Composition Theorems of Stepanov Almost Periodic Functions and Stepanov-Like Pseudo-Almost Periodic Functions

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

(1.1)

Definition 1.2.

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

(1.2)

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

(1.3)

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

(1.4)

Definition 1.5.

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 1.6.

A function is called Stepanov almost periodic if ; that is, for all , there exists a relatively dense set such that

(1.6)

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

(1.7)

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

(1.8)

We also denote by the space of all functions such that

(1.9)

uniformly for in any compact set .

Definition 1.9.

A function is called pseudo-almost periodic if

(1.10)

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

(1.12)

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

(2.1)

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

(2.2)

In addition, we denote by the norm of and .

Lemma 2.1.

Let , be compact, and . Then .

Proof.

For all , there exist such that

(2.3)

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

(2.4)

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

(2.5)

for each and , which gives that

(2.6)

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

(2.7)

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

(2.9)

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

(2.10)

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

(2.11)

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

(2.12)

for all and . Thus, .

Lemma 2.3.

Let be compact, , and . Then , where

(2.13)

Proof.

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

(2.14)

Combining this with , for all , there exists such that

(2.15)

for all and . Thus, we get

(2.16)

which yields that

(2.17)

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

(2.18)

This together with (2.17) implies that

(2.19)

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

(2.21)

It follows from Theorem 2.2 that , that is, .

Next, let us show that . For , we have

(2.22)

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

(2.23)

which yields

(2.24)

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 :

(2.25)

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

(2.28)

Proof.

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

Let

(2.29)

where and . Denote

(2.30)

where

(2.31)

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

(2.32)

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

(2.33)

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.