Existence and Stability of Antiperiodic Solution for a Class of Generalized Neural Networks with Impulses and Arbitrary Delays on Time Scales

Open Access
Research Article

Abstract

By using coincidence degree theory and Lyapunov functions, we study the existence and global exponential stability of antiperiodic solutions for a class of generalized neural networks with impulses and arbitrary delays on time scales. Some completely new sufficient conditions are established. Finally, an example is given to illustrate our results. These results are of great significance in designs and applications of globally stable anti-periodic Cohen-Grossberg neural networks with delays and impulses.

Keywords

Lyapunov Function Discontinuity Point Generalize Exponential Function Hopfield Neural Network Global Exponential Stability 

1. Introduction

In this paper, we consider the following generalized neural networks with impulses and arbitrary delays on time scales:

where Open image in new window is an Open image in new window -periodic time scale and if Open image in new window , then Open image in new window is a subset of Open image in new window , Open image in new window , Open image in new window , Open image in new window represent the right and left limits of Open image in new window in the sense of time scales, Open image in new window is a sequence of real numbers such that Open image in new window as Open image in new window . There exists a positive integer Open image in new window such that Open image in new window . Without loss of generality, we also assume that Open image in new window . For each interval Open image in new window of Open image in new window , we denote that Open image in new window , especially, we denote that Open image in new window .

System (1.1) includes many neural continuous and discrete time networks [1, 2, 3, 4, 5, 6, 7, 8, 9]. For examples, the high-order Hopfield neural networks with impulses and delays (see [8]):
the Cohen-Grossberg neural networks with bounded and unbounded delays (see [9]):

and so on.

Arising from problems in applied sciences, it is well known that anti-periodic problems of nonlinear differential equations have been extensively studied by many authors during the past twenty years; see [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and references cited therein. For example, anti-periodic trigonometric polynomials are important in the study of interpolation problems [22, 23], and anti-periodic wavelets are discussed in [24].

Recently, several authors [25, 26, 27, 28, 29, 30] have investigated the anti-periodic problems of neural networks without impulse by similar analytic skills. However, to the best of our knowledge, there are few papers published on the existence of anti-periodic solutions to neural networks with impulse.

The main purpose of this paper is to study the existence and global exponential stability of anti-periodic solutions of system (1.1) by using the method of coincidence degree theory and Lyapunov functions.

The initial conditions associated with system (1.1) are of the form

Throughout this paper, we assume that

() Open image in new window and there exist positive constants Open image in new window such that Open image in new window for all Open image in new window ;

() Open image in new window . There exist positive constants Open image in new window and Open image in new window such that

for all Open image in new window ;

() Open image in new window , for Open image in new window . There exist positive constants Open image in new window such that

for all Open image in new window and Open image in new window ;

() Open image in new window and there exist positive constants Open image in new window such that

for all Open image in new window

For convenience, we introduce the following notation:

where Open image in new window is an Open image in new window -periodic function.

The organization of this paper is as follows. In Section 2, we introduce some definitions and lemmas. In Section 3, by using the method of coincidence degree theory, we obtain the existence of the anti-periodic solutions of system (1.1). In Section 4, we give the criteria of global exponential stability of the anti-periodic solutions of system (1.1). In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections 3 and 4. The conclusions are drawn in Section 6.

2. Preliminaries

In this section, we will first recall some basic definitions and lemmas which can be found in books [31, 32].

Definition 2.1 (see [31]).

A time scale Open image in new window is an arbitrary nonempty closed subset of real numbers Open image in new window . The forward and backward jump operators Open image in new window , Open image in new window and the graininess Open image in new window are defined, respectively, by

Definition 2.2 (see [31]).

A function Open image in new window is called right-dense continuous provided it is continuous at right-dense point of Open image in new window and left-side limit exists (finite) at left-dense point of Open image in new window . The set of all right-dense continuous functions on Open image in new window will be denoted by Open image in new window . If Open image in new window is continuous at each right-dense and left-dense point, then Open image in new window is said to be a continuous function on Open image in new window , the set of continuous function will be denoted by Open image in new window .

Definition 2.3 (see [31]).

