- Research Article
- Open access
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On Type of Periodicity and Ergodicity to a Class of Fractional Order Differential Equations
Advances in Difference Equations volume 2010, Article number: 179750 (2010)
Abstract
We study several types of periodicity to a class of fractional order differential equations.
1. Introduction
Fractional order differential equations is a very important subject matter. These orders can be complex in viewpoint of pure mathematics. During the last few decades fractional order differential equations have emerged vigorously (cf., [1–8]). We observe that there is much interest in developing the qualitative theory of such equations. Indeed, this has been strongly motivated by their natural and widespread applicability in several fields of sciences and technology. Many real phenomena in those fields can be described very successfully by models using mathematical tools of fractional calculus, such as dielectric polarization, electrode-electrolyte polarization, electromagnetic wave, modeling of earthquake, fluid dynamics, traffic model with fractional derivative, measurement of viscoelastic material properties, modeling of viscoplasticity, Control Theory, and economy (cf., [3, 4, 9–15]). Very recently, some basic theory for initial value problem of fractional differential equations involving the Riemann-Liouville differential operators was discussed by Benchohra et al. [16], Agarwal et al. [17–19], Lakshmikantham [20], and Lakshmikantham and Vatsala [21, 22]. Mophou and N'Guérékata [23] have studied existence of mild solution for fractional semilinear differential equations with nonlocal conditions (more details can be found in [24–29]). El-Sayed and Ibrahim [30] and Benchohra et al. [31] initiated the study of fractional multivalued differential inclusions. In this direction, we refer to the article by Henderson and Ouahab [32] concerning the existence of solutions to fractional functional differential inclusions with finite delay, and existence of solutions for these types of equations in the infinite delay framework (see [16, 31]). In the case that fractional order is , existence results for fractional boundary value problems of differential inclusions were studied by Ouahab [33].
We study in this work some sufficient conditions for the existence and uniqueness of pseudo-almost periodic mild solutions to the following semilinear fractional differential equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ1_HTML.gif)
where ,
is a linear densely defined operator of sectorial type on a complex Banach space
and
is a pseudo-almost periodic function (see Definition 2.10) satisfying suitable conditions in
. The fractional derivative is understood in the Riemann-Liouville sense. Type (1.1) equations are attracting increasing interest. For example, anomalous diffusion in fractals by Eidelman and Kochubei [10] or in macroeconomics by Ahn and McVinisch [1] has been recently studied in the setting of fractional differential equations like (1.1). The study of almost automorphic mild solutions of (1.1) was studied by Cuevas and Lizama in [34] (see also [35]).
As for almost periodic functions, pseudo-almost periodic functions have many applications in several problems, for example, in theory of functional differential equations, integral equations, and partial differential equations. The concept of pseudo-almost periodic was introduced by Zhang [36–39] in the early nineties. Since then, such notion became of great interest to several mathematicians (see [40–49]). To the knowledge of the authors, no results yet exist for pseudo-almost periodic mild solution of (1.1).
We also discuss sufficient conditions for the existence and uniqueness of an asymptotically almost periodic mild solution of the fractional Cauchy problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ2_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ3_HTML.gif)
In a work by Cuevas and de Souza [50] the authors proved existence and uniqueness of an S-asymptotically -periodic solution of problem (1.2)-(1.3) (see also [51]). On the other hand, we give results on existence and uniqueness of an asymptotically almost automorphic mild solution to a class of fractional integrodifferential neutral equations.
We now turn to a summary of this work. The second section provides the definitions and preliminaries results to be used in theorems stated and proved in this article. In particular, we review some of the standard properties of the solution operator generated by a sectorial operator (see Proposition 2.2). We also recall the notion of almost periodicity, asymptotically almost periodicity, asymptotically almost automorphy, and pseudo-almost periodicity. In the third section, we obtain very general results on the existence of pseudo-almost periodic mild solution to equation (1.1). The fourth section is concerned with the existence of an asymptotically almost periodic mild solution to problem (1.2)-(1.3). While in the fifth section we use the machinery developed in the previous sections to obtain new results on existence and uniqueness of an asymptotically almost automorphic solution to a class of fractional integrodifferential neutral equation. To build intuition and throw some light on the power of our results and methods, we give, in the sixth section, a few applications.
