The Existence and Uniqueness of Positive Periodic Solutions for a Class of Nicholson-Type Systems with Impulses and Delays

Abstract

By establishing the equivalence, respectively, to the existence and uniqueness of positive periodic solutions for corresponding delay Nicholson-type systems without impulses, some criteria for the existence and uniqueness of positive periodic solutions for a class of Nicholson-type systems with impulses and delays are established. The results of this paper extend some earlier works reported in the literature.

1. Introduction

As we know, in order to describe the dynamics of Nicholson’s blowflies, Gurney et al. [1] proposed a mathematical model 𝑁^′(𝑡)=−𝛿𝑁(𝑡)+𝑃𝑁(𝑡−𝜏)𝑒^{−𝑎𝑁(𝑡−𝜏)}, (1) where 𝑁(𝑡) is the size of the population at time 𝑡, 𝑃 is the maximum per capita daily egg production, 1/𝑎 is the size at which the population reproduces at its maximum rate, 𝛿 is the per capita daily adult death rate, and 𝜏 is the generation time. Nicholson’s blowflies model described by delay differential equation (1) belongs to a class of biological system, and more and more biology systems have attracted more attention because of their extensively realistic significance [2–6]. In particular, 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. There have been some results in the literature of the problem of the existence of positive periodic solutions for Nicholson’s blowflies equation [7–11].

Recently, in order to describe the models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics which are examples of Nicholson-type delay differential systems, Berezansky et al. [12], Wang et al. [13], and Liu [14] studied the following Nicholson-type delay systems: 𝑁^′₁(𝑡)=−𝛼₁(𝑡)𝑁₁(𝑡)+𝛽₁(𝑡)𝑁₂(𝑡)+^𝑚_𝑗=1𝑐_1𝑗(𝑡)𝑁₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝛾1𝑗(𝑡)𝑁1(𝑡−𝜏1𝑗(𝑡))},𝑁^′₂(𝑡)=−𝛼₂(𝑡)𝑁₂(𝑡)+𝛽₂(𝑡)𝑁₁(𝑡)+^𝑚_𝑗=1𝑐_2𝑗(𝑡)𝑁₂𝑡−𝜏_2𝑗(𝑡)𝑒^{−𝛾2𝑗(𝑡)𝑁2(𝑡−𝜏2𝑗(𝑡))}, (2) where 𝛼_𝑖(𝑡), 𝛽_𝑖(𝑡), 𝑐_𝑖𝑗(𝑡), 𝛾_𝑖𝑗(𝑡), and 𝜏_𝑖𝑗(𝑡)∈𝐶(𝑅,(0,∞)), 𝑖=1,2, 𝑗=1,2,…,𝑚. For coefficients and delays that are constants, Berezansky et al. [12] presented several results for the permanence and globally asymptotic stability of system (2). Supposing that 𝛼_𝑖(𝑡), 𝛽_𝑖(𝑡), 𝑐_𝑖𝑗(𝑡), 𝛾_𝑖𝑗(𝑡), and 𝜏_𝑖𝑗(𝑡) are almost periodic functions, Wang et al. [13] obtained some criteria to ensure that the solutions of system (2) converge locally exponentially to a positive almost periodic solution. Furthermore, Liu [14] established some criteria for existence and uniqueness of positive periodic solutions of system (2) by applying the method of the Lyapunov functional.

However, besides delay effects, impulsive effects likewise exist widely in many evolution processes in which states are changed abruptly at certain moment of time, involving such fields as medicine and biology, economics, mechanics, electronics, and telecommunications. That was the reason for the development of the theory of impulsive differential equations. We refer the reader to the monographs [15–17]. In practical, Yan [18] provided the method for studying a class of impulsive differential equations by changing impulsive equations into corresponding equations without impulses.

Therefore, it is necessary and reasonable to consider impulsive effects on the existence and uniqueness of positive periodic solutions for Nicholson-type delay systems (2). However, to the best of our knowledge, there are few results of this problem. Thus, techniques and methods of existence and uniqueness of positive periodic solutions for system (2) with impulsive effects should be developed and explored.

In this paper, we consider a class of Nicholson-type systems with impulses and delays 𝑦^′₁(𝑡)=−𝛼₁(𝑡)𝑦₁(𝑡)+𝛽₁(𝑡)𝑦₂(𝑡)+^𝑚_𝑗=1𝑐_1𝑗(𝑡)𝑦₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝛾1𝑗(𝑡)𝑦1(𝑡−𝜏1𝑗(𝑡))},𝑦^′₂(𝑡)=−𝛼₂(𝑡)𝑦₂(𝑡)+𝛽₂(𝑡)𝑦₁(𝑡)+^𝑚_𝑗=1𝑐_2𝑗(𝑡)𝑦₂⋅𝑡−𝜏_2𝑗(𝑡)𝑒^{−𝛾2𝑗(𝑡)𝑦2(𝑡−𝜏2𝑗(𝑡))},𝑡≥0,𝑡≠𝑡_𝑘,𝑦_𝑖𝑡⁺_𝑘=1+𝑏_𝑘𝑦_𝑖𝑡_𝑘,𝑖=1,2,𝑘=1,2,…, (3) where 𝛼_𝑖(𝑡), 𝛽_𝑖(𝑡), 𝑐_𝑖𝑗(𝑡), 𝛾_𝑖𝑗(𝑡), 𝜏_𝑖𝑗(𝑡)∈𝐶([0,∞),(0,∞)), 𝑖=1,2, 𝑗=1,2,…,𝑚. Δ𝑦_𝑖(𝑡_𝑘)=𝑦_𝑖(𝑡⁺_𝑘)−𝑦_𝑖(𝑡⁻_𝑘) are the impulses at moments 𝑡_𝑘.

In (3), we will use the following hypotheses:(𝐴₁)0<𝑡₁<𝑡₂<⋯ are fixed impulsive points with lim_𝑘→∞𝑡_𝑘=∞;(𝐴₂){𝑏_𝑘} is a real sequence and 𝑏_𝑘>−1,𝑘=1,2,…;(𝐴₃)𝛼_𝑖(𝑡), 𝛽_𝑖(𝑡), 𝑐_𝑖𝑗(𝑡), 𝛾_𝑖𝑗(𝑡), 𝜏_𝑖𝑗(𝑡), and ∏_{0<𝑡𝑘<𝑡}(1+𝑏_𝑘) are periodic functions with common periodic 𝜔>0, 𝑖=1,2, 𝑗=1,2,…,𝑚, and 𝑘=1,2,….

