Periodic and Almost Periodic Time Scales

Nonauton. Dyn. Syst. 2016; 3:24–41

Research Article Open Access

Chao Wang, Ravi P. Agarwal, and Donal O’Regan Periodicity, almost periodicity for time scales and related functions

DOI 10.1515/msds-2016-0003 Received December 9, 2015; accepted May 4, 2016

Abstract: In this paper, we study almost periodic and changing-periodic time scales considered by Wang and Agarwal in 2015. Some improvements of almost periodic time scales are made. Furthermore, we introduce a new concept of periodic time scales in which the invariance for a time scale is dependent on an translation direction. Also some new results on periodic and changing-periodic time scales are presented.

Keywords: Time scales, Almost periodic time scales, Changing periodic time scales

MSC: 26E70, 43A60, 34N05.

1 Introduction

Recently, many papers were devoted to almost periodic issues on time scales (see for examples [1–5]). In 2014, to study the approximation property of time scales, Wang and Agarwal introduced some new concepts of almost periodic time scales in [6] and the results show that this type of time scales can not only unify the continuous and discrete situations but can also strictly include the concept of periodic time scales. In 2015, some open problems related to this topic were proposed by the authors (see [4]). Wang and Agarwal addressed the concept of changing-periodic time scales in 2015. This type of time scales can contribute to decomposing an arbitrary time scales with bounded graininess function µ into a countable union of periodic time scales i.e, Decomposition Theorem of Time Scales. The concept of periodic time scales which appeared in [7, 8] can be quite limited for the simple reason that it requires inf T = −∞ and sup T = +∞. The rst concepts of periodic time scales and periodic functions were proposed by Kaufmann and Raoul in [7], and they were improved by Adıvar in [9]. However, the concepts of periodic time scales proposed by [7–9] were without considering the shift direction of time scales, which will be quite limited for understanding the recent results from [2, 10] etc. For an example, consider +∞ [ T = [2k, 2k + 1], (1.1) k=0 according to the concept of periodic time scales from [7, 8], (1.1) is not a periodic time scale since inf T = 0. In fact, for any t ∈ T, we have t + 2 ∈ T, but there exists t0 = 0 ∈ T such that t0 − 2 = −2 ̸∈ T. Nevertheless, (1.1) has very nice closedness in translation for time variables if the time scale is only translated towards the positive direction, because for any t ∈ T, we have t + 2 ∈ T. For another example, consider [ T = [22n , 22n+1], (1.2) n∈Z+∪{0}

Chao Wang: Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China, E-mail: [email protected] Ravi P. Agarwal: Department of Mathematics, Texas A&M University-Kingsville, TX 78363-8202, Kingsville, TX, USA, E-mail: [email protected] Donal O’Regan: School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland, E-mail: [email protected]

© 2016 Chao Wang et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Periodicity, almost periodicity for time scales and related functions Ë 25 according to the new concept of periodic time scales from [9], (1.2) is also not a periodic time scales since T has a inmum that equals to 1 with respect to the shift operator δ−. In fact, for any t ∈ T, although δ+(4, t) = 4t ∈ T, there exists t0 = 1 ∈ T such that δ−(4, t0) = t0/4 = 1/4 ̸∈ T. However, (1.2) also has very nice closedness in shift for time variables if the time scale is only shifted towards the positive direction, because for any t ∈ T, we have δ+(4, t) ∈ T. Hence, concepts of periodic time scales from [7, 9] were without considering the shift direction of time scales. In this paper we propose a new concept of periodic time scales with “translation direction” so that we can regard (1.1) as a periodic time scale. Also we improve the results on almost periodic time scales from [6]. In addition, we obtain new results related to changing-periodic time scales from [2].

2 Periodic and almost periodic time scales

To obtain some new results related to periodic and almost periodic time scales, rst, we introduce some new denitions of intersection type for a translation time scale. Let τ Π1 := {τ ∈ R : T ∩ T ≠ ∅} ≠ {0}, τ where T := T + τ = {t + τ : ∀t ∈ T}.

Denition 2.1. We introduce three types of intersection for a translation time scale as follows:

(1) We say T is a positive-direction intersection time scale if for any p > 0, there exists a number P > p and P P ∈ Π1, we call T ∩ T the positive-direction intersection for T. (2) We say T is a negative-direction intersection time scale if for any q < 0, there exists a number Q < q and Q Q ∈ Π1, we call T ∩ T the negative-direction intersection for T. (3) We say T is a bi-direction intersection time scale if for any p > 0 and q < 0, there exists two numbers P Q P > p, Q < q and P, Q ∈ Π1, we call (T ∩ T ) ∪ (T ∩ T ) the bi-direction intersection for T. (4) We say T is an oriented-direction intersection time scale if T is a positive-direction intersection time scale or a negative-direction intersection time scale.

Remark 2.1. From Denition 2.1, we obtain that if sup T = +∞, inf T = −∞, then T is a bi-direction intersection τ time scale since for any τ ∈ R, we have T ∩ T = {−∞, +∞}. If inf T = −∞, then T is a negative-direction τ intersection time scale since for any τ ∈ R, we have T∩T = {−∞}. If sup T = +∞, then T is a positive-direction τ intersection time scale since for any τ ∈ R, we have T ∩ T = {+∞}.

Next, we introduce a new concept of periodic time scales. First we recall the following denition from the literature.

Denition 2.2 ([8]). A time scale T is called a periodic time scale if Π := τ ∈ R : t ± τ ∈ T, ∀t ∈ T ≠ {0}.

Note that T dened in Denition 2.2 is a particular bi-direction intersection time scale according to (3) in ±τ Denition 2.1. In fact, for any τ ∈ Π, we have T ∩ T = T. We now consider some new concepts which are more general than Denition 2.2.

Denition 2.3. We say T is called a periodic time scale if

τ Π2 := {τ ∈ R : T ⊆ T, ∀t ∈ T} ≠ {0}. (2.1)

Furthermore, we can describe it in detail as follows: 26 Ë Chao Wang et al.

