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A matched space for time scales and applications to the study on functions
 Chao Wang^{1, 2}Email authorView ORCID ID profile,
 Ravi P Agarwal†^{2} and
 Donal O’Regan†^{3}
https://doi.org/10.1186/s1366201713660
© The Author(s) 2017
Received: 1 June 2017
Accepted: 19 September 2017
Published: 2 October 2017
Abstract
In this paper, using the algebraic structure of the Abelian group, we introduce the concept of a matched space for time scales, and we construct the algebraic structure of matched spaces to solve the closedness of time scales under nontranslational shifts. Using a matched space for time scales, a new concept of periodic time scales is introduced. Based on it, new concepts of periodic functions, almost periodic functions and almost automorphic functions whose concepts were defined on translations of their arguments are proposed through nontranslational shifts. The results in this paper provide new methods to consider periodic solution, almost periodic solution and almost automorphic solutions for qdifference equations and others on irregular time scales via the background of the algebraic structure.
Keywords
 time scales
 matched space
 periodic functions
 almost periodic functions
 almost automorphic functions
 solutions for dynamic equations
MSC
 26E70
 33E30
 39A13
 34N05
 34C25
 34C27
 43A60
1 Introduction
Periodic, almost periodic and almost automorphic functions are important classes of functions which arise naturally in differential, difference and integral equations (see [1–6]). The theory of time scales is a useful tool for unifying difference and differential equations (see [7–9] and the references therein). Using the classical definition of periodic time scales (see Kaufmann and Raffoul [10], Akhmet and Turan [11]), periodic solutions, almost periodic solutions and almost automorphic solutions for dynamic equations were considered. In 2006, the authors studied almost periodic solutions of differential equations on certain time scales with transition conditions (DETC) through ψsubstitution (see [11]), while the next investigation of almost periodic problems on time scales for impulsive dynamic equations was conducted in 2014 through Δcalculus directly (see [12–14]). As a matter of fact, the shift closedness of time scales plays a very important role in introducing the welldefined functions on time scales, it is a prerequisite to introduce the functions which are defined by shifts, e.g., periodic functions, almost periodic functions and almost automorphic functions, etc. We note that classical periodic time scales have a very nice translation closedness which paves the way for defining functions through translations. We restate the following definitions and introductory examples which can be found in [10, 11].
Definition 1.1
A time scale \(\mathbb{T}\) is said to be periodic if there exists \(P>0\) such that \(t\pm P\in \mathbb{T}\) for all \(t\in \mathbb{T}\). If \(\mathbb{T}\neq \mathbb{R}\), the smallest positive P is called the period of the time scale.
Example 1.1
 (1)
\(\mathbb{T}=h\mathbb{Z}\) has period \(P=h\).
 (2)
\(\mathbb{T}=\{t=kq^{m}: k\in \mathbb{Z}, m\in \mathbb{N}_{0}\}\), where \(0< q<1\), has period \(P=1\).
Definition 1.2
Example 1.2
Wang and Agarwal proposed Example 1.2 in [16] and corrected the concept of almost periodic functions on periodic time scales. Note that Example 1.2 satisfies Definition 1.1 but does not satisfy Definition 1.2 since \(P\notin \mathbb{P}_{a,b}\) though \(t\pm P\in \mathbb{P}_{a,b}\). As a result, Definition 1.2 does not include Definition 1.1. In [17], the authors introduced some new concepts of periodic time scales attached with a “translation direction”. In addition, several examples were provided to show that these new concepts are more general than Definition 1.1, and the composition theorem of time scales from [17, 18] will be guaranteed under these new notions. Moreover, the authors of [19, 20] proposed and solved the almost periodic problems for qdynamic equations on the quantum time scale for the first time, and the authors introduced the theory of almost periodic functions under shift operators which can derive and include the almost periodic theory of qdynamic equations on the quantum time scale. Also, some new almost periodic functions were constructed on irregular time scales under the stochastic background. Furthermore, in the literature [21], the authors made some significant comments and notes on their related works and the recent development of time scales to contribute to solving the closedness problems on various types of time scales.
The closedness of time scales under shifts is a key and difficult problem which needs to be solved since it is closely related to defining and studying functions, investigating delay dynamic equations on time scales and nonlinear periodic or nonperiodic phenomena in the real world. In this paper, we initiate the idea that the shift number set which controls the shifts of a time scale and maintains its shift closedness may be separated from the time scale (e.g., \(\mathbb{T} \cap \Pi =\emptyset \) in Example 1.2). Using the algebraic structure of an Abelian group, the concept of a matched space for time scales is introduced and the algebraic structure of matched spaces is constructed to solve the closedness of time scales under nontranslational shifts. With this algebraic structure, a new concept of periodic time scales is introduced and based on it, new concepts of periodic functions, almost periodic functions and almost automorphic functions are proposed through nontranslational shifts. Furthermore, these notions attached with shift directions are also introduced and studied.
2 A matched space for time scales
In this section, we introduce the algebraic structure of matched spaces for time scales. For details on time scales, we refer the reader to [7–9].
A point \(t\in \mathbb{T}\) is called leftdense if \(t>\inf {\mathbb{T}}\) and \(\rho (t)=t\), leftscattered if \(\rho (t)< t\), rightdense if \(t<\sup {\mathbb{T}}\) and \(\sigma (t)=t\), and rightscattered if \(\sigma (t)>t\). If \(\mathbb{T}\) has a leftscattered maximum M, then \(\mathbb{T}^{k}=\mathbb{T}\backslash \{M\}\); otherwise \(\mathbb{T} ^{k}=\mathbb{T}\). If \(\mathbb{T}\) has a rightscattered minimum m, then \(\mathbb{T}_{k}=\mathbb{T}\backslash \{m\}\); otherwise \(\mathbb{T}_{k}=\mathbb{T}\).
A function \(f: \mathbb{T}\rightarrow \mathbb{R}\) is rightdense continuous provided it is continuous at a rightdense point in \(\mathbb{T}\) and its leftside limits exist at leftdense points in \(\mathbb{T}\). If f is continuous at each rightdense point and each leftdense point, then f is said to be a continuous function on \(\mathbb{T}\).
Now, we construct the algebraic structure for a pair \((\Pi^{*}, \tilde{\delta })\) by using an Abelian group property to introduce the definition of a relatively dense set with respect to the group \((\Pi^{*},\tilde{\delta })\), where \(\Pi^{*}\) is a subset of \(\mathbb{R}\) together with an operation δ̃.
Definition 2.1
 (1)
\(\Pi^{*}\) is closed with respect to an operation δ̃, i.e., for any \(\tau_{1},\tau_{2}\in \Pi^{*}\), we have \(\tilde{\delta }(\tau_{1},\tau_{2})\in \Pi^{*}\).
 (2)
For any \(\tau \in \Pi^{*}\), there exists an identity element \(e_{\Pi^{*}}\in \Pi^{*}\) such that \(\tilde{\delta }(e_{\Pi^{*}}, \tau)=\tau \).
 (3)
For all \(\tau_{1},\tau_{2},\tau_{3}\in \Pi^{*}\), \(\tilde{\delta } (\tau_{1},\tilde{\delta }(\tau_{2},\tau_{3}))=\tilde{\delta } (\tilde{\delta }(\tau_{1},\tau_{2}),\tau_{3} )\) and \(\tilde{\delta }(\tau_{1},\tau_{2})=\tilde{\delta }(\tau _{2},\tau_{1})\).
 (4)
For each \(\tau \in \Pi^{*}\), there exists an element \(\tau^{1}\in \Pi^{*}\) such that \(\tilde{\delta }(\tau,\tau^{1})= \tilde{\delta }(\tau^{1},\tau)=e_{\Pi^{*}}\), where \(e_{\Pi^{*}}\) is the identity element in \(\Pi^{*}\).
 (5)
If \(\tau_{1}>\tau_{2}\), then \(\tilde{\delta }(\cdot,\tau _{1})>\tilde{\delta }(\cdot,\tau_{2})\).
Remark 2.1
Note that from Definition 2.1, one can derive some classical concepts of relatively dense set. Here, we present some classical concepts according to it.
(i) If \(\Pi^{*}=\mathbb{R}\) and \(\tilde{\delta }(\tau_{1},\tau _{2})=\tau_{1}+\tau_{2}\), then \(e_{\Pi^{*}}=0\). One can obtain the following concept:
Definition 2.2
Definition 1.10 from [1]
A subset S of \(\mathbb{R}\) is called relatively dense with respect to the pair \((\mathbb{R},+)\) if there exists a number \(L>0\) such that \([a,a+L]\cap S\neq \emptyset \) for all \(a\in \mathbb{R}\). The number L is called the inclusion length with respect to the group \((\mathbb{R},+)\).
