Some remarks on almost periodic time scales and almost periodic functions on time scales

In this note we communicate some important remarks about the concepts of almost periodic time scales and almost periodic functions on time scales that are proposed by Wang and Agarwal in their recent papers (Adv. Difference Equ. (2015) 2015:312; Adv. Difference Equ. (2015) 2015:296; Math. Meth. Appl. Sci. 2015, DOI: 10.1002/mma.3590).


Introduction
In order to study almost periodic functions, pseudo almost periodic functions, almost automorphic functions, pseudo almost automorphic functions and so on, on time scales, the following concept of almost periodic time scales was proposed in [1,2]:
Since Definition 1.1 requires that time scale T has a global property, that is, there exists at least one τ ∈ R such that t ± τ ∈ T for all t ∈ T, it is very restrictive. This may exclude many interesting time scales. Therefore, it is a challenging and important problem in theory and applications to find new concepts of almost periodic time scales. Recently, Wang and Agarwal in [8,9,10] have made some efforts to introduce some new types of almost periodic time scales and almost periodic functions on time scales. However, unfortunately, there are many flaws and mistakes in [8,9,10].
Our main purpose of this note is to point out some flaws and mistakes in [8,9,10] and give another correction of the concept of almost periodic functions on time scales in [1], which is overcorrected in [8].

Preliminaries
In this section, we shall first recall some definitions and state some results which are used in what follows.
Let T be a nonempty closed subset (time scale) of R. The forward and backward jump operators σ, ρ : T → T and the graininess µ : T → R + are defined, respectively, by A function f : T → R is right-dense continuous provided it is continuous at right-dense point in T and its left-side limits exist at left-dense points in T. If f is continuous at each right-dense point and each left-dense point, then f is said to be continuous function on T.
For y : T → R and t ∈ T k , we define the delta derivative of y(t), y ∆ (t), to be the number (if it exists) with the property that for a given ε > 0, there exists a neighborhood U of t such that If y is continuous, then y is right-dense continuous, and if y is delta differentiable at t, then y is continuous at t.
Let y be right-dense continuous. If Y ∆ (t) = y(t), then we define the delta integral by  As a slightly modified version of Definition 2.4, the following definition of almost periodic time scales is given in [15]:

Definition 2.6. [1] Let T be an invariant under a translation time scale. A function
is a relatively dense set in T for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) contains a τ (ε, S) ∈ E{ε, f, S} such that This τ is called the ε-translation number of f .

Some remarks
By Definition 2.6, we see that E{ε, f, S} ⊂ Π. So, according to Lemma 2.1 and also, as pointed out in [8], T ∩ Π = ∅ if 0 / ∈ T, and so, in this case, E{ε, f, S} can not be a relatively dense set in T. To fix this flaw, the authors of [8] use "E{ε, f, S} is a relatively dense set in Π" to replace "E{ε, f, S} is a relatively dense set in T" in Definition 2.6 to give a correction of Definition 2.6, that is, Definition 1.8 in [8]. Here, we will use "E{ε, f, S} is a relatively dense set in R" to replace "E{ε, f, S} is a relatively dense set in T" in Definition 2.6 to give another correction of Definition 2.6 as follows.
is a relatively dense set in R for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) contains a τ (ε, S) ∈ E{ε, f, S} such that The authors of [8] also pointed out that there exist many integrals in [1] such as the following: t+l t f (s)∆s, but t ∈ T cannot guarantee that t + l ∈ T, where l ∈ R is an inclusion length in Definition 2.1. However, after we have thoroughly and carefully checked the whole paper [1], there is no such integral in [1] at all. Besides, even if there is such integral in paper [1], according to our convention Hence, the integral t+l t f (s)∆s is well defined. In order to extend the concept of almost periodic time scales, authors of [8] give the following result: Unfortunately, the following example shows that Theorem 3.1 is incorrect. In the following, we divide five cases to show that for every τ ∈ R \ {0},T ∩T τ =T, that is, T contains no any sub time scale that is an invariant under a translation time scale.
Case 1: It is easy to see that T ∩T τ = ∅ for τ ∈ R \ Z. Case 2: If τ is a positive even number, then 2l 1 +1 ∈T and 2l 1 +1 / ∈T τ , henceT∩T τ =T. Case 3: If τ is a negative even number, then −2k 1 ∈T and −2k 1 / ∈T τ , henceT ∩T τ =T. Case 4: If τ is a positive odd number, then there are at last only finite many positive odd numbers in {−2k n + τ } ∞ n=1 , that is,T τ only contains at last finite many positive odd numbers. Since there are infinite many positive odd numbers inT, henceT ∩T τ =T.
Case 5: If τ is a negative odd number, then there are at last only finite many negative even numbers in {2l m + 1 + τ } ∞ n=1 , that is,T τ only contains at last finite many negative even numbers. Since there are infinite many negative even numbers inT, henceT ∩T τ =T.

