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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 823535, 6 pages
Mild Solutions of Neutral Semilinear Stochastic Functional Dynamic Systems with Local Non-Lipschitz Coefficients
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
Received 12 March 2013; Accepted 27 June 2013
Academic Editor: H. Srivastava
Copyright © 2013 Feng Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Semilinear stochastic dynamic systems in a separable Hilbert space often model some evolution phenomena arising in physics and engineering. In this paper, we study the existence and uniqueness of mild solutions to neutral semilinear stochastic functional dynamic systems under local non-Lipschitz conditions on the coefficients by means of the stopping time technique. We especially generalize and improve the results that appeared in Govinadan (2005), Bao and Hou (2010), and Jiang and Shen (2011).
Semilinear stochastic dynamic systems in a separable Hilbert space often model some evolution phenomena arising in physics, biology engineering, and so forth . Recently, for the case where the coefficients satisfy the global Lipschitz condition and the linear growth condition, many results are known [1–3]. However, the global Lipschitz condition, even the local Lipschitz condition, is seemed to be considerably strong when one discusses variable applications in real world. Reference  discussed the existence of mild solution to neutral semilinear stochastic functional systems with non-Lipschitz coefficients. Reference  discussed the existence and uniqueness of solutions to neutral stochastic functional systems with infinite delay under the local Lipschitz conditions in the Euclidean space.
We focus on neutral semilinear stochastic functional dynamic systems for the case where the coefficients do not necessarily satisfy the global Lipschitz condition. Thus we study the existence and uniqueness of mild solutions to neutral semilinear stochastic functional systems with the condition, which was investigated by [6–8] as the Carathéodory-type conditions for strong solutions. Motivated by the above papers, in this paper, we will extend the existence and uniqueness of mild solutions of (2) under the non-Lipschitz conditions to the local non-Lipschitz conditions.
Throughout this paper, let be a complete probability space with a normal filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). Moreover, let be two real separable Hilbert spaces, and we denote by their inner products and by their vector norms, respectively. We denote that denotes the space of all bounded linear operators from into , equipped with the usual operator norm . In this paper, we always use the same symbol to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let and denote the family of all continuous -valued functions from to with norm . Here let be the family of all almost surely bounded, -measurable, continuous random variables from to .
Let denote a -valued Wiener process defined on the probability space with covariance operator ; that is, , for all , where is a positive, self-adjoint, trace class operator on . In particular, we denote a -valued -Wiener process with respect to . To define stochastic integrals with respect to the -Wiener process , we introduce the subspace of which is endowed with the inner product and is a Hilbert space. We assume that there exists a complete orthonormal system in , a bounded sequence of nonnegative real numbers such that , and a sequence of independent Brownian motions such that and , where is the -algebra generated by . Let be the space of all Hilbert-Schmidt operators from to . It turns out to be a separable Hilbert space equipped with the norm for any . Obviously, for any bounded operators this norm reduces to .
Suppose that is an analytic semigroup with its infinitesimal generator ; for the literature related to semigroup theory, we suggest Pazy . We suppose , the resolvent set of . For any , it is possible to define the fractional power which is a closed linear operator with its domain .
Consider the following neutral semilinear stochastic functional systems: where can be regarded as a -valued stochastic process; are all Borel measurable; is the infinitesimal generator of an analytic semigroup of bounded linear operators , in .
Definition 1. A process , is called a mild solution of (2) if(i) is adapted to with a.s.;(ii) has càdlàg paths on a.s.: and, for each satisfies the integral equation, for any .
To guarantee the existence and uniqueness of a mild solution to (2), the following much weaker conditions, instead of non-Lipschitz condition, are described.(H1) is the infinitesimal generator of an analytic semigroup of bounded linear operators , in , and is uniformly bounded, for some constant .(H2) (a) There exists a function such that is locally integrable in for any fixed and is continuous monotone nondecreasing and concave in for any fixed . Moreover, for any fixed and , the following inequality is satisfied: (b) For any constant , the differential dynamic system has a global solution for any initial value .(H3) (a) There exists a function such that is locally integrable in for any fixed and is continuous monotone nondecreasing and concave in for any fixed . for any fixed . Moreover, for any fixed and , the following inequality is satisfied: (b) For any constant , if a nonnegative function satisfies that then holds for any .(H4) There exist a number and a positive such that, for any and and Moreover, we assume that .
Remark 2. Let , where is locally integrable and is a concave nondecreasing function from to such that for , and . Then by comparison theorem of differential dynamic systems we know that assumption (-b) holds.
Now let us give some concrete examples of the function . Let and let be sufficiently small. Define where denotes the derivative of function . They are all concave nondecreasing functions satisfying . In particular, we see that Lipschitz condition in  and non-Lipschitz conditions in  are special cases of our proposed condition.
To show our main results, we need the following lemma.
Lemma 3 (see ). If (H1) holds and , then, for any (i)for each , (ii)there exist positive constants and such that
3. Existence and Uniqueness of the Mild Solution
In this section, we will establish the existence and uniqueness of the mild solution. When the coefficients and satisfy the global Lipschitz condition and the linear growth condition, [7, 8] discussed the existence and uniqueness of the mild solution to stochastic differential dynamic systems. In , the existence and uniqueness of the mild solution of (2) under the non-Lipschitz condition are given as follows.
Remark 5. If for some constant , then the condition (H3) implies a global Lipschitz condition, which is studied in . In Remark 2, if , it is studied in . Therefore, some of the results [3, 4] are improved and generalized.
Now, we will replace (H3) by the following local non-Lipschitz condition.(a) For any integer , there exists a function such that is locally integrable in for any fixed and is continuous monotone nondecreasing and concave in with . Moreover, for any fixed and with , the following inequality is satisfied: (b) For any constant , if a nonnegative function satisfies that then holds for any .
Remark 6. Equation (13) is a generalization of the local Lipschitz condition. is the local non-Lipschitz condition of this type for the end of wider applications.
Proof. For any , define the truncation functions and as follows:
and then the functions and satisfy (H2) and
By Theorem 4, there exist the unique mild solutions and , respectively, to the following stochastic systems:
Define the stopping time . Recall that, for , , . Hence, for some ,
From (H4), we have
Applying the Hölder inequality, (H4), and Lemma 3, we have
For , we see that and . Thus, by () and the Jensen inequality, we obtain
By (), Liu [1, Theorem page 14] and the Jensen inequality, there exists a positive constant such that
From (19)–(23), choosing we have
By (22) and the Gronwall inequality, there exists a constant such that
By (-b), we have which means thats for , we obtain
For each , there exists an such that . Define by . Since , then we have
Hence we have that is a mild solution to (2). The proof is complete.
4. Concluding Remarks
In the paper, by means of the stopping time technique, the conditions to assure the existence and uniqueness of mild solutions of neutral semilinear stochastic functional dynamic systems are given under the local non-Lipschitz condition, which is a generalization of the local Lipschitz condition. The paper generalizes the results in [8, 10].
The work is supported by the Fundamental Research Funds for the Central Universities under Grant 2722013JC080, China Postdoctoral Science Foundation funded project under Grant 2012M511615, the Natural Science Foundation of Hubei Province of China and the National Science Foundation of China.
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