Multiple Periodic Solutions to a Kind of Lotka-Volterra Food-Chain System with Delays and Impulses on Time Scales

Abstract

By using Mawhin’s continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of at least 2^𝑛 periodic solutions for a kind of 𝑛-species Lotka-Volterra food-chain system with delays and impulses on time scales. One example is given to illustrate the effectiveness of our results.

1. Introduction

The food-chain phenomenon is universal and interesting in ecosystem. It is one of important methods to investigate this phenomenon by establishing the model of differential equations. The importance of studying the dynamics of food chain was pointed by Rosenzweig. In his famous paper on the paradox of enrichment [1], he wrote “Man must be very careful in attempting to enrich an ecosystem in order to increase its food yield. There is a real chance that such activity may result in decimation of the food species that are wanted in greater abundance.” Hereafter, many scholars investigated various kinds of food-chain systems (see [2–9]).

In [10], Li and Zhao have studied the existence of multiple positive periodic solutions for the following Lotka-Volterra food-chain system: 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑥₁(𝑡)−𝑐₁₂(𝑡)𝑥₂(𝑡)−ℎ₁(𝑡),⋮𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑑_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)+𝑐_{𝑖,𝑖−1}(𝑡)𝑥_𝑖−1(𝑡)−𝑐_𝑖,𝑖+1(𝑡)𝑥_𝑖+1(𝑡)−ℎ_𝑖(𝑡)⋮𝑥̇,2≤𝑖≤𝑛−1,_𝑛(𝑡)=𝑥_𝑛(𝑡)−𝑑_𝑛(𝑡)−𝑏_𝑛(𝑡)𝑥_𝑛(𝑡)+𝑐_{𝑛,𝑛−1}(𝑡)𝑥_𝑛(𝑡)−ℎ_𝑛(𝑡). (1.1) As we know, in population dynamics, many evolutionary processes experience short-time rapid chance after undergoing relatively long sooth variation. Examples include stocking of species and annual immigration. Incorporating these phenomena give us impulsive differential equations. For the theory of impulsive differential equations, we refer the reader to [11, 12]. In addition, there are few results on the existence of multiple periodic solutions for the delay food-chain system with impulsive effects in literatures. This motivates us to modify system (1.1) to the form 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑥₁𝑡−𝜏₁₁(𝑡)−𝑐₁₂(𝑡)𝑥₂𝑡−𝜏₁₂(𝑡)−ℎ₁(𝑡),𝑡≠𝑡_𝑘,⋮𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑑_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖𝑡−𝜏_𝑖𝑖(𝑡)+𝑐_{𝑖,𝑖−1}(𝑡)𝑥_𝑖−1𝑡−𝜏_{𝑖,𝑖−1}(𝑡)−𝑐_𝑖,𝑖+1(𝑡)𝑥_𝑖+1𝑡−𝜏_𝑖,𝑖+1(𝑡)−ℎ_𝑖(𝑡),2≤𝑖≤𝑛−1,𝑡≠𝑡_𝑘,⋮𝑥̇_𝑛(𝑡)=𝑥_𝑛(𝑡)−𝑑_𝑛(𝑡)−𝑏_𝑛(𝑡)𝑥_𝑛𝑡−𝜏_𝑛𝑛(𝑡)+𝑐_{𝑛,𝑛−1}(𝑡)𝑥_𝑛𝑡−𝜏_{𝑛,𝑛−1}(𝑡)−ℎ_𝑛(𝑡),𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝑔_𝑖𝑘𝑥_𝑖𝑡_𝑘,1≤𝑖≤𝑛, (1.2) where 𝑔_𝑖𝑘∈(−1,+∞)(𝑖=1,2,…,𝑛;𝑘∈ℕ={1,2,…,𝑛}).  {𝑡_𝑘}_𝑘∈ℕ is a strictly increasing sequence with 𝑡₁>0 and lim_𝑡→∞𝑡_𝑘=∞.  𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛) is the 𝑖th species population density. 𝑎₁(𝑡)>0 is the growth rate of the only producer. 𝑏_𝑖(𝑡)>0(𝑖=1,2,…,𝑛) and ℎ_𝑖(𝑡)>0(𝑖=1,2,…,𝑛) stand for the 𝑖th species intraspecific competition rate and harvesting rate, respectively. 𝑑_𝑖(𝑡)>0(𝑖=2,3,…,𝑛) is the 𝑖th species death rate. 𝑐_𝑖,𝑖+1(𝑡)>0(𝑖=1,2,…,𝑛−1) represents the (𝑖+1)th species predation rate on the 𝑖th species. 𝑐_{𝑖,𝑖−1}(𝑡)>0(𝑖=2,3,…,𝑛) stands for the transformation rate from the (𝑖−1)th species to the 𝑖th species. 𝜏_𝑖,𝑖+1(𝑡)≥0(𝑖=1,2,…,𝑛−1) stands for the time-lag in the process of the (𝑖+1)th species predation rate on the 𝑖th species. 𝜏_{𝑖,𝑖−1}(𝑡)≥0(𝑖=2,3,…,𝑛) stands for the time-lag in the process of transformation from the (𝑖−1)th species to the 𝑖th species. 𝜏_𝑖𝑖(𝑡)≥0(𝑖=1,2,…,𝑛) represents the time-lag in the process of intraspecific competition. In addition, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Therefore, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (e.g., seasonal effects of weather, food supplies, mating habits, etc.), which leads us to assume that 𝑎_𝑖(𝑡), 𝑏_𝑖(𝑡), 𝑐_𝑖𝑗(𝑡), 𝜏_𝑖𝑗(𝑡), and ℎ_𝑖(𝑡)(𝑖,𝑗=1,2,…,𝑛) are all continuous 𝜔-periodic functions. For impulsive effects, we further assume that there exists a 𝑞∈ℕ such that 𝑔_{𝑖(𝑘+𝑞)}=𝑔_𝑖𝑘(𝑖=1,2,…,𝑛) and 𝑡_𝑘+𝑞=𝑡_𝑘+𝜔.

