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

Abstract

We introduce three equivalent concepts of almost periodic time scales as a further study of the corresponding concept proposed in Li and Wang (2011) 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. Finally, almost periodicity of Cauchy matrix for dynamic equations is proved under these new definitions. Based on these results, the existence of almost periodic solutions to a class of nonlinear dynamic equations is investigated by the almost periodicity of Cauchy matrix on almost periodic time scales. Besides, as an application, we apply our results to a class of high-order Hopfield neural networks.

1. 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 [3–6]). Meanwhile, some mathematicians are interested in this subject, and some relative works appeared (see [7–12]).

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 and then give an application to difference equations on . 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: and . 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: where is an almost periodic matrix-valued function and . denotes the set of all continuous functions from to the Banach space . Consider the following nonlinear dynamic equations: where is an almost periodic matrix-valued function and , and is almost periodic in uniformly for . By the Cauchy matrix of (1), in this paper, we can get a bounded solution of (2) as follows: and the question then arises: for any , whether or not the -almost period of the matrix function is valid such that the following inequality holds: if the Cauchy matrix satisfies the inequality: where , , are positive constants. As everyone knows, if is a -periodic time scale, then so (4) will turn into which seems too special even though its validity can be shown on all periodic time scales, particularly on and . 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 , which will be referred to in the next section.

2. A Further Study of Almost Periodic Time Scales and Some Notes

A time scale is a closed subset of . It follows that the jump operators defined by and (supplemented by and ) are well defined. The point is left-dense, left-scattered, right-dense, and right-scattered if , , , and , respectively. If has a right-scattered minimum , define ; otherwise, set . For the notations , and so on, we will denote time scale intervals where with . For more knowledge of time scales, one can see [15–18].

Firstly, we recall the concept of almost periodic time scales in [1].

Definition 1 (see [1]). A time scale 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 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 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; denote or , denotes an open set in or , and denotes an arbitrary compact subset of .

Definition 3 (see [1]). Let be an almost periodic time scale. A function is called an almost periodic function in uniformly for if the -translation set of is relatively dense set in for all and for each compact subset of ; that is, for any given and each compact subset of , there exists a constant such that each interval of length contains such that is called the -translation number of and and is called the inclusion length of .

For convenience, we denote and introduce some notations: let and be two sequences. Then means that is a subsequence of ; , , and and are common subsequences of and , respectively, means that and for some given function .

We will introduce the translation operator ; means that and is written only when the limit exists.

Definition 4 (see [1]). Let , if, for any given sequence , there exists subsequence such that exists uniformly on ; then, is called an almost periodic function in uniformly for .

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 is periodic if there exists such that if then . For , the smallest positive is called the period of the time scale.

Remark 6 (see [15]). If is a periodic time scale with period , then . Consequently, the graininess function satisfies and so it is a periodic function with period .

Note that Definitions 3 and 4 are proposed based on the set . Although Definition 1 is exactly like Definition 5 since , one has . 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:

Define the distance between two time scales, and , by

Definition 7 (see [19]). A subset of is called relatively dense if there exists a positive number such that for all . The number is called the inclusion length.

Definition 8. We say is an almost periodic time scale if, for any give , there exists a constant such that each interval of length contains such that that is, for any , the following set is relatively dense. is called the -translation number of and is called the inclusion length of , and is called the -translation set of .

Remark 9. From Definition 1, one can easily see that if , then for any , there exists a constant such that each interval of length contains a such that Therefore, Definition 8 includes Definition 1. Particularly, it is worth emphasising that in Definition 8 need not satisfy for all .

Remark 10. According to Definition 8, one can obtain that , , and Furthermore, in Definition 8, one can see that if , then ; that is, if , then . If , , then we have since

Theorem 11. Let be an almost periodic time scale. Then for any given sequence , there exists a subsequence such that converges to a time scale ; that is, for any given , there exists such that implies . Furthermore, is also almost periodic.

Proof. For any . Let be an inclusion length of . For any given subsequence , we denote , where and , . Therefore, there exists a subsequence such that as , .
Also, it is easy to see that there exists so that implies
Since is a convergent sequence, there exists so that implies . Now, one can take , such that , are common with , and then for any integers , we have that is, Hence, we can obtain Thus, we can take sequences ,  , and such that for any integers , the following holds: For all sequences , , we can take a sequence , , and then it is easy to see that for any integers , with the following holds: Therefore, converges to some which is a closed subset of ; that is, as .
Finally, for any given , one can take ; then, the following holds: Letting , we have which implies that is relatively dense. Therefore, is almost periodic. This completes the proof.

