 Research
 Open Access
 Published:
Nonlocal stochastic integrodifferential equations driven by fractional Brownian motion
Advances in Difference Equations volume 2016, Article number: 115 (2016)
Abstract
In this paper, we study the existence of mild solutions for nonlocal stochastic integrodifferential equations driven by fractional Brownian motions with Hurst parameter \(H>\frac{1}{2}\) in a Hibert space. Sufficient conditions for the existence of mild solutions are derived by means of the LeraySchauder nonlinear alternative. A special case of this result is given and an example is provided to illustrate the effectiveness of the proposed result.
Introduction
It is well known that fractional Brownian motion (fBm, for short) is a family of centered Gaussian processes with continuous sample paths indexed by the Hurst parameter \(H\in(0,1)\). fBm admits stationary increments and selfsimilarity, and it has a longmemory when \(H>\frac{1}{2}\). These compact and significant properties make fBm a natural candidate as a model for noise in a wide variety of physical phenomena, such as biological physics, condensed matter physics, telecommunication networks, mathematical finance, and so on (see [1] and the references therein). Therefore, it is interesting and important to investigate stochastic calculus with respect to fBm and related topics (we refer to [2–5] and references therein for a complete presentation of this subject).
Recently, stochastic differential equations driven by fBm have attracted a lot of attentions of works and the theory has been developed in different settings. Some interesting results of finitedimensional stochastic differential equations with fractional noise have been investigated by Hu et al. [6], Fan [7], Liu and Yan [8, 9], and references therein; the case of stochastic differential equations driven by fBm in infinitedimensional Hilbert spaces has also been studied extensively, for example, Boufoussi and Hajji [10] investigated the existence of neutral stochastic functional differential equations driven by fBm, Caraballo et al. [11] proved the existence and exponential behavior of mild solutions to stochastic delay evolution equations with fractional noise, and Duncan et al. [12] established the weak, strong and mild solutions to stochastic equations with multiplicative fractional noise.
Since it was introduced in 1990 by Byszewski and Lakshmikantham [13], the nonlocal Cauchy problems have been extensively studied in differential equations and dynamical systems [14–17]. It is demonstrated that the corresponding differential equations with nonlocal conditions more accurately describe the phenomena. For example, it is discovered in [16] that the nonlocal initial condition \(x(0)+g(t_{1},\ldots,t_{n},x(t_{1}),\ldots,x(t_{n}))=x_{0} \), \(0< t_{1}< t_{2}<\cdots <t_{n}\leq T\), has better effects in characterizing the diffusion phenomenon of a small amount of gas in a transparent tube than the classical Cauchy condition \(x(0)=x_{0}\). In the infinitedimensional framework, stochastic differential equations with nonlocal conditions driven by Brownian motion (i.e., the case \(H=\frac{1}{2}\)) have received a lot of attention during the last years. For example, Muthukumar et al. [18] studied the controllability of fractional stochastic integrodifferential equations with nonlocal conditions, the existence of solutions for stochastic functional differential equations has been discussed in [19, 20], the secondorder stochastic functional differential equations with nonlocal conditions have been investigated in [21, 22] and references therein. For recent important results of stochastic differential equations in Hilbert spaces, we refer to [23, 24] and the references therein.
In contrast, for \(H\neq\frac{1}{2}\), to the best of our knowledge, there is no work concerning the existence of mild solutions for stochastic evolution equations with nonlocal conditions. Therefore, the main objective of this paper is to fill this gap. Further, many existence results of stochastic differential equations with nonlocal conditions are valid only for the Lipschitz or compact assumptions on nonlocal items. The main purpose of this manuscript is to investigate the existence of nonlocal stochastic integrodifferential equations driven by fractional Brownian motion, for which the nonlocal items are valid for nonLipschitz and noncompact assumptions.
In this paper, we consider the existence of mild solutions for a class of stochastic integrodifferential equations of the following form:
in a real separable Hilbert space \(\mathcal{H}\) with inner product \((\cdot,\cdot)\) and norm \(\Vert \cdot \Vert \), where \(A(t)\) is a closed linear operator with a dense domain \(D(A)\), which is independent of t, \(B(t,s)\) is a bounded operator in \(\mathcal{H}\). \(B^{H}\) is a fBm with \(H\in(\frac{1}{2},1)\) on a real separable Hilbert space V. \(f:J\times\mathcal{H}\rightarrow\mathcal{H}\), \(\sigma\in L_{2}^{0}(V, \mathcal{H})\), \(g:C(J,\mathcal{H})\rightarrow \mathcal{H}\) are appropriate functions specified later. Here, \(L_{2}^{0}(V, \mathcal {H})\) denotes the space of all QHilbertSchmidt operators from V into \(\mathcal{H}\) (see Section 2 below). \(x_{0}\) is an \(\mathcal {F}_{0}\)measurable random variable independent of \(B^{H}\) with finite second moment.
A brief outline of this paper is given. In Section 2, we present some basic notations and preliminaries; in Section 3, the existence result of system (1.1) is investigated by means of the LeraySchauder nonlinear alternative, a special case of system (1.1) is also considered. An example is presented to illustrate the effectiveness of the main result.
Preliminaries
Throughout this paper, we assume that \(H\in(\frac{1}{2},1)\) unless otherwise specified. In this section, our goal is to introduce some useful results about fBm and the corresponding stochastic integral taking values in a Hilbert space. For more details on this section, we refer to Hu [2], Mishura [5] and the references therein.
Fractional Brownian motion
We begin by recalling the definition of a fBm. Let \((\Omega,\mathcal {F},P)\) be a complete probability space and \(H\in(0,1)\) be given. A fBm with Hurst parameter H is a continuous centered Gaussian process \(\{\beta^{H}(t); t\in\mathbb {R}\}\) with the covariance function
It is well known that fBm \(\beta^{H}(t)\) with \(H>\frac{1}{2}\) admits the Wiener integral representation of the following form:
where B is a standard Brownian motion and the kernel \(K_{H}(t,s)\) is given by
with \(c_{H}=\sqrt{\frac{H(2H1)}{\beta(22H,H\frac{1}{2})}}\), here \(\beta (\cdot,\cdot)\) denotes the Beta function.
Let \(T>0\) be an arbitrary fixed horizon and \(\phi\in L^{2}(0,T)\) be a deterministic function. It is well known from [25] that the Wiener integral of ϕ with respect to \(\beta^{H}\) is given by
where \(K_{H}^{\star}\phi(s)=\int_{s}^{T}\phi(r)\frac{\partial K_{H}}{\partial r}(r,s)\,dr\).
Stochastic integral in Hilbert space
We now introduce fBms with values in a Hilbert space and the corresponding stochastic integral. Let \(V=(V,(\cdot,\cdot)_{V}, \Vert \cdot \Vert _{V} )\) be a real, separable Hilbert space. \(\mathcal{L}(V,\mathcal{H})\) represents the space of all bounded linear operators from V to \(\mathcal{H}\). Let \(\{e_{n}, n\in N\}\) be a complete orthonormal basis in V and \(Q\in \mathcal{L}(V,\mathcal{H})\) be an operator defined by \(Qe_{n}=\lambda_{n}e_{n}\) with \(\operatorname {Tr}Q=\sum_{n=1}^{\infty}\lambda_{n}<+\infty\), where \(\{\lambda_{n}, n\in N\}\) are nonnegative real numbers. A Vvalued infinitedimensional fBm with covariance Q can be defined by
where \(\beta_{n}^{H}\) are real, independent fBms with the same Hurst parameter \(H\in(\frac{1}{2},1)\).
Let \(L_{2}^{0}(V,\mathcal{H})\) denote the space of all \(\Phi\in\mathcal {L}(V,\mathcal{H})\) such that \(\Phi Q^{\frac{1}{2}}\) is a HilbertSchmidt operator. The norm is defined by
Generally, Φ is called a QHilbertSchmidt operator from V to \(\mathcal{H}\).
Definition 2.1
Let \(\phi:[0,T]\rightarrow L_{2}^{0}(V,\mathcal{H})\) such that
Then the stochastic integral of ϕ with respect to the infinitedimensional fBm \(B^{H}\) is defined by
Lemma 2.1
[11]
If \(\phi:[0,T]\rightarrow L_{2}^{0}(V,\mathcal{H})\) satisfying
and for \(\tau,\sigma\in[0,T]\) with \(\sigma>\tau\), then
where \(C_{H}\) is a constant depending on H. If, in addition, \(\sum_{n=1}^{\infty} \Vert \phi(t) Q^{\frac{1}{2}}e_{n}\Vert \) is uniformly convergent for \(t\in[0,T]\), then
Remark 1
If \(\{\lambda_{n}\}_{n\in\mathbb{N}}\) is a bounded sequence of nonnegative real numbers such that the unclear operator Q satisfies \(Qe_{n}=\lambda e_{n}\), assuming that there exists a positive constant \(k_{\Phi}\) such that
then it is obvious that \(\sum_{n=1}^{\infty} \Vert \Phi(t) Q^{\frac {1}{2}}e_{n}\Vert \) is uniformly convergent for \(t\in[0,T]\).
Deterministic integrodifferential equations in Banach spaces
In this subsection, we recall some basic notations and properties needed in the sequel. For more details on this subsection, we refer to [26].
Let \((X,\Vert \cdot \Vert )\) be a Banach space, \(C(I;X)\) denotes the space of all continuous functions from I into X. We consider the following integraldifferential equation with nonlocal condition:
Definition 2.2
[26]
A resolvent operator for problem (2.1) is a bounded operatorvalued function \(R(t,s)\in\mathcal{L}(X,X)\), \(0\leq s\leq t\leq T\), such that the properties