2. Preliminaries and Basic Results
Let and
be two Banach spaces. The notation
and
stand for the collection of all continuous functions from
into
and the Banach space of all bounded continuous functions from
into
endowed with the uniform convergence topology, respectively. Similarly,
and
stand, respectively, for the class of all jointly continuous functions from
into
and the collection of all jointly bounded continuous functions from
into
. The notation
stands for the space of bounded linear operators from
into
endowed with the uniform operator topology, and we abbreviate it to
whenever
. We set
for the closed ball with center at
radius
in the space
. A closed and linear operator
is said to be sectorial of type
if there exist
,
and
such that its resolvent exists outside the sector
and
,
. Sectorial operators are well studied in the literature. For a recent reference including several examples and properties we refer the reader to [52]. In order to give an operator theoretical approach, we recall the following definition (cf., [50, 51]).
Definition 2.1.
Let be a closed and linear operator with domain
defined on a Banach space
. Recall
the generator of a solution operator if there exist
and a strongly continuous function
such that
and
,
,
. In this case,
is called the solution operator generated by
.
We note that if is sectorial of type
with
, then
is the generator of a solution operator given by
, where
is a suitable path lying outside the sector
(cf., Cuesta's paper [53]). Very recently, Cuesta [53, Theorem
] has 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%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ4_HTML.gif)
Note that is, in fact, integrable. The concept of a solution operator, as defined above, is closely related to the concept of a resolvent family (see Prüss [54, Chapter 1]). For the scalar case, where there is a large bibliography, we refer the reader to the monography by Gripenberg et al. [55] and references therein. Because of the uniqueness of the Laplace transform, in the border case
the family
corresponds to a
-semigroup, whereas in the case
a solution operator corresponds to the concept of a cosine family; see Arendt et al. [56] and Fattorini [57]. We note that solution operators, as well as resolvent families, are a particular case of
-regularized families introduced by Lizama [58]. According to [58] a solution operator
corresponds to a
-regularized family. The following result is a direct consequence of [58, Proposition
and Lemma
].
Proposition 2.2.
Let be a solution operator on
with generator
. Then, one has the following.
-
(a)
and
for all
,
-
(b)
Let
and
. Then
-
(c)
Let
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ5_HTML.gif)
A characterization of generators of solution operators, analogous to the Hille-Yosida Theorem for -semigroup, can be directly deduced from [58, Theorem
]. Results on perturbation, approximation, representation as well as ergodic type theorems can be deduced from the more general context of
-regularized resolvents (see [58–61]).
Let us recall the notions of almost periodicity, asymptotically almost periodicity, asymptotically almost automorphy, and pseudo-almost periodicity which shall come into play later on.
Definition 2.3 (see [62]).
Let be a Banach space. Then
is called almost periodic if
is continuous, and for each
there exists
such that for every interval of length
it contains a number
with the property that
for each
. The number
above is called an
-translation number for
, and the collection of such functions will be denoted by
.
Remark 2.4 (see [63]).
Note that each almost periodic function is bounded and uniformly continuous. It is well known that the range of an almost periodic function
is relatively compact.
endowed with the norm of uniform convergence on
is a Banach space.
Definition 2.5.
Let and
be two Banach spaces. Then
is called almost periodic in
uniformly for
if
is continuous, and for each
and any compact
there exists
such that every interval
of length
it contains a number
with the property that
for all
,
. The collection of such functions will be denoted by
.
It is well known that the study of composition of two functions with special properties is important and basic for deep investigations. We begin with the following standard result in the theory of almost periodic function (see [39, 63]).
Lemma 2.6.
Let and
. Then the function
.
Definition 2.7.
A continuous function (resp.,
) is called asymptotically almost periodic (resp., asymptotically almost periodic in
uniformly in
) if it admits a decomposition
, where
(resp.,
) and
(resp.,
). Here
denotes the subspace of
such that
and
denotes the space of all continuous functions
such that
uniformly for
in any compact subset of
. Denote by
(resp.,
) the set of all such functions.
is a Banach space with the sup norm.
Definition 2.8.
A continuous function is called uniformly continuous on bounded sets uniformly for
if for every
and every bounded subset
of
there exists
such that
for all
and all
so that
.
Lemma 2.9.
Let and let
be uniformly continuous on bounded sets uniformly for
. If
, then
.
