International Journal of Mathematics and Mathematical Sciences 
Volume 2008 (2008), Article ID 843695, 14 pages
doi:10.1155/2008/843695
Research Article

Existence and Global Exponential Stability of Periodic Solutions for General Neural Networks with Time-Varying Delays

Xinsong Yang

Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

Received 28 October 2007; Revised 1 February 2008; Accepted 14 April 2008

Recommended by Attila Gilanyi

Abstract

By using the coincidence degree theorem and differential inequality techniques, sufficient conditions are obtained for the existence and global exponential stability of periodic solutions for general neural networks with time-varying (including bounded and unbounded) delays. Some known results are improved and some new results are obtained. An example is employed to illustrate our feasible results.

1. Introduction

In recent years, the delayed cellular neural networks (DCNNs) have been extensively studied because of their immense potentials of application perspective in different areas such as pattern recognition, optimization, and signal and image processing [1–3]. Hence, they have been the object of intensive analysis by numerous authors, and some interesting results on the existence and stability of periodic and almost periodic solutions have been obtained [4–12]. To our knowledge, few authors have considered global stability of periodic solutions for the neural networks with bounded and unbounded time-varying delays. In theory and application, global stability of periodic solutions of DCNNs is of great importance since the global stability of equilibrium points can be considered as a special case of periodic solution with zero period [8]. Hence, in this paper, we will study the existence and global exponential stability of periodic solutions of the following general neural networks with time-varying delays:(1.1)where is the state of neuron, , and are connection matrices, is the input function, and is the activation function of the neurons.

DCNNs in [4–12] and the references cited therein are special cases of (1.1). In particular, when are constants, the authors of [9] considered the existence and global exponential stability for (1.1) with periodic impulses. The methods used in [9] are Mawhin's coincidence degree theorem [13] and Lyapunov functions. In [14], by using Mawhin's coincidence degree theorem [13], the authors investigated the global existence of positive periodic solutions of mutualism systems with bounded and unbounded time-varying delays, and some sufficient conditions are obtained. In [12], the authors considered (1.1) when are constants.

We assume what follows.

(H1) are continuous -periodic functions, and are continuous -periodic in the first variable. . (H2) There exist positive constants and such that , and , for all (H3) . There are positive constants , , such that , for all (H4) The delay kernels are continuous, integrable, and satisfy(1.2) (H5) There exists a constant such that(1.3)

The organization of this paper is as follows. In Section 2, we introduce some notations and definitions, and state some preliminary results needed in later sections. We then study, in Section 3, the existence of periodic solutions of system (1.1) by using the continuation theorem of coincidence degree theorem proposed by Gains and Mawhin [13]. In Section 4, by constructing Lyapunov function we will derive sufficient conditions for the global exponential stability of the periodic solution of system (1.1). At last, an example is employed to illustrate the feasible results of this paper.

2. Preliminaries

For convenience, we use to denote the maximums of , respectively. We also use symbols to denote a column vector, in which the symbol denotes the transpose of a vector. denotes the identity matrix of size . A matrix or vector means that all entries of are greater than or equal to zero (resp., ). For matrices or vectors and , (resp., ) means that (resp., ).

The initial condition of (1.1) is of the form(2.1)where , are continuous -periodic solutions.

Definition 2.1. Let be an -periodic solution of (1.1) with initial value . If there exist constants and such that for every solution of (1.1) with initial value ,(2.2)where , then is said to be globally exponentially stable.

Definition 2.2. (See [15, 16]). A real matrix is said to be a nonsingular -matrix if , and , where denotes the inverse of .

Lemma 2.3 (See [15, 16]). Let with . Then the following statements are equivalent:

(1) is a nonsingular -matrix, (2) there exists a vector such that , (3) there exists a vector such that .

Lemma 2.4 (See [16]). Let be an matrix and , then there exists a vector such that , where denotes the spectral radius of .

To end this section, we introduce Mawhin's continuation theorem [13, page 40] as follows.

Consider an abstract equation in a Banach space , (2.3)where is a Fredholm operator with index-zero, and is a parameter. Let and denote two projectors, and such that .

Lemma 2.5. Let be a Banach space. Suppose that is a Fredholm operator with index-zero, let be an open bounded set, and let be a continuous operator which is -compact on . Moreover, assume that the following conditions are satisfied:

(a) for each , (b) for each , (c) . Then, has at least one solution in .

3. Existence of Periodic Solutions

Theorem 3.1. Let hold. Assume that the following condition is satisfied:

there exists a vector such that(3.1)where

Then, (1.1) has at least one -periodic solution.

Proof. For convenience, we introduce the following notations:(3.2) In order to use Lemma 2.5, we take ; then is a Banach space with the norm(3.3) Set(3.4) Obviously, and So, is closed in . It is easy to show that and are continuous projectors satisfying Hence, is a Fredholm mapping of index-zero. Furthermore, through an easy computation, we find that the generalized inverse of has the form(3.5)Thus,(3.6) Clearly, and are continuous. Using the Arzela-Ascoli theorem, it is not difficult to show that , are relatively compact for any open bounded set . Therefore, is -compact on for any open bounded set .

