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

- Yongkun Li
^{1}, - Erliang Xu
^{1}and - Tianwei Zhang
^{1}Email author

**2010**:132790

https://doi.org/10.1155/2010/132790

© Yongkun Li et al. 2010

**Received: **14 June 2010

**Accepted: **16 August 2010

**Published: **18 August 2010

## 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

## 1. Introduction

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 .

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.

Throughout this paper, we assume that

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

() . There exist positive constants and such that

() , for . There exist positive constants such that

() and there exist positive constants such that

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]).

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]).

Definition 2.4 (see [31]).

Definition 2.5 (see [33]).

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]).

Then the generalized exponential function has the following properties.

Assume that are two regressive functions, then

Lemma 2.9 (see [31]).

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

Lemma 2.10.

Lemma 2.11.

Definition 2.12.

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

## 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

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

Proof.

respectively, where , , is any norm of .

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

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 .

It is clear that satisfies all the requirements in Lemma 2.13 and condition is 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

()for all , there exist positive constants such that

()there are -periodic functions such that ;

()there exists a positive constant such that

Then the -anti-periodic solution of system (1.1) is globally exponentially stable.

Proof.

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.

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

Proof.

is a nonsingular matrix, thus is satisfied.

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.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- Li X: Existence and global exponential stability of periodic solution for impulsive Cohen-Grossberg-type BAM neural networks with continuously distributed delays.
*Applied Mathematics and Computation*2009, 215(1):292–307. 10.1016/j.amc.2009.05.005MathSciNetView ArticleMATHGoogle Scholar - Bai C: Global exponential stability and existence of periodic solution of Cohen-Grossberg type neural networks with delays and impulses.
*Nonlinear Analysis*2008, 9(3):747–761. 10.1016/j.nonrwa.2006.12.007MathSciNetView ArticleMATHGoogle Scholar - Chen Z, Zhao D, Fu X: Discrete analogue of high-order periodic Cohen-Grossberg neural networks with delay.
*Applied Mathematics and Computation*2009, 214(1):210–217. 10.1016/j.amc.2009.03.083MathSciNetView ArticleMATHGoogle Scholar - Li YK: Global stability and existence of periodic solutions of discrete delayed cellular neural networks.
*Physics Letters. A*2004, 333(1–2):51–61. 10.1016/j.physleta.2004.10.022MathSciNetView ArticleMATHGoogle Scholar - Li YK, Xing Z: Existence and global exponential stability of periodic solution of CNNs with impulses.
*Chaos, Solitons and Fractals*2007, 33(5):1686–1693. 10.1016/j.chaos.2006.03.041MathSciNetView ArticleMATHGoogle Scholar - Li YK, Lu L: Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses.
*Physics Letters. A*2004, 333(1–2):62–71. 10.1016/j.physleta.2004.09.083MathSciNetView ArticleMATHGoogle Scholar - Zhang Z, Zhou D: Global robust exponential stability for second-order Cohen-Grossberg neural networks with multiple delays.
*Neurocomputing*2009, 73(1–3):213–218. 10.1016/j.neucom.2009.09.003View ArticleGoogle Scholar - Zhang J, Gui Z: Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays.
*Journal of Computational and Applied Mathematics*2009, 224(2):602–613. 10.1016/j.cam.2008.05.042MathSciNetView ArticleMATHGoogle Scholar - Li K: Stability analysis for impulsive Cohen-Grossberg neural networks with time-varying delays and distributed delays.
*Nonlinear Analysis*2009, 10(5):2784–2798. 10.1016/j.nonrwa.2008.08.005MathSciNetView ArticleMATHGoogle Scholar - Okochi H: On the existence of periodic solutions to nonlinear abstract parabolic equations.
*Journal of the Mathematical Society of Japan*1988, 40(3):541–553. 10.2969/jmsj/04030541MathSciNetView ArticleMATHGoogle Scholar - Okochi H: On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains.
*Nonlinear Analysis*1990, 14(9):771–783. 10.1016/0362-546X(90)90105-PMathSciNetView ArticleMATHGoogle Scholar - Chen YQ: On Massera's theorem for anti-periodic solution.
*Advances in Mathematical Sciences and Applications*1999, 9(1):125–128.MathSciNetMATHGoogle Scholar - Yin Y: Monotone iterative technique and quasilinearization for some anti-periodic problems.
