Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 1526478, 9 pages
https://doi.org/10.1155/2017/1526478

Research Article

Almost Automorphic Functions on the Quantum Time Scale and Applications

Yongkun Li

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Correspondence should be addressed to Yongkun Li; yklie@ynu.edu.cn

Received 28 September 2017; Accepted 23 November 2017; Published 14 December 2017

Academic Editor: Douglas R. Anderson

Copyright © 2017 Yongkun Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers; by using this transformation we give equivalent definitions of almost automorphic functions on the quantum time scale; following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale. Finally, as an application of our results, we establish the existence of almost automorphic solutions of linear and semilinear dynamic equations on the quantum time scale.

1. Introduction

Because the theory of quantum calculus has important applications in quantum theory (see Kac and Cheung [1]), it has received much attention. For example, since Bohner and Chieochan [2] introduced the concept of the periodicity for functions defined on the quantum time scale, quite a few authors have devoted themselves to the study of the periodicity for dynamic equations on the quantum time scale [3–6].

However, in reality, the almost periodic phenomenon is more common and complicated than the periodic one. In addition, the almost automorphy, which was introduced in the literature by Bochner in 1955 [7, 8], is a generalization of the almost periodicity and plays an important role in understanding the almost periodicity. Therefore, to study the almost automorphy of dynamic equations on the quantum time scale is more interesting and more challenging.

Recently, on almost periodic time scales or called the invariant time scales under translations, papers [9, 10] introduced the concept of weighted pseudo almost automorphic functions and the concept of almost automorphic functions, respectively. Several other works, for instance, papers [11–18] also studied the almost automorphy on almost periodic time scales. The almost periodic time scale is a kind of additive time scales, while the quantum time scale is not an additive time scale; it is a kind of multiplicative time scales. Therefore, the concept of almost automorphic functions on almost periodic time scales is not suitable for dealing with almost automorphic problems on the quantum time scale and all of the results obtained in [9–18] can not be directly applied to the quantum time scale’s case.

Motivated by the above, our main purpose of this paper is to propose two types of definitions of almost automorphic functions on the quantum time scale, study some of their basic properties, and establish the existence of almost automorphic solutions of nonautonomous linear 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 automorphic functions on the quantum time scale and investigate some of their basic properties. In Section 4, we introduce a transformation and give an equivalent definition of almost automorphic functions on the quantum time scale. Moreover, following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale. In Section 5, as an application of the results, we study the existence of almost automorphic solutions for semilinear dynamic equations on the quantum time scale. We draw a conclusion in Section 6.

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 .

Let be a real or complex Banach space. 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 , is called delta differentiable at if there exists such that, for any given , there is an open neighborhood of satisfying

for 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 1. Note that

if is continuously differentiable.

A function is called regressive provided for all . An -matrix-valued function on a time scale is called regressive provided is invertible for all .

Definition 2 (see [19]). A time scale is called an almost periodic time scale or an invariant time scale under translations if

For more details about the theory of time scale calculus and the theory of quantum calculus, the reader may want to consult [1, 20–22].

3. Almost Automorphic Functions on the Quantum Time Scale

In this section, we propose two types of concepts of almost automorphic functions on the quantum time scale and study some of their basic properties. Our first type of concepts of almost automorphic functions on the quantum time scale is as follows.

Definition 3. Let be a real or complex Banach space and a strongly continuous function. We say that is almost automorphic if, for every sequence of integer numbers , there exists a subsequence such that

is well defined for each and

for each .

Remark 4. Since has only one right-dense point and all of the other points of it are isolated points, so is a strongly continuous function if and only if .

Theorem 5. If , and are almost automorphic functions , then the following are true:

(i) is almost automorphic.(ii) is almost automorphic for every scalar c.(iii) is almost automorphic for each fixed (iv); that is, is a bounded function.(v)The range of is relatively compact in .

Proof. The proofs of , , and are obvious.
The proof of : If is no true, then . Hence, there exists a sequence such that

Since is almost automorphic, one can extract a subsequence such that

exists; that is, , which is a contradiction. The proof of is completed.
The proof of : For any sequence in , where , because is almost automorphic, one can extract a subsequence of such that

Thus, is relatively compact in . The proof is complete.

Remark 6. It is easy to see that

and , where is the function that appears in Definition 3.

Theorem 7. If is almost automorphic, define a function by , if exists. Then is almost automorphic.

Proof. For any given sequence , there exists a subsequence of such that

is well defined for each and

for each .
Define a function , and set ; we get

pointwise on . Since exists, is well defined and continuous. Thus, is almost automorphic. The proof is complete.

Theorem 8. Let and be two Banach spaces and an almost automorphic function. If is a continuous function, then the composite function is almost automorphic.

Proof. Since is almost automorphic, for any sequence , we can extract a subsequence of such that

is well defined for each and

for each .
Since is continuous, we have

is well defined for each and

for each .
That is, the composite function is almost automorphic. The proof is complete.

Corollary 9. If is a bounded linear operator in and is an almost automorphic function, then is also almost automorphic.

