Application of Mawhin's Coincidence Degree and Matrix Spectral Theory to a Delayed System

Abstract

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model with M-predators and N-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.

1. Introduction and Motivation

1.1. History and Motivations

Mawhin's coincidence degree theory has been applied extensively to study the existence of periodic solutions for nonlinear differential systems (e.g. see [1–16] and references therein). The most important step of applying Mawhin’s degree theory to nonlinear differential equations is to obtain the priori bounds of unknown solutions to the operator equation 𝐿𝑥=𝜆𝑁𝑥. However, different estimation techniques for the priori bounds of unknown solutions to the equation 𝐿𝑥=𝜆𝑁𝑥 may lead to different results. Most of papers obtained the priori bounds by employing the inequalities: 𝑥(𝑡)≤𝑥(𝜉)+^𝜔₀|𝑥̇(𝑡)|𝑑𝑡,𝑥(𝑡)≥𝑥𝜂−^𝜔₀|𝑥̇(𝑡𝑥(𝜉))|𝑑𝑡,=min_{𝑡∈[0,𝜔]}𝑥(𝑡),𝑥𝜂=max_{𝑡∈[0,𝜔]}𝑥(𝑡). (1.1) These inequalities lead to a relatively strong condition given in terms of algebraic inequality or classic norms (see e.g., [3–16]). Different from standard consideration, in this paper, we employ matrix spectral theory to obtain the priori bounds, not the above inequalities. So in this paper, the existence and stability of periodic solution for a multispecies predator-prey model is studied by jointly employing Mawhin's coincidence degree and matrix spectral theory.

1.2. Model Formulation

One of classical Lotka-Vlterra system is predator-prey models which have been investigated extensively by mathematicians and ecologist. Many good results have been obtained for stability, bifurcations, chaos, uniform persistence, periodic solution, almost periodic solutions. It has been observed that most of works focus on either two or three species model. There are few paper considering the multispecies model. To model the dynamic behavior of multispecie predator-prey system, Yang and Rui [17] proposed a predator-prey model with M-predators and N-preys of the form: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑁_𝑘=1𝑎_𝑖𝑘(𝑡)𝑥_𝑘(𝑡)−^𝑀_𝑙=1𝑐_𝑖𝑙(𝑡)𝑦_𝑙(𝑡𝑦̇),𝑖=1,2,…,𝑁,_𝑗(𝑡)=𝑦_𝑗(𝑡)−𝑟_𝑗(𝑡)+^𝑁_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘(𝑡)−^𝑀_𝑙=1𝑒_𝑗𝑙(𝑡)𝑦_𝑙(𝑡),𝑗=1,2,…,𝑀, (1.2) where 𝑥_𝑖(𝑡) denotes the density of prey species 𝑋_𝑖 at time 𝑡, 𝑦_𝑗(𝑡) denotes the density of predator species 𝑌_𝑗 at time 𝑡. The coefficients 𝑏_𝑖(𝑡),𝑟_𝑗(𝑡),𝑎_𝑖𝑘(𝑡),𝑐_𝑖𝑙(𝑡),𝑑_𝑗𝑘(𝑡), and 𝑒_𝑗𝑙(𝑡),(𝑖,𝑘=1,…,𝑁;𝑗,𝑙=1,…,𝑀) are nonnegative continuous periodic functions defined on 𝑡∈(−∞,+∞). The coefficient 𝑏_𝑖 is the intrinsic growth rate of prey species 𝑋_𝑖, 𝑟_𝑗 is the death rate of the predator species 𝑌_𝑗, 𝑎_𝑖𝑘 measures the amount of competition between the prey species 𝑋_𝑖 and 𝑋_𝑘(𝑘≠𝑖,𝑖,𝑘=1,…,𝑁), 𝑒_𝑗𝑙 measures the amount of competition between the predator species 𝑌_𝑗 and 𝑌_𝑘(𝑘≠𝑗,𝑗,𝑘=1,…,𝑀), and the constant 𝑘^Δ_𝑖𝑗=𝑑_𝑖𝑗/𝑐_𝑖𝑗 denotes the coefficient in conversing prey species 𝑋_𝑖 into new individual of predator species 𝑌_𝑗(𝑖=1,…,𝑁;𝑗=1,…,𝑀). By using the differential inequality, Zhao and Chen [18] improved the results of Yang and Rui [17]. Recently, Xia et al. [19] obtained some sufficient conditions for the existence and global attractivity of a unique almost periodic solution of the system (1.2).

It is more natural to consider the delay model because most of the species start interacting after reaching a maturity period. Hence many scholars think that the delayed models are more realistic and appropriate to be studied than ordinary model. Delayed system is important also because sometimes time delays may lead to oscillation, bifurcation, chaos, instability which may be harmful to a system. Inspired by the above argument, Wen [20] considered a periodic delayed multispecies predator-prey system as follows: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘(𝑡)𝑥_𝑘𝑡−𝜏_𝑖𝑘−^𝑀_𝑙=1𝑐_𝑖𝑙(𝑡)𝑦_𝑙𝑡−𝜂_𝑖𝑙,𝑦̇_𝑗(𝑡)=𝑦_𝑗(𝑡)−𝑟_𝑗(𝑡)+^𝑁_𝑘=1𝑑_𝑗𝑘(𝑡)𝑥_𝑘𝑡−𝛿_𝑗𝑘−𝑒_𝑗𝑗(𝑡)𝑦_𝑗(𝑡)−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙(𝑡)𝑦_𝑙𝑡−𝜉_𝑗𝑙, (1.3) where 𝑏_𝑖(𝑡), 𝑟_𝑗(𝑡), 𝑎_𝑖𝑘(𝑡), 𝑐_𝑖𝑙(𝑡), 𝑑_𝑗𝑘(𝑡), 𝑒_𝑗𝑙(𝑡), 𝑒_𝑗𝑙(𝑡)(𝑖,𝑘=1,2,…,𝑁;𝑗,𝑙=1,2,…,𝑀) are assumed to be continuous 𝜔-periodic functions and the delays 𝜏_𝑖𝑘, 𝛿_𝑗𝑘, 𝜂_𝑖𝑙, 𝜉_𝑗𝑙 are assumed to be positive constants. The system (1.3) is supplemented with the initial condition: 𝑥_𝑖(𝜃)=𝜙_𝑖(𝜃),𝑦_𝑗(𝜃)=𝜓_𝑖(𝜃),𝜃∈[−𝜏,0],𝜙_𝑖(0)>0,𝜓_𝑖(0)>0, (1.4) where 𝜏=maxmax_{1≤𝑖,𝑘≤𝑛}𝜏_𝑖𝑘,max_{1≤𝑖≤𝑁,1≤𝑙≤𝑀}𝜂_𝑖𝑙,max_{1≤𝑗,𝑙≤𝑀}𝜉_𝑗𝑙,max_{1≤𝑗≤𝑀,1≤𝑘≤𝑁}𝛿_𝑗𝑘>0. (1.5) It is easy to see that for such given initial conditions, the corresponding solution of the system (1.3) remains positive for all 𝑡≥0. The purpose of this paper is to obtain some new and interesting criteria for the existence and global asymptotic stability of periodic solution of the system (1.3).

1.3. Comparison with Previous Work

To obtain the periodic solutions of the system (1.3), the method used in [20] is based on employing the differential inequality and Brower fixed point theorem. Different from consideration taken by [20], our method is based on combining matrix spectral theory with Mawhin's degree theory. In our method, we study the global asymptotic stability by combining matrix's spectral theory with Lyapunov functional method. The existence and stability conditions are given in terms of spectral radius of explicit matrices. These conditions are much different from the sufficient conditions obtained in [20].

