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We revisit the notion on almost automorphic functions on time scales given by Lizama and Mesquita (2013). Then we present the notion of almost automorphic functions of order

The concept of time scales was initiated in 1988 by Hilger in his outstanding Ph.D. thesis [

It was natural to study almost periodic time scales as well as almost periodic differential equations on almost periodic time scales [

Recently Lizama and Mesquita introduced the notion of almost automorphic functions on time scales in their work [

Let

However, we observe that the inclusion may be strict. Indeed, let us consider the time scale

For this reason, several results in [

Secondly we would like to study the existence and stability of almost automorphic solutions of the following linear dynamic system with finite delay:

We organized our paper as follows. In Section

In this section we recall some definitions and recent results on time scales.

A time scale is an arbitrary nonempty closed subset of real numbers.

Let

In Definition

Let

A point

A point

A function

A function

We will denote the set of

Let

Moreover, we say that

Next we recall some easy and useful relationships concerning the delta derivative.

Assume

If

If

If

If

Assume

the sum

for any constant

the product

We define higher order derivatives of a function on time scale in the usual way.

For a function

Let

The following results on chain rule can be found in [

Let

Assume that

If

Let

We now present some definitions and results useful for the study of some dynamical systems.

One says that a function

The set of all regressive functions will be denoted by

One defines the set

If

The generalized exponential functions have the following properties.

Assume that

Assume that

if

If

Let

We now present some definitions about matrix-valued functions on

Let

We say that

An

From now on,

A time scale

Let

In view of the definition of

It is clear that if

We have the following properties of the points in

Let

Let

As we pointed out in Section

Let

We denote by

In view of Lemma

We have the following properties.

Let

for each

See [

We have the following remark on the property given in [

Notice that

in order to give a sense to

we need the symmetry of the time scale

The space

If

Let

Let

Let

Let

Let

Let

Let

We can now introduce the notion of almost automorphic functions of order

We denote by

It’s clear that

Since for

Let

We denote by

A function

Denote by

Directly from the above definitions it follows that

We have

It is straightforward from the definition of an almost automorphic function on time scales (see Theorem

A linear combination of

For the proof of (i) and (ii), one proceeds as in [

Now, let us prove (iv). For any

If a sequence

From the assumption, it is clear that

In view of Proposition

The first result in this section gives a sufficient condition which guarantees that the derivative of a function

Let

Assume that the points of

Now, let us suppose that

If

In view of Theorem

Similarly as in [

Let

Consider the following.

Given a fixed sequence

Let us write

Now we are ready to enunciate and prove Bohl-Bohr’s type theorem known from the literature for almost automorphic functions on time scale. The proof is inspired by the proof of Theorem

Let

In view of Lemma

Assume that

We get, for every

This leads to contradiction if

If

If

Theorem

If

If

If

We know that

In this section,

Let

Since

For every

Let

According to Proposition

In Corollary

Now we will consider the superposition of operator (the autonomous case) acting on the space

If

First, we observe that the result holds if for

By Theorem

Let

Consider the following system:

Let

Let

In view of Lemma

Let

Let

By Lemma

In the following we will consider

Set

Let

Then, the solution

In what follows, we will give sufficient condition for the existence of

Let

We make the following assumption.

Assume

There exists a positive constant

For convenience, for a

Assume that

For any

Now, we prove that the following mapping is a contraction on

Let

Let

Assume that

According to Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the referee for his/her careful reading and valuable suggestions. Aril Milce received a Ph.D. scholarship from the French Embassy in Haiti. He is also supported by “Ecole Normale Supérieure,” an entity of the “Université d’Etat d’Haïti.”