(a) if for any p > 0, there exists a number P > p and P ∈ Π2, we say T is a positive-direction periodic time scale; (b) if for any q < 0, there exists a number Q < q and Q ∈ Π2, we say T is a negative-direction periodic time scale. (c) if ±τ ∈ Π2, we say T is a bi-direction periodic time scale; (d) we say T is an oriented-direction periodic time scale if T is a positive-direction periodic time scale or a negative-direction periodic time scale.

Remark 2.2. From Denition 2.3, we obtain that a bi-direction periodic time scale is an oriented-direction periodic time scale. Furthermore, one can easily observe that Denition 2.2 is just a concept of bi-direction periodic time scales, it is just a particular case of Denition 2.3. Under Denition 2.3, (1.1) is a positive-direction periodic time scale.

From Denitions 2.1 and 2.3, and from Zorn’s Lemma, we obtain a sucient condition to guarantee that T is an oriented-direction periodic time scale.

Lemma 2.1 ([11], Zorn’s Lemma). Suppose (P, ) is a partially ordered set. 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 2.1. Let T be an oriented-direction intersection time scale. If for any given ε > 0, there exists a constant l(ε) such that each interval of length l(ε) contains a τ(ε) ∈ Π1 such that

d(T, Tτ) < ε, (2.2) that is, for any ε > 0, the following set

E{T, ε} := {τ ∈ Π1 : d(T, Tτ) < ε} (2.3)

τ is relatively dense in Π1, where d(·, ·) denote a Hausdor distance and Tτ = T ∩ T . Then T is an oriented- direction periodic time scale.

Proof. Note that from the condition of Theorem 2.1, we obtain ⊂ , and let L := sup l( 1 ) . Without Tτ T n∈N n loss of generality, we assume that T is a positive-direction intersection time scale, and then for any p > 0, ∞ there exists some n0 > 0 such that n0L > p. So we can choose a sequence {τn}n=1 ⊂ [p, n0L] such that 1 d( , ) < . (2.4) T Tτn n We denote the set I = Tτ : τ ∈ [p, n0L] , and obviously, I is a closed set. From (2.4), we obtain the totally ordered set {Tτn ∈ I : Tτn ⊂ Tτn+1 , n ∈ N} and limn→∞ Tτn = Tτ∞ = T ∈ I. Hence, T is the maximum element in I. * Now I forms a semi-ordered set with respect to the inclusion relation and I is closed. Denote I an arbitrary subset of I and is totally ordered. We consider two cases: Case (1): * I = Tτn ∈ I : Tτn ⊃ Tτn+1 , n ∈ N , * * then Tτ1 ∈ I ⊂ I and Tτ1 is an upper bound of I in I. Case (2): * I = Tτn ∈ I : Tτn ⊂ Tτn+1 , n ∈ N , S∞ * then limn→∞ Tτn = n=1 Tτn = Tτ∞ is an upper bound of I . Because I is closed, then Tτ∞ ∈ I. According τ0 to Zorn’s Lemma (i.e., Lemma 2.1), there exists some τ0 ∈ [p, n0L] such that T ∩ T := Tτ0 is the maximum element in I. Periodicity, almost periodicity for time scales and related functions Ë 27

τ From the above, we obtain that T ∩ T 0 and T are two maximum elements in I. Therefore, it follows from τ0 τ0 T∩T ⊆ T that T∩T = T, i.e., T is an oriented-direction periodic time scale with τ0-period. This completes the proof.

τ τ Remark 2.3. In [12], to have T ∩ T ≠ ∅, the authors changed the time scale T in Denition 8 from [6] into Tτ, which will lead to an oriented-direction periodicity for a time scale according to Theorem 2.1.

Remark 2.4. From Theorem 2.1, we see that Lemmas 10, 11, Theorems 12, 13, 14 and Corollary 15 in [12] are based on oriented-direction periodic time scales.

Example 2.1. Let a > 1 and consider the following almost periodic time scale

∞ √ √ [ Pa,| sin 3t+sin 7t| = [pm , a + pm], m=1 where m−1 X √ √ √ pm = (m − 1)a + sin 3 ka + | sin 3a + sin 7a|

k=1 √ √ √ +| sin 3(2a + | sin 3a + sin 7a|) √ √ √ + sin 7(2a + | sin 3a + sin 7a|)| √ √ √ + ... + | sin 3((k − 1)a + | sin 3a + sin 7a|) √ √ √ + sin 7((k − 1)a + | sin 3a + sin 7a|)| | {z } k terms √ √ √ + sin 7 ka + | sin 3a + sin 7a| √ √ √ +| sin 3(2a + | sin 3a + sin 7a|) √ √ √ + sin 7(2a + | sin 3a + sin 7a|)| √ √ √ + ... + | sin 3((k − 1)a + | sin 3a + sin 7a|) √ √ √ + sin 7((k − 1)a + | sin 3a + sin 7a|)| . (2.5) | {z } k terms Then we have ( ∞ t, if t ∈ ∪ [pm , a + pm), σ(t) = m=1 √ √ ∞ t + | sin 3t + sin 7t|, if t ∈ ∪m=1{a + pm} and ( ∞ 0, if t ∈ ∪ [pm , a + pm), µ(t) = m=1 √ √ ∞ | sin 3t + sin 7t|, if t ∈ ∪m=1{a + pm}.

The concept from [12] does not include the time scale (2.5).

Remark 2.5. From Theorem 2.1, we know Denition 9 from [12] is an oriented-direction periodic time scale. Note if T and Tτ approximate each other, then this approach is not an approximation between T and the translation of T, but it is an approximation that occurs between T and a subset of T. Note the idea in Denition 9 from [12] does not reect approximation behavior between a time scale and its corresponding translation. We use the distance τ τ d(T, T ) from [6] to describe an approximation that occurs between T and its corresponding translation T (see Figure 1 and Figure 2). 28 Ë Chao Wang et al.

Figure 1: The approximation from [12] that will lead to the oriented-direction periodicity for the time scale T.

Figure 2: The approximation that describes the distance between the time scale T and its translation Tτ from [6].

Remark 2.6. We can use the similar idea of Denition 2.3 and [9] to introduce a new concept of periodic time scales in shift operators (i.e., Denition 2.4).

