# Uniformly Almost Periodic Functions and Almost Periodic Solutions to Dynamic Equations on Time Scales

Authors: Yongkun Li; Chao Wang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Firstly, we propose a concept of uniformly almost periodic functions on almost periodic time scales and investigate some basic properties of them. When time scale𝕋=ℝ or ℤ, our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations. Finally, as an application of our results, we study the existence of almost periodic solutions for an almost periodic nonlinear dynamic equations on time scales.


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## Body

## 1. Introduction

In recent years, researches in many fields on time scales have received much attention. The theory of calculus on time scales (see [1, 2] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [3] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his fundamental work. It has been created in order to unify the study of differential and difference equations. Many papers have been published on the theory of dynamic equations on time scales [4–10]. Also, the existence of almost periodic, asymptotically almost periodic, and pseudo-almost periodic solutions is among the most attractive topics in qualitative theory of differential equations and difference equations due to their applications, especially in biology, economics and physics [11–29]. However, there are no concepts of almost periodic functions on time scales so that it is impossible for us to study almost periodic solutions for dynamic equations on time scales.Motivated by the above, our main purpose of this paper is firstly to propose a concept of uniformly almost periodic functions on time scales and investigate some basic properties of them. Then we study the existence and uniqueness of almost periodic solutions to linear dynamic equations on almost time scales. Finally, as an application of our results, we study the existence of almost periodic solutions for almost periodic nonlinear dynamic equations on time scales.The organization of this paper is as follows. In Section2, we introduce some notations and definitions and state some preliminary results needed in the later sections. In Section 3, we propose the concept of uniformly almost periodic functions on almost periodic time scales and investigate the basic properties of uniformly almost periodic functions on almost periodic time scales. In Section 4, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations on time scales. In Section 5, as an application of our results, we study the existence of almost periodic solutions for almost periodic nonlinear dynamic equations on time scales.


## 2. Preliminaries

In this section, we will first recall some basic definitions lemmas which are used in what follows.Let𝕋 be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators σ,ρ:𝕋→𝕋 and the graininess μ: 𝕋→ℝ+ are defined, respectively, by (2.1)σ(t)=inf⁡{s∈T:s>t},ρ(t)=sup⁡{s∈T:s<t},μ(t)=σ(t)-t.A pointt∈𝕋 is called left-dense if t>inf𝕋 and ρ(t)=t, left-scattered if ρ(t)<t, right-dense if t<sup𝕋 and σ(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.A functionf:𝕋→ℝ is right-dense continuous provided it is continuous at right-dense point in 𝕋 and its left-side limits exist 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.Fory:𝕋→ℝ and t∈𝕋k, we define the delta derivative of y(t),yΔ(t), to be the number (if it exists) with the property that, for a given ɛ>0, there exists a neighborhood U of t such that (2.2)|[y(σ(t))-y(s)]-yΔ(t)[σ(t)-s]|<ɛ|σ(t)-s|
for all s∈U.Lety be right-dense continuous; if YΔ(t)=y(t), then we define the delta integral by (2.3)∫aty(s)Δs=Y(t)-Y(a).A functionp:𝕋→ℝ is called regressive provided 1+μ(t)p(t)≠0 for all t∈𝕋k. The set of all regressive and rd-continuous functions p:𝕋→ℝ will be denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ). We define the set ℛ+=ℛ+(𝕋,ℝ)={p∈ℛ:1+μ(t)p(t)>0,forallt∈𝕋}.Ann×n-matrix-valued function A on a time scale 𝕋 is called regressive provided (2.4)I+μ(t)A(t)isinvertible∀t∈T,
and the class of all such regressive and rd-continuous functions is denoted, similar to the above scalar case, by ℛ=ℝ(𝕋)=ℛ(𝕋,ℝn×n).Ifr is a regressive function, then the generalized exponential function er is defined by (2.5)er(t,s)=exp⁡{∫stξμ(τ)(r(τ))Δτ}
for all s,t∈𝕋, with the cylinder transformation (2.6)ξh(z)={Log(1+hz)hifh≠0,zifh=0.Definition 2.1 (see [1, 2]).
Letp,q:𝕋→ℝ be two regressive functions; define
(2.7)p⊕q=p+q+μpq,⊝p=-p1+μp,p⊝q=p⊕(⊝q).Lemma 2.2 (see [1, 2]).
Assume thatp,q:𝕋→R are two regressive functions, then(i)
e0(t,s)≡1 and ep(t,t)≡1;(ii)
ep(σ(t),s)=(1+μ(t)p(t))ep(t,s);(iii)
ep(t,s)=1/ep(s,t)=e⊝p(s,t);(iv)
ep(t,s)ep(s,r)=ep(t,r);(v)
(e⊝p(t,s))Δ=(⊖p)(t)e⊖p(t,s);(vi)
ifa,b,c∈𝕋,  then    ∫abp(t)ep(c,σ(t))Δt=ep(c,a)-ep(c,b).Lemma 2.3 (see [1, 2]).
Assume that{fn}n∈ℕ is a function sequence on J such that  (i)
{fn}n∈ℕ is a uniformly bounded on J;(ii)
{fnΔ}n∈ℕ is a uniformly bounded on J.Then, there is a subsequence of {fn}n∈ℕ which converges uniformly on J where J is an arbitrary compact subset of 𝕋.


## 3. Uniformly Almost Periodic Functions

Let𝕋 be a given time scale, and 𝕋 is a complete metric space with the metric (distance) d defined by (3.1)d(t,t0)=|t-t0|fort,t0∈T.
For a given δ>0, the δ-neighborhood U(t0,δ) of a given point t0∈𝕋 is the set of all points t∈𝕋 such that d(t,t0)<δ.Throughout this paper,𝔼n denotes ℝn or ℂn, D denotes an open set in 𝔼n or D=𝔼n, and S denotes an arbitrary compact subset of D.Definition 3.1.
f:X→𝔼n is called continuous at t0∈X⊆𝕋 if, and only if for any ɛ>0, there exists U(t0,δ) such that, for any s∈U(t0,δ),
(3.2)|f(s)-f(t0)|<ɛ.f is called continuous on X provided that it is continuous for every t∈X.Definition 3.2.
f:X→𝔼n is called uniformly continuous on X⊆𝕋 if, for any ɛ>0, there exists δ(ε) such that, for any t1,t2∈X with |t1-t2|<δ(ε) it is implied that
(3.3)|f(t1)-f(t2)|<ε.
