Almost Periodic Solutions to Dynamic Equations on Time Scales and Applications

Abstract

We first introduce the concept of admitting an exponential dichotomy to a class of linear dynamic equations on time scales and study the existence and uniqueness of almost periodic solution and its expression form to this class of linear dynamic equations on time scales. Then, as an application, using these concepts and results, we establish sufficient conditions for the existence and exponential stability of almost periodic solution to a class of Hopfield neural networks with delays. Finally, two examples and numerical simulations given to illustrate our results are plausible and meaningful.

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. Also, the existence of almost periodic, asymptotically almost periodic, 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 [4–20].

Motivated by the above, based on the theory of almost periodic functions on time scales in our previous work [21, 22], we first introduce the concept of admitting an exponential dichotomy to a class of linear dynamic equations on time scales and study the existence and uniqueness of almost periodic solution and its expression form to this class of linear dynamic equations on time scales. Then, as an application, using these concepts, results, the fixed point theorem and differential inequality techniques, we establish sufficient conditions for the existence and exponential stability of almost periodic solution to a class of Hopfield neural networks with delays. Finally, two examples given to illustrate our results are plausible and meaningful to unify continuous and discrete models.

The organization of this paper is as follows. In Section 2, we introduce some notations and state some preliminary results needed in the later sections. In Section 3, we introduce the concepts of admitting an exponential dichotomy to a class of linear dynamic equations on time scales under which the existence, uniqueness, and expression form of an almost periodic solution are obtained. Furthermore, some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations are also derived. In Section 4, as an application of our results, we study the existence and exponential stability of the almost periodic solutions of a class of Hopfield neural networks with delays, finally, we give two examples and numerical simulations to show that our unification of continuous and discrete situations is effective.

2. Preliminaries

In this section, we will first recall some basic definitions, lemmas which are used in what follows.

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| 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).

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, for all 𝑡∈𝕋}.

An 𝑛×𝑛-matrix-valued function 𝐴 on a time scale 𝕋 is called regressive provided that 𝐼+𝜇(𝑡)𝐴(𝑡) is invertible for all 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,ℝ^𝑛×𝑛).

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} for all 𝑠,𝑡∈𝕋, with the cylinder transformation 𝜉_ℎ(𝑧)=Log(1+ℎ𝑧)ℎ,ifℎ≠0,𝑧,ifℎ=0. (2.2)

Definition 2.1 (see [1, 2]). Let 𝑝,𝑞∶𝕋→ℝ be two regressive functions, define 𝑝𝑝⊕𝑞=𝑝+𝑞+𝜇𝑝𝑞,⊝𝑝=−.1+𝜇𝑝,𝑝⊝𝑞=𝑝⊕⊝𝑞 (2.3)

If 𝐴 is a matrix, then we let 𝐴^∗ denote its conjugate transpose.

Lemma 2.2 (see [1, 2]). If 𝐴,𝐵∈ℛ are matrix-value functions on 𝕋, then(i)𝑒₀(𝑡,𝑠)≡𝐼 and 𝑒_𝐴(𝑡,𝑡)≡𝐼,(ii)𝑒_𝐴(𝜎(𝑡),𝑠)=(𝐼+𝜇(𝑡)𝐴(𝑡))𝑒_𝐴(𝑡,𝑠), (iii)𝑒_𝐴⁻¹(𝑡,𝑠)=𝑒^∗_⊝𝐴^∗(𝑡,𝑠), (iv)𝑒_𝐴(𝑡,𝑠)=𝑒_𝐴⁻¹(𝑠,𝑡)=𝑒^∗_⊝𝐴^∗(𝑠,𝑡), (v)𝑒_𝐴(𝑡,𝑠)𝑒_𝐴(𝑠,𝑟)=𝑒_𝐴(𝑡,𝑟), (vi)𝑒_𝐴(𝑡,𝑠)𝑒_𝐵(𝑡,𝑠)=𝑒_𝐴⊕𝐵(𝑡,𝑠) if 𝑒_𝐴(𝑡,𝑠) and 𝐵(𝑡) commute.

Lemma 2.3 (see [1, 2]). If 𝐴∈ℛ and 𝑎,𝑏,𝑐∈𝕋, then [𝑒_𝐴(𝑐,⋅)]^Δ=−[𝑒_𝐴(𝑐,⋅)]^𝜎𝐴 and ^𝑏_𝑎𝑒_𝐴(𝑐,𝜎(𝑡))𝐴(𝑡)Δ𝑡=𝑒_𝐴(𝑐,𝑎)−𝑒_𝐴(𝑐,𝑏). (2.4)

Lemma 2.4 (see [1, 2]). Let 𝐴∈ℛ be an 𝑛×𝑛-matrix-valued function on 𝕋 and suppose that 𝑓∶𝕋→ℝ^𝑛 is 𝑟𝑑-contibuous. Let 𝑡₀∈𝕋 and 𝑦₀∈ℝ^𝑛. Then the initial value problem 𝑦^Δ=−𝐴^∗(𝑡)𝑦^𝜎+𝑓(𝑡),𝑦𝑡₀=𝑦₀ (2.5) has a unique solution 𝑦∶𝕋→ℝ^𝑛. Moreover, this solution is given by 𝑦(𝑡)=𝑒_⊝𝐴^∗(𝑡,𝑡₀)𝑦₀+∫^𝑡_𝑡0𝑒_⊝𝐴^∗(𝑡,𝜏)𝑓(𝜏)Δ𝜏.

For convenience, 𝔼^𝑛 denotes ℝ^𝑛 or ℂ^𝑛, 𝐴𝑃(𝕋) denote the set of all almost periodic 𝑛×𝑚-matrix-valued functions on 𝕋.

