Positive Periodic Solutions in a Discrete Time Three Species Competition System

Abstract

A periodic discrete time three species competition system is investigated. With the aid of differential equations with piecewise constant arguments, a discrete analogue of continuous nonautonomous three species competition system is proposed. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive periodic solutions of the model are obtained.

1. Introduction

During the past decades, the dynamical properties of competitive populations have received great attention from both theoretical and mathematical biologists due to their universal prevalence and importance. Numerous excellent results have been reported for a lot of different continuous or impulsive competitive models. For example, Kuang [1] analyzed the permanent coexistence of the following delayed three species competition system: 𝑑𝑥₁𝑑𝑡=𝑥₁(𝑡)1−𝑥₁(𝑡)−^𝑡_−∞𝐾(𝑠−𝑡)𝑥₂(𝑠)−𝑑𝑠_𝑡−∞𝐿(𝑠−𝑡)𝑥₃(𝑠)𝑑𝑠,𝑑𝑥₂𝑑𝑡=𝑥₂(𝑡)1−𝑥₂(𝑡)−^𝑡_−∞𝐿(𝑠−𝑡)𝑥₁(𝑠)−𝑑𝑠_𝑡−∞𝐾(𝑠−𝑡)𝑥₃(𝑠)𝑑𝑠,𝑑𝑥₂𝑑𝑡=𝑥₃(𝑡)1−𝑥₃(𝑡)−^𝑡_−∞𝐾(𝑠−𝑡)𝑥₁(𝑠)−𝑑𝑠_𝑡−∞𝐿(𝑠−𝑡)𝑥₂(𝑠)𝑑𝑠, (1) where 𝑥_𝑖(𝑡)  (𝑖=1,2,3) stands for the density of competing species at time 𝑡. For the biological meaning of model (1), one can see [1]. Tang et al. [2, 3] presented sufficient conditions for the existence and global attractivity of positive periodic solutions of the following periodic 𝑛-species Lotka-Volterra competition system with delays 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡). (2) Bohner et al. [4] focused on the existence of periodic solutions in a predator-prey and competition dynamic systems, Pao [5] considered the global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, and Gopalsamy and Weng [6] made a detailed analysis on the global attractivity for a competition system with feedback controls. For more related work, one can see [7–10].

In 2011, Zhu and Lu [11] investigated the following delayed three species competitive system: 𝑑𝑥₁𝑑𝑡=𝑥₁(𝑡)𝑟₁(𝑡)−𝑎₁₁(𝑡)𝑥₁(𝑡)−𝑎₁₂(𝑡)^𝑡_−∞𝐾₁₂(𝑠−𝑡)𝑥₂(𝑠)d𝑠−𝑎₁₃(𝑡)^𝑡_−∞𝐾₁₃(𝑠−𝑡)𝑥₃(𝑠)𝑑𝑠,𝑑𝑥₂𝑑𝑡=𝑥₂(𝑡)𝑟₂(𝑡)−𝑎₂₂(𝑡)𝑥₂(𝑡)−𝑎₂₁(𝑡)^𝑡_−∞𝐾₂₁(𝑠−𝑡)𝑥₁(𝑠)𝑑𝑠−𝑎₂₃(𝑡)^𝑡_−∞𝐾₂₃(𝑠−𝑡)𝑥₃(𝑠)𝑑𝑠,𝑑𝑥₃𝑑𝑡=𝑥₃(𝑡)𝑟₃(𝑡)−𝑎₃₃(𝑡)𝑥₃(𝑡)−𝑎₃₁(𝑡)^𝑡_−∞𝐾₃₁(𝑠−𝑡)𝑥₁(𝑠)𝑑𝑠−𝑎₃₂(𝑡)^𝑡_−∞𝐾₃₂(𝑠−𝑡)𝑥₂(𝑠)𝑑𝑠, (3) where 𝑥_𝑖(𝑡)  (𝑖=1,2,3) stands for the density of competing species at time 𝑡, 𝑟_𝑖,𝑎_𝑖𝑗∈𝐶(𝑅,[0,+∞))(𝑖,𝑗=1,2,3) are 𝜔-periodic functions (𝜔>0), and 𝐾_𝑖𝑗(𝑖,𝑗=1,2,3) is a nonnegative function in 𝐿₁(−∞,0] with 𝑟_𝑖=1𝜔^𝜔₀𝑟_𝑖(𝑠)𝐾𝑑𝑠>0,𝑖=1,2,3,_𝑖𝑗=⁰_−∞𝐾_𝑖𝑗(𝜃)𝑎𝑑𝜃>0,_𝐾𝑖𝑗=1𝜔^𝜔₀𝑎_𝑖𝑗(𝜎)𝑑𝜎^𝜎_−∞𝐾_𝑖𝑗(𝑠−𝜎)𝑑𝑠>0. (4) For more details about the model, one can see [11]. By applying the theory of coincidence degree theory, Zhu and Lu [11] established the existence of positive periodic solution for system (3).

Numerous researchers have argued that discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have nonoverlapping generations. Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulations. Therefore, it is reasonable and interesting to study discrete time systems governed by difference equations. Recently, a great deal of work has been devoted to this topics; see [12–19]. The principle purpose of this paper is to propose a discrete analogue of system (3) and study the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time three species competition system.

The remainder of the paper is organized as follows. In Section 2, with the help of differential equations with piecewise constant arguments, we first propose a discrete analogue of system (3), modelling the dynamics of time nonautonomous competing system where populations have nonoverlapping generations. In Section 3, based on the coincidence degree and the related continuation theorem, sufficient conditions for the existence of positive solutions of difference equations are given.

2. Discrete Analogue of System (3)

There are several different ways of deriving discrete time version of dynamical systems corresponding to continuous time formulations. One of the ways of deriving difference equations modelling the dynamics of populations with nonoverlapping generations that we will use in the following is based on appropriate modifications of models with overlapping generations. For more details about the approach, we refer to [17, 20].

