Existence of Multiple Positive Periodic Solutions to Two Species Parasitical Model with Impulsive Effects and Harvesting Terms

Abstract

By applying Mawhin’s continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of four positive solutions for two species parasitical system with impulsive effects and harvesting terms. Finally, an example is given to illustrate the effectiveness of our results.

1. Introduction

In recent years, the existence of periodic solutions in biological models has been widely studied. Models with harvesting terms are often considered. Generally, the model with harvesting terms is described as follows: 𝑥̇₁=𝑥₁𝑓𝑥₁,𝑥₂−ℎ,𝑥̇₂=𝑥₂𝑔𝑥₁,𝑥₂−𝑘, (1) where 𝑥₁ and 𝑥₂ are functions of two species, respectively; ℎ and 𝑘 are harvesting terms standing for the harvests (see [1, 2]). Because of the effect of changing environment such as the weather, season, and food, the number of species population periodically varies with the time. The rate of change usually is not a constant. Motivated by this, we consider the periodic nonautonomous population models. For example, two species parasitical system with harvesting terms is as follows [3]: 𝑥̇=𝑥(𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑥(𝑡)−ℎ₁(𝑡),𝑦̇=𝑦(𝑡)𝑎₂(𝑡)−𝑏₂(𝑡)𝑦(𝑡)+𝑐(𝑡)𝑥(𝑡)−ℎ₂(𝑡), (2) where 𝑥(𝑡) and 𝑦(𝑡) denote the densities of the host and the parasites, respectively; 𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐(𝑡), and ℎ_𝑖(𝑡)(𝑖=1,2) are all positive continuous functions and denote the intrinsic growth rate, death rate, obtaining nutriment rate from the host, and harvesting rate, respectively. In the model (2), the parasitical influence on its host is negligible. 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 gives us impulsive differential equations. For the theory of impulsive differential equations, we refer the reader to [4, 5].

However, to the best of our knowledge, there are few results on the existence of multiple periodic solutions for the delay parasitical with impulsive effects in the literatures. This motivates us to consider the existence of multiple periodic solutions for following parasitical with impulsive effects and harvesting terms nonautonomous model: 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑥₁𝑡−𝜏₁₁(𝑡)−ℎ₁(𝑡),𝑡≠𝑡_𝑘,𝑥̇₂(𝑡)=𝑥₂(𝑡)𝑎₂(𝑡)−𝑏₂(𝑡)𝑥₂𝑡−𝜏₂₂(𝑡)+𝑐(𝑡)𝑥₁𝑡−𝜏₂₁(𝑡)−ℎ₂(𝑡),𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝑔_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑡=𝑡_𝑘, (3) where 𝑔_𝑖𝑘∈(−1,+∞)(𝑖=1,2;𝑘∈ℕ={1,2}).    {𝑡_𝑘}_𝑘∈ℕ is a strictly increasing sequence with 𝑡₁>0 and lim_𝑡→∞𝑡_𝑘=∞.  𝑥_𝑖(𝑡)(𝑖=1,2) is the 𝑖th species population density.   𝑎_𝑖(𝑡)>0(𝑖=1,2) denotes the intrinsic growth rate; 𝑏_𝑖(𝑡)>0 and ℎ_𝑖(𝑡)>0(𝑖=1,2) stand for death rate, obtaining nutriment rate from the host, and harvesting rate, respectively. 𝑐(𝑡)>0 represents obtaining nutriment rate from the host; 𝜏₂₁(𝑡)≥0 stands for the time-lag in the process of transformation from the 1th species to the 2th species. 𝜏_𝑖𝑖(𝑡)≥0(𝑖=1,2) 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(𝑡),𝜏_𝑖𝑖(𝑡),𝜏₂₁(𝑡), and ℎ_𝑖(𝑡)(𝑖=1,2) 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) and 𝑡_𝑘+𝑞=𝑡_𝑘+𝜔.

Since a very basic and important problem in the study of a population growth model with a periodic environment is the global existence and stability of a positive periodic solution, which plays a similar role as a globally stable equilibrium does in an autonomous model, also, on the existence of positive periodic solutions to system (3), few results are found in the literatures. This motivates us to investigate the existence of a positive periodic or multiple positive periodic solutions for system (3). In fact, it is more likely for some biological species to take on multiple periodic change regulations and have multiple local stable periodic phenomena. Therefore, it is essential for us to investigate the existence of multiple positive periodic solutions for population models. Our main aim of this paper is by using Mawhin’s continuation theorem of coincidence degree theory to establish the existence of four positive periodic solutions for system (3). For the work concerning the multiple existence of periodic solutions of periodic population models which was done using coincidence degree theory, we refer the reader to [6–8].

This paper is organized as follows. In Section 2, by using the continuation theorem of coincidence degree theory and the skills of inequalities, we establish the existence of at least four positive periodic solutions of system (3). An example is presented in the last section to illustrate the effectiveness of our results.

2. Existence of at Least Four Positive Periodic Solutions

We first summarize a few concepts from the book by Gaines and Mawhin [9].

Let 𝑋 and 𝑍 be real normed vector spaces. Let 𝐿∶Dom𝐿⊂𝑋→𝑍 be a linear mapping and 𝑁∶𝑋×[0,1]→𝑍 a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿 = codimIm𝐿<∞, and Im𝐿 is closed in 𝑍. If 𝐿 is a Fredholm mapping of index zero, then there exist 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], and if 𝑄𝑁(Ω×[0,1]) is bounded, 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω×[0,1]→𝑋 is compact. Because Im𝑄 is isomorphic to Ker𝐿, there exists an isomorphism 𝐽∶Im𝑄→Ker𝐿.

Lemma 1 (see [9]). 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.

Let 𝑇 be a given positive constant, and a finite number of points of the sequence {𝜏_𝑘} lies in the interval [0,𝑇]. Let 𝑃𝐶([0,𝑇],ℝ^𝑛) be the set of functions 𝑥∶[0,𝑇]→ℝ^𝑛 which are piecewise continuous in [0,𝑇] and have points of discontinuous 𝜏_𝑘∈[0,𝑇], where they are continuous from the left. In the set 𝑃𝐶([0,𝑇],ℝ^𝑛) introduce the norm ‖𝑥‖=sup|𝑥(𝑡)|∶𝑡∈[0,𝑇] with which 𝑃𝐶([0,𝑇],ℝ^𝑛) becomes a Banach space with the uniform convergence topology.

