Existence and Global Attractivity of Positive Periodic Solutions for The Neutral Multidelay Logarithmic Population Model with Impulse

Abstract

Suffiicient and realistic conditions are established in this paper for the existence and global attractivity of a positive periodic solution to the neutral multidelay logarithmic population model with impulse by using the theory of abstract continuous theorem of k-set contractive operator and some inequality techniques. The results improve and generalize the known ones in Li 1999, Lu and Ge 2004, Y. Luo and Z. G. Luo 2010, and Wang et al. 2009. As an application, we also give an example to illustrate the feasibility of our main results.

1. Introduction

In this paper, we investigate the existence and uniqueness of the positive periodic solution of the following neutral population system with multiple delays and impulse: 𝑑𝑁(𝑡)(𝑡𝑟(𝑡)(𝑡)(𝑡)−𝑑𝑡=𝑁)−𝑎ln𝑁_𝑛𝑖=1𝑏_𝑖(𝑡)ln𝑁𝑡−𝜎_𝑖(𝑡)−^𝑚_𝑗=1𝑐_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()(𝑠)−𝑡−𝑠ln𝑁𝑑𝑠_𝑝𝑙=1𝑑_𝑙(𝑡)𝑑ln𝑁𝑡−𝜏_𝑙(𝑡)𝑑𝑡,𝑡≠𝑡_𝑘,𝑁𝑡⁺_𝑘=𝑒^(1+𝜃𝑘)𝑁𝑡_𝑘,𝑘=1,2,…, (1) with the following initial conditions: 𝑁(𝜉)=𝜑(𝜉),𝑁^′(𝜉)=𝜑^′(𝜉)[](0)[),𝜉∈−𝜏,0,𝜑>0,𝜑∈𝐶−𝜏,0,𝑅⁺𝐶¹([−𝜏,0)),𝑅⁺, (2) where 𝑟(𝑡),𝑎(𝑡),𝑏_𝑖(𝑡),𝑐_𝑗(𝑡),𝑑_𝑙(𝑡),𝜎_𝑖(𝑡),𝜏_𝑙(𝑡) are positive continuous 𝜔-periodic functions with 𝜎_𝑖(𝑡)≥0,𝜏_𝑙(𝑡)≥0,𝑡∈[0,𝜔],𝜏=max{𝜎_𝑖(𝑡),𝜏_𝑙(𝑡)},𝜎^′_𝑖(𝑡)<1,𝜏^′_𝑙(𝑡)<1,forall𝑖={1,2,…,𝑛},forall𝑙={1,2,…,𝑝}. Furthermore,𝑑_𝑙(𝑡)∈𝐶¹(𝑅,𝑅),𝜏_𝑙(𝑡)∈𝐶²(𝑅,𝑅),∫^∞₀𝐾_𝑗(𝑠)𝑑𝑠=1,∫₀^+∞𝑠𝐾_𝑗(𝑠)𝑑𝑠<+∞,forall𝑗={1,2,…,𝑚}, for all 𝑙={1,2,…,𝑝}. For the ecological justification of (1) and similar types refer to [1–7]. In recent years, Gopalsamy [1] and Kirlinger [2] had proposed the following single species logarithmic model: 𝑁^′(𝑡)=𝑁(𝑡)[𝑎−𝑏ln𝑁(𝑡)−𝑐ln𝑁(𝑡−𝜏)]. (3)

In [3], Li considered the following nonautonomous single species logarithmic model: 𝑁^′(𝑡)=𝑁(𝑡)[𝑎(𝑡)−𝑏(𝑡)ln𝑁(𝑡)−𝑐(𝑡)ln𝑁(𝑡−𝜏(𝑡))]. (4) He used the continuation theorem of the coincidence degree theory to establish sufficient conditions for the existence and attractivity of positive periodic solutions of the system (4).

For more works on the periodic solution of the neutral type logistic model or the Lotka-Volterra model, see [8–12] for details. Only little scholars considered the neutral logarithmic model (see [4–7]). Li [4] had studied the following single species neutral logarithmic model: 𝑁^′(𝑡)(𝑡𝑟(𝑡)(𝑡)()(𝑡=𝑁)−𝑎ln𝑁𝑡−𝜎−𝑏)ln𝑁𝑡−𝜂^′. (5)Lu and Ge [5] and Y. Luo and Z. G. Luo [6] employed an abstract continuous theorem of 𝑘-set contractive operator to investigate the following equation:𝑁^′(𝑡)=𝑁(𝑡)𝑟(𝑡)−^𝑛_𝑗=1𝑎_𝑗(𝑡)ln𝑁𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑏_𝑖(𝑡)ln𝑁𝑡−𝜏_𝑖(𝑡)^′. (6) They established some criteria to guarantee the existence of positive periodic solutions of the system (6), respectively.

In [7], Wang et al. had investigated the existence and uniqueness of the positive periodic solution of the following neutral multispecies logarithmic population model: 𝑁^′(𝑡)(𝑡)𝑟(𝑡)(𝑡)(𝑡)−=𝑁−𝑎ln𝑁_𝑛𝑗=1𝑏_𝑗(𝑡)ln𝑁𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑗(𝑡)^𝑡_−∞𝑘_𝑗()𝑠−𝑡−𝑠ln𝑁(())𝑑𝑠_𝑛𝑗=1𝑑_𝑗(𝑡)ln𝑁𝑡−𝜂_𝑗(𝑡)^′. (7) By using an abstract continuous theorem of a 𝑘-set contractive operator, the criteria are established for the existence and global attractivity of positive periodic solutions for model (7).

On the other hand, there are some other perturbations in the real world such as fires and floods, which 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 [13–19], 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 [20–22].

However, to this day, no scholars had done works on the existence, uniqueness, and global stability of the positive periodic solution of (1). One could easily see that the systems (5)–(7) are all special cases of the system (1). Therefore, we propose and study the system (1) in this paper.

Throughout this paper, we make the following notations.

Let 𝜔>0 be a constant,  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}.

Then, those spaces are both Banach spaces. We also denote that 1ℎ=𝜔^𝜔₀ℎ(𝑡)𝑑𝑡,foranyℎ∈𝑃𝐶_𝜔. (8)

For the sake of generality and convenience, we always make the following fundamental assumptions:(𝐻₁)𝑟(𝑡),𝑎(𝑡),𝑏_𝑖(𝑡),𝑐_𝑗(𝑡),𝑑_𝑙(𝑡),𝜎_𝑖(𝑡),𝜏_𝑙(𝑡)(𝑖=1,2,…,𝑛;𝑗=1,2,…,𝑚;𝑙=1,2,…,𝑝) are all positive periodic continuous functions with period 𝜔>0;(𝐻₂)0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯ are fixed impulsive points with lim_𝑘→∞𝑡_𝑘=+∞;(𝐻₃){𝜃_𝑘} is a real sequence such that 𝜃_𝑘+1>0, and ∏_{0<𝑡𝑘<𝑡}(1+𝜃_𝑘) is an 𝜔-periodic function.

In the following section, some definitions and some useful lemmas are listed. In the third section, by using an abstract continuous theorem of 𝑘-set contractive operator and some inequality techniques, we acquired some sufficient conditions which ensure the existence and uniqueness of the positive periodic solution of the systems (1) and (2). Finally, we give an example to show our results.

2. Preliminaries

In order to obtain the existence and uniqueness of a periodic solution for the systems (1) and (2), we first give some definitions and lemmas.