Next, we will discretize the system (3). Assume that the average growth rates in system (3) change at regular intervals of time; then we can obtain the following modified system: 1𝑥₁(𝑡)𝑥̇₁(𝑡)=𝑟₁([𝑡])−𝑎₁₁([𝑡])𝑥₁([𝑡])−𝑎₁₂([𝑡])^+∞_𝑙=0𝐾₁₂(−𝑙)𝑥₂([𝑡]−𝑙)−𝑎₁₃([𝑡])^+∞_𝑙=0𝐾₁₃(−𝑙)𝑥₃𝑡]),1([−𝑙𝑥₂(𝑡)𝑥̇₂(𝑡)=𝑟₂([𝑡])−𝑎₂₂([𝑡])𝑥₂([𝑡])−𝑎₂₁([𝑡])^+∞_𝑙=0𝐾₂₁(−𝑙)𝑥₁([𝑡]−𝑙)−𝑎₂₃([𝑡])^+∞_𝑙=0𝐾₂₃(−𝑙)𝑥₃𝑡]),1([−𝑙𝑥₃(𝑡)𝑥̇₃(𝑡)=𝑟₃([𝑡])−𝑎₃₃([𝑡])𝑥₃([𝑡])−𝑎₃₁([𝑡])^+∞_𝑙=0𝐾₃₁(−𝑙)𝑥₁([𝑡]−𝑙)−𝑎₃₂([𝑡])^+∞_𝑙=0𝐾₃₂(−𝑙)𝑥₂([𝑡]−𝑙), (5) where [𝑡] denotes the integer part of 𝑡, 𝑡∈(0,+∞)   and  𝑡≠0,1,2,…. Equations of type (5) are known as differential equations with piecewise constant arguments and these equations occupy a position midway between differential equations and difference equations. By a solution of (5), we mean a function 𝑥=(𝑥₁,𝑥₂,𝑥₃)^𝑇, which is defined for 𝑡∈[0,+∞) and has the following properties: (1)𝑥 is continuous on [0,+∞);(2)the derivatives 𝑑𝑥₁(𝑡)/𝑑𝑡,  𝑑𝑥₂(𝑡)/𝑑𝑡,  𝑑𝑥₃(𝑡)/𝑑𝑡 exist at each point 𝑡∈[0,+∞) with the possible exception of the points 𝑡∈{0,1,2,…}, where left-sided derivative exists;(3)the equations in (5) are satisfied on each interval [𝑘,𝑘+1) with 𝑘=0,1,2,….

We integrate (5) on any interval of the form [𝑘,𝑘+1), 𝑘=0,1,2,…, and obtain for 𝑘≤𝑡<𝑘+1, 𝑘=0,1,2,…𝑥₁(𝑡)=𝑥₁(𝑘)exp𝑟₁(𝑘)−𝑎₁₁(𝑘)𝑥₁(𝑘)−𝑎₁₂(𝑘)^+∞_𝑙=0𝐾₁₂(−𝑙)𝑥₂(𝑘−𝑙)−𝑎₁₃(𝑘)^+∞_𝑙=0𝐾₁₃(−𝑙)𝑥₃(×(,𝑥𝑘−𝑙)𝑡−𝑘)₂(𝑡)=𝑥₂(𝑘)exp𝑟₂(𝑘)−𝑎₂₂(𝑘)𝑥₂(𝑘)−𝑎₂₁([𝑡])^+∞_𝑙=0𝐾₂₁(−𝑙)𝑥₁(𝑘−𝑙)−𝑎₂₃(𝑘)^+∞_𝑙=0𝐾₂₃(−𝑙)𝑥₃(×(,𝑥𝑘−𝑙)𝑡−𝑘)₃(𝑡)=𝑥₃(𝑘)exp𝑟₃(𝑘)−𝑎₃₃(𝑘)𝑥₃(𝑘)−𝑎₃₁(𝑘)^+∞_𝑙=0𝐾₃₁(−𝑙)𝑥₁(𝑘−𝑙)−𝑎₃₂(𝑘)^+∞_𝑙=0𝐾₃₂(−𝑙)𝑥₂(×(.𝑘−𝑙)𝑡−𝑘) (6) Let 𝑡→𝑘+1; then (6) reads as 𝑥₁(𝑘+1)=𝑥₁(𝑘)exp𝑟₁(𝑘)−𝑎₁₁(𝑘)𝑥₁(𝑘)−𝑎₁₂(𝑘)^+∞_𝑙=0𝐾₁₂(−𝑙)𝑥₂(𝑘−𝑙)−𝑎₁₃(𝑘)^+∞_𝑙=0𝐾₁₃(−𝑙)𝑥₃(,𝑥𝑘−𝑙)₂(𝑘+1)=𝑥₂(𝑘)exp𝑟₂(𝑘)−𝑎₂₂(𝑘)𝑥₂(𝑘)−𝑎₂₁([𝑡])^+∞_𝑙=0𝐾₂₁(−𝑙)𝑥₁(𝑘−𝑙)−𝑎₂₃(𝑘)^+∞_𝑙=0𝐾₂₃(−𝑙)𝑥₃(,𝑥𝑘−𝑙)₃(𝑘+1)=𝑥₃(𝑘)exp𝑟₃(𝑘)−𝑎₃₃(𝑘)𝑥₃(𝑘)−𝑎₃₁(𝑘)^+∞_𝑙=0𝐾₃₁(−𝑙)𝑥₁(𝑘−𝑙)−𝑎₃₂(𝑘)^+∞_𝑙=0𝐾₃₂(−𝑙)𝑥₂(𝑘−𝑙) (7) which is a discrete time analogue of system (3), where 𝑘=0,1,2,….

In order to obtain our main results, we assume that the following hold.

(H1) 𝑟_𝑖,𝑎_𝑖𝑗∶𝑍→𝑅⁺ are positive 𝜔-periodic; that is, 𝑟_𝑖(𝑘+𝜔)=𝑟_𝑖(𝑘) and 𝑎_𝑖𝑗(𝑘+𝜔)=𝑎_𝑖𝑗(𝑘)(𝑖,𝑗=1,2,3) for any 𝑘∈𝑍, where 𝜔, a fixed positive integer, denotes the common period of the parameters in system (7).

(H2) Consider 0≤^+∞_𝑙=0𝐾₁₂(−𝑙)<+∞,0≤^+∞_𝑙=0𝐾₁₃(−𝑙)<+∞,0≤^+∞_𝑙=0𝐾₂₁(−𝑙)<+∞,0≤^+∞_𝑙=0𝐾₂₃(−𝑙)<+∞,0≤^+∞_𝑙=0𝐾₃₁(−𝑙)<+∞,0≤^+∞_𝑙=0𝐾₃₂(−𝑙)<+∞. (8)

3. Existence of Positive Periodic Solutions

For convenience and simplicity on the following discussion, we always use the notations below throughout the paper: 𝐼_𝜔∶={0,1,2,…,𝜔−1},1𝑓∶=𝜔^𝜔−1_𝑘=0𝑓(𝑘), (9) where 𝑓(𝑘) is an 𝜔-periodic sequence of real numbers defined for 𝑘∈𝑍. In order to explore the existence of positive periodic solutions of (7) and for the reader’s convenience, we will first summarize below a few concepts and results without proof, borrowing from [21].