(ii) If \(\Pi^{*}=h\mathbb{Z}\), \(h>0\) and \(\tilde{\delta }(\tau _{1},\tau_{2})=\tau_{1}+\tau_{2}\), then \(e_{\Pi^{*}}=0\). We can immediately get the following concept:
Definition 2.3
A subset S of \(\mathbb{R}\) is called relatively dense with respect to the pair \((h\mathbb{Z},+)\) if there exists a number \(L\in h\mathbb{Z} ^{+}\) such that \([a,a+L]_{h\mathbb{Z}}\cap S\neq \emptyset \) for all \(a\in h\mathbb{Z}\). The number L is called the inclusion length with respect to the group \((h\mathbb{Z},+)\).
(iii) If \(\Pi^{*}=q^{\mathbb{Z}}:=\{q^{n}:q>1,n\in \mathbb{Z} \}\) and \(\tilde{\delta }(\tau_{1},\tau_{2})=\tau_{1}\cdot \tau_{2}\), then \(e_{\Pi^{*}}=1\). One can establish the following concept:
Definition 2.4
A subset S of \(\mathbb{R}\) is called relatively dense with respect to the pair \((q^{\mathbb{Z}},\cdot)\) if there exists a number \(L\in (1,+\infty)_{q^{\mathbb{Z}}}\) such that \([a,aL]_{q^{\mathbb{Z}}} \cap S\neq \emptyset \) for all \(a\in q^{\mathbb{Z}}\). The number L is called the inclusion length with respect to the group \((q^{\mathbb{Z}}, \cdot)\).
Definition 2.5
A subset S of \(\mathbb{R}\) is called relatively dense with respect to the pair \((\mathbb{N}_{\pm }^{\frac{1}{2}},\tilde{\delta })\) if there exists a number \(L\in (1,+\infty)_{\mathbb{N}_{+}^{\frac{1}{2}}}\) such that \([a,\sqrt{a^{2}+L^{2}}]_{\mathbb{N}_{+}^{\frac{1}{2}}}\cap S \neq \emptyset \) for all \(a\in \mathbb{N}_{+}^{\frac{1}{2}}\) and \([a,\sqrt{a^{2}L^{2}}]_{\mathbb{N}_{}^{\frac{1}{2}}}\cap S\neq \emptyset \) for all \(a\in \mathbb{N}_{}^{\frac{1}{2}}\). The number L is called the inclusion length with respect to the group \((\mathbb{N}_{\pm }^{\frac{1}{2}},\tilde{\delta })\).
From Definitions 2.22.5, one will see that by employing Definition 2.1, some new concepts of relatively dense sets for various types of time scales can be derived. It is easy to observe that Definition 2.2 is suitable for \(\mathbb{T}=\mathbb{R}\), Definition 2.3 is appropriate for \(\mathbb{T}=h\mathbb{Z}\), Definition 2.4 is suitable for \(\mathbb{T}=\overline{q^{ \mathbb{Z}}}\) and Definition 2.5 is applicable to \(\mathbb{T}= \mathbb{N}_{\pm }^{\frac{1}{2}}\).
Similarly, by using the operator δ̃, one can also establish the concepts of relatively dense sets for \(\mathbb{T}=\overline{q ^{\mathbb{Z}}\cup q^{\mathbb{Z}}}\) and \(\overline{(q)^{\mathbb{Z}}}\), \(q>1\), etc. Hence, it is obvious that Definition 2.1 is so general that it can unify the concepts of relatively dense sets for many irregular time scales, it will be a useful tool to describe the almost periodicity of functions on a more comprehensive scope of time scales.
According to Definition 2.1, one can obtain the following property.
Theorem 2.2
Proof
Next, we introduce the concept of an adjoint mapping between \(\mathbb{T}\) and Π to lay the foundation for introducing matched spaces.
Definition 2.6
Example 2.3
 (1)Consider the following time scale \(\mathbb{T}\) and the set Π:and obviously, there exists a bijective map$$\mathbb{T}=\overline{\bigl\{ 2^{i}:i\in \mathbb{Z}\bigr\} },\qquad \Pi = \overline{\bigl\{ 3^{i}:i \in \mathbb{Z}\bigr\} }, $$i.e., \(F(A)=B\). Note that \(\Pi \backslash \{0\}=\Pi^{*}\not \subset \mathbb{T}\) and \(e_{\Pi^{*}}=1\). Here, \(\tilde{\delta }(\tau_{1},\tau _{2})=\tau_{1}\cdot \tau_{2}\).$$\begin{aligned} \textstyle\begin{array}{cccc} F:\quad & \mathbb{T}& \rightarrow& \Pi, \\ &A\in \overline{ \bigl\{ 2^{i}, i\in \mathbb{Z} \bigr\} } &\rightarrow & B\in \overline{ \bigl\{ 3^{i}, i\in \mathbb{Z} \bigr\} } , \\ & A &\rightarrow& A^{\log_{2}3}=B, \\ & A=0 &\rightarrow& B=0, \end{array}\displaystyle \end{aligned}$$
 (2)Consider the following time scale \(\mathbb{T}\) and the set Π:and obviously, there exists a bijective mapping$$\mathbb{T}=\bigcup_{i\in \mathbb{Z}}\bigl[i(a+b),i(a+b)+b\bigr], \quad \mbox{where }a\neq b, \Pi =\bigl\{ i(a+b):i\in \mathbb{Z}\bigr\} , $$i.e., \(F (A )=B\). Note that \(\Pi =\Pi^{*}\subset \mathbb{T}\) and \(e_{\Pi^{*}}=0\). Here, \(\tilde{\delta }(\tau_{1},\tau_{2})=\tau_{1}+ \tau_{2}\).$$\begin{aligned} \textstyle\begin{array}{cccc} F:\quad & \mathbb{T}& \rightarrow& \Pi, \\ &A\in \bigl\{ \bigl[i(a+b),i(a+b)+b\bigr], i\in \mathbb{Z} \bigr\} &\rightarrow& B \in \bigl\{ i(a+b), i\in \mathbb{Z} \bigr\} , \\ & A &\rightarrow& \mbox{the left point of the interval }A, \end{array}\displaystyle \end{aligned}$$
 (3)Consider the following time scale \(\mathbb{T}\) and the set Π:and obviously, there exists a bijective mapping$$\mathbb{T}=\bigcup_{i\in \mathbb{Z}} \bigl[i(a+b),i(a+b)+b \bigr], \quad \mbox{where }a>b, \qquad \Pi = \biggl\{ \pm \frac{i^{2}(a+b)}{\sqrt{3}}:i\in \mathbb{Z} \biggr\} , $$i.e., \(\mathscr{Z}(A)=B_{1}\). Then$$\begin{aligned} \textstyle\begin{array}{cccc} \mathscr{Z}:\quad & \mathbb{T} &\rightarrow& \tilde{\Pi }_{1}, \\ &A\in \bigl\{ \bigl[i(a+b),i(a+b)+b\bigr], i\in \mathbb{Z} \bigr\} &\rightarrow & B_{1}\in \bigl\{ i(a+b):i\in \mathbb{Z} \bigr\} , \\ & A &\rightarrow &\mbox{the left point of the interval }A, \end{array}\displaystyle \end{aligned}$$i.e., \(F(A)=\mathscr{N}\circ \mathscr{Z}(A)=B\). Note that \(\Pi =\Pi ^{*}\not \subset \mathbb{T}\) and \(e_{\Pi^{*}}=0\). Here,$$\begin{aligned} \textstyle\begin{array}{cccc} \mathscr{N}: \quad & \tilde{\Pi }_{1} &\rightarrow& \Pi , \\ &B_{1}\in \bigl\{ i(a+b), i\in \mathbb{Z} \bigr\} &\rightarrow& B\in \biggl\{ \pm \frac{i^{2}(a+b)}{\sqrt{3}}:i\in \mathbb{Z} \biggr\} , \\ & B_{1} &\rightarrow& \frac{\operatorname{sgn}(B_{1})B_{1}^{2}}{\sqrt{3}(a+b)}=B, \end{array}\displaystyle \end{aligned}$$$$\tilde{\delta }(\tau_{1},\tau_{2})=\operatorname{sgn}\bigl( \operatorname{sgn}(\tau _{1})\sqrt{\vert \tau_{1}\vert }+ \operatorname{sgn}(\tau_{2})\sqrt{\vert \tau_{2}\vert } \bigr) \bigl(\operatorname{sgn}(\tau_{1})\sqrt{\vert \tau_{1}\vert }+\operatorname{sgn}(\tau _{2})\sqrt{\vert \tau_{2}\vert } \bigr)^{2}. $$
We introduce an operator δ with the following algebraic structure between \(\mathbb{T}\) and Π.
Definition 2.7
 (\(P_{1}\)):

(Monotonicity) The function δ is strictly increasing with respect to its all arguments, i.e., ifthen \(t< u\) implies \(\delta (T_{0},t)<\delta (T_{0},u)\); if \((T_{1},u), (T_{2},u)\in \mathcal{D}_{\delta }\) with \(T_{1}< T_{2}\), then \(\delta (T_{1},u)<\delta (T_{2},u)\).$$(T_{0},t), (T_{0},u)\in \mathcal{D}_{\delta }:= \bigl\{ (s,t)\in \Pi ^{*}\times \mathbb{T}^{*}:\delta (s,t)\in \mathbb{T}^{*} \bigr\} , $$
 (\(P_{2}\)):

(Existence of inverse elements) The operator δ has the inverse operator \(\delta^{1}:\Pi^{*}\times \mathbb{T}^{*}\rightarrow \mathbb{T}^{*}\) and \(\delta^{1}(\tau,t)= \delta (\tau^{1},t)\), where \(\tau^{1}\in \Pi^{*}\) is the inverse element of τ.