Let denote the space of all bounded continuous functions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ6_HTML.gif)
and denotes the space of all continuous functions such that
is bounded for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ7_HTML.gif)
uniformly in .
Definition 2.10 (see [36, 64]).
A function (resp,
) is called pseudo-almost periodic (resp., pseudo-almost periodic in
uniformly in
) if
where
(
) and
(
).
The functions and
are called the almost periodic component and, respectively, the ergodic perturbation of the function
. The set of all such functions will be denoted by
(resp.,
). Obviously
is a subspace of
. Furthermore, we have that
is a closed subspace of
hence, it is a Banach space with the supremum norm (see [65]).
Lemma 2.11 (see [65]).
Let satisfy the following conditions.
-
(i)
and
is bounded for every bounded subset
.
-
(ii)
is uniformly continuous in each bounded subset of
uniformly in
. More explicitly, given
and
bounded, there exists
such that
and
imply that
for all
.
If , then
.
Lemma 2.12.
Assume that is sectorial of type
. If
is an almost periodic function and
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ8_HTML.gif)
then .
Proof.
For , we take
involved in Definition 2.3, then for every interval of length
contains a number
such that
for each
. The estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ9_HTML.gif)
is responsible for the fact that .
Lemma 2.13.
Assume that is sectorial of type
. If
is an asymptotically almost periodic function and
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ10_HTML.gif)
then .
Proof.
If , where
and
then we have that
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ12_HTML.gif)
By the previous lemma . Next, let us show that
. Since
, for each
there exists a constant
such that
for all
. Then for all
, we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ13_HTML.gif)
Therefore, , that is,
. This completes the proof.
Lemma 2.14.
Assume that is sectorial of type
. If
is pseudo-almost periodic function and
is the function defined in (2.5). Then
.
Proof.
It is clear that . In fact, we get
where
and
are given by (2.1). If
, where
and
, then from Lemma 2.12,
. To complete the proof, we show that
. For
we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ14_HTML.gif)
where ,
.
It is not hard to check that as
. Next, since
is bounded and
is integrable in
, using the Lebesgue dominated convergence theorem, it follows that
. The proof is now completed.
Let be a continuous function such that
as
. We consider the space
endowed with the norm
Lemma 2.15 (see [66]).
A subset is a relatively compact set if it verifies the following conditions.
(c-1) The set is relatively compact in
for each
.
(c-2) The set is equicontinuous.
(c-3) For each there exists
such that
for all
and all
.
Let be a continuous function such that
as
. Consider the space
endowed with the norm
Lemma 2.16 (see [67]).
A subset is a relatively compact set if it verifies the following conditions.
(c-1) The set is relatively compact in
for all
.
(c-2) uniformly for all
.
Definition 2.17.
A continuous function is called almost automorphic if for every sequence of real numbers
there exists a subsequence
such that
is well defined for each
, and
for each
. Denote by
the set of all such functions; it constitutes a Banach space when it is endowed with the sup norm.
Almost automorphic functions were introduced by Bochner [68] as a natural generalization of the concept of almost periodic function. A complete description of the properties and further applications to evolution equations can be found in the monographs [69] and [70] by N'Guérékata.
Definition 2.18.
Let and
be two Banach spaces. A continuous function
is called almost automorphic in
uniformly for
in compact subsets of
if for every compact subset
of
and every real sequence
there exists a subsequence
such that
is well defined for each
,
and
for each
,
. Denote by
the set of all such functions.
Lemma 2.19 (see [34]).
Assume that is sectorial of type
. If
is an almost automorphic function and
is given by (2.5), then
.
In 1980s, N'Guérékata [71] defined asymptotically almost automorphic functions as perturbation of almost automorphic functions by functions vanishing at infinite. Since then, those functions have generated lots of developments and applications; we refer the reader to [69, 72–74] and the references therein.
Definition 2.20 (see [75]).
A continuous function (resp.,
) is called asymptotically almost automorphic (asymptotically almost automorphic in
uniformly for
in compact subsets of
) if it admits a decomposition
,
, where
(resp.,
) and
(resp.,
). Denote by
(resp.,
) the set of all such functions.
is a Banach space with the sup norm (see [75, Lemma
]). We note that the range of an asymptotically almost automorphic function is relatively compact [75].