1.4. Outline of This Work

The structure of this paper is as follows. In Section 2, some new and interesting sufficient conditions for the existence of periodic solution of system (1.3) are obtained. Section 3 is devoted to examining the stability of the periodic solution obtained in the previous section. In Section 4, some corollaries are presented to show the effectiveness of our results. Finally, an example is given to show the feasibility of our results.

2. Existence of Periodic Solutions

In this section, we will obtain some sufficient conditions for the existence of periodic solution of the system (1.3).

2.1. Preliminaries on the Matrix Theory and Degree Theory

For convenience, we introduce some notations, definitions, and lemmas. Throughout this paper, we use the following notations.(i)We always use 𝑖,𝑘=1,…,𝑁;𝑗,𝑙=1,…,𝑀, unless otherwise stated.(ii)If 𝑓(𝑡) is a continuous 𝜔-periodic function defined on 𝑅, then we denote 𝑓=min_{[]𝑡∈0,𝜔}𝑓(𝑡),𝑓=max_{[]𝑡∈0,𝜔}𝑓(𝑡𝑓=1),𝑚𝜔^𝜔₀𝑓(𝑡)𝑑𝑡. (2.1)We use 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈ℝ^𝑛 to denote a column vector, 𝒟=(𝑑_𝑖𝑗)_𝑛×𝑛 is an 𝑛×𝑛 matrix, 𝒟^𝑇 denotes the transpose of 𝒟, and 𝐸_𝑛 is the identity matrix of size 𝑛. A matrix or vector 𝒟>0 (resp., 𝒟≥0) means that all entries of 𝒟 are positive (resp., nonnegative). For matrices or vectors 𝒟 and 𝐸, 𝒟>𝐸 (resp., 𝒟≥𝐸) means that 𝒟−𝐸>0 (resp., 𝒟−𝐸≥0). We denote the spectral radius of the matrix 𝒟 by 𝜌(𝒟).

If 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)^𝑇∈ℝ^𝑛, then we have a choice of vector norms in ℝ^𝑛, for instance ‖𝑣‖₁, ‖𝑣‖₂, and ‖𝑣‖_∞ are the commonly used norms, where ‖𝑣‖₁=^𝑛_𝑗=1𝑣_𝑖,‖𝑣‖₂=^𝑛_𝑗=1𝑣_𝑖^21/2,‖𝑣‖_∞=max_{1≤𝑖≤𝑛}𝑣_𝑖. (2.2) We recall the following norms of matrices induced by respective vector norms. For instance if 𝒜=(𝑎_𝑖𝑗)_𝑛×𝑛, the norm of the matrix ‖𝒜‖ induced by a vector norm ‖⋅‖ is defined by ‖𝒜‖_𝑝=sup_{𝑣∈ℝ𝑛,𝑣≠0}‖𝒜𝑣‖_𝑝‖𝑣‖_𝑝=sup_{‖𝑣‖𝑝=1}‖𝒜𝑣‖_𝑝=sup_{‖𝑣‖𝑝≤1}‖𝒜𝑣‖_𝑝. (2.3) In particular one can show that ‖𝒜‖₁=max_{1≤𝑗≤𝑛}∑^𝑛_𝑖=1|𝑎_𝑖𝑗| (column norm), ‖𝒜‖₂=[𝜆_max(𝒜^𝑇𝒜)]^1/2=[max.eigenvalueof(𝒜^𝑇𝒜)]^1/2and‖𝒜‖_∞=max_{1≤𝑖≤𝑛}∑^𝑛_𝑗=1|𝑎_𝑖𝑗| (row norm).

Definition 2.1 (see [1, 21]). Let 𝑋,𝑍 be normed real Banach spaces, let 𝐿∶Dom𝐿⊂𝑋→𝑍 be a linear mapping, and 𝑁∶𝑋→𝑍 be a continuous mapping. The mapping 𝐿 is called a Fredholm mapping of index zero, if dimKer 𝐿=codimIm𝐿<+∞ and Im 𝐿 is closed in 𝑍. If 𝐿 is a Fredholm mapping of index zero and there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 such that Im 𝑃=Ker𝐿, Ker𝑄=Im𝐿=Im(𝐼−𝑄), then 𝐿∣dom𝐿∩Ker𝑃∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. We denote the inverse of that map by 𝐾_𝑃. If Ω is an open bounded subset of 𝑋, the mapping 𝑁 will be called 𝐿-compact on Ω if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact. Since Im𝑄 is isomorphic to Ker𝐿, there exists an isomorphism 𝐽∶Im𝑄→Ker𝐿.

Definition 2.2 (see [1, 22]). Let Ω⊂ℝ^𝑛 be open and bounded, 𝑓∈𝐶¹(Ω,ℝ^𝑛)∩𝐶(Ω,ℝ^𝑛) and 𝑦∈ℝ^𝑛/𝑓(𝜕Ω∪𝑁_𝑓), that is, 𝑦 is a regular value of 𝑓. Here, 𝑁_𝑓={𝑥∈Ω∶𝐽_𝑓(𝑥)=0}, the critical set of 𝑓 and 𝐽_𝑓 is the Jacobian of 𝑓 at 𝑥. Then the degree deg{𝑓,Ω,𝑦} is defined by deg𝑓,Ω,𝑦=_{𝑥∈𝑓−1𝑦}sgn𝐽_𝑓(𝑥), (2.4) with the agreement that ∑𝜙=0. For more details about Degree Theory, the reader may consult Deimling [22].

Lemma 2.3 (Continuation Theorem [1]). Let Ω⊂𝑋 be an open and bounded set and 𝐿 be a Fredholm mapping of index zero and 𝑁 be 𝐿-compact on Ω (i.e., 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact). Assume(i) for each 𝜆∈(0,1), 𝑥∈𝜕Ω∩Dom𝐿, 𝐿𝑥≠𝜆𝑁𝑥;(ii) for each 𝑥∈𝜕Ω∩Ker𝐿, 𝑄𝑁𝑥≠0 and deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0. Then 𝐿𝑥=𝑁𝑥 has at least one solution in Ω∩Dom𝐿.

Definition 2.4 (see [23, 24]). A real 𝑛×𝑛 matrix 𝒜=(𝑎_𝑖𝑗) is said to be an 𝑀-matrix if 𝑎_𝑖𝑗≤0, 𝑖,𝑗=1,2,…,𝑛, 𝑖≠𝑗, and 𝒜⁻¹≥0.

Lemma 2.5 (see [23, 24]). Let 𝒜≥0 be an 𝑛×𝑛 matrix and 𝜌(𝒜)<1, then (𝐸_𝑛−𝒜)⁻¹≥0, where 𝐸_𝑛 denotes the identity matrix of size 𝑛.

Now we introduce some function spaces and their norms, which will be valid throughout this paper. Denote 𝑋=𝑈(𝑡)=(𝑢(𝑡),𝑣(𝑡))^𝑇∈𝐶¹ℝ,ℝ^𝑁+𝑀()(𝑡),𝑈(𝑡)=(𝑢(𝑡)(𝑡∣𝑈𝑡+𝜔=𝑈∀𝑡∈ℝ𝑍=,𝑣))^𝑇∈𝐶ℝ,ℝ^𝑁+𝑀∣𝑈(𝑡+𝜔)=𝑈(𝑡)∀𝑡∈ℝ. (2.5) The norms are given by 𝑈_𝑛(𝑡)₀=max_{𝑡∈[0,𝜔]}𝑈_𝑛(𝑡),𝑈_𝑛(𝑡)₁=𝑈_𝑛(𝑡)₀+𝑈̇_𝑛(𝑡)₀𝑈,𝑖=1,2,…,𝑁+𝑀,_𝑛(𝑡)₀=max_{1≤𝑛≤𝑁+𝑀}𝑈_𝑛(𝑡)₀,𝑈_𝑛(𝑡)₁=max_{1≤𝑛≤𝑁+𝑀}𝑈_𝑛(𝑡)₁. (2.6) Obviously, 𝑋 and 𝑍, respectively, endowed with the norms ‖⋅‖₁ and ‖⋅‖₀ are Banach spaces.

