Existence of Almost Periodic Solutions to 𝑁th-Order Neutral Differential Equations with Piecewise Constant Arguments

Abstract

We present some conditions for the existence and uniqueness of almost periodic solutions of 𝑁th-order neutral differential equations with piecewise constant arguments of the form (𝑥(𝑡)+𝑝𝑥(𝑡−1))^(𝑁)=𝑞𝑥([𝑡])+𝑓(𝑡), here [⋅] is the greatest integer function, 𝑝 and 𝑞 are nonzero constants, 𝑁 is a positive integer, and 𝑓(𝑡) is almost periodic.

1. Introduction

In this paper we study certain functional differential equations of neutral delay type with piecewise constant arguments of the form𝑥(𝑡)+𝑝𝑥(𝑡−1)^(𝑁)=𝑞𝑥([𝑡])+𝑓(𝑡), (1.1) here [⋅] is the greatest integer function, 𝑝 and 𝑞 are nonzero constants, 𝑁 is a positive integer, and 𝑓(𝑡) is almost periodic. Throughout this paper, we use the following notations: ℝ is the set of reals; ℝ⁺ the set of positive reals; ℤ the set of integers; that is, ℤ={0,±1,±2,…}; ℤ⁺ the set of positive integers; ℂ denotes the set of complex numbers. A function 𝑥∶ℝ→ℝ is called a solution of (1.1) if the following conditions are satisfied:(i)𝑥 is continuous on ℝ;(ii)the 𝑁th-order derivative of 𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−1) exists on ℝ except possibly at the points 𝑡=𝑛, 𝑛∈ℤ, where one-sided 𝑁th-order derivatives of 𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−1) exist;(iii)𝑥 satisfies (1.1) on each interval (𝑛,𝑛+1) with integer 𝑛∈ℤ.

Differential equations with piecewise constant arguments are usually referred to as a hybrid system, and could model certain harmonic oscillators with almost periodic forcing. For some excellent works in this field we refer the reader to [1–5] and references therein, and for a survey of work on differential equations with piecewise constant arguments we refer the reader to [6].

In paper [1, 2], Yuan and Li and He, respectively, studied the existence of almost periodic solutions for second-order equations involving the argument 2[(𝑡+1)/2] in the unknown function. In paper [3], Seifert intensively studied the special case of (1.1) for 𝑁=2 and |𝑝|<1 by using different methods. However, to the best of our knowledge, there are no results regarding the existence of almost periodic solutions for 𝑁th-order neutral differential equations with piecewise constant arguments as (1.1) up to now.

Motivated by the ideas of Yuan [1] and Seifert [3], in this paper we will investigate the existence of almost periodic solutions to (1.1). Both the cases when |𝑝|<1 and |𝑝|>1 are considered.

2. The Main Results

We begin with some definitions, which can be found (or simply deduced from the theory) in any book, say [7], on almost periodic functions.

Definition 2.1. A set 𝐾⊂ℝ is said to be relatively dense if there exists 𝐿>0 such that [𝑎,𝑎+𝐿]∩𝐾≠∅ for all 𝑎∈ℝ.

Definition 2.2. A bounded continuous function 𝑓∶ℝ→ℝ (resp., ℂ) is said to be almost periodic if the 𝜀-translation set of 𝑓𝑇𝑓,𝜀=𝜏∈ℝ∶𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀∀𝑡∈ℝ (2.1) is relatively dense for each 𝜀>0. We denote the set of all such function 𝑓 by AP(ℝ,ℝ) (resp., AP(ℝ,ℂ)).

Definition 2.3. A sequence 𝑥∶ℤ→ℝ^𝑘 (resp., ℂ^𝑘), 𝑘∈ℤ, 𝑘>0, denoted by {𝑥_𝑛}, is called an almost periodic sequence if the 𝜀-translation set of {𝑥_𝑛}𝑇𝑥_𝑛,𝜀=𝜏∈ℤ∶𝑥_𝑛+𝜏−𝑥_𝑛<𝜀∀𝑛∈ℤ (2.2) is relatively dense for each 𝜀>0, here |⋅| is any convenient norm in ℝ^𝑘 (resp., ℂ^𝑘). We denote the set of all such sequences {𝑥_𝑛} by APS(ℤ,ℝ^𝑘)(resp., APS(ℤ,ℂ^𝑘)).

