Abstract and Applied Analysis

Volume 2013 (2013), Article ID 327213, 8 pages

http://dx.doi.org/10.1155/2013/327213

Research Article

Some Properties of Weighted Pseudo almost Periodic Functions

Zhe-Ming Zheng,¹ Hui-Sheng Ding,¹ and Gaston M. N’Guérékata²

¹College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

²Department of Mathematics, Morgan State University, 1700 E Cold Spring Lane, Baltimore, MD 21251, USA

Received 31 August 2013; Revised 14 October 2013; Accepted 5 November 2013

Academic Editor: T. Diagana

Copyright © 2013 Zhe-Ming Zheng 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.

Several interesting and new properties of weighted pseudo almost periodic functions are established. Firstly, we obtain an equivalent definition for weighted pseudo almost periodic functions, which shows a close relationship between asymptotically almost periodic functions and weighted pseudo almost periodic functions; secondly, we prove that the space of asymptotically almost periodic functions is always a proper subspace of the space of weighted pseudo almost periodic functions; thirdly, we show that under some cases, the space of weighted pseudo almost periodic functions equals the classical space of pseudo almost periodic functions.

1. Introduction

The notion of weighted pseudo almost periodic function is introduced by Diagana [1], which is an interesting generalization of the classical Bohr almost periodic function as well as a generalization of the classical pseudo almost periodic function introduced by Zhang [2].

Since the work of Diagana, it has been of great interest for many mathematicians to investigate weighted pseudo almost periodic functions and their applications to evolution equations. There is a large amount of literature on this topic. However, due to the influence of weighted term, the behaviors of weighted pseudo almost periodic functions are more tricky and changeable than those of the classical pseudo almost periodic functions.

It is needed to note that, recently, several authors have made interesting and important contributions on weighted pseudo almost periodic functions. For example, Blot et al. [3, 4] established many results including completeness, translation invariance, and composition theorem of weighted pseudo almost periodic functions in the framework of measure theory; Diagana [5] investigated the weighted mean for almost periodic functions; Liang et al. [6] showed that the decomposition of weighted pseudo almost periodic functions is not unique, in general, and that translation invariance implies unique decomposition; Ji and Zhang [7] obtained several results concerning translation invariance and ergodicity of an almost periodic function under a weight; Zhang et al. [8] proved that the space of weighted pseudo almost periodic functions is complete under a new norm, which is different from the supremum norm; Ding et al. [9] presented several basic properties about vector-valued weighted pseudo almost automorphic functions, including equivalence, completeness, translation invariance, composition theorem, and convolution theorem of these functions. In addition, we would like to refer the reader to [10–13] for some other recent development on weighted pseudo almost periodic functions and related topics.

Throughout the rest of this paper, we denote by the set of positive integers, by the set of real numbers, by a Banach space, by the set of all bounded continuous functions , and by the set of functions (weights) , which are locally integrable over . In addition, for and , we denote

Obviously, , with strict inclusions.

Next, let us recall some notions and basic results about almost periodic functions and pseudo almost periodic functions (for more details, see [14–16]).

Definition 1. A set is called relatively dense in if there exists a number such that, for all , .

Definition 2. A continuous function is called almost periodic if for every there exists a relatively dense set such that

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

Definition 3. A continuous function is called asymptotically almost periodic if it can be expressed as , where and , where

The set of all such functions will be denoted by .

Lemma 4. The following holds true:

(a)if , then is uniformly continuous on ;(b)a necessary and sufficient condition for a continuous function belonging to is that for every there exist a constant and a relatively dense set such that

 for all with and with .

Now, let us recall some notions and basic results about weighted pseudo almost periodic functions. Denote

Definition 5 (see [1]). Let . A function is called weighted pseudo almost periodic or -pseudo almost periodic if it can be expressed as , where and . The set of such functions will be denoted by .

Remark 6. If , then a -pseudo almost periodic function becomes a classical pseudo almost periodic function. So is just the space of all pseudo almost periodic functions. In addition, it is easy to show that provided that .

Definition 7 (see [7]). Let . A set is said to be a -ergodic zero set if

The following lemma is due to [17, Lemma  3.2] (see also [7]).

Lemma 8. Let and . Then if and only if, for every , is a -ergodic zero set, where .

2. The Relationship between AAP(X) and PAP(X,ρ)

In this section, we discuss the relationship between and . Firstly, we will establish an equivalent definition for weighted pseudo almost periodic functions.

Theorem 9. Let and . Then a necessary and sufficient condition for is that there exists a -ergodic zero set such that

Proof. Consider the following.

Sufficiency. It follows from that, for every ,

is a bounded set. Combining this with the fact that is a -ergodic zero set, we conclude that is a -ergodic zero set. Then, by Lemma 8, .

Necessity. Let . We denote

Then, by Lemma 8, every is a -ergodic zero set. Next, we divide the remaining proof into three steps.

Step  1. There exists an increasing sequence with , satisfying that, for every and , there hold

In fact, for , since and are both -ergodic zero sets, there exists such that, for all , there hold

For , by using the fact that and are both -ergodic zero sets, there exists such that, for all , there hold

For , by using the fact that and are both -ergodic zero, there exists such that, for all , there hold

Continuing by this way, we can get an increasing sequence , which satisfies , (10).