Definition 1 (see [21]). A function 𝑁∶𝑅→(0,+∞) is said to be a positive solution of (1) and (2), if the following conditions are satisfied:(a)𝑁(𝑡) is absolutely continuous on each (𝑡_𝑘,𝑡_𝑘+1);(b)for each 𝑘∈𝑍₊,𝑁(𝑡⁺_𝑘) and 𝑁(𝑡⁻_𝑘) exist and 𝑁(𝑡⁻_𝑘)=𝑁(𝑡_𝑘);(c)𝑁(𝑡) satisfies the first equation of (1) and (2) for almost everywhere (for short a.e.) in [0,∞]⧵{𝑡_𝑘} and satisfies 𝑁(𝑡⁺_𝑘)=(1+𝜃_𝑘)𝑁(𝑡_𝑘) for 𝑡=𝑡_𝑘,𝑘∈𝑍₊={1,2,…}.

Definition 2. The system (1) is said to be globally attractive, if there exists a positive solution 𝑥(𝑡) of (1) such that lim_𝑡→+∞|𝑥(𝑡)−𝑦(𝑡)|=0 for any other positive solution 𝑦(𝑡) of (1).

We can easily get the following Lemma 3.

Lemma 3. The region 𝑅={𝑁(𝑡)∶𝑁(0)>0,𝑡≥0} is invariant with respect to (1).

Proof. In view of biological population, we obtain 𝑁(0)>0. By the system (1), we have 𝑁(𝑡)=𝑁(0)×exp^𝑡₀𝑟𝜂𝜂𝜂−−𝑎ln𝑁_𝑛𝑖=1𝑏_𝑖𝜂ln𝑁𝜂−𝜎_𝑖𝜂−^𝑚_𝑗=1𝑐_𝑗𝜂^𝑡_−∞𝐾_𝑗(𝑠)−𝜂−𝑠ln𝑁𝑑𝑠_𝑝𝑙=1𝑑_𝑙𝜂𝑑ln𝑁𝜂−𝜏_𝑙𝜂𝑑𝜂𝑑𝜂>0,𝑡∈0,𝑡₁,𝑁(𝑡)=𝑁𝑡_𝑘×exp^𝑡_𝑡𝑘𝑟𝜂𝜂𝜂−−𝑎ln𝑁_𝑛𝑖=1𝑏_𝑖𝜂ln𝑁𝜂−𝜎_𝑖𝜂−^𝑚_𝑗=1𝑐_𝑗𝜂^𝑡_−∞𝐾_𝑗(𝑠)−𝜂−𝑠ln𝑁𝑑𝑠_𝑝𝑙=1𝑑_𝑙𝜂𝑑ln𝑁𝜂−𝜏_𝑙𝜂𝑡𝑑𝜂𝑑𝜂>0,𝑡∈_𝑘,𝑡_𝑘+1,𝑁𝑡⁺_𝑘=𝑒^(1+𝜃𝑘)𝑁𝑡_𝑘>0,𝑘∈𝑁. (9) Then, the solution of (1) and (2) is positive.
Under the above hypotheses (𝐻₁)--(𝐻₃), we consider the following neutral nonimpulsive system: 𝑑𝑦(𝑡)(𝑡𝑟(𝑡)(𝑡)(𝑡)−𝑑𝑡=𝑦)−𝐴ln𝑦_𝑛𝑖=1𝐵_𝑖(𝑡)ln𝑦𝑡−𝜎_𝑖(𝑡)−^𝑚_𝑗=1𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()(𝑠)−𝑡−𝑠ln𝑦𝑑𝑠_𝑝𝑙=1𝐷_𝑙(𝑡)𝑑ln𝑦𝑡−𝜏_𝑙(𝑡),𝑑𝑡 (10) with the following initial conditions: 𝑦(𝜉)=𝜑(𝜉),𝑦^′(𝜉)=𝜑^′(𝜉)[](0)[),𝜉∈−𝜏,0,𝜑>0,𝜑∈𝐶−𝜏,0,𝑅⁺𝐶¹([−𝜏,0)),𝑅⁺, (11) where𝐴(𝑡)=𝑎(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐵_𝑖(𝑡)=𝑏_𝑖(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑖(𝑡)}1+𝜃_𝑘,𝐶_𝑗(𝑡)=𝑐_𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐷_𝑙(𝑡)=𝑑_𝑙(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑙(𝑡)}1+𝜃_𝑘. (12) By a solution 𝑦(𝑡) of (10) and (11), it means an absolutely continuous function 𝑦(𝑡) defined on [−𝜏,0] that satisfies (10) a.e., for 𝑡≥0, and 𝑦(𝜉)=𝜑(𝜉),𝑦^′(𝜉)=𝜑^′(𝜉) on [−𝜏,0].
The following lemmas will be used in the proofs of our results, and the proof of the lemma is similar to that of Theorem 1 in [13].

Lemma 4. Suppose that (𝐻₁)--(𝐻₄) hold. Then,(1)if 𝑦(𝑡) is a solution of (10) and (11) on [−𝜏,+∞), then 𝑁(𝑡)=∏_{0<𝑡𝑘<𝑡}𝑒^(1+𝜃𝑘)𝑦(𝑡) is a solution of (1) and (2) on [−𝜏,+∞);(2)if 𝑁(𝑡) is a solution of (1) and (2) on [−𝜏,+∞), then 𝑦(𝑡)=∏_{0<𝑡𝑘<𝑡}(𝑒^(1+𝜃𝑘))⁻¹𝑁(𝑡) is a solution of (10) and (11) on [−𝜏,+∞).

