Almost Periodic Solutions for Second Order Dynamic Equations on Time Scales

Abstract

We firstly introduce the concept and the properties of 𝐶^𝑚 almost periodic functions on time scales, which generalizes the concept of almost periodic functions on time scales and the concept of 𝐶^(𝑛)-almost periodic functions. Secondly, we consider the existence and uniqueness of almost periodic solutions for second order dynamic equations on time scales by Schauder’s fixed point theorem and contracting mapping principle. At last, we obtain alternative theorems for second order dynamic equations on time scales.

1. Introduction

The theory of dynamic equations on time scales was first introduced by Hilger [1]. The study of dynamic equations on time scales helps to avoid studying results twice, once for differential equations and once for difference equations. In recent years, the theory of first order and second order dynamic equations on time scales has been studied, and some important results have been presented in [2–6]. However, to the best of our knowledge, there are no results on the existence of almost periodic solutions for the second order dynamic equations on time scales. The aim of this paper is to consider the existence of almost periodic solutions for second order dynamic equations on time scales.

The concept of almost periodicity was first introduced by Bohr [7] and later generalized by Bochner, Fink, N’Guérékata, and Shen and Yi and others (see [8–11]). Recently, Guan and Wang [12] and Li and Wang [13, 14] developed the theory of almost periodic functions on time scales, which do not only unify the almost periodic functions on ℝ and the almost periodic sequences on ℤ but also extend to nontrivial time scales, for example, 𝑞-difference equations.

The existence and uniqueness solutions for second order dynamic equations have become important in recent years in mathematical models and they rise in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. The existence of oscillatory and nonoscillatory solutions for second order equations has been studied in [15–18] (and the references therein).

This paper is concerned with the second order dynamic equation as follows: 𝑥^ΔΔ=𝑔𝑡,𝑥,𝑥^Δ, (1) where 𝑔∶𝕋×ℝ²→ℝ is almost periodic in 𝑡∈𝕋 uniformly. Such a type of equation appears in many problems of applications, such as Brillouin focusing systems [19, 20], nonlinear elasticity [21], and Ermakov-Pinney equations [22, 23].

In this paper, we consider the existence of almost periodic solutions (1) and present alternative theorems for second order dynamic equations. In order to do this, we introduce a new concept called 𝐶^𝑚 almost periodicity, which generalizes the concept of almost periodic functions on time scales and the concept of 𝐶^(𝑛)-almost periodic functions on ℝ introduced by Adamczak [24], Bugajewski and N’Guérékata [25]. The study of almost periodic solutions on time scales has tremendous potential for applications in mathematical models of real processes and phenomena.

This paper is organized as follows. In Section 2, we recall some properties on time scales. In Section 3, we give the concept and properties of 𝐶^𝑚 almost periodicity, and 𝐶^𝑚 uniformly almost periodicity on time scales. The most important part of this paper are Sections 4 and 5. In Section 4, by exponential dichotomy on time scales and fixed point theorems, the existence and uniqueness theorems of almost periodic solutions for second order dynamic equations are obtained. In Section 5, we show the alternative theorems for second order dynamic equations on time scales by topological degree method.

2. Preliminaries on Time Scales

Suppose that 𝕋 is an arbitrary time scale, that is, a nonempty closed subset of ℝ. Set 𝕋⁺=𝕋∩ℝ⁺, [𝑎,𝑏]_𝕋((𝑎,𝑏]_𝕋,[𝑎,𝑏)_𝕋,(𝑎,𝑏)_𝕋, resp.) =𝕋∩[𝑎,𝑏]((𝑎,𝑏],[𝑎,𝑏),(𝑎,𝑏), resp.). First, we recall some results of time scales in [5, 6].

Definition 1 (see [5, 6]). For 𝑡∈𝕋, the forward jump operator 𝜎∶𝕋→𝕋, the graininess function 𝜇∶𝕋→[0,∞), the backward jump operator 𝜌∶𝕋→𝕋, and the backwards graininess function 𝜈∶𝕋→[0,∞) are given by 𝜎(𝑡){}(𝑡)(𝑡)𝜌(𝑡){}(𝑡)(𝑡),∶=inf𝑠∈𝕋∶𝑠>𝑡,𝜇∶=𝜎−𝑡,∶=sup𝑠∈𝕋∶𝑠<𝑡,𝜈=𝑡−𝜌 (2) respectively. Set inf𝜙=sup𝕋 and sup𝜙=inf𝕋, where 𝜙 denotes the empty set.
If 𝜎(𝑡)>𝑡, then the point 𝑡 is called right-scattered, while if 𝜌(𝑡)<𝑡, then the point 𝑡 is termed left-scattered. If 𝑡<sup𝑇 and 𝜎(𝑡)=𝑡, then the point 𝑡 is called right-dense, while if 𝑡>inf𝕋 and 𝜌(𝑡)=𝑡, then 𝑡 is called left-dense. Set 𝕋^𝜅=𝕋⧵𝜌sup𝕋,sup𝕋ifsup𝕋<∞𝕋ifsup𝕋=∞. (3)

Definition 2 (see [5, 6]). (i) Let 𝑓∶𝕋→ℝ be a function. Then 𝑓 is called Δ-differentiable at 𝑡∈𝕋^𝜅 if the limit lim_{𝑠→𝑡𝑠≠𝜎(𝑡)}𝑓(𝜎(𝑡))−𝑓(𝑠)𝜎(𝑡)−𝑠 (4) provided that this limit exists as a finite number 𝜃, and 𝜃 is called the Δ-derivative of 𝑓 at 𝑡∈𝕋^𝜅 and we denote it by 𝜃=𝑓^Δ(𝑡).
(ii) If 𝐹^Δ(𝑡)=𝑓(𝑡), then the Δ-integral is defined by ^𝑡_𝑎𝑓(𝑠)Δ𝑠=𝐹(𝑡)−𝐹(𝑎). (5)

Definition 3 (see [5, 6]). A function 𝑓∶𝕋→ℝ^𝑛 is called rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point. The set of rd-continuous functions 𝑓∶𝕋→ℝ^𝑛 is denoted by 𝐶_rd(𝕋,ℝ^𝑛).
Obviously, 𝐶_rd(𝕋,ℝ^𝑛)⊂𝐶(𝕋,ℝ^𝑛). If 𝑓 is Δ-differential, then 𝑓 is continuous. If 𝑓∈𝐶_rd(𝕋,ℝ^𝑛), then 𝑓 is Δ-integral.

Definition 4 (see [5, 6]). Suppose that 𝑝,𝑞∈ℛ={𝑝∶𝕋→ℝ∶1+𝜇(𝑡)𝑝(𝑡)≠0for𝑡∈𝕋^𝜅}. Then (i)(𝑝⊕𝑞)(𝑡)∶=𝑝(𝑡)+𝜇(𝑡)𝑝(𝑡)𝑞(𝑡) for all 𝑡∈𝕋^𝜅; (ii)⊖𝑝(𝑡)∶=−𝑝(𝑡)/(1+𝜇(𝑡)𝑝(𝑡)) for all 𝑡∈𝕋^𝜅.

Definition 5 (see [5, 6]). Suppose that 𝑝∈ℛ; then 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏for𝑠,𝑡∈𝕋, (6) where 𝜉_𝜇(𝜏)𝑝(𝜏=𝑝(𝜏)(𝜏))1if𝜇=0𝜇(𝜏)log1+𝜇(𝜏)𝑝(𝜏)if𝜇(𝜏)≠0. (7)
The following lemmas are concerned with the properties of 𝑒_𝑝(𝑡,𝑠).

