Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 970120, 13 pages
http://dx.doi.org/10.1155/2014/970120

Research Article

Existence of Random Attractors for a Class of Second-Order Lattice Dynamical Systems with Brownian Motions

Yamin Wang,1,2 Ziqiao Huang,¹ Fuad E. Alsaadi,³ Stanislao Lauria,⁴ and Yurong Liu1,3

¹Department of Mathematics, Yangzhou University, Yangzhou 225002, China
²Department of Basis Course, Lianyungang Technical College, Lianyungang 222006, China
³Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
⁴Department of Computer Science, Brunel University, Uxbridge, Middlesex UB8 3PH, UK

Received 25 September 2014; Accepted 2 November 2014; Published 3 December 2014

Academic Editor: Bo Shen

Copyright © 2014 Yamin Wang et al. 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.

Abstract.

This paper is concerned with the random attractors for a class of second-order stochastic lattice dynamical systems. We first prove the uniqueness and existence of the solutions of second-order stochastic lattice dynamical systems in the space . Then, by proving the asymptotic compactness of the random dynamical systems, we establish the existence of the global random attractor. The system under consideration is quite general, and many existing results can be regarded as the special case of our results.

1. Introduction

We consider the following second-order stochastic lattice dynamical system:

where , are real-value functions on ; and are given vectors satisfying bounded conditions; ; and are nonlinear terms satisfying some growth assumptions to be given later; is the linear operator on . In (1), , where and denotes the element having at position and all the other components and are independent two-side Brownian motions.

Lattice dynamical systems (LDSs) are infinite systems of ordinary differential equations, modeled on an underlying spatial lattice with some regular structure, for example, the integer lattice in the plane. Such systems arise as models in many applications, including image processing and pattern recognition, electrical engineering laser systems, biology, and material science; see [1–8] and the references therein. LDSs in one sense lie between ordinary and partial differential equations, but very often they exhibit new phenomena not found in either of these fields. LDSs raise a host of challenges to the researcher and are of broad interest to scientists and mathematicians.

So far, various properties of solutions about LDSs have been studied by many authors, such as the traveling solutions, the chaotic properties of solutions, and the phenomena of synchronization (see, e.g., [4, 5, 9]). One of the most important problems in mathematical physics is understanding of the asymptotic behavior of dynamical systems. Global attractor theory is an important tool to study the asymptotic behavior of infinite dimensional systems. For dissipative infinite dimensional dynamical systems given by partial differential equations, global attractor theory has been well developed; see [10, 11] and references therein. Recently, the long-term behavior of LDS has gained the extensive attention. For the LDSs without noise, the first result on existence of global attractors was established by Bates et al. in [12]. Since then, much work has been done for either first-order or second-order deterministic LDS (see, e.g., [12, 13]).

On the other hand, when modeling real world systems, stochastic disturbance is probably one of the main resources of the performance degradations of the dynamical systems, since the actual dynamic behavior is very often a noisy process brought on by random fluctuations from probabilistic causes. Stochastic systems have found successful applications in more and more branches of science and engineering. Random attraction as an interesting dynamic behavior has received increasing research attention. For stochastic partial differential equations, Ruelle has initiated the study of global random attractors in [14]. And the fundamental theory of global attractors for stochastic partial differential equations has been established and developed by Crauel, Debussche, Flandoli, Schmalfuss, and others; see, for example, [15–18] and the references therein. Very recently, much attention has been focused on lattice dynamical system with stochastic noises. Bates et al. [19] first studied the existence of global random attractor for a class of first-order dynamical systems driven by white noises on lattice . Then, Lv and Sun [20] have extended the result in [19] to generalized first-order stochastic systems on the lattice . For some latest results on first-order random attractors, we refer readers to, for example, [21] and the references therein.