 (\(P_{3}\)):

(Existence of identity element) \(e_{\Pi^{*}} \in \Pi^{*}\) and \(\delta (e_{\Pi^{*}},t)=t\) for any \(t\in \mathbb{T} ^{*}\), where \(e_{\Pi^{*}}\) is the identity element in \(\Pi^{*}\).
 (\(P_{4}\)):

(Bridge condition) For any \(\tau_{1},\tau_{2} \in \Pi^{*}\) and \(t\in \mathbb{T}^{*}\), \(\delta (\tilde{\delta }( \tau_{1},\tau_{2}),t)=\delta (\tau_{1},\delta (\tau_{2},t))=\delta (\tau_{2},\delta (\tau_{1},t))\).
Example 2.4
 (1)Let \(\mathbb{T}=\overline{q^{\mathbb{Z}}}=\overline{ \{q ^{n}: q>1, n\in \mathbb{Z} \}}\). For such a time scale, take \(e_{\Pi^{*}}=1\), we attach the shift operatorsand \(\Pi^{*}= \{q^{n}: q>1, n\in \mathbb{Z} \}\).$$\begin{aligned}& \delta (s,t)=\frac{t}{s},\\& \tilde{\delta }(s_{1},s_{2})=s_{1} \cdot s_{2} \end{aligned}$$
 (2)Let \(\mathbb{T}=\overline{q^{\mathbb{Z}}}= \{q^{n}: q>1, n \in \mathbb{Z} \}\cup \{0\}\). For this time scale, take \(e_{\Pi^{*}}=1\), we attach the shift operatorsand \(\Pi^{*}= \{q^{n}: q>1, n\in \mathbb{Z} \}\).$$\begin{aligned}& \delta (s,t)=st, \\& \tilde{\delta }(s_{1},s_{2})=s_{1} \cdot s_{2} \end{aligned}$$
 (3)Let \(\mathbb{T}=\overline{(q)^{\mathbb{Z}}}= \{( q)^{n}: q>1, n\in \mathbb{Z} \}\cup \{0\}\). For such a time scale and any \(t\in \mathbb{T}^{*}\), take \(e_{\Pi^{*}}=1\), we attach the shift operatorsand \(\Pi^{*}= \{(q)^{2n}: q>1, n\in \mathbb{Z} \}\).$$\delta (s,t)= \textstyle\begin{cases} st,& t>0, \\ \frac{t}{s}, & t< 0, \end{cases}\displaystyle \qquad \tilde{\delta }(s_{1},s_{2})=s_{1} \cdot s_{2} $$
 (4)Consider \(\mathbb{T}=\overline{q^{\mathbb{Z}}\cup q^{ \mathbb{Z}}}= \{q^{n}: q>1, n\in \mathbb{Z} \}\cup \{q ^{n}: q>1, n\in \mathbb{Z} \}\cup \{0\}\). For this time scale and any \(t\in \mathbb{T}^{*}\), take \(e_{\Pi^{*}}=1\), we attach the shift operatorsand \(\Pi^{*}= \{q^{n}: q>1, n\in \mathbb{Z} \}\).$$\delta (s,t)= \textstyle\begin{cases} st,& t>0, \\ \frac{t}{s}, & t< 0, \end{cases}\displaystyle \qquad \tilde{\delta }(s_{1},s_{2})=s_{1} \cdot s_{2} $$
 (5)Consider \(\mathbb{N}^{\frac{1}{2}}_{\pm }= \{\pm \sqrt{n}, n\in \mathbb{N} \}\). For any \(t\in \mathbb{T}^{*}\), take \(e_{\Pi^{*}}=0\), we attach the shift operatorsand \(\Pi^{*}=\mathbb{N}^{\frac{1}{2}}_{\pm }= \{\pm \sqrt{n}: n \in \mathbb{N} \}\).$$\delta (s,t)= \textstyle\begin{cases} \sqrt{t^{2}+\operatorname{sgn}(s)\cdot s^{2}}, & t>0, \\ \sqrt{t^{2}\operatorname{sgn}(s)\cdot s^{2}}, & t< 0, \end{cases}\displaystyle \qquad \tilde{\delta }(s_{1},s_{2})= \textstyle\begin{cases} \sqrt{s_{1}^{2}+\operatorname{sgn}(s_{2})\cdot s_{2}^{2}}, & s_{1}>0, \\ \sqrt{s_{1}^{2}\operatorname{sgn}(s_{2})\cdot s_{2}^{2}}, & s_{1}< 0, \end{cases} $$
In the next example, one will see that the condition (\(P_{2}\)) from Definition 2.7 may not be satisfied but the complete closedness for some types of time scales still can be guaranteed under the shift δ.
Example 2.5
Let \(\mathbb{T}_{1}=\overline{ \{q^{n}: q>1, n\in \mathbb{Z} ^{+} \}}\) and \(\mathbb{T}_{2}=\overline{ \{q^{n}: q>1, n \in \mathbb{Z}^{} \}}\). For these two time scales, \(\Pi_{1}= \mathbb{T}_{1}\), \(\Pi_{2}=\mathbb{T}_{2}\). Denote the adjoint mapping between \(\mathbb{T}_{1}\) and \(\Pi_{1}\) by \(F_{1}\), and the mapping between \(\mathbb{T}_{2}\) and \(\Pi_{2}\) by \(F_{2}\), then it is easy to obtain \(F_{1}=F_{2}=I\), where I is an identity mapping. Take \(e_{\Pi^{*}}=1\) and \(\Pi^{*}= \{q^{n}: q>1, n\in \mathbb{Z} \}=\Pi_{1}^{*}\cup \Pi_{2}^{*}\), where \(\Pi_{1}^{*}= \{q ^{n}: q>1, n\in \mathbb{Z}^{+} \}\cup \{1\}\), \(\Pi_{2}^{*}= \{q^{n}: q>1, n\in \mathbb{Z}^{} \}\cup \{1\}\), \(\tilde{\delta }_{1}=\tilde{\delta }_{2}=s_{1}s_{2}\).
It is clear that for any \(s_{1}\in \Pi_{1}^{*}\), \(s_{2}\in \Pi_{2}^{*}\), we obtain \(\delta_{1}(s_{1},t_{1})=s_{1}t_{1}\in \mathbb{T}^{*}_{1}\) for all \(t_{1}\in \mathbb{T}_{1}^{*}\) but \(s_{1}^{1}\notin \Pi_{1} ^{*}\backslash \{1\}\), which implies that \(\delta_{1}(s_{1}^{1},t _{1})=\frac{t_{1}}{s_{1}}\notin \mathbb{T}_{1}^{*}\) for \(t_{1}=q\), \(s _{1}=q^{2}\).
Similarly, \(\delta_{2}(s_{2},t_{2})=s_{2}t_{2}\in \mathbb{T}^{*}_{2}\) for all \(t_{2}\in \mathbb{T}_{2}^{*}\) but \(s_{2}^{1}\notin \Pi_{2} ^{*}\backslash \{1\}\), which implies that \(\delta_{2}(s_{2}^{1},t _{2})=\frac{t_{2}}{s_{2}}\notin \mathbb{T}_{2}^{*}\) for \(t_{2}= \frac{1}{q}\), \(s_{2}=\frac{1}{q^{2}}\).
Hence, under the shift operators \(\delta_{1}(s,t)=\delta_{2}(s,t)=st\), one can see that \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\) are completely closed. However, all the elements from \(\Pi_{1}^{*}\backslash \{1\}\) and \(\Pi_{2}^{*}\backslash \{1\}\) have no corresponding inverse elements in \(\Pi_{1}^{*}\) and \(\Pi_{2}^{*}\), respectively, that is, \((\Pi_{1}^{*}, \tilde{\delta }_{1})\) and \((\Pi_{2}^{*},\tilde{\delta }_{2})\) are not Abelian groups but they can guarantee the complete closedness of time scales \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\).
Remark 2.6
From the condition (\(P_{2}\)) in Definition 2.7, one can easily observe that \(\delta^{1}\) exists if and only if \((\Pi^{*}, \tilde{\delta })\) is an Abelian group.
In order to include the cases of the closedness of time scales from Example 2.5, it is significant to propose the following concept of shift operator δ attached with shift directions.
