A New Existence Theory for Positive Periodic Solutions to a Class of Neutral Delay Model with Feedback Control and Impulse

Abstract

We acquire some sufficient and realistic conditions for the existence of positive periodic solution of a general neutral impulsive 𝑛-species competitive model with feedback control by applying some analysis techniques and a new existence theorem, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for 𝑘-set contraction. As applications, we also examine some special cases, which have been studied extensively in the literature, some known results are improved and generalized.

1. Introduction

In this paper, we consider the existence of the positive periodic solution of the following impulsive 𝑛-species competition system with multiple delays and feedback control: 𝑁^′_𝑖(𝑡)=𝑁_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝛾_𝑖𝑗(𝑡−_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝛿_𝑖(𝑡),𝑖=1,2,…,𝑛,𝑡≠𝑡_𝑘,𝑢^′_𝑖(𝑡)=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑁_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑁_𝑖𝑡−𝜎_𝑖(𝑡),𝑡≥0,Δ𝑁_𝑖𝑡_𝑘=𝑝_𝑖𝑘+𝑞_𝑖𝑘𝑁_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…, (1) with the following initial conditions: 𝑁_𝑖(𝜉)=𝜙_𝑖(𝜉),𝑁^′_𝑖(𝜉)=𝜙^′_𝑖(𝜉),𝜉∈[−𝜏,0],𝜙_𝑖(0)𝜙>0,_𝑖∈𝐶[−𝜏,0),𝑅⁺𝐶¹[−𝜏,0),𝑅⁺𝑢_𝑖(𝜉)=𝜑_𝑖(𝜉),𝜉∈[−𝜏,0],𝜑_𝑖(0)𝜑>0,_𝑖∈𝐶[−𝜏,0),𝑅⁺,𝑖=1,2,3,…,𝑛, (2) where 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, 𝑐_𝑖𝑗, 𝑒_𝑖, 𝑓_𝑖, 𝛼_𝑖, 𝛽_𝑖, 𝜃_𝑖∈𝐶(𝑅,[0,+∞)), 𝑑_𝑖𝑗∈𝐶¹(𝑅,[0,+∞)), 𝛾_𝑖𝑗, 𝛿_𝑖∈𝐶¹(𝑅,𝑅), and 𝜏_𝑖𝑗∈𝐶²(𝑅,𝑅) are continuous 𝜔-periodic functions; 𝑟_𝑖∈𝐶(𝑅,𝑅) are continuous  𝜔-periodic functions with ∫^𝜔₀𝑟_𝑖(𝑡)𝑑𝑡>0. The growth functions 𝑟_𝑖 are not necessarily positive; since the environment fluctuates randomly, in some conditions, 𝑟_𝑖 may be negative. Consider the following: 𝜏=max_{𝑡∈[0,𝜔]}{𝛾_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡),𝛿_𝑖(𝑡),𝜎_𝑖(𝑡),1≤𝑖,𝑗≤𝑛}; and ∫^∞₀𝑘_𝑖𝑗(𝑠)𝑑𝑠=1, ∫₀^+∞𝑠𝑘_𝑖𝑗(𝑠)𝑑𝑠<+∞, and 𝑖,𝑗=1,2,…,𝑛. And 𝑝_𝑖𝑘 and 𝑞_𝑖𝑘 represent the birth rate and the harvesting (or stocking) rate of 𝑁_𝑖 at time 𝑡_𝑘, respectively. When 𝑞_𝑖𝑘>0, it stands for harvesting, while 𝑞_𝑖𝑘<0 means stocking. For the ecological justification of (1) and the similar types, refer to [1–14].

In 1991, in [1], Gopalsamy et al. have established the existence of a positive periodic solution for a periodic neutral delay logistic equation 𝑑𝑁(𝑡)𝑁(𝑡𝑁()(𝑡)𝑁𝑑𝑡=𝑟)1−𝑡−𝑚𝑇−𝑐^′(𝑡−𝑚𝑇)𝐾(𝑡), (3) where 𝐾(𝑡),𝑟(𝑡), and 𝑐(𝑡) are positive continuous 𝑇-periodic functions with 𝑇>0 and 𝑚 is a positive integer. In 1993, in [2], Kuang proposed an open problem (Open problem 9.2) to obtain sufficient conditions for the existence of a positive periodic solution of the following equation: 𝑑𝑁(𝑡𝑎(𝑡)(𝑡)𝑁(𝑡)(𝑡)𝑁((𝑡(𝑡)𝑁𝑑𝑡=𝑁)−𝛽−𝑏𝑡−𝜏))−𝑐^′(𝑡−𝜏(𝑡)). (4) In [3], Li tried to give an affirmative answer to the previous open problem; however, there is a mistake in the proof of Theorem 2 in  [3]. With the aim of giving a right answer to the previous open problem,  [4–6] also have investigated the previous question. However, it is more complex to check the sufficient conditions of the system  [5, 6].  Moreover, in  [7], Li studied the existence of positive periodic solution of the neutral Lotka-Volterra equation with several delays 𝑁^′(𝑡)=𝑁(𝑡)𝑎(𝑡)−^𝑛_𝑖=1𝑏_𝑖(𝑡)𝑁𝑡−𝜏_𝑖−^𝑛_𝑖=1𝑐_𝑖(𝑡)𝑁^′𝑡−𝛾_𝑖, (5) where 𝑎(𝑡),𝑏_𝑖(𝑡), and 𝑐_𝑖(𝑡) are positive continuous 𝑇-periodic functions and 𝜏_𝑖,𝛾_𝑖(𝑖=1,…,𝑛) are nonnegative constants. Recently, in [8], Lu and Ge investigated a neutral delay population model with multiple delays: 𝑑𝑁(𝑡)𝑎(𝑡)(𝑡)𝑁(𝑡)−𝑑𝑡=𝑁−𝛽^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑁𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)𝑁^′𝑡−𝜏_𝑖(𝑡). (6) They applied the theory of abstract continuous theorem of 𝑘-set contractive operator and some analysis techniques to obtain some sufficient conditions for the existence of positive periodic solutions of the model (6).

It is of course very interesting to study the neutral delay population model for higher dimensional systems. In fact, in  [9], Li has studied the neutral Lotka-Volterra system with constant delays 𝑁^′_𝑖(𝑡)=𝑁_𝑖(𝑡)𝑎_𝑖(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝜏_𝑖𝑗−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝛾_𝑖𝑗, (7) where 𝑖=1,…,𝑛, and 𝑎_𝑖(𝑡), 𝑏_𝑖𝑗(𝑡), and 𝑐_𝑖𝑗(𝑡) are positive continuous 𝑇-periodic functions, and 𝜏_𝑖𝑗,𝛾_𝑖𝑗 are nonnegative constants. He obtained sufficient conditions that guarantee the existence of positive periodic solution of the system (7), by applying a continuation theorem based on Gaines and Mawhin's coincidence degree. Noticing that delays arise frequently in practical applications, it is difficult to measure them precisely. In population dynamics, it is clear that a constant delay is only a special case. In most situations, delays are variable, and so in  [10], Liu and Chen investigated the following general neutral Lotka-Volterra system with unbounded delays: 𝑁^′_𝑖(𝑡)=𝑁_𝑖(𝑡)𝑎_𝑖(𝑡)−^𝑛_𝑗=1𝛽_𝑖𝑗(𝑡)𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝛾_𝑖𝑗(𝑡),𝑖=1,2,…,𝑛. (8) They introduced a new existence theorem to obtain a set of sufficient conditions for the existence of positive periodic solutions for the system (8), and their results improved and generalized some known results.

Moreover, in some situations,  people may wish to change the position of the existing periodic solution but keep its stability. This is of significance in the control of ecology balance. One of the methods for its realization is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control schemes or by harvesting procedure. In fact, during the last decade, the qualitative behaviors of the population dynamics with feedback control have been studied extensively; see [11, 15–23]. Recently, in [11], Chen considered the following neutral Lotka-Volterra competition model with feedback control of the form: 𝑦^′_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑦^′_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖𝑡−𝜎_𝑖(𝑡),𝑢^′_𝑖(𝑡)=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑦_𝑖(𝑡)+𝛾_𝑖(𝑡)𝑦_𝑖𝑡−𝛿_𝑖(𝑡),𝑖=1,2,…,𝑛. (9) With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, he established easily verifiable criteria for the global existence of positive periodic solutions of the system (9), and his results extended and improved existing results.

On the other hand, there are some other perturbations in the real world, such as fires and floods, that are not suitable to be considered continually. These perturbations bring sudden changes to the system. Systems with such sudden perturbations  involving impulsive differential equations have attracted the interest of many researchers in the past twenty years, see [12–14, 24–30], since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, and optimal control. For details, see [31–33].

In  [12], Huo studied the following neutral impulsive delay Lotka-Volterra system: 𝑁^′_𝑖(𝑡)=𝑁_𝑖(𝑡)𝛼_𝑖(𝑡)−^𝑛_𝑗=1𝛽_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝛾_𝑖𝑗(𝑡),𝑖=1,2,…,𝑛,𝑡≠𝑡_𝑘,Δ𝑁_𝑖(𝑡)=𝑁_𝑖𝑡⁺−𝑁_𝑖𝑡_𝑘=𝑏_𝑖𝑘𝑁_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…. (10) By using some techniques of Mawhin’s coincidence degree theory, he obtained sufficient conditions for the existence of periodic positive solutions of the system (10).

In  [13], Wang and Dai investigated the following periodic neutral population model with delays and impulse: 𝑁^′(𝑡)=𝑁(𝑡)𝑎(𝑡)−𝑒(𝑡)𝑁(𝑡)−^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑁𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)𝑁^′𝑡−𝜏_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑁𝑡⁺=1+𝜃_𝑘𝑁𝑡_𝑘,𝑘=1,2,… (11) They obtained some sufficient conditions for the existence of positive periodic solutions of the model (11) by using the theory of abstract continuous theorem of 𝑘-set contractive operator and some analysis techniques.

