# Almost Periodic Functions on the Quantum Time Scale and Applications

Authors: Yongkun Li
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

The concept of almost periodicity was initiated by the Bohr during the period 1923-1925 [1, 2]. Bohr’s theory quickly attracted the attention of very famous mathematicians of that time. And since then the questions of the theory of almost periodic functions and almost periodic solutions of differential equations have been very interesting and challenge problems of great importance. The interaction between these two theories has enriched both. On one hand, it is well known that, in Celestial Mechanics, almost periodic solutions and stable solutions are intimately related. In the same way, stable electronic circuits exhibit almost periodic behavior. On the other hand, certain problems in differential equations have led to new definitions and results in the theory of almost periodic functions.In recent years, the theory of quantum calculus has received much attention, due to its tremendous applications in several fields of physics, such as cosmic strings and black holes, conformal quantum mechanics, nuclear and high energy physics, fractional quantum Hall effect, and high-Tc superconductors. For interested reader, we refer to, for example, [3–6] and the references cited therein.On one hand, recently, Bohner and Chieochan [7] introduced in the literature the concept of periodicity for functions defined on the quantum time scale owing to the fact that taking into account the periodicity of q-difference equations is important in order to better understand several physics phenomena. However, in reality, the almost periodic phenomenon is more common and complicated than the periodic one. Therefore, investigating the almost periodicity of dynamic equations on the quantum time scale is more interesting and more challenge.On the other hand, in order to study the almost periodicity on time scales, a concept of almost periodic time scales was proposed in [8]. Based on this concept, a series of concepts of almost periodic function classes, such as almost periodic functions [8], pseudo almost periodic functions [9], almost automorphic functions [10], weighted pseudo almost automorphic functions [11], almost periodic set-valued functions [12], and almost periodic functions in the sense of Stepanov on time scales [13] were defined successively. Although the concept of almost periodic time scales in [8] can unify the continuous and discrete situations effectively, it is very restrictive because it requires the time scale with certain global additivity. This excludes some interesting time scales, for instance, the quantum time scale, which has no such global additivity.Motivated by the above discussion, our main purpose of this paper is to propose two types of definitions of almost periodic functions on the quantum time scale, to study some basic properties of almost periodic functions on the quantum time scale, to give an equivalent definition of almost periodic functions on the quantum time scale, and to explore some of their applications to periodic dynamic equations on the quantum time scale.The organization of this paper is as follows. In Section2, we introduce some notations and definitions of time scale calculus. In Section 3, we propose the concepts of almost periodic functions on the quantum time scale and investigate some of their basic properties. In Section 4, we study the existence and uniqueness of almost periodic solutions of dynamic equations on the quantum time scale by Lyapunov method. In Section 5, we give an equivalent definition of almost periodic functions on the quantum time scale. In Section 6, as an application of our results, we first propose a class of high-order Hopfield neural networks on the quantum time scale; then by the exponential dichotomy of linear dynamic equations on time scales and the Banach fixed point theorem, we establish the existence and global exponential stability of almost periodic solutions of this class of neural networks. In Section 7, we draw a conclusion.


## 2. Preliminaries

In this section, we shall recall some basic definitions of time scale calculus.A time scaleT 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(1)σt≔inf⁡s∈T:s>t,ρt≔sup⁡s∈T:s<t,μt=σt-t.A pointt 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 Tk=T∖m; otherwise Tk=T. If T has a right-scattered minimum m, then Tk=T∖m; otherwise Tk=T.Definition 1 (see [8]).
A time scale is called an almost periodic time scale if(2)Π≔τ∈R:t+τ∈T,∀t∈T≠0.A functionf:T→R 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.Forf:T→R and t∈Tk, then f is called delta differentiable at t∈T if there exists c∈R such that, for given any ε≥0, there is an open neighborhood U of t satisfying(3)fσt-fs-cσt-s≤εσt-sfor all s∈U. In this case, c is called the delta derivative of f at t∈T and is denoted by c=fΔ(t). For T=R, we have fΔ=f′, the usual derivative, for T=Z, we have the backward difference operator, fΔ(t)=Δf(t)≔f(t+1)-f(t), and, for T=qZ¯(q>1), the quantum time scale, we have the q-derivative:(4)fΔt≔Dqft=fqt-ftq-1t,t≠0,limt→0⁡fqt)-ftq-1t,t=0.Remark 2.
Note that(5)Dqf0=df0dtif f is continuously differentiable.Define forV∈Crd[qZ¯×Rn,R], Dq+VΔ(t,x(t)) to mean that, given ε>0, there exists a right neighborhood Nε⊂N such that(6)1μt,sVσt,xσt-Vs,xσt-μt,sft,xt<Dq+VΔt,xt+εfor each s∈Nε, s>t, where μ(t,s)≡σ(t)-s. If t is right-scattered and Vt,xt is continuous at t; this reduces to(7)Dq+VΔt,xt=Vσt,xσt-Vt,xtσt-t.Lemma 3 (see [14]).
Lety,f∈Crd and p∈R+; then(8)Dqyt≤ptyt+ftforallt∈qZ¯implies(9)yt≤yt0ept,t0+∫t0tept,στfτdqτforallt∈qZ¯,where t0∈qZ¯.For more details about the theory of time scale calculus and the theory of quantum calculus, the reader may want to consult [3, 14, 15].