Definition 2.8
 (\(P_{1}\)):

(Monotonicity) The function δ is strictly increasing with respect to all its arguments, i.e., ifthen \(t< u\) implies \(\delta (T_{0},t)<\delta (T_{0},u)\); if \((T_{1},u), (T_{2},u)\in \mathcal{D}_{\delta }\) with \(T_{1}< T_{2}\), then \(\delta (T_{1},u)<\delta (T_{2},u)\).$$(T_{0},t), (T_{0},u)\in \mathcal{D}_{\delta }:= \bigl\{ (s,t)\in \Pi ^{*}\times \mathbb{T}^{*}:\delta (s,t)\in \mathbb{T}^{*} \bigr\} , $$
 (\(P_{2}\)):

(Existence of identity element) \(e_{\Pi^{*}} \in \Pi^{*}\) and \(\delta (e_{\Pi^{*}},t)=t\) for any \(t\in \mathbb{T} ^{*}\), where \(e_{\Pi^{*}}\) is the identity element in \(\Pi^{*}\).
 (\(P_{3}\)):

(Bridge condition) For any \(\tau_{1},\tau_{2} \in \Pi^{*}\) and \(t\in \mathbb{T}^{*}\), \(\delta (\tilde{\delta }( \tau_{1},\tau_{2}),t)=\delta (\tau_{1},\delta (\tau_{2},t))=\delta (\tau_{2},\delta (\tau_{1},t))\).
Remark 2.7
Note that the condition (\(P_{2}\)) from Definition 2.7 is not satisfied in Definition 2.8 since \((\Pi^{*},\tilde{\delta })\) may not be an Abelian group in Definition 2.8, i.e., \(\delta^{1}\) may not exist.
Definition 2.9
 (1)
We say δ is a positivedirection shift operator if for any \(p>e_{\Pi^{*}}\) and \(p\in \Pi^{*}\), there exists a number \(P>p\) and \(P\in \Pi^{*}\) such that \(\delta (P,t)\in \mathbb{T}^{*}\) for all \(t\in \mathbb{T}^{*}\).
 (2)
We say δ is a negativedirection shift operator if for any \(q< e_{\Pi^{*}}\) and \(q\in \Pi^{*}\), there exists a number \(Q< q\) and \(Q\in \Pi^{*}\) such that \(\delta (Q,t)\in \mathbb{T}^{*}\) for all \(t\in \mathbb{T}^{*}\).
 (3)
We say δ is a bidirection shift operator if for any \(p>e_{\Pi^{*}}\) and \(q< e_{\Pi^{*}}\), where \(p,q\in \Pi^{*}\), there exist two numbers \(P>p\), \(Q< q\) and \(P,Q\in \Pi^{*}\) such that \(\delta (P,t), \delta (Q,t)\in \mathbb{T}^{*}\) for all \(t\in \mathbb{T}^{*}\).
 (4)
We say δ is an orienteddirection shift operator if δ is a positivedirection shift operator or a negativedirection shift operator.
Remark 2.8
Under Definition 2.9, the complete closedness of time scales in Example 2.5 can be well described, that is, δ for \(\mathbb{T}_{1}\) is a positivedirection shift operator, δ for \(\mathbb{T}_{2}\) is a negativedirection shift operator. In addition, all the operators δ in Example 2.4 are bidirection shift operators. Also, one can observe that the sets \((\Pi^{*}, \tilde{\delta })\) in Example 2.4 are Abelian groups, but \((\Pi_{1}^{*},\tilde{\delta }_{1})\) and \((\Pi_{2}^{*},\tilde{\delta } _{2})\) in Example 2.5 are not Abelian groups.
Remark 2.9
 (i)
If \(\mathbb{T}=\mathbb{R}\), then \(\delta (s,t)=t+s\), where \(s\in \Pi^{*}=\mathbb{R}\) and \(\tilde{\delta }(s_{1},s_{2})=s_{1}+s _{2}\). It is easy to observe that \((\Pi^{*},\tilde{\delta })\) forms an Abelian group, so one can easily obtain that δ is a bidirection shift operator.
 (ii)
If \(\mathbb{T}=h\mathbb{Z}\), \(h>0\), then \(\delta (s,t)=t+s\), where \(s\in \Pi^{*}=h\mathbb{Z}\) and \(\tilde{\delta }(s_{1},s_{2})=s _{1}+s_{2}\). Also, one can easily get that δ is a bidirection shift operator because \((\Pi^{*},\tilde{\delta })\) forms an Abelian group.
 (iii)
If \(\mathbb{T}=\bigcup_{k=0}^{+\infty }[2k,2k+1]\), then \(\delta (s,t)=t+s\), where \(s\in \Pi^{*}=\{2n, n\in \mathbb{N}\}\) and \(\tilde{\delta }(s_{1},s_{2})=s_{1}+s_{2}\). Since each element from \(\Pi^{*}\) has no inverse element in \(\Pi^{*}\backslash \{0\}\) with respect to the operator δ̃, i.e., \((\Pi^{*}, \tilde{\delta })\) is not an Abelian group. In fact, one can easily see that δ is a positivedirection shift operator.
In the literature [17], the authors proposed the concept of time scales attached with translation direction to discuss the closedness of time scales under translations. In this paper, one can observe that the translation of time scales is just a particular case of the shift of time scales. Hence, it is significant to consider the closedness of time scales under shifts because not only can it unify the discrete and continuous time scales (i.e., \(\mathbb{T}=h\mathbb{Z}\) or \(\mathbb{R}\)) but also the quantum time scale (i.e., \(\mathbb{T}=\overline{q^{ \mathbb{Z}}}\)) and other irregular types of time scales such as \(\overline{(q)^{\mathbb{Z}}}\), \(\mathbb{N}_{\pm }^{\frac{1}{2}}\), etc.
For convenience, we introduce the following concepts of semigroup of \(\Pi^{*}\) attached with shift direction.
Definition 2.10
According to Definition 2.10, the following theorem is obvious.
Theorem 2.10
If \(\Pi^{*}\) is a semigroup with negativedirection (or positivedirection), then δ is a negativedirection shift operator (or a positivedirection shift operator).
In what follows, we assume that δ is a bidirection shift operator, i.e., \((\Pi^{*},\tilde{\delta })\) forms an Abelian group. From Definition 2.7, we introduce the concept of matched spaces for time scales and establish some related properties.
Definition 2.11
Let the pair \((\Pi^{*},\tilde{\delta })\) be an Abelian group and \(\Pi^{*}\), \(\mathbb{T}^{*}\) be the largest subsets of the time scales Π and \(\mathbb{T}\), respectively. Further, let Π be an adjoint set of \(\mathbb{T}\) and F be the adjoint mapping between \(\mathbb{T}\) and Π. If there exists the shift operator δ satisfying Definition 2.7, then we say the group \((\mathbb{T}, \Pi,F,\delta)\) is a matched space for the time scale \(\mathbb{T}\).
Remark 2.11
Note that Definition 2.11 reflects the algebraic structure of matched spaces, i.e, a matched space for the time scale is the group \((\mathbb{T},\Pi,F,\delta)\).
Definition 2.12
If \(\mathbb{T}=\Pi \), then we say the group \((\mathbb{T},\Pi,F, \delta)\) is a standard matched space.
Remark 2.12
It is easy to see that for a standard matched space, the adjoint mapping F is a unit operator I, for simplicity, \((\mathbb{T},\Pi,F, \delta):=(\mathbb{T},\delta)\). Under the standard matched space \((\mathbb{T},\delta)\), one can employ the algebraic structure to include all the results from [15]. In fact, let \(\Pi = \mathbb{T}\), \(e_{\Pi^{*}}=t_{0}\), \(\delta (s,t)=\delta_{+}(s,t)\) and \(\delta (s^{1},t)=\delta^{1}(s,t)=\delta_{}(s,t)\), where \(s\in [t_{0},+\infty)_{\mathbb{T}^{*}}\) and \(t_{0}\) is an initial point, then all the results from [15] become a particular case of the established results in this paper.
Lemma 2.13
 (1)
\(\delta^{1}(e_{\Pi^{*}},t)=\delta (e_{\Pi^{*}},t)=t\) for all \(t\in \mathbb{T}^{*}\).
 (2)
If \((s,u)\in \mathcal{D}_{\delta^{1}}\) and \((s,t)\in \mathcal{D}_{\delta }\), then \(\delta (s,t)=u\) implies \(\delta^{1}(s,u)=t\) and \(\delta^{1}(s,u)=t\) implies \(\delta (s,t)=u\).
 (3)
If \((T_{1},u),(T_{2},u)\in \mathcal{D}_{\delta^{1}}\) and \(T_{1}T_{2}>0\), \(T_{1}< T_{2}\), then \(\delta^{1}(T_{1},u)>\delta^{1}(T _{2},u)\).
 (4)
\(\delta (\tau,\delta^{1}(s,t_{0} ))=\delta^{1} (s,\delta (\tau,t_{0}))\) for any \(t_{0}\in \mathbb{T}^{*}\).
 (5)
\(\delta (\tilde{\delta }(u,s),\delta^{1}(s,v))= \delta (u,v)\); \(\delta^{1} (\tilde{\delta }(u,s),\delta (s,v))=\delta^{1}(u,v)\).
 (6)
If \(\delta (\tau,\cdot)\) is Δdifferentiable in its second variable, then \(\delta^{\Delta }(\tau,\cdot)>0\).
