Journal of Function Spaces
Volume 2018 (2018), Article ID 8208323, 9 pages
https://doi.org/10.1155/2018/8208323

Research Article

Stepanov-Like Asymptotical Almost Periodic Functions and an Application

Yongkun Li, Yaolu Wang, and Jianglian Xiang

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Correspondence should be addressed to Yongkun Li; yklie@ynu.edu.cn

Received 20 September 2018; Revised 12 November 2018; Accepted 21 November 2018; Published 4 December 2018

Academic Editor: Richard I. Avery

Copyright © 2018 Yongkun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we first study some basic properties of Stepanov-like asymptotical almost periodic functions including the completeness of the space of Stepanov-like asymptotical almost periodic functions. Then, as an application, based on these and the contraction mapping principle, we obtain sufficient conditions for the existence and uniqueness of Stepanov-like asymptotical almost periodic solutions for a class of semilinear delay differential equations.

1. Introduction

Almost periodic functions, which are an important generalization of periodic functions, were introduced into the field of mathematics by Bohr [1, 2]. From the very beginning, the concept of almost periodic function has attracted extensive attention of mathematicians and has led to various extensions and variations of this concept. For example, Stepanov proposed a weaker concept of almost periodic functions in the sense of Bohr. For more details about Stepanov’s almost periodic functions, see [3–11].

On the one hand, due to the fact that almost periodic phenomena exist in the real world, more and more scholars are interested in the almost periodicity and its various generalizations. For example, Diagana [12] introduced Stepanov-like pseudo almost periodicity in 2007. The Stepanov-like pseudo almost periodicity is a generalization of the classical pseudo almost periodicity [13]. The concept of Stepanov-like weighted pseudo almost periodicity was introduced by Diagana et al. [14]. This notion is more extensive than Stepanov-like pseudo almost periodicity. Moreover, Diagana also introduced Stepanov-like almost automorphic functions which are a generalization of the classical almost automorphic functions; for more details, see [15]. In 2009, Diagana introduced the notion of Stepanov-like pseudo almost automorphy which generalizes the concept of pseudo almost automorphy [16].

On the other hand, the concept of the asymptotically almost periodicity was introduced into the research field by French mathematician Frechet [17, 18]. Such a notion is a natural generalization of the concept of the almost periodicity in the sense of Bohr. Since then, asymptotically almost periodic functions have become a very important function class and to find asymptotically almost periodic solutions for differential equations has been a hot topic for researchers. For the basic properties of asymptotical almost periodic functions, we refer the reader to [19] and for some recent papers about the existence of asymptotically almost periodic solutions for differential equations arising in theory and application, we refer the reader to [20–25]. However, up to now, few studies have been done on Stepanov-like asymptotical almost periodic functions [26], but these studies are necessary.

Motivated by the above discussions, in this paper, we first study some basic properties of Stepanov-like asymptotical almost periodic functions. Then, based on these properties and by using the contraction mapping principle, we investigate the existence and uniqueness of Stepanov-like asymptotical almost periodic solutions for a class of semilinear delay differential equations.

2. Preliminaries

In this section, we recall some basis definitions and lemmas of Bohr almost periodic functions and Stepanov’s almost periodic functions which are used throughout this paper.

Let be a Banach space and be the collection of bounded continuous functions from to with the norm .

Definition 1 (see [27]). A function is said to be almost periodic in Bohr sense if for each there exists such that in every interval of length of one can find a number with the property

We denote the space of all such functions by ; the norm of the space is

Definition 2 (see [28]). Let be a set of some almost periodic functions in Bohr sense. Then is uniformly almost periodic family if it is uniformly bounded, equicontinuous and for every there exists a number such that every interval of length contains a number such that

Lemma 3 (see [27]). Function is equivalent to the property of relative compactness for the family .

Definition 4 (see [27]). The space is defined as follows:

with the norm defined by

Lemma 5 (see [27]). The space is a Banach space.

Definition 6 (see [27]). A function is said to be Stepanov’s almost periodic if for every there exists such that each interval contains a point with the property

We denote the space of all such functions by and the norm of is

Lemma 7 (see [27]). The space is a Banach space.

Lemma 8 (see [27]). A function if and only if , where

Lemma 9 (see [27]). A function if and only if is relatively compact in .

Lemma 10 (see [29]). Let be almost periodic functions in Bohr sense from into Banach space , respectively. Then for each , all the functions , have a common set of -almost periods.

3. Stepanov-Like Asymptotic Almost Periodic Functions and Their Basic Properties

Let

then we give the following definition.

Definition 11 (see [26]). A function is said to be a Stepanov-like asymptotical almost periodic function if it can be expressed as

where , . The collection of all such functions will be denoted by .

Remark 12. Several equivalent statements of Definition 11 are given by Theorem 1.6.2 in [26].

Remark 13. Obviously, if and , then .

Lemma 14. The space of is a Banach space endowed with the norm

Proof. Let be a Cauchy sequence, then we can find a function such that

Hence

Since , we deduce that

that is, . Thus, is a Banach space. The proof is complete.