Similar to the finite covering theorem in functional analysis (see [30]), one can easily show that the following.Lemma 3.3.
Let([a,b]∩𝕋)⊂𝕋 be a closed interval. If ([a,b]∩𝕋)⊆⋃α∈I(Gα∩𝕋), where I is an index set, and for every α∈I, Gα is an open set in ℝ, then there exist α1,α2,…,αn∈I, such that ([a,b]∩𝕋)⊆⋃k=1n(Gαk∩𝕋).Also, one can easily prove the following two lemmas.Lemma 3.4.
If{Sn(x)} converges uniformly to S(x) on [a,b]∩𝕋 and each Sn(x) is continuous on [a,b]∩𝕋, then S(x) is continuous on [a,b]∩𝕋, limn→∞∫abSn(x)Δx=∫abS(x)Δx=∫ablimn→∞Sn(x)Δx, and {∫axSn(t)Δt} converges uniformly to ∫axS(t)Δt.Lemma 3.5.
If a sequence{Sn(x)} converges to S(x) on [a,b]∩𝕋 and, for each n∈ℕ, Sn(x) has continuous derivative SnΔ(x) and {SnΔ(x)} converges uniformly to Q(x), then SΔ(x)=Q(x),thatis,limn→∞SnΔ(x)=(limn→∞Sn(x))Δ, and {Sn(x)} converges to S(x) uniformly on [a,b]∩𝕋.By using Lemma3.3, one can easily show that the following.Lemma 3.6.
Letf∈C([a,b]∩𝕋,𝔼n), then f is uniformly continuous on [a,b]∩𝕋.Definition 3.7.
A time scale𝕋 is called an almost periodic time scale if
(3.4)Π:={τ∈R:t±τ∈T,∀t∈T}≠{0}.
Obviously, ifτ1,τ2∈Π, then τ1±τ2∈Π and if T is an almost periodic time scale, then inf𝕋=-∞ and sup𝕋=+∞.Example 3.8.
If𝕋=⋃k∈ℤ[k(a+b),k(a+b)+b], where a≠-b, then
(3.5)σ(t)={tift∈⋃k=0∞[k(a+b),k(a+b)+b),t+aift∈⋃k=0∞{k(a+b)+b},μ(t)={0ift∈⋃k=0∞[k(a+b),k(a+b)+b),aift∈⋃k=0∞{k(a+b)+b}.a+b∈Π≠{0}. Hence, it is an almost periodic time scale. Obviously, if b=0,a=1, then 𝕋=ℤ. If b=1,a=0, then 𝕋=ℝ.Definition 3.9.
Let𝕋 be an almost periodic time scale. A function f∈C(𝕋,𝔼n) is called an almost periodic function if the ɛ-translation set of f(3.6)E{ɛ,f}={τ∈Π:|f(t+τ)-f(t)|<ɛ,∀t∈T}
is a relatively dense set in 𝕋 for all ɛ>0; that is, for any given ɛ>0, there exists a constant l(ɛ)>0 such that each interval of length l(ɛ,S) contains a τ(ɛ)∈E{ɛ,f} such that
(3.7)|f(t+τ)-f(t)|<ɛ,∀t∈T.τ is called the ɛ-translation number of f and l(ɛ) is called the inclusion length of E{ɛ,f}.Definition 3.10.
Let𝕋 be an almost periodic time scale. A function f∈C(𝕋×D,𝔼n) is called an almost periodic function in t∈𝕋 uniformly for x∈D if the ɛ-translation set of f(3.8)E{ɛ,f,S}={τ∈Π:|f(t+τ,x)-f(t,x)|<ɛ,∀(t,x)∈T×S}
is a relatively dense set in 𝕋 for all ɛ>0 and for each compact subset S of D; that is, for any given ɛ>0 and each compact subset S of D, there exists a constant l(ɛ,S)>0 such that each interval of length l(ɛ,S) contains a τ(ɛ,S)∈E{ɛ,f,S} such that
(3.9)|f(t+τ,x)-f(t,x)|<ɛ,∀t∈T×S.τ is called the ɛ-translation number of f and l(ɛ,S) is called the inclusion length of E{ɛ,f,S}.Obviously, an almost periodic function can be regarded as a special case of a uniformly almost periodic function. So, in the following, we mainly discuss the basic properties of uniformly almost periodic functions. The basic properties of almost periodic functions can be derived directly from the corresponding ones of uniformly almost periodic functions.Remark 3.11.
If𝕋=ℝ, then Π=ℝ in this case, Definitions 3.4 and 3.5 are equivalent to the definitions of the almost periodic functions and the uniformly almost periodic functions in [16], respectively. If 𝕋=ℤ, then Π=ℤ, in this case, Definitions 3.4 and 3.5 are equivalent to the definitions of the almost periodic functions and the uniformly almost periodic sequences in [17, 18].For convenience, we denoteAP(𝕋)={f:f∈C(𝕋,𝔼n),f is almost periodic} and introduce some notations: let α={αn} and β={βn} be two sequences. Then, β⊂α means that β is a subsequence of α, α+β={αn+βn},-α={-αn}, and α and β are common subsequences of α′ and β′, respectively, which means that αn=αn(k)′ and βn=βn(k)′ for some given function n(k).We will introduce the translation operatorT,Tαf(t,x)=g(t,x) which means that g(t,x)=limn→+∞f(t+αn,x) and is written only when the limit exists. The mode of convergence, for example, pointwise, uniform, and so forth, will be specified at each use of the symbol.Similar to the proof of Theorem 1.13 in [16], one can show that.Theorem 3.12.
Letf∈C(𝕋×D,𝔼n) be almost periodic in t uniformly for x∈D, then it is uniformly continuous and bounded on 𝕋×S.Theorem 3.13.
Letf∈C(𝕋×D,𝔼n) be almost periodic in t uniformly for x∈D, then, for any given sequence α′⊂Π, there exist a subsequence β⊂α′ and g∈C(𝕋×D,𝔼n) such that Tβf(t,x)=g(t,x) holds uniformly on 𝕋×S and g(t,x) is almost periodic in t uniformly for x∈D.Proof.
