Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 143706, 8 pages

http://dx.doi.org/10.1155/2013/143706

Research Article

Existence of Periodic Solutions for Shunting Inhibitory Cellular Neural Networks with Neutral Delays

Ninghua Chen

Fujian Preschool Education College, Fuzhou, Fujian 350007, China

Received 3 July 2013; Accepted 16 September 2013

Academic Editor: Cecile Piret

Copyright © 2013 Ninghua Chen. 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.

This paper considers the existence of periodic solutions for shunting inhibitory cellular neural networks (SICNNs) with neutral delays. By applying the theory of abstract continuation theorem of -set contractive operator and some analysis technique, a new result on the existence of periodic solutions is obtained.

1. Introduction

It is well known that SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. The dynamical behaviors of SICNNs with delays have been widely investigated in recent years. A large number of important results on the dynamical behaviors of SICNNs have been established and successfully applied to signal processing, pattern recognition, associative memories, and so on. In particular, there exist many results on the existence and stability of periodic and almost periodic solutions for SICNNs with delays. We refer the reader to [1–4] and references cited therein.

On the other hand, the stability analysis of various neutral delay-differential systems has drawn much research attention [5–7]. The theory of neutral delay-differential systems is of both theoretical and practical interest. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral delay-differential equations; such networks arise in high speed computers where nearly lossless transmission lines are used to interconnect switching circuits. Also, the neutral systems often appear in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar.

Motivated by the above, we consider shunting inhibitory cellular neural networks with neutral delays described by

where , is the cell at the position of the lattice and the -neighborhood of is given as

is the activity of the cell, , is the passive decay rate of the cell activity, is the external input to , is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell , and the activity function is a continuous function representing the output or firing rate of cell .

By using the Lyapunov functions and linear matrix inequality approach (LMI), most authors studied the asymptotic stability or exponential stability of the equilibrium point (see [7–10]). However, few papers have been published on the existence of periodic solutions or almost periodic solutions for neutral type neural networks ([11–14]). Thus, it is worthwhile to study the existence of periodic solutions for neutral type neural networks with delays. And to the author’s best knowledge, there is no published paper considering the existence of periodic solutions for SICNNs with neutral delays.

The initial conditions of (1) are of the form

where .

For the sake of convenience, we denote

where is a continuous -periodic function.

The main purpose of this paper is to obtain some sufficient conditions for the existence of the periodic solutions for system (1). By applying the theory of abstract continuation theorem of -set contractive operator and some analysis technique, we derive some new sufficient conditions ensuring the existence of periodic solutions for system (1). Moreover, numerical examples are provided to illustrate the feasibility of the new results.

2. Conditions and Lemma

Throughout this paper, for , we assume that

(H₁) are continuous -periodic functions;(H₂) function is nonnegative, bounded, -periodic, and continuously differential defined on and , where expresses the derivative of ;(H₃) there exists positive constant such that , for all .

In order to study (1), we should make some preparations.

Let be a Banach space. For a bounded subset , let

denote the (Kuratowski) measure of noncompactness, where diam denotes the diameter of the set . Let , be two Banach spaces and be a bounded open subset of . A continuous and bounded map is called -set contractive if for any bounded set we have , where is a constant. In addition, for a Fredholm operator with index zero, according to [15], we define

Lemma 1 (see [16]). Let be a Fredholm operator with index zero, and let be a fixed point. Suppose that is a -set contractive with , where is bounded, open, and symmetric about . Furthermore, we suppose that

(1) for ,(2) for , where is some bilinear form on and is the project of onto Coker .

Then there is a such that .

In order to use Lemma 1 to establish the existence of periodic solutions of (1), we set

with the norm defined by , where and

with the norm defined by , where , . Then , are Banach spaces.

Let defined by

and defined by the following:

for .

It is easy to see that is a Fredholm operator with index zero. Clearly, (1) has a -periodic solution if and only if for some , where

Lemma 2 (see [17]). The differential operator is a Fredholm operator with index zero and satisfies .

Lemma 3. If , then is a -set contractive map, where .

Proof. The proof of Lemma 3 is similar to Lemma  3 in [14] and will be omitted here.

Lemma 4 (see [13]). Let . If is -periodic, then

Lemma 5 (see [18]). Suppose that and for all , then the function has an inverse with , for all .

Throughout this paper, we assume that and . So that has a unique inverse, and we set to represent the inverse of function .

3. Main Results

Set

Theorem 6. Suppose that hold, furthermore, assume that

() and ,() is nonsingular -matrix, then system (1) has at least one -periodic solution.

Proof. We consider the operator equation , , where defined by (9) and (10), respectively. Corresponding to system (1), for , we have

Suppose that is a solution of system (14) for a parameter . Multiplying both sides of system (14) by and integrating over , from and Cauchy-Schwarz inequality, we obtain

and according to Lemma 5, we have

That is,

From the above inequality, we obtain

which gives

Multiplying both sides of (14) by and integrating over , for , we have

Thus

that is,

where . Then we may rewrite (22) as

Substituting (23) into (19), we get

By (H₅), we have . That is,

where . Substituting (25) into (23), we have . That is,

where .

Integrating both sides of (14), from to , we obtain

Using Cauchy-Schwarz inequality, we have

In view of (14), we have

From Lemma 4, for any , , we have

Dividing by on the two sides of the above inequalities, we obtain that

Let , such that , then by (25)–(31), we have

Similarly,

Thus, for all , , which yields that . Again from (14), we get

which gives .

Denote , , . Let , , and define a bounded bilinear form on by . Also we define by . Obviously, . Without loss of generality, we may assume that . Let , . Thus

Considering assumption (H₄), we know , and it follows that . So we have , for all .

Therefore, by using Lemma 1, we obtain that (1) has at least one -periodic solution. The proof is complete.

4. Examples

In this section, we give two examples to demonstrate the results obtained in previous sections.

Consider the following shunting inhibitory cellular neural networks with neutral delays:

where .

Example 1. Let , . We take

Obviously, (H₁)–(H₃) hold and

, , , and .

Let , we have ,

It is easy to see that (H₄) and (H₅) hold. Therefore, all conditions of Theorem 6 are satisfied, then system (36) has at least one -periodic solution.

Example 2. Let , . We take

Obviously, (H₁)–(H₃) hold and

, , , , , , , , and .

Let , we have ,

It is easy to see that (H₄) and (H₅) hold. Therefore, all conditions of Theorem 6 are satisfied, then system (36) has at least one -periodic solution.

5. Conclusion

In this paper, a class of neutral type shunting inhibitory cellular neural networks is investigated by means of an abstract continuous theorem of -set contractive operator and some analysis technique. Sufficient conditions are derived to ensure the existence of periodic solutions. To the best of our knowledge, the results presented here have not appeared in the related literature. Also, illustrative examples are given to demonstrate the effectiveness of the results obtained.

Acknowledgments

The author is grateful to the referees for their suggestions on the first draft of the paper. This work was supported by the scientific research fund of Fujian Provincial Education Department (JA11309).

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