Definition 2.5 (see [22]). A time scale 𝕋 is called an almost periodic time scale if Π∶={𝜏∈ℝ∶𝑡±𝜏∈𝕋,∀𝑡∈𝕋}≠{0}. (2.6)

Definition 2.6 (see [22]). Let 𝕋 be an almost periodic time scale. A continuous 𝑛×𝑚-matrix-valued function 𝑓 on 𝕋 is called almost periodic on 𝕋 if the 𝜀-translation set of 𝑓𝐸𝜀,𝑓=𝜏∈Π∶𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀,∀𝑡∈𝕋 (2.7) is relatively dense set in 𝕋 for all 𝜀>0; that is, for any given 𝜀>0, there exists a constant 𝑙(𝜀)>0 such that each interval of length 𝑙(𝜀) contains a 𝜏(𝜀)∈𝐸{𝜀,𝑓} such that 𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀,∀𝑡∈𝕋. (2.8)𝜏 is called the 𝜀-translation number of 𝑓 and 𝑙(𝜀) is called the inclusion length of 𝐸{𝜀,𝑓}, where ‖⋅‖ is a matrix norm on 𝕋, (say, e.g., if 𝐴=(𝑎_𝑖𝑗(𝑡))_𝑛×𝑚, then we can take e.g., ‖𝐴‖=sup_𝑡∈𝕋(∑^𝑛_𝑖=1∑^𝑚_𝑗=1|𝑎_𝑖𝑗(𝑡)|²)^1/2).

For convenience, let 𝛼={𝛼_𝑛}, 𝛽={𝛽_𝑛} be two sequences. Then 𝛽⊂𝛼 means that 𝛽 is a subsequence of 𝛼; 𝛼+𝛽={𝛼_𝑛+𝛽_𝑛}; −𝛼={−𝛼_𝑛}; and 𝛼 and 𝛽 are common subsequences of 𝛼^′ and 𝛽^′, respectively, means that 𝛼_𝑛=𝛼^′_𝑛(𝑘) and 𝛽_𝑛=𝛽^′_𝑛(𝑘) for some given function 𝑛(𝑘).

Let 𝑓, 𝑔 be 𝑛×𝑚-matrix-valued functions on 𝕋, we will introduce the translation operator 𝑇, 𝑇_𝛼𝑓=𝑔 means that 𝑔(𝑡)=lim_𝑛→+∞𝑓(𝑡+𝛼_𝑛) and is written only when the limit exists.

Definition 2.7. Let 𝑓 be an 𝑛×𝑚-matrix-valued function on 𝕋, 𝐻(𝑓)={𝑔: there exits 𝛼⊂Π such that 𝑇_𝛼𝑓=𝑔 exists uniformly on 𝕋} is called the hull of 𝑓.

Similar to the proof of Theorem  3.14 in [22], one can easily get a more general version of the following.

Theorem 2.8. Let 𝑓 be an 𝑛×𝑚-matrix-valued function on 𝕋, if for any 𝛼^′⊂Π, there exists 𝛼⊂𝛼^′ such that 𝑇_𝛼𝑓(𝑡) exists uniformly on 𝕋, then 𝑓(𝑡) is almost periodic.

Also, from Theorem  3.30 in [22], one can easily show a more general version of the following.

Lemma 2.9. An 𝑛×𝑚-matrix-valued function 𝑓 is almost periodic on 𝕋, if and only if for every pair of sequences 𝛼^′,𝛽^′⊆Π, there exist common subsequences 𝛼⊂𝛼^′,𝛽⊂𝛽^′ such that 𝑇_𝛼+𝛽𝑓(𝑡)=𝑇_𝛼𝑇_𝛽𝑓(𝑡). (2.9)

3. Almost Periodic Dynamic Equations on Time Scales

Consider the linear almost periodic equation 𝑥^Δ=−𝐴^∗(𝑡)𝑥^𝜎+𝑓(𝑡) (3.1) and its associated homogeneous equation 𝑥^Δ=−𝐴^∗(𝑡)𝑥^𝜎, (3.2) where 𝐴(𝑡) is an almost periodic matrix function and 𝑓(𝑡) is an almost periodic vector function.

Definition 3.1. If 𝐵∈𝐻(𝐴^∗), we say that 𝑦^Δ=−𝐵(𝑡)𝑦^𝜎 (3.3) is a homogeneous equation in the hull of (3.1).

Definition 3.2. If 𝐵∈𝐻(𝐴^∗), and 𝑔∈𝐻(𝑓), we say that 𝑦^Δ=−𝐵(𝑡)𝑦^𝜎+𝑔(𝑡) (3.4) is an equation in hull of (3.1).

Definition 3.3. Let 𝐴(𝑡) be 𝑛×𝑛𝑟𝑑-continuous matrix function on 𝕋, the linear equation 𝑥^Δ=−𝐴^∗(𝑡)𝑥^𝜎 (3.5) 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 (3.5), satisfying 𝑋(𝑡)𝑃𝑋⁻¹(𝑠)≤𝐾𝑒_⊝𝛼((𝑠(𝑠),𝑋(𝑡)𝑋𝑡,𝜎)),𝑠,𝑡∈𝕋,𝑡≥𝜎)(𝐼−𝑃⁻¹(𝑠)≤𝐾𝑒_⊝𝛼(𝜎(𝑠),𝑡),𝑠,𝑡∈𝕋,𝑡≤𝜎(𝑠). (3.6)

From Theorem 2.8 and Lemma 2.9 for 𝛽=−𝛼 and Lemma 2.4, one can also easily get the following Favard theorem.

Lemma 3.4. If 𝐴(𝑡) is almost periodic matrix function and 𝑥(𝑡) is an almost periodic solution to the homogeneous linear almost periodic equation 𝑥^Δ=−𝐴^∗(𝑡)𝑥^𝜎, then inf_𝑡∈𝕋|𝑥(𝑡)|>0 or 𝑥(𝑡)≡0.

Similar to the proof of Lemma  4.16 in [22], one can easily show the following lemma.

Lemma 3.5. Suppose that 𝐴(𝑡) is an almost periodic matrix function and (3.2) satisfies an exponential dichotomy (3.6), then for every 𝐵(𝑡)∈𝐻(𝐴^∗), (3.3) satisfies an exponential dichotomy with same projection 𝑃 and same constants 𝐾,𝛼.

Similar to the proof of Lemma  4.17 in [22], one can easily show the following:

Lemma 3.6. If the homogeneous equation (3.2) satisfies an exponential dichotomy (3.6), then (3.2) has only one bounded solution 𝑥(𝑡)≡0.

Lemma 3.7. If the homogeneous equation (3.2) satisfies an exponential dichotomy (3.6), then all equations in the hull of (3.2) have only one bounded solution 𝑥(𝑡)≡0.

Proof. By Lemma 3.5, all equations in the hull of (3.2) satisfy an exponential dichotomy (3.6), according to Lemma 3.6, all equations in the hull of (3.2) have only one bounded solution 𝑥(𝑡)≡0. This completes the proof.