Lemma 6 (see [6, 26]). Suppose that 𝑝,𝑞∈ℛ, 𝑠,𝑡∈𝕋. Then  (i) 𝑒₀(𝑡,𝑠)=1, 𝑒_𝑝(𝑡,𝑡)=1;  (ii) 𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠);  (iii) 1/𝑒_𝑝(𝑡,𝑠)=𝑒_⊖𝑝(𝑡,𝑠)=𝑒_𝑝(𝑠,𝑡);  (iv) 𝑒_𝑝(𝑡,𝑠)𝑒_𝑝(𝑠,𝑟)=𝑒_𝑝(𝑡,𝑟);  (v) 𝑒_𝑝(𝑡,𝑠)𝑒_𝑞(𝑡,𝑠)=𝑒_𝑝⊕𝑞(𝑡,𝑠);  (vi) 𝑒^Δ_𝑝(⋅,𝑠)=𝑝𝑒_𝑝(⋅,𝑠);  (vii) (1/𝑒_𝑝(⋅,𝑠))^Δ=−𝑝(⋅)/𝑒^𝜎_𝑝(⋅,𝑠);  (viii) 𝑒_𝑝(𝑡,⋅)^Δ=−𝑝𝑒^𝜎_𝑝(𝑡,⋅);  (ix) let 𝑝>0 be a constant, and let the graininess function 𝜇 be uniformly bounded with sup_𝑡∈𝕋𝜇(𝑡)=𝑀>0. Then for 𝑡≥𝑠, 𝑒^{𝑝(𝑡−𝑠)}≥𝑒_𝑝(𝑡,𝑠)≥1+𝑝𝑀^{(𝑡−𝑠)/𝑀},𝑒^{−𝑝(𝑡−𝑠)}≤𝑒_⊖𝑝()≤1𝑡,𝑠1+𝑝𝑀^{(𝑡−𝑠)/𝑀}. (8)

Lemma 7 (see [5]). Let 𝑓∶𝕋→ℝ be function. Then (i) ∫_𝑡^𝜎(𝑡)𝑓(𝑠)Δ𝑠=𝜇(𝑡)𝑓(𝑡); (ii) ∫^𝑡_𝜌(𝑡)𝑓(𝑠)Δ𝑠=𝜈(𝑡)𝑓(𝜌(𝑡)).

3. 𝐶^𝑚 Almost Periodic Functions

In this section, we will introduce a new concept called 𝐶^𝑚 almost periodic and 𝐶^𝑚 uniformly almost periodic on time scales. First, we recall the conceptions and the properties of almost periodic functions and uniformly almost periodic functions in the works [12–14].

Definition 8 (see [12]). Suppose that 𝕋 is a time scale.(i)A real number 𝜏 is called a translation invariant for 𝕋, if 𝜏+𝑡∈𝕋for𝑡∈𝕋. (9) The notation 𝛾(𝕋) denotes the set of all translation invariants of 𝕋. (ii)If there exist 𝛼,𝛽∈𝛾(𝕋)=𝛾(𝕋)−{0} with 𝛼𝛽<0, then 𝕋 is called two-way translation invariant time scale or almost periodic time scale.

There exist lots of two-way translation invariant time scales. For example, ℝ, ℤ, ℎℤ,ℙ_𝑎,𝑏=⋃_𝑘∈ℤ[𝑘(𝑎+𝑏),𝑘(𝑎+𝑏)+𝑎](𝑎,𝑏>0) and so on.

Lemma 9 (see [12]). If 𝕋 is a two-way translation invariant time scale, then sup𝕋=∞, inf𝕋=−∞.

Thus, 𝕋^𝜅=𝕋 follows from 𝕋 being a two-way translation invariant time scale.

Lemma 10. Suppose that 𝕋 is a two-way translation invariant time scale. Then there exists a 𝑀>0 such that 𝜇(𝑡),𝜈(𝑡)≤𝑀 for all 𝑡∈𝕋.

Proof. Since 𝕋 is a two-way translation invariant time scale, there is a 𝑀>0 with 𝑀∈𝛾(𝕋) such that 𝑡±𝑀∈𝕋 for all 𝑡∈𝕋. Thus 𝜎(𝑡)≤𝑀+𝑡 and 𝜌(𝑡)≥𝑡−𝑀 for all 𝑡∈𝕋. Furthermore, 𝜇(𝑡),𝜈(𝑡)≤𝑀 for all 𝑡∈𝕋.

If 𝕋 is a two-way translation invariant time scale, set 𝜗⁺∶=min[0,∞)_𝕋,𝑀∶=sup_𝑡∈𝕋𝜇(𝑡),‖𝑥‖∶=max_{1≤𝑖≤𝑛}sup_𝑡∈𝕋𝑥^𝑖(𝑡), (10) where 𝑥(𝑡)=(𝑥¹(𝑡),…,𝑥^𝑛(𝑡))^𝜏∈𝐵𝐶(𝕋,ℝ^𝑛) with 𝐵𝐶(𝕋,ℝ^𝑛) denoting the set of all bounded functions in 𝐶(𝕋,ℝ^𝑛).

Definition 11 (see [12]). Suppose that 𝕋 is a two-way translation invariant time scale. A function 𝑓∈𝐶(𝕋,ℝ^𝑛) is called (Bohr) almost periodic if for every 𝜀>0 there exists a 𝑙_𝜀>0 such that every interval of length 𝑙_𝜀 contains at least one 𝜏∈𝛾(𝕋) satisfying 𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀for𝑡∈𝕋. (11)

Definition 12 (see [13]). Suppose that 𝕋 is a two-way translation invariant time scale. A function 𝑓∈𝐶(𝕋,ℝ^𝑛) is called (Bochner) almost periodic if any sequence {𝑠^′_𝑛}⊂𝛾(𝕋); there exists a subsequence {𝑠_𝑛}⊂{𝑠^′_𝑛} such that 𝑓(𝑡+𝑠_𝑛) converges uniformly for 𝑡∈𝕋.

In [14], Li and Wang show that Definition 11 is equivalent to Definition 12. In the following, the notation AP(𝕋,ℝ^𝑛) denotes the set of all almost periodic functions in 𝐶(𝕋,ℝ^𝑛).

Lemma 13 (see [13]). Suppose that 𝑓,𝑔∈𝐴𝑃(𝕋,ℝ^𝑛) and 𝛼∈ℝ is a constant. Then  (i) 𝑓+𝑔,𝛼𝑓∈𝐴𝑃(𝕋,ℝ^𝑛);  (ii) 𝑓 is bounded on 𝕋;  (iii) 𝑓 is uniformly continuous on 𝕋;  (iv) 𝐹=∫^𝑡₀𝑓(𝑠)Δ𝑠 is almost periodic if and only if 𝑓 is bounded on 𝕋.

Definition 14. A function 𝑓∈𝐶(𝕋×ℝ^𝑚,ℝ^𝑛) is called almost periodic in 𝑡∈𝕋 uniformly for all 𝑥∈ℝ^𝑚 if 𝑓(⋅,𝑥) is almost periodic for all 𝑥 in each compact subset of ℝ^𝑚.

The notation UAP(𝕋×ℝ^𝑚,ℝ^𝑛) denotes the set of all uniformly almost periodic functions in 𝐶(𝕋×ℝ^𝑚,ℝ^𝑛).

Similar to the proofs in [14], Lemmas 15 and 16 hold.

Lemma 15. Suppose that 𝑓,𝑔∈𝑈𝐴𝑃(𝕋×ℝ^𝑚,ℝ^𝑛) and 𝛼∈ℝ is a constant. Then  (i) 𝑓+𝑔,𝛼𝑓∈𝑈𝐴𝑃(𝕋×ℝ^𝑚,ℝ^𝑛);  (ii) for each compact subset 𝑆⊂ℝ^𝑚, 𝑓 is bounded and uniformly continuous on 𝕋×𝑆;  (iii) 𝐹(𝑡,𝑥)=∫^𝑡₀𝑓(𝑠,𝑥)Δ𝑠∈𝑈𝐴𝑃(𝕋×ℝ^𝑚,ℝ^𝑛) if and only if 𝐹(𝑡,𝑥) is bounded on 𝕋×𝑆 with 𝑆⊂ℝ^𝑚 being compact.