Definition 2. The set ℱ⊂𝑃𝐶([0,𝑇],ℝ^𝑛) is said to be quasiequicontinuous in [0,𝑇], if for any 𝜖>0, there exists 𝛿>0 such that if 𝑥∈ℱ;𝑘∈ℤ;𝑡₁,𝑡₂∈(𝜏_𝑘−1,𝜏_𝑘]∩[0,𝑇] and |𝑡₁−𝑡₂|<𝛿, then |𝑥(𝑡₁)−𝑥(𝑡₂)|<𝜖.

The following result called compactness criterion gives a necessary and sufficient condition for relative compactness in 𝑃𝐶([0,𝑇],ℝ^𝑛).

Lemma 3 (see [4]). The set ℱ⊂𝑃𝐶([0,𝑇],ℝ^𝑛) is relatively compact if and only if(a)ℱ is bounded, namely, ‖𝑥‖≤𝑐, for each 𝑥∈ℱ and some 𝑐>0;(b)ℱ is quasiequicontinuous in [0,𝑇].

For the sake of convenience, we denote 𝑓=(1/𝜔)∫^𝜔₀𝑓(𝑡)d𝑡; here 𝑓(𝑡) is a continuous 𝜔-periodic function. Let 𝑃𝐶_𝜔 denote the space of 𝜔-periodic functions 𝜓∶ℝ→ℝ which are continuous for 𝑡≠𝑡_𝑘, are continuous from the left for 𝑡∈ℝ, and have possible discontinuities of the first kind at points 𝑡=𝑡_𝑘; that is, the limit from the right of 𝑡_k exists but may be different from the value at 𝑡_𝑘. We also denote 𝑃𝐶¹_𝜔={𝜓∈𝑃𝐶_𝜔∶𝜓̇∈𝑃𝐶_𝜔}. Obviously, 𝜓 is continuous if 𝜓∈𝑃𝐶¹_𝜔.

For simplicity, we need to introduce some notations as follows:𝑔_𝑖=12^𝑞_𝑘=1ln1+𝑔_𝑖𝑘,𝐺_𝑖=12^𝑞_𝑘=1ln1+𝑔_𝑖𝑘+ln1+𝑔_𝑖𝑘,𝜌_𝑖=𝑒^{𝑎𝑖𝜔+𝐺𝑖},𝑙^±₁=𝑎₁+2𝑔₁/𝜔𝜌₁±(𝑎₁+2𝑔₁/𝜔)𝜌₁²−4𝑏₁ℎ₁𝜌₁2𝑏₁,𝑙^±₂=𝑐𝑙⁺₁+𝑎₂+2𝑔₂𝜔𝜌₂±𝑐𝑙⁺₁+𝑎₂+2𝑔₂𝜔𝜌₂²−4𝑏₂ℎ₂𝜌₂2𝑏₂⁻¹,𝐴^±_𝑖=𝑎_𝑖+2𝑔_𝑖/𝜔±𝑎_𝑖+2𝑔_𝑖/𝜔²−4𝑏_𝑖ℎ_𝑖𝜌_𝑖2𝑏_𝑖,𝐵^±_𝑖=𝐴^±_𝑖𝜌_𝑖,𝐻_𝑖=𝑙⁻_𝑖𝜌_𝑖, (4) where 𝑖=1,2.

Throughout this paper, we need the following assumptions: (𝑇₁) (𝑎_𝑖+2𝑔_𝑖/𝜔)𝜌_𝑖^−1/2>2𝑏_𝑖ℎ_𝑖(𝑖=1,2);(𝑇₂)  𝜌_𝑖>1(𝑖=1,2).

The following results will play an important role in the proof of our main result.

Lemma 4 (see [5]). Suppose 𝜓∈𝑃𝐶¹_𝜔, then0≤sup_{[]𝑠∈0,𝜔}𝜓(𝑠)−inf_{[]𝑠∈0,𝜔}𝜓(𝑠)≤12^𝜔₀𝜓^′(𝑠)d𝑠+^𝑞_𝑘=1Δ𝜓𝑡_𝑘. (5)

Lemma 5. 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, and 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. (6) By the relationship of the derivative and the monotonicity, the above assertions obviously hold. The proof of Lemma 5 is complete.

Lemma 6. Assume that (𝑇₁) and (𝑇₂) hold, then one has the following inequalities: (1)ln𝑙⁺_𝑖>ln𝐴⁺_𝑖>ln𝐴⁻_𝑖>lnl⁻_𝑖>ln𝐻_𝑖(𝑖=1,2); (2)ln𝑙⁺_𝑖>ln𝐵⁺_𝑖>ln𝐵⁻_𝑖>ln𝑙⁻_𝑖>ln𝐻_𝑖(𝑖=1,2); (3)min{ln𝐴⁺_𝑖,ln𝐵⁺_𝑖}>max{ln𝐴⁻_𝑖,ln𝐵⁻_𝑖}(𝑖=1,2).

