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This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations

The almost periodic function introduced seminally by Bohr in 1925 plays an important role in describing the phenomena that are similar to the periodic oscillations which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, and ecosphere. The concept of almost automorphy, which is an important generalization of the classical almost periodicity, was first introduced in the literature [

As a natural extension of almost automorphy, the concept of asymptotic almost automorphy, which is the central issue to be discussed in this paper, was introduced in the literature [

With motivation coming from a wide range of engineering and physical applications, fractional differential equations have recently attracted great attention of mathematicians and scientists. This kind of equations is a generalization of ordinary differential equations to arbitrary noninteger orders. Fractional differential equations find numerous applications in the field of viscoelasticity, feedback amplifiers, electrical circuits, electro analytical chemistry, fractional multipoles, neuron modelling encompassing different branches of physics, chemistry, and biological sciences [

The study of almost periodic and almost automorphic type solutions to fractional differential equations was initiated by Araya and Lizama [

Recently, Xia et al. [

Equation (

To the best of our knowledge, much less is known about the existence of asymptotically almost automorphic mild solutions to (

The rest of this paper is organized as follows. In Section

This section is concerned with some notations, definitions, lemmas, and preliminary facts which are used in what follows.

From now on, let

First, let us recall some basic definitions and results on almost automorphic and asymptotically almost automorphic functions.

A continuous function

Denote by

By the point-wise convergence, the function

A continuous function

The function

Similar to Lemma 2.2 of [

Let

Suppose that

If

If

A continuous function

Denote by

The function

A continuous function

Denote by

The function

Next we give some basic definitions and properties of the fractional calculus theory which are used further in this paper.

The fractional integral of order

Riemann-Liouville derivative of order

The first and maybe the most important property of Riemann-Liouville fractional derivative is that, for

It is important to define sectorial operator for the definition of mild solution of any fractional abstract equations. So, let us now give the definitions of sectorial linear operators and their associated solution operators.

A closed and linear operator

Sectorial operators are well studied in the literature, usually for the case

Let

Note that if

Very recently, Cuesta in [

In the following, we present the following compactness criterion, which is a special case of the general compactness result of Theorem 2.1 in [

A set

the set

The following Krasnoselskii’s fixed point theorem plays a key role in the proofs of our main results, which can be found in many books.

Let

In this section, we study the existence of asymptotically almost automorphic mild solutions for the semilinear fractional differential equations of the form

We recall the following definition that will be essential for us.

Assume that

In the proofs of our results, we need the following auxiliary result.

Given

Firstly, note that

By a similar argument one can obtain

Since

Now we are in position to state and prove our first main result. To prove our main result, let us introduce the following assumptions:

(

Assuming that

Let

Since

Assume that

The proof is divided into the following five steps.

Firstly, since the function

In the sequel, we verify that

Let

Next, we prove that

In fact, let

Firstly, from (

On the other hand, in view of (

In fact, for any

Given

In the sequel, we consider the compactness of

Set

Next, we verify the equicontinuity of the set

Let

Now an application of Lemma

Firstly, the complete continuity of

Then, consider the following coupled system of integral equations:

Taking

Let

It is interesting to note that the function

Theorem

Assume that

The proof is divided into the following five steps.

Firstly, similar to the proof in Step 1 of Theorem

In the sequel, we verify that

Let

Next, we prove that

In fact, for

Firstly, from (

On the other hand, in view of (

In fact, for any

The proof is similar to the proof in Step 4 of Theorem

The proof is similar to the proof in Step 5 of Theorem

Taking

Let

Now we consider a more general case of equations introducing a new class of functions

(

Assume that

The proof is divided into the following five steps.

Firstly, similar to the proof in Step 1 of Theorem

Next, we prove that

In fact, for

Firstly, from (

On the other hand, it is not difficult to see that there exists a constant