Recently, in  [14], Luo et al. studied the following 𝑛-species competition system with general periodic neutral delay and impulse: 𝑁^′_𝑖(𝑡)=𝑁_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)×^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝜏_𝑖𝑗(𝑡−_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝛾_𝑖𝑗(𝑡),𝑖=1,2,…,𝑛,𝑡≠𝑡_𝑘,Δ𝑁_𝑖𝑡_𝑘=𝑁_𝑖𝑡⁺_𝑘−𝑁_𝑖𝑡_𝑘=𝜃_𝑖𝑘𝑁_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…. (12) They obtained some sufficient and realistic conditions for the existence of positive periodic solutions of the system (12), by using a new existence theorem, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for 𝑘-set contraction.

However, to this day, no scholars had done works on the existence of positive periodic solution of the system (1). One could easily see that systems (3)–(12) are all special cases of the system (1). Therefore, we propose and study the system (1) in this paper.

For the sake of generality and convenience, we make the following notations and assumptions: let 𝜔>0 be a constant and 𝐶_𝜔={𝑥∣𝑥∈𝐶(𝑅,𝑅),𝑥(𝑡+𝜔)=𝑥(𝑡)}, with the norm defined by |𝑥|₀=max_{𝑡∈[0,𝜔]}|𝑥(𝑡)|; 𝐶¹_𝜔={𝑥∣𝑥∈𝐶¹(𝑅,𝑅),𝑥(𝑡+𝜔)=𝑥(𝑡)}, with the norm defined by ‖𝑥‖=max_{𝑡∈[0,𝜔]}{|𝑥|₀,|𝑥^′|₀}; 𝑃𝐶={𝑥∣𝑥∶𝑅→𝑅⁺,lim_𝑠→𝑡𝑥(𝑠) = 𝑥(𝑡),if𝑡≠𝑡_𝑘,lim_𝑡→𝑡⁻_𝑘𝑥(𝑡) = 𝑥(𝑡_𝑘),lim_𝑡→𝑡⁺_𝑘𝑥(𝑡)exists,𝑘∈𝑍⁺}; 𝑃𝐶¹={𝑥∣𝑥∶𝑅→𝑅⁺,𝑥^′∈𝑃𝐶}; 𝑃𝐶_𝜔={𝑥∣𝑥∈𝑃𝐶,𝑥(𝑡+𝜔)=𝑥(𝑡)}, with the norm defined by |𝑥|₀=max_{𝑡∈[0,𝜔]}|𝑥(𝑡)|; 𝑃𝐶¹_𝜔={𝑥∣𝑥∈𝑃𝐶¹,𝑥(𝑡+𝜔)=𝑥(𝑡)}, with the norm defined by ‖𝑥‖=max_{𝑡∈[0,𝜔]}{|𝑥|₀,|𝑥^′|₀}.Then, the previous spaces are all Banach spaces. We also denote that 1𝑔=𝜔^𝜔₀𝑔(𝑡)𝑔𝑑𝑡,_𝐿=min_{[]𝑡∈0,𝜔}𝑔(𝑡),forany𝑔∈𝑃𝐶_𝜔,Δ_𝑖𝑘=1+𝑝_𝑖𝑘+𝑞_𝑖𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…, (13) and make the following assumptions: (A)𝑎_𝑖𝑗, 𝑏_𝑖𝑗, 𝑐_𝑖𝑗, 𝑒_𝑖, 𝑓_𝑖, 𝛼_𝑖, 𝛽_𝑖, 𝜃_𝑖∈𝐶(𝑅,[0,+∞)), 𝑑_𝑖𝑗∈𝐶¹(𝑅,[0,+∞)), 𝛾_𝑖𝑗,𝛿_𝑖∈𝐶¹(𝑅,𝑅), and 𝜏_𝑖𝑗∈𝐶²(𝑅,𝑅) are continuous 𝜔-periodic functions, and ∫^∞₀𝑘_𝑖𝑗(𝑠)𝑑𝑠=1, ∫₀^+∞𝑠𝑘_𝑖𝑗(𝑠)𝑑𝑠<+∞,  and  𝑖,𝑗=1,2,…,𝑛;(B)[𝑡_𝑘]_𝑘∈𝑁 satisfies 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯ and lim_𝑘→∞𝑡_𝑘=+∞;(C){Δ_𝑖𝑘} is a real sequence such that Δ_𝑖𝑘>0, ∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘(𝑖=1,2,…,𝑛,𝑘=1,2,…) are 𝜔-periodic functions.

The organization of this paper is as follows. In the following section, we introduce some lemmas and an important existence theorem developed in [34, 35]. In the third section, we derive some sufficient conditions, which ensure the existence of positive periodic solution of system (1) by applying this theorem and some other techniques. Finally, we study some special cases of system (1), which have been studied extensively in the literature. These examples show that our sufficient conditions are new, and some known results can be improved and generalized.

2. Preliminaries

In this section, in order to obtain the existence of a periodic solution for system (1) and (2), we will give some concepts and results from [35], and we will state an existence theorem and some lemmas.

For a fixed 𝜏>0, let 𝐶=∶𝐶([−𝜏,0];𝑅^𝑛), if 𝑥∈𝐶([−𝜏,0];𝑅^𝑛) for some 𝛿>0 and 𝜂∈𝑅, then 𝑥_𝑡∈𝐶 for 𝑡∈[𝜂,𝜂+𝛿] is defined by 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃) for 𝜃∈[−𝜏,0]. The supremum norm in 𝐶 is denoted by ‖⋅‖, that is, ‖𝜙‖=max_{𝑡∈[−𝜏,0]}|𝜙(𝜃)| for 𝜙∈𝐶, where |⋅| denotes the norm in 𝑅^𝑛 and |𝑢|=∑^𝑛_𝑗=1|𝑢_𝑖| for 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑅^𝑛. Consider the following neutral functional differential equation: 𝑑𝑥(𝑡)𝑑𝑡−𝑏𝑡,𝑥_𝑡=𝑓𝑡,𝑥_𝑡, (14) where 𝑓∶𝑅×𝐶→𝑅^𝑛 is completely continuous, and 𝑏∶𝑅×𝐶→𝑅^𝑛 is continuous. Moreover, we assume the following: (1) there exists 𝜔>0 such that for every (𝑡,𝜙)∈𝑅×𝐶, we have 𝑏(𝑡+𝜔,𝜙)=𝑏(𝑡,𝜙) and 𝑓(𝑡+𝜔,𝜙)=𝑓(𝑡,𝜙);(2) there exists a constant 𝑘<1 such that |𝑏(𝑡,𝜙)−𝑏(𝑡,𝜑)|≤𝑘‖𝜙−𝜑‖, for 𝑡∈𝑅 and 𝜙,𝜑∈𝐶.

By using the continuation theorem for composite coincidence degree, in [34], Erbe et al. proved the following existence theorem (see also Theorem 4.7.1 in [35]).

Lemma 1. Assume that there exists a constant 𝑀>0 such that(i) for any 𝜆∈(0,1) and any 𝜔-periodic solution 𝑥 of the system 𝑑𝑥(𝑡)𝑑𝑡−𝜆𝑏𝑡,𝑥_𝑡=𝑓𝑡,𝑥_𝑡. (15) One has that |𝑥(𝑡)|<𝑀 for 𝑡∈𝑅;(ii)ℎ(𝑢)=∶∫^𝜔₀𝑓(𝑠,𝑢)𝑑𝑠≠0 for 𝑢∈𝜕𝐵_𝑀(𝑅^𝑛), where 𝐵_𝑀(𝑅^𝑛)={𝑢∈𝑅^𝑛∶|𝑢|<𝑀}, and 𝑢 denotes the constant mapping from [−𝜏,0] to 𝑅^𝑛 with the value 𝑢∈𝑅^𝑛;(iii)deg(ℎ,𝐵_𝑀(𝑅^𝑛))≠0. Then, there exists at least one 𝜔-periodic solution of the system (14) that satisfies sup_𝑡∈𝑅|𝑥(𝑡)|<𝑀.

The following remark is introduced by Fang (see Remark 1 in [36]).

Remark 2. Lemma 1 remains valid if the assumption (ii) is replaced by the following:
(ii*) there exists a constant 𝑘<1 such that |𝑏(𝑡,𝜙)−𝑏(𝑡,𝜑)|≤𝑘‖𝜙−𝜑‖ for 𝑡∈𝑅 and 𝜙,𝜑∈{𝜙∈𝐶∶‖𝜙‖<𝑀} with 𝑀 as given in condition (i) of Lemma 1.

We will also need the following lemmas.

Lemma 3 (see [8, 13]). Suppose that 𝜎∈𝐶¹_𝜔 and 𝜎^′(𝑡)<1,𝑡∈[0,𝜔]. Then, the function 𝑡−𝜎(𝑡) has a unique inverse 𝜇(𝑡) satisfying 𝜇∈𝐶(𝑅,𝑅) with 𝜇(𝑎+𝜔)=𝜇(𝑎)+𝜔, ∀𝑎∈𝑅, and if ℎ∈𝑃𝐶_𝜔,𝜎^′(𝑡)<1, and 𝑡∈[0,𝜔], then ℎ(𝜇(𝑡))∈𝑃𝐶_𝜔.

Lemma 4 (see [27]). Suppose that 𝑥(𝑡) is a differently continuous 𝜔-periodic function on 𝑅 with (𝜔>0). Then, to any 𝑡_∗∈𝑅,max_{𝑡∗≤𝑡≤𝑡∗+𝜔}|𝑥(𝑡)|≤|𝑥(𝑡_∗)|+(1/2)∫^𝜔₀|𝑥^′(𝑡)|𝑑𝑡.

Lemma 5. Consider that  𝑅₊^2𝑛={(𝑁_𝑖(𝑡),𝑢_𝑖(𝑡))∶𝑁_𝑖(0)>0,𝑢_𝑖(0)>0,𝑖=1,2,…,𝑛} is the positive invariable region of the system (1) and (2).