Denition 2.4. Let δ : R × T → T be a shift operator associated with the initial point t0, so δ(t0, t) = t. Denote

δ T τ := δ(τ, T) = {δ(τ, t), ∀t ∈ T}, we say T is a periodic time scale under the shift operator δ if

δτ Πδ := {τ ∈ R : T ⊆ T} ≠ {t0}.

Furthermore, denote

Πδ± := {τ ∈ R : δ±(τ, t) ∈ T, ∀t ∈ T}, if Πδ+ ≠ {t0}, we say T is a positive-direction periodic time scale under the shift operator δ+; if Πδ− ≠ {t0}, we say T is a negative-direction periodic time scale under the shift operator δ−; if Πδ± ≠ {t0}, we say T is a bi- direction periodic time scale under the shift operators δ±, where the shift operators δ± satisfy all the conditions from [9].

Note that under the above Denition 2.4, the time scale (1.2) can be regarded as a positive-direction periodic time scale under the shift operator δ+ since Πδ+ ≠ {t0} = {1}, but (1.2) is not a bi-direction periodic time scale under the shift operator δ± since for the shift operator δ+, there not exists the corresponding shift operator δ− for this time scale. Hence, the new concept of periodic time scales from [9] is still a bi-direction periodic time scale under the shift operators, it is a particular case of Denition 2.4.

In what follows, we emphasize that the time scales distance is measured by the Hausdor distance. Periodicity, almost periodicity for time scales and related functions Ë 29

Denition 2.5 ([13]). Let X and Y be two non-empty subsets of a metric space (M, d). We dene their Hausdor distance d(X, Y) by d(X, Y) = max sup inf d˜(x, y), sup inf d˜(x, y) , (2.6) x∈X y∈Y y∈Y x∈X where d˜(·, ·) denotes the distance between two points (see Figure 3).

Hence, from Denition 2.5, if we assume that X = T1 and Y = T2, this well-known denition is ˜ ˜ d(T1, T2) = max sup inf d(t, s), sup inf d(t, s) , s∈ t∈ t∈T1 T2 s∈T2 T1 which is Denition 7 in [12]. In [6], the authors let τ be a number and set the time scales:

+∞ +∞ [ τ [ τ τ T := [αi , βi], T := T + τ = {t + τ : ∀t ∈ T} := [αi , βi ]. (2.7) i=−∞ i=−∞

Figure 3: Components of the calculation of the Hausdor distance between the green line X and the blue line Y.

τ Dene the distance between two time scales, T and T by τ τ τ d(T, T ) = max sup |αi − αi |, sup |βi − βi | , (2.8) i∈Z i∈Z where τ τ τ τ αi := inf α ∈ T : |αi − α| and βi := inf β ∈ T : |βi − β| . (2.9) τ Note if we let X = T and Y = T in Denition 2.5, then (2.6) immediately turns into (2.8) and we can τ calculate the distance between T and T from the distance of their intervals, i.e, the formula (2.8) (see Figure 4). Now we introduce some notations to make the denitions clearer in [6].

Denition 2.6. Let T be an oriented-direction intersection time scale. We say T is an almost periodic time scale if for any given ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ(ε) ∈ Π1 such that τ d(T, T ) < ε, 30 Ë Chao Wang et al.

τ τ Figure 4: The Hausdor distance between the interval [αi , βi] and the interval [αi , βi ]. i.e., for any ε > 0, the following set

τ E{T, ε} = {τ ∈ Π1 : d(T , T) < ε} is relatively dense in Π1. Here τ is called the ε-translation number of T and l(ε) is called the inclusion length of E{T, ε}, E{T, ε} is called the ε-translation set of T, and for simplicity, we use the notation E{T, ε} := Πε .

τ Remark 2.7. From the property of the set Π1, we no longer worry about the case T ∩ T = ∅ in Denition 2.6. We observe that such a revision does not have any impact on the results in [6].

Remark 2.8. According to Denition 2.6, if T is a bi-direction intersection time scale, then one can obtain that sup T = +∞, inf T = −∞ and

τ τ sup T ∩ T = +∞, inf T ∩ T = −∞.

τ −τ Furthermore, for a bi-direction intersection time scale, one can see that if d(T, T ) < ε, then d(T, T ) < ε, i.e., if τ ∈ E{T, ε}, then −τ ∈ E{T, ε}. If τ1 ∈ E{T, ε}, τ2 ∈ E{T, ε}, then we have τ1 + τ2 ∈ E{T, 2ε} since

τ +τ τ τ τ +τ τ τ d(T, T 1 2 ) ≤ d(T, T 1 ) + d(T 1 , T 1 2 ) = d(T, T 1 ) + d(T, T 2 ) < 2ε.

Remark 2.9. In Denition 2.6, if for any p > 0, τ ∈ E{T, ε} ∩ (p, +∞), then we say T is a positive-direction almost periodic time scale; if for any q < 0, τ ∈ E{T, ε} ∩ (−∞, q), then we say T is a negative-direction almost periodic time scale; if for any p > 0, q < 0, ±τ ∈ E{T, ε} ∩ (−∞, q) ∪ (p, +∞) , then we say T is a bi-direction almost periodic time scale.

From Denition 2.6, we note the following theorems.

′ Theorem 2.2 ([6]). Let T be an almost periodic time scale. Then for any given sequence α ⊂ Π1, there exists ′ αn a subsequence α ⊂ α such that {T } converges to a time scale T0, i.e., for any given ε > 0, there exists N0 > 0 αn such that n > N0 implies d(T , T0) < ε. Furthermore, T0 is also almost periodic. Periodicity, almost periodicity for time scales and related functions Ë 31

′ ′ αn Theorem 2.3 ([6]). Let T be a time scale. If for any sequence α ⊂ Π1, there exists α ⊂ α such that {T } converges to a time scale T0, then T is almost periodic.

From Theorems 2.2 and 2.3, we can obtain the following equivalent denition of almost periodic time scales.

′ ′ Denition 2.7 ([6]). Let T be a time scale. If for any given sequence α ⊂ Π1, there exists a subsequence α ⊂ α αn such that {T } converges to a time scale T0, then T is called an almost periodic time scale.

Remark 2.10. From Theorems 2.2-2.3 we note from Theorem 2.1, that the results in [6] are valid for an oriented- direction periodic time scale.