Proposition 2.4. {𝑥_𝑛}={(𝑥_𝑛1,𝑥_𝑛2,…,𝑥_𝑛𝑘)}∈APS(ℤ,ℝ^𝑘) (resp., APS(ℤ,ℂ^𝑘)) if and only if {𝑥_𝑛𝑖}∈APS(ℤ,ℝ) (resp., APS(ℤ,ℂ)), 𝑖=1,2,…,𝑘.

Proposition 2.5. Suppose that {𝑥_𝑛}∈APS(ℤ,ℝ), 𝑓∈AP(ℝ,ℝ). Then the sets 𝑇(𝑓,𝜀)∩ℤ and 𝑇({𝑥_𝑛},𝜀)∩𝑇(𝑓,𝜀) are relatively dense.

Now one rewrites (1.1) as the following equivalent system 𝑥(𝑡)+𝑝𝑥(𝑡−1)^′=𝑦₁(𝑡),2.3₁𝑦^′₁(𝑡)=𝑦₂(𝑡),2.3₂𝑦⋮⋮_′𝑁−2(𝑡)=𝑦_𝑁−1(𝑡),2.3_𝑁−1𝑦^′_𝑁−1(𝑡)=𝑞𝑥([𝑡])+𝑓(𝑡).2.3_𝑁 (2.3) Let (𝑥(𝑡),𝑦₁(𝑡),…,𝑦_𝑁−1(𝑡))be solutions of system (2.3) on ℝ, for 𝑛≤𝑡<𝑛+1, 𝑛∈ℤ, using (2.3_𝑁) we obtain 𝑦_𝑁−1(𝑡)=𝑦_𝑁−1(𝑛)+𝑞𝑥(𝑛)(𝑡−𝑛)+^𝑡_𝑛𝑓𝑡₁𝑑𝑡₁, (2.4) and using this with (2.3_𝑁−1) we obtain 𝑦_𝑁−2(𝑡)=𝑦_𝑁−2(𝑛)+𝑦_𝑁−1(𝑛)+1)(𝑡−𝑛2𝑞𝑥(𝑛)(𝑡−𝑛)²+^𝑡_𝑛^𝑡2_𝑛𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂. (2.5) Continuing this way, and, at last, we get 𝑥(𝑡)+𝑝𝑥(𝑡−1)=𝑥(𝑛)+𝑝𝑥(𝑛−1)+𝑦₁(𝑛)+1)(𝑡−𝑛2𝑦₂(𝑛)(𝑡−𝑛)²+1+⋯()!𝑦𝑁−1^𝑁−1(𝑛)(𝑡−𝑛)^𝑁−1+1(𝑛)𝑁!𝑞𝑥)(𝑡−𝑛^𝑁+^𝑡_𝑛^𝑡𝑁_𝑛⋯^𝑡2_𝑛𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂⋯𝑑𝑡_𝑁. (2.6) Since 𝑥(𝑡) must be continuous at 𝑛+1, using these equations we get for 𝑛∈ℤ, 𝑥()=𝑞𝑛+11−𝑝+𝑥(𝑛)𝑁!+𝑦₁(𝑛)+1𝑦2!²(𝑛)1+⋯+()!𝑦𝑁−1^𝑁−1(𝑛)+𝑝𝑥(𝑛−1)+𝑓_𝑛⁽¹⁾,2.7₁𝑦₁()=𝑞𝑛+1()!𝑥(𝑛)𝑁−1+𝑦₁(𝑛)+𝑦₂(𝑛)+1𝑦2!³(𝑛)1+⋅⋅⋅+()!𝑦𝑁−2^𝑁−1(𝑛)+𝑓_𝑛⁽²⁾,2.7₂𝑦⋮⋮_𝑁−2()=𝑞𝑛+12𝑥(𝑛)+𝑦_𝑁−2(𝑛)+𝑦_𝑁−1(𝑛)+𝑓_𝑛^(𝑁−1),2.7_𝑁−1𝑦_𝑁−1(𝑛+1)=𝑞𝑥(𝑛)+𝑦_𝑁−1(𝑛)+𝑓_𝑛^(𝑁),2.7_𝑁 (2.7) where 𝑓_𝑛⁽¹⁾=_𝑛^{𝑛+1𝑡𝑁}_𝑛⋯^𝑡2_𝑛𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂⋯𝑑𝑡_𝑁,…,𝑓_𝑛^(𝑁−1)=_𝑛^𝑛+1𝑡2_𝑛𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂,𝑓_𝑛^(𝑁)=_𝑛^𝑛+1𝑓𝑡₁𝑑𝑡₁. (2.8)