Proof. In view of biological population, we obtain 𝑁_𝑖(0)>0,𝑢_𝑖(0)>0. By the system (1) and (2), we have 𝑁_𝑖(𝑡)=𝑁_𝑖(0)exp^𝑡₀𝑟_𝑖(𝜉)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝜉)𝑁_𝑗(𝜉)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝜉)×^𝑡_−∞𝐾_𝑖𝑗(𝜉−𝑠)𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝜉)𝑁_𝑗𝜉−𝛾_𝑖𝑗(𝜉)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝜉)𝑁^′_𝑗𝜉−𝜏_𝑖𝑗(𝜉)−𝑒_𝑖(𝜉)𝑢_𝑖(𝜉)−𝑓_𝑖(𝜉)𝑢_𝑖×𝜉−𝜎_𝑖(𝜉)𝑑𝜉,𝑡∈0,𝑡₁𝑁,𝑖=1,2,…,𝑛,_𝑖(𝑡)=𝑁_𝑖𝑡_𝑘exp^𝑡_𝑡𝑘𝑟_𝑖(𝜉)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝜉)𝑁_𝑗(𝜉)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝜉)×^𝑡_−∞𝐾_𝑖𝑗(𝜉−𝑠)𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝜉)𝑁_𝑗𝜉−𝛾_𝑖𝑗(𝜉)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝜉)𝑁^′_𝑗𝜉−𝜏_𝑖𝑗(𝜉)−𝑒_𝑖(𝜉)𝑢_𝑖(𝜉)−𝑓_𝑖(𝜉)𝑢_𝑖×𝜉−𝜎_𝑖(𝜉),𝑡𝑑𝜉𝑡∈_𝑘,𝑡_𝑘+1𝑁,𝑖=1,2,…,𝑛,𝑘≥1,_𝑖𝑡⁺_𝑘=𝑒^{(𝑝𝑖𝑘+𝑞𝑖𝑘)}𝑁_𝑖𝑡_𝑘𝑢>0,𝑘∈𝑁,𝑖=1,2,…,𝑛,_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝛽_𝑖(𝑠)𝑁_𝑖(𝑠)+𝜗_𝑖(𝑠)𝑁_𝑖×𝑠−𝛾_𝑖(𝑠)𝑑𝑠∶=𝜙_𝑖𝑁_𝑖(𝑡), (16) where 𝐺_𝑖()=∫𝑡,𝑠exp^𝑠_𝑡𝛼_𝑖(𝜉)𝑑𝜉exp∫^𝑠_𝑡𝛼_𝑖(𝜉)𝑑𝜉−1. (17) Then, the solution of the systems (1) and (2) is positive.

Definition 6. A function 𝑁_𝑖∶[−𝜏,0]→[0,+∞)(𝑖=1,2,…,𝑛) is said to be a positive solution of (1) and (2) on [−𝜏,∞], if the following conditions are satisfied: (a)𝑁_𝑖(𝑡) is absolutely continuous on each (𝑡_𝑘,𝑡_𝑘+1);(b) for each 𝑘∈𝑍₊,𝑁_𝑖(𝑡⁺_𝑘) and 𝑁_𝑖(𝑡⁻_𝑘) exist, and 𝑁_𝑖(𝑡⁻_𝑘)=𝑁_𝑖(𝑡_𝑘);(c)𝑁_𝑖(𝑡) satisfies the first equation of (1) and (2) for almost everywhere (for short a.e.) in [0,∞]⧵{𝑡_𝑘} and satisfies 𝑁_𝑖(𝑡⁺_𝑘)=Δ_𝑖𝑘𝑁_𝑖(𝑡_𝑘) for 𝑡=𝑡_𝑘,𝑘∈𝑍₊={1,2,…}.

Consider the following nonimpulsive delay differential equation: 𝑦^′_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)×^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑦_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡−_𝑛𝑗=1)𝐷_𝑖𝑗(𝑡)𝑦^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝛿_𝑖(𝑡),𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽^∗_𝑖(𝑡)𝑦_𝑖(𝑡)+𝜃^∗_𝑖(𝑡)𝑦_𝑖𝑡−𝜎_𝑖(𝑡), (18) with the following initial conditions: 𝑦_𝑖(𝜉)=𝜙_𝑖(𝜉),𝑦^′_𝑖(𝜉)=𝜙^′_𝑖(𝜉),𝜉∈[−𝜏,0],𝜙_𝑖(0)𝜙>0,_𝑖∈𝐶[−𝜏,0],𝑅⁺𝐶¹[−𝜏,0],𝑅⁺,𝑢_𝑖(𝜉)=𝜑_𝑖(𝜉),𝜉∈[−𝜏,0],𝜑_𝑖(0)𝜑>0,_𝑖∈𝐶[−𝜏,0),𝑅⁺,𝑖=1,2,3,…,𝑛, (19) where 𝐴_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘,𝐵_𝑖𝑗(𝑡)=𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘,𝐶_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝛾𝑖𝑗(𝑡)}Δ_𝑖𝑘,𝐷_𝑖𝑗(𝑡)=𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}Δ_𝑖𝑘,𝜏=max_{𝑡∈[0,𝜔]}𝛾_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡),𝛿_𝑖(𝑡),𝜎_𝑖(𝑡),𝛽,1≤𝑖,𝑗≤𝑛_∗𝑖(𝑡)=𝛽_𝑖(𝑡)_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘,𝜃^∗_𝑖(𝑡)=𝜃_𝑖(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑖(𝑡)}Δ_𝑖𝑘,𝑖,𝑗=1,2,…,𝑛. (20) The following lemmas will be used in the proofs of our results. The proof of Lemma 7 is similar to that of Theorem 1 in [24].

Lemma 7. Suppose that (A)–(C) hold, then (i) if (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is a solution of (18) and (19) on [−𝜏,+∞), then (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is a solution of (1) and (2) on [−𝜏,+∞), where 𝑁_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡); (ii) if (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is a solution of (1) and (2) on [−𝜏,+∞), then (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is a solution of (18) and (19) on [−𝜏,+∞), where 𝑦_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ⁻¹_𝑖𝑘𝑁_𝑖(𝑡).

Proof. (i) It is easy to see that 𝑁_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) is absolutely continuous on every interval (𝑡_𝑘,𝑡_𝑘+1],𝑡≠𝑡_𝑘,𝑘=1,2,…, 𝑁^′_𝑖(𝑡)−𝑁_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝛾_𝑖𝑗(𝑡−_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝛿_𝑖(𝑡)=_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦^′_𝑖(𝑡)−_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)×𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑦_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝛾𝑖𝑗(𝑡)}Δ_𝑖𝑘𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝛾𝑖𝑗(𝑡)}Δ_𝑖𝑘𝑦^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)×𝑢_𝑖𝑡−𝛿_𝑖(𝑡)=_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦^′_𝑖(𝑡)−𝑦_𝑖(𝑡)×𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑦_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖×𝑡−𝛿_𝑖(𝑡)𝑢=0,_′𝑖(𝑡)+𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)−𝛽_𝑖(𝑡)𝑁_𝑖(𝑡)−𝜃_𝑖(𝑡)𝑁_𝑖𝑡−𝜎_𝑖(𝑡)=𝑢^′_𝑖(𝑡)+𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)−𝛽^∗_𝑖(𝑡)𝑦_𝑖(𝑡)−𝜃^∗_𝑖(𝑡)𝑦_𝑖𝑡−𝛾_𝑖(𝑡)=0. (21) On the other hand, for any 𝑡=𝑡_𝑘, 𝑘=1,2,…, 𝑁_𝑖𝑡⁺_𝑘=lim_{𝑡→𝑡+𝑘0<𝑡𝑗<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)=_{0<𝑡𝑗≤𝑡𝑘}Δ_𝑖𝑘𝑦_𝑖𝑡_𝑘,𝑁_𝑖𝑡_𝑘=_{0<𝑡𝑗<𝑡𝑘}Δ_𝑖𝑘𝑦_𝑖𝑡_𝑘, (22) thus, Δ𝑁_𝑖𝑡⁺_𝑘=Δ_𝑖𝑘𝑦_𝑖𝑡_𝑘, (23) which implies that (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is a solution of the system (1) and (2). Therefore, if (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is a solution of the system (18) and (19) on [−𝜏,+∞), we can prove that (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) are solutions of the system (1) and (2) on [−𝜏,+∞).
(ii) Since 𝑁_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) is absolutely continuous on every interval (𝑡_𝑘,𝑡_𝑘+1],𝑡≠𝑡_𝑘,𝑘=1,2,…, and in view of (23), it follows that for any 𝑘=1,2,…, 𝑦_𝑖𝑡⁺_𝑘=_{0<𝑡𝑗≤𝑡𝑘}Δ⁻¹_𝑖𝑘𝑁_𝑖𝑡⁺_𝑘=_{0<𝑡𝑗<𝑡𝑘}Δ⁻¹_𝑖𝑘𝑁_𝑖𝑡_𝑘=𝑦_𝑖𝑡_𝑘,𝑦_𝑖𝑡⁻_𝑘=_{0<𝑡𝑗<𝑡𝑘}Δ⁻¹_𝑖𝑘𝑁_𝑖𝑡⁻_𝑘=_{0<𝑡𝑗≤𝑡−𝑘}Δ⁻¹_𝑖𝑘𝑁_𝑖𝑡⁻_𝑘=𝑦_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛, (24) which implies that (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is continuous on [−𝜏,+∞). It is easy to prove that (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is absolutely continuous on [−𝜏,+∞). Similar to the proof of (i), we can check that (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡))^𝑇(𝑖=1,2,…,𝑛) is a solution of the system (18) and (19) on [−𝜏,+∞). The proof of Lemma 7 is completed.

Lemma 8. Consider that (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡)) is a 𝜔-periodic solution of (18) and (19) if and only if 𝑦_𝑖(𝑡) is a 𝜔-periodic solution of the following system: 𝑑𝑦_𝑖(𝑡)𝑑𝑡=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑦_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡−_𝑛𝑗=1)𝐷_𝑖𝑗(𝑡)𝑦^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝜓_𝑖𝑦_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜓_𝑖𝑦_𝑖×𝑡−𝛿_𝑖(𝑡), (25) where 𝜓_𝑖𝑦_𝑖(𝑡)∶=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝛽^∗_𝑖(𝑠)𝑦_𝑖(𝑠)+𝜃^∗_𝑖(𝑠)𝑦_𝑖×𝑠−𝜎_𝑖(𝑠)𝑑𝑠, (26) and 𝐺_𝑖(𝑡,𝑠) is defined by (17).

Proof. The proof of Lemma 8 is similar to that of Lemma  2.2 in [11], and we omit the details here.

