Existence of Positive Periodic Solutions for Periodic Neutral Lotka-Volterra System with Distributed Delays and Impulses

Abstract

By using a fixed-point theorem of strict-set-contraction, we investigate the existence of positive periodic solutions for a class of the following impulsive neutral Lotka-Volterra system with distributed delays: 𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)[𝑟_𝑖(𝑡)−∑^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−∑^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)∫⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗(𝑡+𝜉)𝑑𝜉−∑^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)∫⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉],Δ𝑥_𝑖(𝑡_𝑘)=−𝐼_𝑖𝑘(𝑥_𝑖(𝑡_𝑘)),𝑖=1,2,…,𝑛,𝑘=1,2,…. Some verifiable criteria are established easily.

1. Introduction

It is well known that natural environments are physically highly variable, and in response, birth rates, death rates, and other vital rates of populations vary greatly in time. Theoretical evidence suggests that many population and community patterns represent intricate interactions between biology and variation in the physical environment (see [1–4]). Thus, the focus in theoretical models of population and community dynamics must be not only on how populations depend on their own population densities or the population densities of other organisms, but also on how populations change in response to the physical environment. It is reasonable to study the models of population with periodic coefficients. In addition to the theoretical and practical significance, the Lotka-Volterra model is one of the famous models for dynamics of population; therefore it has been studied extensively [5–9]. In view of the above effects, by applying a fixed-point theorem of strict-set-contraction, Li [10] established criteria to guarantee the existence of positive periodic solutions of the following neutral Lotka-Volterra system with distributed delays: 𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝑇𝑖𝑗}𝐾_𝑖𝑗(𝜃)𝑥_𝑗()−𝑡+𝜃𝑑𝜃_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰₋𝑇_𝑖𝑗𝐾_𝑖𝑗(𝜃)𝑥^′_𝑗(𝑡+𝜃)𝑑𝜃,𝑖=1,2,…,𝑛, (1) where 𝑎_𝑖,𝑏_𝑖𝑗,𝑐_𝑖𝑗∈𝐶(𝑅,𝑅⁺)(𝑖,𝑗=1,2,…,𝑛) are 𝜔-periodic functions and 𝑇_𝑖𝑗,𝑇_𝑖𝑗∈(0,+∞)(𝑖,𝑗=1,2,…,𝑛) and 𝐾_𝑖𝑗,𝐾_𝑖𝑗∈(𝑅,𝑅⁺) satisfying ∫⁰_{−𝑇𝑖𝑗}𝐾_𝑖𝑗(𝜃)𝑑𝜃=1, ∫⁰₋𝑇_𝑖𝑗𝐾_𝑖𝑗(𝜃)𝑑𝜃, 𝑖,𝑗=1,2,…,𝑛. On the other side, birth of many species is an annual birth pulse or harvesting. To have a more accurate description of many mathematical ecology systems, we need to consider the use of impulsive differential equations [11–13]. Some qualitative properties such as oscillation, periodicity, asymptotic behavior, and stability properties have been investigated extensively by many authors over the past few years [14–18]. However, to our knowledge, there are few published papers discussing the existence of periodic solutions for neutral Lotka-Volterra system with distributed delays and impulses. In this paper, we are concerned with the following neutral Lotka-Volterra system with distributed delays and impulses: 𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()−𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉,𝑖=1,2,…,𝑛,𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=−𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…, (2) where 𝑟_𝑖,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗∈𝐶(𝑅,𝑅⁺)(𝑖,𝑗=1,2,…,𝑛) are 𝜔-periodic functions and 𝜏_𝑖𝑗,𝜎_𝑖𝑗∈(0,+∞)(𝑖,𝑗=1,2,…,𝑛) and 𝑓_𝑖𝑗,𝑔_𝑖𝑗∈𝐶(𝑅,𝑅⁺) satisfying ∫⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑑𝜉=1, ∫⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑑𝜉=1,𝑖,𝑗=1,2,…,𝑛. Moreover, Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘) (here 𝑥(𝑡⁺_𝑘) represents the right limit of 𝑥(𝑡) at the point 𝑡_𝑘), 𝐼_𝑖𝑘∈𝐶(𝑅⁺,𝑅⁺); that is, 𝑥 changes decreasingly suddenly at times 𝑡_𝑘;  𝜔>0 is a constant, 𝑅=(−∞,+∞), 𝑅⁺=[0,+∞). We assume that there exists an integer 𝑞>0 such that 𝑡_𝑘+𝑞=𝑡_𝑘+𝜔,𝐼_{𝑖(𝑘+𝑞)}=𝐼_𝑖𝑘, where 0<𝑡₁<𝑡₂<⋯<𝑡_𝑞<𝜔.

The main purpose of this paper is by using a fixed-point theorem of strict-set-contraction [19, 20] to establish new criteria to guarantee the existence of positive periodic solutions of the system (2).

For convenience, we introduce the notation 𝑓^𝑀=max_{[]𝑡∈0,𝜔}𝑓(𝑡),𝑓^𝐿=min_{[]𝑡∈0,𝜔}𝑓(𝑡,𝛾_𝑖)=lim_𝑢→0sup_{𝑡≤𝑡𝑘≤𝑡+𝜔}𝐼_𝑖𝑘(𝑢)𝑢𝛿,𝑖=1,2,…,𝑛,_𝑖=𝑒^{−∫𝜔0𝑟𝑖(𝑡)𝑑𝑡}𝛼,𝑖=1,2,…,𝑛,_𝑖𝑗=^𝜔₀𝛿_𝑗𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)−𝑐_𝑖𝑗(𝑡𝛽)𝑑𝑡,_𝑖𝑗=^𝜔₀𝑎_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)+𝑐_𝑖𝑗(𝑡)𝑑𝑡,𝑖,𝑗=1,2,…,𝑛, (3) where 𝑓 is a continuous 𝜔-periodic function.