Based on Denitions 2.6 and 2.7, we give two equivalent concepts of almost periodic functions on time scales. n n n n n We shall use the following notation: E denotes R or C , D denotes an open set in E or D = E , and S denotes an arbitrary compact subset of D.

n Denition 2.8. Let T be an almost periodic time scale, i.e., T satises Denition 2.6. A function f ∈ C(T×D, E ) is called an almost periodic function in t ∈ T uniformly for x ∈ D if the ε2-translation set of f

−τ E{ε2, f , S} = {τ ∈ Πε1 : |f (t + τ, x) − f (t, x)| < ε2, for all (t, x) ∈ (T ∩ T ) × S}

is a relatively dense set in Πε1 for all ε2 > ε1 > 0 and for each compact subset S of D; that is, for any given ε2 > ε1 > 0 and each compact subset S of D, there exists a constant l(ε2, S) > 0 such that each interval of length l(ε2, S) contains a τ(ε2, S) ∈ E{ε2, f , S} such that

−τ |f (t + τ, x) − f (t, x)| < ε2, for all (t, x) ∈ (T ∩ T ) × S.

This τ is called the ε2-translation number of f and l(ε2, S) is called the inclusion length of E{ε2, f , S}.

Figure 5: The functions approximate each other along with the approximation of their respective time scales. 32 Ë Chao Wang et al.

Remark 2.11. The positive number ε1 in Denition 2.6 should be suciently small. Clearly, if ε1 = 0, then Denition 2.8 is equivalent to Denition 1.8 in [10], i.e., the concept of almost periodic functions on periodic time scales. It is worth noting that because ε2 > ε1 > 0, if ε1 > 0 is an arbitrarily small positive number, then ε2 can be an arbitrarily small positive number, i.e., the approximation of the translation of time scales is the prerequisite that guarantees the approximation of the translation of functions dened on time scales (see Figure 5).

Remark 2.12. Note in Denition 16 of [12], the ε for the time scales equals the ε for the functions, and ε is an −τ arbitrary positive number, which will lead to that if ε = 0, then T = T∩T and f(t) = f (t+τ). Hence, in Denition 16 in [12] we have a τ-periodic function on a τ-periodic time scale i.e, in fact, f(t) dened in Denition 16 of [12] is a periodic function on a periodic time scale in nature.

Now, let Πε −τ Πε = E{T, ε}, T = T : −τ ∈ E{T, ε} . Then, Denition 2.8 can be stated as follows:

n Denition 2.9 ([4]). Let T be an almost periodic time scale. A function f ∈ C(T × D, E ) is called an almost periodic function in t ∈ T uniformly for x ∈ D if the ε2-translation set of f

E{ε2, f , S} = {τ ∈ Πε1 : |f(t + τ, x) − f(t, x)| < ε2, Π for all (t, x) ∈ (T ∩ (∪T ε1 )) × S}

Πε |f (t + τ, x) − f (t, x)| < ε2, for all (t, x) ∈ (T ∩ (∪T 1 )) × S.

Here τ is called the ε2-translation number of f and l(ε2, S) is called the inclusion length of E{ε2, f , S}.

Based on Denitions 2.6 and 2.8, we will give the second equivalent concept of almost periodic functions on time scales:

n Denition 2.10. Assume that T is an almost periodic time scale. Let f (t, x) ∈ C(T × D, E ). If for any given ′ ′ −αn sequence α ⊂ Π1, there exists a subsequence α ⊂ α such that the limit set T0 of {T } exists and Tα f(t, x) exists uniformly on T0 × S, then f (t, x) is called an almost periodic function in t uniformly for x ∈ D.

Lemma 2.2. If T is an almost periodic time scale, then for any ε > 0 and any right dense point t ∈ T, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ(ε) ∈ Π1 such that there exists a τ sequence {tn} ⊆ T ∩ T and limn→∞ tn = t, tn > t, n ∈ N.

Proof. Since T is an almost periodic time scale, then for any ε > 0, there exists a constant l(ε) > 0 such that τ each interval of length l(ε) contains a τ(ε) ∈ Π such that d(T, T ) < ε. Because t ∈ T is a right dense point in T, then there exists a sequence {tn} ⊆ T and tn > t, n ∈ N such that limn→∞ tn = t. We argue by contradiction. τ τ * τ τ * If {tn} ̸⊂ T , then t ̸∈ T , which implies that there exists some t ∈ T such that d(T, T ) ≥ |t − t | > 0, τ τ and this is a contradiction. Hence, we have {tn} ⊆ T . Therefore, there exists a sequence {tn} ⊆ T ∩ T and limn→∞ tn = t, tn > t, n ∈ N. This completes the proof.

Theorem 2.4. If T is an almost periodic time scale, then for any ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ(ε) ∈ E{ε, µ} such that

−τ |µ(t + τ) − µ(t)| < ε, for all t ∈ T ∩ T . (2.10) Periodicity, almost periodicity for time scales and related functions Ë 33

Figure 6: The relationship between d(T, Tτ) and the graininess function µ.

Proof. First, let T be an almost periodic time scale. We claim that: for any τ ∈ Πε, if t is a right scattered point in T, then t + τ are right scattered points in T; if t is a right dense point in T, then t + τ are right dense points in T. −τ In fact, if t ∈ T ∩ T is a right scattered point in T, then t + τ are right scattered points in T. We argue by contradiction. Assume that t + τ ∈ T is a right dense point in T, then by Lemma 2.2, there exists a sequence τ {tn} ⊆ (T ∩ T ) and tn ≥ t + τ, n ∈ N such that limn→∞ tn = t + τ. Hence, we have limn→∞ tn − τ = t, i.e., −τ {tn − τ} ⊆ (T ∩ T ) ⊆ T. So t is a right dense point in T, which is a contradiction. Thus, t + τ is a right scattered point in T. −τ On the other hand, if t ∈ T ∩ T is a right dense point in T, then t + τ are right dense points in T. In fact, −τ by Lemma 2.2, there exists a sequence {tn} ⊆ T ∩ T and tn ≥ t, n ∈ N such that limn→∞ tn = t. Then we τ have limn→∞ tn + τ = t + τ and {tn + τ} ⊆ T ∩ T ⊂ T. Thus, t + τ is a right dense point in T. −τ Hence, t ∈ T ∩ T implies that t, t + τ ∈ T. When t is a right dense point, then t + τ is also a right dense point, so we have µ(t) = µ(t + τ) = 0. When t is a right scattered point, then t + τ is also a right scattered point and τ |µ(t) − µ(t + τ)| ≤ d(T, T ) < ε. This completes the proof.