Theorem 12. Let be a time scale, if, for any sequence , there exists such that converges to a time scale , then is almost periodic.

Proof. For contradiction, if this is not true, then there exists such that for any sufficiently large , we can find an interval with length of and there is no -translation numbers of in this interval; that is, every point in this interval is not in .
One can take a number and find an interval with , where , satisfy that there is no -translation numbers of in the interval (, ). Next, taking , obviously, , so ; then, one can find an interval with , where , satisfy that there is no -translation numbers of in the interval (, ). Next, taking , obviously, , . One can repeat these processes again and again and find , such that , . Hence, for any ,  , without loss of generality, letting , we have Therefore, there is no convergent subsequence of , a contradiction. Hence, 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 be a time scale, and if, for any given sequence , there exists a subsequence such that converges to a time scale , then 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.

Definition 14. Let be an almost periodic time scale. A function is called an almost periodic function in uniformly for if the -translation set of is a relatively dense set for all and for each compact subset of ; that is, for any given and each compact subset of , there exists a constant such that each interval of length contains a such that is called the -translation number of and is called the inclusion length of .

Remark 15. From Definition 14, one can easily see that if is a periodic time scale, that is, satisfies Definition 1, then in Definition 14, we have . Hence, Definition 14 strictly includes Definition 3.

Definition 16. Assume that is an almost periodic time scale. Let , if for any given sequence , there exists a subsequence such that the limit set of exists and exists uniformly on , then is called an almost periodic function in uniformly for .

Remark 17. Noting that is almost periodic, according to Definition 13, we have Therefore, we can also substitute or for in Definition 16. Furthermore, one can see that if is periodic, then we have , that is, Definition 16 strictly includes Definition 4.

From Definition 8 and the definition of the graininess function , one can have the following.

Theorem 18. If is an almost periodic time scale, then for any there exists a constant such that each interval of length contains such that

Remark 19. The inequality (31) can also be written as which indicates that if is -periodic, we have ; then, is an almost periodic time scale.

Remark 20. Conversely, if the graininess function is an almost periodic function, from the definition of the function , that is, , one can obviously see that there must exist at least in the each interval of length such that . Therefore, we can easily get that 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 , . One can say that is an almost periodic time scale if, for any , the set is relatively dense; that is, is an almost periodic function on .

By Theorem 18 and Definition 21, we can get the following corollaries.

Corollary 22. If and is a periodic time scale, then 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. is an -periodic time scale if and only if the graininess function is a -periodic function.

Next, we will show some examples of almost periodic time scales.

Example 25. If , where , then Thus, is almost periodic. Obviously, if , , then . if , then .

Remark 26. One can easily see that Example 25 is a periodic time scale with periodicity , and by Corollary 23, it is also an almost periodic time scale.(1)As everyone knows, the graininess function defined by  can describe the construction of a time scale. From Definition 21, one can see that is an almost periodic function if and only if is an almost periodic time scale; on the other hand, from Definition 5 and Corollary 24, is a periodic time scale if and only if 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.(2)In this point, we will show some examples of time scales which are almost periodic but not periodic.

Example 27. Let 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, is an almost periodic time scale. It is worth noting that there is not any such that for all ; thus, is not a periodic time scale by Definitions 1 or 5.

Example 28. Let and consider the the following time scale: where 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, is an almost periodic time scale. It is worth noting that there is not any such that for all ; thus, is not a periodic time scale by Definitions 1 or 5.

Example 29. Let 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, is an almost periodic time scale. It is worth noting that there is not any such that for all ; thus, is not a periodic time scale by Definitions 1 or 5.

Example 30. Let and consider the the following time scale: where 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, is an almost periodic time scale. It is worth noting that there is not any such that for all ; thus, is not a periodic time scale by Definition 1 or Definition 5.

Remark 31. From Example 27 to Example 30, there is not any such that for all . Therefore, all examples show that the concepts of almost periodic time scales proposed in this paper strictly include all periodic time scales and they are more general and accurate.

By Definitions 8 and 21, we can give the following sufficient and necessary condition to guarantee that is almost periodic.

Theorem 32. Let be a time scale and is almost periodic if and only if is relatively dense in ; that is, is an almost periodic function on .