Proof
(1) Since \(\delta^{1}(e_{\Pi^{*}},t)=\delta (e_{\Pi^{*}}^{1},t)= \delta (e_{\Pi^{*}},t)\) for all \(t\in \mathbb{T}^{*}\), we can obtain the result.
(6) Since \(\delta (\tau,\cdot)\) is strictly increasing in its second variable, we have (6) by Corollary 1.16 from [8]. This completes the proof. □
Corollary 2.14
Let \(\mathbb{T}^{*}\) be the largest subset of the time scale \(\mathbb{T}\). For a standard matched space \((\mathbb{T},\delta)\), \(\delta (\tau,t)=\delta (t,\tau)\) holds for all \((\tau,t)\in \mathbb{T}^{*}\times \mathbb{T}^{*}\).
Proof
Next, we provide several examples to show the algebraic structure of matched spaces for time scales.
Example 2.15
 (1)\(\mathbb{T}=\{\pm n^{2}:n\in \mathbb{Z}\}\), \(\Pi =\{ \tau:\tau =n, n\in \mathbb{Z}\}\), thus, the shift operator δ can be taken as follows:Thus, \(\forall \tau_{1},\tau_{2}\in \Pi^{*}=\Pi \), \(\tilde{\delta }( \tau_{1},\tau_{2})=\tau_{1}+\tau_{2}\). Obviously, \(\tilde{\delta }( \tau_{1},\tau_{2})\in \Pi^{*}\). On the other hand, there exists a bijective mapping$$\delta (\tau,t)= \textstyle\begin{cases} \operatorname{sgn}(\sqrt{t}+ \tau)\cdot (\sqrt{t}+ \tau)^{2}, & \mbox{if }t>0, \\ \operatorname{sgn}(\tau)\tau^{2}, & \mbox{if }t=0, \\ \operatorname{sgn}(\sqrt{t} \tau)\cdot (\sqrt{t} \tau)^{2}, &\mbox{if }t< 0. \end{cases}\displaystyle \qquad \tau =n_{0},\quad \forall n_{0}\in \mathbb{Z}. $$i.e., \(F(A)=\operatorname{sgn}(A)\sqrt{\vert A\vert }\). Hence, we can obtain a matched space \((\mathbb{T},\Pi,F,\delta)\) for the time scale \(\mathbb{T}\).$$\begin{aligned} \textstyle\begin{array}{cccc} F:\quad & \mathbb{T} &\rightarrow& \Pi, \\ & A\in \bigl\{ \pm n^{2}, n\in \mathbb{Z} \bigr\} &\rightarrow& B \in \{n, n\in \mathbb{Z} \}, \\ & A &\rightarrow& \operatorname{sgn}(A)\sqrt{\vert A\vert }=B, \end{array}\displaystyle \end{aligned}$$
 (2)\(\mathbb{T}=\overline{q^{\mathbb{Z}}}\) and \(\Pi =\overline{q ^{\mathbb{Z}}}\). \(\delta (\tau,t)=\tau t\), \(\tau =q^{n_{0}}\), \(\forall n_{0}\in \mathbb{Z}\). Thus, \(\forall \tau_{1},\tau_{2}\in \Pi^{*}= \Pi \backslash \{0\}\), \(\tilde{\delta }(\tau_{1},\tau_{2})=\tau_{1} \cdot \tau_{2}\). Obviously, \(\tilde{\delta }(\tau_{1},\tau_{2})\in \Pi ^{*}\). On the other hand, there exists a bijective mappingi.e., \(F(A)=A\). Hence, we can obtain a matched space \((\mathbb{T}, \Pi,F,\delta)\) for the time scale \(\mathbb{T}\).$$\begin{aligned} \textstyle\begin{array}{cccc} F:\quad & \mathbb{T} &\rightarrow& \Pi , \\ &A\in \overline{ \bigl\{ q^{n}, n\in \mathbb{Z} \bigr\} } &\rightarrow& B \in \overline{ \bigl\{ q^{n}, n\in \mathbb{Z} \bigr\} } , \\ & A &\rightarrow& A=B, \end{array}\displaystyle \end{aligned}$$
 (3)\(\mathbb{T}= \overline{\bigcup_{n\in \mathbb{Z}}[2^{2n},2^{2n+1}]}\) and \(\Pi =\overline{ \{\tau:\tau =2^{2n}, n\in \mathbb{Z}\}}\). \(\delta (\tau,t)=\tau t\), \(\tau =2^{2n_{0}}\), \(\forall n_{0}\in \mathbb{Z}\). Thus, \(\forall \tau _{1},\tau_{2}\in \Pi^{*}=\Pi \backslash \{0\}\), \(\tilde{\delta }(\tau _{1},\tau_{2})=\tau_{1}\cdot \tau_{2}\). Obviously, \(\tilde{\delta }( \tau_{1},\tau_{2})\in \Pi^{*}\). On the other hand, there exists a bijective mappingi.e., \(F(A)=B\). Hence, we can obtain a matched space \((\mathbb{T}, \Pi,F,\delta)\) for \(\mathbb{T}\).$$\begin{aligned} \textstyle\begin{array}{cccc}F:\quad & \mathbb{T} &\rightarrow& \Pi, \\ &A\in \overline{\bigcup_{n\in \mathbb{Z}}\bigl[2^{2n},2^{2n+1} \bigr]} &\rightarrow& B\in \overline{\bigl\{ \tau:\tau =2^{2n}, n\in \mathbb{Z}\bigr\} } , \\ & A &\rightarrow& \mbox{the left point of the interval }A \\ & A=0 &\rightarrow& B=0, \end{array}\displaystyle \end{aligned}$$
 (4)\(\mathbb{T}= \{\frac{q^{n}}{1+q^{n}}:q>1 \mbox{is a constant and }n\in \mathbb{Z} \}\cup \{0,1\}\) and \(\Pi =\overline{ \{\tau:\tau =q^{n}, n\in \mathbb{Z}\}}\). Thus, the shift operators can be as follows:Thus, we can obtain \(\tilde{\delta }(\tau_{1},\tau_{2})=\tau_{1} \cdot \tau_{2}\). Obviously, \(\tilde{\delta }(\tau_{1},\tau_{2}) \in \Pi^{*}=\Pi \backslash \{0\}\). On the other hand, there exists a bijective mapping$$\delta (\tau,t)=\frac{q^{\frac{1}{\ln q}(\ln \frac{t}{1t}+\ln \tau)}}{1+q^{\frac{1}{\ln q}(\ln \frac{t}{1t}+\ln \tau)}},\quad \tau =q ^{n_{0}}, \forall n_{0}\in \mathbb{Z}. $$i.e., \(F(A)=B\). Hence, we can obtain a matched space \((\mathbb{T}, \Pi,F,\delta)\) for \(\mathbb{T}\).$$\begin{aligned} \textstyle\begin{array}{cccc}F:\quad & \mathbb{T} &\rightarrow& \Pi, \\ &\hspace{106pt}A\in \biggl\{ \frac{q^{n}}{1+q^{n}}: \\ &q>1 \mbox{ is a constant and }n\in \mathbb{Z} \biggr\} \cup \{0,1\} &\rightarrow& B\in \overline{\bigl\{ \tau:\tau =q^{n}, n\in \mathbb{Z} \bigr\} }, \\ & A &\rightarrow& \frac{A}{1A} , \\ & A=0 &\rightarrow& B=0, \\ & A=1 &\rightarrow& B=1, \end{array}\displaystyle \end{aligned}$$
 (5)If \(\mathbb{T}=\bigcup_{k\in \mathbb{Z}}[k(a+b),k(a+b)+b]\), where \(a\neq b\) and \(\Pi =\{\tau:\tau =n(a+b), n\in \mathbb{Z}\}\). \(\delta (\tau,t)=t+\tau\), \(\tau =n_{0}(a+b)\), \(\forall n_{0}\in \mathbb{Z}\). Thus, we can obtain \(\tilde{\delta }(\tau_{1},\tau_{2})= \tau_{1}+\tau_{2}\). Obviously, \(\tilde{\delta }(\tau_{1},\tau_{2}) \in \Pi^{*}=\Pi \). On the other hand, there exists a bijective mappingi.e., \(F(A)=B\). Hence, we can obtain a matched space \((\mathbb{T}, \Pi,F,\delta)\) for \(\mathbb{T}\).$$\begin{aligned} \textstyle\begin{array}{cccc}F:\quad & \mathbb{T} &\rightarrow& \Pi, \\ &A\in \bigcup_{k\in \mathbb{Z}}\bigl[k(a+b),k(a+b)+b\bigr] &\rightarrow& B\in \bigl\{ \tau:\tau =n(a+b), n\in \mathbb{Z}\bigr\} , \\ & A &\rightarrow& \mbox{the left point of the interval }A, \end{array}\displaystyle \end{aligned}$$
Using the algebraic structure of matched spaces, we introduce the following new concept of periodic time scales.