Lemma 2.6. If 𝑓∈AP(ℝ,ℝ), then sequences {𝑓_𝑛^(𝑖)}∈APS(ℤ,ℝ),𝑖=1,2,…,𝑁.

Proof. We typically consider {𝑓_𝑛⁽¹⁾}forall𝜀>0 and 𝜏∈𝑇(𝑓,𝜀)∩ℤ, we have 𝑓⁽¹⁾_𝑛+𝜏−𝑓_𝑛⁽¹⁾=^𝑛+𝜏+1_𝑛+𝜏^𝑡𝑁_𝑛+𝜏⋯^𝑡2_𝑛+𝜏𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂⋯𝑑𝑡_𝑁−_𝑛^{𝑛+1𝑡𝑁}_𝑛⋯^𝑡2_𝑛𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂⋯𝑑𝑡_𝑁≤_𝑛^{𝑛+1𝑡𝑁}_𝑛⋯^𝑡2_𝑛𝑓𝑡₁+𝜏−𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂⋯𝑑𝑡_𝑁≤𝜀.𝑁! (2.9) From Definition 2.3, it follows that {𝑓_𝑛⁽¹⁾} is an almost periodic sequence. In a manner similar to the proof just completed, we know that {𝑓_𝑛⁽²⁾},{𝑓_𝑛⁽³⁾},…,{𝑓_𝑛^(𝑁)} are also almost periodic sequences. This completes the proof of the lemma.

Lemma 2.7. The system of difference equations 𝑐_𝑛+1=𝑞1−𝑝+𝑐𝑁!^𝑛+𝑑_𝑛⁽¹⁾+1𝑑2!^𝑛(2)1+⋯+()!𝑑𝑁−1^{𝑛(𝑁−1)}+𝑝𝑐_𝑛−1+𝑓_𝑛⁽¹⁾,2.10₁𝑑⁽¹⁾_𝑛+1=𝑞(𝑁−1)!c_𝑛+𝑑_𝑛⁽¹⁾+𝑑_𝑛⁽²⁾+1𝑑2!^𝑛(3)1+⋅⋅⋅+()!𝑑𝑁−2^{𝑛(𝑁−1)}+𝑓_𝑛⁽²⁾,2.10₂𝑑⋮⋮_{(𝑁−2)𝑛+1}=𝑞2𝑐_𝑛+𝑑_𝑛^(𝑁−2)+𝑑_𝑛^(𝑁−1)+𝑓_𝑛^(𝑁−1),2.10_𝑁−1𝑑^(𝑁−1)_𝑛+1=𝑞𝑐_𝑛+𝑑_𝑛^(𝑁−1)+𝑓_𝑛^(𝑁),2.10_𝑁 (2.10) has solutions on ℤ; these are in fact uniquely determined by 𝑐₀,𝑐₋₁,𝑑₀⁽¹⁾,…,𝑑₀^(𝑁−1).

