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Advances in Difference Equations

, 2010:508374 | Cite as

Almost Automorphic Solutions to Abstract Fractional Differential Equations

  • Hui-Sheng Ding
  • Jin Liang
  • Ti-Jun Xiao
Open Access
Research Article
Part of the following topical collections:
  1. Abstract Differential and Difference Equations

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

A continuous function Open image in new window is called almost automorphic if for every real sequence Open image in new window , there exists a subsequence Open image in new window such that
is well defined for each Open image in new window and

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.

The space Open image in new window of all Stepanov bounded functions, with the exponent Open image in new window , consists of all measurable functions Open image in new window on Open image in new window with values in Open image in new window such that

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

Definition 1.4.

The space Open image in new window of Open image in new window -almost automorphic functions ( Open image in new window -a.a. for short) consists of all Open image in new window such that Open image in new window . In other words, a function Open image in new window is said to be Open image in new window -almost automorphic if its Bochner transform Open image in new window is almost automorphic in the sense that 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 .

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.

A function Open image in new window , Open image in new window with Open image in new window for each Open image in new window is said to be Open image in new window -almost automorphic in Open image in new window uniformly for Open image in new window , if for every sequence of real numbers Open image in new window , there exists a subsequence Open image in new window and a function Open image in new window with Open image in new window such that

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.

A function Open image in new window is called a mild solution of (1.1) if Open image in new window is integrable on Open image in new window for each Open image in new window and

Lemma 2.2.

Let Open image in new window be a strongly continuous family of bounded and linear operators such that
where Open image in new window is nonincreasing. Then, for each Open image in new window ,

Proof.

In addition, for each Open image in new window , by the principle of uniform boundedness,

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

which yields that

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 . Combining this with
for each Open image in new window . Similar to the above proof, one can show that

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

Noticing that

we know that Open image in new window is uniformly convergent on Open image in new window . Thus

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 ;

(ii)there exists a nonnegative function Open image in new window with Open image in new window such that for all Open image in new window and 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.

Assume that Open image in new window is sectorial operator for some Open image in new window , Open image in new window and Open image in new window ; and the assumptions (i) and (ii) of Theorem 2.4 hold. Then (1.1) has a unique almost automorphic mild solution provided that

Proof.

In view of Open image in new window which is compact in Open image in new window , by Theorem 2.4, there exists Open image in new window such that Open image in new window . On the other hand, by (2.2), we have

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

which gives
In view of (2.19), Open image in new window is a contraction mapping. On the other hand, it is well known that Open image in new window is a Banach space under the supremum norm. Thus, Open image in new window has a unique fixed point Open image in new window , which satisfies

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.

Assume that Open image in new window is sectorial operator for some Open image in new window , Open image in new window and Open image in new window ; and the assumptions (i) and (ii) of Theorem 2.4 hold with Open image in new window , then (1.1) has a unique almost automorphic mild solution provided that

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.

Let us consider the following fractional relaxation-oscillation equation given by
with boundary conditions

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.

In the above example, for any Open image in new window , Open image in new window is Lipschitz continuous about Open image in new window uniformly in Open image in new window with Lipschitz constant Open image in new window , this means that Open image in new window has a better Lipschitz continuity than (2.30). However, one cannot ensure the unique existence of almost automorphic mild solution to (2.26) when

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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Copyright information

© Hui-Sheng Ding et al. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceJiangxi Normal UniversityNanchangChina
  2. 2.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  3. 3.School of Mathematical SciencesFudan UniversityShanghaiChina

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