Proof. (1) It is easy to see that 𝑁(𝑡)=∏_{0<𝑡𝑘<𝑡}𝑒^(1+𝜃𝑘)𝑦(𝑡) is absolutely continuous on every interval (𝑡_𝑘,𝑡_𝑘+1],𝑡≠𝑡_𝑘,𝑘=1,2,…, 𝑁^′(𝑡)(𝑡𝑟(𝑡)(𝑡)(𝑡)−−𝑁)−𝑎ln𝑁_𝑛𝑖=1𝑏_𝑖(𝑡)ln𝑁𝑡−𝜎_𝑖(𝑡)−^𝑚_𝑗=1𝑐_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()(𝑠)−𝑡−𝑠ln𝑁𝑑𝑠_𝑝𝑙=1𝑑_𝑙(𝑡)𝑑𝑁𝑡−𝜏_𝑙(𝑡)𝑑𝑡=_{0<𝑡𝑘<𝑡}𝑒^(1+𝜃𝑘)𝑦^′(𝑡)−_{0<𝑡𝑘<𝑡}𝑒^(1+𝜃𝑘)𝑦(𝑡)×𝑟(𝑡)−𝑎(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘(𝑡)−ln𝑦_𝑛𝑖=1𝑏_𝑖(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑖(𝑡)}1+𝜃_𝑘×ln𝑦𝑡−𝜎_𝑖(𝑡)−^𝑚_𝑗=1𝑐_𝑗(𝑡)^𝑡_−∞𝐾_𝑗(𝑡−𝑠)×_{0<𝑡𝑘<𝑡}1+𝜃_𝑘(𝑠)−ln𝑦𝑑𝑠_𝑝𝑙=1𝑑_𝑙(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑙(𝑡)}1+𝜃_𝑘×𝑑𝑦𝑡−𝜏_𝑙(𝑡)𝑑𝑡=_{0<𝑡𝑘<𝑡}𝑒^(1+𝜃𝑘)×𝑦^′(𝑡)(𝑡)×𝑟(𝑡)(𝑡)(𝑡)−−𝑦−𝐴ln𝑦_𝑛𝑖=1𝐵_𝑖(𝑡)ln𝑦𝑡−𝜎_𝑖(𝑡−_𝑚𝑗=1)𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()(𝑠)−𝑡−𝑠ln𝑦𝑑𝑠_𝑛𝑗=1𝐷_𝑙(𝑡)𝑑𝑦𝑡−𝜏_𝑙(𝑡)𝑑𝑡=0. (13) On the other hand, for any 𝑡=𝑡_𝑘,𝑘=1,2,…, 𝑁𝑡⁺_𝑘=lim_{𝑡→𝑡+𝑘0<𝑡𝑗<𝑡}𝑒^(1+𝜃𝑘)𝑦(𝑡)=_{0<𝑡𝑗≤𝑡𝑘}𝑒^(1+𝜃𝑘)𝑦𝑡_𝑘,𝑁𝑡_𝑘=_{0<𝑡𝑗<𝑡𝑘}𝑒^(1+𝜃𝑘)𝑦𝑡_𝑘, (14) thus, 𝑁𝑡⁺_𝑘=𝑒^(1+𝜃𝑘)𝑁𝑡_𝑘,𝑘=1,2,…. (15) It follows from (13)–(15) that 𝑁_𝑖(𝑡) is a solution of (1) and (2).
(2) Since 𝑁(𝑡)=∏_{0<𝑡𝑘<𝑡}𝑒^(1+𝜃𝑘)𝑦(𝑡) is absolutely continuous on every interval (𝑡_𝑘,𝑡_𝑘+1],𝑡≠𝑡_𝑘, 𝑘=1,2,…, and in view of (15), it follows that for any 𝑘=1,2,…, 𝑦𝑡⁺_𝑘=_{0<𝑡𝑗≤𝑡𝑘}𝑒^{(1+𝜃𝑘)−1}𝑁𝑡⁺_𝑘=_{0<𝑡𝑗<𝑡𝑘}𝑒^{(1+𝜃𝑘)−1}𝑁𝑡_𝑘=𝑦𝑡_𝑘,𝑦𝑡⁻_𝑘=_{0<𝑡𝑗<𝑡𝑘}𝑒^{(1+𝜃𝑘)−1}𝑁𝑡⁻_𝑘=_{0<𝑡𝑗≤𝑡−𝑘}𝑒^{(1+𝜃𝑘)−1}𝑁𝑡⁻_𝑘=𝑦𝑡_𝑘, (16) which implies that 𝑦(𝑡) is continuous on [−𝜏,+∞). It is easy to prove that 𝑦(𝑡) is absolutely continuous on [−𝜏,+∞). Similar to the proof of (1), one can check that 𝑦(𝑡)=∏_{0<𝑡𝑘<𝑡}(𝑒^(1+𝜃𝑘))⁻¹𝑁(𝑡) are solutions of (10) and (11) on [−𝜏,+∞). The proof of Lemma 3 is completed.

Here, we take the transformation 𝑦(𝑡)=𝑒^𝑥(𝑡); then, (10) can be rewritten in the following form: 𝑑𝑥(𝑡)(𝑡)(𝑡)𝑥(𝑡)−𝑑𝑡=𝑟−𝐴_𝑛𝑖=1𝐵_𝑖(𝑡)𝑥𝑡−𝜎_𝑖(𝑡)−^𝑚_𝑗=1𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()𝑥(𝑠)−𝑡−𝑠𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑡)𝑥^′𝑡−𝜏_𝑙(𝑡), (17) where 𝐷_0𝑙(𝑡)=𝐷_𝑙(𝑡)(1−𝜏^′_𝑙(𝑡)),𝑙=1,2,…,𝑝. Therefore, we only discuss the existence and uniqueness of a periodic solution for (17).

Definition 5 (see [23]). Let 𝑈 be a bounded subset in 𝑋. Define that 𝛼_𝑋(𝑈)𝑈=inf𝛿>0∶thereisafinitenumberofsubsets_𝑖⊂𝑈suchthat𝑈=_𝑖𝑈_𝑖anddiam𝑈_𝑖≤𝛿, (18) where diam(𝑈_𝑖) denotes the diameter of the set 𝑈_𝑖, obviously, 0≤𝛼_𝑋(𝑈)<∞. So, 𝛼_𝑋(𝑈) is called the (Kuratowski) measure of noncompactness of 𝑋.

Definition 6 (see [23]). 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 𝑈⊂𝐷 one has 𝛼_𝑌(𝑇(𝑈))≤𝑘𝛼_𝑋(𝑈), (19)𝑇 is called strict-set-contractive if it is 𝑘-set contractive for some 0≤𝑘<1.
For a Fredholm operator 𝐿∶𝑋→𝑌 with index zero, according to [9, 24], we define 𝑙(𝐿)=sup𝑟𝛼_𝑋(𝑈)≤𝛼_𝑌(𝐿(𝑈.))foranyboundedset𝑈⊂𝑋 (20)

Lemma 7 (see [9, 24]). Let 𝐿∶𝑋→𝑌 be a Fredholm operator with zero index and 𝑎∈𝑌 be a fixed point. Suppose that 𝑁∶Ω→𝑌 is called a 𝑘-set contractive with 𝑘<𝑙(𝐿), where Ω⊂𝑋 is bounded, open, and symmetric about 0∈Ω. Further, one also assumes that(1)𝐿𝑥≠𝜆𝑁𝑥+𝜆𝑎,  for 𝑥∈𝜕Ω,𝜆∈(0,1); (2)[𝑄𝑁(𝑥)+𝑄𝑎,𝑥][𝑄𝑁(−𝑥)+𝑄𝑎,𝑥]<0, for 𝑥∈Ker𝐿⋂𝜕Ω;
where [⋅,⋅] is a bilinear form on 𝑌×𝑋 and Q is the projection of Y onto Coker(𝐿), where Coker(𝐿) is the cokernel of the operator 𝐿. Then, there is an 𝑥∈Ω such that 𝐿𝑥−𝑁𝑥=𝑎.

In order to use Lemma 7 to study (17), we set 𝑋=𝐶¹_𝜔,𝑌=𝐶_𝜔, 𝐿∶𝐶¹_𝜔⟶𝐶_𝜔,𝐿𝑥=𝑑𝑥,𝑑𝑡 (21) and 𝑁∶𝐶¹_𝜔→𝐶_𝜔 defined by 𝑁𝑥=−𝐴(𝑡)(𝑡)𝑥(𝑡)−^𝑛_𝑖=1𝐵_𝑖(𝑡)𝑥𝑡−𝜎_𝑖(𝑡−_𝑚𝑗=1)𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()𝑥(𝑠)−𝑡−𝑠𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑡)𝑥^′𝑡−𝜏_𝑙(𝑡). (22) It is easy to see from [25] that 𝐿 is a Fredholm operator with index zero. Thus, (17) has a positive 𝜔-periodic solution if and only if 𝐿𝑥=𝑁𝑥+𝑟 has a solution 𝑥∈𝐶¹_𝜔, where 𝑟=𝑟(𝑡).

Lemma 8 (see [25]). The differential operator 𝐿 is a Fredholm operator with index zero and satisfies 𝑙(𝐿)≥1.

Lemma 9. Let 𝑟₀,𝑟₁ be two positive constants and Ω={𝑥_𝑖∈𝐶¹_𝜔,|𝑥_𝑖|_𝑜<𝑟₀,|𝑥^′_𝑖|_𝑜<𝑟₁}. If 𝑘=∑^𝑝_𝑙=1|𝐷_0𝑙|₀, then 𝑁∶Ω→𝐶¹_𝜔 is a k-set contractive map.

As Lemma 9 can be proved in the same way as in the proof of Lemma 2.4 in [7], we omit it here.