2.2. Result on the Existence of Periodic Solutions

Theorem 2.6. Assume that the following conditions hold:(𝐻₁): the system of algebraic equations: 𝑚𝑏_𝑖−𝑚𝑎_𝑖𝑖𝑢_𝑖−^𝑁_{𝑘=1,𝑘≠𝑖}𝑚𝑎_𝑖𝑘𝑢_𝑘−^𝑀_𝑙=1𝑚𝑐_𝑖𝑙𝑤_𝑙𝑚=0,𝑖=1,2,…,𝑁,−𝑟_𝑗+^𝑁_𝑘=1𝑚𝑑_𝑗𝑘𝑢_𝑘−𝑚𝑒_𝑗𝑗𝑤_𝑗−^𝑀_{𝑙=1,𝑙≠𝑗}𝑚𝑒_𝑗𝑙𝑤_𝑙=0,𝑗=1,2,…,𝑀, (2.7) has finite solution (𝑢^∗₁,…,𝑢^∗_𝑁,𝑤^∗₁,…,𝑤^∗_𝑀)^𝑇∈ℝ₊^𝑁+𝑀 with 𝑢^∗>0,𝑤^∗>0; (𝐻₂):𝜌(𝒦)<1, where 𝒦=^{𝒫𝑁×𝑁𝒬𝑁×𝑀}_{ℳ𝑀×𝑁𝒩𝑀×𝑀(𝑁+𝑀)×(𝑁+𝑀)}, 𝒫_𝑁×𝑁=𝑝_{𝑖𝑘𝑁×𝑁},𝑝_𝑖𝑘=0,𝑖=𝑘,𝑎_𝑖𝑘𝑎⁻¹_𝑘𝑘𝒬,𝑖≠𝑘._𝑁×𝑀=𝑞_{𝑖𝑙𝑁×𝑀},𝑞_𝑖𝑙=𝑐_𝑖𝑙𝑒⁻¹_𝑙𝑙,ℳ_𝑀×𝑁=𝑚_{𝑗𝑘𝑀×𝑁},𝑚_𝑗𝑘=𝑑_𝑗𝑘𝑎⁻¹_𝑘𝑘,𝒩_𝑀×𝑀=𝑛_{𝑗𝑙𝑀×𝑀},𝑛_𝑗𝑙=0,𝑗=𝑙,𝑒_𝑗𝑙𝑒⁻¹_𝑙𝑙,𝑗≠𝑙. (2.8)
Then system (1.3) has at least one positive 𝜔-periodic solution.

