [beta] satisfy the following inequality k := [L.sub.h][parallel][(-A).sup.-[beta]][parallel] < 1.

(C.4) The mapping [(-A).sup.[beta]]h is continuous in mean square: for all x [member of] C([0, T]; [L.sup.2]([OMEGA]; X)), [lim.sub.t[right arrow]s] E[[parallel][(-A).sup.[beta]]h(t, x(t)) - [(- A).sup.[beta]]h(s, x(s))[parallel].sub.2.sub.X] = 0.

(C.5) The mapping [mathematical expression not reproducible] satisfies [mathematical expression not reproducible].

(C.6) The mapping u(x) takes values in [L.sup.2]([0, T]; U) and B is a bounded linear operator from U to X.

Theorem 6. Let conditions (C.1)-(C.6) be satisfied. Then system (1) has a unique mild solution on [-[tau], T].

Proof. Denote [B.sub.T] := C([-[tau], T]; [L.sup.2]([OMEGA]; X)) by subspace of all continuous functions form [-[tau], T] into [L.sup.2]([OMEGA]; X) equipped with the norm [mathematical expression not reproducible]. Fix T > 0 and consider the space [S.sub.T] := {x [member of] [B.sub.T] : x(s) = [phi](s), for s [member of] [-[tau],0]}. [S.sub.T] is a closed subspace of [B.sub.T] equipped with the norm [mathematical expression not reproducible]. Obviously, [S.sub.T] and [B.sub.T] are two Banach spaces. Define an operator [pi] : [S.sub.T] [right arrow] [S.sub.T] by [pi](x)(t) = [phi](t) for t [member of] [-[tau], 0] and for t [member of] [0,T],

[mathematical expression not reproducible]. (17)

We first verify that [pi] is continuous in mean square on [0, T]. Let x [member of] [S.sub.T], 0 [less than or equal to] t < T, and let [epsilon] be positive and sufficiently small (similar estimates hold for [epsilon] < 0), then

[mathematical expression not reproducible]. (18)

It is easy to know that E[[parallel][K.sub.i]([epsilon])[parallel].sup.2.sub.X] [right arrow] 0 as [epsilon] [right arrow] 0, i = 1, 2, 4, 7.

Further, by using Cauchy-Schwarz inequality, we get

[mathematical expression not reproducible]. (19)

Applying conditions (C.1) and (C.3) to E[[parallel][K.sub.31]([epsilon])[parallel].sup.2.sub.X], we can obtain

[mathematical expression not reproducible] (20)

as [epsilon] [right arrow] 0. Applying condition (C.1) and (C.3) to E[[parallel][I.sub.32]([epsilon])[parallel].sup.2.sub.X], we have

[mathematical expression not reproducible] (21)

as [epsilon] [right arrow] 0. That is to say, E[[parallel][K.sub.3]([epsilon])[parallel].sup.2.sub.U] [right arrow] 0 as [epsilon] [right arrow] 0.

For the fifth term, by using Lemma 3, we can obtain

[mathematical expression not reproducible] (22)

as [epsilon] [right arrow] 0.

For the sixth term, we get

[mathematical expression not reproducible]. (23)

Firstly, apply Lemma 2 to E[[parallel][K.sub.61]([epsilon])[parallel].sup.2.sub.X], and we have

[mathematical expression not reproducible] (24)

as [epsilon] [right arrow] 0. For every s fixed, we get S([epsilon])[sigma](s) [right arrow] [sigma](s) as [epsilon] [right arrow] 0 and [mathematical expression not reproducible].

By Lemma 2 and the Lebesgue dominated convergence theorem, we can obtain

[mathematical expression not reproducible] (25)

as [epsilon] [right arrow] 0. Therefore E[[parallel][K.sub.6]([epsilon])[parallel].sup.2.sub.X] [right arrow] 0 as [epsilon] [right arrow] 0. Thus, [pi] is continuous in mean square on [0, T]. So we conclude that [pi]([S.sub.T]) [subset] [S.sub.T].

Moreover, we shall show that [pi] is contractive in [mathematical expression not reproducible] with some 0 < [T.sub.1] [less than or equal to] T. Let x, y [member of] [S.sub.T] and for any fixed t [member of] [0,T], we get

[mathematical expression not reproducible]. (26)

By conditions (C.1)-(C.4) and Lemma 3, we have

[mathematical expression not reproducible]. (27)

Hence

[mathematical expression not reproducible], (28)

where

[mathematical expression not reproducible]. (29)

By condition (C.3), we have [gamma](0) = k < 1, then there exists 0 < [T.sub.1] [less than or equal to] T such that 0 < [gamma]([T.sub.1]) < 1 which shows that [pi] is a contraction mapping in [mathematical expression not reproducible]; therefore [pi] has a unique fixed point in [mathematical expression not reproducible], which is a unique mild solution of (1) on [-[tau], [T.sub.1]]. Repeat this procedure to extend the mild solution to [[tau], T]. The proof is completed.