By Lemmas 3.4 and 3.7, one can easily have the existence and uniqueness theorem for an almost periodic solution to (3.1):

Theorem 3.8. Let 𝐴(𝑡) be an almost periodic matrix function, 𝑓(𝑡) is an almost periodic vector function. If (3.2) admits an exponential dichotomy, then (3.1) has a unique almost periodic solution 𝑥(𝑡)=^𝑡_−∞𝑋(𝑡)𝑃𝑋⁻¹(𝑠)𝑓(𝑠)Δ𝑠−_𝑡^+∞𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝑠)𝑓(𝑠)Δ𝑠, (3.7) where 𝑋(𝑡) is the fundamental solution matrix of (3.2).

Similar to the proof of Lemma  2.15 in [21], one can easily show the following.

Lemma 3.9. Let 𝑐_𝑖(𝑡) be an almost periodic function on 𝕋, where 𝑐_𝑖(𝑡)>0, −𝑐_𝑖(𝑡)∈ℛ⁺, 𝑡∈𝕋, 𝑖=1,2,…,𝑛 and min_{1≤𝑖≤𝑛}{inf_𝑡∈𝕋𝑐_𝑖(𝑡)}=𝑚>0, then the linear equation 𝑥^Δ=diag−𝑐₁(𝑡),−𝑐₂(𝑡),…,−𝑐_𝑛(𝑡)𝑥^𝜎 (3.8) admits an exponential dichotomy on 𝕋.

4. Applications

In the real world, both continuous and discrete systems are very important in implementation and applications. Therefore, it is meaningful to study almost periodic problems on time scales which can unify the continuous and discrete situations.

In this section, we consider the following model for the delayed Hopfield neutral networks (HNNs): 𝑥^Δ_𝑖=−𝑐_𝑖(𝑡)𝑥^𝜎_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛, (4.1) in which 𝑛 is the number of units in a neural network, 𝑥_𝑖(𝑡) is the state vector of the 𝑖th unit at the time 𝑡, 𝑐_𝑖(𝑡) represents the rate at time 𝑡 with which the 𝑖th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, 𝑔_𝑗(𝑥_𝑗) denotes the conversion of the membrane potential of the 𝑗th unit into its firing rate, 𝑏_𝑖𝑗(𝑡) denotes the strength of the 𝑗th unit on the ith unit at time 𝑡−𝜏_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡)≥0 corresponds to the transmission delay of the ith unit along the axon of the 𝑗th unit at time 𝑡, and 𝐼_𝑖(𝑡) denotes the external bias on the 𝑖th unit at time 𝑡.

It is well known that the HNNs have been successfully applied to signal and image processing, pattern recognition, and optimization. Hence, they have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of periodic solutions of system (4.1) in the literature. We refer the reader to [23–30] and the references cited therein. In order to unify continuous and discrete situations, by using results in Sections 3 and 4, one can discuss almost periodic problems on time scales.

The main purpose of this section is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (4.1). By applying fixed point theorem and differential inequality techniques, we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential stability of the almost periodic solution on time scales. Our results are new even if the time scale 𝕋=ℎℤ and other types of almost periodic time scales such as ∪^∞_𝑘=0[2𝑘,2𝑘+1], 𝑘∈ℤ. Numerical examples and simulations are given to illustrate our feasible results and effectiveness of our methods.

For (4.1), we assume that 𝑡−𝜏_𝑖𝑗(𝑡)∈𝕋 for 𝑡∈𝕋 and 𝑐_𝑖>0,𝐼_𝑖,𝑏_𝑖𝑗,𝜏_𝑖𝑗∶𝕋→ℝ are almost periodic functions, where 𝑐_𝑖 and 𝜏≥0 are constants, 𝑖,𝑗=1,2,…,𝑛, and we use the following notation: inf_𝑡∈𝕋𝑐_𝑖(𝑡)∶=𝑐_𝑖,sup_𝑡∈𝕋𝑏_𝑖𝑗(𝑡)∶=𝑏_𝑖𝑗,sup_𝑡∈𝕋𝐼_𝑖(𝑡)∶=𝐼_𝑖,𝜏=∶max_{1≤𝑖,𝑗≤𝑛}sup_𝑡∈𝕋𝜏_𝑖𝑗(𝑡),𝑖,𝑗=1,2,…,𝑛. (4.2)

We also assume that the following condition (𝐻₀) holds.

(𝐻₀) for each 𝑗∈{1,2,…,𝑛}, 𝑔_𝑗∶ℝ→ℝ is Lipschitz with Lipschitz constant 𝐿_𝑗, that is, 𝑔_𝑗𝑢_𝑗−𝑔_𝑗𝑣_𝑗≤𝐿_𝑗𝑢_𝑗−𝑣_𝑗,∀𝑢_𝑗,𝑣_𝑗∈ℝ. (4.3)

For convenience, we will use 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈ℝ^𝑛 to denote a column vector, in which the symbol (⋅)^𝑇 denotes the transpose of a vector. We let |𝑥| denote the absolute-value vector given by |𝑥|=(|𝑥₁|,|𝑥₂|,…,|𝑥_𝑛|)^𝑇, and define ‖𝑥‖=max_{1≤𝑖≤𝑛}|𝑥_𝑖|. For matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛,𝐴^𝑇 denotes the transpose of 𝐴, 𝐴⁻¹ denotes the inverse of 𝐴, |𝐴| denotes the absolute-value matrix given by |𝐴|=(|𝑎_𝑖𝑗|)_𝑛×𝑛, and 𝜚(𝐴) denotes the spectral radius of 𝐴. A matrix or vector 𝐴≥0 means that all entries of 𝐴 are greater than or equal to zero. 𝐴>0 is defined similarly. For matrices or vectors 𝐴 and 𝐵, 𝐴≥𝐵 (resp. 𝐴>𝐵) means that 𝐴−𝐵≥0 (resp. 𝐴−𝐵>0). Let 𝐷=diag𝑐₁,𝑐₂,…,𝑐_𝑛,𝐸=𝑏_{𝑖𝑗𝑛×𝑛},𝐿=diag𝐿₁,𝐿₂,…,𝐿_𝑛. (4.4)