## 3. Almost Periodic Functions

From now on, we useX to denote the complex numbers C or the real numbers R, use D to denote an open set in X or D=X, and use S to denote an arbitrary compact subset of D.Definition 4.
A functionf∈C(qZ¯×D,X) is called an almost periodic function in t∈qZ¯ uniformly in x∈D if for any given ε>0 and each compact subset S of D; there exists l(ε,S)>0 such that each interval [qp,qpql]∩qZ¯ contains a qτ(p,ε,S)∈[qp,qpql]∩qZ¯(p∈Z) such that(10)ftqτ,x-ft,x<ε,∀t,x∈qZ¯×S.This qτ is called the ε-translation number of f and l(ε,S) is called the inclusion length of the set Eε,f,S={qτ∈qZ¯:ftqτ,x-ft,x<ε,∀(t,x)∈qZ¯×S}.Definition 5.
A functionf∈C(qZ¯×D,X) is called an almost periodic function in t∈qZ¯ uniformly in x∈D if, for any given ε>0 and each compact subset S of D, there exists l(ε,S)>0 such that each interval [qp,qpql]∩qZ¯ contains a qτ(p,ε,S)∈[qp,qpql]∩qZ¯(p∈Z) such that (11)qτftqτ,x-ft,x<ε,∀t,x∈qZ¯∖0×S.This qτ is called the ε-translation number of f and l(ε,S) is called the inclusion length of the set E{ε,f,S}={qτ∈qZ¯:qτftqτ,x-ft,x<ε,∀(t,x)∈(qZ¯∖{0})×S}.Definition 6.
A functionf∈C(qZ¯,X) is called an almost periodic function if, for any given ε>0, there exists l(ε)>0 such that each interval [qp,qpql]∩qZ¯ contains a qτ(p,ε)∈[qp,qpql]∩qZ¯(p∈Z) such that(12)ftqτ-ft<ε,∀t∈qZ¯.This qτ is called the ε-translation number of f and l(ε) is called the inclusion length of the set E{ε,f}={qτ∈qZ¯:ftqτ-ft<ε,∀t∈qZ¯}.Definition 7.
A functionf∈C(qZ¯,X) is called an almost periodic function if, for any given ε>0, there exists l(ε)>0 such that each interval [qp,qpql]∩qZ¯ contains a qτ(p,ε)∈[qp,qpql]∩qZ¯(p∈Z) such that (13)qτftqτ-ft<ε,∀t∈qZ¯∖0.This qτ is called the ε-translation number of f and l(ε) is called the inclusion length of the set E{ε,f}={qτ∈qZ¯:qτftqτ-ft<ε,∀t∈qZ¯}.Definition 8.
A functionf∈C(qZ¯,En) is called an asymptotically almost periodic function if(14)ft=pt+qt,where p is an almost periodic function on qZ¯, and q(t)→0 as t→∞.For convenience, we introduce some notations.LetBC(qZ¯×D,X)={f∈C(qZ¯×D,X) is bounded on qZ¯×D}, AP(qZ¯×D,X)={f∈C(qZ¯×D,X) is almost periodic in t∈qZ¯ uniformly in x∈D} and AP(qZ¯,X)={f∈C(qZ¯,X)isalmostperiodic}.Letα={αn}⊂Z and β={βn}⊂Z be two sequences of integer numbers. We denote α+β≔{αn+βn},-α≔{-αn} and use β⊂α to denote that β is a subsequence of α. We say α and β are common subsequences of α′ and β′, respectively, if there is some given function n(ν) such that αn=αn(ν)′ and βn=βn(ν)′.We introduce two translation operatorsT and T^, Tαf(t,x)=g(t,x) and T^f(t,x)=g(t,x) mean that g(t,x)=limn→+∞⁡f(qαnt,x) and g(t,x)=limn→+∞⁡qαnf(qαnt,x)(t≠0), respectively, and are written only when the limits exist. The mode of convergence will be specified at each use of the symbols.Remark 9.
When study the properties of almost periodic functions defined by Definitions4 and 6, we use the translation operator T. When studying the properties of almost periodic functions defined by Definitions 5 and 7, we use the translation operator T¯.Remark 10.
All the results of this section hold for almost periodic functions defined by Definitions4, 5, 6, and 7. Since their proofs are similar, we only prove them under Definitions 4 and 6.Remark 11.
A functionf:qZ¯×D→X is continuous if and only if limn→-∞f(qn,x)=f(0,x) exists uniformly in x∈S.Theorem 12.
Letf∈AP(qZ¯×D,X); then it is uniformly continuous and bounded on qZ¯×S.Proof.
For a givenε0≤1 and some compact subset S⊂D, there exists l(ε,S) such that, in each interval [qp,qpql]∩qZ¯(p∈Z), there exists a qτ∈E{ε0,f,S} such that(15)fqτt,x-ft,x<ε0≤1,∀t,x∈qZ¯×S.It follows from f∈C(qZ¯×D,X) that, for any (t,x)∈([1,ql]∩qZ¯)×S, there exists M>0 such that ft,x<M. For any given t∈qZ¯ and t≠0, take qτ∈E(ε,f,S)∩[t-1,t-1ql]; then tqτ∈[1,ql]∩qZ¯. Hence, for x∈S, we get(16)ftqτ,x<M,ftqτ,x-ft,x<1.Therefore, we obtain(17)ft,x<max⁡maxx∈S⁡f0,x,M+1,∀t,x∈qZ¯×S.
Furthermore, because of the continuity off at (0,x),x∈S, for any given ε>0, there exists a positive integer N0(ε) such that fqn,x-f0,x<ε/2 for all n<-N0 and x∈S. Thus, for any (qn1,x),(qn2,x)∈([0,q-N0]∩qZ¯)×S, we find(18)fqn1,x-fqn2,x<fqn1,x-f0,x+fqn2,x-f0,x<ε;therefore, f(t,x) is uniformly continuous on ([0,q-N0]∩qZ¯)×S. Since all the points in the interval [q-N0,+∞)∩qZ¯ are isolated points and μ(t)≥(q-1)q-N0 for all t∈[q-N0,+∞)∩qZ¯, so f(t,x) is uniformly continuous on ([q-N0,+∞)∩qZ¯)×S. The proof is completed.Theorem 13.
Letf∈AP(qZ¯×D,X); then, for any given sequence α′⊂Z, there exist a subsequence β⊂α′ and g∈C(qZ¯×D,X) such that Tβf=g holds uniformly on qZ¯×S and g∈AP(qZ¯×D,X).Proof.
For any givenε>0 and sequence α′={αn′}⊂Z, we denote αn′=τn′+γn′, where qτn′∈E{ε/2,f,S}, γn′∈Z, and 0≤γn′≤l(ε/2,S),n=1,2,…. Hence, it is easy to see that there exists a subsequence γ={γn}⊂γ′={γn′} such that γn=s for n∈N, 0≤s≤l. We can take α⊂α′, τ⊂τ′ such that α,τ common with γ and αn=τn+s for all n∈N; then(19)ftqτp-τm,x-ft,x≤ftqτp-τm,x-ftqτp,x+ftqτp,x-ft,x<ε,which implies that(20)qαp-αm=qτp-τm∈Eε,f,S.Hence, we can get(21)ftqαp,x-ftqαm,x≤supt,x∈qZ¯×S⁡ftqαp,x-ftqαm,x≤supt,x∈qZ¯×S⁡ftqαp-αm,x-ft,x<ε.Thus, we can take subsequences α(k)={αn(k)},k=1,2,…, and α(k+1)⊂α(k)⊂α such that, for any integers m,p, and all (t,x)∈qZ¯×S,(22)ftqαpk,x-ftqαmk,x<1k,k=1,2,….For all sequences α(k),k=1,2,…, we can take a sequence β={βn}, βn=αn(n); then it is easy to see that, for any integers p,m with p<m and all (t,x)∈qZ¯×S,(23)ftqβp,x-ftqβm,x<1p.Therefore, {f(tqβn,x)} converges uniformly on qZ¯×S; that is, Tβf=g holds uniformly on qZ¯×S.