Lemma 10 (see [7, 9]). Suppose that 𝜏∈𝐶¹_𝜔 and 𝜏^′(𝑡)<1, 𝑡∈[0,𝜔], then the function 𝑡−𝜏(𝑡) has a unique inverse 𝜇(𝑡) satisfying 𝜇∈𝐶(𝑅,𝑅) with 𝜇(𝑎+𝜔)=𝜇(𝑎)+𝜔 for all 𝑎∈𝑅, and if 𝑔∈𝑃𝐶_𝜔,𝜏^′(𝑡)<1,𝑡∈[0,𝜔], then 𝑔(𝜇(𝑡))∈𝑃𝐶_𝜔.

Lemma 11 11 (see [26, 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.

3. Main Theorem

Since 𝜎^′_𝑖(𝑡)<1, 𝜏^′_𝑙(𝑡)<1,𝑡∈[0,𝜔], we see that 𝜎_𝑖(𝑡),𝜏_𝑙(𝑡) all have its inverse function. Throughout the following part, we set 𝜁_𝑖(𝑡),𝜉_𝑙(𝑡) that represent the inverse function of 𝑡−𝜎_𝑖(𝑡), 𝑡−𝜏_𝑙(𝑡), respectively. We also denote Γ(𝑡)∶=𝐴(𝑡)+^𝑛_𝑖=1𝐵_𝑖𝜁_𝑖(𝑡)1−𝜎^′_𝑖𝜁_𝑖(𝑡)+^𝑚_𝑗=1𝐶_𝑗(𝑡)−^𝑝_𝑙=1𝐷^′_𝑙𝜉_𝑙(𝑡)1−𝜏^′_𝑙𝜉_𝑙(𝑡.) (23)

Remark 12. From Lemma 9, we get that 𝜁_𝑖(𝜔)=𝜁_𝑖(0)+𝜔,𝜉_𝑙(𝜔)=𝜉_𝑙(0)+𝜔,𝑖=1,2,…,𝑛; 𝑙=1,2,…,𝑝; then, ^𝜔₀𝐵_𝑖𝜁_𝑖(𝑠)1−𝜎^′_𝑖𝜁_𝑖(𝑠)=𝑑𝑠_{𝜁𝑖𝜁(𝜔)𝑖(0)}𝐵_𝑖(𝑡)1−𝜎^′_𝑖(𝑡)1−𝜎^′_𝑖(𝑡)=𝑑𝑡_{𝜁𝑖𝜁(0)+𝜔𝑖(0)}𝐵_𝑖(𝑡)𝑑𝑡=𝐵_𝑖𝜔,𝑖=1,2,…,𝑛. (24) Similarly, ^𝜔₀𝐷^′_𝑙𝜉_𝑙(𝑠)1−𝜏^′_𝑙𝜉_𝑙(𝑠)=𝑑𝑠_{𝜉𝑙𝜉(𝜔)𝑙(0)}𝐷^′_𝑙(𝑡)1−𝜏^′_𝑙(𝑡)1−𝜏^′_𝑙(𝑡)=𝑑𝑡_{𝜉𝑙𝜉(0)+𝜔𝑙(0)}𝐷^′_𝑙(𝑡)𝑑𝑡=0,𝑙=1,2,…,𝑝. (25) Thus, Γ𝜔=^𝜔₀Γ(𝑡)𝑑𝑡=𝐴+^𝑛_𝑖=1𝐵_𝑖+^𝑚_𝑗=1𝐶_𝑗𝜔. (26)

Theorem 13. In addition to (𝐻₁)--(𝐻₃), suppose that the following conditions hold: (𝐻₄) there exists a constant 𝜂>0 such that |Γ(𝑡)|≥𝜂, for all 𝑡∈[0,𝜔], where Γ(𝑡) is defined by (23);(𝐻₅) if (1/2)(|𝐴|₀𝜔+∑^𝑛_𝑖=1|𝐵_𝑖|₀𝜔+∑^𝑚_𝑗=1|𝐶_𝑗|₀𝜔^1/2)+∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀^1/2|𝐷_𝑙|₀<1, and ∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀|𝐷_𝑙|₀<1.
Then, (1) has at least one positive 𝜔-periodic solution.