Lemma 16. Let 𝑓∈𝑈𝐴𝑃(𝕋×ℝ^𝑚,ℝ^𝑛) and 𝜙∈𝐴𝑃(𝕋,ℝ^𝑚). Then 𝐹(𝑡)=𝑓(𝑡,𝜙(𝑡))∈𝐴𝑃(𝕋,ℝ^𝑛).

Lemma 17. Let 𝑓(𝑡)=(𝑓¹(𝑡),…,𝑓^𝑛(𝑡))^𝜏∈𝐶(𝕋,ℝ^𝑚𝑛) with 𝑓^𝑖∈𝐶(𝕋,ℝ^𝑚) for 𝑖=1,…,𝑛. Then 𝑓∈𝐴𝑃(𝕋,ℝ^𝑚𝑛) if and only if 𝑓^𝑖∈𝐴𝑃(𝕋,ℝ^𝑚) for 𝑖=1,…,𝑛.

Proof. If 𝑓∈AP(𝕋,ℝ^𝑚𝑛), then for each 𝜀>0, there exists a 𝑙_𝜀>0 such that every interval of length 𝑙_𝜀 contains a 𝜏 implying that max_{1≤𝑖≤𝑛}sup_𝑡∈𝕋𝑓^𝑖(𝑡+𝜏)−𝑓^𝑖(𝑡)=𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀. (12) Thus for 𝑖=1,…,𝑛, ‖𝑓^𝑖(𝑡+𝜏)−𝑓^𝑖(𝑡)‖<𝜀; that is, 𝑓^𝑖∈AP(𝕋,ℝ^𝑚) for 𝑖=1,…,𝑛.
If 𝑓^𝑖∈AP(𝕋,ℝ^𝑚) for 𝑖=1,…,𝑛, then for each sequence {𝑡_𝑘^′′}⊂𝛾(𝕋), there exists a subsequence {𝑡^′_𝑘} such that 𝑓¹(𝑡+𝑡^′_𝑘) converges uniformly as 𝑘→∞. Furthermore, there exists a subsequence {𝑡_𝑘}⊂{𝑡^′_𝑘} such that 𝑓²(𝑡+𝑡_𝑘) converges uniformly as 𝑘→∞. Inductively, there exists a subsequence {𝑠_𝑘}⊂{𝑡_𝑘^′′} such that 𝑓^𝑖(𝑡+𝑠_𝑘) converges uniformly as 𝑘→∞ for 𝑖=1,…,𝑛. Moreover, (𝑓¹(𝑡+𝑠_𝑘),…,𝑓^𝑚(𝑡+𝑠_𝑘))^𝜏 converges uniformly as 𝑘→∞ which implies that 𝑓∈AP(𝕋,ℝ^𝑚𝑛).

Set 𝑦_𝑚=^𝑚_𝑗=0𝑦^Δ𝑗(𝑡), (13) where 𝑦^Δ0=𝑦. Then (𝐵𝐶^𝑚(𝕋,ℝ^𝑛),‖⋅‖_𝑚) is a Banach space, where 𝐵𝐶^𝑚(𝕋,ℝ^𝑛) denotes the set of all bounded functions in 𝐶^𝑚(𝕋,ℝ^𝑛), and ‖⋅‖₀=‖⋅‖.

Now we introduce a new concept called 𝐶^𝑚 almost periodic.

Definition 18. Suppose that 𝕋 is a two-way translation invariant time scale. A function 𝑓∈𝐶^𝑚(𝕋,ℝ^𝑛) is called 𝐶^𝑚 (Bohr) almost periodic if for every 𝜀>0 there exists a 𝑙_𝜀>0 such that every interval of length 𝑙_𝜀 contains at least one 𝜏∈𝛾(𝕋) satisfying 𝑓(𝑡+𝜏)−𝑓(𝑡)_𝑚<𝜀for𝑡∈𝕋. (14)

Definition 19. Suppose that 𝕋 is a two-way translation invariant time scale. A function 𝑓∈𝐶^𝑚(𝕋,ℝ^𝑛) is called 𝐶^𝑚 (Bochner) almost periodic if any sequence {𝑠^′_𝑛}⊂𝛾(𝕋); there exists a subsequence {𝑠_𝑛}⊂{𝑠^′_𝑛} such that 𝑓(𝑡+𝑠_𝑛) converges uniformly for 𝑡∈𝕋.

Theorem 20. Let 𝑓∶𝕋→ℝ^𝑛. Then 𝑓 is 𝐶^𝑚 (Bohr) almost periodic if and only if 𝑓,𝑓^Δ,…,𝑓^Δ𝑚 are (Bohr) almost periodic.

Proof. If 𝑓∈AP^𝑚(𝕋,ℝ^𝑛), then for each 𝜀>0, there is a 𝑙_𝜀>0 such that for every interval of length 𝑙_𝜀>0 contains a 𝜏∈𝛾(𝕋) satisfying 𝑓^Δ𝑖(𝑡+𝜏)−𝑓^Δ𝑖(𝑡<_𝑚𝑗=0)𝑓^Δ𝑗(𝑡+𝜏)−𝑓^Δ𝑗(𝑡=𝑓()(𝑡)𝑡+𝜏−𝑓)_𝑚<𝜀for𝑖=0,1,…,𝑚, (15) which implies 𝑓,𝑓^Δ,…,𝑓^Δ𝑚 are (Bohr) almost periodic.
If 𝑓,𝑓^Δ,…,𝑓^Δ𝑚 are almost periodic, by Lemma 17, (𝑓,𝑓^Δ,…,𝑓^Δ𝑚)^𝜏∈AP(𝕋,ℝ^(𝑚+1)𝑛). That is, for each 𝜀>0, there is a 𝑙_𝜀>0 such that for every interval of length 𝑙_𝜀>0 contains a 𝜏 satisfying 𝑓^Δ𝑗(𝑡+𝜏)−𝑓^Δ𝑗(𝑡<𝜀)𝑚+1for𝑗=0,…,𝑚. (16) Thus, 𝑓(𝑡+𝜏)−𝑓(𝑡)_𝑚=^𝑚_𝑗=0𝑓^Δ𝑗(𝑡+𝜏)−𝑓^Δ𝑗(𝑡)<𝜀. (17) Therefore, 𝑓∈AP^𝑚(𝕋,ℝ^𝑛).

If 𝑓∶𝕋→ℝ^𝑛 is 𝐶^𝑚 (Bochner) almost periodic, the following theorem also holds.

Theorem 21. Let 𝑓∶𝕋→ℝ^𝑛. Then 𝑓 is 𝐶^𝑚 (Bochner) almost periodic if and only if 𝑓,𝑓^Δ,…,𝑓^Δ𝑚 are (Bochner) almost periodic.

Since 𝑓∶𝕋→ℝ^𝑛 is (Bohr) almost periodic if and only if 𝑓∶𝕋→ℝ^𝑛 is (Bochner) almost periodic, according to Theorems 20 and 21, Definition 18 is equivalent to Definition 19. In the following, the notations AP^𝑚(𝕋,ℝ^𝑛), denotes the set of all almost periodic functions in 𝐶^𝑚(𝕋,ℝ^𝑛). Obviously, AP⁰(𝕋,ℝ^𝑛)=AP(𝕋,ℝ^𝑛), and AP^𝑚+1𝕋,ℝ^𝑛⊂AP^𝑚𝕋,ℝ^𝑛⊂𝐵𝐶^𝑚𝕋,ℝ^𝑛. (18)

According to Definition 19, we obtain the following theorem.

Theorem 22. Suppose that 𝑓,𝑔∈𝐴𝑃^𝑚(𝕋,ℝ^𝑛) and 𝛼∈ℝ is a constant. Then 𝑓+𝑔,𝛼𝑓,𝑓^∨∈𝐴𝑃^𝑚(𝕋,ℝ^𝑛), where 𝑓^∨(𝑡)=𝑓(−𝑡).

