Almost Periodic Functions on Time Scales and Applications

Abstract

We first propose the concept of almost periodic time scales and then give the definition of almost periodic functions on almost periodic time scales, then by using the theory of calculus on time scales and some mathematical methods, some basic results about almost periodic differential equations on almost periodic time scales are established. Based on these results, a class of high-order Hopfield neural networks with variable delays are studied on almost periodic time scales, and some sufficient conditions are established for the existence and global asymptotic stability of the almost periodic solution. Finally, two examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

1. Introduction

It is well known that in celestial mechanics, almost periodic solutions and stable solutions to differential equations or difference equations are intimately related. In the same way, stable electronic circuits, ecological systems, neural networks, and so forth exhibit almost periodic behavior. A vast amount of researches have been directed toward studying these phenomena (see [1–6]). Also, the theory of calculus on time scales (see [7] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [8] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his foundational work. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations. However, there are no concepts of almost periodic time scales and almost periodic functions on time scales, so that it is impossible for us to study almost periodic solutions to differential equations on time scales.

Motivated by the above, the main purpose of this paper is to propose the concept of almost periodic time scales and then give the definition of almost periodic functions on almost periodic time scales, then establish some basic results about almost periodic differential equations on almost periodic time scales by using the theory of calculus on time scales and some mathematical methods. Furthermore, based on these results, as an application, we consider the following high-order Hopfield neural networks with variable delays on time scales: 𝑥^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝛾_𝑖𝑗(𝑡+_{𝑛𝑛𝑗=1𝑙=1})𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝑥_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛, (1.1) where 𝑛 corresponds to the number of units in a neural network, 𝑥_𝑖(𝑡) corresponds to the state vector of the 𝑖th unit at the time 𝑡, 𝑐_𝑖(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, 𝑎_𝑖𝑗(𝑡) and 𝑏_𝑖𝑗𝑙(𝑡) are the first- and second-order connection weights of the neural network, 𝛾_𝑖𝑗(𝑡)≥0,𝜎_𝑖𝑗𝑙(𝑡)≥0 and 𝑣_𝑖𝑗𝑙(𝑡)≥0 correspond to the transmission delays, 𝐼_𝑖(𝑡) denote the external inputs at time 𝑡, and 𝑓_𝑗 and 𝑔_𝑗 are the activation functions of signal transmission.

2. Almost Periodic Differential Equations on Time Scales

In this section, we will establish some basic results about almost periodic differential equations on almost periodic time scales.

Let 𝕋 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 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡},𝜇(𝑡)=𝜎(𝑡)−𝑡. (2.1)

A point 𝑡∈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 𝕋^𝑘=𝕋⧵{𝑚}, otherwise 𝕋^𝑘=𝕋. If 𝕋 has a right-scattered minimum 𝑚, then 𝕋_𝑘=𝕋⧵{𝑚}, otherwise 𝕋_𝑘=𝕋.

A function 𝑓∶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.

For 𝑦∶𝕋→ℝ and 𝑡∈𝕋^𝑘, we define the delta derivative of 𝑦(𝑡),𝑦^Δ(𝑡), to be the number (if it exists) with the property that for a given 𝜀>0, there exists a neighborhood 𝑈 of 𝑡 such that 𝑦(𝜎(𝑡))−𝑦(𝑠)−𝑦^Δ(𝑡)[𝜎(𝑡)−𝑠]<𝜀|𝜎(𝑡)−𝑠|, (2.2) for all 𝑠∈𝑈.

Let 𝑦 be right-dense continuous, if 𝑌^Δ(𝑡)=𝑦(𝑡), then we define the delta integral by ^𝑡_𝑎𝑦(𝑠)Δ𝑠=𝑌(𝑡)−𝑌(𝑎). (2.3)

A function 𝑝∶𝕋→ℝ is called regressive provided 1+𝜇(𝑡)𝑝(𝑡)≠0 for all 𝑡∈𝕋^𝑘. 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,∀𝑡∈𝕋}.

If 𝑟 is a regressive function, then the generalized exponential function 𝑒_𝑟 is defined by 𝑒_𝑟(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)(𝑟(𝜏))Δ𝜏, (2.4) for all 𝑠,𝑡∈𝕋, with the cylinder transformation 𝜉_ℎ(𝑧)=Log(1+hz)ℎ,ifℎ≠0,𝑧,ifℎ=0. (2.5)

Definition 2.1 (see [7]). Let 𝑝,𝑞∶𝕋→ℝ are two regressive functions, define 𝑝𝑝⊕𝑞=𝑝+𝑞+𝜇𝑝𝑞,⊖𝑝=−.1+𝜇𝑝,𝑝⊖𝑞=𝑝⊕⊖𝑞 (2.6)

Lemma 2.2 (see [7]). Assume that 𝑝,𝑞∶𝕋→𝑅 be two regressive functions, then(i)𝑒₀(𝑡,𝑠)≡1 and 𝑒_𝑝(𝑡,𝑡)≡1;(ii)𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠);(iii)𝑒_𝑝(𝑡,𝑠)=1/𝑒_𝑝(𝑠,𝑡)=𝑒_⊖𝑝(𝑠,𝑡);(iv)𝑒_𝑝(𝑡,𝑠)𝑒_𝑝(𝑠,𝑟)=𝑒_𝑝(𝑡,𝑟);(v)(𝑒_⊖𝑝(𝑡,𝑠))^Δ=(⊖𝑝)(𝑡)𝑒_⊖𝑝(𝑡,𝑠);(vi)if 𝑎,𝑏,𝑐∈𝕋, then ∫^𝑏_𝑎𝑝(𝑡)𝑒_𝑝(𝑐,𝜎(𝑡))Δ𝑡=𝑒_𝑝(𝑐,𝑎)−𝑒_𝑝(𝑐,𝑏).