Proof. Let 𝑢(𝑡) be an arbitrary 𝜔-periodic solution of the operator equation as follows: 𝐿𝑢=𝜆𝑁𝑢+𝜆𝑟,𝜆∈(0,1), (27) where 𝐿, 𝑁 defined by (21) and (22), respectively. Then, 𝑢(𝑡) satisfies the following operator equation: 𝑑𝑢(𝑡)𝑟(𝑡)(𝑡)𝑢(𝑡)−𝑑𝑡=𝜆−𝐴_𝑛𝑖=1𝐵_𝑖(𝑡)𝑢𝑡−𝜎_𝑖(𝑡)−^𝑚_𝑗=1𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()𝑢(𝑠)−𝑡−𝑠𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑡)𝑢^′𝑡−𝜏_𝑙(𝑡). (28) Integrating both sides of (28) over [0,𝜔], we have 𝑟𝜔=^𝜔₀𝐴(𝑡)𝑢(𝑡)+^𝑛_𝑖=1𝐵_𝑖(𝑡)𝑢𝑡−𝜎_𝑖(𝑡)+^𝑚_𝑗=1𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()𝑢(𝑠)+𝑡−𝑠𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑡)𝑢^′𝑡−𝜏_𝑙(𝑡=)𝑑𝑡_𝜔0𝐴(𝑡)𝑢(𝑡)+𝑑𝑡_{𝜔0𝑛𝑖=1}𝐵_𝑖(𝑡)𝑢𝑡−𝜎_𝑖(𝑡)+𝑑𝑡_{𝜔0𝑚𝑗=1}𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗(𝑡−𝑠)𝑢(𝑠)𝑑𝑠𝑑𝑡−^𝜔₀^𝑝_𝑙=1𝐷^′_𝑙(𝑡)𝑢𝑡−𝜏_𝑙(𝑡)𝑑𝑡. (29) Let 𝑡−𝜎_𝑖(𝑡)=𝑠, then 𝑡=𝜁_𝑖(𝑠) and ^𝜔₀𝐵_𝑖(𝑡)𝑢𝑡−𝜎_𝑖(𝑡)=𝑑𝑡_{𝜔−𝜎𝑖(𝜔)−𝜎𝑖(0)}𝐵_𝑖𝜁_𝑖(𝑠)1−𝜎^′_𝑖𝜁_𝑖(𝑠)𝑢(𝑠)𝑑𝑠. (30) By Lemma 10, we have ∫^𝜔₀𝐵_𝑖(𝑡)𝑢𝑡−𝜎_𝑖(𝑡)𝑑𝑡=∫^𝜔₀𝐵_𝑖𝜁_𝑖(𝑠)1−𝜎^′_𝑖𝜁_𝑖(𝑠)𝑢(𝑠)𝑑𝑠. (31) Similarly, we have ^𝜔₀𝐷^′_𝑙(𝑡)𝑢_𝑖𝑡−𝜏_𝑙(𝑡)=𝑑𝑡_𝜔0𝐷^′_𝑙𝜉_𝑙(𝑠)1−𝜏^′_𝑙𝜉_𝑙(𝑠)𝑢(𝑠)𝑑𝑠. (32) Substituting (31) and (32) into (29), we have ^𝜔₀Γ(𝑠)𝑢(𝑠)𝑑𝑠=𝑟𝜔. (33) Considering assumption (𝐻₄), we know that |Γ(𝑡)|≥𝜂>0, and then, it follows from the integro mean value theorem that there exists a 𝛼∈[0,𝜔] satisfying |𝑢(𝛼)|=𝑟_𝑖|Γ(𝛼≤)|𝑟_𝑖𝜂. (34) By Lemma 11, we can get that |𝑢|₀≤𝑟_𝑖𝜂+12^𝜔₀𝑢^′(𝑡)𝑑𝑡. (35) Multiplying both sides of (28) by 𝑢^′(𝑡) and integrating them over [0,𝜔], we have ^𝜔₀𝑢^′(𝑡)²𝑑𝑡=𝜆^𝜔₀𝑟(𝑡)𝑢^′(𝑡)𝑑𝑡−^𝜔₀𝐴(𝑡)𝑢(𝑡)𝑢^′(𝑡)−𝑑𝑡_{𝜔0𝑛𝑖=1}𝐵_𝑖(𝑡)𝑢𝑡−𝜎_𝑖(𝑡)𝑢^′(𝑡)−𝑑𝑡_{𝜔0𝑚𝑗=1}𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗(𝑡−𝑠)𝑢(𝑠)𝑢^′(𝑡)−𝑑𝑠𝑑𝑡_{𝜔0𝑝𝑙=1}𝐷_0𝑙(𝑡)𝑢^′𝑡−𝜏_𝑙(𝑡)𝑢^′(𝑡)≤𝑟|𝑑𝑡|₀+|𝐴|₀+^𝑛_𝑖=1𝐵_𝑖0|𝑢|₀^𝜔₀𝑢^′(𝑡+)𝑑𝑡_{𝜔0𝑚𝑗=1}𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗(𝑡−𝑠)𝑢(𝑠)𝑑𝑠𝑢^′(𝑡)𝑑𝑡+^𝜔₀^𝑝_𝑙=1𝐷_0𝑙(𝑡)𝑢^′𝑡−𝜏_𝑙(𝑡)𝑢^′(𝑡)𝑑𝑡. (36) By using the Cauchy-Schwarz inequality, we get that ^𝜔₀|𝑢^′(𝑡)|²≤𝑟|𝑑𝑡|₀+|𝐴|₀+^𝑛_𝑖=1𝐵_𝑖0|𝑢|₀×^𝜔₀|𝑢^′(𝑡)|²𝑑𝑡^1/2𝜔^1/2+^𝑚_𝑗=1𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗(𝑡−𝑠)𝑢(𝑠)𝑑𝑠²𝑑𝑡^1/2×^𝜔₀𝑢^′(𝑡)²𝑑𝑡^1/2+^𝑝_𝑙=1^𝜔₀𝐷_0𝑙(𝑡)𝑢^′(𝑡−𝜏_𝑙(𝑡))²𝑑𝑡^1/2×^𝜔₀𝑢^′(𝑡)²𝑑𝑡^1/2=|𝑟|₀𝜔^1/2+|𝐴|₀𝜔^1/2+^𝑛_𝑖=1|𝐵_𝑖|₀𝜔^1/2+^𝑚_𝑗=1|𝐶_𝑗|₀|𝑢|₀×^𝜔₀|𝑢^′(𝑡)|²𝑑𝑡^1/2+^𝑝_𝑙=1^𝜔₀𝐷_0𝑙(𝑡)𝑢^′𝑡−𝜏_𝑙(𝑡)²𝑑𝑡^1/2×^𝜔₀𝑢^′(𝑡)²𝑑𝑡^1/2. (37) Meanwhile, we see that ^𝜔₀𝐷_0𝑙(𝑡)𝑢^′(𝑡−𝜏_𝑙(𝑡))²𝑑𝑡^1/2=^𝜔₀11−𝜏^′_𝑙𝜉_𝑙(𝑡)𝐷_0𝑙𝜉_𝑙(𝑡)𝑢^′(𝑡)²𝑑𝑡^1/2=^𝜔₀(1−𝜏^′_𝑙(𝜉_𝑙(𝑡)))𝐷_𝑙𝜉_𝑙(𝑡)𝑢^′(𝑡)²𝑑𝑡^1/2≤1−𝜏^′_𝑙0^1/2𝐷_𝑙0^𝜔₀𝑢^′(𝑡)²𝑑𝑡. (38) Substituting (38) and (35) into (37), we can find that ^𝜔₀𝑢^′(𝑡)²≤𝑟|𝑑𝑡|₀𝜔^1/2+|𝐴|₀𝜔^1/2+^𝑛_𝑖=1𝐵_𝑖0𝜔^1/2+^𝑚_𝑗=1𝐶_𝑗0×𝑟_𝑖𝜂+12^𝜔₀𝑢^′(𝑡)𝑑𝑡^𝜔₀𝑢^′(𝑡)²𝑑𝑡^1/2+^𝑝_𝑙=11−𝜏^′_𝑙0^1/2𝐷_𝑙0^𝜔₀𝑢^′(𝑡)²≤𝑑𝑡|𝑟|₀𝜔^1/2+|𝐴|₀𝜔^1/2+^𝑛_𝑖=1|𝐵_𝑖|₀𝜔^1/2+^𝑚_𝑗=1𝐶_𝑗0𝑟_𝑖𝜂×^𝜔₀𝑢^′(𝑡)𝑑𝑡^1/2+12|𝐴|₀𝜔^1/2+^𝑛_𝑖=1|𝐵_𝑖|₀𝜔^1/2+^𝑚_𝑗=1|𝐶_𝑗|₀𝜔^1/2+^𝑝_𝑙=11−𝜏^′_𝑙0^1/2𝐷_𝑙0×^𝜔₀𝑢^′(𝑡)²𝑑𝑡, (39) which gives that ^𝜔₀𝑢^′(𝑡)²𝑑𝑡^1/2≤|𝑟|₀𝜔^1/2+|𝐴|₀𝜔^1/2+^𝑛_𝑖=1𝐵_𝑖0𝜔^1/2+^𝑚_𝑗=1𝐶_𝑗0𝑟_𝑖𝜂+12|𝐴|₀𝜔+^𝑛_𝑖=1𝐵_𝑖0𝜔+^𝑚_𝑗=1𝐶_𝑗0𝜔^1/2+^𝑝_𝑙=11−𝜏^′_𝑙0^1/2𝐷_𝑙0^𝜔₀𝑢^′(𝑡)𝑑𝑡^1/2. (40) From (𝐻₅), we get that (1/2)(|𝐴|₀𝜔+∑^𝑛_𝑖=1|𝐵_𝑖|₀𝜔+∑^𝑚_𝑗=1|𝐶_𝑗|₀𝜔^1/2)+∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀^1/2|𝐷_𝑙|₀<1. Then, there exists a constant 𝑁>0 such that ^𝜔₀𝑢^′_𝑖(𝑡)²𝑑𝑡^1/2≤𝑁. (41) From (35) and the 𝐻𝑜𝑙̈𝑑𝑒𝑟 inequality, we obtain that |𝑢|₀≤𝑟𝜂+12^𝜔₀𝑢^′(𝑡≤)𝑑𝑡𝑟𝜂+𝜔^1/22^𝜔₀𝑢^′(𝑡)²𝑑𝑡^1/2≤𝑟𝜂+𝜔^1/22𝑁∶=𝑁₁. (42) Again from (28), we get that 𝑢^′₀≤|𝑟|₀+|𝐴|₀+^𝑛_𝑖=1𝐵_𝑖0+^𝑚_𝑗=1𝐶_𝑗0|𝑢|₀+^𝑝_𝑙=1𝐷_0𝑙0𝑢^′₀. (43) From condition ∑^𝑝_𝑙=1|𝐷_0𝑙|₀≤∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀|𝐷_𝑙|₀<1, it is easy to see that 𝑢^′₀≤|𝑟|₀+|𝐴|₀+∑^𝑛_𝑖=1𝐵_𝑖0+∑^𝑚_𝑗=1𝐶_𝑗0𝑁₁1−∑^𝑝_𝑙=1|𝐷_0𝑙|₀∶=𝑁₂. (44) Now, we take 𝑁₃>max{𝑁₁,𝑁₂,|𝑟/(𝐴+∑^𝑛_𝑖=1𝐵_𝑖+∑^𝑚_𝑗=1𝐶_𝑗)|} and Ω={𝑢∣𝑢∈𝐶¹_𝜔,‖𝑢‖<𝑁₃}. Then, 𝑘=∑^𝑝_𝑙=1|𝐷_0𝑙|₀<1≤𝑙(𝐿). So by (42) and (44), we can find that all the conditions of Lemma 4 except (2) hold. In what follows, we will prove that condition (2) of Lemma 4 is also satisfied. In order to do this, we defined a bounded bilinear form on 𝐶_𝜔,𝐶^!_𝜔 by [⋅,⋅] as the following [𝑦,𝑥]=∫^𝜔₀𝑦(𝑡)𝑥(𝑡)𝑑𝑡. Also, we defined 𝑄∶𝑦→Coker(𝐿) as 𝑦∶→(1/𝜔)∫^𝜔₀𝑦(𝑡)𝑑𝑡. It is obvious that 𝑢∣𝑢∈Ker𝐿𝜕Ω=𝑢∣𝑢≡𝑁₃,or−𝑁₃. (45) Without loss of generality, suppose that 𝑢≡𝑁₃; then, [𝑄𝑁(𝑢)+𝑄𝑟,𝑢][𝑄𝑁(−𝑢)+𝑄𝑟,𝑢]=𝑁²₃^𝜔₀𝑟(𝑡)𝑑𝑡−𝑁₃^𝜔₀𝐴(𝑡)+^𝑛_𝑖=1𝐵_𝑖(𝑡)+^𝑚_𝑗=1𝐶_𝑗(𝑡)×𝑑𝑡_𝜔0𝑟(𝑡)𝑑𝑡+𝑁₃^𝜔₀𝐴(𝑡)+^𝑛_𝑖=1𝐵_𝑖(𝑡)+^𝑚_𝑗=1𝐶_𝑗(𝑡)𝑑𝑡=𝜔²𝑁²₃𝑟−𝑁₃𝐴+^𝑛_𝑖=1𝐵_𝑖+^𝑚_𝑗=1𝐶_𝑗×𝑟+𝑁₃𝐴+^𝑛_𝑖=1𝐵_𝑖+^𝑚_𝑗=1𝐶_𝑗, (46) since 𝑁₃>|𝑟/(𝐴+∑^𝑛_𝑖=1𝐵_𝑖+∑^𝑚_𝑗=1𝐶_𝑗)|, then 𝑁₃>𝑟𝐴+∑^𝑛_𝑖=1𝐵_𝑖+∑^𝑚_𝑗=1𝐶_𝑗,−𝑁₃<−𝑟𝐴+∑^𝑛_𝑖=1𝐵_𝑖+∑^𝑚_𝑗=1𝐶_𝑗. (47) By (46), we get that [𝑄𝑁(𝑢)+𝑄𝑟,𝑢][𝑄𝑁(−𝑢)+𝑄𝑟,𝑢]<0. (48) Therefore, by using Lemma 4, we obtain that (1) has at least one positive 𝜔-periodic solution; the proof of Theorem 13 is completed.

