#### Abstract

In this paper, we first propose two types of concepts of almost periodic functions on the quantum time scale. Secondly, we study some basic properties of almost periodic functions on the quantum time scale. Thirdly, based on these, we study the existence and uniqueness of almost periodic solutions of dynamic equations on the quantum time scale by Lyapunov method. Then, we give an equivalent definition of almost periodic functions on the quantum time scale. Finally, as an application, we propose a class of high-order Hopfield neural networks on the quantum time scale and establish the existence and global exponential stability of almost periodic solutions of this class of neural networks.

#### 1. Introduction

The concept of almost periodicity was initiated by the Bohr during the period 1923-1925 [1, 2]. Bohr’s theory quickly attracted the attention of very famous mathematicians of that time. And since then the questions of the theory of almost periodic functions and almost periodic solutions of differential equations have been very interesting and challenge problems of great importance. The interaction between these two theories has enriched both. On one hand, it is well known that, in Celestial Mechanics, almost periodic solutions and stable solutions are intimately related. In the same way, stable electronic circuits exhibit almost periodic behavior. On the other hand, certain problems in differential equations have led to new definitions and results in the theory of almost periodic functions.

In recent years, the theory of quantum calculus has received much attention, due to its tremendous applications in several fields of physics, such as cosmic strings and black holes, conformal quantum mechanics, nuclear and high energy physics, fractional quantum Hall effect, and high- superconductors. For interested reader, we refer to, for example, [3–6] and the references cited therein.

On one hand, recently, Bohner and Chieochan [7] introduced in the literature the concept of periodicity for functions defined on the quantum time scale owing to the fact that taking into account the periodicity of -difference equations is important in order to better understand several physics phenomena. However, in reality, the almost periodic phenomenon is more common and complicated than the periodic one. Therefore, investigating the almost periodicity of dynamic equations on the quantum time scale is more interesting and more challenge.

On the other hand, in order to study the almost periodicity on time scales, a concept of almost periodic time scales was proposed in [8]. Based on this concept, a series of concepts of almost periodic function classes, such as almost periodic functions [8], pseudo almost periodic functions [9], almost automorphic functions [10], weighted pseudo almost automorphic functions [11], almost periodic set-valued functions [12], and almost periodic functions in the sense of Stepanov on time scales [13] were defined successively. Although the concept of almost periodic time scales in [8] can unify the continuous and discrete situations effectively, it is very restrictive because it requires the time scale with certain global additivity. This excludes some interesting time scales, for instance, the quantum time scale, which has no such global additivity.

Motivated by the above discussion, our main purpose of this paper is to propose two types of definitions of almost periodic functions on the quantum time scale, to study some basic properties of almost periodic functions on the quantum time scale, to give an equivalent definition of almost periodic functions on the quantum time scale, and to explore some of their applications to periodic dynamic equations on the quantum time scale.

The organization of this paper is as follows. In Section 2, we introduce some notations and definitions of time scale calculus. In Section 3, we propose the concepts of almost periodic functions on the quantum time scale and investigate some of their basic properties. In Section 4, we study the existence and uniqueness of almost periodic solutions of dynamic equations on the quantum time scale by Lyapunov method. In Section 5, we give an equivalent definition of almost periodic functions on the quantum time scale. In Section 6, as an application of our results, we first propose a class of high-order Hopfield neural networks on the quantum time scale; then by the exponential dichotomy of linear dynamic equations on time scales and the Banach fixed point theorem, we establish the existence and global exponential stability of almost periodic solutions of this class of neural networks. In Section 7, we draw a conclusion.

#### 2. Preliminaries

In this section, we shall recall some basic definitions of time scale calculus.

A time scale is an arbitrary nonempty closed subset of the real numbers, the forward and backward jump operators , and the forward graininess are defined, respectively, by

A point is said to be left-dense if and , right-dense if and , left-scattered if , and right-scattered if . If has a left-scattered maximum , then ; otherwise . If has a right-scattered minimum , then ; otherwise .

*Definition 1 (see [8]). *A time scale is called an almost periodic time scale if

A function is right-dense continuous or rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . If is continuous at each right-dense point and each left-dense point, then is said to be a continuous function on .

For and , then is called delta differentiable at if there exists such that, for given any , there is an open neighborhood of satisfyingfor all . In this case, is called the delta derivative of at and is denoted by . For , we have , the usual derivative, for , we have the backward difference operator, , and, for , the quantum time scale, we have the -derivative:

*Remark 2. *Note thatif is continuously differentiable.

Define for , to mean that, given , there exists a right neighborhood such thatfor each , , where . If is right-scattered and is continuous at ; this reduces to

Lemma 3 (see [14]). *Let and ; thenimplieswhere .*

For more details about the theory of time scale calculus and the theory of quantum calculus, the reader may want to consult [3, 14, 15].

#### 3. Almost Periodic Functions

From now on, we use to denote the complex numbers or the real numbers , use to denote an open set in or , and use to denote an arbitrary compact subset of .

*Definition 4. *A function is called an almost periodic function in uniformly in if for any given and each compact subset of ; there exists such that each interval contains a such thatThis is called the -translation number of and is called the inclusion length of the set

*Definition 5. *A function is called an almost periodic function in uniformly in if, for any given and each compact subset of , there exists such that each interval contains a such that This is called the -translation number of and is called the inclusion length of the set

*Definition 6. *A function is called an almost periodic function if, for any given , there exists such that each interval contains a such thatThis is called the -translation number of and is called the inclusion length of the set

*Definition 7. *A function is called an almost periodic function if, for any given , there exists such that each interval contains a such that This is called the -translation number of and is called the inclusion length of the set

*Definition 8. *A function is called an asymptotically almost periodic function ifwhere is an almost periodic function on , and as .