From Lemmas 7 and  8, if we want to discuss the existence of positive periodic solutions of systems (1) and (2), we only discuss the existence of positive periodic solutions of systems (25) and (26).

3. The Main Result

Since 𝛾^′_𝑖𝑗(𝑡)<1,𝜏^′_𝑖𝑗(𝑡)<1,𝑡∈[0,𝜔], we see that 𝛾_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡) all have their inverse function. Throughout the following part, we set to 𝜗_𝑖𝑗(𝑡),𝜈_𝑖𝑗(𝑡) that represent the inverse function of 𝑡−𝛾_𝑖𝑗(𝑡),𝑡−𝜏_𝑖𝑗(𝑡), respectively. We denote that Γ_𝑖𝑗(𝑡)=𝐴_𝑖𝑗(𝑡)+𝐵_𝑖𝑗(𝑡)+𝐶_𝑖𝑗𝜗_𝑖𝑗(𝑡)1−𝛾^′_𝑖𝑗𝜗_𝑖𝑗(𝑡)−𝐷^′_𝑖𝑗𝜈_𝑖𝑗(𝑡)1−𝜏^′_𝑖𝑗𝜈_𝑖𝑗(𝑡),Γ¹_𝑖(𝑡)=𝑒_𝑖(𝑡)𝜓_𝑖1(𝑡),Γ²_𝑖(𝑡)=𝑓_𝑖(𝑡)𝜓_𝑖1𝑡−𝜎_𝑖(𝑡). (27)

Remark 9. From Lemma 3, we get that 𝜗_𝑖𝑗(𝜔)=𝜗_𝑖𝑗(0)+𝜔,𝜈_𝑖𝑗(𝜔)=𝜈_𝑖𝑗(0)+𝜔,𝑖,𝑗=1,2,…,𝑛, then ^𝜔₀𝐶_𝑖𝑗𝜗_𝑖𝑗(𝑠)1−𝛾^′_𝑖𝑗𝜗_𝑖𝑗(𝑠)𝑑𝑠=^𝜗𝑖𝑗_𝜗^(𝜔)_𝑖𝑗(0)𝐶_𝑖𝑗(𝑡)1−𝛾^′_𝑖𝑗(𝑡)1−𝛾^′_𝑖𝑗(𝑡)=𝑑𝑡_{𝜗𝑖𝑗𝜗(0)+𝜔𝑖𝑗(0)}𝐶_𝑖𝑗(𝑡)𝑑𝑡=𝐶_𝑖𝑗𝜔,𝑖,𝑗=1,2,…,𝑛. (28)
Similarly, ^𝜔₀𝐷^′_𝑖𝑗𝜈_𝑖𝑗(𝑠)1−𝜏^′_𝑖𝑗𝜈_𝑖𝑗(𝑠)𝑑𝑠=^𝜈𝑖𝑗_𝜈^(𝜔)_𝑖𝑗(0)𝐷^′_𝑖𝑗(𝑡)1−𝜏^′_𝑖𝑗(𝑡)1−𝜏^′_𝑖𝑗(𝑡)𝑑𝑡=^𝜈𝑖𝑗_𝜈^(0)+𝜔_𝑖𝑗(0)𝐷^′_𝑖𝑗(𝑡)𝑑𝑡=0,𝑖,𝑗=1,2,…,𝑛. (29) Thus, Γ_𝑖𝑗𝜔=^𝜔₀Γ_𝑖𝑗(𝑡)𝑑𝑡=𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝜔. (30)

Here, we have the following notations: 𝜌_𝑖𝑗=Γ^𝐿_𝑖𝑗1−𝛾^′_𝑖𝑗^𝐿1−𝛾^′_𝑖𝑗^𝐿+𝐷_𝑖𝑗0,𝑅_𝑖=1𝜔^𝜔₀𝑟_𝑖(𝑡𝐿)𝑑𝑡,₀=max^𝑛𝑛_𝑖=1𝑗=1𝐷_𝑖𝑗0,^𝑛𝑛_𝑖=1𝑗=1𝐷_0,𝑖𝑗0,𝑀₀=max^𝑛_𝑖=1ln𝜇^∗_𝑖01,𝐻,2𝜔Λ^∗+^𝑛_𝑖=1Λ_𝑖,𝐻=max_{𝑖∈[1,𝑛]}𝐻_𝑖,𝐻_𝑖=ln𝑟_𝑖𝜌_𝑖𝑖+^𝑛_𝑗=1𝑟_𝑖𝜌_𝑖𝑗+𝑅_𝑖+𝑟_𝑖Λ𝜔,_∗=^𝑛_𝑖=1𝑟_𝑖0+^𝑛𝑛_𝑖=1𝑗=1𝐴_𝑖𝑗0𝑒^𝐻𝑗+^𝑛𝑛_𝑖=1𝑗=1𝐵_𝑖𝑗0𝑒^𝐻𝑗+^𝑛𝑛_𝑖=1𝑗=1𝐶_𝑖𝑗0𝑒^𝐻𝑗+𝑒_𝑖0+𝑓_𝑖0𝛽^∗_𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑒^𝐻𝑖×1−^𝑛𝑛_𝑖=1𝑗=1𝐷_0,𝑖𝑗0𝑒^𝐻𝑗−1,Λ_𝑖𝑟=maxln^𝑖𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖,ln𝑟_𝑖−∑^𝑛_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝑒^𝐻𝑗𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖+𝑒_𝑖+𝑓_𝑖𝛽^∗_𝑖0+𝜃^∗_𝑖0/𝛼^𝐿_𝑖, (31) where Γ_𝑖𝑗(𝑡),Γ¹_𝑖(𝑡), and Γ²_𝑖(𝑡) are defined by (27), and 𝐷_0,𝑖𝑗(𝑡)=𝐷_𝑖𝑗(𝑡)(1−𝛾^′_𝑖𝑗(𝑡)).

Theorem 10. Suppose that the following conditions hold:(1) the system of algebraic equations 𝑓^∗𝜇=𝑟_𝑖−^𝑛_𝑗=1𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝜇_𝑗+Γ¹_𝑖+Γ²_𝑖𝜇_𝑖𝑛×1=0 (32) has a unique positive solution 𝜇^∗=(𝜇^∗₁,…,𝜇^∗_𝑛);(2)𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗>0, 𝑟_𝑖>∑^𝑛_{𝑗=1,𝑗≠𝑖}(𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗)𝑒^𝐻𝑗, 𝛾^′_𝑖𝑗(𝑡)<1, 𝜏^′_𝑖𝑗(𝑡)<1 and Γ_𝑖𝑗(𝑡)>0;(3)𝐾₀=∶𝐿₀𝑒^𝑀0<1. Then the system (1) and (2) has at least one positive 𝜔-periodic solution.

To prove the previous theorem, we make the change of variables 𝑦_𝑖(𝑡)=𝑒^{𝑥𝑖(𝑡)},𝑖=1,2,…,𝑛. (33) Then, the system (25) can be rewritten in the following form: 𝑥^′_𝑖(𝑡)=𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑒^{𝑥𝑗(𝑠)}−𝑑𝑠_𝑛𝑗=1𝐶_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝛾𝑖𝑗(𝑡))}−^𝑛_𝑗=1𝐷_0𝑖𝑗(𝑡)𝑥^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}−𝑒_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)−𝑓_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)𝑡−𝛿_𝑖(𝑡). (34) Let 𝑋 denote the linear space of real value continuous 𝜔-periodic functions on 𝑅. The linear space 𝑋 is a Banach space with the usual norm ‖𝑥‖₀=max_𝑡∈𝑅|𝑥(𝑡)|=max_𝑡∈𝑅∑^𝑛_𝑗=1|𝑥_𝑖(𝑡)| for a given 𝑥=(𝑥₁,…,𝑥_𝑛)∈𝑋.

We define the following maps: 𝑏∶𝑅×𝐶⟶𝑅^𝑛,𝑏𝑡,𝜙=𝑏₁𝑡,𝜙,𝑏₂𝑡,𝜙,…,𝑏_𝑛,𝑏𝑡,𝜙_𝑖𝑡,𝜙=−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝜙𝑗(−𝛾𝑖𝑗(𝑡))},𝑓∶𝑅×𝐶⟶𝑅^𝑛,𝑓𝑡,𝜙=𝑓₁𝑡,𝜙,𝑓₂𝑡,𝜙,…,𝑓_𝑛,𝑓𝑡,𝜙_𝑖𝑡,𝜙=𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑒^𝜙𝑗(0)−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)⁰_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑒^{𝜙𝑗(𝑠)}−𝑑𝑠_𝑛𝑗=1𝐶_𝑖𝑗(𝑡)𝑒^{𝜙𝑗(−𝜏𝑖𝑗(𝑡))}+^𝑛_𝑗=1𝐷^′_𝑖𝑗(𝑡)𝑒^{𝜙𝑗(−𝛾𝑖𝑗(𝑡))}−𝑒_𝑖(𝑡)𝜓_𝑖𝑒^𝜙𝑖(𝑡)−𝑓_𝑖(𝑡)𝜓_𝑖𝑒^𝜙𝑖𝑡−𝛿_𝑖(𝑡),𝑖=1,2,…,𝑛,𝜙=𝜙₁,𝜙₂,…,𝜙_𝑛∈𝐶,𝑡∈𝑅. (35) Clearly, 𝑏∶𝑅×𝐶→𝑅^𝑛 and 𝑓∶𝑅×𝐶→𝑅^𝑛 are complete continuation functions, and system (34) takes the form𝑑𝑥(𝑡)𝑑𝑡−𝑏𝑡,𝑥_𝑡=𝑓𝑡,𝑥_𝑡. (36) In the proof of our main result below, we will use the following two important lemmas.

Lemma 11. If the assumptions of Theorem 10 are satisfied and if Ω={𝜙∈𝐶∶‖𝜙‖<𝑀}, where 𝑀>𝑀₀ such that 𝑘=𝐿₀𝑒^𝑀<1, then |𝑏(𝑡,𝜙)−𝑏(𝑡,𝜑)|≤𝑘‖𝜙−𝜑‖, for 𝑡∈𝑅 and 𝜙,𝜑∈Ω.