However, dynamics in each equally spaced time interval may vary continuously. So, it may be more realistic to assume that the population dynamics involves the hybrid discrete-continuous processes. For example, Gamarra and Solé pointed out that such hybrid processes appear in the population dynamics of certain species that feature nonoverlapping generations: the change in population from one generation to the next is discrete and so is modeled by a difference equation, while within-generation dynamics vary continuously (due to mortality rates, resource consumption, predation, interaction, etc.) and thus are described by a differential equation [13]. The theory of calculus on time scales (see [14, 15] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [16] in order to unify continuous and discrete analysis, and it has become an effective approach to the study of mathematical models involving the hybrid discrete-continuous processes. This motivates us to consider Lotka-Volterra food-chain system with delays and impulses on time scales 𝕋 of the following form: 𝑢^Δ₁(𝑡)=𝑎₁(𝑡)−𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}−𝑐₁₂(𝑡)𝑒^{𝑢2(𝑡−𝜏12(𝑡))}−ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)},𝑡∈𝕋,𝑡≠𝑡_𝑘,⋮𝑢^Δ_𝑖(𝑡)=−𝑑_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑒^{𝑢𝑖(𝑡−𝜏𝑖𝑖(𝑡))}+𝑐_{𝑖,𝑖−1}(𝑡)𝑒^{𝑢𝑖−1(𝑡−𝜏𝑖,𝑖−1(𝑡))}−𝑐_𝑖,𝑖+1(𝑡)𝑒^{𝑢𝑖+1(𝑡−𝜏𝑖,𝑖+1(𝑡))}−ℎ_𝑖(𝑡)𝑒^{−𝑢𝑖(𝑡)},2≤𝑖≤𝑛−1,𝑡∈𝕋,𝑡≠𝑡_𝑘,⋮𝑢^Δ_𝑛(𝑡)=−𝑑_𝑛(𝑡)−𝑏_𝑛(𝑡)𝑒^{𝑢𝑛(𝑡−𝜏𝑛𝑛(𝑡))}+𝑐_{𝑛,𝑛−1}(𝑡)𝑒^{𝑢𝑛(𝑡−𝜏𝑛,𝑛−1(𝑡))}−ℎ_𝑛(𝑡)𝑒^{−𝑢𝑛(𝑡)},𝑡∈𝕋,𝑡≠𝑡_𝑘,Δ𝑢_𝑖𝑡_𝑘=𝑢_𝑖𝑡⁺_𝑘−𝑢_𝑖𝑡⁻_𝑘=ln1+𝑔_𝑖𝑘,1≤𝑖≤𝑛, (1.3) where 𝕋 is a 𝜔-periodic time scale. 𝑔_𝑖𝑘∈(−1,+∞)(𝑖=1,2,…,𝑛;𝑘∈ℕ={1,2,…,𝑛}) is a constant. {𝑡_𝑘}_𝑘∈ℕ is a strictly increasing sequence with 𝑡₁>0 and lim_𝑘→∞𝑡_𝑘=∞.  𝑎₁(𝑡)>0, 𝑏_𝑖(𝑡)>0, ℎ_𝑖(𝑡)>0, 𝑑_𝑖(𝑡)>0, 𝑐_𝑖,𝑖+1(𝑡)>0, 𝑐_{𝑖,𝑖−1}(𝑡)>0, 𝜏_𝑖,𝑖+1(𝑡), 𝜏_{𝑖,𝑖−1}(𝑡), 𝜏_𝑖𝑖(𝑡)≥0 are rd-continuous 𝜔-periodic functions. For impulsive effects, we further assume that there exists a integer 𝑞∈ℕ such that 𝑔_{𝑖(𝑘+𝑞)}=𝑔_𝑖𝑘(𝑖=1,2,…,𝑛) and 𝑡_𝑘+𝑞=𝑡_𝑘+𝜔. We denote by 𝜅=min{[0,+∞)∩𝕋}, 𝐼_𝜔=[𝜅,𝜅+𝜔]∩𝕋. Without loss of generality, we also assume that 𝐼_𝜔∩{𝑡_𝑘∶𝑘∈ℕ}={𝑡₁,𝑡₂,…,𝑡_𝑞}.

Remark 1.1. In (1.3), let 𝑥_𝑖(𝑡)=𝑒^{𝑢𝑖(𝑡)},  𝑖=1,2,…,𝑛. If 𝕋=ℝ (the set of all real numbers), then (1.3) reduces to (1.2).

To the best our knowledge, few papers have been published on the existence of multiple periodic solutions for this model. Our main purpose of this paper is by using Mawhin’s continuation theorem of coincidence degree theory [17], to establish the existence of at least 2^𝑛 periodic solutions for system (1.3). For the work concerning the multiple existence of periodic solutions of periodic population models which was done using coincidence degree theory, we refer to [18–25].

The organization of the rest of this paper is as follows. In the Section 2, we will introduce some basic notations and lemmas which are used in what follows. In Section 3, by employing the continuation theorem of coincidence degree theory and the skills of inequalities, we establish the existence of at least 2^𝑛 periodic solutions of system (1.3). In the final Section, one example is given to illustrate the effectiveness of our results.

2. Preliminaries on Time Scales

In this section, we briefly recall some basic definitions and lemmas on time scales which are used in what follows. For more details, one can see [14–16].

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.

Let 𝜔>0. Throughout this paper, the time scale 𝕋 is assumed to be 𝜔-periodic, that is, 𝑡∈𝕋 implies 𝑡+𝜔∈𝕋. In particular, the time scale 𝕋 under consideration is unbounded above and below.

Definition 2.1. A function 𝑓∶𝕋→ℝ is called regulated provided its right-side limits exist (finite) at all right-side points in 𝕋 and its left-side limits exist (finite) at all left-side points in 𝕋.

Definition 2.2. A function 𝑓∶𝕋→ℝ is called rd-continuous provided it is continuous at right-dense point in 𝕋 and its left-side limits exist (finite) at left-dense points in 𝕋. The set of rd-continuous functions 𝑓∶𝕋→ℝ will be denoted by 𝐶_rd=𝐶_rd(𝕋)=𝐶_rd(𝕋,ℝ).

Definition 2.3. Assume 𝑓∶𝕋→ℝ and 𝑡∈𝕋^𝑘. Then we define 𝑓^Δ(𝑡) to be the number (if it exists) with the property that given any 𝜀>0 there exists a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡−𝛿,𝑡+𝛿)∩𝕋 for some 𝛿>0) such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]<𝜀|𝜎(𝑡)−𝑠| (2.2) for all 𝑠∈𝑈. we call 𝑓^Δ(𝑡) the delta (or Hilger) derivative of 𝑓 at 𝑡. The set of functions 𝑓∶𝕋→ℝ that are differentiable and whose derivative is rd-continuous is denoted by 𝐶¹_rd=𝐶¹_rd(𝕋)=𝐶¹_rd(𝕋,ℝ).

If 𝑓 is continuous, then 𝑓 is rd-continuous. If 𝑓 is rd-continuous, the 𝑓 is regulated. If 𝑓 is delta differentiable at 𝑡, then 𝑓 is continuous at 𝑡.

