Permanence and Almost Periodic Solution for an Enterprise Cluster Model Based on Ecology Theory with Feedback Controls on Time Scales

Abstract

We present a model with feedback controls based on ecology theory, which effectively describes the competition and cooperation of enterprise cluster in real economic environments. Applying the comparison theorem of dynamic equations on time scales and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence and the existence of uniformly asymptotically stable almost periodic solution of the system are obtained.

1. Introduction

In recent years, a few researchers have presented some models about enterprise clusters based on ecology theory, which arouse growing interest in applying the methods of ecology and dynamic system theory to study enterprise clusters, for example [1–9] and references cited therein. In [1], two models from biology were given and explained by economic view, and sufficient conditions were obtained to guarantee the coexistence and stability of enterprise clusters. In [3], the developing strategy of enterprise clusters was analyzed based on the logistic model, and the suggestions of constructing cooperative relation and choosing generalization or specialization tactics for commodity were put forward. In addition, based on the theoretical model of ecological population science, Wang and Pan [6] made a detailed analysis to the equilibrium mechanism of enterprise clusters, including net model and center halfback model and drew a conclusion that the relationship of pierce competition and beneficial cooperation among enterprise clusters was the crucial factor for them to keep stability. More related research about enterprises cluster one can refer to the literatures [10–13]. Recently, the literature [5] considered the competition and cooperation system of two enterprises based on ecosystem: d𝑥₁(𝑡)d𝑡=𝑟₁𝑥₁(𝑡)𝑥1−¹(𝑡)𝐾−𝛼𝑥₂(𝑡)−𝑐₂²𝐾,d𝑥₂(𝑡)d𝑡=𝑟₂𝑥₂(𝑡)𝑥1−²(𝑡)𝐾−𝛼𝑥₁(𝑡)−𝑐₁²𝐾, (1) where 𝑥₁(𝑡), 𝑥₂(𝑡) represent the output of enterprises 𝐴 and 𝐵, 𝑟₁, 𝑟₂ are the intrinsic growth rate, 𝐾 denotes the carrying capacity of market under nature unlimited conditions, 𝛼, 𝛽 are the competitive coefficients of two enterprises, 𝑐₁, 𝑐₂ are the initial production of two enterprises. Accordingly, we consider now the equation with nonconstant coefficients, which can be obtained as a modified system (1) with variable coefficients (Letting 𝑎₁=𝑟₁/𝐾, 𝑎₂=𝑟₂/𝐾, 𝑏₁=𝑟₁𝛼/𝐾, 𝑏₂=𝑟₂𝛽/𝐾 in system (1)): d𝑥₁(𝑡)d𝑡=𝑥₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)−𝑏₁(𝑡)𝑥₂(𝑡)−𝑐₂(𝑡)²,d𝑥₂(𝑡)d𝑡=𝑥₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)−𝑏₁(𝑡)𝑥₁(𝑡)−𝑐₁(𝑡)². (2)

In real world, the situation of enterprises is often distributed by unpredictable forces which can result in changes in enterprises' parameters such as intrinsic growth rates. So it is necessary to study models with control variables which are so-called disturbance functions [14–17].

As well known, both continuous and discrete systems are very important in implementation and applications, but it is troublesome to study the permanence and the existence of almost periodic solutions for continuous system and discrete system, respectively. It is very important to study that on time scales which can unify the continuous and discrete situations.

Motivated by the above statements, we consider the following competitive and cooperation model of a satellite enterprises and a dominant enterprise with feedback controls on time scale 𝕋: 𝑥^Δ(𝑡)=𝑟₁(𝑡)−𝑎₁(𝑡)exp{𝑥(𝑡)}−𝑏₁(𝑡)exp𝑦(𝑡)−𝑐₂(𝑡)²−𝑒₁(𝑡)𝑢(𝑡),𝑢^Δ(𝑡)=−𝛼₁(𝑡)𝑢(𝑡)+𝛽₁(𝑡){𝑥(𝑡,𝑦exp)}_Δ(𝑡)=𝑟₂(𝑡)−𝑎₂(𝑡)exp𝑦(𝑡)+𝑏₂(𝑡)exp{𝑥(𝑡)}−𝑐₁(𝑡)²−𝑒₂(𝑡)𝑣(𝑡),𝑣^Δ(𝑡)=−𝛼₂(𝑡)𝑣(𝑡)+𝛽₂(𝑡)exp𝑦(𝑡). (3)

As pointed in [18, 19], periodic phenomenon and almost periodic phenomenon are widespread in nature and almost periodic phenomena is more frequent than periodic phenomenon. Hence, they have been the object of intensive analysis by numerous authors. In particular, there have been extensive results on existence of almost periodic solutions of differential equations in the literature. Some of these results can be found in [20–24]. Our main purpose of this paper is by using the comparison theorem of dynamic equations on time scales and constructing a suitable Lyapunov functional to study the permanence and the existence of almost periodic solutions of (3).

Remark 1. Let 𝑁₁(𝑡)=exp{𝑥(𝑡)}, 𝑁₂(𝑡)=exp{𝑦(𝑡)}; if 𝕋=ℝ (the set of all real numbers), then (3) reduces to d𝑁₁(𝑡)d𝑡=𝑁₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑁₁(𝑡)−𝑏₁(𝑡)𝑁₂(𝑡)−𝑐₂(𝑡)²−𝑒₁(𝑡)𝑢(𝑡),d𝑁₂(𝑡)d𝑡=𝑁₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑁₂(𝑡)−𝑏₁(𝑡)𝑁₁(𝑡)−𝑐₁(𝑡)²−𝑒₂(𝑡)𝑣(𝑡),d𝑢(𝑡)d𝑡=−𝛼₁(𝑡)𝑢(𝑡)+𝛽₁(𝑡)𝑁₁(𝑡),d𝑣(𝑡)d𝑡=−𝛼₂(𝑡)𝑣(𝑡)+𝛽₂(𝑡)𝑁₂(𝑡). (4) If 𝕋=ℤ (the set of all integers), then (3) reduces to 𝑁₁(𝑡+1)=𝑁₁(𝑡)exp𝑟₁(𝑡)−𝑎₁(𝑡)𝑁₁(𝑡)−𝑏₁(𝑡)𝑁₂(𝑡)−𝑐₂(𝑡)²−𝑒₁(𝑡)𝑢(𝑡,𝑁₂)(𝑡+1)=𝑁₂(𝑡)exp𝑟₂(𝑡)−𝑎₂(𝑡)𝑁₂(𝑡)+𝑏₂(𝑡)𝑁₁(𝑡)−𝑐₁(𝑡)²−𝑒₂(𝑡)𝑣(𝑡),Δ𝑢(𝑡)=−𝛼₁(𝑡)𝑢(𝑡)+𝛽₁(𝑡)𝑁₁(𝑡),Δ𝑣(𝑡)=−𝛼₂(𝑡)𝑣(𝑡)+𝛽₂(𝑡)𝑁₂(𝑡). (5)

Therefore, we only consider the permanence and existence of nonnegative almost periodic solution of system (3), then the permanence and almost periodic solution of (4) and (5) can be obtained as direct results of (3).

Let 𝕋 and 𝕋⁺ denote the nonempty closed subset (time scales) of ℝ. Let 𝑓 be a continuous bounded function defined on 𝕋, and we denote 𝑓^𝑀=sup_𝑡∈𝕋𝑓(𝑡),𝑓^𝑚=inf_𝑡∈𝕋𝑓(𝑡). (6) Throughout this paper, we assume that (H₁)  𝑟_𝑖(𝑡), 𝑎_𝑖(𝑡), 𝑏_𝑖(𝑡), 𝑐_𝑖(𝑡), 𝑒_𝑖(𝑡), 𝛼_𝑖(𝑡), and 𝛽_𝑖(𝑡) are almost periodic functions on time scale 𝕋 such that 0<𝑟^𝑚_𝑖≤𝑟_𝑖(𝑡)≤𝑟^𝑀_𝑖,0<𝑎^𝑚_𝑖≤𝑎_𝑖(𝑡)≤𝑎^𝑀_𝑖,0<𝑏^𝑚_𝑖≤𝑏_𝑖(𝑡)≤𝑏^𝑀_𝑖,0<𝑐^𝑚_𝑖≤𝑐_𝑖(𝑡)≤𝑐^𝑀_𝑖,0<𝑒^𝑚_𝑖≤𝑒_𝑖(𝑡)≤𝑒^𝑀_𝑖,0<𝛼^𝑚_𝑖≤𝛼_𝑖(𝑡)≤𝛼^𝑀_𝑖,0<𝛽^𝑚_𝑖≤𝛽_𝑖(𝑡)≤𝛽^𝑀_𝑖,−𝑎^𝑚_𝑖∈ℛ⁺,−𝛼^𝑚_𝑖∈ℛ⁺,𝑖=1,2, (7) where ℛ⁺ is the set of positively regressive functions from 𝕋 to ℝ.

2. Preliminaries

In this section, we will recall some definitions and lemmas which will be used in the proof of our main results.