Next, we show thatg∈C(qZ¯×D,X). Otherwise, there exists a point (t0,x0)∈qZ¯×D such that g is not continuous at this point. Then there exist ε0>0 and sequences {δm}⊂R+, {tm}⊂qZ¯, {xm}⊂D, where δm>0, δm→0 as m→+∞, |t0-tm|+|x0-xm|<δm and(24)gt0,x0-gtm,xm≥ε0.Letting X={xm}∪{x0}, obviously, X is a compact subset of D. Since Tβf=g holds uniformly on qZ¯×S, there exists a positive integer N=N(ε0,X) such that, for n>N,(25)ftmqβn,xm-gtm,xm<ε03∀m∈Z+and(26)ft0qβn,x0-gt0,x0<ε03.According to the uniform continuity of f on qZ¯×X, for sufficiently large m, we have(27)ft0qβn,x0-ftmqβn,xm<ε03.From (25)-(27), we get(28)gt0,x0-gtm,xm<ε0,which contradicts (24). Therefore, g is continuous on qZ¯×D.
Finally, for any compact subsetS⊂D and given ε>0, we take qτ∈E{ε,f,S}; then(29)ftqβn+τ,x-ftqβn,x<ε,∀t,x∈qZ¯×S.Letting n→+∞, we have(30)gtqτ,x-gt,x≤ε,∀t,x∈qZ¯×S,which implies that g∈AP(qZ¯×D,X). This completes the proof.Theorem 14.
Lettingf∈BC(qZ¯×D,X), if, for any sequence α′⊂Z, there exists a subsequence α⊂α′ such that Tαf exists uniformly on qZ¯×S; then f∈AP(qZ¯×D,X).Proof (proof by contradiction).
Iff∉AP(qZ¯×D,X), then there exist ε0>0 and S⊂D such that, for no matter how large l>0, we can always find an interval [qp,qpql]∩qZ¯ such that [qp,qpql]∩E{ε0,f,S}=∅.
To this end, we take a numberα1′∈Z and find an interval [qa1,qb1]∩qZ¯ with b1-a1>2α1′ and a1,b1∈2Z such that [qa1,qb1]∩E{ε0,f,S}=∅. Next, we take α2′=1/2(a1+b1), and it is easy to see that qα2′-α1′∈[qa1,qb1]∩qZ¯, and so qα2′-α1′∉E{ε0,f,S}; then we find an interval [qa2,qb2]∩qZ¯ with b2-a2>2(α1′+α2′) and a2,b2∈2Z such that [qa2,qb2]∩E{ε0,f,S}=∅. Next, we take α3′=1/2(a2+b2), obviously, qα3′-α2′,qα3′-α1′∉E{ε0,f,S}. We can repeat these process again and again and can find α4′,α5′,…, such that qαi′-αj′∉Eε0,f,S,i>j. Thus, for any i≠j,i,j=1,2,…, without loss of generality, let i>j; we find (31)supt,x∈qZ¯×S⁡ftqαi′,x-ftqαj′,x=supt,x∈qZ¯×S⁡ftqαi′-αj′,x-ft,x≥ε0;which implies that there is no uniformly convergent subsequence of {f(tqαn′,x)} for (t,x)∈qZ¯×S. This is a contradiction. Hence, f∈AP(qZ¯×D,X). This completes the proof.In view of Definitions4 and 5, by Theorems 13 and 14, one can easily get the following two results.Theorem 15.
A functionf∈C(qZ¯×D,X) is almost periodic in t uniformly in x∈D, if and only if, for any given sequence α′⊂Z, there exists a subsequence α⊂α′ such that Tαf exists uniformly on qZ¯×S.Theorem 16.
A functionf∈C(qZ¯×D,X) is almost periodic function in t uniformly in x∈D, if and only if, for any given sequence α′⊂Z, there exists a subsequence α⊂α′ such that T^αf exists uniformly on qZ¯×S.Theorem 17.
Iff∈AP(qZ¯×D,X) and φ∈AP(qZ¯,X) with φ(t)⊂S for all t∈qZ¯. Then f(·,φ(·))∈AP(qZ¯,X).Proof.
It is easy to see that for any sequenceα′⊂Z, and there exist a subsequence α⊂α′, and functions ψ and g such that Tαφ=ψ exists uniformly on qZ¯ and Tαf=g exists uniformly on qZ¯×S, where ψ∈AP(qZ¯,X) and g∈AP(qZ¯×D,X). Hence, g is uniformly continuous on qZ¯×S; then, for any given ε>0, there exists δ(ε/2)>0 such that, for any x1,x2∈S and x1-x2<δ(ε/2), we have(32)gt,x1-gt,x2<ε2,∀t∈qZ¯.In addition, there exists an N0(ε)>0 such that, for n≥N0(ε),(33)ftqαn,x-gt,x<ε2,∀t,x∈qZ¯×Sand(34)φtqαn-ψt<δε2,∀t∈qZ¯,where φ(tqnα)⊂S,ψ(t)⊂S for all t∈qZ¯. Therefore, for n≥N0(ε),(35)ftqαn,φtqαn-gt,ψt≤ftqαn,φtqαn-gt,φtqαn+gt,φtqαn-gt,ψt<ε,which implies that Tαf(t,φ(t))=g(t,ψ(t)) exists uniformly on qZ¯. Thus, f(·,φ(·))∈AP(qZ¯,X). The proof is complete.Definition 18.
Letf∈BC(qZ¯×D,X); then H(f)={g:qZ¯×D→X∣ there exits α⊂Z such that Tαf=g existing uniformly on qZ¯×S} is called the hull of f.Theorem 19.
H ( f ) is compact in uniform norm if and only if f∈AP(qZ¯×D,X).Proof.
IfH(f) is a compact set, then, for each sequence α′⊂Z, there exists a subsequence α⊂α′ such that Tαf exists uniformly on qZ¯×D.
Conversely, iff∈AP(qZ¯×D,X) and {gn}⊂H(f), then we can choose α′={αn′} such that(36)ftqαn′,x-gnt,x<1n,∀t,x∈qZ¯×S.So, we can find a subsequence α⊂α′ such that Tαf exists uniformly on qZ¯×D. Let β⊂γ={n} such that β and α are common subsequences; then(37)ftqαn,x-gβnt,x→0n→∞,∀t,x∈qZ¯×S;hence(38)limn→∞⁡gβnt,x=Tαft,x∈Hf,∀t,x∈qZ¯×S;therefore, H(f) is a compact set. The proof is complete.Theorem 20.
Iff∈AP(qZ¯×D,X), then, for any g∈H(f), H(f)=H(g).Proof.
For anyh∈H(g), there exists α′⊂Z such that Tα′g=h exists uniformly on qZ¯×S. Since f∈AP(qZ¯×D,X), one can extract a subsequence α⊂α′ such that Tαf(t,x)=limn→∞⁡f(tqαn,x) exists uniformly on qZ¯×S. Because g∈H(f), so there exists α(1)⊂Z such that(39)limn→+∞⁡ftqαn1,x=gt,x,∀t,x∈qZ¯×S;hence,(40)limn→+∞⁡ftqαn1+αn,x=gtqαn,x,∀t,x∈qZ¯×S;then, we can take β=βn=αn1+αn such that(41)ftqβn,x-gtqαn,x<1n,∀t,x∈qZ¯×S.It follows from Tβf=Tαg=Tα′g=h that h∈H(f). Thus, H(g)⊆H(f).
On the other hand, for anyg∈H(f), there exists α such that Tαf=g exists uniformly on qZ¯×S; hence,(42)ftqαn,x-gt,x→0,n→∞,∀t,x∈qZ¯×S.Let tqαn=s; we obtain(43)fs,x-gsq-αn,x→0,n→∞,∀s,x∈qZ¯×S;that is, T-αg=f exists uniformly on qZ¯×S. Thus, f∈H(g). Therefore, H(f)⊆H(g). The proof is complete.From Definition18 and Theorem 20, one can easily show the following.Theorem 21.
