Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations
- Chao Wang^{1}Email authorView ORCID ID profile and
- Ravi P Agarwal^{2, 3}
https://doi.org/10.1186/s13662-015-0650-0
© Wang and Agarwal 2015
Received: 9 June 2015
Accepted: 28 September 2015
Published: 9 October 2015
Abstract
In this paper, we use the analysis of relatively dense sets to point out some deficiencies and inaccuracies in the definition of uniformly almost periodic functions which has been proposed in recent works, and we correct it. Some new generalizations of invariance under translation time scales and almost periodic functions are established. Our study will ensure that now we can study almost periodic problems precisely on time scales.
Keywords
MSC
1 Problems and corrections
Nowadays, the study of dynamic equations on time scales is a leading topic of research in different directions including directions for boundary value problems, periodic and almost periodic problems (see [1–7]), etc. As one of the most important research directions, the occurrence of almost periodic phenomenon is very common in nature. Thus the existence of almost periodic solutions for a wide variety of dynamical systems has been studied extensively. In 2011, the authors of [8, 9] extended this concept to uniformly almost periodic functions on time scales. This concept has been extensively used to study almost periodic solutions of functional dynamic equations, neural networks, biological dynamic models, and so on. However, the basic concept of uniformly almost periodic functions on time scales introduced in [8, 9] is not accurate. Unfortunately, in several recent works [10–18] this false concept has been cited and applied.
The main purpose of this paper is to point out the deficiencies and inaccuracies in the works of [8, 9], and then to give a correct definition of uniformly almost periodic functions. We begin with the following basic definitions.
Definition 1.1
Definition 1.2
The set Π in Definition 1.1 is called an invariant translation set for \(\mathbb{T}\).
Remark 1.1
It follows from Definition 1.1 that Π is also a closed subset of \(\mathbb{R}\), so Π is a time scale.
Definition 1.3
(Problem 1, Definition 3.7 in [8])
Note that the above definition requires \(E\{\varepsilon,f,S\}\) to be a relatively dense set in \(\mathbb{T}\), which makes Definition 1.3 inaccurate and false. To justify our claim, we will give a detailed explanation and provide a counter-example. For this, we recall the concept of a relatively dense set.
Definition 1.4
(Fink (1974) from [20])
Let \(A\subset B\subset\mathbb{R}\), we say that A is relatively dense in B if there exists a positive number l such that for all \(a\in B\) we have \([a,a+l]_{B}\cap A\neq\emptyset\), where \([a,a+l]_{B}=[a,a+l]\cap B\), and l is called the inclusion length.
Let \(\mathbb{T}=B\) in Definition 1.4, then the concept of relative density in \(\mathbb{T}\) from Definition 1.4 can be stated as follows.
Definition 1.5
(Problem 2)
Let \(A\subset\mathbb{T}\), we say that A is relatively dense in \(\mathbb{T}\) if there exists a positive real number l such that for all \(a\in\mathbb{T}\) we have \([a,a+l]_{\mathbb{T}}\cap A\neq\emptyset\), here l is called the inclusion length.
Definition 1.5 is directly used to introduce Definition 1.3. However, this definition is not accurate. In fact, to some extent, false. To explain this, we claim that the closed interval \([a,b]_{\mathbb{T}}\) must have \(a,b\in\mathbb{T}\). Recall that the definition of a Cauchy integral on time scales (see Definition 1.71 in [21]) is completely based on this fact.
Definition 1.6
(Bohner (2001) from [21])
We also note that, in Definition 1.3, it is required that \(E\{\varepsilon,f,S\}\subset\Pi\) is relatively dense in \(\mathbb{T}\), which means that the set \(A=E\{\varepsilon,f,S\}\) in Definition 1.5 is relatively dense. But then in Definition 1.5, we need \([a,a+l]_{\mathbb{T}}\cap E\{\varepsilon,f,S\}\neq\emptyset\), which implies that \([a,a+l]_{\mathbb{T}}\cap\Pi\neq\emptyset\), and hence \(\mathbb{T}\cap\Pi\neq\emptyset\). However, this condition is too restrictive. In fact, the following counter-example shows that \(\mathbb{T}\cap\Pi=\emptyset\).
Example 1.2
Remark 1.3
Remark 1.4
From Example 1.2 and Remark 1.3, it follows that the invariant translation set Π for \(\mathbb{T}\) may be separated from \(\mathbb{T}\), i.e., \(\mathbb{T}\cap\Pi=\emptyset\). Furthermore, the time scale \(\mathbb{T}\) from Example 1.2 is periodic, which means that Definition 1.3 does not include the situation for periodic time scales like Example 1.2. Hence, to some extent, Definition 1.3 is false.
From Example 1.2 and Remark 1.3, it is clear that it is too particular if we require that \(E\{\varepsilon,f,S\}\) is relatively dense in \(\mathbb{T}\) in Definition 1.3 because it implies that \(\mathbb{T}\cap\Pi\neq\emptyset\). Through our investigations, we find that the relatively dense set should not be for \(\mathbb{T}\) rather it should be for the set Π defined in Definition 1.1.
Definition 1.7
(Correction of Definition 1.5)
Let Π be as in Definition 1.1 for an invariant under a translation time scale. Let \(A\subset\Pi\), we say that A is relatively dense in Π if there exists a positive number \(l\in\Pi\) such that for all \(a\in\Pi\) we have \([a,a+l]_{\Pi}\cap A\neq\emptyset\), here l is called the inclusion length.
Remark 1.5
Using Definition 1.7 and Remark 1.5, we can introduce a precise definition of uniformly almost periodic functions on time scales as follows.
