- Research Article
- Open Access

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

- Wei Long
^{1}and - Hui-Sheng Ding
^{1}Email author

**2011**:654695

https://doi.org/10.1155/2011/654695

© W. Long and H.-S. Ding. 2011

**Received:**31 December 2010**Accepted:**20 February 2011**Published:**13 March 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).

## Keywords

- Banach Space
- Periodic Function
- Mild Solution
- Real Banach Space
- Unique Fixed Point

## 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.

Definition 1.2.

We denote the set of all such functions by or .

Definition 1.3.

We denote by the set of all such functions.

Definition 1.4.

Definition 1.5.

It is obvious that and whenever .

Definition 1.6.

We denote the set of all such functions by or .

Remark 1.7.

It is clear that for .

Definition 1.8.

for each and each . We denote by the set of all such functions.

It is also easy to show that for .

uniformly for in any compact set .

Definition 1.9.

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)
, for and ,

- (b)
is strongly continuous.

Definition 1.13.

- (a)
for all ,

- (b)
the restriction is invertible for all ,

- (c)

Remark 1.14.

## 2. Main Results

In addition, we denote by the norm of and .

Lemma 2.1.

Let , be compact, and . Then .

Proof.

for all , , and . On the other hand, since , there exists a function such that (2.1) holds.

which means that .

Theorem 2.2.

- (a)
with , and with .

- (b)

Then there exists such that .

Proof.

Then and . On the other hand, since , there is a function such that (2.1) holds.

Thus, .

for all and . Thus, .

Lemma 2.3.

Proof.

Hence, .

Theorem 2.4.

- (a)
with and . Moreover, with ;

- (b)

Then there exists such that .

Proof.

It follows from Theorem 2.2 that , that is, .

that is, . Now, we get .

Theorem 2.5.

- (a)
- (b)
the evolution family generated by has an exponential dichotomy with constants , dichotomy projections , , and Green's function ;

- (c)

for all and with .

Proof.

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

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.

## Declarations

### 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).

## Authors’ Affiliations

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