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⁺ (the set of all nonnegative real numbers) are defined, respectively, by 𝜎(𝑡){},𝜌(𝑡){},𝜇(𝑡)(𝑡)=inf𝑠∈𝕋∶𝑠>𝑡=sup𝑠∈𝕋∶𝑠<𝑡=𝜎−𝑡. (8)

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 called rd-continuous provided it is continuous at right-dense points in 𝕋 (i.e., lim_{𝑠∈𝕋,𝑠→𝑡}𝑓(𝑠)=𝑓(𝑡)) and its left-sided 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. The set of continuous functions 𝑓∶𝕋→ℝ will be denoted by 𝐶(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 any 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| (9) 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)Δ(𝑠)=𝑌(𝑡)−𝑌(𝑎). (10)

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). If 1+𝜇(𝑡)𝑝(𝑡)>0, ∀𝑡∈𝕋, then 𝑝 is a positively regressive function from 𝕋 to ℝ. 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,𝑠,𝑡∈𝕋 (11) with the cylinder transformation 𝜉_ℎ(𝑧)=log(1+ℎ𝑧)ℎ,ifℎ≠0,𝑧,ifℎ=0. (12)

Definition 2 (see [25, 26]). Let 𝑝,𝑞∶𝕋→ℛ be two regressive functions, we define 𝑝𝑝⊕𝑞=𝑝+𝑞+𝜇𝑝𝑞,⊖𝑝=−,.1+𝜇𝑝𝑝⊖𝑞=𝑝⊕⊖𝑞 (13) Then the generalized exponential function has the following properties.

Lemma 3 (see [25, 26]). Assume that 𝑝,𝑞∶𝕋→ℝ are 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)𝑒_𝑝(𝑠,𝑟)=𝑒_𝑝(𝑡,𝑟); (vi)If 𝑎,𝑏,𝑐∈𝕋, then ∫^𝑏_𝑎𝑒_𝑝(𝑐,𝜎(𝑡))Δ(𝑡)=𝑒_𝑝(𝑐,𝑎)−𝑒_𝑝(𝑐,𝑏).

Definition 4 (see [27]). A time scale 𝕋 is called an almost periodic time scale if Π∶={𝜏∈ℝ∶𝑡+𝜏∈𝕋,∀𝑡∈𝕋}≠{0}. (14)

Throughout this paper, we restrict our discussion on almost periodic time scales.

Definition 5 (see [27]). A function 𝑓∶𝕋→ℝ is called an almost periodic function if the 𝜖-translation number set of 𝑓: 𝑇𝑓,𝜖,𝕋=𝜏∈𝕋∶𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜖,∀𝑡∈𝕋 (15) is a relatively dense set in 𝕋 for all 𝜖>0; that is, for any given 𝜖>0, there exists a number 𝑙=𝑙(𝜖)>0 such that for any interval with length 𝑙(𝜖), there is a number 𝜏=𝜏(𝜖)∈𝑇(𝑓,𝜖,𝕋)>0 in this interval such that 𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜖,𝑡∈𝕋. (16)

Definition 6 (see [27]). Let 𝑓∶𝕋×𝔻→ℝ, where 𝔻 is an open set in ℝ and 𝑓(𝑡,𝑥) is said to be almost periodic in 𝑡 uniformly for 𝑥∈𝔻 or uniformly almost periodic for short. If for any compact set 𝕊 in 𝔻, there exists a positive number 𝑙(𝜖,𝑠) such that any interval of length 𝑙(𝜖,𝑠) contains a number 𝜏 for which |𝑓(𝑡+𝜏,𝑥)−𝑓(𝑡,𝑥)|<𝜖 for all 𝑡∈𝕋 and 𝑥∈𝕊, 𝜏 is called the 𝜖-translation number of 𝑓(𝑡,𝑥).

Lemma 7 (see [27]). A 𝑓(𝑡) is an almost periodic function if and only if for any sequence {𝜏^′_𝑛}⊂𝕋 there exists a subsequence {𝜏_𝑛}⊂{𝜏^′_𝑛} such that 𝑓(𝑡+𝜏_𝑛) converges uniformly on 𝑡∈𝕋 as 𝑛→∞. Furthermore, the limit function is also an almost periodic function.

Definition 8. System (3) is said to be permanent if there exist positive constants 𝑀_𝑖, 𝑁_𝑖, 𝑚_𝑖, 𝑛_𝑖(𝑖=1,2,3,4) which are independent of the solutions of the system, such that any solution (𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇 of system (3) satisfies 𝑚₁≤liminf_𝑡→∞𝑥(𝑡)≤limsup_𝑡→∞𝑥(𝑡)≤𝑀₁,𝑚₂≤liminf_𝑡→∞𝑦(𝑡)≤limsup_𝑡→∞𝑦(𝑡)≤𝑀₂,𝑚₃≤liminf_𝑡→∞𝑢(𝑡)≤limsup_𝑡→∞𝑢(𝑡)≤𝑀₃,𝑚₄≤liminf_𝑡→∞𝑣(𝑡)≤limsup_𝑡→∞𝑣(𝑡)≤𝑀₄. (17)
Consider the following nonlinear almost periodic differential system on time scales: 𝑥^Δ=𝑓(𝑡,𝑥) (18) and the product system of (18) 𝑥^Δ=𝑓(𝑡,𝑥),𝑦^Δ=𝑓𝑡,𝑦. (19)

Lemma 9 (see [28, 29]). Suppose that there exists a Lyapunov functional 𝑉(𝑡,𝑥,𝑦) defined for 𝑉(𝑡,𝑥,𝑦)∈(𝕋⁺×𝔻×𝔻,ℝ) satisfying the following conditions: (i)𝑎(|𝑥−𝑦|)≤𝑉(𝑡,𝑥,𝑦)≤𝑏(|𝑥−𝑦|), where 𝑎,𝑏∈𝑘,𝑘={𝑎∈𝑐(ℛ⁺,ℛ⁺)∶𝑎(0)=0 and 𝑎 is increasing}; (ii)|𝑉(𝑡,𝑥₁,𝑦₁)−𝑉(𝑡,𝑥₂,𝑦₂)|≤𝐿[|𝑥₁−𝑥₂|+|𝑦₁−𝑦₂|], where 𝐿>0 is a constant; (iii)𝐷⁺𝑉^Δ₍₁₉₎(𝑡,𝑥,𝑦)≤−𝑐𝑉(𝑡,𝑥,𝑦), where −𝑐∈ℛ⁺ and 𝑐>0.
In addition, if there exists a solution 𝑥(𝑡) of system (18) such that 𝑥(𝑡)∈𝕊 for 𝑡∈𝕋⁺, where 𝕊⊂𝔻 is any compact subset of  𝔻, then there exists a unique uniformly asymptotically stable almost periodic solution 𝑝(𝑡) of system (18). In particular, if 𝑓(𝑡,𝑥) is periodic of period 𝜔, then there exists a unique uniformly asymptotically stable periodic solution of (18) of period 𝜔.

3. Persistence

We need the following lemma to obtain the permanence of system (3).

Lemma 10 (see [28, 29]). Let −𝑎∈ℛ⁺. (i) If 𝑥^Δ(𝑡)≤𝑏−𝑎𝑥(𝑡), then for 𝑡>𝑡₀𝑥(𝑡)≤𝑥𝑡₀𝑒_(−𝑎)𝑡,𝑡₀+𝑏𝑎1−𝑒_(−𝑎)𝑡,𝑡₀. (20) In particular, if 𝑎>0,  𝑏>0, we have limsup_𝑡→+∞𝑥(𝑡)≤𝑏/𝑎. (ii) If 𝑥^Δ(𝑡)≥𝑏−𝑎𝑥(𝑡), then for 𝑡>𝑡₀𝑥(𝑡)≥𝑥𝑡₀𝑒_(−𝑎)𝑡,𝑡₀+𝑏𝑎1−𝑒_(−𝑎)𝑡,𝑡₀. (21) In particular, if 𝑎>0,  𝑏>0, we have liminf_𝑡→+∞𝑥(𝑡)≥𝑏/𝑎.

Proposition 11. Assume that (H₁) holds; then every solution (𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇 of system (3) satisfies limsup_𝑡→∞𝑥(𝑡)≤𝑥^∗,limsup_𝑡→∞𝑦(𝑡)≤𝑦^∗,limsup_𝑡→∞𝑢(𝑡)≤𝑢^∗,limsup_𝑡→∞𝑣(𝑡)≤𝑣^∗, (22) where 𝑥^∗=𝑟^𝑀₁−𝑎^𝑚₁𝑎^𝑚₁,𝑦^∗=𝑟^𝑀₂−𝑎^𝑚₂+𝑏^𝑀₂𝑒^2𝑥∗𝑎^𝑚₂,𝑢^∗=𝛽^𝑀₁𝑒^𝑥∗𝛼^𝑚₁,𝑣^∗=𝛽^𝑀₂𝑒^𝑦∗𝛼^𝑚₂. (23)