Remark 7. Boufoussi and Hajji in  and Liu and Luo in , respectively, considered the existence and uniqueness of solutions for special cases of (1). In (20) and (21), we use the Holder inequality and linear growth condition to obtain the continuity of [K.sub.3]([epsilon]) and obtain a weaker result than those of [8,13]. The necessary conditions both in [8,13] are

1/2 < [beta] < 1. (30)

However, our condition is

0 < [beta] < 1. (31)

In this sense, this paper improves and generalizes the results in [8,13].

4. Controllability Result

In this section, we focus on the controllability problem of (1). Now we introduce the concept of controllability of neutral stochastic delay partial differential equations.

Definition 8. System(1) is said to be controllable on the finite interval [-[tau], T], if, for each initial stochastic process [phi] defined on [-[tau], 0] and [x.sub.1] [member of] X, there exists a stochastic control u [member of] [L.sup.2]([0, T]; U) which is adapted to the filtration [{[F.sub.t]}.sub.t[greater than or equal to]0] such that the mild solution x(x) of (1) satisfies x(T) = [x.sub.1], where [x.sub.1] and T are the preassigned terminal state and time, respectively.

In order to set the controllability problem, we assume that the following conditions hold:

(C.7) The linear operator [omega] from U to X defined by [omega]u = [[integral].sup.T.sub.0] S(T - s)Bu(s)ds has an inverse operator [[omega].sup.-1] with values in [L.sup.2]([0, T]; U) \ ker[omega], where ker[omega] = {x [member of] [L.sup.2]([0, T]); U) : [omega]x = 0} (see ), and there exists a pair of finite positive constants [M.sub.B] and [M.sub.[omega]] such that [parallel]B[parallel] [less than or equal to] [M.sub.B] and [parallel][[omega].sup.-1][parallel] [less than or equal to] [M.sub.[omega]].

Theorem 9. Let conditions (C.1)-(C.7) be satisfied. Then, system (1) is controllable on [-[tau], T].

Proof. Using the condition (C.7) for an arbitrary mapping x(x), define the stochastic control by

[mathematical expression not reproducible]. (32)

Applying this control to the operator [pi]. Obviously, to find a fixed point for the operator [pi] is equivalent to prove the existence of mild solutions of equation. Clearly, [pi](T) = [x.sub.1], which implies that the stochastic control u steers the system from the initial state [phi] to [x.sub.1] in time T, provided we can find a fixed point of the operator [pi] which means that the system is controllable on [-[tau], T].

First, we shall prove that [pi] is continuous in mean square on [0, T]. Let x [member of] [S.sub.T], 0 [less than or equal to] t < T, and [epsilon] be positive and sufficiently small (similar estimates hold for [epsilon] < 0), then

[mathematical expression not reproducible]. (33)

From Section 3, we can obtain that E[[parallel][K.sub.i]([epsilon])[parallel].sup.2.sub.X] [right arrow] 0 as [epsilon] [right arrow] 0, i = 1, 2,..., 6. Now we estimate the term E[[parallel][K.sub.7]([epsilon])[parallel].sup.2.sub.X], and we have

[mathematical expression not reproducible]. (34)

Applying conditions (C.1)-(C.7) and Lemmas 2-4 to E[[parallel][K.sub.71]([epsilon])[parallel].sup.2.sub.X], we get

[mathematical expression not reproducible]. (35)

Then E[[parallel][K.sub.71]([epsilon])[parallel].sup.2.sub.X] [right arrow] 0 as [epsilon] [right arrow] 0. Using the similar technique to E[[parallel][K.sub.72]([epsilon])[parallel].sup.2.sub.X], we have

[mathematical expression not reproducible]. (36)

Then E[[parallel][K.sub.72]([epsilon])[parallel].sup.2.sub.X] [right arrow] 0 as [epsilon] [right arrow] 0.

Thus, [pi] is indeed continuous in mean square on [0, T]. So we conclude that [pi]([S.sub.T]) [subset] [S.sub.T].

Last, we will show that [pi] is a contraction mapping in [mathematical expression not reproducible] with some 0 < [T.sub.1] [less than or equal to] T. Let x, y [member of] [S.sub.T], as proceeding as we did previously; we can obtain

[mathematical expression not reproducible]. (37)

Thus

[mathematical expression not reproducible]. (38)

Hence

[mathematical expression not reproducible], (39)

where

[mathematical expression not reproducible]. (40)

By condition (C.3), we know that b(0) = k < 1 then there exists 0 < [T.sub.1] [less than or equal to] T such that 0 < [gamma]([T.sub.1]) < 1 which shows that [pi] is a contraction mapping in [mathematical expression not reproducible]. Then [pi] has a unique fixed point in [mathematical expression not reproducible], which is a unique mild solution of (1) on [-[tau], [T.sub.1]]. Repeating this procedure can be used to extend the mild solution to the entire interval [-[tau], T]. Obviously, [pi](x)(T) = [x.sub.1] which means that system (1) is controllable on [-[tau], T]. This completes the proof.

5. An Example

In this section, we provide an example to illustrate our main results. Consider the following NSDPDE with finite delays [[tau].sub.1], [[tau].sub.2], and [[tau].sub.3] (0 [less than or equal to] [[tau].sub.i] [less than or equal to] [tau] < [infinity], i = 1, 2, 3):

[mathematical expression not reproducible], (41)

where W(t) is a cylindrical Brownian motion and [B.sup.H.sub.Q](t) is a cylindrical fractional Brownian motion in X.

