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 Open Access
Nonlocal stochastic integrodifferential equations driven by fractional Brownian motion
 Jing Cui^{1}Email author and
 Zhi Wang^{2}
https://doi.org/10.1186/s1366201608431
© Cui and Wang 2016
Received: 20 January 2016
Accepted: 15 April 2016
Published: 26 April 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.
Keywords
 stochastic integrodifferential equations
 fractional Brownian motion
 nonlocal condition
MSC
 60H15
 60H20
 34K45
1 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.
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.
2 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.
2.1 Fractional Brownian motion
2.2 Stochastic integral in Hilbert space
Definition 2.1
Lemma 2.1
[11]
Remark 1
2.3 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].
Definition 2.2
[26]
 (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 \(s\in[0,T]\), \(t\in[s,T]\)
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).
2.4 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.
 \((A_{1})\) :

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)
At the end of this section, we recall the wellknown LeraySchauder nonlinear alternative, which is employed to prove our results.
Lemma 2.2
[27]
 (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 \(u\in\lambda\phi(u)\).
3 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
Theorem 3.1
Proof
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.
Step 4. We show that there exists an open set \(\mathbf {D}\subseteq\mathcal{C}_{\alpha}\) such that, for all \(v\in\partial\mathbf{D}\), \(v\notin\{\lambda(\Phi v):\lambda\in (0,1)\}\).
 \((A_{5})\) :

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 thatwhere \(K^{\star}=e^{6M^{2}T^{2}K_{f}+2\gamma T}\).$$\bigl\vert g_{i}(x) \bigr\vert ^{2}\leq m_{i} \Vert x\Vert ^{2}+n_{i},\quad\quad 6kM^{2}K^{\star} \sum_{i=1}^{k}m_{i}< 1, $$
 (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.
4 An example
5 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 \(H\in(0,\frac{1}{2})\).
Declarations
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).
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.
Authors’ Affiliations
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