Proof. Let (𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇 be any solution of system (3); it follows from the first equation of system (3) and the Bernoulli inequality 𝑒^𝑥≥1+𝑥 for 𝑥∈ℝ, that 𝑥^Δ(𝑡)≤𝑟₁(𝑡)−𝑎₁(𝑡)exp𝑥(𝑡)≤𝑟₁(𝑡)−𝑎₁(𝑡(𝑡≤𝑟)(1+𝑥))_𝑀1−𝑎^𝑚₁−𝑎^𝑚₁𝑥(𝑡). (24) By applying (i) of Lemma 10 to the differential inequality above, we have limsup_𝑡→∞𝑥(𝑡)≤𝑟^𝑀₁−𝑎^𝑚₁𝑎^𝑚₁∶=𝑥^∗. (25) For any positive constant 𝜖 small enough, it follows from (25) that there exists a large enough 𝑇₀>0 such that 𝑥(𝑡)≤𝑥^∗+𝜖 for all 𝑡>𝑇₀. Then the second equation and the third equation of system (3) lead to 𝑢^Δ(𝑡)≤−𝛼^𝑚₁𝑢(𝑡)+𝛽^𝑀₁𝑒^𝑥∗+𝜖,𝑦^Δ(𝑡)≤𝑟₂(𝑡)−𝑎₂(𝑡)exp𝑦(𝑡)+𝑏₂(𝑡)𝑒^{2(𝑥∗+𝜖)}≤𝑟₂(𝑡)−𝑎₂(𝑡)𝑦(𝑡)+1+𝑏₂(𝑡)𝑒^{2(𝑥∗+𝜖)}. (26) That is, 𝑢^Δ(𝑡)≤−𝛼^𝑚₁𝑢(𝑡)+𝛽^𝑀₁𝑒^𝑥∗+𝜖,𝑦^Δ(𝑡)≤𝑟^𝑀₂+𝑏^𝑀₂𝑒^{2(𝑥∗+𝜖)}−𝑎^𝑚₂−𝑎^𝑚₂𝑦(𝑡). (27) By applying Lemma 10, it follows from (27) that limsup_𝑡→∞𝑦(𝑡)≤𝑟^𝑀₂−𝑎^𝑚₂+𝑏^𝑀₂𝑒^{2𝑥∗+𝜖}𝑎^𝑚₂,limsup_𝑡→∞𝑢(𝑡)≤𝛽^𝑀₁𝑒^𝑥∗+𝜖𝛼^𝑚₁. (28) Letting 𝜖→0 in the above inequality leads to limsup_𝑡→∞𝑦(𝑡)≤𝑟^𝑀₂−𝑎^𝑚₂+𝑏^𝑀₂𝑒^2𝑥∗𝑎^𝑚₂∶=𝑦^∗,limsup_𝑡→∞𝑢(𝑡)≤𝛽^𝑀₁𝑒^𝑥∗𝛼^𝑚₁∶=𝑢^∗. (29) By using similar arguments as those in the proofs of (28), it follows that limsup_𝑡→∞𝑣(𝑡)≤𝛽^𝑀₂𝑒^𝑦∗𝛼^𝑚₂∶=𝑣^∗. (30) This completes the proof.

Proposition 12. Assume that (H₁) holds; suppose further that (H₂)  𝑟^𝑚₁−𝑏^𝑀₁(𝑒^𝑦∗−𝑐^𝑚₂)²−𝑒^𝑀₁𝑢^∗>0,𝑟^𝑚₂−𝑒^𝑀₂𝑣^∗>0, then every solution  (𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇  of system  (3)  satisfiesliminf_𝑡→∞𝑥(𝑡)≥𝑥_∗,liminf_𝑡→∞𝑦(𝑡)≥𝑦_∗,liminf_𝑡→∞𝑢(𝑡)≥𝑢_∗,liminf_𝑡→∞𝑣(𝑡)≥𝑣_∗, (31) where𝑥_∗𝑟=ln^𝑚1−𝑏^𝑀₁𝑒^𝑦∗−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗𝑎^𝑀₁,𝑦_∗𝑟=ln^𝑚2−𝑒^𝑀₂𝑣^∗+𝑏^𝑚₂𝑒^𝑥∗−𝑐^𝑀₁²𝑎^𝑀₂,𝑢_∗=𝛽^𝑚₁𝑒^𝑥∗𝛼^𝑀₁,𝑣_∗=𝛽^𝑚₂𝑒^𝑦∗𝛼^𝑀₂. (32)

Proof. Let (𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇 be any positive solution of system (3). We first prove that liminf_𝑡→∞𝑥(𝑡)≥𝑥_∗. (33) For any small enough constant 𝜖≥0, there exists a point 𝑇₁∈𝕋 and 𝑇₁≥𝑇₀ such that 𝑥(𝑡)≤𝑥^∗+𝜖,𝑦(𝑡)≤𝑦^∗𝑢(𝑡)+𝜖,≤𝑢^∗+𝜖,𝑣(𝑡)≤𝑣^∗+𝜖,𝑡>𝑇₁. (34) By the first equation of system (3) and condition (H₁), we have 𝑥^Δ(𝑡)≥𝑟^𝑚₁−𝑎^𝑀₁exp{𝑥(𝑡)}−𝑏^𝑀₁exp𝑦^∗+𝜖−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗+𝜖≥𝑟^𝑚₁−𝑏^𝑀₁𝑒^𝑦∗+𝜖−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗+𝜖−𝑎^𝑀₁exp{𝑥(𝑡)},𝑡>𝑇₁. (35) We assert 𝑟^𝑚₁−𝑏^𝑀₁𝑒^𝑦∗+𝜖−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗+𝜖−𝑎^𝑀₁exp{𝑥(𝑡)}≤0,𝑡>𝑇₁; (36) otherwise, we assume that there exists 𝑡̃>𝑇₁ such that 𝑟^𝑚₁−𝑏^𝑀₁𝑒^𝑦∗+𝜖−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗+𝜖−𝑎^𝑀₁𝑥𝑡̃𝑟exp>0,_𝑚1−𝑏^𝑀₁𝑒^𝑦∗+𝜖−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗+𝜖−𝑎^𝑀₁{𝑥(𝑡𝑇exp)}≤0,∀𝑡∈₁,𝑡̃, (37) then we get 𝑥𝑡̃𝑟<ln^𝑚1−𝑏^𝑀₁𝑒^𝑦∗+𝜖−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗+𝜖𝑎^𝑀₁,𝑥(𝑡)𝑟≥ln^𝑚1−𝑏^𝑀₁𝑒^𝑦∗+𝜖−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗+𝜖𝑎^𝑀₁,∀𝑡∈𝑇₁,𝑡̃. (38) Equation (38) implies 𝑥^Δ(𝑡)̃≤0, which is a contradiction; therefore (36) holds for ∀𝑡≥𝑇₁.
Consequently, 𝑥(𝑡)𝑟≥ln^𝑚1−𝑏^𝑀₁𝑒^𝑦∗+𝜖−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗+𝜖𝑎^𝑀₁,∀𝑡≥𝑇₁, (39) then liminf_𝑡→+∞𝑥(𝑡)𝑟≥ln^𝑚1−𝑏^𝑀₁𝑒^𝑦∗−𝑐^𝑚₂²−𝑒^𝑀₁𝑢^∗𝑎^𝑀₁=𝑥_∗. (40) Now, for any small enough 𝜖>0, there exists a point 𝑇₂∈𝕋 such that 𝑥(𝑡)≥𝑥_∗−𝜖,𝑡≥𝑇₂; from the third equation of system (3), we have 𝑦^Δ(𝑡)≥𝑟^𝑚₂−𝑎^𝑀₂exp𝑦(𝑡)+𝑏^𝑚₂exp𝑥_∗−𝜖−𝑐^𝑀₁²−𝑒^𝑀₂𝑣^∗+𝜖; (41) by using a similar argument as that in (40), we can get liminf_𝑡→+∞𝑦(𝑡)𝑟≥ln^𝑚2+𝑏^𝑚₂𝑒^𝑥∗−𝑐^𝑀₁²−𝑒^𝑀₂𝑣^∗𝑎^𝑀₂∶=𝑦_∗. (42)
Next we prove liminf_𝑡→∞𝑢(𝑡)≥𝑢_∗,liminf_𝑡→∞𝑣(𝑡)≥𝑣_∗. (43)
Since liminf_𝑡→∞𝑥(𝑡)≥𝑥_∗ then there exists 𝑇∈𝕋 such that 𝑥(𝑡)≥𝑥_∗−𝜖 for any 𝜖>0 and 𝑡>𝑇. By the second equation of system (3), we have 𝑢^Δ(𝑡)≥−𝛼^𝑀₁𝑢(𝑡)+𝛽^𝑚₁𝑒^{𝑥∗−𝜀}, (44) Applying (ii) of Lemma 10, we have liminf_𝑡→∞𝑢(𝑡)≥𝛽^𝑚₁𝑒^𝑥∗𝛼^𝑀₁∶=𝑢_∗; (45) similar to the above proof, it is easy to obtain that liminf_𝑡→∞𝑣(𝑡)≥𝛽^𝑚₂𝑒^𝑦∗𝛼^𝑀₂∶=𝑣_∗. (46) So the proof of Proposition 12 is complete.

Theorem 13. Assume that (H₁)-(H₂) hold; then system (3) is persistent.
We denote by Ω the set of all solutions (𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇 of system (3) satisfying 𝑥_∗≤𝑥(𝑡)≤𝑥^∗,𝑦_∗≤𝑦(𝑡)≤𝑦^∗,𝑢_∗≤𝑢(𝑡)≤𝑢^∗,𝑣_∗≤𝑣(𝑡)≤𝑣^∗, (47) for all 𝑡∈𝕋. It should be noticed that, from the proofs of Propositions 11 and 12, we know that the conditions of Theorem 13 hold and Ω is an invariant set of system (3).

4. Existence of Uniformly Asymptotically Stable Almost Periodic Solution

According to Lemma 9, we first prove that there exists a bounded solution of system (3) and then construct an adaptive Lyapunov functional for system (3). The next result tells us that there exists a bounded solution of (3). To be precise, consider the following.

Proposition 14. Assume that (H₁)-(H₂) hold, then Ω≠∅.