Lemma 2.4. Let 𝑓 be regulated, then there exists a function 𝐹 which is delta differentiable with region of differentiation 𝐷 such that 𝐹^Δ(𝑡)=𝑓(𝑡)∀𝑡∈𝐷. (2.3)

Definition 2.5. Assume 𝑓∶𝕋→ℝ is a regulated function. Any function 𝐹 as in Lemma 2.4 is called a Δ-antiderivative of 𝑓. We define the indefinite integral of a regulated function 𝑓 by 𝑓(𝑡)Δ𝑡=𝐹(𝑡)+𝐶, (2.4) where 𝐶 is an arbitrary constant and 𝐹 is a Δ-antiderivative of 𝑓. We define the Cauchy integral by ^𝑏_𝑎𝑓(𝑠)Δ𝑠=𝐹(𝑏)−𝐹(𝑎)∀𝑎,𝑏∈𝕋. (2.5) A function 𝐹∶𝕋→ℝ is called an antiderivative of 𝑓∶𝕋→ℝ provided 𝐹^Δ(𝑡)=𝑓(𝑡)∀𝑡∈𝕋^𝑘. (2.6)

Lemma 2.6. If 𝑎,𝑏∈𝕋, 𝛼,𝛽∈ℝ and 𝑓,𝑔∈𝐶(𝕋,ℝ), then (i)∫^𝑏_𝑎[𝛼𝑓(𝑡)+𝛽𝑔(𝑡)]Δ𝑡=𝛼∫^𝑏_𝑎𝑓(𝑡)Δ𝑡+𝛽∫^𝑏_𝑎𝑔(𝑡)Δ𝑡; (ii) if 𝑓(𝑡)≥0 for all 𝑎≤𝑡<𝑏, then ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡≥0; (iii) if |𝑓(𝑡)|≤𝑔(𝑡) on [𝑎,𝑏)∶={𝑡∈𝕋∶𝑎≤𝑡<𝑏}, then |∫^𝑏_𝑎𝑓(𝑡)Δ𝑡|≤∫^𝑏_𝑎𝑔(𝑡)Δ𝑡.

For convenience, we now introduce some notation to be used throughout this paper. Let 𝜅=min{[0,+∞)∩𝕋},𝐼_𝜔=[𝜅,𝜅+𝜔]∩𝕋,𝑔^𝑀=sup_{𝑡∈𝐼𝜔}𝑔(𝑡),𝑔^𝑙=inf_{𝑡∈𝐼𝜔}𝑔(𝑡),1𝑔=𝜔_𝐼𝜔𝑔(𝑠)1Δ𝑠=𝜔_𝜅^𝜅+𝜔𝑔(𝑠)Δ𝑠, (2.7) where 𝑔∈𝐶_rd(𝕋) is an 𝜔-periodic real function, that is, 𝑔(𝑡+𝜔)=𝑔(𝑡) for all 𝑡∈𝕋.

Lemma 2.7 (see [26]). Assume that {𝑓_𝑛}_𝑛∈𝑁 is a function on 𝐽 such that (i){𝑓_𝑛}_𝑛∈𝑁 is uniformly bounded on 𝐽, (ii){𝑓^Δ_𝑛}_𝑛∈𝑁 is uniformly bounded on 𝐽. Then there is a subsequence of {𝑓_𝑛}_𝑛∈𝑁 which converges uniformly on 𝐽.

3. Existence of at Least 2^𝑛 Periodic Solutions

In this section, by using Mawhin’s continuation theorem and the skills of inequalities, we will show the existence of periodic solutions of (1.3). To do so, we need to make some preparations.

Let 𝑋 and 𝑍 be real normed vector spaces. Let 𝐿∶Dom𝐿⊂𝑋→𝑍 be a linear mapping and 𝑁∶𝑋×[0,1]→𝑍 be a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dim Ker𝐿=codimIm𝐿<∞ and Im𝐿 is closed in 𝑍. If 𝐿 is a Fredholm mapping of index zero, then there exists continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 such that Im𝑃=Ker𝐿 and Ker𝑄=Im𝐿=Im(𝐼−𝑄), and 𝑋=Ker𝐿⊕Ker𝑃, 𝑍=Im𝐿⊕Im𝑄. It follows that 𝐿|_{Dom𝐿∩Ker𝑃}∶(𝐼−𝑃)𝑋→Im𝐿 is invertible and its inverse is denoted by 𝐾_𝑃. If Ω is a bounded open subset of 𝑋, the mapping 𝑁 is called 𝐿-compact on Ω×[0,1], if 𝑄𝑁(Ω×[0,1]) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω×[0,1]→𝑋 is compact. Because Im𝑄 is isomorphic to Ker𝐿, there exists an isomorphism 𝐽∶Im𝑄→Ker𝐿.

The Mawhin’s continuous theorem [17, page 40] is given as follows.

Lemma 3.1 (see [17]). Let 𝐿 be a Fredholm mapping of index zero and let 𝑁 be 𝐿-compact on Ω×[0,1]. Assume (a) for each 𝜆∈(0,1), every solution 𝑥 of 𝐿𝑥=𝜆𝑁(𝑥,𝜆) is such that 𝑥∉𝜕Ω∩Dom𝐿; (b)𝑄𝑁(𝑥,0)𝑥≠0 for each 𝑥∈𝜕Ω∩Ker𝐿; (c)deg(𝐽𝑄𝑁(𝑥,0),Ω∩Ker𝐿,0)≠0. Then 𝐿𝑥=𝑁(𝑥,1) has at least one solution in Ω∩Dom𝐿.