τ + Remark 2.13. From Figure 6, let T be an almost periodic time scale and µτ : T → R be the graininess function τ on T , and then we obtain that |µ(t) − µτ(t + τ)| ≤ max |µτ(t + τ) − h3|, |µ(t) − h3| τ = max{h2, h1} ≤ d(T, T ) < ε for all t ∈ T. (2.11)

−τ Particularly, if t ∈ (T ∩ T ) ⊂ T, then t + τ ∈ T and µτ(t + τ) = µ(t + τ). Hence, from (2.11), we obtain (2.10). −τ Furthermore, if t + τ ̸∈ T, i.e., t ̸∈ T ∩ T , we get µτ(t + τ) = µ(t). 34 Ë Chao Wang et al.

Remark 2.14. The inequality (2.10) can also be written as

−τ |σ(t + τ) − σ(t) − τ| < ε, for all t ∈ T ∩ T , which indicates that if T is τ-periodic, then σ(t + τ) = σ(t) + τ.

Now, from Denition 2.6 and the Hausdor distance (2.8), we can give the third concept of almost periodic time scales by the graininess function µ as follows:

+ τ + Denition 2.11. Let µ : T → R , µ(t) = σ(t) − t and µτ : T → R . We say T is an almost periodic time scale if for any ε > 0, the set * Π = {τ ∈ Π1 : |µτ(t + τ) − µ(t)| < ε, ∀t ∈ T} (2.12) is relatively dense in Π1.

Remark 2.15. Note that Denition 2.11 implies the following:

−τ (1) If t ∈ (T∩T ) ⊂ T, then t+τ ∈ T and µτ(t+τ) = µ(t+τ). Hence, from (2.12), we obtain (2.10). Furthermore, −τ if t ∈ T ∩ T is a right dense point in T, it follows from (2.12) that t + τ ∈ T is also a right dense point. −τ Similarly, if t ∈ T ∩ T is a right scattered point in T, by (2.12), then t + τ ∈ T is also a right scattered point. −τ In fact, if t ∈ T ∩ T is a right dense point in T, then µ(t) = 0. From (2.12), we know that µ(t + τ) = 0, so −τ t + τ ∈ T is a right dense point. Similarly, if t ∈ T ∩ T is a right scattered point, then µ(t) > 0. From (2.12), we have

µ(t + τ) = |µ(t) − µ(t) + µ(t + τ)| ≥ |µ(t)| − |µ(t + τ) − µ(t)| ≥ µ(t) − ε > 0. Hence, t + τ is a right scattered point. (2) From Figure 6 and (2.12), we obtain that

τ d(T, T ) ≤ sup |µτ(t + τ) − µ(t)| < ε, t∈T

which indicates that T is an oriented-direction almost periodic time scale. τ τ + (3) From Figure 6 and Remark 2.13, T is a τ-translation of T. Let µτ : T → R be the graininess function of τ T , and we obtain that ( µ(t), t + τ ̸∈ T, µτ(t + τ) = (2.13) µ(t + τ), t + τ ∈ T. Thus, from (2.13), we can simplify Denition 2.11 as follows: + Let µ : T → R be a graininess function of T. We say T is an almost periodic time scale if for any ε > 0, the set * −τ Π = {τ ∈ Π1 : |µ(t + τ) − µ(t)| < ε, ∀t ∈ T ∩ T }

is relatively dense in Π1.

Remark 2.16. From Example 2.1, we know µ(t) has almost periodicity at all its right scattered points. Clearly, µ(t) satises all the conditions in Denition 2.11, and thus, the time scale T from Example 2.1 is an almost periodic time scale. Although | inf T| < +∞, (1.1) and Example 2.1 can be regarded as a periodic time scale and an almost periodic time scale with sup T = +∞, respectively (i.e., they are positive-direction translation time scales).

3 Changing-periodic time scales

In [2], Wang and Agarwal proposed a new concept called changing-periodic time scales. Now, in this section, considering Denition 2.3, we will improve and obtain some new results about changing-periodic time scales. Periodicity, almost periodicity for time scales and related functions Ë 35

Denition 3.1 ([2]). Let T be an innite time scale. We say T is a changing-periodic or a piecewise-periodic time scale if the following conditions are fullled:

∞ k S S S { } + (a) T = Ti Tr and Ti i∈Z is a well connected timescale sequence, where Tr = [αi , βi] and k is i=1 i=1 some nite number, and [αi , βi] are closed intervals for i = 1, 2, ... , k or Tr = ∅; + S∞ S (b) Si is a nonempty subsets of R with 0 ̸∈ Si for each i ∈ Z and Π = i=1 Si R0, where R0 = {0} or R0 = ∅; (c) for all t ∈ Ti and all ω ∈ Si, we have t + ω ∈ Ti, i.e., Ti is an ω-periodic time scale; k k (d) for i ≠ j, for all t ∈ Ti\{tij} and all ω ∈ Sj, we have t + ω ̸∈ T, where {tij} is the connected points set of the { } + timescale sequence Ti i∈Z ; (e) R0 = {0} if and only if Tr is a zero-periodic time scale and R0 = ∅ if and only if Tr = ∅; and the set Π is called a changing-periods set of T, Ti is called the periodic sub-timescale of T and Si is called the periods subset of T or the periods set of Ti, Tr is called the remain timescale of T and R0 the remain periods set of T.

Remark 3.1. Note that from condition (c) in Denition 3.1, we should know that Ti is an oriented-direction periodic time scale. If T is a bi-direction periodic time scale, then it is Denition 3.7 in [8]. We adopt Denition 2.3 to understand it. Now, (2.1) can be written as

Π2 = {τ ∈ R : t + τ ∈ T, ∀t ∈ T} ≠ {0}, (3.1) rather than Π2 = {τ ∈ R : t ± τ ∈ T, ∀t ∈ T} ≠ {0}, (3.2) since (3.2) is a bi-direction periodic time scale and sup T = +∞, inf T = −∞, and it is quite limited for understanding the decomposition theorem of time scales. From condition (c) in Denition 3.1, we see that Ti is a periodic time scale implies Ti satises (3.1) rather than (3.2), i.e., Ti is an oriented-direction periodic time scale.