Proof. It is easy to check that 𝑐_𝑛,𝑑_𝑛^(𝑖), 𝑖=1,2,…,𝑁−1 are uniquely determined in term of 𝑐₀,𝑐₋₁, 𝑑₀⁽¹⁾,𝑑₀⁽²⁾,…,𝑑₀^(𝑁−1) for 𝑛∈ℤ⁺. For 𝑛=−1, (2.10_𝑁) uniquely determines 𝑑^(𝑁−1)₋₁, (2.10_𝑁−1) uniquely determines 𝑑^(𝑁−2)₋₁,…,(2.10₂) uniquely determines 𝑑⁽¹⁾₋₁, and thus since 𝑝≠0, (2.10₁) uniquely determines 𝑐₋₂. So 𝑐₋₁,𝑐₋₂, 𝑑⁽¹⁾₋₁,𝑑⁽²⁾₋₁,…,𝑑^(𝑁−1)₋₁ are determined. Continuing in this way, we establish the lemma.

Lemma 2.8. For any solution (𝑐_𝑛,𝑑_𝑛⁽¹⁾,𝑑_𝑛⁽²⁾,…,𝑑_𝑛^(𝑁−1)), 𝑛∈ℤ, of system (2.10), there exists a solution (𝑥(𝑡),𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑁−1(𝑡)), 𝑡∈𝑅, of (2.3) such that 𝑥(𝑛)=𝑐_𝑛, 𝑦₁(𝑛)=𝑑_𝑛⁽¹⁾,…,𝑦_𝑁−1(𝑛)=𝑑_𝑛^(𝑁−1), 𝑛∈ℤ.

Proof . Define 𝑤(𝑡)=𝑐_𝑛+𝑝𝑐_𝑛−1+𝑑_𝑛⁽¹⁾()+1𝑡−𝑛𝑑2!^𝑛(2)(𝑡−𝑛)²+1+⋯()!𝑑𝑁−1^{𝑛(𝑁−1)}(𝑡−𝑛)^𝑁−1+1𝑁!𝑞𝑐_𝑛(𝑡−𝑛)^𝑁+^𝑡_𝑛^𝑡𝑁_𝑛⋯^𝑡2_𝑛𝑓𝑡₁𝑑𝑡₁𝑑𝑡₁⋯𝑑𝑡_𝑁, (2.11) for 𝑛≤𝑡<𝑛+1, 𝑛∈ℤ. It can easily be verified that 𝑤(𝑡) is continuous on ℝ; we omit the details.
Define 𝑥(𝑡)=𝜑(𝑡), −1≤𝑡≤0, where 𝜑(𝑡) is continuous, and 𝜑(0)=𝑐₀, 𝜑(−1)=𝑐₋₁; 𝑥(𝑡)=𝑤(𝑡+1)−𝜑(𝑡+1)𝑝𝑥(𝑡)=[𝑤()(,−2≤𝑡<−1,𝑡+1−𝑥𝑡+1)]𝑝,−3≤𝑡<−2. (2.12) Continuing this way, we can define 𝑥(𝑡) for 𝑡<0. Similarly, define 𝑥(𝑡)()(𝑡)𝑥(𝑡)()(𝑡)=−𝑝𝜑𝑡−1+𝑤,0≤𝑡<1,=−𝑝𝑥𝑡−1+𝑤,1≤𝑡<2, (2.13) continuing in this way 𝑥(𝑡) is defined for 𝑡≥0, and so 𝑥(𝑡) is defined for all 𝑡∈ℝ.
Next, define 𝑦₁(𝑡)=𝑤^′(𝑡),𝑦₂(𝑡)=𝑤^′′(𝑡),…,𝑦_𝑁−1(𝑡)=𝑤^(𝑁−1)(𝑡), 𝑡≠𝑛∈ℤ, and by the appropriate one-sided derivative of 𝑤^′(𝑡),𝑤^′′(𝑡),…,𝑤^(𝑁−1)(𝑡) at 𝑛∈ℤ. It is easy to see that 𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑁−1(𝑡) are continuous on ℝ, and (𝑥(𝑛),𝑦₁(𝑛),𝑦₂(𝑛),…,𝑦_𝑁−1(𝑛))=(𝑐_𝑛,𝑑_𝑛⁽¹⁾,𝑑_𝑛⁽²⁾,…,𝑑_𝑛^(𝑁−1)) for 𝑛∈ℤ; we omit the details.