Now, we reach the position to search for an appropriate open bounded subset for the application of Lemma 2.5. Corresponding to the operator equation (2.3), we have(3.7)Let be a solution of system (3.7) for some . Then, for any are all continuously differentiable. Thus, there exist such that . Hence, From (3.7), we have(3.8)In view of , we have(3.9) Set . Clearly, (3.9) implies that(3.10)Thus,(3.11)together with , we have(3.12)Therefore,(3.13)Again from , it follows from Lemma 2.3 that is a nonsingular -matrix, and there exists a vector such that , which implies that we can choose a constant such that and and(3.14) We take(3.15)which satisfies Condition (a) of Lemma 2.5. If , then is a constant vector in , and there exists some such that . We claim that(3.16)By way of contradiction, suppose that then there exists some such that(3.17)which implies that(3.18)hence(3.19)This implies that , which contradicts (3.14). Therefore, (3.16) holds, and hence, Condition of Lemma 2.5 is satisfied.

Furthermore, we define a continuous function by(3.20)for all and . It follows that(3.21) If , then is a constant vector in , and there exists some such that . We claim that(3.22)By way of contradiction, suppose that then there exists some such that(3.23)that is,(3.24)Now, we will consider the following two cases.

Case 1. If , from , we have(3.25)Then, from (3.24), we have(3.26)which implies that(3.27) Hence,(3.28)this implies that , which contradicts (3.14). Therefore, (3.22) holds.

Case 2. If , from we have(3.29)Then, from (3.24), we have(3.30)The later proof is similar to that of Case 1. We can also show that (3.22) holds. It follows that for , Hence, by homotopy invariance theorem and , , we obtain for . Till now, we have proved that satisfies all conditions of Lemma 2.5. Therefore, (1.1) has a periodic solution . This completes the proof.

Corollary 3.2. Let hold. Assume that is a nonsingular -matrix, then (1.1) has at least one -periodic solution, where is defined as above.

Corollary 3.3. Let hold. Assume that , then (1.1) has at least one -periodic solution, where is defined as above.

Proof. Obviously, . By and from Lemma 2.4, there exists a vector such that . The remaining part of the proof is the same as that of Theorem 3.1.

4. Global Exponential Stability of Periodic Solutions

In this section, we will construct some suitable Lyapunov functionals to derive sufficient conditions which ensure that (1.1) has a unique -periodic solution, and all solutions of (1.1) exponentially converge to its unique -periodic solution.

Theorem 4.1. Assume that hold and

, for all , where are the positive constants of Hypothesis . Then, (1.1) has exactly one -periodic solution, which is globally exponentially stable.

Proof. By Theorem 3.1, there exists an -periodic solution of (1.1). Suppose that is an arbitrary solution of (1.1). Set . Then,(4.1)where(4.2) From , it follows from Lemma 2.3 that is a nonsingular -matrix, and there exists a vector such that . Then,(4.3)Set(4.4)Clearly, , are continuous functions on , where is the positive constant of Hypothesis . Since(4.5)we can choose a positive constant such that(4.6) Now, we choose a positive constant such that(4.7) Define a Lyapunov function by In view of (4.1), we obtain(4.8)From (4.7), we have(4.9)which implies that(4.10)where is defined as that in Definition 2.1.

We claim that(4.11)Contrarily, there must exist and such that (4.12)Together with (4.8) and (4.12), we obtain(4.13)Hence,(4.14)which contradicts (4.6). Therefore, (4.11) holds. It follows that(4.15)Let . Then, from (4.15), we get(4.16)In view of Definition 2.1, the -periodic solution of system (1.1) is globally exponentially stable. This completes the proof.

Corollary 4.2. Let hold. Assume that is a nonsingular -matrix or . Then, system (1.1) has exactly one -periodic solution, which is globally exponentially stable, where is defined as that in .

Remark 4.3. As a special case, and . Let , then . Obviously, and hold.

Remark 4.4. When are constants, and , one can easily know that [11, Theorems 1 and 2] are direct corollaries of Theorems 3.1 and 4.1 of this paper, respectively. Moreover, we need not the following assumption:

Very recently, Zhou and Hu [12] considered the global exponential periodicity and stability of the following cellular neural networks:(4.17)where are constants, , and . The assumptions in [12] on the delay kernels of (4.17) are as follows.

(F1) The delay kernels are real-valued nonnegative continuous functions and (F2) (F3) There exists a positive number such that Obviously, (4.17) is a special case of (1.1). Moreover, (3.1) of Theorem 3.1 and (3.15) of Theorem 3.2 in [12] are special cases of (4.3) in this paper, that is, , and . So, [12, Theorems 3.1 and 3.2] are special cases of Theorem 4.1 in this paper. To summarize, the results of this paper are completely new and generalize the results of [4–12] and the references cited therein.

5. Application

In this section, we give an example to illustrate that our results are feasible. Consider the following simple DCNNs with time-varying delays:(5.1)where and Then, we have (5.2) Hence,(5.3) We take , then(5.4)It is easy to check that all the conditions needed in Theorem 4.1 are satisfied. Therefore, (5.1) has a unique global exponential 1/20-periodic solution.

Remark 5.1. Because is not linear about , thus none of the results in [4–12, 14] can be applied to (5.1). This implies that the results of this paper are new.

Acknowledgments

The author would like to express his gratitude to some anonymous reviewers and the editor for their valuable suggestions and comments. This work was supported by the Scientific Research Fund of Yunnan Provincial Education Department (07Y10085) and the NSF of Honghe University (XSS06009) of China.

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