Abstract and Applied AnalysisVolume 2012 (2012), Article ID 805846, 21 pagesdoi:10.1155/2012/805846

Research Article

Existence and Global Exponential Stability of Periodic Solution to Cohen-Grossberg BAM Neural Networks with Time-Varying Delays

Kaiyu Liu, Zhengqiu Zhang, and Liping Wang

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received 16 December 2011; Revised 4 February 2012; Accepted 6 February 2012

Academic Editor: Agacik Zafer

Copyright © 2012 Kaiyu Liu 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.

We investigate first the existence of periodic solution in general Cohen-Grossberg BAM neural networks with multiple time-varying delays by means of using degree theory. Then using the existence result of periodic solution and constructing a Lyapunov functional, we discuss global exponential stability of periodic solution for the above neural networks. Our result on global exponential stability of periodic solution is different from the existing results. In our result, the hypothesis for monotonicity ineqiality conditions in the works of Xia (2010) Chen and Cao (2007) on the behaved functions is removed and the assumption for boundedness in the works of Zhang et al. (2011) and Li et al. (2009) is also removed. We just require that the behaved functions satisfy sign conditions and activation functions are globally Lipschitz continuous.

1. Introduction

In 1983, Cohen and Grossberg [1] constructed a kind of simplified neural networks that are now called Cohen-Grossberg neural networks (CGNNs); they have received increasing interesting due to their promising potential applications in many fields such as pattern recognition, parallel computing, associative memory, and combinatorial optimization. Such applications heavily depend on the dynamical behaviors. Thus, the qualitative analysis of the dynamical behaviors is a necessary step for the practical design and application of neural networks (or neural system [2–4]). The stability of Cohen-Grossberg neural network with or without delays has been widely studied by many researchers, and various interesting results have been reported [5–14].

On the other hand, since the pioneering work of Kosko [15, 16], a series of neural networks related to bidirectional associative memory models have been proposed. These models generalized the single-layer autoassociative Hebbian correlator to a class of two-layer pattern-matched heteroassociative circuits. Bidirectional associative memory neural networks have also been used in many fields such as pattern recognition and automatic control and image and signal processing. During the last years, many authors have discussed the existence and global stability of BAM neural networks [17–20]. In recent years, a few authors [17, 21–26] discussed global stability of Cohen-Grossberg BAM neural networks.

As is well known, the studies on neural dynamical system not only involve a discussion of stability properties but also involve other dynamic behavior, such as periodic oscillatory behavior, chaos, and bifurcation. In many applications, periodic oscillatory behavior is of great interest; it has been found in applications in learning theory. Hence, it is of prime importance to study periodic oscillatory solutions of neural networks.

This motivates us to consider periodic solutions of Cohen-Grossberg BAM neural networks. Recently, a few authors discussed the existence and stability of periodic solution to Cohen-Grossberg BAM neural networks with delays [27–31].

In [27], the authors proposed a class of bidirectional Cohen-Grossberg neural networks with distributed delays as follows: By using the Lyapunov functional method and some analytical techniques, some sufficient conditions were obtained for global exponential stability of periodic solutions to these networks.

In [28], the authors discussed the following Cohen-Grossberg-type BAM neural networks with time-varying delays: where are the number of neurons in the networks with initial value conditions: where , , , , , are continuous functions, are continuous functions, are parameters, and are continuous functions, and denote the state variables of the th neurons from the neural field and the th neurons from the neural field at time , respectively, represent amplification functions of the th neurons from the neural field and the th neurons from the neural field , respectively, are appropriately behaved functions of the th neurons from the neural field and the th neurons from the neural field , respectively, are the activation functions of the th neurons from the neural field and the th neurons from the neural field , respectively, are the exogenous inputs of the th neurons from the neural field and the jth neurons from the neural field , respectively, and are the connection weights, which denote the strengths of connectivity between the neuron from the neural field and the neuron from the neural field , and correspond to the transmission time delays.

By using the analysis method and inequality technique, some sufficient conditions were obtained to ensure the existence, uniqueness, global attractivity, and exponential stability of the periodic solution to this neural networks.

In [29, 30], the authors discussed, respectively, two Cohen-Grossberg BAM neural networks on time scales. When time scale becomes , the existence and global exponential stability of periodic solution are obtained in [29, 30] under the assumptions that activation functions satisfy global Lipschitz conditions and boundedness conditions and behaved functions satisfy some inequality conditions.

In [31], the authors discussed the following Cohen-Grossberg BAM neural networks of neutral type with delays: Under the assumptions that activation functions satisfy global Lipschitz conditions and behaved functions satisfy some inequality conditions, global exponential stability of periodic solution is obtained for system (1.4).

In this paper, our purpose is to obtain a new sufficient condition for the existence and global exponential stability of periodic solution of system (1.2). The paper is organized as follows. In Section 2, we discuss the existence of periodic solution of system (1.2) by using coincidence degree theory and inequality technique. In Section 3, we study the global exponential stability of periodic solution of system (1.2) by using the existence result of periodic solution and constituting Lyapunov functional. Our result on global exponential stability of periodic solution is different from the existing results. In our result, the hypotheses for monotonicity inequalities in [27, 28] on behaved functions are replaced with sign conditions and the assumption for boundedness in [29, 30] on activation functions is removed.

2. Existence of Periodic Solution

In this section, we first establish the existence of at least a periodic solution by applying the coincidence degree theory. To establish the existence of at least a periodic solution by applying the coincidence degree theory, we recall some basic tools in the frame work of Mawhin’s coincidence degree [32] that will be used to investigate the existence of periodic solutions.

Let , be Banach spaces, : a linear mapping, and a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and Im is closed in . If is a Fredholm mapping of index zero, then there exist continuous projectors and such that and . It follows that is invertible. We denote the inverse of the map by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact. Since Im is isomorphic to Ker , there exists an isomorphism .