Definition 2.3. A subset 𝑆 of 𝕋 is called relatively dense if there exists a positive number 𝐿 such that [𝑎,𝑎+𝐿]∩𝑠≠∅ for all 𝑎∈𝕋. The number 𝐿 is called the inclusion length.

Definition 2.4. Let 𝒞 be a collection of sets which is constructed by subsets of ℝ. A time scale 𝕋 is called an almost periodic time scale with respect to 𝒞, if 𝒞^∗=±𝜏∈_𝑐∈𝒞𝑐∶𝑡±𝜏∈𝕋,∀𝑡∈𝕋≠∅, (2.7) and 𝒞^∗ is called the smallest almost periodic set of 𝕋.

Remark 2.5. If 𝒞={ℝ}, then 𝒞^∗={±𝜏∈ℝ∶𝑡±𝜏∈𝕋,∀𝑡∈𝕋}, 𝒞^∗ is called the smallest almost periodic set of 𝕋. If ℬ is a set which is constructed by absolute values of all the elements in 𝒞^∗, that is ℬ={|𝜏|∶𝜏∈𝒞^∗}, obviously, ℬ is the smallest positive almost periodic set of the time scale 𝕋. Let 𝑝=inf_𝜏∈ℬ{𝜏}>0, 𝑝 is called the smallest positive period of a time scale 𝕋 with respect to 𝒞. It is easy to see that this definition includes the concept of periodic time scale and is proper (see [9]).
Throughout this paper, we always restrict our discussion on almost periodic time scales. In this section, we use the notation |⋅| to denote a norm of ℝ^𝑛.

Definition 2.6. Let 𝕋 be an almost periodic time scale with respect to 𝒞. A function 𝑓(𝑡)∈𝐶(𝕋,𝔼^𝑛) is called almost periodic if for any given 𝜀>0, the set 𝐸𝜀,𝑓=𝜏∈𝒞^∗∶𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀,∀𝑡∈𝕋 (2.8) is relatively dense in 𝕋; that is, for any given 𝜀>0, there exists an 𝑙=𝑙(𝜀)>0 such that each interval of length 𝑙 contains at least one 𝜏=𝜏(𝜀)∈𝑇(𝑓,𝜀,𝕋) satisfying 𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀,∀𝑡∈𝕋. (2.9) The set 𝐸(𝜀,𝑓) is called 𝜀-translation set of 𝑓(𝑡), 𝜏 is called 𝜀-translation number of 𝑓(𝑡), and 𝑙(𝜀) is called contain interval length of 𝐸(𝜀,𝑓).
Obviously, 𝐸{𝜀,𝑓}⊆𝒞^∗. So if 𝐸{𝜀,𝑓}≠∅, then we can discuss almost periodic problems on an almost periodic time scale and it is meaningful. We denote 𝐴𝑃(𝕋) as a set constructed by all almost periodic functions on an almost time scale 𝕋.

Remark 2.7. If 𝒞={ℝ} and 𝕋=ℝ, then 𝒞^∗=ℝ, in this case, Definition 2.6 is equivalent to Definition 1.1 in [10]. If 𝒞={ℤ} and 𝕋=ℤ, then 𝒞^∗=ℤ, in this case, Definition 2.6 is equivalent to the definition of the almost periodic sequences in [11, 12].

Lemma 2.8. Let 𝑓∈𝐶(𝕋,ℝ^𝑛) be an almost periodic function, then 𝑓(𝑡) is bounded on 𝕋.

Proof. For given 𝜀≤1, there exists a constant 𝑙, such that in any interval of length 𝑙(𝜀), there exists 𝜏∈𝐸{𝜀,𝑓}, such that the inequality |𝑓(𝑡+𝜏)−𝑓(𝑡)|<𝜀,∀𝑡∈𝕋 holds. And noticing that 𝑓∈𝐶(𝕋,ℝ^𝑛), then in the limited interval [0,𝑙]_𝕋, there exists a number 𝑀>0, such that |𝑓(𝑡)|<𝑀. For any given 𝑡∈𝕋, we can take 𝜏∈𝐸{𝜀,𝑓}∩[−𝑡,−𝑡+𝑙]_𝕋, then we have 𝑡+𝜏∈[0,𝑙]_𝕋. Hence, we can obtain |𝑓(𝑡+𝜏)|<𝑀 and |𝑓(𝑡+𝜏)−𝑓(𝑡)|<1. So for all 𝑡∈𝕋, we have |𝑓(𝑡)|<𝑀+1. This completes the proof.

Similar to the case of 𝕋=ℝ, one can easily show the following theorems.

Theorem 2.9. If 𝑓,𝑔∈𝐶(𝕋,ℝ^𝑛) are almost periodic, then 𝑓+𝑔,𝑓𝑔 are almost periodic.

Theorem 2.10. If 𝑓(𝑡)∈𝐶(𝕋,ℝ^𝑛) is almost periodic, then 𝐹(𝑡,𝑥) is almost periodic if and only if 𝐹(𝑡) is bounded on 𝕋, where 𝐹(𝑡)=∫^𝑡₀𝑓(𝑠)Δ𝑠.

Theorem 2.11. If 𝑓(𝑡) is almost periodic, 𝐹(⋅) is uniformly continuous on the value field of 𝑓(𝑡), then 𝐹∘𝑓 is almost periodic.