Corollary 33. The time scales are invariant under translations if and only if is periodic.

Remark 34. The Examples 3.7, 3.8, 3.9, 3.10, and 3.11 in [7] are invariant time scales under translations, and obviously, all of them are periodic time scales.

3. Cauchy Matrix for Dynamic Equations

In this section, by the new concepts proposed in Section 2, we will prove some useful theorems and lemmas of Cauchy matrix of (1), and using these results, we can obtain the existence and uniqueness of almost periodic solutions of (2) straightly. These theorems and lemmas can be applied to study almost periodic solutions of many other types of mathematical models on time scales.

Theorem 35. Let be a fundamental matrix of system (1). Then, for , every solution of system (2) is given by In particular, if is a Cauchy matrix of system (1), then, for , any solution of (2) with initial condition can be written as

Proof. Since is a nonsingular matrix and -differentiable. Then, under the linear change of variables, then, system (2) turns into the following: From (51) we can find that, for , where is a constant vector; that is, then, we can get (48) and (49). This completes the proof.

Remark 36. By Theorem 35, one can easily check that (2) has a bounded solution as follows: Noting that if (1) admits an exponential dichotomy, then we can take the projection in Lemma 2.13 of [2] and to get this bounded solution. Furthermore, by the concept of exponential dichotomies on time scales, one can easily see that there exist , such that where is the fundamental solution matrix of (1); that is, (5) holds.

Definition 37 (see [16]). Let be a function and let . Then define to be the number (provided it exists) with the property that given any there exists a neighborhood of with for such that is called the partial delta derivative of at with respect to the variable .

Theorem 38. For system (1), let the matrix 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 , there exists a relatively dense set of almost periods such that, for , we have where is a positive constant.

Proof. Since we have Further we have where ; then, we can get where . 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 , is positive regressive and is almost periodic; then, there exists a relatively dense set of almost periods such that, for , we have where is a positive constant.

Lemma 40. Let be regressive and let be almost periodic; then, for any , there exists such that implies

Proof. For any , we have Since is almost periodic time scale, is bounded on . Denoting that we can take ; then, This completes the proof.

We can prove the following theorem exactly like Lemma 40, so we give it straightly.

Theorem 41. Let be the Cauchy matrix of (1) and is regressive. If is continuous on , then, for any , there exists such that implies If satisfies then .

Using the above results, one can show the following theorem.

Theorem 42. If is almost periodic in uniformly for and the Cauchy matrix of (1) satisfies (57), is continuous on and Then (2) has a unique continuous almost periodic solution as follows:

Proof. Let be the space formed by all almost periodic functions on an almost periodic time scale . Define an operator : Consider the following difference: where . Noting that , that is, , by Lemma 40, we have then, from (74) and Theorem 39, we have and since is continuous on , by Theorems 38, 39, and 41, one has where .
By (73), we can obtain Hence, is almost periodic. Now, if , then we can obtain Since , then is a contraction mapping. Hence, has a fixed point in ; that is, (2) has a unique almost periodic solution. This completes the proof.

4. 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.

Lemma 43 (see [2]). Let be an almost periodic function on , where , , , and then, the linear system admits an exponential dichotomy on .

Remark 44. By Remark 36, it is easy to see that one can take 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 , 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 second-order 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.(), , , are almost periodic functions, and for every .()There exist positive constants such that and for , .()Functions , , satisfy the Lipschitz condition; that is, there exist constants , such that , , .() where Then, by hypotheses , , , and 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.

5. 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 , 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 be a Banach space and let be an almost periodic time scale. (i)Let be a bounded continuous function. We say that is almost automorphic if, for every sequence of real numbers , we can extract a subsequence such that the limit set of exists and:  is well defined for each . Furthermore, the limit set of is and  for each . Denote by the set of all such functions.(ii)A continuous function is said to be almost automorphic if is almost automorphic in uniformly for all , where is any bounded subset of or . Denote by 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.

Remark 46. In Definition 45, if , then . Similarly, if , then . Hence, the (i) of Definition 45 makes sense.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally to the paper and typed, read, and approved the final paper.

Acknowledgments

The authors would like to express their sincere thanks to the editor Professor Dr. Elena Braverman for handling our paper during the reviewing process and to the referees for suggesting some corrections that help making the content of the paper more accurate. This work is supported by the foundation of Yunnan University in China (no. 2013CG020).