Proof. By the almost periodicity of 𝑟_𝑖(𝑡), 𝑎_𝑖(𝑡), 𝑏_𝑖(𝑡), 𝑐_𝑖(𝑡), 𝑒_𝑖(𝑡), 𝛼_𝑖(𝑡), 𝛽_𝑖(𝑡), 𝑖=1,2, there exists a sequence {𝜏_𝑛}⊆𝕋 with 𝜏_𝑛→∞ as 𝑛→∞ such that 𝑟_𝑖𝑡+𝜏_𝑛⟶𝑟_𝑖(𝑡),𝑎_𝑖𝑡+𝜏_𝑛⟶𝑎_𝑖(𝑡),𝑏_𝑖𝑡+𝜏_𝑛⟶𝑏_𝑖(𝑡),𝑐_𝑖𝑡+𝜏_𝑛⟶𝑐_𝑖(𝑡),𝑒_𝑖𝑡+𝜏_𝑛⟶𝑒_𝑖(𝑡),𝛼_𝑖𝑡+𝜏_𝑛⟶𝛼_𝑖(𝑡),𝛽_𝑖𝑡+𝜏_𝑛⟶𝛽_𝑖(𝑡),𝑛⟶∞. (48) Let 𝜖 be an arbitrary small number; it follows from Propositions 11 and 12 that there exists a positive number 𝑇₃ such that 𝑥_∗−𝜖≤𝑥(𝑡)≤𝑥^∗𝑦+𝜖,_∗−𝜖≤𝑦(𝑡)≤𝑦^∗𝑢+𝜖,_∗−𝜖≤𝑢(𝑡)≤𝑢^∗𝑣+𝜖,_∗−𝜖≤𝑣(𝑡)≤𝑣^∗+𝜖∀𝑡>𝑇₃. (49) Denote 𝑥_𝑛(𝑡)∶=𝑥𝑡+𝜏_𝑛,𝑦_𝑛(𝑡)∶=𝑦𝑡+𝜏_𝑛,𝑢_𝑛(𝑡)∶=𝑢𝑡+𝜏_𝑛,𝑣_𝑛(𝑡)∶=𝑣𝑡+𝜏_𝑛,𝑡≥𝑇₃−𝜏_𝑛,𝑛=1,2,…. (50) For any positive integer 𝑚, it is easy to see that there exist sequences 𝑥_𝑛(𝑡)∶𝑛≥𝑚,𝑦_𝑛(𝑡),𝑢∶𝑛≥𝑚_𝑛(𝑡)∶𝑛≥𝑚,𝑣_𝑛(𝑡)∶𝑛≥𝑚 (51) such that the sequences {𝑥_𝑛(𝑡)},{𝑦_𝑛(𝑡)},{𝑢_𝑛(𝑡)},{𝑣_𝑛(𝑡)} have subsequences, denoted by {𝑥_𝑛(𝑡)},{𝑦_𝑛(𝑡)},{𝑢_𝑛(𝑡)},{𝑣_𝑛(𝑡)} again, converging on any finite interval of 𝕋 as 𝑛→∞, respectively. Thus we have functions 𝑥^∗(𝑡),𝑦^∗(𝑡),𝑢^∗(𝑡),𝑣^∗(𝑡) such that 𝑥_𝑛(𝑡)⟶𝑥^∗(𝑡),𝑦_𝑛(𝑡)⟶𝑦^∗(𝑡),𝑢_𝑛(𝑡)⟶𝑢^∗(𝑡),𝑣_𝑛(𝑡)⟶𝑣^∗(𝑡),𝑡∈𝕋,𝑛⟶∞. (52) Combining with𝑥^Δ(𝑡)=𝑟₁𝑡+𝜏_𝑛−𝑎₁𝑡+𝜏_𝑛exp𝑥𝑡+𝜏_𝑛−𝑏₁𝑡+𝜏_𝑛exp𝑦𝑡+𝜏_𝑛−𝑐₂𝑡+𝜏_𝑛²−𝑒₁𝑡+𝜏_𝑛𝑢𝑡+𝜏_𝑛,𝑢^Δ(𝑡)=−𝛼₁𝑡+𝜏_𝑛𝑢𝑡+𝜏_𝑛+𝛽₁𝑡+𝜏_𝑛exp𝑥𝑡+𝜏_𝑛,𝑦^Δ(𝑡)=𝑟₂𝑡+𝜏_𝑛−𝑎₂𝑡+𝜏_𝑛exp𝑦𝑡+𝜏_𝑛+𝑏₂𝑡+𝜏_𝑛exp𝑥𝑡+𝜏_𝑛−𝑐₁𝑡+𝜏_𝑛²−𝑒₂𝑡+𝜏_𝑛𝑣𝑡+𝜏_𝑛,𝑣^Δ(𝑡)=−𝛼₂𝑡+𝜏_𝑛𝑣𝑡+𝜏_𝑛+𝛽₂𝑡+𝜏_𝑛exp𝑦𝑡+𝜏_𝑛 (53)gives 𝑥^∗Δ(𝑡)=𝑟₁(𝑡)−𝑎₁(𝑡)exp𝑥^∗(𝑡)−𝑏₁(𝑡)exp𝑦^∗(𝑡)−𝑐₂(𝑡)²−𝑒₁(𝑡)𝑢^∗(𝑡),𝑢^∗Δ(𝑡)=−𝛼₁(𝑡)𝑢^∗(𝑡)+𝛽₁(𝑡)exp𝑥^∗(𝑡,𝑦_∗Δ)(𝑡)=𝑟₂(𝑡)−𝑎₂(𝑡)exp𝑦^∗(𝑡)+𝑏₂(𝑡)exp𝑥^∗(𝑡)−𝑐₁(𝑡)²−𝑒₂(𝑡)𝑣^∗(𝑡),𝑣^∗Δ(𝑡)=−𝛼₂(𝑡)𝑣^∗(𝑡)+𝛽₂(𝑡)exp𝑦^∗(𝑡). (54) We can easily see that (𝑥^∗(𝑡),𝑦^∗(𝑡),𝑢^∗(𝑡),𝑣^∗(𝑡))^𝑇 is a solution of system (3) and 𝑥_∗−𝜖≤𝑥^∗(𝑡)≤𝑥^∗𝑦+𝜖,_∗−𝜖≤𝑦^∗(𝑡)≤𝑦^∗𝑢+𝜖,_∗−𝜖≤𝑢^∗(𝑡)≤𝑢^∗𝑣+𝜖,_∗−𝜖≤𝑣^∗(𝑡)≤𝑣^∗+𝜖, (55) for 𝑡∈𝕋. Since 𝜖 is an arbitrary small positive number, it follows that 𝑥_∗≤𝑥^∗(𝑡)≤𝑥^∗,𝑦_∗≤𝑦^∗(𝑡)≤𝑦^∗,𝑢_∗≤𝑢^∗(𝑡)≤𝑢^∗,𝑣_∗≤𝑣^∗(𝑡)≤𝑣^∗ (56) for 𝑡∈𝕋. This completes the proof.

Theorem 15. Assume that (H₁)- (H₂) hold. Suppose further that (H₃)𝑐>0 and −𝑐∈ℛ⁺, where 𝑐=min{𝑃₁−𝑄₁𝜇,𝑃₂−𝑄₂𝜇,𝑅₁−𝑆₁𝜇,𝑅₂−𝑆₂𝜇}, 𝑃₁=2𝑎^𝑚₁𝜉^𝑚₁−𝑏^𝑚₁𝜉^𝑚₂2𝑒^𝑦∗−2𝑐^𝑀₂−𝑒^𝑚₁−𝑏^𝑀₂𝜉^𝑀₂2𝑒^𝑥∗−2𝑐^𝑚₁−𝛽^𝑀₁𝜉^𝑀₁,𝑄₁=𝑎^𝑀2₁𝜉^𝑀2₁+𝑏^𝑀₁𝑎^𝑀₁𝜉^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝑎^𝑀₁𝜉^𝑀₁𝑒^𝑀₁+𝑏^𝑀2₂𝜉^𝑀2₁2𝑒^𝑥∗−2𝑐^𝑚₁²+𝑏^𝑀₂𝑒^𝑀₂𝜉^𝑀₁2𝑒^𝑥∗−2𝑐^𝑚₁+𝛽^𝑀2₁𝜉^𝑀2₂+𝑎^𝑚₂𝑏^𝑚₂𝜉^𝑚₁𝜉^𝑚₂2𝑒^𝑥∗−2𝑐^𝑀₁+𝛽^𝑚₁𝜉^𝑚₁𝛼^𝑚₁,𝑃₂=2𝑎^𝑚₂𝜉^𝑚₂−𝑏^𝑚₁𝜉^𝑚₂2𝑒^𝑦∗−2𝑐^𝑀₂−𝑒^𝑚₂−𝑏^𝑀₂𝜉^𝑀₂2𝑒^𝑥∗−2𝑐^𝑚₁−𝛽^𝑀₂𝜉^𝑀₂,𝑄₂=𝑎^𝑀2₂𝜉^𝑀2₂+𝑏^𝑀₁𝑎^𝑀₁𝜉^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝑏^𝑀2₁𝜉^𝑀2₂2𝑒^𝑦∗−2𝑐^𝑚₂²+𝑏^𝑀₁𝑒^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝑎^𝑚₂𝜉^𝑚₂𝑏^𝑚₂𝜉^𝑚₁2𝑒^𝑥∗−2𝑐^𝑀₁+𝑎^𝑀₂𝜉^𝑀₂𝑒^𝑀₂+𝛽^𝑀2₂𝜉^𝑀2₂+𝛽^𝑚₂𝜉^𝑚₂𝛼^𝑚₂,𝑅_𝑖=2𝛼^𝑚_𝑖−𝛽^𝑀_𝑖𝜉^𝑀_𝑖−𝑒^𝑚_𝑖,(),𝑆𝑖=1,2₁=𝑒^𝑀2₁+𝑒^𝑀₁𝑏^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝑎^𝑀₁𝜉^𝑀₁𝑒^𝑀₁+𝛼^𝑀2₁+𝛽^𝑚₁𝜉^𝑚₁𝛼^𝑚₁,𝑆₂=𝑒^𝑀2₂+𝑒^𝑀₂𝑏^𝑀₂𝜉^𝑀₁2𝑒^𝑥∗−2𝑐^𝑚₁+𝑎^𝑀₂𝜉^𝑀₂𝑒^𝑀₂+𝛼^𝑀2₂+𝛽^𝑚₂𝜉^𝑚₂𝛼^𝑚₂,𝜇=sup_𝑡∈𝕋𝜇(𝑡),𝜉^𝑚₁=𝑒^𝑥∗,𝜉^𝑀₁=𝑒^𝑥∗,𝜉^𝑚₂=𝑒^𝑦∗,𝜉^𝑀₂=𝑒^𝑦∗; (57) then (3) has a unique almost periodic solution 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇 that is uniformly asymptotically stable and satisfies 𝑥_∗≤𝑥(𝑡)≤𝑥^∗,𝑦_∗≤𝑦(𝑡)≤𝑦^∗,𝑢_∗≤𝑢(𝑡)≤𝑢^∗,𝑣_∗≤𝑣(𝑡)≤𝑣^∗ (58) for all 𝑡∈𝕋.