Next we express system (2.7) in terms of an equivalent system in ℝ^𝑁+1 give by𝑣_𝑛+1=𝐴𝑣_𝑛+ℎ_𝑛, (2.14) where 𝑞𝐴=1−𝑝+11𝑁!12!⋅⋅⋅()!𝑝𝑞𝑁−1()!1𝑁−111⋅⋅⋅()!0𝑞𝑁−2⋅⋅⋅⋅⋅⋅,𝑣2!00⋯10𝑞00⋅⋅⋅10100⋅⋅⋅00_𝑛=𝑥(𝑛),𝑦₁(𝑛),𝑦₂(𝑛),…,𝑦_𝑁−1,𝑥(𝑛−1)^𝑇,ℎ_𝑛=𝑓_𝑛⁽¹⁾,𝑓_𝑛⁽²⁾,…,𝑓_𝑛^(𝑁),0^𝑇. (2.15)

Lemma 2.9. Suppose that all eigenvalues of 𝐴 are simple (denoted by 𝜆₁,𝜆₂,…,𝜆_𝑁+1) and |𝜆_𝑖|≠1, 1≤𝑖≤𝑁+1. Then system (2.14) has a unique almost periodic solution.

Proof. From our hypotheses, there exists a (𝑁+1)×(𝑁+1) nonsingular matrix 𝑃 such that 𝑃𝐴𝑃⁻¹=Λ, where Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑁+1) and 𝜆₁,𝜆₂,…,𝜆_𝑁+1 are the distinct eigenvalues of 𝐴. Define 𝑣_𝑛=𝑃𝑣_𝑛, then (2.14) becomes 𝑣_𝑛+1=Λ𝑣_𝑛+ℎ_𝑛, (2.16) where ℎ_𝑛=𝑃ℎ_𝑛.
For the sake of simplicity, we consider first the case |𝜆₁|<1. Define 𝑣_𝑛1=_𝑚≤𝑛𝜆₁^𝑛−𝑚ℎ_(𝑚−1)1, (2.17) where ℎ_𝑛=(ℎ_𝑛1,ℎ_𝑛2,…,ℎ_𝑛(𝑁+1))^𝑇, 𝑛∈ℤ. Clearly {ℎ_𝑛1} is almost periodic, since ℎ_𝑛=𝑃ℎ_𝑛, and {ℎ_𝑛} is. For 𝜏∈𝑇({ℎ_𝑛1},𝜀), we have 𝑣_(𝑛+𝜏)1−𝑣_𝑛1=_{𝑚≤𝑛+𝜏}𝜆₁^{𝑛+𝜏−𝑚}ℎ_(𝑚−1)1−_𝑚≤𝑛𝜆₁^𝑛−𝑚ℎ_(𝑚−1)1letting𝑚=𝑚′+𝜏,thenreplacing𝑚′by𝑚=_𝑚≤𝑛𝜆₁^𝑛−𝑚ℎ_{(𝑚+𝜏−1)1}−_𝑚≤𝑛𝜆₁^𝑛−𝑚ℎ_(𝑚−1)1=_𝑚≤𝑛𝜆₁^𝑛−𝑚ℎ_{(𝑚+𝜏−1)1}−ℎ_(𝑚−1)1≤𝜀1−𝜆₁, (2.18) this shows that {𝑣_𝑛1}∈APS(ℤ,ℂ).
If |𝜆_𝑖|<1, 2≤𝑖≤𝑁+1, in a manner similar to the proof just completed for 𝜆₁, we know that {𝑣_𝑛𝑖}∈𝑃𝐴𝑆(ℤ,ℂ), 2≤𝑖≤𝑁+1, and so {𝑣_𝑛}∈APS(ℤ,ℂ^𝑁+1). It follows easily that then {𝑃⁻¹𝑣_𝑛}={𝑣_𝑛}∈APS(ℤ,ℝ^𝑁+1) and our lemma follows.
Assume now |𝜆₁|>1. Now define 𝑣_𝑛1=_𝑚≤𝑛𝜆₁^𝑚−𝑛ℎ_(𝑚−1)1,𝑛∈ℤ. (2.19) As before, the fact that {𝑣_𝑛1}∈APS(ℤ,ℂ) follows easily from the fact that {ℎ_𝑛1}∈APS(ℤ,ℂ). So in every possible case, we see that each component 𝑣_𝑛𝑖, 𝑖=1,2,…,𝑁+1, of 𝑣_𝑛 is almost periodic and so {𝑣_𝑛}∈APS(ℤ,ℝ^𝑁+1).
The uniqueness of this almost periodic solution {𝑣_𝑛} of (2.14) follows from the uniqueness of the solution 𝑣_𝑛 of (2.16) since 𝑃⁻¹𝑣_𝑛=𝑣_𝑛, and the uniqueness of 𝑣_𝑛 of (2.16) follows, since if 𝑣_𝑛 were a solution of (2.16) distinct from 𝑣_𝑛, 𝑢_𝑛=𝑣_𝑛−𝑣_𝑛 would also be almost periodic and solve 𝑢_𝑛+1=Λ𝑢_𝑛,𝑛∈ℤ. But by our condition on Λ, it follows that each component of 𝑢_𝑛 must become unbounded either as 𝑛→∞ or as 𝑛→−∞, and that is impossible, since it must be almost periodic. This proves the lemma.

