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Composition Theorems of Stepanov Almost Periodic Functions and Stepanov-Like Pseudo-Almost Periodic Functions
Advances in Difference Equations volume 2011, Article number: 654695 (2011)
Abstract
We establish a composition theorem of Stepanov almost periodic functions, and, with its help, a composition theorem of Stepanov-like pseudo almost periodic functions is obtained. In addition, we apply our composition theorem to study the existence and uniqueness of pseudo-almost periodic solutions to a class of abstract semilinear evolution equation in a Banach space. Our results complement a recent work due to Diagana (2008).
1. Introduction
Recently, in [1, 2], Diagana introduced the concept of Stepanov-like pseudo-almost periodicity, which is a generalization of the classical notion of pseudo-almost periodicity, and established some properties for Stepanov-like pseudo-almost periodic functions. Moreover, Diagana studied the existence of pseudo-almost periodic solutions to the abstract semilinear evolution equation . The existence theorems obtained in [1, 2] are interesting since
is only Stepanov-like pseudo-almost periodic, which is different from earlier works. In addition, Diagana et al. [3] introduced and studied Stepanov-like weighted pseudo-almost periodic functions and their applications to abstract evolution equations.
On the other hand, due to the work of [4] by N'Guérékata and Pankov, Stepanov-like almost automorphic problems have widely been investigated. We refer the reader to [5–11] for some recent developments on this topic.
Since Stepanov-like almost-periodic (almost automorphic) type functions are not necessarily continuous, the study of such functions will be more difficult considering complexity and more interesting in terms of applications.
Very recently, in [12], Li and Zhang obtained a new composition theorem of Stepanov-like pseudo-almost periodic functions; the authors in [13] established a composition theorem of vector-valued Stepanov almost-periodic functions. Motivated by [2, 12, 13], in this paper, we will make further study on the composition theorems of Stepanov almost-periodic functions and Stepanov-like pseudo-almost periodic functions. As one will see, our main results extend and complement some results in [2, 13].
Throughout this paper, let be the set of real numbers, let
be the Lebesgue measure for any subset
, and
be two arbitrary real Banach spaces. Moreover, we assume that
if there is no special statement. First, let us recall some definitions and basic results of almost periodic functions, Stepanov almost periodic functions, pseudo-almost periodic functions, and Stepanov-like pseudo-almost periodic functions (for more details, see [2, 14, 15]).
Definition 1.1.
A set is called relatively dense if there exists a number
such that

Definition 1.2.
A continuous function is called almost periodic if for each
there exists a relatively dense set
such that

We denote the set of all such functions by or
.
Definition 1.3.
A continuous function is called almost periodic in
uniformly for
if, for each
and each compact subset
, there exists a relatively dense set

We denote by the set of all such functions.
Definition 1.4.
The Bochner transform ,
,
, of a function
on
, with values in
, is defined by

Definition 1.5.
The space of all Stepanov bounded functions, with the exponent
, consists of all measurable functions
on
with values in
such that

It is obvious that and
whenever
.
Definition 1.6.
A function is called Stepanov almost periodic if
; that is, for all
, there exists a relatively dense set
such that

We denote the set of all such functions by or
.
Remark 1.7.
It is clear that for
.
Definition 1.8.
A function with
, for each
, is called Stepanov almost periodic in
uniformly for
if, for each
and each compact set
, there exists a relatively dense set
such that

for each and each
. We denote by
the set of all such functions.
It is also easy to show that for
.
Throughout the rest of this paper, let (resp.,
) be the space of bounded continuous (resp., jointly bounded continuous) functions with supremum norm, and

We also denote by the space of all functions
such that

uniformly for in any compact set
.
Definition 1.9.
A function is called pseudo-almost periodic if

with and
. We denote by
the set of all such functions.
It is well-known that is a closed subspace of
, and thus
is a Banach space under the supremum norm.
Definition 1.10.
A function is called Stepanov-like pseudo-almost periodic if it can be decomposed as
with
and
. We denote the set of all such functions by
or
.
It follows from [2] that for all
.
Definition 1.11.
A function with
, for each
, is called Stepanov-like pseud-almost periodic in
uniformly for
if it can be decomposed as
with
and
. We denote by
the set of all such functions.
Next, let us recall some notations about evolution family and exponential dichotomy. For more details, we refer the reader to [16].
Definition 1.12.
A set of bounded linear operator on
is called an evolution family if
-
(a)
,
for
and
,
-
(b)
is strongly continuous.
Definition 1.13.
An evolution family is called hyperbolic (or has exponential dichotomy) if there are projections
,
, being uniformly bounded and strongly continuous in
, and constants
,
such that
-
(a)
for all
,
-
(b)
the restriction
is invertible for all
  