Let 𝑋,𝑌 be normed vector spaces, 𝐿∶Dom𝐿⊂𝑋→𝑌 a linear mapping, and 𝑁∶𝑋→𝑌 a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=codimIm𝐿<+∞ and Im𝐿 is closed in 𝑌. If 𝐿 is a Fredholm mapping of index zero and there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑌→𝑌 such that Im𝑃=Ker𝐿, Im𝐿=Ker𝑄=Im(𝐼−𝑄), it follows that 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 exist isomorphisms 𝐽∶Im𝑄→Ker𝐿.

Lemma 1 (see [21] continuation theorem). Let 𝐿 be a Fredholm mapping of index zero and let 𝑁 be 𝐿-compact on Ω. Suppose(a)for each 𝜆∈(0,1), every solution 𝑥 of 𝐿𝑥=𝜆𝑁𝑥 is such that 𝑥∉𝜕Ω;(b)𝑄𝑁𝑥≠0 for each 𝑥∈Ker𝐿⋂𝜕Ω and deg{𝐽𝑄𝑁,Ω⋂Ker𝐿,0}≠0.
Then the equation 𝐿𝑥=𝑁𝑥 has at least one solution lying in Dom𝐿⋂Ω.

Lemma 2 (see [17]). Let 𝑔∶𝑍→𝑅 be 𝜔 periodic; that is, 𝑔(𝑘+𝜔)=𝑔(𝑘); then for any fixed 𝑘₁,𝑘₂∈𝐼_𝜔 and any 𝑘∈𝑍, one has 𝑔(𝑘)≤𝑔𝑘₁+^𝜔−1_𝑠=0𝑔()(𝑠,𝑔(𝑘)𝑘𝑠+1−𝑔)≥𝑔₂−^𝜔−1_𝑠=0𝑔(𝑠+1)−𝑔(𝑠). (10)

Lemma 3. (𝑥₁(𝑘),𝑥₂(𝑘),𝑥₃(𝑘)) is an 𝜔 periodic solution of (7) with strictly positive components if and only if (ln{𝑥₁(𝑘)},ln{𝑥₂(𝑘)},ln{𝑥₃(𝑘)}) is an 𝜔 periodic solution of 𝑥₁(𝑘+1)−𝑥₁(𝑘)=𝑟₁(𝑘)−𝑎₁₁(𝑘)exp𝑥₁(𝑘)−𝑎₁₂(𝑘)^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝑥₂(𝑘−𝑙)−𝑎₁₃(𝑘)^+∞_𝑙=0K₁₃(−𝑙)exp𝑥₃(),𝑥𝑘−𝑙₂(𝑘+1)−𝑥₂(𝑘)=𝑟₂(𝑘)−𝑎₂₂(𝑘)exp𝑥₂(𝑘)−𝑎₂₁(𝑘)^+∞_𝑙=0𝐾₂₁(−𝑙)exp𝑥₁(𝑘−𝑙)−𝑎₂₃(𝑘)^+∞_𝑙=0𝐾₂₃(−𝑙)exp𝑥₃(,𝑥𝑘−𝑙)₃(𝑘+1)−𝑥₃(𝑘)=𝑟₃(𝑘)−𝑎₃₃(𝑘)exp𝑥₃(𝑘)−𝑎₃₁(𝑘)^+∞_𝑙=0𝐾₃₁(−𝑙)exp𝑥₁(𝑘−𝑙)−𝑎₃₂(𝑘)^+∞_𝑙=0𝐾₃₂(−𝑙)exp𝑥₂(𝑘−𝑙). (11)

The proofs of Lemma 3 are trivial, so we omitted the details here.

Define 𝑙₃=𝑧={𝑧(𝑘)}∶𝑧(𝑘)∈𝑅³,𝑘∈𝑍. (12)

For 𝑎=(𝑎₁,𝑎₂,𝑎₃)^𝑇∈𝑅³, define |𝑎|=max{|𝑎₁|,|𝑎₂|,|𝑎₃|}. Let 𝑙^𝜔⊂𝑙₃ denote the subspace of all 𝜔 periodic sequences equipped with the usual supremum norm ‖⋅‖; that is, ‖𝑧‖=max_{𝑘∈𝐼𝜔}|𝑧(𝑘)|, for any 𝑧={𝑧(𝑘)∶𝑘∈𝑍}∈𝑙^𝜔. It is easy to show that 𝑙_𝜔 is a finite-dimensional Banach space.

Let 𝑙^𝜔₀=𝑧={𝑧(𝑘)}∈𝑙^𝜔∶^𝜔−1_𝑘=0𝑧(𝑘),𝑙=0_𝜔𝑐=𝑧={𝑧(𝑘)}∈𝑙^𝜔∶𝑧(𝑘)=ℎ∈𝑅³,𝑘∈𝑍, (13) then it follows that 𝑙^𝜔₀ and 𝑙^𝜔_𝑐 are both closed linear subspaces of 𝑙^𝜔 and 𝑙^𝜔=𝑙^𝜔₀+𝑙^𝜔_𝑐,dim𝑙^𝜔_𝑐=3. (14) Next, we will be ready to establish our result.

Theorem 4. Let 𝐵₃, 𝐵₆, 𝐵₁₁, 𝐵₁₅, 𝐵₁₇, 𝐵₂₂, 𝐵₃₃, 𝐵₃₅, 𝐵₃₉, and 𝐵₄₁ be defined by (33), (37), (49), (57), (61), (73), (81), (86), (94), and (98), respectively, and set 𝜃₁=𝑎₁₁expB₃+𝑎^∞_12𝑙=1𝐾₁₂(−𝑙)exp𝐵₆,𝜃₂=𝑎₁₁exp𝐵₃+𝑎^∞_13𝑙=1𝐾₁₃(−𝑙)exp𝐵₁₁,𝜃₃=𝑎₁₁exp𝐵₁₇+𝑎^∞_12𝑙=1𝐾₁₂(−𝑙)exp𝐵₁₅,𝜃₄=𝑎^∞_12𝑙=1𝐾₁₂(−𝑙)exp𝐵₁₅+𝑎^∞_13𝑙=1𝐾₁₃(−𝑙)exp𝐵₂₂,𝜃₅=𝑎₁₁exp𝐵₃₅+𝑎^∞_13𝑙=1𝐾₁₃(−𝑙)exp𝐵₃₃,𝜃₆=𝑎^∞_13𝑙=1𝐾₁₃(−𝑙)exp𝐵₃₉,𝜃₇=𝑎^∞_12𝑙=1𝐾₁₂(−𝑙)exp𝐵₃₉+𝑎^∞_13𝑙=1𝐾₁₃(−𝑙)exp𝐵₄₁. (15) Suppose that (H1), (H2) and (H3)  𝑟₁>{𝜃₁,𝜃₂,𝜃₃,𝜃₄,𝜃₅,𝜃₆,𝜃₇} hold, then system (7) has at least an 𝜔 periodic solution with positive components.