Lemma 2.10. Suppose that conditions of Lemma 2.9 hold, 𝑤(𝑡) is as defined in the proof of Lemma 2.8 with (𝑐_𝑛,𝑑_𝑛⁽¹⁾,𝑑_𝑛⁽²⁾,…,𝑑_𝑛^(𝑁−1)) the unique first 𝑁 components of the almost periodic solution of (2.14) given by Lemma 2.9, then 𝑤(𝑡) is almost periodic.

Proof. For 𝜏∈𝑇({𝑐_𝑛},𝜀)∩𝑇({𝑑_𝑛⁽¹⁾},𝜀)∩𝑇({𝑑_𝑛⁽²⁾},𝜀)∩⋯∩𝑇({𝑑_𝑛^(𝑁−1)},𝜀)∩𝑇(𝑓,𝜀), |𝑤()(𝑡=𝑐𝑡+𝜏−𝑤)|_𝑛+𝜏−𝑐_𝑛+𝑝𝑐_{𝑛+𝜏−1}−𝑐_𝑛−1+𝑑⁽¹⁾_𝑛+𝜏−𝑑_𝑛⁽¹⁾()+1𝑡−𝑛𝑑2!^(2)𝑛+𝜏−𝑑_𝑛⁽²⁾(𝑡−𝑛)²+1+⋯()!𝑑𝑁−1^{(𝑁−1)𝑛+𝜏}−𝑑_𝑛^(𝑁−1)(𝑡−𝑛)^𝑁−1+𝑞𝑐𝑁!^𝑛+𝜏−𝑐_𝑛(𝑡−𝑛)^𝑁+^𝑡+𝜏_𝑛+𝜏^𝑡𝑁_𝑛+𝜏⋯^𝑡2_𝑛+𝜏𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂⋯𝑑𝑡_𝑁−^𝑡_𝑛^𝑡𝑁_𝑛⋯^𝑡2_𝑛𝑓𝑡₁𝑑𝑡₁𝑑𝑡₂⋯𝑑𝑡_𝑁≤𝑝+𝑞1++𝑁!^𝑁−1_𝑖=01𝑖!𝜀. (2.20) It follows from definition that 𝑤(𝑡) is almost periodic.