In the proof of our existence theorem, we will use the continuation theorem of Gaines and Mawhin [32].

Lemma 2.1 (continuation theorem). Let be a Fredholm mapping of index zero, and let be -compact on . Suppose (a),(b),(c). Then, has at least one solution in .

For the sake of convenience, we introduce some notations.

denotes the norm in , , where is a continuously periodic function with common period . Our main result on the existence of at least a periodic solution for system (1.2) is stated in the following theorem.

Theorem 2.2. One assume that the following conditions holds: (i) are continuously periodic functions on with common period ;(ii) and are continuously bounded, that is, there exist positive constants such that (iii) and are continuous and there exist positive constants such that for all , (iv) there exist positive constants such that for all , (v) there exist two positive constants with and such that for , Then, system (1.2) has at least one -periodic solution.

Proof. In order to apply Lemma 2.1 to system (1.2), let Define Equipped with the above norm and are Banach spaces.Let for Then, it follows that , is closed in , , and are continuous projectors such that Hence, is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) is given by Then, Obviously, and are continuous. It is not difficult to show that is compact for any open bounded set by using the Arzela-Ascoli theorem. Moreover, is clearly bounded. Thus, is -compact on with any open bounded set . Condition (iii) in Theorem 2.2 implies that for all Condition (iv) in Theorem 2.2 implies that for all Corresponding to the operator equation , we have for Assume that is a solution of system (2.13) for some . Multiplying the first equation of system (2.13) by and integrating over , we have Multiplying the second equation of system (2.13) by and integrating over , we have From (2.14) and (2.15), we obtain Hence, Denoting , then Substituting (2.20) into (2.18) and substituting (2.21) into (2.19) give for Denoting for the sake of convenience where, , and from (2.22) and (2.23), we obtain Now we consider two possible cases for (2.26) and (2.25): When , from (2.25), we have Thus, Therefore, (ii) When , from (2.26), we have Thus, Therefore, Hence, from (2.30) and (2.33), we have for Multiplying the first equation of system (2.13) by and integrating over , from (2.20) and (2.35) and the fact that it follows that Similarly, multiplying the second equation of system (2.13) by and integrating over , from (2.21) and (2.34) and the fact that it follows that there exists a positive constant such that From (2.34) and (2.35), it follows that there exist points and such that Since for all , then from (2.40)–(2.43), we have for Obviously, , and are all independent of . Now let where are two chosen positive constants such that the bound of is larger. Then, is bounded open subset of . Hence, satisfies requirement (a) in Lemma 2.1. We prove that (b) in Lemma 2.1 holds. If it is not true, then when we have Therefore, there exist points and such that From this and following the arguments of (2.40) and (2.41), we have for forall Hence, Thus, . This contradicts the fact that . Hence, this proves that (b) in Lemma 2.1 holds. Finally, we show that (c) in Lemma 2.1 holds. We only need to prove that . Now, we show that To this end, we define a mapping by where is a parameter. We show that when , . If it is not true, then when , . Thus, constant vector with satisfies for , That is, Denote .

Claim 1. We claim that , otherwise, . We consider two possible cases: (i) and (ii).(i)When , we have which contradicts (2.53).(ii)When , we have which contradicts (2.54). From the discussion of (i) and (ii), Claim 1 holds.

Claim 2. We claim that , otherwise, . We consider two possible cases: (i) and (ii) .The proofs of (i) and (ii) are similar to those of (ii) and (1) in Claim 1, respectively, therefore Claim 2 holds.

Thus, and . Thus, . This contradicts the fact that . According to the topological degree theory and by taking since , we obtain So far, we have proved that satisfies all the assumptions in Lemma 2.1. Therefore, system (1.2) has at least one -periodic solution.

3. Global Exponential Stability of Periodic Solution

In this section, by constructing a Lyapunov functional, we derive new sufficient conditions for global exponential stability of a periodic solution of system (1.2).

Theorem 3.1. In addition to all conditions in Theorem 2.2, one assumes further that the following conditions hold: (H₁) there exists two positive constants with and such that and ;(H₂) there exist constants and , such that Then, the periodic solution of system (1.2) is globally exponentially stable.

Proof. By Theorem 2.2, system (1.2) has at least one periodic solution, say, . Suppose that is an arbitrary periodic solution of system (1.2). From (H₁), we can choose a suitable such that We define a Lyapunov functional as follows for where . Calculating the upper right derivative of along the solutions of system (1.2), we obtain Since there exist points such that from (3.4), we have In view of (3.2), it follows that . Therefore, Equation (3.3) implies that Substituting (3.8) into (3.7) gives where The proof of Theorem 3.1 is complete.

4. An Example

Example 4.1. Consider the following Cohen-Grossberg BAM neural networks with time-varying delays:

In Theorem 3.1, Since then .

Since then conditions (H₁), (H₂), and (v) are satisfied. It is easy to prove that the rest of the conditions in Theorem 3.1 are satisfied. By Theorem (3.2), system (4.1) has a unique periodic solution that is globally exponentially stable.

5. Conclusion

We investigate first the existence of the periodic solution in general Cohen-Grossberg BAM neural networks with multiple time-varying delays by means of using degree theory. Then, using the existence result of periodic solution and constructing a Lyapunov functional, we discuss global exponential stability of periodic solution for the above neural networks. In our result, the hypotheses for monotonicity in [27, 28] on the behaved functions are replaced with sign conditions and the assumption for boundedness on activation functions is removed. We just require that the behaved functions satisfy sign conditions and activation functions are globally Lipschitz continuous.

Acknowledgment

This paper is supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (no. 20091341, 2011508) and the Fund of National Natural Science of China (no. 61065008).

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