For simplicity, we need to introduce some notations as follows: 𝑔_𝑖=12^𝑞_𝑘=1ln1+𝑔_𝑖𝑘(1≤𝑖≤𝑛),𝐺_𝑖=12^𝑞_𝑘=1ln1+𝑔_𝑖𝑘+ln1+𝑔_𝑖𝑘),𝑙(1≤𝑖≤𝑛_±1=𝑎₁+2𝑔₁/𝜔𝑒^{𝑎1𝜔+𝐺1}±𝑎₁+2𝑔₁/𝜔𝑒^{𝑎1𝜔+𝐺12}−4𝑏₁ℎ₁𝑒^{𝑎1𝜔+𝐺1}2𝑏₁,𝐴^±₁=𝑎₁+2𝑔₁/𝜔−𝑐₁₂𝑙⁺₂±𝑎₁+2𝑔₁/𝜔−𝑐₁₂𝑙⁺₂²−4𝑏₁ℎ₁𝑒^{𝑎1𝜔+𝐺1}2𝑏₁,𝐵^±₁=𝑎₁+2𝑔₁/𝜔−𝑐₁₂𝑙⁺₂𝑒^{−(𝑎1𝜔+𝐺1)}±𝑎₁+2𝑔₁/𝜔−𝑐₁₂𝑙⁺₂𝑒^{−(𝑎1𝜔+𝐺1)2}−4𝑏₁ℎ₁𝑒^{−(𝑎1𝜔+𝐺1)}2𝑏₁,𝑙^±_𝑖=𝑐_{𝑖,𝑖−1}+2𝑔_𝑖/𝜔𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖}±𝑐_{𝑖,𝑖−1}+2𝑔_𝑖/𝜔𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖2}−4𝑏_𝑖ℎ_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖}2𝑏_𝑖(),𝐴2≤𝑖≤𝑛_±𝑖=2𝑔_𝑖/𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_i+1±2𝑔_𝑖/𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1²−4𝑏_𝑖ℎ_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖}2𝑏_𝑖(),𝐵2≤𝑖≤𝑛−1_±𝑖=12𝑏_𝑖±2𝑔_𝑖𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1𝑒^{−(𝑐𝑖,−1𝜔+𝐺𝑖)2}−4𝑏_𝑖ℎ_𝑖𝑒^{−(𝑐𝑖,𝑖−1𝜔+𝐺𝑖)}+2𝑔_𝑖𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1𝑒^{−(𝑐𝑖,𝑖−1𝜔+𝐺𝑖)}(),𝐵2≤𝑖≤𝑛−1_±𝑛=2𝑔_𝑛/𝜔−𝑑_𝑛𝑒^{−(𝑐𝑛,𝑛−1𝜔+𝐺𝑛)}±2𝑔_𝑛/𝜔−𝑑_𝑛𝑒^{−(𝑐𝑛,𝑛−1𝜔+𝐺𝑛)2}−4𝑏_𝑛ℎ_𝑛𝑒^{−(𝑐𝑛,𝑛−1𝜔+𝐺𝑛)}2𝑏_𝑛,𝐴^±_𝑛=2𝑔_𝑛/𝜔−𝑑_𝑛±2𝑔_𝑛/𝜔−𝑑_𝑛²−4𝑏_𝑛ℎ_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛}2𝑏_𝑛,𝐻₁=𝑙⁻₁𝑒^{−(𝑎1𝜔+𝐺1)},𝐻_𝑖=𝑙⁻_𝑖𝑒^{−(𝑐𝑖,𝑖−1𝜔+𝐺𝑖)}(2≤𝑖≤𝑛). (3.1) Throughout this paper, we need the following assumptions:(𝐻₁)(𝑎₁+(2𝑔₁/𝜔)−𝑐₁₂𝑙⁺₂)𝑒^{−(1/2)(𝑎1𝜔+𝐺1)}>2𝑏₁ℎ₁, (𝐻₂)((2𝑔_𝑖/𝜔)−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1)𝑒^{−(1/2)(𝑐𝑖,𝑖−1𝜔+𝐺𝑖)}>2𝑏_𝑖ℎ_𝑖(2≤𝑖≤𝑛−1), (𝐻₃)((2𝑔_𝑛/𝜔)−𝑑_𝑛)𝑒^{−(1/2)(𝑐𝑛,𝑛−1𝜔+𝐺𝑛)}>2𝑏_𝑛ℎ_𝑛. The following results will play an important role in the proof of our main result.

Lemma 3.2. Let 𝕋 be a 𝜔-periodic time scale. {𝑡_𝑘}^∞₁⊂𝕋 is a strictly increasing sequence with 𝑡₁>0 and lim_𝑘→∞𝑡_𝑘=∞.  𝜅=min{[0,∞)∩𝕋}, 𝐼_𝜔=[𝜅,𝜅+𝜔]∩𝕋, 𝐼_𝜔∩{𝑡_𝑘}^∞_𝑘=1={𝑡₁,𝑡₂,…,𝑡_𝑞}. Suppose 𝜓∶𝕋→ℝ be a 𝜔-periodic function which is rd-continuous for 𝑡≠𝑡_𝑘, exists the right limit 𝜓(𝑡⁺_𝑘) and left limit 𝜓(𝑡⁻_𝑘)=𝜓(𝑡_𝑘) in the sense of time scales. Then 0≤sup_{𝑠∈𝐼𝜔}𝜓(𝑠)−inf_{𝑠∈𝐼𝜔}𝜓(𝑠)≤12_𝜅^𝜅+𝜔𝜓^Δ(𝑠)Δ𝑠+^𝑞_𝑘=1Δ𝜓𝑡_𝑘. (3.2)

Proof. First, take 𝑡_∗∈𝐼_𝜔 such that 𝜓(𝑡_∗)=inf_{𝑠∈𝐼𝜔}𝜓(𝑠). Next, take 𝑡^∗∈[𝑡_∗,𝑡_∗+𝜔]∩𝕋 such that 𝜓(𝑡^∗)=sup_{𝑠∈𝐼𝜔}𝜓(𝑠). Then we have 𝜓𝑡^∗=𝜓𝑡_∗+𝜓𝑡^∗−𝜓𝑡_∗=𝜓𝑡_∗+𝜓𝑡^∗−_{𝑡∗≤𝑡𝑘<𝑡∗}Δ𝜓𝑡_𝑘−𝜓𝑡_∗+_{𝑡∗≤𝑡𝑘<𝑡∗}Δ𝜓𝑡_𝑘=𝜓𝑡_∗+^𝑡∗_𝑡∗𝜓(𝑠)Δ𝑠+_{𝑡∗≤𝑡𝑘<𝑡∗}Δ𝜓𝑡_𝑘≤𝜓𝑡_∗+^𝑡∗_𝑡∗𝜓(𝑠)Δ𝑠+_{𝑡∗≤𝑡𝑘<𝑡∗}Δ𝜓𝑡_𝑘. (3.3) Noticing that 𝜓(𝑡^∗)=𝜓(𝑡^∗−𝜔)=𝜓(𝑡_∗)+(𝜓(𝑡^∗)−𝜓(𝑡_∗−𝜔)), similar to the above, we can get 𝜓𝑡^∗≤𝜓𝑡_∗+^𝑡∗_{𝑡∗−𝜔}𝜓(𝑠)Δ𝑠+_{𝑡∗−𝜔≤𝑡𝑘<𝑡∗}Δ𝜓𝑡_𝑘. (3.4) From (3.3) and (3.4), we get 2𝜓𝑡^∗≤2𝜓𝑡_∗+^𝑡∗_𝑡∗𝜓(𝑠)Δ𝑠+^𝑡∗_{𝑡∗−𝜔}𝜓(𝑠)Δ𝑠+_{𝑡∗≤𝑡𝑘<𝑡∗}Δ𝜓𝑡_𝑘+_{𝑡∗−𝜔≤𝑡𝑘<𝑡∗}Δ𝜓𝑡_𝑘=2𝜓𝑡_∗+^𝜔₀𝜓(𝑠)Δ𝑠+_{0≤𝑡𝑘<𝜔}Δ𝜓𝑡_𝑘, (3.5) that is 𝜓𝑡^∗≤𝜓𝑡_∗+12_𝜅^𝜅+𝜔𝜓(𝑠)Δ𝑠+^𝑞_𝑘=1Δ𝜓𝑡_𝑘, (3.6) which implies that (3.2) holds. This completes the proof.