Lemma 2.21.
Assume that is sectorial of type
. If
is an asymptotically almost automorphic function and
is given by (2.7), then
.
Proof.
, where
and
. We have that
, where
and
are the functions given by (2.8) and (2.9), respectively. By previous lemma
and by the proof of Lemma 2.13
. This ends the proof.
Lemma 2.22 (see [75]).
Let and let
be uniformly continuous on bounded sets uniformly for
. If
, then
.
3. Pseudo-Almost Periodic Mild Solutions
We recall the following definition that will be essential for us.
Definition 3.1 (see [34]).
Suppose that generates an integrable solution operator
. A continuous function
satisfying the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ15_HTML.gif)
is called a mild solution to the equation (1.1).
The following are the main results of this section.
Theorem 3.2.
Assume that is sectorial of type
. Let
be a function pseudo-almost periodic in
, uniformly in
and assume that there exists an integrable bounded function
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ16_HTML.gif)
Then equation (1.1) has a unique pseudo-almost periodic mild solution.
Proof.
We define the operator by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ17_HTML.gif)
Given , in view of Lemma 2.11, we have that
is a pseudo-almost periodic function, and hence bounded in
. Since the function
is integrable on
(
), we get that
exists. Now, by Lemma 2.14, we obtain that
and hence
is well defined. It suffices to show that the operator
has a unique fixed point in
. For this, consider
. We can deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ18_HTML.gif)
where . Since
for
sufficiently large, by the contraction principle,
has a unique fixed point
. This completes the proof.
We can establish the following existence result.
Proposition 3.3.
Assume that is sectorial of type
. Let
be a function pseudo-almost periodic in
uniformly in
that satisfies the Lipschitz condition (3.2) with
. Let
. If
, where
and
are the constants in (2.1), then equation (1.1) has a unique pseudo-almost periodic mild solution.
Proof.
Let be the map defined in the previous theorem. For
we can estimate that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ19_HTML.gif)
which finishes the proof.
Corollary 3.4.
Assume that is sectorial of type
. Let
be a function pseudo-almost periodic in
uniformlies in
that satisfy the Lipschitz condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ20_HTML.gif)
If , where
and
are the constants given in (2.1), then equation (1.1) has a unique pseudo-almost periodic mild solution.
To establish our next result we consider perturbations of (1.1) that satisfy the following boundedness condition.
(H1) There exists a continuous nondecreasing function such that
for all
and
.
We have the following result.
Theorem 3.5.
Assume that is sectorial of type
. Let
be a function pseudo-almost periodic in
uniformly in
that satisfies assumption (H1) and the following conditions.
(H2) is uniformly continuous on bounded subset of
uniformly in
.
(H3) For each ,
, where
is given by Lemma 2.15. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ21_HTML.gif)
where and
are constants given in (2.1).
(H4) For each there is
such that, for every
,
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ22_HTML.gif)
for all .
(H5) For all ,
, and
, the set
is relatively compact in
.
(H6) .
Then equation (1.1) has a pseudo-almost periodic mild solution.
Proof.
We define the operator on
as in (3.3). We show that
has a fixed point in
.
-
(1)
For
, we have that
(3.9)
It follows from condition (H3) that . From condition (H4) it follows that
is a continuous map.
-
(ii)
We next show that
is completely continuous. The argument comes from Lemma 2.15. In fact, let
and
for
. Initially, we will prove that
is a relatively compact subset of
for each
. It follows from condition (H3) that the function
is integrable on
. Hence, for
, we can choose
such that
. Hence
, where
denotes the convex hull of
. Using that
is strongly continuous and the property (H5), we infer that
is relatively compact set, and
, which establishes our assertion.
We next show that the set
is equicontinuous. In fact, we can decompose
(3.10)For each
, we can choose
and
such that
(3.11)
for . Moreover, since
is relatively compact set and
is strongly continuous, we can choose
such that
for
. Combining these estimates, we get
for
small enough and independent of
.
Finally, applying condition (H3), we can show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ26_HTML.gif)
and this convergence is independent of . Taking into account Lemma 2.15,
is a relatively compact set in
.
-
(iii)
If
is a solution of equation
for some
, then we can check that
and, combining with condition (H6), we conclude that the set
is bounded.