Proof. From Proposition 14, we know that system (3) has a bounded solution satisfying 𝑥_∗≤𝑥(𝑡)≤𝑥^∗,𝑦_∗≤𝑦(𝑡)≤𝑦^∗,𝑢_∗≤𝑢(𝑡)≤𝑢^∗,𝑣_∗≤𝑣(𝑡)≤𝑣^∗,𝑡∈𝕋. (59) Hence, |𝑥(𝑡)|<𝐴₁,𝑦(𝑡)<𝐴₂,|𝑢(𝑡)|<𝐴₃,|𝑣(𝑡)|<𝐴₄, (60) where 𝐴₁=max𝑥_∗,𝑥^∗,𝐴₂=max𝑦_∗,𝑦^∗,𝐴₃=max𝑢_∗,𝑢^∗,𝐴₄=max𝑣_∗,𝑣^∗. (61) For all (𝑥,𝑦,𝑢,𝑣)∈ℝ⁴, we define the norm 𝑥,𝑦,𝑢,𝑣=|𝑥|+𝑦+|𝑢|+|𝑣|. (62) Suppose that 𝑋=(𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡)) and 𝑌=(𝑝(𝑡),𝑞(𝑡),𝑤(𝑡),𝑧(𝑡)) are any two solutions of system (3) defined on 𝕋⁺×Ω²; then ‖𝑋‖≤𝐶, ‖𝑌‖≤𝐶, where 𝐶=∑⁴₁𝐴_𝑖.
Consider the product system of system (3) 𝑥^Δ(𝑡)=𝑟₁(𝑡)−𝑎₁(𝑡)exp{𝑥(𝑡)}−𝑏₁(𝑡)exp𝑦(𝑡)−𝑐₂(𝑡)²−𝑒₁(𝑡)𝑢(𝑡),𝑢^Δ(𝑡)=−𝛼₁(𝑡)𝑢(𝑡)+𝛽₁(𝑡){𝑥(𝑡,𝑦exp)}_Δ(𝑡)=𝑟₂(𝑡)−𝑎₂(𝑡)exp𝑦(𝑡)+𝑏₂(𝑡)exp{𝑥(𝑡)}−𝑐₁(𝑡)²−𝑒₂(𝑡)𝑣(𝑡),𝑣^Δ(𝑡)=−𝛼₂(𝑡)𝑣(𝑡)+𝛽₂(𝑡)𝑦(𝑡,𝑝exp)_Δ(𝑡)=𝑟₁(𝑡)−𝑎₁(𝑡)exp𝑝(𝑡)−𝑏₁(𝑡)exp𝑞(𝑡)−𝑐₂(𝑡)²−𝑒₁(𝑡)𝑤(𝑡),𝑤^Δ(𝑡)=−𝛼₁(𝑡)𝑤(𝑡)+𝛽₁(𝑡)𝑝(𝑡,𝑞exp)_Δ(𝑡)=𝑟₂(𝑡)−𝑎₂(𝑡)exp𝑞(𝑡)+𝑏₂(𝑡)exp𝑝(𝑡)−𝑐₁(𝑡)²−𝑒₂(𝑡)𝑧(𝑡),𝑧^Δ(𝑡)=−𝛼₂(𝑡)𝑧(𝑡)+𝛽₂(𝑡)exp𝑞(𝑡). (63) We construct a Lyapunov function defined on 𝕋⁺×Ω² as follows: 𝑉(𝑡,𝑋,𝑌)=𝑥(𝑡)−𝑝(𝑡)²+𝑦(𝑡)−𝑞(𝑡)²+(𝑢(𝑡)−𝑤(𝑡))²+(𝑣(𝑡)−𝑧(𝑡))²; (64) it is easy to see that the norm ‖‖=𝑥(𝑡)(𝑡+𝑦(𝑡)(𝑡+|𝑢(𝑡)(𝑡+|𝑣(𝑡)(𝑡𝑋−𝑌−𝑝)−𝑞)−𝑤)|−𝑧)| (65) and the norm ‖𝑋−𝑌‖_∗=𝑥(𝑡)−𝑝(𝑡)²+𝑦(𝑡)−𝑞(𝑡)²+(𝑢(𝑡)−𝑤(𝑡))²+(𝑣(𝑡)−𝑧(𝑡))^21/2 (66) are equivalent; that is, there exist two constants 𝐶₁>0,  𝐶₂>0 such that 𝐶₁‖𝑋−𝑌‖≤‖𝑋−𝑌‖_∗≤𝐶₂‖𝑋−𝑌‖, (67) thus 𝐶₁‖𝑋−𝑌‖²≤𝑉(𝑡,𝑋,𝑌)≤𝐶₂‖𝑋−𝑌‖². (68) Let 𝑎,𝑏∈𝐶(ℝ⁺,ℝ⁺), 𝑎(𝑥)=𝐶²₁𝑥², 𝑏(𝑥)=𝐶²₂𝑥²; thus condition (i) of Lemma 9 is satisfied. In addition, 𝑉(𝑡,𝑋,𝑌)−𝑉𝑡,𝑋₁,𝑌₁=𝑥(𝑡)−𝑝(𝑡)²+𝑦(𝑡)−𝑞(𝑡)²+(𝑢(𝑡)−𝑤(𝑡))²+(𝑣(𝑡)−𝑧(𝑡))²−𝑥₁(𝑡)−𝑝₁(𝑡)²+𝑦₁(𝑡)−𝑞₁(𝑡)²+𝑢₁(𝑡)−𝑤₁(𝑡)²+𝑣₁(𝑡)−𝑧₁(𝑡)²≤𝑥(𝑡)−𝑝(𝑡)²−𝑥₁(𝑡)−𝑝₁(𝑡)²+𝑦(𝑡)−𝑞(𝑡)²−𝑦₁(𝑡)−𝑞₁(𝑡)²+(𝑢(𝑡)−𝑤(𝑡))²−𝑢₁(𝑡)−𝑤₁(𝑡)²+(𝑣(𝑡)−𝑧(𝑡))²−𝑣₁(𝑡)−𝑧₁(𝑡)²=𝑥(𝑡)−𝑝(𝑡)+𝑥₁(𝑡)−𝑝₁(𝑡)×𝑥(𝑡)−𝑝(𝑡)−𝑥₁(𝑡)−𝑝₁(𝑡+𝑦(𝑡)(𝑡)+𝑦)−𝑞₁(𝑡)−𝑞₁(𝑡)×𝑦(𝑡)−𝑞(𝑡)−𝑦₁(𝑡)−𝑞₁(𝑡)+(𝑢(𝑡)−𝑤(𝑡))+𝑢₁(𝑡)−𝑤₁(𝑡)×(𝑢(𝑡)−𝑤(𝑡))−𝑢₁(𝑡)−𝑤₁(𝑡)+(𝑣(𝑡)−𝑧(𝑡))+𝑣₁(𝑡)−𝑧₁(𝑡)×(𝑣(𝑡)−𝑧(𝑡))−𝑣₁(𝑡)−𝑧₁(𝑡)≤𝐿𝑥(𝑡)−𝑥₁(𝑡)+𝑦(𝑡)−𝑦₁(𝑡+𝑢(𝑡))−𝑢₁(𝑡)+𝑣(𝑡)−𝑣₁(𝑡+𝑝(𝑡))−𝑝₁(𝑡)+𝑞(𝑡)−𝑞₁(𝑡+𝑤(𝑡))−𝑤₁(𝑡)+𝑧(𝑡)−𝑧₁(𝑡)=𝐿𝑋−𝑋₁+𝑌−𝑌₁, (69) where 𝑋₁=𝑥₁(𝑡),𝑦₁(𝑡),𝑢₁(𝑡),𝑣₁(𝑡),𝑌₁=𝑝₁(𝑡),𝑞₁(𝑡),𝑤₁(𝑡),𝑧₁(𝑡),𝐿=4max𝐴₁,𝐴₂,𝐴₃,𝐴₄. (70) Therefore, condition (ii) of Lemma 9 is satisfied. Finally, calculating the 𝑉^Δ(𝑡,𝑋,𝑌) along system (63), we can obtain 𝑉^Δ()=2𝑥(𝑡)(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑡,𝑋,𝑌−𝑝+𝜇−𝑝^Δ𝑥(𝑡)−𝑝(𝑡)^Δ+2𝑦(𝑡)−𝑞(𝑡)+𝜇(𝑡)𝑦(𝑡)−𝑞(𝑡)^Δ𝑦(𝑡)−𝑞(𝑡)^Δ+2(𝑢(𝑡)−𝑤(𝑡))+𝜇(𝑡)(𝑢(𝑡)−𝑤(𝑡))^Δ(𝑢(𝑡)−𝑤(𝑡))^Δ+2(𝑣(𝑡)−𝑧(𝑡))+𝜇(𝑡)(𝑣(𝑡)−𝑧(𝑡))^Δ(𝑣(𝑡)−𝑧(𝑡))^Δ. (71) Let 𝑥(𝑡)=𝑥(𝑡)−𝑝(𝑡),𝑦(𝑡)=𝑦(𝑡)−𝑞(𝑡),𝑢(𝑡)=𝑢(𝑡)−𝑤(𝑡),𝑣(𝑡)=𝑣(𝑡)−𝑧(𝑡). (72) Then 𝑉(𝑡,𝑋,𝑌)=𝑥²(𝑡)+𝑦²(𝑡)+𝑢²(𝑡)+𝑣²(𝑡) (73) and (71) is transformed into 𝑉^Δ(𝑡,𝑋,𝑌)=2𝑥(𝑡)+𝜇(𝑡)𝑥^Δ(𝑡)𝑥^Δ(𝑡)+2𝑦(𝑡)+𝜇(𝑡)𝑦^Δ(𝑡)𝑦^Δ(𝑡)+2𝑢(𝑡)+𝜇(𝑡)𝑢^Δ(𝑡)𝑢^Δ(𝑡)+2𝑣(𝑡)+𝜇(𝑡)𝑣^Δ(𝑡)𝑣^Δ(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡), (74) where 𝑉₁(𝑡)=[2𝑥(𝑡)+𝜇(𝑡)𝑥^Δ(𝑡)]𝑥^Δ(𝑡), 𝑉₂(𝑡)=[2𝑦(𝑡)+𝜇(𝑡)𝑦^Δ(𝑡)]𝑦^Δ(𝑡), 𝑉₃(𝑡)=[2𝑢(𝑡)+𝜇(𝑡)𝑢^Δ(𝑡)]𝑢^Δ(𝑡), 𝑉₄(𝑡)=[2𝑣(𝑡)+𝜇(𝑡)𝑣^Δ(𝑡)]𝑣^Δ(𝑡). Using the mean value theorem, we have exp{𝑥(𝑡)}−exp𝑝(𝑡)=𝜉₁(𝑡)𝑥(𝑡)(𝑡),𝑦(𝑡𝑞(𝑡−𝑝exp)−exp)=𝜉₂(𝑡)𝑦(𝑡)−𝑞(𝑡), (75) where 𝜉₁(𝑡) lies between exp{𝑥(𝑡)} and exp{𝑝(𝑡)},𝜉₂(𝑡) lies between exp{𝑦(𝑡)} and exp{𝑞(𝑡)}. Substituting (75) into (63), we have 