Here and in the sequel we assume that a product equals unit if the number of factor is equal to zero. We will only consider the solutions of (3) with initial values given by 𝑦_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑠∈[−𝜏,0],𝜑_𝑖(0)>0, (4) where 𝜑_𝑖(𝑠)∈𝐶([−𝜏,0],[0,∞)), and 𝜏=max{𝜏⁺_𝑖𝑗}, 𝜏⁺_𝑖𝑗=max_{0≤𝑡≤𝜔}𝜏_𝑖𝑗(𝑡), 𝑖=1,2, 𝑗=1,2,…,𝑚.

The organization of this paper is as follows. In Section 2, we introduce some notations, definitions, and lemmas. In Section 3, some new sufficient conditions ensuring the existence and uniqueness of positive periodic solutions of (3) are derived by changing (3) into a corresponding equation without impulses. We study the existence of positive periodic solutions for the corresponding equation without impulses by coincidence degree theory, and we study the uniqueness of positive periodic solutions for the corresponding equation without impulses by the Lyapunov function. Finally, some conclusions are drawn in Section 4.

2. Preliminaries

For the sake of convenience, throughout this paper, we adopt the following notations: 𝛼⁻_𝑖=min_{0≤𝑡≤𝜔}𝛼_𝑖(𝑡),𝛼⁺_𝑖=max_{0≤𝑡≤𝜔}𝛼_𝑖(𝑡),𝛽⁻_𝑖=min_{0≤𝑡≤𝜔}𝛽_𝑖(𝑡),𝛽⁺_𝑖=max_{0≤𝑡≤𝜔}𝛽_𝑖(𝑡),𝑝⁻_𝑖𝑗=min_{0≤𝑡≤𝜔}𝑝_𝑖𝑗(𝑡),𝑝⁺_𝑖𝑗=max_{0≤𝑡≤𝜔}𝑝_𝑖𝑗(𝑡),𝑞⁻_𝑖𝑗=min_{0≤𝑡≤𝜔}𝑞_𝑖𝑗(𝑡),𝑞⁺_𝑖𝑗=max_{0≤𝑡≤𝜔}𝑞_𝑖𝑗(𝑡),𝑖=1,2,𝑗=1,2,…,𝑚. (5)

Definition 1. A map 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡))^𝑇∶[−𝜏,∞)→(0,∞)×(0,∞) is said to be a solution of (3) on [𝜏,∞) satisfying the initial value condition (4), if(i)𝑦_𝑖(𝑡) is absolutely continuous on each interval (0,𝑡₁] and (𝑡_𝑘,𝑡_𝑘+1],𝑘=1,2,…;(ii)for any 𝑡_𝑘, 𝑘=1,2,…, 𝑦_𝑖(𝑡⁺_𝑘) and 𝑦_𝑖(𝑡⁻_𝑘) exist, and 𝑦_𝑖(𝑡⁻_𝑘)=𝑦_𝑖(𝑡_𝑘);(iii)𝑦_𝑖(𝑡) satisfies (3), 𝑖=1,2.
Under the previous hypotheses (𝐴₁)--(𝐴₃), we consider the following delay differential equation without impulses: 𝑥^′₁(𝑡)=−𝛼₁(𝑡)𝑥₁(𝑡)+𝛽₁(𝑡)𝑥₂(𝑡)+^𝑚_𝑗=1𝑝_1𝑗(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))},𝑥^′₂(𝑡)=−𝛼₂(𝑡)𝑥₂(𝑡)+𝛽₂(𝑡)𝑥₁(𝑡)+^𝑚_𝑗=1𝑝_2𝑗(𝑡)𝑥₂𝑡−𝜏_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2(𝑡−𝜏2j(𝑡))},𝑡≥0, (6) with initial condition (4) 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠)[],],[,for𝑠∈−𝜏,0𝜑∈𝐶([−𝜏,00,∞)),𝜑_𝑖(0)>0, (7) where 𝑝_𝑖𝑗(𝑡)=_{𝑡−𝜏𝑖𝑗(𝑡)≤𝑡𝑘<𝑡}1+𝑏_𝑘⁻¹𝑐_𝑖𝑗(𝑡),𝑞_𝑖𝑗(𝑡)=_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+𝑏_𝑘𝛾_𝑖𝑗(𝑡),𝑖=1,2,𝑗=1,2,…,𝑚. (8)
By a solution 𝑦(𝑡) of (6) on [−𝜏,∞), we mean that an absolutely continuous function on [−𝜏,∞) satisfies (6) for 𝑡≥0 and initial condition (4). Similar to the method of [18], we have the following.

Lemma 2. Assume that (𝐴₁)--(𝐴₃) hold. Then(i)if 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇 is a solution (or positive 𝜔-periodic solution) of (6) on [−𝜏,∞), then 𝑦(𝑡)=(∏_{0<𝑡𝑘<𝑡}(1+𝑏_𝑘)𝑥₁(𝑡),∏_{0<𝑡𝑘<𝑡}(1+𝑏_𝑘)𝑥₂(𝑡))^𝑇 is a solution (or positive 𝜔-periodic solution) of (3) on [−𝜏,∞);(ii)if 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡))^𝑇 is a solution (or positive 𝜔-periodic solution) of (3) on [−𝜏,∞), then 𝑥(𝑡)=(∏_{0<𝑡𝑘<𝑡}(1+𝑏_𝑘)⁻¹𝑦₁(𝑡),∏_{0<𝑡𝑘<𝑡}(1+𝑏_𝑘)⁻¹𝑦₂(𝑡))^𝑇 is a solution (or positive 𝜔-periodic solution) of (6) on [−𝜏,∞).