Iff∈AP(qZ¯×D,X), then, for any g∈H(f), g∈AP(qZ¯×D,X).Theorem 22.
Iff,g∈AP(qZ¯×D,X), then we have the following:
(a)f+g,fg∈AP(qZ¯×D,X).
(b) Ifinft∈qZ¯|g(t,x)|≥M>0, then f/g∈AP(qZ¯×D,X).Proof.
(a) Since f,g∈AP(qZ¯×D,X), from any sequence {αn′}⊂Z one can extract a subsequence {αn} such that Tαf=limn→∞⁡f(tqαn,x) exists uniformly on qZ¯×S. Then from the {αn}⊂Z we can extract a subsequence {βn′} such that Tβ′g=limn→∞g⁡(tqβn′,x) exists uniformly on qZ¯×S. Consequently, we can extract a subsequence {βn}⊂{βn′} such that Tβ(f+g)=limn→∞⁡(f(tqβn,x)+g(tqβn,x)) and Tβ(fg)=limn→∞⁡(f(tqβn,x)g(tqβn,x)) exists uniformly on qZ¯×S. That is, f+g,fg∈AP(qZ¯×D,X).
( b ) Because g∈AP(qZ¯×D,X), so, for any given ε>0 and each compact subset S of D, there exists l(ε,S)>0 such that each interval [qp,qpql]∩qZ¯ contains a qτ(p,ε,S)∈[qp,qpql]∩qZ¯(p∈Z) such that(44)gtqτ,x-gt,x<M2ε,∀t∈qZ¯×S.Now(45)1gtqτ,x-1gt,x=gtqτ,x-gt,xgtqτ,xgt,x≤gtqτ,x-gt,xM2<εfor all (t,x)∈qZ¯×S. That is, 1/g∈AP(qZ¯×D,X). Now using (a) it follows that f/g∈AP(qZ¯×D,X). The proof is complete.Theorem 23.
Iffn∈AP(qZ¯×D,X),n=1,2,… and the sequence {fn} converges uniformly to f on qZ¯×S, then f∈AP(qZ¯×D,X).Proof.
For anyε>0, there exists sufficiently large n0 such that for all (t,x)∈qZ¯×S,(46)ft,x-fn0t,x<ε3.Take qτ∈E{fn0,ε/3,S}; then, for all (t,x)∈qZ¯×S, we have (47)ftqτ,x-ft,x≤ftqτ,x-fn0tqτ,x+fn0tqτ,x-fn0t,x+fn0t,x-ft,x<ε;that is, qτ∈E(f,ε,S). Therefore, f∈AP(qZ¯×D,X). This completes the proof.Theorem 24.
Letf∈AP(qZ¯×D,X) and F(t,x)≔∫0tf(s,x)dqs, then F∈AP(qZ¯×D,X) if and only if F is bounded on qZ¯×S.Proof.
IfF∈AP(qZ¯×D,X), then F is bounded on qZ¯×S.
IfF is bounded, without loss of generality, we can assume that F is a real-valued function. Denote (48)G≔supt,x∈qZ¯×S⁡Ft,x>g≔inft,x∈qZ¯×S⁡Ft,x;then, for any ε>0, there exist t1=qn1 and t2=qn2 such that (49)Ft1,x<g+ε6,Ft2,x>G-ε6,∀x∈S.Let l=l(ε1,S) be an inclusion length of E(f,ε1,S), where ε1=ε/6d,  d=t1-t2. For any α∈qZ¯, take τ∈E(f,ε1,S)∩[αt1-1,αt1-1ql]. Let si=tiqτ(i=1,2) and L=l+n1-n2; then s1,s2∈[α,αqL]∩qZ¯. For all x∈S, we have(50)Fs2,x-Fs1,x=Ft2,x-Ft1,x-∫t1t2ft,xdqt+∫t1qτt2qτft,xdqt=Ft2,x-Ft1,x+∫t1t2ftqτ,x-ft,xdqt>G-g-ε3-ε1d=G-g-ε2;that is,(51)Fs1,x-g+G-Fs2,x<ε2.Since (52)Fs1,x-g≥0,G-Fs2,x≥0,we have proved that, in any interval [α,αqL]∩qZ¯, there exist s1 and s2 such that(53)Fs1,x<g+ε2,Fs2,x>G-ε2.
Now, letε2=ε/2L; we will show that if qτ∈E(f,ε2,S), then qτ∈E(F,ε,S). In fact, for any (t,x)∈qZ¯×S, we can take s1,s2∈[t,tqL]∩qZ¯ such that (54)Fs1,x<g+ε2,Fs2,x>G-ε2;thus, for qτ∈E(f,ε2,S), we have(55)Ftqτ,x-Ft,x=Fs1qτ,x-Fs1,x+∫ts1ft,xdqt-∫tqτs1qτft,xdqt>g-g+ε2-∫tt1ftqτ,x-ft,xdqt>-ε2-ε2L=-εand(56)Ftqτ,x-Ft,x=Fs2qτ,x-Fs2,x+∫ts2ft,xdqt-∫tqτs2qτft,xdqt<ε2+ε2L=ε.That is, for qτ∈E(f,ε2,S), qτ∈E(F,ε,S); therefore, F∈AP(qZ¯×D,X). The proof is complete.Theorem 25.
Iff∈AP(qZ¯×D,X), F(·) is uniformly continuous on the value field of f; then F∘f∈AP(qZ¯×D,X).Proof.
SinceF is uniformly continuous on the value field of f and f∈AP(qZ¯×D,X), there exists a sequence α={αn}⊂Z such that(57)TαF∘f=limn→+∞⁡Fftqαn,x=Flimn→+∞⁡ftqαn,x=FTαfholds uniformly on qZ¯×S. Hence, F∘f∈AP(qZ¯×D,X). The proof is complete.Theorem 26.
A functionf∈AP(qZ¯×D,X) if and only if, for every pair of sequences α′,β′⊂Z, there exist common subsequences α⊂α′, β⊂β′ such that(58)Tα+βf=TαTβf,∀t,x∈qZ¯×S.Proof.
Iff∈AP(qZ¯×D,X), then, for every pair of sequences α′,β′⊂Z, there exists subsequence β(2)⊂β′ such that Tβ(2)f=g∈AP(qZ¯×D,X) uniformly holds on qZ¯×S. One can take that α(2)⊂α′ and α(2),β(2) are the common subsequences of α′,β′, respectively. Hence, there exists α(3)⊂α(2) such that Tα(3)g=h∈AP(qZ¯×D,X) uniformly holds on qZ¯×S.
Similarly, one can takeβ(3)⊂β(2) and β(3),α(3) are the common subsequences of β(2),α(2), respectively. Hence, there exist common subsequence α⊂α(3),β⊂β(3) such that Tα+βf=k∈AP(qZ¯×D,X) holds uniformly on qZ¯×S. According to the above, it is easy to see that Tβf=g and Tαg=h hold uniformly on qZ¯×S.
Hence, for anyε>0, if n is sufficiently large, then, for any (t,x)∈qZ¯×S, we have (59)ftqαn+βn,x-kt,x<ε3,gt,x-ftqβn,x<ε3,ht,x-gtqαn,x<ε3.Therefore,(60)ht,x-kt,x≤ht,x-gtqαn,x+gtqαn,x-ftqαn+βn,x+ftqαn+βn,x-kt,x<εholds for any (t,x)∈qZ¯×S. Since ε>0 is arbitrary, we have h(t,x)≡k(t,x); that is, Tα+βf=TαTβf holds uniformly on qZ¯×S.
On the other hand, if (58) holds, then, for any sequence γ′⊂Z, there exists subsequence γ⊂γ′ such that Tγf exists uniformly on qZ¯×S.
Now, we will prove thatf∈AP(qZ¯×D,X). If this is not true, that is, Tγf(t,x) does not converge uniformly on qZ¯×S, then there exist ε0>0 and subsequences α′⊂γ, β′⊂γ, and s′. α′={αn′}, β′={βn′}, s={sn′} such that(61)fqsn′+αn′,x-fqsn′+βn′,x≥ε0>0. From (58) it follows that there exist common subsequences α(2)⊂α′, s(2)⊂s′ such that(62)Ts2+α2ft,x=Ts2Tα2ft,x,∀t,x∈qZ¯×S.Take β(2)⊂β′ and β(2),α(2),s(2) are common subsequences of β′,α′,s′, respectively, such that(63)Ts+βft,x=TsTβft,x,∀t,x∈qZ¯×S.Similarly, we can take α⊂α(2) satisfying α,β,s is common subsequences of α(2),β(2),s(2), respectively; by (62), we have(64)Ts+αft,x=TsTαft,x,∀t,x∈qZ¯×S.Since Tαf(t,x)=Tβf(t,x)=Tγf(t,x), by (63) and (64), for all (t,x)∈qZ¯×S, Ts+βft,x=Ts+αft,x; that is, for all (t,x)∈qZ¯×S, limn→+∞⁡f(tqsn+βn,x)=limn→+∞⁡f(tqsn+αn,x). Taking t=0, this contradicts (61). Therefore, f∈AP(qZ¯×D,X). The proof is complete.


