A Further Study of Almost Periodic Time Scales with Some Notes and Applications

and Applied Analysis 3 examples and applications will be shown in which our results can be applied to iron out the flaws of the proposed definition in [1]. We give some notations; E denote R or C, D denotes an open set in E or D = E, and S denotes an arbitrary compact subset ofD. Definition 3 (see [1]). 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 ε-translation set of f E {ε, f, S} = {τ ∈ Π : 󵄨 󵄨 󵄨 󵄨 f (t + τ, x) − f (t, x) 󵄨 󵄨 󵄨 󵄨 < ε, ∀ (t, x) ∈ T × S} (10) is 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 ofD, 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 󵄨 󵄨 󵄨 󵄨 f (t + τ, x) − f (t, x) 󵄨 󵄨 󵄨 󵄨 < ε, ∀ (t, x) ∈ T × S. (11) τ is called the ε-translation number of f and and l(ε) is called the inclusion length of E{ε, f, S}. For convenience, we denote AP(T) = {f ∈ C(T ,E) : f is almost periodic} and introduce some notations: let α = {α n } and β = {β n } be two sequences. Then β ⊂ α means that β is a subsequence of α; α + β = {α n + β n }, −α = {−α n }, and α and β are common subsequences of α and β, respectively, means that α n = α 󸀠 n(k) and β n = β 󸀠 n(k) for some given function n(k). We will introduce the translation operator T; T α f(t, x) = g(t, x) means that g(t, x) = lim n→+∞ f(t + α n , x) and is written only when the limit exists. Definition 4 (see [1]). Let f(t, x) ∈ C(T × D,E), if, for any given sequence α ⊂ Π, there exists a subsequence α ⊂ α such that T α f(t, x) exists uniformly on T × S; then, f(t, x) is called an almost periodic function in t uniformly for x ∈ D. However, in [15], the authors propose the definition of periodic time scales and give the remark as follows. Definition 5 (see [15]). One can say that a time scale T is periodic if there exists p > 0 such that if t ∈ T then t ± p ∈ T . For T ̸ = R, the smallest positive p is called the period of the time scale. Remark 6 (see [15]). If T is a periodic time scale with period p, then σ(t + np) = σ(t) + np. Consequently, the graininess function μ satisfies μ(t+np) = σ(t+np)−(t+np) = σ(t)−t = μ(t) and so it is a periodic function with period p. Note that Definitions 3 and 4 are proposed based on the set Π. Although Definition 1 is exactly like Definition 5 since ∀τ ∈ Π, one has t ± τ ∈ T . In order to clarify some theoretical ambiguities between periodic time scales in [15] and almost periodic time scales in [1], in the following, we will propose a more general and accurate concept of almost periodic time scales instead of Definition 1 and give some examples of time scales which are almost periodic but not periodic. Let τ be a number. We set the time scales as follows:


Introduction
Almost periodicity is a recent concept in the literature of time scales.It was formally introduced by Li and Wang in [1,2], and based on this, some results concerning almost periodicity for dynamic equations on time scales were proved and a series of relative applications were published (see [3][4][5][6]).Meanwhile, some mathematicians are interested in this subject, and some relative works appeared (see [7][8][9][10][11][12]).
As everyone knows, the almost periodic time scales play a very important and fundamental role in redefining some classical functions on time scales such as almost periodic functions [1], pseudo almost periodic functions [6] and almost automorphic functions [7], and even weighted pseudo almost automorphic functions [3].In [13], by using the concept and properties of almost periodic time scales, Lizama et al. prove a strong connection between almost periodic functions on time scales and almost periodic functions on R and then give an application to difference equations on T = ℎZ.Besides, some works have been done under the concept of almost periodic time scales; see [7,13,14].
However, some mathematicians find that the concept of almost periodic time scales in [1] is exactly like the concept of periodic time scales in [15].Furthermore, in Section 3 of [7], indeed, all invariant under translations time scales are periodic time scales, that is, from Example 3.9 to Example 3.11, which indicate that we investigated almost periodic problems of dynamic equations under the periodic time scales in the past, and all the obtained results are valid for all periodic time scales, particularly, for two special periodic time scales: T = R and T = Z.Although this method can unify the continuous and discrete situations effectively, whether or not there exists a time scale which is almost periodic but not periodic if we introduce a new concept of almost periodic time scales.Therefore, it is very necessary to investigate the almost periodic time scales and introduce a more general and accurate definition that can strictly include all periodic time scales to overcome some difficulties in this research field.
It is known to all that the Cauchy matrix is very important in the research of dynamic equations.However, by using the almost periodicity of Cauchy matrix to discuss almost periodic problems of dynamic equations, we will encounter a problem.Let (, ) be the Cauchy matrix of the following dynamic equations: and the question then arises: for any  > 0, whether or not the -almost period  of the matrix function () is valid such that the following inequality holds: ‖ ( + ,  ( + )) −  (,  ())‖ ≤ Γ 0  ⊖ (,  ()) ,  ≥ , if the Cauchy matrix satisfies the inequality: ‖ (, )‖ ≤  ⊖ (, ) ,  ≥ , where , Γ 0 ,  are positive constants.As everyone knows, if T is a -periodic time scale, then  ( + ) =  () + , so (4) will turn into ‖ ( + ,  () + ) −  (,  ())‖ ≤ Γ 0  ⊖ (,  ()) ,  ≥ , which seems too special even though its validity can be shown on all periodic time scales, particularly on T = R and T = Z.Nevertheless, if we can introduce a new concept of almost periodic time scales which strictly includes the periodic time scales such that (4), rather than (7), is valid under the condition (5), that is, the almost periodicity of Cauchy matrix can be guaranteed without considering (6) on this kind of general time scales, then the almost periodicity of (3) can easily be shown under (4).Motivated by the above, the almost periodic time scales need a further study since the concept proposed in the past [1] is actually periodic, which will lead to some research difficulties and specificity of the obtained results.In this paper, we will introduce three equivalent concepts of almost periodic time scales as a revision of the corresponding concept proposed in [1], and several examples of almost periodic time scales which are not periodic are provided.Furthermore, the concepts of almost periodic functions are redefined under the sense of this new timescale concept.
The present paper is organized as follows.In Section 2, we will introduce three equivalent concepts of almost periodic time scales and give some key notes; then, the concepts of almost periodic functions are redefined under the sense of this new timescale definition.Furthermore, several examples of almost periodic time scales which are not periodic are provided.In Section 3, the almost periodicity of Cauchy matrix is analyzed under these new definitions; then, the almost periodicity of (3) is easily shown under the condition (5).In Section 4, our results are applied to investigate the existence of almost periodic solutions to a class of high-order Hopfield neural networks on time scales.In Section 5, we conduct a further discussion of almost periodic time scales, on which the concept of almost automorphic functions is introduced, and some relative works will appear in our future research.
It is worth noting that the three new equivalent definitions of almost periodic time scales proposed in this paper will play an important role in analyzing almost periodicity, pseudo almost periodicity, and weighted pseudo almost periodicity of Cauchy matrix for dynamic equations on time scales.All results obtained in [1] and their proof processes are valid under these new concepts without considering the set Π, which will be referred to in the next section.

A Further Study of Almost Periodic Time Scales and Some Notes
where ,  ∈ T with  < ().For more knowledge of time scales, one can see [15][16][17][18].Firstly, we recall the concept of almost periodic time scales in [1].
Definition 1 (see [1]).A time scale T is called an almost periodic time scale if Remark 2. The concept of almost periodic time scales proposed in [1] was cited by [7] to introduce the definition of almost automorphic functions on time scales which can be applied to study almost automorphic solutions of dynamic equations on time scales, and T is also called invariant time scale under translations in Definition 3.1 of [7].In fact, we find that Definition 1 is equivalent to Definition 1.1 proposed in [15]; that is, the almost periodic time scale T proposed in [1] is a periodic time scale.In this paper, we will give three more general and accurate equivalent concepts of almost periodic time scales and redefine the concepts of almost periodic functions on this new time scale concept.Furthermore, some examples and applications will be shown in which our results can be applied to iron out the flaws of the proposed definition in [1].
We give some notations; E  denote R  or C  ,  denotes an open set in E  or  = E  , and  denotes an arbitrary compact subset of .
Definition 3 (see [1]).Let is called the -translation number of  and and () is called the inclusion length of {, , }.
However, in [15], the authors propose the definition of periodic time scales and give the remark as follows.
Definition 5 (see [15]).One can say that a time scale T is periodic if there exists  > 0 such that if  ∈ T then  ±  ∈ T. For T ̸ = R, the smallest positive  is called the period of the time scale.
Note that Definitions 3 and 4 are proposed based on the set Π.Although Definition 1 is exactly like Definition 5 since ∀ ∈ Π, one has  ±  ∈ T. In order to clarify some theoretical ambiguities between periodic time scales in [15] and almost periodic time scales in [1], in the following, we will propose a more general and accurate concept of almost periodic time scales instead of Definition 1 and give some examples of time scales which are almost periodic but not periodic.
Let  be a number.We set the time scales as follows: T  := T +  = { +  : ∀ ∈ T} : Define the distance between two time scales, T and T  , by Definition 7 (see [19]).A subset  of R is called relatively dense if there exists a positive number  such that [,  + ] ∩  ̸ = 0 for all  ∈ R. The number  is called the inclusion length.
Definition 8. We say T is an almost periodic time scale if, for any give  > 0, there exists a constant () > 0 such that each interval of length () contains  () such that that is, for any  > 0, the following set is relatively dense. is called the -translation number of T and () is called the inclusion length of {T, }, and {T, } is called the -translation set of T.

