Journal of Applied Mathematics

Volume 2012 (2012), Article ID 463913, 19 pages

doi:10.1155/2012/463913

Research Article

Almost Periodic Solutions to Dynamic Equations on Time Scales and Applications

Yongkun Li and Chao Wang

Department of Mathematics, Yunnan University, Yunnan, Kunming 650091, China

Received 13 January 2012; Accepted 19 August 2012

Academic Editor: Shiping Lu

Copyright © 2012 Yongkun Li and Chao Wang. 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.

We first introduce the concept of admitting an exponential dichotomy to a class of linear dynamic equations on time scales and study the existence and uniqueness of almost periodic solution and its expression form to this class of linear dynamic equations on time scales. Then, as an application, using these concepts and results, we establish sufficient conditions for the existence and exponential stability of almost periodic solution to a class of Hopfield neural networks with delays. Finally, two examples and numerical simulations given to illustrate our results are plausible and meaningful.

1. Introduction

In recent years, researches in many fields on time scales have received much attention. The theory of calculus on time scales (see [1, 2] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [3] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his fundamental work. It has been created in order to unify the study of differential and difference equations. Also, the existence of almost periodic, asymptotically almost periodic, pseudo-almost periodic solutions is among the most attractive topics in qualitative theory of differential equations and difference equations due to their applications, especially in biology, economics, and physics [4–20].

Motivated by the above, based on the theory of almost periodic functions on time scales in our previous work [21, 22], we first introduce the concept of admitting an exponential dichotomy to a class of linear dynamic equations on time scales and study the existence and uniqueness of almost periodic solution and its expression form to this class of linear dynamic equations on time scales. Then, as an application, using these concepts, results, the fixed point theorem and differential inequality techniques, we establish sufficient conditions for the existence and exponential stability of almost periodic solution to a class of Hopfield neural networks with delays. Finally, two examples given to illustrate our results are plausible and meaningful to unify continuous and discrete models.

The organization of this paper is as follows. In Section 2, we introduce some notations and state some preliminary results needed in the later sections. In Section 3, we introduce the concepts of admitting an exponential dichotomy to a class of linear dynamic equations on time scales under which the existence, uniqueness, and expression form of an almost periodic solution are obtained. Furthermore, some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations are also derived. In Section 4, as an application of our results, we study the existence and exponential stability of the almost periodic solutions of a class of Hopfield neural networks with delays, finally, we give two examples and numerical simulations to show that our unification of continuous and discrete situations is effective.

2. Preliminaries

In this section, we will first recall some basic definitions, lemmas which are used in what follows.

Let be a nonempty closed subset (time scale) of . The forward and backward jump operators and the graininess are defined, respectively, by

A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise . If has a right-scattered minimum , then ; otherwise .

A function is right-dense continuous provided it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If is continuous at each right-dense point and each left-dense point, then is said to be a continuous function on .

For and , we define 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 for all .

Let be right-dense continuous, if , then we define the delta integral by .

A function is called regressive provided for all . The set of all regressive and -continuous functions will be denoted by . We define the set , for all .

An -matrix-valued function on a time scale is called regressive provided that is invertible for all and the class of all such regressive and -continuous functions is denoted, similar to the above scalar case, by .

If is a regressive function, then the generalized exponential function is defined by for all , with the cylinder transformation

Definition 2.1 (see [1, 2]). Let be two regressive functions, define

If is a matrix, then we let denote its conjugate transpose.

Lemma 2.2 (see [1, 2]). If are matrix-value functions on , then

(i) and ,(ii), (iii), (iv), (v), (vi) if and commute.

Lemma 2.3 (see [1, 2]). If and , then and

Lemma 2.4 (see [1, 2]). Let be an -matrix-valued function on and suppose that is -contibuous. Let and . Then the initial value problem

has a unique solution . Moreover, this solution is given by .

For convenience, denotes or , denote the set of all almost periodic -matrix-valued functions on .

Definition 2.5 (see [22]). A time scale is called an almost periodic time scale if

Definition 2.6 (see [22]). Let be an almost periodic time scale. A continuous -matrix-valued function on is called almost periodic on if the -translation set of

is relatively dense set in for all ; that is, for any given , there exists a constant such that each interval of length contains a such that

is called the -translation number of and is called the inclusion length of , where is a matrix norm on , (say, e.g., if , then we can take e.g., ).

For convenience, let , be two sequences. Then means that is a subsequence of ; ; ; and and are common subsequences of and , respectively, means that and for some given function .

Let , be -matrix-valued functions on , we will introduce the translation operator , means that and is written only when the limit exists.

Definition 2.7. Let be an -matrix-valued function on , : there exits such that exists uniformly on is called the hull of .

