Weighted -Asymptotically -Periodic Solutions of a Class of Fractional Differential Equations

Abstract

We study the existence of weighted -asymptotically -periodic mild solutions for a class of abstract fractional differential equations of the form , where is a linear sectorial operator of negative type.

1. Introduction

-asymptotically -periodic functions have applications to several problems, for example in the theory of functional differential equations, fractional differential equations, integral equations and partial differential equations. The concept of -asymptotic -periodicity was introduced in the literature by Henríquez et al. [1, 2]. Since then, it attracted the attention of many researchers (see [1–10]). In Pierri [10] a new -asymptotically -periodic space was introduced. It is called the space of weighted -asymptotically -periodic (or -asymptotically -periodic) functions. In particular, the author has established conditions under which a -asymptotically -periodic function is asymptotically -periodic and also discusses the existence of -asymptotically -periodic solutions for an integral abstract Cauchy problem. The author has applied the results to partial integrodifferential equations.

We study in this paper sufficient conditions for the existence and uniqueness of a weighted -asymptotically -periodic (mild) solution to the following semi-linear integrodifferential equation of fractional order

(1.1)

(1.2)

where , is a linear densely defined operator of sectorial type on a complex Banach space and is an appropriate function. Note that the convolution integral in (1.1) is known as the Riemann-Liouville fractional integral [11]. We remark that there is much interest in developing theoretical analysis and numerical methods for fractional integrodifferential equations because they have recently proved to be valuable in various fields of sciences and engineering. For details, including some applications and recent results, see the monographs of Ahn and MacVinish [12], Gorenflo and Mainardi [13] and Trujillo et al. [14–16] and the papers of Agarwal et al. [17–23], Cuesta [11, 24], Cuevas et al. [5, 6], dos Santos and Cuevas [25], Eidelman and Kochubei [26], Lakshmikantham et al. [27–30], Mophou and N'Guérékata [31], Ahmed and Nieto [32], and N'Guérékata [33]. In particular equations of type (1.1) are attracting increasing interest (cf. [5, 11, 24, 34]).

The existence of weighted -asymptotically -periodic (mild) solutions for integrodifferential equation of fractional order of type (1.1) remains an untreated topic in the literature. Anticipating a wide interest in the subject, this paper contributes in filling this important gap. In particular, to illustrate our main results, we examine sufficient conditions for the existence and uniqueness of a weighted -asymptotically -periodic mild solution to a fractional oscillation equation.

2. Preliminaries and Basic Results

In this section, we introduce notations, definitions and preliminary facts which are used throughout this paper. Let and be Banach spaces. The notation stands for the space of bounded linear operators from into endowed with the uniform operator topology denoted , and we abbreviate to and whenever . In this paper denotes the Banach space consisting of all continuous and bounded functions from into with the norm of the uniform convergence. For a closed linear operator we denote by the resolvent set and by the spectrum of (that is, the complement of in the complex plane). Set the resolvent of for .

2.1. Sectorial Linear Operators and the Solution Operator for Fractional Equations

A closed and linear operator is said sectorial of type μ if there are and such that the spectrum of is contained in the sector and , for all .

In order to give an operator theoretical approach for the study of the abstract system we recall the following definition.

Definition 2.1 (see [17]).

Let be a closed linear operator with domain in a Banach space . One calls the generator of a solution operator for (1.1)-(1.2) if there are and a strongly continuous function such that and , for all . In this case, is called the solution operator generated by . By [35, Proposition 2.6], . We observe that the power function is uniquely defined as , with .

We note that if is a sectorial of type μ with , then is the generator of a solution operator given by , , where is a suitable path lying outside the sector (cf. [11]). Recently, Cuesta [11, Theorem 1] proved that if is a sectorial operator of type for some and , then there exists such that

(2.1)

Remark 2.2.

In the remainder of this paper, we always assume that is a a sectorial of type and , are the constants introduced above.

2.2. Weighted -Asymptotically -Periodic Functions

We recall the following definitions.

Definition 2.3 (see [1]).

A function is called -asymptotically -periodic if there exists such that . In this case, we say that is an asymptotic period of .

Throughout this paper, represents the space formed for all the -valued -asymptotically -periodic functions endowed with the uniform convergence norm denoted . It is clear that is a Banach space (see [1, Proposition 3.5]).

Definition 2.4 (see [10]).

Let . A function is called weighted -asymptotically -periodic (or -asymptotically -periodic) if .

In this paper, represents the space formed by all the -asymptotically -periodic functions endowed with the norm

(2.2)

Proposition 2.5.

The space is a Banach space.

Proof.

Let be a Cauchy sequence in . From the definition of , there exists such that in . Next, we prove that in .