Proof. Note that every solution 𝑈(𝑡)=(𝑢(𝑡),𝑣(𝑡))^𝑇=𝑢₁(𝑡),…,𝑢_𝑁(𝑡),𝑣₁(𝑡),…,𝑣_𝑀(𝑡)^𝑇∈𝑋 (2.9) of the system (1.3) with the initial condition is positive. By using the following changes of variables: 𝑢_𝑖(𝑡)=ln𝑥_𝑖(𝑡),𝑣_𝑗(𝑡)=ln𝑦_𝑗(𝑡),𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑀, (2.10) the system (1.3) can be rewritten as 𝑢̇_𝑖(𝑡)=𝑏_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑒^{𝑢𝑖(𝑡)}−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘(𝑡)𝑒^{𝑢𝑘(𝑡−𝜏𝑖𝑘)}−^𝑀_𝑙=1𝑐_𝑖𝑙(𝑡)𝑒^{𝑣𝑙(𝑡−𝜂𝑖𝑙)}𝑣̇,𝑖=1,2,…,𝑁,_𝑗(𝑡)=−𝑟_𝑗(𝑡)+^𝑁_𝑘=1𝑑_𝑗𝑘(𝑡)𝑒^{𝑢𝑘(𝑡−𝛿𝑗𝑘)}−𝑒_𝑗𝑗(𝑡)𝑒^{𝑣𝑗(𝑡)}−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙(𝑡)𝑒^{𝑣𝑙(𝑡−𝜉𝑗𝑙)},𝑗=1,2,…,𝑀. (2.11) Obviously, system (1.3) has at least one 𝜔-periodic solution which is equivalent to the system (2.11) having at least one 𝜔-periodic solution. To prove Theorem 2.6, our main tasks are to construct the operators (i.e., 𝐿, 𝑁, 𝑃, and 𝑄) appearing in Lemma 2.3 and to find an appropriate open set Ω satisfying conditions (i), (ii) in Lemma 2.3.
For any 𝑈(𝑡)∈𝑋, in view of the periodicity, it is easy to check that Δ_𝑖(𝑈,𝑡)=𝑏_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑒^{𝑢𝑖(𝑡)}−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘(𝑡)𝑒^{𝑢𝑘(𝑡−𝜏𝑖𝑘)}−^𝑀_𝑙=1𝑐_𝑖𝑙(𝑡)𝑒^{𝑣𝑙(𝑡−𝜂𝑖𝑙)}Δ∈𝑍,_𝑗(𝑈,𝑡)=−𝑟_𝑗(𝑡)+^𝑁_𝑘=1𝑑_𝑗𝑘(𝑡)𝑒^{𝑢𝑘(𝑡−𝛿𝑗𝑘)}−𝑒_𝑗𝑗(𝑡)𝑒^{𝑣𝑗(𝑡)}−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙(𝑡)𝑒^{𝑣𝑙(𝑡−𝜉𝑗𝑙)}∈𝑍. (2.12) Now, we define the operators 𝐿,𝑁 as follows: 𝐿∶Dom𝐿⊂𝑋→𝑍,𝐿(𝑢(𝑡),𝑣(𝑡))=𝑑𝑢_𝑖(𝑡),𝑑𝑡𝑑𝑣_𝑗(𝑡)Δ𝑑𝑡∈𝑍,𝑁∶𝑋→𝑍isdefinedby𝑁𝑈=^𝑖()Δ𝑈,𝑡_𝑗().𝑈,𝑡 (2.13) Define, respectively, the projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 by 1𝑃𝑈=𝜔^𝜔₀𝑈(𝑡)1𝑑𝑡,𝑈∈𝑋,𝑄𝑈=𝜔^𝜔₀𝑈(𝑡)𝑑𝑡,𝑈∈𝑍. (2.14) It is obvious that the domain of 𝐿 in 𝑋 is actually the whole space, and Ker𝐿={𝑥(𝑡)∈𝑋∣𝐿𝑥(𝑡)=0,i.e.𝑥̇(𝑡)=0}=ℝ^𝑁+𝑀,Im𝐿=𝑧(𝑡)∈𝑍∣^𝜔₀𝑧(𝑡)𝑑𝑡=0isclosedin𝑍. (2.15) Moreover, 𝑃,𝑄 are continuous operators such that Im𝑃=ℝ^𝑁=Ker𝐿,Im𝐿=Ker𝑄=Im(𝐼−𝑄),dimKer𝐿=codimIm𝐿=𝑁+𝑀<+∞. (2.16) It follows that 𝐿 is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to 𝐿) 𝐾_𝑃∶Im𝐿→Dom𝐿∩Ker𝑃 exists, which is given by 𝐾_𝑃𝑦=^𝑡₀𝑦(𝑠)1𝑑𝑠−𝜔^𝜔₀^𝑡₀𝑦(𝑠)𝑑𝑠𝑑𝑡. (2.17) Then 𝑄𝑁∶𝑋→𝑍 and 𝐾_𝑃(𝐼−𝑄)𝑁∶𝑋→𝑋 are defined by 1𝑄𝑁𝑈=𝜔∫^𝜔₀Δ_𝑖()1𝑈,𝑡𝑑𝑡𝜔∫^𝜔₀Δ_𝑗(),𝐾𝑈,𝑡𝑑𝑡_𝑃(𝐼−𝑄)𝑁𝑥=^𝑡₀(𝑠)1𝑁𝑈𝑑𝑠−𝜔^𝜔₀^𝑡₀(𝑠)𝑡𝑁𝑈𝑑𝑠𝑑𝑡−𝜔−12^𝜔₀𝑁𝑈(𝑠)𝑑𝑠. (2.18) Clearly, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. By using the generalized Arzela-Ascoli theorem, it is not difficult to prove that (𝐾_𝑃(𝐼−𝑄)𝑁)(Ω) is relatively compact in the space (𝑋,‖⋅‖₁). The proof of this step is complete.
Then, in order to apply condition (i) of Lemma 2.3, we need to search for an appropriate open bounded subset Ω, denoted by Ω=𝑈_𝑛(𝑡)∈𝑋∣𝑈_𝑛(𝑡)₁=𝑈_𝑛(𝑡)₀+𝑈̇_𝑛(𝑡)₀<ℎ_𝑛. (2.19) Specifically, our aim is to find an appropriate ℎ_𝑛. Corresponding to the operator equation 𝐿𝑥=𝜆𝑁𝑥 for each 𝜆∈(0,1), we have 𝑢̇_𝑖(𝑡)=𝜆𝑏_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑒^{𝑢𝑖(𝑡)}−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘(𝑡)𝑒^{𝑢𝑘(𝑡−𝜏𝑖𝑘)}−^𝑀_𝑙=1𝑐_𝑖𝑙(𝑡)𝑒^{𝑣𝑙(𝑡−𝜂𝑖𝑙)},𝑣̇_𝑗(𝑡)=𝜆−𝑟_𝑗(𝑡)+^𝑁_𝑘=1𝑑_𝑗𝑘(𝑡)𝑒^{𝑢𝑘(𝑡−𝛿𝑗𝑘)}−𝑒_𝑗𝑗(𝑡)𝑒^{𝑣𝑗(𝑡)}−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙(𝑡)𝑒^{𝑣𝑙(𝑡−𝜉𝑗𝑙)}. (2.20) Since 𝑈(𝑡)∈𝑋, each 𝑈_𝑛(𝑡),𝑛=1,2,…,𝑁+𝑀, as components of 𝑈(𝑡), is continuously differentiable and 𝜔-periodic. In view of continuity and periodicity, there exists 𝑡_𝑖∈[0,𝜔] such that 𝑢_𝑖(𝑡_𝑖)=max_{𝑡∈[0,𝜔]}|𝑢_𝑖(𝑡)|, 