Remark 3.3. The set T of Definition 2.5 is the sub-invariant under the translation unit in T of Definitions 2.2 and 2.3 in [8].
In [9], the following definitions are given: and k is some finite number, and [α i , β i ] are closed intervals for i = 1, 2, . . . , k or T r = ∅; (c) for all t ∈ T i and all ω ∈ S i , we have t + ω ∈ T i , i.e., T i is an ω-periodic time scale; (d) for i = j, for all t ∈ T i \ {t k ij } and all ω ∈ S j , we have t + ω / ∈ T, where {t k ij } is the connected points set of the timescale sequence {T i } i∈Z + ; (e) R 0 = {0} if and only if T r is a zero-periodic time scale and R 0 = ∅ if and only if T r = ∅; and the set Π is called a changing-periods set of T, T i is called the periodic sub-timescale of T and S i is called the periods subset of T or the periods set of T i , T r is called the remain timescale of T and R 0 the remain periods set of T.
In [9], the following results are given:   The following concept of an index function for changing-periodic time scales in [9] has flaws.

. (Theorem 2.11 in [9]) If T is an infinite time scale and the graininess function
Definition 3.5. (Definition 2.8 in [9]) Let T be a changing-periodic time scale, then the function τ is called an index function for T, where the corresponding periods set of T τt is denoted as S τt . In what follows we shall call S τt the adaption set generated by t, and all the elements in S τt will be called the adaption factors for t.

Remark 3.5. According to Definition 3.3 and Definition 3.4, for
} is a countable points set or an empty set, and T r ∩ T i may not be an empty set. Therefore, the concept of an index function for changing-periodic time scales is not well defined.
is a relatively dense set for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) containsτ (ε, S) ∈ E{ε, f, S} such that here,τ is called the ε-local translation number of f and l(ε, S) is called the local inclusion length of E{ε, f, S}. Definition 3.7. (Definition 3.2 in [9]) Assume that T is a changing-periodic time scale. Let f (t, x) ∈ C(T × D, E n ) if for any given adaption factors sequence (α τ ) ′ ⊂ S τt , there exists a subsequence α τ ⊂ (α τ ) ′ such that T α τ f (t, x) exists uniformly on T × S, then f (t, x) is called a local-almost periodic function in t uniformly for x ∈ D.
Remark 3.6. Since Definition 3.6 and Definition 3.7 are based on the concept of an index function for changing-periodic time scales, they are not well defined.
In [16,10], the following definitions of almost periodic time scales and almost periodic functions on time scales are given: Definition 3.8. [10,16] We say that T is an almost periodic time scale if for any given ε 1 > 0, there exists a constant l(ε 1 ) > 0 such that each interval of length l(ε 1 ) contains a τ (ε 1 ) such that that is, for any ε 1 > 0, the following set is relatively dense. This τ is called the ε 1 -translation number of T, l(ε 1 ) is called the inclusion length of E{T, ε 1 }, and E{T, ε 1 } is called the ε 1 -translation set of T.
is a relatively dense set for all ε 2 > ε 1 > 0 and for each compact subset S of D; that is, for any given ε 2 > ε 1 > 0 and each compact subset S of D, there exists a constant l(ε 2 , S) > 0 such that each interval of length l(ε 2 , S) contains τ (ε 2 , S) ∈ E{ε 2 , f, S} such that This τ is called the ε 2 -translation number of f and l(ε 2 , S) is called the inclusion length of E{ε 2 , f, S}.
Remark 3.7. Since the fact that T is an almost periodic time scale under Definition 3.8 may do not guarantee that the set {τ ∈ Π ε 1 = E{T, ε 1 } : T∩T τ = ∅} is relatively dense. Therefore, Definition 3.9 is not well defined. An correction for this, we refer to [13]. Definition 3.10. (Definition 6.2 in [10]) Let T be an almost periodic time scale. A function f ∈ C(T × D, E n ) is called an ε * -local almost periodic function in t ∈ T uniformly for x ∈ D if there exists some fixed ε * > 0, such that for all ε 1 > ε * and ε * < d(T, T τ ) < ε 1 , the ε 2 -translation set of f E{ε 2 , f, S} = {τ ∈ Π ε 1 : |f (t + τ, x) − f (t, x)| < ε 2 , f or all (t, x) ∈ (T ∩ T −τ ) × S} is a relatively dense set for all ε 2 > ε 1 > 0 and for each compact subset S of D; that is, for any given ε 2 > 0 and each compact subset S of D, there exists a constant l(ε 2 , S) > 0 such that each interval of length l(ε 2 , S) contains a τ (ε 2 , S) ∈ E{ε 2 , f, S} such that This τ is called the ε 2 -translation number of f and l(ε 2 , S) is called the inclusion length of E{ε 2 , f, S}.
As a closing of this note, we shall point out that, in order to define local-almost periodic functions, local-almost automorphic functions and so on, on time scales, one needs a proper definition of almost periodic time scales, which can support these classes of functions on time scales, the almost periodic time scale under Definition 2.3 may be the most general almost periodic time scale. However, since a time scale under Definition 2.4 has the concrete property (ii) of Definition 2.4, for conveniens, the almost periodic time scale under Definition 2.4 may be the best one on which one can define and investigate local-almost periodic functions, local-almost automorphic functions and so on.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.