Almost automorphic functions on the quantum time scale and applications

In this paper, 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. 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.


Introduction
Since 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 periodicity for functions defined on the quantum time scale, quite a few authors have devoted themselves to the study of periodicity for dynamic equations on the quantum time scale ( [3,4,5,6]).
However, in reality, almost periodic phenomenon is more common and complicate than periodic one. In addition, the almost automorphy is a generalization of 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 challenge.
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 non-autonomous 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. 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.

Preliminaries
In this section, we shall recall some basic definitions of time scale calculus. A time scale T is an arbitrary nonempty closed subset of the real numbers, the forward and backward jump operators σ, ρ : T → T and the forward graininess µ : T → R + are defined, respectively, by σ(t) := inf{s ∈ T : s > t}, ρ(t) := sup{s ∈ T : s < t} and µ(t) = σ(t) − t.
A point t is said to be left-dense if t > inf T and ρ(t) = t, right-dense if t < sup T and σ(t) = t, left-scattered if ρ(t) < t and right-scattered if σ(t) > t. If T has a left-scattered maximum m, then T κ = T\m, otherwise T κ = T. If T has a right-scattered minimum m, then T κ = T\m, otherwise T k = T.
Let X be a (real or complex) Banach space. A function f : T → X is right-dense continuous or rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. If f is continuous at each right-dense point and each left-dense point, then f is said to be a continuous function on T.
For f : T → X and t ∈ T k , then f is called delta differentiable at t ∈ T if there exists c ∈ X such that for given any ε ≥ 0, there is an open neighborhood U of t satisfying In this case, c is called the delta derivative of f at t ∈ T, and is denoted by A function p : T → R is called regressive provided 1 + µ(t)p(t) = 0 for all t ∈ T κ . An n × n-matrix-valued function A on a time scale T is called regressive provided I + µ(t)A(t) is invertible for all t ∈ T κ .
For more details about the theory of time scale calculus and the theory of quantum calculus, the reader may want to consult [1,8,9,10].

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 3.1. Let X be a (real or complex) Banach space and f : q Z → X a (strongly) continuous function. We say that f is almost automorphic if for every sequence of integer numbers {s ′ n } ⊂ Z, there exists a subsequence {s n } such that: is well defined for each t ∈ q Z and lim n→∞ g(tq −sn ) = f (t) for each t ∈ q Z .  (i) f 1 + f 2 is almost automorphic.
(ii) cf is almost automorphic for every scalar c.
(iii) f a (t) ≡ f (tq a ) is almost automorphic for each fixed a ∈ Z.
(iv) sup t∈R f (t) < ∞, that is, f is a bounded function.
(v) The range R f = {f (t)|t ∈ q Z } of f is relatively compact in X.
Proof. The proofs of (i), (ii), and (iii) are obvious. The proof of (iv). If (iv) is no true, then sup t∈q Z f (t) = ∞. Hence, there exists a sequence Since f is almost automorphic, one can extract a subsequence {s n } ⊂ {s ′ n } such that lim n⇀∞ f (q sn ) = ξ exists, that is, lim n⇀∞ f (q sn ) = ξ < ∞, which is a contradiction. The proof of (iv) is completed.
The proof of (v). For any sequence .
Thus, R f is relatively compact in X. The proof is complete.

