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

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.

1. Introduction

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

(1.1)

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

System (1.1) includes many neural continuous and discrete time networks [1–9]. For examples, the high-order Hopfield neural networks with impulses and delays (see [8]):

(1.2)

(1.3)

the Cohen-Grossberg neural networks with bounded and unbounded delays (see [9]):

(1.4)

(1.5)

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 [10–21] 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 [25–30] 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

(1.6)

Throughout this paper, we assume that

() and there exist positive constants such that for all ;

(). There exist positive constants and such that

(1.7)

for all ;

(), for . There exist positive constants such that

(1.8)

for all and ;

() and there exist positive constants such that

(1.9)

for all

For convenience, we introduce the following notation:

(1.10)

where is an -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 is an arbitrary nonempty closed subset of real numbers . The forward and backward jump operators , and the graininess are defined, respectively, by

(2.1)

Definition 2.2 (see [31]).

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

Definition 2.3 (see [31]).

For , one defines 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

(2.2)

for all

Definition 2.4 (see [31]).

If , then one defines the delta integral by

(2.3)

Definition 2.5 (see [33]).

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

(2.4)

for each , , where .

In case is right-scattered and is continuous at , this reduces to

(2.5)

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

Definition 2.6.

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

If , one says that is -anti-periodic if is the smallest positive number such that for all .

Definition 2.7 (see [31]).

A function is called regressive if for all , where is the graininess function. If is regressive and right-dense continuous function, then the generalized exponential function is defined by

(2.6)

for , with the cylinder transformation

(2.7)

Let be two regressive functions, we define

(2.8)

Then the generalized exponential function has the following properties.

Lemma 2.8 (see [31, 32]).

Assume that are two regressive functions, then

(i) and ;

(ii);

(iii);

(iv);

(v);

(vi);

(vii).

Lemma 2.9 (see [31]).

Assume that , are delta differentiable at . Then

(2.9)

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

Lemma 2.10.

Let . If is -periodic, then

(2.10)

Lemma 2.11.

Let . For rd-continuous functions one has

(2.11)

Definition 2.12.

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

(2.12)

where

(2.13)

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

Lemma 2.13 (see [37]).

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

()

Then the equation has at least one solution on .

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 hold. Suppose further that

() is a nonsingular matrix, where, for

(3.1)

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

Proof.

Let is a piecewise continuous map with first-class discontinuity points in , and at each discontinuity point it is continuous on the left. Take

(3.2)

are two Banach spaces with the norms

(3.3)

respectively, where , , is any norm of .

Set

(3.4)

where

(3.5)

where

(3.6)

It is easy to see that

(3.7)

Thus, dim Ker  codim Im, and is a linear Fredholm mapping of index zero.

Define the projectors and by

(3.8)

(3.9)

respectively. It is not difficult to show that and are continuous projectors such that

(3.10)

Further, let and the generalized inverse is given by

(3.11)

in which for all .

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

Corresponding to the operator equation , we have

(3.12)

or

(3.13)

where

(3.14)

Set , in view of (3.13), and Lemma 2.11, we obtain that

(3.15)

where . Integrating (3.13) from to , we have from that

(3.16)

by , we obtain that

(3.17)

where . From Lemma 2.10, for any , we have

(3.18)

(3.19)

Dividing by on the both sides of (3.18) and (3.19), respectively, we obtain that

(3.20)

Let , such that , by the arbitrariness of in view of (3.15), (3.17), (3.20), we have

(3.21)

where . Thus, we have from (3.21) that

(3.22)

where . In addition, we have that

(3.23)

By (3.22), we obtain that,

(3.24)

where . That is,

(3.25)

Denote that,

(3.26)

Then (3.25) can be rewritten in the matrix form

(3.27)

From the conditions of Theorem 3.1, is a nonsingular matrix, therefore,

(3.28)

Let

(3.29)

Take

(3.30)

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

4. Global Exponential Stability of Antiperiodic Solution

Suppose that is an -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 hold. Suppose further that

()there exist positive constants such that

(41)

()for all , there exist positive constants such that

(42)

()there are -periodic functions such that ;

()there exists a positive constant such that

(43)

()impulsive operator satisfy

(44)

Then the -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 -anti-periodic solution with initial value , suppose that is an arbitrary solution of system (1.1) with initial value . Then it follows from system (1.1) that

(45)

In view of system (4.5), for , we have

(46)

Hence, we can obtain from that

(47)

for , and we have from that

(48)

For any , we construct the Lyapunov functional

(49)

For , calculating the delta derivative of along solutions of system (4.5), we can get

(410)

(411)

By assumption , it concludes that

(412)

Also,

(413)

It follows that for all .

On the other hand, we have

(414)

where

(415)

It is obvious that

(416)

So we can finally get

(417)

Since , from Definition 2.12, the 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:

(51)

where

(52)

when , system (5.1) has at least one exponentially stable -anti-periodic solution.

Proof.

By calculation, we have , . It is obvious that , and are satisfied. Furthermore, we can easily calculate that

(53)

is a nonsingular matrix, thus is satisfied.

When . Take , we have that

(54)

Hence holds. By Theorems 3.1 and 4.1, system (5.1) has at least one exponentially stable -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 or and are of great significance in designs and applications of globally stable anti-periodic Cohen-Grossberg neural networks with delays and impulses.