Definition 4.1 (see [31]). For 𝐴∈𝐶_𝑟𝑑(𝕋,ℝ^𝑛), 𝑡∈𝕋 and 𝐹∈𝐶_𝑟𝑑(𝕋×ℝ^𝑛,ℝ^𝑛), 𝑉∈𝐶_𝑟𝑑(𝕋×ℝ^𝑛,ℝ⁺), we call Δ^𝑟𝑉(𝑡,𝐴(𝑡)) and Δ_𝑟𝑉(𝑡,𝐴(𝑡)) the right upper and right lower derivatives of the function 𝑉 at (𝑡,𝐴(𝑡)), respectively, if Δ^𝑟𝑉((𝑡=𝑉(𝜎(𝑡)(𝜎(𝑡((𝑡𝑡,𝐴)),𝐴)))−𝑉𝑡,𝐴))𝜇(𝑡),𝜎(𝑡)>𝑡,limsup_𝑠→𝑡+𝑉(𝑠,𝐴(𝑡)+(𝑠−𝑡)𝐹(𝑡,𝐴(𝑡)))−𝑉(𝑡,𝐴(𝑡))(𝑡)Δ𝑠−𝑡,𝜎=𝑡,_𝑟𝑉((𝑡=𝑉(𝜎(𝑡)(𝜎(𝑡((𝑡𝑡,𝐴)),𝐴)))−𝑉𝑡,𝐴))𝜇(𝑡),𝜎(𝑡)>𝑡,liminf_𝑠→𝑡+𝑉(𝑠,𝐴(𝑡)+(𝑠−𝑡)𝐹(𝑡,𝐴(𝑡)))−𝑉(𝑡,𝐴(𝑡))(𝑡)𝑠−𝑡,𝜎=𝑡. (4.5)

Lemma 4.2. Let 𝑓∈𝐶(𝕋,ℝ) is Δ-differentiable at 𝑡. Then Δ^𝑟𝑓(𝑡)≤sign𝑓^𝜎(𝑡)𝑓^Δ(𝑡),𝑤ℎ𝑒𝑟𝑒𝑓^𝜎(𝑡)=𝑓(𝜎(𝑡)). (4.6)

Proof. Case (i). If 𝑡 is a right dense point, that is, 𝜎(𝑡)=𝑡. Δ^𝑟𝑓(𝑡)≤sign𝑓(𝑡)𝑓^Δ(𝑡)=sign𝑓^𝜎(𝑡)𝑓^Δ(𝑡). (4.7)
Case (ii). If 𝑡 is a right scattered point, that is, 𝜎(𝑡)>𝑡. If 𝑓(𝑡)𝑓^𝜎(𝑡)>0, one can easily have sign(𝑓(𝑡))=sign(𝑓^𝜎(𝑡)), so we can obtain Δ^𝑟𝑓(𝑡=𝑓)^𝜎(𝑡)−𝑓(𝑡)𝜇(𝑡)=sign𝑓^𝜎(𝑡)𝑓^𝜎(𝑡)−sign𝑓(𝑡)𝑓(𝑡)𝜇(𝑡)=sign𝑓^𝜎(𝑡)𝑓^𝜎(𝑡)−𝑓(𝑡)𝜇(𝑡)=sign𝑓^𝜎(𝑡)𝑓^Δ(𝑡). (4.8)
If 𝑓(𝑡)𝑓^𝜎(𝑡)≤0, then one can get |𝑓(𝑡)|≥sign(𝑓^𝜎(𝑡))𝑓(𝑡). Then Δ^𝑟𝑓(𝑡=𝑓)^𝜎(𝑡)−𝑓(𝑡)𝜇(𝑡)=sign𝑓^𝜎(𝑡)𝑓^𝜎(𝑡)−𝑓(𝑡)𝜇(𝑡)≤sign𝑓^𝜎(𝑡)𝑓^𝜎(𝑡)−sign𝑓^𝜎(𝑡)𝑓(𝑡)𝜇(𝑡)=sign𝑓^𝜎(𝑡)𝑓^𝜎(𝑡)−𝑓(𝑡)𝜇(𝑡)=sign𝑓^𝜎(𝑡)𝑓^Δ(𝑡). (4.9)
Therefore, by (4.7), (4.8), (4.9), one can get Δ^𝑟𝑓(𝑡)≤sign𝑓^𝜎(𝑡)𝑓^Δ(𝑡). (4.10)
This completes the proof.

As usual, we introduce the phase space 𝐶([𝑡₀−𝜏,𝑡₀]∩𝕋,ℝ^𝑛) as a Banach space of continuous mappings from [𝑡₀−𝜏,𝑡₀]∩𝕋 to ℝ^𝑛 equipped with the supremum norm defined by ‖𝜑‖=max_{1≤𝑖≤𝑛}sup_{𝑡∈[𝑡0−𝜏,𝑡0]∩𝕋}|𝜑_𝑖(𝑡)| for all 𝜑=(𝜑₁(𝑡),𝜑₂(𝑡),…,𝜑_𝑛(𝑡))^𝑇∈𝐶([𝑡₀−𝜏,𝑡₀]∩𝕋,ℝ^𝑛).

The initial conditions associated with system (4.1) are of the following form: 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑠∈𝑡₀−𝜏,𝑡₀∩𝕋,𝑖=1,2,…,𝑛, (4.11) where 𝜑=(𝜑₁(𝑡),𝜑₂(𝑡),…,𝜑_𝑛(𝑡))^𝑇∈𝐶([𝑡₀−𝜏,𝑡₀]∩𝕋,ℝ^𝑛),  𝑡₀∈𝕋.

Definition 4.3. The almost periodic solution 𝑧^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇 of (4.1) is said to be globally exponentially stable, if there exists a positive 𝛼 such that for any 𝛿∈[𝑡₀−𝜏₀,𝑡₀]∩𝕋, there exists 𝑁=𝑁(𝛿)≥1 such that for any solution 𝑧=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 of (4.1), it is valid that 𝑧−𝑧^∗≤𝑁𝑧₀−𝑧^∗𝑒_⊝𝛼(𝑡,𝛿),𝑡∈𝑡₀,+∞∩𝕋, (4.12) where 𝑧₀(𝑠),𝑠∈[𝑡₀−𝜏₀,𝑡₀]∩𝕋 is the initial condition.