Throughout this paper, we assume that(𝐴₁)𝛿_𝑖=𝑒^{−∫𝜔0𝑟𝑖(𝑡)𝑑𝑡}<1,𝑖=1,2,…,𝑛; (𝐴₂)𝛿_𝑗𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)−𝑐_𝑖𝑗(𝑡)≥0,𝑖,𝑗=1,2,…,𝑛; (𝐴₃)(1+𝑟^𝐿_𝑖)(𝛿²_𝑖/(1−𝛿_𝑖))𝛼_𝑖𝑗≥max_{𝑡∈[0,𝜔]}{𝑎_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)+𝑐_𝑖𝑗(𝑡)},𝑖,𝑗=1,2,…,𝑛; (𝐴₄)(𝑟^𝑀_𝑖−1)(1/𝛿_𝑖(1−𝛿_𝑖))𝛽_𝑖𝑗≤min_{𝑡∈[0,𝜔]}{𝛿_𝑗𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)−𝑐_𝑖𝑗(𝑡)},𝑖,𝑗=1,2,…,𝑛; (𝐴₅)max_{1≤𝑖≤𝑛}{∑^𝑛_𝑗=1𝑐^𝑀_𝑖𝑗}≤min_{1≤𝑖≤𝑛}{(𝛿²_𝑖/(1−𝛿_𝑖))min_{1≤𝑗≤𝑛}{𝛼_𝑖𝑗}}.

The paper is organized as follows. In Section 2, we give some definitions and lemmas to prove the main results of this paper. In Section 3, by using a fixed-point theorem of strict-set-contraction, we established some criteria to guarantee the existence of at least one positive periodic solution of system (2). Finally, in Section 4, we give an example to show the validity of our result.

2. Preliminaries

In order to obtain the existence of a periodic solution of system (2), we first introduce some definitions and lemmas.

Definition 1 (see [13]). A function 𝑥_𝑖∶𝑅→(0,+∞) is said to be a positive solution of (2), if the following conditions are satisfied:(a)𝑥_𝑖(𝑡) is absolutely continuous on each (𝑡_𝑘,𝑡_𝑘+1);(b)for each 𝑘∈𝑍₊, 𝑥_𝑖(𝑡⁺_𝑘) and 𝑥_𝑖(𝑡⁻_𝑘) exist and 𝑥_𝑖(𝑡⁻_𝑘)=𝑥_𝑖(𝑡_𝑘);(c)𝑥_𝑖(𝑡) satisfies the first equation of (2) for almost everywhere in 𝑅 and 𝑥_𝑖(𝑡_𝑘) satisfies the second equation of (2) at impulsive point 𝑡_𝑘,𝑘∈𝑍₊.

Definition 2 (see [21]). Let 𝑋 be a real Banach space and 𝐸 a closed, nonempty subset of 𝑋. 𝐸 is a cone provided(i)𝛼𝑥+𝛽𝑦∈𝐸 for all 𝑥,𝑦∈𝐸 and all 𝛼,𝛽≥0,(ii)𝑥,−𝑥∈𝐸 imply 𝑥=0.

Definition 3 (see [21]). Let 𝐴 be a bounded subset in 𝑋. Define 𝛼_𝑋(𝐴)=inf𝛿>0∶thereisafinitenumberofsubsets𝐴_𝑖⊂𝐴suchthat𝐴=_𝑖𝐴_𝑖anddiam𝐴_𝑖≤𝛿, (4) where diam(𝐴_𝑖) denotes the diameter of the set 𝐴_𝑖, obviously,  0≤𝛼_𝑋(𝐴)<∞. So 𝛼_𝑋(𝐴) is called the Kuratowski measure of noncompactness of  𝑋.

Definition 4 (see [21]). Let 𝑋,𝑌 be two Banach spaces and 𝐷⊂𝑋, a continuous and bounded map 𝑇∶𝐷→𝑌, is called k-set contractive if, for any bounded set 𝑆⊂𝐷, we have 𝛼_𝑌(𝑇(𝑆))≤𝑘𝛼_𝑋(𝑆); (5)𝜙is called strict-set-contractive if it is 𝑘-set-contractive for some 0≤𝑘<1.

Definition 5 (see [22]). The set 𝐹∈𝑃𝐶_𝜔 is said to be quasi-equicontinuous in [0,𝜔] if for any 𝜖>0 there exists 𝛿>0 such that if 𝑥∈𝐹,  𝑘∈𝑁⁺,  𝑡₁,𝑡₂∈(𝑡_𝑘−1,𝑡_𝑘)∩[0,𝜔],  and |𝑡₁−𝑡₂|<𝛿, then |𝑥(𝑡₁)−𝑥(𝑡₂)|<𝜖.

Lemma 6 (see [22]). The set 𝐹⊂𝑃𝐶_𝜔 is relatively compact if and only if(1)𝐹 is bounded; that is, ‖𝑥‖≤𝑀,  for each 𝑥∈𝐹, and some 𝑀>0;(2)𝐹 is quasi-equicontinuous in [0,𝜔].

Lemma 7. 𝑥_𝑖(𝑡) is an𝜔-periodic solution of (2) is equivalent to 𝑥_𝑖(𝑡) is an𝜔-periodic solution of the following equation: 𝑥_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘, (6) where 𝐺_𝑖()=𝑒𝑡,𝑠^{−∫𝑠𝑡𝑟𝑖(𝜃)𝑑𝜃}1−𝑒^{−∫𝜔0𝑟𝑖(𝜃)𝑑𝜃},𝑠∈[𝑡,𝑡+𝜔],𝑖=1,2,…,𝑛. (7)