Lemma 3.3. Let 𝑥>0, 𝑦>0, 𝑧>0 and 𝑥>2√𝑦𝑧, for the functions 𝑓(𝑥,𝑦,𝑧)=(𝑥+𝑥²−4𝑦𝑧)/2𝑧 and 𝑔(𝑥,𝑦,𝑧)=(𝑥−𝑥²−4𝑦𝑧)/2𝑧, the following assertions hold. (1)𝑓(𝑥,𝑦,𝑧) and 𝑔(𝑥,𝑦,𝑧) are monotonically increasing and monotonically decreasing on the variable 𝑥∈(0,∞), respectively. (2)𝑓(𝑥,𝑦,𝑧) and 𝑔(𝑥,𝑦,𝑧) are monotonically decreasing and monotonically increasing on the variable 𝑦∈(0,∞), respectively. (3)𝑓(𝑥,𝑦,𝑧) and 𝑔(𝑥,𝑦,𝑧) are monotonically decreasing and monotonically increasing on the variable 𝑧∈(0,∞), respectively.

Proof. In fact, for all 𝑥>0, 𝑦>0, 𝑧>0, we have 𝜕𝑓=𝜕𝑥𝑥+𝑥²−4𝑦𝑧2𝑧𝑥²−4𝑦𝑧>0,𝜕𝑔=𝜕𝑥𝑥²−4𝑦𝑧−𝑥2𝑧𝑥²−4𝑦𝑧<0,𝜕𝑓=𝜕𝑦−1𝑥²−4𝑦𝑧<0,𝜕𝑔=1𝜕𝑦𝑥²−4𝑦𝑧>0,𝜕𝑓=−𝜕𝑧𝑥+𝑥²−4𝑦𝑧²4𝑧²𝑥²−4𝑦𝑧<0,𝜕𝑔=𝜕𝑧𝑥−𝑥²−4𝑦𝑧²4𝑧²𝑥²−4𝑦𝑧>0. (3.7) By the relationship of the derivative and the monotonicity, the above assertions obviously hold. The proof of Lemma 3.3 is complete.

Lemma 3.4. Assume that (𝐻₁), (𝐻₂), and (𝐻₃) hold, then we have the following inequalities: (1)ln𝑙⁺_𝑖>ln𝐴⁺_𝑖>ln𝐴⁻_𝑖>ln𝑙⁻_𝑖>ln𝐻_𝑖(1≤𝑖≤𝑛); (2)ln𝑙⁺_𝑖>ln𝐵⁺_𝑖>ln𝐵⁻_𝑖>ln𝑙⁻_𝑖>ln𝐻_𝑖(1≤𝑖≤𝑛); (3)min{ln𝐴⁺_𝑖,ln𝐵⁺_𝑖}>max{ln𝐴⁻_𝑖,ln𝐵⁻_𝑖}(1≤𝑖≤𝑛).