Step  2.   is a -ergodic zero set, where

In fact, for every , there exists such that . Thus, we have

Combining this with (10), we conclude that

which means that is a -ergodic zero set.

Step  3. .

In fact, for every and with , there exists such that

which yields that , and thus

This completes the proof.

Remark 10. In Theorem 9, we can choose to be an open set. In fact, letting

where every is a sufficiently small positive constant satisfying

it is easy to prove that is an open set and a -ergodic zero set. Moreover, obviously, we have .

Next, we present an equivalent definition of weighted pseudo almost periodic functions, which establishes a close relationship between asymptotically almost periodic functions and weighted pseudo almost periodic functions.

Theorem 11. Let and . Then a necessary and sufficient condition for is that there exist and a -ergodic zero set such that

Proof. The sufficiency part is easy to prove. We only give the proof for the necessity part. Let . Then, there exist and such that . It follows from Theorem 9 that there exists a -ergodic zero set such that

In addition, by Remark 10, without loss of generality, we can assume that is an open set. Then, we can conclude that there exists a function such that for all . Letting , we have , and for all . This completes the proof.

Combining Lemma 4 and Theorem 11, we can get the following.

Corollary 12. Let and . Then, there exists a -ergodic zero set satisfying that, for every , there are two constants and a relatively dense set such that

Remark 13. For the case of , Zhang [18] established a similar result to Corollary 12. But here we use a different approach, and even for the case of , Corollary 12 improves the “if part” of [18, Theorem  11]. In fact, in [18], for every , there exists a -ergodic zero set . Here, we find a common ergodic zero set .

In the above, we show that there is a close relationship between asymptotically almost periodic functions and weighted pseudo almost periodic functions. Next, we will show that is always a proper subspace of .

Theorem 14. Let . Then

Proof. It is easy to show that since . So we only need to show that . Without loss for generality, we only give the proof for the case of . We divide the remaining proof into three steps.

Step  1. Let , where

where

It is easy to see that every is open. Moreover, there are infinitely many nonempty , and thus we can assume that every is nonempty without loss for generality. In addition, we claim that is a -ergodic zero set. In fact, we have the following two cases.

Case  I. If , then , which yields that

where is a positive integer. Then, we get

Case  II. If , then , which yields that

Then, similar to Case  I, one can also obtain that

Thus, is a -ergodic zero set.

Step  2. For every , noting that is open, there exist and such that . Now, we construct a bounded and continuous function on by

Step  3. .

Since and is a -ergodic zero set, we have . It remains to show that . We prove it by contradiction, assuming that there exist and such that . For sufficiently large , since , we can choose

such that

Moreover, since , for sufficiently large , we also have

So, we get

which contradicts the fact that

This completes the proof.

3. Equivalence

Just as noted in Remark 6, we know that provided that . Then, there is a natural question:

Does imply that ?

In fact, the above question has a negative answer. For example, recently, it is proved in [7] and [9] (by a different method) that

In this section, we will make further study on this question. We will prove that for some other , there still holds . Firstly, we recall a theorem, which is due to [7, Theorem  4.3].

Theorem 15. Let and

Then .

Theorem 16. Let be a periodic function with almost everywhere on . Then , and thus .

Proof. It suffices to prove that . We divide the remaining proof into two steps.

Step  1. Every -ergodic zero set is a -ergodic zero set.

Let be a -ergodic zero set. Then, we have

Assuming that

then there exist and a sequence of positive numbers , which satisfies , and

where is a positive periodic of . On the other hand, since almost everywhere on , by Lusin's Theorem, there exists a closed set with , such that is continuous on and for all . Let

Then . Then, we have

which yields that

Combining this with (41), we get

which contradicts (39). Therefore, is a -ergodic zero set.

Step  2. Every -ergodic zero set is a -ergodic zero set.

Let be a -ergodic zero set. For every , there exists such that, for all set with , there holds

In addition, again by Lusin's Theorem, for the above , there exists a closed set with , such that is continuous on . Let

Then . Recalling that is a -ergodic zero set, for the above and , there exists such that

Combining (46) and (48), by using the periodicity of , we conclude that, for all , there holds

where

and, similarly,

Noting that

we get

Then, by the arbitrariness of , we conclude that

that is, is a -ergodic zero set.

Example 17. Let , where . Then by Theorem 16, .

Except for the case in Theorem 16, there is some other , which satisfies that . For example, we have the following.

Theorem 18. Let

Then .

Proof. We first show that every -ergodic zero set is a -ergodic zero set. Let be a -ergodic zero set. For every , there exists such that, for all ,

Combining this with the fact that is even and increasing on , we get

which yields that

Then, by the arbitrariness of , we know that

that is, is a -ergodic zero set.

Next, let us show that every -ergodic zero set is a -ergodic zero set. Let be a -ergodic zero set. Assuming that

then there exist and a sequence of positive numbers , which satisfies , and

By (61) and the fact that is even and increasing on , we conclude that

Letting , we get

which contradicts the fact that is a -ergodic zero set. Thus, is a -ergodic zero set. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Ding acknowledges support from the NSF of China (11101192), the Program for Cultivating Young Scientist of Jiangxi Province (20133BCB23009), and the NSF of Jiangxi Province; Zheng acknowledges support from the Graduate Innovation Fund of Jiangxi Province.

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