Proof. Assume that 𝑥_𝑖(𝑡)∈𝑋,𝑖=1,2,…,𝑛, is a periodic solution of (2). Then, we have 𝑑𝑥𝑑𝑡^𝑖(𝑡)𝑒^{−∫𝑡0𝑟𝑖(𝜃)𝑑𝜃}=𝑒^{−∫𝑡0𝑟𝑖(𝜃)𝑑𝜃}𝑥_𝑖(𝑡)×−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()−𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉,𝑡≠𝑡_𝑘,𝑖=1,2,…,𝑛. (8) Integrating the previous equation with [𝑡,𝑡+𝜔], we can have 𝑥_𝑖(𝑠)𝑒^{−∫𝑠0𝑟𝑖(𝜃)𝑑𝜃𝑡𝑚1}_𝑡^+𝑛𝜔+𝑥_𝑖(𝑠)𝑒^{−∫𝑠0𝑟𝑖(𝜃)𝑑𝜃𝑡𝑚2}_𝑡^+𝑛𝜔_{𝑚1+𝑛𝜔}+⋯+𝑥_𝑖(𝑠)𝑒^{−∫𝑠0𝑟𝑖(𝜃)𝑑𝜃}_𝑡^𝑡+𝜔_{𝑚𝑞+𝑛𝜔}=_𝑡^𝑡+𝜔𝑒^{−∫𝑠0𝑟𝑖(𝜃)𝑑𝜃}𝑥_𝑖(𝑠)×−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()−𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠, (9) where 𝑡_𝑚𝑘+𝑛𝜔∈(𝑡,𝑡+𝜔), 𝑚_𝑘∈{1,2,…,𝑞}, 𝑘=1,2,…,𝑞, 𝑛∈𝑍₊. Therefore, 𝑥_𝑖(𝑡)𝑒^{−∫𝑡0𝑟𝑖(𝜃)𝑑𝜃}1−𝑒^{−∫𝑡𝑡+𝜔𝑟𝑖(𝜃)𝑑𝜃}+_{𝑡≤𝑡𝑘<𝑡+𝜔}Δ𝑥_𝑖𝑡_𝑚𝑘𝑒^{−∫𝑡𝑚𝑘0+𝑛𝜔𝑟𝑖(𝜃)𝑑𝜃}=_𝑡^𝑡+𝜔𝑒^{−∫𝑠0𝑟𝑖(𝜃)𝑑𝜃}𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠, (10) which can be transformed into 𝑥_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛. (11)
Thus, 𝑥_𝑖 is a periodic solution of (6). If 𝑥_𝑖(𝑡)∈𝐸,𝑖=1,2,…,𝑛, is a periodic solution of (6), for any 𝑡=𝑡_𝑘,  from (6) we have 𝑥^′(𝑡)=𝑑𝑑𝑡_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗()=𝐺𝑠+𝜉𝑑𝜉𝑑𝑠_𝑖(𝑡,𝑡+𝜔)𝑥_𝑖()×𝑡+𝜔_𝑛𝑗=1𝑎_𝑖𝑗(𝑡+𝜔)𝑥_𝑗()+𝑡+𝜔_𝑛𝑗=1𝑏_𝑖𝑗(𝑡+𝜔)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜔+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡+𝜔)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜔+𝜉)𝑑𝜉−𝐺_𝑖(𝑡,𝑡)𝑥_𝑖(𝑡)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)×⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉+𝑟_𝑖(𝑡)𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()−𝑡−𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉. (12) For any 𝑡=𝑡_𝑗, 𝑗∈𝑍₊, we have from (6) that 𝑥_𝑖𝑡⁺_𝑗−𝑥_𝑖𝑡_𝑗=^𝑡𝑗_𝑡^+𝜔_𝑗𝐺_𝑖𝑡⁺_𝑗,𝑠−𝐺_𝑖𝑡_𝑗,𝑠𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_−𝜎𝑗𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡+𝑗≤𝑡𝑘<𝑡𝑗+𝜔}𝐺_𝑖𝑡⁺_𝑗,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘−_{𝑡𝑗≤𝑡𝑘<𝑡𝑗+𝜔}𝐺_𝑖𝑡_𝑗,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘=−𝐼_𝑖𝑘𝑥𝑡_𝑘. (13)
Hence 𝑥_𝑖(𝑡) is a positive 𝜔-periodic solution of (2). Thus we complete the proof of Lemma 7.

Lemma 8 (see [20, 23]). Let 𝐸 be a cone of the real Banach space 𝑋 and 𝐸_𝑟,𝑅={𝑥∈𝐸∶𝑟≤‖𝑥‖≤𝑅} with 0<𝑟<𝑅. Assume that 𝐴∶𝐸_𝑟,𝑅→𝐸 is strict-set-contractive such that one of the following two conditions is satisfied:(a)𝐴𝑥≰𝑥,  for all  𝑥∈𝐸,‖𝑥‖=𝑟 and 𝐴𝑥≱𝑥,  for all  𝑥∈𝐸,‖𝑥‖=𝑅;(b)𝐴𝑥≱𝑥,  for all  𝑥∈𝐸,‖𝑥‖=𝑟 and 𝐴𝑥≰𝑥,  for all  𝑥∈𝐸,‖𝑥‖=𝑅.
Then 𝐴 has at least one fixed point in 𝐸_𝑟,𝑅.
In order to apply Lemma 7 to system (1), one sets 𝑃𝐶(𝑅)=𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛^𝑇∶𝑅⟶𝑅∣𝑥∈𝐶𝑡_𝑘,𝑡_𝑘+1,𝑡,𝑅∃𝑥⁻_𝑘=𝑥𝑡_𝑘,𝑥𝑡⁺_𝑘,𝑘∈𝑁,𝑃𝐶¹(𝑅)=𝑥=𝑥₁,𝑥₂,⋯,𝑥_𝑛^𝑇∶𝑅⟶𝑅∣𝑥∈𝐶¹𝑡_𝑘,𝑡_𝑘+1,𝑅,∃𝑥^′𝑡⁻_𝑘=𝑥^′𝑡_𝑘,𝑥𝑡⁺_𝑘,𝑘∈𝑁,𝑡∈𝑅. (14) Define 𝑋=𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛^𝑇∶𝑥∈𝑃𝐶(𝑅)∣𝑥(𝑡+𝜔)=𝑥(𝑡) (15) with the norm defined by ‖𝑥‖₀=∑^𝑛_𝑖=1|𝑥_𝑖|₀, where |𝑥_𝑖|₀=sup_{𝑡∈[0,𝜔]}{|𝑥_𝑖(𝑡)|},𝑖=1,2,…,𝑛, and 𝑌=𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛^𝑇∶𝑥∈𝑃𝐶¹(𝑅)()(𝑡)∣𝑥𝑡+𝜔=𝑥,𝑡∈𝑅 (16) with the norm defined by ‖𝑥‖₁=∑^𝑛_𝑖=1|𝑥_𝑖|₁, where |𝑥_𝑖|₁=max{|𝑥_𝑖|₀,|𝑥^′_𝑖|₀}. Then 𝑋 and 𝑌 are both Banach spaces. Define the cone 𝐸 in 𝑌 by 𝐸=𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛^𝑇∶𝑥∈𝑃𝐶¹(𝑅)∣𝑥_𝑖(𝑡)=𝛿_𝑗𝑥_𝑖(𝑡)₁,𝑡∈[0,𝜔],𝑖,𝑗=1,2…,𝑛. (17) Let the map 𝑇 be defined by (𝑇𝑥)(𝑡)=𝑇₁𝑥(𝑡),𝑇₂𝑥(𝑡),…,𝑇_𝑛𝑥(𝑡)^𝑇, (18) where 𝑥∈𝐸, 𝑡∈𝑅, 𝑇_𝑖𝑥(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗()+𝑠+𝜉𝑑𝜉𝑑𝑠_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘, (19) where 𝐺_𝑖()=𝑒𝑡,𝑠^{−∫𝑠𝑡𝑟𝑖(𝜃)𝑑𝜃}1−𝑒^{−∫𝜔0𝑟𝑖(𝜃)𝑑𝜃},𝑠∈[𝑡,𝑡+𝜔],𝑖=1,2,…,𝑛. (20) It is obvious to see that 𝐺_𝑖(𝑡+𝜔,𝑠+𝜔)=𝐺_𝑖(𝑡,𝑠),𝜕𝐺_𝑖(𝑡,𝑠)/𝜕𝑡=𝑟_𝑖(𝑡)𝐺_𝑖(𝑡,𝑠),𝐺_𝑖(𝑡,𝑡+𝜔)−𝐺_𝑖(𝑡,𝑡)=−1, and 𝛿_𝑖1−𝛿_𝑖≤𝐺_𝑖()≤1𝑡,𝑠1−𝛿_𝑖,𝑠∈[𝑡,𝑡+𝜔],𝑖=1,2,…,𝑛. (21)