Proof. Let 𝑋=𝑌=𝑙^𝜔, (𝑘)()(𝑘),(𝑘)=𝑓𝐿𝑧)(=𝑧𝑘+1−𝑧𝑁𝑧)(¹(𝑘)𝑓₂(𝑘)𝑓₃(𝑘), (16) where 𝑧∈𝑋,  𝑘∈𝑍, and 𝑓₁(𝑘)=𝑟₁(𝑘)−𝑎₁₁(𝑘)exp𝑥₁(𝑘)−𝑎₁₂(𝑘)^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝑥₂(𝑘−𝑙)−𝑎₁₃(𝑘)^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝑥₃(),𝑓𝑘−𝑙₂(𝑘)=𝑟₂(𝑘)−𝑎₂₂(𝑘)exp𝑥₂(𝑘)−𝑎₂₁(𝑘)^+∞_𝑙=0𝐾₂₁(−𝑙)exp𝑥₁(𝑘−𝑙)−𝑎₂₃(𝑘)^+∞_𝑙=0𝐾₂₃(−𝑙)exp𝑥₃(),𝑓𝑘−𝑙₃(𝑘)=𝑟₃(𝑘)−𝑎₃₃(𝑘)exp𝑥₃(𝑘)−𝑎₃₁(𝑘)^+∞_𝑙=0𝐾₃₁(−𝑙)exp𝑥₁(𝑘−𝑙)−𝑎₃₂(𝑘)^+∞_𝑙=0𝐾₃₂(−𝑙)exp𝑥₂(𝑘−𝑙). (17) Then it is trivial to see that 𝐿 is a bounded linear operator and Ker𝐿=𝑙^𝜔_𝑐,Im𝐿=𝑙^𝜔₀,𝑑𝑖𝑚𝐾𝑒𝑟𝐿=3=codimIm𝐿, (18) then it follows that 𝐿 is a Fredholm mapping of index zero. Define 1𝑃𝑦=𝜔^𝜔−1_𝑠=0𝑦(𝑠)1,𝑦∈𝑋,𝑄𝑧=𝜔^𝜔−1_𝑠=0𝑧(𝑠),𝑧∈𝑌. (19) It is not difficult to show that 𝑃 and 𝑄 are continuous projectors such that Im𝑃=Ker𝐿,Im𝐿=Ker𝑄=Im(𝐼−𝑄). (20) Furthermore, the generalized inverse (to 𝐿) 𝐾_𝑃∶Im𝐿→Ker𝑃⋂Dom𝐿 exists and is given by 𝐾_𝑃(𝑧)=^𝜔−1_𝑠=0𝑧(𝑠)−1𝜔^𝜔−1_𝑠=0(𝜔−𝑠)𝑧(𝑠). (21) Obviously, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. Since 𝑋 is a finite-dimensional Banach space, it is not difficult to show that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is compact for any open bounded set Ω⊂𝑋. Moreover, 𝑄𝑁(Ω) is bounded. Thus, 𝑁 is 𝐿-compact on Ω with any open bounded set Ω⊂𝑋.
Now we are at the point to search for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation 𝐿𝑧=𝜆𝑁𝑧, 𝜆∈(0,1), we have 𝑥₁(𝑘+1)−𝑥₁(𝑘)=𝜆𝑟₁(𝑘)−𝑎₁₁(𝑘)exp𝑥₁(𝑘)−𝑎₁₂(𝑘)^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝑥₂(𝑘−𝑙)−𝑎₁₃(𝑘)^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝑥₃(,𝑥𝑘−𝑙)₂(𝑘+1)−𝑥₂(𝑘)=𝜆𝑟₂(𝑘)−𝑎₂₂(𝑘)exp𝑥₂(𝑘)−𝑎₂₁(𝑘)^+∞_𝑙=0𝐾₂₁(−𝑙)exp𝑥₁(𝑘−𝑙)−𝑎₂₃(𝑘)^+∞_𝑙=0𝐾₂₃(−𝑙)exp𝑥₃(,𝑥𝑘−𝑙)₃(𝑘+1)−𝑥₃(𝑘)=𝜆𝑟₃(𝑘)−𝑎₃₃(𝑘)exp𝑥₃(𝑘)−𝑎₃₁(𝑘)^+∞_𝑙=0𝐾₃₁(−𝑙)exp𝑥₁(𝑘−𝑙)−𝑎₃₂(𝑘)^+∞_𝑙=0𝐾₃₂(−𝑙)exp𝑥₂(𝑘−𝑙). (22) Suppose that 𝑧(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘),𝑥₃(𝑘))^𝑇∈𝑋 is an arbitrary solution of system (22) for a certain 𝜆∈(0,1); summing both sides of (22) from 0 to 𝜔−1 with respect to 𝑘, respectively, we obtain ^𝜔−1_𝑘=0𝑎₁₁(𝑘)exp𝑥₁(𝑘)+𝑎₁₂(𝑘)^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝑥₂(𝑘−𝑙)+𝑎₁₃(𝑘)^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝑥₃(𝑘−𝑙)=𝑟₁𝜔,_{𝜔−1𝑘=0}𝑎₂₂(𝑘)exp𝑥₂(𝑘)+𝑎₂₁(𝑘)^+∞_𝑙=0𝐾₂₁(−𝑙)exp𝑥₁(𝑘−𝑙)+𝑎₂₃(𝑘)^+∞_𝑙=0𝐾₂₃(−𝑙)exp𝑥₃(𝑘−𝑙)=𝑟₂𝜔,_{𝜔−1𝑘=0}𝑎₃₃(𝑘)exp𝑥₃(𝑘)+𝑎₃₁(𝑘)^+∞_𝑙=0𝐾₃₁(−𝑙)exp𝑥₁(𝑘−𝑙)+𝑎₃₂(𝑘)^+∞_𝑙=0𝐾₃₂(−𝑙)exp𝑥₂(𝑘−𝑙)=𝑟₃𝜔. (23)(24)(25) It follows from (22), (23), (24), and (25) that ^𝜔−1_𝑘=0𝑥₁(𝑘+1)−𝑥₁(𝑘)≤2𝑟₁𝜔,_{𝜔−1𝑘=0}𝑥₂(𝑘+1)−𝑥₂(𝑘)≤2𝑟₂𝜔,_{𝜔−1𝑘=0}𝑥₃(𝑘+1)−𝑥₃(𝑘)≤2𝑟₃𝜔. (26) In view of the hypothesis that 𝑧={𝑧(𝑘)}∈𝑋, there exist 𝜉_𝑖,𝜂_𝑖∈𝐼_𝜔 such that 𝑥_𝑖𝜉_𝑖=min_{𝑘∈𝐼𝜔}𝑥_𝑖(𝑘,𝑥_𝑖)𝜂_𝑖=max_{𝑘∈𝐼𝜔}𝑥_𝑖(𝑘)(𝑖=1,2,3). (27) By (23), (24), and (25), we have 𝑎₁₁(𝑘)exp𝑥₁𝜉₁≤^𝜔−1_𝑘=0𝑎₁₁(𝑘)exp𝑥₁(𝑘)<𝑟₁𝜔,𝑎₂₂(𝑘)exp𝑥₂𝜉₂≤^𝜔−1_𝑘=0𝑎₂₂(𝑘)exp𝑥₂(𝑘)<𝑟₂𝜔,𝑎₃₃(𝑘)exp𝑥₃𝜉₃≤^𝜔−1_𝑘=0𝑎₃₃(𝑘)exp𝑥₃(𝑘)<𝑟₃𝜔. (28) Thus 𝑥₁𝜉₁<ln𝑟₁𝑎₁₁,𝑥₂𝜉₂<ln𝑟₂𝑎₂₂,𝑥₃𝜉₃<ln𝑟₃𝑎₃₃. (29) In the sequel, we consider six cases.