Theorem 2.11. Suppose that |𝑝|≠1 and all eigenvalues of 𝐴 in (2.14) are simple (denoted by 𝜆₁,𝜆₂,…,𝜆_𝑁+1) and satisfy |𝜆_𝑖|≠1, 1≤𝑖≤𝑁+1. Then (1.1) has a unique almost periodic solution 𝑥(𝑡), which can, in fact be determined explicitly in terms of 𝑤(𝑡) as defined in the proof of Lemma 2.8.

Proof. Consider the following.Case 1 (|𝑝|<1). For each 𝑚∈ℤ⁺ define 𝑥_𝑚(𝑡) as follows: 𝑥_𝑚(𝑡)=𝑤(𝑡)−𝑝𝑥_𝑚(𝑡−1),𝑡>−𝑚,𝑥_𝑚(𝑡)=𝜙(𝑡),𝑡≤−𝑚, (2.21)(2.22) here 𝑤(𝑡) is as defined in the proof of Lemma 2.8, and 𝜙(𝑡)=𝑐_𝑛+𝑐_𝑛+1−𝑐_𝑛(𝑡−𝑛),𝑛≤𝑡<𝑛+1,𝑛∈ℤ, (2.23) where 𝑐_𝑛 is the first component of the solution 𝑣_𝑛 of (2.14) given by Lemma 2.9. Let 𝑙∈ℤ⁺, then from (2.21) we get −𝑝^𝑙𝑥_𝑚(𝑡−𝑙)=−𝑝^𝑙𝑤(𝑡−𝑙)+−𝑝^𝑙+1𝑥_𝑚(𝑡−𝑙−1),𝑡>−𝑚. (2.24) It follows that 𝑥_𝑚(𝑡)=^𝑙−1_𝑗=0−𝑝^𝑗𝑤𝑡−𝑗+−𝑝^𝑙𝑥_𝑚(𝑡−𝑙),𝑡>−𝑚. (2.25) If 𝑙>𝑡+𝑚, 𝑥_𝑚(𝑡−𝑙)=𝜙(𝑡−𝑙), and so for such 𝑙, 𝑥_𝑚(𝑡)−^𝑙−1_𝑗=0−𝑝^𝑗𝑤𝑡−𝑗≤𝑝^𝑙𝜙(𝑡−𝑙). (2.26)
Let 𝑙→∞, we get 𝑥_𝑚(𝑡)=^∞_𝑗=0−𝑝^𝑗𝑤𝜙(𝑡)𝑡−𝑗,𝑡>−𝑚,,𝑡≤−𝑚. (2.27)
Since 𝑤(𝑡) and 𝜙(𝑡) are uniformly continuous on ℝ, it follows that {𝑥_𝑚(𝑡)∶𝑚∈ℤ⁺} is equicontinuous on each interval [−𝐿,𝐿],𝐿∈ℤ⁺, and by the Ascoli-Arzelá Theorem, there exists a subsequence, which we again denote by 𝑥_𝑚(𝑡), and a function 𝑥(𝑡) such that 𝑥_𝑚(𝑡)→𝑥(𝑡) uniformly on [−𝐿,𝐿], and by a familiar diagonalization procedure, can find a subsequence, again denoted by 𝑥_𝑚(𝑡) which is such that 𝑥_𝑚(𝑡)→𝑥(𝑡) for each 𝑡∈ℝ. From (2.27) it follows that 𝑥_𝑚(𝑡)=^∞_𝑗=0−𝑝^𝑗𝑤𝑡−𝑗, (2.28) and so 𝑥(𝑡) is almost periodic since 𝑤(𝑡−𝑗) is almost periodic in 𝑡 for each 𝑗≥0, and |𝑝|<1. From (2.21), letting 𝑚→∞, we get 𝑥(𝑡)+𝑝𝑥(𝑡−1)=𝑤(𝑡),𝑡∈ℝ, and since 𝑤(𝑡) solves (1.1), 𝑥(𝑡) does also. The uniqueness of 𝑥(𝑡) as an almost periodic solution of (1.1) follows from the uniqueness of the almost periodic solution 𝑣_𝑛∶ℤ→ℝ^𝑁+1 of (2.14) given by Lemma 2.9, which determines the uniqueness of 𝑤(𝑡), and therefore from (2.21) the uniqueness of 𝑥(𝑡).
Case 2 (|𝑝|>1). Rewriting (2.24) as −1𝑝^𝑙𝑥_𝑚(𝑡−𝑙)=−1𝑝^𝑙𝑤(𝑡−𝑙)+−1𝑝^𝑙+1𝑥_𝑚(𝑡−𝑙−1),𝑡>−𝑚, (2.29) we deduce in a similar manner that 𝑥_𝑚(𝑡)=^∞_𝑗=0−1𝑝^𝑗𝑤𝜙(𝑡)𝑡−𝑗,𝑡>−𝑚,,𝑡≤−𝑚. (2.30)
The remainder of the proof is similar to that of Case 1, we omit the details.