Proof. For 𝑡∈𝑅 and 𝜙,𝜑∈Ω, we have 𝑏_𝑖𝑡,𝜙−𝑏_𝑖𝑡,𝜑≤^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝜙𝑗(−𝛾𝑖𝑗(𝑡))}−𝑒^{𝜑𝑗(−𝛾𝑖𝑗(𝑡))}≤^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝜎𝑖𝑗𝜙𝑗(−𝛾𝑖𝑗(𝑡))+(1−𝜎𝑖𝑗)𝜑𝑗(−𝛾𝑖𝑗(𝑡))}×𝜙_𝑗−𝛾_𝑖𝑗(𝑡)−𝜑_𝑗−𝛾_𝑖𝑗(𝑡), (37) for some 𝜎_𝑖𝑗∈(0,1). Then, we get 𝑏_𝑖𝑡,𝜙−𝑏_𝑖𝑡,𝜑≤^𝑛_𝑗=1𝐷_𝑖𝑗0𝑒^𝑀𝜙−𝜑. (38) Hence, 𝑏𝑡,𝜙−𝑏𝑡,𝜑≤^𝑛𝑛_𝑖=1𝑗=1𝐷_𝑖𝑗0𝑒^𝑀𝜙−𝜑≤𝐿₀𝑒^𝑀𝜙−𝜑=𝑘𝜙−𝜑. (39) The proof of Lemma 11 is thus completed.

Lemma 12. If the assumptions of Theorem 10 are satisfied, then every solution 𝑥∈𝑋 of the system 𝑑𝑥(𝑡)𝑑𝑡−𝜆𝑏𝑡,𝑥_𝑡=𝑓𝑡,𝑥_𝑡,𝜆∈(0,1) (40) satisfies ‖𝑥‖₀≤𝑀₀.

Proof. Let (𝑑/𝑑𝑡)[𝑥(𝑡)−𝜆𝑏(𝑡,𝑥_𝑡)]=𝑓(𝑡,𝑥_𝑡), for 𝑥∈𝑋, that is, 𝑥_𝑖(𝑡)+𝜆^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))′}=𝜆𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)×^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑒^{𝑥𝑗(𝑠)}−𝑑𝑠_𝑛𝑗=1𝐶_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝛾𝑖𝑗(𝑡))}+^𝑛_𝑗=1𝐷^′_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}−𝑒_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)−𝑓_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖×(𝑡)𝑡−𝛿_𝑖(𝑡),𝑖=1,2,…,𝑛;𝜆∈(0,1), (41) which yields, after integrating from 0 to 𝜔, that ^𝜔₀^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}+𝐵_𝑖𝑗(𝑡)×^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑒^{𝑥𝑗(𝑠)}𝑑𝑠+𝐶_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝛾𝑖𝑗(𝑡))}−𝐷^′_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}+𝑒_𝑖(𝑡)×𝜓_𝑖𝑒^𝑥𝑖(𝑡)+𝑓_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)𝑡−𝛿_𝑖(𝑡)=𝑑𝑡_{𝜔0𝑛𝑗=1}Γ_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}𝑑𝑡+^𝜔₀𝑒_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)+𝑓_𝑖(𝑡)×𝜓_𝑖𝑒^𝑥𝑖(𝑡)𝑡−𝛿_𝑖(𝑡=)𝑑𝑡_𝜔0𝑟_𝑖(𝑡)𝑑𝑡=𝑟_𝑖𝜔,𝑖=1,2,…,𝑛, (42) where Γ_𝑖𝑗(𝑡) is defined by (27). From (41), we derive ^𝜔₀𝑥_𝑖(𝑡)+𝜆^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(−𝜏𝑖𝑗(𝑡))′}𝑑𝑡=𝜆^𝜔₀𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)×^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑒^{𝑥𝑗(𝑠)}−𝑑𝑠_𝑛𝑗=1𝐶_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝛾𝑖𝑗(𝑡))}+^𝑛_𝑗=1𝐷^′_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}−𝑒_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)−𝑓_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖×(𝑡)𝑡−𝛿_𝑖(𝑡)≤𝑑𝑡_𝜔0𝑟_𝑖(𝑡)𝑑𝑡+^𝜔₀^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}+𝐵_𝑖𝑗(𝑡)×^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑒^{𝑥𝑗(𝑠)}𝑑𝑠+𝐶_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝛾𝑖𝑗(𝑡))}−𝐷^′_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}+𝑒_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)+𝑓_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)×𝑡−𝛿_𝑖(𝑡)𝑑𝑡. (43) It follows from (41)–(43) that ^𝜔₀𝑥_𝑖(𝑡)+𝜆^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(−𝜏𝑖𝑗(𝑡))′}𝑑𝑡≤𝑅_𝑖+𝑟_𝑖𝜔,𝑖=1,2,…,𝑛. (44) By amplification, it follows from (42) that 𝑟_𝑖𝜔≥^𝑛_𝑗=1^𝜔₀Γ_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}=𝑑𝑡_{𝑛𝑗=1𝜔0}Γ_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}−𝜌_𝑖𝑗𝑒^{𝑥𝑗(𝑡)}+𝜌_𝑖𝑗𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}+𝜌_𝑖𝑗𝑒^{𝑥𝑗(𝑡)}+𝜌_𝑖𝑗𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}=𝑑𝑡_{𝑛𝑗=1𝜔0}Γ_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}−𝜌_𝑖𝑗𝑒^{𝑥𝑗(𝑡)}+𝜌_𝑖𝑗𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}+𝑑𝑡_{𝑛𝑗=1𝜔0}𝜌_𝑖𝑗𝑒^{𝑥𝑗(𝑡)}+𝜌_𝑖𝑗𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}𝑑𝑡. (45) In view of Remark 9 and by a similar analysis, we obtain ^𝑛_𝑗=1^𝜔₀Γ_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}−𝜌_𝑖𝑗𝑒^{𝑥𝑗(𝑡)}+𝜌_𝑖𝑗𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}=𝑑𝑡_{𝑛𝑗=1𝜔0}Γ_𝑖𝑗(𝑠)−𝜌_𝑖𝑗−𝜌_𝑖𝑗𝐷_𝑖𝑗𝜈_𝑖𝑗(𝑠)1−𝜏^′_𝑖𝑗𝜈_𝑖𝑗(𝑠)𝑒^{𝑥𝑗(𝑠)}𝑑𝑡. (46) As 𝜌_𝑖𝑗=Γ^𝐿_𝑖𝑗(1−𝜏^′_𝑖𝑗)^𝐿/((1−𝜏^′_𝑖𝑗)^𝐿+|𝐷_𝑖𝑗|₀), it follows that Γ_𝑖𝑗(𝑠)−𝜌_𝑖𝑗−𝜌_𝑖𝑗(𝐷_𝑖𝑗(𝜈_𝑖𝑗(𝑠))/(1−𝜏^′_𝑖𝑗(𝜈_𝑖𝑗(𝑠))))≥0. So we find from (45) that 𝑟_𝑖𝜔≥^𝑛_𝑗=1^𝜔₀𝜌_𝑖𝑗𝑒^{𝑥𝑗(𝑡)}+𝜌_𝑖𝑗𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}𝑑𝑡. (47) That is, 𝑟_𝑖𝜔≥^𝜔₀^𝑛_𝑗=1𝜌_𝑖𝑗𝑒^{𝑥𝑗(𝑡)}+𝜌_𝑖𝑗𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}𝑑𝑡. (48) By the mean value theorem, we see that there exist points 𝜁_𝑖∈[0,𝜔],(𝑖=1,…,𝑛) such that𝑟_𝑖≥^𝑛_𝑗=1𝜌_𝑖𝑗𝑒^{𝑥𝑗(𝜁𝑖)}+^𝑛_𝑗=1𝜌_𝑖𝑗𝐷_𝑖𝑗𝜁_𝑖𝑒^{𝑥𝑗(𝜁𝑖−𝜏𝑖𝑗(𝜁𝑖))},𝑖=1,…,𝑛, (49) which implies that 𝑥_𝑖𝜁_𝑖≤ln𝑟_𝑖𝜌_𝑖𝑖,𝐷_𝑖𝑗𝜁_𝑖𝑒^{𝑥𝑗(𝜁𝑖−𝜏𝑖𝑗(𝜁𝑖))}≤𝑟_𝑖𝜌_𝑖𝑗,𝑖=1,…,𝑛. (50) By (44) and (50), we can see that 𝑥_𝑖(𝑡)+𝜆^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}≤𝑥_𝑖𝜁_𝑖+𝜆^𝑛_𝑗=1𝐷_𝑖𝑗𝜁_𝑖𝑒^{𝑥𝑗(𝜁𝑖−𝜏𝑖𝑗(𝜁𝑖))}+^𝜔₀𝑥_𝑖(𝑡)+𝜆^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))′}𝑑𝑡≤ln𝑟_𝑖𝜌_𝑖𝑖+^𝑛_𝑗=1𝑟_𝑖𝜌_𝑖𝑗+𝑅_𝑖+𝑟_𝑖𝜔=∶𝐻_𝑖,𝑖=1,2,…,𝑛. (51) For 𝜆∑^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}≥0, one can find that 𝑥_𝑖(𝑡)≤𝐻_𝑖,𝑖=1,…,𝑛. (52) Besides, from (41), we have 𝑥^′_𝑖(𝑡)=𝜆𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)×^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑒^{𝑥𝑗(𝑠)}𝑑𝑠−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}−^𝑛_𝑗=1𝐷_0,𝑖𝑗(𝑡)𝑥^′_𝑗𝑡−𝛾_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝛾𝑖𝑗(𝑡))}−𝑒_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)−𝑓_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)×𝑡−𝛿_𝑖(𝑡),𝑖=1,2,…,𝑛. (53) Notice that for all 𝑡∈𝑅, one has 𝜓_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝛽^∗_𝑖(𝑠)+𝜃^∗_𝑖(𝑠=)𝑑𝑠_{𝑡𝑡+𝜔}𝐺_𝑖(𝑡,𝑠)𝛼_𝑖(𝑠)𝛽^∗_𝑖(𝑠)+𝜃^∗_𝑖(𝑠)𝛼_𝑖(𝑠)≤𝛽𝑑𝑠_∗𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝛼_𝑖(𝑠)=𝛽𝑑𝑠_∗𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖,𝑖=1,…,𝑛. (54) Then, by (52) and (54), we get 𝑥^′_𝑖0≤𝑟_𝑖(𝑡)+^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}+^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)×^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑒^{𝑥𝑗(𝑠)}𝑑𝑠+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝛾𝑖𝑗(𝑡))}+^𝑛_𝑗=1𝐷_0,𝑖𝑗(𝑡)𝑥^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡−𝜏𝑖𝑗(𝑡))}+𝑒_𝑖(𝑡)×𝜓_𝑖𝑒^𝑥𝑖(𝑡)+𝑓_𝑖(𝑡)𝜓_𝑖𝑒^𝑥𝑖(𝑡)𝑡−𝛿_𝑖(𝑡)≤𝑟_𝑖0+^𝑛_𝑗=1𝐴_𝑖𝑗0𝑒^𝐻𝑗+^𝑛_𝑗=1𝐵_𝑖𝑗0𝑒^𝐻𝑗+^𝑛_𝑗=1𝐶_𝑖𝑗0𝑒^𝐻𝑗+^𝑛_𝑗=1𝐷_0,𝑖𝑗0𝑥^′_𝑗0𝑒^𝐻𝑗+𝑒_𝑖0+𝑓_𝑖0×𝛽^∗_𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑒^𝐻𝑖,𝑖=1,2,…,𝑛. (55) Furthermore, we have 𝑥^′₀=^𝑛_𝑖=1𝑥^′_𝑖0≤^𝑛_𝑖=1𝑟_𝑖0+^𝑛𝑛_𝑖=1𝑗=1𝐴_𝑖𝑗0𝑒^𝐻𝑗+^𝑛𝑛_𝑖=1𝑗=1𝐵_𝑖𝑗0𝑒^𝐻𝑗+^𝑛𝑛_𝑖=1𝑗=1𝐶_𝑖𝑗0𝑒^𝐻𝑗+𝑒_𝑖0+𝑓_𝑖0𝛽^∗_𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑒^𝐻𝑖+^𝑛𝑛_𝑖=1𝑗=1𝐷_0,𝑖𝑗0𝑥^′₀𝑒^𝐻𝑗. (56) By the assumption (3) of Theorem 10, we see that ^𝑛𝑛_𝑖=1𝑗=1𝐷_𝑖𝑗0𝑒^𝐻𝑗≤^𝑛𝑛_𝑖=1𝑗=1𝐷_0,𝑖𝑗0𝑒^𝐻≤^𝑛𝑛_𝑖=1𝑗=1𝐷_0,𝑖𝑗0𝑒^𝑀0<1. (57) Then, we have 𝑥^′₀≤^𝑛_𝑖=1𝑟_𝑖0+^𝑛𝑛_𝑖=1𝑗=1𝐴_𝑖𝑗0𝑒^𝐻𝑗+^𝑛𝑛_𝑖=1𝑗=1𝐵_𝑖𝑗0𝑒^𝐻𝑗+^𝑛𝑛_𝑖=1𝑗=1𝐶_𝑖𝑗0𝑒^𝐻𝑗+𝑒_𝑖0+𝑓_𝑖0𝛽^∗_𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑒^𝐻𝑖×1−^𝑛𝑛_𝑖=1𝑗=1𝐷_0,𝑖𝑗0𝑒^𝐻𝑗−1=∶Λ^∗. (58) Since 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))∈𝑋, there exists a 𝜉_𝑖∈[0,𝜔] such that 𝑥_𝑖𝜉_𝑖=inf_{[]𝑡∈0,𝜔}𝑥_𝑖(𝑡),𝑖=1,…,𝑛. (59) It follows from (42) that 𝑟_𝑖𝜔≥^𝑛_𝑗=1^𝜔₀Γ_𝑖𝑗(𝑡)𝑒^{𝑥𝑗(𝑡)}=𝑑𝑡_𝑛𝑗=1𝑒^{𝑥𝑗(𝜉𝑗)𝜔}₀Γ_𝑖𝑗(𝑡)𝑑𝑡,𝑖=1,…,𝑛. (60) It follows from (30) and (60) that 𝑟_𝑖≥^𝑛_𝑗=1𝑒^{𝑥𝑗(𝜉𝑗)}𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗,𝑖=1,…,𝑛. (61) From (61), we obtain 𝑒^{𝑥𝑖(𝜉𝑖)}𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖≤𝑟_𝑖,𝑖=1,…,𝑛. (62) As 𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖>0, it follows from the previous formula that 𝑥_𝑖𝜉_𝑖≤ln𝑟_𝑖𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖,𝑖=1,…,𝑛. (63) On the other hand, there also exists a 𝜂_𝑖∈[0,𝜔] such that 𝑥_𝑖𝜂_𝑖=sup_{[]𝑡∈0,𝜔}𝑥_𝑖(𝑡),𝑖=1,…,𝑛. (64) It follows from (42), (54), and (64) that 𝑟_𝑖≤^𝑛_𝑗=1Γ_𝑖𝑗𝑒^{𝑥𝑗(𝜂𝑖)}+𝑒_𝑖+𝑓_𝑖𝛽^∗_𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑒^{𝑥𝑖(𝜂𝑖)}=^𝑛_{𝑗=1,𝑗≠𝑖}Γ_𝑖𝑗𝑒^{𝑥𝑗(𝜂𝑖)}+Γ_𝑖𝑖+𝑒_𝑖+𝑓_𝑖𝛽^∗_𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑒^{𝑥𝑖(𝜂𝑖)},𝑖=1,…,𝑛. (65) From (41), (54), and (64), we can have 𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖+𝑒_𝑖+𝑓_𝑖𝛽^∗_𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑒^{𝑥𝑖(𝜂𝑖)}≥𝑟_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝑒^{𝑥𝑗(𝜂𝑖)}≥𝑟_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝑒^𝐻𝑗,𝑖=1,…,𝑛. (66) That is, 𝑥_𝑖𝜂_𝑖≥ln𝑟_𝑖−∑^𝑛_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝑒^𝐻𝑗𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖+𝑒_𝑖+𝑓_𝑖𝛽^∗_𝑖0+𝜃^∗_𝑖0/𝛼^𝐿_𝑖,𝑖=1,…,𝑛. (67) Now, from (60) and (63) we know that there exist 𝜍_𝑖∈[0,𝜔](𝑖=1,…,𝑛) such that 𝑥_𝑖𝜍_𝑖𝑟≤maxln^𝑖𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖,ln𝑟_𝑖−∑^𝑛_{𝑗=1,𝑗≠𝑖}𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝑒^𝐻𝑗𝐴_𝑖𝑖+𝐵_𝑖𝑖+𝐶_𝑖𝑖+𝑒_𝑖+𝑓_𝑖𝛽^∗_𝑖0+𝜃^∗_𝑖0/𝛼^𝐿_𝑖=∶Λ_𝑖,𝑖=1,…,𝑛. (68) From (54), (64), and Lemma 3, we have 𝑥_𝑖≤𝑥_𝑖𝜍_𝑖+12^𝜔₀𝑥^′_𝑖(𝑡)𝑑𝑡≤Λ_𝑖+12^𝜔₀𝑥^′_𝑖𝑑𝑡,𝑖=1,…,𝑛. (69) Then, ‖𝑥‖₀≤^𝑛_𝑖=1𝑥_𝑖≤^𝑛_𝑖=1Λ_𝑖+12^𝜔₀𝑥^′₀<𝑑𝑡_𝑛𝑖=1Λ_𝑖+12Λ^∗𝜔≤𝑀₀. (70) Obviously, 𝑀₀ is independent of 𝜆; the proof of Lemma 12 is completed.

