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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 942309, 13 pages
Almost Periodic Solutions for Neutral-Type BAM Neural Networks with Delays on Time Scales
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Received 5 May 2013; Accepted 17 June 2013
Academic Editor: Sabri Arik
Copyright © 2013 Yongkun Li and Li Yang. 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.
Using the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, we obtain some sufficient conditions for the existence and exponential stability of almost periodic solutions for a class of neutral-type BAM neural networks with delays on time scales. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new even if the time scale or and complementary to the previously known results.
The bidirectional associative memory (BAM) neural network, which was introduced by Kosko (see ), is a special recurrent neural network that can store bipolar vector pairs and is 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.
Recently, due to its wide range of applications, for instance, pattern recognition, associative memory, and combinatorial optimization, BAM neural network has received much attention. For example, in [2–4], some sufficient conditions were obtained for the stability of the equilibrium points of BAM neural networks; in [5, 6], authors investigated the periodic solutions of BAM neural networks by using the continuation theorem of coincidence degree theory; in [7–9], authors studied the almost periodic solution for BAM neural networks by using the exponential dichotomy and fixed point theorems; for other results about BAM neural networks, the reader may see [10–13] and reference therein.
Since it is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions , many authors investigated the dynamical behaviors of neutral-type neural networks with delays [15–26]. For example, in , under the assumptions that the activation functions satisfy boundedness and Lipschitz conditions, authors discussed global asymptotic stability of neutral-type BAM neural networks with delays as follows:
Also, it is well known that the study of dynamical systems on time scales is now an active area of research. One of the reasons for this is the fact that the study on time scales unifies the study of both discrete and continuous processes, besides many others. The pioneering works in this direction are [27–30]. The theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988, providing a rich theory that unifies and extends discrete and continuous analysis [31, 32]. The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, and social sciences.
In fact, both continuous and discrete systems are very important in implementation and applications. But, it is troublesome to study the the dynamical properties for continuous and discrete systems, respectively. Therefore, it is meaningful to study that on time scale which can unify the continuous and discrete situations (see [13, 31, 33–40]).
Motivated by the above, in this paper, we propose a neutral-type BAM neural network with delays on time scales as follows: where is an almost periodic time scale; , are the number 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 th neuron will reset their potential to the resting state in isolation when they are disconnected from the network and the external inputs at time ; , , , and are the input-output functions (the activation functions); , , , and are transmission delays at time and satisfy , , , and for ; are elements of feedback templates at time ; , are elements of feed-forward templates at time ; and , denote biases of the th neuron and the th neuron at time , , .
Our main purpose of this paper is, using the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, to study the existence and exponential stability of almost periodic solutions for (2). Our results of this paper are new and complementary to the previously known results even if the time scale or .
For convenience, we denote . For an almost periodic function , denote , . Set that ,, are almost periodic functions on with the norm , where , , , , , and , and is the set of continuous functions with continuous derivatives on , then is a Banach space.
The initial condition of (2) is where , , , , .
Throughout this paper, we assume that the following conditions hold:, with , , and are all almost periodic functions, where denotes the set of positively regressive functions from to , ;, and there exist positive constants , , , and such that for all , , .
This paper is organized as follows. In Section 2, we introduce some notations and definitions and state some preliminary results which are needed in later sections. In Section 3, we establish some sufficient conditions for the existence of almost periodic solutions of (2). In Section 4, we prove that the almost periodic solution obtained in Section 3 is exponentially stable. In Section 5, we give an example to illustrate the feasibility of our results obtained in previous sections. We draw a conclusion in Section 6.
In this section, we introduce some definitions and state some preliminary results.
Definition 1 (see ). Let be a nonempty closed subset (time scale) of . The forward and backward jump operators and the graininess are defined, respectively, by
Lemma 2 (see ). Assume that are two regressive functions, then(i) and ;(ii);(iii);(iv).
Lemma 3 (see ). Let , be -differentiable functions on . Then(i), for any constants , ;(ii).
Lemma 4 (see ). Assume that for . Then .
Definition 5 (see ). A function is positively regressive if for all .
Lemma 6 (see ). Suppose that . Then(i), for all ; (ii)if for all , , then for all .
Lemma 7 (see ). If and , then
Lemma 8 (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 9 (see ). A time scale is called an almost periodic time scale if
Definition 10 (see ). Let be an almost periodic time scale. A function is said to be almost periodic on , if for any , the set is relatively dense in ; that is, for any , there exists a constant such that each interval of length contains at least one such that The set is called the -translation set of , is called the -translation number of , and is called the inclusion of .
Lemma 11 (see ). If is an almost periodic function, then is bounded on .
Lemma 12 (see ). If are almost periodic functions, then are also almost periodic.
Definition 13 (see ). Letting and be a matrix-valued function on , the linear system is said to admit an exponential dichotomy on if there exist positive constants , , , projection , and the fundamental solution matrix of (11) satisfying where is a matrix norm on ; that is, if , then we can take .
Definition 16. Let , be an almost periodic solution of (2) with initial value . If there exists a positive constant with such that for , there exists such that for an arbitrary solution of (2) with initial value , satisfies where , , , , , , , . Then, the solution is said to be exponentially stable.
3. Existence of Almost Periodic Solutions
In this section, we will state and prove the sufficient conditions for the existence of almost periodic solutions of (2).
Let , where ,, , and let be a constant satisfying , ,. We have the following theorem.
Theorem 17. Let and hold. Suppose that() there exists a positive constant such that where (), , ,, where , , .Then, (2) has a unique almost periodic solution in , , ,.
Proof. For any given , we consider the following almost periodic system:
Since holds, it follows from Lemma 15 that the linear system
admits an exponential dichotomy on . Thus, by Lemma 14, we obtain that (19) has an almost periodic solution, which is expressed as follows:
For , then . Define the following operator:
We will show that is a contraction.
First, we show that for any , we have . Note that and similarly, Therefore, we have On the other hand, for , we have and for , In view of , we have that is, . Next, we show that is a contraction. For , , for , denote by Then, we have In a similar way, we have By , we have where and , . It implies that . Hence, is a contraction. Therefore, has a fixed point in ; that is, (2) has a unique almost periodic solution in . This completes the proof of Theorem 17.
4. Exponential Stability of Almost Periodic Solution
In this section, we will study the exponential stability of almost periodic solution of (2).
Theorem 18. Let (H1)–(H4) hold. Suppose further that() and , where and .Then, the almost periodic solution of (2) is exponentially stable.
Proof. By Theorem 17, (2) has an almost periodic solution with initial condition . Suppose that is an arbitrary solution of (2) with initial condition . Denote , where , , , . Then, it follows from (2) that
The initial condition of (34) and (35) is
Multiplying both sides of (34) by and integrating on , where , we get Similarly, multiplying both sides of (35) by and integrating on , we get For positive constant with , we have , where . Take where By , we have In view of , we have . Hence, it is obvious that 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 (44) does not hold. Firstly, we consider the following four cases.
Case 1. Equation (45) is not true and (46)–(48) are all true. Then, there exists and such that Therefore, there must be a constant such that Note that, in view of (37), we have where ,. In the proof, we use the inequality . Thus, we get a contradiction.
Case 2. Equation (46) is not true and (45), (47), and (48) are all true. Then, there exists and such that Hence, there must be a constant such that Note that, in view of (37), we have Thus, we have