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Existence and exponential stability of an equilibrium point for fuzzy BAM neural networks with time-varying delays in leakage terms on time scales
Advances in Difference Equations volume 2013, Article number: 218 (2013)
Abstract
In this paper, by using a fixed point theorem and differential inequality techniques, we consider the existence and global exponential stability of an equilibrium point for a class of fuzzy bidirectional associative memory neural networks with time-varying delays in leakage terms on time scales. We also present a numerical example to show the feasibility of obtained results. The results of this paper are completely new and complementary to the previously known results.
MSC:34B37, 34N05.
1 Introduction
The bidirectional associative memory (BAM) neural networks were first introduced by Kosto in 1988 [1]. These are special recurrent neural networks that can store bipolar vector pairs and are composed of neurons arranged in two layers. The neurons in one layer are fully interconnected to the neurons in the other layer, while there are no interconnections among neurons in the same layer.
In recent years, due to their wide range of applications, for example, pattern recognition, associative memory, and combinatorial optimization, BAM neural networks have received much attention. There are lots of results on the existence and stability of an equilibrium point, periodic solutions or almost periodic solutions of BAM neural networks [2–10].
Based on traditional cellular neural networks, Yang and Yang proposed a fuzzy cellular neural network, which integrates fuzzy logic into the structure of traditional cellular neural networks and maintains local connectedness among cells [11]. The fuzzy neural network has fuzzy logic between its template input and/or output besides the sum of product operation. Studies have revealed that the fuzzy neural network is very useful for image processing problems, which is a cornerstone in image processing and pattern recognition. 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 [12–17]. Moreover, time delay in the leakage term also has a great impact on the dynamics of neural networks. As pointed out by the author in [18], time delay in the stabilizing negative feedback term has a tendency to destabilize a system. Therefore, it is meaningful to consider fuzzy neural networks with time delays in the leakage terms [19–24].
In fact, both continuous and discrete systems are very important in implementation and applications. To avoid the trouble of studying the dynamical properties for continuous and discrete systems respectively, it is meaningful to study those on time scales, which was initiated by Stefan Hilger in his PhD thesis, in order to unify continuous and discrete analyses. Lots of scholars have studied neural networks on time scales and obtained many good results [25–34]. However, to the best of our knowledge, there is no paper published on the stability of fuzzy BAM neural networks with time delays in the leakage terms on time scales.
Motivated by the above, in this paper, we integrate fuzzy operations into BAM neural networks with time delays in the leakage terms and study the stability of considered neural networks on time scales. By using a fixed point theorem and differential inequality techniques, we consider the existence and global exponential stability of an equilibrium point for the following BAM neural network with time-varying delays in leakage terms on time scales:
where is a time scale; n, m are the number of neurons in layers; and denote the activations of the i th neuron and the j th neuron at time t; and represent the rate at which the i th neuron and the j th neuron will reset their potential to the resting state in isolation when they are disconnected from the network and the external inputs; and denote the leakage delays; , are the input-output functions (the activation functions); and are transmission delays; , , and ; , are elements of feedback templates; , denote the elements of fuzzy feedback MIN templates and , are the elements of fuzzy feedback MAX templates; , are fuzzy feed-forward MIN templates and , are fuzzy feed-forward MAX templates; , denote the input of the i th neuron and the j th neuron; , denote biases of the i th neuron and the j th neuron, , , ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively.
The initial condition of (1.1) is of the form
where , denote positive real-valued continuous functions on and , respectively.
For the sake of convenience, we introduce some notations. For matrix D, denotes the transpose of D, denotes the spectral radius of D. A matrix or a vector means that all entries of D are greater than or equal to zero, can be defined similarly. For matrices or vectors D and E, (respectively ) means that (respectively ).
Throughout this paper, we assume that the following condition holds:
-
(H)
and there exist positive constants , such that
for all , , .
The organization of the rest of this paper is as follows. In Section 2, we introduce some preliminary results which are needed in the later sections. In Section 3, we establish some sufficient conditions for the existence and uniqueness of the equilibrium point of (1.1). In Section 4, we prove the equilibrium point of (1.1) is globally exponentially stable. In Section 5, we give an example to illustrate the feasibility and effectiveness of our results obtained in previous sections.
2 Preliminaries
In this section, we state some preliminary results.
Definition 2.1 [25]
Let be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators and the graininess are defined, respectively, by
Lemma 2.1 [25]
Assume that are two regressive functions, then
-
(i)
and ;
-
(ii)
;
-
(iii)
;
-
(iv)
.
Lemma 2.2 [25]
Let f, g be Δ-differentiable functions on T, then
-
(i)
for any constants , ;
-
(ii)
.
Lemma 2.3 [35]
Assume that for , then .
Definition 2.2 [35]
A function is called regressive if
for all . The set of all regressive and rd-continuous functions will be denoted by ℛ. We define the set .