For Open image in new window , one defines the delta derivative of Open image in new window to be the number (if it exists) with the property that for a given Open image in new window , there exists a neighborhood Open image in new window of Open image in new window such that

for all Open image in new window

Definition 2.4 (see [31]).

If Open image in new window , then one defines the delta integral by

Definition 2.5 (see [33]).

For each Open image in new window , let Open image in new window be a neighborhood of Open image in new window . Then, one defines the generalized derivative (or dini derivative), Open image in new window to mean that, given Open image in new window , there exists a right neighborhood Open image in new window of Open image in new window such that

for each Open image in new window , Open image in new window , where Open image in new window .

In case Open image in new window is right-scattered and Open image in new window is continuous at Open image in new window , this reduces to

Similar to [34], we will give the definition of anti-periodic function on a time scale as following.

Definition 2.6.

Let Open image in new window be a periodic time scale with period Open image in new window . One says that the function Open image in new window is Open image in new window -anti-periodic if there exists a natural number Open image in new window such that Open image in new window , Open image in new window for all Open image in new window and Open image in new window is the smallest number such that Open image in new window .

If Open image in new window , one says that Open image in new window is Open image in new window -anti-periodic if Open image in new window is the smallest positive number such that Open image in new window for all Open image in new window .

Definition 2.7 (see [31]).

A function Open image in new window is called regressive if Open image in new window for all Open image in new window , where Open image in new window is the graininess function. If Open image in new window is regressive and right-dense continuous function, then the generalized exponential function Open image in new window is defined by
for Open image in new window , with the cylinder transformation
Let Open image in new window be two regressive functions, we define

Then the generalized exponential function has the following properties.

Lemma 2.8 (see [31, 32]).

Assume that Open image in new window are two regressive functions, then

(i) Open image in new window and Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window ;

(iv) Open image in new window ;

(v) Open image in new window ;

(vi) Open image in new window ;

(vii) Open image in new window .

Lemma 2.9 (see [31]).

The following lemmas can be found in [35, 36], respectively.

Lemma 2.10.

Lemma 2.11.

Let Open image in new window . For rd-continuous functions Open image in new window one has

Definition 2.12.

The anti-periodic solution Open image in new window of system (1.1) is said to be globally exponentially stable if there exist positive constants Open image in new window and Open image in new window , for any solution Open image in new window of system (1.1) with the initial value Open image in new window , such that

The following continuation theorem of coincidence degree theory is crucial in the arguments of our main results.

Lemma 2.13 (see [37]).

Let Open image in new window , Open image in new window be two Banach spaces, Open image in new window be open bounded and symmetric with Open image in new window . Suppose that Open image in new window is a linear Fredholm operator of index zero with Open image in new window and Open image in new window is L-compact. Further, one also assumes that

() Open image in new window

Then the equation Open image in new window has at least one solution on Open image in new window .

3. Existence of Antiperiodic Solutions

In this section, by using Lemma 2.13, we will study the existence of at least one anti-periodic solution of (1.1).

Theorem 3.1.

Assume that Open image in new window hold. Suppose further that

() Open image in new window is a nonsingular Open image in new window matrix, where, for Open image in new window

Then system (1.1) has at least one Open image in new window -anti-periodic solution.

Proof.

Let Open image in new window is a piecewise continuous map with first-class discontinuity points in Open image in new window , and at each discontinuity point it is continuous on the left Open image in new window . Take
are two Banach spaces with the norms

respectively, where Open image in new window , Open image in new window , Open image in new window is any norm of Open image in new window .

It is easy to see that

Thus, dim Ker Open image in new window   codim Im Open image in new window , and Open image in new window is a linear Fredholm mapping of index zero.

respectively. It is not difficult to show that Open image in new window and Open image in new window are continuous projectors such that
Further, let Open image in new window and the generalized inverse Open image in new window is given by

in which Open image in new window for all Open image in new window .

Similar to the proof of Theorem Open image in new window in [38], it is not difficult to show that Open image in new window , Open image in new window are relatively compact for any open bounded set Open image in new window . Therefore, Open image in new window is Open image in new window -compact on Open image in new window for any open bounded set Open image in new window .