For anyɛ>0 and S⊂D, let l=l(ɛ/4,S) be an inclusion length of E{ɛ/4,f,S}. For any given subsequence α′={αn′}⊂Π, we denote αn′=τn′+γn′, where τn′∈E{ɛ/4,f,S},γn′∈Π, and 0≤γn′≤l,n=1,2,…. (In fact, for any interval with length of l, there exists τn′∈E{ɛ/4,f,S}, thus, we can choose a proper interval with length of l such that 0≤αn′-τn′≤l, from the definition of Π, it is easy to see that γn′=αn′-τn′∈Π.) Therefore, there exists a subsequence γ={γn}⊂γ′={γn′} such that γn→s as n→∞, 0≤s≤l.
Also, it follows from Theorem3.12 that f(t,x) is uniformly continuous on 𝕋×S. Hence, there exists δ(ɛ,S)>0 so that |t1-t2|<δ, for x∈S, implies(3.10)|f(t1,x)-f(t2,x)|<ɛ2.
Sinceγ is a convergent sequence, there exists N=N(δ) so that p,m≥N implies |γp-γm|<δ. Now, one can take α⊂α′,τ⊂τ′={τn′} such that α,τ common with γ then; for any integers p,m≥N, we have(3.11)|f(t+τp-τm,x)-f(t,x)|≤|f(t+τp-τm,x)-f(t+τp,x)|+|f(t+τp,x)-f(t,x)|<ɛ4+ɛ4=ɛ2,
that is,
(3.12)(αp-αm)-(γp-γm)=τp-τm∈E{ɛ2,f,S}.
Hence, we can obtain
(3.13)|f(t+αp,x)-f(t+αm,x)|≤sup⁡(t,x)∈T×S|f(t+αp,x)-f(t+αm,x)|≤sup⁡(t,x)∈T×S|f(t+αp-αm,x)-f(t,x)|≤sup⁡(t,x)∈T×S|f(t+αp-αm,x)-f(t+γp-γm,x)|+sup⁡(t,x)∈T×S|f(t+γp-γm,x)-f(t,x)|<ɛ2+ɛ2=ɛ.
Thus, we can take sequences 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)∈𝕋×S, the following holds:
(3.14)|f(t+αp(k),x)-f(t+αm(k),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 {f(t+βn,x)}⊂{f(t+αn,x)} for any integers p,m with p<m and all (t,x)∈𝕋×S the following holds:
(3.15)|f(t+βp,x)-f(t+βm,x)|<1p.
Therefore, {f(t+βn,x)} converges uniformly on 𝕋×S that is, Tβf(t,x)=g(t,x) holds uniformly on 𝕋×S, where β={βn}⊂α.
Next, we will prove thatg(t,x) is continuous on 𝕋×D. If this is not true,then  there must exist (t0,x0)∈𝕋×D such that g(t,x) is not continuous at this point. Then there exist ɛ0>0 and sequences {δm},{tm},and{xm}, where δm>0,δm→0 as m→+∞, |t0-tm|+|x0-xm|<δm and(3.16)|g(t0,x0)-g(tm,xm)|≥ɛ0.
Let X={xm}⋃{x0}; obviously, X is a compact subset of D. Hence, there exists positive integer N=N(ɛ0,X) so that n>N implies
(3.17)|f(tm+βn,xm)-g(tm,xm)|<ɛ03∀m∈Z+,|f(t0+βn,x0)-g(t0,x0)|<ɛ03.
According to the uniform continuity of f(t,x) on 𝕋×X, for sufficiently large m, we have
(3.18)|f(t0+βn,x0)-f(tm+βn,xm)|<ɛ03.
From (3.17)–(3.18), we get
(3.19)|g(t0,x0)-g(tm,xm)|<ɛ0,
this contradicts (3.16). Therefore, g(t,x) is continuous on 𝕋×D.
Finally, for any compact setS⊂D and given ɛ>0, one can take τ∈E{ɛ,f,S; then, for all (t,x)∈𝕋×S, the following holds: (3.20)|f(t+βn+τ,x)-f(t+βn,x)|<ɛ.
Let n→+∞, for all (t,x)∈𝕋×S; we have
(3.21)|g(t+τ,x)-g(t,x)|≤ɛ,
which implies that E{ɛ,g,S} is relatively dense. Therefore, g(t,x) is almost periodic in t uniformly for x∈D. This completes the proof.Theorem 3.14.
Letf∈C(𝕋×D,𝔼n); if, for any sequence α′⊂Π, there exists α⊂α′ such that Tαf(t,x) exists uniformly on 𝕋×S, then f(t,x) is almost periodic in t uniformly for x∈D.Proof.
For contradiction, if this is not true, then there existɛ0>0 and S⊂D such that, for any sufficiently large l>0, we can find an interval with length of l and there is no ɛ0-translation numbers of f(t,x) in this interval; that is, every point in this interval is not in E{ɛ0,f,S}.
One can take a numberα1′∈Π and find an interval (a1,b1) with b1-a1>2|α1′|, where a1,b1∈Π such that there is no ɛ0-translation numbers of f(t,x) in this interval. Next, taking α2′=(1/2)(a1+b1), obviously, α2′-α1′∈(a1,b1), so α2′-α1′∉E{ɛ0,f,S}; then, one can find an interval (a2,b2) with b2-a2>2(|α1′|+|α2′|), where a2,b2∈Π such that there is no ɛ0-translation numbers of f(t,x) in this interval. Next, taking α3′=(1/2)(a2+b2), obviously, α3′-α2′,α3′-α1′∉E{ɛ0,f,S}. One can repeat these processes, again and again one can find α4′,α5′,…, such that αi′-αj′∉E{ɛ0,f,S},i>j. Hence, for any i≠j,i,j=1,2,…, without loss of generality, let i>j, for x∈S; we have(3.22)sup⁡(t,x)∈T×S|f(t+αi′,x)-f(t+αj′,x)|=sup⁡(t,x)∈T×S|f(t+αi′-αj′,x)-f(t,x)|≥ɛ0.
Therefore, there is no uniformly convergent subsequence of {f(t+αn′,x)} for (t,x)∈𝕋×S; this is a contradiction. Thus, f(t,x) is almost periodic in t uniformly for x∈D. This completes the proof.From Theorems3.13 and 3.14, we can obtain the following equivalent definition of uniformly almost periodic functions.Definition 3.15.
Letf∈C(𝕋×D,𝔼n); if, for any given sequence α′⊂Π, there exists a subsequence α⊂α′ such that Tαf(t,x) exists uniformly on 𝕋×S, then f(t,x) is called an almost periodic function in t uniformly for x∈D.Similar to the proof of Theorem 2.11 in [16], one can prove that the following.Theorem 3.16.