Definition 2.12. Let 𝑥∈ℝ^𝑛, and 𝐴(𝑡) be an 𝑛×𝑛 rd-continuous matrix on 𝕋, the linear system 𝑥^Δ(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑡∈𝕋 (2.10) is said to admit an exponential dichotomy on 𝕋 if there exist positive constant 𝑘,𝛼, projection 𝑃, and the fundamental solution matrix 𝑋(𝑡) of (2.10), satisfying 𝑋(𝑡)𝑃𝑋⁻¹(𝜎(𝑠))₀≤𝑘𝑒_⊖𝛼((𝑠(𝑠),𝑋(𝑡)𝑋𝑡,𝜎)),𝑠,𝑡∈𝕋,𝑡≥𝜎)(𝐼−𝑃⁻¹(𝜎(𝑠))₀≤𝑘𝑒_⊖𝛼(𝜎(𝑠),𝑡),𝑠,𝑡∈𝕋,𝑡≤𝜎(𝑠), (2.11) where |⋅|₀ is a matrix norm on 𝕋, (say, e.g., if𝐴=(𝑎_𝑖𝑗)_𝑛×𝑚, then we can take |𝐴|₀=(∑^𝑛_𝑖=1∑^𝑚_𝑗=1|𝑎_𝑖𝑗|²)^1/2).
Consider the following almost periodic system 𝑥^Δ(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡∈𝕋, (2.12) where 𝐴(𝑡) is an almost periodic matrix function, 𝑓(𝑡) is an almost periodic vector function.

Lemma 2.13. If the linear system (2.10) admits exponential dichotomy, then system (2.12) has a bounded solution 𝑥(𝑡) as follows: 𝑥(𝑡)=^𝑡_−∞𝑋(𝑡)𝑃𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠−_𝑡^+∞𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠, (2.13) where 𝑋(𝑡) is the fundamental solution matrix of (2.10).

Proof. In fact, 𝑥^Δ(𝑡)−𝐴(𝑡)𝑥(𝑡)=𝑋^Δ(𝑡)^𝑡_−∞𝑃𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠+𝑋(𝜎(𝑡))𝑃𝑋⁻¹(𝜎(𝑡))𝑓(𝑡)−𝑋^Δ(𝑡)_𝑡^+∞(𝐼−𝑃)𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠+𝑋(𝜎(𝑡))(𝐼−𝑃)𝑋⁻¹(𝜎(𝑡𝑓(𝑡)(𝑡)𝑋(𝑡)))−𝐴^𝑡_−∞𝑃𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠+𝐴(𝑡)𝑋(𝑡)_𝑡^+∞(𝐼−𝑃)𝑋⁻¹(𝜎(𝑠𝑓(𝑠)(𝜎(𝑡)𝑋))Δ𝑠=𝑋))(𝑃+𝐼−𝑃⁻¹(𝜎(𝑡𝑓(𝑡)(𝑡),))=𝑓‖𝑥‖=sup_𝑡∈𝕋^𝑡_−∞𝑋(𝑡)𝑃𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠−_𝑡^+∞𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠≤sup_𝑡∈𝕋^𝑡_−∞𝑒_⊖𝛼(𝑡,𝜎(𝑠))Δ𝑠+_𝑡^+∞𝑒_⊖𝛼(𝜎(𝑠))≤1,𝑡Δ𝑠𝑘‖𝑓‖𝛼−1𝑘𝑓=⊖𝛼2+𝜇𝛼𝛼𝑘𝑓, (2.14) where ‖⋅‖=sup_𝑡∈𝕋|⋅|. So, by Lemma 2.8, 𝑥(𝑡) is a bounded solution of system (2.12). The proof is complete.

Lemma 2.14 (see [7]). Let 𝐴 be a regressive 𝑛×𝑛-matrix-valued function on 𝕋. Let 𝑡₀∈𝕋 and 𝑦₀∈ℝ^𝑛. Then the initial value problem 𝑦^Δ(𝑡)=𝐴(𝑡)𝑦(𝑡),𝑦𝑡₀=𝑦₀ (2.15) has a unique solution 𝑦∶𝕋→ℝ^𝑛. Moreover, the solution is given by 𝑦(𝑡)=𝑒_𝐴𝑡,𝑡₀𝑦₀. (2.16)

Lemma 2.15. Let 𝑐_𝑖(𝑡) be an almost periodic function on 𝕋, where 𝑐_𝑖(𝑡)>0, −𝑐_𝑖(𝑡)∈ℛ⁺, ∀𝑡∈𝕋 and min_{1≤𝑖≤𝑛}inf_𝑡∈𝕋𝑐_𝑖(𝑡)=𝑚>0, (2.17) then the linear system 𝑥^Δ(𝑡)=diag−𝑐₁(𝑡),−𝑐₂(𝑡),…,−𝑐_𝑛(𝑡)𝑥(𝑡) (2.18) admits an exponential dichotomy on 𝕋.