Theorem 23. Let {𝑓_𝑘}⊂𝐶^𝑚(𝕋,ℝ^𝑛) be a sequence of 𝐶^𝑚 almost periodic functions. If lim_𝑘→∞𝑓_𝑘(𝑡)=𝑓(𝑡) holds uniformly on 𝕋, then 𝑓∈𝐴𝑃^𝑚(𝕋,ℝ^𝑛).

Proof. Noting that {𝑓_𝑘} converge uniformly, thus for any 𝜀>0, there exists a 𝑁=𝑁(𝜀)>0 such that 𝑓_𝑁(𝑡)−𝑓(𝑡)_𝑚<𝜀3,𝑓_𝑁(𝑡+𝜏)−𝑓(𝑡+𝜏)_𝑚<𝜀3∀𝜏∈𝛾(𝕋). (19)
Since 𝑓_𝑁∈AP^𝑚(𝕋,ℝ^𝑛), there exists a 𝑙_𝜀/3 such that every interval of length 𝑙_𝜀/3 contains a 𝜏₀∈𝛾(𝕋) such that 𝑓_𝑁(𝑡+𝜏₀)−𝑓_𝑁(𝑡)_𝑚<𝜀3. (20) Then 𝑓𝑡+𝜏₀−𝑓(𝑡)_𝑚≤𝑓(𝑡+𝜏₀)−𝑓_𝑁(𝑡+𝜏₀)_𝑚+𝑓_𝑁𝑡+𝜏₀−𝑓_𝑁(𝑡)_𝑚+𝑓_𝑁(𝑡)−𝑓(𝑡)_𝑚<𝜀, (21) which yields that 𝑓∈AP^𝑚(𝕋,ℝ^𝑛).

Therefore, AP^𝑚(𝕋,ℝ^𝑛) is a Banach space with the norm ‖⋅‖_𝑚.

Similar to the proof of Lemma 13, Theorem 24 holds.

Theorem 24. Let 𝑓∈𝐴𝑃^𝑚(𝕋,ℝ^𝑛). If 𝐹(𝑡)=∫^𝑡_𝑎𝑓(𝑠)Δ𝑠 is bounded, then 𝐹∈𝐴𝑃^𝑚+1(𝕋,ℝ^𝑛).

Theorem 25. Suppose that 𝑓∈𝑈𝐴𝑃(𝕋×ℝ^𝑚𝑛,ℝ^𝑛), 𝜙∈𝐴𝑃^𝑚−1(𝕋,ℝ^𝑛). Then 𝐹(𝑡)=𝑓(𝑡,𝜙(𝑡),…,𝜙^Δ𝑚−1(𝑡))∈𝐴𝑃(𝕋,ℝ^𝑛).

Proof. Since 𝜙∈AP^𝑚−1(𝕋,ℝ^𝑛), then for each sequence {𝑡_𝑛^′′}⊂𝛾(𝕋) there exists a subsequence {𝑡^′_𝑛}⊂{𝑡_𝑛^′′} and function 𝜓=(𝜓₀,…,𝜓_𝑚−1)^𝜏∈𝐶(𝕋,ℝ^𝑚𝑛) such that lim_𝑛→∞𝜙^Δ𝑖𝑡+𝑡^′_𝑛=𝜓_𝑖(𝑡)for𝑡∈𝕋,𝑖=0,…,𝑚−1, (22) holds uniformly.
By Theorem 25 and Theorem  3.20 in [14], 𝜓∈𝐻((𝜙,…,𝜙^Δ𝑚−1)^𝜏) with 𝐻((𝜙,…,𝜙^Δ𝑚−1)^𝜏) being hull of (𝜙,…,𝜙^Δ𝑚−1)^𝜏 and 𝐻((𝜙,…,𝜙^Δ𝑚−1)^𝜏) is compact. Thus there exists {𝑡_𝑛}⊂{𝑡^′_𝑛} such that lim_𝑛→∞𝑓𝑡+𝑡_𝑛,𝑥₁,…,𝑥_𝑚=𝑔𝑡,𝑥₁,…,𝑥_𝑚for𝑡∈𝕋,𝑥₁,…,𝑥_𝑛^𝜏∈𝐻𝜙,…,𝜙^{Δ𝑚−1𝜏} (23) holds uniformly.
Set 𝐺(𝑡)=𝑔(𝑡,𝜓₁(𝑡),…,𝜓_𝑚−1(𝑡)). Then, 𝐹𝑡+𝑡_𝑛(𝑡≤𝑓−𝐺)𝑡+𝑡_𝑛,𝜙𝑡+𝑡_𝑛,…,𝜙^Δ𝑚−1𝑡+𝑡_𝑛−𝑓𝑡+𝑡_𝑛,𝜓₀(𝑡),…,𝜓_𝑚−1(𝑡)+𝑓𝑡+𝑡_𝑛,𝜓₀(𝑡),…,𝜓_𝑚−1(𝑡)−𝑔𝑡,𝜓₀(𝑡),…,𝜓_𝑚−1(𝑡)<𝜀, (24) by the uniform continuity of 𝑓, (22), and (23). Therefore 𝐹∈AP(𝕋,ℝ^𝑛).

4. Almost Periodic Solutions

In this section, we will consider the existence and uniqueness of almost periodic solutions of (1). To investigate (1), we will consider the following auxiliary equation: 𝑥^ΔΔ=𝑥+𝑓𝑡,𝑥,𝑥^Δ, (25) where 𝑓(𝑡,𝑥,𝑦)=𝑔(𝑡,𝑥,𝑦)−𝑥. Obviously, 𝑓∈UAP(𝕋×ℝ²,ℝ) follows from 𝑔∈UAP(𝕋×ℝ²,ℝ).

Lemma 26 (see [13, 14]). Let 𝐴 be a 𝑛×𝑛-matrix in ℛ(𝕋,ℝ^𝑛×𝑛)={𝐴∈𝐺𝑙(𝑛,𝑛)∶det(𝐼_𝑛+𝜇(𝑡)𝐴)≠0}, where 𝐼_𝑛 is 𝑛×𝑛 identity matrix. If the linear system 𝑥^Δ=𝐴(𝑡)𝑥 (26) admits an exponential dichotomy on 𝕋, that is, there exist a projection 𝑃 on ℝ^𝑛 and positive constants 𝐾_𝑖 and 𝛼_𝑖 for 𝑖=1,2 such that Φ(𝑡)𝑃Φ⁻¹(𝑠)≤𝐾₁𝑒_⊖𝛼1()Φ(𝑡𝐼𝑡,𝑠𝑓𝑜𝑟𝑡≥𝑠)_𝑛−𝑃Φ⁻¹(𝑠)≤𝐾₂𝑒_⊖𝛼2(𝑠,𝑡)𝑓𝑜𝑟𝑡≤𝑠, (27) then the system 𝑥^Δ=𝐴(𝑡)𝑥+𝑓(𝑡) (28) has a bounded solution 𝑥(𝑡) as follows: 𝑥(𝑡)=^𝑡_−∞Φ(𝑡)𝑃Φ⁻¹(𝜎(𝑠𝑓(𝑠)−))Δ𝑠_∞𝑡Φ(𝑡)(𝐼−𝑃)Φ⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠. (29) Moreover, if 𝐴 and 𝑓 are almost periodic functions, then (28) admits a unique almost periodic solution as (29).

We will rewrite (25) by the following lemma.

Lemma 27. Suppose that 𝑘∈ℝ with 𝑘≠0 is chosen. If 𝑥 is the solution of (25), then (𝑥,(𝑥^Δ−𝑥)/𝑘)^𝜏 is the solution of vector equation 𝑋^Δ=𝐴𝑋+𝐹(𝑡,𝑋), (30) where 𝑥𝑋=¹𝑥₂,𝐹()=0𝑓,𝐴=1𝑘0−1𝑡,𝑋𝑡,𝑥₁,𝑥₁+𝑘𝑥₂𝑘. (31) However, if 𝑋=(𝑥₁,𝑥₂)^𝜏 is the solution of (30), then 𝑥₁ is the solution of (25).