Example 3.1. Let k ∈ Z, and consider the following time scale:

+∞ +∞ √ √ √ [ 3 3 1 [ [ 3 2 3 2 3 = (2k + 1), (2k + 1) + (2k + 1), (2k + 1) + . T 2 2 12 2 2 5 k=−∞ k=−∞ We denote

+∞ +∞ √ √ √ [ 3 3 1 [ 3 2 3 2 3 = (2k + 1), (2k + 1) + and = (2k + 1), (2k + 1) + . T1 2 2 12 T2 2 2 5 k=−∞ k=−∞

Then, by a direct calculation the set Π2 is [ √ Π2 = 3n, n ∈ Z 3 2n, n ∈ Z := S1 ∪ S2.

This time scale is a changing-periodic time scale according to Denition 3.1.

Example 3.2. Let k ∈ Z, and consider the following time scale:

+∞ +∞ √ √ √ [ 3 3 1 [ [ 3 2 3 2 3 = − (2k + 1), − (2k + 1) − (2k + 1), (2k + 1) + . T 2 2 12 2 2 5 k=0 k=0 We denote

+∞ +∞ √ √ √ [ 3 3 1 [ 3 2 3 2 3 = − (2k + 1), − (2k + 1) − and = (2k + 1), (2k + 1) + . T1 2 2 12 T2 2 2 5 k=0 k=0 36 Ë Chao Wang et al.

Then, by direct calculation the set Π2 is [ √ Π2 = − 3n, n ∈ N 3 2n, n ∈ N := S1 ∪ S2.

According to Denition 3.1, this time scale is a changing-periodic time scale in which T1 is a negative-direction periodic sub-timescale and T2 is a positive-direction periodic sub-timescale.

Example 3.3. Consider T = {−4k, 4k +3 : k ∈ N}. Note that T1 = {−4k : k ∈ N} and T2 = {4k +3 : k ∈ N} are oriented-direction periodic time scales, i.e., T1 is a negative-direction periodic time scale and T2 is a positive- direction periodic time scale. Hence, T is a changing-periodic time scale.

+ Theorem 3.1. If T is an innite time scale and the graininess function µ : T → R is bounded, then T is a changing-periodic time scale.

Proof. Without loss of generality, we assume that sup T = +∞ and inf T = −∞. From Remark 2.1, T is an oriented-direction intersection time scale. For any p > 0, q < 0, we take some n1, n2 such that p < n1µ¯, q > −n µ¯, where µ¯ = sup µ(t) and max{n µ¯, n µ¯} µ¯ + L, L > 0 is the length of a closed nite interval with 2 t∈T 1 2 the largest length in T. We denote the set τ I = T ∩ T : τ ∈ [−n2µ¯, q] ∪ [p, n1µ¯] .

* Clearly, I forms a semi-ordered set with respect to the inclusion relation and I is closed. Denote I any subset of I and is totally ordered. We consider two cases: Case (1): * τn I = Tτn ∈ I : Tτn ⊃ Tτn+1 , n ∈ N , where Tτn := T ∩ T , * * then Tτ1 ∈ I ⊂ I and Tτ1 is an upper bound of I in I. Case (2): * I = Tτn ∈ I : Tτn ⊂ Tτn+1 , n ∈ N ,

τn S∞ τn τ∞ * τ∞ then limn→∞(T∩T ) = n=1(T∩T ) = T∩T is an upper bound of I . Because I is closed, then T∩T ∈ I. τ0 According to Zorn’s lemma, there exists some τ0 ∈ [−n2µ¯, q] ∪ [p, n1µ¯] such that T ∩ T is the maximum τ τ τ element in I. Note that since µ is bounded, T ∩ T 0 ≠ ∅ and sup(T ∩ T 0 ) = +∞, inf(T ∩ T 0 ) = −∞. Now we show that T is a changing-periodic time scales. We divide the proof into three steps, We divide the proof into the following steps: 1 1 Step I. We can nd a time scale T0 such that T0 ⊂ T is the largest periodic sub-timescale in T. For this, τ1 we make a continuous translation of T to nd a number τ1 such that T ∩ T := T1 is the maximum. Next, τ2 consider a translation of T1 again to nd a number τ2 such that T1 ∩ T1 := T2 is the maximum. Continue τn this process n times to nd a number τn such that Tn−1 ∩ Tn−1 := Tn is the maximum. This process leads to a decreasing sequence of timescale sets:

T ⊃ T1 ⊃ T2 ⊃ ... ⊃ Tn ⊃ ... .

1 1 1 1 τ0 1 Hence, it follows that limn→∞ Tn = T0. This shows that there exists some τ0 such that (T0) ⊆ T0 through 1 1 1 τ0 1 τ0 1 2 the translation of this number and (T0) is maximum (in fact, if (T0) ̸⊆ T0, then there exists T0 such that 1 2 1 τ0 1 1 1 1 T0 = (T0) ∩ T0 ⊂ T0, this is a contradiction since limn→∞ Tn = T0), obviously, this also implies that T0 is not a nite union of the closed intervals. 1 1 Now we claim that T0 ≠ ∅. In fact, if T0 = ∅, then there exists some suciently large n0 such that Tn0 = ∅, i.e., there exists some sub-timescale Tn0−1 ⊂ T such that Tn0−1 has no intersection with itself through the τn0 translation of the number τn and ∩ is the maximum element in 0 Tn0−1 Tn0−1 τ In0 = {Tn0−1 ∩ Tn0−1 : ∀τ ≠ 0} = ∅, which means that Tn0−1 is a single point set, but this is a contradiction since sup Tn0−1 = +∞, inf Tn0−1 = −∞. 1 1 Therefore, T0 is a τ0-periodic sub-timescale whose translation direction is determined by the sign of the 1 number τ0. Periodicity, almost periodicity for time scales and related functions Ë 37

