Dynamics of Mutualism-Competition-Predator System with Beddington-DeAngelis Functional Responses and Impulsive Perturbations

Abstract

We explore the dynamics of a class of mutualism-competition-predator interaction models with Beddington-DeAngelis functional responses and impulsive perturbations. Sufficient conditions for existence of positive periodic solution are established by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are given for the global stability and the globally exponential stability of system by employing comparison principle and Lyapunov method.

1. Introduction

The ecological predator-prey systems and impulsive functional differential equations have been studied extensively by many authors [1–7]. A predator's per capita feeding rate on prey, or its functional response, provides a foundation for predator-prey theory. Since 1959, Holling's prey-dependent type II functional response, a model that is a function of prey abundance only, has served as the basis for a large literature on predator-prey theory. The traditional Kolmogorov type predator-prey model with Holling's type II functional response: 𝐱^′𝐱=𝑟𝐱1−𝐾𝐱−𝑐𝐲𝑚+𝐱,𝐲^′𝐱=−𝑑𝐲+𝑓𝐲,𝑚+𝐱 (1.1) and its various generalized forms have received great attention from both theoretical and mathematical biologists and have been well studied [1, 8, 9]. In (1.1), 𝐱 and 𝐲 represent the densities of prey specie and predator specie at time 𝑡, respectively. Predator-prey model with Holling's type II functional response assumes that predators do not interfere with one another's activities; thus competition among predators for food occurs only via the depletion of prey. However, when predators have to search for food (and therefore, have to share or compete for food), the functional response in a prey-predator model should be predator-dependent. The predator-prey system with the Beddington-DeAngelis functional response: 𝐱𝐱′=𝑟𝐱1−𝐾−𝛼𝐱𝐲𝑎+𝑏𝐱+𝑐𝐲,𝐲′=−𝑑𝐲+𝛽𝐱𝐲𝑎+𝑏𝐱+𝑐𝐲 (1.2) was originally introduced by Beddington [10] and DeAngelis et al. [11], independently, where 𝐱 and 𝐲 represent the densities of prey specie and predator specie at time 𝑡, respectively. The main difference of this functional response from Holling's type II functional response is that it contains an extra term presenting mutual interference by predators. In this model, individuals from a population of two or more predators not only allocate time to searching for and processing prey, but also spend some time engaging in encounters with other predators, the Beddington-DeAngelis type can provide better descriptions of predator feeding over a range of predator-prey abundances.

On the other hand, differential equations with impulsive effects form a wide set of different problems. During the last three decades those problems were intensively studied. Some authors devote themselves to the study of impulsive differential equation [12–22]. The main definitions and results of the theory of systems of ordinary differential equations with impulse effects were given in [12, 13, 15]. Similarity and differentiality of such problems of applied mathematics with corresponding problems of ordinary differential equations (and without the conditions of impulsive effects) were demonstrated, and general characteristics of these systems were described. Periodic and almost-periodic solutions of differential equations with impulsive effects were studied in [18]. De La Sen investigated time-varying systems with nonnecessarily bounded everywhere continuous time-differentiable time-varying point delays [19]. The delay-free and delayed dynamics are assumed to be time-varying and impulsive, the constructed solution trajectories of both the unforced and forced systems are obtained from different (input-state space/output space and state space to output space) operators. The system stability and the compactness of the operators describing the solution trajectories are well investigated. De La Sen and Luo also obtain sufficiency-type stability results for time-delay linear systems with constant point delays under impulsive inputs of impulses of state-dependent amplitudes occurring separately through time. They proved that the amplitudes of the impulses and the time intervals between impulses may be chosen sufficiently large if the delay-free dynamics is sufficiently stable compared to the delayed one [20]. Xu and Sun investigated the problem of finite-time stability of linear time-varying singular systems with impulses at fixed times, they have proposed a sufficient condition for linear singular impulsive systems to be finite-time stable in terms of a set of coupled matrix inequalities [21]. Zhang and Sun also considered the stability of impulsive linear differential equations with time delay. By using Lyapunov functions and analysis technique, they get some results for the stability of impulsive linear differential equations with time delay [22].

Some impulsive factors have also great impact on the growth of a population. For example, we notice that the births of many species are not continuous but happen at some regular time (e.g., the births of some wildlife are seasonal). Moreover, the human beings have been harvesting or stocking species at some time, then the species is affected by another type of impulse. If we incorporate these impulsive factors into the models of population interactions, the models must be governed by impulsive ordinary differential equations.

Population communities are embedded in periodically varying environments. Therefore, this study should take into account the biological and environmental periodicity (e.g., seasonal effects of weather, food supplies, and mating habits), we focus on the existence of periodic solution with strictly positive components. In real world, any biological or environmental parameters are naturally subject to fluctuation in time, so it is reasonable to study the corresponding nonautonomous system.

The first author investigated the existence of positive periodic solutions of a nonautonomous competitive Lotka-Volterra system with impulse and Holling type III functional response [23]. An 𝑁-dimensional Lotka-Volterra system with fixed moments of impulsive perturbations is given in [24] by Ahmad and Stamova. By means of piecewise continuous functions which are modifications of classical Lyapunov’ functions they give sufficient conditions for asymptotic stability of the solutions. Inspired by [23, 24], in this paper, we focus our attention on the existence of periodic solution and globally asymptotic stability of solutions for multispecies mutualism-competition-predator system with impulses in which the competition among predator species and the mutualism among prey species are simultaneously considered. In general, mutualism is relevant to two species [25, 26], thus we focus on two prey species and assume that there exists the relation of mutualism between two prey species. The primary approach is based on the coincidence degree and its related continuation theorem [27], which has been widely used in dealing with the existence of periodic solutions of differential equations. We will investigate the following nonautonomous system: 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑏₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)−𝑥₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥₂(𝑡)−^𝑛_𝑘=1𝑐_1𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥₁(𝑡)+𝛾_1𝑘(𝑡)𝑦_𝑘(𝑡),𝑥̇₂(𝑡)=𝑥₂(𝑡)𝑏₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)−𝑥₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥₁(𝑡)−^𝑛_𝑘=1𝑐_2𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥₂(𝑡)+𝛾_2𝑘(𝑡)𝑦_𝑘(𝑡),𝑦_𝑗̇(𝑡)=𝑦_𝑗(𝑡)−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑘=1𝛿_𝑗𝑘(𝑡)𝑦_𝑘(𝑡),𝑡≠𝑡_𝑘,𝑘∈ℤ⁺,𝑗=1,…,𝑛,Δ𝑥_𝑖(𝑡)=𝑥_𝑖𝑡⁺−𝑥_𝑖(𝑡⁻)=𝑏_𝑖𝑘𝑥_𝑖(𝑡),Δ𝑦_𝑗(𝑡)=𝑦_𝑗𝑡⁺−𝑦_𝑗(𝑡⁻)=𝑏̃_𝑗𝑘𝑦_𝑗(𝑡),𝑡=𝑡_𝑘,𝑘∈ℤ⁺,𝑖=1,2,𝑗=1,…,𝑛, (1.3) where(i)𝑥_𝑖(𝑡)  (𝑖=1,2) denote the densities of prey species at time 𝑡, respectively;(ii)𝑦_𝑗(𝑡)  (𝑗=1,…,𝑛) denote the density of predator species at time 𝑡, respectively;(iii)𝑏_𝑖𝑘 represent the sum of the birth rate and the harvesting (or stocking) rate of 𝑥_𝑖(𝑡) at time 𝑡, respectively;(iv)𝑏̃_𝑗𝑘 represent the sum of the birth rate and the harvesting (or stocking) rate of 𝑦_𝑗(𝑡) at time 𝑡, respectively; (v)𝑥_𝑖(𝑡⁺) and 𝑥_𝑖(𝑡⁻) represent the right and left limit of 𝑥_𝑖(𝑡) at 𝑡, 𝑦_𝑗(𝑡⁺) and 𝑦_𝑗(𝑡⁻) represent the right and left limit of 𝑦_𝑗(𝑡) at 𝑡.

Let 𝑔(𝑡) be a bounded continuous function on ℝ. Define 𝑔^ℓ=inf_𝑡∈ℝ𝑔(𝑡),𝑔^𝑢=sup_𝑡∈ℝ𝑔(𝑡). (1.4) Particularly, if 𝑔(𝑡) are 𝑇-periodic functions with respect to 𝑡, we denote 1𝑔=𝑇^𝑇₀𝑔(𝑡)𝑑𝑡. (1.5) The range of the indices 𝑖∈{1,2} and 𝑗,𝑘∈{1,…,𝑛} are used in this paper unless otherwise stated.

Throughout the paper, we give the hypotheses as follows(A1) for any 𝑡∈ℝ, 𝑏_𝑖(𝑡), 𝑎_𝑖(𝑡), 𝑟_𝑗(𝑡), 𝜇_𝑖(𝑡), 𝜔_𝑖(𝑡), 𝑑_𝑗𝑖(𝑡), 𝛿_𝑗𝑘(𝑡), 𝑐_𝑖𝑘(𝑡), 𝛼_𝑖𝑘, 𝛽_𝑖𝑘, 𝛾_𝑖𝑘 are nonnegative continuous 𝑇-periodic functions and 𝛼_𝑖𝑗=𝛼_𝑗𝑖, 𝛽_𝑖𝑗=𝛽_𝑗𝑖, 𝛾_𝑖𝑗=𝛾_𝑗𝑖;(A2)𝑏_𝑖𝑘, 𝑏̃_𝑗𝑘>0, 1+𝑏_𝑖𝑘>0, 1+𝑏̃_𝑗𝑘>0, 𝑏_𝑖𝑘, 𝑏̃_𝑗𝑘(𝑘∈ℤ⁺) are constants. There exists a positive integer 𝑞, such that 𝑡_𝑘+𝑞=𝑡_𝑘+𝑇,𝑏_{𝑖(𝑘+𝑞)}=𝑏_𝑖𝑘,𝑏̃_{𝑗(𝑘+𝑞)}=𝑏̃_𝑗𝑘,(𝑘∈ℤ⁺). Without loss of generality, we also suppose that 𝑡_𝑘≠0 and [0,𝑇]∩{𝑡_𝑘∶𝑘∈ℤ⁺}={𝑡₁,𝑡₂,…,𝑡_𝑠}, then it follows that 𝑞=𝑠;(A3)𝑥_𝑖(𝑡),𝑦_𝑗(𝑡) is left-continuous at 𝑡_𝑘, that is, the following relations are satisfied: 𝑥_𝑖𝑡⁻_𝑘=𝑥_𝑖𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=1+𝑏_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘∈ℤ⁺,𝑦_𝑗𝑡⁻_𝑘=𝑦_𝑗𝑡_𝑘,𝑦_𝑗𝑡⁺_𝑘=1+𝑏̃_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝑘∈ℤ⁺; (1.6)(A4)𝑡₁<𝑡₂<⋯ and lim_𝑘→∞𝑡_𝑘=∞;(A5)𝑥_𝑖(𝑡⁺₀)>0,𝑦_𝑗(𝑡⁺₀)>0.

A brief description of the organization of this paper is as follows. The basic concepts and lemmas are given in Section 2. The main results in this paper are stated in Theorems 3.1, 4.5, and 4.6.

2. Basic Concepts and Lemma

Let 𝜑∶[0,𝑇]⊂𝐽→ℝ be a piecewise continuous function with points of discontinuity 𝑡₁,…,𝑡_𝑠, we denote 𝑃𝐶_𝑇=𝜑∶𝜑(0)=𝜑(𝑇),𝑃𝐶¹_𝑇=𝜑∈𝑃𝐶_𝑇∶𝑑𝜑𝑑𝑡∈𝑃𝐶_𝑇. (2.1) Suppose 𝜑_𝑃𝐶𝑇=sup_{[]𝑡∈0,𝑇}𝜑(𝑡),𝜑_{𝑃𝐶1𝑇}=max𝜑_𝑃𝐶𝑇,𝜑̇_𝑃𝐶𝑇. (2.2) It can be easily proved that 𝑃𝐶_𝑇 and 𝑃𝐶¹_𝑇 are Banach spaces under the condition endowed the above norms.