Remark 9.
From Definition 1, one can easily see that if Π ̸ = 0, then for any  > 0, there exists a constant () > 0 such that each interval of length () contains a () ∈ Π such that Therefore, Definition 8 includes Definition 1. Particularly, it is worth emphasising that () in Definition 8 need not satisfy  ± () ∈ T for all  ∈ T.
Remark 10.According to Definition 8, one can obtain that sup T = +∞, inf T = −∞, and Furthermore, in Definition 8, one can see that if (T, T  ) < , then (T, Theorem 11.Let T be an almost periodic time scale.Then for any given sequence   , there exists a subsequence  ⊂   such that {T   } converges to a time scale T 0 ; that is, for any given  > 0, there exists  0 > 0 such that  >  0 implies (T   , T 0 ) < .Furthermore, T 0 is also almost periodic.
Finally, for any given  > 0, one can take  ∈ {T, }; then, the following holds: Letting  → +∞, we have which implies that {T 0 , } is relatively dense.Therefore, T 0 is almost periodic.This completes the proof.
Theorem 12. Let T be a time scale, if, for any sequence   , there exists  ⊂   such that {T   } converges to a time scale T 0 , then T is almost periodic.
Proof.For contradiction, if this is not true, then there exists  0 > 0 such that for any sufficiently large  > 0, we can find an interval with length of  and there is no  0 -translation numbers of T in this interval; that is, every point in this interval is not in {T,  0 }.One can take a number   1 and find an interval Therefore, there is no convergent subsequence of {T    }, a contradiction.Hence, T is almost periodic.This completes the proof.
From Theorems 11 and 12, we can obtain the following equivalent definition of almost periodic time scales.Definition 13.Let T be a time scale, and if, for any given sequence   , there exists a subsequence  ⊂   such that {T   } converges to a time scale T 0 , then T is called an almost periodic time scale.
In the sequel, based on Definitions 8 and 13, we will give the two equivalent concepts of almost periodic functions on time scales.
Therefore, we can also substitute Thus, T is almost periodic.Obviously, if  = 0,  = 1, then (2) In this point, we will show some examples of time scales which are almost periodic but not periodic.
Example 27.Let  > 1 and consider the the following time scale: where ) . (37) Then, we have One can see that this kind of time scale has the graininess function  which is an almost periodic function, and by Definition 21, T is an almost periodic time scale.It is worth noting that there is not any  ∈ R such that  ±  ∈ T for all  ∈ T; thus, T is not a periodic time scale by Definitions 1 or 5.
Example 28.Let  > 1 and consider the the following time scale: where ) . (40) Then, we have We see that this kind of time scale has the graininess function  which is an almost periodic function, and by Definition 21, T is an almost periodic time scale.It is worth noting that there is not any  ∈ R such that  ±  ∈ T for all  ∈ T; thus, T is not a periodic time scale by Definitions 1 or 5.
Example 29.Let  > 1 and consider the the following time scale: where Then, we have One can see that this kind of time scale has the graininess function  which is an almost periodic function, and by Definition 21, T is an almost periodic time scale.It is worth noting that there is not any  ∈ R such that  ±  ∈ T for all  ∈ T; thus, T is not a periodic time scale by Definitions 1 or 5.
Example 30.Let  > 1 and consider the the following time scale: Definition 37 (see [16]).Let  : T → R be a function and let  = ( 1 ,  2 , . . .,   ) ∈ T  .Then define  Δ  () to be the number (provided it exists) with the property that given any  > 0 there exists a neighborhood  of   with  = (  −,   +)∩T  for  > 0 such that Δ  is called the partial delta derivative of  at  with respect to the variable   .
Theorem 38.For system (1), let the matrix  ∈ (T, R × ) be almost periodic.If the Cauchy matrix (, ) satisfies the inequality where  and  are positive real numbers and  is positive regressive, then the diagonal of the matrix (, ) is almost periodic; that is, for any  > 0, there exists a relatively dense set Γ of almost periods such that, for  ∈ Γ, we have where Γ 0 is a positive constant. Proof.Since where  = sup ∈T (); then, we can get where Γ 0 =  2 (1 + )/.This completes the proof.
We can prove the following theorem exactly like Theorem 38 if we let (, ) =   (, ), so we give it straightly.
Theorem 39.For any  > 0,  ∈ R + is positive regressive and  is almost periodic; then, there exists a relatively dense set Γ of almost periods such that, for  ∈ Γ, we have This completes the proof.
We can prove the following theorem exactly like Lemma 40, so we give it straightly.
Hence,  has a fixed point in D; that is, (2) has a unique almost periodic solution.This completes the proof.