Proof. Since 𝑎₁+2𝑔₁𝜔𝜌₁>𝑎₁+2𝑔₁𝜔>𝑎₁+2𝑔₁/𝜔√𝜌₁>2𝑏₁ℎ₁>0,𝑐𝑙⁺₂+2𝑎₂+2𝑔₂𝜔𝜌₂>𝑎₂+2𝑔₂𝜔>2𝑔₂/𝜔+𝑎₂√𝜌₂>2𝑏₂ℎ₂>0,𝑏_𝑖>0,ℎ_𝑖>0,𝑖=1,2. (7) Applying Lemma 5, we have 0<𝐻₁=𝑙⁻₁𝜌₁<𝑙⁻₁=𝑔𝑎₁+2𝑔₁𝜔𝜌₁,𝑏₁,ℎ₁𝜌₁<𝑔𝑎₁+2𝑔₁𝜔,𝑏₁,ℎ₁𝜌₁=𝐴⁻₁<𝐴⁺₁=𝑓𝑎₁+2𝑔₁𝜔,𝑏₁,ℎ₁𝜌₁<𝑓𝑎₁+2𝑔₁𝜔𝜌₁,𝑏₁,ℎ₁𝜌₁=𝑙⁺₁,0<𝐻₂=𝑙⁻₂√𝜌₂<𝑙⁻₂=𝑔𝑐𝑙⁺₂+𝑎₂+2𝑔₂𝜔𝜌₂,𝑏₂,ℎ₂𝜌₂<𝑔2𝑔₂𝜔+𝑎₂,𝑏₂,ℎ₂𝜌₂=𝐴⁻₂<𝐴⁺₂=𝑓2𝑔₂𝜔+𝑎₂,𝑏₂,ℎ₂𝜌₂<𝑓𝑐𝑙⁺₁+𝑎₂+2𝑔₂𝜔𝜌₂,𝑏₂,ℎ₂𝜌₂=𝑙⁺₂. (8)
Thus, we have ln𝐻_𝑖<ln𝑙⁻_𝑖<ln𝐴⁻_𝑖<ln𝐴⁺_𝑖<ln𝑙⁺_𝑖,𝑖=1,2. The proof of (1) of Lemma 6 is complete. For (2) of Lemma 6, we similarly have 0<𝐻₁=𝑙⁻₁𝜌₁<𝑙⁻₁=𝑔𝑎₁+2𝑔₁𝜔𝜌₁,𝑏₁,ℎ₁𝜌₁<𝑔𝑎₁+2𝑔₁𝜔,𝑏₁,ℎ₁𝜌₁=𝐵⁻₁<𝐵⁺₁=𝑓𝑎₁+2𝑔₁𝜔,𝑏₁,ℎ₁𝜌₁<𝑓𝑎₁+2𝑔₁𝜔𝜌₁,𝑏₁,ℎ₁𝜌₁=𝑙⁺₁,0<𝐻₂=𝑙⁻₂𝜌₂<𝑙⁻₂=𝑔𝑎₂+2𝑔₂𝜔𝜌₂,𝑏₂,ℎ₂𝜌₂<𝑔2𝑔₂𝜔+𝑎₂𝜌₂,𝑏₂,ℎ₂𝜌₂=𝐵⁻₂<𝐵⁺₂=𝑓2𝑔₂𝜔+𝑎₂𝜌₂,𝑏₂,ℎ₂𝜌₂<𝑓𝑐𝑙⁺₁+𝑎₂+2𝑔₂𝜔𝜌₂,𝑏₂,ℎ₂𝜌₂=𝑙⁺₂, (9) which imply that ln𝐻_𝑖<ln𝑙⁻_𝑖<ln𝐵⁻_𝑖<ln𝐵⁺_𝑖<ln𝑙⁺_𝑖(𝑖=1,2); that is, the inequality (2) of Lemma 6 holds. Let us now prove the inequality (3) of Lemma 6. In fact, since maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖=ln𝐴⁻_𝑖+ln𝐵⁻_𝑖+ln𝐴⁻_𝑖−ln𝐵⁻_𝑖2(𝑖=1,2),minln𝐴⁺_𝑖,ln𝐵⁺_𝑖=ln𝐴⁺_𝑖+ln𝐵⁺_𝑖+ln𝐴⁺_𝑖−ln𝐵⁺_𝑖2(𝑖=1,2), (10) we have minln𝐴⁺_𝑖,ln𝐵⁺_𝑖−maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖=ln𝐴⁺_𝑖+ln𝐵⁺_𝑖−ln𝐴⁻_𝑖−ln𝐵⁻_𝑖−ln𝐴⁻_𝑖−ln𝐵⁻_𝑖−ln𝐴⁺_𝑖−ln𝐵⁺_𝑖(2)⁻¹. (11)
If 𝐴⁺_𝑖≥𝐵⁺_𝑖, then 𝐴⁻_𝑖≤𝐵⁻_𝑖, and minln𝐴⁺_𝑖,ln𝐵⁺_𝑖−maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖=ln𝐵⁺_𝑖−ln𝐵⁻_𝑖>0. (12)
If 𝐴⁺_𝑖<𝐵⁺_𝑖, then 𝐴⁻_𝑖>𝐵⁻_𝑖, and minln𝐴⁺_𝑖,ln𝐵⁺_𝑖−maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖=ln𝐴⁺_𝑖−ln𝐴⁻_𝑖>0. (13) Hence, we drive min{ln𝐴⁺_𝑖,ln𝐵⁺_𝑖}>max{ln𝐴⁻_𝑖,ln𝐵⁻_𝑖}. From the above, we have completed the proof of Lemma 6.

Theorem 7. Assume that (𝑇₁) and (𝑇₂) hold. Then system (3) has at least four positive 𝜔-periodic solutions.