Make the change of variables 𝑥_𝑖(𝑡)=exp{𝑢_𝑖(𝑡)},𝑦_𝑗(𝑡)=exp{𝑣_𝑗(𝑡)}, then (1.1) can be reformulated as 𝑢̇₁(𝑡)=𝑏₁(𝑡)−𝑎₁(𝑡)exp𝑢₁(𝑡−𝑢)exp₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₂(𝑡−_𝑛𝑘=1)𝑐_1𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)exp𝑢₁(𝑡)+𝛾_1𝑘(𝑡)exp𝑣_𝑘(𝑡,)𝑢̇₂(𝑡)=𝑏₂(𝑡)−𝑎₂(𝑡)exp𝑢₂(𝑡−𝑢)exp₂(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₁(𝑡−_𝑛𝑘=1)𝑐_2𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)exp𝑢₂(𝑡)+𝛾_2𝑘(𝑡)exp𝑣_𝑘(𝑡,)𝑦_𝑗̇(𝑡)=−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)exp𝑣_𝑗(𝑡−_𝑛𝑘=1)𝛿_𝑗𝑘(𝑡)exp𝑣_𝑘(𝑡),𝑡≠𝑡_𝑘,𝑘∈ℤ⁺,𝑗=1,…,𝑛,Δ𝑢_𝑖(𝑡)=𝑢_𝑖𝑡⁺−𝑢_𝑖(𝑡⁻)=ln1+𝑏_𝑖𝑘,Δ𝑣_𝑗(𝑡)=𝑣_𝑗𝑡⁺−𝑣_𝑗(𝑡⁻)=ln1+𝑏̃_𝑗𝑘,𝑡=𝑡_𝑘𝑘∈ℤ⁺,𝑖=1,2,𝑗=1,…,𝑛. (2.3) For (1.3) and (2.3), we have similar lemma and definitions. So we only relate such results for (1.3).

Definition 2.1. The mapping 𝐱∶[0,𝑇]→ℝ^𝑛+2,𝐱(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),…,𝑦_𝑛(𝑡)) is called a solution of system (1.1) in [0,𝑇], if (1)𝐱(𝑡) is piecewise continuous in [0,𝑇] and [0,𝑇]∩{𝑡_𝑘∶𝑘∈ℤ⁺} are discontinuous points of the first kind of 𝐱(𝑡), and 𝐱(𝑡) are left continuous at [0,𝑇]∩{𝑡_𝑘∶𝑘∈ℤ⁺};(2)𝐱(𝑡) satisfies system in [0,𝑇].

Definition 2.2. The mapping 𝐱∶𝐽→ℝ^𝑛+2 is called a 𝑇-periodic solution of system (1.1) if (1)𝐱(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),…,𝑦_𝑛(𝑡)) is a solution of system (1.1);(2)𝐱(𝑡) satisfies 𝐱(𝑡+𝑇−0)=𝐱(𝑡−0),𝑡∈ℝ.

Obviously, if 𝐱(𝑡) is a solution of (1.1) satisfying 𝐱(0)=𝐱(𝑇) in [0,𝑇], then from the periodicity of the vector field of (1.1), we know that 𝐱^∗(𝑡)=𝐱(𝑡−ℓ𝑇),𝑡∈[ℓ𝑇,(ℓ+1)𝑇]⧵𝑡_𝑘∶𝑘∈ℤ⁺𝐱,ℓ∈ℕ,_∗,(𝑡)isleftcountiousat𝑡_𝑘 (2.4) is an 𝑇-periodic solution for (1.1). Thus, the problem which discusses the existence of solution of (1.1) will be transformed to discuss the existence of periodic solution for (1.1) in [0,𝑇] satisfying 𝐱(0)=𝐱(𝑇). In order to explore the existence of positive periodic solutions of (1.1), we recall some concepts and results on coincidence degree from [27, pages 39-40], borrowing notations there.

Let 𝕏,ℤ be normed vector spaces, ℒ∶Domℒ⊆𝕏→ℤ a linear mapping, and 𝒩∶𝕏→ℤ a continuous mapping. If dimkerℒ=comdimImℒ<+∞ and Imℒ is a closed in ℤ, then the mapping ℒ will be called a Fredholm mapping of index zero. If ℒ is a Fredholm mapping of index zero, there exists continuous projects 𝒫∶𝕏→𝕏 and 𝒬∶ℤ→ℤ such that Im𝒫=Kerℒ,Imℒ=Ker𝒬=Im(𝐼−𝒬). It follows that ℒ|_{Domℒ∩Ker𝒫}∶(𝐼−𝒫)𝕏→Imℒ has an inverse which is denoted by 𝐾_𝑝. If Ω be an open bounded subset of 𝕏, the mapping 𝒩 will be called ℒ-compact on Ω provided that 𝒬𝒩(Ω) is bounded, and 𝐾_𝑝(𝐼−𝒬)𝒩∶Ω→𝕏 is compact. Since Im𝒬 is isomorphic to Kerℒ, there exists an isomorphism 𝒥∶Im𝒬→Kerℒ.

The following continuation theorem is due to Gaines and Mawhin [27].

Lemma 2.3 2.3 (Continuation Theorem in [27]). Let ℒ a Fredholm mapping of index zero and 𝒩 be ℒ-compact on Ω. Suppose that (a)for each 𝜆∈(0,1), every solution 𝐱 of ℒ𝐱=𝜆𝒩𝐱 such that 𝐱∉𝜕Ω;(b)𝒬𝒩𝐱≠0 for each 𝐱∈Kerℒ∩𝜕Ω and deg{𝒥𝒬𝒩,Ω∩Kerℒ,0}≠0.Then the operator equation ℒ𝐱=𝒩𝐱 has at least one solution lying in Domℒ∩Ω.

We can show that the solution of (1.1) with positive initial value remains positive too, that is, the following Lemma 2.4 holds.

Lemma 2.4. Suppose the hypotheses (A1)–(A5) hold. 𝑥_𝑖(𝑡),𝑦_𝑗(𝑡) is a solution of (1.1), then 𝑥_𝑖(𝑡),𝑦_𝑗(𝑡)>0,𝑡∈𝐽. (2.5)

Proof. By integrating of (1.1) in the interval (𝑡₀,𝑡₁], we have 𝑥₁(𝑡)=𝑥_𝑖𝑡⁺₀exp^𝑡_𝑡0𝑏₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)−𝑥₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥₂(𝑡)−^𝑛_𝑘=1𝑐_1𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥₁(𝑡)+𝛾_1𝑘(𝑡)𝑦_𝑘(𝑡),𝑥𝑑𝑡₂(𝑡)=𝑥_𝑖𝑡⁺₀exp^𝑡_𝑡0𝑏₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)−𝑥₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥₁(𝑡)−^𝑛_𝑘=1𝑐_2𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥₂(𝑡)+𝛾_2𝑘(𝑡)𝑦_𝑘(𝑡),𝑦𝑑𝑡_𝑗(𝑡)=𝑦_𝑗𝑡⁺₀exp^𝑡_𝑡0−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑘=1𝛿_𝑗𝑘(𝑡)𝑦_𝑘(𝑡)𝑑𝑡, (2.6) for 𝑡∈(𝑡₀,𝑡₁].
There does not exist point of discontinuity of 𝑥_𝑖(𝑡), 𝑦_𝑗(𝑡) in the interval (𝑡₀,𝑡₁], it is obvious that 𝑥_𝑖(𝑡)>0, 𝑦_𝑗(𝑡)>0 for 𝑡∈(𝑡₀,𝑡₁], hence 𝐱(𝑡₁)>0. We have from (1.1) that 𝑥_𝑖𝑡⁺₁=1+𝑏_𝑖𝑘𝑥_𝑖𝑡₁,𝑦_𝑗𝑡⁺₁=1+𝑏̃_𝑗𝑘𝑦_𝑗𝑡₁. (2.7) It follows from (A2) that 𝑥_𝑖𝑡⁺₁=1+𝑏_𝑖𝑘𝑥_𝑖𝑡₁>0,𝑦_𝑗𝑡⁺₁=1+𝑏̃_𝑗𝑘𝑦_𝑗𝑡₁>0. (2.8) We now integrate (1.1) in the interval (𝑡₁,𝑡₂], and we have 𝑥₁(𝑡)=𝑥_𝑖𝑡⁺₁exp^𝑡_𝑡0𝑏₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)−𝑥₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥₂(𝑡)−^𝑛_𝑘=1𝑐_1𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥₁(𝑡)+𝛾_1𝑘(𝑡)𝑦_𝑘(𝑡),𝑥𝑑𝑡₂(𝑡)=𝑥_𝑖𝑡⁺₁exp^𝑡_𝑡0𝑏₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)−𝑥₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥₁(𝑡)−^𝑛_𝑘=1𝑐_2𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥₂(𝑡)+𝛾_2𝑘(𝑡)𝑦_𝑘(𝑡),𝑦𝑑𝑡_𝑗(𝑡)=𝑦_𝑗𝑡⁺₁exp^𝑡_𝑡0−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑘=1𝛿_𝑗𝑘(𝑡)𝑦_𝑘(𝑡)𝑑𝑡, (2.9) for 𝑡∈(𝑡₁,𝑡₂]. From the above relation it follows that 𝑥_𝑖(𝑡)>0 for 𝑡∈(𝑡₁,𝑡₂].
By similar arguments, we can obtain that 𝑥₁(𝑡)=𝑥_𝑖𝑡⁺_𝑘exp^𝑡_𝑡0𝑏₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)−𝑥₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥₂(𝑡)−^𝑛_𝑘=1𝑐_1𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥₁(𝑡)+𝛾_1𝑘(𝑡)𝑦_𝑘(𝑡),𝑥𝑑𝑡₂(𝑡)=𝑥_𝑖𝑡⁺_𝑘exp^𝑡_𝑡0𝑏₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)−𝑥₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥₁(𝑡)−^𝑛_𝑘=1𝑐_2𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥₂(𝑡)+𝛾_2𝑘(𝑡)𝑦_𝑘(𝑡),𝑦𝑑𝑡_𝑗(𝑡)=𝑦_𝑗𝑡⁺_𝑘exp^𝑡_𝑡0−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑘=1𝛿_𝑗𝑘(𝑡)𝑦_𝑘(𝑡)𝑑𝑡, (2.10) for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1]. So 𝑥_𝑖(𝑡)>0,𝑦_𝑗(𝑡)>0 for 𝑡∈𝐽.

3. Existence of Positive Periodic Solution

We denote Δ_𝑖=𝑏_𝑖+1𝑇^𝑞_𝑘=1ln1+𝑏_𝑖𝑘,Δ_𝑗=1𝑇^𝑞_𝑘=1ln1+𝑏̃_𝑗𝑘−𝑟_𝑗,𝐵=𝑎_{𝑖∗𝑗∗}𝟎𝟎𝛿_{𝑖′𝑗′}Δ,Δ=^𝑖Δ_𝑗, (3.1) where 𝑖^∗,𝑗^∗∈{1,…,𝑛},𝑖′,𝑗′∈{1,…,𝑛}. We also denote 𝐵_𝑘(𝑘∈{1,…,𝑛+2}) the matrix obtained by replacing the 𝑘th column of 𝐵 with Δ.

Theorem 3.1. If system (2.3) satisfies (A1)–(A5) and following conditions(A6)Δ₁>((exp{𝐻₁}/𝐾₂)(1/𝜔₁)−∑^𝑛_𝑘=1(𝑐_1𝑘/𝛾_1𝑘))/𝑎₁,Δ₂>((exp{𝐻₂}/𝐾₁)(1/𝜔₂)−∑^𝑛_𝑘=1(𝑐_2𝑘/𝛾_2𝑘))/𝑎₂,Δ_𝑗>∑^𝑛_{𝑘=1,𝑘≠𝑗}𝛿_𝑗𝑘exp{𝐻_𝑘}/𝛿_𝑗𝑗, where 𝐻_𝑖Δ=maxln^𝑖𝑎_𝑖𝑖+2^𝑞_𝑘=1ln1+𝑏_𝑖𝑘,𝐻_𝑗Δ=maxln^𝑗+∑^𝑛_𝑘=1𝑑_𝑗𝑘/𝛽_𝑗𝑘𝛿_𝑗𝑗+2^𝑞_𝑘=1ln1+𝑏_𝑗𝑘, (3.2)(A7)𝐵>0,𝐵_𝑘>0,𝑘∈{1,…,𝑛+2}. Then system (1.1) has at least one positive 𝑇-periodic solution.