Iff∈C(𝕋×D,𝔼n) is almost periodic in t uniformly for x∈D and φ(t) is almost periodic with {φ(t):t∈𝕋}⊂S, then f(t,φ(t)) is almost periodic.Definition 3.17.
Letf∈C(𝕋×D,𝔼n); H(f)={g:𝕋×D→𝔼n∣ there exits α∈Π such that Tαf(t,x)=g(t,x) exists uniformly on 𝕋×S} is called the hull of f.Similar to the proofs of Theorems 1.6 and 1.8 in [19], one can prove the following two theorems, respectively.Theorem 3.18.
H(f) is compact if and only if f is almost periodic in t uniformly for x∈D.Theorem 3.19.
Iff(t,x) is almost periodic in t uniformly for x∈D, then for any g(t,x)∈H(f),H(f)=H(g).From Definition3.17 and Theorem 3.19, one can easily show that.Theorem 3.20.
Iff(t,x) is almost periodic in t uniformly for x∈D, then, for any g(t,x)∈H(f), g(t,x) is almost periodic in t uniformly for x∈D.Theorem 3.21.
Iff(t,x) is almost periodic in t uniformly for x∈D, then, for any ɛ>0, there exists a positive constant L=L(ɛ,S) and for any a∈ℝ, there exist a constant η>0 and α∈ℝ such that ([α,α+η]∩Π)⊂[a,a+L] and ([α,α+η]∩Π)⊂E(ɛ,f,S).Proof.
Sincef(t,x) is uniformly continuous on 𝕋×S, for any ɛ>0, there exists δ(ɛ1,S)>0 so that |t1-t2|<δ(ɛ1,S) implies
(3.23)|f(t1,x)-f(t2,x)|<ɛ1,∀x∈S,
where ɛ1=ɛ/2.
We takeη=δ(ɛ/2,S)=δ(ɛ1,S), and L=l(ɛ1,S)+η, where l(ɛ1,S) is the inclusion length of E(ɛ1,f,S).
For anya∈ℝ, consider an interval [a,a+L], take(3.24)τ∈E(f,ɛ1,S)∩[a-η2,a+η2+l(ɛ1,S)],
and we have
(3.25)([τ-η2,τ+η2]∩Π)⊂[a,a+L].
Hence, for all ξ∈[τ-(η/2),τ+(η/2)]∩Π, we have |ξ-τ|≤η. Therefore, for any (t,x)∈𝕋×S,
(3.26)|f(t+ξ,x)-f(t,x)|≤|f(t+ξ,x)-f(t+τ,x)|+|f(t+τ,x)-f(t,x)|≤ɛ.
So, we let α=τ-(η/2), then ([α,α+η]∩Π)⊂E(ɛ,f,S). This completes the proof.Theorem 3.22.
Iff,g are almost periodic in t uniformly for x∈D, then, for any ɛ>0, E(f,ɛ,S)∩E(g,ɛ,S) is a nonempty relatively dense set in 𝕋.Proof.
Sincef,g are almost periodic in t uniformly for x∈D, they are uniformly continuous on 𝕋×S. For any given ɛ>0, one can take δi=δi(ɛ/2,S)(i=1,2), and l1=l1(ɛ/2,S),l2=l2(ɛ/2,S) are inclusion lengths of E(f,ɛ/2,S),E(g,ɛ/2,S), respectively.
According to Theorem3.18, we can take (3.27)η=η(ɛ,S)=min⁡(δ1,δ2)∈Π,Li=li+η(i=1,2),L=max⁡(L1,L2).
Hence, we can find ɛ/2-translation numbers of f(t,x) and g(t,x): τ1=mη and τ2=nη, respectively, where τ1,τ2∈[a,a+L]∩Π, m,n are integers, and |τ1-τ2|≤L.
Letm-n=s, then s can only be taken from a finite number set {s1,s2,…,sp}. When m-n=sj,j=1,2,…,p, denote the ɛ/2-translation numbers of f(t) and g(t) by τ1j, τ2j, respectively, that is, τ1j-τ2j=sjη,j=1,2,…,p, and we take T=maxj{|τ1j|,|τ2j|}.
For anya∈ℝ, on the interval [a+T,a+T+L], we can take ɛ/2-translation numbers of f(t,x) and g(t,x): τ1 and τ2, respectively; there must exist some integer sj such that (3.28)τ1-τ2=sjη=τ1j-τ2j.
Set
(3.29)τ(ɛ,S)=τ1-τ1j=τ2-τ2j,
then τ(ɛ,S)∈[a,a+L+2T]∩Π, and, for any (t,x)∈𝕋×S, we have
(3.30)|f(t+τ,x)-f(t,x)|≤|f(t+τ1-τ1j,x)-f(t-τ1j,x)|+|f(t-τ1j,x)-f(t,x)|<ɛ,|g(t+τ,x)-g(t,x)|≤|g(t+τ2-τ2j,x)-g(t-τ2j,x)|+|g(t-τ2j,x)-g(t,x)|<ɛ.
Therefore, there exists at least a τ=τ(ɛ,S) on any interval [a,a+L+2T] with the length (L+2T) such that τ∈E(f,ɛ,S)∩E(g,ɛ,S). The proof is complete.According to Definition3.10, one can easily prove the following.Theorem 3.23.
Iff(t,x) is almost periodic in t uniformly for x∈D, then, for any α∈ℝ,b∈Π, functions αf(t,x),f(t+b,x) are almost periodic in t uniformly for x∈D.Theorem 3.24.
Iff,g are almost periodic in t uniformly for x∈D, then f+g,fg are almost periodic in t uniformly for x∈D and if   inft∈𝕋|g(t,x)|>0, then f(t,x)/g(t,x) are almost periodic in t uniformly for x∈D.Proof.
From Theorem3.22, for any ɛ>0, E(f,ɛ/2,S)∩E(g,ɛ/2,S) is a nonempty relatively dense set. It is easy to see that if τ∈E(f,ɛ/2,S)∩E(g,ɛ/2,S), then τ∈E(f+g,ɛ,S). Hence,
(3.31)(E(f,ɛ2,S)∩E(g,ɛ2,S))⊂E(f+g,ɛ,S).
Therefore, E(f+g,ɛ,S) is a relatively dense set, so f+g is almost periodic in t uniformly for x∈D.
