Research Article | Open Access
Existence and Exponential Stability of Equilibrium Point for Fuzzy BAM Neural Networks with Infinitely Distributed Delays and Impulses on Time Scales
By using the fixed point theorem and constructing a Lyapunov functional, we establish some sufficient conditions on the existence, uniqueness, and exponential stability of equilibrium point for a class of fuzzy BAM neural networks with infinitely distributed delays and impulses on time scales. We also present a numerical example to show the feasibility of obtained results. Our example also shows that the described time and continuous neural time networks have the same dynamic behaviours for the stability.
The bidirectional associative memory (BAM) neural network models were first introduced by Kosko (see ). BAM neural network is a special class of recurrent neural networks that can store bipolar vector pairs. It is composed of neurons arranged in two layers, the -layer and -layer. This class of networks possesses good applications prospects in areas of pattern recognition, signal and image process, and automatic control. Such applications heavily depend on the dynamical behaviors of neural networks. Thus, the analysis of the dynamical behaviors is a necessary step for practical design neural networks. In particular, many researchers have studied the dynamics of BAM neural networks with or without delays including stability and existence of periodic solutions or almost periodic solutions. For the results on BAM neural networks, the reader may see [2–11] and reference therein.
In mathematical modeling of real world problems, we will encounter some inconveniences, for example, the complexity and the uncertainty or vagueness. For the sake of taking vagueness into consideration, fuzzy theory is considered as a suitable method. Yang et al. proposed fuzzy cellular neural network, which integrates fuzzy logic into the structure of traditional cellular neural networks and maintains local connectedness among cells . Fuzzy neural network has fuzzy logic between its template input and/or output besides the sum of product operation. Studies have been revealed that fuzzy neural network has its potential in imagine processing and pattern recognition and some results have reported on the stability and periodicity of fuzzy neural networks. Besides, in reality, time delays often occur due to finite switching speeds of the amplifiers and communication time and can destroy a stable network or cause sustained oscillations, bifurcation, or chaos. Hence, it is important to consider both the fuzzy logic and delay effect on dynamical behaviors of neural networks. There have been many results on the fuzzy neural networks with time delays [13–21]. For example, in , under the assumption that the activation function is a second differentiable bounded function, , the authors proved the exponential stability and obtained the domain of robust attraction of the equilibrium point of the following interval fuzzy neural network:
In fact, both continuous and discrete systems are very important in implementation and applications. To avoid the troublesomeness of studying the dynamical properties for continuous and discrete systems, respectively, it is meaningful to study that on time scales, which was initiated by Stefan Hilger in his Ph.D. thesis in order to unify continuous and discrete analysis. Lots of scholars have studied neural networks on time scales and obtained many good results [23–29]. For example, in , the authors considered the existence and global exponential stability of an equilibrium point for a class of fuzzy BAM neural networks with time-varying delays in leakage terms on time scales. Moreover, many systems also undergo abrupt changes at certain moments due to instantaneous perturbations, which lead to impulsive effects. Therefore, it is significant to study the dynamics of impulsive systems. For example, in , the authors studied the exponential stability of the following impulsive system on time scales: where and is a time scale, and in , authors studied a class of fuzzy BAM neural networks with finite distributed delays and impulses, by using fixed point theorem and differential inequality techniques; they established the existence and global exponential stability of unique equilibrium point to the networks.
However, to the best of our knowledge, few papers were published on the exponential stability of fuzzy BAM neural networks with distributed delays and impulses on time scales. Motivated by the above, in this paper, we study the following fuzzy BAM neural networks with infinitely distributed delays and impulses on time scales: where is a time scale that is closed under addition; , are the numbers of neurons in layers; and denote the activations of the th neuron and the th neuron at time ; and represent the rate with which the th neuron and the th neuron will reset their potential to the resting state in isolation when they are disconnected from the network and the external inputs; , are the input-output functions (the activation functions); , are elements of feedback templates; , denote elements of fuzzy feedback MIN templates and , are elements of fuzzy feedback MAX templates; , are fuzzy feed-forward MIN templates and , are fuzzy feed-forward MAX templates; , denote the input of the th neuron and the th neuron; the delayed feedback and are real valued nonnegative continuous functions defined on with and , where and are nonnegative constants; , denote biases of the th neuron and the th neuron, , ; and denote the fuzzy AND and fuzzy OR operations, respectively; the impulsive moments satisfy and .