-
(iv)
It follows, from Lemmas 2.11 and 2.14, that
and, consequently,
is completely continuous. Since
is bounded and using Leray-Schauder alternative theorem, we infer that
has a fixed point
. Let
be a sequence in
that converges to
. We see that
converges to
uniformly in
. This implies that
and completes the proof.
It is particularly interesting to note that the next result is not covered by the results by Cuevas and Lizama [34].
Corollary 3.6.
Assume that conditions (H1)–(H6) hold and that is sectorial of type
. If
is almost periodic in
uniformly for
, then equation (1.1) has an almost periodic mild solution.
Proof.
It is a consequence of Lemmas 2.6 and 2.12.
4. Asymptotically Almost Periodic Mild Solutions
We recall the following definition.
Definition 4.1 (see [50]).
Suppose that generates an integrable solution operator
. A function
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ27_HTML.gif)
is called a mild solution of the problem (1.2)-(1.3).
Theorem 4.2.
Assume that is sectorial of type
. Let
be a function asymptotically almost periodic in
uniformly in
and assume that there exists an integrable bounded function
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ28_HTML.gif)
Then the problem (1.2)-(1.3) has a unique asymptotically almost periodic mild solution.
Proof.
We define the operator on the space
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ29_HTML.gif)
We show initially that . In fact, we observe that the estimate (2.1) implies that
. It follows from Lemma 2.9 that the function
is asymptotically almost periodic; then by Lemma 2.13,
and hence
is well defined. Let
be in
and define
. We have the following estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ30_HTML.gif)
which is responsible for the fact that has a unique fixed point in
.
Corollary 4.3.
Assume that is sectorial of type
. Let
be a function asymptotically almost periodic in
uniformly in
that satisfies the Lipschitz condition (4.2) with
. If
, where
and
are the constants given in (2.1), then the problem (1.2)-(1.3) has a unique asymptotically almost periodic mild solution.
Taking with
and
in (1.2), the above result produces the following corollary.
Corollary 4.4.
Let be a function asymptotically almost periodic in
uniformly in
that satisfies the Lipschitz condition (4.2) with
. Then problem (1.2)-(1.3) has a unique asymptotically almost periodic solution whenever
.
Remark 4.5.
A similar result as that of the previous corollary was obtained by Cuevas and de Souza [50] for obtaining an S-asymptotically -periodic mild solution for problem (1.2)-(1.3) (see [50, Remark 3.6] for complementary comments).
Next, we establish a version of Theorem 4.2 which enable us to consider locally Lipschitz perturbations for equation (1.2). We have the following result.
Theorem 4.6.
Assume that is sectorial of type
. Let
be a function asymptotically almost periodic in
uniformly in
and assume that there is a continuous and nondecreasing function
such that for each positive number
, and
,
,
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ31_HTML.gif)
where and
for
; then there is
such that for each
with
there exists a unique asymptotically almost periodic mild solution of (1.2)-(1.3).
Proof.
Let and
be such that
. We affirm that the assertion holds for
. In fact, we consider
such that
. We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ32_HTML.gif)
endowed with the metric . We define the operator
on the space
by (4.3). Let
we next show that
We have the estimate
that is,
.
On the other hand, for we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ33_HTML.gif)
To conclude, we note that , which means that
is a
-contraction. This completes the proof.
Theorem 4.7.
Assume that is sectorial of type
. Let
be an asymptotically almost periodic in
uniformly in
that satisfies the following conditions.
(H*1) There is a continuous nondecreasing function such that
for all
and
.
(H*2) is uniformly continuous on bounded sets of
uniformly in
.
(H*3) For each ,
, where
is given by Lemma 2.16. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ34_HTML.gif)
where and
are constants given in (2.1).
(H*4) For each there is
such that, for every
,
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ35_HTML.gif)
for all .
(H*5) For all , and
, the set
is relatively compact in
.
(H*6) .
Then problem (1.2)-(1.3) has an asymptotically almost periodic mild solution.
Proof.
We define the operator on
as in (4.3). We show that
has a fixed point in
.
-
(i)
For
, we have that
(4.10)It follows from (H*3) that
. From condition (H*4) it follows that
is a continuous map.
