Abstract
In this paper, a kind of BAM neural networks with leakage delays in the negative feedback terms and time-varying delays in activation functions was considered. By constructing a suitable Lyapunov function and using inequality techniques, some sufficient conditions to ensure the existence and exponential stability of antiperiodic solutions of these neural networks were derived. These conditions extend some results recently appearing in recent papers. Lastly, an example is given to show the feasibility of these conditions.
1. Introduction
It is known that neural networks have been applied in numerous fields, such as pattern recognition, classification, associative memory, optimization, signal and image processing, parallel computation, and nonlinear optimization problems. Up to now, there are many works focusing on the dynamical nature of various kinds of neural networks, such as stability, periodic solution, almost periodic solution, bifurcation, and chaos (see [1–11]). However, since significant time delays are ubiquitous, it is necessary to introduce delays into communication channels which leads to delayed neural networks models. Kosko [12] proposed a new kind of neural networks named bidirectional associative memory (BAM) neural networks with time delays which was given by the following:where is the number of neurons in layers. In model (1), and stand for the activations of th neurons; positive constants and represent the rate with which the th neuron will reset its potential to the resting state in isolation when they are disconnected from the network and the external inputs at time ; time delays and are nonnegative functions; and and denote the components of external input source introduced from outside the network. The authors in this paper applied this model to image processing.
There are many papers about BAM neural networks and stochastic BAM neural networks; for details, see [13–18]. It is known that the existence-uniqueness and stability is an important theoretical problem in the field of dynamics systems and one can find this topic in these papers [19–22]. Always, leakage delays appear in the negative feedback term of neural networks. Based on this, Gopalsmay [23] proposed the following BAM neural networks and studied the stability of the equilibrium and periodic solutionsIn this system, time delays and are positive constants, where . By constructing Lyapunov-Kravsovskii functionals, using inequalities and -matrices technology, the author gave two sets of delay dependent sufficient conditions on the existence of a unique equilibrium as well as its asymptotic and exponential stability to system (2). Because of time-varying delays in the real world, Liu [24] proposed the following BAM neural networks and discussed the global exponential stability of the network:By constructing a Lyapunov functional, some sufficient conditions on the global exponential stability of the equilibrium for system (3) were established. There are few papers considering the variable external input. It is significative to consider time-varying delays and external input in neural networks.
Recently, Li et al. [25] considered the following BAM neural networks with time-varying external input:To the best of our knowledge, there are few papers focusing on the existence and stability of antiperiodic solution to BAM neural networks with time-varying delays in the leakage terms in the negative feedback term. Motivated by the above discussions, in this paper, we propose a kind of BAM neural networks with time-varying delays and external input as follows:We can easily find that model (5) extends the above models. In model (5), we consider the time-varying leakage delays in negative feedback terms and time-varying delays in activation functions. And the activation functions in model (5) are more general. If the parameters are all equal to 0, then model (5) will deduce to special model in (4). And if are all constants, then model (5) will extend the models in (2) and (3). Therefore, it is important to investigate the stability of model (5). In this paper, by constructing a suitable Lyapunov functional, we give some sufficient conditions to ensure the existence and global exponential stability of antiperiodic solutions of system (5).
The rest of this paper is organized as follows. In Section 2, we introduce some notations and give some lemmas. In Section 3, we obtain some sufficient conditions on the existence and global exponential stability of antiperiodic solution of system (5). In Section 4, we give some examples to illustrate the efficiency. In Section 5, some conclusions are given.
2. Notations and Preliminary Results
First, we give some notations. We denote For vector and matrix , we define the following norms: respectively. For where , we denote The initial conditions of the system (5) are given by
Let be the solution of system (5) with initial conditions (10), where , . We say the solution is -antiperiodic if , for all , where is a positive constant.
Throughout this paper, we assume that the following conditions hold.
(H1) For , there exist positive constants and such that for all .
(H2) For all , where and is a positive constant.
Definition 1. The solution of system (5) is said to be globally exponentially stable if there exist constants and such that for each solution of system (5).