Similar to the proof of Theorem  3.14 in [22], one can easily get a more general version of the following.

Theorem 2.8. Let be an -matrix-valued function on , if for any , there exists such that exists uniformly on , then is almost periodic.

Also, from Theorem  3.30 in [22], one can easily show a more general version of the following.

Lemma 2.9. An -matrix-valued function is almost periodic on , if and only if for every pair of sequences , there exist common subsequences such that

3. Almost Periodic Dynamic Equations on Time Scales

Consider the linear almost periodic equation

and its associated homogeneous equation

where is an almost periodic matrix function and is an almost periodic vector function.

Definition 3.1. If , we say that

is a homogeneous equation in the hull of (3.1).

Definition 3.2. If , and , we say that

is an equation in hull of (3.1).

Definition 3.3. Let be -continuous matrix function on , the linear equation

is said to admit an exponential dichotomy on if there exist positive constants , projection and the fundamental solution matrix of (3.5), satisfying

From Theorem 2.8 and Lemma 2.9 for and Lemma 2.4, one can also easily get the following Favard theorem.

Lemma 3.4. If is almost periodic matrix function and is an almost periodic solution to the homogeneous linear almost periodic equation , then or .

Similar to the proof of Lemma  4.16 in [22], one can easily show the following lemma.

Lemma 3.5. Suppose that is an almost periodic matrix function and (3.2) satisfies an exponential dichotomy (3.6), then for every , (3.3) satisfies an exponential dichotomy with same projection and same constants .

Similar to the proof of Lemma  4.17 in [22], one can easily show the following:

Lemma 3.6. If the homogeneous equation (3.2) satisfies an exponential dichotomy (3.6), then (3.2) has only one bounded solution .

Lemma 3.7. If the homogeneous equation (3.2) satisfies an exponential dichotomy (3.6), then all equations in the hull of (3.2) have only one bounded solution .

Proof. By Lemma 3.5, all equations in the hull of (3.2) satisfy an exponential dichotomy (3.6), according to Lemma 3.6, all equations in the hull of (3.2) have only one bounded solution . This completes the proof.

By Lemmas 3.4 and 3.7, one can easily have the existence and uniqueness theorem for an almost periodic solution to (3.1):

Theorem 3.8. Let be an almost periodic matrix function, is an almost periodic vector function. If (3.2) admits an exponential dichotomy, then (3.1) has a unique almost periodic solution

where is the fundamental solution matrix of (3.2).

Similar to the proof of Lemma  2.15 in [21], one can easily show the following.

Lemma 3.9. Let be an almost periodic function on , where , , , and , then the linear equation

admits an exponential dichotomy on .

4. Applications

In the real world, both continuous and discrete systems are very important in implementation and applications. Therefore, it is meaningful to study almost periodic problems on time scales which can unify the continuous and discrete situations.

In this section, we consider the following model for the delayed Hopfield neutral networks (HNNs):

in which is the number of units in a neural network, () is the state vector of the th unit at the time , () represents the rate at time with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, () denotes the conversion of the membrane potential of the th unit into its firing rate, () denotes the strength of the th unit on the ith unit at time corresponds to the transmission delay of the ith unit along the axon of the th unit at time , and () denotes the external bias on the th unit at time .

It is well known that the HNNs have been successfully applied to signal and image processing, pattern recognition, and optimization. Hence, they have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of periodic solutions of system (4.1) in the literature. We refer the reader to [23–30] and the references cited therein. In order to unify continuous and discrete situations, by using results in Sections 3 and 4, one can discuss almost periodic problems on time scales.

The main purpose of this section is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (4.1). By applying fixed point theorem and differential inequality techniques, we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential stability of the almost periodic solution on time scales. Our results are new even if the time scale and other types of almost periodic time scales such as , . Numerical examples and simulations are given to illustrate our feasible results and effectiveness of our methods.

For (4.1), we assume that for and are almost periodic functions, where and are constants, , and we use the following notation:

We also assume that the following condition holds.

for each , is Lipschitz with Lipschitz constant , that is,

For convenience, we will use to denote a column vector, in which the symbol denotes the transpose of a vector. We let denote the absolute-value vector given by , and define . For matrix denotes the transpose of , denotes the inverse of , denotes the absolute-value matrix given by , and denotes the spectral radius of . A matrix or vector means that all entries of are greater than or equal to zero. is defined similarly. For matrices or vectors and , (resp. ) means that (resp. ). Let

Definition 4.1 (see [31]). For , and , , we call and the right upper and right lower derivatives of the function at , respectively, if

Lemma 4.2. Let is -differentiable at . Then

Proof. Case (i). If is a right dense point, that is, .