𝑖=1,2,…,𝑁, and there also exists 𝑡_𝑁+𝑗∈[0,𝜔] such that 𝑣_𝑗(𝑡_𝑁+𝑗)=max_{𝑡∈[0,𝜔]}|𝑣_𝑗(𝑡)|, 𝑗=1,2,…,𝑀. Accordingly, 𝑢̇_𝑖(𝑡_𝑖)=0, 𝑣̇_𝑗(𝑡_𝑁+𝑗)=0, and we get 𝑏_𝑖𝑡_𝑖−𝑎_𝑖𝑖𝑡_𝑖𝑒^{𝑢𝑖(𝑡𝑖)}−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑡_𝑖𝑒^{𝑢𝑘(𝑡𝑖−𝜏𝑖𝑘)}−^𝑀_𝑙=1𝑐_𝑖𝑙𝑡_𝑖𝑒^{𝑣𝑙(𝑡𝑁+𝑗−𝜂𝑖𝑙)}=0,−𝑟_𝑗𝑡_𝑁+𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑡_𝑁+𝑗𝑒^{𝑢𝑘(𝑡𝑖−𝛿𝑗𝑘)}−𝑒_𝑗𝑗𝑡_𝑁+𝑗𝑒^{𝑣𝑗(𝑡𝑁+𝑗)}−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑡_𝑁+𝑗𝑒^{𝑣𝑙(𝑡𝑁+𝑗−𝜉𝑗𝑙)}=0. (2.21) That is, 𝑎_𝑖𝑖𝑡_𝑖𝑒^{𝑢𝑖(𝑡𝑖)}=𝑏_𝑖𝑡_𝑖−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑡_𝑖𝑒^{𝑢𝑘(𝑡𝑖−𝜏𝑖𝑘)}−^𝑀_𝑙=1𝑐_𝑖𝑙𝑡_𝑖𝑒^{𝑣𝑙(𝑡𝑁+𝑗−𝜂𝑖𝑙)},𝑒_𝑗𝑗𝑡_𝑁+𝑗𝑒^{𝑣𝑗(𝑡𝑁+𝑗)}=−𝑟_𝑗𝑡_𝑁+𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑡_𝑁+𝑗𝑒^{𝑢𝑘(𝑡𝑖−𝛿𝑗𝑘)}−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑡_𝑁+𝑗𝑒^{𝑣𝑙(𝑡𝑁+𝑗−𝜉𝑗𝑙)}. (2.22) Note that 𝑢_𝑘(𝑡_𝑘)=max_{𝑡∈[0,𝜔]}|𝑢_𝑘(𝑡)| and 𝑣_𝑙(𝑡_𝑁+𝑙)=max_{𝑡∈[0,𝜔]}|𝑣_𝑙(𝑡)|, which implies 𝑢_𝑘𝑡_𝑖≤𝑢_𝑘𝑡_𝑘,𝑢_𝑘𝑡_𝑖−𝜏_𝑖𝑘≤𝑢_𝑘𝑡_𝑘,𝑢_𝑘𝑡_𝑖−𝛿_𝑗𝑘≤𝑢_𝑘𝑡_𝑘;𝑣_𝑙𝑡_𝑁+𝑗≤𝑣_𝑙𝑡_𝑁+𝑙,𝑣_𝑙𝑡_𝑁+𝑗−𝜂_𝑖𝑙≤𝑣_𝑙𝑡_𝑁+𝑙,𝑣_𝑙𝑡_𝑁+𝑗−𝜉_𝑗𝑙≤𝑣_𝑙𝑡_𝑁+𝑙. (2.23) It follows that 𝑎_𝑖𝑖𝑒^{𝑢𝑖(𝑡𝑖)}≤𝑎_𝑖𝑖𝑡_𝑖𝑒^{𝑢𝑖(𝑡𝑖)}=𝑏_𝑖𝑡_𝑖−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑡_𝑖𝑒^{𝑢𝑘(𝑡𝑖−𝜏𝑖𝑘)}−^𝑀_𝑙=1𝑐_𝑖𝑙𝑡_𝑖𝑒^{𝑣𝑙(𝑡𝑁+𝑗−𝜂𝑖𝑙)}≤𝑏_𝑖+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑒^{𝑢𝑘(𝑡𝑖−𝜏𝑖𝑘)}+^𝑀_𝑙=1𝑐_𝑖𝑙𝑒^{𝑣𝑙(𝑡𝑁+𝑗−𝜂𝑖𝑙)}≤𝑏_𝑖+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑒^{𝑢𝑘(𝑡𝑘)}+^𝑀_𝑙=1𝑐_𝑖𝑙𝑒^{𝑣𝑙(𝑡𝑁+𝑙)},𝑒_𝑗𝑗𝑒^{𝑣𝑗(𝑡𝑁+𝑗)}≤𝑒_𝑗𝑗𝑡_𝑁+𝑗𝑒^{𝑣𝑗(𝑡𝑁+𝑗)}=−𝑟_𝑗𝑡_𝑁+𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑡_𝑁+𝑗𝑒^{𝑢𝑘(𝑡𝑖−𝛿𝑗𝑘)}−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑡_𝑁+𝑗𝑒^{𝑣𝑙(𝑡𝑁+𝑗−𝜉𝑗𝑙)}≤𝑟_𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑒^{𝑢𝑘(𝑡𝑖−𝛿𝑗𝑘)}+^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑒^{𝑣𝑙(𝑡𝑁+𝑗−𝜉𝑗𝑙)}≤𝑟_𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑒^{𝑢𝑘(𝑡𝑘)}+^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑒^{𝑣𝑙(𝑡𝑁+𝑙)}. (2.24) Let 𝑎_𝑖𝑖𝑒^{𝑢𝑖(𝑡𝑖)}=𝑧_𝑖𝑡_𝑖,𝑒_𝑗𝑗𝑒^{𝑣𝑗(𝑡𝑁+𝑗)}=𝑧_𝑗𝑡_𝑁+𝑗. (2.25) Using (2.25), the inequalities (2.24) become 𝑧_𝑖𝑡_𝑖≤𝑏_𝑖+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑎⁻¹_𝑘𝑘𝑧_𝑘𝑡_𝑘+^𝑀_𝑙=1𝑐_𝑖𝑙𝑒⁻¹_𝑙𝑙𝑧_𝑙𝑡_𝑁+𝑙,𝑧_𝑗𝑡_𝑁+𝑗≤𝑟_𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑎⁻¹_𝑘𝑘𝑧_𝑘𝑡_𝑘+^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑒⁻¹_𝑙𝑙𝑧_𝑙𝑡_𝑁+𝑙, (2.26) or 𝑧_𝑖𝑡_𝑖−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑎⁻¹_𝑘𝑘𝑧_𝑘𝑡_𝑘−^𝑀_𝑙=1𝑐_𝑖𝑙𝑒⁻¹_𝑙𝑙𝑧_𝑙𝑡_𝑁+𝑙≤𝑏_𝑖,𝑧_𝑗𝑡_𝑁+𝑗−^𝑁_𝑘=1𝑑_𝑗𝑘𝑎⁻¹_𝑘𝑘𝑧_𝑘𝑡_𝑘−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑒⁻¹_𝑙𝑙𝑧_𝑙𝑡_𝑁+𝑙≤𝑟_𝑗, (2.27) which implies 𝐸_𝑁×𝑁−𝒫_𝑁×𝑁−𝒬_𝑁×𝑀−ℳ_𝑀×𝑁𝐸_𝑀×𝑀−𝒩_{𝑀×𝑀(𝑁+𝑀)×(𝑁+𝑀)}×𝑧₁𝑡₁,𝑧…,_𝑁𝑡_𝑁,𝑧₁𝑡_𝑁+1,𝑧…,_𝑀𝑡_𝑁+𝑀≤𝑏₁,…,𝑏_𝑁,𝑟₁,…,𝑟_𝑀, (2.28) where 𝒫_𝑁×𝑁=𝑝_{𝑖𝑘𝑁×𝑁},𝑝_𝑖𝑘=0,𝑖=𝑘,𝑎_𝑖𝑘𝑎⁻¹_𝑘𝑘𝒬,𝑖≠𝑘,_𝑁×𝑀=𝑞_{𝑖𝑙𝑁×𝑀},𝑞_𝑖𝑙=𝑐_𝑖𝑙𝑒⁻¹_𝑙𝑙,ℳ_𝑀×𝑁=𝑚_{𝑗𝑘𝑀×𝑁},𝑚_𝑗𝑘=𝑑_𝑗𝑘𝑎⁻¹_𝑘𝑘,𝒩_𝑀×𝑀=𝑛_{𝑗𝑙𝑀×𝑀},𝑛_𝑗𝑙=0,𝑗=𝑙,𝑒_𝑗𝑙𝑒⁻¹_𝑙𝑙,𝑗≠𝑙. (2.29) Set 𝐷=(𝑏₁,…,𝑏_𝑁,𝑟₁,…,𝑟_𝑀)^𝑇. It follows from (2.28) and (𝐻₂) that (𝐸−𝒦)𝑧₁𝑡₁,…,𝑧_𝑁𝑡_𝑁,𝑧₁𝑡_𝑁+1,…,𝑧_𝑀𝑡_𝑁+𝑀^𝑇≤𝐷. (2.30) In view of 𝜌(𝒦)<1 and Lemma 2.5, we get (𝐸_𝑁+𝑀−𝒦)⁻¹≥0. Let 𝐻=ℎ₁,…,ℎ_𝑁,ℎ_𝑁+1,…,ℎ_𝑁+𝑀^𝑇∶=(𝐸−𝒦)⁻¹𝐷≥0. (2.31) Using (2.30) and (2.31), we get 𝑧₁𝑡₁,…,𝑧_𝑁𝑡_𝑁,𝑧₁𝑡_𝑁+1,…,𝑧_𝑀𝑡_𝑁+𝑀^𝑇≤𝐻, (2.32) or 𝑧_𝑖𝑡_𝑖≤ℎ_𝑖,𝑖=1,2,…,𝑁,𝑧_𝑗𝑡_𝑁+𝑗≤ℎ_𝑁+𝑗,𝑗=1,2,…,𝑀. (2.33) Then 𝑢_𝑖𝑡_𝑖ℎ≤ln^𝑖𝑎_𝑖𝑖,𝑣_𝑗𝑡_𝑁+𝑗ℎ≤ln^𝑁+𝑗𝑒_𝑗𝑗, (2.34) which implies 𝑢_𝑛(𝑡)₀=max_{𝑡∈[0,𝜔]}𝑢_𝑛(𝑡)=max_{𝑡∈[0,𝜔]}𝑢_𝑖𝑡_𝑖,𝑣_𝑗𝑡_𝑁+𝑗ℎ=maxln^𝑖𝑎_𝑖𝑖ℎ,ln^𝑁+𝑗𝑒_𝑗𝑗. (2.35) On the other hand, it follows from (2.31) that (𝐸−𝒦)𝐻=𝐷,or𝐻=𝒦𝐻+𝐷, (2.36) that is ℎ_𝑖=^𝑁_{𝑘=1,𝑘≠𝑖}𝑝_𝑖𝑘ℎ_𝑘+^𝑀_𝑙=1𝑞_𝑖𝑙ℎ_𝑁+𝑙+𝑏_𝑖,ℎ_𝑁+𝑗=^𝑁_𝑘=1𝑚_𝑗𝑘ℎ_𝑘+^𝑀_{𝑙=1,𝑙≠𝑗}𝑛_𝑗𝑙ℎ_𝑁+𝑙+𝑟_𝑗. (2.37) Estimating (2.20), by using (2.25), (2.33), and (2.37), we have 𝑢̇_𝑖(𝑡)₀=𝜆𝑏_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑒^{𝑢𝑖(𝑡)}−^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘(𝑡)𝑒^{𝑢𝑘(𝑡−𝜏𝑖𝑘)}−^𝑀_𝑙=1𝑐_𝑖𝑙(𝑡)𝑒^{𝑣𝑙(𝑡−𝜂𝑖𝑙)}₀≤𝑏_𝑖+𝑎_𝑖𝑖𝑒^{𝑢𝑖(𝑡)}₀+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑒^{𝑢𝑘(𝑡−𝜏𝑖𝑘)}₀+^𝑀_𝑙=1𝑐_𝑖𝑙𝑒^{𝑣𝑙(𝑡−𝜂𝑖𝑙)}₀=𝑏_𝑖+𝑎_𝑖𝑖𝑒^{𝑢𝑖(𝑡𝑖)}+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑒^{𝑢𝑘(𝑡𝑘)}+^𝑀_𝑙=1𝑐_𝑖𝑙𝑒^{𝑣𝑙(𝑡𝑁+𝑙)}=𝑏_𝑖+𝑎_𝑖𝑖𝑎⁻¹_𝑖𝑖𝑧_𝑖𝑡_𝑖+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑎⁻¹_𝑘𝑘𝑧_𝑘𝑡_𝑘+^𝑀_𝑙=1𝑐_𝑖𝑙𝑒⁻¹_𝑙𝑙𝑧_𝑙𝑡_𝑁+𝑙=𝑏_𝑖+𝑧_𝑖𝑡_𝑖+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑎⁻¹_𝑘𝑘𝑧_𝑘𝑡_𝑘+^𝑀_𝑙=1𝑐_𝑖𝑙𝑒⁻¹_𝑙𝑙𝑧_𝑙𝑡_𝑁+𝑙≤𝑏_𝑖+ℎ_𝑖+^𝑁_{𝑘=1,𝑘≠𝑖}𝑝_𝑖𝑘ℎ_𝑘+^𝑀_𝑙=1𝑞_𝑖𝑙ℎ_𝑁+𝑙=ℎ_𝑖+ℎ_𝑖=2ℎ_𝑖,𝑣̇_𝑗(𝑡)₀=𝜆−𝑟_𝑗(𝑡)+^𝑁_𝑘=1𝑑_𝑗𝑘(𝑡)𝑒^{𝑢𝑘(𝑡−𝛿𝑗𝑘)}−𝑒_𝑗𝑗(𝑡)𝑒^{𝑣𝑗(𝑡)}−^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙(𝑡)𝑒^{𝑣𝑙(𝑡−𝜉𝑗𝑙)}₀≤𝑟_𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑒^{𝑢𝑘(𝑡−𝛿𝑗𝑘)}₀+^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑒^{𝑣𝑙(𝑡−𝜉𝑗𝑙)}₀+𝑒_𝑗𝑗𝑒^{𝑣𝑗(𝑡)}₀=𝑟_𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑒^{𝑢𝑘(𝑡𝑘)}+^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑒^{𝑣𝑙(𝑡𝑁+𝑙)}+𝑒_𝑗𝑗𝑒^{𝑣𝑗(𝑡𝑁+𝑗)}=𝑟_𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑎⁻¹_𝑘𝑘𝑧_𝑘𝑡_𝑘+^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑒⁻¹_𝑙𝑙𝑧_𝑙𝑡_𝑁+𝑙+𝑒_𝑗𝑗𝑒⁻¹_𝑗𝑗𝑧_𝑗𝑡_𝑁+𝑗=𝑟_𝑗+^𝑁_𝑘=1𝑑_𝑗𝑘𝑎⁻¹_𝑘𝑘𝑧_𝑘𝑡_𝑘+^𝑀_{𝑙=1,𝑙≠𝑗}𝑒_𝑗𝑙𝑒⁻¹_𝑙𝑙𝑧_𝑙𝑡_𝑁+𝑙+𝑧_𝑗𝑡_𝑁+𝑗≤𝑟_𝑗+^𝑁_𝑘=1𝑚_𝑗𝑘ℎ_𝑘+^𝑀_{𝑙=1,𝑙≠𝑗}𝑛_𝑗𝑙ℎ_𝑁+𝑙+ℎ_𝑁+𝑗=ℎ_𝑁+𝑗+ℎ_𝑁+𝑗=2ℎ_𝑁+𝑗. (2.38) The above relations imply 𝑢̇_𝑛(𝑡)₀=max_{𝑡∈[0,𝜔]}𝑢̇_𝑛(𝑡)=max_{𝑡∈[0,𝜔]}𝑢̇_𝑖(𝑡),𝑣̇_𝑗(𝑡)=max2ℎ_𝑖,2ℎ_𝑁+𝑗. (2.39) Further, it follows form the definition of norm that 𝑢_𝑛(𝑡)₁=𝑢_𝑛(𝑡)₀+𝑢̇_𝑛(𝑡)₀ℎ=maxln^𝑖𝑎_𝑖𝑖ℎ,ln^𝑁+𝑗𝑒_𝑗𝑗+max2ℎ_𝑖,2ℎ_𝑁+𝑗. (2.40) Let us set the following: ℎ_𝑛ℎ=maxln^𝑖𝑎_𝑖𝑖ℎ,ln^𝑁+𝑗𝑒_𝑗𝑗+max2ℎ_𝑖,2ℎ_𝑁+𝑗+𝑑, (2.41) where 𝑑 is any positive constant.
Then for any solution of 𝐿𝑥=𝜆𝑁𝑥, we have |𝑢_𝑛(𝑡)|₁=|𝑢_𝑛(𝑡)|₀+|𝑢̇_𝑛(𝑡)|₀<ℎ_𝑛 for all 𝑛=1,2,…,𝑁+𝑀. Obviously, ℎ_𝑛 are independent of 𝜆 and the choice of 𝑈(𝑡). Consequently, by taking this ℎ_𝑛, the open subset Ω satisfies that Ω∩Dom𝐿, that is, the open subset Ω satisfies the assumption (i) of Lemma 2.3.
Now in the last step of the proof, we need to verify that for the given open bounded set Ω obtained in Step 2, the assumption (ii) of Lemma 2.3 also holds. That is, for each 𝑈∈𝜕Ω∩Ker𝐿, 𝑄𝑁𝑈≠0 and deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0.
Take 𝑈∈𝜕Ω∩Ker𝐿. Then, in view of Ker𝐿=ℝ^𝑁+𝑀, 𝑈 is a constant vector in ℝ^𝑁+𝑀, denoted by 𝑈=(𝑢₁,…,𝑢_𝑁,𝑣₁,…,𝑣_𝑀)^𝑇 and with the property 𝑈_𝑛=𝑈_𝑛0=𝑈_𝑛1=ℎ_𝑛. (2.42) By operating 𝑈 by 𝑄𝑁 gives (𝑄𝑁𝑈)_𝑛=𝑚𝑏_𝑖−𝑚𝑎_𝑖𝑖𝑒^𝑢𝑖−∑^𝑁_{𝑘=1,𝑘≠𝑖}𝑚𝑎_𝑖𝑘𝑒^𝑢𝑘−∑^𝑀_𝑙=1𝑚𝑐_𝑖𝑙𝑒^𝑣𝑙𝑚−𝑟_𝑗+∑^𝑁_𝑘=1𝑚𝑑_𝑗𝑘𝑒^𝑢𝑘−𝑚𝑒_𝑗𝑗𝑒^𝑣𝑗−∑^𝑀_{𝑙=1,𝑙≠𝑗}𝑚𝑒_𝑗𝑙𝑒^𝑣𝑙. (2.43) It is easy to obtain that (𝑄𝑁𝑈)_𝑛 and deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0, where deg(⋅) is the Brouwer degree and 𝐽 is the identity mapping since Im𝑄=Ker𝐿. We have shown that the open subset Ω⊂𝑋 satisfies all the assumptions of Lemma 2.3. Hence, by Lemma 2.3, the system (2.11) has at least one positive 𝜔-periodic solution in Dom𝐿∩Ω. By (2.10), the system (1.3) has at least one positive 𝜔-periodic solution. This completes the proof of Theorem 2.6.