By noting that is a Cauchy sequence, for given there exists such that , for all , which implies

(2.3)

Under the above conditions, for and we see that

(2.4)

which implies that for and as .

To conclude the proof we need to show that . Let as above. Since , there exits such that for all . Now, by using that , for we get

(2.5)

which implies that . This completes the proof.

Definition 2.6.

A function is called uniformly -asymptotically -periodic on bounded sets if for every bounded subset , the set is bounded and , uniformly for . If we say that is uniformly -asymptotically -periodic on bounded sets (see [1]).

To prove some of our results, we need the following lemma.

Lemma 2.7.

Let . Assume is uniformly -asymptotically -periodic on bounded sets and there is such that

(2.6)

If , then the function belongs to .

Proof.

Using the fact that is bounded, it follows that . For be given, we select such that

(2.7)

for all and . Then, for we see that

(2.8)

which proves the assertion.

Lemma 2.8.

Let . Let and be the function defined by

(2.9)

If as and , then .

Proof.

From the estimate , it follows that . For be given we select such that

(2.10)

for all . Under these conditions, for we have that

(2.11)

which completes the proof.

3. Existence of Weighted -Asymptotically -Periodic Solutions

In this section we discuss the existence of weighted -asymptotically -periodic solutions for the abstract system (1.1)-(1.2). To begin, we recall the definition of mild solution for (1.1)-(1.2).

Definition 3.1 (see [5]).

A function is called a mild solution of the abstract Cauchy problem (1.1)-(1.2) if

(3.1)

Now, we can establish our first existence result.

Theorem 3.2.

Assume is a uniformly -asymptotically -periodic on bounded sets function and there is a mesurable bounded function such that

(3.2)

If , then there exits a unique -asymptotically -periodic mild solution of (1.1)-(1.2). Suppose, there is a function such that and , for every and all . If is such that as , then is weighted -asymptotically -periodic.

Proof.

Let be the operator defined by

(3.3)

We show initially that is -valued. Since , as , it is sufficient to show that the function is -valued. Let . Using the fact that is a bounded function, it follows that . For be given, we select a constant such that

(3.4)

Then, for we see that

(3.5)

which implies that as , and hence . Moreover, from the above estimate it is easy to infer that , for all , is a contraction and there exists a unique -asymptotically -periodic mild solution of (1.1)-(1.2).

Next, we prove that last assertion. Let be the function defined by . For , we get

(3.6)

Concerning the quantities and , we note that

(3.7)

Using the estimates (3.7) in (3.6), we see that

(3.8)

where is a positive constant independent of . Finally, by using the Gronwall-Bellman inequality we infer that

(3.9)

which shows that . This completes the proof.

Example 3.3.

We set , with . Let be a function such that , for all and let be defined by , . We observe that

(3.10)

whence is -asymptotically -periodic on bounded sets. By Theorem 3.2 we conclude that if , then there is a unique -asymptotically -periodic mild solution of (1.1)-(1.2). Moreover .

Theorem 3.4.

Let . Assume , as and

(3.11)

where is the constant introduced in Lemma 2.8.Then there is a unique weighted -asymptotically -periodic mild solution of

(3.12)

Proof.

The proof is based in Lemmas 2.7 and 2.8. Let be the map defined by

(3.13)

We show initially that is -valued. From the estimate

(3.14)

we have that .

Let . From Lemma 2.7, we have that is a weighted -asymptotically -periodic function and by Lemma 2.8 we obtain that . Thus, the map is -valued. In order to prove that is a contraction, we note that for and ,

(3.15)

so that,

(3.16)

On the another hand, for we see that

(3.17)

from which we obtain that

(3.18)

By noting that is a linear operator for all and combining (3.16) and (3.18) we obtain that

(3.19)

for all , which shows that is a contraction on and hence there is a unique -asymptotically -periodic mild solution. The proof is complete.

To complete this paper, we examine the existence and uniqueness of weighted -asymptotically -periodic mild solutions for the following fractional differential equation

(3.20)

with boundary conditions

(3.21)

(3.22)

where and . In what follows we consider the space and let be the operator given by with domain , . It is well known that is sectorial of type negative.

Proposition 3.5.

Let satisfying conditions of Lemma 2.8 and let . If is small enough, then the problems (3.20)–(3.22) has a unique -asymptotically -periodic mild solution.

Proof.

Problem (3.20)–(3.22) can be expressed as an abstract fractional differential equation of the form (3.12), where , for . We define

(3.23)

We have the following estimates:

(3.24)

(3.25)

estimate (3.25), we get

(3.26)

Since we obtain that . Moreover, we have the inequality

(3.27)

If we choose small enough, we have that condition (3.11) is fulfilled. By Theorem 3.4, the problems (3.20)–(3.22) has a unique -asymptotically -periodic (mild) solution. This finishes the proof.