## 4. Almost Periodic Dynamic Equations on the Quantum Time Scale

Consider the following nonlinear quantum dynamic equation:(65)Dqx=ft,x,where f∈C(qZ¯×Xn,Xn).Now, we establish existence theorem for almost periodic solution of (65) by using Theorem 20.Theorem 27.
Ifϕ is an asymptotically almost periodic solution of (65) on qZ+, then the almost periodic part of ϕ is an almost periodic solution of (65).Proof.
Letϕ=p+q, where p is the almost periodic part and q(t)→0 as t→∞. By Theorem 20, for any sequence α′={αn′}⊂Z, there exists a subsequence α⊂α′ such that Tα′f=g holds uniformly on qZ¯×S and Tαp=ψ holds uniformly on qZ¯. Therefore, Tαϕ=Tαp=ψ is an almost periodic solution of the corresponding equation in the hull(66)Dqxt=gt,xon T. Now that T-αg=f holds uniformly on qZ¯×S and T-αψ=p holds uniformly on qZ¯; hence p is an almost periodic solution of (65). The proof is complete.Theorem 28.
If for eachg∈H(f), the equation Dqx(t)=g(t,x(t)) in the hull has a solution in S, where S⊂En is a compact set; then these solutions are almost periodic. In particular, system (65) has an almost periodic solution in S.Proof.
Letϕ be a solution of Dqx(t)=gt,xt in S with g∈H(f). For any given sequences α′⊂Z and β′⊂Z, we can extract common subsequences α⊂α′ and β⊂β′ such that Tα+βf(t,x)=TαTβf(t,x) holds uniformly on qZ¯×S, and Tα+βφ and TαTβφ exist on any compact subset of qZ¯. Therefore, both Tα+βφ(t) and TαTβφ(t) are solutions of the same equation in the hull(67)Dqxt=Tα+βft,xin S. Hence, Tα+βφ=TαTβφ, so, φ is an almost periodic solution. The proof is complete.For system (65), we consider its product system: (68)xΔ=ft,x,yΔ=ft,y.Theorem 29.
Suppose that there exists a Lyapunov functionV(t,x,y)∈C(qZ+×D×D,R) satisfying the following conditions:(i)
ax-y≤V(t,x,y)≤bx-y, where a,b∈K with K={a∈C(R+,R+):a(0)=0 and aisincreasing};(ii)
Vt,x1,y1-vt,x2,y2≤Lx1-x2+y1-y2, where L>0 is a constant;(iii)
D+V(68)Δ(t,x,y)≤-cV(t,x,y), where -c∈R+ and c>0.
Moreover, if there exists a solutionx(t) of (65) such that x(t)∈S for t∈qZ+, where S⊂D is a compact set; then there exists a unique uniformly asymptotically stable almost periodic solution p(t) of (65) in S. Furthermore, if f(t,x) is periodic with period ω in t, then p(t) is a periodic solution of (65) with period ω.Proof.
Letα={αn}⊂Z be a sequence such that αn→∞ as n→∞ and Tαf=f uniformly on qZ¯×S. Assume that ϕ(t)⊂S is a solution of (65) for t∈qZ+. Then ϕ(tqαn) is a solution of the equation(69)Dqx=ftqαn,x,which is also in S. For a given ε>0, choose an integer k0(ε) so large that if m≥k>k0(ε), we have(70)b2Be-cαk,0<aε2and(71)ftqαk,x-ftqαm,x<caε2L,where B is a positive constant such that S⊂{x:x≤B}. Then it follows from (ii) and (iii) that(72)D+VΔt,ϕt,ϕtq-αm-αk≤-cVt,ϕt,ϕtq-αm-αk+Lftqαm-αk,ϕtqαm-αk)-ft,ϕtqαm-αk.In view of (71), we have(73)D+VΔt,ϕt,ϕtq-αm-αk≤-cVt,ϕt,ϕtq-αm-αk+caε2, If m≥k≥k0(ε), by Lemma 3, condition (i), and (70), (73) implies that (74)Vtqαk,ϕtqαk,ϕtq-αm≤e-ctqαm,0V0,ϕ0,ϕqαm-αk+aε21-e-ctqαk,0≤e-ctqαm,0V0,ϕ0,ϕqαm-αk+aε2<aε,fort∈qZ+.Therefore, by condition (i), for m≥k≥k0 and t∈qZ+, we have(75)ϕtqαk-ϕtqαm<ε,which implies that ϕ is asymptotically almost periodic. Hence, it follows from Theorem 27 that the almost periodic part p(t)⊂S of ϕ is an almost periodic solution of system (65).
By using the Lyapunov functionV(t,x,y), it is easy to show that p(t) is uniformly asymptotically stable and every solution remaining in D approaches p(t) as t→∞, which also implies the uniqueness of p(t).
In particular, iff(t,x) is periodic in t with period ω, then p(tqω) is also a solution of (65) which remains in S. By the uniqueness of the solutions, we know that p(tqω)=p(t). This completes the proof.