Proof. By making the substitution 𝑥_𝑖(𝑡)=𝑒^{𝑢𝑖(𝑡)},𝑖=1,2, (14) system (3) can be reformulated as 𝑢̇₁(𝑡)=𝑎₁(𝑡)−𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}−ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)},𝑡≠𝑡_𝑘,𝑢̇₂(𝑡)=𝑎₂(𝑡)−𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}+𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}−ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)},𝑡≠𝑡_𝑘,Δ𝑢_𝑖𝑡_𝑘=𝑢_𝑖𝑡⁺_𝑘−𝑢_𝑖𝑡⁻_𝑘=ln1+𝑔_𝑖𝑘,𝑖=1,2,𝑡=𝑡_𝑘. (15)
Let 𝑋=𝑢=𝑢₁,𝑢₂^𝑇∶𝑢_𝑖∈𝑃𝐶¹_𝜔,𝑖=1,2,𝑍=𝑋×ℝ^2𝑞, (16) and define ‖𝑢‖₀=²_𝑖=1sup_{[]𝑡∈0,𝜔}𝑢_𝑖(𝑡‖𝑧‖),𝑢∈𝑋,₁=‖𝑢‖₀+^𝑞_𝑗=1𝜉_𝑗,𝑧=𝑢,𝜉₁,𝜉₂,…,𝜉_𝑞∈𝑍. (17) Then both (𝑋,‖⋅‖₀) and (𝑍,‖⋅‖₁) are Banach spaces. Let {̇}𝐿𝑢Dom𝐿=𝑢∈𝑋∶𝑥∈𝑋,𝐿∶Dom𝐿⟶𝑍,𝑢=₁,𝑢₂^𝑇=𝑢̇₁𝑢̇₂,Δ𝑢₁𝑡_𝑘Δ𝑢₂𝑡_𝑘^𝑞_𝑘=1,𝑁()=𝐹𝑁∶𝑋⟶𝑍,𝑢,𝜆¹()𝐹𝑡,𝜆₂(),𝑡,𝜆ln1+𝑔_1𝑘ln1+𝑔_2𝑘^𝑞_𝑘=1, (18) where 𝐹₁(𝑡,𝜆)=𝑎₁(𝑡)−𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}−ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)},𝐹₂(𝑡,𝜆)=𝑎₂(𝑡)−𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}+𝜆𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}−ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)},𝑃𝑢𝑃∶𝑋→𝑋,𝑄∶𝑍→𝑍,𝑢=₁,𝑢₂^𝑇=𝑢₁,𝑢₂^𝑇𝑄𝑢,𝑢∈𝑋,₁𝑢₂,𝑚_1𝑘𝑚_2𝑘^𝑞_𝑘=1=𝑢₁+1𝜔^𝑞_𝑘∑₌₁𝑚_1𝑘𝑢₂+1𝜔^𝑞_𝑘∑₌₁𝑚_2𝑘,00^𝑞_𝑘=1. (19) Obviously, Ker𝐿=𝑢=𝑢₁,𝑢₂^𝑇∈𝑋∶𝑢₁(𝑡),𝑢₂(𝑡𝑡)=𝑐)(₁,𝑐₂∈ℝ²,=𝑢,𝑡∈ℝIm𝐿₁𝑢₂,𝑚_1𝑘𝑚_2𝑘^𝑞_𝑘=1𝜔∈𝑍∶𝑢₁+∑^𝑞_𝑘=1𝑚_1𝑘𝜔=0𝑢₂+∑^𝑞_𝑘=1𝑚_2𝑘.=0 (20) Since Im𝐿 is closed in 𝑍,Im𝑃=Ker𝐿,Ker𝑄=Im𝐿, and dimKer𝐿=2=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 by𝐾_𝑃𝑢𝑧=¹𝑢₂,𝑚_1𝑘𝑚_2𝑘^𝑞_𝑘=1=∫^𝑡₀𝑢₁(𝑠)d𝑠+∑_{0<𝑡𝑘<𝑡}𝑚_1𝑘−1𝜔^𝑞_𝑘∑₌₁𝑚_1𝑘−1𝜔∫^𝜔₀∫^𝑡₀𝑢₁(𝑠)∫d𝑠d𝑡_𝑡0𝑢₂(𝑠)d𝑠+∑_{0<𝑡𝑘<𝑡}𝑚_2𝑘−1𝜔^𝑞_𝑘∑₌₁𝑚_2𝑘−1𝜔∫^𝜔₀∫^𝑡₀𝑢₂(𝑠).d𝑠d𝑡 (21)Then ()=1𝑄𝑁𝑢,𝜆𝜔∫^𝜔₀𝐹₁(𝑠,𝜆)d𝑠+2𝑔₁𝜔1𝜔∫^𝜔₀𝐹₂(𝑠,𝜆)d𝑠+2𝑔₂𝜔,00^𝑞_𝑘=1,𝐾_𝑃()𝑁()=∫𝐼−𝑄𝑢,𝜆_𝑡0𝐹₁(𝑠,𝜆)d𝑠+∑_{0<𝑡𝑘<𝑡}ln1+𝑔_1𝑘∫^𝑡₀𝐹₁(𝑠,𝜆)d𝑠+∑_{0<𝑡𝑘<𝑡}ln1+𝑔_1𝑘+12−𝑡𝜔∫^𝜔₀𝐹₁(𝑠,𝜆)d𝑠+2𝑔₁∫^𝜔₀𝐹₂(𝑠,𝜆)d𝑠+2𝑔₂−1𝜔∫^𝜔₀∫^𝑡₀𝐹₁(𝑠,𝜆)d𝑠d𝑡+2𝑔₁∫^𝜔₀∫^𝑡₀𝐹₂(𝑠,𝜆)d𝑠d𝑡+2𝑔₂, (22) where 𝐹_𝑖(𝑠,𝜆)(𝑖=1,2) are very similar to that of 𝐹_𝑖(𝑡,𝜆)(𝑖=1,2). By the Lebesgue convergence theorem, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. Moreover, because of periodicity, it follows from Lemma 3 that 𝑄𝑁(Ω),𝐾_𝑃(𝐼−𝑄)𝑁(Ω) are relatively compact for any open bounded set Ω⊂𝑋. Thus, 𝑁 is 𝐿-compact on Ω for any open bounded set Ω⊂𝑋.
In order to use Lemma 1, we have to find at least four appropriate open bounded subsets of 𝑋. Considering the operator equation 𝐿𝑢=𝜆𝑁(𝑢,𝜆),𝜆∈(0,1), we have 𝑢̇₁(𝑡)=𝜆𝑎₁(𝑡)−𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}−ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)},𝑢̇₂(𝑡)=𝜆𝑎₂(𝑡)−𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}+𝜆𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}−ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)},Δ𝑢_𝑗𝑡_𝑘=𝜆ln1+𝑔_𝑗𝑘,𝑗=1,2. (23) Assume