Proof. The proof is obvious.

Theorem 10. Let be almost automorphic. If  for all for some integer number , then  for all .

Proof. It suffices to prove that for . Since is almost automorphic, for the sequence of natural numbers , one can extract a subsequence such that

It is clear that, for any , we can find with for all . Thus, for all . By (19),   for . Hence, according to formula (20), we obtain for . Since is continuous at , Therefore, for . The proof is complete.

Theorem 11. Let be a sequence of almost automorphic functions such that uniformly in . Then is almost automorphic.

Proof. For any given sequence , by the diagonal procedure one can extract a subsequence of such that

for each and each .
We claim that the sequence of function is a Cauchy sequence. In fact, for any , we have

and hence

For each , from the uniform convergence of , there exists a positive integer such that, for all ,

for all and all .
It follows from (21) and the completeness of the space that the sequence converges pointwise on to a function, say to function .
Now, we will prove

pointwise on .
Indeed, for each , we have

For any , we can find some positive integer such that

for every , and for every . Hence, by formula (26), we get

for every .
In view of (21), for every , there is some positive integer such that

for every . From this and (28), we obtain

for .
Similarly, we can prove that

The proof is complete.

Remark 12. If we denote by the set of all almost automorphic functions , then by Theorem 5, we see that is a vector space, and according to Theorem 11, this vector space equipped with the norm

is a Banach space.

Definition 13. A continuous function is said to be almost automorphic in for each , if, for each sequence of integer numbers , there exists a subsequence such that

exists for each and each , and

exists for each and each .

Theorem 14. If are almost automorphic functions in for each , then the following functions are also almost automorphic in for each :

(i)(ii): is an arbitrary scalar.

Proof. The proof is obvious. We omit it here. The proof is complete.

Theorem 15. If are almost automorphic in for each , then

for each .

Proof. Suppose the opposite. Assume, to the contrary, that

for some . Thus, there exists a sequence of integer numbers such that

Since is almost automorphic in , one can extract a subsequence from such that

which is a contradiction. The proof is complete.

Theorem 16. If is almost automorphic in for each , then the function in Definition 13 satisfies

for each .

Proof. The proof is obvious. We omit it here. The proof is complete.

Theorem 17. If is almost automorphic in for each and if satisfies the Lipschitzian condition in uniformly in , that is, there exists a positive constant such that, for each pair ,

uniformly in , then satisfies the same Lipschitz condition in uniformly in .

Proof. Because for each sequence of integer numbers , there exists a subsequence such that

exists for each and each , for any and any given , we have

for sufficiently large.
Hence, for sufficiently large we find

Letting , we get

for each . The proof is complete.

Theorem 18. Let be almost automorphic in for each and assume that satisfies a Lipschitz condition in uniformly in . Let be almost automorphic. Then the function defined by is almost automorphic.

Proof. It is easy to see that, for any given sequence , there exists a subsequence such that

for each and ,

for each ,

for each and , and

for each .
Consider the function defined by , . We will show that , for each and , for each .
In fact, noting that

by (45) and formula (46), we get

Similarly we can prove that for each . This completes the proof.

Before ending this section, we give the second type of concepts of almost automorphic functions on the quantum time scale as follows.

Definition 19. Let be a real or complex Banach space and a strongly continuous function. We say that is almost automorphic if, for every sequence of integer numbers , there exists a subsequence such that

is well defined for each and

for each .

Definition 20. A continuous function is said to be almost automorphic in for each , if, for each sequence of integer numbers , there exists a subsequence such that

exists for each and each , and

exists for each and each .

Remark 21. It is easy to check that all the results of this section hold for almost automorphic functions defined by Definitions 3 and 13 which are also valid for almost automorphic functions defined by Definitions 19 and 20.

4. An Equivalent Definition of Almost Automorphic Functions on the Quantum Time Scale

In this section, we will give an equivalent definition of almost automorphic functions on the quantum time scale . To this end, we introduce a notation and stipulate , and for all . Let ; we define a function by

that is,

Since is right continuous at , it is clear that the above definition is well defined.

Moreover, for , we define a function by

that is,

Since is continuous at , it is clear that the above definition is well defined.

Definition 22. A function is called almost automorphic if for every sequence there exists a subsequence such that

is well defined for each , and

for each .

Definition 23. A function is called almost automorphic if for every sequence there exists a subsequence such that

is well defined for each , and

for each and .

Remark 24. We can view as a kind of generalized integer number set. Obviously, the automorphic functions defined by Definitions 22 and 23 (which are defined on or ) share the same properties as the ordinary automorphic functions defined on or .

Definition 25. A function is called almost automorphic if and only if the function defined by (55) is almost automorphic.

Definition 26. A function is called almost automorphic in for each if and only if the function defined by (57) is almost automorphic in for each .

Obviously, Definitions 25 and 26 are equivalent to Definitions 3 and 13, respectively. Moreover, by Remark 24, all of the properties of almost automorphic functions on the quantum time scale can be directly obtained from the corresponding properties of the ordinary almost automorphic functions defined on or .