On the other hand, denotesup(t,x)∈𝕋×S|f(t,x)|=M1,sup(t,x)∈𝕋×S|g(t,x)|=M2, take τ∈E(f,ɛ/2,S)⋂E(g,ɛ/2,S), then, for all (t,x)∈𝕋×S, we have(3.32)|f(t+τ,x)g(t+τ,x)-f(t,x)g(t,x)|≤|g(t+τ,x)‖f(t+τ,x)-f(t,x)|+|f(t,x)‖g(t+τ,x)-g(t,x)|≤(M1+M2)ɛ≡ɛ1.
Therefore, τ∈E(fg,ɛ1,S),E(fg,ɛ1,S) is a relatively dense set, so fg is almost periodic in t uniformly for x∈D.
Faunally, denoteinf(t,x)∈𝕋×S|g(t,x)|=N and take τ∈E(g,ɛ,S), then, for all (t,x)∈𝕋×S we have (3.33)|1g(t+τ,x)-1g(t,x)|=|g(t+τ,x)-g(t,x)g(t+τ,x)g(t,x)|<ɛN2≡ɛ2,
that is, τ∈E(1/g,ɛ2,S). Hence, 1/g is almost periodic in t uniformly for x∈D, so f/g is almost periodic in t uniformly for x∈D. The proof is complete.Theorem 3.25.
IfF∈C(ℝ×D,𝔼n) is almost periodic in t uniformly for x∈D, then F(t,x) is also continuous on 𝕋×D and almost periodic in t uniformly for x∈D.Proof.
LetF(r,x)∈C(ℝ×D,𝔼n) be uniformly almost periodic, then, for any sequence α′⊂𝕋p, there exists a subsequence α⊂α′ such that TaF(t+αn,x) exists uniformly on ℝ×S, where S is any compact set in D. Consequently, Tαf(t+αn,x)=TαF(t+αn,x) exists uniformly on 𝕋×S. In view of Theorem 3.14, this shows that f(t,x) is uniformly almost periodic.Corollary 3.26.
IfF∈C(ℝ,𝔼n) is an almost periodic function, then F(t) is an almost periodic function on 𝕋.Theorem 3.27.
Iffn∈C(𝕋×D,𝔼n),n=1,2,… are almost periodic in t for x∈D and the sequence {fn(t,x)} uniformly converges to f(t,x) on 𝕋×S, then f(t,x) is almost periodic in t uniformly for x∈D.Proof.
For anyɛ>0, there exists sufficiently large n0 such that, for all (t,x)∈𝕋×S,
(3.34)|f(t,x)-fn0(t,x)|<ɛ3.
Take τ∈E{fn0,ɛ/3,S}, then, for all (t,x)∈𝕋×S, we have
(3.35)|f(t+τ,x)-f(t,x)|≤|f(t+τ,x)-fn0(t+τ,x)|+|fn0(t+τ,x)-fn0(t,x)|+|fn0(t,x)-f(t,x)|<ɛ,
that is, τ∈E(f,ɛ,S). Therefore, E(f,ɛ,S) is also a relatively dense set; f(t,x) is almost periodic in t uniformly for x∈D. This completes the proof.Theorem 3.28.
Iff(t,x) is almost periodic in t uniformly for x∈D, denote F(t,x)=∫0tf(s,x)Δs, then F(t,x) is almost periodic in t uniformly for x∈D if and only if F(t,x) is bounded on 𝕋×S.Proof.
IfF(t,x) is almost periodic in t uniformly for x∈D, then it is easy to see that F(t,x) is bounded on 𝕋×S.
IfF(t,x) is bounded, without loss of generality, then we can assume that F(t,x) is a real-valued function. Denote (3.36)G:=sup⁡(t,x)∈T×SF(t,x)>g:=inf⁡(t,x)∈T×SF(t,x),
for any ɛ>0, there exist t1 and t2 such that
(3.37)F(t1,x)<g+ɛ6,F(t2,x)>G-ɛ6,∀x∈S.
Let l=l(ɛ1,S) be an inclusion length of E(f,ɛ1,S), where ɛ1=ɛ/6d,d=|t1-t2|. For any α∈𝕋, take τ∈E(f,ɛ1,S)∩[α-t1,α-t1+l]. Denote si=ti+τ,(i=1,2),L=l+d, so s1,s2∈[α,α+L]∩𝕋, for all x∈s,
(3.38)F(s2,x)-F(s1,x)=F(t2,x)-F(t1,x)-∫t1t2f(t,x)Δt+∫t1+τt2+τf(t,x)Δt=F(t2,x)-F(t1,x)+∫t1t2[f(t+τ,x)-f(t,x)]Δt>G-g-ɛ3-ɛ1d=G-g-ɛ2,
that is
(3.39)(F(s1,x)-g)+(G-F(s2,x))<ɛ2.
Since
(3.40)F(s1,x)-g≥0,G-F(s2,x)≥0,
in any interval with length L, there exist s1,s2 such that
(3.41)F(s1,x)<g+ɛ2,F(s2,x)>G-ɛ2.
Now, we denoteɛ2=ɛ/2L; in the following, we will prove that if τ∈E(f,ɛ2,S), then τ∈E(F,ɛ,S). In fact, for any (t,x)∈𝕋×S, one can take s1,s2∈[t,t+L]∩𝕋 such that(3.42)F(s1,x)<g+ɛ2,F(s2,x)>G-ɛ2,
so for τ∈E(f,ɛ2,S), we have
(3.43)F(t+τ,x)-F(t,x)=F(s1+τ,x)-F(s1,x)+∫ts1f(t,x)Δt-∫t+τs1+τf(t,x)Δt>g-(g+ɛ2)-∫tt1[f(t+τ,x)-f(t,x)]Δt>-ɛ2-ɛ2L=-ɛ,F(t+τ,x)-F(t,x)=F(s2+τ,x)-F(s2,x)+∫ts2f(t,x)Δt-∫t+τs2+τf(t,x)Δt<ɛ2+ɛ2L=ɛ.
That is, for τ∈E(f,ɛ2,S), we have τ∈E(F,ɛ,S), so F(t,x) is almost periodic in t uniformly for x∈D. The proof is complete.Theorem 3.29.
Iff(t,x) is almost periodic in t uniformly for x∈D and F(·) is uniformly continuous on the value field of f(t,x), then F∘f is almost periodic in t uniformly for x∈D.Proof.