*Nonlinear World*1996, 3(2):253–266.MathSciNetMATHGoogle Scholar - Yin Y: Remarks on first order differential equations with anti-periodic boundary conditions.
*Nonlinear Times and Digest*1995, 2(1):83–94.MathSciNetMATHGoogle Scholar - Aftabizadeh AR, Aizicovici S, Pavel NH: On a class of second-order anti-periodic boundary value problems.
*Journal of Mathematical Analysis and Applications*1992, 171(2):301–320. 10.1016/0022-247X(92)90345-EMathSciNetView ArticleMATHGoogle Scholar - Aizicovici S, McKibben M, Reich S: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities.
*Nonlinear Analysis*2001, 43: 233–251. 10.1016/S0362-546X(99)00192-3MathSciNetView ArticleMATHGoogle Scholar - Chen Y, Nieto JJ, O'Regan D: Anti-periodic solutions for fully nonlinear first-order differential equations.
*Mathematical and Computer Modelling*2007, 46(9–10):1183–1190. 10.1016/j.mcm.2006.12.006MathSciNetView ArticleMATHGoogle Scholar - Chen TY, Liu WB, Zhang JJ, Zhang MY: Existence of anti-periodic solutions for Liénard equations.
*Journal of Mathematical Study*2007, 40(2):187–195.MathSciNetMATHGoogle Scholar - Liu B: Anti-periodic solutions for forced Rayleigh-type equations.
*Nonlinear Analysis*2009, 10(5):2850–2856. 10.1016/j.nonrwa.2008.08.011MathSciNetView ArticleMATHGoogle Scholar - Wang W, Shen J: Existence of solutions for anti-periodic boundary value problems.
*Nonlinear Analysis*2009, 70(2):598–605. 10.1016/j.na.2007.12.031MathSciNetView ArticleMATHGoogle Scholar - Li Y, Huang L: Anti-periodic solutions for a class of Liénard-type systems with continuously distributed delays.
*Nonlinear Analysis*2009, 10(4):2127–2132. 10.1016/j.nonrwa.2008.03.020MathSciNetView ArticleMATHGoogle Scholar - Delvos F-J, Knoche L: Lacunary interpolation by antiperiodic trigonometric polynomials.
*BIT*1999, 39(3):439–450. 10.1023/A:1022314518264MathSciNetView ArticleMATHGoogle Scholar - Du JY, Han HL, Jin GX: On trigonometric and paratrigonometric Hermite interpolation.
*Journal of Approximation Theory*2004, 131(1):74–99. 10.1016/j.jat.2004.09.005MathSciNetView ArticleMATHGoogle Scholar - Chen HL: Antiperiodic wavelets.
*Journal of Computational Mathematics*1996, 14(1):32–39.MathSciNetMATHGoogle Scholar - Peng G, Huang L: Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays.
*Nonlinear Analysis*2009, 10(4):2434–2440. 10.1016/j.nonrwa.2008.05.001MathSciNetView ArticleMATHGoogle Scholar - Ou C: Anti-periodic solutions for high-order Hopfield neural networks.
*Computers & Mathematics with Applications*2008, 56(7):1838–1844. 10.1016/j.camwa.2008.04.029MathSciNetView ArticleMATHGoogle Scholar - Aizicovici S, McKibben M, Reich S: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities.
*Nonlinear Analysis*2001, 43: 233–251. 10.1016/S0362-546X(99)00192-3MathSciNetView ArticleMATHGoogle Scholar - Shao JY: Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays.
*Physics Letters, Section A*2008, 372(30):5011–5016. 10.1016/j.physleta.2008.05.064View ArticleMATHGoogle Scholar - Li YK, Yang L: Anti-periodic solutions for Cohen-Grossberg neural networks with bounded and unbounded delays.
*Communications in Nonlinear Science and Numerical Simulation*2009, 14(7):3134–3140. 10.1016/j.cnsns.2008.12.002MathSciNetView ArticleMATHGoogle Scholar - Gong S: Anti-periodic solutions for a class of Cohen-Grossberg neural networks.
*Computers & Mathematics with Applications*2009, 58(2):341–347. 10.1016/j.camwa.2009.03.105MathSciNetView ArticleMATHGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar - Bohner M, Peterson A:
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.View ArticleMATHGoogle Scholar - Lakshmikantham V, Vatsala AS: Hybrid systems on time scales.
*Journal of Computational and Applied Mathematics*2002, 141(1–2):227–235. 10.1016/S0377-0427(01)00448-4MathSciNetView ArticleMATHGoogle Scholar - Kaufmann ER, Raffoul YN: Periodic solutions for a neutral nonlinear dynamical equation on a time scale.
*Journal of Mathematical Analysis and Applications*2006, 319(1):315–325. 10.1016/j.jmaa.2006.01.063MathSciNetView ArticleMATHGoogle Scholar - Agarwal R, Bohner M, Peterson A: Inequalities on time scales: a survey.
*Mathematical Inequalities & Applications*2001, 4(4):535–557.MathSciNetView ArticleMATHGoogle Scholar - Bohner M, Fan M, Zhang J: Existence of periodic solutions in predator-prey and competition dynamic systems.
*Nonlinear Analysis*2006, 7(5):1193–1204. 10.1016/j.nonrwa.2005.11.002MathSciNetView ArticleMATHGoogle Scholar - Oregan D, Cho YJ, Chen YQ:
*Topological Degree Theory and Application*. Taylor & Francis, London, UK; 2006.Google Scholar - Li YK, Chen XR, Zhao L: Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales.
*Neurocomputing*2009, 72(7–9):1621–1630.View ArticleGoogle Scholar

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