- Research Article
- Open access
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Weighted
-Asymptotically
-Periodic Solutions of a Class of Fractional Differential Equations
Advances in Difference Equations volume 2011, Article number: 584874 (2011)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ5_HTML.gif)
Under the above conditions, for and
we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ8_HTML.gif)
If , then the function
belongs to
.
Proof.
Using the fact that is bounded, it follows that
. For
be given, we select
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ9_HTML.gif)
for all and
. Then, for
we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ10_HTML.gif)
which proves the assertion.
Lemma 2.8.
Let . Let
and
be the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ11_HTML.gif)
If as
and
, then
.
Proof.
From the estimate , it follows that
. For
be given we select
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ12_HTML.gif)
for all . Under these conditions, for
we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ17_HTML.gif)
Then, for we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ19_HTML.gif)
Concerning the quantities and
, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ20_HTML.gif)
Using the estimates (3.7) in (3.6), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ21_HTML.gif)
where is a positive constant independent of
. Finally, by using the Gronwall-Bellman inequality we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ24_HTML.gif)
where is the constant introduced in Lemma 2.8.Then there is a unique weighted
-asymptotically
-periodic mild solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ25_HTML.gif)
Proof.
The proof is based in Lemmas 2.7 and 2.8. Let be the map defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ26_HTML.gif)
We show initially that is
-valued. From the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ27_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ28_HTML.gif)
so that,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ29_HTML.gif)
On the another hand, for we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ30_HTML.gif)
from which we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ31_HTML.gif)
By noting that is a linear operator for all
and combining (3.16) and (3.18) we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ32_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ33_HTML.gif)
with boundary conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ34_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ36_HTML.gif)
We have the following estimates:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ38_HTML.gif)
estimate (3.25), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ39_HTML.gif)
Since we obtain that
. Moreover, we have the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F584874/MediaObjects/13662_2010_Article_58_Equ40_HTML.gif)
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.
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Acknowledgments
C. Cuevas thanks the Department of Mathematics of Universidad de La Frontera, where this project was started. The authors are grateful to the referees for their valuable comments and suggestions. C. Cuevas is partially supported by CNPQ/Brazil under Grant 300365/2008-0.
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Cuevas, C., Pierri, M. & Sepulveda, A. Weighted -Asymptotically
-Periodic Solutions of a Class of Fractional Differential Equations.
Adv Differ Equ 2011, 584874 (2011). https://doi.org/10.1155/2011/584874
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DOI: https://doi.org/10.1155/2011/584874