𝑥^Δ(𝑡)=−𝑎₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)−𝑏₁(𝑡)exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)𝜉₂(𝑡)𝑦(𝑡)−𝑒₁(𝑡)𝑢(𝑡),𝑦^Δ(𝑡)=−𝑎₂(𝑡)𝜉₂(𝑡)𝑦(𝑡)+𝑏₂(𝑡)exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)−𝑒₂(𝑡)𝑣(𝑡),𝑢^Δ(𝑡)=−𝛼₁(𝑡)𝑢(𝑡)+𝛽₁(𝑡)𝜉₁(𝑡)𝑥(𝑡),𝑣^Δ(𝑡)=−𝛼₂(𝑡)𝑣(𝑡)+𝛽₂(𝑡)𝜉₂(𝑡)𝑦(𝑡), (76) then 𝑉₁(𝑡)=2𝑥(𝑡)(𝑡)×+𝜇−𝑎₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)−𝑏₁(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)𝜉₂(𝑡)𝑦(𝑡)−𝑒₁(𝑡)𝑢(𝑡)×−𝑎₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)−𝑏₁(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)𝜉₂(𝑡)𝑦(𝑡)−𝑒₁(𝑡)𝑢(𝑡)=−2𝑎₁(𝑡)𝜉₁(𝑡)𝑥²(𝑡)+𝜇(𝑡)𝑎²₁(𝑡)𝜉²₁(𝑡)𝑥²(𝑡)+𝜇(𝑡)𝑏₁(𝑡)𝑎₁(𝑡)𝜉₁(𝑡)𝜉₂(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)𝑦(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑎₁(𝑡)𝑒₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)𝑢(𝑡)−2𝑏₁(𝑡)𝜉₂(𝑡)𝑥(𝑡)𝑦(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)+𝜇(𝑡)𝑏₁(𝑡)𝑎₁(𝑡)𝜉₁(𝑡)𝜉₂(𝑡)𝑥(𝑡)𝑦(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)+𝜇(𝑡)𝑏²₁(𝑡)𝜉²₂(𝑡)𝑦²(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)²+𝜇(𝑡)𝑒₁(𝑡)𝑏₁(𝑡)𝜉₂(𝑡)𝑦(𝑡)𝑢(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)−2𝑒₁(𝑡)𝑥(𝑡)𝑢(𝑡)+𝜇(𝑡)𝑎₁(𝑡)𝜉₁(𝑡)𝑒₁(𝑡)𝑥(𝑡)𝑢(𝑡)+𝜇(𝑡)𝑒₁(𝑡)𝑏₁(𝑡)𝜉₂(𝑡)𝑦(𝑡)𝑢(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)+𝜇(𝑡)𝑒²₁(𝑡)𝑢²(𝑡)=−2𝑎₁(𝑡)𝜉₁(𝑡)+𝜇(𝑡)𝑎²₁(𝑡)𝜉²₁(𝑡)𝑥²(𝑡)+𝜇(𝑡)𝑒²₁(𝑡)𝑢²(𝑡)+𝜇(𝑡)𝑏²₁(𝑡)𝜉²₂(𝑡)𝑦²(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)²+2𝜇(𝑡)𝑏₁(𝑡)𝑎₁(𝑡)𝜉₁(𝑡)𝜉₂(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)−2𝑏₁(𝑡)𝜉₂(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)𝑥(𝑡)𝑦(𝑡)+2𝜇(𝑡)𝑏₁(𝑡)𝑒₁(𝑡)𝜉₂(𝑡)×exp𝑦(𝑡)+exp𝑞(𝑡)−2𝑐₂(𝑡)𝑦(𝑡)𝑢(𝑡)+2𝜇(𝑡)𝑎₁(𝑡)𝜉₁(𝑡)𝑒₁(𝑡)−2𝑒₁(𝑡)𝑥(𝑡)𝑢(𝑡)≤𝜇(𝑡)𝑎^𝑀2₁𝜉^𝑀2₁−2𝑎^𝑀₁𝜉^𝑀₁𝑥²(𝑡)+𝜇(𝑡)𝑒^𝑀2₁𝑢²(𝑡)+𝜇(𝑡)𝑏^𝑀2₁𝜉^𝑀2₂2𝑒^𝑦∗−2𝑐^𝑚₂²𝑦²(𝑡)+𝜇(𝑡)𝑏^𝑀₁𝑎^𝑀₁𝜉^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂−𝑏^𝑚₁𝜉^𝑚₂2𝑒^𝑦∗−2𝑐^𝑀₂𝑥²(𝑡)+𝑦²(𝑡)+𝜇(𝑡)𝑒^𝑀₁𝑏^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂×𝑢²(𝑡)+𝑦²(𝑡)+𝜇(𝑡)𝑎^𝑀₁𝜉^𝑀₁𝑒^𝑀₁−𝑒^𝑚₁𝑢²(𝑡)+𝑥²(𝑡),𝑉₂(𝑡)=2𝑦(𝑡)(𝑡)×+𝜇−𝑎₂(𝑡)𝜉₂(𝑡)𝑦(𝑡)+𝑏₂(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)×𝜉₁(𝑡)𝑥(𝑡)−𝑒₂(𝑡)𝑣(𝑡)×−𝑎₂(𝑡)𝜉₂(𝑡)𝑦(𝑡)+𝑏₂(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)−𝑒₂(𝑡)𝑣(𝑡)=−2𝑎₂(𝑡)𝜉₂(𝑡)𝑦²(𝑡)+𝜇(𝑡)𝑎²₂(𝑡)𝜉²₂(𝑡)𝑦²(𝑡)−𝜇(𝑡)𝑏₂(𝑡)𝜉₁(𝑡)𝑎₂(𝑡)𝜉₂(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝑦(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑎₂(𝑡)𝑒₂(𝑡)𝜉₂(𝑡)𝑦(𝑡)𝑣(𝑡)+2𝑏₂(𝑡)𝜉₁(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝑦(𝑡)𝑥(𝑡)−𝜇(𝑡)𝑎₂(𝑡)𝑏₂(𝑡)𝜉₂(𝑡)𝜉₁(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝑦(𝑡)𝑥(𝑡)+𝜇(𝑡)𝑏²₂(𝑡)𝜉²₁(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)²𝑥²(𝑡)−𝜇(𝑡)𝑒₂(𝑡)𝑏₂(𝑡)𝜉₁(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝑣(𝑡)𝑥(𝑡)−2𝑒₂(𝑡)𝑣(𝑡)𝑦(𝑡)+𝜇(𝑡)𝑎₂(𝑡)𝑒₂(𝑡)𝜉₂(𝑡)𝑣(𝑡)𝑦(𝑡)+𝜇(𝑡)𝑒²₂(𝑡)𝑣²(𝑡)−𝜇(𝑡)𝑒₂(𝑡)𝑏₂(𝑡)𝜉₁(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝑣(𝑡)𝑥(𝑡)=𝜇(𝑡)𝑎²₂(𝑡)𝜉²₂(𝑡)−2𝑎₂(𝑡)𝜉₂(𝑡)𝑦²(𝑡)+2𝑏₂(𝑡)𝜉₂(𝑡)exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)−2𝜇(𝑡)𝑏₂(𝑡)𝜉₁(𝑡)𝑎₂(𝑡)𝜉₂(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝑥(𝑡)𝑦(𝑡)+2𝜇(𝑡)𝑎₂(𝑡)𝜉₂(𝑡)𝑒₂(𝑡)−2𝑒₂(𝑡)𝑣(𝑡)𝑦(𝑡)+𝜇(𝑡)𝑏²₂(𝑡)𝜉²₁(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)²𝑥²(𝑡)−2𝜇(𝑡)𝑏₂(𝑡)𝜉₁(𝑡)𝑒₂(𝑡)×exp{𝑥(𝑡)}+exp𝑝(𝑡)−2𝑐₁(𝑡)𝑥(𝑡)𝑣(𝑡)+𝜇(𝑡)𝑒²₂(𝑡)𝑣²(𝑡)+2𝜇(𝑡)𝑎₂(𝑡)𝜉₂(𝑡)𝑒₂(𝑡)−2𝑒₂(𝑡)𝑦(𝑡)𝑣(𝑡)≤𝜇(𝑡)𝑎^𝑀2₂𝜉^𝑀2₂−2𝑎^𝑚₂𝜉^𝑚₂𝑦²(𝑡)+𝜇(𝑡)𝑒^𝑀2₂𝑣²(𝑡)+𝜇(𝑡)𝑏^𝑀2₂𝜉^𝑀2₁×2𝑒