Almost Automorphic Solutions to Abstract Fractional Differential Equations
© Hui-Sheng Ding et al. 2010
Received: 24 October 2009
Accepted: 12 January 2010
Published: 2 March 2010
A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equation , in complex Banach spaces, with Stepanov-like almost automorphic coefficients. Moreover, an application to a fractional relaxation-oscillation equation is given.
In this paper, we investigate the existence and uniqueness of almost automorphic solutions to the following semilinear abstract fractional differential equation:
where , is a sectorial operator of type in a Banach space , and is Stepanov-like almost automorphic in satisfying some kind of Lipschitz conditions in . In addition, the fractional derivative is understood in the Riemann-Liouville's sense.
Recently, fractional differential equations have attracted more and more attentions (cf. [1–8] 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 the set of positive integers, by the set of real numbers, and by a complex Banach space. In addition, we assume 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., [9–11]).
for each . Denote by the set of all such functions.
It is obvious that and whenever .
for each .
It is clear that if and is -almost automorphic, then is -almost automorphic. Also if , then is -almost automorphic for any .
for each and for each . We denote by the set of all such functions.
2. Almost Automorphic Solution
First, let us recall that a closed and densely defined linear operator is called sectorial if there exist , , and such that its resolvent exists outside the sector
Recently, in , Cuesta proved that if is sectorial operator for some ( ), , and , then there exits such that
where is a suitable path lying outside the sector .
In addition, by , we have the following definition.
Fix and . We have
This means that is continuous.
Fix . By the definition of , for every sequence of real numbers , there exist a subsequence and a function such that
for each . Therefore, for each .
For the case of , the conclusion of Lemma 2.2 was given in [1, Lemma ].
The following theorem will play a key role in the proof of our existence and uniqueness theorem.
Theorem 2.4 (see ).
(i) with ;
(iii) and is compact in .
Then there exists such that
Now, we are ready to present the existence and uniqueness theorem of almost automorphic solutions to (1.1).
Since , and is nonincreasing. So Lemma 2.2 yields that . This means that maps into .
For each and , we have
for all . Thus (1.1) has a unique almost automorphic mild solution.
In the case of , by following the proof of Theorem 2.5 and using the standard contraction principle, one can get the following conclusion.
At last, we give an application to illustrate the abstract result.
for some .
Let , with
by Theorem 2.5, there exists a unique almost automorphic mild solution to (2.26) provided that and is sufficiently small.
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.
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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