Lemma 2.4 [35]
Suppose that , then
-
(i)
for all ;
-
(ii)
if for all , , then for all .
Lemma 2.5 [35]
If and , then
and
Lemma 2.6 [35]
Let , and assume that is continuous at , where with . Also assume that is rd-continuous on . Suppose that for each , there exists a neighborhood U of such that
where denotes the derivative of f with respect to the first variable. Then
-
(i)
implies ;
-
(ii)
implies .
Definition 2.3 A point is said to be an equilibrium point of (1.1) if is a solution of (1.1).
Lemma 2.7 [17]
Let be defined on R, . Then, for any , , , we have the following estimations:
and
where , .
Definition 2.4 [36]
A real matrix is said to be an M-matrix if , , and all successive principal minors of A are positive.
Lemma 2.8 [36]
Let be a matrix with nonpositive off-diagonal elements, then the following statements are equivalent:
-
(i)
A is an M-matrix;
-
(ii)
there exists a vector such that ;
-
(iii)
there exists a vector such that ;
-
(iv)
there exists a positive definite diagonal matrix D such that .
Lemma 2.9 [36]
Let be an matrix with , then , where denotes the spectral radius of A and is the identity matrix of size l.
Definition 2.5 Let be an equilibrium point of (1.1). If there exists a positive constant λ with such that for , there exists such that for an arbitrary solution of (1.1) with initial value satisfies
where , , . Then the equilibrium point is said to be globally exponentially stable.
3 Existence and uniqueness of an equilibrium point
In this section, we study the existence and uniqueness of an equilibrium point of (1.1).
Theorem 3.1 Let (H) hold. Suppose further that , where , , and
, . Then (1.1) has one unique equilibrium point.
Proof Let be an equilibrium point of (1.1), then we have
where , . Define a mapping as follows:
where
for , . Obviously, we need to show that is a contraction mapping on . In fact, for any and , we have
and
It follows that
Let N be a positive integer. In view of (3.1), we have
Since , we obtain , which implies that there exist a positive integer M and a positive constant such that
Hence, we have
which implies that . Since , it is obvious that the mapping is a contraction mapping. By the fixed point theorem of a Banach space, Φ possesses a unique fixed point in , that is, there exists a unique equilibrium point of (1.1). The proof of Theorem 3.1 is completed. □
4 Global exponential stability of an equilibrium point
In this section, we study the global exponential stability of the equilibrium point of (1.1).
Theorem 4.1 Let (H) and hold. Suppose further that
(H5) , where
and
Then the equilibrium point of (1.1) is globally exponentially stable.
Proof By Theorem 3.1, (1.1) has a unique equilibrium point . Suppose that is an arbitrary solution of (1.1) with the initial condition . Let , where , , , . Then (1.1) can be rewritten as
where and
The initial condition of (4.1) is the following:
We rewrite (4.1) as follows:
and
Multiplying both sides of (4.2) by and integrating on , where , we get
Similarly, multiplying both sides of (4.3) by and integrating on , we get
For a positive constant with , we have for . Take . In view of (H5), it is obvious that . Hence, we have
We claim that
To prove this claim, we show that for any , the following inequality holds:
which means that for , we have
and for , we have
By way of contradiction, assume that (4.7) does not hold. Firstly, we consider the following two cases.
Case One: (4.8) is not true and (4.9) is true. Then there exist and such that
Hence, there must be a constant such that
Note that in view of (4.4), we have
which is a contradiction.
Case Two: (4.8) is true and (4.9) is not true. Then there exist and such that
Hence, there must be a constant such that
Note that in view of (4.5), we have
which is also a contradiction.
By the above two cases, for other cases of negative proposition of (4.7), we can obtain a contradiction. Therefore, (4.7) holds. Let , then (4.6) holds. Hence, we have that
which means that the equilibrium point of (1.1) is globally exponentially stable. The proof of Theorem 4.1 is completed. □
5 Example
In this section, we present an example to illustrate the feasibility of our results.
Example 5.1 Let . Consider the following fuzzy BAM system with delays in leakage terms on a time scale :
where time delays , , , , are defined as those in system (1.1) and the coefficients are as follows:
and are identity matrices. By calculating, we have . We can verify that for and , all the conditions of Theorem 3.1 and Theorem 4.1 are satisfied. Hence, for or , (5.1) always has one unique equilibrium point, which is globally exponentially stable.
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Acknowledgements
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.
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Li, Y., Yang, L. & Sun, L. Existence and exponential stability of an equilibrium point for fuzzy BAM neural networks with time-varying delays in leakage terms on time scales. Adv Differ Equ 2013, 218 (2013). https://doi.org/10.1186/1687-1847-2013-218
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DOI: https://doi.org/10.1186/1687-1847-2013-218