Proof. Let 𝕏=𝑃𝐶¹_𝑇×𝑃𝐶¹_𝑇×⋯×𝑃𝐶¹_𝑇 be the Cartesian production of (𝑛+2)-tuples, ℤ=𝕏×ℝ^(𝑛+2)𝑞, where ℝ^(𝑛+2)𝑞 is (𝑛+2)𝑞-dimensional Euclidean space. It is clear that 𝐱(0)=𝐱(𝑇) for any 𝐱∈𝕏.
We define ‖𝐱‖=²_𝑖=1𝑥_{𝑖𝑃𝐶1𝑇}+^𝑛_𝑗=1𝑦_{𝑗𝑃𝐶1𝑇}=²_𝑖=1sup_[0,𝑇]𝑥_𝑖(𝑡)+^𝑛_𝑗=1sup_[]0,𝑇𝑦_𝑗(𝑡,‖𝐳‖_ℤ)=‖𝐱‖+‖𝐜‖_∗, (3.3) for any 𝐱=(𝑥₁,𝑥₂,𝑦₁,…,𝑦_𝑛)∈𝕏,𝐜=(𝑐₁,…,𝑐_(𝑛+2)𝑞)∈ℝ^(𝑛+2)𝑞, and 𝐳=(𝐱,𝐜)∈ℤ, where ‖⋅‖ is the norm of 𝕏, and ‖⋅‖_∗ is any norm of ℝ^(𝑛+2)𝑞. Then it is trivial to check that 𝕏,ℤ are both Banach spaces when they are endowed with the above norms ‖⋅‖ and ‖⋅‖_ℤ, respectively.
Let 𝑢ℒ∶Domℒ⊂𝕏→ℤ,^𝑖(𝑡)𝑣_𝑗(𝑡)⟶𝑢̇_𝑖(𝑡)𝑣_𝑗̇(𝑡),Δ𝑢_𝑖𝑡_𝑘Δ𝑣_𝑗𝑡_𝑘^𝑞_𝑘=1,𝑢𝒩∶𝕏→ℤ,𝒩^𝑖(𝑡)𝑣_𝑗(𝑡)=𝑢̇_𝑖(𝑡)𝑣_𝑗̇(𝑡),Δ𝑢_𝑖(𝑡_𝑘)Δ𝑣_𝑗(𝑡_𝑘)^𝑞_𝑘=1. (3.4) Then 𝑢Kerℒ=^𝑖(𝑡)𝑣_𝑗(𝑡)∶𝑢_𝑖(𝑡)𝑣_𝑗(𝑡)=𝑐₁⋮𝑐_𝑛+2∈ℝ^𝑛+2,𝑓Imℒ=𝐳=ⁱ𝑓_𝑗,𝑙_𝑖𝑘𝑙̃_𝑗𝑘^𝑞_𝑘=1∈ℤ∶^𝑇₀𝑓_𝑖𝑑𝑡+^𝑞_𝑘=1𝑙_𝑖𝑘=0,^𝑇₀𝑓_𝑗𝑑𝑡+^𝑞_𝑘=1𝑙̃_𝑗𝑘=0,dimKerℒ=𝑛+2=codimImℒ. (3.5) Since Imℒ is closed in ℤ,ℒ is a Fredholm mapping of index zero. We respectively define 𝒫 and 𝒬 in the following: 𝒫𝑢_𝑖(𝑡)𝑣_𝑗(𝑡)=1𝑇^𝑇0𝑢_𝑖(𝑡)𝑑𝑡_𝑇0𝑣_𝑗(𝑡),𝑓𝑑𝑡𝒬z=𝒬^𝑖𝑓_𝑗,𝑙_𝑖𝑘𝑙̃_𝑗𝑘^𝑞_𝑘=1=1𝑇^𝑇0𝑓_𝑖𝑑𝑡+^𝑞_𝑘=1𝑙_{𝑖𝑘𝑇0}𝑓_𝑗𝑑𝑡+^𝑞_𝑘=1𝑙̃_𝑗,𝑘,0⋮0^𝑞_𝑘=1. (3.6) It is easy to show that 𝒫, 𝒬 are continuous projectors such that Imℒ=Ker𝒬=Im(𝐼−𝒬). Furthermore, the generalized inverse (ℒ)𝐾_𝑝∶Imℒ→Domℒ⋂Ker𝒫 exists. That follows one will calculate 𝐾_𝑝.
Let 𝑓𝐳=^𝑖𝑓_𝑗,𝑙_𝑖𝑘𝑙̃_𝑗𝑘^𝑞_𝑘=1∈Imℒ, (3.7) then 𝑢̇_𝑖(𝑡)𝑣_𝑗̇(𝑡)=𝑓_𝑖𝑓(𝑡)_𝑗(𝑡)_{𝑡≠𝑡𝑘},Δ𝑢_𝑖(𝑡)Δ𝑣_𝑗(𝑡)_{𝑡=𝑡𝑘}=𝑙_𝑖𝑘𝑙̃_𝑗𝑘, (3.8) that is, 𝑢_𝑖(𝑡)𝑣_𝑗(𝑡)=^𝑡0𝑓_𝑖(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}𝑙_𝑖𝑘+𝑢_𝑖(0)_𝑡0𝑓_𝑗(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}𝑙̃_𝑗𝑘+𝑣_𝑗(0). (3.9) Note that 𝑢_𝑖(𝑡)𝑣_𝑗(𝑡)1∈Ker𝒫,thatis,𝑇^𝑇₀𝑢_𝑖(𝑠)1𝑑𝑠𝑇^𝑇₀𝑣_𝑗(𝑠)=𝟎𝑑𝑠𝟎. (3.10) From(3.9), we obtain 1𝑇^𝑇₀^𝑡₀𝑢_𝑖(𝑠)𝑑𝑠𝑑𝑡+^𝑇_{0𝑡>𝑡𝑘}𝑙_𝑖𝑘𝑑𝑡+𝑇𝑢_𝑖(0)1𝑇^𝑇₀^𝑡₀𝑣_𝑗(𝑠)𝑑𝑠𝑑𝑡+^𝑇_0t>𝑡𝑘𝑙̃_𝑗𝑘𝑑𝑡+𝑇𝑣_𝑗(0)=𝟎𝟎, (3.11) and hence, 𝑢_𝑖(𝑡)𝑣_𝑗(𝑡)=^𝑡0𝑢_𝑖(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}𝑙_𝑖𝑘−1𝑇^𝑇₀^𝑡₀𝑢_𝑖(𝑠)𝑑𝑠𝑑𝑡−^𝑞_𝑘=1𝑙_𝑖𝑘+1𝑇^𝑞_𝑘=1𝑙_𝑖𝑘𝑡_𝑘𝑡0𝑣_𝑗(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}𝑙̃_𝑗𝑘−1𝑇^𝑇₀^𝑡₀𝑣_𝑗(𝑠)𝑑𝑠𝑑𝑡−^𝑞_𝑘=1𝑙̃_𝑗𝑘+1𝑇^𝑞_𝑘=1𝑙̃_𝑗𝑘𝑡_𝑘, (3.12) that is, 𝐾_𝑝z=^𝑡0𝑓_𝑖(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}𝑙_𝑖𝑘−1𝑇^𝑇₀^𝑡₀𝑓_𝑖(𝑠)𝑑𝑠𝑑𝑡−^𝑞_𝑘=1𝑙_{𝑖𝑘𝑡0}𝑓_𝑗(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}𝑙̃_𝑗𝑘−1𝑇^𝑇₀^𝑡₀𝑓_𝑗(𝑠)𝑑𝑠𝑑𝑡−^𝑞_𝑘=1𝑙̃_𝑗𝑘. (3.13) Thus 𝑢𝒬𝒩=^𝑖(𝑡)𝑣_𝑗(𝑡)=ℎ_𝑖ℎ_𝑗,00^𝑞_𝑘=1, (3.14) where ℎ₁=Δ₁−1𝑇^𝑇₀𝑎₁(𝑡)exp𝑢₁(𝑡+𝑢)exp₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₂(𝑡+_𝑛𝑘=1)𝑐_1𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)exp𝑢₁(𝑡)+𝛾_1𝑘(𝑡)exp𝑣_𝑘(𝑡ℎ)𝑑𝑡,₂=Δ₂−1𝑇^𝑇₀𝑎₂(𝑡)exp𝑢₂(𝑡+𝑢)exp₂(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₁(𝑡+_𝑛𝑘=1)𝑐_2𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)exp𝑢₂(𝑡)+𝛾_2𝑘(𝑡)exp𝑣_𝑘(𝑡ℎ)𝑑𝑡,_𝑗=Δ_𝑗+1𝑇^𝑇₀²_𝑘=1𝑑_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)exp𝑣_𝑗(𝑡−_𝑛𝑘=1)𝛿_𝑗𝑘(𝑡)exp𝑣_𝑘(𝑡)𝑑𝑡. (3.15) Clearly, 𝒬𝒩 and 𝐾_𝑝(𝐼−𝒬)𝒩 are continuous. Using the Arzela-Ascoli theorem, it is not diffcult to show that 𝐾_𝑝(𝐼−𝒬)𝒩(Ω) is compact for any open bounded set Ω⊂𝕏. Moreover, 𝒬𝒩(Ω) is bounded. Thus, 𝒩 is ℒ-compact on Ω with any open bounded set Ω⊂𝕏. A isomorphism 𝒥 of Im𝒬 onto Kerℒ can be chosen to be the identity mapping; since Im𝒬=Kerℒ, there exists an isomorphism 𝒥∶Im𝒬→𝕏 given by 𝑔𝒥∶Im𝒬⟶𝕏,⋮¹𝑔_𝑛+2,0⋮0^𝑞_𝑘=1⟶𝑔₁⋮𝑔_𝑛+2. (3.16) Corresponding to operator equation ℒ𝐮=𝜆𝒩𝐮,𝐮(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),𝑣₁(𝑡),…,𝑣_𝑛(𝑡))^𝑇,𝜆∈(0,1), we have 𝑢̇₁(𝑡)=𝜆𝑏₁(𝑡)−𝑎₁(𝑡)exp𝑢₁(𝑡−𝑢)exp₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₂(𝑡−_𝑛𝑘=1)𝑐_1𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)exp𝑢₁(𝑡)+𝛾_1𝑘(𝑡)exp𝑣_𝑘(𝑡,)𝑢̇₂(𝑡)=𝜆𝑏₂(𝑡)−𝑎₂(𝑡)exp𝑢₂(𝑡−𝑢)exp₂(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₁(𝑡−_𝑛𝑘=1)𝑐_2𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)exp𝑢₂(𝑡)+𝛾_2𝑘(𝑡)exp𝑣_𝑘(𝑡,)𝑣_𝑗̇(𝑡)=𝜆−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)exp𝑣_𝑗(𝑡−_𝑛𝑘=1)𝛿_𝑗𝑘(𝑡)exp𝑣_𝑘(𝑡),𝑡≠𝑡_𝑘,𝑘∈𝑁⁺,Δ𝑢_𝑖(𝑡)=𝑢_𝑖𝑡⁺−𝑢_𝑖(𝑡⁻)=𝜆ln1+𝑏_𝑖𝑘,Δ𝑣_𝑗(𝑡)=𝑣_𝑗𝑡⁺−𝑣_𝑗(𝑡⁻)=𝜆ln1+𝑏̃_𝑗𝑘,𝑡=𝑡_𝑘,𝑘∈𝑁⁺. (3.17)
Suppose that 𝐮(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),𝑣₁(𝑡),…,𝑣_𝑛(𝑡))^𝑇 is a solution of system (3.17) for a fixed 𝜆∈(0,1). Integrating on both sides of (3.17) from 0 to 𝑇, we obtain ^𝑇₀−𝑏₁(𝑡)+𝑎₁(𝑡)exp𝑢₁(𝑡+𝑢)exp₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₂(𝑡+_𝑛𝑘=1)𝑐_1𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)exp𝑢₁(𝑡)+𝛾_1𝑘(𝑡)exp𝑣_𝑘(𝑡=)𝑑𝑡_𝑞𝑘=1ln1+𝑏_1𝑘,_𝑇0−𝑏₂(𝑡)+𝑎₂(𝑡)exp𝑢₂(𝑡+𝑢)exp₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)exp𝑢₁(𝑡+_𝑛𝑘=1)𝑐_2𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)exp𝑢₂(𝑡)+𝛾_2𝑘(𝑡)exp𝑣_𝑘(𝑡=)𝑑𝑡_𝑞𝑘=1ln1+𝑏_2𝑘,_𝑇0𝑟_𝑗(𝑡)−²_𝑘=1𝑑_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)exp𝑣_𝑗(𝑡+_𝑛𝑘=1)𝛿_𝑗𝑘(𝑡)exp𝑣_𝑘(𝑡)𝑑𝑡=^𝑞_𝑘=1ln1+𝑏̃_𝑗𝑘. (3.18) Since 𝐮∈𝕏, there exist 𝜉_𝑖,𝜉_𝑗∈[0,𝑇] such that 𝑢_𝑖𝜉_𝑖=min_{[]𝑡∈0,𝑇}𝑢_𝑖(𝑡),𝑣_𝑗𝜉_𝑗=min_{[]𝑡∈0,𝑇}𝑣_𝑗(𝑡). (3.19) On the other hand, note that there exist 𝜁⁺_𝑖,𝜁⁺_𝑗∈[0,𝑇] such that 𝑢_𝑖𝜁⁺_𝑖=sup_{[]𝑡∈0,𝑇}𝑢_𝑖(𝑡),𝑣_𝑗𝜁⁺_𝑗=sup_{[]𝑡∈0,𝑇}𝑣_𝑗(𝑡). (3.20) If 𝜁_𝑖≠𝑡_𝑘, then 𝑢_𝑖(𝜁⁺_𝑖)=𝑢_𝑖(𝜁_𝑖),𝑣_𝑗(𝜁⁺_𝑗)=𝑣_𝑗(𝜁_𝑗). While if 𝜁_𝑗=𝑡_𝑘, we have 𝑢_𝑖(𝜁⁺_𝑖)=𝑢_𝑖(𝑡⁺_𝑘),𝑣_𝑗(𝜁⁺_𝑗)=𝑣_𝑗(𝑡⁺_𝑘). Thus we can obtain from (3.18) and (3.19) that ^𝑇₀𝑎_𝑖(𝑡)exp𝑢_𝑖𝜉_𝑖𝑑𝑡⩽^𝑇₀𝑎_𝑖(𝑡)𝑒^{𝑢𝑖(𝑡)}⩽𝑑𝑡_𝑞𝑘=1ln1+𝑏_𝑖𝑘+^𝑇₀𝑏_𝑖(𝑡)𝑑𝑡,_𝑇0𝛿_𝑗𝑗(𝑡)exp𝑣_𝑗𝜉_𝑗𝑑𝑡⩽^𝑇₀𝛿_𝑗𝑗(𝑡)exp𝑣_𝑗(𝑡⩽)𝑑𝑡_𝑞𝑘=1ln1+𝑏̃_𝑗𝑘−^𝑇₀𝑟_𝑗(𝑡)𝑑𝑡+^𝑛_𝑘=1^𝑇₀𝑑_𝑗𝑘(𝑡)𝛽_𝑗𝑘(𝑡)𝑑𝑡. (3.21) It follows that 𝑢_𝑖𝜉_𝑖Δ⩽maxln^𝑖𝑎_𝑖∶=𝑝,𝑣_𝑗𝜉_𝑗Δ⩽maxln^𝑗+∑^𝑛_𝑘=1𝑑_𝑗𝑘/𝛽_𝑗𝑘𝛿_𝑗𝑗∶=𝑔. (3.22) From (3.17) and (3.18), we obtain 𝑢̇^𝑇_0𝑖(𝑡)𝑑𝑡⩽2^𝑞_𝑘=1ln1+𝑏_𝑖𝑘,^𝑇₀𝑣_𝑗̇(𝑡)𝑑𝑡⩽2^𝑞_𝑘=1ln1+𝑏̃_𝑗𝑘. (3.23) By (3.22) and (3.23), we have that 𝑢_𝑖(𝑡)⩽𝑢_𝑖𝜉_𝑖+^𝑇₀𝑢̇_𝑖(𝑡)𝑑𝑡⩽𝑝+2^𝑞_𝑘=1ln1+𝑏_𝑖𝑘∶=𝐻_𝑖,𝑣_𝑗(𝑡)⩽𝑣_𝑗𝜉_𝑗+^𝑇₀𝑣_𝑗̇(𝑡)𝑑𝑡⩽𝑔+2^𝑞_𝑘=1ln1+𝑏̃_𝑗𝑘∶=𝐻_𝑗. (3.24) On the other hand, by (3.18) and (3.20), we also have ^𝑇₀𝛿_𝑗𝑗(𝑡)exp𝑣_𝑗𝜁⁺_𝑗𝑑𝑡⩾^𝑇₀𝛿_𝑗𝑗(𝑡)exp𝑣_𝑗(𝑡⩾)𝑑𝑡_𝑞𝑘=1ln1+𝑏̃_𝑗𝑘−^𝑇₀𝑟_𝑗(𝑡)𝑑𝑡−^𝑛_{𝑘=1,𝑘≠𝑗}^𝑇₀𝛿_𝑗𝑘(𝑡)exp𝐻_𝑘𝑣𝑑𝑡,_𝑗𝜁⁺_𝑗Δ⩾ln^𝑗−∑^𝑛_{𝑘=1,𝑘≠𝑗}𝛿_𝑗𝑘exp𝐻_𝑘𝛿_𝑗𝑗, (3.25) and hence, 𝑣_𝑗(𝑡)⩾𝑣_𝑗𝜁⁺_𝑗−^𝑇₀𝑣_𝑗̇(𝑡Δ)𝑑𝑡⩾ln^𝑗−∑^𝑛_{𝑘=1,𝑘≠𝑗}𝛿_𝑗𝑘exp𝐻_𝑘𝛿_𝑗𝑗−2^𝑞_𝑘=1ln1+𝑏̃_𝑗𝑘∶=𝑀_𝑗. (3.26)By (3.18), we have^𝑇₀𝑎₁(𝑡)exp𝑢₁𝜁⁺₁𝑑𝑡⩾^𝑇₀𝑎₁(𝑡)exp𝑢₁(𝑡⩾)𝑑𝑡_𝑞𝑘=1ln1+𝑏_1𝑘+^𝑇₀𝑏₁(𝑡)𝑑𝑡−^𝑇₀exp𝑢₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₂(𝑡−)𝑑𝑡_{𝑛𝑘=1𝑇0}𝑐_1𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)exp𝑢₁(𝑡)+𝛾_1𝑘(𝑡)exp𝑣_𝑘(𝑡⩾)𝑑𝑡_𝑞𝑘=1ln1+𝑏_1𝑘+^𝑇₀𝑏₁(𝑡)𝐻𝑑𝑡−exp₁𝐾₂^𝑇₀1𝜔₁(𝑡)−𝑑𝑡_{𝑛𝑘=1𝑇0}𝑐_1𝑘(𝑡)𝛾_1𝑘(𝑡)𝑑𝑡,_𝑇0𝑎₂(𝑡)exp𝑢₂𝜁⁺₂𝑑𝑡⩾^𝑇₀𝑎₂(𝑡)exp𝑢₂(𝑡⩾)𝑑𝑡_𝑞𝑘=1ln1+𝑏_2𝑘+^𝑇₀𝑏₂(𝑡)𝑑𝑡−^𝑇₀exp𝑢₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)exp𝑢₁(𝑡−)𝑑𝑡_{𝑛𝑘=1𝑇0}𝑐_2𝑘(𝑡)exp𝑣_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_1𝑘(𝑡)exp𝑢₂(𝑡)+𝛾_2𝑘(𝑡)exp𝑣_𝑘(𝑡⩾)𝑑𝑡_𝑞𝑘=1ln1+𝑏_2𝑘+^𝑇₀𝑏₂(𝑡)𝐻𝑑𝑡−exp₂𝐾₁^𝑇₀1𝜔₂(𝑡)−𝑑𝑡_{𝑛𝑘=1𝑇0}𝑐_2𝑘(𝑡)𝛾_2𝑘(𝑡)𝑑𝑡, (3.27) where 𝐾_𝑖 is a positive number satisfied exp{𝑢_𝑖(𝑡)}⩾𝐾_𝑖 for 𝑡∈[0,𝑇]; therefore, 𝑢₁𝜁⁺₁Δ⩾ln¹−exp𝐻₁/𝐾₂1/𝜔₁−∑^𝑛_𝑘=1𝑐_1𝑘/𝛾_1𝑘𝑎₁,𝑢₂𝜁⁺₂Δ⩾ln²−exp𝐻₂/𝐾₁1/𝜔₂−∑^𝑛_𝑘=1𝑐_2𝑘/𝛾_2𝑘𝑎₂. (3.28) So 𝑢₁(𝑡)⩾𝑢₁𝜁⁺₁−^𝑇₀𝑢̇₁(𝑡Δ)𝑑𝑡⩾ln¹−exp𝐻₁/𝐾₂1/𝜔₁−∑^𝑛_𝑘=1𝑐_1𝑘/𝛾_1𝑘𝑎₁−2^𝑞_𝑘=1ln1+𝑏_1𝑘∶=𝑁₁,𝑢₂(𝑡)⩾𝑢₂𝜁⁺₂−^𝑇₀𝑢̇₂(𝑡Δ)𝑑𝑡⩾ln²−exp𝐻₂/𝐾₁1/𝜔₂−∑^𝑛_𝑘=1𝑐_2𝑘/𝛾_2𝑘𝑎₂−2^𝑞_𝑘=1ln1+𝑏_2𝑘∶=𝑁₂. (3.29) Take 𝑀_𝑖=min{𝐾_𝑖,𝑁_𝑖}. By (3.24)–(3.29), we have𝑢_𝑖(𝑡)<max𝑀_𝑖,𝐻_𝑖∶=𝐺_𝑖,𝑣_𝑗(𝑡)<max𝑀_𝑗,𝐻_𝑗∶=𝐺_𝑗. (3.30) Here, 𝐺_𝑖,𝐺_𝑗 are independent of the choice of 𝜆.
Let 𝐮=(𝑢₁(𝑡),𝑢₂(𝑡),𝑣₁(𝑡),…,𝑣_𝑛(𝑡))^𝑇∈ℝ^𝑛+2, set 𝐺=‖𝐮‖+𝐺₀, where 𝐺₀ is taken sufficiently large, such that the solution 𝐮^∗=(𝑢^∗₁,𝑢^∗₂,𝑣^∗₁,…,𝑣^∗_𝑛)^𝑇 of the equation 𝑏₁−𝑎₁exp𝑢₁(𝑡−1)𝑇^𝑇₀exp𝑢₁(𝑡)𝑑𝑡𝜇₁(𝑡)+𝜔₁(𝑡)exp𝑢₂(𝑡−1)𝑇^𝑛_𝑘=1^𝑇₀𝑐_1𝑘(𝑡)exp𝑣_𝑘(𝑡)𝑑𝑡𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)exp𝑢₁(𝑡)+𝛾_1𝑘(𝑡)exp𝑣_𝑘(𝑡+1)𝑇^𝑞_𝑘=1ln1+𝑏_1𝑘=0,𝑏₂−𝑎₂exp𝑢₂(𝑡−1)𝑇^𝑇₀exp𝑢₂(𝑡)𝑑𝑡𝜇₂(𝑡)+𝜔₂(𝑡)exp𝑢₁(𝑡−1)𝑇^𝑛_𝑘=1^𝑇₀𝑐_2𝑘(𝑡)exp𝑣_𝑘(𝑡)𝑑𝑡𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)exp𝑢₂(𝑡)+𝛾_2𝑘(𝑡)exp𝑣_𝑘(𝑡+1)𝑇^𝑞_𝑘=1ln1+𝑏_2𝑘−=0,𝑟_𝑗+1𝑇²_𝑘=1^𝑇₀𝑑_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)𝑑𝑡𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)exp𝑢_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)exp𝑣_𝑗(𝑡−_𝑛𝑘=1)𝛿_𝑗𝑘exp𝑣_𝑘(𝑡+1)𝑇^𝑞_𝑘=1ln1+𝑏̃_𝑗𝑘=0 (3.31) satisfies 𝑢^∗₁,𝑢^∗₂,𝑣^∗₁,…,𝑣^∗_𝑛^𝑇<𝐺₀,𝐮^∗𝑡_𝑘+0<𝐺,𝑘=1,2,…,𝑞. (3.32)
Let Ω={𝐮=(𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛)^𝑇∈𝕏∣‖(𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛)^𝑇‖<𝐺}. Then it is clear that Ω satisfies the requirement (a) of Lemma 2.3.
When 𝐮∈𝜕Ω∩Kerℒ=𝜕Ω∩ℝ^𝑛+2, 𝐮 is a constant vector in ℝ^𝑛+2 with ‖𝐮‖=𝐺, then 𝒬𝒩𝐮≠0, we check 𝒬𝒩𝐮 and according to mean value theorem, there exist 𝜃₁,𝜃₂,𝜃_𝑗,𝜂₁,𝜂₂∈[0,𝑇] such that Δ𝒬𝒩𝐮=¹−𝑎₁exp𝑢₁(𝑡−𝑢)exp₁(𝑡)𝜇₁𝜃₁+𝜔₁𝜃₁exp𝑢₂(𝑡−_𝑛𝑘=1)𝑐_1𝑘exp𝑣_𝑘(𝑡)𝛼_1𝑘𝜂₁+𝛽_1𝑘𝜂₁exp𝑢₁(𝑡)+𝛾_1𝑘𝜂₁exp𝑣_𝑘(𝑡Δ₂)−𝑎₂exp𝑢₂(𝑡−𝑢)exp₂(𝑡)𝜇₂𝜃₂+𝜔₂𝜃₂exp𝑢₁(𝑡−_𝑛𝑘=1)𝑐_2𝑘exp𝑣_𝑘(𝑡)𝛼_2𝑘𝜂₂+𝛽_2𝑘𝜂₂exp𝑢₂(𝑡)+𝛾_2𝑘𝜂₂exp𝑣_𝑘(𝑡Δ_𝑗)+²_𝑘=1𝑑_𝑗𝑘exp𝑢_𝑘(𝑡)𝛼_𝑗𝑘𝜃_𝑗+𝛽_𝑗𝑘𝜃_𝑗exp𝑢_𝑘(𝑡)+𝛾_𝑗𝑘𝜃_𝑗exp𝑣_𝑗(𝑡−_𝑛𝑘=1)𝛿_𝑗𝑘exp𝑣_𝑘(𝑡0⋮0)^𝑞𝑘=1. (3.33) For any 𝐮∈Ω∩Kerℒ, we have Δ𝒥𝒬𝒩𝐮=¹−𝑎₁exp𝑢₁(𝑡−𝑢)exp₁(𝑡)𝜇₁𝜃₁+𝜔₁𝜃₁exp𝑢₂(𝑡−_𝑛𝑘=1)𝑐_1𝑘exp𝑣_𝑘(𝑡)𝛼_1𝑘𝜂₁+𝛽_1𝑘𝜂₁exp𝑢₁(𝑡)+𝛾_1𝑘𝜂₁exp𝑣_𝑘(𝑡Δ₂)−𝑎₂exp𝑢₂(𝑡−𝑢)exp₂(𝑡)𝜇₂𝜃₂+𝜔₂𝜃₂exp𝑢₁(𝑡−_𝑛𝑘=1)𝑐_2𝑘exp𝑣_𝑘(𝑡)𝛼_2𝑘𝜂₂+𝛽_2𝑘𝜂₂exp𝑢₂(𝑡)+𝛾_2𝑘𝜂₂exp𝑣_𝑘(𝑡Δ_𝑗)+²_𝑘=1𝑑_𝑗𝑘exp𝑢_𝑘(𝑡)𝛼_𝑗𝑘𝜃_𝑗+𝛽_𝑗𝑘𝜃_𝑗exp𝑢_𝑘(𝑡)+𝛾_𝑗𝑘𝜃_𝑗exp𝑣_𝑗(𝑡−_𝑛𝑘=1)𝛿_𝑗𝑘exp𝑣_𝑘(𝑡). (3.34) Define the map 𝜓∶(Ω∩ℝ^𝑛+2)×[0,1]→ℝ^𝑛+2 by 𝜓𝑢₁𝑢₂𝑣_𝑗=Δ₁−𝑎₁exp𝑢₁(𝑡Δ₂)−𝑎₂exp𝑢₂(𝑡Δ_𝑗)−^𝑛_𝑘=1𝛿_𝑗𝑘exp𝑣_𝑘(𝑡𝑢)−𝜇exp₁(𝑡)𝜇₁𝜃₁+𝜔₁𝜃₁exp𝑢₂(𝑡+_𝑛𝑘=1)𝑐_1𝑘exp𝑣_𝑘(𝑡)𝛼_1𝑘𝜂₁+𝛽_1𝑘𝜂₁exp𝑢₁(𝑡)+𝛾_1𝑘𝜂₁exp𝑣_𝑘(𝑡𝑢)exp₂(𝑡)𝜇₂𝜃₂+𝜔₂𝜃₂exp𝑢₁(𝑡+_𝑛𝑘=1)𝑐_2𝑘exp𝑣_𝑘(𝑡)𝛼_2𝑘𝜂₂+𝛽_2𝑘𝜂₂exp𝑢₂(𝑡)+𝛾_2𝑘𝜂₂exp𝑣_𝑘(𝑡−_2𝑘=1)𝑑_𝑗𝑘exp𝑢_𝑘(𝑡)𝛼_𝑗𝑘𝜃_𝑗+𝛽_𝑗𝑘𝜃_𝑗exp𝑢_𝑘(𝑡)+𝛾_𝑗𝑘𝜃_𝑗exp𝑣_𝑗(𝑡,) (3.35) where (𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛)∈ℝ^𝑛+2,𝜇∈[0,1]. We can show that 𝜓𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛,𝜇≠0, (3.36) for any 𝐮=(𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛)^𝑇∈𝜕Ω∩ℝ^𝑛+2, 𝜇∈[0,1]. Otherwise, it is similar to above mentioned discussion, we obtain that the solution 𝐮=(𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛)^𝑇 of 𝜓(𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛,𝜇)=0 satisfies ‖𝐮‖<𝐺; therefore, this contradicts the fact that 𝐮∈𝜕Ω∩ℝ^𝑛+2.
Since 𝜓(𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛,𝜇) is a homotopic mapping and topological degree is invariant under homotopic mapping, thus we can show the topological degree as folows: deg𝒥𝒬𝒩,Ω∩Kerℒ,(0,…,0)^𝑇=deg𝜓_𝜇=1,Ω∩Kerℒ,(0,…,0)^𝑇=deg𝜓_𝜇=0,Ω∩Kerℒ,(0,…,0)^𝑇. (3.37) On the other hand, the differential equations 𝑏_𝑖−𝑎_𝑖𝑥^∗_𝑘+1𝑇^𝑞_𝑘=1ln1+𝑏_𝑖𝑘−=0,𝑟_𝑗−^𝑛_𝑘=1𝛿_𝑗𝑘𝑦^∗_𝑘+1𝑇^𝑞_𝑘=1ln1+𝑏̃_𝑗𝑘=0 (3.38) satisfies condition (A7), by the law of Crammer, one can easily obtain that (3.38) has unique solution 𝑥^∗₁,𝑥^∗₂,𝑦^∗₁,…,𝑦^∗_𝑛, (3.39) where 𝑥^∗_𝑖=det(𝐵_𝑖)/det(𝐵),𝑦^∗_𝑗=det(𝐵_𝑖)/det(𝐵). Therefore, one can check deg𝒥𝒬𝒩,Ω∩Kerℒ,(0,…,0)^𝑇=deg𝜓_𝜇=0,Ω∩Kerℒ,(0,…,0)^𝑇=sgn(−1)^𝑛+2𝑥^∗₁𝑥^∗₂^𝑛_𝑗=1𝑦^∗_𝑗det(𝐵)≠0. (3.40)
By now we have proved that Ω satisfies all requirements of Lemma 2.3, then ℒ𝐱=𝒩𝐱 has at least one solution in Domℒ∩Ω, that is, (2.3) has at least one 𝑇-periodic solution in Domℒ∩Ω and denote 𝐮=(𝑢₁,𝑢₂,𝑣₁,…,𝑣_𝑛)^𝑇 by the solution of system (2.3). Set 𝑥_𝑖=exp{𝑢_𝑖},𝑦_𝑗=exp{𝑣_𝑗}, then 𝐱=(𝑥₁,𝑥₂,𝑦₁,…,𝑦_𝑛)^𝑇 is one positive 𝑇-periodic solution of system (1.3). The proof is complete.