## 5. The Relationship of Almost Periodic Functions on the Quantum Time Scale and on the Integer Set

In this section, we will give an equivalent definition of almost periodic functions on the quantum time scaleqZ¯. To this end, we introduce a notation -∞q and stipulate q-∞q=0,t±(-∞q)=t and t>-∞q for any t∈Z. Let f∈C(qZ¯,X); we define a function f~:Z∪{-∞q}→X by(76)f~t=fqt,t∈Z,f0,t=-∞q;that is, (77)ft=f~logq⁡t,t∈qZ,limt→0+⁡ft,t=0.Since f(t) is right continuous at t=0, it is clear that the above definition is well defined.Moreover, forf∈C(qZ¯×X,X), we define a function f~:Z∪{-∞q}×X→X by(78)f~t,x=fqt,x,t,x∈Z×X,f0,x,t=-∞q,x∈X;that is, (79)ft,x=f~logq⁡t,x,t,x∈qZ×X,limt→0⁡ft,x,t=0,x∈X.Since f(t,x) is continuous at (0,x), it is clear that the above definition is well defined.Definition 30.
A functionf∈C((Z∪{-∞q})×X,X) is called almost periodic in t∈Z∪{-∞q} uniformly in x∈X if for every ε>0 and each compact subset S of X there exists an integer l(ε)>0 such that each interval of length l(ε,S) contains an integer τ such that(80)ft+τ,x-ft,x<ε,∀t,x∈Z∪-∞q×S.This τ is called the ε-translation number or ε-almost period.Definition 31.
A functionf∈C(Z∪{-∞q},X) is called almost periodic if, for every ε>0, there exists an integer l(ε)>0 such that each interval of length l(ε) contains an integer τ such that(81)ft+τ-ft<ε,∀t∈Z∪-∞q.This τ is called the ε-translation number or ε-almost period.Remark 32.
According to the operation rule about-∞q, Z∪{-∞q} is an almost periodic time scale under the Definition 1 and when t=-∞q, the inequalities (80) and (81) hold naturally. Hence, ∀t∈Z∪{-∞q} and ∀(t,x)∈(Z∪{-∞q})×S in (80) and (81) can be replaced by ∀t∈Z and ∀(t,x)∈Z×S, respectively.Remark 33.
Obviously, the almost periodic functions defined by Definitions30 and 31 (which are defined on Z∪{-∞q}×X) or Z∪{-∞q} have the same properties as the ordinary almost periodic functions defined on Z×X or Z, respectively.By Definition4 and (78), one has the following.Theorem 34.
A functionf∈C(qZ¯×X,X) is almost periodic in t∈qZ¯ for each x∈X if and only if the function f~(t,x) defined by (78) is almost periodic in t∈qZ¯ for each x∈X.By Definition6 and (76), one has the following.Theorem 35.
A functionf∈C(qZ¯,X) is almost periodic if and only if the function f~(t) defined by (76) is almost periodic.Obviously, by Theorems34 and 35 and Remark 33, all of the properties of almost periodic functions on the quantum time scale can be directly obtained from the corresponding properties of the ordinary almost periodic functions defined on Z or Z×X.