3. Globally Asymptotic Stability

Under the assumption of Theorem 2.6, we know that system (1.3) has at least one positive 𝜔-periodic solution, denoted by 𝑋^∗(𝑡)=(𝑥^∗₁(𝑡),…,𝑥^∗_𝑁(𝑡),𝑦^∗₁(𝑡),…,𝑦^∗_𝑀(𝑡))^𝑇. The aim of this section is to derive a set of sufficient conditions which guarantee the existence and global asymptotic stability of the positive 𝜔-periodic solution 𝑋^∗(𝑡).

Before the formal analysis, we recall some facts which will be used in the proof.

Lemma 3.1 (see [25]). Let 𝑓 be a nonnegative function defined on [0,+∞] such that 𝑓 is integrable on [0,+∞] and is uniformly continuous on [0,+∞]. Then lim_𝑡→+∞𝑓(𝑡)=0.

Lemma 3.2 (see [23, 24]). Let 𝒦=(Γ_𝑖𝑗)_𝑛×𝑛 be a matrix with nonpositive off-diagonal elements. 𝒦 is an 𝑀-matrix if and only if there exists a positive diagonal matrix 𝜉=diag(𝜉₁,𝜉₂,…,𝜉_𝑛) such that 𝜉_𝑖𝑎_𝑖𝑖>_𝑗≠𝑖𝜉_𝑗𝑎_𝑖𝑗,𝑖=1,2,…,𝑛. (3.1)

Theorem 3.3. Assume that all the assumptions in Theorem 2.6 hold. Then system (1.3) has a unique positive 𝜔-periodic solution 𝑋^∗(𝑡) which is globally asymptotically stable.

Proof. Let 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡))^𝑇=(𝑥₁(𝑡),…,𝑥_𝑁(𝑡),𝑦₁(𝑡),…,𝑦_𝑀(𝑡))^𝑇 be any positive solution of system (1.3). It is easy to see that 𝜌(𝒦^𝑇)=𝜌(𝒦)<1. Thus, in view of Lemma 2.5 and Definition 2.4, (𝐸−𝒦^𝑇) is an 𝑀-matrix, where 𝐸 denotes an identity matrix of size 𝑁+𝑀. Therefore, by Lemma 3.2, there exists a diagonal matrix 𝐿=diag(𝛼₁,…,𝛼_𝑁,𝛽₁,…,𝛽_𝑀) with positive diagonal elements such that the product (𝐸−𝒦^𝑇)𝐿 is strictly diagonally dominant with positive diagonal entries, namely, 𝛼_𝑖𝑎_𝑖𝑖>^𝑁_{𝑘=1,𝑘≠𝑖}𝛼_𝑘𝑎_𝑘𝑖+^𝑀_𝑙=1𝛽_𝑙𝑑_𝑙𝑖𝛽,𝑖=1,…,𝑁,_𝑗𝑒_𝑗𝑗>^𝑁_𝑘=1𝛼_𝑘𝑐_𝑘𝑗+^𝑀_{𝑙=1,𝑙≠𝑗}𝛽_𝑙𝑒_𝑙𝑗,𝑗=1,…,𝑀. (3.2) Now, we define a Lyapunov function 𝑉(𝑡) as follows: 𝑉(𝑡)=^𝑁_𝑖=1𝛼_𝑖ln𝑥_𝑖(𝑡)−ln𝑥^∗_𝑖(𝑡)+^𝑁_{𝐾=1,𝐾≠𝑖}^𝑡_{𝑡−𝜏𝑖𝑘}𝑎_𝑖𝑘𝑠+𝜏_𝑖𝑘𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡+)𝑑𝑠_{𝑀𝑙=1𝑡𝑡−𝜂𝑖𝑙}𝑐_𝑖𝑙𝑠+𝜂_𝑖𝑙𝑦_𝑙(𝑡)−𝑦^∗_𝑙(𝑡+)𝑑𝑠_𝑀𝑗=1𝛽_𝑗ln𝑦_𝑗(𝑡)−ln𝑦^∗_𝑗(𝑡)+^𝑁_𝐾=1^𝑡_{𝑡−𝛿𝑗𝑘}𝑑_𝑗𝑘𝑠+𝛿_𝑗𝑘𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡+)𝑑𝑠_{𝑀𝑙=1,𝑙≠𝑗𝑡𝑡−𝜉𝑗𝑙}𝑒_𝑗𝑙𝑠+𝜉_𝑗𝑙𝑦_𝑙(𝑡)−𝑦^∗_𝑙(𝑡)𝑑𝑠,𝑡≥𝑡₀. (3.3) Calculating the upper right derivative of 𝑉(𝑡) and using (3.2), we get 𝐷⁺𝑉(𝑡)≤^𝑁_𝑖=1𝛼_𝑖−𝑎_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑡+𝜏_𝑖𝑘𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡+_𝑀𝑙=1)𝑐_𝑖𝑙𝑡+𝜂_𝑖𝑙𝑦_𝑙(𝑡)−𝑦^∗_𝑙(𝑡+_𝑀𝑗=1)𝛽_𝑗−𝑒_𝑗𝑗(𝑡)𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)+^𝑁_𝑘=1𝑑_𝑗𝑘𝑡+𝛿_𝑗𝑘𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡+_{𝑀𝑙=1,𝑙≠𝑗})𝑒_𝑗𝑙𝑡+𝜉_𝑗𝑙𝑦_𝑙(𝑡)−𝑦^∗_𝑙(𝑡)≤−^𝑁_𝑖=1𝛼_𝑖−𝑎_𝑖𝑖𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡+_𝑀𝑙=1)𝑐_𝑖𝑙𝑦_𝑙(𝑡)−𝑦^∗_𝑙(𝑡+_𝑀𝑗=1)𝛽_𝑗−𝑒_𝑗𝑗𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)+^𝑁_𝑘=1𝑑_𝑗𝑘𝑥_𝑘(𝑡)−𝑥^∗_𝑘(𝑡+_{𝑀𝑙=1,𝑙≠𝑗})𝑒_𝑗𝑙𝑦_𝑙(𝑡)−𝑦^∗_𝑙(𝑡)=−^𝑁_𝑖=1𝛼_𝑖𝑎_𝑖𝑖−^𝑁_{𝑘=1,𝑘≠𝑖}𝛼_𝑘𝑎_𝑘𝑖−^𝑀_𝑙=1𝛽_𝑙𝑑_𝑙𝑖𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡−_𝑀𝑗=1)𝛽_𝑗𝑒_𝑗𝑗−^𝑁_𝑘=1𝛼_𝑘𝑐_𝑘𝑗−^𝑀_{𝑙=1,𝑙≠𝑗}𝛽_𝑙𝑒_𝑙𝑗𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)=−𝑐^𝑁_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)+^𝑀_𝑗=1𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡), (3.4) where 𝑐=min𝛼_𝑖𝑎_𝑖𝑖−^𝑁_{𝑘=1,𝑘≠𝑖}𝛼_𝑘𝑎_𝑘𝑖−^𝑀_𝑙=1𝛽_𝑙𝑑_𝑙𝑖,𝛽_𝑗𝑒_𝑗𝑗−^𝑁_𝑘=1𝛼_𝑘𝑐_𝑘𝑗−^𝑀_{𝑙=1,𝑙≠𝑗}𝛽_𝑙𝑒_𝑙𝑗>0. (3.5) It follows from (3.4) that 𝐷⁺𝑉(𝑡)≤0. Obviously, the zero solution of (1.3) is Lyapunov stable. On the other hand, integrating (3.4) over [𝑡₀,𝑡] leads to 𝑉(𝑡)−𝑉𝑡₀⩽−𝑐^𝑡_𝑡0^𝑁_𝑖=1𝑥_𝑖(𝑠)−𝑥^∗_𝑖(𝑠)+^𝑀_𝑗=1𝑦_𝑗(𝑠)−𝑦^∗_𝑗(𝑠)𝑑𝑠,𝑡⩾0, (3.6) or 𝑉(𝑡)+𝑐^𝑡_𝑡0^𝑁_𝑖=1𝑥_𝑖(𝑠)−𝑥^∗_𝑖(𝑠)+^𝑀_𝑗=1𝑦_𝑗(𝑠)−𝑦^∗_𝑗(𝑠)𝑑𝑠⩽𝑉𝑡₀<+∞,𝑡⩾𝑡₀. (3.7) Noting that 𝑉(𝑡)⩾0, it follows that ^𝑡_𝑡0^𝑁_𝑖=1𝑥_𝑖(𝑠)−𝑥^∗_𝑖(𝑠)+^𝑀_𝑗=1𝑦_𝑗(𝑠)−𝑦^∗_𝑗(𝑠𝑉𝑡)𝑑𝑠⩽⁰𝑐<+∞,𝑡⩾𝑡₀. (3.8) Therefore, by Lemma 3.1, it is not difficult to conclude that lim_𝑡→+∞𝑋_𝑖(𝑡)−𝑋^∗_𝑖(𝑡)=0. (3.9) Theorem 3.3 follows.

