# Almost Automorphic Solutions to Abstract Fractional Differential Equations

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## Abstract

A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equation Open image in new window , in complex Banach spaces, with *Stepanov-like almost automorphic coefficients*. Moreover, an application to a fractional relaxation-oscillation equation is given.

### Keywords

Abstract Result Uniqueness Theorem Mild Solution Fractional Differential Equation Lipschitz Constant## 1. Introduction

In this paper, we investigate the existence and uniqueness of almost automorphic solutions to the following semilinear abstract fractional differential equation:

where Open image in new window , Open image in new window is a sectorial operator of type Open image in new window in a Banach space Open image in new window , and Open image in new window is Stepanov-like almost automorphic in Open image in new window satisfying some kind of Lipschitz conditions in Open image in new window . In addition, the fractional derivative is understood in the Riemann-Liouville's sense.

Recently, fractional differential equations have attracted more and more attentions (cf. [1, 2, 3, 4, 5, 6, 7, 8] and references therein). On the other hand, the Stepanov-like almost automorphic problems have been studied by many authors (cf., e.g., [9, 10] and references therein). Stimulated by these works, in this paper, we study the almost automorphy of solutions to the fractional differential equation (1.1) with Stepanov-like almost automorphic coefficients. A new and general existence and uniqueness theorem of almost automorphic solutions to the equation is established. Moreover, an application to fractional relaxation-oscillation equation is given to illustrate the abstract result.

Throughout this paper, we denote by Open image in new window the set of positive integers, by Open image in new window the set of real numbers, and by Open image in new window a complex Banach space. In addition, we assume Open image in new window if there is no special statement. Next, let us recall some definitions of almost automorphic functions and Stepanov-like almost automorphic functions (for more details, see, e.g., [9, 10, 11]).

Definition 1.1.

for each Open image in new window . Denote by Open image in new window the set of all such functions.

Definition 1.2.

Definition 1.3.

It is obvious that Open image in new window and Open image in new window whenever Open image in new window .

Definition 1.4.

for each Open image in new window .

Remark 1.5.

It is clear that if Open image in new window and Open image in new window is Open image in new window -almost automorphic, then Open image in new window is Open image in new window -almost automorphic. Also if Open image in new window , then Open image in new window is Open image in new window -almost automorphic for any Open image in new window .

Definition 1.6.

for each Open image in new window and for each Open image in new window . We denote by Open image in new window the set of all such functions.

## 2. Almost Automorphic Solution

First, let us recall that a closed and densely defined linear operator Open image in new window is called sectorial if there exist Open image in new window , Open image in new window , and Open image in new window such that its resolvent exists outside the sector

Recently, in [3], Cuesta proved that if Open image in new window is sectorial operator for some Open image in new window ( Open image in new window ), Open image in new window , and Open image in new window , then there exits Open image in new window such that

where

where Open image in new window is a suitable path lying outside the sector Open image in new window .

In addition, by [2], we have the following definition.

Definition 2.1.

Lemma 2.2.

Proof.

Fix Open image in new window and Open image in new window . We have

This means that Open image in new window is continuous.

Fix Open image in new window . By the definition of Open image in new window , for every sequence of real numbers Open image in new window , there exist a subsequence Open image in new window and a function Open image in new window such that

for each Open image in new window . Therefore, Open image in new window for each Open image in new window .

Noticing that

Remark 2.3.

For the case of Open image in new window , the conclusion of Lemma 2.2 was given in [1, Lemma Open image in new window ].

The following theorem will play a key role in the proof of our existence and uniqueness theorem.

Theorem 2.4 (see [11]).

Assume that

(i) Open image in new window with Open image in new window ;

(iii) Open image in new window and Open image in new window is compact in Open image in new window .

Then there exists Open image in new window such that Open image in new window

Now, we are ready to present the existence and uniqueness theorem of almost automorphic solutions to (1.1).

Theorem 2.5.

Proof.

Since Open image in new window , Open image in new window and is nonincreasing. So Lemma 2.2 yields that Open image in new window . This means that Open image in new window maps Open image in new window into Open image in new window .

For each Open image in new window and Open image in new window , we have

for all Open image in new window . Thus (1.1) has a unique almost automorphic mild solution.

In the case of Open image in new window , by following the proof of Theorem 2.5 and using the standard contraction principle, one can get the following conclusion.

Theorem 2.6.

Remark 2.7.

Theorem 2.6 is due to [2, Theroem Open image in new window ] in the case of Open image in new window being almost automorphic in Open image in new window . Thus, Theorem 2.6 is a generalization of [2, Theroem Open image in new window ].

At last, we give an application to illustrate the abstract result.

Example 2.8.

for some Open image in new window .

Let Open image in new window , Open image in new window with

by Theorem 2.5, there exists a unique almost automorphic mild solution to (2.26) provided that Open image in new window and Open image in new window is sufficiently small.

Remark 2.9.

by using Theorem 2.6. On the other hand, it is interesting to note that one can use Theorem 2.5 to obtain the existence in many cases under this restriction.

## Notes

### Acknowledgments

The authors are grateful to the referee for valuable suggestions and comments, which improved the quality of this paper. H. Ding acknowledges the support from the NSF of China, the NSF of Jiangxi Province of China (2008GQS0057), and the Youth Foundation of Jiangxi Provincial Education Department(GJJ09456). J. Liang and T. Xiao acknowledge the support from the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory of Modern Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).

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