In fact, sinceF is uniformly continuous on the value field of f(t,x) and f(t,x) is almost periodic in t uniformly for x∈D, there exists a real sequence α={αn}⊆Π such that
(3.44)Tα(F∘f)=lim⁡n→+∞F(f(t+αn,x))=F(lim⁡n→+∞f(t+αn,x))=F(Tαf)
holds uniformly on 𝕋×S. Hence, F∘f is almost periodic in t uniformly for x∈D. The proof is complete.Similar to the proof of Theorem 1.17 in [19], one can easily get the following.Theorem 3.30.
A functionf(t,x) is almost periodic in t uniformly for x∈D if and only if, for every pair of sequences α′,β′⊆Π, there exist common subsequences α⊂α′,β⊂β′ such that
(3.45)Tα+βf(t,x)=TαTβf(t,x).Definition 3.31.
If every element of matrix functionM(t,x)=(mij(t,x))n×m, where mij(t,x)∈C(𝕋,𝔼)(1,2,…,n;j=1,2,…,m) is almost periodic in t uniformly for x∈D, then M(t,x) is called almost periodic in t uniformly for x∈D.If we use matrix norm:|M(t,x)|=∑ijmij2(t,x), then the definition above can be rewritten as.Definition 3.32.
A matrix functionM(t,x) is almost periodic in t uniformly for x∈D if and only if, for any ɛ>0, the translation set
(3.46)E(M,ɛ,S)={τ∈Π:|M(t+τ,x)-M(t,x)|<ɛ,∀(t,x)∈T×S}
is a relatively dense set.Theorem 3.33.
Definition3.31 is equivalent to Definition 3.32.Proof.
In fact, ifM(t,x) is almost periodic in t uniformly for x∈D, by Definition 3.31, then every element mij(t,x) is almost periodic in t uniformly for x∈D. Thus, for any ɛ>0, there exists nonempty relatively dense set θ=⋂i,jE(mij(t,x),ɛ/mn,S). For any τ∈θ, we have
(3.47)|M(t+τ,x)-M(t,x)|=[∑i,j|mij(t+τ,x)-mij(t,x)|2]1/2<ɛ.
On the other hand, if, for anyɛ>0, E(M,ɛ,S) is a relatively dense set, then, for any i=1,2,…,n,j=1,2,…,m and τ∈E(M,ɛ,S), we have(3.48)|mij(t+τ,x)-mij(t,x)|<|M(t+τ,x)-M(t,x)|<ɛ,∀(t,x)∈T×S.
Hence, every element mij(t,x) is almost periodic in t uniformly for x∈D that is, M(t,x) is almost periodic in t uniformly for x∈D. The proof is complete.Definition 3.34.
A continuous matrix functionM(t,x) is called normal if, for any sequence α′⊆Π, thenthere exists subsequence α⊂α′ such that TαM(t,x) exists uniformly on 𝕋×S.Theorem 3.35.
A continuous matrix functionM(t,x) is normal if and only if M(t,x) is almost periodic in t uniformly for x∈D.Proof.
IfM(t,x) is normal, then every element mij(t,x) satisfies Definition 3.15, so M(t) is almost periodic.
On the other hand, ifM(t,x) is almost periodic in t uniformly for x∈D, by Definition 3.31, for any sequence α′⊆Π, thenthere exists subsequence α1⊂α′ such that Tα1m11(t,x) exists uniformly on 𝕋×S. Hence, there exists α2⊂α1, such that Tα2m12(t,x) exists uniformly on 𝕋×S; we can repeat this step mn times, then we can get a series of subsequences satisfying(3.49)α={αk}=αmn⊂αmn-1⊂…⊂α2⊂α1⊂α′
such that
(3.50)Tαmij(t,x),i=1,2,…,n,j=1,2,…,m
exist uniformly on 𝕋×S. Therefore, there exists subsequence α⊂α′ such that TαM(t,x) exists uniformly on 𝕋×S; that is, M(t,x) is normal. The proof is complete.


## 4. Almost Periodic Dynamic Equations on Almost Periodic Time Scales

Consider the following nonlinear dynamic equation(4.1)xΔ=f(t,x),
where f∈C(𝕋×𝔼n,𝔼n); let Ω={x(t):x(t) is a bounded solution to (4.1)}.Definition 4.1.
IfΩ≠ϕ, then λ=infx∈Ω∥x∥ exists λ is called the least-value of solutions to (4.1). If there exists φ(t)∈Ω such that ∥φ∥=λ, then φ(t) is called a minimum norm solution to (4.1), where ∥·∥=supt∈𝕋|·|.Similar to the proof of Theorem 5.1 in [19], one can easily get the following.Lemma 4.2.
Iff(t,x)∈C(𝕋×S,𝔼n) is bounded on 𝕋×S and (4.1) has a bounded solution φ(t) such that {φ(t),t∈𝕋}⊂S and 0∈S, then (4.1) must have a minimum norm solution.Lemma 4.3.
Iff(t,x) is almost periodic in t uniformly for x∈𝔼n, S={φ(t):t≥t0}¯ and (4.1) has a bounded solution φ(t) on [t0,∞)∩𝕋, then (4.1) has a bounded solution ψ(t) on 𝕋 and {ψ(t),t∈𝕋}⊂S.Proof.
In fact, we may takeα′={αk′}⊂Π such that limn→+∞αk′=+∞ and Tα′f(t,x)=f(t,x) holds uniformly on 𝕋×S. For any fixed a, consider the interval (a,∞)∩𝕋 and φk(t)=φ(t+αk′). It is easy to see that, for k sufficiently large, {φk} is defined on (a,∞)∩𝕋 and is a solution to xΔ=f(t+αk′,x) and {φk(t)} is uniformly bounded and equicontinuous on (a,∞)∩𝕋. Then let α be a sequence which goes to -∞, according to Lemma 2.3, there must exist α⊂α′ such that Tαφ(t)=ψ(t) holds uniformly on any compact subset of 𝕋 and, for all t∈𝕋, we have ψ(t)∈S. Since Tαf(t,x)=f(t,x), by Lemma 3.5, ψ(t) is a solution to (4.1). This completes the proof.Lemma 4.4.
Letf(t,x)∈C(𝕋×𝔼n,𝔼n) be almost periodic in t uniformly for x∈𝔼n. If (4.1) has a minimum norm solution, then, for any g(t,x)∈H(f), the following equation:
(4.2)xΔ=g(t,x)
has the same least-value of solutions as that to (4.1).Proof.