Proof. According to Lemma 2.14, one can see that 𝑋(𝑡)=𝑒_−𝑐𝑡,𝑡₀, (2.19) where −𝑐=diag(−𝑐₁(𝑡),−𝑐₂(𝑡),…,−𝑐_𝑛(𝑡)), is a fundamental solution matrix of (2.18).
Now, we prove that 𝑋(𝑡) admits an exponential dichotomy on 𝕋. In fact, noticing that −𝑐_𝑖(𝑡)∈ℛ⁺, then for 𝑡≥𝜎(𝑠),𝑡,𝑠∈𝕋,∀𝑖=1,2,…,𝑛.
If 𝜇(𝜃)>0,𝜃∈[𝜎(𝑠),𝑡)_𝕋,𝑠,𝑡∈𝕋, we have 𝜇(𝜃)1−)(𝑚/2(𝜃)(𝜃)𝑚1+𝜇)(𝑚/2>1−𝜇2>1−𝜇(𝜃)𝑐_𝑖(𝜃)>0, (2.20) then Log1−𝜇(^𝑡_𝜎(𝑠)𝜃)𝑐_𝑖(𝜃)𝜇(𝜃)Δ𝜃≤^𝑡_𝜎(𝑠)Log1−𝜇(𝜃)(𝑚/2)/1+𝜇(𝜃)(𝑚/2)𝜇(𝜃)Δ𝜃, (2.21) therefore exp^𝑡_𝜎(𝑠)Log1−𝜇(𝜃)𝑐_𝑖(𝜃)𝜇(𝜃)Δ𝜃≤exp^𝑡_𝜎(𝑠)Log1−𝜇(𝜃)(𝑚/2)/1+𝜇(𝜃)(𝑚/2)𝜇(𝜃)Δ𝜃, (2.22) then, we can get 𝑒_−𝑐𝑖(𝑡,𝜎(𝑠))≤𝑒_⊖(𝑚_/2)(𝑡,𝜎(𝑠)). (2.23)
If 𝜇(𝜃)=0,𝜃∈[𝜎(𝑠),𝑡)_𝕋,𝑠,𝑡∈𝕋, we can get 𝑒_−𝑐𝑖(𝑡,𝜎(𝑠))=exp^𝑡_𝜎(𝑠)−𝑐_𝑖(𝜃)Δ𝜃≤exp^𝑡_𝜎(𝑠)−𝑚2Δ𝜃=𝑒_⊖(𝑚_/2)(𝑡,𝜎(𝑠)). (2.24) Hence, set 𝑃=𝐼, then 𝑋(𝑡)𝑃𝑋⁻¹(𝜎(𝑠))₀=𝑒_−𝑐𝑡,𝑡₀𝐼𝑒⁻¹_−𝑐𝜎(𝑠),𝑡₀₀=𝑒_−𝑐𝑡,𝑡₀𝑒_−𝑐𝑡₀,𝜎(𝑠)₀=𝑒_−𝑐(𝑡,𝜎(𝑠))₀≤𝑛^1/2𝑒_⊖(𝑚_/2)(𝑡,𝜎(𝑠)), (2.25) where 𝑚=min_{1≤𝑖≤𝑛}{inf_𝑡∈𝕋𝑐_𝑖(𝑡)}. We can take 𝑘=𝑛^1/2,𝛼=𝑚/2, therefore, 𝑋(𝑡) admits an exponential dichotomy on 𝕋 with 𝑃=𝐼. This completes the proof.

3. An Application

It is well known that high-order Hopfield neural networks (HHNNs) have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. There exist many results on the existence and stability of periodic and almost periodic solutions for the neural networks with delays. We refer the reader to [13–27] and references therein.

In fact, both continuous and discrete systems are very important in implementing and applications. But it is troublesome to study the existence and stability of almost periodic solutions for continuous and discrete systems, respectively. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations (see [28, 29]). In this section, by using the concepts and results developed in previous sections, we will study the existence and global asymptotic stability of almost periodic solution of (1.1).

The system (1.1) is supplemented with initial values given by 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑠∈[−𝜃,0]_𝕋,𝜃=max𝛾,𝜎,𝑣,𝛾=max_{1≤𝑖,𝑗≤𝑛}𝛾_𝑖𝑗,𝜎=max_{1≤𝑖,𝑗,𝑙≤𝑛}𝜎_𝑖𝑗𝑙,𝑣=max_{1≤𝑖,𝑗,𝑙≤𝑛}𝑣_𝑖𝑗𝑙𝛾,𝑖,𝑗,𝑙=1,2,…,𝑛,_𝑖𝑗=sup_𝑡∈𝕋𝛾_𝑖𝑗(𝑡),𝜎_𝑖𝑗𝑙=sup_𝑡∈𝕋𝜎_𝑖𝑗𝑙(𝑡),𝑣_𝑖𝑗𝑙=sup_𝑡∈𝕋𝑣_𝑖𝑗𝑙(𝑡), (3.1) where 𝜑_𝑖(⋅)∈𝐶([−𝜃,0]_𝕋,ℝ).

For the sake of convenience, we introduce the following notations:𝑐_𝑖=inf_𝑡∈𝕋𝑐_𝑖(𝑡),𝑐_𝑖=sup_𝑡∈𝕋𝑐_𝑖(𝑡),𝑎_𝑖𝑗=sup_𝑡∈𝕋𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗𝑙=sup_𝑡∈𝕋𝑏_𝑖𝑗𝑙(𝑡),𝐼_𝑖=sup_𝑡∈𝕋𝐼_𝑖(𝑡). (3.2)

In this section, we assume the follwing. (𝐻₁)𝑐_𝑖,𝑎_𝑖𝑗,𝑏_𝑖𝑗𝑙,𝐼_𝑖,𝑡−𝛾_𝑖𝑗,𝑡−𝜎_𝑖𝑗𝑙,𝑡−𝑣_𝑖𝑗𝑙∈𝐶(𝕋,ℝ) are almost periodic, −𝑐_𝑖∈ℛ⁺ and 𝑐_𝑖>0, for 𝑖,𝑗,𝑙=1,2,…,𝑛. (𝐻₂) There exist positive constants 𝑀_𝑗,𝑁_𝑗,𝑗=1,2,…,𝑛 such that |𝑓_𝑗(𝑥)|≤𝑀_𝑗,|𝑔_𝑗(𝑥)|≤𝑁_𝑗 for 𝑗=1,2,…,𝑛,𝑥∈ℝ.(𝐻₃) Functions 𝑓_𝑗(𝑢),𝑔_𝑗(𝑢)(𝑗=1,2,…,𝑛) satisfy the Lipschitz condition, that is, there exist constants 𝐿_𝑗,𝐻_𝑗>0 such that |𝑓_𝑗(𝑢₁)−𝑓_𝑗(𝑢₂)|≤𝐿_𝑗|𝑢₁−𝑢₂|,|𝑔_𝑗(𝑢₁)−𝑔_𝑗(𝑢₂)|≤𝐻_𝑗|𝑢₁−𝑢₂|,𝑗=1,2,…,𝑛.