(a)
\(R(s,s)=I \), \(0\leq s\leq T\), \(\Vert R(t,s)\Vert \leq Me^{\gamma(ts)}\) for some constants M and γ;

(b)
for each \(x\in X\), \(R(t,s)x\) is strongly continuous in s and t;

(c)
for each \(x\in X\), \(R(t,s)x\) is continuous differentiable in \(s\in[0,T]\), \(t\in[s,T]\)
and
with \(\frac{\partial R}{\partial t}(t,s)x\), \(\frac{\partial R}{\partial s}(t,s)x\) are strongly continuous on \(0\leq s\leq t\leq T\).
We mention here that the resolvent operator \(R(t,s)\) can be reduced from the evolution operator of the generator \(A(t)\) under some suitable conditions (see [26] for the details).
Definition 2.3
[26]
A continuous function \(u(\cdot):J\rightarrow X\) is said to be a mild solution to problem (2.1) if, for all \(u_{0}\in X\), it satisfies the following integral equation:
Hypotheses on stochastic system (1.1)
Let \(C(J,\mathcal{H}):= \{x:J\rightarrow\mathcal{H}, x \text{ is continuous almost surely}\}\). The collection of all strongly measurable, square integrable, \(\mathcal{H}\)valued random variables, denoted by \(L^{2}(\Omega,\mathcal{H})\), is a Banach space with the norm \(\Vert x\Vert _{2}=(E\Vert x\Vert ^{2})^{\frac{1}{2}}\). Let \(C(J,L^{2}(\Omega,H))\) be the Banach space of all continuous mappings from J into \(L^{2}(\Omega,H)\) such that \(\sup_{t\in J}E\vert x(t)\vert _{H}^{2}<\infty\). We denote by \(\mathcal{C}\) the space of all \(\mathcal{F}_{t}\)adapted stochastic processes \(x\in C(J,L^{2}(\Omega,H))\) equipped with the norm \(\Vert x\Vert _{\mathcal{C}}=(\sup_{s\in J}E\vert x(s)\vert _{H}^{2})^{\frac{1}{2}}\), it is clear that \((\mathcal{C},\Vert \cdot \Vert _{\mathcal{C}})\) is a Banach space.
In this work, we need the following assumptions:
 \((A_{1})\) :

The resolvent operator \(R(t,s)\), \(0\leq s\leq t\), is compact and there exist some positive constants M, γ such that \(\Vert R(t,s)\Vert \leq Me^{\gamma( ts)}\).
 \((A_{2})\) :

There exists a constant \(K_{f}>0\) such that for all \(x,y\in\mathcal{H}\), \(t\in[0,T]\),
$$\bigl\Vert f(t,x)f(t,y) \bigr\Vert ^{2}\leq K_{f} \Vert xy\Vert ^{2}. $$  \((A_{3})\) :

\(\sigma\in L_{2}^{0}(V,\mathcal{H})\) with \(\sum_{n=1}^{\infty} \Vert \sigma Q^{\frac{1}{2}}e_{n}\Vert _{L_{2}^{0}}<\infty\).
 \((A_{4})\) :

\(g:C(J,\mathcal{H})\rightarrow\mathcal{H}\) is square integrable and satisfies:

(4a)
there exists a constant \(\alpha\in(0,T)\) such that \(g(\xi)=g(\eta)\) for any \(\xi,\eta\in C(J,\mathcal{H})\) with \(\xi=\eta\) on \([\alpha, T]\);