Letφ(t) be the minimum norm solution to (4.1) and λ is the least-value. Since g(t,x)∈H(f), there exists a sequence α′∈Π such that Tα′f(t,x)=g(t,x) holds uniformly on 𝕋×S. From Lemma 2.3, there exists α⊂α′ such that Tαφ(t)=ψ(t) holds uniformly on any compact subset of 𝕋. By Lemma 3.5, ψ(t) is a solution to (4.2). For |φ(t)|≤λ, we have |ψ(t)|≤λ; thus, λ′=∥ψ(t)∥≤λ. Since φ(t)=T-αψ(t) and |ψ(t)|≤λ′, we have |φ(t)|≤λ′; thus, λ=∥φ(t)∥≤λ′. Therefore, λ=λ′. The proof is complete.From the process of the proof of Lemma4.4, one can easily get the following.Lemma 4.5.
Ifφ(t) is a minimum norm solution to (4.1) and there exists a sequence α′⊆Π such that Tα′f(t,x)=g(t,x) exists uniformly on 𝕋×S,   furthermore, if there exists a subsequence α⊂α′ such that Tαφ(t)=ψ(t) holds uniformly on any compact set of 𝕋, then ψ(t) is a minimum norm solution to (4.2).Lemma 4.6.
Letf(t,x)∈C(𝕋×𝔼n,𝔼n) be almost periodic in t uniformly for x∈𝔼n and, for every g(t,x)∈H(f), (4.2) has a unique minimum norm solution; then theses minimum norm solutions are almost periodic on 𝕋.Proof.
For a fixedg(t,x)∈H(f), (4.2) has the unique minimum norm solution ψ(t). Since g(t,x) is almost periodic in t uniformly for x∈𝔼n, we have that for any sequences α′,β′⊆Π, there exist common subsequences α⊂α′,β⊂β′ such that Tα+βg(t,x)=TαTβg(t,x) holds uniformly on 𝕋×S and TαTβψ(t), Tα+βψ(t) hold uniformly on 𝕋. It follows from Lemmas 4.4 and 4.5 that TαTβψ(t) and Tα+βψ(t) are minimum norm solutions to the following equation:
(4.3)xΔ=Tα+βg(t,x).
Since the minimum norm solution is unique, we have TαTβψ(t)=Tα+βψ(t). Therefore, ψ(t) is almost periodic. The proof is complete.We will now discuss the linear almost periodic dynamic equation on an almost periodic time scale𝕋:(4.4)xΔ=A(t)x+f(t)
and its associated homogeneous equation(4.5)xΔ=A(t)x,
where A(t) is an almost periodic matrix function and f(t) is an almost periodic vector function.Definition 4.7.
IfB∈H(A), thenwe say that
(4.6)yΔ=B(t)y
is a homogeneous equation in the hull of (4.4).Definition 4.8.
IfB∈H(A) and g∈H(f), then we say that
(4.7)yΔ=B(t)y+g(t)
is an equation in the hull of (4.4).Definition 4.9 (see [31]).
LetA(t) be n×n rd-continuous matrix function on 𝕋; the linear system
(4.8)xΔ(t)=A(t)x(t)
is said to admit an exponential dichotomy on 𝕋 if there exist positive constants K,α,  projection P, and the fundamental solution matrix X(t) of (4.8), satisfying
(4.9)|X(t)PX-1(s)|≤Ke⊖α(t,s),s,t∈T,t≥s,|X(t)(I-P)X-1(s)|≤Ke⊖α(s,t),s,t∈T,t≤s.Similar to the proof of Theorem 5.7 (Favard's Theorem) in [19], one can obtain that the following.Lemma 4.10.
IfA(t) is an almost periodic matrix function and x(t) is an almost periodic solution of the homogeneous linear almost periodic dynamic equation xΔ=A(t)x, then inft∈𝕋|x(t)|>0 or x(t)≡0.Similar to the proof of Theorems 6.3 and 5.8 in [19], one can easily get.Lemma 4.11.
Suppose that (4.5) has an almost periodic solution x(t) and inft∈𝕋|x(t)|>0. If (4.4) has bounded solution on [t0,∞)∩𝕋, then (4.4) has an almost periodic solution.Lemma 4.12.
If every bounded solution of a homogeneous equation in the hull of (4.4) is almost periodic, then all bounded solutions of (4.4) are almost periodic.Proof.
According to Lemma4.10, we know that every nontrivial bounded solution of equations in the hull of (4.4) satisfies inft∈𝕋|x(t)|>0. From Lemma 4.11, it follows that if (4.4) has bounded solutions on 𝕋, then (4.4) must have an almost periodic solution ψ(t). If φ(t) is an arbitrary bounded solution of (4.4), then η(t)=ψ(t)-φ(t) is a bounded solution of its associated homogeneous equation (4.5) and it is almost periodic. Thus, φ(t) is almost periodic. This completes the proof.Lemma 4.13.
If a homogeneous equation in the hull of (4.4) has the unique bounded solution x(t)≡0, then (4.4) has a unique almost periodic solution.Proof.
Letψ(t),andφ(t) be two bounded solutions to (4.4), then x(t)=φ(t)-ψ(t) is a solution of a homogeneous equation in the hull of (4.4), since x(t)≡0, we have that φ(t)≡ψ(t). Thus, by Lemma 4.12, (4.4) has a unique almost periodic solution. This completes the proof.Similar to the proof of Lemma 7.4 in [16], one can easily prove that the following.Lemma 4.14.
LetP be a projection and X a differentiable invertible matrix such that XPX-1 is bounded on 𝕋. Then, there exists a differentiable matrix S such that XPX-1=SPS-1 for all t∈𝕋 and S,S-1 are bounded on 𝕋. In fact, there is an S of the form S=XQ-1, where Q commutes with P.Similar to the proof of Lemma 7.5, in [19], one can easily get.Lemma 4.15.
If (4.5) has an exponential dichotomy and X(t) is the fundamental solution matrix of (4.5), C non-singular, then X(t)C has an exponential dichotomy with the same projection P if and only if CP=PC.Similar to the proof of Theorem 7.6 in [19], we can easily obtain.Lemma 4.16.
Suppose thatA(t) is an almost periodic matrix function and (4.5) has an exponential dichotomy, then for every B(t)∈H(A), (4.6) has an exponential dichotomy with the same projection P and the same constants K,α.Lemma 4.17.
If the homogeneous equation (4.5) has an exponential dichotomy, then (4.5) has only one bounded solution x(t)≡0.Proof.