Let 𝑋={𝜓∣𝜓∶𝕋→ℝ^𝑛,isacontinuousalmostperiodicfunction} with the norm ‖𝜓‖_𝑋=sup_𝑡∈𝕋‖𝜓(𝑠)‖. Clearly, 𝑋 is a Banach space.

Definition 3.1. The almost periodic solution 𝑥^∗(𝑡) of system (1.1) is said to be globally asymptotically stable if for any 𝜀>0 and 𝑡₀∈[−𝜃,+∞)_𝕋, there exists 𝛿(𝜀)>0 and 𝜎=𝜎(𝑡₀,𝜀,𝜑)>0 such that ‖𝜑(𝑡)−𝑥^∗(𝑡)‖<𝛿 for 𝑡∈[−𝜃,0]_𝕋 implies ‖𝑥(𝑡,𝜑)−𝑥^∗(𝑡)‖<𝜀 for all 𝑡∈[𝑡₀+𝜎,+∞)_𝕋.

Theorem 3.2. Assume that (𝐻₁)–(𝐻₃) hold, and suppose that(𝐻₄)max_{1≤𝑖≤𝑛}{∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝐻_𝑙+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗/𝑐_𝑖}<1,
then (1.1) has a unique almost periodic solution.

Proof. For any given 𝜑∈𝑋, we consider the following almost periodic differential system: 𝑥^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝜑_𝑗𝑡−𝛾_𝑖𝑗(𝑡+_{𝑛𝑛𝑗=1𝑙=1})𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗𝜑_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝜑_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛. (3.3)
Since min_{1≤𝑖≤𝑛}{inf𝑐_𝑖(𝑡)}>0,𝑖=1,2,…,𝑛,𝑡∈𝕋, it follows from Lemma 2.15 that the linear system 𝑥^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡),𝑖=1,2,…,𝑛 (3.4) admits an exponential dichotomy on 𝕋. Thus, by Lemmas 2.13 and 2.15, we obtain that system (1.1) has a bounded solution: 𝑥_𝜑𝑖(𝑡)=^𝑡_−∞𝑒_−𝑐𝑖((𝑠×𝑡,𝜎))_𝑛𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝜑_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗𝜑_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝜑_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)+𝐼_𝑖(𝑡)Δ𝑠,𝑖=1,2,…,𝑛 (3.5) and it follows from Theorems 2.9–2.11 and 𝑒_−𝑐𝑖(𝑡,𝜎(𝑠)) being almost periodic that 𝑥_𝜑 is also almost periodic.
Denote max_𝑖∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑀_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙𝑐_𝑖+max_𝑖𝐼_𝑖𝑐_𝑖∶=𝐿 (3.6) and define a mapping 𝑇∶𝑋→𝑋,𝑇𝜑(𝑡)=𝑥_𝜑(𝑡),∀𝜑∈𝑋. Set 𝑋^∗=𝜑∈𝑋𝜑_𝑋≤𝐿. (3.7)
Next, let us check that 𝑇𝜑∈𝑋^∗. For any given 𝜑∈𝑋^∗, it suffices to prove that ‖𝑇(𝜑)‖≤𝐿; 𝑇𝜑_𝑋=sup_𝑡∈𝕋max_𝑖^𝑡_−∞𝑒_−𝑐𝑖(𝑡,𝜎(𝑠))^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝜑_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗𝜑_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝜑_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)+𝐼_𝑖(𝑡)Δ𝑠≤sup_𝑡∈𝕋max_𝑖^𝑡_−∞𝑒_−𝑐𝑖(𝑡,𝜎(𝑠))^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝜑_𝑗𝑡−𝛾_𝑖𝑗(𝑡+_{𝑛𝑛𝑗=1𝑙=1})𝑏_𝑖𝑗𝑙𝑔_𝑗𝜑_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝜑_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)+max_𝑖𝐼_𝑖𝑐_𝑖Δ𝑠≤sup_𝑡∈𝕋max_𝑖^𝑡_−∞𝑒_−𝑐𝑖(𝑡,𝜎(𝑠))^𝑛_𝑗=1𝑎_𝑖𝑗𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙+max_𝑖𝐼_𝑖𝑐_𝑖Δ𝑠≤max_𝑖∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑀_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙𝑐_𝑖+max_𝑖𝐼_𝑖𝑐_𝑖=𝐿, (3.8) which shows that 𝑇𝜑∈𝑋^∗. So 𝑇 is a self-mapping from 𝑋^∗ to 𝑋^∗.
Next, we shall prove that 𝑇 is a contraction of 𝑋^∗.
For any 𝜑,𝜓∈𝑋^∗, 𝑇𝜑−𝑇𝜓_𝑋=sup_𝑡∈𝕋𝑇𝜑(𝑡)−𝑇𝜓(𝑡)≤sup_𝑡∈𝕋max_𝑖^𝑡_−∞𝑒_−𝑐𝑖(𝑡,𝜎(𝑠))^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝜑_j𝑡−𝛾_𝑖𝑗(𝑡)−𝑓_𝑗𝜓_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗𝜑_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝜑_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)−𝑔_𝑗𝜓_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝜓_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)Δ𝑠≤sup_𝑡∈𝕋max_𝑖^𝑡_−∞𝑒_−𝑐𝑖(𝑡,𝜎(𝑠))^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝜑_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−𝑓_𝑗𝜓_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗𝜑_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝜑_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)−𝑔_𝑙𝜓_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)+𝑔_𝑗𝜑_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)−𝑔_𝑗𝜓_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝜓_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)Δ𝑠≤sup_𝑡∈𝕋max_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝐻_𝑙+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗Δ𝑠𝜑−𝜓_𝑋≤max_𝑖∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝐻_𝑙+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗𝑐_𝑖𝜑−𝜓_𝑋=𝜌𝜑−𝜓_𝑋. (3.9) Because 𝜌<1, so 𝑇 is a contraction of 𝑋^∗.
By the fixed point theorem of Banach space, 𝑇 has a unique fixed point 𝜑 in 𝑋^∗ such that 𝑇𝜑=𝜑, 𝜑 is an almost periodic solution of system (1.1) in 𝑋^∗. The proof is complete.