Definition 2.13
Remark 2.16
From Definition 2.13, one should note that \(\tilde{\Pi }\subseteq \Pi^{*}\subseteq \Pi \) and for every \(\tau \in \tilde{\Pi }\), there exists \(\tau^{1}\in \tilde{\Pi }\), i.e., there exists an inverse element for every element in Π̃.
Remark 2.17
For any \(t\in \mathbb{T}^{*}\), if \((\tau_{2},t)\in \mathcal{D}_{ \delta }\), \((\tau_{1},t)\in \mathcal{D}_{\delta }\), then by Definition 2.7, there exists a function \(\tilde{\delta }:\Pi^{*}\times \Pi ^{*}\rightarrow \Pi^{*}\) such that \(\tilde{\delta }(\tau_{1},\tau_{2}) \in \Pi^{*}\). Obviously, if \(\mathbb{T}\) is a periodic time scale in the sense of Definition 1.1, i.e., \(e_{\Pi^{*}}=0\) and \(\tilde{\Pi }= \Pi^{*}=\Pi = \{\tau \in \mathbb{R}:t\pm \tau \in \mathcal{D}_{ \delta },\forall t\in \mathbb{T} \}\notin \{\{0\},\emptyset \}\), we have \(\tau_{1},\tau_{2}\in \tilde{ \Pi }\), then \(\tilde{\delta }(\tau_{1},\tau_{2})=\tau_{1}+\tau_{2}\in \tilde{\Pi }\).
Remark 2.18
From (3), one can see \((\tau^{\pm 1},t)\in \mathcal{D}_{ \delta }\) implies that \(\delta (\tau^{1},t)=\delta^{1}(\tau,t)\) exists.
In the following, we present some periodic time scales in the sense of Definition 2.13.
Example 2.19
 (1)$$\delta (\tau,t)= \textstyle\begin{cases} \operatorname{sgn}(\sqrt{t}+ \tau)\cdot (\sqrt{t}+ \tau)^{2}, & \mbox{if }t>0, \tau =\pm 1, \\ \operatorname{sgn}(\tau), & \mbox{if }t=0, \tau =\pm 1, \\ \operatorname{sgn}(\sqrt{t}\tau)\cdot (\sqrt{t}\tau)^{2}, & \mbox{if }t< 0, \tau =\pm 1. \end{cases} $$
 (2)
\(\mathbb{T}=\overline{q^{\mathbb{Z}}}\), \(\delta (\tau,t)= \tau t\), \(\tau =q^{\pm 1}\).
 (3)
\(\mathbb{T}= \overline{\bigcup_{n\in \mathbb{Z}}[2^{2n},2^{2n+1}]}\), \(\delta ( \tau,t)=\tau t\), \(\tau =4^{\pm 1}\).
 (4)
\(\delta (\tau,t)=\frac{q^{\frac{1}{\ln q}(\ln \frac{t}{1t}+\ln \tau)}}{1+q^{\frac{1}{\ln q}(\ln \frac{t}{1t}+ \ln \tau)}}\), \(\tau =q^{\pm 1}\).
 (5)
\(\delta (\tau,t)=t+\tau\), \(\tau =\pm (a+b)\).
Remark 2.20
 (i)
for (1) from Example 2.3, let \(\delta (\tau,t)=t ^{\log_{3}\tau }\), \(\tau =3^{\pm 1}\in \Pi^{*}\), and one can obtain \(\delta (\tau,t)\in \mathbb{T}^{*}\) under the matched space \((\mathbb{T},\Pi,F,\delta)\).
 (ii)for (3) from Example 2.3, letand one can obtain \(\delta (\tau,t)\in \mathbb{T}^{*}\) under the matched space \((\mathbb{T},\Pi,F,\delta)\).$$\delta (\tau,t)=t+\operatorname{sgn}(\tau)\sqrt{\sqrt{3}(a+b)\vert \tau \vert },\quad \tau =\pm \frac{a+b}{\sqrt{3}}\in \Pi^{*}, $$
Lemma 2.21
Proof
If \(\sigma (t^{*})=t^{*}\), then (5) implies \(\delta_{\tau }(t ^{*})=\delta_{\tau } (\sigma (t^{*}))\). That is, \(\delta_{ \tau }(t^{*})=\sigma (\delta_{\tau }(t^{*}))\).
Corollary 2.22
Proof
From the proof process of Lemma 2.21, we immediately obtain (7). By (7), we obtain that \(\delta_{\tau } (\sigma (s))=\sigma (\delta_{\tau }(s))\) for all \(s\in \mathbb{T} ^{*}\). Substituting \(s=\delta^{1}_{\tau }(t)\), we obtain \(\delta_{ \tau } (\sigma (\delta^{1}_{\tau }(t)))=\sigma (\delta _{\tau }(\delta^{1}_{\tau }(t)))\), and it follows from (2) in Lemma 2.13 that \(\delta^{1}_{\tau } (\sigma (t))=\sigma (\delta^{1}_{\tau }(t))\) for all \(t\in \mathbb{T}^{*}\). This completes the proof. □
Observe that (7) along with (8) yields (4).
Definition 2.14
Example 2.23
Example 2.24
In the following, we introduce the concept of Δperiodic function under the matched space \((\mathbb{T},\Pi,F,\delta)\).
Definition 2.15
Example 2.25
Theorem 2.26
Proof
Remark 2.27
From Definition 2.26, if the time scale is in the sense of Definition 1.1, then one can obtain \(\int_{t_{0}\pm \tau }^{t \pm \tau }f(s)\Delta s=\int_{t_{0}}^{t}f(s)\Delta s\) by taking \(\delta_{\tau^{\pm 1}}(t)=t\pm \tau \).
In the following, according to Definitions 2.8 and 2.9, one can also introduce the concept of matched spaces for time scales under shift operators attached with the shift directions.
Definition 2.16
 (i)
if δ is a bidirection shift operator, then we say \((\mathbb{T},\Pi,F,\delta)\) is a bidirection matched space;
 (ii)
if δ is a positivedirection shift operator, then we say \((\mathbb{T},\Pi,F,\delta)\) is a positivedirection matched space;
 (iii)
if δ is a negativedirection shift operator, then we say \((\mathbb{T},\Pi,F,\delta)\) is a negativedirection matched space;
 (iv)
a positivedirection or negativedirection matched space is called an orienteddirection matched space.
Remark 2.28
Note that the matched space \((\mathbb{T},\Pi,F,\delta)\) established in the sense of Definition 2.7 is the bidirection matched space since δ is the bidirection shift operator.
Definition 2.17
 (i)
if \(\tilde{\Pi }^{+}:=\tilde{\Pi }\cap [e_{\Pi^{*}},+ \infty)\notin \{\{e_{\Pi^{*}}\},\emptyset \}\), then we say \(\mathbb{T}\) is a positivedirection complete closedness time scale;
 (ii)
if \(\tilde{\Pi }^{}:=\tilde{\Pi }\cap (\infty,e_{\Pi ^{*}}]\notin \{\{e_{\Pi^{*}}\},\emptyset \}\), then we say \(\mathbb{T}\) is a negativedirection complete closedness time scale;
 (iii)
if \(\tilde{\Pi }^{\pm }:=\tilde{\Pi }^{+}\cup \tilde{\Pi }^{}\) and \(\tilde{\Pi }^{+},\tilde{\Pi }^{}\notin \{\{e_{\Pi^{*}}\},\emptyset \}\), then we say \(\mathbb{T}\) is a bidirection complete closedness time scale;
 (iv)
a positivedirection or negativedirection complete closedness time scale is called an orienteddirection complete closedness time scale.
Remark 2.29
For Example 2.5, according to Definition 2.17, one can observe that \(\mathbb{T}_{1}\) is a positivedirection complete closedness time scale under the matched space \((\mathbb{T}_{1},\Pi _{1},F_{1},\delta_{1})\), and \(\mathbb{T}_{2}\) is a negativedirection complete closedness time scale under the matched space \((\mathbb{T} _{2},\Pi_{2},F_{2},\delta_{2})\). Moreover, a bidirection complete closedness time scale is actually equivalent to the periodic time scale under Definition 2.13.
3 Applications
In this section, we introduce some new concepts of almost periodic functions and almost automorphic functions based on the algebraic structure of matched spaces for time scales. Throughout this section, we assume that \(\mathbb{X}\) is a Banach space and \(D\subseteq \mathbb{X}\) is an open set.
In the following, we assume that \((\mathbb{T},\Pi,F,\delta)\) is a bidirection matched space, then all the elements from \(\Pi^{*}\) have the corresponding inverse elements in \(\Pi^{*}\).
Definition 3.1
Example 3.1
Remark 3.2
From Definition 3.1, one can easily establish the following classical concepts of almost periodic functions on \(\mathbb{R}\) and \(h\mathbb{Z}\), \(h>0\), etc. Here we present some of them.