For the second-order SLDS with stochastic noises on the lattice or , the existence of the random attractor is receiving the attention from research community [22, 23]. For example, [22] investigated the asymptotic behavior for a class of second-order stochastic lattice dynamical systems and proved the existence of the random attractor for the concerned second-order SLDS. Paper [23] addressed the asymptotic behavior of solutions to second-order SLDS with random coupled coefficients and multiplicative white noises in weighted spaces of infinite sequences and discussed the existence of a tempered random bounded absorbing set and a random attractor for the SLDS. However, the asymptotic behavior of second-order SLDS has not yet been fully investigated because of the technical complexity and remains open and challenging. In this paper, based on the idea of [13, 19], we aim to prove the existence of a global random attractor for a class of second-order SLDS (1). It is worth pointing out that the second-order SLDS considered in this paper is quite general, and many existing results can be viewed as the special cases of our results.

This paper is organized as follows. In Section 2, we introduce some basic concepts related to stochastic dynamical systems and the global random attractor. Meanwhile, we present some notations and give a simple description of our system. In Section 3, Some bounded conditions and assumptions of nonlinear terms are given, and the existence and uniqueness of solutions of system (1) are established. In Section 4, we prove the existence of an absorbing set. In Section 5, we establish the existence conditions for a global random attractor of system (1), and some concluding remarks are given in Section 6.

2. Preliminaries and Equivalent Norm

2.1. Preliminaries

In this subsection, we recall some basic concepts about random dynamical systems and the definition of random global attractor (see [17, 19, 24] for details).

Let be a Hilbert space and a probability space. Denote as a collection of random subsets of . A continuous random dynamical system over , , is defined as follows.

Definition 1. A stochastic process is a continuous random dynamical system over , , , if is -measurable and, for all ,

(S1)the mapping is continuous;(S2) is the identity operator on ;(S3) for all (cocycle property).

Definition 2. A random bounded set is called tempered with respect to , if, for all ,

where .

Definition 3. A random set is called an absorbing set in if, for all and a.e. , there exists such that

Definition 4. A random set is called a global random attractor for if the following hold:

(A1) is a random compact set; that is, is measurable for every , and is compact for a.e. ;(A2) is a strictly invariant set;(A3) attracts all sets in ; that is, for all and a.e. one has

where is Hausdorff semimetric (for any ). Also, the collection is called the domain of attraction of .

The following theorem is needed to prove the existence of global random attractor of system (1).

Theorem 5. Let be an absorbing set for the continuous random dynamical system . Suppose the set is closed and, for a.e. , satisfies the following asymptotic compactness condition: each sequence (when ) has a convergent subsequence in . Then, has a unique global random attractor

Proof. The proof of this theorem is similar to [12], so it is omitted here.

2.2. Equivalent Norm

Let

For all , we define the inner product and norm as follows:

In this paper, we assume that the linear operator in (1) can be decomposed as

where and are bounded linear operators defined by

It follows readily that

For all , , define the bilinear forms by

where ( and are given constants).

Now, we prove that the spaces and are equivalent spaces.

Lemma 6. The two bilinear forms , in (7) and (11) are both the inner products, and the resulting norms in (7) and in (11) are equivalent.

Proof. It is easy to check that the two bilinear forms and are both the inner products. We now only need to show that the norms and are equivalent. Noticing that

it follows that the norms and are equivalent. The proof is completed.

Let , where . From Lemma 6, we know that is a Hilbert space.

3. Existence and Uniqueness of Solutions

In this section, we will deal with the existence and uniqueness of solutions of system (1). For system (1), we make the following assumption (bounded conditions).

(C1):

where , , , and are known positive constants;

(C2): For all , and for each of bounded sets , ; moreover,

where , , and are positive constants and ;

(C3): Assume . For all , . Furthermore, there exist constants and such that

Also, satisfies cocycle property

(C4):

(C5): is a Brownian motion with values in defined on the probability space , where , is a complete -algebra, and is the corresponding Wiener measure on . To be specific, , where and denotes the element being at position and all the other components being and are independent of the two-side Brownian motions.