Theorem 3.3. Assume that (𝐻₁)–(𝐻₄) hold. Suppose further that (𝐻₅). Let 𝛿_𝑖=2𝑐_𝑖−2∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗−2∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙𝑐_𝑖+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙², (3.10) and for any 𝑡₀∈[−𝜃,+∞)_𝕋, the following holds ^𝑡_𝑡0𝛿_𝑖−𝜇(𝑠)Δ𝑠⟶+∞,𝑡⟶+∞,𝑖=1,2,…,𝑛. (3.11) Then the almost periodic solution of system (1.1) is globally asymptotically stable.

Proof. According to Theorem 3.2, we know that (1.1) has an almost periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡))^𝑇. Suppose that 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is an arbitrary solution of (1.1). Then it follows from system (1.1) that 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)^Δ=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−𝑓_𝑗𝑥^∗_𝑗𝑡−𝛾_𝑖𝑗(𝑡+_{𝑛𝑛𝑗=1𝑙=1})𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝑥_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)−𝑔_𝑗𝑥^∗_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝑥^∗_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡), (3.12) for 𝑖=1,2,…,𝑛, the initial condition of (3.12) is 𝜓_𝑖(𝑠)=𝜑_𝑖(𝑠)−𝑥^∗_𝑖(s),𝑠∈[−𝜃,0]_𝕋,𝑖=1,2,…,𝑛. (3.13) Let 𝑦_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡), we use the Lyapunov function 𝑉(𝑦)=𝑦^𝑇𝑦, where 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇, from (3.12) we have 𝑦²_𝑖(𝑡)^Δ=2𝑦_𝑖(𝑡)𝑦^Δ_𝑖(𝑡)+𝜇(𝑡)𝑦^Δ_𝑖(𝑡)²≤2𝑦_𝑖(𝑡)−𝑐_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗𝑦_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)+𝐻_𝑙𝑁_𝑗𝑦_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)+𝜇(𝑡)−𝑐_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗𝑦_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)+𝐻_𝑙𝑁_𝑗𝑦_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)²≤−2𝑐_𝑖+2^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+2^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙𝑦²+𝜇(𝑡)𝑐_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²𝑦²=−2𝑐_𝑖−2^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗−2^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙−𝜇(𝑡)𝑐_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²𝑦²=−𝑐_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²×2𝑐_𝑖−2∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗−2∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙𝑐_𝑖+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²−𝜇(𝑡)𝑦²=−𝑐_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²𝛿_𝑖−𝜇(𝑡)𝑦². (3.14)
Then we can easily get 𝑉^Δ𝑦(𝑡)=^𝑛_𝑖=1𝑦²_𝑖(𝑡)^Δ≤−^𝑛_𝑖=1𝑐_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²𝛿_𝑖−𝜇(𝑡)𝑦²≤−𝜉(𝑡)𝜙𝑦, (3.15) where 𝜉(𝑡)=min_{1≤𝑖≤𝑛}{𝛿_𝑖}−𝜇(𝑡), 𝜙(𝑠)=∑^𝑛_𝑖=1(𝑐_𝑖+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_𝑖𝑗𝑙(𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙))²𝑠².
Let 𝑎(𝑠)=(1/𝑛)𝑠² and 𝑏=𝑠² where 𝑎,𝑏∈𝐶([0,∞),ℝ⁺), it is easy to see that (i)𝑎(‖𝑦‖)≤𝑉(𝑦(𝑡))≤𝑏(‖𝑦‖); (ii)𝑉^Δ(𝑦(𝑡))|_(1⋅1)≤−𝜉(𝑡)𝜙(‖𝑦‖).
Let Ω be a domain in the space ℝ^𝑛 that contains the origin of coordinates. We choose a constant 𝛼>0 and set Γ={𝑦∈Ω∶𝑉(𝑦(𝑡))≤𝑎(𝛼)}. Assume that statement 𝑆₀ has the following form: for any 𝜑∈Γ and 𝜀>0, 𝑡₀∈[−𝜃,+∞)_𝕋, there exists a constant 𝜍=𝜍(𝑡₀,𝜀,𝜑)>0 such that ‖𝑦(𝑡₀+𝜍,𝜑)‖<𝜀.
Assume that statement 𝑆₀ is not true, that is, there exist 𝜑^′∈Γ and 𝜀₁>0 such that for any constant 𝜍>0, one has ‖𝑦(𝑡₀+𝜍,𝜑^′)‖≥𝜀₁. Since ^𝑡_𝑡0𝜉(𝑠)Δ𝑠⟶+∞,𝑡⟶+∞, (3.16) we conclude that, for 𝜂>𝑎(𝛼)/𝜙(𝜀₁), there exists a constant 𝜍₁=𝜍₁(𝑡₀,𝜀₁,𝜑^′) such that ^𝑡0+𝜍1_𝑡0𝜉(𝑠)Δ𝑠>𝜂. (3.17) Every 𝑡∈[𝑡₀,𝑡₀+𝜍₁)_𝕋 can be represented in the form 𝑡=𝑡₀+𝜍₂. Hence, for all 𝑡∈[𝑡₀,𝑡₀+𝜍₁)_𝕋, we have ‖𝑦(𝑡,𝜑^′)‖≥𝜀₁. Integrating the inequality from (ii), we get 𝑉𝑦𝑡₀+𝜍₁≤𝑉𝑦𝑡₀−^𝑡0+𝜍1_𝑡0𝜉(𝑡)𝜙𝑦Δ𝑠<𝑎(𝛼)−𝜙𝜀₁^𝑡0+𝜍1_𝑡0𝜉(𝑡)Δ𝑠≤𝑎(𝛼)−𝜙𝜀₁𝜂<0, (3.18) which is impossible. Thus, 𝑆₀ is true.
We choose an arbitrary 𝜀>0 and set 𝜀₁=𝑏⁻¹(𝑎(𝜀))>0. Since statement 𝑆₀ is true, for any 𝜑∈Γ and 𝜀>0 there exists a constant 𝜍=𝜍(𝑡₀,𝜀,𝜑)>0 such that ‖𝑦(𝑡₀+𝜍,𝜑)‖<𝜀₁.
Since 𝑡>𝑡₀+𝜍 for all 𝑡∈[𝑡₀+𝜍,+∞)_𝕋, we obtain the following inequality for the solution 𝑦(𝑡)=𝑦(𝑡,𝜑): 𝑎𝑦≤𝑉𝑦(𝑡)≤𝑉𝑦𝑡₀+𝜍≤𝑏𝑦𝑡₀+𝜍<𝑏𝜀₁. (3.19) Hence, ‖𝑦(𝑡)‖≤𝑎⁻¹𝑏(𝜀₁)=𝜀 for all 𝑡∈[𝑡₀+𝜍,+∞)_𝕋.
Thus, for any 𝜑∈Γ and 𝜀>0, there is a constant 𝜍=𝜍(𝑡₀,𝜀,𝜑)>0 such that ‖𝑦(𝑡,𝜑)‖<𝜀 for all 𝑡∈[𝑡₀+𝜍,+∞)_𝕋. Choosing a constant 𝛿>0 so that 𝑏(𝛿)≤𝑎(𝛼), we obtain 𝐵_𝛿∶={𝑦∈ℝ^𝑛∶‖𝑦‖<𝛿}⊂Γ. Therefore, the almost periodic solution 𝑥^∗(𝑡) of system (1.1) is globally asymptotically stable. This completes the proof.