Proof. First, we prove (i). If 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇 is a solution (or positive 𝜔-periodic solution) of (6) on [−𝜏,∞), then it is easy to see that 𝑦_𝑖(𝑡)  (𝑖=1,2) is absolutely continuous on each interval (0,𝑡₁] and (𝑡_𝑘,𝑡_𝑘+1], 𝑘=1,2,…, and, for any 𝑡≠𝑡_𝑘, 𝑦^′₁(𝑡)+𝛼₁(𝑡)𝑦₁(𝑡)−𝛽₁(𝑡)𝑦₂(𝑡)−^𝑚_𝑗=1𝑐_1𝑗(𝑡)𝑦₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝛾1𝑗(𝑡)𝑦1(𝑡−𝜏1𝑗(𝑡))}=_{0<𝑡𝑘<𝑡}1+𝑏_𝑘𝑥^′₁(𝑡)+𝛼₁(𝑡)_{0<𝑡𝑘<𝑡}1+𝑏_𝑘𝑥₁(𝑡)−𝛽₁(𝑡)_{0<𝑡𝑘<𝑡}1+𝑏_𝑘𝑥₂(𝑡)−^𝑚_𝑗=1𝑐_1𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏1𝑗(𝑡)}1+𝑏_𝑘𝑥₁⋅𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝛾1𝑗(𝑡)∏𝑘0<𝑡<𝑡−𝜏1𝑗(𝑡)(1+𝑏𝑘)𝑥1(𝑡−𝜏1𝑗(𝑡))}=_{0<𝑡𝑘<𝑡}1+𝑏_𝑘𝑥^′₁(𝑡)+𝛼₁(𝑡)𝑥₁(𝑡)−𝛽₁(𝑡)𝑥₂(𝑡)−^𝑚_𝑗=1𝑐_1𝑗(𝑡)_{𝑡−𝜏1𝑗(𝑡)≤𝑡𝑘<𝑡}1+𝑏_𝑘⁻¹𝑥₁⋅𝑡−𝜏_1𝑗(𝑡)⋅𝑒^{−𝛾1𝑗(𝑡)∏𝑘0<𝑡<𝑡−𝜏1𝑗(𝑡)(1+𝑏𝑘)𝑥1(𝑡−𝜏1𝑗(𝑡))}=_{0<𝑡𝑘<𝑡}1+𝑏_𝑘𝑥^′₁(𝑡)+𝛼₁(𝑡)𝑥₁(𝑡)−𝛽₁(𝑡)𝑥₂(𝑡)−^𝑚_𝑗=1𝑝_1𝑗(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)⋅𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}=0. (9) Similarly, we have 𝑦^′₂(𝑡)+𝛼₂(𝑡)𝑦₂(𝑡)−𝛽₂(𝑡)𝑦₁(𝑡)−^𝑚_𝑗=1𝑐_2𝑗(𝑡)𝑦₂𝑡−𝜏_2𝑗(𝑡)𝑒^{−𝛾2𝑗(𝑡)𝑦2(𝑡−𝜏2𝑗(𝑡))}=0. (10) On the other hand, for every 𝑡_𝑘∈{𝑡_𝑘}, 𝑖=1,2, 𝑦_𝑖𝑡⁺_𝑘=lim_{𝑡→𝑡+𝑘0<𝑡𝑗<𝑡}1+𝑏_𝑗𝑥_𝑖(𝑡)=_{0<𝑡𝑗≤𝑡𝑘}1+𝑏_𝑗𝑥_𝑖𝑡_𝑘,𝑦_𝑖𝑡_𝑘=_{0<𝑡𝑗<𝑡𝑘}1+𝑏_𝑗𝑥_𝑖𝑡_𝑘. (11) Thus, for every 𝑘=1,2,…, 𝑦_𝑖𝑡⁺_𝑘=1+𝑏_𝑘𝑦_𝑖𝑡_𝑘. (12) It follows from (9) and (12) that 𝑥(𝑡) is the solution (or positive 𝜔-periodic solution) of (3) corresponding to initial condition (4).
Next, we prove (ii). If 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡))^𝑇 is a solution (or positive 𝜔-periodic solution) of (3) on [−𝜏,∞), then 𝑦_𝑖(𝑡)  (𝑖=1,2) is absolutely continuous on each interval (0,𝑡₁] and (𝑡_𝑘,𝑡_𝑘+1], and in view of (12), it follows that, for any 𝑘=1,2,…, 𝑖=1,2, 𝑥_𝑖𝑡⁺_𝑘=_{0<𝑡𝑗≤𝑡𝑘}1+𝑏_𝑗⁻¹𝑦_𝑖𝑡⁺_𝑘=_{0<𝑡𝑗<𝑡𝑘}1+𝑏_𝑗⁻¹𝑦_𝑖𝑡_𝑘=𝑥_𝑖𝑡_𝑘,𝑥_𝑖𝑡⁻_𝑘=_{0<𝑡𝑗≤𝑡𝑘−1}1+𝑏_𝑗⁻¹𝑦_𝑖𝑡⁻_𝑘=_{0<𝑡𝑗<𝑡𝑘}1+𝑏_𝑗⁻¹𝑦_𝑖𝑡_𝑘=𝑥_𝑖𝑡_𝑘. (13) Equations in (13) imply that 𝑥_𝑖(𝑡) is continuous on [0,∞). It is easy to prove that 𝑥(𝑡) is also absolutely continuous on [0,∞). Now, one can easily check that 𝑥(𝑡)=∏_{0<𝑡𝑘<𝑡}(1+𝑏_𝑘)⁻¹𝑦(𝑡) is the solution (or positive 𝜔-periodic solution) of (6) corresponding to initial condition (4). This completes the proof.

Lemma 3. Assume that (𝐴₁)--(𝐴₃) hold. Then every solution of (3) is defined and positive on [−𝜏,∞).

Proof. Clearly, by Lemma 2, we only need to prove that every solution of (6) is defined and positive on [−𝜏,∞). In order to show that, we only need to see Lemma 2.3 in [13].

In the next section, we will study the existence of positive periodic solution of system (3). The method to be used in this paper involves the applications of the Mawhin’s continuous theorem of the coincidence degree theory. We introduce some concepts and results concerning the coincidence degree as follows. Let 𝑋 and 𝑍 be real Banach spaces, let 𝐿∶Dom𝐿⊂𝑋→𝑍 be a linear mapping, and let 𝑁∶𝑋→𝑍 be a continuous mapping. The mapping 𝐿 is 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, there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 such that Im𝑃=Ker𝐿; and Ker𝑄=Im𝐿=Im(𝐼−𝑄).