For convenience, we introduce some notations.

Let is bounded on , is almost periodic in uniformly in and .

Let and be two sequences of integer numbers. We denote and use to denote that is a subsequence of . We say and are common subsequences of and , respectively, if there is some given function such that and .

We introduce two translation operators and , and mean that and , respectively, and are written only when the limits exist. The mode of convergence will be specified at each use of the symbols.

*Remark 9. *When study the properties of almost periodic functions defined by Definitions 4 and 6, we use the translation operator . When studying the properties of almost periodic functions defined by Definitions 5 and 7, we use the translation operator .

*Remark 10. *All the results of this section hold for almost periodic functions defined by Definitions 4, 5, 6, and 7. Since their proofs are similar, we only prove them under Definitions 4 and 6.

*Remark 11. *A function is continuous if and only if exists uniformly in .

Theorem 12. *Let ; then it is uniformly continuous and bounded on .*

*Proof. *For a given and some compact subset , there exists such that, in each interval , there exists a such thatIt follows from that, for any , there exists such that . For any given and , take ; then . Hence, for , we getTherefore, we obtainFurthermore, because of the continuity of at , for any given , there exists a positive integer such that for all and . Thus, for any , we findtherefore, is uniformly continuous on . Since all the points in the interval are isolated points and for all , so is uniformly continuous on . The proof is completed.

Theorem 13. *Let ; then, for any given sequence , there exist a subsequence and such that holds uniformly on and .*

*Proof. *For any given and sequence , we denote , where , , and . Hence, it is easy to see that there exists a subsequence such that for , We can take , such that common with and for all ; thenwhich implies thatHence, we can getThus, we can take subsequences , and such that, for any integers , and all ,For all sequences , we can take a sequence , ; then it is easy to see that, for any integers with and all ,Therefore, converges uniformly on ; that is, holds uniformly on .

Next, we show that . Otherwise, there exists a point such that is not continuous at this point. Then there exist and sequences , , , where , as , andLetting , obviously, is a compact subset of . Since holds uniformly on , there exists a positive integer such that, for ,andAccording to the uniform continuity of on , for sufficiently large , we haveFrom (25)-(27), we getwhich contradicts (24). Therefore, is continuous on .

Finally, for any compact subset and given , we take ; thenLetting , we havewhich implies that . This completes the proof.

Theorem 14. *Letting , if, for any sequence , there exists a subsequence such that exists uniformly on ; then .*

*Proof (proof by contradiction). *If , then there exist and such that, for no matter how large , we can always find an interval such that

To this end, we take a number and find an interval with and such that . Next, we take , and it is easy to see that , and so ; then we find an interval with and such that . Next, we take , obviously, . We can repeat these process again and again and can find , such that . Thus, for any , without loss of generality, let ; we find which implies that there is no uniformly convergent subsequence of for . This is a contradiction. Hence, . This completes the proof.

In view of Definitions 4 and 5, by Theorems 13 and 14, one can easily get the following two results.

Theorem 15. *A function is almost periodic in uniformly in , if and only if, for any given sequence , there exists a subsequence such that exists uniformly on *

Theorem 16. *A function is almost periodic function in uniformly in , if and only if, for any given sequence , there exists a subsequence such that exists uniformly on *

Theorem 17. *If and with for all . Then .*

*Proof. *It is easy to see that for any sequence , and there exist a subsequence , and functions and such that exists uniformly on and exists uniformly on , where and . Hence, is uniformly continuous on ; then, for any given , there exists such that, for any and , we haveIn addition, there exists an such that, for ,andwhere for all Therefore, for ,which implies that exists uniformly on . Thus, . The proof is complete.

*Definition 18. *Let ; then there exits such that existing uniformly on is called the hull of .

Theorem 19. * is compact in uniform norm if and only if .*

*Proof. *If is a compact set, then, for each sequence , there exists a subsequence such that exists uniformly on .

Conversely, if and , then we can choose such thatSo, we can find a subsequence such that exists uniformly on . Let such that and are common subsequences; thenhencetherefore, is a compact set. The proof is complete.

Theorem 20. *If , then, for any , .*

*Proof. *For any , there exists such that exists uniformly on . Since , one can extract a subsequence such that exists uniformly on . Because , so there exists such thathence,then, we can take such thatIt follows from that . Thus, .

On the other hand, for any , there exists such that exists uniformly on ; hence,Let ; we obtainthat is, exists uniformly on . Thus, . Therefore, . The proof is complete.

From Definition 18 and Theorem 20, one can easily show the following.

Theorem 21. *If , then, for any , .*

Theorem 22. *If , then we have the following:**(a) .**(b) If , then .*

*Proof. *() Since , from any sequence one can extract a subsequence such that exists uniformly on . Then from the we can extract a subsequence such that exists uniformly on . Consequently, we can extract a subsequence such that and exists uniformly on . That is, .

Because , so, for any given and each compact subset of , there exists such that each interval contains a such thatNowfor all . That is, . Now using () it follows that . The proof is complete.

Theorem 23. *If and the sequence converges uniformly to on , then .*

*Proof. *For any , there exists sufficiently large such that for all ,Take ; then, for all , we have that is, . Therefore, . This completes the proof.

Theorem 24. *Let and , then if and only if is bounded on .*

*Proof. *If , then is bounded on .

If is bounded, without loss of generality, we can assume that is a real-valued function. Denote then, for any , there exist and such that Let be an inclusion length of , where , . For any , take . Let and ; then . For all , we havethat is,Since we have proved that, in any interval , there exist and such thatNow, let ; we will show that if , then . In fact, for any , we can take such that thus, for , we have