(4b)
there exists a nondecreasing function \(\Gamma:[0,+\infty)\rightarrow(0,+\infty)\) such that
$$E \bigl\vert g(x) \bigr\vert _{H}^{2}\leq\Gamma \bigl( \Vert x\Vert _{\mathcal{C}}^{2} \bigr). $$

(4a)
For simplicity, we also assume that \(f(t,0)=0\) for all \(t\in J\).
At the end of this section, we recall the wellknown LeraySchauder nonlinear alternative, which is employed to prove our results.
Lemma 2.2
[27]
Let X be a closed and convex subset of a Banach space B. Assume that \(\mathcal{U}\) is a relatively open subset of X with \(0\in\mathcal {U}\) and \(\phi: \bar{\mathcal{U}}\rightarrow X\) is a compact map, then either

(a)
ϕ has a fixed point in \(\bar{\mathcal{U}}\), or

(b)
there is a point \(u\in\partial\mathcal{U}\) and \(\lambda\in(0,1)\) with \(u\in\lambda\phi(u)\).
Main results
In this section, we state and establish our main result. We first present the definition of the mild solution for system (1.1).
Definition 3.1
A \(\mathcal{H}\)valued, \(\mathcal{F}_{t}\)adapted stochastic process \(x(t)\) is called a mild solution of (1.1), if, for all \(t\in[0,T]\), \(x(t)\) satisfies
Theorem 3.1
Let assumptions \((A_{1})\)\((A_{4})\) be satisfied. Then the system (1.1) has at least one mild solution provided that there exists a constant \(N_{0}>0\) such that
Proof
We first introduce an equivalent norm \(\Vert \cdot \Vert _{\tilde{\mathcal{C}}}\) in the space \(\mathcal{C}\) defined by
It is routine to check that \(\tilde{\mathcal{C}}:=(\mathcal{C},\Vert \cdot \Vert _{\tilde{\mathcal{C}}})\) is a Banach space. Let \(u\in\mathcal{C}\) be fixed. For \(t\in J\), \(x\in\tilde{\mathcal{C}}\), define a mapping \(P_{u}\) on \(\tilde{\mathcal{C}}\) by
Since \(R(t,s)\) is strongly continuous, by the assumptions on f, g, and σ, it is easy to check that \(P_{u}\) maps \(\tilde{\mathcal{C}}\) into itself. For \(x,y\in\tilde{\mathcal{C}}\), by \((A_{1})\) and Hölder’s inequality, we have
recalling the choice of L, we have
which implies that \(\Vert P_{u}xP_{u}y\Vert _{\tilde{\mathcal{C}}}^{2}\leq\frac {1}{2}\Vert xy\Vert _{\tilde{\mathcal{C}}}^{2}\). By the Banach contraction mapping principle, it follows that \(P_{u}\) has a unique fixed point \(x_{u}\in \tilde{\mathcal{C}}\) such that
Based on this fact, for some \(\alpha\in(0,T)\), we set
From (3.2), it follows that
Let \(\mathcal{C}_{\alpha}:=C([\alpha,T],L^{2}(\Omega,H))\). Define a map \(\Phi:\mathcal{C}_{\alpha}\rightarrow\mathcal{C}_{\alpha}\) by
We will show that the operator Φ satisfies all the conditions of Lemma 2.2. The proof will be divided into the following steps.
Step 1. Φ maps bounded sets into bounded sets in \(C_{\alpha}\). Let \(r>0\) and
It is obvious that \(B_{r}(\alpha)\) is a bounded closed convex subset in \(\mathcal{C}_{\alpha}\). We now show that there exists a constant \(L_{0}>0\) such that \(\Vert \Phi v\Vert _{\mathcal{C}}^{2}\leq L_{0}\) for each \(v\in B_{r}(\alpha)\).
Let \(v\in B_{r}(\alpha)\) and \(t\in J\). Recalling the definition of Φ, by the assumptions \((A_{1})\)\((A_{4})\) together with Hölder’s inequality we have
where \(L_{1}=6M^{2}[E\Vert x_{0}\Vert ^{2}+\Gamma(r)]+\frac{3M^{2}C_{H}T^{2H1}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}}{2\gamma}\). This gives, by Gronwall’s inequality,
Step 2. Φ maps bounded sets into equicontinuous sets of \(\mathcal{C}_{\alpha}\). Let \(v\in B_{r}(\alpha)\) and \(\alpha\leq t_{1}< t_{2}\leq T\), we have