^𝑥∗−2𝑐^𝑚₁²𝑥²(𝑡)+𝜇(𝑡)𝑏^𝑀₂𝑒^𝑀₂𝜉^𝑀₁2𝑒^𝑥∗−2𝑐^𝑚₁𝑥²(𝑡)+𝑣²(𝑡)+𝑏^𝑀₂𝜉^𝑀₂2𝑒^𝑥∗−2𝑐^𝑚₁−𝜇(𝑡)𝑎^𝑚₂𝑏^𝑚₂𝜉^𝑚₁𝜉^𝑚₂2𝑒^𝑥∗−2𝑐^𝑀₁×𝑥²(𝑡)+𝑦²(𝑡)+𝜇(𝑡)𝑎^𝑀₂𝜉^𝑀₂𝑒^𝑀₂−𝑒^𝑚₂𝑦²(𝑡)+𝑣²(𝑡),𝑉₃(𝑡)=2𝑢(𝑡)+𝜇(𝑡)𝑢^Δ(𝑡)𝑢^Δ(𝑡)=2𝑢(𝑡)+𝜇(𝑡)−𝛼₁(𝑡)𝑢(𝑡)+𝛽₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)×−𝛼₁(𝑡)𝑢(𝑡)+𝛽₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)=−2𝛼₁(𝑡)𝑢²(𝑡)+𝜇(𝑡)𝛼²₁(𝑡)𝑢²(𝑡)−𝜇(𝑡)𝛽₁(𝑡)𝜉₁(𝑡)𝛼₁(𝑡)𝑥(𝑡)𝑢(𝑡)+2𝛽₁(𝑡)𝜉₁(𝑡)𝑥(𝑡)𝑢(𝑡)−𝜇(𝑡)𝛽₁(𝑡)𝜉₁(𝑡)𝛼₁(𝑡)𝑥(𝑡)𝑢(𝑡)+𝜇(𝑡)𝛽²₁(𝑡)𝜉²₁(𝑡)𝑥²(𝑡)=𝜇(𝑡)𝛼²₁(𝑡)−2𝛼₁(𝑡)𝑢²(𝑡)+𝜇(𝑡)𝛽²₁(𝑡)𝜉²₁(𝑡)𝑥²(𝑡)+2𝛽₁(𝑡)𝜉₁(𝑡)−2𝜇(𝑡)𝛽₁(𝑡)𝜉₁(𝑡)𝛼₁(𝑡)𝑥(𝑡)𝑢(𝑡)≤𝜇(𝑡)𝛼^𝑀2₁−2𝛼^𝑚₁𝑢²(𝑡)+𝜇(𝑡)𝛽^𝑀2₁𝜉^𝑀2₁𝑥²(𝑡)+𝛽^𝑀₁𝜉^𝑀₁−𝜇(𝑡)𝛽^𝑚₁𝜉^𝑚₁𝛼^𝑚₁𝑥²(𝑡)+𝑢²(𝑡,𝑉₄)(𝑡)=2𝑣(𝑡)+𝜇(𝑡)𝑣^Δ(𝑡)𝑣^Δ(𝑡)=2𝑣(𝑡)+𝜇(𝑡)−𝛼₂(𝑡)𝑣(𝑡)+𝛽₂(𝑡)𝜉₂(𝑡)𝑦(𝑡×)−𝛼₂(𝑡)𝑣(𝑡)+𝛽₂(𝑡)𝜉₂(𝑡)𝑦(𝑡)=𝜇(𝑡)𝛼²₂(𝑡)−2𝛼₂(𝑡)𝑣²(𝑡)+𝜇(𝑡)𝛽²₂(𝑡)𝜉²₂(𝑡)𝑦²(𝑡)+2𝛽₂(𝑡)𝜉₂(𝑡)−2𝜇(𝑡)𝛽₂(𝑡)𝜉₂(𝑡)𝛼₂(𝑡)𝑦(𝑡)𝑣(𝑡)≤𝜇(𝑡)𝛼^𝑀2₂−2𝛼^𝑚₂𝑣²(𝑡)+𝜇(𝑡)𝛽^𝑀2₂𝜉^𝑀2₂𝑦²(𝑡)+𝛽^𝑀₂𝜉^𝑀₂−𝜇(𝑡)𝛽^𝑚₂𝜉^𝑚₂𝛼^𝑚₂𝑦²(𝑡)+𝑣²(𝑡), (77) where 𝜉^𝑚₁=𝑒^𝑥∗,𝜉^𝑀₁=𝑒^𝑥∗,𝜉^𝑚₂=𝑒^𝑦∗,𝜉^𝑀₂=𝑒^𝑦∗; (78) hence 𝑉^Δ=𝑉₁+𝑉₂+𝑉₃+𝑉₄≤𝜇(𝑡)𝑎^𝑀2₁𝜉^𝑀2₁−2𝑎^𝑀₁𝜉^𝑀₁+𝜇(𝑡)𝑏^𝑀₁𝑎^𝑀₁𝜉^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂−𝑏^𝑚₁𝜉^𝑚₂2𝑒^𝑦∗−2𝑐^𝑀₂+𝜇(𝑡)𝑎^𝑀₁𝜉^𝑀₁𝑒^𝑀₁−𝑒^𝑚₁+𝜇(𝑡)𝑏^𝑀2₂𝜉^𝑀2₁2𝑒^𝑥∗−2𝑐^𝑚₁²+𝜇(𝑡)𝑏^𝑀₂𝑒^𝑀₂𝜉^𝑀₁2𝑒^𝑥∗−2𝑐^𝑚₁+𝑏^𝑀₂𝜉^𝑀₂2𝑒^𝑥∗−2𝑐^𝑚₁−𝜇(𝑡)𝑎^𝑚₂𝑏^𝑚₂𝜉^𝑚₁𝜉^𝑚₂2𝑒^𝑥∗−2𝑐^𝑀₁+𝜇(𝑡)𝛽^𝑀2₁𝜉^𝑀2₁+𝛽^𝑀₁𝜉^𝑀₁−𝜇(𝑡)𝛽^𝑚₁𝜉^𝑚₁𝛼^𝑚₁𝑥²(𝑡)+𝜇(𝑡)𝑏^𝑀2₁𝜉^𝑀2₂2𝑒^𝑦∗−2𝑐^𝑚₂²+𝜇(𝑡)𝑏^𝑀₁𝑎^𝑀₁𝜉^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂−𝑏^𝑚₁𝜉^𝑚₂2𝑒^𝑦∗−2𝑐^𝑀₂+𝜇(𝑡)𝑒^𝑀₁𝑏^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝜇(𝑡)𝑎^𝑀2₂𝜉^𝑀2₂−2𝑎^𝑚₂𝜉^𝑚₂+𝑏^𝑀₂𝜉^𝑀₂2𝑒^𝑥∗−2𝑐^𝑚₁−𝜇(𝑡)𝑎^𝑚₂𝑏^𝑚₂𝜉^𝑚₁𝜉^𝑚₂2𝑒^𝑥∗−2𝑐^𝑀₁+𝜇(𝑡)𝑎^𝑀₂𝜉^𝑀₂𝑒^𝑀₂−𝑒^𝑚₂+𝜇(𝑡)𝛽^𝑀2₂𝜉^𝑀2₂+𝛽^𝑀₂𝜉^𝑀₂−𝜇(𝑡)𝛽^𝑚₂𝜉^𝑚₂𝛼^𝑚₂𝑦²(𝑡)+𝜇(𝑡)𝑒^𝑀2₁+𝜇(𝑡)𝑒^𝑀₁𝑏^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝜇(𝑡)𝑎^𝑀₁𝜉^𝑀₁𝑒^𝑀₁−𝑒^𝑚₁+𝜇(𝑡)𝛼^𝑀2₁−2𝛼^𝑚₁+𝛽^𝑀₁𝜉^𝑀₁−𝜇(𝑡)𝛽^𝑚₁𝜉^𝑚₁𝛼^𝑚₁𝑢²(𝑡)+𝜇(𝑡)𝑒^𝑀2₂+𝜇(𝑡)𝑏^𝑀₂𝑒^𝑀₂𝜉^𝑀₁2𝑒^𝑥∗−2𝑐^𝑚₁+𝜇(𝑡)𝑎^𝑀₂𝜉^𝑀₂𝑒^𝑀₂−𝑒^𝑚₂+𝜇(𝑡)𝛼^𝑀2₂−2𝛼^𝑚₂+𝛽^𝑀₂𝜉^𝑀₂−𝜇(𝑡)𝛽^𝑚₂𝜉^𝑚₂𝛼^𝑚₂𝑣²(𝑡)≤−𝑃₁−𝑄₁𝜇(𝑡)𝑥²(𝑡)+𝑃₂−𝑄₂𝜇(𝑡)𝑦²(𝑡)+𝑅₁−𝑆₁𝜇(𝑡)𝑢²(𝑡)+𝑅₂−𝑆₂𝜇(𝑡)𝑣²(𝑡𝑃)≤−₁−𝑄₁𝜇𝑥²(𝑡)+𝑃₂−𝑄₂𝜇𝑦²(𝑡)+𝑅₁−𝑆₁𝜇𝑢²(𝑡)+𝑅₂−𝑆₂𝜇𝑣²(𝑡)≤−𝑐𝑥²(𝑡)+𝑦²(𝑡)+𝑢²(𝑡)+𝑣²(𝑡)=−𝑐𝑉(𝑡,𝑋,𝑌), (79) where 𝑐=min𝑃₁−𝑄₁𝜇,𝑃₂−𝑄₂𝜇,𝑅₁−𝑆₁𝜇,𝑅₂−𝑆₂𝜇,𝑃₁=2𝑎^𝑚₁𝜉^𝑚₁−𝑏^𝑚₁𝜉^𝑚₂2𝑒^𝑦∗−2𝑐^𝑀₂−𝑒^𝑚₁−𝑏^𝑀₂𝜉^𝑀₂2𝑒^𝑥∗−2𝑐^𝑚₁−𝛽^𝑀₁𝜉^𝑀₁,𝑄₁=𝑎^𝑀2₁𝜉^𝑀2₁+𝑏^𝑀₁𝑎^𝑀₁𝜉^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝑎^𝑀₁𝜉^𝑀₁𝑒^𝑀₁+𝑏^𝑀2₂𝜉^𝑀2₁2𝑒^𝑥∗−2𝑐^𝑚₁²+𝑏^𝑀₂𝑒^𝑀₂𝜉^𝑀₁2𝑒^𝑥∗−2𝑐^𝑚₁+𝛽^𝑀2₁𝜉^𝑀2₂+𝑎^𝑚₂𝑏^𝑚₂𝜉^𝑚₁𝜉^𝑚₂2𝑒^𝑥∗−2𝑐^𝑀₁+𝛽^𝑚₁𝜉^𝑚₁𝛼^𝑚₁,𝑃₂=2𝑎^𝑚₂𝜉^𝑚₂−𝑏^𝑚₁𝜉^𝑚₂2𝑒^𝑦∗−2𝑐^𝑀₂−𝑒^𝑚₂−𝑏^𝑀₂𝜉^𝑀₂2𝑒^𝑥∗−2𝑐^𝑚₁−𝛽^𝑀₂𝜉^𝑀₂,𝑄₂=𝑎^𝑀2₂𝜉^𝑀2₂+𝑏^𝑀₁𝑎^𝑀₁𝜉^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝑏^𝑀2₁𝜉^𝑀2₂2𝑒^𝑦∗−2𝑐^𝑚₂²+𝑏^𝑀₁𝑒^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝑎^𝑚₂𝜉^𝑚₂𝑏^𝑚₂𝜉^𝑚₁2𝑒^𝑥∗−2𝑐^𝑀₁+𝑎^𝑀₂𝜉^𝑀₂𝑒^𝑀₂+𝛽^𝑀2₂𝜉^𝑀2₂+𝛽^𝑚₂𝜉^𝑚₂𝛼^𝑚₂,𝑅_𝑖=2𝛼^𝑚_𝑖−𝛽^𝑀_𝑖𝜉^𝑀_𝑖−𝑒^𝑚_𝑖,(),𝑆𝑖=1,2₁=𝑒^𝑀2₁+𝑒^𝑀₁𝑏^𝑀₁𝜉^𝑀₂2𝑒^𝑦∗−2𝑐^𝑚₂+𝑎^𝑀₁𝜉^𝑀₁𝑒^𝑀₁+𝛼^𝑀2₁+𝛽^𝑚₁𝜉^𝑚₁𝛼^𝑚₁,𝑆₂=𝑒^𝑀2₂+𝑒^𝑀₂𝑏^𝑀₂𝜉^𝑀₁2𝑒^𝑥∗−2𝑐^𝑚₁+𝑎^𝑀₂𝜉^𝑀₂𝑒^𝑀₂+𝛼^𝑀2₂+𝛽^𝑚₂𝜉^𝑚₂𝛼^𝑚₂. (80) From the condition (H₃), 𝑐>0 and −𝑐∈ℛ⁺, we see that the condition (iii) of Lemma 9 is satisfied. So all the conditions of Lemma 9 are satisfied. By Lemma 9, there exists a unique uniformly asymptotically stable almost periodic solution 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇 of system (3) which satisfies 𝑥_∗≤𝑥(𝑡)≤𝑥^∗,𝑦_∗≤𝑦(𝑡)≤𝑦^∗,𝑢_∗≤𝑢(𝑡)≤𝑢^∗,𝑣_∗≤𝑣(𝑡)≤𝑣^∗ (81) and 𝑋(𝑡)∈Ω for all 𝑡∈𝕋. This completes the proof.

