The Scientific World Journal
Volume 2015 (2015), Article ID 727329, 15 pages
http://dx.doi.org/10.1155/2015/727329

Research Article

-Almost Periodic Solutions of BAM Neural Networks with Time-Varying Delays on Time Scales

Yongkun Li, Lili Zhao, and Li Yang

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 5 May 2014; Accepted 5 August 2014

Academic Editor: P. Balasubramaniam

Copyright © 2015 Yongkun Li 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.

On a new type of almost periodic time scales, a class of BAM neural networks is considered. By employing a fixed point theorem and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability of -almost periodic solutions for this class of networks with time-varying delays are established. Two examples are given to show the effectiveness of the proposed method and results.

1. Introduction

It is well known that bidirectional associative memory (BAM) neural networks have been extensively applied within various engineering and scientific fields such as pattern recognition, signal and image processing, artificial intelligence, and combinatorial optimization [1–3]. Since all these applications closely relate to the dynamics, the dynamical behaviors of BAM neural networks have been widely investigated. There have been extensive results on the problem of the existence and stability of equilibrium points, periodic solutions, and antiperiodic solutions of BAM neural networks in the literature. We refer the reader to [4–16] and the references cited therein. Moreover, it is known that the existence and stability of almost periodic solutions play a key role in characterizing the behavior of dynamical systems (see [17–26]) and the -almost periodic function is an important subclass of almost periodic functions. However, to the best of our knowledge, few authors have studied problems of -almost periodic solutions of BAM neural networks.

On the other hand, the theory of calculus on time scales (see [27, 28] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analyses, and it helps avoid proving twice results, once for differential equations and once for difference equations. Therefore, it is significant to study neural networks on time scales (see [5, 29, 30]). In fact, both continuous-time and discrete-time BAM-type neural networks have equal importance in various applications. But it is troublesome to study the existence and stability of almost periodic and -almost periodic solutions for continuous and discrete systems, respectively. Motivated by the above, our purpose of this paper is to study the existence and stability of -almost periodic solutions for the following BAM neural networks on time scales:

where is an almost periodic time scale which will be defined in the next section; and correspond to the activation of the th neurons and the th neurons at the time , respectively; are positive functions and they denote the rates with which the cells and reset their potential to the resting state when isolated from the other cells and inputs at time ; and are the connection weights at time ; , are nonnegative, which correspond to the finite speed of the axonal signal transmission at time ; , denote the external inputs at time ; and and are the activation functions of signal transmission. For each interval of , we denote .

Throughout this paper, we assume the following:

and there exist positive constants such that

 where , , ;, , , , , , , are bounded almost periodic functions on , , .

System (1) is supplemented with the initial values given by

where denotes a real-valued bounded rd-continuous function defined on , and

2. Preliminaries

In this section, we will first recall some basic definitions and lemmas which are used in what follows.

Let be a nonempty closed subset (time scale) of . The forward and backward jump operators and the graininess are defined, respectively, by

A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise, . If has a right-scattered minimum , then ; otherwise, .

A function is right-dense continuous provided it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If is continuous at each right-dense point and each left-dense point, then is said to be continuous function on .

For and , we define the delta derivative of , , to be the number (if it exists) with the property that for a given there exists a neighborhood of such that

for all .

If is continuous, then is right-dense continuous, and if is delta differentiable at , then is continuous at .

Let be right-dense continuous. If , then we define the delta integral by

A function is called regressive if

for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set .

If is a regressive function, then the generalized exponential function is defined by

with the cylinder transformation

Let be two regressive functions; we define

Then the generalized exponential function has the following properties.

Lemma 1 (see [31]). Assume that are two regressive functions; then,

(i) and ;(ii);(iii);(iv);(v).

In this section, denotes or , denotes an open set in or , and denotes an arbitrary compact subset of .

Definition 2. A time scale is called an almost periodic time scale if

satisfies that, for any , one has , where .

Definition 3. Let be an almost periodic time scale. For any , we define

where .

Obviously, if is an almost periodic time scale, then and . If there exists a such that , then Definition 2 is equivalent to Definition 3.7 in [31]; otherwise, Definition 2 is more general than Definition 3.7 in [31].

Definition 4. Let be an almost periodic time scale. A function is called an almost periodic function in uniformly for if the -translation set of

is a relatively dense set in for all and for each compact subset of ; that is, for any given and each compact subset of , there exists a constant such that each interval of length contains a such that

is called the -translation number of and and is called the inclusion length of .

For convenience, we introduce some notations. Let and be two sequences. Then means that is a subsequence of . We introduce the translation operator , and means that . From Definitions 2 and 4, one can easily see that all the results obtained in [31] are still valid under the new concepts of almost periodic time scales and almost periodic functions on time scales. For example, similar to Theorems 3.13 and 3.14 in [31], we can obtain the following equivalent definition of uniformly almost periodic functions.

Definition 5. Let , and if for any given sequence and each compact subset of there exists a subsequence such that exists uniformly on , then is called an almost periodic function in uniformly for .

Definition 6. A function is said to be a -almost periodic function, if are two almost periodic functions on .

Definition 7 (see [31]). Let , and let be an rd-continuous matrix on ; the linear system

is said to admit an exponential dichotomy on if there exist positive constants and , projection , and the fundamental solution matrix of (15), satisfying

where is a matrix norm on .