-
(ii)
We next show that
is completely continuous. Let
and
for
. Initially, we can infer that
is a relatively compact subset of
for each
. In fact, using condition (H*5) we get that
is relatively compact. It is easy to see that
, which establishes our assertion. From the decomposition of
given by
, it follows that the set
is equicontinuous. We can show that
uniformly for all
. From Lemma 2.16, we deduce that
is relatively compact set in
.
We note that the set is bounded. In fact, it follows from condition (H*6) and the estimate
. It follows, from Lemmas 2.9 and 2.13, that
. The remaining of proof makes use of a similar argument already done in the proof of Theorem 3.5.
5. Asymptotically Almost Automorphic Solutions of Fractional Integro Differential Neutral Equations
This section is mainly concerned with the existence and uniqueness of an asymptotically almost automorphic mild solution to the fractional integro differential neutral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ37_HTML.gif)
where ,
is a linear densely defined operator of sectorial type, and
are functions subject to some additional conditions.
Definition 5.1.
Suppose that generates an integrable solution operator
. A function
satisfying the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ38_HTML.gif)
is called a mild solution of problem (5.1).
We have the following result.
Theorem 5.2.
Assume that is sectorial of type
. Let
be two functions asymptotically almost automorphic in
uniformly for
in compact subsets of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ39_HTML.gif)
If , where
and
are the constants given in (2.1), then problem (5.1) has a unique asymptotically almost automorphic mild solution.
Proof.
We define the operator on the space
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ40_HTML.gif)
Applying Lemma 2.22, we infer that and
belong to
. By Lemma 2.21, we obtain that
is
-valued. Furthermore, we have the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ41_HTML.gif)
which proves that is a contraction we conclude that
has a unique fixed point in
. This completes the proof.
Next, we establish a local version of the previous result.
Theorem 5.3.
Assume that is sectorial of type
. Let
be two functions asymptotically almost automorphic in
uniformly for
in compact subsets of
and assume that there are continuous and nondecreasing functions
such that for each positive number
, and
,
,
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ42_HTML.gif)
for all , where
and
for every
. Then there is
such that
satisfies
then there is a unique asymptotically almost automorphic mild solution of (5.1).
Proof.
Let and
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ43_HTML.gif)
where and
are the constants given in (2.1). We consider
such that
, with
; we define the space
endowed with the metric
. We also define the operator
on the space
by (5.4). Let
be in
in a similar way as that of proof of Theorem 5.2; we have that
. Moreover, we obtain the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ44_HTML.gif)
Therefore . On the other hand, for
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ45_HTML.gif)
which shows that is a contraction from
into
. The assertion is now a consequence of the contraction mapping principle.
Remark 5.4.
A similar result was obtained by Diagana et al. [76] for the existence of asymptotically almost automorphic solutions to some abstract partial neutral integrodifferential equations.
Theorem 5.5.
Assume that is sectorial of type
and that conditions (H*1),(H*3), (H*4) and (H*5) hold. In addition, suppose that the following properties hold.
(A1) The functions are asymptotically almost automorphic in
and uniformly for
in compact subsets of
and uniformly continuous on bounded sets of
uniformly in
.
(A2) There is a constant such that
for all
and
(here
is given in Lemma 2.16). Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ46_HTML.gif)
where and
are the constants given in (2.1).
(A3) .
Then problem (5.1) has an asymptotically almost automorphic mild solution.
Proof.
We define the operator on
as in (5.4); we consider the decomposition
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ47_HTML.gif)
For , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ48_HTML.gif)
Hence is
-valued. On the other hand,
is an
-contraction. It follows from the proof of the Theorem 4.7 that
is completely continuous. From Lemmas 2.21 and 2.22, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ49_HTML.gif)
Hence and
is completely continuous. Putting
we claim that there is
such that
. In fact, if we assume that this assertion is false, then for all
we can choose
and
such that
. We observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ50_HTML.gif)
Thus ), which is contrary to assumption (A3). We have that
is a contraction on
and
is a compact set. It follows from [77, Corollary
] that
has a fixed point
. More precisely,
, and this finishes the proof.