Lemma 28. Suppose that 𝕋 is a two-way translation invariant time scale. If 𝑘²𝜇(𝑡)+2𝑘≤2/3 for 𝑡∈𝕋, then the homogeneous equation 𝑋^Δ=𝐴𝑋=1𝑘0−1𝑋 (32) admits an exponential dichotomy on 𝕋. Furthermore, the solution of (32) with the initial value 𝑋(𝜗⁺)=𝐼 is 𝑒_𝐴𝑡,𝜗⁺=𝑒₁𝑡,𝜗⁺1𝑘∫^𝑡_𝜗+1+𝜇(𝜏)⁻¹Δ𝜏01. (33)

Proof. For 𝑖=1,2, the matrix 𝐴 satisfies 𝑎_𝑖𝑖(𝑡)−_𝑗≠𝑖𝑎_𝑗𝑖(𝑡−1)2𝜇(𝑡)_𝑗≠𝑖𝑎_𝑗𝑖(𝑡)²≥2𝑘+𝑘²𝜇(𝑡). (34) Thus by Theorem  5.1 in [27], (32) admits an exponential dichotomy on 𝕋.
According to Theorem  5.35 in [5], we have 𝑒_𝐴𝑡,𝜗⁺=𝑟₁𝑃₀+𝑟₂𝑃₁=𝑟₁𝑃₀+𝑟₂𝐴−𝐼₂𝑃₀, (35) where (𝑟₁,𝑟₂)^𝜏 is the solution of 𝑟^Δ₁𝑟^Δ₂=10𝑟11𝑟¹₂,𝑟₁𝜗⁺𝑟₂𝜗⁺=10. (36) Thus (33) holds.

Lemma 29. Suppose that 𝑘∈𝕋 with 𝑘≠0. Then 𝑥∈𝐴𝑃¹(𝕋,ℝ) if and only if (𝑥,(𝑥^Δ−𝑥)/𝑘)^𝜏∈𝐴𝑃(𝕋,ℝ²).

Proof. If 𝑥∈AP¹(𝕋,ℝ) holds, from Theorem 20 (or Theorem 21), 𝑥,𝑥^Δ∈AP(𝕋,ℝ). According to Lemmas 13 and 17, we have (𝑥,(𝑥^Δ−𝑥)/𝑘)^𝜏∈AP(𝕋,ℝ²).
If (𝑥,(𝑥^Δ−𝑥)/𝑘)^𝜏∈AP(𝕋,ℝ²) holds, then 𝑥,(𝑥^Δ−𝑥)/𝑘∈AP(𝕋,ℝ). Thus 𝑥^Δ∈AP(𝕋,ℝ) and 𝑥∈AP¹(𝕋,ℝ).

Lemma 30. Suppose that 𝕋 is a two-way translation invariant time scale. Assume that (26) admits an exponential dichotomy on 𝕋, with a projection 𝑃 on ℝ^𝑛 and positive constants 𝐾_𝑖 and 𝛼_𝑖 for 𝑖=1,2 such that (27) holds. If ‖𝐴‖=max_{1≤𝑖,𝑗≤𝑛}sup_𝕋|𝑎_𝑖𝑗(𝑡)|<∞ is bounded, where 𝐴(𝑡)=(𝑎_𝑖𝑗(𝑡))_𝑛×𝑛, then for any fixed ℎ∈ℝ, the families of functions {Φ(⋅)𝑃Φ⁻¹(𝑠)∶𝑠∈(−∞,𝑡−ℎ]_𝕋} and {Φ(⋅)(𝐼−𝑃)Φ⁻¹(𝑠)∶𝑠∈[𝑡+ℎ,∞)_𝕋} are equicontinuous on [𝜗⁺,∞)_𝕋 and (−∞,𝜗⁺]_𝕋, respectively.

Proof. The proof of equicontinuity of {Φ(⋅)(𝐼−𝑃)Φ⁻¹(𝑠)∶𝑠∈[𝑡+ℎ,∞)_𝕋} on (−∞,𝜗⁺]_𝕋 is similar to that of {Φ(⋅)𝑃Φ⁻¹(𝑠)∶𝑠∈(−∞,𝑡−ℎ]_𝕋}. Thus, we only prove the equi-continuity of {Φ(⋅)𝑃Φ⁻¹(𝑠)∶𝑠∈(−∞,𝑡−ℎ]_𝕋}.
Since Φ is the fundamental solutions of (26), the function Φ(⋅)𝑃Φ⁻¹(𝑠) is Δ-differential on [𝜗⁺,∞). Moreover, for fixed ℎ>0 and 𝜔∈𝛾(𝕋) chosen, we have ^∞_𝜗+Φ(𝑡)𝑃Φ⁻¹(𝑠)^Δ≤Δ𝑡_∞𝜗+𝐾₁𝑒_⊖𝛼1(𝐴‖≤‖𝐴‖𝑡,𝑠)‖Δ𝑡_{∞𝑛=0(𝑛+1)𝜔𝑛𝜔}𝐾₁11+𝑀𝛼₁^{(𝑡−𝑠)/𝑀}=‖𝐴‖𝐾Δ𝑡₁𝜔1+𝑀𝛼₁^−ℎ/𝑀1−1+𝑀𝛼₁^−𝜔/𝑀<∞, (37) which implies that (Φ(𝑡)𝑃Φ⁻¹(𝑠))^Δ∈𝐿¹([0,∞)_𝕋). By Φ(⋅)𝑃Φ⁻¹(𝑠) being Δ-differential Δ-a.e. for 𝑠∈(−∞,𝑡−ℎ]_𝕋 and Theorem  4.1 in [28], Φ(⋅)𝑃Φ(𝑠) is absolutely continuous; thus the family {Φ(⋅)𝑃Φ⁻¹(𝑠)∶𝑠∈(−∞,𝑡−ℎ]_𝕋} is equi-continuous on (−∞,𝜗⁺]_𝕋.

Theorem 31. Suppose that there is a 𝑘>0 such that 𝑘²𝜇(𝑡)+2𝑘≤2/3 for all 𝑡∈𝕋. Assume 𝑓∈𝑈𝐴𝑃(𝕋×ℝ²,ℝ) takes bounded sets into bounded sets. Then (25) admits a almost periodic solution.