In what follows, we will give some lemmas concerning 𝐸 and 𝐴 defined by (17) and (18), respectively.

Lemma 9. Assume that (𝐴₁)--(𝐴₃) hold.(i)If max{𝑟^𝑀_𝑖,𝑖=1,2,…,𝑛}≤1, then 𝑇∶𝐸→𝐸 is well defined.(ii)If (𝐴₄) holds and max{𝑟^𝑀_𝑖,𝑖=1,2,…,𝑛}>1, then 𝑇∶𝐸→𝐸 is well defined.

Proof. For any 𝑥∈𝐸, it is clear that 𝑇𝑥∈𝑃𝐶¹(𝑅). From (18), for 𝑡∈[0,𝜔], we have 𝑇_𝑖𝑥()=𝑡+𝜔_{𝑡+2𝜔𝑡+𝜔}𝐺_𝑖(𝑡+𝜔,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡+𝜔≤𝑡𝑘<𝑡+2𝜔}𝐺_𝑖𝑡+𝜔,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡+𝜔,𝑢+𝜔)𝑥_𝑖()×𝑢+𝜔_𝑛𝑗=1𝑎_𝑖𝑗(𝑢+𝜔)𝑥_𝑗(𝑢+𝜔)+^𝑛_𝑗=1𝑏_𝑖𝑗()×𝑢+𝜔_{0−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑢+𝜔+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑢+𝜔)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑢+𝜔+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑢)𝑥_𝑖(𝑢)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑢)𝑥_𝑗(𝑢)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑢)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑢+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑢)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗()+𝑢+𝜉𝑑𝜉𝑑𝑠_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘=𝑇_𝑖𝑥(𝑡),𝑖=1,2,…,𝑛. (22)
That is, (𝑇_𝑖𝑥)(𝑡+𝜔)=(𝑇_𝑖𝑥)(𝑡),𝑡∈[0,𝜔]. So 𝑇𝑥∈𝑌. In view of (𝐴₂), for 𝑥∈𝐸,𝑡∈[0,𝜔], we have ^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗()≥𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑎_𝑖𝑗(𝑡)𝛿_𝑗𝑥_𝑗1−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥_𝑗1−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑗1=^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝛿_𝑗−𝑏_𝑖𝑗(𝑡)−𝑐_𝑖𝑗(𝑡)𝑥^′_𝑗1≥0,𝑖=1,2,…,𝑛. (23)
Therefore, for 𝑥∈𝐸,𝑡∈[0,𝜔], we find 𝑇_𝑖𝑥₀≤11−𝛿_𝑖^𝜔₀𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)×⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑇_𝑖𝑥𝑡)≥𝛿(_𝑖1−𝛿_𝑖𝑡^𝑡+𝜔𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘=𝛿_𝑖1−𝛿_𝑖×^𝜔₀𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘≥𝛿_𝑖𝑇_𝑖𝑥₀,𝑖=1,2,…,𝑛. (24)(25) Now, we show that (𝑇_𝑖𝑥)(𝑡)≥𝛿_𝑖|𝑇_𝑖𝑥|₁, 𝑖=1,2,…,𝑛, 𝑡∈[0,𝜔]. From (18), we obtain 