Corollary 16. Assume that (H₁)–(H₃) hold. Suppose that the nonnegative coefficients 𝑟_𝑖(𝑡), 𝑎_𝑖(𝑡), 𝑏_𝑖(𝑡), 𝑐_𝑖(𝑡), 𝑒_𝑖(𝑡), 𝛼_𝑖(𝑡), 𝛽_𝑖(𝑡), 𝑖=1,2 are periodic of period 𝜔; then system (3) has a unique uniformly asymptotically stable periodic solution of period 𝜔.

5. An Example

In this section, we present an example to illustrate the feasibility and effectiveness of our results.

Consider the system (3) with the following coefficients 𝑟₁(𝑡)=14.5+0.5sin𝑡+0.5cos√𝑟2𝑡,₂(𝑡)=14.5+0.4cos𝑡+0.6sin√𝑎2𝑡,₁(𝑡)𝑎=10.5+0.2sin𝑡+0.3cos𝜋𝑡,₂(𝑡)=10.5+0.4cos√𝑏2𝑡+0.1cos𝑡,₁(𝑡)=0.3+0.1sin√2𝑡+0.1sin√𝑏3𝑡,₂(𝑡)=0.3+0.1cos𝑡+0.1sin√𝑐2𝑡,₁(𝑡)=1.05+0.02sin2𝑡+0.03cos√𝑐2𝑡,₂(𝑡)=1.05+0.03sin2𝑡+0.02cos√𝛼2𝑡,₁(𝑡)=1.1+0.05sin𝑡+0.05cos√𝛼2𝑡,₂(𝑡)=1.1+0.03sin𝑡+0.07cos√𝛽2𝑡,₁(𝑡)=0.5+0.1cos√𝛽2𝑡,₂(𝑡)=0.5+0.1sin√𝑒3𝑡,₁(𝑡)=0.5+0.4cos√𝑒2𝑡,₂(𝑡)=0.5+0.4sin√3𝑡. (82) Let 𝕋=⋃^∞_𝑘=0[𝑘,𝑘+0.75]. Then system (3) is persistent and has a unique uniformly asymptotically stable almost periodic solution.

Proof. By calculation, we have that 𝜇(𝑡)=0,𝑡∈^∞_𝑘=0[𝑘,𝑘+0.75),0.25,𝑡∈^∞_𝑘=0{},𝑥𝑘+0.75_∗=0.5,𝑦^∗𝑢≈0.63591,_∗≈0.9892,𝑣^∗𝑥≈1.1332,_∗≈0.0961,𝑦_∗𝑢≈0.1195,_∗≈0.3670,𝑣_∗≈0.3756,𝑐≈1.0478,𝜇=0.25. (83) So all conditions are satisfied and the results follow from Theorems 13 and 15. Therefore, system (3) is persistent and has a unique uniformly asymptotically stable almost periodic solution. This completes the proof.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11361072 and the YNU Postdoctoral Science Foundation Project under Grant W4030002.