Proof. (1) Since 𝑎₁+2𝑔₁𝜔𝑒^{𝑎1𝜔+𝐺1}>𝑎₁+2𝑔₁𝜔−𝑐₁₂𝑙⁺₂>𝑎₁+2𝑔₁𝜔−𝑐₁₂𝑙⁺₂𝑒^{−(1/2)(𝑎1𝜔+𝐺1)}>2𝑏₁ℎ₁>0,𝑐_{𝑖,𝑖−1}+2𝑔_𝑖𝜔𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖}>2𝑔_𝑖𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1>2𝑔_𝑖𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1𝑒^{−(1/2)(𝑐𝑖,𝑖−1𝜔+𝐺𝑖)}>2𝑏_𝑖ℎ_𝑖>0,𝑐_{𝑛,𝑛−1}+2𝑔_𝑛𝜔𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛}>2𝑔_𝑛𝜔−𝑑_𝑛>2𝑔_𝑛𝜔−𝑑_𝑛𝑒^{−(1/2)(𝑐𝑛,𝑛−1𝜔+𝐺𝑛)}>2𝑏_𝑛ℎ_𝑛>0,0<𝑏_𝑖,0<ℎ_𝑖,𝑖=1,2,…,𝑛. (3.8) Applying Lemma 3.3, we have 0<𝐻₁=𝑙⁻₁𝑒^{−(𝑎1𝜔+𝐺1)}<𝑙⁻₁=𝑔𝑎₁+2𝑔₁𝜔𝑒^{𝑎1𝜔+𝐺1},𝑏₁,ℎ₁𝑒^{𝑎1𝜔+𝐺1}<𝑔𝑎₁+2𝑔₁𝜔−𝑐₁₂𝑙⁺₂,𝑏₁,ℎ₁𝑒^{𝑎1𝜔+𝐺1}=𝐴⁻₁<𝐴⁺₁=𝑓𝑎₁+2𝑔₁𝜔−𝑐₁₂𝑙⁺₂,𝑏₁,ℎ₁𝑒^{𝑎1𝜔+𝐺1}<𝑓𝑎₁+2𝑔₁𝜔𝑒^{𝑎1𝜔+𝐺1},𝑏₁,ℎ₁𝑒^{𝑎1𝜔+𝐺1}=𝑙⁺₁,0<𝐻_𝑖=𝑙⁻_𝑖𝑒^{−(𝑐𝑖,𝑖−1𝜔+𝐺𝑖)}<𝑙⁻_𝑖=𝑔𝑐_{𝑖,𝑖−1}+2𝑔_𝑖𝜔𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖},𝑏_𝑖,ℎ_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖}<𝑔2𝑔_𝑖𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1,𝑏_𝑖,ℎ_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖}=𝐴⁻_𝑖<𝐴⁺_𝑖=𝑓2𝑔_𝑖𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1,𝑏_𝑖,ℎ_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖}<𝑓𝑐_{𝑖,𝑖−1}+2𝑔_𝑖𝜔𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖},𝑏_𝑖,ℎ_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖}=𝑙⁺_𝑖,𝑖=2,3,…,𝑛−1,0<𝐻_𝑛=𝑙⁻_𝑛𝑒^{−(𝑐𝑛,𝑛−1𝜔+𝐺𝑛)}<𝑙⁻_𝑛=𝑔𝑐_{𝑛,𝑛−1}+2𝑔_𝑛𝜔𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛},𝑏_𝑛,ℎ_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺n}<𝑔2𝑔_𝑛𝜔−𝑑_𝑛,𝑏_𝑛,ℎ_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛}=𝐴⁻_𝑛<𝐴⁺_𝑛=𝑓2𝑔_𝑛𝜔−𝑑_𝑛,𝑏_𝑛,ℎ_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛}<𝑓𝑐_{𝑛,𝑛−1}+2𝑔_𝑛𝜔𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛},𝑏_𝑛,ℎ_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛}=𝑙⁺_𝑛. (3.9)
Thus, we have ln𝐻_𝑖<ln𝑙⁻_𝑖<ln𝐴⁻_𝑖<ln𝐴⁺_𝑖<ln𝑙⁺_𝑖, 𝑖=1,2,…,𝑛. The proof of (1) of Lemma 3.4 is complete. For (2) of Lemma 3.4, we similarly have 0<𝐻₁=𝑙⁻₁𝑒^{−(𝑎1𝜔+𝐺1)}<𝑙⁻₁=𝑔𝑎₁+2𝑔₁𝜔𝑒^{𝑎1𝜔+𝐺1},𝑏₁𝑒^{𝑎1𝜔+𝐺1},ℎ₁<𝑔𝑎₁+2𝑔₁𝜔−𝑐₁₂𝑙⁺₂,𝑏₁𝑒^{𝑎1𝜔+𝐺1},ℎ₁=𝐵⁻₁<𝐵⁺₁=𝑓𝑎₁+2𝑔₁𝜔−𝑐₁₂𝑙⁺₂,𝑏₁𝑒^{𝑎1𝜔+𝐺1},ℎ₁<𝑓𝑎₁+2𝑔₁𝜔𝑒^{𝑎1𝜔+𝐺1},𝑏₁𝑒^{𝑎1𝜔+𝐺1},ℎ₁=𝑙⁺₁,0<𝐻_𝑖=𝑙⁻_𝑖𝑒^{−(𝑐𝑖,𝑖−1𝜔+𝐺𝑖)}<𝑙⁻_𝑖=𝑔𝑐_{𝑖,𝑖−1}+2𝑔_𝑖𝜔𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖},𝑏_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖},ℎ_𝑖<𝑔2𝑔_𝑖𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1,𝑏_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖},ℎ_𝑖=𝐵⁻_𝑖<𝐵⁺_𝑖=𝑓2𝑔_𝑖𝜔−𝑑_𝑖−𝑐_𝑖,𝑖+1𝑙⁺_𝑖+1,𝑏_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖},ℎ_𝑖<𝑓𝑐_{𝑖,𝑖−1}+2𝑔_𝑖𝜔𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖},𝑏_𝑖𝑒^{𝑐𝑖,𝑖−1𝜔+𝐺𝑖},ℎ_𝑖=𝑙⁺_𝑖,𝑖=2,3,…,𝑛−1,0<𝐻_𝑛=𝑙⁻_𝑛𝑒^{−(𝑐𝑛,𝑛−1𝜔+𝐺𝑛)}<𝑙⁻_𝑛=𝑔𝑐_{𝑛,𝑛−1}+2𝑔_𝑛𝜔𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛},𝑏_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛},ℎ_𝑛<𝑔2𝑔_𝑛𝜔−𝑑_𝑛,𝑏_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛},ℎ_𝑛=𝐵⁻_𝑛<𝐵⁺_𝑛=𝑓2𝑔_𝑛𝜔−𝑑_𝑛,𝑏_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛},ℎ_𝑛<𝑓𝑐_{𝑛,𝑛−1}+2𝑔_𝑛𝜔𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛},𝑏_𝑛𝑒^{𝑐𝑛,𝑛−1𝜔+𝐺𝑛},ℎ_𝑛=𝑙⁺_𝑛, (3.10) which imply that ln𝐻_𝑖<ln𝑙⁻_𝑖<ln𝐵⁻_𝑖<ln𝐵⁺_𝑖<ln𝑙⁺_𝑖, 𝑖=1,2,…,𝑛; that is, the inequality (2) of Lemma 3.4 holds. Let us now to prove the inequality (3) of Lemma 3.4. In fact, since maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖=ln𝐴⁻_𝑖+ln𝐵⁻_𝑖+ln𝐴⁻_𝑖−ln𝐵⁻_𝑖2(1≤𝑖≤𝑛),minln𝐴⁺_𝑖,ln𝐵⁺_𝑖=ln𝐴⁺_𝑖+ln𝐵⁺_𝑖−ln𝐴⁺_𝑖−ln𝐵⁺_𝑖2(1≤𝑖≤𝑛), (3.11) we have minln𝐴⁺_𝑖,ln𝐵⁺_𝑖−maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖=12ln𝐴⁺_𝑖+ln𝐵⁺_𝑖−ln𝐴⁻_𝑖−ln𝐵⁻_𝑖−ln𝐴⁻_𝑖−ln𝐵⁻_𝑖−ln𝐴⁺_𝑖−ln𝐵⁺_𝑖. (3.12) If 𝐴⁺_𝑖≥𝐵⁺_𝑖, then 𝐴⁻_𝑖≤𝐵⁻_𝑖, and minln𝐴⁺_𝑖,ln𝐵⁺_𝑖−maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖=ln𝐵⁺_𝑖−ln𝐵⁻_𝑖>0; (3.13) If 𝐴⁺_𝑖≤𝐵⁺_𝑖, then 𝐴⁻_𝑖≥𝐵⁻_𝑖, and minln𝐴⁺_𝑖,ln𝐵⁺_𝑖−maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖=ln𝐴⁺_𝑖−ln𝐴⁻_𝑖>0. (3.14) So, we drive min{ln𝐴⁺_𝑖,ln𝐵⁺_𝑖}>max{ln𝐴⁻_𝑖,ln𝐵⁻_𝑖}. From the above, we have completed the proof of Lemma 3.4.

Theorem 3.5. Assume that (𝐻₁), (𝐻₂), and (𝐻₃) hold. Then system (1.3) has at least 2^𝑛  𝜔-periodic solutions.