Remark 3.2.
It is easy to see that and R g ⊆R f , where g is the function that appears in Definition 3.1.
exists. Then f ⋆ : q Z → X is almost automorphic.
Proof. For any given sequence {s ′ n } ⊂ Z, there exists a subsequence {s n } of {s ′ n } such that is well defined for each t ∈ q Z and Thus, f ⋆ (t) is almost automorphic. The proof is complete.
. Let X and Y be two Banach spaces and f : q Z → X an almost automorphic function. If φ : X → Y is a continuous function, then the composite function φ(f ) : q Z → Y is almost automorphic.
Proof. Since f is almost automorphic, for any sequence {s ′ n } ⊂ Z, we can extract a subse- is well defined for each t ∈ q Z and is well defined for each t ∈ q Z and lim n→∞ ϕ(g(tq −sn )) = ϕ lim Corollary 3.1. If A is a bounded linear operator in X and f : q Z → X an almost aytomorphic function, then A(f )(t) is also almost automophic.
Theorem 3.4. Let f be almost automorphic. If f (q n ) = 0 for all n > n 0 for some integer number n 0 , then f (t) ≡ 0 for all t ∈ q Z .
Proof. It suffices to prove that f (t) = 0 for t ≤ q n 0 . Since f is almost automorphic, for the sequence of natural numbers N = {n}, one can extract a subsequence {n k } ⊂ N such that It is clear that for any t ≤ q n 0 , we can find {n kj } ⊂ {n k } with tq n kj > q n 0 for all j = 1, 2, . . ..
Hence, according to (3.2), we obtain The proof is complete.
Proof. For any given sequence {s ′ n } ⊂ Z, by the diagonal procedure one can extract a subse- for each i = 1, 2, ... and each t ∈ q Z . We claim that the sequence of function {g i (t)} is a Cauchy sequence. In fact, for any i, j ∈ N, we have hence, For each ε > 0, from the uniform convergence of {f n }, there exists a positive integer N(ε) such that for all i, j > N, for all t ∈ q Z , and all n = 1, 2, . . .. It follows from (3.3) and the completeness of the space X that the sequence {g i (t)} converges pointwisely on q Z to a function, say to function g(t).
for every n > M. From this and (3.5), we obtain Similarly, we can prove that The proof is complete.
Remark 3.3. If we denote by AA(X), the set of all almost automorphic functions f : q Z → X, then by Theorem 3.1, we see that AA(X) is a vector space, and according to Theorem 3.5, this vector space equipped with the norm is a Banach space.

Definition 3.2.
A continuous function f : R × X → X is said to be almost automorphic in t ∈ q Z for each x ∈ X, if for each sequence of integer numbers {s ′ n }, there exists a subsequence {s n } such that lim n→∞ f (tq sn , x) = g(t, x) exists for each t ∈ q Z and each x ∈ X, and exists for each t ∈ q Z and each x ∈ X.
Theorem 3.6. If f 1 , f 2 : q Z × X → X are almost automorphic functions in t for each x ∈ X, then the following functions are also almost automorphic in t for each x ∈ X: (ii) cf 1 , c is an arbitrary scalar.
Proof. The proof is obvious. We omit it here. The proof is complete.
Proof. Suppose not. Assume, to the contrary, that for some x 0 ∈ X. Thus, there exists a sequence of integer numbers {s ′ n } such that Since f (t, x 0 ) is almost automorphic in t, one can extract a subsequence {s n } from {s ′ n } such that sup which is a contradiction. The proof is complete.
Theorem 3.8. If f is almost automorphic in t for each x ∈ X, then the function g in Definition 3.2 satisfies sup Proof. The proof is obvious. We omit it here. The proof is complete.
Theorem 3.9. If f is almost automorphic in t for each x ∈ X and if f is Lipschitzian in x uniformly in t, that is, there exists a positive constant L > 0 such that for each pair x, y ∈ X, uniformly in t ∈ q Z , then g satisfies the same Lipschitz condition in x uniformly in t. exists for each t ∈ q Z and each x ∈ X, so for any t ∈ q Z and any given ε > 0, we have x) − f (tq sn , x) < ε 2 and g(t, y) − f (tq sn , y) < ε 2 for n sufficiently large.
Hence, for n sufficiently large we find Letting ε → 0 + , we get g(t, x) − g(t, y) ≤ L x − y for each x, y ∈ X. The proof is complete.
Theorem 3.10. Let f : q Z × X → X be almost automorphic in t for each x ∈ X and assume that f satisfies a Lipschitz in x uniformly in t ∈ q Z . Let ϕ : q Z → X be almost automorphyic. Then the function F : q Z → X defined by F (t) = f (t, ϕ(t)) is almost automorphic.
Before ending this section, we give the second type of concepts of almost automorphic functions on the quantum time scale as follows: Definition 3.3. Let X be a (real or complex) Banach space and f : q Z → X a (strongly) continuous function. We say that f is almost automorphic if for every sequence of integer numbers {s ′ n } ⊂ Z, there exists a subsequence {s n } such that: is well defined for each t ∈ q Z and for each t ∈ q Z .