Corresponding to the operator equation Open image in new window , we have
Set Open image in new window , in view of (3.13), Open image in new window and Lemma 2.11, we obtain that
where Open image in new window . From Lemma 2.10, for any Open image in new window , we have
Dividing by Open image in new window on the both sides of (3.18) and (3.19), respectively, we obtain that
Let Open image in new window , such that Open image in new window , by the arbitrariness of Open image in new window in view of (3.15), (3.17), (3.20), we have
where Open image in new window . Thus, we have from (3.21) that
where Open image in new window . In addition, we have that
By (3.22), we obtain that,
Denote that,
Then (3.25) can be rewritten in the matrix form
From the conditions of Theorem 3.1, Open image in new window is a nonsingular Open image in new window matrix, therefore,

It is clear that Open image in new window satisfies all the requirements in Lemma 2.13 and condition Open image in new window is satisfied. In view of all the discussions above, we conclude from Lemma 2.13that system (1.1) has at least one Open image in new window -anti-periodic solution. This completes the proof.

4. Global Exponential Stability of Antiperiodic Solution

Suppose that Open image in new window is an Open image in new window -anti-periodic solution of system (1.1). In this section, we will construct some suitable Lyapunov functions to study the global exponential stability of this anti-periodic solution.

Theorem 4.1.

Assume that Open image in new window hold. Suppose further that

()there exist positive constants Open image in new window such that

()for all Open image in new window , there exist positive constants Open image in new window such that

()there are Open image in new window -periodic functions Open image in new window such that Open image in new window ;

()there exists a positive constant Open image in new window such that

()impulsive operator Open image in new window satisfy

Then the Open image in new window -anti-periodic solution of system (1.1) is globally exponentially stable.

Proof.

According to Theorem 3.1, we know that system (1.1) has an Open image in new window -anti-periodic solution Open image in new window with initial value Open image in new window , suppose that Open image in new window is an arbitrary solution of system (1.1) with initial value Open image in new window . Then it follows from system (1.1) that
In view of system (4.5), for Open image in new window , we have
Hence, we can obtain from Open image in new window that
For any Open image in new window , we construct the Lyapunov functional
For Open image in new window , calculating the delta derivative Open image in new window of Open image in new window along solutions of system (4.5), we can get
By assumption Open image in new window , it concludes that

It follows that Open image in new window for all Open image in new window .

On the other hand, we have
It is obvious that
So we can finally get

Since Open image in new window , from Definition 2.12, the Open image in new window solution of system (1.1) is globally exponential stable. This completes the proof.

5. An Example

Example 5.1.

Consider the following impulsive generalized neural networks:

when Open image in new window , system (5.1) has at least one exponentially stable Open image in new window -anti-periodic solution.

Proof.

By calculation, we have Open image in new window , Open image in new window . It is obvious that Open image in new window , Open image in new window and Open image in new window are satisfied. Furthermore, we can easily calculate that

is a nonsingular Open image in new window matrix, thus Open image in new window is satisfied.

Hence Open image in new window holds. By Theorems 3.1 and 4.1, system (5.1) has at least one exponentially stable Open image in new window -anti-periodic solution. This completes the proof.

6. Conclusions

Using the time scales calculus theory, the coincidence degree theory, and the Lyapunov functional method, we obtain sufficient conditions for the existence and global exponential stability of anti-periodic solutions for a class of generalized neural networks with impulses and arbitrary delays. This class of generalized neural networks include many continuous or discrete time neural networks such as, Hopfield type neural networks, cellular neural networks, Cohen-Grossberg neural networks, and so on. To the best of our knowledge, the known results about the existence of anti-periodic solutions for neural networks are all done by a similar analytic method, and only good for neural networks without impulse. Our results obtained in this paper are completely new even if the time scale Open image in new window or Open image in new window and are of great significance in designs and applications of globally stable anti-periodic Cohen-Grossberg neural networks with delays and impulses.

Notes

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant 10971183.

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Copyright information

© Yongkun Li et al. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of MathematicsYunnan UniversityKunmingChina

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