Definition 4.4 (see [30]). A real 𝑛×𝑛 matrix 𝐾=(𝑘_𝑖𝑗)_𝑛×𝑛 is said to be an 𝑀-matrix if 𝑘_𝑖𝑗≤0,𝑖,𝑗=1,2,…,𝑛, 𝑖≠𝑗, and 𝐾⁻¹≥0.

Lemma 4.5 (see [30]). Let 𝐴≥0 be an 𝑛×𝑛 matrix and 𝜚(𝐴)<1, then (𝐸_𝑛−𝐴)⁻¹≥0, where 𝐸_𝑛 denotes the identity matrix of size 𝑛.

In the following, we will show the existence and uniqueness of almost periodic solution to (4.1) on time scales.

Theorem 4.6. Let (𝐻₀) hold and 𝜚(𝐷⁻¹𝐸𝐿)<1. Then, there exists exactly one almost periodic solution of system (4.1).

Proof. Let 𝑋={𝜙∣𝜙=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡))^𝑇, where 𝜙_𝑖∶𝕋→ℝ is an almost periodic function, 𝑖=1,2,…,𝑛}. Then 𝑋 is a Banach space with the norm defined by ‖𝜙‖_𝑋=sup_𝑡∈𝕋max_{1≤𝑖≤𝑛}|𝜙_𝑖(𝑡)|.
To proceed further, we need to introduce an auxiliary equation 𝑥^Δ_𝑖=−𝑐_𝑖(𝑡)𝑥^𝜎_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝜙_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛, (4.13) where 𝜙(𝑡)=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡))^𝑇∈𝑋. Notice that 𝜏_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡) and 𝐼_𝑖(𝑡) are almost periodic functions, by Lemma 3.9 and Theorem 3.8, we know that the auxiliary (4.13) has exactly one almost periodic solution 𝑥^𝜙(𝑡)=𝑥^𝜙₁(𝑡),𝑥^𝜙₂(𝑡),…,𝑥^𝜙_𝑛(𝑡)^𝑇=^𝑡_−∞𝑒_−𝑐1(𝑡,𝑠)^𝑛_𝑗=1𝑏_1𝑗(𝑠)𝑔_𝑗𝜙_𝑗𝑡−𝜏_1𝑗(𝑠)+𝐼₁(𝑠)Δ𝑠,…,_𝑡−∞𝑒_−𝑐𝑛(𝑡,𝑠)^𝑛_𝑗=1𝑏_𝑛𝑗(𝑠)𝑔_𝑗𝜙_𝑗𝑡−𝜏_𝑛𝑗(𝑠)+𝐼_𝑛(𝑠)Δ𝑠^𝑇. (4.14)
Define a mapping Φ∶𝑋→𝑋 by setting Φ(𝜙(𝑡))=𝑥^𝜙(𝑡)forall𝑋. Let 𝜙,𝜓∈𝑋, then by (𝐻₀), we have Φ𝜙(𝑡)𝜓(𝑡)=Φ𝜙(𝑡)𝜓(𝑡)−Φ−Φ₁,Φ𝜙(𝑡)−Φ𝜓(𝑡)₂,…,Φ𝜙(𝑡)−Φ𝜓(𝑡)_𝑛^𝑇=^𝑡_−∞𝑒_−𝑐1(𝑡,𝑠)^𝑛_𝑗=1𝑏_1𝑗(𝑠)𝑔_𝑗𝜙_𝑗𝑡−𝜏_1𝑗(𝑠)−𝑔_𝑗𝜓_𝑗𝑡−𝜏_1𝑗(𝑠)Δ𝑠,…,_𝑡−∞𝑒_−𝑐𝑛(𝑡,𝑠)^𝑛_𝑗=1𝑏_𝑛𝑗(𝑠)𝑔_𝑗𝜙_𝑗𝑡−𝜏_𝑛𝑗(𝑠)−𝑔_𝑗𝜓_𝑗𝑡−𝜏_𝑛𝑗(𝑠)Δ𝑠^𝑇≤^𝑡_−∞𝑒_−𝑐1(𝑡,𝜎(𝑠))^𝑛_𝑗=1𝑏_1𝑗𝐿_𝑗𝜙_𝑗𝑠−𝜏_1𝑗(𝑠)−𝜓_𝑗𝑠−𝜏_1𝑗(𝑠)Δ𝑠,…,_𝑡−∞𝑒_−𝑐𝑛(𝑡,𝜎(𝑠))^𝑛_𝑗=1𝑏_𝑛𝑗𝐿_𝑗𝜙_𝑗𝑠−𝜏_𝑛𝑗(𝑠)−𝜓_𝑗𝑠−𝜏_𝑛𝑗(𝑠)Δ𝑠^𝑇≤^𝑛_𝑗=1𝑐₁⁻¹𝑏_1𝑗𝐿_𝑗sup_𝑡∈𝕋𝜙_𝑗(𝑡)−𝜓_𝑗(𝑡),…,^𝑛_𝑗=1𝑐_𝑛⁻¹𝑏_𝑛𝑗𝐿_𝑗sup_𝑡∈𝕋𝜙_𝑗(𝑡)−𝜓_𝑗(𝑡)^𝑇, (4.15) which implies that sup_𝑡∈𝕋Φ𝜙(𝑡)−Φ𝜓(𝑡)₁,sup_𝑡∈𝕋Φ𝜙(𝑡)−Φ𝜓(𝑡)₂,…,sup_𝑡∈𝕋Φ𝜙(𝑡)−Φ𝜓(𝑡)_𝑛^𝑇≤^𝑛_𝑗=1𝑐₁⁻¹𝑏_1𝑗𝐿_𝑗sup_𝑡∈𝕋𝜙_𝑗(𝑡)−𝜓_𝑗(𝑡),…,^𝑛_𝑗=1𝑐_𝑛⁻¹𝑏_𝑛𝑗𝐿_𝑗sup_𝑡∈𝕋𝜙_𝑗(𝑡)−𝜓_𝑗(𝑡)^𝑇≤𝐹sup_𝑡∈𝕋𝜙₁(𝑡)−𝜓₁(𝑡),…,sup_𝑡∈𝕋𝜙_𝑛(𝑡)−𝜓_𝑛(𝑡)^𝑇, (4.16) where 𝐹=𝐷⁻¹𝐸𝐿. Let 𝑚 be a positive integer. Then, from (4.16), we get sup_𝑡∈𝕋Φ^𝑚𝜙(𝑡)−Φ^𝑚𝜓(𝑡)₁,…,sup_𝑡∈𝕋Φ^𝑚𝜙(𝑡)−Φ^𝑚𝜓(𝑡)_𝑛^𝑇=sup_𝑡∈𝕋ΦΦ^𝑚−1𝜙(𝑡)−ΦΦ^𝑚−1𝜓(𝑡)₁,…,sup_𝑡∈𝕋ΦΦ^𝑚−1𝜙(𝑡)−ΦΦ^𝑚−1𝜓(𝑡)_𝑛^𝑇≤𝐹sup_𝑡∈𝕋Φ^𝑚−1𝜙(𝑡)−Φ^𝑚−1𝜓(𝑡)₁,…,sup_𝑡∈𝕋Φ^𝑚−1𝜙(𝑡)−Φ^𝑚−1𝜓(𝑡)_𝑛^𝑇⋮≤𝐹^𝑚sup_𝑡∈𝕋𝜙(𝑡)−𝜓(𝑡)₁,…,sup_𝑡∈𝕋𝜙(𝑡)−𝜓(𝑡)_𝑛^𝑇=𝐹^𝑚sup_𝑡∈𝕋𝜙₁(𝑡)−𝜓₁(𝑡),…,sup_𝑡∈𝕋𝜙_𝑛(𝑡)−𝜓_𝑛(𝑡)^𝑇. (4.17) Since 𝜚(𝐹)<1, we obtain lim_𝑚→+∞𝐹^𝑚=0, which implies that there exist a positive integer 𝑁 and a positive constant 𝑟<1 such that 𝐹^𝑁=𝐷⁻¹𝐸𝐿^𝑁=ℎ_{𝑖𝑗𝑛×𝑛},^𝑛_𝑗=1ℎ_𝑖𝑗≤𝑟,𝑖=1,2,…,𝑛. (4.18)
In view of (4.17), (4.18), we have Φ^𝑁𝜙(𝑡)−Φ^𝑁𝜓(𝑡)_𝑖≤sup_𝑡∈𝕋Φ^𝑁𝜙(𝑡)−Φ^𝑁𝜓(𝑡)_𝑖≤^𝑛_𝑗=1ℎ_𝑖𝑗sup_𝑡∈𝕋𝜙_𝑗(𝑡)−𝜓_𝑗(𝑡)≤sup_𝑡∈𝕋max_{1≤𝑗≤𝑛}𝜙_𝑗(𝑡)−𝜓_𝑗(𝑡)^𝑛_𝑗=1ℎ_𝑖𝑗≤𝑟𝜙−𝜓_𝑋 (4.19)
for all 𝑡∈𝕋, 𝑖=1,2,…,𝑛. It follows that Φ^𝑁𝜙(𝑡)−Φ^𝑁𝜓(𝑡)_𝑋=sup_𝑡∈𝕋max_{1≤𝑖≤𝑛}Φ^𝑁𝜙(𝑡)−Φ^𝑁𝜓(𝑡)_𝑖≤𝑟𝜙−𝜓_𝑋. (4.20)
This implies that the mapping Φ^𝑁∶𝑋→𝑋 is a contraction mapping.
By the fixed point theorem of Banach space, Φ possesses a unique fixed point 𝑍^∗ in 𝑋 such that Φ𝑍^∗=𝑍^∗. We know from (4.14) that 𝑍^∗ satisfies system (4.1), and therefore, it is the unique almost periodic solution of system (4.1). This completes the proof.