Since |1−𝜏^′_𝑙|₀<1, forall𝑙=1,2,…,𝑝, it follows that ∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀|𝐷_𝑙|₀≤∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀^1/2|𝐷_𝑙|₀<1. So from Theorem 13, we have the following result.

Corollary 14. Suppose that (𝐻₁)--(𝐻₃) and the following conditions hold:(𝐻^′₄) there exists a constant 𝜂>0 such that |Γ(𝑡)|≥𝜂, for all 𝑡∈[0,𝜔], where Γ(𝑡) is defined by (23);(𝐻^′₅) if (1/2)(|𝐴|₀𝜔+∑^𝑛_𝑖=1|𝐵|₀𝜔+∑^𝑚_𝑗=1|𝐶_𝑗|₀𝜔^1/2)+∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀^1/2|𝐷_𝑙|₀<1.
Then (1) has at least one positive 𝜔-periodic solution.

On the other hand, if |1−𝜏^′_𝑙|₀>1, for all 𝑙=1,2,…,𝑝, it follows that ∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀|𝐷_𝑙|₀≥∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀^1/2|𝐷_𝑙|₀. So from Theorem 13, we also have the following result.

Corollary 15. Suppose that (𝐻₁)--(𝐻₃) and the following conditions hold:(𝐻^∗₄) there exists a constant 𝜂>0 such that |Γ(𝑡)|≥𝜂, for all 𝑡∈[0,𝜔], where Γ(𝑡) is defined by (23);(𝐻^∗₅) if (1/2)(|𝐴|₀𝜔+∑^𝑛_𝑖=1|𝐵|₀𝜔+∑^𝑚_𝑗=1|𝐶_𝑗|₀𝜔^1/2)+∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀|𝐷_𝑙|₀<1.
Then, (1) has at least one positive 𝜔-periodic solution.

Our next theorem is concerned with the global attractivity of periodic solution of the system (1).

Theorem 16. Suppose that (𝐻₁)--(𝐻₅) and the following conditions hold:(𝐻₆) there is a positive constant 𝜇 such that ^𝑝_𝑙=1𝐷_𝑙0+^𝑡₀^𝑝_𝑙=1|𝐻_𝑙|₀+^𝑛_𝑖=1|𝐵_𝑖|₀+^𝑚_𝑗=1|𝐶_𝑗|₀×exp−∫^𝑡_𝑢𝐴(𝜉)𝑑𝜉𝑑𝑢≤𝜇<1; (49)(𝐻₇)exp(−∫^𝑡₀𝐴(𝜉)𝑑𝜉)→0, as 𝑡→+∞.
Then, the positive 𝜔-periodic solution of (1) is globally attractive, where 𝐻_𝑙(𝑡)=[𝐷^′_𝑙(𝑡)+𝐷_𝑙(𝑡)𝐴(𝑡)][1−𝜏^′_𝑙(𝑡)]+𝐷_𝑙(𝑡)𝜏_𝑙^′′(𝑡)/[1−𝜏^′_𝑙(𝑡)]².