Proof. Set the operator Γ∶AP(𝕋,ℝ²)→AP(𝕋,ℝ²) by Γ𝑋(𝑡)=^𝑡_−∞Φ(𝑡)𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠−))𝑠,𝑋))Δ𝑠_∞𝑡Φ(𝑡)(𝐼−𝑃)Φ⁻¹(𝜎(𝑠))𝐹(𝑠,𝑋(𝑠))Δ𝑠, (38) where Φ is a fundamental solution matrix of (32). From Theorem 25 and Lemma 26, the operator Γ∶AP(𝕋,ℝ²)→AP(𝕋,ℝ²) is well defined, ‖Γ𝑋‖ is bounded, and the solution of (30) is equivalent to the fixed point of Γ.
Since 𝑓∈UAP(𝕋×ℝ²,ℝ), then for each compact subset 𝑆⊂AP(𝕋,ℝ²), 𝑓 is bounded and uniformly continuous on 𝕋×𝑆. Thus, for each 𝜀>0 and compact subset 𝑆⊂ℝ², there is a compact subset 𝑆₁⊂𝑆 such that 𝑋=(𝑥₁,𝑥₂)^𝜏, 𝑌=(𝑦₁,𝑦₂)^𝜏∈𝑆₁ with ‖𝑋−𝑌‖<𝛿 implies that ‖𝐹(𝑡,𝑋)−𝐹(𝑡,𝑌)‖=‖𝑓(𝑡,𝑥₁,𝑥₁+𝑘𝑥₂)−𝑔(𝑡,𝑦₁,𝑦₁+𝑘𝑦₂)‖/𝑘<𝜀. Then ‖Γ𝑋−Γ𝑌‖≤^𝑡_−∞Φ(𝑡)𝑃Φ⁻¹(𝜎(𝑠×‖𝐹((𝑠((𝑠+))𝑠,𝑋))−𝐹𝑠,𝑌))‖Δ𝑠_∞𝑡Φ(𝑡)𝐼₂−𝑃Φ⁻¹(𝜎(𝑠×‖𝐹((𝑠((𝑠≤))𝑠,𝑋))−𝐹𝑠,𝑌))‖Δ𝑠_𝑡−∞𝐾₁𝑒_⊖𝛼1(𝑡,𝜎(𝑠))Δ𝑠+^∞_𝑡𝐾₂𝑒_⊖𝛼2(𝜎(𝑠))𝜀≤𝐾,𝑡Δ𝑠₁1+𝑀𝛼₁𝛼₁+𝐾₂𝛼₂𝜀, (39) which implies that Γ is continuous.
In the following, we will show that Γ∶AP(𝕋,ℝ²)→AP(𝕋,ℝ²) is equi-continuous. If 𝑋=(𝑥,𝑥^Δ)∈AP(𝕋,ℝ²) and there exists a 𝐶>0 such that ‖𝑋‖≤𝐶, then there exists 𝐵>0 such that sup_{𝑡∈𝕋,‖𝑋‖≤𝐶}‖𝐹(𝑡,𝑋)‖=sup_{𝑡∈𝕋,‖𝑋‖≤𝐶}𝑓𝑡,𝑥,𝑥+𝑘𝑥^Δ𝑘≤𝐵. (40) Without loss of generality, we suppose that 𝑡₁>𝑡₂, and Φ(𝜗⁺)=𝐼₂. Let 𝜀>0, and let 𝑋∈AP(𝕋,ℝ²). We note that Γ𝑋𝑡₁−Γ𝑋𝑡₂=^𝑡2_−∞Φ𝑡₁−Φ𝑡₂𝑃Φ⁻¹×(𝜎(𝑠𝐹((𝑠+))𝑠,𝑋))Δ𝑠_𝑡1𝑡2Φ𝑡₁𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠+))𝑠,𝑋))Δ𝑠_𝑡1𝑡2Φ𝑡₂(𝐼−𝑃)Φ⁻¹(𝜎(𝑠𝐹((𝑠−))𝑠,𝑋))Δ𝑠_∞𝑡1Φ𝑡₁−Φ𝑡₂(𝐼−𝑃)Φ⁻¹(𝜎(𝑠((𝑠))×𝐹𝑠,𝑋))Δ𝑠=𝐽₁+𝐽₂+𝐽₃−𝐽₄. (41)
From Lemma 6, we have 𝐽₂=^𝑡1_𝑡2Φ𝑡₁𝑃Φ⁻¹(𝜎(𝑠))𝐹(𝑠,𝑋(𝑠))Δ𝑠≤𝐵^𝑡1_𝑡2Φ𝑡₁𝑃Φ⁻¹(𝜎(𝑠))Δ𝑠≤𝐵^𝑡1_𝑡2𝐾₁𝑒_⊖𝛼1𝑡₁,𝜎(𝑠)Δ𝑠=𝐵𝐾₁^𝑡1_𝑡21+𝜇(𝑠)𝛼₁𝑒_𝛼1𝑠,𝑡₁≤Δ𝑠𝐵𝐾₁1+𝑀𝛼₁𝛼₁1−𝑒_𝛼1𝑡₂,𝑡₁,𝐽₃≤𝐵^𝑡1_𝑡2𝐾₂𝑒_⊖𝛼2𝜎(𝑠),𝑡₂=Δ𝑠𝐵𝐾₂𝛼₂1−𝑒_⊖𝛼2𝑡₁,𝑡₂. (42) Thus, there is a 𝛿₁>0 such that 0<𝑡₁−𝑡₂<𝛿₁ implies that ‖𝐽₂‖,‖𝐽₃‖<𝜀/4.
In the following, we will prove there exists a 𝛿₂>0 such that 0<𝑡₁−𝑡₂<𝛿₂ implies that ‖𝐽₁‖<𝜀/4 and ‖𝐽₄‖<𝜀/4. We will prove that by the following cases: (i)𝑡₁>𝑡₂>𝜗⁺; (ii)𝑡₁≥𝜗⁺≥𝑡₂; (iii)𝜗⁺>𝑡₁>𝑡₂.
Suppose that case (i) holds. By Theorem  4.1 in [28] Φ(𝑡) is absolutely continuous; that is, there exists a 𝛿₃>0 such that 0<𝑡₁−𝑡₂<𝛿₃ implies that Φ𝑡₁−Φ𝑡₂<𝛼₂𝜀4𝐵𝐾₂. (43) Then 𝐽₄=^∞_𝑡1Φ𝑡₁−Φ𝑡₂Φ𝜗⁺×(𝐼−𝑃)Φ⁻¹(𝜎(𝑠𝐹((𝑠Φ𝑡))𝑠,𝑋))Δ𝑠≤𝐵₁−Φ𝑡₂×^∞_𝑡1Φ𝜗⁺(𝐼−𝑃)Φ⁻¹(𝜎(𝑠))Δ𝑠≤𝐵𝐾₂Φ𝑡₁−Φ𝑡₂^∞_𝑡1𝑒_⊖𝛼2𝜎(𝑠),𝜗⁺≤Δ𝑠𝐵𝐾₂𝛼₂𝑒_⊖𝛼2𝑡₁,𝜗⁺Φ𝑡₁−Φ𝑡₂≤𝐵𝐾₂𝛼₂Φ𝑡₁−Φ𝑡₂<𝜀4. (44)