(a) If 𝑥₁(𝜂₁)≥𝑥₂(𝜂₂)≥𝑥₃(𝜂₃), then it follows from (23) that 𝑎₁₁+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃^+∞_𝑙=0𝐾₁₃(𝑥−𝑙)×exp₁𝜂₁𝜔≥𝑟₁𝜔 (30) which leads to 𝑥₁𝜂₁>ln𝑟₁𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)∶=𝑀₁, (31) It follows from (29), (31), and Lemma 2 that 𝑥₁(𝑘)≤𝑥₁𝜉₁+^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≤ln𝑟₁𝑎₁₁+2𝑟₁𝜔∶=𝐵₁,𝑥₁(𝑘)≥𝑥₁𝜂₁−^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≥𝑀₁−2𝑟₁𝜔∶=𝐵₂. (32) By (32), we derive max_{𝑘∈𝐼𝜔}𝑥₁(𝑘)≤max𝐵₁,𝐵₂∶=𝐵₃. (33) From (23) and (33), we obtain that 𝑎₁₁𝜔exp𝐵₃+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃^+∞_𝑙=0𝐾₁₃(𝑥−𝑙)×𝜔exp₂𝜂₂≥𝑟₁𝜔. (34) Then 𝑥₂𝜂₂≥ln𝑟₁−𝑎₁₁exp𝐵₃𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (35) Thus by (29), (35), and Lemma 2, we get 𝑥₂(𝑘)≤𝑥₂𝜉₂+^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≤ln𝑟₂𝑎₂₂+2𝑟₂𝜔∶=𝐵₄,𝑥₂(𝑘)≥𝑥₂𝜂₂−^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≥ln𝑟₁−𝑎₁₁exp𝐵₃𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₂𝜔∶=𝐵₅. (36) It follows from (36) that max_{𝑘∈𝐼𝜔}𝑥₂(𝑘)≤max𝐵₄,𝐵₅∶=𝐵₆. (37) In view of (33), (37), and (23), we get 𝑎₁₁𝜔exp𝐵₃+𝑎₁₂𝜔^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₆+𝑎₁₃𝜔^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝑥₃𝜂₃≥𝑟₁𝜔. (38) Then 𝑥₃𝜂₃≥ln𝑟₁−𝑎₁₁exp𝐵₃−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₆𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (39) Thus by (29), (39) and Lemma 2, we get 𝑥₃(𝑘)≤𝑥₃𝜉₃+^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≤ln𝑟₃𝑎₃₃+2𝑟₃𝜔∶=𝐵₇,𝑥₃(𝑘)≥𝑥₃𝜂₃−^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≥ln𝑟₁−𝑎₁₁exp𝐵₃−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₆𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₃𝜔∶=𝐵₈. (40) It follows from (40) that max_{𝑘∈𝐼𝜔}𝑥₃(𝑘)≤max𝐵₇,𝐵₈∶=𝐵₉. (41)
(b) If 𝑥₁(𝜂₁)≥𝑥₃(𝜂₃)≥𝑥₂(𝜂₂), then it follows from (23) that 𝑎₁₁+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃^+∞_𝑙=0𝐾₁₃(𝑥−𝑙)×exp₁𝜂₁𝜔≥𝑟₁𝜔 (42) which leads to 𝑥₁𝜂₁>ln𝑟₁𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)∶=𝑀₁. (43) It follows from (29), (43), and Lemma 2 that 𝑥₁(𝑘)≤𝑥₁𝜉₁+^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≤ln𝑟₁𝑎₁₁+2𝑟₁𝜔∶=𝐵₁,𝑥₁(𝑘)≥𝑥₁𝜂₁−^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≥𝑀₁−2𝑟₁𝜔∶=𝐵₂. (44) By (44), we derive max_{𝑘∈𝐼𝜔}𝑥₁(𝑘)<max𝐵₁,𝐵₂∶=𝐵₃. (45) From (23) and (45), we obtain that 𝑎₁₁𝜔exp𝐵₃+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃^+∞_𝑙=0𝐾₁₃(𝑥−𝑙)×𝜔exp₃𝜂₃≥𝑟₁𝜔. (46) Then 𝑥₃𝜂₃≥ln𝑟₁−𝑎₁₁exp𝐵₃𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(),−𝑙 (47) Thus by (29), (47), and Lemma 2, we get 𝑥₃(𝑘)≤𝑥₃𝜉₃+^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≤ln𝑟₃𝑎₃₃+2𝑟₃𝜔∶=𝐵₇,𝑥₃(𝑘)≥𝑥₃𝜂₃−^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≥ln𝑟₁−𝑎₁₁exp𝐵₃𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₃𝜔∶=𝐵₁₀. (48) It follows from (48) that max_{𝑘∈𝐼𝜔}𝑥₃(𝑘)≤max𝐵₇,𝐵₁₀∶=𝐵₁₁. (49) In view of (45), (49), and (23), we get 𝑎₁₁𝜔exp𝐵₃+𝑎₁₂𝜔^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝑥₂𝜂₂+𝑎₁₃𝜔^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₁₁≥𝑟₁𝜔. (50) Then 𝑥₂𝜂₂≥ln𝑟₁−𝑎₁₁exp𝐵₃−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₁₁𝑎₁₂∑^+∞_𝑙=0𝐾₁₂().−𝑙 (51) Thus by (29), (51), and Lemma 2, we get 𝑥₃(𝑘)≤𝑥₃𝜉₃+^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≤ln𝑟₃𝑎₃₃+2𝑟₃𝜔∶=𝐵₇,𝑥₃(𝑘)≥𝑥₃𝜂₃−^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≥ln𝑟₁−𝑎₁₁exp𝐵₃−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₁₁𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)−2𝑟₃𝜔∶=𝐵₁₂. (52) It follows from (52) that max_{𝑘∈𝐼𝜔}𝑥₃(𝑘)≤max𝐵₇,𝐵₁₂∶=𝐵₁₃. (53)