𝑇_𝑖𝑥^′(𝑡)=𝐺_𝑖(𝑡,𝑡+𝜔)𝑥_𝑖()×𝑡+𝜔_𝑛𝑗=1𝑎_𝑖𝑗(𝑡+𝜔)𝑥_𝑗()+𝑡+𝜔_𝑛𝑗=1𝑏_𝑖𝑗(𝑡+𝜔)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜔+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡+𝜔)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜔+𝜉)𝑑𝜉−𝐺_𝑖(𝑡,𝑡)𝑥_𝑖(𝑡)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)×⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉+𝑟_𝑖(𝑡)𝑇_𝑖𝑥(𝑡)=𝑟_𝑖(𝑡)𝑇_𝑖𝑥(𝑡)−𝑥_𝑖(𝑡)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉,𝑖=1,2,…,𝑛. (26) It follows from (23) and (25) that if (𝑇_𝑖𝑥)^′(𝑡)≥0,𝑖=1,2,…,𝑛, then 𝑇_𝑖𝑥^′(𝑡)≤𝑟_𝑖(𝑡)𝑇_𝑖𝑥(𝑡)≤𝑟^𝑀_𝑖𝑇_𝑖𝑥(𝑡)≤𝑇_𝑖𝑥(𝑡),𝑖=1,2,…,𝑛. (27) On the other hand, from (25) and (𝐴₃), if (𝑇_𝑖𝑥)^′(𝑡)<0,𝑖=1,2,…,𝑛, then −𝑇_𝑖𝑥^′(𝑡)=−𝑟_𝑖(𝑡)𝑇_𝑖𝑥(𝑡)+𝑥_𝑖(𝑡)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗()≤𝑥𝑡+𝜉𝑑𝜉_{𝑖1𝑛𝑗=1}𝑎_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)+𝑐_𝑖𝑗(𝑡)𝑥_𝑖1−𝑟^𝐿_𝑖𝑇_𝑖𝑥(𝑡)≤1+𝑟^𝐿_𝑖𝛿²_𝑖1−𝛿_𝑖𝑥_𝑖1×^𝑛_𝑗=1^𝜔₀𝛿_𝑗𝑎_𝑖𝑗(𝑠)−𝑏_𝑖𝑗(𝑠)−𝑐_𝑖𝑗(𝑠)𝑥_𝑖1𝑑𝑠−𝑟^𝐿_𝑖𝑇_𝑖𝑥(𝑡)=1+𝑟^𝐿_𝑖𝑡^𝑡+𝜔𝛿_𝑖1−𝛿_𝑖𝛿_𝑖𝑥_𝑖1×^𝑛_𝑗=1𝛿_𝑗𝑎_𝑖𝑗(𝑠)−𝑏_𝑖𝑗(𝑠)−𝑐_𝑖𝑗(𝑠)𝑥_𝑖1𝑑𝑠−𝑟^𝐿_𝑖𝑇_𝑖𝑥𝑡)≤(1+𝑟^𝐿_𝑖𝑡^𝑡+𝜔_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)×⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗()+𝑠+𝜉𝑑𝜉𝑑𝑠_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘−𝑟^𝐿_𝑖𝑇_𝑖𝑥(𝑡)=1+𝑟^𝐿_𝑖𝑇_𝑖𝑥(𝑡)−𝑟^𝐿_𝑖𝑇_𝑖𝑥(𝑡)=𝑇_𝑖𝑥(𝑡),𝑖=1,2,…,𝑛. (28)
It follows from (27) and (28) that |(𝑇_𝑖𝑥)^′|₀≤|𝑇_𝑖𝑥|₀,𝑖=1,2,…,𝑛. So |𝑇_𝑖𝑥|₁=|𝑇_𝑖𝑥|₀,𝑖=1,2,…,𝑛. By (25), we have (𝑇_𝑖𝑥)(𝑡)≥𝛿_𝑖|𝑇_𝑖𝑥|₁,𝑖=1,2,…,𝑛. Hence, 𝑇𝑥∈𝐸. This completes the proof of (i). In view of the proof of (ii), we only need to prove that (𝑇_𝑖𝑥)^′(𝑡)≥0,𝑖=1,2,…,𝑛, implies 𝑇_𝑖𝑥^′(𝑡)≤𝑇_𝑖𝑥(𝑡),𝑖=1,2,…,𝑛. (29) From (23), (26),  (𝐴₂), and (𝐴₄), we have 𝑇_𝑖𝑥^′(𝑡)=𝑟_𝑖(𝑡)𝑇_𝑖𝑥(𝑡)−𝑥_𝑖(𝑡)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()−𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉≤𝑟_𝑖(𝑡)𝑇_𝑖𝑥(𝑡)−𝛿_𝑖𝑥_𝑖1×^𝑛_𝑗=1𝛿_𝑗𝑎_𝑖𝑗(𝑡)−𝑏_𝑖𝑗(𝑡)−𝑐_𝑖𝑗(𝑡)𝑥_𝑗1≤𝑟^𝑀_𝑖𝑇_𝑖𝑥(𝑡)−𝛿_𝑖𝑥_𝑖1𝑟^𝑀_𝑖−1𝛿_𝑖1−𝛿_𝑖×^𝑛_𝑗=1^𝜔₀𝑎_𝑖𝑗(𝑠)+𝑏_𝑖𝑗(𝑠)+𝑐_𝑖𝑗(𝑠)𝑥_𝑗1𝑑𝑠=𝑟^𝑀_𝑖𝑇_𝑖𝑥(𝑡)−𝑟^𝑀_𝑖−1_𝑡^𝑡+𝜔11−𝛿_𝑖𝑥_𝑖1×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)+𝑏_𝑖𝑗(𝑠)+𝑐_𝑖𝑗(𝑠)𝑥_𝑗1𝑑𝑠≤𝑟^𝑀_𝑖𝑇_𝑖𝑥(𝑡)−𝑟^𝑀_𝑖×−1_{𝑡𝑡+𝜔}𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)×⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘=𝑟^𝑀_𝑖𝑇_𝑖𝑥(𝑡)−𝑟^𝑀_𝑖−1𝑇_𝑖𝑥(𝑡)=𝑇_𝑖𝑥(𝑡),𝑖=1,2,…,𝑛. (30) The proof of (ii) is complete. Thus we complete the proof of Lemma 9.