6. Applications
To illustrate our results, initially we examine sufficient conditions for the existence and uniqueness of pseudo-almost periodic mild solutions to the fractional relaxation-oscillation equation given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ51_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ52_HTML.gif)
where . To study this system in the abstract form (1.1), we choose the space
and the operator
defined by
, with domain
. It is well known that
is generator of an analytic semigroup on
. Hence,
is sectorial of type
. (6.1) can be formulated by the inhomogeneous problem (1.1), where
. Let us consider the nonlinearity
for all
and
,
with
,
. We observe that
. Hence
. We observe that
is pseudo-almost periodic in
, uniformly in
such that (3.6) holds for
. If we assume that
, then by Corollary 3.4, the fractional relaxation-oscillation equation (6.1) has a unique pseudo-almost periodic mild solution.
Taking and
, we define the function
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ53_HTML.gif)
We consider the following fractional relaxation-oscillation equation given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ54_HTML.gif)
Equation (6.4) can be expressed as an abstract equation of the form (1.1), where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ55_HTML.gif)
Proposition 6.1.
Problem (6.4) has a pseudo-almost periodic mild solution.
Proof.
Let us briefly discuss the proof of this proposition. We get without difficulties the following two estimates:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ56_HTML.gif)
which are responsible for the fact that and that
is uniformly continuous on bounded sets of
uniformly in
.
It is straightforward to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ57_HTML.gif)
and . Hence, we can define
in (H1) by
. Taking
,
;
. From the discussion above, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ58_HTML.gif)
which means that conditions (H3) and (H4) of Theorem 3.5 are satisfied. An easy computation leads to . An argument involving Simon's theorem (see [78, Theorem
, pages 71–74]) proves that the set
is relatively compact in
. In fact, we can verify that
,
. Hence, for
,
is bounded uniformly in
and
. On the other hand, we can infer the following estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ59_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ60_HTML.gif)
uniformly in and
. Finally, Simon's theorem leads to the conclusion that
is relatively compact. Using Theorem 3.5, equation (6.4) has a pseudo-almost periodic mild solution.
Next, we examine the existence and uniqueness of an asymptotically almost automorphic mild solution to the fractional differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ61_HTML.gif)
where ,
are appropriate functions, and
. From Corollary 4.3, we can deduce the following result.
Proposition 6.2.
Assume that is an asymptotically almost periodic function and that there exists a constant
such that
for all
. If
, then (6.11) has a unique asymptotically almost periodic mild solution.
We consider the fractional differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ62_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ63_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ64_HTML.gif)
From Theorem 4.6, we deduce the following result.
Proposition 6.3.
Assume that is an asymptotically almost periodic function, then there is
such that for each
with
there exists a unique asymptotically almost periodic mild solution of (6.12)–(6.14).
Proof.
The proof is straightforward. Indeed, (6.12) can be expressed as an abstract equation of form (1.2), where ,
,
. We observe that
, for all
and
. Hence the perturbation is locally Lipschitz. We remark that
is asymptotically almost periodic in
uniformly in
, as we mentioned before, by using Theorem 4.6.
Take and
. We define the function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ65_HTML.gif)
We examine asymptotically almost periodic mild solution to the fractional relaxation-oscillation equation given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ66_HTML.gif)
where .
Proposition. 6.4
Problem (6.16) has an asymptotically almost periodic mild solution.
Proof.
We briefly recall some argument of the proof. Problem (6.16) can be written as an abstract problem of the form (1.2)-(1.3) in , where the perturbation associated is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ67_HTML.gif)
We can choose the function in (H*1) by
. From the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ68_HTML.gif)
we get conditions (H*2) and (H*4), the latter being considered with ,
.
We can infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ69_HTML.gif)
Hence condition (H*3) is fulfilled. By looking at the estimates
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F179750/MediaObjects/13662_2009_Article_1253_Equ70_HTML.gif)
and using Simon's theorem, we conclude that condition (H*5) holds. Consequently, by Theorem 4.7 we can assert that problem (6.16) has an asymptotically almost periodic mild solution. This completes the proof of Proposition 6.4.
Remark 6.5.
It is easy to check that results in Section 5 are applicable to similar fractional differential equations as those treated in this section. For the sake of shortness, the details are left to the reader.
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Acknowledgment
Claudio Cuevas is partially supported by CNPQ/Brazil under Grant no. 300365/2008-0.
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Agarwal, R., Andrade, B. & Cuevas, C. On Type of Periodicity and Ergodicity to a Class of Fractional Order Differential Equations. Adv Differ Equ 2010, 179750 (2010). https://doi.org/10.1155/2010/179750
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DOI: https://doi.org/10.1155/2010/179750