(c) If 𝑥₂(𝜂₂)≥𝑥₁(𝜂₁)≥𝑥₃(𝜂₃), then it follows from (23) that 𝑎₁₁+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃^+∞_𝑙=0𝐾₁₃(𝑥−𝑙)×exp₂𝜂₂𝜔≥𝑟₁𝜔 (54) which leads to 𝑥₂𝜂₂>ln𝑟₁𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)∶=𝑀₁. (55) It follows from (29), (55), and Lemma 2 that 𝑥₂(𝑘)≤𝑥₂𝜉₂+^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≤ln𝑟₂𝑎₂₂+2𝑟₂𝜔∶=𝐵₄,𝑥₂(𝑘)≥𝑥₂𝜂₂−^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≥𝑀₁−2𝑟₂𝜔∶=𝐵₁₄. (56) By (56), we derive max_{𝑘∈𝐼𝜔}𝑥₂(𝑘)≤max𝐵₄,𝐵₁₄∶=𝐵₁₅. (57) From (23) and (57), we obtain that 𝑎₁₁+𝑎₁₃^+∞_𝑙=0𝐾₁₃(−𝑙)𝜔exp𝑥₁𝜂₁+𝑎₁₂^+∞_𝑙=0K₁₂(−𝑙)𝜔exp𝐵₁₅≥𝑟₁𝜔. (58) Then 𝑥₁𝜂₁≥ln𝑟₁−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₁₅𝑎₁₁+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (59) Thus by (29), (59), and Lemma 2, we get 𝑥₁(𝑘)≤𝑥₁𝜉₁+^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≤ln𝑟₁𝑎₁₁+2𝑟₁𝜔∶=𝐵₁,𝑥₁(𝑘)≥𝑥₁𝜂₁−^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≥ln𝑟₁−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₁₅𝑎₁₁+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₁𝜔∶=𝐵₁₆. (60) It follows from (60) that max_{𝑘∈𝐼𝜔}𝑥₁(𝑘)≤max𝐵₁,𝐵₁₆∶=𝐵₁₇. (61) In view of (57), (61), and (23), we get 𝑎₁₁𝜔exp𝐵₁₇+𝑎₁₂𝜔^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₁₅+𝑎₁₃𝜔^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝑥₃𝜂₃≥𝑟₁𝜔. (62) Then 𝑥₃𝜂₃≥ln𝑟₁−𝑎₁₁exp𝐵₁₇−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₁₅𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (63) Thus by (29), (63), and Lemma 2, we get 𝑥₃(𝑘)≤𝑥₃𝜉₃+^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠2)≤ln𝑟₃𝑎₃₃+2𝑟₃𝜔∶=𝐵₇,𝑥₃(𝑘)≥𝑦𝜂₃−^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≥ln𝑟₁−𝑎₁₁exp𝐵₁₇𝑎₁₃∑^+∞_𝑙=0𝐾₁₃()−−𝑙𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₁₅𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₃𝜔∶=𝐵₁₈. (64) It follows from (64) that max_{𝑘∈𝐼𝜔}𝑥₃(𝑘)≤max𝐵₇,𝐵₁₈∶=𝐵₁₉. (65)
(d) If 𝑥₂(𝜂₂)≥𝑥₃(𝜂₃)≥𝑥₁(𝜂₁), then it follows from (23) that 𝑎₁₁+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃^+∞_𝑙=0𝐾₁₃(𝑥−𝑙)×exp₂𝜂₂𝜔≥𝑟₁𝜔 (66) which leads to 𝑥₂𝜂₂>ln𝑟₁𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)∶=𝑀₁. (67) It follows from (29), (67), and Lemma 2 that 𝑥₂(𝑘)≤𝑥₂𝜉₂+^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≤ln𝑟₂𝑎₂₂+2𝑟₂𝜔∶=𝐵₄,𝑥₂(𝑘)≥𝑥₂𝜂₂−^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≥𝑀₁−2𝑟₂𝜔∶=𝐵₁₄. (68) By (56), we derive max_{𝑘∈𝐼𝜔}𝑥₂(𝑘)≤max𝐵₄,𝐵₁₄∶=𝐵₂₀. (69) From (23) and (57), we obtain that 𝑎₁₁+𝑎₁₃^+∞_𝑙=0𝐾₁₃(−𝑙)𝜔exp𝑥₃𝜂₃+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)𝜔exp𝐵₂₀≥𝑟₁𝜔. (70) Then 𝑥₃𝜂₃≥ln𝑟₁−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₂₀𝑎₁₁+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (71) Thus by (29), (71), and Lemma 2, we get 𝑥₃(𝑘)≤𝑥₃𝜉₃+^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≤ln𝑟₃𝑎₃₃+2𝑟₃𝜔∶=𝐵₇,𝑥₃(𝑘)≥𝑥₃𝜂₃−^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≥ln𝑟₁−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₂₀𝑎₁₁+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₃𝜔∶=𝐵₂₁. (72) It follows from (72) that max_{𝑘∈𝐼𝜔}𝑥₃(𝑘)≤max𝐵₇,𝐵₂₀∶=𝐵₂₂. (73) In view of (69), (73), and (23), we get 𝑎₁₁𝜔exp𝑥₁𝜂₁+𝑎₁₂𝜔^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₂₀+𝑎₁₃𝜔^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₂₂≥𝑟₁𝜔. (74) Then 𝑥₁𝜂₁≥ln𝑟₁−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₂₀𝑎₁₃∑^+∞_𝑙=0𝐾₁₃()−−𝑙𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₂₂𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (75) Thus