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.

In recent years, researches in many fields on time scales have received much attention. The theory of calculus on time scales (see [

Motivated by the above, based on the theory of almost periodic functions on time scales in our previous work [

The organization of this paper is as follows. In Section

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

Let

A point

A function

For

Let

A function

An

If

Let

If

If

If

Let

For convenience,

A time scale

Let

For convenience, let

Let

Let

Similar to the proof of Theorem 3.14 in [

Let

Also, from Theorem 3.30 in [

An

Consider the linear almost periodic equation

If

If

Let

From Theorem

If

Similar to the proof of Lemma 4.16 in [

Suppose that

Similar to the proof of Lemma 4.17 in [

If the homogeneous equation (

If the homogeneous equation (

By Lemma

By Lemmas

Let

Similar to the proof of Lemma 2.15 in [

Let

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):

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 (

The main purpose of this section is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (

For (

We also assume that the following condition

For convenience, we will use

For

Let

If

Therefore, by (

This completes the proof.

As usual, we introduce the phase space

The initial conditions associated with system (

The almost periodic solution

A real

Let

In the following, we will show the existence and uniqueness of almost periodic solution to (

Let

Let

To proceed further, we need to introduce an auxiliary equation

Define a mapping

In view of (

for all

This implies that the mapping

By the fixed point theorem of Banach space,

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

Suppose that all the conditions of Theorem

Since

we get

Thus, for

If

If

Let

Note that

Transient response of state variables

Phase response of state variables

Let

Note that

Transient response of state variables

Phase response of state variables

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.

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