Next, we will establish a result for the exponential stability of the almost periodic solution of system (4.1).

Theorem 4.7. Suppose that all the conditions of Theorem 4.6 hold. Then system (4.1) has exactly one almost periodic solution 𝑍^∗(𝑡). Moreover, 𝑍^∗(𝑡) is globally exponentially stable.

Proof. Since 𝜚(𝐷⁻¹𝐸𝐿)<1, it follows from Theorem 4.6 that system (4.1) has a unique almost periodic solution 𝑍^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡))^𝑇. Let 𝑍(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 be an arbitrary solution of system (4.1) and define 𝑦(𝑡)=𝑍(𝑡)−𝑍^∗(𝑡). Then, set 𝑓_𝑗𝑡,𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)=𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑥^∗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑔_𝑗𝑥^∗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)𝑗=1,2,…,𝑛, (4.21)
we get 𝑦^Δ_𝑖=−𝑐_𝑖(𝑡)𝑦^𝜎_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑡,𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑖=1,2,…,𝑛. (4.22)
Thus, for 𝑖=1,2,…,𝑛, by Lemma 4.2, we have Δ^𝑟𝑦_𝑖(𝑡)≤sign𝑦^𝜎_𝑖𝑦^Δ_𝑖(𝑡)≤−𝑐_𝑖𝑦^𝜎_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑡,𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)≤−𝑐_𝑖𝑦^𝜎_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗sup_{[]𝑠∈𝑡−𝜏,𝑡∩𝕋}𝑦_𝑗(𝑠)=−𝑐_𝑖𝑦^𝜎_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝑦_𝑗(𝑡), (4.23) where 𝑦_𝑗(𝑡)=sup_{𝑠∈[𝑡−𝜏,𝑡]∩𝕋}|𝑦_𝑗(𝑠)|, 𝑗=1,2,…,𝑛. Again from 𝜚(𝐷⁻¹𝐸𝐿)<1, it follows from Lemma 4.5 that 𝐸_𝑛−𝐷⁻¹𝐸𝐿 is an 𝑀-matrix, we obtain that there exist a constant 𝜎₀>0 and a vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>(0,0,…,0)^𝑇 such that (𝐸_𝑛−𝐷⁻¹𝐸𝐿)𝜉>(𝜎₀,𝜎₀,…,𝜎₀). Therefore, 𝜉_𝑖−∑^𝑛_𝑗=1𝑐_𝑖⁻¹𝑏_𝑖𝑗𝐿_𝑗𝜉_𝑗>𝜎₀, 𝑖=1,2,…,𝑛, which imply that −𝑐_𝑖𝜉_𝑖+∑^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝜉_𝑗<−𝑐_𝑖𝜎₀, 𝑖=1,2,…,𝑛. Hence, we can choose a positive constant 𝛼<1 such that 𝛼𝜉_𝑖+−𝑐_𝑖𝜉_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝜉_𝑗𝑒_𝛼(𝜏,0)<0,𝑖=1,2,…,𝑛. (4.24) Choose a constant 𝛽>1 such that 