by (29), (75), and Lemma 2, we get 𝑥₁(𝑘)≤𝑥₁𝜉₁+^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≤ln𝑟₁𝑎₁₁+2𝑟₁𝜔∶=𝐵₁,𝑥₁(𝑘)≥𝑥₁𝜂₁−^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≥ln𝑟₁−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₂₀𝑎₁₃∑^+∞_𝑙=0𝐾₁₃()−−𝑙𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₂₂𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₁𝜔∶=𝐵₂₃. (76) It follows from (76) that max_{𝑘∈𝐼𝜔}𝑥₃(𝑘)≤max𝐵₁,𝐵₂₃∶=𝐵₂₄. (77)
(e) If 𝑥₃(𝜂₃)≥𝑥₁(𝜂₁)≥𝑥₂(𝜂₂), then it follows from (23) that 𝑎₁₁+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃^+∞_𝑙=0𝐾₁₃(𝑥−𝑙)×exp₃𝜂₃𝜔≥𝑟₁𝜔 (78) which leads to 𝑥₃𝜂₃>ln𝑟₁𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (79) It follows from (29), (79), and Lemma 2 that 𝑥₃(𝑘)≤𝑥₃𝜉₃+^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≤ln𝑟₃𝑎₃₃+2𝑟₃𝜔∶=𝐵₇,𝑥₃(𝑘)≥𝑥₃𝜂₃−^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≥ln𝑟₁𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₃𝜔∶=𝐵₃₂. (80) By (80), we derive max_{𝑘∈𝐼𝜔}𝑥₃(𝑘)≤max𝐵₇,𝐵₃₂∶=𝐵₃₃. (81) From (23) and (33), we obtain that 𝑎₁₁+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)𝜔exp𝑥₁𝜂₁+𝑎₁₃^+∞_𝑙=0𝐾₁₃(−𝑙)𝜔exp𝐵₃₃≥𝑟₁𝜔. (82) Then 𝑥₁𝜂₁≥ln𝑟₁−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₃₃𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂().−𝑙 (83) Thus by (29), (83), and Lemma 2, we get 𝑥₁(𝑘)≤𝑥₁𝜉₁+^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≤ln𝑟₁𝑎₁₁+2𝑟₁𝜔∶=𝐵₁,𝑥₁(𝑘)≥𝑥₁𝜂₁−^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≥ln𝑟₁−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₃₃𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)−2𝑟₁𝜔∶=𝐵₃₄. (84)(85) It follows from (83) and (84) that max_{𝑘∈𝐼𝜔}𝑥₁(𝑘)≤max𝐵₁,𝐵₃₄∶=𝐵₃₅. (86) In view of (83), (84), and (23), we get 𝑎₁₁𝜔exp𝐵₃₅+𝑎₁₂𝜔^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝑥₂𝜂₂+𝑎₁₃𝜔^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₃₃≥𝑟₁𝜔. (87) Then 𝑥₂𝜂₂≥ln𝑟₁−𝑎₁₁exp𝐵₁−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₃₃𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (88) Thus by (29), (88), and Lemma 2, we get 𝑥₂(𝑘)≤𝑥₂𝜉₂+^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≤ln𝑟₂𝑎₂₂+2𝑟₂𝜔∶=𝐵₄,𝑥₂(𝑘)≥𝑥₂𝜂₃−^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≥ln𝑟₁−𝑎₁₁exp𝐵₄−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₃₃𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₁𝜔∶=𝐵₃₆. (89) It follows from (89) that max_{𝑘∈𝐼𝜔}𝑥₂(𝑘)≤max𝐵₄,𝐵₃₆∶=𝐵₃₇. (90)
(f) If 𝑥₃(𝜂₃)≥𝑥₂(𝜂₂)≥𝑥₁(𝜂₁), then it follows from (23) that 𝑎₁₁+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃^+∞_𝑙=0𝐾₁₃(𝑥−𝑙)×exp₃𝜂₃𝜔≥𝑟₁𝜔 (91) which leads to 𝑥₃𝜂₃>ln𝑟₁𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃().−𝑙 (92) It follows from (29), (92), and Lemma 2 that 𝑥₃(𝑘)≤𝑥₃𝜉₃+^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≤ln𝑟₃𝑎₃₃+2𝑟₃𝜔∶=𝐵₇,𝑥₃(𝑘)≥𝑥₃𝜂₃−^𝜔−1_𝑠=0𝑥₃(𝑠+1)−𝑥₃(𝑠)≥ln𝑟₁𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)+𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)−2𝑟₃𝜔∶=𝐵₃₈. (93) By (93), we derive max_{𝑘∈𝐼𝜔}𝑥₃(𝑘)<max𝐵₇,𝐵₃₈∶=𝐵₃₉. (94) From (23) and (94), we obtain that 𝑎₁₁+𝑎₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)𝜔exp𝑥₂𝜂₂+𝑎₁₃^+∞_𝑙=0𝐾₁₃(−𝑙)𝜔exp𝐵₃₉≥𝑟₁𝜔. (95) Then 𝑥₂𝜂₂≥ln𝑟₁−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₃₉𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂().−𝑙 (96) Thus by (29), (96), and Lemma 2, we get 𝑥₂(𝑘)≤𝑥₂𝜉₂+∑^𝜔−1_𝑠=0𝑥₂(𝑠+1)−𝑥₂(𝑠)≤ln𝑟₂𝑎₂₂+2𝑟₂𝜔∶=𝐵₄,𝑥₂(𝑘)≥𝑥₂𝜂₂−∑^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≥ln𝑟₁−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₃₉𝑎₁₁+𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)−2𝑟₂𝜔∶=𝐵₄₀. (97) It follows from (97) that max_{𝑘∈𝐼𝜔}𝑥₂(𝑘)≤max𝐵₄,𝐵₄₀∶=𝐵₄₁. (98) In view of (94), (98), and (23), we get 𝑎₁₁𝜔exp𝑥₁𝜂₁+𝑎₁₂𝜔^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₃₉+𝑎₁₃𝜔^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₄₁≥𝑟₁𝜔. (99) Then 𝑥₁𝜂₁≥ln𝑟₁−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₃₉𝑎₁₁−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₄₁𝑎₁₁. (100) Thus by (29), (100), and Lemma 2, we get 𝑥₁(𝑘)≤𝑥₁𝜉₁+^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≤ln𝑟₁𝑎₁₁+2𝑟₁𝜔∶=𝐵₁,𝑥₁(𝑘)≥𝑥₁𝜂₁−^𝜔−1_𝑠=0𝑥₁(𝑠+1)−𝑥₁(𝑠)≥ln𝑟₁−𝑎₁₂∑^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝐵₃₉𝑎₁₁−𝑎₁₃∑^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝐵₄₁𝑎₁₁−2𝑟₁𝜔∶=𝐵₄₂. (101) It follows from (101) that max_{𝑘∈𝐼𝜔}𝑥₂(𝑘)≤max𝐵₁,𝐵₄₂∶=𝐵₄₃. (102) Obviously, 𝐵_𝑖(𝑖=1,2,…,43) are independent of 𝜆∈(0,1). Take 𝑀=max{𝐵₃,𝐵₆,𝐵₈,𝐵₁₁,𝐵₁₂,𝐵₁₃,𝐵₁₅,𝐵₁₇,𝐵₁₉,𝐵₂₀,𝐵₂₂,𝐵₂₄,𝐵₃₃,𝐵₃₅,𝐵₃₇,𝐵₃₉,𝐵₄₀,𝐵₄₃,}+𝐵₀, where 𝐵₀ is taken sufficiently large such that max{|ln{𝑥^∗₁}|,|ln{𝑥^∗₂}|,|ln{𝑥^∗₃}|}<𝐵₀, where (𝑥^∗₁,𝑥^∗₂,𝑥^∗₃)^𝑇 is the unique positive solution of (11). Now we have proved that any solution 𝑧={𝑧(𝑘)}={(𝑥₁(𝑘),𝑥₂(𝑘),𝑥₃(𝑘))^𝑇} of (22) in 𝑋 satisfies ‖𝑧‖<𝑀, 𝑘∈𝑍.
Let Ω∶={𝑧={𝑧(𝑘)}∈𝑋∶‖𝑧‖<𝑀}, then it is easy to see that Ω is an open, bounded set in 𝑋 and verifies requirement (a) of Lemma 1. When 𝑧∈𝜕Ω∩Ker𝐿, 𝑧={(𝑥₁,𝑥₂,𝑥₃)^𝑇} is a constant vector in 𝑅³ with ‖𝑧‖=max{|𝑥₁|,|𝑥₂|,|𝑥₃|}=𝑀. Then𝑄𝑁𝑧=𝑟₁−𝑎₁₁exp𝑥₁−𝑎∑₁₂^+∞_𝑙=0𝐾₁₂(−𝑙)exp𝑥₂−𝑎∑₁₃^+∞_𝑙=0𝐾₁₃(−𝑙)exp𝑥₃𝑟₂−𝑎₂₂exp𝑥₂−𝑎∑₂₁^+∞_𝑙=0𝐾₂₁(−𝑙)exp𝑥₁−𝑎∑₂₃^+∞_𝑙=0𝐾₂₃(−𝑙)exp𝑥₃𝑟₃−𝑎₃₃exp𝑥₃−𝑎∑₃₁^+∞_𝑙=0𝐾₃₁(−𝑙)exp𝑥₁−𝑎∑₃₂^+∞_𝑙=0𝐾₃₂(−𝑙)exp𝑥₂≠0. (103)Now let us consider homotopic 𝜙(𝑥₁,𝑥₂,𝑥₃𝜇)=𝜇𝑄𝑁𝑧+(1−𝜇)𝐺𝑧, 𝜇∈[0,1], where 𝐺𝑧=𝑟₁−𝑎₁₁exp𝑥₁𝑟₂−𝑎₂₂exp𝑥₂𝑟₃−𝑎₃₃exp𝑥₃. (104) Letting 𝐽 be the identity mapping and by direct calculation, we getdeg𝐽𝑄𝑁𝑥₁,𝑥₂,𝑥₃^𝑇𝑥;Ωker𝐿;0=deg𝑄𝑁₁,𝑥₂,𝑥₃^𝑇𝜙𝑥;Ωker𝐿;0=deg₁,𝑥₂,𝑥₃𝜙𝑥,1;Ωker𝐿;0=deg₁,𝑥₂,𝑥₃−,0;Ωker𝐿;0=signdet𝑎₁₁exp𝑥^∗₁000−𝑎₂₂exp𝑥^∗₂000−𝑎₃₃exp𝑥^∗₃=sign−𝑎₁₁𝑎₂₂𝑎₃₃exp𝑥^∗₁+𝑥^∗₂+𝑥^∗₃=−1≠0. (105)By now, we have proved that Ω verifies all requirements of Lemma 1, then it follows that 𝐿𝑧=𝑁𝑧 has at least one solution in Dom𝐿∩Ω; that is, to say, (11) has at least one 𝜔 periodic solution in Dom𝐿∩Ω say 𝑧^∗={𝑧^∗(𝑘)}={(𝑥^∗₁(𝑘),𝑥^∗₂(𝑘),𝑥^∗₃(𝑘))^𝑇}. Let 𝑥^∗₁(𝑘)=exp{𝑥^∗₁(𝑘)}, 𝑥^∗₂(𝑘)=exp{𝑥^∗₂(𝑘)}, and 𝑥^∗₃(𝑘)=exp{𝑥^∗₃(𝑘)}; then by Lemma 3 we know that 𝑧^∗={𝑥^∗(𝑘)}={𝑥^∗₁(𝑘),𝑥^∗₂(𝑘),𝑥^∗₃(𝑘))^𝑇} is an 𝜔 periodic solution of system (7) with strictly positive components. The proof is complete.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 11261010 and no. 11101126), the Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), the Natural Science and Technology Foundation of Guizhou Province (J[2012] 2100), the Governor Foundation of Guizhou Province ([2012] 53), and the Doctoral Foundation of Guizhou University of Finance and Economics (2010).