## 6. An Application

Based on the equivalent definition of almost periodic functions on the quantum time scale, Remark32, the transformations (76) and (78), we can transform the almost periodic problem of dynamic equations on the quantum time scale into the almost periodic problem of dynamic equations on almost periodic time scales. For example, consider the dynamic equation on the quantum time scale:(82)Dqxt=ft,xt,xtq-σt),t∈qZ¯,where σ:Z→Z is a scalar delay function, f∈C(qT¯×E2n,En). Under transformation (78), (82) is transformed to(83)Δx~n=Fn,x~n,x~n-σ~n,n∈Z∪-∞q, and vice visa, where F(n)=(q-1)qnf~(n,x~(n),x~(n-σ~(n)))).Clearly, ifx(t) is a solution of (82) if and only if x~(n) is a solution of (83). Therefore, applying the theory of difference equations or the theory of almost periodic dynamic equations on time scales [8], we can first discuss the almost periodic problem of (83), and then by transformation (78) we can obtain the corresponding result about (82). Besides, since when n={-∞}q, both sides of (83) are equal to 0, so, if ϕ(n) is a solution of the equation, (84)Δx~n=Fn,x~n,x~n-σ~n,n∈Z,then (85)φn=ϕn,n∈Z,ϕ0,n=-∞qis a solution of (83).As an application, we consider the following high-order Hopfield neural network on the quantum time scaleqT¯:(86)Dqxit=-citxit+∑j=1maijtfjxjtq-γijt+∑j=1m∑l=1mbijltgjxjtq-ωijltglxltq-vijlt+Iit,t∈qT¯,where i=1,2,…,m, m corresponds to the number of units in a neural network, xi(t) corresponds to the state vector of the ith unit at the time t, ci(t) represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t, aij(t), αij(t), and bijl(t) are the first- and second-order connection weights of the neural network at time t, Ii(t) denotes the external inputs at time t, fj and gj are the activation functions of signal transmission, and γij(t),ωijl(t),vijl(t):Z→Z+ correspond to the transmission delays at time t.Under transformation (78), (86) is transformed to(87)Δx~in=-q-1qnc~inx~in+q-1qn∑j=1ma~ijnf~jx~jn-γ~ijn+q-1qn∑j=1m∑l=1mb~ijlng~jx~jn-ω~ijlng~lx~ln-v~ijln+q-1qnI~in,n∈Z∪-∞q,i=1,2,…,m and vice visa. If we take T=Z∪{-∞q}, then (87) can be written as(88)x~iΔt=-c^itx~it+∑j=1ma^ijtf~jx~jt-γ~ijt+∑j=1m∑l=1mb^ijltg~jx~jt-ω~ijltg~lx~lt-v~ijlt+I^it,t∈T,i=1,2,…,m,where c^i(t)=(q-1)qtc~i(t), a^ij(t)=(q-1)qta~ij(t), b^ijl(t)=(q-1)qtb~ijl(t), I^i(t)=(q-1)qtI~i(t),t∈T,i,j,l=1,2,…,m.Remark 36.
From the discussion above, we see that if (88) has an almost periodic solution x~=(x~1,x~2,…,x~m)T, then (86) has an almost periodic solution x=(x1,x2,…,xm)T.Now, we shall establish the existence and global exponential stability of almost periodic solutions of (88). First, for convenience, we introduce some notations. We will use x=(x1,x2,…,xn)T∈Rn to denote a column vector, in which the symbol T denotes the transpose of vector. We let x denote the absolute-value vector given by x=(x1,x2,…,xn)T and define x=max1≤i≤n⁡xi.LetAP(T)={x∈C(T,R):x(t) is a real valued, almost periodic function on T} with the norm φ0=supt∈T⁡φ(t)0, where φ(t)0=max1≤i≤n⁡φit and φ∈AP(T); then AP(T) is an Banach space.Theorem 37.
Assume the following:(H1)
f~j,g~j∈C(R,R), and there exist positive constants ϵj and εj such that(89)f~ju-f~jv≤ϵju-v,g~ju-g~jv≤εju-v,whereu,v∈R,j=1,2,…,m;(H2)
there exist constantsNj>0 such that g~ju≤Nj,u∈R,j=1,2,…,m;(H3)
fori,j,l=1,2,…,m, c^i is positive regressive and inft∈T⁡c^i(t)>0,c^i,a^ij,b^ijl,I^i,γ~ij,ω~ijl,v~ijl∈AP(T);(H4)
there exists a constantr0 such that(90)max1≤i≤m⁡ηic^i-+L≤r0,max1≤i≤m⁡η¯i<min1≤i≤m⁡c^i-,where, fori,j,l=1,2,…,n,(91)ηi=∑j=1ma^ij+fj0+ϵjr0+gj0+εjr0∑l=1mb^ijl+gl0+εlr0,η¯i=∑j=1ma^ij+ϵj+∑j=1m∑l=1mb^ijl+Nlεj+Njεl,L=max1≤i≤n⁡Ii¯c^i-,c^i+=supt∈T⁡cit,c^i-=inft∈T⁡c^it,a^ij+=supt∈T⁡a^ijt,b^ijl+=supt∈T⁡b^ijlt,I^i+=supt∈T⁡I^it.Then system (88) has a unique almost periodic solution in the region(92)E=φ∈APT:φ0≤r0.Proof.
For any givenφ∈AP(T), we consider the following almost dynamic equation:(93)x~iΔt=-c^itx~it+∑j=1ma^ijtf~jφjt-γ~ijt+∑j=1m∑l=1mb^ijltg~jφjt-ω~ijltg~lφlt-v~ijlt+I^it,i=1,2,…,m.In view of (H3), it follows from Lemma 2.15 in [16] that the linear system(94)x~iΔt=-c^itx~it,i=1,2,…,madmits an exponential dichotomy on T. Thus, by Theorem 4.19 in [8], we know that system (93) has exactly one almost periodic solution(95)x~φit=∫-∞te-c^it,σs∑j=1ma^ijsf~jφjs-γ~ijs+∑j=1m∑l=1mb^ijlsg~jφjs-ω~ijlsg~lφls-v~ijls+I^isΔs,i=1,2,…,m.First, we define a nonlinear operator on AP(T) by(96)Φφt=x~φt,∀φ∈APT.Next, we check that Φ(E)⊂E. For any given φ∈E, it suffices to prove that Φ(φ)0≤r0. By conditions (H1)–(H4), we have(97)Φφ0=supt∈Tmax1≤i≤m⁡∫-∞te-c^it,σs∑j=1ma^ijsf~jφjs-γ~ijs+∑j=1m∑l=1mb^ijlsg~jφjs-ω~ijlsg~lφls-v~ijls+I^isΔs≤supt∈Tmax1≤i≤m⁡∫-∞te-c^i-t,σs∑j=1ma^ij+f~jφjs-γ~ijs+∑j=1m∑l=1mb^ijl+g~jφjs-ω~ijlsg~lφls-v~ijlsΔs+max1≤i≤n⁡I^i+c^i-≤supt∈Tmax1≤i≤m⁡∫-∞te-c^i-t,σs∑j=1ma^ij+f~j0+ϵjφjs-γ~ijs+∑j=1m∑l=1mb^ijl+g~j0+εjφjs-ω~ijlsg~l0+εlφls-v~ijlsΔs+L≤supt∈Tmax1≤i≤m⁡∫-∞te-c^i-t,σs∑j=1ma^ij+f~j0+ϵjr0+∑j=1m∑l=1mb^ijl+g~j0+εjr0g~l0+εlr0Δs+L≤max1≤i≤m⁡ηic^i-+L≤r0,which implies that Φ(E)⊂E. For any φ,ψ∈E, according to (H1) and (H4), we have(98)Φφ-Φψ0=supt∈Tmax1≤i≤m⁡∫-∞te-c^it,σs∑j=1ma^ijsf~jφjs-γ~ijs-f~jψjs-γ~ijs+∑j=1m∑l=1mb^ijlsg~jφjs-ω~ijlsg~lφls-v~ijls-∑j=1mbijlsg~jψjs-ω~ijlsg~lψls-v~ijls≤supt∈Tmax1≤i≤m⁡∫-∞te-c^i-t,σs∑j=1ma^ij+ϵjφjs-γ~ijs-ψjs-γ~ijs+∑j=1m∑l=1mb^ijl+εjφjs-ω~ijls-ψjs-ω~ijlsg~lφls-v~ijls+∑j=1m∑l=1mb^ijl+εlφls-v~ijls-ψls-v~ijlsg~jψjs-ω~ijlsΔs≤supt∈Tmax1≤i≤m⁡∫-∞te-c^i-t,σs∑j=1ma^ij+ϵj+∑j=1m∑l=1mb^ijl+Njεl+Nlεjφ-ψ0Δs≤max1≤i≤m⁡η¯imin1≤i≤m⁡c^i-φ-ψ0,which implies that Φ is a contraction mapping. Hence, Φ has a unique fixed point in E; that is, (88) has a unique almost periodic solution in the region E. This completes the proof.Theorem 38.
Assume that(H1)-(H4) hold; then system (88) has a unique almost periodic solution that is globally exponentially stable.Proof.
From Theorem37, we see that system (88) has a unique almost periodic solution x~∗(t)=(x~1∗(t),x~2∗(t),…,x~m∗(t))T with initial value φ∗(t)=(φ1∗(t),φ2∗(t),…,φm∗(t))T. Suppose that x~(t)=(x~1(t),x~2(t),…,x~m(t))T is an arbitrary solution of (88) with initial value φ(t)=(φ1(t),φ2(t),…,φm(t))T. Then it follows from system (88) that(99)z~iΔt=-c^itz~it+∑j=1ma^ijtf~jz~jt-γ~ijt+x~j∗t-γ~ijt-f~jx~j∗t-γ~ijt+∑j=1m∑l=1mb^ijltg~jz~jt-ω~ijlt+x~j∗t-ω~ijlt-g~jx~j∗t-ω~ijlt×g~lz~lt-v~ijlt+x~l∗t-v~ijlt+g~lz~lt-v~ijlt+x~l∗t-v~ijlt-g~lx~l∗s-v~ijltg~jx~j∗t-ω~ijlt,where z~i(t)=x~i(t)-x~i∗(t),i=1,2,…,m.
The initial condition of (99) is(100)ϕis=φis-φi∗s,s∈-τ,0Twhere τ=max⁡{γ~,ω~,v~}, γ~=max1≤i,j≤m⁡{supt∈T⁡γ~ijt}, ω~=max1≤i,j,l≤m⁡{supt∈T⁡ω~ijlt}, v~=max1≤i,j,l≤m⁡{supt∈T⁡v~ijlt}, and ϕi(·) denotes a real-valued bounded Δ-differentiable function defined on [-τ,0]T.
Multiplying both sides of (99) by e-c^i(t0,σ(t)) and integrating over [t0,t]T, where t0∈[-τ,0]T, for i=1,2,…,m, we have(101)z~it=z~it0e-c^it,t0+∫t0te-c^it,σs∑j=1ma^ijsf~jz~js-γ~ijs+x~j∗s-γ~ijs-f~jx~j∗s-γ~ijs+∑j=1m∑l=1mb^ijlsg~jz~js-ω~ijls+x~j∗s-ω~ijls-g~jx~j∗s-ω~ijlsg~lz~ls-v~ijls+x~l∗s-v~ijls+g~lz~ls-v~ijls+x~l∗s-v~ijls-g~lx~l∗s-v~ijlsg~jx~j∗s-ω~ijlsΔs.For i=1,2,…,m, let Γi be defined as follows: (102)Γiζ=c^i--ζ-exp⁡ζsups∈T⁡μs∑j=1ma^ij+ϵjexp⁡ζγ~+∑j=1m∑l=1mb^ijl+εjNlexp⁡ζω~+εlNjexp⁡ζv~.
By(H4), we get (103)Γi0=c^i--η¯i>0,i=1,2,…,m.Since Γi is continuous on [0,+∞) and Γi(ζ)→-∞ as γ→+∞, there exists ξi>0 such that Γi(ξi)=0 and Γi(ζ)>0 for ζ∈(0,ξi). Take β=min1≤i≤m⁡{ξi}; we have Γi(β)≥0. Hence, we can choose a positive constant 0<λ<min⁡{β,min1≤i≤m⁡{c^i-}} such that(104)Γiλ>0,i=1,2,…,m,which implies that (105)c^i+exp⁡λsups∈T⁡μsc^i--λ∑j=1ma^ij+ϵjexp⁡λγ~+∑j=1m∑l=1mb^ijl+εjNlexp⁡λω~+εlNjexp⁡λv~<1,i=1,2,…,m.Taking(106)M=max1≤i≤m⁡c^i-∑j=1ma^ij+ϵj+∑j=1m∑l=1mb^ijl+εjNl+εlNj,then by (H4) we have M>1. Besides, we can obtain that (107)1M<c^i+exp⁡λsups∈T⁡μsc^i--λ∑j=1ma^ij+ϵjexp⁡λγ~+∑j=1m∑l=1mb^ijl+εjNlexp⁡λω~+εlNjexp⁡λv~.Let z^(t)=max1≤i≤m⁡z^it, ϕ0=max1≤i≤m⁡{sups∈[t0-τ¯,t0]T⁡φis-φi∗s}. In addition, noticing that e⊖λ(t,t0)≥1 for t∈[t0-τ,t0]T, thus, for any ϵ>0, it is obvious that(108)z~t0<ϕ0+ϵand (109)z~t<ϕ0+ϵe⊖λt,t0<Mϕ0+ϵe⊖λt,t0,∀t∈t0-τ,t0T.We claim that(110)z~t<Mϕ0+ϵe⊖λt,t0,∀t∈t0,+∞TIf (110) is not true, then there must be some t1∈(t0,+∞)T such that (111)z~t1≥Mϕ0+ϵe⊖λt1,t0,z~t<Mϕ0+ϵe⊖λt,t0,t∈t0-τ,t1T.Therefore, there must exist a constant c≥1 such that(112)z~t1=cM∗ϕ0+ϵe⊖λt1,t0,z~t<cM∗ϕ0+ϵe⊖λt,t0,t∈t0-τ,t1T.In view of (101), we have(113)z~it1=z~it0e-c^it1,t0∫t0te-c^it,σs∑j=1ma^ijsf~jz~js-γ~ijs+x~j∗s-γ~ijs-f~jx~j∗s-γ~ijs+∑j=1m∑l=1mb^ijlsg~jz~js-ω~ijls+x~j∗s-ω~ijls-g~jx~j∗s-ω~ijlsg~lz~ls-v~ijls+x~l∗s-v~ijls+g~lz~ls-v~ijls+x~l∗s-v~ijls-g~lx~l∗s-v~ijlsg~jx~j∗s-ω~ijlsΔs≤e-c^it1,t0ϕ0+ϵ+cMe⊖λt1,t0ϕ0+ϵ∫t0t1e-c^it1,σseλt1,σs×∑j=1ma^ij+ϵjeλσs,s-γ~ijs+∑j=1m∑l=1mb^ijl+εjNleλσs,s-ω~ijls+εlNjeλσs,s-v~ijlsΔs≤e-c^it1,t0ϕ0+ϵ+cMe⊖λt1,t0ϕ0+ϵ∫t0t1e-c^i⊕λt1,σs×∑j=1ma^ij+ϵjexp⁡λγ~+sups∈T⁡μs+∑j=1m∑l=1mb^ijl+εjNlexp⁡λω~+sups∈T⁡μs+εlNjexp⁡λv~+sups∈T⁡μsΔs=cMe⊖λt1,t0ϕ0+ϵ1cMe-c^i⊕λt1,t0+exp⁡λsups∈T⁡μs×∑j=1ma^ij+ϵjexp⁡λγ~+∑j=1m∑l=1mb^ijl+εjNlexp⁡λω~+εlNjexp⁡λv~×∫t0t1e-c^i⊕λt1,σsΔs≤cMe⊖λt1,t0ϕ0+ϵ1cMe-c^i⊕λt1,t0+exp⁡λsups∈T⁡μs×∑j=1ma^ij+ϵjexp⁡λγ~+∑j=1m∑l=1mb^ijl+εjNlexp⁡λω~+εlNjexp⁡λv~×1-e-c^i⊕λt1,t0c^i--λ≤cMe⊖λt1,t0ϕ0+ϵ1M-exp⁡λsups∈T⁡μsc^i--λ∑j=1ma^ij+ϵjexp⁡λγ~+∑j=1m∑l=1mb^ijl+εjNlexp⁡λω~+εlNjexp⁡λv~e-c^i⊕λt1,t0+exp⁡λsups∈T⁡μsc^i--λ∑j=1ma^ij+ϵjexp⁡λγ~+∑j=1m∑l=1mb^ijl+εjNlexp⁡λω~+εlNjexp⁡λv~<cMe⊖λt1,t0ϕ0+ϵ,which contradicts the first equation (112). Therefore, (110) holds. Letting ϵ→0+ leads to (114)z~t≤Mϕ0e⊖λt,0,∀t∈t0,+∞T.Hence, the almost periodic solution of system (88) is globally exponentially stable. The proof is complete.