4. Numerical Examples and Simulations

Consider the following neural networks system on time scales: 𝑥^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+²_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝛾_𝑖𝑗(𝑡+_{22𝑗=1𝑙=1})𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜎_𝑖𝑗𝑙(𝑡)𝑔_𝑙𝑥_𝑙𝑡−𝑣_𝑖𝑗𝑙(𝑡)+𝐼_𝑖(𝑡),𝑖=1,2,𝑡>0, (4.1) where𝑓₁𝑥₁=𝑔₁𝑥₁3=sin4𝑥₁,𝑓₂𝑥₂=𝑔₂𝑥₂2=cos5𝑥₂. (4.2) Obviously, 𝑓_𝑖(𝑥_𝑖),𝑔_𝑖(𝑥_𝑖)(𝑖=1,2) satisfy (𝐻₂) and (𝐻₃), and𝐿₁=𝐿₂=𝐻₁=𝐻₂=𝑀₁=𝑀₂=𝑁₁=𝑁₂=1. (4.3)

Example 4.1. 𝕋=ℝ,𝜇(𝑡)≡0, 𝑎₁₁(𝑡)=1+12020cos𝑡,𝑎₁₂(𝑡)=1+340√40cos2𝑡,𝑎₂₁(𝑡)=3+120420cos3𝑡,𝑎₂₂(𝑡)=3+140140cos4𝑡,𝑐₁(𝑡)1=1+620sin5𝑡,𝑐₂(𝑡)1=1+125sin3𝑡,𝐼₁(𝑡)=3+140√40sin3𝑡,𝐼₂(𝑡)=1+340340cos4𝑡,𝑏₁₁₁(𝑡)=𝑏₂₂₂(𝑡)=1+180√80sin2𝑡,𝑏₁₁₂(𝑡)=𝑏₂₁₂(𝑡)=1+140440cos3𝑡,𝑏₁₂₁(𝑡)=𝑏₂₂₁(𝑡)=1+180√80cos3𝑡,𝑏₁₂₂(𝑡)=𝑏₂₁₁(𝑡)=1+180380sin4𝑡. (4.4) We get that (𝐻₁) is satisfied, and 𝑐₁=21,20𝑐₂=2625,𝑐₁=1920,𝑐₂=24,25𝑎₁₁=1,10𝑎₁₂=1,10𝑎₂₁=15,𝑎₂₂=1,10𝑏₁₁₁=𝑏₂₂₂=1,40𝑏₁₁₂=𝑏₂₁₂=1,20𝑏₁₂₁=𝑏₂₂₁=1,40𝑏₁₂₂=𝑏₂₁₁=1,40 (4.5) so, we have ∑²_𝑗=1𝑎_1𝑗𝐿_𝑗+∑²_𝑗=1∑²_𝑙=1𝑏_1𝑗𝑙𝑁_𝑗𝐻_𝑙+∑²_𝑗=1∑²_𝑙=1𝑏_1𝑗𝑙𝑁_𝑙𝐻_𝑗𝑐₁=9∑19<1,_2𝑗=1𝑎_2𝑗𝐿_𝑗+∑²_𝑗=1∑²_𝑙=1𝑏_2𝑗𝑙𝑁_𝑗𝐻_𝑙+∑²_𝑗=1∑²_𝑙=1𝑏_2𝑗𝑙𝑁_𝑙𝐻_𝑗𝑐₂=55𝛿96<1,₁=2𝑐₁−2∑^𝑛_𝑗=1𝑎_1𝑗𝐿_𝑗−2∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_1𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙𝑐₁+∑^𝑛_𝑗=1𝑎_1𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_1𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²𝛿=0.444>0,₂=2𝑐₂−2∑^𝑛_𝑗=1𝑎_2𝑗𝐿_𝑗−2∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_2𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙𝑐₂+∑^𝑛_𝑗=1𝑎_2𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_2𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²=0.3164>0. (4.6) Thus, it is easy to see that ^𝑡_𝑡0𝛿_𝑖−𝜇(𝑠)Δ𝑠⟶+∞,𝑡⟶+∞,𝑖=1,2. (4.7) The conditions of Theorems 3.2 and 3.3 is satisfied. Hence, we know that system (4.1) has an almost periodic solution, which is asymptotically stable.
We take 𝛾_𝑖𝑗=𝜎_𝑖𝑗𝑙=𝑣_𝑖𝑗𝑙=0.1,𝑖,𝑗,𝑙=1,2, and the initial condition 𝜑₁(𝜃)=−0.13,𝜑₂(𝜃)=0.03,𝜃∈[−0.1,0], we can give the following numerical simulation figures to show our results are plausible and effective on time scales (see Figures 1, 2, and 3).
The numerical simulations of Figures 1, 2, and 3 in Example 4.1 show that the unique almost periodic solution is asymptotically stable, our results are effective on time scales.