Let X = [L.sup.2]([0, [pi]]) and define the operator A : D(A) [subset] X [right arrow] X by A = [[partial derivative].sup.2]/[partial derivative][[xi].sup.2] with the domain

D(A) := {z [member of] X, z, z' are absolutely continuous on, z" [member of] X, z(0) = z([pi]) = 0}. (42)

It is well known that an analytic semigroup [{S(t)}.sub.t[greater than or equal to]0] is generated by the operator A in X satisfying [[parallel]S(t)[parallel].sub.X] [less than or equal to] [e.sup.-t]. Furthermore,

Au = [[infinity].summation over (i=1)] [n.sup.2] <u, [e.sub.n]> [e.sub.n], u [member of] D(A), (43)

where [e.sub.n](x) = [square root of (2/[pi])] sin(nx) is a complete orthonormal set of eigenvectors of A.Then, the following properties hold:

(i) If y [member of] D(A), then Ay = [[SIGMA].sup.[infinity].sub.n=1] [n.sup.2] <y, [e.sub.n]>[e.sub.n].

(ii) For each y [member of] X, [A.sup.-1/2] y = [[SIGMA].sup.[infinity].sub.n=1](1/n) <y, [e.sub.n]>[e.sub.n] . In particular, [[parallel][A.sup.-1/2][parallel].sub.X] = 1.

(iii) The operator [A.sup.1/2] is given by

[A.sup.1/2] y = [[infinity].summation over (n=1)] n <y, [x.sub.n]> [e.sub.n] (44)

on the space D[[A.sup.1/2]] = {y(x) [member of] X, [[SIGMA].sup.[infinity].sub.n=1] n <y, [e.sub.n]> [e.sub.n][SIGMA]}.

We assume that the following conditions hold:

(1) Let B : U [right arrow] X be a bounded linear operator defined by

Bu(t) ([xi]) = [mu](t, [xi]) , 0 [less than or equal to] [xi] < [pi], u [member of] [L.sup.2] ([0, [pi]], U). (45)

(2) The operator [omega] : [member of] [L.sup.2]([0,[pi]],U) [right arrow] X is given by

[omega]u ([xi]) = [[integral].sup.T.sub.0] S(T - s) [mu](s, [xi]) ds, 0 [less than or equal to] [xi] [less than or equal to] [pi], (46)

has a bounded invertible operator [[omega].sup.-1], and satisfies condition (C.7). For the construction of the operator [omega] and its inverse, see  for more details.

(3) There exist nonnegative constants [L.sub.h] < 1, [L.sub.f], and [L.sub.g] such that, for all t [member of] [-[tau],T], x, y [member of] X, and [xi] [member of] [0,[pi]],

[mathematical expression not reproducible]. (47)

(4) There exist nonnegative constants [C.sub.f], [C.sub.g], and [C.sub.h], such that, for all t [member of] [-[tau],T], x [member of] X, and [xi] [member of] [0,[pi]],

[mathematical expression not reproducible]. (48)

(5) The mapping [(-A).sup.1/2] H is continuous in mean square on [-[tau], T]: for all t [member of] [-[tau],T], x [member of] X, and [xi] [member of] [0,[pi]], we have

[mathematical expression not reproducible]. (49)

(6) The mapping [mathematical expression not reproducible] satisfies

[mathematical expression not reproducible]. (50)

Define the operators F, G, and H : [R.sup.+] x [L.sup.2]([0, [pi]]) [right arrow] [L.sup.2]([0, [pi]]) by

[mathematical expression not reproducible]. (51)

If we put

x(t)([xi]) = x(t, [xi]), t [member of] [0, T], [xi] [member of] [0, [pi]], x(t, [xi]) = [phi](t, [xi]), t [member of] [-[tau], 0], [xi] [member of] [0, [pi]], (52)

then problem (41) can be written in the abstract form

[mathematical expression not reproducible] (53)

Thus all the conditions of Theorems 6 and 9 are fulfilled. Therefore, we conclude that the system (41) has a unique mild solution which is controllable on [-[tau], T].

6. Conclusion

In this paper, by using the Banach fixed point theorem, we obtain some sufficient conditions ensuring the existence, uniqueness, and controllability of neutral stochastic delay partial differential equations driven by standard Brownian motion and fractional Brownian motion. In our next paper, we will explore the existence, uniqueness, and controllability problems with non-Lipschitz condition.

https://doi.org/10.1155/2018/7502514

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by the National Natural Science Foundation of China under Grant no. 11271093 and the Innovation Research for the Postgraduates of Guangzhou University under Grant no. 2017GDJC-D09.

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Dehao Ruan and Jiaowan Luo (iD)

Department of Probability and Statistics, School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong 510006, China

Correspondence should be addressed to Jiaowan Luo; jluo@gzhu.edu.cn

Received 30 November 2017; Revised 5 January 2018; Accepted 7 February 2018; Published 2 April 2018

Academic Editor: Chris Goodrich
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