Definition 1.8
(Correction of Definition 1.3)
2 Local-uniformly almost periodic functions
In what follows we shall introduce a generalized concept of uniformly almost periodic functions on time scales, and we present some interesting results.
For convenience, we denote \(\mathbb{T}^{\tau}=\{t+\tau:t\in\mathbb{T}\}\). If we choose a nonzero real number \(\tau\in\Pi\), then it follows that \(\mathbb{T}=\mathbb{T}^{\tau}\) if and only if \(\mathbb{T}\) is invariant under translations, i.e., \(\mathbb{T}\) coincides exactly with \(\mathbb{T}^{\tau}\) if \(\mathbb{T}\) is invariant under translations. Thus, Definition 1.1 has the following equivalent form.
Definition 2.1
The following result guarantees that for any arbitrary time scale \(\mathbb{T}\), there exists at least one invariant under a translation time scale \(\mathbb{T}_{0*}\subset\mathbb{T}\).
Theorem 2.1
Let \(\tilde{\Pi}:=\{\tau\in\mathbb{R}:\mathbb{T}\cap\mathbb{T}^{\tau}\neq \emptyset\}\neq\{0\}\) and \(\mathbb{T}\cap\mathbb{T}^{\tau}:=\mathbb{T}_{0}^{\tau}\), if \(\mathbb{T}_{0*}:=\bigcap_{\tau\in\tilde{\Pi}}\mathbb{T}_{0}^{\tau}\neq \emptyset\), then \(\mathbb{T}_{0*}\) is an invariant under a translation time scale.
Proof
Remark 2.2
According to Theorem 2.1, we know that \(\mathbb{T}_{0*}\) is a periodic time scale, i.e., there exists \(\hat{\tau}\neq0\), \(\hat{\tau}\in\tilde{\Pi}\) such that \(t+\hat{\tau}\in\mathbb{T}_{0*}\subset\mathbb{T}\). We denote \(\hat{\Pi}=\{n\hat{\tau}:n\in\mathbb{Z}\}\subset\tilde{\Pi}\) the invariant translation set for \(\mathbb{T}_{0*}\).
Now we state Zorn’s lemma which will be needed to prove an interesting theorem.
Lemma 2.3
([22], Zorn’s Lemma)
Suppose \((P,\preceq)\) is a partially ordered set. A subset T is totally ordered if for any s, t in T we have \(s\preceq t\) or \(t\preceq s\). Such a set T has an upper bound u in P if \(t\preceq u\) for all t in T. Suppose a non-empty partially ordered set P has the property that every non-empty chain has an upper bound in P. Then the set P contains at least one maximal element.
Theorem 2.4
Let \(\mathbb{T}\) be an arbitrary time scale with \(\sup\mathbb{T}=+\infty\), \(\inf\mathbb{T}=-\infty\). If \(\mu:\mathbb{T}\rightarrow\mathbb{R}^{+}\) is bounded, then \(\mathbb{T}\) contains at least one invariant under the translation unit.
Proof
Example 2.5
In view of Theorem 2.1, we can introduce the following concept.
Definition 2.2
Let \(\mathbb{T}\) be an arbitrary time scale. If all conditions of Theorem 2.1 are satisfied, then we say the invariant under a translation time scale \(\mathbb{T}_{0*}\) is a sub-invariant under the translation unit in \(\mathbb{T}\).
Remark 2.6
It follows from Theorem 2.1 that if we define functions on \(\mathbb{T}_{0*}\), then it will lead to the concept of functions invariant under translation time scales. Since \(\mathbb{T}_{0*}\) is a subset of \(\mathbb{T}\) (in other words, \(\mathbb{T}_{0*}\) is a local part of \(\mathbb{T}\)), it opens up a new avenue to investigate the local properties of functions on an arbitrary time scale. Further, we can introduce the concept of local-almost periodic functions on time scales, then local-almost periodic solutions of dynamic equations on time scales can be studied exactly as in [8]. In fact, all the results established in [8] can also be obtained for \(\mathbb{T}_{0*}\) by simply replacing invariant under translation time scales \(\mathbb{T}\) in [8] by \(\mathbb{T}_{0*}\) (since \(\mathbb{T}_{0*}\) is a sub-invariant under the translation unit).
Using Remark 2.6, we can give a generalization of Definition 1.8 so called local-uniformly almost periodic functions on time scales.
Definition 2.3
Remark 2.7
From Definition 2.3, we know that if \(f(t)\) is a local-uniformly almost periodic functions on \(\mathbb{T}\), then \(f(t)\) is uniformly almost periodic on \(\mathbb{T}_{0*}\).
Following [8, 9], we can also give a generalization of Definition 1.1, namely, we can introduce the concept of invariance under translation time scales with respect to a family of sets.
Definition 2.4
Finally, we note that the assumption \(\mathfrak{C}^{*}=\{0\}\) makes all results obtained in [9] false. In fact, the following revision is needed.
Definition 2.5
(Correction and generalization of Definition 2.6 in [9])
To conclude we emphasize that the precise definition of almost (or local-almost) periodic functions on time scales occupies a fundamental position in establishing some key results for dynamical systems on time scales. Thus, the above corrections not only fill the gaps in the existing literature, but they will also help to pursue further research in the right direction.
Declarations
Acknowledgements
The authors would like to express their sincere thanks to the referees for suggesting some corrections that helped making the content of the paper more accurate. This work is supported by Yunnan University Scientific Research Fund Project of China (No. 2013CG020), Yunnan Province Education Department Scientific Research Fund Project of China (No. 2014Y008), and Yunnan Province Science and Technology Department Applied Basic Research Project of China (No. 2014FB102).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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