By the strong continuity of \(R(t,s)\) and the assumption of g, we have
For the term \(I_{2}\), by Hölder’s inequality and \((A_{1})\) we have
applying properties (b) of Definition 2.2 we get
by the assumption on f, we further derive that
we conclude, by the dominated convergence theorem, that the first term of (3.4) tends to 0 independently of v. Similarly, we can deduce that the second term of (3.4) also tends to 0 independently of v.
For \(I_{3}\), by Lemma 2.1 and the properties of \(R(t,s)\) we have
where \(C_{H}\) is the constant in Lemma 2.1.
Thus, the right hand side of (3.3) tends to 0 independently of v when \(t_{1}\rightarrow t_{2}\). From the assumption \((A_{1})\), the compactness of \(R(t,s)\), \(t>s>0\) implies the continuity in the uniform operator topology. Therefore, the set \(\{\Phi v,v\in B_{r}(\alpha)\}\) is equicontinuous.
Step 3. Φ maps \(B_{r}(\alpha)\) into a relatively compact set in \(\mathcal{H}\). That is, for every \(t\in[\alpha,T]\), the set \(\Pi (t):=\{(\Phi v)(t),v\in B_{r}(\alpha)\}\) is relatively compact in H.
Let \(t\in[\alpha,T]\) be fixed and \(0<\epsilon<t\). For \(v\in B_{r}(\alpha )\), we define an operator \(\Phi^{\epsilon}\) on \(B_{r}(\alpha)\) by
By the compactness of \(R(t,s)\), \(t,s>0\), one sees that the set \(\Phi^{\epsilon}(t)=\{( \Phi^{\epsilon}v)(t),v\in B_{r}(\alpha)\}\) is relatively compact in H for each \(0<\epsilon<t\). Moreover, applying Hölder’s inequality together with Lemma 2.1 we have
which implies that the relatively compact set \(\{(\Phi^{\epsilon }v):v\in B_{r}(\alpha)\}\) arbitrarily close to the set \(\{(\Phi v):v\in B_{r}(\alpha)\}\). Thus, the set \(\{(\Phi v),v\in B_{r}(\alpha)\}\) is relatively compact in H.
Step 4. We show that there exists an open set \(\mathbf {D}\subseteq\mathcal{C}_{\alpha}\) such that, for all \(v\in\partial\mathbf{D}\), \(v\notin\{\lambda(\Phi v):\lambda\in (0,1)\}\).
Let \(v\in\mathcal{C}_{\alpha}\) be a possible solution of \(v=\lambda(\Phi v)\) for some \(\lambda\in(0,1)\). Then, for each \(t\in J\), we have
It is clear that
On the other hand, by the assumptions on \(R(t,s)\), f, and g, we can use a similar argument to before, yielding
Applying Gronwall’s inequality it follows that
Recalling (3.5), we have
That is,
By assumption (3.1), there exists a constant \(M^{\star}\) such that \(\Vert v\Vert _{\mathcal {C}}^{2}\neq M^{\star}\).
Set
It is obvious that D is an open subset of \(B_{r}(\alpha)\) for all \(r\geq M^{\star}\). We infer that, for all \(v\in\partial\mathbf {D}\), \(v\notin\{\lambda(\Phi v):\lambda\in(0,1)\}\).
From step 1 to step 4 together with the ArzelaAscoli theorem, it suffices to show that \(\Phi:\bar{\mathbf{D}}\rightarrow\mathcal{C}_{\alpha}\) is a compact operator, where \(\bar{\mathbf{D}}\) is the closure of D. As a consequence of Lemma 2.2, it follows that Φ has a fixed point \(v_{0}\in\bar{\mathbf{D}}\). Let \(y=x_{v_{0}}\). By (3.2), we have
Noting that for \(t\in[\alpha,T]\),
which indicates, by the assumptions on g, that \(g(v_{0})=g(y)\). We conclude that \(y(t)\) is a mild solution of system (1.1). The proof is completed. □
We conclude this section with a comment on a special case of the nonlocal Cauchy problem, namely, where \(g(x)\) is given by \(g(x)=\sum_{i=0}^{k}g_{i}(x(t_{i}))\). More precisely, consider the following stochastic integraldifferential equation with nonlocal condition:
We need the following assumptions.
 \((A_{5})\) :