* 1 * Step II. For the time scale T1 := T\T0, where A denotes the closure of the set A, by replacing T with T1, 2 * 1 2 and repeating the Step I, we can obtain the periodic sub-timescale T0. For the time scale T2 := T\(T0 ∪ T0), by * 3 replacing T with T2, and repeating the Step I, we can obtain the periodic sub-timescale T0. Similarly, we can 4 n i i j k { } + ∩ \{ } ∅ obtain T0, ... , T0 .... Obviously, the timescale sequence T0 i∈Z is well connected and T0 T0 tij0 = k i j for i ≠ j, where {tij0} is the connected points set between T0 and T0. If for some suciently large n0, still * Sn0 i Tn0 = T\ i=1 T0 is an innite time scale, then we repeat the Step I again until the remaining timescale S∞ i T\ i=1 T0 = ∅, or a nite union of the closed intervals. Step III. Letting the set Π of T be as ∞ ∞ [ [ [ i [ Π = Si R0 = {nτ0, n ∈ Z} R0, i=1 i=1 where R0 = {0} or ∅, from Steps I and II, it follows that Si ∩ Sj = ∅ if i ≠ j. ∞ k S i S S From Steps I, II, III, we nd T0 [αi , βi] = T, where k is some nite number and [αi , βi] are i=1 i=1 ∞ S i closed intervals for i = 1, 2, ... , k, or T0 = T. Therefore, T is a changing-periodic time scale. This i=1 completes the proof.

Remark 3.2. In [2, 10], the authors emphasize that “periodic time scale” implies ”oriented-direction periodic time scale”, all “periodic time scale” is with “translation direction” . The denition we should adopt is Deni- tion 2.3. Note Denition 2.3 in this paper includes Denition 1.1 in [7]. Note Denition 3.7 in [8] is equivalent to Denition 1.1 in [7].

Remark 3.3. For any oriented-direction periodic time scale T, we can add {−∞, +∞} to the time scale such that inf T = −∞ and sup T = +∞. For example, for the time scale (1.1), we can add {−∞} to the time scale such that inf T = −∞ if we need, and it is still a positive-direction periodic time scale.

Remark 3.4. From Theorem 3.1, Theorem 2.21 in [2] is true.

+ Theorem 3.2. Let T be an arbitrary time scale with sup T = +∞, inf T = −∞. If µ : T → R is bounded, then T contains at least one oriented-direction periodic time scale.

Proof. From Remark 2.1, T is a bi-direction intersection time scale. For any p > 0, q < 0, we take some n1, n2 such that p < n µ¯, q > −n µ¯, where µ¯ = sup µ(t) and max{n µ¯, n µ¯} µ¯ + L, L > 0 is the length of a 1 2 t∈T 1 2 closed nite interval with the largest length in T. We denote the set

τ I = T ∩ T : τ ∈ [−n2µ¯, q] ∪ [p, n1µ¯] .

τn S∞ τn τ∞ * τ∞ then limn→∞(T∩T ) = n=1(T∩T ) = T∩T is an upper bound of I . Because I is closed, then T∩T ∈ I. τ0 According to Zorn’s lemma, there exists some τ0 ∈ [−n2µ¯, q] ∪ [p, n1µ¯] such that T ∩ T is the maximum τ τ τ element in I. Note that since µ is bounded, T ∩ T 0 ≠ ∅ and sup(T ∩ T 0 ) = +∞ or inf(T ∩ T 0 ) = −∞. 1 1 We can nd a time scale T0 such that T0 ⊂ T is the largest periodic sub-time scale in T. For this, we make τ1 a continuous translation of T to nd a number τ1 such that T ∩ T := T1 is the maximum. Next, consider a τ2 translation of T1 again to nd a number τ2 such that T1 ∩ T1 := T2 is the maximum. Continue this process 38 Ë Chao Wang et al.

τn n times to nd a number τn such that Tn−1 ∩ Tn−1 := Tn is the maximum. This process leads to a decreasing sequence of time scale sets: T ⊃ T1 ⊃ T2 ⊃ ... ⊃ Tn ⊃ ... . 1 1 1 1 τ0 1 Hence, it follows that limn→∞ Tn = T0. This shows that there exists some τ0 such that (T0) ⊆ T0 through 1 1 1 τ0 1 τ0 1 2 the translation of this number and (T0) is maximum (in fact, if (T0) ̸⊆ T0, then there exists T0 such that 1 2 1 τ0 1 1 1 1 T0 = (T0) ∩ T0 ⊂ T0, this is a contradiction since limn→∞ Tn = T0), obviously, this also implies that T0 is not a nite union of the closed intervals. 1 1 Now we claim that T0 ≠ ∅. In fact, if T0 = ∅, then there exists some suciently large n0 such that Tn0 = ∅, i.e., there exists some sub-timescale Tn0−1 ⊂ T such that Tn0−1 has no intersection with itself through the τn0 translation of the number τn and ∩ is the maximum element in 0 Tn0−1 Tn0−1

τ In0 = {Tn0−1 ∩ Tn0−1 : τ ≠ 0} = ∅,

which means that Tn0−1 is a single point set, but this is a contradiction since sup Tn0−1 = +∞, inf Tn0−1 = −∞. 1 1 Therefore, T0 is a τ0-periodic sub-timescale whose translation direction is determined by the sign of the 1 1 number τ0, that is, T0 is an invariant under translations unit in T. This completes the proof.

Remark 3.5. From Theorem 2.4 in [10], under the conditions in Theorem 3.2, we obtain that T contains at least one invariant under translations unit. Note that this invariant under translations unit is with direction (i.e. oriented-periodic time scale).

Example 3.4. Consider the time scale T = {−6k, 6k + 5 : k ∈ N}. We see that T contains a positive-direction periodic time scale T1 = {6k + 5 : k ∈ N} and a negative-direction periodic time scale T2 = {−6k : k ∈ N}, respectively. Note that T is a changing-periodic time scale.