If 𝑝=0, the system of difference equations (2.10) of Lemma 2.7 now becomes𝑐_𝑛+1=11+𝑞𝑐𝑁!^𝑛+𝑑_𝑛⁽¹⁾+1𝑑2!^𝑛(2)1+⋯+()!𝑑𝑁−1^{𝑛(𝑁−1)}+𝑓_𝑛⁽¹⁾,𝑑⁽¹⁾_𝑛+1=1(𝑁−1)!𝑞𝑐_𝑛+𝑑_𝑛⁽¹⁾+𝑑_𝑛⁽²⁾+1𝑑2!^𝑛(3)1+⋯+()!𝑑𝑁−2^{𝑛(𝑁−1)}+𝑓_𝑛⁽²⁾,⋮𝑑^(𝑁−2)_𝑛+1=𝑞2𝑐_𝑛+𝑑_𝑛^(𝑁−1)+𝑑_𝑛^(𝑁−2)+𝑓_𝑛^(𝑁−1),𝑑^(𝑁−1)_𝑛+1=𝑞𝑐_𝑛+𝑑_𝑛^(𝑁−1)+𝑓_𝑛^(𝑁), (2.31) and system (2.14) reduces to 𝑣^∗_𝑛+1=𝐴^∗𝑣^∗_𝑛+ℎ^∗_𝑛, (2.32) where 𝐴^∗=𝑞1+11𝑁!12!⋅⋅⋅()!1𝑁−2()!𝑞𝑁−1()!1𝑁−111⋅⋅⋅()!1𝑁−3()!𝑞𝑁−2()!1𝑁−201⋅⋅⋅()!1𝑁−4()!𝑞𝑁−3⋅⋅⋅⋅⋅⋅2!00⋅⋅⋅11𝑞00⋅⋅⋅01 (2.33) and 𝑣^∗_𝑛=(𝑥(𝑛),𝑦₁(𝑛),𝑦₂(𝑛),…,𝑦_𝑁−1)^𝑇, ℎ^∗_𝑛=(𝑓_𝑛⁽¹⁾,𝑓_𝑛⁽²⁾,…,𝑓_𝑛^(𝑁))^𝑇. Then we have the following theorem.

Theorem 2.12. Let 𝑝=0 and 𝑞≠(−1)^𝑁𝑁!, if all eigenvalues of 𝐴^∗ in (2.32) are simple (denoted by 𝜆₁,𝜆₂,…,𝜆_𝑁) and satisfy |𝜆_𝑖|≠1,1≤𝑖≤𝑁, then (1.1) has a unique almost periodic solution 𝑥(𝑡).

Proof. System (2.32) has a solution on ℤ since 𝐴^∗ is nonsingular because 𝑞≠(−1)^𝑁𝑁!. The rest of the proof follows in the same way as the proof of Theorem 2.11 and is omitted.

Funding

This paper was supported by NNSF of China and NSF of Guangdong Province (1015160150100003).