For every i, \(i=1,2,\ldots,k\), \(g_{i}(\cdot ):C(J,\mathcal{H})\rightarrow\mathcal{H}\) satisfy:

(5a)
there exists a constant \(\alpha\in(0,T)\) such that \(g_{i}(\xi)=g_{i}(\eta)\) for any \(\xi,\eta\in C(J,\mathcal{H})\) with \(\xi =\eta\) on \([\alpha, T]\);

(5b)
there exist some positive constants \(m_{i}\), \(n_{i}\), \(i=1,2,\ldots,k\) such that
$$\bigl\vert g_{i}(x) \bigr\vert ^{2}\leq m_{i} \Vert x\Vert ^{2}+n_{i},\quad\quad 6kM^{2}K^{\star} \sum_{i=1}^{k}m_{i}< 1, $$where \(K^{\star}=e^{6M^{2}T^{2}K_{f}+2\gamma T}\).

(5a)
Theorem 3.2
Under the assumptions \((A_{1})\)\((A_{3})\) and \((A_{5})\), the system (3.6) has at least one mild solution.
Proof
Since the proof resembles the arguments of Theorem 3.1, we sketch it only.
We define a map g on \(\mathcal{H}\) by
Let L, \(\tilde{\mathcal{C}}\), \(P_{u}\), ū, Φ, and \(B_{r}(\alpha)\) be as in Theorem 3.1. Thus, \(P_{u}\) has a unique fixed point \(x_{u}\). By (3.2), we have
By using the assumptions on f, g, and σ, we can deduce that
By Gronwall’s inequality it follows that
where
Noting that
we further derive that
Let \(\epsilon>0\) be fixed. Set \(M^{\star}=\frac{K^{\star}\bar {L}}{16M^{2}kK^{\star}\sum_{i=1}^{k}m_{i}}+\epsilon\) and
Then D is an open subset of \(B_{r}(\alpha)\) for all \(r\geq M^{\star}\). In the sequel, we can employ similar arguments to step 1step 4 of Theorem 3.1 and prove the existence of mild solutions for system (3.6), we omit it here. This completes the proof. □
An example
As an application, we present an example to illustrate our results. Considering the following stochastic integrodifferential equation with nonlocal condition:
where \(0< t_{1}< t_{2}<\cdots<t_{k}<T\), \(B^{H}(t)\) stands for a cylindrical fBm defined on a complete probability space \((\Omega, \mathcal{F},P,\{{\mathscr {F}}_{t}\} )\), \(h_{i}\in L^{2}([0,\pi])\), and \(a(t)\) are continuous functions.
To rewrite the stochastic differential equation in the abstract form (1.1), let \(\mathcal{H}=K=V=L^{2}([0,\pi])\) with the norm \(\Vert \cdot \Vert \). The operator \(A(t)\) is defined by \(A(t)x(z)=a(t)\frac{\partial ^{2}x}{\partial z^{2}}\) with the domain
Then \(A(t)\) generates an evolution operator and \(R(t,s)\) can be reduced by this evolution operator such that \((A_{1})\) is satisfied (see [26] for details).
Let
Then system (4.1) has an abstract formulation given by (3.6) and the assumptions \((A_{2})\), \((A_{3})\), and \((A_{5})\) are satisfied with \(K_{f}=1\), \(m_{i}=\sup_{z\in[0,\pi]}\Vert h_{i}(z)\Vert ^{2}\), \(n_{i}=0\). By Theorem 3.2, the system (4.1) has a mild solution on \([0,T]\) provided that \(6kM^{2}K^{\star }\sum_{i=1}^{k}m_{i}<1\).
Conclusion
In this paper, we have investigated the existence of mild solutions for a class of nonlocal stochastic integrodifferential equations driven by fractional Brownian motion in a Hilbert space. By employing the LeraySchauder nonlinear alternative, the existence of mild solutions is proved, a special case of this result is given. An example is presented to illustrate our theoretical results. In a sequel of this paper we will study the asymptotic behavior of the mild solutions, and we are also interested in studying the existence and asymptotic stability of stochastic evolution equations driven by fractional Brownian motion with the Hurst parameter \(H\in(0,\frac{1}{2})\).
References
ChristodoulouVolos, C, Siokis, FM: Long range dependence in stock market returns. Appl. Financ. Econ. 16, 13311338 (2006)
Biagini, F, Hu, YZ, et al.: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London (2010)
Hu, YZ: Integral transformations and anticipative calculus for fBms. Mem. Am. Math. Soc. 175, 825 (2005)
Hu, YZ, Nualart, D, Xu, FJ: Central limit theorem for an additive functional of the fractional Brownian motion. Ann. Probab. 32(1A), 168203 (2014)
Mishura, YS: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, Berlin (2008)
Hu, YZ, Lu, F, Nualart, D: FeynmanKac formula for the heat equation driven by fractional noise with Hurst parameter \(H < \frac{1}{2}\). Ann. Probab. 40(3), 10481063 (2012)