Definition 3.4.
A continuous function f : R × X → X is said to be almost automorphic in t ∈ q Z for each x ∈ X, if for each sequence of integer numbers {s ′ n }, there exists a subsequence {s n } such that lim n→∞ q sn f (tq sn , x) = g(t, x) exists for each t ∈ q Z and each x ∈ X, and exists for each t ∈ q Z and each x ∈ X.

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 q Z . To this end, we introduce a notation −∞ q and stipulate q −∞q = 0, t ± (−∞ q ) = t and t > −∞ q for all t ∈ Z. Let f ∈ C(q Z , X), we define a functioñ f : that is, is right continuous at t = 0, it is clear that the above definition is well defined.
Moreover, for f ∈ C(q Z × X, X), we define a functionf : that is, Since f (t, x) is continuous at (0, x), it is clear that the above definition is well defined.
is well defined for each t ∈ Z ∪ {−∞ q }, and for each t ∈ Z ∪ {−∞ q } and x ∈ X. Definition 4.4. A function f ∈ C(q Z × X, X) is called almost automorphic in t ∈ q Z for each x ∈ X if and only if the functionf (t, x) defined by (4.2) is almost automorphic in t ∈ q Z for each x ∈ X.
Obviously, Definitions 4.3 and 4.4 are equivalent to Definitions 3.1 and 3.2, respectively. Moreover, by Remark 4.1, 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 Z or Z × X.

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 E to stand for either R or C.
Theorem 5.1. [11] Let T be an almost periodic time scale. Suppose that the linear homogeneous system (5.3) admits an exponential dichotomy with the positive constants K 1 , K 2 and α 1 , α 2 and invertible projection P commuting with X(t), where X(t) is principal fundamental matrix solution of (5.3), then the nonhomogeneous system has a solution x(t) of the form Moreover, we have Consider the following semilinear dynamic equation on almost periodic time scale T: where τ : T → [0, ∞) T is a scalar delay function and satisfies t − τ (t) ∈ T for all t ∈ T, A(t) is a regressive, rd-continuous n × n matrix valued function, f ∈ C rd (T × E 2n , E n ). The corresponding linear homogeneous system of (5.6) is We make the following assumptions: (A 1 ) Functions τ (t), A(t) and f (t, u, v) are almost automorphic in t.
(A 2 ) There exists a constant L 1 , L 2 > 0 such that for all t ∈ T and for any vector valued functions u and v defined on T.
(A 3 ) The linear homogeneous system (5.7) admits an exponential dichotomy with the positive constants K 1 , K 2 and α 1 , α 2 and invertible projection P commuting with X(t), where X(t) is principal fundamental matrix solution of (5.7). Now, define the mapping Ψ by The following result can be proven similar to Lemma 6 in [11], hence we omit it.
Then (5.6) has a unique almost automorphic solution.
Proof. For any x, y ∈ AA(E n ), we have Hence, Φ is a contraction. Therefore, Φ has a unique fixed point in AA(E n ), so, (5.6) has a unique almost automorphic solution.
In Theorem 5.2, if we take T = Z ∪ {−∞ q }, then we have Consider a linear quantum difference equation where A is an n × n matrix valued function and f is an n-dimensional vector valued function. Under the transformation (4.1), (5.8) transforms to and vice versa. Consider the following non-autonomous linear difference equation where A(k) are given non-singular n×n matrices with elements a ij (k), 1 ≤ i, j ≤ n, f : Z → E n is a given n × 1 vector function and x(k) is an unknown n × 1 vector with components x i (k), 1 ≤ i ≤ n. Its associated homogeneous equation is given by Similar to Definition 2.11 in [12], we give the following definition: Definition 5.2. Let U(k) be the principal fundamental matrix of the difference system (5.11).
Similar to the proof of Theorem 3.1 in [12], one can easily show that Theorem 5.4. Suppose A(k) is discrete almost automorphic and a non-singular matrix and the set {A −1 (k)} k∈Z∪{−∞q} is bounded. Also, suppose the function f : Z ∪ {−∞ q } → E n is a discrete almost automorphic function and Eq. (5.11) admits an exponential dichotomy with positive constants ν, η, β and α. Then, the system (5.10) has an almost automorphic solution on Z ∪ {−∞ q }.

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, 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.