Lemma 15. Let and , then

Proof. Assume that (15) does not hold, then there exist and such that

Since , there exists and for every , there exists such that

By using the uniform continuity on of the almost periodic function , there exists such that and for all ,

From (16)-(18), it follows that

Since , by (19), we have

Thus,

which contradicts

Consequently, (15) holds. The proof is complete.

Definition 16. A function if the following three conditions are true:

(i)for every , ,(ii)the set is uniformly bounded in the -norm and equicontinuous in the -norm,(iii)for every , there exists a number such that every interval of length contains a number with the property

Lemma 17. For a bounded continuous function , denote

Then satisfies the following properties:

(a).(b)(c)(d) if and only if .

Proof. Properties through are easy to show. We only prove the property . If , then for each there exists such that in any interval of length of one can find a number with the property

From , we have and , then . That is,

Hence,

It follows from (25) and (27) that

Since , it is uniformly continuous. So, we know that, for every , there exists such that if then . Therefore, and by (27) we get , which implies that is continuous. Thus, .
Conversely, if , then for each there exists such that in any interval of length of one can find a number with the property

Hence, . Since is continuous and , then for every , there exists such that if we have Therefore, , which implies is continuous. Thus, .

Lemma 18. Let , where consists of some almost periodic functions in Bohr sense from to , is finite. Then the family is uniformly almost periodic if and only if .

Proof. It is easy to find that satisfies the following properties:

(i),(ii).

Moreover, by in Lemma 17 we have

Similar to the proof of (27), we have

If , then for each , there exists such that in any interval of length of one can find a number with the property

Noticing that , hence, . Therefore, for all . So, the family is a uniformly almost periodic family.
Conversely, suppose that is a uniformly almost periodic family. Then for each there exists such that in any interval of length of one can find a number with the property

From (33), we obtain that and . By (31) we obtain that

Besides, since is continuous,

By (31) we obtain that as , which implies that is continuous. Therefore, . The proof is completed.

Lemma 19. Let be a uniformly almost periodic family in Bohr sense. Then given a sequence , there exists a subsequence satisfying the following property: for every , there exists a constant such that

Proof. According to Lemma 18, we know that is almost periodic. By Lemma 3 we obtain that, for every sequence , there exists a subsequence satisfying the following property: for each , there exists a constant such that

When , we obtain . Thus, for all and . According to the definition of , we have

Hence, it is easy to see that

The proof is completed.

Theorem 20. Let . Then for every sequence , there exists a subsequence such that is convergent uniformly with respect to .

Proof. Since , we know that, for every , there exists a number such that every interval of length contains a number satisfying

According to (40), we have

Thus,

From Lemma 8, we know that, for every fixed , . Therefore, (42) implies that is a uniformly almost periodic family. From Lemma 19, we have that, for every sequence , there exists a subsequence such that, for every ,

Hence, is convergent uniformly with respect to .

Definition 21. A function if the following two conditions are true:

(i)for every , ,(ii)for every , there exists a constant such that

Definition 22. A function if it can be expressed as

where , .

Theorem 23. The space of is a Banach space endowed with the norm

Proof. Let be a Cauchy sequence; i.e., for each , there exists a natural number such that

Let , where . By using Lemma 15, we have

Then, for every ,

According to the arbitrariness of , we obtain

Thus,

which means that is a Cauchy sequence. So . Similarly, we can obtain

According to Lemma 14, we obtain and . The proof is completed.

4. Stepanov-Like Asymptotical Almost Periodic Solutions of Semilinear Delay Differential Equations

In this section, we investigate the existence and uniqueness of Stepanov-like asymptotical almost periodic solutions for the following semilinear differential equation:

where , , and is a constant.

We make some assumption:

and , where , .There exist constants such that, for all and for all ,

Lemma 24. Let and hold; then .

Proof. Since , we have , where . Then function can be written in the form:

Step 1. We prove . Let . By Lemma 9, for every sequence , there exists a subsequence such that the sequence is convergent. From Theorem 20, it follows that is also convergent uniformly with respect to . Therefore, for any , there exist positive integers and such that

and

Hence

Therefore, .
Step 2. We prove that

and

From , we have

According to the definition of and Definition 21, we have

and

Hence, . The proof is complete.

Theorem 25. Assume - hold. If , where , then system (53) has a unique Stepanov-like asymptotical almost periodic solution.

Proof. For any , consider the linear differential equation

Since holds, by the exponential dichotomy of linear differential equation, (64) has a unique bounded solution

Define an operator by setting for every .
Step 1 ( is self-mapping). From Lemma 24, we have . Let , where , and define

By Lemmas 8 and 10, there exists such that every interval of length contains a number such that

Then

where . Hence, .
Since , we have

where . According to Lebesgue’s dominated convergence theorem, we obtain . Therefore, is self-mapping.
Step 2 ( is a contraction mapping). , we have

Hence, has a unique fixed point in . Therefore, system (53) has a unique Stepanov-like asymptotical almost periodic solution. The proof is complete.

An example: consider the following equation:

where is a positive constant. In this case, ,

Obviously, ,

Hence, . Thus, all the conditions of Theorem 25 are satisfied. By Theorem 25, system (71) has a unique Stepanov-like asymptotical almost periodic solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of China under Grants No. 11861072 and No. 11361072.

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