that 𝑢∈𝑋 is a 𝜔-periodic solution of system (23) for some 𝜆∈(0,1). Integrating (23) from 0 to 𝜔, we have −2𝑔₁=^𝜔₀𝑎₁(𝑡)−𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}−ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)}d𝑡,−2𝑔₂=^𝜔₀𝑎₂(𝑡)−𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}+𝜆𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}−ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)}d𝑡. (24) Then ^𝜔₀𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}+ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)}d𝑡=𝑎₁𝜔+2𝑔₁,_𝜔0𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}−𝜆𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}+ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)}=d𝑡𝑎₂𝜔+2𝑔₂. (25) Furthermore, note that 𝑢=(𝑢₁,𝑢₂)^𝑇∈𝑋. There exists 𝜉_𝑖,𝜂_𝑖∈[0,𝜔],𝑖=1,2, such that 𝑢_𝑖(𝜉_𝑖)=sup_{𝑡∈[0,𝜔]}𝑢_𝑖(𝑡),𝑢_𝑖(𝜂_𝑖)=inf_{𝑡∈[0,𝜔]}𝑢_𝑖(𝑡).
On the one hand, according to the first equation of (23) and (25), we have 𝑢̇^𝜔₀₁(𝑡)d𝑡<2𝑎₁𝜔+2𝑔₁. (26)In the light of the first equation of (25) and (26)  and Lemma 4, we get 𝜔𝑏₁𝑒^{𝑢1(𝜉1)−(𝑎1𝜔+𝐺1)}+𝜔ℎ₁𝑒^{−𝑢1(𝜉1)}≤𝜔𝑏₁𝑒^{𝑢1(𝜉1)−(1/2)[∫𝜔0|̇𝑢1(𝑠)|d𝑠+∑𝑞𝑘=1|Δ𝑢1(𝑡𝑘)|]}+𝜔ℎ₁𝑒^{−𝑢1(𝜉1)}<𝜔𝑏₁𝑒^𝑢1(𝜂1)+𝜔ℎ₁𝑒^{−𝑢1(𝜉1)}≤^𝜔₀𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}+ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)}=d𝑡𝑎₁𝜔+2𝑔₁, (27) namely, 𝑏₁𝑒^{2𝑢1(𝜉1)}−𝑎₁+2𝑔₁𝜔𝑒^{𝑎1𝜔+𝐺1}𝑒^𝑢1(𝜉1)+ℎ₁𝑒^{𝑎1𝜔+𝐺1}<0, (28) which implies that ln𝑙⁻₁<𝑢₁𝜉₁<ln𝑙⁺₁. (29) Applying Lemma (23), we drive ln𝐻₁=ln𝑙⁻₁−𝑎₁𝜔+𝐺₁<𝑢₁𝜂≤𝑢₁𝜉₁<ln𝑙⁺₁. (30) According to (23) and (25), we get 𝑢̇^𝜔₀₂(𝑡)d𝑡<2𝑎₂𝜔+2𝑔₂. (31)
By (25) and (31), we have 𝜔𝑏₂𝑒^{𝑢2(𝜉2)−(𝑎2𝜔+𝐺2)}−𝜔𝑐𝑙⁺₁+𝜔ℎ₂𝑒^{−𝑢2(𝜉2)}≤𝜔𝑏₂𝑒^{𝑢2(𝜉2)−(1/2)[∫𝜔0|̇𝑢2(𝑠)|d𝑠+∑𝑞𝑘=1|Δ𝑢2(𝑡𝑘)|]}−𝜔𝑐𝑙⁺₁+𝜔ℎ₂𝑒^{−𝑢2(𝜉2)}<𝜔𝑏₂𝑒^𝑢2(𝜂2)−𝜔𝑐𝑙⁺₁+𝜔ℎ₂𝑒^{−𝑢2(𝜉2)}<^𝜔₀𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}−𝜆𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}+ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)}=d𝑡𝑎₂𝜔+2𝑔₂, (32) that is, 𝑏₂𝑒^{2𝑢2(𝜉2)}−𝑐𝑙⁺₁+𝑎₂+2𝑔₂𝜔𝑒^{𝑎2𝜔+𝐺2}𝑒^𝑢2(𝜉2)+ℎ₂𝑒^{𝑎2𝜔+𝐺2}<0, (33) which implies that ln𝑙⁻₂<𝑢₂𝜉₂<ln𝑙⁺₂. (34) Applying Lemma 4, we drive ln𝐻₂=ln𝑙⁻₂−𝑎₂𝜔+𝐺₂<𝑢₂𝜂≤𝑢₂𝜉₂<ln𝑙⁺₂. (35)
On the other hand, in the light of the first equation of (25) and (26), we have 𝑎₁𝜔+2𝑔₁=^𝜔₀𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}+ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)}≤d𝑡𝑏₁𝜔𝑒^𝑢1(𝜉1)+ℎ₁𝜔𝑒^{−𝑢1(𝜂1)}<𝑏₁𝜔𝑒^𝑢1(𝜉1)+ℎ₁𝜔𝑒^{−𝑢1(𝜉1)+(1/2)[∫𝜔0|̇𝑢1(𝑠)|d𝑠+∑𝑞𝑘=1|Δ𝑢1(𝑡𝑘)|]}<𝑏₁𝜔𝑒^𝑢1(𝜉1)+ℎ₁𝜔𝑒^{−𝑢1(𝜉1)+𝑎1𝜔+𝐺1}, (36) namely, 𝑏₁𝑒^{2𝑢1(𝜉1)}−𝑎₁+2𝑔₁𝜔𝑒^𝑢1(𝜉1)+ℎ₁𝑒^{𝑎1𝜔+𝐺1}>0, (37) which implies that