,
-
(c)
and
for all
,
where
. We call that
(1.11)is the Green's function corresponding to
and
.
Remark 1.14.
Exponential dichotomy is a classical concept in the study of long-term behaviour of evolution equations; see, for example, [16]. It is easy to see that

2. Main Results
Throughout the rest of this paper, for , we denote by
the set of all the functions
satisfying that there exists a function
such that

and, for any compact set , we denote by
the set of all the functions
such that (1.7) is replaced by

In addition, we denote by the norm of
and
.
Lemma 2.1.
Let ,
be compact, and
. Then
.
Proof.
For all , there exist
such that

Since , for the above
, there exists a relatively dense set
such that

for all ,
, and
. On the other hand, since
, there exists a function
such that (2.1) holds.
Fix ,
. For each
, there exists
such that
. Thus, we have

for each and
, which gives that

Now, by Minkowski's inequality and (2.4), we get

which means that .
Theorem 2.2.
Assume that the following conditions hold:
-
(a)
with
, and
with
.
-
(b)
, and there exists a set
with
such that
(2.8)is compact in
.
Then there exists such that
.
Proof.
Since , there exists
such that
. Let

Then and
. On the other hand, since
, there is a function
such that (2.1) holds.
It is easy to see that is measurable. By using (2.1), for each
, we have

Thus, .
Next, let us show that . By Lemma 2.1,
. In addition, we have
. Thus, for all
, there exists a relatively dense set
such that

for all and
. By using (2.11), we deduce that

for all and
. Thus,
.
Lemma 2.3.
Let be compact,
, and
. Then
, where

Proof.
Noticing that is a compact set, for all
, there exist
such that

Combining this with , for all
, there exists
such that

for all and
. Thus, we get

which yields that

On the other hand, since , for the above
, there exists
such that, for all
,

This together with (2.17) implies that

Hence, .
Theorem 2.4.
Assume that and the following conditions hold:
-
(a)
with
and
. Moreover,
with
;
-
(b)
with
and
, and there exists a set
with
such that
(2.20)is compact in
.
Then there exists such that
.
Proof.
Let be as in the proof of Theorem 2.2. In addition, let
, where

It follows from Theorem 2.2 that , that is,
.
Next, let us show that . For
, we have

where was used. For
, since
, by Lemma 2.3, we know that

which yields

that is, . Now, we get
.
Next, let us discuss the existence and uniqueness of pseudo-almost periodic solutions for the following abstract semilinear evolution equation in :

Theorem 2.5.
Assume that and the following conditions hold:
-
(a)
with
and
. Moreover,
with
(2.26) -
(b)
the evolution family
generated by
has an exponential dichotomy with constants
, dichotomy projections
,
, and Green's function
;
-
(c)
for all
, for all
, and for all
there exists a relatively dense set
such that
and
(2.27)
for all and
with
.
Then (2.25) has a unique pseudo-almost periodic mild solution provided that

Proof.
Let , where
and
. Then
and
is compact in
. By the proof of Theorem 2.4, there exists
such that
.
Let

where and
. Denote

where

By [13, Theorem 2.3] we have . In addition, by a similar proof to that of [2, Theorem 3.2], one can obtain that
. So
maps
into
. For
, by using the Hölder's inequality, we obtain

for all , which yields that
has a unique fixed point
and

This completes the proof.
Remark 2.6.
For some general conditions which can ensure that the assumption (c) in Theorem 2.5 holds, we refer the reader to [17, Theorem 4.5]. In addition, in the case of and
generating an exponential stable semigroup
, the assumption (c) obviously holds.
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Acknowledgments
The work was supported by the NSF of China, the Key Project of Chinese Ministry of Education, the NSF of Jiangxi Province of China, the Youth Foundation of Jiangxi Provincial Education Department (GJJ09456), and the Youth Foundation of Jiangxi Normal University (2010-96).
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Long, W., Ding, HS. Composition Theorems of Stepanov Almost Periodic Functions and Stepanov-Like Pseudo-Almost Periodic Functions. Adv Differ Equ 2011, 654695 (2011). https://doi.org/10.1155/2011/654695
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DOI: https://doi.org/10.1155/2011/654695
Keywords
- Banach Space
- Periodic Function
- Mild Solution
- Real Banach Space
- Unique Fixed Point