Proof. Let 𝐶[𝐼_𝜔;𝑡₁,…,𝑡_𝑞]={𝑢∶𝐼_𝜔→ℝ^𝑛 is a piecewise continuous map with first-class discontinuity points in 𝐼_𝜔∩{𝑡_𝑘}^∞_𝑘=1, and at each discontinuity point, it is continuous on the left}. Take 𝑋=𝑢=𝑢₁,𝑢₂,…,𝑢_𝑛^𝑇∈𝐶𝐼_𝜔;𝑡₁,…,𝑡_𝑞∶𝑢_𝑖(𝑡+𝜔)=𝑢_𝑖(𝑡),𝑍=𝑋×ℝ^{𝑛×(𝑞+1)} (3.15) and define ‖𝑢‖₀=^𝑛_𝑖=1sup_𝑡∈I𝜔𝑢_𝑖(𝑡‖𝑧‖),𝑢∈𝑋,₁=‖𝑢‖₀+^𝑞_𝑗=1𝜉_𝑗,𝑧=𝑢,𝜉₁,𝜉₂,…,𝜉_𝑞∈𝑍. (3.16) Then both (𝑋,‖⋅‖₀) and (𝑍,‖⋅‖₁) are Banach spaces. Let Dom𝐿=𝑢∈𝑋∶𝑢^Δ𝑢∈𝑋,𝐿∶Dom𝐿→𝑍,𝑁∶𝑋→𝑍,𝐿𝑢=^Δ1𝑢^Δ₂⋮𝑢^Δ_𝑛,Δ𝑢₁𝑡₁Δ𝑢₂𝑡₁⋮Δ𝑢_𝑛𝑡₁,…,Δ𝑢₁𝑡_𝑞Δ𝑢₂𝑡_𝑞⋮Δ𝑢_𝑛𝑡_𝑞,𝑁()=𝐹𝑢,𝜆¹()⋮𝐹𝑡,𝜆_𝑖()⋮𝐹𝑡,𝜆_𝑛(),𝑡,𝜆ln1+𝑔₁₁⋮ln1+𝑔_𝑖1⋮ln1+𝑔_𝑛1,…,ln1+𝑔_1𝑞⋮ln1+𝑔_𝑖𝑞⋮ln1+𝑔_𝑛𝑞, (3.17) where 𝐹₁(𝑡,𝜆)=𝑎₁(𝑡)−𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}−𝜆𝑐₁₂(𝑡)𝑒^{𝑢2(𝑡−𝜏12(𝑡))}−ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)},𝐹_𝑖(𝑡,𝜆)=−𝜆𝑑_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑒^{𝑢𝑖(𝑡−𝜏𝑖𝑖(𝑡))}+𝑐_{𝑖,𝑖−1}(𝑡)𝑒^{𝑢𝑖−1(𝑡−𝜏𝑖,𝑖−1(𝑡))}−𝜆𝑐_𝑖,𝑖+1(𝑡)𝑒^{𝑢𝑖+1(𝑡−𝜏𝑖,𝑖+1(𝑡))}−ℎ_𝑖(𝑡)𝑒^{−𝑢𝑖(𝑡)}𝐹,𝑖=2,3,…,𝑛−1,_𝑛(𝑡,𝜆)=−𝜆𝑑_𝑛(𝑡)−𝑏_𝑛(𝑡)𝑒^{𝑢𝑛(𝑡−𝜏𝑛𝑛(𝑡))}+𝑐_{𝑛,𝑛−1}(𝑡)𝑒^{𝑢𝑛−1(𝑡−𝜏𝑛,𝑛−1(𝑡))}−ℎ_𝑛(𝑡)𝑒^{−𝑢𝑛(𝑡)}.𝑃𝑢𝑃∶𝑋→𝑋,𝑄∶𝑍→𝑍,𝑢=₁,𝑢₂,…,𝑢_𝑛^𝑇=𝑢₁,𝑢₂,…,𝑢_𝑛^𝑇𝑄𝑢,𝑢∈𝑋,₁𝑢₂⋮𝑢_𝑛,𝑚_1𝑘𝑚_2𝑘⋮𝑚_𝑛𝑘^𝑞_𝑘=1=𝑢₁+1𝜔^𝑞_𝑘∑₌₁𝑚_1𝑘𝑢₂+1𝜔^𝑞_𝑘∑₌₁𝑚_2𝑘⋮𝑢_𝑛+1𝜔^𝑞_𝑘∑₌₁𝑚_𝑛𝑘,00⋮0^𝑞_𝑘=1. (3.18) Obviously, Ker𝐿=𝑢=𝑢₁,𝑢₂,…,𝑢_𝑛^𝑇∈𝑋∶𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡)=𝑐₁,𝑐₂,…,𝑐_𝑛∈ℝ^𝑛,𝑢,𝑡∈𝕋Im𝐿=¹𝑢₂⋮𝑢_𝑛,𝑚_1𝑘𝑚_2𝑘⋮𝑚_𝑛𝑘^𝑞_𝑘=1𝜔∈𝑍∶𝑢₁+∑^𝑞_𝑘=1𝑚_1𝑘𝜔=0𝑢₂+∑^𝑞_𝑘=1𝑚_2𝑘⋮𝜔=0𝑢_𝑛+∑^𝑞_𝑘=1𝑚_𝑛𝑘.=0 (3.19) Since Im𝐿 is closed in 𝑍, Im𝑃=Ker𝐿, Ker𝑄=Im𝐿, and dimKer𝐿=𝑛=codimIm𝐿, we know that 𝐿 is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to 𝐿) 𝐾_𝑃∶Im𝐿→Ker𝑃∩Dom𝐿 is given by 𝐾_𝑃𝑢𝑧=¹𝑢₂⋮𝑢_𝑛,𝑚_1𝑘𝑚_2𝑘⋮𝑚_𝑛𝑘^𝑞_𝑘=1=∫^𝑡_𝜅𝑢₁(𝑠)Δ𝑠+∑_{𝜅<𝑡𝑘<𝑡}𝑚_1𝑘−1𝜔^𝑞_𝑘∑₌₁𝑚_1𝑘−1𝜔∫_𝜅^𝜅+𝜔∫^𝑡_𝜅𝑢₁(𝑠)∫Δ𝑠Δ𝑡_𝑡𝜅𝑢₂(𝑠)Δ𝑠+∑_{𝜅<𝑡𝑘<𝑡}𝑚_2𝑘−1𝜔^𝑞_𝑘∑₌₁𝑚_2𝑘−1𝜔∫_𝜅^𝜅+𝜔∫^𝑡_𝜅𝑢₂(𝑠)⋮∫Δ𝑠Δ𝑡_𝑡𝜅𝑢₁(𝑠)Δ𝑠+∑_{𝜅<𝑡𝑘<𝑡}𝑚_1𝑘−1𝜔^𝑞_𝑘∑₌₁𝑚_𝑛𝑘−1𝜔∫_𝜅^𝜅+𝜔∫^𝑡_𝜅𝑢_𝑛(𝑠).