ln𝐴⁺₁<𝑢₁𝜉₁or𝑢₁𝜉₁<ln𝐴⁻₁,𝑎₁𝜔+2𝑔₁=^𝜔₀𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}+ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)}d𝑡≤𝜔𝑏₁𝑒^𝑢1(𝜉1)+ℎ₁𝑒^{−𝑢1(𝜂1)}<𝜔𝑏₁𝑒^{𝑢1(𝜂1)+(1/2)[∫𝜔0|̇𝑢1(𝑠)|d𝑠+∑𝑞𝑘=1|Δ𝑢1(𝑡𝑘)|]}+ℎ₁𝑒^{−𝑢1(𝜂1)}<𝑏₁𝜔𝑒^{𝑢1(𝜂1)+𝑎1𝜔+𝐺1}+ℎ₁𝜔𝑒^{−𝑢1(𝜂1)}, (38) that is, 𝑏₁𝑒^{2𝑢1(𝜂1)}−𝑎₁+2𝑔₁𝜔𝑒^{−(𝑎1𝜔+𝐺1)}𝑒^𝑢1(𝜂1)+ℎ₁𝑒^{−(𝑎1𝜔+𝐺1)}>0, (39) which implies that ln𝐵⁺₁<𝑢₁𝜂₁or𝑢₁𝜂₁<ln𝐵⁻₁. (40) By (25) and (31), we have 2𝑔₂<𝑎₂𝜔+2𝑔₂=^𝜔₀𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}−𝜆𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}+ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)}<d𝑡𝑏₂𝜔𝑒^𝑢2(𝜉2)+ℎ₂𝜔𝑒^{−𝑢2(𝜂2)}<𝑏₂𝜔𝑒^𝑢2(𝜉2)+ℎ₂𝜔𝑒^{−𝑢2(𝜉2)+(1/2)[∫𝜔0|̇𝑢2(𝑠)|d𝑠+∑𝑞𝑘=1|Δ𝑢2(𝑡𝑘)|]}<𝑏₂𝜔𝑒^𝑢2(𝜉2)+ℎ₂𝜔𝑒^{−𝑢2(𝜉2)+𝑎2𝜔+𝐺2}, (41) namely, 𝑏₂𝑒^{2𝑢2(𝜉2)}−𝑎₂+2𝑔₂𝜔𝑒^𝑢2(𝜉2)+ℎ₂𝑒^{𝑎2𝜔+𝐺2}>0, (42) which implies that ln𝐴⁺₂<𝑢₂𝜉₂or𝑢₂𝜉₂<ln𝐴⁻₂,2𝑔₂<𝑎₂𝜔+2𝑔₂=^𝜔₀𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}−𝜆𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}+ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)}<d𝑡𝑏₂𝜔𝑒^𝑢2(𝜉2)+ℎ₂𝜔𝑒^{−𝑢2(𝜂2)}<𝑏₂𝜔𝑒^{𝑢2(𝜂2)+(1/2)[∫𝜔0|̇𝑢2(𝑠)|d𝑠+∑𝑞𝑘=1|Δ𝑢2(𝑡𝑘)|]}+ℎ₂𝜔𝑒^{−𝑢2(𝜂2)}<𝑏₂𝜔𝑒^{𝑢2(𝜂2)+𝑎2𝜔+𝐺2}+ℎ₂𝜔𝑒^{−𝑢2(𝜂2)}, (43) that is, 𝑏₂𝑒^{2𝑢2(𝜂2)}−2𝑔₂𝜔+𝑎₂𝑒^{−(𝑎2𝜔+𝐺2)}𝑒^𝑢2(𝜂2)+ℎ₂𝑒^{−(𝑎2𝜔+𝐺2)}>0, (44) which implies that ln𝐵⁺₂<𝑢₂𝜂₂or𝑢₂𝜂₂<ln𝐵⁻₂. (45)
In view of (30)–(45) and Lemma 6, we have minln𝐴⁺₁,ln𝐵⁺₁<𝑢₁𝜂₁<𝑢₁𝜉₁<ln𝑙⁺₁,minln𝐴⁺₂,ln𝐵⁺₂<𝑢₂𝜂₂<𝑢₂𝜉₂<ln𝑙⁺₂ (46) or ln𝐻₁<𝑢₁𝜂₁<𝑢₁𝜉₁<maxln𝐴⁻₁,ln𝐵⁻₁,ln𝐻₂<𝑢₂𝜂₂<𝑢₂𝜉₂<maxln𝐴⁻₂,ln𝐵⁻₂, (47) which imply that, for all 𝑡∈𝑅, minln𝐴⁺₁,ln𝐵⁺₁<𝑢₁(𝑡)<ln𝑙⁺₁orln𝐻⁻₁<𝑢₁(𝑡)<maxln𝐴⁻₁,ln𝐵⁻₁,minln𝐴⁺₂,ln𝐵⁺₂<𝑢₂(𝑡)<ln𝑙⁺₂orln𝐻⁻₂<𝑢₂(𝑡)<maxln𝐴⁻₂,ln𝐵⁻₂. (48)
Clearly, ln𝑙^±₁,ln𝑙^±₂,ln𝐴^±₁,ln𝐴^±₂,ln𝐵^±₁, and ln𝐵^±₂ are independent of 𝜆. Now let Ω₁=𝑢=𝑢₁,𝑢₂^𝑇∈𝑋∶minln𝐴⁺₁,ln𝐵⁺₁<𝑢₁(𝑡)<ln𝑙⁺₁,minln𝐴⁺₂,ln𝐵⁺₂<𝑢₂(𝑡)<ln𝑙⁺₂,Ω₂=𝑢=𝑢₁,𝑢₂^𝑇∈𝑋∶minln𝐴⁺₁,ln𝐵⁺₁<𝑢₁(𝑡)ln𝐻₂<𝑢₂(𝑡)<maxln𝐴⁻₂,ln𝐵⁻₂,Ω₃=𝑢=𝑢₁,𝑢₂^𝑇∈𝑋∶ln𝐻₁<𝑢₁(𝑡)<maxln𝐴⁻₁,ln𝐵⁻₁,minln𝐴⁺₂,ln𝐵⁺₂<𝑢₂(𝑡)<ln𝑙⁺₂,Ω₄=𝑢=𝑢₁,𝑢₂^𝑇∈𝑋∶ln𝐻₁<𝑢₁(𝑡)<maxln𝐴⁻₁,ln𝐵⁻₁,ln𝐻₂<𝑢₂(𝑡)<maxln𝐴⁻₂,ln𝐵⁻₂. (49) Obviously, the number of the above sets is four. We denote these sets as Ω_𝑘,𝑘=1,2,3, and 4.   Ω_𝑘 are bounded open subsets of 𝑋,Ω_𝑖∩Ω_𝑗=𝜙,𝑖≠𝑗. Thus Ω_𝑘(𝑘=1,2,3, and 4) satisfies the requirement (a) in Lemma 1.
Now we show that (b) of Lemma 1 holds; that is, we prove when 𝑢∈𝜕Ω_𝑘∩Ker𝐿=𝜕Ω_𝑘∩𝑅²,𝑄𝑁(𝑢,0)≠(0,0,)^𝑇,𝑘=1,2,3, and 4. If it is not true, then when 𝑢∈𝜕Ω_𝑘∩Ker𝐿=𝜕Ω_𝑘∩𝑅²,𝑖=1,2,3, and 4, constant vector 𝑢=(𝑢₁,𝑢₂)^𝑇 with 𝑢∈𝜕Ω_𝑘(𝑘=1,2,3, and 4) satisfies −2𝑔₁=^𝜔₀𝑎₁(𝑡)−𝑏₁(𝑡)𝑒^{𝑢1(𝑡−𝜏11(𝑡))}−ℎ₁(𝑡)𝑒^{−𝑢1(𝑡)}d𝑡,−2𝑔₂=^𝜔₀𝑎₂(𝑡)−𝑏₂(𝑡)𝑒^{𝑢2(𝑡−𝜏22(𝑡))}+𝜆𝑐(𝑡)𝑒^{𝑢1(𝑡−𝜏21(𝑡))}−ℎ₂(𝑡)𝑒^{−𝑢2(𝑡)}d𝑡, (50) that is, −2𝑔₁𝜔=𝑎₁−𝑏₁𝑒^𝑢1−ℎ₁𝑒^−𝑢1,−2𝑔₂𝜔=𝑎₂−𝑏₂𝑒^𝑢2−ℎ₂𝑒^−𝑢2. (51) Similar to the process of (25)–(48), we obtain ln𝐻_𝑖<𝑢⁻_𝑖<maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖<minln𝐴⁺_𝑖,ln𝐵⁺_𝑖<𝑢⁺_𝑖<ln𝑙⁺_𝑖(1≤𝑖≤2). (52) Then 𝑢 belongs to one of Ω_𝑘∩𝑅²,𝑘=1,2,3, and 4. This contradicts the fact that 𝑢∈𝜕Ω_𝑘∩𝑅²,𝑘=1,2,3, and 4. This proves that (b) in Lemma 1 holds.