Based on the previous results, we can now apply Lemma 1 and Remark 2 to (34) and obtain a proof of Theorem 10.

Proof. Obviously, for 𝑀 as given in Lemma 11, condition (i) in Lemma 1 is satisfied. Let ℎ(𝜇)=(ℎ₁(𝜇),…,ℎ_𝑛(𝜇)). Since ℎ_𝑖𝜇=^𝜔₀𝑓_𝑖𝜇=𝑠,𝑑𝑠_𝜔0𝑟_𝑖(𝑡)𝑑𝑡−^𝑛_𝑖=1^𝜔₀𝐴_𝑖𝑗(𝑡)𝑑𝑡𝑒^𝜇𝑗−^𝑛_𝑖=1^𝜔₀𝐵_𝑖𝑗(𝑡)𝑑𝑡𝑒^𝜇𝑗−^𝑛_𝑖=1^𝜔₀𝐶_𝑖𝑗(𝑡)𝑑𝑡𝑒^𝜇𝑗−^𝜔₀𝑒_𝑖(𝑡)+𝑓_𝑖(𝑡)𝑑𝑡𝑒^𝜇𝑖=𝑟_𝑖−^𝑛_𝑗=1𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝑒^𝜇𝑗+𝑒_𝑖+𝑓_𝑖𝜔 (71) and 𝑀>∑^𝑛_𝑖=1|ln𝜇^∗_𝑖|, we have ℎ(𝜇)≠0 for any 𝜇∈𝜕𝐵_𝑀(𝑅^𝑛). That is, condition (ii) in Lemma 1 holds. At last, we verify that condition (iii) of Lemma 1 also holds. By assumption (1) of Theorem 10 and the formula for the Brouwer degree (see Theorem 2.2.3 in [35, 36]), a straightforward calculation shows that degℎ,𝐵_𝑀𝑅^𝑛=_{𝜇∈ℎ−1(0)⋂𝐵𝑀(𝑅𝑛)}𝜇)signdet𝐷ℎ=sign(−1^𝑛×det𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗𝑒^{∑𝑛𝑗=1𝜇∗𝑗}+𝑒_𝑖+𝑓_𝑖𝑒^𝜇∗𝑖≠0. (72) By now, all the assumptions required in Lemma 1 hold. It follows from Lemma 1 and Remark 2 that system (34) has an 𝜔-periodic solution. Returning to 𝑦_𝑖(𝑡)=𝑒^{𝑥𝑖(𝑡)}, we infer that systems (18) and (19) have at least one positive 𝜔-periodic solution. By Lemmas 7 and 8, (𝑁^∗(𝑡),𝑢^∗(𝑡))^𝑇=(𝑁^∗₁(𝑡),…,𝑁^∗_𝑛(𝑡),𝑢^∗₁(𝑡),…,𝑢^∗_𝑛(𝑡))^𝑇 is the unique positive periodic solution of the system (1) and (2), where 𝑁^∗_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦^∗_𝑖(𝑡)(𝑖=1,2,…,𝑛). The proof of Theorem 10 is complete.