Lemma 10. Assume that (𝐴₁)--(𝐴₃) hold and 𝑅max_{1≤𝑗≤𝑛}{∑^𝑛_𝑗=1𝑏^𝑀_𝑖𝑗}<1.(i)If max{𝑟^𝑀_𝑖,𝑖=1,2,…,𝑛}≤1, then 𝑇∶𝐸⋂Ω_𝑅→𝐸 is strict-set-contractive.(ii)If (𝐴₄) holds and min{𝑟^𝑀_𝑖,𝑖=1,2,…,𝑛}>1, then 𝑇∶𝐸⋂Ω_𝑅→𝐸 is strict-set-contractive, where Ω_𝑅={𝑥∈𝑌∶|𝑥|₁<𝑅}.

Proof. We only need to prove (i), since the proof of (ii) is similar. It is easy to see that 𝑇 is continuous and bounded. Now we prove that a 𝛼_𝑌(𝑇(𝑆))≤𝑅max_{1≤𝑗≤𝑛}{∑^𝑛_𝑗=1𝑏^𝑀_𝑖𝑗}𝛼_𝑌(𝑆) for any bounded set 𝑆∈Ω_𝑅. Let 𝜂=𝛼_𝑌(𝑆). Then, for any positive number 𝜖<𝑅max_{1≤𝑗≤𝑛}{∑^𝑛_𝑗=1𝑏^𝑀_𝑖𝑗}𝜂, there is a finite family of subsets {𝑆_𝑖} satisfying 𝑆=⋃_𝑖𝑆_𝑖 with diam(𝑆_𝑖)≤𝜂+𝜖. Therefore 𝑥−𝑦₁≤𝜂+𝜖,forany𝑥,𝑦∈𝑆_𝑖. (31) As 𝑆 and 𝑆_𝑖 are precompact in 𝑋, it follows that there is a finite family of subsets {𝑆_𝑖𝑗} of 𝑆_𝑖 such that 𝑆_𝑖=⋃_𝑗𝑆_𝑖𝑗 and 𝑥−𝑦₀≤𝜖,forany𝑥,𝑦∈𝑆_𝑖𝑗. (32) In addition, for any 𝑥∈𝑆 and 𝑡∈[0,𝜔], we have 𝑇_𝑖𝑥(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘≤𝑅²1−𝛿_𝑖^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑠)+1𝑑𝑠1−𝛿_{𝑖𝑡≤𝑡𝑘<𝑡+𝜔}𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘∶=𝐵_𝑖𝑇,𝑖=1,2,…,𝑛,_𝑖𝑥^′(𝑡=𝑟_𝑖)(𝑡)𝑇_𝑖𝑥(𝑡)−𝑥_𝑖(𝑡)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉≤𝑟^𝑀_𝑖𝐵_𝑖+𝑅^2𝑛_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗+𝑐^𝑀_𝑖𝑗,𝑖=1,2,…,𝑛. (33)(34) Hence, ‖(𝑇𝑥)‖₀≤^𝑛_𝑖=1𝐵_𝑖,(𝑇𝑥)^′₀≤^𝑛_𝑖=1𝐵_𝑖𝑟^𝑀_𝑖𝐵_𝑖+𝑅^2𝑛_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗. (35) Applying the Arzela-Ascoli theorem, we know that 𝑇(𝑆) is precompact in 𝑋. Then, there is a finite family of subsets {𝑆_𝑖𝑗𝑘} of 𝑆_𝑖𝑗 such that 𝑆_𝑖𝑗=⋃_𝑘𝑆_𝑖𝑗𝑘 and 𝑇𝑥−𝑇𝑦₀≤𝜖,forany𝑥,𝑦∈𝑆_𝑖𝑗𝑘. (36) From (23), (26), (31)–(34), and (𝐴₂), for any 𝑥,𝑦∈𝑆_𝑖𝑗𝑘, we have 𝑇_𝑖𝑥^′−𝑇_𝑖𝑦^′₀=max_{[]𝑡∈0,𝜔}𝑟_𝑖(𝑡)𝑇_𝑖𝑥(𝑡)−𝑟_𝑖(𝑡)𝑇_𝑖𝑦(𝑡)−𝑥_𝑖(𝑡)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑡+𝜉)𝑑𝜉+𝑦_𝑖(𝑡)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑦_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑦^′_𝑗(𝑡+𝜉)𝑑𝜉≤max_{𝑡∈[0,𝜔]}𝑟_𝑖(𝑡)𝑇_𝑖𝑥(𝑡)−𝑇_𝑖𝑦(𝑡)+max_{[]𝑡∈0,𝜔}𝑥_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗()−𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑎_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)×⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑦_𝑗(𝑡+𝜉)𝑑𝜉+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)×⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑦^′_𝑗(𝑡+𝜉)𝑑𝜉+max_{[]𝑡∈0,𝜔}𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡×_𝑛𝑗=1)𝑎_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)×⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑦_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑦^′_𝑗(𝑡+𝜉)𝑑𝜉≤𝑟^𝑀_𝑖𝑇_𝑖𝑥−𝑇_𝑖𝑦₀+𝑅max_{[]𝑡∈0,𝜔}^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑦_𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)×⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)×𝑥_𝑖(𝑡+𝜉)−𝑦_𝑖(𝑡+𝜉)𝑑𝜉+𝑐_𝑖𝑗(𝑡)×⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)×𝑥^′_𝑗(𝑡+𝜉)−𝑦^′(𝑡−𝜉)𝑑𝜉+max_{[]𝑡∈0,𝜔}𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡×_𝑛𝑗=1)𝑎_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑦_𝑗()+𝑡+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)×⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑦^′_𝑗(𝑡+𝜉)𝑑𝜉≤𝑟^𝑀_𝑖𝜖+𝑅𝜖^𝑛_𝑗=1𝑎^𝑀_𝑖𝑗+𝑅𝜖^𝑛_𝑗=1𝑏^𝑀_𝑖𝑗+𝑥_𝑖0×𝜂+𝜖_𝑛𝑗=1𝑐^𝑀_𝑖𝑗+𝑅𝜖^𝑛_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗+𝑐^𝑀_𝑖𝑗=𝑥_𝑖0^𝑛_𝑗=1𝑐^𝑀_𝑖𝑗𝜂+𝐵_𝑖𝜖, (37) where 𝐵_𝑖=𝑟^𝑀_𝑖+2𝑅^𝑛_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗+𝑐^𝑀_𝑖𝑗,𝑖=1,2,…,𝑛. (38) From (36) and (37) we obtain 𝑇𝑥−𝑇𝑦₁≤^𝑛_𝑖=1𝑥_𝑖0^𝑛_𝑗=1𝑐^𝑀_𝑖𝑗𝜂+𝜖^𝑛_𝑖=1𝐵_𝑖≤𝑅max_{1≤𝑖≤𝑛}^𝑛_𝑗=1𝑐^𝑀_𝑖𝑗𝜂+𝜖^𝑛_𝑖=1𝐵_𝑖,forany𝑥,𝑦∈𝑆_𝑖𝑗𝑘. (39) As 𝜖 is arbitrarily small, it follows that 𝛼_𝑌(𝑇(𝑆))≤𝑅max_{1≤𝑗≤𝑛}^𝑛_𝑗=1𝑐^𝑀_𝑖𝑗𝛼_𝑌(𝑆). (40) Therefore, 𝑇 is strict-set-contractive. The proof of Lemma 10 is complete.

3. Main Results

Our main result of this paper is as follows.

Theorem 11. Assume that (𝐴₁)--(𝐴₃), (𝐴₅) hold.(i)If max{𝑟^𝑀_𝑖,𝑖=1,2,…,𝑛}≤1, then system (2) has at least one positive 𝜔-periodic solution.(ii)If (𝐴₄) holds and min{𝑟^𝑀_𝑖,𝑖=1,2,…,𝑛}>1, then system (2) has at least one positive 𝜔-periodic solution.