Lemma 4 (see [19]). Let 𝑋 be a Banach space. Suppose that 𝐿∶𝐷(𝐿)⊂𝑋→𝑋 is a Fredholm operator with index zero and 𝑁∶Ω→𝑋 is L-compact on Ω with Ω open bounded in 𝑋. Moreover, assume that all the following conditions are satisfied:(i)𝐿𝑥≠𝜆𝑁𝑥,  for all 𝑥∈𝜕Ω∩𝐷(𝐿),𝜆∈(0,1);(ii)𝑄𝑁𝑥≠0, for all 𝑥∈𝜕Ω∩Ker𝐿;(iii)the Brouwer degree deg{𝑄𝑁,Ω∩Ker𝐿,0}≠0. (14)Then equation 𝐿𝑥=𝑁𝑥 has at least one solution in Ω.

For ease of exposition, let 𝑥_𝑖∞=max_{0≤𝑡≤𝜔}𝑥_𝑖(𝑡),𝑢(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡)^𝑇,𝑥_𝑖2=^𝜔₀𝑥_𝑖(𝑡)^21/2,𝑖=1,2,𝑗=1,2,…,𝑚. (15) We denote 𝑋 as the set of all continuously positive 𝜔-periodic functions 𝑥(𝑡) defined on [0,∞) and denote ‖𝑥‖_𝑋=max𝑥_1∞,𝑥_2∞. (16) Then, 𝑋 is a Banach space when it is endowed with the norm ‖𝑥‖_𝑋.

For 𝑢(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇∈𝑋, Let (𝑁𝑢)(𝑡)=−𝛼₁(𝑡)𝑥₁(𝑡)+𝛽₁(𝑡)𝑥₂(𝑡)+∑^𝑚_𝑗=1𝑝_1𝑗(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}−𝛼₂(𝑡)𝑥₂(𝑡)+𝛽₂(𝑡)𝑥₁(𝑡)+∑^𝑚_𝑗=1𝑝_2𝑗(𝑡)𝑦₂𝑡−𝜏_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2(𝑡−𝜏2𝑗(𝑡))}, (17)(𝐿𝑢)(𝑡)=𝑥^′₁(𝑡),𝑥^′₂(𝑡)^𝑇,1𝑃𝑢=𝑄𝑢=𝜔^𝜔₀𝑢(𝑡)=1𝑑𝑡𝜔^𝜔₀𝑥₁(𝑡)1𝑑𝑡,𝜔^𝜔₀𝑥₂(𝑡)𝑑𝑡^𝑇. (18)(19) In view of (17) and (18), the operator equation 𝐿𝑢=𝜆𝑁𝑢 (20) is equivalent to the following: 𝑥^′₁(𝑡)𝑥^′₂(𝑡)=𝜆−𝛼₁(𝑡)𝑥₁(𝑡)+𝛽₁(𝑡)𝑥₂(𝑡)+∑^𝑚_𝑗=1𝑝_1𝑗(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}−𝛼₂(𝑡)𝑥₂(𝑡)+𝛽₂(𝑡)𝑥₁(𝑡)+∑^𝑚_𝑗=1𝑝_2𝑗(𝑡)𝑦₂𝑡−𝜏_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2(𝑡−𝜏2𝑗(𝑡))}, (21)where 𝜆∈(0,1).

Again from (17) and (18), it is not difficult to show that Ker𝐿=𝑅², Im𝐿={𝑢(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇∈𝑋,∫^𝜔₀𝑥₁(𝑡)𝑑𝑡=∫^𝜔₀𝑥₂(𝑡)𝑑𝑡=0} is closed in 𝑋, and dimKer𝐿=codimIm𝐿. From the definitions of continuous projectors 𝑃 and 𝑄, we can easily to get Im𝑃=Ker𝐿,Ker𝑄=Im𝐿. (22) It follows that the operator 𝐿 is a Fredholm operator with index zero. Furthermore, the generalized inverse (of 𝐿) 𝐾_𝑃∶Im𝐿→𝐷(𝐿)∩Ker𝑃 reads as 𝐾_𝑃𝑢(𝑡)=∫^𝑡₀𝑥₁(𝑠)1𝑑𝑠−𝜔∫^𝜔₀∫^𝑡₀𝑥₁(𝑠)∫𝑑𝑠𝑑𝑡_𝑡0𝑥₂(𝑠)1𝑑𝑠−𝜔∫^𝜔₀∫^𝑡₀𝑥₂(𝑠)(𝑡)=𝑥𝑑𝑠𝑑𝑡for𝑢¹(𝑡)𝑥₂(𝑡)∈Im𝐿. (23) Therefore, it is easy to see from (17) and (23) that 𝑁 is 𝐿-compact on Ω, where Ω is any open bounded set in 𝑋.

3. Main Results

Theorem 5. Assume that (𝐴₁)--(𝐴₃) hold. Moreover, the following condition is satisfied:(𝐴₄)  𝐷=min{𝛼⁻₁−𝛽⁺₂,𝛼⁻₂−𝛽⁺₁}>0.
Then (3) has at least one positive 𝜔-periodic solution.