Δ𝑠Δ𝑡 (3.20) Then ()=1𝑄𝑁𝑢,𝜆𝜔∫_𝜅^𝜅+𝜔𝐹₁(𝑠,𝜆)Δ𝑠+2𝑔₁𝜔⋮1𝜔∫_𝜅^𝜅+𝜔𝐹_𝑛(𝑠,𝜆)Δ𝑠+2𝑔_𝑛𝜔,0⋮0^𝑞_𝑘=1,𝐾_𝑃()𝑁()=∫𝐼−𝑄𝑢,𝜆^𝑡𝜅𝐹₁(𝑠,𝜆)Δ𝑠+∑_{𝜅<𝑡𝑘<𝑡}ln1+𝑔_1𝑘⋮∫^𝑡_𝜅𝐹_𝑛(𝑠,𝜆)Δ𝑠+∑_{𝜅<𝑡𝑘<𝑡}ln1+𝑔_1𝑘+12−𝑡𝜔∫_𝜅^𝜅+𝜔𝐹₁(𝑠,𝜆)Δ𝑠+2𝑔₁⋮∫_𝜅^𝜅+𝜔𝐹_𝑛(𝑠,𝜆)Δ𝑠+2𝑔_𝑛−1𝜔∫_𝜅^𝜅+𝜔∫^𝑡_𝜅𝐹₁(𝑠,𝜆)Δ𝑠Δ𝑡+2𝑔₁⋮∫_𝜅^𝜅+𝜔∫^𝑡_𝜅𝐹_𝑛(𝑠,𝜆)Δ𝑠Δ𝑡+2𝑔_𝑛, (3.21) where 𝐹₁(𝑠,𝜆)=𝑎₁(𝑠)−𝑏₁(𝑠)𝑒^{𝑢1(𝑠−𝜏11(𝑠))}−𝜆𝑐₁₂(𝑠)𝑒^{𝑢2(𝑠−𝜏12(𝑠))}−ℎ₁(𝑠)𝑒^{−𝑢1(𝑠)},𝐹_𝑖(𝑠,𝜆)=−𝜆𝑑_𝑖(𝑠)−𝑏_𝑖(𝑠)𝑒^{𝑢𝑖(𝑠−𝜏𝑖𝑖(𝑠))}+𝑐_{𝑖,𝑖−1}(𝑠)𝑒^{𝑢𝑖−1(𝑠−𝜏𝑖,𝑖−1(𝑠))}−𝜆𝑐_𝑖,𝑖+1(𝑠)𝑒^{𝑢𝑖+1(𝑠−𝜏𝑖,𝑖+1(𝑠))}−ℎ_𝑖(𝑠)𝑒^{−𝑢𝑖(𝑠)}𝐹,𝑖=2,3,…,𝑛−1,_𝑛(𝑠,𝜆)=−𝜆𝑑_𝑛(𝑠)−𝑏_𝑛(𝑠)𝑒^{𝑢𝑛(𝑠−𝜏𝑛𝑛(𝑠))}+𝑐_{𝑛,𝑛−1}(𝑠)𝑒^{𝑢𝑛−1(𝑠−𝜏𝑛,𝑛−1(𝑠))}−ℎ_𝑛(𝑠)𝑒^{−𝑢𝑛(𝑠)}. (3.22) By the Lebesgue convergence theorem, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. Next, we show that 𝐾_𝑃(𝐼−𝑄)𝑁 maps bounded sets into relatively compact sets. Let Ω⊂𝑋 be an arbitrary bounded set in 𝑋, then there exists a number 𝑅>0 such that ‖𝑢‖<𝑅 for any 𝑢⊂Ω. We prove that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is relatively compact. In fact, for any {𝑢_𝜈=(𝑢^𝜈₁,𝑢^𝜈₂,…,𝑢^𝜈_𝑛)^𝑇}_𝜈∈ℕ∈Ω and 𝑡∈𝐼_𝜔, we have 𝐾_𝑃(𝐼−𝑄)𝑁𝑢_𝜈,𝜆=^𝑛_𝑖=1sup_{𝑡∈𝐼𝜔}^𝑡_𝜅𝐹_𝑖(𝑠,𝜆)Δ𝑠+_{𝜅<𝑡𝑘<𝑡}ln1+𝑔_𝑖𝑘+12−𝑡𝜔×_𝜅^𝜅+𝜔𝐹_𝑖(𝑠,𝜆)Δ𝑠+2𝑔_𝑖−1𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝐹_𝑖(𝑠,𝜆)Δ𝑠Δ𝑡+2𝑔₁. (3.23) Since ‖𝑢_𝜈‖<𝑅, it follows from the periodicity that there exists a constant 𝑊₁ such that ‖𝐾_𝑃(𝐼−𝑄)𝑁(𝑢_𝜈,𝜆)‖≤𝑊₁. For any {𝑢_𝜈=(𝑢^𝜈₁,𝑢^𝜈₂,…,𝑢^𝜈_𝑛)^𝑇}_𝜈∈ℕ∈Ω and 𝑡∈𝐼_𝜔∩(𝑡_𝑘,𝑡_𝑘+1], 𝑘=1,2,…,𝑞−1, we have 𝐾_𝑃(𝐼−𝑄)𝑁𝑢_𝜈,𝜆^Δ=𝐹₁()⋮𝐹𝑠,𝜆_𝑛()−1𝑠,𝜆𝜔∫_𝜅^𝜅+𝜔𝐹₁(𝑠,𝜆)Δ𝑠+2𝑔₁⋮∫_𝜅^𝜅+𝜔𝐹_𝑛(𝑠,𝜆)Δ𝑠+2𝑔_𝑛. (3.24) Similarly, there has a constant 𝑊₂ such that ^𝑛_𝑖=1sup_{𝑡∈𝐼𝜔∩(𝑡𝑘,𝑡𝑘+1]}𝐾_𝑃(𝐼−𝑄)𝑁𝑢_𝜈,𝜆^Δ≤𝑊₂,𝑘=1,2,…,𝑞−1. (3.25) It follows from the definition of 𝑋 that the right limit of 𝑢_𝜈(𝑡) exists at each