Proof. We only need to prove (i), since the proof of (ii) is similar. Let 𝑅=min_{[]𝑖∈1,𝑛}𝛿²_𝑖1−𝛿_𝑖min_{[]𝑗∈1,𝑛}𝛼_𝑖𝑗⁻¹,0<𝑟<min_{[]𝑖∈1,𝑛}𝛿_𝑖1−𝛿_𝑖−𝛾_𝑖max_{𝑗∈[1,𝑛]}𝛽_𝑖𝑗. (41) Then it is easy to see that 0<𝑟<𝑅. From Lemmas 9 and 10, we know that 𝑇 is strict-set-contractive on 𝐸_𝑟,𝑅. In view of (26), we see that if there exists 𝑥^∗∈𝐸 such that 𝑇𝑥^∗=𝑥^∗, then 𝑥^∗ is one positive 𝜔-periodic solution of system (2). Now, we will prove that condition (ii) of Lemma 7 holds. First, we prove that 𝑇𝑥≱𝑥, for all 𝑥∈𝐸,‖𝑥‖₁<𝑟. Otherwise, there exists 𝑥∈𝐸,‖𝑥‖₁<𝑟, such that 𝑇𝑥≥𝑥. So ‖𝑥‖₀>0 and 𝑇𝑥−𝑥≥0, which implies that 𝑇_𝑖𝑥(𝑡)−𝑥_𝑖(𝑡)≥𝛿_𝑖𝑇_𝑖𝑥−𝑥_𝑖1[]≥0,forany𝑡∈0,𝜔,𝑖=1,2,…,𝑛. (42) Moreover, for 𝑡∈[0,𝜔], we have 𝑇_𝑖𝑥(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘≤11−𝛿_𝑖𝑥_𝑖0^𝑛_𝑗=1^𝜔₀𝑎_𝑖𝑗(𝑠)+𝑏_𝑖𝑗(𝑠)+𝑐_𝑖𝑗(𝑠)𝑥_𝑗1𝑑𝑠+𝛾_𝑖≤𝛽_𝑖𝑗𝑟+𝛾_𝑖1−𝛿_𝑖𝑥_𝑖0≤𝛿_𝑖𝑥_𝑖0,𝑖=1,2,…,𝑛. (43) In view of (42) and (43), we obtain ‖𝑥‖₀≤‖𝑇𝑥‖₀≤max_{1≤𝑖≤𝑛}𝛿_𝑖‖𝑥‖₀<‖𝑥‖₀, (44) which is a contradiction. Finally, we prove that 𝑇𝑥≰𝑥,  for all 𝑥∈𝐸,  and‖𝑥‖₁=𝑅 also hold. For this case, suppose, for the sake of contradiction, that there exist 𝑥∈𝐸and‖𝑥‖₁=𝑅 such that 𝑇𝑥≤𝑥. Furthermore, for any 𝑡∈[0,𝜔], we have 𝑥_𝑖(𝑡)−𝑇_𝑖𝑥(𝑡)≥𝛿_𝑖𝑥_𝑖−𝑇_𝑖𝑥₁[]≥0,forany𝑡∈0,𝜔,𝑖=1,2,…,𝑛. (45) In addition, for any 𝑡∈[0,𝜔], we find 𝑇_𝑖𝑥(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑥_𝑖(𝑠)×^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑥_𝑗()+𝑠+𝜉𝑑𝜉_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑥^′_𝑗(𝑠+𝜉)𝑑𝜉𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}𝐺_𝑖𝑡,𝑡_𝑘𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘>𝛿²_𝑖1−𝛿_𝑖𝑥_𝑖1^𝑛_𝑗=1^𝜔₀𝛿_𝑗𝑎_𝑖𝑗(𝑠)−𝑏_𝑖𝑗(𝑠)−𝑐_𝑖𝑗(𝑠)𝑑𝑠𝑥_𝑗1=𝛿²_𝑖1−𝛿_𝑖𝑥_𝑖1^𝑛_𝑗=1𝛼_𝑖𝑗𝑥_𝑗1,𝑖=1,2,…,𝑛. (46) Thus, we have ‖𝑇𝑥‖₀=^𝑛_𝑖=1𝑇_𝑖𝑥₀>^𝑛_𝑖=1𝛿²_𝑖1−𝛿_𝑖𝑥_𝑖1^𝑛_𝑗=1𝛼_𝑖𝑗𝑥_𝑗1≥min_{1≤𝑖≤𝑛}𝛿²_𝑖min_{1≤𝑗≤𝑛}𝛼_𝑖𝑗1−𝛿_𝑖^𝑛_𝑖=1𝑥_𝑖1^𝑛_𝑗=1𝑥_𝑗1≥min_{1≤𝑖≤𝑛}𝛿²_𝑖min_{1≤𝑗≤𝑛}𝛼_𝑖𝑗1−𝛿_𝑖𝑅²=𝑅. (47) From (45) and (47), we obtain ‖𝑥‖₁≥‖𝑇𝑥‖₁≥‖𝑇𝑥‖₀>𝑅, (48) which is a contradiction. Therefore, condition (ii) of Lemma 7 holds. By Lemma 7, we see that 𝑇 has at least one nonzero fixed point in 𝐸. Thus, the system (6) has at least one positive 𝜔-periodic solution. Therefore, it follows from Lemma 6 that system (2) has a positive 𝜔-periodic solution. The proof of Theorem 11 is complete.

Remark 12. If 𝐼_𝑖𝑘(𝑥(𝑡_𝑘))=0, we can easily derive the corresponding results in [10]. So we extend the corresponding results in [10].

4. Example

In this section, we give an example to show the effectiveness of our result.