Fan, X: Integration by parts formula and applications for SDEs driven by fractional Brownian motions. Stoch. Anal. Appl. 33(2), 199212 (2015)
Liu, JF, Yan, LT: Solving a nonlinear fractional stochastic partial differential equation with fractional noise. J. Theor. Probab. 75(16), 141 (2014)
Liu, JF, Yan, LT: On a semilinear stochastic partial differential equation with doubleparameter fractional noises. Sci. China Math. 57(4), 855872 (2014)
Boufoussi, B, Hajji, S: Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Stat. Probab. Lett. 82(8), 15491558 (2012)
Caraballo, T, GarridoAtienza, MJ, Taniguchi, T: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74(11), 36713684 (2011)
Duncan, TE, Maslowski, B, PasikDuncan, B: Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stoch. Process. Appl. 115(8), 13571383 (2005)
Byszewski, L, Lakshmikantham, V: Theorem about the existence and uniqueness of a solution of a nonlocal Cauchy problem in a Banach space. Appl. Anal. 40, 1119 (1990)
Byszewski, L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494505 (1991)
Byszewski, L, Akca, H: Existence of solutions of a semilinear functional differential evolution nonlocal problem. Nonlinear Anal. 34, 6572 (1998)
Deng, K: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179, 630637 (1993)
Sobczyk, K: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic, London (1991)
Muthukumar, P, Rajivganthi, C: Approximate controllability of fractional order neutral stochastic integrodifferential system with nonlocal conditions and infinite delay. Taiwan. J. Math. 17(5), 16931713 (2013)
Balasubramaniam, P, Park, JY, Kumara, AVA: Existence of solutions for semilinear neutral stochastic functional differential equations with nonlocal conditions. Nonlinear Anal. 71, 10491058 (2009)
Yan, ZM: On a nonlocal problems for partial stochastic functional integrodifferential equations in Hilbert spaces. Electron. J. Math. Anal. Appl. 1(2), 212229 (2013)
Balasubramaniam, P, Park, JY: Nonlocal Cauchy problem for second order stochastic evolution equations in Hilbert space. Dyn. Syst. Appl. 16, 713728 (2007)
Cui, J, Yan, LT: Existence results for impulsive neutral secondorder stochastic evolution equations with nonlocal conditions. Math. Comput. Model. 57, 23782387 (2013)
Balasubramaniam, P, Tamilalagan, P: Approximate controllability of a class of fractional neutral stochastic integrodifferential inclusions with infinite delay by using Mainardis function. Appl. Math. Comput. 256, 232246 (2015)
Balasubramaniam, P, Kumaresan, N, et al.: Local and global existence of mild solution for impulsive fractional stochastic differential equation. Bull. Malays. Math. Sci. Soc. 38, 867884 (2015)
Nualart, D: Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)
Grimmer, R: Resolvent operators for integral equation in a Banach space. Transl. Am. Math. Soc. 273(1), 333349 (1982)
Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003)
Acknowledgements
The authors are grateful to the anonymous referees and the associate editor for valuable comments and suggestions to improve this paper. The Project is sponsored by NSFC (11571071, 11271020, 11401010), Mathematical Tianyuan Foundation of China (11426036), Zhejiang Provincial Natural Science Foundation (LQ16A010006) and Natural Science Foundation of Ningbo Municipality (2015A610158).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare to have no competing interests.
Authors’ contributions
All authors participated in drafting and checking the manuscript, and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Cui, J., Wang, Z. Nonlocal stochastic integrodifferential equations driven by fractional Brownian motion. Adv Differ Equ 2016, 115 (2016). https://doi.org/10.1186/s1366201608431
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366201608431
MSC
 60H15
 60H20
 34K45
Keywords
 stochastic integrodifferential equations
 fractional Brownian motion
 nonlocal condition