4. Global Stability and Globally Exponential Stability of Solutions

Let 𝐱₀=(𝑥₁₀,𝑥₂₀,𝑦_1,0,…,𝑦_𝑛,0) and 𝑥_𝑖0,𝑦_𝑗0∈𝑅₊. We denote by 𝐱(𝑡)=𝐱𝑡;𝑡₀,𝑥₀=𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),…,𝑦_𝑛(𝑡) (4.1) the solution of system (1.3) satisfying the initial conditions: 𝐱𝑡₀+0;𝑡₀,𝑥₀=𝐱₀ (4.2) and by 𝐽=𝐽(𝑡₀,𝐱₀) the maximal interval of type [𝑡₀,𝜂) in which the solution 𝐱(𝑡;𝑡₀,𝑥₀) is defined.

Let 𝐱(𝑡;𝑡₀,𝑥₀),𝐱(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),…,𝑦_𝑛(𝑡)) and 𝐱^∗(𝑡;𝑡₀,𝑥^∗₀), 𝐱^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),𝑦^∗₁(𝑡),…,𝑦^∗_𝑛(𝑡)) be any two solutions of (1.1) with initial conditions: 𝐱𝑡₀+0;𝑡₀,𝑥₀=𝐱₀,𝐱^∗𝑡₀+0;𝑡₀,𝑥^∗₀=𝐱^∗₀,𝑡₀∈𝑅₊. (4.3) We introduce the following notations: 𝐺_𝑘=𝑡_𝑘−1,𝑡_𝑘×ℝ₊^𝑛+2,𝑘∈ℤ⁺,𝐺=^∞_𝑘=1𝐺_𝑘. (4.4) We put forward two definitions in [24, 28].

Definition 4.1 4.1 (Ahmad and Stamova [24]). The system (1.3) is said to be(a)globally stable if for all 𝜀>0, there exists 𝛿=𝛿(𝜀,𝑡₀)>0 such that if 𝐱(𝑡),𝐱^∗(𝑡)∈ℝ₊^𝑛+2, with ‖𝐱₀−𝐱^∗₀‖⩽𝛿 then for all 𝑡⩾𝑡₀‖𝐱(𝑡;𝑡₀,𝑥₀)−𝐱^∗(𝑡;𝑡₀,𝑥^∗₀)‖<𝜀; (b)globally asymptotically stable if it is globally stable and lim_𝑡→∞𝐱𝑡;𝑡₀,𝑥₀−𝐱^∗𝑡;𝑡₀,𝑥^∗₀=0; (4.5)(c)globally exponentially stable if for all 𝛼>0, there exists 𝛾=𝛾(𝛼)>0 such that 𝐱(𝑡),𝐱^∗(𝑡)∈ℝ₊^𝑛+2, with ‖𝐱₀−𝐱^∗₀‖⩽𝛼, then for all 𝑡⩾𝑡₀𝐱𝑡;𝑡₀,𝑥₀−𝐱^∗𝑡;𝑡₀,𝑥^∗₀<𝛾𝐱₀−𝐱^∗₀exp−𝜑𝑡−𝑡₀. (4.6)

Definition 4.2 4.2 (Ahmad and Stamova [24, 28]). We say that the function 𝑉(𝑡,𝐱),𝑉∶[𝑡₀,∞)×ℝ₊^𝑛+2 belongs to the class 𝑉₀ if the following conditions are satisfied: (1)the function 𝑉 is continuous in ⋃^∞_𝑖=1𝐺_𝑖, and 𝑉(𝑡,0)=0 for 𝑡∈[𝑡₀,∞);(2)the function V satisfies locally the Lipschitz condition with respect to 𝐱 on each of the sets 𝐺_𝑖;(3)for each 𝑘∈ℤ⁺, there exist the finite limits: 𝑉𝑡_𝑘−0,𝐱=lim_{𝑡→𝑡𝑘,𝑡<𝑡𝑘}𝑉(𝑡,𝐱),𝑉𝑡_𝑘+0,𝐱=lim_{𝑡→𝑡𝑘,𝑡>𝑡𝑘}𝑉(𝑡,𝐱); (4.7)(4)for each 𝑘∈ℤ⁺, the following equalities are valid: 𝑉𝑡_𝑘−1,𝐱=𝑉𝑡_𝑘,𝐱. (4.8)
In the proofs of the main theorems, we will use the following comparison results.

Lemma 4.3. Suppose the hypotheses (A1)–(A5) hold. There exist functions 𝑃_𝑖,𝑄_𝑖,𝑃_𝑗,𝑄_𝑗∈ℝ^𝑛+2 such that 𝑃_𝑖(𝑡)⩽𝑥_𝑖(𝑡)⩽𝑄_𝑖(𝑡),𝑃_𝑗(𝑡)⩽𝑦_𝑗(𝑡)⩽𝑄_𝑗(𝑡) for all 𝑡⩾𝑡₀.