Example 1. Consider the following nonimpulsive system: 𝑥^′₁(𝑡)=𝑥₁(𝑡)1−sin𝑡−()𝑥2410+sin𝑡₁(𝑡)−(9−cos𝑡)𝑥₂(𝑡)−()×3−2cos𝑡_0−𝜏11𝑓₁₁(𝜉)𝑥₁(𝑡+𝜉)𝑑𝜉−5+sin𝑡×15_0−𝜏12𝑓₁₂(𝜉)𝑥₂()()×𝑡+𝜉𝑑𝜉−1+sin𝑡_0−𝜎11𝑔₁₁(𝜉)𝑥^′₁(𝑡+𝜉)𝑑𝜉−4−3cos𝑡6×⁰_−𝜎12𝑔₁₂(𝜉)𝑥^′₂(),𝑥𝑡+𝜉𝑑𝜉_′2(𝑡)=𝑥₂(𝑡)1+cos𝑡−()𝑥8𝜋12+3cos𝑡₁(𝑡)−(7−sin𝑡)𝑥₂(𝑡)−()×2+3sin𝑡_0−𝜏21𝑓₂₁(𝜉)𝑥₂()()×𝑡+𝜉𝑑𝜉−1−cos𝑡_0−𝜏22𝑓₂₂(𝜉)𝑥₂(𝑡+𝜉)𝑑𝜉−4+sin𝑡×12_0−𝜎21𝑔₂₁(𝜉)𝑥^′₂(𝑡+𝜉)𝑑𝜉−5−cos𝑡×10_0−𝜎22𝑔₂₂(𝜉)𝑥^′₂(𝑡+𝜉)𝑑𝜉, (49) where 𝜏_𝑖𝑗,𝜎_𝑖𝑗∈(0,+∞)(𝑖,𝑗=1,2), and 𝑓_𝑖𝑗,𝑔_𝑖𝑗∈𝐶(𝑅,𝑅⁺) satisfying ∫⁰_{−𝜏𝑖𝑗}𝑓_𝑖𝑗(𝜉)𝑑𝜉=1, ∫⁰_{−𝜎𝑖𝑗}𝑔_𝑖𝑗(𝜉)𝑑𝜉,𝑖,𝑗=1,2. Obviously 𝑟₁(𝑡)=1−sin𝑡24,𝑟₂=1+cos𝑡,𝑎8𝜋₁₁(𝑡)𝑎=10+sin𝑡,₁₂(𝑡)=9−cos𝑡,𝑎₂₁(𝑡)𝑎=12+3cos𝑡,₂₂(𝑡)=7−sin𝑡,𝑏₁₁(𝑡)𝑏=3−2cos𝑡,₁₂(𝑡)=5+sin𝑡,𝑏15₂₁(𝑡)=2+3sin𝑡,𝑏₂₂(𝑡)𝑐=1−cos𝑡,₁₁(𝑡)=1+sin𝑡,𝑐₁₂(𝑡)=4−3cos𝑡6,𝑐₂₁(𝑡)=4+sin𝑡,𝑐12₂₂(𝑡)=5−cos𝑡10,𝑟^𝐿₁=𝑟^𝐿₂=0. (50) Furthermore, we obtain 𝛿₁=𝑒^−(𝜋/12)<1,𝛿₂=𝑒^−(1/4)𝛼<1,₁₁=20𝜋𝛿₁𝛼−8𝜋≈23.8007,₁₂=18𝜋𝛿₂𝛼−2𝜋≈37.2423,₂₁=24𝜋𝛿₁−14𝜋3𝛼≈44.0594,₂₂=14𝜋𝛿₂−3𝜋≈24.4284,min_1≤𝑗≤2𝛼_1𝑗=𝛼₁₁=20𝜋𝛿₁−8𝜋≈23.8007,min_1≤𝑗≤2𝛼_2𝑗=𝛼₂₂=14𝜋𝛿₂𝛿−3𝜋≈24.4284,₁𝑎₁₁(𝑡)−𝑏₁₁(𝑡)−𝑐₁₁(𝑡)𝛿>0.009>0,₂𝑎₁₂(𝑡)−𝑏₁₂(𝑡)−𝑐₁₂(𝑡)𝛿>1>0,₁𝑎₂₁(𝑡)−𝑏₂₁(𝑡)−𝑐₂₁(𝑡)𝛿>4>0,₂𝑎₂₂(𝑡)−𝑏₂₂(𝑡)−𝑐₂₂(𝑡)>2>0,1+𝑟^𝐿₁𝛿²₁𝛼₁₁1−𝛿₁=𝑒^−(𝜋/6)20𝜋𝛿₁−8𝜋1−𝑒^−(𝜋/12)≈65.2614>18≥max_{0≤𝑡≤2𝜋}𝑎₁₁(𝑡)+𝑏₁₁(𝑡)+𝑐₁₁(𝑡),1+𝑟^𝐿₁𝛿²₁𝛼₁₂1−𝛿₁=𝑒^−(𝜋/6)18𝜋𝛿₂−2𝜋1−𝑒^−(𝜋/12)≈102.1184>34730≥max_{0≤𝑡≤2𝜋}𝑎₁₂(𝑡)+𝑏₁₂(𝑡)+𝑐₁₂(𝑡),1+𝑟^𝐿₂𝛿²₂𝛼₂₁1−𝛿₂=𝑒^−(1/2)24𝜋𝛿₁−(14𝜋/3)1−𝑒^−(1/4)≈113.3411>24512>max_{0≤𝑡≤2𝜋}𝑎₂₁(𝑡)+𝑏₂₁(𝑡)+𝑐₂₁(𝑡),1+𝑟^𝐿₂𝛿²₂𝛼₂₂1−𝛿₂=𝑒^−(1/2)14𝜋𝛿₂−3𝜋1−𝑒^−(1/4)≈62.8411>535>max_{0≤𝑡≤2𝜋}𝑎₂₂(𝑡)+𝑏₂₂(𝑡)+𝑐₂₂(𝑡,𝛿₂₁)min_1≤𝑗≤2𝛼_1𝑗1−𝛿₁=𝑒^−(𝜋/6)20𝜋𝛿₁−6𝜋1−𝑒^−(𝜋/12)𝛿≈65.2614,₂₂min_1≤𝑗≤2𝛼_2𝑗1−𝛿₂=𝑒^−(1/2)14𝜋𝛿₂−3𝜋1−𝑒^−(1/4)≈62.8411,min_1≤𝑖≤2𝛿²_𝑖min_1≤𝑗≤2𝛼_𝑖𝑗1−𝛿_𝑖=𝛿²₂min_1≤𝑗≤2𝛼_2𝑗1−𝛿₁𝑐≈62.8411,_𝑀11+𝑐^𝑀₁₂=196,𝑐^𝑀₂₁+𝑐^𝑀₂₂=61,60max_1≤𝑖≤2²_𝑗=1𝑑^𝑀_𝑖𝑗=196. (51) Therefore, 196=max_1≤𝑖≤2²_𝑖=1𝑑^𝑀_𝑖𝑗<min_1≤𝑖≤2𝛿²_𝑖min_1≤𝑗≤2𝛼_𝑖𝑗1−𝛿_𝑖≈62.8411. (52) Hence, (𝐴₁)--(𝐴₃), (𝐴₅) hold and 𝑎^𝑀_𝑖≤1,𝑖=1,2. According to Theorem 11, system (49) has at least one positive 2𝜋-periodic solution.

Acknowledgments

The authors are thankful to the referees and editor for the improvement of the paper. This work was supported by the Construct Program of the Key Discipline in Hunan Province. Research was supported by the National Natural Science Foundation of China (10971229, 11161015), the China Postdoctoral Science Foundation (2012M512162), and Hunan Provincial Natural Science-Hengyang United Foundation of China (11JJ9002).