Figure 1 

Transient response of state 𝑥1,𝑥2 in Example 4.1.

Figure 2 

Phase response of state variables 𝑥1,𝑥2 in Example 4.1.

Figure 3 

Phase response of state variables 𝑥1,𝑥2 with time 𝑡 in Example 4.1.

Example 4.2. 𝕋=ℤ,𝜇(𝑡)≡1, 𝑎₁₁(𝑡)=0.003+0.002cos𝑡,𝑎₁₂(𝑡)=0.002+0.001cos√,𝑎2𝑡₂₁(𝑡)4=0.001+0.003cos3𝑡,𝑎₂₂(𝑡)1=0.003+0.001cos4𝑡,𝑐₁(𝑡)6=0.7+0.05sin5𝑡,𝑐₂(𝑡)1=0.8+0.02sin3𝑡,𝐼₁(𝑡)=0.001+0.002sin√3𝑡,𝐼₂(𝑡)3=0.002+0.001cos4𝑡,𝑏₁₁₁(𝑡)=𝑏₂₂₂(𝑡)=0.001+0.002sin√2𝑡,𝑏₁₁₂(𝑡)=𝑏₂₁₂(𝑡)4=0.002+0.003cos3𝑡,𝑏₁₂₁(𝑡)=𝑏₂₂₁(𝑡)=0.001+0.001cos√3𝑡,𝑏₁₂₂(𝑡)=𝑏₂₁₁(𝑡)3=0.003+0.002sin4𝑡. (4.8)
We get that (𝐻₁) is satisfied, and 𝑐₁=0.75,𝑐₂=0.82,𝑐₁=0.65,𝑐₂=0.78,𝑎₁₁=0.005,𝑎₁₂=0.003,𝑎₂₁=0.004,𝑎₂₂=0.004,𝑏₁₁₁=𝑏₂₂₂=0.003,𝑏₁₁₂=𝑏₂₁₂=0.005,𝑏₁₂₁=𝑏₂₂₁=0.002,𝑏₁₂₂=𝑏₂₁₁=0.005, (4.9) so, we have ∑²_𝑗=1𝑎_1𝑗𝐿_𝑗+∑²_𝑗=1∑²_𝑙=1𝑏_1𝑗𝑙𝑁_𝑗𝐻_𝑙+∑²_𝑗=1∑²_𝑙=1𝑏_1𝑗𝑙𝑁_𝑙𝐻_𝑗𝑐₁∑=0.058<1,_2𝑗=1𝑎_2𝑗𝐿_𝑗+∑²_𝑗=1∑²_𝑙=1𝑏_2𝑗𝑙𝑁_𝑗𝐻_𝑙+∑²_𝑗=1∑²_𝑙=1𝑏_2𝑗𝑙𝑁_𝑙𝐻_𝑗𝑐₂𝛿=0.049<1,₁=2𝑐₁−2∑^𝑛_𝑗=1𝑎_1𝑗𝐿_𝑗−2∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_1𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙𝑐₁+∑^𝑛_𝑗=1𝑎_1𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_1𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²𝛿=1.9712>1,₂=2𝑐₂−2∑^𝑛_𝑗=1𝑎_2𝑗𝐿_𝑗−2∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_2𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙𝑐₂+∑^𝑛_𝑗=1𝑎_2𝑗𝐿_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏_2𝑗𝑙𝑁_𝑙𝐻_𝑗+𝑁_𝑗𝐻_𝑙²=1.8202>1. (4.10) Thus, it is easy to see that ^𝑡_𝑡0𝛿_𝑖−𝜇(𝑠)Δ𝑠⟶+∞,𝑡⟶+∞,𝑖=1,2. (4.11) The conditions of Theorems 3.2 and 3.3 is satisfied. Hence, we know that system (4.1) has an almost periodic solution, which is asymptotically stable.
We take 𝛾_𝑖𝑗=𝜎_𝑖𝑗𝑙=𝑣_𝑖𝑗𝑙=0.01,𝑖,𝑗,𝑙=1,2, and the initial condition 𝜑₁(𝜃)=−0.013,𝜑₂(𝜃)=0.026,𝜃∈[−0.01,0], we can give the following numerical simulation figures to show our results are plausible and effective on time scales (see Figures 3, 4, and 5).
The numerical simulations of Figures 4, 5, and 6 in Example 4.2 show that the unique almost periodic solution is asymptotically stable, our results are effective on time scales.

Figure 4 

Transient response of state 𝑥1,𝑥2 in Example 4.2.

Figure 5 

Phase response of state variables 𝑥1,𝑥2 in Example 4.2.

Figure 6 

Phase response of state variables 𝑥1,𝑥2 with time 𝑡 in Example 4.2.

5. Conclusion

In this paper, some basic results about almost periodic differential equations on almost periodic time scales are established, and the existence and global asymptotic stability of an almost periodic solution for a class of high-order Hopfield neural networks on almost periodic time scales is investigated. The results derived in this paper are meaningful.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.