Lemma 2 (see [25]). Letand then
Lemma 3. Suppose that (H3)where are any constants. Then there exists such that
Proof. LetClearly, are continuously differential functions and satisfy that According to the intermediate value theorem, it is clear that there exist constants such that Let ; then it follows that and The proof of Lemma 3 is complete.
Lemma 4. Suppose that (H1) holds true. Then, for any solution of system (5), there exists a constant such that
Proof. From system (5), we conclude that Let where and then system (5) has the following form: By (27), we get In view of Lemma 2, we haveLet and it follows from (29) that . That is, and This completes the proof of Lemma 4.
3. Main Results
In this section, we give our main results for system (5).
Theorem 5. Suppose that (H1)-(H3) are satisfied. Then any solution of system (5) is globally exponentially stable.
Proof. First, we denote By system (5), we have which leads to Then With the same method, we havewhere Now we define a Lyapunov function as follows: where is defined by Lemma 3. Differentiating along solutions to system (5), together with (33) and (34), we have In view of Lemma 3, we have . That is to say, for all . ThusLet By (37), one has for all . Then for all . Thus the solution of system (5) is globally exponentially stable.
Theorem 6. Suppose that (H1)–(H3) hold. Then system (5) has exactly one -antiperiodic solution which is globally stable.
Proof. By system (5) and (H2), for each , we get In a similar way, we have Let It follows that, for any is also the solution of system (5). If the initial functions , are bounded, we conclude from Theorem 5 that there exists a constant such that where , and is a positive constant. For any we have Then In view of Lemma 4, we know that the solutions of system (5) are bounded. In view of (4) and (5), we can easily know that uniformly converges to a continuous function on any compact set of . In a similar way, we can easily prove that uniformly converges to a continuous function on any compact set of . Now we will show that is -antiperiodic solution of system (5). Since thus is -antiperiodic solution. Similarly, is also -antiperiodic solution. Thus we know that is a solution of system (5). In fact, together with the continuity of the right side of system (5), letting , we can easily get Therefore, is the -periodic solution of system (5). Finally, by applying Theorem 5, it is easy to check that is globally exponentially stable. The proof of Theorem 6 is completed.
Remark 7. There are a large number of papers about neural networks with delays. The main topic of these papers is exponential stability. For example, the authors in [26–28] considered the exponential stability of neural networks with constants delays. The authors in [29–31] considered the exponential stability of neural networks with time-varying delays and [32–35] with distributed delays. But these are few results on stability of BAM neural networks with leakage terms. This paper constructs a kind of BAM neural networks with time-varying delay and leakage terms and the results in our paper extend the results in the above papers.
4. An Example
In this section, to illustrate the efficiency of our results obtained in Section 3, we give an example. Consider the following BAM neural networks with time-varying delays and external input: where We also set Then we find that , and Also we have for
It is easy to verify that Then all the conditions (H1)–(H3) hold. Thus system (49) has exactly one -antiperiodic solution which is globally exponentially stable. The numerical result is shown in Figure 1. And system (49) is exponentially stable when all inputs are equal to 0, which can also be seen from Figure 2.
5. Conclusions
In this paper, a kind of BAM neural networks with time-varying delays and external input has been dealt with. By constructing a suitable Lyapunov-Kravsovskii functional and using matrix theory and inequality technique, a series of sufficient criteria to guarantee the existence and global exponential stability of antiperiodic solutions for this system have been established. And the criteria are easy to check in practice. We also give an example to illustrate the feasibility and effectiveness. In real life, many BAM neural networks are affected by external perturbation. There are many papers concerned with stochastic BAM neural networks, e.g., [36–41]. In the future, we will give our research on these networks.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The authors have made the same contribution. All authors read and approved the final manuscript.
Acknowledgments
This work was jointly supported by the National Natural Science Foundation of China (61773217, 61374080), the Natural Science Foundation of Jiangsu Province (BK20161552), Qing Lan Project of Jiangsu Province, Research Foundation of Hubei Province (B2018413), Development Funding from Yangtze University (18100200018), and Yangtze University College of Technology and Engineering (2017KY09, 2017KY10).