For , we define the norm as . For the sake of convenience, we denote the -interval as . The initial condition of (3) is of the form where , denote positive real-valued continuous functions on .
Throughout this paper, we make the following assumptions:(H1)for , ;(H2) and there exist positive constants , such that for all , , ;(H3)there exists a positive constant such that for , ,
The organization of the rest of this paper is as follows. In Section 2, we introduce some preliminary results which are needed in later sections. In Section 3, we establish some sufficient conditions for the existence and uniqueness of the equilibrium point of (3). In Section 4, we prove the equilibrium point of (3) is exponentially stable. In Section 5, we give an example to illustrate the feasibility of our results obtained in previous sections.
In this section, we state some preliminary results.
Definition 2 (see ). Let be a nonempty closed subset (time scale) of . The forward and backward jump operators and the graininess are defined, respectively, by
Definition 3 (see ). A function is called regressive if for all . The set of all regressive and -continuous functions will be denoted by . We define the set .
Lemma 4 (see ). Assume that are two regressive functions; then (i) and ;(ii);(iii);(iv).
Definition 5 (see ). A function is called a delta antiderivative of provided holds for all . In this case we defined the integral of by and we have the following formula:
Lemma 6 (see ). Let be -differentiable functions on ; then (i), for any constants ;(ii).
Lemma 7 (see ). Assume that for ; then .
Lemma 8 (see ). Suppose that ; then (i), for all ;(ii)if for all , then for all .
Lemma 9 (see ). If and , then
Lemma 10 (see ). Let , and assume that is continuous at , where with . Also assume that is -continuous on . Suppose that for each , there exists a neighborhood of such that where denotes the derivative of with respect to the first variable. Then (i) implies ;(ii) implies .
Definition 11 (see ). For each , let be a neighborhood of ; then we define the generalized derivative (or Dini derivative), , to mean that, given , there exists a right neighborhood of such that for each , . In case is right-scattered and is continuous at , this reduces to
Definition 12 (see ). If , , and is rd-continuous on , then we define the improper integral by provided this limit exists, and we say that the improper integral converges in this case. If this limit does not exist, then we say that the improper integral diverges.
Definition 13 (see ). If , , and is rd-continuous on , then we define the improper integral by provided this limit exists, and we say that the improper integral converges in this case. If this limit does not exist, then we say that the improper integral diverges.
Lemma 14 (see ). Let and with ; then
Lemma 16 (see ). Let be defined on , . Then for any , , , we have the following estimations: where , .
Definition 17. Let be an equilibrium point of (3). If there exists a positive constant with such that, for , there exists such that for an arbitrary solution of (3) with initial value satisfies where , . Then the equilibrium point is said to be exponentially stable.
3. Existence and Uniqueness of the Equilibrium Point
In this section, we discuss the existence and uniqueness of the equilibrium point of (3).
Without loss of generality, we assume that the impulsive jump vectors and satisfy That is, if is an equilibrium point of the following nonimpulsive system then it is also the equilibrium point of impulsive system (3).
Theorem 18. Let and hold. Suppose further that , where then (3) has one unique equilibrium point.
Proof. If is an equilibrium point of (3), then we have where , . To finish the proof, it suffices to prove that (23) has a unique solution. Define a mapping as follows: where where , . Obviously, we need to show that is a contraction mapping on . In fact, for any and , we have Therefore, we have By , we obtain that is a contraction mapping. By the fixed point theorem of Banach space, there exists a unique fixed point of , which is a solution of (21). Therefore, (3) has exactly one equilibrium point. The proof of Theorem 18 is completed.
4. Exponential Stability of Equilibrium Point
In this section, we consider the impulsive fuzzy BAM neural networks of the following type: where , , , , , , , , , , , are defined as those in (3). The initial conditions associated with (28) are given by (4). In the following, we study the exponential stability of the unique equilibrium point for (28) on time scales by using Lyapunov method.
Theorem 19. Let hold. Suppose further that there exists a constant such that
and , , , .
Then the equilibrium point of (28) is exponentially stable.
Proof. By Theorem 18, (28) has one unique equilibrium point . Let be an arbitrary solution of (28). Denote , , , . Then from (28), we have the following:
For , construct Lyapunov functional , where Calculating the -derivative of along the solution of (30), we have