Consider the following linear almost periodic system:

where is an almost periodic matrix function and is an almost periodic vector function.

Lemma 8 (see [31]). If the linear system (15) admits exponential dichotomy, then system (16) has a unique almost periodic solution :

where is the fundamental solution matrix of (15).

Lemma 9 (see [24]). Let be an almost periodic function on , where , , and

then, the linear system

admits an exponential dichotomy on .

Lemma 10 (see [27]). Every rd-continuous function has an antiderivative. In particular, if , then defined by

is an antiderivative of .

Lemma 11 (see [27]). If and , then

By Lemmas 10 and 11, it is easy to get the following lemma.

Lemma 12. Suppose that is an -continuous function and is a positive -continuous function which satisfies that . Let

where ; then,

Lemma 13 (see [31]). If is a real-valued almost periodic function on and is a Lipschitz function, then is an almost periodic function on .

Lemma 14 (see [31]). If are almost periodic functions, then the following hold:

(1) is almost periodic function;(2) is almost periodic function.

3. Existence of -Almost Periodic Solutions

First, for convenience, we introduce some notations. We will use to denote a column vector, in which the symbol denotes the transpose of vector. We let denote the absolute-value vector given by and define .

Let be a bounded real-valued, almost periodic function on , , and

For , if we define induced modulus

where

then is a Banach space.

Theorem 15. Assume that , , and the following hold:

, , , , , , ;there exists a constant such that

where

then, system (1) has a unique -almost periodic solution in the region

Proof. For any given , we consider the following almost periodic differential equation:

Since , it follows from Lemma 10 that the linear system

admits an exponential dichotomy on . Thus, by Lemma 9, we know that system (32) has exactly one almost periodic solution:

By Lemmas 13 and 14, we have that

are almost periodic functions on ; that is, (34) is not only an almost periodic solution of system (32), but also a -almost periodic solution of system (32). First, we define a nonlinear operator on by

Next, we check that . For any given , it suffices to prove that . By conditions , we have

then, it follows from (37) that

Therefore, .
Taking and combining conditions and , we obtain that

Similarly, from (39) it follows that

By (40), we obtain that is a contraction mapping from to . Since is a closed subset of , has a fixed point in , which means that (32) has a unique -almost periodic solution in . Then system (1) has a unique -almost periodic solution in the region

This completes the proof.

4. Exponential Stability of the -Almost Periodic Solution

Definition 16. The -almost periodic solution of system (1) with initial value is said to be globally exponentially stable. There exist a positive constant with and such that every solution

of system (1) with any initial value

satisfies

where

Theorem 17. Suppose that hold and ; then, system (1) has a unique -almost periodic solution which is globally exponentially stable.

Proof. According to Theorem 15, we know that (1) has a -almost periodic solution

with initial value . Suppose that

is an arbitrary solution of (1) with initial value

Then it follows from system (1) that

where , and , , and the initial conditions of (49) and (50) are

Let and be defined by

By , we get

Since are continuous on and , as , there exist such that , , and for and for . By choosing , we have

So, we can choose a positive constant such that

which imply that

Multiplying (49) by and integrating on , by Lemma 12, we get

Similarly, multiplying (50) by and integrating on , we have

Take ; then by we have . Thus, there exists such that, for ,

It is easy to see that and

which imply that

Next, we claim that

If (62) is not true, then there must be some , and some such that

By (57)–(64) and , we obtain

Equations in (65) imply that

which contradicts (63), and so (62) holds. Hence, the -almost periodic solution of system (1) is globally exponentially stable. Global exponential stability implies that the -almost periodic solution is unique.

Remark 18. In [17, 25, 26, 29], the existence and stability of almost periodic solutions are studied for several classes of neural networks on almost periodic time scales. However, the almost periodic time scales used in [17, 25, 26, 29] are a kind of periodic time scales. So, the methods and the results of this paper are essentially new.

5. Some Examples

Consider the following neural network:

where

Example 1. In (67), take :

Let be arbitrary almost periodic functions; then, hold. Let ; then, holds. Next, let us check ; if we take , then

Thus, holds for . By Theorems 15 and 17, system (67) has a unique -almost periodic solution in the region

which is globally exponentially stable (see Figures 1–4).

Figure 1: Responses of with continuous time .

Figure 2: Responses of .

Figure 3: Responses of with time .

Figure 4: Responses of .

Example 2. In (67), take :

Let be arbitrary almost periodic functions; then, - hold. Let ; then, holds. Next, let us check ; if we take , then

Thus, holds for . By Theorems 15 and 17, system (67) has a unique -almost periodic solution in the region

which is globally exponentially stable (see Figures 5–8).

Figure 5: Responses of with discrete time .

Figure 6: Responses of .

Figure 7: Responses of with time .

Figure 8: Responses of .

6. Conclusion

In this paper, by using calculus theory on time scales, a fixed point theorem, and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability of -almost periodic solutions for a class of neural networks with time-varying delays on a new type of almost periodic time scales are established. To the best of our knowledge, this is the first time to study the existence of -almost periodic solutions of BAM neural networks on time scales. Our methods that are used in this paper can be used to study other types of neural networks, such as Cohen-Grossberg neural networks and fuzzy cellular neural networks.

Conflict of Interests

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

Acknowledgments

The authors thank the referees for their careful reading of the paper, support, and insightful comments. This work is supported by the National Natural Sciences Foundation of China under Grant 11361072.

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