Proof. Based on Lemma 4, what we need to do is just to search for an appropriate open bounded subset Ω for applying Mawhin’s continuous theorem. To do this, it suffices to prove that the set of all possible positive 𝜔-periodic solution of (6) is bounded.
Let 𝑢(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇 be an arbitrary positive 𝜔-periodic solution of (6). Corresponding to the operator equation (21), we have 𝑥^′₁(𝑡)=𝜆−𝛼₁(𝑡)𝑥₁(𝑡)+𝛽₁(𝑡)𝑥₂(𝑡)+^𝑚_𝑗=1𝑝_1𝑗(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))},𝑥^′₂(𝑡)=𝜆−𝛼₂(𝑡)𝑥₂(𝑡)+𝛽₂(𝑡)𝑥₁(𝑡)+^𝑚_𝑗=1𝑝_2𝑗(𝑡)𝑥₂𝑡−𝜏_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2(𝑡−𝜏2𝑗(𝑡))}. (24) Multiplying 𝑥₁(𝑡) and the first formula of (24), and then integrating from 0 to 𝜔, we obtain ^𝜔₀𝑥^′₁(𝑡)𝑥₁(𝑡)𝑑𝑡=𝜆^𝜔₀−𝛼₁(𝑡)𝑥²₁(𝑡)𝑑𝑡+^𝜔₀𝛽₁(𝑡)𝑥₁(𝑡)𝑥₂(𝑡)+𝑑𝑡_{𝑚𝑗=1𝜔0}𝑝_1𝑗(𝑡)𝑥₁(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)⋅𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}𝑑𝑡; (25) hence, ^𝜔₀𝛼₁(𝑡)𝑥²₁(𝑡)≤𝑑𝑡_𝜔0𝛽₁(𝑡)𝑥₁(𝑡)𝑥₂(𝑡)+𝑑𝑡_{𝑚𝑗=1𝜔0}𝑝_1𝑗(𝑡)𝑥₁(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}𝑑𝑡. (26) Furthermore, from the Hölder inequality and 𝑒^𝑥>𝑥 (or 𝑥⋅𝑒^−𝑥<1) for 𝑥>0, we have 𝛼⁻₁𝑥₁²₂≤^𝜔₀𝛼₁(𝑡)𝑥²₁(𝑡)𝑑𝑡≤𝛽⁺₁^𝜔₀𝑥₁(𝑡)²𝑑𝑡^1/2⋅^𝜔₀𝑥₂(𝑡)²𝑑𝑡^1/2+^𝑚_𝑗=1^𝜔₀𝑝_1𝑗(𝑡)𝑥₁(𝑡)𝑞_1𝑗(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑞_1𝑗(𝑡)⋅𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}𝑑𝑡≤𝛽⁺₁𝑥₁₂⋅𝑥₂₂+^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗^𝜔₀𝑥₁(𝑡)𝑑𝑡≤𝛽⁺₁𝑥₁₂⋅𝑥₂₂+^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗√𝜔⋅𝑥₁₂; (27) that is, 𝛼⁻₁𝑥₁₂≤𝛽⁺₁𝑥₂₂+^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗√𝜔. (28) Similarly, we have 𝛼⁻₂𝑥₂₂≤𝛽⁺₂𝑥₁₂+^𝑚_𝑗=1𝑝⁺_2𝑗𝑞⁻_2𝑗√𝜔. (29) Combining (28) and (29), we get 𝛼⁻₁−𝛽⁺₂𝑥₁₂+𝛼⁻₂−𝛽⁺₁𝑥₂₂≤^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗+𝑝⁺_2𝑗𝑞⁻_2𝑗√𝜔. (30) Since 𝐷=min{𝛼⁻₁−𝛽⁺₂,𝛼⁻₂−𝛽⁺₁}>0, (30) implies that |𝑥₁|₂ and |𝑥₂|₂ are bounded. Therefore, according to the previous proof, there exists a positive constant 𝐷₁ (independent of 𝜆) such that 𝑥₁₂≤𝐷₁,𝑥₂₂≤𝐷₁. (31) From (24), together with the Hölder inequality and 𝑥𝑒^−𝑥<1 for 𝑥>0, we can obtain ^𝜔₀𝑥^′₁(𝑡)𝑑𝑡≤𝛼⁺₁^𝜔₀𝑥₁(𝑡)𝑑𝑡+𝛽⁺₁^𝜔₀𝑥₂(𝑡+)𝑑𝑡_𝑚𝑗=1𝑝⁺_1𝑗^𝜔₀𝑞_1𝑗(𝑡)𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑞_1𝑗(𝑡)⋅𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}𝑑𝑡≤𝛼⁺₁√𝜔𝑥₁₂+𝛽⁺₁√𝜔𝑥₂₂+^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗𝜔≜𝐷₂,_𝜔0𝑥^′₂(𝑡)𝑑𝑡≤𝛼⁺₂^𝜔₀𝑥₂(𝑡)𝑑𝑡+𝛽⁺₂^𝜔₀𝑥₁(𝑡+)𝑑𝑡_𝑚𝑗=1𝑝⁺_2𝑗^𝜔₀𝑞_2𝑗(𝑡)𝑥₂𝑡−𝜏_2𝑗(𝑡)𝑞_2𝑗(𝑡)⋅𝑒^{−𝑞2𝑗(𝑡)𝑥2(𝑡−𝜏2𝑗(𝑡))}𝑑𝑡≤𝛼⁺₂√𝜔𝑥₂₂+𝛽⁺₂√𝜔𝑥₁₂+^𝑚_𝑗=1𝑝⁺_2𝑗𝑞⁻_2𝑗𝜔≜𝐷₃. (32) (31)–(32) imply that there exist two positive constants 𝜇_𝑖, 𝑖=1,2, such that ^𝜔₀𝑥^′_𝑖(𝑡)𝑑𝑡≤𝜇_𝑖,𝑖=1,2. (33) Equations in (31) imply that there exist two points 𝜉_𝑖∈[0,𝜔], 𝑖=1,2, and two positive constants 𝑑_𝑖, 𝑖=1,2, such that 𝑥_𝑖𝜉_𝑖≤𝑑_𝑖,𝑖=1,2. (34) Since, for ∀𝑡∈[0,𝜔], 𝑥_𝑖(𝑡)=𝑥_𝑖𝜉_𝑖+^𝑡_𝜉𝑖𝑥^′_𝑖(𝑠)𝑑𝑠≤𝑑_𝑖+^𝜔₀𝑥^′_𝑖(𝑡)𝑑𝑡,𝑖=1,2, (35) from (33), it follows that there exists a positive constant 𝜁 such that 𝑥_𝑖∞≤𝜁,𝑖=1,2. (36) Clearly, 𝜁 is independent of 𝜆. Let 𝐻^∗=𝜁+𝐶, where 𝐶>1 is taken sufficiently large so that 𝐷𝐻^∗>^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗+𝑝⁺_2𝑗𝑞⁻_2𝑗; (37) Now, we take Ω={𝑢(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇∈𝑋,‖𝑢‖_𝑋=max{|𝑥₁|_∞,|𝑥₂|_∞}<𝐻^∗}. This satisfied condition (i) of Lemma 4.
When 𝑢(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇∈𝜕Ω⋂Ker𝐿, (𝑥₁(𝑡),𝑥₂(𝑡))^𝑇 is a constant vector (𝑥₁,𝑥₂)^𝑇 in 𝑅² with ‖𝑢‖_𝑋=max{|𝑥₁|_∞,|𝑥₂|_∞}=𝐻^∗, then𝑥𝑄𝑁¹𝑥₂=−𝑥₁𝜔∫^𝜔₀𝛼₁(𝑡)𝑥𝑑𝑡+²𝜔∫^𝜔₀𝛽₁(𝑡)𝑑𝑡+∑^𝑚_𝑗=1𝑥₁𝜔∫^𝜔₀𝑝_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1}−𝑥𝑑𝑡₂𝜔∫^𝜔₀𝛼₂(𝑡)𝑥𝑑𝑡+¹𝜔∫^𝜔₀𝛽₂(𝑡)𝑑𝑡+∑^𝑚_𝑗=1𝑥₂𝜔∫^𝜔₀𝑝_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2}𝑑𝑡. (38)In view of (38), we have max_{0≤𝑡≤𝜔}𝑄𝑁𝑥₁,𝑥₂^𝑇₁=max_{0≤𝑡≤𝜔}−𝑥₁𝜔^𝜔₀𝛼₁(𝑡)𝑥𝑑𝑡+²𝜔^𝜔₀𝛽₁(𝑡)+𝑑𝑡_𝑚𝑗=1𝑥₁𝜔^𝜔₀𝑝_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1}≥𝑥𝑑𝑡₁𝜔^𝜔₀𝛼₁(𝑡)𝑥𝑑𝑡−²𝜔^𝜔₀𝛽₁(𝑡)−𝑑𝑡_𝑚𝑗=1𝑥₁𝜔^𝜔₀𝑝_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1}𝑑𝑡≥𝛼⁻₁𝑥₁−𝛽⁺₁𝑥₂−^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗⋅𝑞⁻_1𝑗𝑥₁𝑒^{−𝑞−1𝑗𝑥1}≥𝛼⁻₁𝑥₁−𝛽⁺₁𝑥₂−^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗,max_{0≤𝑡≤𝜔}𝑄𝑁𝑥₁,𝑥₂^𝑇₂=max_{0≤𝑡≤𝜔}−𝑥₂𝜔^𝜔₀𝛼₂(𝑡)𝑥𝑑𝑡+¹𝜔^𝜔₀𝛽₂(𝑡)+𝑑𝑡_𝑚𝑗=1𝑥₂𝜔^𝜔₀𝑝_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2}≥𝑥𝑑𝑡₂𝜔^𝜔₀𝛼₂(𝑡)𝑥𝑑𝑡−¹𝜔^𝜔₀𝛽₂(𝑡)−𝑑𝑡_𝑚𝑗=1𝑥₂𝜔^𝜔₀𝑝_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2}𝑑𝑡≥𝛼⁻₂𝑥₂−𝛽⁺₂𝑥₁−^𝑚_𝑗=1𝑝⁺_2𝑗𝑞⁻_2𝑗⋅𝑞⁻_2𝑗𝑥₂𝑒^{−𝑞−2𝑗𝑥2}≥𝛼⁻₂𝑥₂−𝛽⁺₂𝑥₁−^𝑚_𝑗=1𝑝⁺_2𝑗𝑞⁻_2𝑗, (39) Equations