Next, from Denition 2.3, we can apply our results to dynamic equations and obtain some new results related to changing-periodic time scales under the new concept of periodic time scales (i.e., in the sense of Denition 2.3). In order to make this paper more concise, we will not restate the results from [2] here. One can consult [2] for details. Consider the following linear dynamic equation on a changing-periodic time scale T:

x∆(t) = A(t)x(t) + f(t) (3.3) and its associated homogeneous equation

x∆(t) = A(t)x(t), (3.4) where A(t) is a local-almost periodic matrix function, and f(t) is a local-almost periodic vector function. Fur- ther, we assume that f (t) and A(t) are synchronously local-almost periodic functions.

Denition 3.2. We say the following dynamic equation

∆τt x (t) = A(t)x(t) + f(t), t ∈ Tτt (3.5) and its associated homogeneous equation

∆τt x (t) = A(t)x(t), t ∈ Tτt (3.6)

are the local-almost periodic dynamic equations generated by (3.3) and (3.4), respectively, where ∆τt implies the

∆-derivative on the time scale Tτt . Furthermore, we say all the solutions for (3.5) and (3.6) are the local solutions for (3.3) and (3.4), respectively.

From Theorem 3.5, Corollaries 3.7-3.9, and by considering Denition 2.3, we obtain the following corollaries. Periodicity, almost periodicity for time scales and related functions Ë 39

Corollary 3.1. Let T be a changing-periodic time scale and τt be an index function. If (3.4) admits an exponential dichotomy on the local part Tτt with Pτt = 0 for all t ∈ T and Tτt is a positive-direction periodic time scale, then (3.3) has a unique local-almost periodic solution on Tτt given by +∞ Z −1 x(t) = X(t)X στs (s) f(s)∆τs s. (3.7) t

Remark 3.6. In Corollary 3.1, if (3.4) admits an exponential dichotomy on the local part Tτt with Pτt = I for all t ∈ T and Tτt is a negative-direction periodic time scale, then (3.3) has a unique local-almost periodic solution on Tτt given by t Z −1 x(t) = X(t)X στs (s) f(s)∆τs s. (3.8) −∞

If Tτt is a bi-direction periodic time scale, then (3.3) has a unique local-almost periodic solution on Tτt of the form (3) in Theorem 3.5 in [2].

Remark 3.7. In fact, one can observe that (3.7) and (3.8) are almost periodic solutions for (3.5), according to Denition 3.2, they are the local-almost periodic solutions for (3.3).

From Denition 2.3, we will give the concept of combinable-almost periodic functions on changing-periodic time scales.

Denition 3.3 ([2]). Let T be a changing-periodic time scale. If there exists an ωi0 -periodic sub-timescale set

{Ti0 }i0∈I with the same direction such that the periods set {ωi0 }i0∈I have a lowest common multiple ω and f is almost periodic on Ti0 for each i0, then f is called a combinable-almost periodic function on T and I is called a combinable index number set. In fact, f is almost periodic on the ω-periodic sub-timescale S . Further, if i0∈I Ti0 S = , then f is called the globally combinable-almost periodic function on . i0∈I Ti0 T T

Remark 3.8. In Denition 3.3, {Ti0 }i0∈I with the same direction implies that, Ti0 are positive-direction periodic time scales for all i0 ∈ I; or Ti0 are negative-direction periodic time scales for all i0 ∈ I. Remark 3.9. Denote I = τt : Tτt are with the same direction and a common periodicity ω for all t ∈ T and S TI := Tτt . One can easily observe that TI is an ω oriented-direction periodic sub-timescale of T and TI is τt∈I dependent on t.

Denition 3.4. We say the following dynamic equation

∆c x (t) = A(t)x(t) + f (t), t ∈ TI (3.9) and its associated homogeneous equation

∆c x (t) = A(t)x(t), t ∈ TI (3.10) are the combinable-almost periodic dynamic equations generated by (3.3) and (3.4), respectively, where ∆c implies the ∆-derivative on the time scale TI. Furthermore, we say all the solutions for (3.9) and (3.10) are the combinable solutions for (3.3) and (3.4), respectively.

Corollary 3.2. Let T be a changing-periodic time scale and f be a combinable-almost periodic function on T, and I be a combinable index number set. If (3.4) admits an exponential dichotomy on the local part Tτt with S Pτt = 0 for all t ∈ T and τ ∈I Tτt is a positive-direction periodic time scale, then (3.3) has a unique combinable- t S almost periodic solution on τ given by τt∈I T t +∞ Z −1 x(t) = X(t)X σc(s) f (s)∆c s. (3.11) t 40 Ë Chao Wang et al.

S Remark 3.10. In Corollary 3.2, if τ ∈I Tτt is a negative-direction periodic time scale, then (3.3) has a unique t S combinable-almost periodic solution on τ given by τt∈I T t

t Z −1 x(t) = X(t)X σc(s) f (s)∆c s. (3.12) −∞ S If τ ∈I Tτt is a bi-direction periodic time scale, then (3.3) has a unique combinable-almost periodic solution on S t τ of the form in Corollary 3.8 in [2]. τt∈I T t

Remark 3.11. In fact, one can observe that (3.11) and (3.12) are almost periodic solutions for (3.9), according to Denition 3.4, they are the combinable-almost periodic solutions for (3.3).

4 Conclusion

In this paper, we introduce the concept of periodic time scales with a translation direction, i.e., Denition 2.3. This paper not only provides some new improvements for periodic time scales but also improves the results for almost periodic time scales. In addition, new results related to changing-periodic time scales are obtained. Also, we make several remarks to clarify our new theory.

5 Further discussion and future work

Recently, Wang and Agarwal have introduced a series of new concepts on time scales. It is necessary to attach directions to periodic and almost periodic time scales during their translations so that some new results can be obtained. We introduce Denition 2.3 in this paper and update some important results on time scales. From Denition 2.3, we can divide an arbitrary time scale with bounded graininess function µ into a countable union of periodic time scales. In our future work, we will study periodicity and almost periodicity for dynamic equations on various types of time scales.

Competing interests: The authors declare that they have no competing interests.

Author’s contributions: All authors contributed equally to the manuscript and typed, read and approved the nal manuscript.

Acknowledgement: This work is supported by Tian Yuan Fund of NSFC (No. 11526181), Yunnan Province Ed- ucation Department Scientic Research Fund Project in China (No. 2014Y008), and Yunnan Province Science and Technology Department Applied Basic Research Project in China (No. 2014FB102).

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