Case (ii). If is a right scattered point, that is, . If , one can easily have , so we can obtain

If , then one can get . Then

Therefore, by (4.7), (4.8), (4.9), one can get

This completes the proof.

As usual, we introduce the phase space as a Banach space of continuous mappings from to equipped with the supremum norm defined by for all .

The initial conditions associated with system (4.1) are of the following form:

where ,  .

Definition 4.3. The almost periodic solution of (4.1) is said to be globally exponentially stable, if there exists a positive such that for any , there exists such that for any solution of (4.1), it is valid that

where is the initial condition.

Definition 4.4 (see [30]). A real matrix is said to be an -matrix if , , and .

Lemma 4.5 (see [30]). Let be an matrix and , then , where denotes the identity matrix of size .

In the following, we will show the existence and uniqueness of almost periodic solution to (4.1) on time scales.

Theorem 4.6. Let hold and . Then, there exists exactly one almost periodic solution of system (4.1).

Proof. Let , where is an almost periodic function, . Then is a Banach space with the norm defined by .

To proceed further, we need to introduce an auxiliary equation

where . Notice that and are almost periodic functions, by Lemma 3.9 and Theorem 3.8, we know that the auxiliary (4.13) has exactly one almost periodic solution

Define a mapping by setting . Let , then by , we have

which implies that

where . Let be a positive integer. Then, from (4.16), we get

Since , we obtain , which implies that there exist a positive integer and a positive constant such that

In view of (4.17), (4.18), we have

for all , . It follows that

This implies that the mapping is a contraction mapping.

By the fixed point theorem of Banach space, possesses a unique fixed point in such that . We know from (4.14) that satisfies system (4.1), and therefore, it is the unique almost periodic solution of system (4.1). This completes the proof.

Next, we will establish a result for the exponential stability of the almost periodic solution of system (4.1).

Theorem 4.7. Suppose that all the conditions of Theorem 4.6 hold. Then system (4.1) has exactly one almost periodic solution . Moreover, is globally exponentially stable.

Proof. Since , it follows from Theorem 4.6 that system (4.1) has a unique almost periodic solution . Let be an arbitrary solution of system (4.1) and define . Then, set

we get

Thus, for , by Lemma 4.2, we have

where , . Again from , it follows from Lemma 4.5 that is an -matrix, we obtain that there exist a constant and a vector such that . Therefore, , , which imply that , . Hence, we can choose a positive constant such that

Choose a constant such that

For any , let

From (4.24), (4.26), noticing that , we obtain

where , . In view of (4.25) and (4.26), for , we have

We claim that

Contrarily, there must exist and such that

which implies that

Case I. If is right dense point, that is, , we have

Case II. If is right scattered point, that is, , we obtain

Thus,

From (4.23), (4.27), (4.30), we can obtain

which contradicts (4.34). Hence, (4.29) holds. Letting and , we have from (4.26) and (4.29) that

for all . This completes the proof.

Example 4.8. Let , consider the following HNN:

where and , , .

Note that , , , , and , , , , , . Then we have

and . Thus, from Theorems 4.6 and 4.7, (4.37) has exactly one almost periodic solution and it is exponentially stable. We can take the initial value , , , , we can give the following numerical simulation figures to show our results are plausible and effective on time scales (see Figures 1 and 2).

Figure 1: Transient response of state variables , and in Example 4.8.

Figure 2: Phase response of state variables , and in Example 4.8.

Example 4.9. Let , consider the following HNN:

where and , , .

Note that , and , , , , , . Then we have

and . Thus, from Theorems 4.6 and 4.7, (4.39) has exactly one almost periodic solution and it is exponentially stable. We can take the initial value , , , , we can give the following numerical simulation figures to show our results are plausible and effective on time scales (see Figures 3 and 4).

Figure 3: Transient response of state variables and in Example 4.9.

Figure 4: Phase response of state variables and in Example 4.9.

5. Conclusion

The existence and uniqueness of almost periodic solution and its expression form to a class of linear dynamic equations on time scales are obtained. As an application, sufficient conditions for the existence and exponential stability of almost periodic solution to a class of Hopfield neural networks with delays are established. To the best of our knowledge, the results presented here have not appeared in the related literature. In fact, both continuous and discrete systems are very important in implementation and applications. But it is troublesome to study the existence and stability of almost periodic solutions for continuous and discrete systems, respectively. Therefore, it is meaningful to study that on timescales which can unify the continuous and discrete situations. Also, the results and methods used in this paper can be used to study many other types neural networks and population models.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of China under no Grant 10971183 and this work was also supported by IRTSTYN.

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