Consider the following: 𝑁^′_𝑖(𝑡)=𝑁_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝛾_𝑖𝑗(𝑡−_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝛿_𝑖(𝑡),𝑢^′_𝑖(𝑡)=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑁_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑁_𝑖𝑡−𝜎_𝑖(𝑡),𝑖=1,2,…,𝑛, (73) which is a special case of system (1) without impulse. We get easily the following result. Here, we have the following notations: 𝜌^∗_𝑖𝑗=Γ^∗𝐿_𝑖𝑗1−𝛾^′_𝑖𝑗^𝐿1−𝛾^′_𝑖𝑗^𝐿+𝑑_𝑖𝑗0,𝑅_𝑖=1𝜔^𝜔₀𝑟_𝑖(𝑡𝐿)𝑑𝑡,_∗=max^𝑛𝑛_𝑖=1𝑗=1𝑑_𝑖𝑗0,^𝑛𝑛_𝑖=1𝑗=1𝑑_0,𝑖𝑗0,𝑀^∗=max^𝑛_𝑖=1ln𝜇^∗_𝑖0,𝐻^∗,12𝜔Θ^∗+^𝑛_𝑖=1Δ_𝑖,𝐻^∗=max_{𝑖∈[1,𝑛]}𝐻^∗_𝑖,𝐻^∗_𝑖=ln𝑟_𝑖𝜌^∗_𝑖𝑖+^𝑛_𝑗=1𝑟_𝑖𝜌^∗_𝑖𝑗+𝑅_𝑖+𝑟_𝑖Θ𝜔,_∗=^𝑛_𝑖=1𝑟_𝑖0+^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗0𝑒^𝐻∗𝑗+^𝑛𝑛_𝑖=1𝑗=1𝑏_𝑖𝑗0𝑒^𝐻∗𝑗+^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗0𝑒^𝐻∗𝑗+𝑒_𝑖0+𝑓_𝑖0𝛽_𝑖0+𝜃_𝑖0𝛼^𝐿_𝑖𝑒^𝐻∗𝑖×1−^𝑛𝑛_𝑖=1𝑗=1𝑑_0,𝑖𝑗0𝑒^{𝐻∗𝑗−1},Θ_𝑖𝑟=maxln^𝑖𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖,ln𝑟_𝑖−∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗𝑒^𝐻∗𝑗𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖+𝑒_𝑖+𝑓_𝑖𝛽_𝑖0+𝜃_𝑖0/𝛼^𝐿_𝑖,Γ^∗_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)+𝑐_𝑖𝑗𝜗_𝑖𝑗(𝑡)1−𝛾^′_𝑖𝑗𝜗_𝑖𝑗(𝑡)−𝑑^′_𝑖𝑗𝜈_𝑖𝑗(𝑡)1−𝜏^′_𝑖𝑗𝜈_𝑖𝑗(𝑡),Γ¹_𝑖(𝑡)=𝑒_𝑖(𝑡)𝜓_𝑖1(𝑡),Γ²_𝑖(𝑡)=𝑓_𝑖(𝑡)𝜓_𝑖1𝑡−𝜎_𝑖(𝑡),𝑑_0,𝑖𝑗(𝑡)=𝑑_𝑖𝑗(𝑡)1−𝜏^′_𝑖𝑗(𝑡), (74) and 𝜗_𝑖𝑗(𝑡),𝜈_𝑖𝑗(𝑡) represent the inverse function of 𝑡−𝛾_𝑖𝑗(𝑡),𝑡−𝜏_𝑖𝑗(𝑡)(𝑖,𝑗=1,2,…,𝑛), respectively.

Corollary 13. Suppose that the following conditions hold;(1) the system of algebraic equations 𝑓^∗𝜇=𝑟_𝑖−^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗𝜇_𝑗+Γ¹_𝑖+Γ²_𝑖𝜇_𝑖𝑛×1=0 (75) has a unique positive solution 𝜇^∗=(𝜇^∗₁,…,𝜇^∗_𝑛);(2)𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗>0, 𝑟_𝑖>∑^𝑛_{𝑗=1,𝑗≠𝑖}(𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗)𝑒^𝐻∗𝑗, 𝜏^′_𝑖𝑗(𝑡)<1, 𝛾^′_𝑖𝑗(𝑡)<1 and Γ^∗_𝑖𝑗(𝑡)>0;(3)𝐾^∗=∶𝐿^∗𝑒^𝑀∗<1. Then, (73) has at least one positive 𝜔-periodic solution.

Proof. Its proof is similar to the proof of Theorem 10. Here, we omit it.

Similarly, we can get the following results.

Theorem 14. Assume that conditions of Theorem 10 hold, and then, the conclusion of Theorem 10 holds for the following system: 𝑁^′_𝑖(𝑡)=−𝑁_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝛾_𝑖𝑗(𝑡−_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝛿_𝑖(𝑡),𝑖=1,2,…,𝑛,𝑡≠𝑡_𝑘,𝑢^′_𝑖(𝑡)=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑁_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑁_𝑖𝑡−𝜎_𝑖(𝑡),𝑡≥0,Δ𝑁_𝑖𝑡_𝑘=𝑝_𝑖𝑘+𝑞_𝑖𝑘𝑁_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…. (76)

Proof. Its proof is similar to the proof of Theorem 10. Here, we omit it.

Corollary 15. Assume that conditions of Corollary 13 hold, and then, the conclusion of Corollary 13 holds for the following system: 𝑁^′_𝑖(𝑡)=−𝑁_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝜏_𝑖𝑗(𝑡−_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝛾_𝑖𝑗(𝑡),𝑢^′_𝑖(𝑡)=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑁_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑁_𝑖𝑡−𝜎_𝑖(𝑡),𝑖=1,2,…,𝑛. (77)

Proof. Its proof is similar to the proof of Theorem 10. Here, we omit it.

Remark 16. When 𝛼_𝑖(𝑡)=𝛽_𝑖(𝑡)=𝜃_𝑖(𝑡)=0, we can derive some immediate corollaries from Theorems 10 and 14; thus, our results generalize the corresponding results in [14].

4. Applications

In order to illustrate some features of our main result, in the following, we will apply Theorem 10 to some special cases, which have been studied extensively in the literature.

Application 17. We consider an 𝑛-species neutral delay competition system in a periodic environment with impulse: 𝑁^′_𝑖(𝑡)=±𝑁_𝑖(𝑡)𝑎_𝑖(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑁_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑁^′_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖×𝑡−𝛿_𝑖(𝑡),𝑖=1,2,…,𝑛,𝑡≠𝑡_𝑘,𝑢^′_𝑖(𝑡)=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑁_𝑖(𝑡)+𝜃_𝑖(𝑡)𝑁_𝑖𝑡−𝜎_𝑖(𝑡),𝑡≥0,Δ𝑁_𝑖𝑡_𝑘=𝑝_𝑖𝑘+𝑞_𝑖𝑘𝑁_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…, (78) where 𝑏_𝑖𝑗, 𝑐_𝑖𝑗, 𝑒_𝑖, 𝑓_𝑖, 𝛼_𝑖, 𝛽_𝑖, 𝜃_𝑖∈𝐶(𝑅,[0,+∞)), 𝑑_𝑖𝑗∈𝐶¹(𝑅,[0,+∞)), 𝛾_𝑖𝑗∈𝐶¹(𝑅,𝑅), and 𝜏_𝑖𝑗∈𝐶²(𝑅,𝑅) are continuous 𝜔-periodic functions. And 𝑎_𝑖∈𝐶(𝑅,𝑅) are continuous 𝜔-periodic functions with ∫^𝜔₀𝑎_𝑖(𝑡)𝑑𝑡>0. Here, we give the following notations: 𝐴_𝑖=1𝜔^𝜔₀𝑎_𝑖(𝑡𝐵)𝑑𝑡,_𝑖𝑗(𝑡)=𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑖𝑘+𝑞_𝑖𝑘,𝐶_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+𝑝_𝑖𝑘+𝑞_𝑖𝑘,𝐷_𝑖𝑗(𝑡)=𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝛾𝑖𝑗(𝑡)}1+𝑝_𝑖𝑘+𝑞_𝑖𝑘,𝐷_0,𝑖𝑗(𝑡)=𝐷_𝑖𝑗(𝑡)1−𝛾^′_𝑖𝑗(𝑡),𝛽^∗_𝑖(𝑡)=𝛽_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑖𝑘+𝑞_𝑖𝑘,𝜃^∗_𝑖(𝑡)=𝜃_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑖𝑘+𝑞_𝑖𝑘,Γ_𝑖𝑗(𝑡)=𝐴_𝑖𝑗(𝑡)+𝐵_𝑖𝑗(𝑡)+𝐶_𝑖𝑗𝜗_𝑖𝑗(𝑡)1−𝛾^′_𝑖𝑗𝜗_𝑖𝑗(𝑡)−𝐷^′_𝑖𝑗𝜈_𝑖𝑗(𝑡)1−𝜏^′_𝑖𝑗𝜈_𝑖𝑗(𝑡),Γ¹_𝑖(𝑡)=𝑒_𝑖(𝑡)𝜓_𝑖1(𝑡),Γ²_𝑖(𝑡)=𝑓_𝑖(𝑡)𝜓_𝑖1𝑡−𝜎_𝑖(𝑡),𝐿₁=max^𝑛𝑛_𝑖=1𝑗=1𝐷_𝑖𝑗0,^𝑛𝑛_𝑖=1𝑗=1𝐷_0,𝑖𝑗0,𝑀₁=max^𝑛_𝑖=1ln𝜇^∗_𝑖0,𝐻^∗,12𝜔Λ^∗+^𝑛_𝑖=1Λ_𝑖,𝐻^∗=max_{𝑖∈[1,𝑛]}𝐻^∗_𝑖,𝐻^∗_𝑖=ln𝑎_𝑖𝜌^∗_𝑖𝑖+^𝑛_𝑗=1𝑎_𝑖𝜌^∗_𝑖𝑗+𝐴_𝑖+𝑎_𝑖𝜌𝜔,_∗𝑖𝑗=Γ^∗𝐿_𝑖𝑗1−𝛾^′_𝑖𝑗^𝐿1−𝛾^′_𝑖𝑗^𝐿+𝐷_𝑖𝑗0,Λ^∗=^𝑛_𝑖=1𝑎_𝑖0+^𝑛𝑛_𝑖=1𝑗=1𝐵_𝑖𝑗0𝑒^𝐻∗𝑗+^𝑛𝑛_𝑖=1𝑗=1𝐶_𝑖𝑗0𝑒^𝐻∗𝑗+𝑒_𝑖0+𝑓_𝑖0𝛽^∗_𝑖0+𝜃^∗_𝑖0𝛼^𝐿_𝑖𝑒^𝐻∗𝑖×1−^𝑛𝑛_𝑖=1𝑗=1𝐷_0,𝑖𝑗0𝑒^{𝐻∗𝑗−1},Λ_𝑖𝑎=maxln^𝑖𝐵_𝑖𝑖+𝐶_𝑖𝑖,ln𝑎_𝑖−∑^𝑛_{𝑗=1,𝑗≠𝑖}𝐵_𝑖𝑗+𝐶_𝑖𝑗𝑒^𝐻∗𝑗𝐵_𝑖𝑖+𝐶_𝑖𝑖+𝑒_𝑖+𝑓_𝑖𝛽^∗_𝑖0+𝜃^∗_𝑖0/𝛼^𝐿_𝑖,𝑖,𝑗=1,2,…,𝑛, (79) where 𝜗_𝑖𝑗(𝑡),𝜈_𝑖𝑗(𝑡) represent the inverse function of 𝑡−𝛾_𝑖𝑗(𝑡),𝑡−𝜏_𝑖𝑗(𝑡), respectively. Applying Theorem 10 to (78), we can obtain the following theorem.