4. Corollaries and Remarks

In order to illustrate some features of our main results, we will present some corollaries and remarks in this section. From the proofs of Theorems 2.6 and 3.3, one can easily deduce the following corollary.

Corollary 4.1. In addition to (𝐻₁), further suppose that 𝐸−𝒦 or 𝐸−𝒦^𝑇 is an 𝑀-matrix. Then system (1.3) has a unique positive 𝜔-periodic solution which is globally asymptotically stable.

Now recall that for a given matrix 𝒦, its spectral radius 𝜌(𝒦) is equal to the minimum of all matrix norms of 𝒦, that is, for any matrix norm ‖⋅‖,𝜌(𝒦)≤‖𝒦‖. Therefore, we have the following corollary.

Corollary 4.2. In addition to (𝐻₁), if one further supposes that there exist positive constants 𝜉_𝑖,𝑖=1,2,…,𝑛,𝜂_𝑗,𝑗=1,2,…,𝑚 such that one of the following inequalities holds. (1)max{max_{1≤𝑘≤𝑛}{𝑎⁻¹_𝑘𝑘𝜉_𝑘⁻¹[∑^𝑛_{𝑖=1,𝑖≠𝑘}𝜉_𝑖𝑎_𝑖𝑘+∑^𝑚_𝑙=1𝜂_𝑗𝑑_𝑗𝑘]},max_{1≤𝑙≤𝑚}{𝑒⁻¹_𝑙𝑙𝜂_𝑙⁻¹[∑^𝑛_𝑖=1𝜉_𝑖𝑐_𝑙𝑖+∑^𝑚_{𝑗=1,𝑗≠𝑙}𝜂_𝑗𝑒_𝑗𝑙]}} < 1, or equivalently, for all 𝑘=1,…,𝑛,𝑙=1,2,…,𝑛, 𝜉_𝑘𝑎_𝑘𝑘>^𝑛_{𝑖=1,𝑖≠𝑘}𝜉_𝑖𝑎_𝑖𝑘+^𝑚_𝑙=1𝜂_𝑗𝑑_𝑗𝑘,𝜂_𝑙𝑒_𝑙𝑙>^𝑛_𝑖=1𝜉_𝑖𝑐_𝑙𝑖+^𝑚_{𝑗=1,𝑗≠𝑙}𝜂_𝑗𝑒_𝑗𝑙. (4.1)(2)∑^𝑛+𝑚_𝑖=1∑^𝑛+𝑚_𝑗=1(𝜉_𝑖⁻¹𝜉_𝑗𝑘_𝑖𝑗)²<1, where 𝒦=(𝑘_𝑖𝑗)_{(𝑛+𝑚)×(𝑛+𝑚)} has been defined in Theorem 2.6.(3)max{max_{1≤𝑖≤𝑛}{𝑎⁻¹_𝑖𝑖𝜉_𝑖⁻¹[∑^𝑛_{𝑘=1,𝑘≠𝑖}𝜉_𝑘𝑎_𝑘𝑖+∑^𝑚_𝑙=1𝜂_𝑙𝑑_𝑙𝑖]},max_{1≤𝑗≤𝑚}{𝑒⁻¹_𝑗𝑗𝜂_𝑗⁻¹[∑^𝑛_𝑘=1𝜉_𝑘𝑐_𝑘𝑗+∑^𝑚_{𝑙=1,𝑙≠𝑗}𝜂_𝑙𝑒_𝑙𝑗]}} < 1, or equivalently, for all 𝑖=1,…,𝑛,𝑗=1,2,…,𝑛, 𝜉_𝑖𝑎_𝑖𝑖>^𝑛_{𝑘=1,𝑘≠𝑖}𝜉_𝑘𝑎_𝑘𝑖+^𝑚_𝑙=1𝜂_𝑙𝑑_𝑙i,𝜂_𝑗𝑒_𝑗𝑗>^𝑛_𝑘=1𝜉_𝑘𝑐_𝑘𝑗+^𝑚_{𝑙=1,𝑙≠𝑗}𝜂_𝑙𝑒_𝑙𝑗. (4.2) Then system (1.3) has a unique positive 𝜔-periodic solution which is globally asymptotically stable.

Proof. For any matrix norm ‖⋅‖ and any nonsingular matrix 𝑆, ‖𝒦‖_𝑆=‖𝑆⁻¹𝒦𝑆‖ also defines a matrix norm. Let us denote 𝐷=diag(𝜉₁,𝜉₂,…,𝜉_𝑛), then the conditions (1.2) and (1.3) correspond to the column norms and Frobenius norm of matrix 𝐷𝒦𝐷⁻¹, respectively. Condition (2.10) corresponds to the row norms of 𝐷𝒦^𝑇𝐷⁻¹ and note that 𝜌(𝐷𝒦^𝑇𝐷⁻¹)=𝜌(𝐷𝒦𝐷⁻¹). Now Corollary 4.2 follows immediately.

5. Example

In this section, an example and its simulations are presented to illustrate the feasibility and effectiveness of our results.

Example 5.1. Consider the following periodic predator-prey model with 2-predators and 1-prey: 𝑥̇₁(𝑡)=𝑥₁(𝑡)7+sin𝑡−𝑥₁(𝑡)−14𝑥₂(𝑡)−1𝑦10¹(,𝑥̇𝑡−1)₂(𝑡)=𝑥₂(𝑡1)7+cos𝑡−4𝑥₁(𝑡)−𝑥₂(𝑡)−14𝑦₁(𝑡,𝑦̇₁)(𝑡)=𝑦₁(𝑡−1)()+3201+cos𝑡2𝑥₁()+1𝑡−14𝑥₂(𝑡)−12𝑦₁(𝑡). (5.1) Simple computation leads to 𝒦=0𝑎⁻¹₂₂𝑎₁₂𝑒⁻¹₁₁𝑐₁₁𝑎⁻¹₁₁𝑎₂₁0𝑒⁻¹₁₁𝑐₂₁𝑎⁻¹₁₁𝑑₁₁𝑎⁻¹₂₂𝑑₁₂0=01415143002140. (5.2) By using mathematica, we see that 𝜌(𝒦)=0.633982<1. Thus, the system (5.1) has a periodic solution which is globally asymptotically stable. Figure 1 shows the asymptotic behavior of the periodic solution.

Figure 1 

Asymptotic behavior of system (5.1) with initial values (𝑥1(0),𝑥2(0),𝑦1(0))=(1,1,1),(0.3,0.3,0.3),(7,7,7), respectively, 𝑡∈[0,25].

Remark 5.2. In this example, one can observe that though the spectral 𝜌(𝒦)<1, the matrix norms of the matrix 𝒦 are all bigger than 1. For instance, the column norm: is‖𝒦‖₁=max_1≤𝑗≤3𝑎³⁻¹_{𝑗𝑗𝑖=1,𝑖≠𝑗}𝑎_𝑖𝑗3=0+2+14>1. (5.3)

Acknowledgments

This work was supported by NNSFC (no. 11271333 and no. 11171090). The first author also acknowledges the supports by the foundation of Slovene human resources development and scholarship fund. He thanks Professor Valery G. Romanovski and Professor Marko Robnik for warm hospitality during his stay at CAMTP in University of Maribor as a Research Fellow.