in (39), together with (37), imply that 𝑄𝑁(𝑥₁,𝑥₂)^𝑇_𝑋≥12max_{0≤𝑡≤𝜔}𝑄𝑁𝑥₁,𝑥₂^𝑇₁+max_{0≤𝑡≤𝜔}𝑄𝑁𝑥₁,𝑥₂^𝑇₂≥12𝛼⁻₁−𝛽⁺₂𝑥₁+𝛼⁻₂−𝛽⁺₁𝑥₂−^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗+𝑝⁺_2𝑗𝑞⁻_2𝑗≥12𝐷𝐻^∗−^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗+𝑝⁺_2𝑗𝑞⁻_2𝑗>0. (40) Consequently, condition (ii) of Lemma 4 is satisfied.
Furthermore, we define a continuous function 𝜓∶Dom𝐿×[0,1]→𝑋 by 𝜓𝑥₁,𝑥₂−𝛼,𝜇=𝜇⁻¹−𝛽⁺₂𝑥₁−𝛼⁻₂−𝛽⁺₁𝑥₂+⋅−𝑥1−𝜇₁𝜔∫^𝜔₀𝛼₁(𝑡)𝑥𝑑𝑡+²𝜔∫^𝜔₀𝛽₁(𝑡)𝑑𝑡+∑^𝑚_𝑗=1𝑥₁𝜔∫^𝜔₀𝑝_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1}−𝑥𝑑𝑡₂𝜔∫^𝜔₀𝛼₂(𝑡)𝑥𝑑𝑡+¹𝜔∫^𝜔₀𝛽₂(𝑡)𝑑𝑡+∑^𝑚_𝑗=1𝑥₂𝜔∫^𝜔₀𝑝_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2}.𝑑𝑡 (41)When (𝑥₁,𝑥₂)^𝑇∈𝜕Ω⋂Ker𝐿 and 𝜇∈[0,1], (𝑥₁(𝑡),𝑥₂(𝑡))^𝑇 is a constant vector (𝑥₁,𝑥₂)^𝑇 in 𝑅² with ‖(𝑥₁,𝑥₂)^𝑇‖_𝑋=max{|𝑥₁|_∞,|𝑥₂|_∞}=𝐻^∗, then, from (37), we obtain 𝜓(𝑥₁,𝑥₂,𝜇)_𝑋≥12−𝜇𝛼⁻₁−𝛽⁺₂𝑥₁+⋅−𝑥1−𝜇₁𝜔^𝜔₀𝛼₁(𝑡)𝑥𝑑𝑡+²𝜔^𝜔₀𝛽₁(𝑡)+𝑑𝑡_𝑚𝑗=1𝑥₁𝜔^𝜔₀𝑝_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1}+𝛼𝑑𝑡−𝜇⁻₂−𝛽⁺₁𝑥₂+⋅−𝑥1−𝜇₂𝜔^𝜔₀𝛼₂(𝑡)𝑥𝑑𝑡+¹𝜔^𝜔₀𝛽₂(𝑡)+𝑑𝑡_𝑚𝑗=1𝑥₂𝜔^𝜔₀𝑝_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥2}≥1𝑑𝑡2𝜇𝛼⁻₁−𝛽⁺₂𝑥₁+𝑥1−𝜇¹𝜔^𝜔₀𝛼₁(𝑡)−𝑥𝑑𝑡1−𝜇²𝜔^𝜔₀𝛽₁(𝑡)−𝑑𝑡1−𝜇^𝑚_𝑗=1𝑥₁𝜔^𝜔₀𝑝_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1}𝛼𝑑𝑡+𝜇⁻₂−𝛽⁺₁𝑥₂+𝑥1−𝜇²𝜔^𝜔₀𝛼₂(𝑡)−𝑥𝑑𝑡1−𝜇¹𝜔^𝜔₀𝛽₂(𝑡)−𝑑𝑡1−𝜇^𝑚_𝑗=1𝑥₂𝜔^𝜔₀𝑝_2𝑗(𝑡)𝑒^{−𝑞2𝑗(𝑡)𝑥1}≥1𝑑𝑡2𝜇𝛼⁻₁−𝛽⁺₂𝑥₁+1−𝜇𝛼⁻₁𝑥₁−1−𝜇𝛽⁺₁𝑥₂−1−𝜇^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗+𝜇𝛼⁻₂−𝛽⁺₁𝑥₂+1−𝜇𝛼⁻₂𝑥₂−1−𝜇𝛽⁺₂𝑥₁−1−𝜇^𝑚_𝑗=1𝑝⁺_2𝑗𝑞⁻_2𝑗≥12𝛼⁻₁−𝛽⁺₂𝑥₁+𝛼⁻₂−𝛽⁺₁𝑥₂−1−𝜇^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗+𝑝⁺_2𝑗𝑞⁻_2𝑗≥12𝐷𝐻^∗−^𝑚_𝑗=1𝑝⁺_1𝑗𝑞⁻_1𝑗+𝑝⁺_2𝑗𝑞⁻_2𝑗>0. (42) It follows that 𝜓𝑥₁,𝑥₂,𝜇≠(0,0)^𝑇,∀𝑥₁,𝑥₂^𝑇∈𝜕Ω∩Ker𝐿,𝜇∈[0,1]. (43) Hence, using the homotopy invariance theorem, we obtain deg𝑄𝑁,Ω∩Ker𝐿,(0,0)^𝑇=deg−(𝛼⁻₁−𝛽⁺₂)𝑥₁,−(𝛼⁻₂−𝛽⁺₁)𝑥₂^𝑇,Ω∩Ker𝐿,(0,0)^𝑇≠0. (44) Condition (iii) of Lemma 4 is also satisfied. Thus, by Lemma 4, we conclude that 𝐿𝑢=𝑁𝑢,𝑢=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇; that is, (6) has at least one solution in 𝑋. Then, by Lemma 2, we immediately obtain that (3) has at least one positive 𝜔-periodic solution. This completes the proof.