Finally, in order to show that (c) in Lemma 1 holds, we only prove that for 𝑢∈𝜕Ω_𝑘∩Ker𝐿=𝜕Ω_𝑘∩𝑅²,𝑘=1,2,3, and 4, then it holds that deg{𝐽𝑄𝑁(𝑢,0),Ω_𝑘∩Ker𝐿,(0,0)^𝑇}≠0. To this end, we define the mapping 𝜙∶Dom𝐿×[0,1]→𝑋 by 𝜙𝑢,𝜇=𝜇𝑄𝑁(𝑢,0)+1−𝜇𝐺(𝑢), (53) here 𝜇∈[0,1] is a parameter, and 𝐺(𝑢) is defined by 𝐺(𝑢)=1𝜔∫^𝜔₀𝑎₁(𝑠)−𝑏₁(𝑠)𝑒^𝑢1(𝑠)−ℎ₁(𝑠)𝑒^{−𝑢1(𝑠)}d𝑠+2𝑔₁𝜔1𝜔∫^𝜔₀𝑎₂(𝑠)−𝑏₂(𝑠)𝑒^𝑢2(𝑠)−ℎ₂(𝑠)𝑒^{−𝑢2(𝑠)}d𝑠+2𝑔₂𝜔,00^𝑞_𝑘=1. (54)We show that for 𝑢∈𝜕Ω_𝑘∩Ker𝐿=𝜕Ω_𝑘∩𝑅²,𝑘=1,2,3, and 4,𝜇∈[0,1], then it holds that 𝜙(𝑢,𝜇)≠(0,0,)^𝑇. Otherwise, parameter 𝜇 and constant vector 𝑢=(𝑢₁,𝑢₂)^𝑇∈𝑅² satisfy 𝜙(𝑢,𝜇)=(0,0)^𝑇, that is, 0=𝜇^𝜔₀𝑎₁(𝑠)−𝑏₁(𝑠)𝑒^𝑢1−ℎ₁(𝑠)𝑒^−𝑢1+2𝑔₁𝜔+d𝑠1−𝜇^𝜔₀𝑎₁(𝑠)−𝑏₁(𝑠)𝑒^𝑢1−ℎ₁(𝑠)𝑒^−𝑢1+2𝑔₁𝜔d𝑠,0=𝜇^𝜔₀𝑎₂(𝑠)−𝑏₂(𝑠)𝑒^𝑢2−ℎ₂(𝑠)𝑒^−𝑢2+2𝑔₂𝜔+d𝑠1−𝜇^𝜔₀𝑎₂(𝑠)−𝑏₂(𝑠)𝑒^𝑢2−ℎ₂(𝑠)𝑒^−𝑢2+2𝑔₂𝜔d𝑠, (55) that is, 𝑎₁−𝑏₁𝑒^𝑢1−ℎ₁𝑒^−𝑢1=−2𝑔₁𝜔,𝑎₂−𝑏₂𝑒^𝑢2−ℎ₂𝑒^−𝑢2=−2𝑔₂𝜔. (56) Following the argument of (25)–(48), we obtain ln𝐻_𝑖<𝑢⁻_𝑖<maxln𝐴⁻_𝑖,ln𝐵⁻_𝑖<minln𝐴⁺_𝑖,ln𝐵⁺_𝑖<𝑢⁺_𝑖<ln𝑙⁺_𝑖(1=1,2). (57) Equation (57) gives that 𝑢 belongs to one of Ω_𝑘∩𝑅²,𝑘=1,2,3, and 4. This contradicts the fact that 𝑢∈𝜕Ω_𝑘∩𝑅²,𝑘=1,2,3, and 4. This proves 𝜙(𝑢,𝜇)≠(0,0)^𝑇 holds. Note that the system of algebraic equations, 𝑎₁−𝑏₁𝑒^𝑥−ℎ₁𝑒^−𝑥=−2𝑔₁𝜔,𝑎₂−𝑏₂𝑒^𝑦−ℎ₂𝑒^−𝑦=−2𝑔₂𝜔, (58) has four distinct solutions since (𝑇₁) and (𝑇₂) hold: 𝑥^∗₁,𝑦^∗₁=ln𝑥⁺,ln𝑦⁺,𝑥^∗₂,𝑦^∗₂=lnx⁺,ln𝑦⁻,𝑥^∗₃,𝑦^∗₃=ln𝑥⁻,ln𝑦⁺,𝑥^∗₄,𝑦^∗₄=ln𝑥⁻,ln𝑦⁻, (59) where 𝑥^±=𝑎₁+2𝑔₁/𝜔±𝑎₁+2𝑔₁/𝜔²−4𝑏₁ℎ₁2𝑏₁,𝑦^±=𝑎₂+2𝑔₂/𝜔±𝑎₂+2𝑔₂/𝜔²−4𝑏₂ℎ₂2𝑏₂. (60) Similar to the proof of Lemma 6, it is easy to verify that ln𝐻⁻₁<ln𝑥⁻<maxln𝐵⁻₁,ln𝐴⁻₁<minln𝐴⁺₁,ln𝐵⁺₁<ln𝑥⁺<ln𝑙⁺₁,ln𝐻⁻₂<ln𝑦⁻<maxln𝐵⁻₂,ln𝐴⁻₂<minln𝐴⁺₂,ln𝐵⁺₂<ln𝑦⁺<ln𝑙⁺₂. (61)
Therefore 𝑥^∗₁,𝑦^∗₁∈Ω₁,𝑥^∗₂,𝑦^∗₂∈Ω₂,𝑥^∗₃,𝑦^∗₃∈Ω₃,𝑥^∗₄,𝑦^∗₄∈Ω₄, (62) uniquely belongs to the corresponding Ω_𝑘. Since Ker𝐿=ImQ, we can take 𝐽=𝐼. A direct computation gives, for 𝑘=1,2,3, and 4, deg𝐽𝑄𝑁(𝑢,0),Ω_𝑘∩Ker𝐿,(0,0)^𝑇−=sign𝑏₁𝑥^∗+ℎ₁𝑥^∗00−𝑏₂𝑦^∗+ℎ₂𝑦^∗=sign−𝑏₁𝑥^∗2+ℎ₁−𝑏₂𝑦^∗2+ℎ₂. (63) Since (𝑎₁+2𝑔₁/𝜔)−𝑏₁𝑥^∗−ℎ₁/𝑥^∗=0,(𝑎₂+2𝑔₂/𝜔)−𝑏₂𝑦^∗−ℎ₁/𝑦^∗=0.
Hence, deg𝐽𝑄𝑁(𝑢,0),Ω₁∩Ker𝐿,(0,0)^𝑇=sign−𝑏₁𝑥^∗₁²+ℎ₁−𝑏₂𝑦^∗₁²+ℎ₂()=1,deg𝐽𝑄𝑁𝑢,0,Ω₂∩Ker𝐿,(0,0)^𝑇=sign−𝑏₁𝑥^∗₂²+ℎ₁−𝑏₂𝑦^∗₂²+ℎ₂()=−1,deg𝐽𝑄𝑁𝑢,0,Ω₃∩Ker𝐿,(0,0)^𝑇=sign−𝑏₁𝑥^∗₃²+ℎ₁−𝑏₂𝑦^∗₃²+ℎ₂()=1,deg𝐽𝑄𝑁𝑢,0,Ω₄∩Ker𝐿,(0,0)^𝑇=sign−𝑏₁𝑥^∗₄²+ℎ₁−𝑏₂𝑦^∗₄²+ℎ₂=−1. (64)
So far, we have proved that Ω_𝑘(𝑘=1,2,3, and 4) satisfies all the assumptions in Lemma 1. Hence, system (15) has at least four different 𝜔-periodic solutions. Thus by Lemma 1, system (3) has at least four different positive 𝜔-periodic solutions. This completes the proof of Theorem 7.