𝛽𝜉_𝑖𝑒_⊝𝛼(𝑡,𝛿)>1,∀𝑡∈𝑡₀−𝜏,𝑡₀∩𝕋,𝛿∈𝑡₀−𝜏,𝑡₀∩𝕋,𝑖=1,2,…,𝑛. (4.25) For any 𝜀>0, let 𝑍_𝑖(𝑡)=𝛽𝜉_𝑖^𝑛_𝑗=1𝑦_𝑗𝑡₀+𝜀𝑒_⊝𝛼(𝑡,𝛿),𝑖=1,2,…,𝑛. (4.26) From (4.24), (4.26), noticing that (1+𝜇(𝑡)(⊝𝛼))<1, we obtain Δ_𝑟𝑍_𝑖(𝑡)=(⊝𝛼)𝛽𝜉_𝑖^𝑛_𝑗=1𝑦_𝑗𝑡₀+𝜀𝑒_⊝𝛼()>𝑡,𝛿−𝑐_𝑖𝜉_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝜉_𝑗𝑒_𝛼(𝜏,0)𝛽^𝑛_𝑗=1𝑦_𝑗𝑡₀+𝜀𝑒_⊝𝛼(𝑡,𝛿)1+𝜇(𝑡)(⊝𝛼)=−𝑐_𝑖𝛽𝜉_𝑖^𝑛_𝑗=1𝑦_𝑗𝑡₀+𝜀𝑒_⊝𝛼(𝜎(𝑡))+,𝛿_𝑛𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝜉_𝑗𝛽^𝑛_𝑗=1𝑦_𝑗𝑡₀+𝜀𝑒_⊝𝛼(𝜎(𝑡),𝛿)𝑒_𝛼(𝜏,0)>−𝑐_𝑖𝑍_𝑖(𝜎(𝑡))+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝑍_𝑗(𝑡),𝑖=1,2,…,𝑛, (4.27) where 𝑍_𝑗(𝑡)=sup_{𝑠∈[𝜎(𝑡)−𝜏,𝜎(𝑡)]∩𝕋}𝑍_𝑗(𝑠), 𝑗=1,2,…,𝑛. In view of (4.25) and (4.26), for 𝑖=1,2,…,𝑛, we have 𝑍_𝑖(𝑡)=𝛽𝜉_𝑖^𝑛_𝑗=1𝑦_𝑗𝑡₀+𝜀𝑒_⊝𝛼()>𝑡,𝛿_𝑛𝑗=1𝑦_𝑗𝑡₀+𝜀>𝑦_𝑖(𝑡),∀𝑡∈𝑡₀−𝜏,𝑡₀∩𝕋. (4.28) We claim that 𝑦_𝑖(𝑡)<𝑍_𝑖(𝑡)∀𝑡∈𝑡₀,+∞∩𝕋,𝑖=1,2,…,𝑛. (4.29) Contrarily, there must exist 𝑖∈{1,2,…,𝑛} and 𝑡_𝑖>0 such that 𝑦_𝑖𝑡_𝑖≥𝑍_𝑖𝑡_𝑖,𝑦_𝑗(𝑡)<𝑍_𝑗(𝑡),∀𝑡∈𝑡₀−𝜏,𝑡_i∩𝕋,𝑗=1,2,…,𝑛, (4.30) which implies that 𝑦_𝑖𝑡_𝑖−𝑍_𝑖𝑡_𝑖≥0,𝑦_𝑗(𝑡)−𝑍_𝑗(𝑡)<0,∀𝑡∈𝑡₀−𝜏,𝑡_i∩𝕋,𝑗=1,2,…,𝑛. (4.31)

Case I. If 𝜌(𝑡_𝑖) is right dense point, that is, 𝜎(𝜌(𝑡_𝑖))=𝜌(𝑡_𝑖)=𝑡_𝑖, we have 0≤Δ^𝑟𝑦_𝑖𝜌𝑡_𝑖−𝑍_𝑖𝜌𝑡_𝑖=limsup_ℎ→0−𝑦_𝑖𝑡_𝑖+ℎ−𝑍_𝑖𝑡_𝑖+ℎ−𝑦_𝑖𝑡_𝑖−𝑍_𝑖𝑡_𝑖ℎ≤limsup_ℎ→0−𝑦_𝑖𝑡_𝑖+ℎ−𝑦_𝑖𝑡_𝑖ℎ−liminf_ℎ→0−𝑍_𝑖𝑡_𝑖+ℎ−𝑍_𝑖𝑡_𝑖ℎ=Δ^𝑟𝑦_𝑖𝑡_𝑖−Δ_𝑟𝑍_𝑖𝑡_𝑖. (4.32)

Case II. If 𝜌(𝑡_𝑖) is right scattered point, that is, 𝜎(𝜌(𝑡_𝑖))=𝑡_𝑖>𝜌(𝑡_𝑖), we obtain 0≤Δ^𝑟𝑦_𝑖𝜌𝑡_𝑖−𝑍_𝑖𝜌𝑡_𝑖=𝑦_𝑖𝑡_𝑖−𝑦_𝑖𝜌𝑡_𝑖𝜇𝜌𝑡_𝑖−𝑍_𝑖𝑡_𝑖−𝑍_𝑖𝜌𝑡_𝑖𝜇𝜌𝑡_𝑖=Δ^𝑟𝑦_𝑖𝜌𝑡_𝑖−Δ_𝑟𝑍_𝑖𝜌𝑡_𝑖. (4.33) Thus, Δ^𝑟𝑦_𝑖𝜌𝑡_𝑖≥Δ_𝑟𝑍_𝑖𝜌𝑡_𝑖. (4.34) From (4.23), (4.27), (4.30), we can obtain Δ^𝑟𝑦_𝑖𝜌𝑡_𝑖≤−𝑐_𝑖𝑦_𝑖𝑡_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝑦_𝑗𝜌𝑡_𝑖≤−𝑐_𝑖𝑍_𝑖𝑡_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝐿_𝑗𝑦_𝑗𝜌𝑡_𝑖<Δ_𝑟𝑍_𝑖𝜌𝑡_𝑖, (4.35) which contradicts (4.34). Hence, (4.29) holds. Letting 𝜀→0⁺ and 𝑀=𝑛max_{1≤𝑖≤𝑛}{𝛽𝜉_𝑖+1}, we have from (4.26) and (4.29) that 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)=𝑦_𝑖(𝑡)≤𝛽𝜉_𝑖^𝑛_𝑗=1𝑦_𝑗𝑡₀𝑒_⊝𝛼(𝑡,𝛿)≤𝛽𝜉_𝑖𝑍−𝑍^∗𝑛𝑒_⊝𝛼(𝑡,𝛿)<max_{1≤𝑖≤𝑛}𝛽𝜉_𝑖+1𝑛𝑍−𝑍^∗𝑒_⊝𝛼(𝑡,𝛿)≤𝑀𝑍−𝑍^∗𝑒_⊝𝛼(𝑡,𝛿),𝑖=1,2,…,𝑛 (4.36) for all 𝑡∈[𝑡₀,+∞)∩𝕋. This completes the proof.