Theorem 6. Let (𝐴₁)--(𝐴₃) hold; furthermore, assume that the following conditions are satisfied:(𝐴₅)min{(𝛼⁻₁−𝛽⁺₂)−∑^𝑚_𝑗=12𝑝⁺_1𝑗,(𝛼⁻₂−𝛽⁺₁)−∑^𝑚_𝑗=12𝑝⁺_2𝑗}>0, (𝐴₆)𝜏_𝑖𝑗(𝑡)∈𝐶¹([0,∞),(0,∞)) and 𝜏^′_𝑖𝑗(𝑡)≤0, 𝑖=1,2, 𝑗=1,2,…,𝑚.
Then (3) has a unique positive 𝜔-periodic solution.

Proof. By Lemma 2, it suffices to prove the uniqueness of positive 𝜔-periodic solutions for system (6). According to Theorem 5, we know that (6) has at least a positive 𝜔-periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))^𝑇 with initial condition (4). Suppose that 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇 is an arbitrary positive 𝜔-periodic solution of (6) with initial condition (4). Then it follows from (6) that 𝑑𝑥𝑑𝑡¹(𝑡)−𝑥^∗₁(𝑡)=−𝛼₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝛽₁(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡)+^𝑚_𝑗=1𝑝_1𝑗(𝑡)⋅𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}−𝑥^∗₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥∗1(𝑡−𝜏1𝑗(𝑡))}. (45) Calculating the upper-right derivative, we have 𝑑⁺𝑥𝑑𝑡¹(𝑡)−𝑥^∗₁(𝑡)≤−𝛼⁻₁𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝛽⁺₁𝑥₂(𝑡)−𝑥^∗₂(𝑡+_𝑚𝑗=1)𝑝⁺_1𝑗𝑥₁𝑡−𝜏_1𝑗(𝑡)⋅𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1j(𝑡))}−𝑥^∗₁𝑡−𝜏_1𝑗(𝑡)⋅𝑒^{−𝑞1𝑗(𝑡)𝑥∗1(𝑡−𝜏1𝑗(𝑡))}. (46) For 𝑦=𝑥𝑒^−𝑟𝑥, 𝑟>0 is a real number, 0<𝑎≤𝑥≤𝑏, and by mean value theorem, we have 𝑏𝑒^−𝑟𝑏−𝑎𝑒^−𝑟𝑎=𝑎𝑒^−𝑟𝑎−𝑏𝑒^−𝑟𝑏=𝑎𝑒^−𝑟𝑎−𝑎𝑒^−𝑟𝑏+𝑎𝑒^−𝑟𝑏−𝑏𝑒^−𝑟𝑏≤−𝑎𝑟𝑒^−𝑟𝜉|𝑏−𝑎|+𝑒^−𝑟𝑏|=|𝑏−𝑎𝑎𝑟𝑒^−𝑟𝑎⋅𝑒^{−𝑟(𝜉−𝑎)}|𝑏−𝑎|+𝑒^−𝑟𝑏|||,|𝑏−𝑎≤2𝑏−𝑎 (47) where 𝑎<𝜉<𝑏. Thus, for any fixed 𝑗=1,2,…,𝑚, we also have 𝑥^∗₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥∗1(𝑡−𝜏1𝑗(𝑡))}−𝑥₁𝑡−𝜏_1𝑗(𝑡)𝑒^{−𝑞1𝑗(𝑡)𝑥1(𝑡−𝜏1𝑗(𝑡))}≤2𝑥₁𝑡−𝜏_1𝑗(𝑡)−𝑥^∗₁𝑡−𝜏_1𝑗(𝑡). (48) Hence, 𝑑⁺𝑥𝑑𝑡¹(𝑡)−𝑥^∗₁(𝑡)≤−𝛼⁻₁𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝛽⁺₁𝑥₂(𝑡)−𝑥^∗₂(𝑡+_𝑚𝑗=1)2𝑝⁺_1𝑗𝑥₁𝑡−𝜏_1𝑗(𝑡)−𝑥^∗₁𝑡−𝜏_1𝑗(𝑡). (49) Similarly, we have 𝑑⁺𝑥𝑑𝑡²(𝑡)−𝑥^∗₂(𝑡)≤−𝛼⁻₂𝑥₂(𝑡)−𝑥^∗₂(𝑡)+𝛽⁺₂𝑥₁(𝑡)−𝑥^∗₁(𝑡+_𝑚𝑗=1)2𝑝⁺_2𝑗𝑥₂𝑡−𝜏_2𝑗(𝑡)−𝑥^∗₂𝑡−𝜏_2𝑗(𝑡). (50) We define a Lyapunov function 𝑉(⋅) by 𝑉(𝑡)=𝑉𝑥₁,𝑥₂(𝑡)=²_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡+_{𝑚𝑗=1𝑡𝑡−𝜏1𝑗(𝑡)})2𝑝⁺_1𝑗𝑥₁(𝑠)−𝑥^∗₁(𝑠+)𝑑𝑠_{𝑚𝑗=1𝑡𝑡−𝜏2𝑗(𝑡)}2𝑝⁺_2𝑗𝑥₂(𝑠)−𝑥^∗₂(𝑠)𝑑𝑠, (51) for 𝑡>0, and by virtue of (49), (50), and assumption (𝐴₆), we get 𝑑⁺𝑉(𝑡)𝛼𝑑𝑡≤−⁻₁−𝛽⁺₂𝑥₁(𝑡)−𝑥^∗₁(𝑡−𝛼₋₂)−𝛽⁺₁𝑥₂(𝑡)−𝑥^∗₂(𝑡+_𝑚𝑗=1)2𝑝⁺_1𝑗𝑥₁𝑡−𝜏_1𝑗(𝑡)−𝑥^∗₁𝑡−𝜏_1𝑗(𝑡)+^𝑚_𝑗=12𝑝⁺_2𝑗𝑥₂𝑡−𝜏_2𝑗(𝑡)−𝑥^∗₂𝑡−𝜏_2𝑗(𝑡)+^𝑚_𝑗=12𝑝⁺_1𝑗𝑥₁(𝑡)−𝑥^∗₁(𝑡)+^𝑚_𝑗=12𝑝⁺_2𝑗𝑥₂(𝑡)−𝑥^∗₂(𝑡−_𝑚𝑗=1)2𝑝⁺_1𝑗𝑥₁𝑡−𝜏_1𝑗(𝑡)−𝑥^∗₁𝑡−𝜏_1𝑗(𝑡⋅)1−𝜏^′_1𝑗(𝑡)−^𝑚_𝑗=12𝑝⁺_2𝑗𝑥₂𝑡−𝜏_2𝑗(𝑡)−𝑥^∗₂𝑡−𝜏_2𝑗(𝑡)⋅1−𝜏^′_2𝑗(𝑡)≤−𝛼⁻₁−𝛽⁺₂−^𝑚_𝑗=12𝑝⁺_1𝑗𝑥₁(𝑡)−𝑥^∗₁(𝑡−𝛼₋₂)−𝛽⁺₁−^𝑚_𝑗=12𝑝⁺_2𝑗𝑥₂(𝑡)−𝑥^∗₂(𝑡𝛼)≤−min⁻₁−𝛽⁺₂−^𝑚_𝑗=12𝑝⁺_1𝑗,𝛼⁻₂−𝛽⁺₁−^𝑚_𝑗=12𝑝⁺_2𝑗⋅𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝑥₂(𝑡)−𝑥^∗₂(𝑡). (52) According to (𝐴₅), min𝛼⁻₁−𝛽⁺₂−^𝑚_𝑗=12𝑝⁺_1𝑗,𝛼⁻₂−𝛽⁺₁−^𝑚_𝑗=12𝑝⁺_2𝑗>0, (53) and it follows that 𝑑⁺𝑉(𝑡)𝑑𝑡≤0,𝑡>0. (54) Hence, we obtain 𝑉(𝑡)≤𝑉(0),∀𝑡>0. (55) So ²_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡(𝑡)(0)=)≤𝑉≤𝑉_2𝑖=1𝑥_𝑖(0)−𝑥^∗_𝑖(0)=0,∀𝑡>0. (56) In view of (56) and periodicity of 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡), we have 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)=0,∀𝑡>0,𝑖=1,2. (57) Then by Lemma 2, we conclude that (3) has a unique positive 𝜔-periodic solution. This completes the proof.

4. Conclusion

In this paper, a class of Nicholson-type systems with impulses and delays are investigated. We conquer the difficulty of coexistence of impulsive and delay factors in a dynamic system and give some results of the existence and uniqueness of positive periodic solutions. The results in this paper extend some earlier works reported in the literature. Moreover, our results are easy to test and important in applications of periodic oscillatory delayed Nicholson-type systems with impulsive control.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11171374) and the Scientific Research Fund of Shandong Provincial of China (Grant no. ZR2011AZ001).