Theorem 18. Assume that the following conditions are satisfied: (1) the system of algebraic equations 𝑓^∗𝜇=𝑎_𝑖−^𝑛_𝑗=1𝐵_𝑖𝑗+𝐶_𝑖𝑗𝜇_𝑗+Γ¹_𝑖+Γ²_𝑖𝜇_𝑖𝑛×1=0 (80) has a unique positive solution 𝜇^∗=(𝜇^∗₁,…,𝜇^∗_𝑛);(2)𝐵_𝑖𝑗+𝐶_𝑖𝑗>0, 𝑎_𝑖>∑^𝑛_{𝑗=1,𝑗≠𝑖}(𝐵_𝑖𝑗+𝐶_𝑖𝑗)𝑒^𝐻𝑗, 𝛾^′_𝑖𝑗(𝑡)<1, 𝜏^′_𝑖𝑗(𝑡)<1, and Γ_𝑖𝑗(𝑡)>0;(3)𝐾₁=∶𝐿₁𝑒^𝑀1<1.
Then, (78) has at least one positive 𝜔-periodic solution.

Remark 19. When 𝛼_𝑖(𝑡)=𝛽_𝑖(𝑡)=𝜃_𝑖(𝑡)=𝑢_𝑖(𝑡)=0, 𝑝_𝑖𝑘+𝑞_𝑖𝑘=0, we can derive some immediate corollaries of Theorem 18, then, Theorem 18 generalizes the corresponding results in [9, 10]. On the other hand, when 𝑝_𝑖𝑘+𝑞_𝑖𝑘=0, we can derive an immediate corollary of Theorem 18, then, Theorem 18 generalizes the corresponding results in  [11].

Application 20. We consider the single specie neutral delay logistic equation with impulse: 𝑁^′(𝑡)(𝑡𝑎(𝑡)(𝑡)𝑁(𝑡)−=±𝑁)[−𝑏_𝑛𝑖=1𝑐_𝑖(𝑡)^𝑡_−∞𝑘_𝑖()𝑁(𝑠)−𝑡−𝑠𝑑𝑠_𝑚𝑗=1𝑑_𝑗(𝑡)𝑁𝑡−𝛾_𝑗(𝑡)−^𝑝_𝑙=1𝑒_𝑙(𝑡)𝑁^′𝑡−𝜏_𝑙(𝑡)(𝑡)𝑢(𝑡)(𝑡)𝑢((𝑡−𝑓−𝑔𝑡−𝛿)),𝑡≠𝑡_𝑘,𝑢^′(𝑡)(𝑡)𝑢(𝑡)(𝑡)𝑁(𝑡)(𝑡)𝑁((𝑡𝑁𝑡=−𝛼+𝛽+𝜃𝑡−𝜎)),𝑡≥0,_+𝑘=𝑝_𝑘+𝑞_𝑘𝑁𝑡_𝑘,𝑘=1,2,…., (81) where 𝑏, 𝑐_𝑖, 𝑑_𝑗, 𝑓, 𝑔, 𝛼, 𝛽, 𝜃∈𝐶(𝑅,[0,+∞)), 𝑒_𝑙∈𝐶¹(𝑅,[0,+∞)), 𝛾_𝑗∈𝐶¹(𝑅,𝑅), and 𝜏_𝑙∈𝐶²(𝑅,𝑅) are continuous 𝜔-periodic functions. And 𝑎∈𝐶(𝑅,𝑅) are continuous 𝜔-periodic functions with ∫^𝜔₀𝑎(𝑡)𝑑𝑡>0. Here, we have the following notations: 1𝐴=𝜔^𝜔₀|𝑎(𝑡𝐵(𝑡)(𝑡))|𝑑𝑡,=𝑏_{0<𝑡𝑘<𝑡}1+𝑝_𝑘+𝑞_𝑘,𝐶_𝑖(𝑡)=𝑐_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑘+𝑞_𝑘,𝐷_𝑗(𝑡)=𝑑_𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑗(𝑡)}1+𝑝_𝑘+𝑞_𝑘,𝐸_𝑙(𝑡)=𝑒_𝑙(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑙(𝑡)}1+𝑝_𝑘+𝑞_𝑘,𝐸_0,𝑙(𝑡)=𝐸_𝑙(𝑡)1−𝜏^′_𝑙(𝑡),𝛽^∗(𝑡)=𝛽(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑖𝑘+𝑞_𝑖𝑘,𝜃^∗(𝑡)=𝜃(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑖𝑘+𝑞_𝑖𝑘,Γ𝜌=^𝐿11−𝜏^′_𝑙^𝐿1−𝜏^′_𝑙^𝐿+𝐸_𝑙0,𝐿₂=max^𝑝_𝑙=1𝐸_𝑙0,^𝑝_𝑙=1𝐸_0,𝑙0,𝑀₂=max𝐻^∗,Δ^∗,𝐻^∗=ln𝑎𝜌+𝑎𝜌+𝐴+|𝑎Δ𝜔,_∗=𝜔2|𝑎|₀+|𝐵|₀+^𝑛_𝑖=1𝐶_𝑖0+^𝑚_𝑗=1𝐷_𝑗0+𝑓₀+𝑔₀×𝛽^∗₀+𝜃^∗₀𝛼^𝐿𝑒^𝐻∗×1−𝐸_0,𝑙0𝑒^𝐻∗−1+ln𝑎𝐵+∑^𝑛_𝑖=1𝐶_𝑖+∑^𝑚_𝑗=1𝐷_𝑗+Γ₂+Γ₃,Γ₁(𝑡)=𝐵(𝑡)+^𝑛_𝑖=1𝐶_𝑖(𝑡)+^𝑚_𝑗=1𝐷_𝑗𝜇_𝑗(𝑡)1−𝜎^′𝜇_𝑗(𝑡−)^𝑝_𝑙=1𝐸^′_𝑙𝜈_𝑙(𝑡)1−𝜏^′_𝑙𝜈_𝑙(𝑡),Γ₂(𝑡)(𝑡)(𝑡),Γ=𝑓𝜓1₃(𝑡)=𝑔(𝑡)𝜓1(𝑡−𝜎(𝑡)), (82)

and 𝜇_𝑗(𝑡),𝜈_𝑙(𝑡) represent the inverse function of 𝑡−𝛾_𝑗(𝑡),𝑡−𝜏_𝑙(𝑡), respectively. Applying Theorem 10 to (81), we can obtain the following theorem.

Theorem 21. Assume that the following conditions are satisfied: (1)𝐵+∑^𝑛_𝑖=1𝐶_𝑖+∑^𝑚_𝑗=1𝐷_𝑗+Γ₂+Γ₃>0, 𝛾^′_𝑗(𝑡)<1, 𝜏^′_𝑙(𝑡)<1 and Γ₁(𝑡)>0;(2)𝐾₂=∶𝐿₂𝑒^𝑀2<1.
Then, system (81) has at least one positive 𝜔-periodic solution.

Remark 22. When 𝑖=0, 𝑗=𝑙=1, 𝛼(𝑡)=𝛽(𝑡)=𝜃(𝑡)=𝑢(0)=0, 𝜎_𝑗(𝑡)=𝜏_𝑙(𝑡), and 𝑝_𝑘+𝑞_𝑘=0, we can derive an immediate corollary of Theorem 21, which is also an answer to the open problem 9.2 due to Kuang  [2]. On the other hand, when 𝑖=0, 𝛼(𝑡)=𝛽(𝑡)=𝜃(𝑡)=𝑢(𝑡)=0, and 𝑝_𝑘+𝑞_𝑘=0, we can derive some immediate corollaries of Theorem 21 (that is the corresponding results in  [7, 8]), therefore, our result improves and generalizes the corresponding result in  [7, 8]. Moreover, when 𝑙=0, 𝛼(𝑡)=𝛽(𝑡)=𝜃(𝑡)=𝑢(𝑡)=0, we can see that our Theorem 21 can hold without the assumption 𝑎>0. When 𝑎<0, Wang's main result (see Theorem 3.1 in  [13]) cannot be applied. Therefore, in comparison with [13], our result improves and generalizes the result in  [13].

Acknowledgments

This research is supported by NSF of China (nos. 10971229, 11161015, and 11371367), PSF of China (no. 2012M512162), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, and 13C084), and the construct program of the key discipline in Hunan province.