Proof. First we will prove that 𝑥_𝑖(𝑡)⩽𝑄_𝑖(𝑡),𝑦_𝑗(𝑡)⩽𝑄_𝑗(𝑡), (4.9) for all 𝑡⩾𝑡₀, where (𝑄_𝑖(𝑡),𝑄_𝑗(𝑡))=(𝑞₁(𝑡),𝑞₂(𝑡),𝑞₁(𝑡),…,𝑞_𝑛(𝑡)) is the maximal solution of the system: 𝑞_𝑖̇(𝑡)=𝑞_𝑖(𝑡)𝑏_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑞_𝑖(𝑡),𝑞̇_𝑗(𝑡)=𝑞_𝑗(𝑡)−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑞_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑞_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑞_𝑗(𝑡),𝑡≠𝑡_𝑘,𝑘∈ℤ⁺,𝑞_𝑖𝑡⁺₀=𝑞_𝑖0>0,𝑞_𝑗𝑡⁺₀=𝑞_𝑗0𝑞>0,_𝑖𝑡⁺_𝑘=1+𝑏_𝑖𝑘𝑞_𝑖𝑡_𝑘,𝑞_𝑗𝑡⁺_𝑘=1+𝑏̃_𝑗𝑘𝑞_𝑗𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈ℤ⁺. (4.10) The maximal solution 𝑄_𝑖(𝑡)=𝑄_𝑖𝑡;𝑡₀,𝑞₀,𝑄_𝑗(𝑡)=𝑄_𝑖𝑡;𝑡₀,𝑞₀,𝑞₀=𝑞₁₀,𝑞₂₀,𝑞₃₀,…,𝑞_(𝑛+2)0 (4.11) of (4.10) is defined by the equality: 𝑄_𝑖𝑡;𝑡₀,𝑞₀=𝑄⁰_𝑖𝑡;𝑡₀,𝑄⁰_𝑖+0,𝑡₀<𝑡⩽𝑡₁,𝑄¹_𝑖𝑡;𝑡₁,𝑄¹_𝑖+0,𝑡₁<𝑡⩽𝑡₂,𝑄…,_𝑘𝑖𝑡;𝑡_𝑘,𝑄^𝑘_𝑖+0,𝑡_𝑘<𝑡⩽𝑡_𝑘+1,𝑄…,_𝑗𝑡;𝑡₀,𝑞₀=𝑄⁰_𝑗𝑡;𝑡₀,𝑄⁰_𝑗+0,𝑡₀<𝑡⩽𝑡₁,𝑄¹_𝑗𝑡;𝑡₁,𝑄¹_𝑗+0,𝑡₁<𝑡⩽𝑡₂,𝑄…,_𝑘𝑗𝑡;𝑡_𝑘,𝑄^𝑘_𝑗+0,𝑡_𝑘<𝑡⩽𝑡_𝑘+1,…, (4.12) where 𝑄^𝑖_𝑘(𝑡;𝑡_𝑘,𝑄^𝑘_𝑖+0), 𝑄^𝑗_𝑘(𝑡;𝑡_𝑘,𝑄^𝑘_𝑗+0) is the solution of the equation without impulses 𝑞_𝑖̇(𝑡)=𝑞_𝑖(𝑡)𝑏_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑞_𝑖(𝑡),𝑞̇_𝑗(𝑡)=𝑞_j(𝑡)−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑞_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑞_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑞_𝑗(𝑡) (4.13) in the interval (𝑡_𝑘,𝑡_𝑘+1], 𝑘∈ℤ⁺, for which 𝑄^𝑘_𝑖+0=(1+𝑏_𝑖𝑘)𝑄^𝑘_𝑖(𝑡_𝑘;𝑡_𝑘−1,𝑄_𝑖^𝑘−1+0), 𝑄^𝑘_𝑗+0=(1+𝑏̃_𝑗𝑘), 𝑄^𝑘_𝑗(𝑡_𝑘;𝑡_𝑘−1,𝑄_𝑗^𝑘−1+0), 𝑘∈ℤ⁺, and 𝑄⁰_𝑖+0=𝑞_𝑖0, 𝑄⁰_𝑗+0=𝑞_𝑗0.
We note that the solutions 𝑥_𝑖(𝑡),𝑦_𝑗(𝑡) of (1.1) are functions which for 𝑡≠𝑡_𝑘,𝑘∈ℤ⁺ satisfy 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑏₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)−𝑥₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥₂(𝑡)−^𝑛_𝑘=1𝑐_1𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥₁(𝑡)+𝛾_1𝑘(𝑡)𝑦_𝑘(𝑡),𝑥̇₂(𝑡)=𝑥₂(𝑡)𝑏₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)−𝑥₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥₁(𝑡)−^𝑛_𝑘=1𝑐_2𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥₂(𝑡)+𝛾_2𝑘(𝑡)𝑦_𝑘(𝑡),𝑦_𝑗̇(𝑡)=𝑦_𝑗(𝑡)−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑘=1𝛿_𝑗𝑘(𝑡)𝑦_𝑘(𝑡), (4.14) and for 𝑡=𝑡_𝑘, 𝑘∈ℤ⁺, satisfying the conditions 𝑥_𝑖(𝑡⁺_𝑘)=(1+𝑏_𝑖𝑘)𝑥_𝑖(𝑡_𝑘), 𝑦_𝑗(𝑡⁺_𝑘)=(1+𝑏̃_𝑗𝑘)𝑦_𝑗(𝑡_𝑘), By hypothesis (A1), it follows from (1.1) that 𝑥̇_𝑖(𝑡)⩽𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡),𝑦_𝑗̇(𝑡)⩽𝑦_𝑗(𝑡)−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑞_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑞_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑞_𝑗(𝑡), (4.15) for 𝑡≠𝑡_𝑘. The elementary differential inequality (4.15) yields that 𝑥_𝑖(𝑡)⩽𝑄_𝑖(𝑡),𝑦_𝑗(𝑡)⩽𝑄_𝑗(𝑡), (4.16) for all 𝑡∈(𝑡₀,𝑡₁], that is, the inequality (4.9) is valid for 𝑡∈(𝑡₀,𝑡₁]. Suppose that (4.9) is satisfied for 𝑡∈(𝑡_𝑘−1,𝑡_𝑘],𝑘>1. Then using hypotheses (A2)–(A4) and the fact that (4.9) is satisfied for 𝑡=𝑡_𝑘, we obtain 𝑥_𝑖𝑡⁺_𝑘=1+𝑏_𝑖𝑘𝑥_𝑖𝑡_𝑘⩽1+𝑏_𝑖𝑘𝑄_𝑖𝑡_𝑘=𝑄_𝑖𝑡⁺_𝑘,𝑦_𝑗𝑡⁺_𝑘=1+𝑏̃_𝑗𝑘𝑦_𝑗𝑡_𝑘⩽1+𝑏̃_𝑗𝑘𝑄_𝑗𝑡_𝑘=𝑄_𝑗𝑡⁺_𝑘. (4.17) We apply again the comparison result (4.15) in the interval (𝑡_𝑘,𝑡_𝑘+1] and obtain 𝑥_𝑖𝑡;𝑡₀,𝑥_𝑖0⩽𝑄^𝑘_𝑖𝑡;𝑡_𝑘,𝑄^𝑘_𝑖+0=𝑄_𝑖𝑡;𝑡₀,𝑞₀,𝑦_𝑗𝑡;𝑡₀,𝑦_𝑗0⩽𝑄^𝑘_𝑗𝑡;𝑡_𝑘,𝑄^𝑘_𝑗+0=𝑄_𝑗𝑡;𝑡₀,𝑞₀, (4.18) that is, the inequality (4.9) is valid for (𝑡_𝑘,𝑡_𝑘+1]. The proof of (4.9) is completed by induction. Further, by analogous arguments, using (A1)–(A4), we obtain from (1.1) and (4.15) the following: 𝑥̇₁(𝑡)⩾𝑥₁(𝑡)𝑏₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)−𝑄₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥₂(𝑡)−^𝑛_𝑘=1𝑐_1𝑘(𝑡)𝑄_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥₁(𝑡)+𝛾_1𝑘(𝑡)𝑦_𝑘(𝑡),𝑥̇₂(𝑡)⩾𝑥₂(𝑡)𝑏₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)−𝑄₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥₁(𝑡)−^𝑛_𝑘=1𝑐_2𝑘(𝑡)𝑄_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥₁(𝑡)+𝛾_2𝑘(𝑡)𝑦_𝑘(𝑡),𝑦_𝑗̇(𝑡)⩾𝑦_𝑗(𝑡)−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑘=1𝛿_𝑗𝑘(𝑡)𝑄_𝑘(𝑡),𝑡≠𝑡_𝑘,𝑘∈ℤ⁺,𝑥_𝑖𝑡⁺_𝑘=1+𝑏_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑦_𝑗𝑡⁺_𝑘=1+𝑏̃_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈ℤ⁺, (4.19) and hence 𝑥_𝑖(𝑡)⩾𝑃_𝑖(𝑡),𝑦_𝑗(𝑡)⩾𝑃_𝑗(𝑡), (4.20) for all 𝑡⩾𝑡₀, where (𝑃_𝑖(𝑡),𝑃_𝑗(𝑡))=(𝑝₁(𝑡),𝑝₂(𝑡),𝑝₁(𝑡),…,𝑝_𝑛(𝑡)) is the minimal solution of the system: 𝑝₁̇(𝑡)=𝑝₁(𝑡)𝑏₁(𝑡)−𝑎₁(𝑡)𝑥₁(𝑡)−𝑄₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥₂(𝑡)−^𝑛_𝑘=1𝑐_1𝑘(𝑡)𝑄_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥₁(𝑡)+𝛾_1𝑘(𝑡)𝑦_𝑘(𝑡),𝑝₂̇(𝑡)=𝑝₂(𝑡)𝑏₂(𝑡)−𝑎₂(𝑡)𝑥₂(𝑡)−𝑄₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥₁(𝑡)−^𝑛_𝑘=1𝑐_2𝑘(𝑡)𝑄_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥₁(𝑡)+𝛾_2𝑘(𝑡)𝑦_𝑘(𝑡),𝑝̇_𝑗(𝑡)=𝑝_𝑗(𝑡)−𝑟_𝑗(𝑡)+²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑝_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑝_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑝_𝑗(𝑡)−^𝑛_𝑘=1𝛿_𝑗𝑘(𝑡)𝑄_𝑘(𝑡),𝑡≠𝑡_𝑘,𝑘∈ℤ⁺,𝑝_𝑖𝑡⁺₀=𝑝_𝑖0>0,𝑝_𝑗𝑡⁺₀=𝑝_𝑗0𝑝>0,_𝑖𝑡⁺_𝑘=1+𝑏_𝑖𝑘𝑝_𝑖𝑡_𝑘,𝑝_𝑗𝑡⁺_𝑘=1+𝑏̃_𝑗𝑘𝑝_𝑗𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈ℤ⁺. (4.21) Thus, the proof follows from (4.9) and (4.20).

Lemma 4.4. Suppose the hypotheses (A1)–(A5) hold. 𝐱(𝑡)=𝐱(𝑡;𝑡₀,𝑥₀)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),…,𝑦_𝑛(𝑡)) is a solution of (1.1), then there exist positive constants: 𝛼_𝑖,𝛽_𝑖,𝛼_𝑗, and 𝛽_𝑗 such that 𝛼_𝑖⩽𝑥_𝑖(𝑡)⩽𝛽_𝑖,𝛼_𝑗⩽𝑦_𝑗(𝑡)⩽𝛽_𝑗, (4.22) for all 𝑡∈(𝑡_𝑘−1,𝑡_𝑘],𝑘∈ℤ⁺ and if in addition 0<1+𝑏_𝑖𝑘⩽1,0<1+𝑏̃_𝑗𝑘⩽1, (4.23) then 𝛼_𝑖⩽𝑥_𝑖(𝑡)⩽𝛽_𝑖,𝛼_𝑗⩽𝑦_𝑗(𝑡)⩽𝛽_𝑗, (4.24) for all 𝑡∈𝐽.