If 𝑡₂=𝜌(𝑡₂)=sup{𝑠∈𝕋∶𝑠<𝑡₂}=sup{𝑡₂−ℎ∈𝕋∶ℎ∈ℝ⁺}, then there exist ℎ₁,ℎ₂ such that 0<ℎ₂<ℎ₁<𝜀/16𝐵𝐾₁(1+𝑀𝛼₁) and 𝑡₂−ℎ₁,𝑡₂−ℎ₂∈𝕋. By the equi-continuity of Φ(⋅)𝑃Φ(𝑠), there exists a 𝛿₄>0 such that 0<𝑡₁−𝑡₂<𝛿₄ implies that for given 𝜔∈𝛾(𝕋) we have Φ𝑡₁𝑃Φ⁻¹𝑡₂−𝜏−Φ𝑡₂𝑃Φ⁻¹𝑡₂<𝜀−𝜏1+𝑀𝛼₁^𝜔/𝑀−18𝐵𝐾₁𝜔1+𝑀𝛼₁^{1+𝜔/𝑀+ℎ2/𝑀}for𝜏≥ℎ₁. (45) Thus, 𝐽₁≤^𝑡2−ℎ1_−∞Φ𝑡₁−Φ𝑡₂×𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠+))𝑠,𝑋))Δ𝑠_{𝑡2𝑡2−ℎ1}Φ𝑡₁−Φ𝑡₂×𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠≤))𝑠,𝑋))Δ𝑠_{[ℎ1,∞)𝕋}Φ𝑡₁−Φ𝑡₂𝑃Φ⁻¹𝑡₂−ℎ₂×Φ𝑡₂−ℎ₂𝑃Φ⁻¹𝜎𝑡₂×𝐹𝑡−𝜏₂−𝜏,𝑋𝑡₂+−𝜏Δ𝜏_{𝑡2𝑡2−ℎ1}Φ𝑡₁𝑃Φ⁻¹(𝜎(𝑠+Φ𝑡))₂𝑃Φ⁻¹(𝜎(𝑠×‖𝐹((𝑠<𝜀))𝑠,𝑋))‖Δ𝑠1+𝑀𝛼₁^𝜔/𝑀−18𝜔1+𝑀𝛼₁^{1+𝜔/𝑀+ℎ2/𝑀}×_{[ℎ1,∞)𝕋}𝑒_⊖𝛼1𝑡₂−ℎ₂,𝜎𝑡₂−𝜏Δ𝜏+𝐵𝐾₁1+𝑀𝛼₁×^𝑡2_𝑡2−ℎ1𝑒_𝛼1𝑠,𝑡₁+𝑒_𝛼1𝑠,𝑡₂≤𝜀Δ𝑠1+𝑀𝛼₁^𝜔/𝑀−18𝜔1+𝑀𝛼₁^{1+𝜔/𝑀+ℎ2/𝑀}×^∞_𝑛=0^(𝑛+1)𝜔_𝑛𝜔1+𝜇𝑡₂−𝜏𝛼₁𝑒_⊖𝛼1×𝑡₂−ℎ₂,𝑡₂−𝜏Δ𝜏+𝐵𝐾₁1+𝑀𝛼₁ℎ₁×1+𝑀𝛼₁^{(𝑡2−𝑡1)/𝑀}<𝜀+11+𝑀𝛼₁^𝜔/𝑀−18𝜔1+𝑀𝛼₁^{1+𝜔/𝑀+ℎ2/𝑀}×^∞_𝑛=0^(𝑛+1)𝜔_𝑛𝜔1+𝑀𝛼₁×11+𝑀𝛼₁^{(𝜏−ℎ2)/𝑀}𝜀Δ𝜏+8≤𝜀4. (46)
If 𝑡₂>𝜌(𝑡₂), then 𝐽₁=^𝜌(𝑡2)_−∞Φ𝑡₁−Φ𝑡₂×𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠+))𝑠,𝑋))Δ𝑠_{𝑡2𝜌𝑡2}Φ𝑡₁−Φ𝑡₂×𝑃Φ⁻¹(𝜎(𝑠))𝐹(𝑠,𝑋(𝑠))Δ𝑠. (47) Similar to the proof above, we can show that there is a 𝛿^′₃>0 such that 𝑡₁−𝑡₂<𝛿^′₃ implies that ^𝜌(𝑡2)_−∞Φ𝑡₁−Φ𝑡₂×𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠≤))𝑠,𝑋))Δ𝑠_∞𝜗+Φ𝑡₁−Φ𝑡₂𝑃Φ⁻¹𝜌𝑡₂×Φ𝜌𝑡₂𝑃Φ⁻¹𝜎𝜌𝑡₂<𝜀−𝜏8. (48)
Note that ^𝑡2_𝜌𝑡2Φ𝑡₁−Φ𝑡₂×𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠=))𝑠,𝑋))Δ𝑠_{𝑡2𝜌𝑡2}Φ𝑡₁−Φ𝑡₂𝑃Φ⁻¹(𝑠)×𝐼+𝜇(𝑠)𝐴⁻¹𝐹((𝑠≤𝜈𝑡𝑠,𝑋))Δ𝑠₂Φ𝑡₁−Φ𝑡₂𝑃‖×Φ₋₁‖𝜌𝑡₂𝐼+𝜇𝜌𝑡₂𝐴⁻¹𝐵, (49) where Φ⁻¹𝜌𝑡₂𝐼+𝜇𝜌𝑡₂𝐴⁻¹=𝑒₁⁻¹𝜌𝑡₂,𝜗⁺1−𝑘∫^𝜌(𝑡2)_𝜗+1+𝜇(𝜏)⁻¹×𝑡Δ𝜏011+𝜈₂⁻¹−𝑘𝜈𝑡₂1+𝜈𝑡₂⁻²01+𝜈𝑡₂⁻¹≤𝑒₁⁻¹𝜌𝑡₂,𝜗⁺1−𝑘∫^𝜌(𝑡2)_𝜗+1+𝜇(𝜏)⁻¹𝑒Δ𝜏01≤max₁⁻¹𝜌𝑡₂,𝜗⁺,𝑘𝜌𝑡₂−𝜗⁺𝑒₁⁻¹𝜌𝑡₂,𝜗⁺. (50) Since 𝑡^Δ/𝑒^Δ₁(𝑡,𝜗⁺)=1/𝑒₁(𝑡,𝜗⁺) for 𝑡∈𝕋 is decreasing and 𝑒^Δ₁(𝑡,𝜗⁺)>0, then (𝑡−𝜗⁺)/(𝑒₁(𝑡,𝜗⁺)−𝑒₁(𝜗⁺,𝜗⁺))>0 is decreasing by Theorem  2.1 in [29] and therefore uniformly bounded on (𝜗⁺,∞). Thus (𝑡−𝜗⁺)/𝑒₁(𝑡,𝜗⁺) is bounded as follows. By the boundedness of 𝜈(𝑡₂), 𝑃, Φ⁻¹(𝜌(𝑡₂))(𝐼+𝜇(𝜌(𝑡₂))𝐴)⁻¹ and absolute continuity of Φ, there is a 𝛿^′₄>0 such that 𝑡₁−𝑡₂<𝛿^′₄ implies that ^𝑡2_𝜌𝑡2Φ𝑡₁−Φ𝑡₂𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠<𝜀))𝑠,𝑋))Δ𝑠8. (51)
Set 𝛿₂=min{𝛿₃,𝛿₄,𝛿^′₃,𝛿^′₄}; then ‖𝐽₁‖,‖𝐽₄‖<𝜀/4 for 0<𝑡₁−𝑡₂<𝛿₂.
Therefore, for 0<𝑡₁−𝑡₂<min{𝛿₁,𝛿₂}, Γ𝑋𝑡₁−Γ𝑋𝑡₂<𝜀. (52)
As the method in the above proof, we can show that for 𝜀>0 there is a 𝛿>0 such that if 𝑡₁≤𝜗⁺≤𝑡₂ or 𝜗⁺>𝑡₁>𝑡₂ with 𝑡₁−𝑡₂<𝛿, the inequality (52) holds. Thus, Γ is equi-continuous.
According to Arzela-Ascoli theorem and the continuity of Γ, Γ is completely continuous. Therefore, by Schauder’s fixed point theorem there exists a fixed point 𝑋_∗=(𝑥_∗,𝑥^Δ_∗)^𝜏∈AP(𝕋,ℝ²) of Γ such that Γ𝑋_∗=𝑋_∗. Furthermore, 𝑥_∗ is almost periodic solution of (25).