LetX(t) be the fundamental solution matrix to (4.5). For any sequence α⊂Π, denote An=A(t+αn),Xn(t)=X(t+αn). Since the homogeneous equation (4.5) has an exponential dichotomy, it is easy to see that there exists a constant M such that ∥Xn(t)∥≤M and ∥XΔ(t)∥=∥An(t)Xn(t)∥≤A¯M, where A¯=supt∈𝕋∥A(t)∥. Therefore, by Lemma 2.3, there exists {αnk}:=α′⊂α such that {Xnk} converges uniformly on any compact subset of 𝕋 and limn→+∞X(t+αn) exists uniformly on 𝕋. So, X(t) is almost periodic. Since the homogeneous equation (4.5) has an exponential dichotomy, inft∈𝕋x(t)=0, from Lemma 4.10, x(t)≡0. This completes the proof.Lemma 4.18.
If the homogeneous equation (4.5) has an exponential dichotomy, then all equations in the hull of (4.5) have only one bounded solution x(t)≡0.Proof.
By Lemma4.16, all equations in the hull of (4.5) have an exponential dichotomy; according to Lemma 4.17, all equations in the hull of (4.5) have only one bounded solution x(t)≡0. This completes the proof.Similar to the proof of Theorem 7.7 in [19], we have the following.Theorem 4.19.
LetA(t) be an almost periodic matrix function and f(t) be an almost periodic vector function. If (4.5) admits an exponential dichotomy, then (4.4) has a unique almost periodic solution:
(4.10)x(t)=∫-∞tX(t)PX-1(σ(s))f(s)Δs-∫t+∞X(t)(I-P)X-1(σ(s))f(s)Δs,
where X(t) is the fundamental solution matrix of (4.5).Example 4.20.
Consider the following dynamic equation on an almost periodic time scale𝕋(4.11)xΔ(t)=Ax(t)+f(t),
where
(4.12)A=(-600-6),f(t)=(sin⁡3tcos⁡2t),μ(t)≠16.
Obviously, I+μ(t)A is invertible for all 𝕋, so A∈ℛ. We claim that the homogeneous equation of (4.11) admits an exponential dichotomy. In fact, the eigenvalues of the coefficient matrix in (4.11) are λ1=λ2=-6, according to Theorem 5.35 (Putzer Algorithm) in [1], the P-matrices are given by
(4.13)P0=I=(1001),P1=(A-λ1I)P0=A+6I=(0000).
We want to choose
(4.14)r1Δ=-6r1,r1(t0)=1,r2Δ=r1-6r2,r2(t0)=0.
Solving the first IVP for r1, we get that r1(t)=e-6(t,t0). Solving the second IVP, that is,
(4.15)r2Δ=-6r2+e-6(t,t0),r2(t0)=0,
we obtain
(4.16)r2=e-6(t,t0)∫t0tΔs1-6μ(s).
Using Theorem 5.35 (Putzer Algorithm) in [1], we get
(4.17)eA(t,t0)=r1(t)P0+r2(t)P1=e-6(t,t0)(1001).
Thus,
(4.18)|X(t)P0X-1(s)|=‖e-6(t,t0)(1001)e⊖-6(s,t0)(1001)‖≤2e⊖3(t,s).
Then, we can take K=2,α=3 such that the homogeneous equation of (4.11) admits an exponential dichotomy. Finally, according to Theorem 4.19, we can get the unique almost periodic solution of (4.11):
(4.19)x(t)=∫-∞tX(t)P0X-1(σ(s))f(s)Δs-∫t+∞X(t)(I-P0)X-1(σ(s))f(s)Δs=∫-∞te-6(t,σ(s))(1001)(sin⁡3scos⁡2s)Δs.


## 5. Almost Periodic Solutions to a Nonlinear Dynamic Equation on Time Scales

As an application of our results obtained in the previous sections, in this section, we consider the following dynamic equation with delays on almost periodic time scale𝕋:(5.1)xΔ(t)=A(t)x(t)+∑i=1nf(t,x(t-τi(t))).Theorem 5.1.
Suppose that the following hold:(H2)
A(t) is an almost periodic matrix function on 𝕋, for every i=1,2,…,n,τi(t) is almost periodic on 𝕋,t-τi(t)∈𝕋, for t∈𝕋, and f∈C(T×ℝn,ℝn) is almost periodic uniformly in t for x∈ℝn;(H1)
xΔ(t)=A(t)x(t) admits an exponential dichotomy on 𝕋 with positive constants K and α.;(H2)
There existsM<α/(2+μα)Kn such that |f(t,x)-f(t,y)|≤M|x-y| for t∈𝕋,x,y∈ℝn.Then system (5.1) has a unique almost periodic solution.Proof.
For anyφ∈AP(𝕋), consider the following equation:
(5.2)xΔ(t)=A(t)x(t)+∑i=1nf(t,φ(t-τi(t))).
According to Theorem 4.19, (5.2) has a unique solution Tφ∈AP(𝕋) and
(5.3)Tφ(t)=∫-∞tX(t)PX-1(σ(s))∑i=1nf(s,φ(s-τi(s)))Δs-∫t+∞X(t)(I-P)X-1(σ(s))∑i=1nf(s,φ(s-τi(s)))Δs.
Define a mapping T:AP(𝕋)→AP(𝕋) by setting (Tφ)(t)=xφ(t),forallx∈AP(𝕋). From (H1), we have
(5.4)|X(t)PX-1(s)|≤Ke⊖α(t,s),s,t∈T,t≥s,|X(t)(I-P)X-1(s)|≤Ke⊖α(s,t),s,t∈T,t≤s.
For any φ,ψ∈AP(𝕋), we have
(5.5)‖Tφ-Tψ‖≤∫-∞tX(t)PX-1(σ(s))∑i=1n|f(s,φ(s-τi(s)))-f(s,ψ(s-τi(s)))|Δs-∫t+∞X(t)(I-P)X-1(σ(s))∑i=1n|f(s,φ(s-τi(s)))-f(s,ψ(s-τi(s)))|Δs≤[∫-∞tK⊖α(t,σ(s))Δs+∫t+∞K⊖α(σ(s),t)Δs]∑i=1nM‖φ-ψ‖≤2+μ¯ααKnM‖φ-ψ‖,
where μ¯=supt∈𝕋μ(t). Since (H2), T is a contractive operator. Therefore, (5.1) has a unique almost periodic solution.



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*Source: 341520-2011-10-20.xml*