Before ending this section, following the idea of the transformation of this section, we can propose a concept of almost automorphy on a more general time scale.

Definition 27. Let be a time scale and be an almost periodic time scale defined by Definition 2. A continuous function is said to be almost automorphic in for each , if there exists a one-to-one transformation such that and, for each sequence of integer numbers , there exists a subsequence such that

exists for each and each , and

exists for each and each , where .

Remark 28. Obviously, in Definition 27, if is an almost periodic time scale defined by Definition 2, by taking , the identity mapping, then Definition 27 coincides with Definition in [9] and Definition in [10], respectively, which are the definitions of almost automorphic functions on almost periodic time scales. If , by taking the transformation defined by (57), then Definition 27 agrees with Definition 13. Therefore, Definition 27 can unify the cases of almost periodic time scales and the quantum time scale.

5. Automorphic Solutions for Semilinear Dynamic Equations on the Quantum Time Scale

In this section, we will study the existence of automorphic solutions of semilinear dynamic equations on the quantum time scale. Throughout this section, we use the letter to stand for either or .

Consider the semilinear dynamic equation on the quantum time scale:

where is a scalar delay function and satisfies for all , is a regressive, rd-continuous matrix valued function, and . Under transformation (57), (65) is transformed to

and vice visa, where .

Clearly, is a solution of (65) if and only if is a solution of (66).

Definition 29 (see [14]). Let be an rd-continuous matrix value function on ; the linear system

is said to admit an exponential dichotomy on if there exist positive constants and and an invertible projection commuting with , where is principal fundamental matrix solution of (67) satisfying

Theorem 30 (see [14]). Let be an almost periodic time scale. Supposing that linear homogeneous system (67) admits an exponential dichotomy with the positive constants and and invertible projection commuting with , where is principal fundamental matrix solution of (67), then the nonhomogeneous system

has a solution of the form

Moreover, we have

Consider the following semilinear dynamic equation on almost periodic time scale :

where is a scalar delay function and satisfies for all , is a regressive, rd-continuous matrix valued function, and . The corresponding linear homogeneous system of (72) is

We make the following assumptions:

Functions , and are almost automorphic in .There exists a constant such that

 for all and for any vector valued functions and defined on .Linear homogeneous system (73) admits an exponential dichotomy with the positive constants and and invertible projection commuting with , where is principal fundamental matrix solution of (73).

Now, define the mapping by

The following result can be proven similar to Lemma in [11]; hence we omit it.

Lemma 31. Suppose ()–() hold. Then the mapping maps into .

Theorem 32. Suppose ()–() hold. Assume further that

().

Then (72) has a unique almost automorphic solution.

Proof. For any , we have

Hence, is a contraction. Therefore, has a unique fixed point in , so (72) has a unique almost automorphic solution.

In Theorem 32, if we take , then we have the following.

Theorem 33. Suppose ()–() hold. Then (66) has a unique almost automorphic solution, and so (65) has a unique almost automorphic solution.

Consider a linear quantum difference equation

where is an matrix valued function and is an -dimensional vector valued function. Under transformation (55), (77) transforms to

and vice versa.

Consider the following nonautonomous linear difference equation:

where are given nonsingular matrices with elements is a given vector function, and is an unknown vector with components . Its associated homogeneous equation is given by

Similar to Definition in [23], we give the following definition.

Definition 34. Let be the principal fundamental matrix of difference system (80). System (80) is said to possess an exponential dichotomy if there exist a projection , which commutes with , and positive constants such that, for all , we have

Similar to the proof of Theorem in [12], one can easily show the following.

Theorem 35. Suppose is discrete almost automorphic and a nonsingular matrix and the set is bounded. Also, suppose the function is a discrete almost automorphic function and (80) admits an exponential dichotomy with positive constants , and . Then, system (79) has an almost automorphic solution on .

Corollary 36. Suppose is discrete almost automorphic and a nonsingular matrix and the set is bounded. Also, suppose the function is a discrete almost automorphic function and equation

admits an exponential dichotomy with positive constants , and . Then, system (77) has an almost automorphic solution on .

6. Conclusion

In this paper, we proposed two types of concepts of almost automorphic functions on the quantum time scale and studied some of their basic properties. Moreover, based on the transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers, we gave equivalent definitions of almost automorphic functions on the quantum time scale. As an application of our results, we established the existence of almost automorphic solutions for semilinear dynamic equations on the quantum time scale. By using the methods and results of this paper, for example, one can study the almost automorphy of neural networks on the quantum time scale and population dynamical models on the quantum time scale and so on. Furthermore, by using the transformation and the set of generalized integer numbers introduced in Section 3 of this paper, or similar to Definition 27, one can propose concepts of almost periodic functions, pseudo almost periodic functions, weighted pseudo almost automorphic functions, almost periodic set-valued functions, almost periodic functions in the sense of Stepanov on the quantum time scale, and so on.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 11361072.

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