(i) Let \(\mathbb{T}=\bigcup_{k\in \mathbb{Z}}[k(a+b),k(a+b)+b]\), where \(a\neq b\). Obviously, if \(b=0\), \(a=1\), then \(\mathbb{T}=\mathbb{Z}\). If \(b=1\), \(a=0\), then \(\mathbb{T}=\mathbb{R}\). Let \(\Pi^{*}=\{n(a+b):n \in \mathbb{Z}\}\), then \(\tilde{\Pi }=\Pi^{*}\) and \(\tilde{\delta }( \tau_{1},\tau_{2})=\tau_{1}+\tau_{2}\), where \(\tau_{1},\tau_{2}\in \Pi ^{*}\). Thus, let \(\delta_{\tau^{\pm 1}}(t)=t\pm \tau \), then we can easily obtain the following concept of almost periodic functions on this periodic time scale.
Definition 3.2
(ii) Let \(\mathbb{T}=\overline{q^{\mathbb{Z}}}\) and \(\Pi^{*}=q^{\mathbb{Z}}\). Then it is easy to get \(\tilde{\Pi }=\Pi ^{*}=q^{\mathbb{Z}}\), \(\tilde{\delta }(\tau_{1},\tau_{2})=\tau_{1} \cdot \tau_{2}\), where \(\tau_{1},\tau_{2}\in \Pi^{*}\). Now, let \(\delta_{\tau^{\pm 1}}(t)=t\tau^{\pm 1}\), then we can easily obtain the following concept of almost periodic functions on the quantum time scale.
Definition 3.3
Definition 3.4
Note that Definition 3.2 is accurately suitable for any periodic time scale in the sense of Definition 1.1, which implies that the almost periodic functions investigated in [13, 14, 16–18] are just some particular cases of Definition 3.1. In fact, the almost periodic phenomenon reflected by Example 3.1 would be impossible to be studied in the previous literature works because there have been no concepts of almost periodic functions under nontranslational shifts until now.
Remark 3.3
According to Definition 2.16, if we attach a shift direction to the matched space, then we can derive some concepts of almost periodic functions attached with shift directions. We present some of them here, the readers can also establish the similar concepts on other irregular time scales.
(i) Let \(\mathbb{T}=\overline{ \{q^{n}: q>1, n\in \mathbb{Z}^{+} \}}\) and \(\Pi^{*}= \{q^{n}: q>1, n\in \mathbb{Z}^{+} \}\cup \{1\}\). Then it is easy to get \(\tilde{\Pi }=\Pi^{*}\), \(\tilde{\delta }(\tau_{1},\tau_{2})=\tau_{1} \cdot \tau_{2}\), where \(\tau_{1},\tau_{2}\in \Pi^{*}\). Now, let \(\delta_{\tau }(t)=t\tau \), then we can easily obtain the following concept of positivedirection almost periodic functions on the time scale \(\mathbb{T}\).
Definition 3.5
(ii) Let \(\mathbb{T}=\overline{ \{q^{n}: q>1, n\in \mathbb{Z}^{} \}}\) and \(\Pi^{*}= \{q^{n}: q>1, n\in \mathbb{Z}^{} \}\cup \{1\}\). Then it is easy to get \(\tilde{\Pi }=\Pi^{*}\), \(\tilde{\delta }(\tau_{1},\tau_{2})=\tau_{1} \cdot \tau_{2}\), where \(\tau_{1},\tau_{2}\in \Pi^{*}\). Now, let \(\delta_{\tau }(t)=t\tau \), then we can easily obtain the following concept of negativedirection almost periodic functions on the time scale \(\mathbb{T}\).
Definition 3.6
From Definitions 3.5 and 3.6, one can observe that \((\Pi^{*},\tilde{\delta })\) are not Abelian groups but are just a semigroup towards the operators δ̃. We can unify them as follows.
Definition 3.7
 (i)
if \((\mathbb{T},\Pi,F,\delta)\) is a positivedirection matched space (or \(\Pi^{*}\) is a positivedirection semigroup), then f is called the positivedirection almost periodic function;
 (ii)
if \((\mathbb{T},\Pi,F,\delta)\) is a negativedirection matched space (or \(\Pi^{*}\) is a negativedirection semigroup), then f is called the negativedirection almost periodic function;
 (iii)
if \((\mathbb{T},\Pi,F,\delta)\) is a bidirection matched space (or \(\Pi^{*}\) is an Abelian group), then f is called the bidirection almost periodic function.
In the sequel, for simplicity and comprehensive statement, we assume that a matched space is an orienteddirection matched space, we can extend Definition 3.7 to the orienteddirection Δalmost periodic functions.
Definition 3.8
 (i)
if \((\mathbb{T},\Pi,F,\delta)\) is a positivedirection matched space (or \(\Pi^{*}\) is a positivedirection semigroup), then f is called the positivedirection Δalmost periodic function;
 (ii)
if \((\mathbb{T},\Pi,F,\delta)\) is a negativedirection matched space (or \(\Pi^{*}\) is a negativedirection semigroup), then f is called the negativedirection Δalmost periodic function;
 (iii)
if \((\mathbb{T},\Pi,F,\delta)\) is a bidirection matched space (or \(\Pi^{*}\) is an Abelian group), then f is called the bidirection Δalmost periodic function.
Example 3.4
Remark 3.5
The Δalmost periodicity from Definition 3.8 mainly focuses on the almost periodic functions on the “quantumlike” time scales such as \(q^{\mathbb{Z}}\), \((q)^{\mathbb{Z}}\) and \(\overline{q ^{\mathbb{Z}}\cup q^{\mathbb{Z}}}\), etc. Note that the almost periodicity and the Δalmost periodicity are equivalent on periodic time scales defined by Definition 1.1 since \(\delta_{ \tau }^{\Delta }(t)=(t+\tau)^{\Delta }\equiv 1\). Hence, the almost periodicity investigated in [13, 14, 16–18] is actually equivalent to the Δalmost periodicity.
Next, we will introduce the concept of almost automorphic functions under bidirection matched spaces for time scales.
Definition 3.9
 (i)Let \(f:\mathbb{T}\rightarrow \mathbb{X}\) be a bounded continuous function. We say that f is almost automorphic if from every sequence \(\{s_{n}\}\subset \tilde{\Pi }\), we can extract a subsequence \(\{\tau_{n}\}_{n=1}^{\infty }\) such thatis well defined for each \(t\in \mathbb{T}^{*}\) and$$g(t)=\lim_{n\rightarrow \infty }f \bigl(\delta_{\tau_{n}}(t) \bigr) $$for each \(t\in \mathbb{T}^{*}\). Denote by \(AA_{\delta }(\mathbb{T}, \mathbb{X})\) the set of all such functions.$$\lim_{n\rightarrow \infty }g \bigl(\delta_{\tau_{n}^{1}}(t) \bigr)= \lim _{n\rightarrow \infty }g \bigl(\delta^{1}_{\tau_{n}}(t) \bigr)=f(t) $$
 (ii)
A continuous function \(f:\mathbb{T}\times \mathbb{X} \rightarrow \mathbb{X}\) is said to be almost automorphic if \(f(t,x)\) is almost automorphic in \(t\in \mathbb{T}^{*}\) uniformly for \(x\in B\), where B is any bounded subset of \(\mathbb{X}\). Denote by \(AA_{\delta }(\mathbb{T}\times \mathbb{X},\mathbb{X})\) the set of all such functions.
Example 3.6
Remark 3.7
Note that Definition 3.9 is the concept of bidirection almost automorphic functions under a bidirection matched space. Moreover, the almost automorphy reflected by Example 3.6 has never been considered in any literature until now. This is a completely new almost automorphic function under nontranslational shifts.
Remark 3.8
If we attach the shift direction to a matched space, then one can obtain the following orienteddirection almost automorphic functions.
Definition 3.10
 (i)Let \(f:\mathbb{T}\rightarrow \mathbb{X}\) be a bounded continuous function. We say that f is orienteddirection almost automorphic if from every sequence \(\{s_{n}\}\subset \tilde{\Pi }\), we can extract a subsequence \(\{\tau_{n}\}_{n=1}^{\infty }\) such thatis well defined for each \(t\in \mathbb{T}^{*}\).$$g(t)=\lim_{n\rightarrow \infty }f \bigl(\delta_{\tau_{n}}(t) \bigr) $$
 (ii)
A continuous function \(f:\mathbb{T}\times \mathbb{X} \rightarrow \mathbb{X}\) is said to be orienteddirection almost automorphic if \(f(t,x)\) is orienteddirection almost automorphic in \(t\in \mathbb{T}^{*}\) uniformly for \(x\in B\), where B is any bounded subset of \(\mathbb{X}\).
We can extend Definition 3.9 to bidirection Δalmost automorphic functions.
