Almost Automorphic Solutions to Abstract Fractional Differential Equations

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

(1.1)

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]).

Definition 1.1.

A continuous function is called almost automorphic if for every real sequence , there exists a subsequence such that

(1.2)

is well defined for each and

(1.3)

for each . Denote by the set of all such functions.

Definition 1.2.

The Bochner transform , , , of a function on , with values in , is defined by

(1.4)

Definition 1.3.

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

(1.5)

It is obvious that and whenever .

Definition 1.4.

The space of -almost automorphic functions ( -a.a. for short) consists of all such that . In other words, a function is said to be -almost automorphic if its Bochner transform is almost automorphic in the sense that for every sequence of real numbers , there exist a subsequence and a function such that

(1.6)

for each .

Remark 1.5.

It is clear that if and is -almost automorphic, then is -almost automorphic. Also if , then is -almost automorphic for any .

Definition 1.6.

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

(1.7)

for each and for each . We denote by the set of all such functions.

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

(2.1)

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

(2.2)

where

(2.3)

where is a suitable path lying outside the sector .

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

Definition 2.1.

A function is called a mild solution of (1.1) if is integrable on for each and

(2.4)

Lemma 2.2.

Let be a strongly continuous family of bounded and linear operators such that

(2.5)

where is nonincreasing. Then, for each ,

(2.6)

Proof.

For each , let

(2.7)

In addition, for each , by the principle of uniform boundedness,

(2.8)

Fix and . We have

(2.9)

In view of , we get

(2.10)

which yields that

(2.11)

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

(2.12)

for each . Combining this with

(2.13)

we get

(2.14)

for each . Similar to the above proof, one can show that

(2.15)

for each . Therefore, for each .

Noticing that

(2.16)

we know that is uniformly convergent on . Thus

(2.17)

Remark 2.3.

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 [11]).

Assume that

(i) with ;

(ii)there exists a nonnegative function with such that for all and ,

(2.18)

(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).

Theorem 2.5.

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

(2.19)

Proof.

For each , let

(2.20)

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

(2.21)

Since , and is nonincreasing. So Lemma 2.2 yields that . This means that maps into .

For each and , we have

(2.22)

which gives

(2.23)

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

(2.24)

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.

Theorem 2.6.

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

(2.25)

Remark 2.7.

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

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

(2.26)

with boundary conditions

(2.27)

where , , and

(2.28)

for some .

Let , with

(2.29)

and for and . Then (2.26) is transformed into (1.1). It is well known that is a sectorial operator for some , and . By [10, Example ], . Then . In addition, for each and ,

(2.30)

Since

(2.31)

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

Remark 2.9.

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

(2.32)

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.