3. An Example

Example 8. Consider the following time-delay parasitical system with impulsive effectiveness and harvesting terms: 𝑥̇₁(𝑡)=𝑥₁(𝑡)3+sin𝑡2−2+sin𝑡10𝑒^2𝜋𝑥₁(𝑡−|sin𝑡|)−4+cos𝑡20𝑒^2𝜋,𝑥̇₂(𝑡)=𝑥₂(𝑡)×2+cos𝑡−255+cos𝑡10𝑒^𝜋𝑥₂(|+𝑡−sin2𝑡|)2+sin2𝑡8𝑥₁(𝑡−|cos2𝑡|)−2+cos𝑡25𝑒^𝜋,Δ𝑥₁𝜋2=𝑒^−𝜋−1,Δ𝑥₁3𝜋2=𝑒^𝜋−1,Δ𝑥₂𝜋2=𝑒⁻𝜋2−1,Δ𝑥₂3𝜋2=𝑒^3𝜋/2−1. (65) In this case, 𝑎₁(𝑡)=(3+sin𝑡)/2,  𝑏₁(𝑡)=(2+sin𝑡)/10𝑒²𝜋,  ℎ₁(𝑡)=(4+cos𝑡)/20𝑒²𝜋,   𝜏₁₁(𝑡)=|sin𝑡|,  𝑎₂(𝑡)=(2+cos𝑡)/25,  𝑏₂(𝑡)=(5+cos𝑡)/10,  ℎ₂(𝑡)=(2+cos𝑡)/25𝑒^𝜋,  𝑐(𝑡)=(2+sin2𝑡)/8,  𝜏₂₂(𝑡)=|cos2𝑡|, and 𝜏₂₁(𝑡)=|sin2𝑡|. It is not difficult to calculate 𝜌₁>𝑒^4𝜋>1,  𝜌₂=𝑒^2𝜋>1. Since 𝑙^±₁=𝑎₁+2𝑔₁/𝜔𝜌₁±𝑎₁+2𝑔₁/𝜔𝜌₁²−4𝑏₁ℎ₁𝜌₁2𝑏₁=15𝑒^4𝜋±3√225𝑒^8𝜋−164𝑒^−2𝜋,𝑙^±₂=𝑐𝑙⁺₁+𝑎₂+2𝑔₂/𝜔𝜌₂±𝑐𝑙⁺₁+𝑎₂+2𝑔₂/𝜔𝜌₂²−4𝑏₂ℎ₂𝜌₂2𝑏₂⁻¹=15𝑒^2𝜋±√225𝑒^4𝜋−6420𝑒^−𝜋. (66) Then 𝑎₁+2𝑔₁/𝜔√𝜌₁=12𝑒^−2𝜋>25𝑒^−2𝜋=2𝑏₁ℎ₁,2𝑔₂/𝜔+𝑎₂√𝜌₂=21𝑒25^−𝜋>25𝑒^−𝜋=2𝑏₂ℎ₂. (67)
Therefore, all conditions of Theorem 7 are satisfied. By Theorem 7, system (65) has at least four positive 2𝜋-periodic solutions.

Aknowledgment

This work is supported by Yunnan Province Education Department Scientific Research Fund Project (no. 2012Z065).