An Application
In the following, we present a result which can be found in [[2, Lemma 2.15] which will be essential to our purposes.
Remark 44.By Remark 36, it is easy to see that one can take (, ()) = () −1 (()) and there exist positive constants ,  such that That is to say, the inequality (57) in Theorem 38 holds.
Consider the following high-order Hopfield neural networks on time scales: for  = 1, 2, . . ., , where  corresponds to the number of units in a neural network,   () corresponds to the state vector of the th unite at the time ,   () represents the rate with which the th unite will reset its potential to the resting state in isolation when disconnected from the network external inputs,   () and   () are the first-and secondorder connection weights of neural network,   () denotes the external inputs at time , and   and   are the activation functions of signal transmission.Now, we assume the following conditions are fulfilled. ( Then, by hypotheses ( 1 ), ( 2 ), ( 3 ), and ( 4 ) and using Lemma 43 and Remark 44, we obtain that all hypotheses of Theorem 42 are satisfied; then, the system (82) possesses a unique almost periodic solution.

Conclusion and Further Discussion
In this paper, we introduce three equivalent concepts of almost periodic time scales which can strictly include the concept of periodic time scales.Several examples are given to show that there exists a class of time scales which is almost periodic but not periodic according to the new proposed definitions.Furthermore, all the results obtained in [1] are valid without considering the set Π, which will bring more generality of the obtained results in our relative previous works.Furthermore, using the almost periodicity of Cauchy matrix for dynamic equations on time scales, by fixed point theorems in Banach space, one can find some new sufficient conditions for the existence of almost periodic solutions for dynamic equations under the sense of these new definitions.Finally, all new concepts proposed in this paper will play an important and fundamental role in establishing almost periodic theory of dynamic equations on time scales.Furthermore, according to Definition 13, one can introduce the concept of almost automorphic functions as follows.
Definition 45.Let X be a Banach space and let T be an almost periodic time scale.
(i) Let  : T → X be a bounded continuous function.
We say that  is almost automorphic if, for every sequence of real numbers {  } ∞ =1 , we can extract a subsequence {  } ∞ =1 such that the limit set T 0 of {T −  } exists and: is well defined for each  ∈ T 0 .Furthermore, the limit set of {T (ii) A continuous function  : T ×  → X is said to be almost automorphic if (, ) is almost automorphic in  ∈ T uniformly for all  ∈ , where  is any bounded subset of X or  = X.Denote by (T × X, X) the set of all such functions.
Under Definition 45, all the obtained results and proof process in [3] are valid and this new kind of time scales proposed in this paper will bring more general sense to our future research works.

Theorem 18 .Remark 19 .
that is, Definition 16 strictly includes Definition 4. From Definition 8 and the definition of the graininess function , one can have the following.If T is an almost periodic time scale, then for any  > 0 there exists a constant () > 0 such that each interval of length () contains  () ∈ {, } such that      ( + ) −  ()     < , ∀ ∈ T ∩ T − .(31) The inequality (31) can also be written as | ( + ) −  () − | < , ∀ ∈ T ∩ T − , (32) which indicates that if T is -periodic, we have ( + ) = () + ; then, T is an almost periodic time scale.Remark 20.Conversely, if the graininess function  is an almost periodic function, from the definition of the function  : T → R, that is, () = () − , one can obviously see that there must exist at least   ∈ {, } in the each interval of length () such that (T, T − ) < .Therefore, we can easily get that T is an almost periodic time scale by Definition 8.According to this, in the following, we will introduce the third definition of almost periodic time scales which is equivalent to Definition 8. Now, we give the third concept of almost periodic time scales by the graininess function  as follows.Definition 21.Let  : T → R, () = () − .One can say that T is an almost periodic time scale if, for any  > 0, the set Π * = { ∈ R :      ( + ) −  ()     < , ∀ ∈ T ∩ T − } (33) is relatively dense; that is,  is an almost periodic function on T. By Theorem 18 and Definition 21, we can get the following corollaries.Corollary 22.If T ̸ = R and T is a periodic time scale, then T has the smallest positive period  and the graininess function  is a periodic function with period .Corollary 23.All periodic time scales are almost periodic.Corollary 24.T is an -periodic time scale if and only if the graininess function  : T → R + is a -periodic function.Next, we will show some examples of almost periodic time scales.Example 25.If T = ⋃ ∈Z [(+), (+)+], where  ̸ = −, then
. T is a periodic time scale if and only if  : T → R + is a periodic function.Hence, from the graininess function , we can see that these two time scales are different and we will show some examples in the next point.