Proof. From Lemma 4.3, we have 𝑃_𝑖(𝑡)⩽𝑥_𝑖(𝑡)⩽𝑄_𝑖(𝑡),𝑃_𝑗(𝑡)⩽𝑦_𝑗(𝑡)⩽𝑄_𝑗(𝑡) for all 𝑡⩾𝑡₀, where 𝑃_𝑖(𝑡),𝑃_𝑗(𝑡) is the minimal solution of the system (4.21), and 𝑄_𝑖(𝑡),𝑄_𝑗(𝑡) is the maximal solution of the system (4.10). Under the conditions of Lemma 4.4 for the solutions of (4.10) and (4.21) with initial functions, it is valid that 𝛼_𝑖⩽𝑃_𝑖(𝑡),𝑄_𝑖(𝑡)⩽𝛽_𝑖,𝛼_𝑗⩽𝑃_𝑗(𝑡),𝑄_𝑗(𝑡)⩽𝛽_𝑗, (4.25)𝛼_𝑖,𝛼_𝑗>0, 0<𝛽_𝑖,𝛽_𝑗<∞, for all 𝑡⩾𝑡₀, 𝑡∈(𝑡_𝑘−1,𝑡_𝑘], 𝑘∈ℤ⁺, then 𝛼_𝑖⩽𝑥_𝑖(𝑡)⩽𝛽_𝑖, 𝛼_𝑗⩽𝑦_𝑗(𝑡)⩽𝛽_𝑗, for all 𝑡⩾𝑡₀, 𝑡∈(𝑡_𝑘−1,𝑡_𝑘], 𝑘∈ℤ⁺. If in addition 0<1+𝑏_𝑖𝑘⩽1, 0<1+𝑏̃_𝑗𝑘⩽1, then from the left continuity of 𝑥_𝑖(𝑡) at the points 𝑡_𝑘, we have 𝛼_𝑖=1+𝑏_𝑖𝑘𝛼_𝑖−𝑏_𝑖𝑘𝛼_𝑖⩽1+𝑏_𝑖𝑘𝑥_𝑖(𝑡)−𝑏_𝑖𝑘𝑥_𝑖(𝑡)⩽1+𝑏_𝑖𝑘𝛽_𝑖−𝑏_𝑖𝑘𝛽_𝑖=𝛽_𝑖,𝛼_𝑗=1+𝑏̃_𝑗𝑘𝛼_𝑗−𝑏̃_𝑗𝑘𝛼_𝑗⩽1+𝑏̃_𝑗𝑘𝑦_𝑗(𝑡)−𝑏̃_𝑗𝑘𝑦_𝑗(𝑡)⩽1+𝑏̃_𝑗𝑘𝛽_𝑗−𝑏̃_𝑗𝑘𝛽_𝑗=𝛽_𝑗, (4.26) that is, 𝛼_𝑖⩽𝑥_𝑗𝑡⁺_𝑘⩽𝛽_𝑖,𝛼_𝑗⩽𝑦_𝑗𝑡⁺_𝑘⩽𝛽_𝑗, (4.27) hence 𝛼_𝑖⩽𝑥_𝑖(𝑡)⩽𝛽_𝑖,𝛼_𝑗⩽𝑦_𝑗(𝑡)⩽𝛽_𝑗, (4.28) for all 𝑡⩾𝑡₀.

Let 𝑉∈𝑉₀, for any (𝑡,𝐱)∈[𝑡_𝑘−1,𝑡_𝑘)×ℝ₊^𝑛+2, the right-hand derivative 𝐷⁺𝑉(𝑡,𝐱(𝑡)) along the solution 𝐱(𝑡,𝑡₀,𝑥₀) of (1.1) is defined by 𝐷⁺𝑉(𝑡,𝐱(𝑡))=lim_ℎ→0+1infℎ[𝑉(𝑡+ℎ,𝐱(𝑡+ℎ))−𝑉(𝑡,𝐱(𝑡))]. (4.29) Define 𝑚(𝑡)=²_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑛_𝑗=1𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡) (4.30) and consider a Lyapunov function: 𝑉𝐱(𝑡),𝐱^∗(𝑡)=^𝑛+2_𝑖=1𝑉_𝑖(𝑡)=²_𝑖=1𝑥ln^𝑖(𝑡)𝑥^∗_𝑖(𝑡)+^𝑛_𝑗=1𝑦ln^𝑗(𝑡)𝑦^∗_𝑗(𝑡). (4.31)

Theorem 4.5. Let the following conditions hold: (1) the hypotheses (A1)–(A5) hold;(2) there exist nonnegative continuous functions 𝛿_𝑖(𝑡), and 𝛿_𝑗(𝑡) such that (A8)𝑎^ℓ₁+𝜇^ℓ₁+𝜔^ℓ₁𝛼^∗₂𝜇^𝑢₁+𝜔^𝑢₁𝛽₂𝜇^𝑢₁+𝜔^𝑢₁𝛽^∗₂>^𝑛_𝑘=1𝑐^𝑢_1𝑘𝛽^𝑢_1𝑘𝛽_𝑘𝛼^ℓ_1𝑘+𝛽^ℓ_1𝑘𝛼₁+𝛾^ℓ_1𝑘𝛼_𝑘𝛼^ℓ_1𝑘+𝛽^ℓ_1𝑘𝛼^∗₁+𝛾^ℓ_1𝑘𝛼^∗_𝑘+𝜔^𝑢₂𝛽^∗₁𝜇^ℓ₂+𝜔^ℓ₂𝛼₁𝜇^ℓ₂+𝜔^ℓ₂𝛼^∗₁+^𝑛_𝑗=1𝑑^𝑢_𝑗1𝛼^𝑢_𝑗1+𝛾^𝑢_𝑗1𝛽_𝑗𝛼^ℓ_𝑗1+𝛽^ℓ_𝑗1𝛼₁+𝛾^ℓ_𝑗1𝛼_𝑗𝛼^ℓ_𝑗1+𝛽^ℓ_𝑗1𝛼^∗₁+𝛾^ℓ_𝑗1𝛼^∗_𝑗,𝑎^ℓ₂+𝜇^ℓ₂+𝜔^ℓ₂𝛼^∗₁𝜇^𝑢₂+𝜔^𝑢₁𝛽₁𝜇^𝑢₂+𝜔^𝑢₂𝛽^∗₁>^𝑛_𝑘=1𝑐^𝑢_2𝑘𝛽^𝑢_2𝑘𝛽_𝑘𝛼^ℓ_2𝑘+𝛽^ℓ_2𝑘𝛼₂+𝛾^ℓ_2𝑘𝛼_𝑘𝛼^ℓ_2𝑘+𝛽^ℓ_2𝑘𝛼^∗₂+𝛾^ℓ_2𝑘𝛼^∗_𝑘+𝜔^𝑢₁𝛽^∗₂𝜇^ℓ₁+𝜔^ℓ₁𝛼₂𝜇^ℓ₁+𝜔^ℓ₁𝛼^∗₂+^𝑛_𝑗=1𝑑^𝑢_𝑗2𝛼^𝑢_𝑗2+𝛾^𝑢_𝑗2𝛽_𝑗𝛼^ℓ_𝑗2+𝛽^ℓ_𝑗2𝛼₂+𝛾^ℓ_𝑗2𝛼_𝑗𝛼^ℓ_𝑗2+𝛽^ℓ_𝑗2𝛼^∗₂+𝛾^ℓ_𝑗2𝛼^∗_𝑗,𝛿^ℓ_𝑗𝑗+²_𝑘=1𝑑^ℓ_𝑗𝑘𝛾^ℓ_𝑗𝑘𝛼_𝑘𝛼^𝑢_𝑗𝑘+𝛽^𝑢_𝑗𝑘𝛽_𝑘+𝛾^𝑢_𝑗𝑘𝛽_𝑗𝛼^𝑢_𝑗𝑘+𝛽^𝑢_𝑗𝑘𝛽^∗_𝑘+𝛾^𝑢_𝑗𝑘𝛽^∗_𝑗>^𝑛_{𝑘=1,𝑘≠𝑗}𝛿^𝑢_𝑗𝑘+²_𝑖=1𝑐^𝑢_𝑖𝑗𝛼^𝑢_𝑖𝑗+𝛽^𝑢_𝑖𝑗𝛽_𝑖𝛼^ℓ_𝑖𝑗+𝛽^ℓ_𝑖𝑗𝛼_𝑖+𝛾^ℓ_𝑖𝑗𝛼_𝑗𝛼^ℓ_𝑖𝑗+𝛽^ℓ_𝑖𝑗𝛼^∗_𝑖+𝛾^ℓ_𝑖𝑗𝛼^∗_𝑗; (4.32)(3)0<1+𝑏_𝑖𝑘⩽1, 0<1+𝑏̃_𝑗𝑘⩽1.
Then, the solution x(𝑡) of (1.1) is globally stable.