## 7. Conclusion

In this paper, we proposed two types of concepts of almost periodic functions on the quantum time scale and investigated some their basic properties. Moreover, we established a result about the existence and uniqueness of almost periodic solutions of dynamic equations on the quantum time scale by Lyapunov method and gave an equivalent concept of almost periodic functions on the quantum time scale. In addition, as an application, we proposed a class of high-order Hopfield neural network on the quantum time scale and established the existence of almost periodic solutions of the neural networks. Based on the concepts introduced and the results obtained in this paper, we can further study almost periodic problems for dynamic equations on the quantum time scale. For example, we can study the almost periodicity of population dynamical models on the quantum time scale. Furthermore, by using the idea of transformations of (76) and (78), we can give a new concept of almost periodic functions on time scales as follows.Definition 39.
LetT be a time scale and T~ be an almost periodic time scale defined by Definition 1 and there exists a mapping χ:T~→T. A function f∈C(T×D,En) is called an almost periodic function in t∈T uniformly in x∈D if, for every ε>0 and for each compact subset S of D, there exists a constant l(ε,S)>0 such that each interval of length l(ε,S) contains a τ(ε,S)∈Π~ such that(115)fχt+τ,x-fχt,x<ε,∀t,x∈T~×S,where Π~={τ∈R:t+τ∈T~,∀t∈T~}. ε is called the ε-translation number of f.Obliviously, Definition39 greatly extends the concept of almost periodic functions on time scales that was introduced in [8] and it unifies the concepts of almost periodic functions on time scales for the cases of T=R,T=Z,T=qZ¯ and the case T is an almost periodic time scale defined by Definition 1. Similarly, we can give new definitions of almost automorphic functions on time scales. These are our future goals.



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