Remark 7. As pointed out in [19], the function satisfies condition (C2).

Next, we show the existence and uniqueness of the solution to system (1) under assumptions (C1)–(C5).

Let where , and , and . Equation (1) can be rewritten as the following equation with initial condition:

where

Lemma 8. For all , if conditions (C1)–(C5) hold, then the operators and in (19) map into themselves, and they are locally Lipschitz on .

Proof. From the above assumptions, it follows that

By the definition of , for all , we have

where , . Hence, we can infer that , for all , .
Let be a bounded set in , , . Similar to the derivation of (21) and (22), there exists a constant dependent on the bounded set such that

The above two inequalities imply that and are locally Lipschitz on , and the proof is then complete.

Theorem 9. If (C1)–(C5) hold, then, for any initial data , there exists a unique local solution of (19) such that , where is a positive constant. In addition, for all , we have the following estimate:

where

Moreover, the solution of (19) depends continuously on the initial data ; that is, for each , the mapping is continuous.

Proof. Taking the inner product of (19) with , we have

Denote and .
It is easy to check that

Now, let us estimate the terms of (27). First, we get

By condition (C3), we have

Noticing that , it follows from (29) that

Letting , we can see that , and then

From Young’s inequality, it follows that

By using Young’s inequality again, we have

Based on assumption (C2) and the definition of , it follows that

Combining (31)–(35) with (27), we can calculate that

Since , it is easy to check that . Also by condition (C1), one has

Substituting (37) into (36) yields

From (14) it follows that

and then a combination of the above inequality and (38) leads to

Hence, one has

which implies that, for all , is bounded. So, for any , (19) has a global solution on any interval , and therefore for all , we have

which indicates that (22) has a global solution .
Next, we show that the solutions of (19) are dependent continuously on initial conditions. Let and and assume are the solutions of (19), where is a constant.
Set . Since and are the solutions of (19), we have

Taking the inner product of (43) with in , we get

By Lemma 8, we have

It is easy to see the operator in (19) is a linear operator, and then it follows from assumption (C1) that there exists a positive constant such that , where depends only on the constants , , , and . Hence, (45) implies that

Furthermore, by Grownwall inequality, it is clear that

Hence, we have

which implies that the solutions of system (19) depend continuously on the initial data. The proof is now complete.

From assumption (C5) and noticing , it is easy to see that is a metric dynamical system. Also, from the definition of , we have

Now, for any , we introduce the map from into as follows:

where is the solution of (19) with initial data and is continuous for from to since the solution of (19) is dependent on initial data continuously.

By (49), (C3), and Theorem 9, it is easy to check that (50) defines a continuous random dynamical system over .

4. Existence of the Absorbing Set

In this section, we are concerned with the existence of an absorbing set for random dynamical system generated by the stochastic system (19). We first introduce an Ornstein-Uhlenbeck process in on the metric dynamic system (see [19, 24] for details).

Letting () where , then solves the Itô equation

From the properties of the Ornstein-Uhlenbeck process, we know that there exists a -invariant set of full measure, and the following properties hold:

(Y1)the mapping is continuous for each ;(Y2)the random variable is tempered;(Y3)there exists a tempered function such that

Theorem 10. There exist a -invariant set of full measure and an absorbing set for . That is, for all and all , there exists such that

Moreover, ; that is, for all , there exists such that

Proof. Letting , where is a solution of (19), then, for any , has properties (Y1), (Y2), and (Y3). By the Itô equation (51), it can be inferred that satisfies

Taking the inner product of (55) with in , we obtain

Similar to the derivation of (31) and (32), we have

where , , and are defined in (29) and (26), respectively.
Notice that

Similar to the derivation of (35), by condition (C2), (17), and , one has

and, by Young’s inequality, we also have

Substituting (57)–(61) into (56), we obtain

By the definition of constant , we find that , and then

Thus, we arrive at

Furthermore, from Grownwall inequality, it follows that

where

Therefore, it follows from (65) and property (Y3) that

where and is tempered thanks to the tempered function . Then, is an absorbing set for ; that is, for all and all , there exists such that for all .
Let ; then is an absorbing set for . Moreover, . The proof is now complete.