Definition 3.11
 (i)Let \(f:\mathbb{T}\rightarrow \mathbb{X}\) be a bounded continuous function and the shift \(\delta_{\tau }(t)\) is Δdifferentiable with rdcontinuous bounded derivatives \(\delta^{\Delta }_{\tau }(t):=\delta^{\Delta }(\tau,t)\) for all \(t\in \mathbb{T}^{*}\). We say that f is Δalmost automorphic if from every sequence \(\{s_{n}\}\subset \tilde{\Pi }\), we can extract a subsequence \(\{\tau_{n}\}_{n=1}^{\infty }\) such thatis well defined for each \(t\in \mathbb{T}^{*}\) and$$g(t)=\lim_{n\rightarrow \infty }f \bigl(\delta_{\tau_{n}}(t) \bigr) \delta_{\tau_{n}}^{\Delta }(t) $$for each \(t\in \mathbb{T}^{*}\). Denote by \(AA_{\delta }(\mathbb{T}, \mathbb{X})\) the set of all such functions.$$\lim_{n\rightarrow \infty }g \bigl(\delta_{\tau_{n}^{1}}(t) \bigr) \delta_{\tau_{n}^{1}}^{\Delta }(t)=f(t) $$
 (ii)
A continuous function \(f:\mathbb{T}\times \mathbb{X} \rightarrow \mathbb{X}\) is said to be Δalmost automorphic if \(f(t,x)\) is Δalmost automorphic in \(t\in \mathbb{T}^{*}\) uniformly for \(x\in B\), where B is any bounded subset of \(\mathbb{X}\). Denote by \(AA_{\delta }(\mathbb{T}\times \mathbb{X}, \mathbb{X})\) the set of all such functions.
Example 3.9
Remark 3.10
It is obvious that Definition 3.11 is the concept of bidirection Δalmost automorphic functions under a bidirection matched space. Note that Δalmost automorphy of functions mainly focuses on almost automorphic functions on the “quantumlike” time scales such as \(q^{\mathbb{Z}}\), \((q)^{\mathbb{Z}}\) and \(\overline{q^{\mathbb{Z}} \cup q^{\mathbb{Z}}}\), etc. Moreover, the almost automorphy and Δalmost automorphy are equivalent on periodic time scales defined by Definition 1.1 since \(\delta_{\tau }^{\Delta }(t)=(t+ \tau)^{\Delta }\equiv 1\).
On the other hand, for ∇calculus theory is an analogy of Δcalculus theory on time scales, all the results established in this paper can be naturally extended to “∇cases”. Similar to Definition 3.11, one can also adopt the same idea to introduce the concept of ∇almost automorphic functions. Furthermore, the almost automorphy and the ∇almost automorphy are also equivalent on periodic time scales defined by Definition 1.1 since \(\delta_{\tau }^{\nabla }(t)=(t+\tau)^{\nabla }\equiv 1\). Hence, the almost automorphy investigated in [12] is actually equivalent to the ∇almost automorphy.
Remark 3.11
Similar to Definition 3.10, one can also introduce the concept of Δalmost automorphic functions attached with shift directions as follows.
Definition 3.12
 (i)Let \(f:\mathbb{T}\rightarrow \mathbb{X}\) be a bounded continuous function and the shift \(\delta_{\tau }(t)\) is Δdifferentiable with rdcontinuous bounded derivatives \(\delta^{\Delta }_{\tau }(t):=\delta^{\Delta }(\tau,t)\) for all \(t\in \mathbb{T}^{*}\). We say that f is orienteddirection Δalmost automorphic if from every sequence \(\{s_{n}\}\subset \tilde{\Pi }\), we can extract a subsequence \(\{\tau_{n}\}_{n=1}^{ \infty }\) such thatis well defined for each \(t\in \mathbb{T}^{*}\).$$g(t)=\lim_{n\rightarrow \infty }f \bigl(\delta_{\tau_{n}}(t) \bigr) \delta_{\tau_{n}}^{\Delta }(t) $$
 (ii)
A continuous function \(f:\mathbb{T}\times \mathbb{X} \rightarrow \mathbb{X}\) is said to be orienteddirection Δalmost automorphic if \(f(t,x)\) is orienteddirection Δalmost automorphic in \(t\in \mathbb{T}^{*}\) uniformly for \(x\in B\), where B is any bounded subset of \(\mathbb{X}\).
Remark 3.12
Because an orienteddirection matched space is either positivedirection or negativedirection, then Definitions 3.10 and 3.12 can be attached with specific shift direction to describe the concept of almost automorphic functions and Δalmost automorphic functions with specific directions, respectively. It is easy to derive them by substituting a specific direction for “orienteddirection” in these two definitions, so we omit them here.
4 Conclusion and discussion
This paper is concerned with the problems of complete closedness of time scales under translational and nontranslational shifts. By introducing the algebraic structure of matched spaces attached with shift directions, some basic concepts of complete closedness of time scales with different shift directions are introduced. Based on these, some new definitions of almost periodic functions and almost automorphic functions including their generalizations such as Δalmost periodic functions and Δalmost automorphic functions on irregular time scales, are proposed. Moreover, some novel almost periodicity and almost automorphy of some functions under nontranslational shift are proposed and studied, which would be impossible to be investigated on periodic time scales under translations before. The results established in this paper will mainly contribute to solving the problems of closedness of time scales under nontranslational shifts, which will be significant to studying functions on a more comprehensive scope of time scales such as the “quantumlike” time scales, e.g., \(q^{\mathbb{Z}}\), \(\overline{(q)^{ \mathbb{Z}}}\), \(\overline{q^{\mathbb{Z}}\cup q^{\mathbb{Z}}}\) and other irregular time scales, e.g., \(\mathbb{N}_{+}^{\frac{1}{2}}\), \(\mathbb{N}_{\pm }^{\frac{1}{2}}\), etc.
As an important type of functions defined by translations, almost periodic functions on time scales have been paid much attention to. In the literature works [11, 22], the authors studied almost periodic solutions of differential equations on certain time scales with transition conditions (DETC) on the basis of reduction to the impulsive differential equations (ITE). The authors employed the reversible translation transform of time scales to turn the time scale into the real line, then the differential equations on time scales can be changed into a traditional impulsive differential equations on \(\mathbb{R}\) simultaneously. Finally, they established an equivalence between DETC and traditional ITE on \(\mathbb{R}\). Their results mainly contribute to building a relationship between DETC and ITE, and through investigating the almost periodic solutions for ITE, the corresponding almost periodic solutions for DETC can be indirectly studied. However, they did not consider to directly use the Δcalculus theory on time scales to solve the almost periodic problems of differential equations on time scales since many time scales are not completely closed under translations, which will lead to the functions defined by translations being not well defined on time scales. Therefore, it was difficult to consider almost periodic problem on time scales directly through Δcalculus theory then.
 (1)
\(\mathbb{T}=\overline{q^{\mathbb{Z}}}\), it follows that \(\mu (t)\rightarrow +\infty \) as \(t\rightarrow +\infty \);
 (2)
\(\mathbb{T}=\mathbb{N}_{0}^{2}=\{n^{2},n\in \mathbb{N} _{0}\}\), it follows that \(\mu (t)\rightarrow +\infty \) as \(t\rightarrow +\infty \);
 (3)
\(\mathbb{T}=\overline{q^{\mathbb{Z}}\cup q^{\mathbb{Z}}} \cup \mathbb{N}_{0}^{2}\cup \mathbb{N}_{0}^{3}\), it follows that \(\mu (t)\rightarrow +\infty \) as \(t\rightarrow +\infty \).
On the other hand, the occurrence and change of many natural phenomena may directly depend on time scales, but not depend on the real line. The period of the occurrence and change of these phenomena can be directly and accurately described by the time scale rather than the real line. For this, Wang, Agarwal and O’Regan proposed the concept of time scales with almost periodicity and studied the doublealmost periodic solutions for dynamic equations (see [23–25]) since the recurrence of many natural phenomena may take approximative periods of time rather than absolutely equivalent time intervals. Therefore, it is more practical to consider dynamic equations on almost periodic time scales by using Δcalculus theory. To investigate some important problems on time scales directly, the important and first step is to establish welldefined classical functions on time scales. In the literature [16, 17], the authors solved the problems of closedness of time scales under translations and introduced some concepts of almost periodic functions defined by translations, which realized that one is able to solve almost periodic problems through Δcalculus theory directly without considering any relationship between the time scale and \(\mathbb{R}\). Moreover, in the literature [18], the authors introduced the concept of changingperiodic time scales and proposed the decomposition theorem of time scales to decompose an arbitrary time scale with the bounded graininess function μ into countable periodic time scales attached with directions, which will guarantee the local complete closedness of a time scale with the bounded graininess function μ under translations. Then the authors made the important supplements and comments on changingperiodic time scales and almost periodic functions (see [17, 21]). In this work, the authors construct the algebraic structure of matched spaces to solve the closedness of time scales under nontranslational shifts, which lays a foundation for considering almost periodic problems, almost automorphic problems and other related topics for dynamic equations on irregular time scales by directly using Δcalculus theory in the future.
Notes
Declarations
Acknowledgements
We express our sincere thanks to all the reviewers’ insightful comments to improve this manuscript. This work was done while Chao Wang was a visiting scholar at Texas A&M UniversityKingsville. This work is supported by Youth Fund of NSFC (No. 11601470), Tian Yuan Fund of NSFC (No. 11526181), Yunnan Province Science and Technology Department Applied Basic Research Project of China (No. 2014FB102), and Dong Lu Youth Excellent Teachers Development Program of Yunnan University (No. wx069051).
Authors’ contributions
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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