Theorem 32. Suppose that  (H1) 𝑓∈𝑈𝐴𝑃(𝕋×ℝ²,ℝ) is Lipschitz continuous in (𝑥,𝑦)∈ℝ² uniformly on 𝕋; that is, there is a 𝐿>0 such that 𝑓𝑡,𝑥,𝑦−𝑓𝑡,𝑥,𝑦≤𝐿‖𝑥−𝑥‖+𝑦−𝑦, (53) for 𝑡∈𝕋, and 𝑥, 𝑦, 𝑥, 𝑦∈ℝ; (H2) there exists 𝑘∈ℝ⁺ with sup_𝑡∈𝕋{𝑘²𝜇(𝑡)+2𝑘}≤2/3 such that 𝐿21+𝑘1+𝑀𝛼₁𝛼₁+1𝛼₂<1. (54)Then (25) admits a unique almost periodic solution.

Proof. Let 𝑋=(𝑥₁,𝑥₂)^𝜏, 𝑌=(𝑦₁,𝑦₂)^𝜏∈ℝ². Note ‖𝐹()(=1𝑡,𝑋−𝐹𝑡,𝑌)‖𝑘𝑓𝑡,𝑥₁,𝑥₁+𝑘𝑥₂−𝑓𝑡,𝑦₁,𝑦₁+𝑘𝑦₂≤𝐿𝑘𝑥₁−𝑦₁+𝑥₁+𝑘𝑥₂−𝑦₁−𝑘𝑦₂≤𝐿𝑘2𝑥₁−𝑦₁+𝑘𝑥₂−𝑦₂2≤𝐿1+𝑘‖𝑋−𝑌‖. (55) Set Γ∶AP(𝕋,ℝ²)→AP(𝕋,ℝ²) as (38). Then ‖‖≤Γ𝑋−Γ𝑌_𝑡−∞Φ(𝑡)𝑃Φ⁻¹(𝜎(𝑠𝐹((𝑠((𝑠+))‖𝑠,𝑋))−𝐹𝑠,𝑌))‖Δ𝑠_∞𝑡Φ(𝑡)(𝐼−𝑃)Φ⁻¹(𝜎(𝑠×‖𝐹((𝑠((𝑠2))𝑠,𝑋))−𝐹𝑠,𝑌))‖Δ𝑠≤𝐿1+𝑘^𝑡_−∞𝐾₁𝑒_⊖𝛼1(𝑡,𝜎(𝑠))Δ𝑠+^∞_𝑡𝐾₂𝑒_⊖𝛼2(𝜎(𝑠))2,𝑡Δ𝑠≤𝐿1+𝑘1+𝑀𝛼₁𝛼₁+1𝛼₂‖𝑋−𝑌‖, (56) which implies that Γ is continuous. And by (H2), Γ is a contraction. Therefore, by contraction mapping principle, there exists a unique fixed point 𝑋^∗∈AP(𝕋,ℝ²) such that Γ𝑋^∗=𝑋^∗. By Lemma 27, 𝑋^∗=(𝑥^∗₁,𝑥^∗₂)^𝜏∈AP(𝕋,ℝ²) implies that 𝑥^∗₁∈AP¹(𝕋,ℝ) is the solution of (25).

Set 𝑓(𝑡,𝑥,𝑦)=𝑔(𝑡,𝑥,𝑦)−𝑥. Then 𝑔∈UAP(𝕋×ℝ²,ℝ) yields that 𝑓∈UAP(𝕋×ℝ²,ℝ).

Theorem 33. Suppose that there is a 𝑘>0 such that 𝑘²𝜇(𝑡)+2𝑘≤2/3 for all 𝑡∈𝕋. Assume 𝑔∈𝑈𝐴𝑃(𝕋×ℝ²,ℝ) takes bounded sets into bounded sets. Then (1) admits a almost periodic solution.

Theorem 34. Let 𝕋 be a two-way translation invariant time scale. Suppose that  (H1) 𝑔∈𝑈𝐴𝑃(𝕋×ℝ²,ℝ) is Lipschitz continuous in (𝑥,𝑦)∈ℝ² uniformly on 𝕋; that is, there is a 𝐿>0 such that 𝑔𝑡,𝑥,𝑦−𝑔𝑡,𝑥,𝑦≤𝐿‖𝑥−𝑥‖+𝑦−𝑦, (57) for 𝑡∈𝕋, and 𝑥,𝑦,𝑥,𝑦∈ℝ; (H2) there exists 𝑘∈ℝ⁺ with sup_𝑡∈𝕋{𝑘²𝜇(𝑡)+2𝑘}≤2/3 such that 𝐿+2𝐿+1𝑘1+𝑀𝛼₁𝛼₁+1𝛼₂<1. (58)Then (25) admits a unique almost periodic solution.

5. Alternative Theorem for Second Order Dynamic Equations on Time Scales

In this section, we consider the following equation: 𝑥^ΔΔ=𝑥+𝑓𝑡,𝑥,𝑥^Δ+𝜆𝑔𝑡,𝑥,𝑥^Δ. (59)

Theorem 35. Assume the following conditions hold:  (i) 𝑓,𝑔∈𝑈𝐴𝑃(𝕋×ℝ²,ℝ) take bounded sets into bounded sets; (ii) there exists an open bounded subset Ω⊂𝐴𝑃(𝕋,ℝ²) such that every possible solution 𝑥 of (59) satisfies (𝑥,𝑥^Δ)^𝜏∉𝜕Ω; (iii) deg(𝐼−Γ,Ω,0)≠0. Then (59) has an almost periodic solution 𝑥 with (𝑥,𝑥^Δ)^𝜏∈Ω.

Proof. Let 𝑋(𝑡)=(𝑥(𝑡),𝑥^Δ(𝑡))^𝜏, and let 𝐺(𝑡,𝑋)=(0,𝑔(𝑡,𝑥,𝑥+𝑘𝑥^Δ)/𝑘)^𝜏. Then 𝑋∈AP(𝕋,ℝ²) yields that 𝐺(𝑡)=𝐺(𝑡,𝑋(𝑡))∈AP(𝕋,ℝ²).
For all (𝑋,𝜆)∈Ω×[0,1], set 𝐻(𝑋,𝜆)=Γ(𝑋)+𝜆^⋅_−∞Φ(⋅)𝑃Φ⁻¹(𝜎(𝑠))𝐺(𝑠,𝑋(𝑠))Δ𝑠+𝜆^∞_⋅Φ(⋅)(𝐼−𝑃)Φ⁻¹(𝜎(𝑠))𝐺(𝑠,𝑋(𝑠))Δ𝑠. (60) Obviously, 𝐻 is a map from AP(𝕋,ℝ²)×[0,1] to AP(𝕋,ℝ²).
As the proof of Theorem 31, 𝐻(⋅,𝜆) for all 𝜆∈[0,1] is completely continuous. Thus deg(𝐼−𝐻(⋅,𝜆),Ω,0)=deg(𝐼−Γ,Ω,0)≠0; (61) that is, (59) has an almost periodic solution 𝑥 with (𝑥,𝑥^Δ)^𝜏∈Ω.

Theorem 36. Consider the following problems: 𝑥^ΔΔ(𝑡)𝑥=𝑥+𝑓,𝑡∈ℝ,_ΔΔ=𝑥+𝑓(𝑡)+𝜆𝑔𝑡,𝑥,𝑥^Δ𝑓𝑜𝑟𝑡∈ℝ,𝜆∈[0,1], (62)(63) where 𝑓∈𝐴𝑃(𝕋,ℝ) and 𝑔∈𝑈𝐴𝑃(𝕋×ℝ²,ℝ) takes bounded sets into bounded sets. Assume the following conditions hold. (i) There exists an open bounded subset Ω⊂𝐴𝑃(𝕋,ℝ²) such that every possible solution 𝑢 of (63) satisfies 𝑢∉𝜕Ω.  (ii) There exists a unique almost periodic solution 𝑥_∗ of (62) such that (𝑥_∗,𝑥^Δ_∗)^𝜏∈Ω. Then (63) has an almost periodic solution 𝑥 with (𝑥,𝑥^Δ)^𝜏∈Ω.

Proof. Set 𝐻(𝑋,𝜆)=𝑋_∗(⋅)+𝜆^⋅_−∞Φ(⋅)𝑃Φ⁻¹(𝜎(𝑠))𝐺(𝑠,𝑋(𝑠))Δ𝑠+𝜆^∞_⋅Φ(⋅)(𝐼−𝑃)Φ⁻¹(𝜎(𝑠𝐺((𝑠∀()[],))𝑠,𝑋))Δ𝑠,𝑢,𝜆∈Ω×0,1 (64) where 𝑋_∗(𝑡) is the almost periodic solution of (62). As the proof of Theorem 31, 𝐻(⋅,𝜆)∶Ω→AP(𝕋×ℝ²,ℝ²) for all 𝜆∈[0,1] is completely continuous. Note (see [30]) deg𝐼,Ω,𝑋_∗=1. (65) Therefore, (())(())deg𝐼−𝐻⋅,𝜆,Ω,0=deg𝐼−𝐻⋅,0,Ω,0=deg𝐼,Ω,𝑋_∗=1; (66) that is, (63) has an almost periodic solution 𝑥 with (𝑥,𝑥^Δ)^𝜏∈Ω.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author’s work was supported NSFC (11026168, 11201046) and the Fundamental Research Funds for the Central Universities in China (DUT12LK32, DUT13JS02). The second author’s work was supported by NSFC (11126070, 11201309) and Natural Science Foundation of SZU (201111). The authors thank the referee and the editor for their valuable suggestions and comments, which improved the presentation of this paper.