5. Existence of a Global Random Attractor

In this section, we will show the existence of global random attractor related to the random lattice dynamical system generated by system (19). In order to apply the result of Theorem 5, we need to prove the following lemmas.

Lemma 11. Let (C1)–(C5) hold and . Then, for any , there exist and such that the solution of (19) satisfies

where , for any .

Proof. Assume is a smooth function satisfying

and there exists a constant such that .
Let be a solution of (55), where .
Suppose is a suitable large constant to be defined later. Set , where

for all .
Taking the inner product of (55) with , we have

It is easy to see

where the last inequality follows from assumption (C1), (9), and the definition of operator in (8).
For simplicity, we denote and . Since , by the definition of , we have

Also, similar to the derivation of (75), we get

where .
It is not difficult to see

Substituting (74)–(80) into (73) yields

Also, it is obvious that

Next, we estimate the term in (71) as follows:

By assumption (C2), we have

Furthermore, by Young’s inequality, we know

Substituting (72), (81)–(88) into (71) leads to

where only depends on , , , , , and .
Now, setting , we estimate

By Grownwall inequality, for all , we can deduce that

where , , , and are defined above.
Next, let us estimate each term on the right hand side of (91). From (65) in Theorem 10 and property (Y3), we find that

Hence, for any given constant , there exists such that, for all ,

Using (65) in Theorem 10 and property (Y3) again, we have

Since , from Theorem 10, we have . So, for any , there exist , , and such that

By assumption (C4), there exists such that, for all , the following inequality holds:

Let be a positive constant to be determined later. When , we have

Choosing and by property , we have

For the fixed , from Lebesgue’s theorem, there exist and such that

By setting

then, for and , we obtain that

It means that if constant is large enough,

holds. The proof of the lemma is now completed.

Finally, we prove the asymptotic compactness of the absorbing set .

Theorem 12. If assumptions (C1)–(C5) hold, then for any the set is asymptotically compact.

Proof. For any , consider the sequence in , where and is an increasing sequence in with as .
First, let us show that has a convergent subsequence. Since is a bounded absorbing set, holds for large . Then, there exists a subsequence of (still denoted by ) such that

In what follows, we prove that the weak convergence (103) is actually strong convergence. In other words, we will show that, for every , there exists such that when ,

By Lemma 11, there exist and such that, for ,

Also, since , there exists such that

Let . By (103), we infer that

which implies that there exists such that when ,

Setting , we get from (105)–(108) that for

Therefore, we arrive at

The proof is therefore completed.

Finally, the main conclusion follows from Theorem 5, Theorem 10, and Theorem 12.

Theorem 13. Assume that (C1)–(C5) hold. Then, for any , the continuous random lattice dynamical system generated by the general second-order stochastic lattice dynamical system (1) has a unique global random attractor.

6. Conclusions

In this paper, we have investigated the random attractors in second-order stochastic lattice dynamical systems. First, we first proved the uniqueness and existence of the solutions of second-order stochastic lattice dynamical systems in the space . Then, by proving the asymptotic compactness of the random dynamical systems, we have established the existence of the global random attractor with the set of tempered bounded sets. These results could be further extended to more general nonlinear systems with uncertainties as in [25–27].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant nos. 61374010, 61074129, and 61175111, the Natural Science Foundation of Jiangsu Province of China under Grant BK2012682, the Qing Lan Project of Jiangsu Province (2010), the 333 Project of Jiangsu Province (2011), and the Six Talents Peak Project of Jiangsu Province (DZXX-047).

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