Proof. Suppose that 𝑦(𝑡)=𝑒^𝑣(𝑡) is a positive 𝜔-periodic solution of (10), 𝑦^∗(𝑡)=𝑒^{𝑣∗(𝑡)} is another positive solution of (10). Similar to (17), we have 𝑑𝑣(𝑡)(𝑡)𝑣(𝑡)−𝑑𝑡=𝑟−𝐴^𝑛_𝑖=1𝐵_𝑖(𝑡)𝑣𝑡−𝜎_𝑖(𝑡−_𝑚𝑗=1)𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()𝑣(𝑠)−𝑡−𝑠𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑡)𝑣^′𝑡−𝜏_𝑙(𝑡),𝑑𝑣^∗(𝑡)(𝑡)𝑣𝑑𝑡=𝑟−𝐴^∗(𝑡)−^𝑛_𝑖=1𝐵_𝑖(𝑡)𝑣^∗𝑡−𝜎_𝑖(𝑡)−^𝑚_𝑗=1𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗(𝑡−𝑠)𝑣^∗(𝑠)−𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑡)𝑣^′∗𝑡−𝜏_𝑙(𝑡). (50)(51) Let 𝑣(𝑡)−𝑣^∗(𝑡)=𝑤(𝑡); then, 𝑑𝑤(𝑡)𝑤(𝑡)−𝑑𝑡=−𝐴^𝑛_𝑖=1𝐵_𝑖(𝑡)𝑤𝑡−𝜎_𝑖(𝑡−_𝑚𝑗=1)𝐶_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()𝑤(𝑠)−𝑡−𝑠𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑡)𝑤^′𝑡−𝜏_𝑙(𝑡). (52) Multiply both sides of (51) with exp(∫^𝑡₀𝐴(𝜉)𝑑𝜉) and then integrate from 0 to 𝑡 to obtain that ^𝑡₀𝑤(𝑢)exp^𝑢₀𝐴(𝜉)𝑑𝜉^′𝑑𝑢=−^𝑡₀^𝑛_𝑖=1𝐵_𝑖(𝑢)𝑤𝑡−𝜎_𝑖(𝑢+_𝑚𝑗=1)𝐶_𝑗(𝑢)^𝑡_−∞𝐾_𝑗()𝑤(𝑠)+𝑢−𝑠𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑢)𝑤^′𝑢−𝜏_𝑙(𝑢)×exp^𝑢₀𝐴(𝜉)𝑑𝜉𝑑𝑢, (53) Then, 𝑤(𝑡)=𝑤(0)exp−^𝑡₀𝐴(𝜉)−𝑑𝜉_{𝑡0𝑛𝑖=1}𝐵_𝑖(𝑢)𝑤𝑡−𝜎_𝑖(𝑢)+^𝑚_𝑗=1𝐶_𝑗(𝑢)^𝑡_−∞𝐾_𝑗()𝑤(𝑠)+𝑢−𝑠𝑑𝑠_𝑝𝑙=1𝐷_0𝑙(𝑢)𝑤^′𝑢−𝜏_𝑙(𝑢)×exp−^𝑡_𝑢𝐴(𝜉)𝑑𝜉𝑑𝑢. (54) Meanwhile, we see that ^𝑡₀^𝑝_𝑙=1𝐷_0𝑙(𝑢)𝑤^′𝑢−𝜏_𝑙(𝑢)exp−^𝑡_𝑢𝐴(𝜉)=𝑑𝜉𝑑𝑢_{𝑡0𝑝𝑙=1}𝐷_0𝑙(𝑢)𝑤^′𝑢−𝜏_𝑙(𝑢)1−𝜏^′_𝑙(𝑢)1−𝜏^′_𝑙(𝑢)×exp−^𝑡_𝑢𝐴(𝜉)=𝑑𝜉𝑑𝑢_{𝑡0𝑝𝑙=1}𝐷_0l(𝑢)exp−∫^𝑡_𝑢𝐴(𝜉)𝑑𝜉1−𝜏^′_𝑙(𝑢)×𝑤^′𝑢−𝜏_𝑙(𝑢)1−𝜏^′_𝑙(𝑢)=𝑑𝑢_𝑝𝑙=1𝐷_0𝑙(𝑡)1−𝜏^′_𝑙(𝑡)𝑤𝑡−𝜏_𝑙(𝑡)−𝐷_0𝑙(0)1−𝜏^′_𝑙(0)𝑤−𝜏_𝑙(0)exp−^𝑡₀𝐴(𝜉)−𝑑𝜉_{𝑝𝑙=1𝑡0}𝐻_𝑙(𝑢)exp−^𝑡_𝑢𝐴(𝜉)𝑑𝜉𝑤𝑢−𝜏_𝑙(𝑢)=𝑑𝑢_𝑝𝑙=1𝐷_𝑙(𝑡)𝑤𝑡−𝜏_𝑙(𝑡)−𝐷_𝑙(0)𝑤−𝜏_𝑙(0)exp−^𝑡₀𝐴(𝜉)−𝑑𝜉_{𝑝𝑙=1𝑡0}𝐻_𝑙(𝑢)exp−^𝑡_𝑢𝐴(𝜉)𝑑𝜉𝑤𝑢−𝜏_𝑙(𝑢)𝑑𝑢, (55) where 𝐻_𝑙(𝑢)=𝐷^′_𝑙(𝑢)+𝐷_𝑙(𝑡)𝐴(𝑡)1−𝜏^′_𝑙(𝑡)+𝐷_𝑙(𝑡)𝜏_𝑙^′′(𝑡)1−𝜏^′_𝑙(𝑡)². (56) Substituting (55) into (54), we get that 𝑤(𝑡)=𝑤(0)+^𝑝_𝑙=1𝐷_𝑙(0)𝑤−𝜏_𝑙(0)×exp−^𝑡₀𝐴(𝜉)+𝑑𝜉_{𝑡0𝑝𝑙=1}𝐻_𝑙(𝑢)𝑤𝑢−𝜏_𝑙(𝑢)−^𝑛_𝑖=1𝐵_𝑖(𝑢)𝑤𝑢−𝜎_𝑖(𝑢)−^𝑚_𝑗=1𝐶_𝑗(𝑢)^𝑢_−∞𝐾_𝑗()𝑤(𝑠)−𝑢−𝑠𝑑𝑠×exp^𝑡_𝑢𝐴(𝜉)𝑑𝜉𝑑𝑢−^𝑝_𝑙=1𝐷_𝑙(𝑡)𝑤𝑡−𝜏_𝑙(𝑡). (57) Therefore, we have |𝑤|₀≤𝑤(0)+^𝑝_𝑙=1𝐷_𝑙(0)𝑤−𝜏_𝑙(0)×exp−^𝑡₀𝐴(𝜉)+𝑑𝜉_{𝑡0𝑝𝑙=1}|𝐻_𝑙|₀+^𝑛_𝑖=1|𝐵_𝑖|₀+^𝑚_𝑗=1𝐶_𝑗0×|𝑤|₀exp−^𝑡_𝑢𝐴(𝜉)𝑑𝜉𝑑𝑢+^𝑝_𝑙=1|𝐷_𝑙|₀|𝑤|₀. (58) From (𝐻₆), we have |𝑤|₀≤𝑤(0)+^𝑝_𝑙=1𝐷_𝑙(0)𝑤−𝜏_𝑙(0)exp−^𝑡₀𝐴(𝜉)×𝑑𝜉1−^𝑝_𝑙=1𝐷_𝑙0−^𝑡₀^𝑝_𝑙=1|𝐻_𝑙|₀+^𝑛_𝑖=1|𝐵_𝑖|₀+^𝑚_𝑗=1|𝐶_𝑗|₀×exp−^𝑡_𝑢𝐴(𝜉)𝑑𝜉𝑑𝑢⁻¹≤𝑤(0)+∑^𝑝_𝑙=1𝐷_𝑙(0)𝑤−𝜏_𝑙(0)exp−∫^𝑡₀𝐴(𝜉)𝑑𝜉.1−𝜇 (59) From (𝐻₇), we have|𝑤|₀=|𝑣−𝑣^∗|₀=0,as𝑡⟶+∞, (60) thus, 𝑣(𝑡)→𝑣^∗(𝑡), as 𝑡→+∞; that is, the positive 𝜔-periodic solution of (10) is globally attractive; by Definition 2, the positive 𝜔-periodic solution of (1) is globally attractive. The proof is completed.

Consider the following equation: 𝑑𝑁(𝑡)(𝑡𝑟(𝑡)(𝑡)(𝑡)−𝑑𝑡=𝑁)−𝑎ln𝑁_𝑛𝑖=1𝑏_𝑖(𝑡)ln𝑁𝑡−𝜎_𝑖(𝑡)−^𝑚_𝑗=1𝑐_𝑗(𝑡)^𝑡_−∞𝐾_𝑗()(𝑠)−𝑡−𝑠ln𝑁𝑑𝑠_𝑝𝑙=1𝑑_𝑙(𝑡)𝑑ln𝑁𝑡−𝜏_𝑙(𝑡),𝑑𝑡 (61) which is a special case of the system (1) without impulse. Similarly, we can get the following conclusions.