Proof. Consider the upper right derivative 𝐷⁺𝑉(𝐱(𝑡),𝐱^∗(𝑡)) along the solution of system (1.1). For 𝑡⩾𝑡₀ and 𝑡≠𝑡_𝑘,𝑘∈ℤ⁺, we derive the estimate as follows: 𝐷⁺𝑉𝐱(𝑡),𝐱^∗(𝑡=)−𝑎₁(𝑡)𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡)−^𝑛_𝑘=1𝑐_1𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥₁(𝑡)+𝛾_1𝑘(𝑡)𝑦_𝑘(𝑡)−𝑐_1𝑘(𝑡)𝑦^∗_𝑘(𝑡)𝛼_1𝑘(𝑡)+𝛽_1𝑘(𝑡)𝑥^∗₁(𝑡)+𝛾_1𝑘(𝑡)𝑦^∗_𝑘(𝑡)−𝑥₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥₂(𝑡)−𝑥^∗₁(𝑡)𝜇₁(𝑡)+𝜔₁(𝑡)𝑥^∗₂(𝑡)×sgn𝑥₁−𝑥^∗₁+−𝑎₂(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡)−^𝑛_𝑘=1𝑐_2𝑘(𝑡)𝑦_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥₂(𝑡)+𝛾_2𝑘(𝑡)𝑦_𝑘(𝑡)−𝑐_2𝑘(𝑡)𝑦^∗_𝑘(𝑡)𝛼_2𝑘(𝑡)+𝛽_2𝑘(𝑡)𝑥^∗₂(𝑡)+𝛾_2𝑘(𝑡)𝑦^∗_𝑘(𝑡)−𝑥₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥₁(𝑡)−𝑥^∗₂(𝑡)𝜇₂(𝑡)+𝜔₂(𝑡)𝑥^∗₁(𝑡)×sgn𝑥₂−𝑥^∗₂+^𝑛_𝑗=1²_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑦_𝑗(𝑡)−𝑑_𝑗𝑘(𝑡)𝑥^∗_𝑘(𝑡)𝛼_𝑗𝑘(𝑡)+𝛽_𝑗𝑘(𝑡)𝑥^∗_𝑘(𝑡)+𝛾_𝑗𝑘(𝑡)𝑦^∗_𝑗(𝑡)−^𝑛_𝑘=1𝛿_𝑗𝑘(𝑡)𝑦_𝑘(𝑡)−𝑦^∗_𝑘(𝑡)sgn𝑦_𝑗−𝑦^∗_𝑗,⩽²_𝑖=1−𝑎^ℓ_𝑖𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑛_𝑘=1𝑐^𝑢_𝑖𝑘𝛼^𝑢_𝑖𝑘𝛼^ℓ_𝑖𝑘+𝛽^ℓ_𝑖𝑘𝑥_𝑖(𝑡)+𝛾^ℓ_𝑖𝑘𝑦_𝑘(𝑡)𝛼^ℓ_𝑖𝑘+𝛽^ℓ_𝑖𝑘𝑥^∗_𝑖(𝑡)+𝛾^ℓ_𝑖𝑘𝑦^∗_𝑘(𝑡)×𝑦_𝑘(𝑡)−𝑦^∗_𝑘(𝑡+_𝑛𝑘=1)𝑐^𝑢_𝑖𝑘𝛽^𝑢_𝑖𝑘𝑦_𝑘(𝑡)𝛼^ℓ_𝑖𝑘+𝛽^ℓ_𝑖𝑘𝑥_𝑖(𝑡)+𝛾^ℓ_𝑖𝑘𝑦_𝑘(𝑡)𝛼^ℓ_𝑖𝑘+𝛽^ℓ_𝑖𝑘𝑥^∗_𝑖(𝑡)+𝛾^ℓ_𝑖𝑘𝑦^∗_𝑘(𝑡)𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡+_𝑛𝑘=1)𝑐^𝑢_𝑖𝑘𝛽^𝑢_𝑖𝑘𝑥_𝑖(𝑡)𝛼^ℓ_𝑖𝑘+𝛽^ℓ_𝑖𝑘𝑥_𝑖(𝑡)+𝛾^ℓ_𝑖𝑘𝑦_𝑘(𝑡)𝛼^ℓ_𝑖𝑘+𝛽^ℓ_𝑖𝑘𝑥^∗_𝑖(𝑡)+𝛾^ℓ_𝑖𝑘𝑦^∗_𝑘(𝑡)𝑦_𝑘(𝑡)−𝑦^∗_𝑘(𝑡−𝜇_ℓ1)+𝜔^ℓ₁𝑥^∗₂(𝑡)𝜇^𝑢₁+𝜔^𝑢₁𝑥₂(𝑡)𝜇^𝑢₁+𝜔^𝑢₁𝑥^∗₂(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡+𝜔_𝑢1)𝑥^∗₂(𝑡)𝜇^ℓ₁+𝜔^ℓ₁𝑥₂(𝑡)𝜇^ℓ₁+𝜔^ℓ₁𝑥^∗₂(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡−𝜇_ℓ2)+𝜔^ℓ₂𝑥^∗₁(𝑡)𝜇^𝑢₂+𝜔^𝑢₂𝑥₁(𝑡)𝜇^𝑢₂+𝜔^𝑢₂𝑥^∗₁(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡+𝜔_𝑢2)𝑥^∗₁(𝑡)𝜇^ℓ₂+𝜔^ℓ₂𝑥₁(𝑡)𝜇^ℓ₂+𝜔^ℓ₂𝑥^∗₁(𝑡𝑥)¹(𝑡)−𝑥^∗₁(𝑡+_{𝑛𝑗=12𝑘=1})𝑑^𝑢_𝑗𝑘𝛼^𝑢_𝑗𝑘𝛼^ℓ_𝑗𝑘+𝛽^ℓ_𝑗𝑘𝑥_𝑘(𝑡)+𝛾^ℓ_𝑗𝑘𝑦_𝑗(𝑡)𝛼^ℓ_𝑗𝑘+𝛽^ℓ_𝑗𝑘𝑥^∗_𝑘(𝑡)+𝛾^ℓ_𝑗𝑘𝑦^∗_𝑗(𝑡)𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡+_2𝑘=1)𝑑^𝑢_𝑗𝑘𝛾^𝑢_𝑗𝑘𝑦_𝑗(𝑡)𝛼^ℓ_𝑗𝑘+𝛽^ℓ_𝑗𝑘𝑥_𝑘(𝑡)+𝛾^ℓ_𝑗𝑘𝑦_𝑗(𝑡)𝛼^ℓ_𝑗𝑘+𝛽^ℓ_𝑗𝑘𝑥^∗_𝑘(𝑡)+𝛾^ℓ_𝑗𝑘𝑦^∗_𝑗(𝑡)𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡−_2𝑘=1)𝑑^ℓ_𝑗𝑘𝛾^ℓ_𝑗𝑘𝑥_𝑘(𝑡)𝛼^𝑢_𝑗𝑘+𝛽^𝑢_𝑗𝑘𝑥_𝑘(𝑡)+𝛾^𝑢_𝑗𝑘𝑦_𝑗(𝑡)𝛼^𝑢_𝑗𝑘+𝛽^𝑢_𝑗𝑘𝑥^∗_𝑘(𝑡)+𝛾^𝑢_𝑗𝑘𝑦^∗_𝑗(𝑡)𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)−𝛿^ℓ_𝑗𝑗𝑦_𝑗−𝑦^∗_𝑗+^𝑛_{𝑘=1,𝑘≠𝑗}𝛿^𝑢_𝑗𝑘𝑦_𝑘(𝑡)−𝑦^∗_𝑘=−𝑎^ℓ₁+𝜇^ℓ₁+𝜔^ℓ₁𝛼^∗₂𝜇^𝑢₁+𝜔^𝑢₁𝛽₂𝜇^𝑢₁+𝜔^𝑢₁𝛽^∗₂−^𝑛_𝑘=1𝑐^𝑢_1𝑘𝛽^𝑢_1𝑘𝛽_𝑘𝛼^ℓ_1𝑘+𝛽^ℓ_1𝑘𝛼₁+𝛾^ℓ_1𝑘𝛼_𝑘𝛼^ℓ_1𝑘+𝛽^ℓ_1𝑘𝛼^∗₁+𝛾^ℓ_1𝑘𝛼^∗_𝑘−𝜔^𝑢₂𝛽^∗₁𝜇^ℓ₂+𝜔^ℓ₂𝛼₁𝜇^ℓ₂+𝜔^ℓ₂𝛼^∗₁−^𝑛_𝑗=1𝑑^𝑢_𝑗1𝛼^𝑢_𝑗1+𝛾^𝑢_𝑗1𝛽_𝑗𝛼^ℓ_𝑗1+𝛽^ℓ_𝑗1𝛼₁+𝛾^ℓ_𝑗1𝛼_𝑗𝛼^ℓ_𝑗1+𝛽^ℓ_𝑗1𝛼^∗₁+𝛾^ℓ_𝑗1𝛼^∗_𝑗×𝑥₁(𝑡)−𝑥^∗₁(𝑡)−𝑎^ℓ₂+𝜇^ℓ₂+𝜔^ℓ₂𝛼^∗₁𝜇^𝑢₂+𝜔^𝑢₁𝛽₁𝜇^𝑢₂+𝜔^𝑢₂𝛽^∗₁−^𝑛_𝑘=1𝑐^𝑢_2𝑘𝛽^𝑢_2𝑘𝛽_𝑘𝛼^ℓ_2𝑘+𝛽^ℓ_2𝑘𝛼₂+𝛾^ℓ_2𝑘𝛼_𝑘𝛼^ℓ_2𝑘+𝛽^ℓ_2𝑘𝛼^∗₂+𝛾^ℓ_2𝑘𝛼^∗_𝑘−𝜔^𝑢₁𝛽^∗₂𝜇^ℓ₁+𝜔^ℓ₁𝛼₂𝜇^ℓ₁+𝜔^ℓ₁𝛼^∗₂−^𝑛_𝑗=1𝑑^𝑢_𝑗2𝛼^𝑢_𝑗2+𝛾^𝑢_𝑗2𝛽_𝑗𝛼^ℓ_𝑗2+𝛽^ℓ_𝑗2𝛼₂+𝛾^ℓ_𝑗2𝛼_𝑗𝛼^ℓ_𝑗2+𝛽^ℓ_𝑗2𝛼^∗₂+𝛾^ℓ_𝑗2𝛼^∗_𝑗×𝑥₂(𝑡)−𝑥^∗₂(𝑡−_𝑛𝑗=1)𝛿^ℓ_𝑗𝑗+²_𝑘=1𝑑^ℓ_𝑗𝑘𝛾^ℓ_𝑗𝑘𝛼_𝑘𝛼^𝑢_𝑗𝑘+𝛽^𝑢_𝑗𝑘𝛽_𝑘+𝛾^𝑢_𝑗𝑘𝛽_𝑗𝛼^𝑢_𝑗𝑘+𝛽^𝑢_𝑗𝑘𝛽^∗_𝑘+𝛾^𝑢_𝑗𝑘𝛽^∗_𝑗−^𝑛_{𝑘=1,𝑘≠𝑗}𝛿^𝑢_𝑗𝑘−²_𝑖=1𝑐^𝑢_𝑖𝑗𝛼^𝑢_𝑖𝑗+𝛽^𝑢_𝑖𝑗𝛽_𝑖𝛼^ℓ_𝑖𝑗+𝛽^ℓ_𝑖𝑗𝛼_𝑖+𝛾^ℓ_𝑖𝑗𝛼_𝑗𝛼^ℓ_𝑖𝑗+𝛽^ℓ_𝑖𝑗𝛼^∗_𝑖+𝛾^ℓ_𝑖𝑗𝛼^∗_𝑗×𝑦_𝑗(𝑡)−𝑦^∗_𝑗. (4.33) Thus in view of hypothesis (A8), we obtain 𝐷⁺𝑉𝐱(𝑡),𝐱^∗(𝑡)⩽−𝛿(𝑡)𝑚(𝑡),𝑡⩾𝑡₀,𝑡≠𝑡_𝑘,𝑘∈ℤ⁺, (4.34) where 𝛿(𝑡)=min{𝛿_𝑖,𝛿_𝑗}.
For 𝑡⩾𝑡₀,𝑡=𝑡_𝑘,(𝑘∈ℤ⁺), we have 𝑉𝐱𝑡⁺_𝑘,𝐱^∗𝑡⁺_𝑘=²_𝑖=1𝑥ln^𝑖𝑡⁺_𝑘𝑥^∗_𝑖𝑡⁺_𝑘+^𝑛_𝑗=1𝑦ln^𝑗𝑡⁺_𝑘𝑦^∗_𝑗𝑡⁺_𝑘=²_𝑖=1ln1+𝑏_𝑖𝑘𝑥_𝑖𝑡_𝑘1+𝑏_𝑖𝑘𝑥^∗_𝑖𝑡_𝑘+^𝑛_𝑗=1𝑏̃ln1+_𝑗𝑘𝑦_𝑗t_𝑘1+𝑏̃_𝑗𝑘𝑦^∗_𝑗𝑡_𝑘=𝑉𝐱𝑡_𝑘,𝐱^∗𝑡_𝑘. (4.35) Then the inequality, 𝑉𝐱(𝑡),𝐱^∗(𝑡)⩽𝑉𝐱𝑡⁺₀,𝐱^∗𝑡⁺₀−^𝑡_𝑡0𝛿(𝑠)𝑚(𝑠)𝑑𝑠,𝑡⩾𝑡₀ (4.36) holds.
By Mean Value Theorem and by Lemma 4.4 it follows that for any closed interval contained in 𝑡∈(𝑡_𝑘−1,𝑡_𝑘],𝑘∈ℤ⁺, there exist positive numbers 𝑟 and 𝑅 such that for every 𝑖,𝑗, 𝑟⩽𝑥_𝑖(𝑡),𝑦_𝑗(𝑡),𝑥^∗_𝑖(𝑡),𝑦^∗_𝑗(𝑡)⩽𝑅 and 1𝑅𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)⩽ln𝑥_𝑖(𝑡)−ln𝑥^∗_𝑖(𝑡⩽1)𝑟𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡,1)𝑅𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)⩽ln𝑦_𝑗(𝑡)−ln𝑦^∗_𝑗(𝑡⩽1)𝑟𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡). (4.37) Hence, we obtain 𝑉𝐱₀,𝐱^∗₀=²_𝑖=1ln𝑥_𝑖𝑡⁺₀−ln𝑥^∗_𝑖𝑡⁺₀+^𝑛_𝑗=1ln𝑦_𝑗𝑡⁺₀−ln𝑦^∗_𝑗𝑡⁺₀⩽1𝑟𝐱₀−x^∗₀. (4.38) Further, from (4.34) and (4.35) we have 𝐷⁺𝑉𝐱(𝑡),𝐱^∗(𝑡)⩽0,𝑡⩾𝑡₀,𝑡≠𝑡_𝑘,Δ𝑉𝐱𝑡_𝑘,𝐱^∗𝑡_𝑘=0, (4.39) and hence 𝑉𝐱(𝑡),𝐱^∗(𝑡)⩽𝑉𝐱₀,𝐱^∗₀, (4.40) for all 𝑡⩾𝑡₀. Given 0<𝜀<𝑅, choose 𝛿=𝜀𝑟/2𝑅. Then from (4.37)–(4.40) it follows that ²_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑛_𝑗=1𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)<𝜀, (4.41) for all 𝑡⩾𝑡₀, whenever ‖𝐱₀−𝐱^∗₀‖⩽𝛿 and 𝑡₀∈ℝ₊. Since 𝑡₀∈ℝ₊ is arbitrary, by Definition 4.1(a), the system (1.1) is globally stable. This proves the theorem.

Theorem 4.6. In addition to the assumptions of Theorem 4.5, suppose that there exists a constant 𝑐 such that ^𝑡_𝑡0𝛿(𝑠)𝑑𝑠=𝑐𝑡−𝑡₀. (4.42) Then the system (1.1) is globally exponentially stable.

Proof. We consider again the Lyapunov function (4.31). From (4.34) and (4.37), we obtain 𝐷⁺𝑉𝐱(𝑡),𝐱^∗(𝑡)⩽−𝛿(𝑡)𝑚(𝑡)⩽−𝑟𝛿(𝑡)𝑉𝐱(𝑡),𝐱^∗(𝑡). (4.43) From the above mentioned estimate and (4.38), we have 𝑉𝐱(𝑡),𝐱^∗(𝑡)⩽𝑉𝐱₀,𝐱^∗₀exp−𝑟^𝑡_𝑡0𝛿(𝑠)𝑑𝑠, (4.44) for all 𝑡⩾𝑡₀. Then from (4.37),(4.40), (4.44), and (4.38), we deduce the inequality ²_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑛_𝑗=1𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡⩽𝑅)𝑟𝐱₀−𝐱^∗₀𝑒^{−𝑟𝑐(𝑡−𝑡0)} (4.45) for 𝑡⩾𝑡₀. This shows that the system (1.1) is globally exponentially stable. This proves the theorem.

Acknowledgment

The second author was partially supported by the Project of Science and Technology of Heilongjiang Province of China no. 12521151.