Example 4.8. Let 𝕋=ℤ, consider the following HNN: 𝑥^Δ₁=−𝑐₁(𝑡)𝑥^𝜎₁+0.04(sin𝑡)𝑔₁𝑥₁𝑡−sin²𝑡+0.016(cos𝑡)𝑔₂𝑥₂𝑡−2sin²𝑡𝑥+2cos𝑡,_Δ2=−𝑐₂(𝑡)𝑥^𝜎₂+0.02(sin2𝑡)𝑔₁𝑥₁𝑡−3cos²𝑡+0.014(cos4𝑡)𝑔₂𝑥₂𝑡−4sin²𝑡𝑥+sin𝑡,_Δ3=−𝑐₃(𝑡)𝑥^𝜎₃+0.013(sin2𝑡)𝑔₃𝑥₃𝑡−cos²𝑡+cos²𝑡, (4.37) where 𝑔₁(𝑥)=𝑔₂(𝑥)=𝑔₃(𝑥)=(1/2)(|𝑥+1|−|𝑥−1|) and 𝑐₁(𝑡)=1+0.05sin𝑡, 𝑐₂(𝑡)=1+0.03cos𝑡, 𝑐₃(𝑡)=1+0.01sin𝑡.

Note that 𝑐₁=0.05, 𝑐₂=0.07, 𝑐₃=0.09, 𝐿₁=𝐿₂=𝐿₃=1, and 𝑏₁₁=0.04, 𝑏₁₂=0.016, 𝑏₂₁=0.02, 𝑏₂₂=0.014, 𝑏₃₃=0.013, 𝑏₁₃=𝑏₂₃=𝑏₃₁=𝑏₃₂=0. Then we have 𝐷=𝑐_𝑖⁻¹𝑏_𝑖𝑗𝐿_𝑗3×3=0.80000.320000.28570.20000000.1444 (4.38) and 𝜚(𝐷)=0.9259<1. Thus, from Theorems 4.6 and 4.7, (4.37) has exactly one almost periodic solution and it is exponentially stable. We can take the initial value 𝜑₁(𝑠)=−0.3, 𝜑₃(𝑠)=−0.1, 𝜑₃(𝑠)=−0.4, 𝑠∈[−4,0]∩ℤ, we can give the following numerical simulation figures to show our results are plausible and effective on time scales (see Figures 1 and 2).

Figure 1 

Transient response of state variables 𝑥1,𝑥2, and 𝑥3 in Example 4.8.

Figure 2 

Phase response of state variables 𝑥1,𝑥2, and 𝑥3 in Example 4.8.

Example 4.9. Let 𝕋=ℝ, consider the following HNN: 𝑥^Δ₁=−𝑐₁(𝑡)𝑥^𝜎₁+12(sin𝑡)𝑔₁𝑥₁𝑡−sin²𝑡+1()𝑔18cos𝑡₂𝑥₂𝑡−2sin²𝑡𝑥+2cos𝑡,_Δ2=−𝑐₂(𝑡)𝑥^𝜎₂+2(sin2𝑡)𝑔₁𝑥₁𝑡−3cos²𝑡+12(cos4𝑡)𝑔₂𝑥₂𝑡−4sin²𝑡𝑥+sin𝑡,_Δ3=−𝑐₃(𝑡)𝑥^𝜎₃+67(sin2𝑡)𝑔₃𝑥₃𝑡−cos²𝑡+cos²𝑡, (4.39) where 𝑔₁(𝑥)=𝑔₂(𝑥)=𝑔₃(𝑥)=(1/2)(|𝑥+1|−|𝑥−1|) and 𝑐₁(𝑡)=2+sin𝑡, 𝑐₂(𝑡)=2+cos𝑡, 𝑐₃(𝑡)=2+sin𝑡.
Note that 𝑐₁=𝑐₂=𝑐₃=𝐿₁=𝐿₂=𝐿₃=1, and 𝑏₁₁=1/2, 𝑏₁₂=1/18, 𝑏₂₁=2, 𝑏₂₂=1/2, 𝑏₃₃=6/7, 𝑏₁₃=𝑏₂₃=𝑏₃₁=𝑏₃₂=0. Then we have 𝐷=𝑐_𝑖⁻¹𝑏_𝑖𝑗𝐿_𝑗3×3=12102118206007 (4.40) and 𝜌(𝐷)=0.8571<1. Thus, from Theorems 4.6 and 4.7, (4.39) has exactly one almost periodic solution and it is exponentially stable. We can take the initial value 𝜑₁(𝑠)=−1.3, 𝜑₃(𝑠)=−0.9, 𝜑₃(𝑠)=−0.7, 𝑠∈[−4,0], we can give the following numerical simulation figures to show our results are plausible and effective on time scales (see Figures 3 and 4).

Figure 3 

Transient response of state variables 𝑥1,𝑥2 and 𝑥3 in Example 4.9.

Figure 4 

Phase response of state variables 𝑥1,𝑥2 and 𝑥3 in Example 4.9.

5. Conclusion

The existence and uniqueness of almost periodic solution and its expression form to a class of linear dynamic equations on time scales are obtained. As an application, sufficient conditions for the existence and exponential stability of almost periodic solution to a class of Hopfield neural networks with delays are established. To the best of our knowledge, the results presented here have not appeared in the related literature. In fact, both continuous and discrete systems are very important in implementation 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 timescales which can unify the continuous and discrete situations. Also, the results and methods used in this paper can be used to study many other types neural networks and population models.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of China under no Grant 10971183 and this work was also supported by IRTSTYN.