Corollary 17. Suppose that the following conditions hold:(𝐻^∗₁)𝑟(𝑡),𝑎(𝑡),𝑏_𝑖(𝑡),𝑐_𝑗(𝑡),𝑑_𝑙(𝑡),𝜎_𝑖(𝑡),𝜏_𝑙(𝑡)(𝑖=1,2,…,𝑛; 𝑗=1,2,…,𝑚; 𝑙=1,2,…,𝑝) are all positive 𝜔-periodic continuous functions with 𝜎_𝑖(𝑡)≥0,𝜏_𝑙(𝑡)≥0, 𝑡∈[0,𝜔], 𝜎^′_𝑖(𝑡)<1, 𝜏^′_𝑙(𝑡)<1, for all 𝑖={1,2,…,𝑛}, for all 𝑙={1,2,…,𝑝}; furthermore, 𝑑_𝑙(𝑡)∈𝐶¹(𝑅,𝑅), 𝜏_𝑙(𝑡)∈𝐶²(𝑅,𝑅),∫^∞₀𝐾_𝑗(𝑠)𝑑𝑠=1,∫₀^+∞𝑠𝐾_𝑗(𝑠)𝑑𝑠<+∞, for all 𝑗={1,2,…,𝑚}, for all 𝑙={1,2,…,𝑝};(𝐻^∗₂) there exists a constant 𝜂>0 such that |Γ^∗(𝑡)|≥𝜂, for all 𝑡∈[0,𝜔], where Γ^∗(𝑡) is defined by the following: Γ^∗(𝑡)∶=𝑎(𝑡)+^𝑛_𝑖=1𝑎_𝑖𝜁_𝑖(𝑡)1−𝜎^′_𝑖𝜁_𝑖(𝑡)+^𝑚_𝑗=1𝑐_𝑗(𝑡)−^𝑝_𝑙=1𝑑^′_𝑙𝜉_𝑙(𝑡)1−𝜏^′_𝑙𝜉_𝑙(𝑡;) (62)(𝐻^∗₃) if (1/2)(|𝑎|₀𝜔+∑^𝑛_𝑖=1|𝑏_𝑖|₀𝜔+∑^𝑚_𝑗=1|𝑐_𝑗|₀𝜔^1/2)+∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀^1/2|𝑑_𝑙|₀<1, and ∑^𝑝_𝑙=1|1−𝜏^′_𝑙|₀|𝑑_𝑙|₀<1.
Then, (61) has at least one positive 𝜔-periodic solution.

Corollary 18. Suppose that (𝐻^∗₁) and the following conditions hold:(𝐻^∗₄) there is a positive constant 𝜇 such that ^𝑝_𝑙=1𝑑_𝑙0+^𝑡₀^𝑝_𝑙=1ℎ_𝑙0+^𝑛_𝑖=1𝑏_𝑖0+^𝑚_𝑗=1𝑐_𝑗0×exp−^𝑡_𝑢𝑎(𝜉)𝑑𝜉𝑑𝑢≤𝜇<1; (63)(𝐻^∗₅)exp(−∫^𝑡₀𝐴(𝜉)𝑑𝜉)→0, as 𝑡→+∞.
Then, the positive 𝜔-periodic solution of (61) is globally attractive, where ℎ_𝑙(𝑡)∶=[𝑑^′_𝑙(𝑡)+𝑑_𝑙(𝑡)𝑎(𝑡)][1−𝜏^′_𝑙(𝑡)]+𝑑_𝑙(𝑡)𝜏_𝑙^′′(𝑡)/[1−𝜏^′_𝑙(𝑡)]².

Remark 19. One could easily see that the systems (5)–(7) are all special cases of the system (61); we can get the similar results, and we omit them here. Hence, our results improve and generalize the corresponding results in [4–7].

4. An Example

Consider the following impulsive model: 𝑑𝑁(𝑡)(𝑡𝑟(𝑡)(𝑡)(𝑡)(𝑡)((𝑡(𝑡)𝑑𝑡=𝑁)−𝑎ln𝑁−𝑏ln𝑁𝑡−𝜎))−𝑐^𝑡_−∞𝐾()(𝑠)(𝑡)((𝑡𝑡−𝑠ln𝑁𝑑𝑠−𝑑𝑑ln𝑁𝑡−𝜏)),𝑑𝑡𝑡≠𝑡_𝑘,Δ𝑁𝑡_𝑘=𝑁𝑡⁺_𝑘−𝑁𝑡_𝑘=𝜃_𝑘𝑁𝑡_𝑘,𝑘=1,2,…, (64) where 𝑟(𝑡),𝑎(𝑡),𝑏(𝑡),𝑐(𝑡),𝑑(𝑡),𝜎(𝑡),𝜏(𝑡) are all positive 𝜔-periodic continuous functions with 𝜎(𝑡)≥0,𝜏(𝑡)≥0,𝑡∈[0,𝜔],𝜎^′(𝑡)<1,𝜏^′(𝑡)<1; furthermore, 𝑑(𝑡)∈𝐶¹(𝑅,𝑅), 𝜏(𝑡)∈𝐶²(𝑅,𝑅), ∫^∞₀𝐾(𝑠)𝑑𝑠=1, ∫₀^+∞𝑠𝐾(𝑠)𝑑𝑠<+∞.

Corollary 20. Suppose that (𝐻₁)--(𝐻₃) and the following conditions hold:(𝐻₄) there exists a constant 𝜂>0 such that |Γ(𝑡)|≥𝜂, for all 𝑡∈[0,𝜔], where Γ(𝑡) is defined by the following: Γ(𝑡)(𝑡)+𝐵(𝜁(𝑡∶=𝐴))1−𝜎^′(𝜁(𝑡(𝑡)−𝐷))+𝐶^′(𝜉(𝑡))1−𝜏^′(𝜉(𝑡;)) (65)(𝐻₅) if (1/2)[(|𝐴|₀+|𝐵|₀)𝜔+|𝐶|₀𝜔^1/2)+|1−𝜏^′|₀^1/2|𝐷|₀<1, and |1−𝜏^′|₀|𝐷|₀<1.
Then, (64) has at least one positive 𝜔-periodic solution, where 𝐴(𝑡)=𝑎(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐵(𝑡)=𝑏(𝑡)_{0<𝑡𝑘<𝑡−𝜎(𝑡)}1+𝜃_𝑘,𝐶(𝑡)=𝑐(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐷(𝑡)=𝑑(𝑡)_{0<𝑡𝑘<𝑡−𝜏(𝑡)}1+𝜃_𝑘. (66)

Corollary 21. Suppose that (𝐻₁)--(𝐻₅) and the following conditions hold:(𝐻₆) there is a positive constant 𝜇 such that |𝐷|₀+^𝑡₀|𝐻|₀+|𝐵|₀+|𝐶|₀×exp−^𝑡_𝑢𝐴(𝜉)𝑑𝜉𝑑𝑢≤𝜇<1; (67)(𝐻₇)exp(−∫^𝑡₀𝐴(𝜉)𝑑𝜉)→0, as 𝑡→+∞.
Then, the positive 𝜔-periodic solution of (64) is globally attractive, where 𝐻(𝑡)∶=[𝐷^′(𝑡)+𝐷(𝑡)𝐴(𝑡)][1−𝜏^′(𝑡)]+𝐷(𝑡)𝜏^′′(𝑡)/[1−𝜏^′(𝑡)]².

Remark 22. The results in the work show that by means of appropriate impulsive perturbations, we can control the dynamics of these equations.

Acknowledgments

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).