# Positive Periodic Solutions in a Discrete Time Three Species Competition System

Authors: Changjin Xu; Yusen Wu
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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:
(1)dx1dt=x1(t)[1-x1(t)-∫-∞tK(s-t)x2(s)dsaaaaaaaaaa-∫-∞tL(s-t)x3(s)ds],dx2dt=x2(t)[1-x2(t)-∫-∞t‍L(s-t)x1(s)dsaaaaaaaaaa-∫-∞t‍K(s-t)x3(s)ds],dx2dt=x3(t)[1-x3(t)-∫-∞t‍K(s-t)x1(s)dsaaaaaaaaaa-∫-∞t‍L(s-t)x2(s)ds],
where xi(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
(2)x˙i(t)=xi(t)[ri(t)-∑j=1naij(t)xj(t-τij(t))].
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:
(3)dx1dt=x1(t)[∫-∞tK12(s-t)x2(s)dsr1(t)-a11(t)x1(t)-a12(t)∫-∞tK12(s-t)x2(s)ds-a13(t)∫-∞t‍K13(s-t)x3(s)ds],dx2dt=x2(t)[∫-∞tK12(s-t)x2(s)dsr2(t)-a22(t)x2(t)aaaaaaaaaa-a21(t)∫-∞t‍K21(s-t)x1(s)dsaaaaaaaaaa-a23(t)∫-∞t‍K23(s-t)x3(s)ds],dx3dt=x3(t)[∫-∞tK12(s-t)x2(s)dsr3(t)-a33(t)x3(t)aaaaaaaaaa-a31(t)∫-∞t‍K31(s-t)x1(s)dsaaaaaaaaaa-a32(t)∫-∞t‍K32(s-t)x2(s)ds],
where xi(t)  (i=1,2,3) stands for the density of competing species at time t, ri,aij∈C(R,[0,+∞))(i,j=1,2,3) are ω-periodic functions (ω>0), and Kij(i,j=1,2,3) is a nonnegative function in L1(-∞,0] with
(4)r¯i=1ω∫0ω‍ri(s)ds>0,i=1,2,3,Kij=∫-∞0‍Kij(θ)dθ>0,aijK=1ω∫0ω‍aij(σ)dσ∫-∞σ‍Kij(s-σ)ds>0.
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 Section2, 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:
(5)1x1(t)x˙1(t)=r1([t])-a11([t])x1([t])aaaaaaaaaa-a12([t])∑l=0+∞‍K12(-l)x2([t]-l)aaaaaaaaaa-a13([t])∑l=0+∞‍K13(-l)x3([t]-l),1x2(t)x˙2(t)=r2([t])-a22([t])x2([t])aaaaaaaaaa-a21([t])∑l=0+∞‍K21(-l)x1([t]-l)aaaaaaaaaa-a23([t])∑l=0+∞‍K23(-l)x3([t]-l),1x3(t)x˙3(t)=r3([t])-a33([t])x3([t])aaaaaaaaaa-a31([t])∑l=0+∞‍K31(-l)x1([t]-l)aaaaaaaaaa-a32([t])∑l=0+∞‍K32(-l)x2([t]-l),
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 derivativesdx1(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,…(6)x1(t)=x1(k)exp{-a13(k)∑l=0+∞‍K13(-l)x3(k-l)][∑l=0+∞K13(-l)x3(k-l)r1(k)-a11(k)x1(k)aaaaaaaaaaaaaaaaa-a12(k)∑l=0+∞‍K12(-l)x2(k-l)aaaaaaaaaaaaaaaaa-a13(k)∑l=0+∞K13(-l)x3(k-l)]aaaaaaaaaaaaaaaaa×(t-k)-a13(k)∑l=0+∞‍K13(-l)x3(k-l)]},x2(t)=x2(k)exp{-a13(k)∑l=0+∞‍K13(-l)x3(k-l)][-a13(k)∑l=0+∞‍K13(-l)x3(k-l)]r2(k)-a22(k)x2(k)aaaaaaaaaaaaaaaaa-a21([t])∑l=0+∞‍K21(-l)x1(k-l)aaaaaaaaaaaaaaaaa-a23(k)∑l=0+∞‍K23(-l)x3(k-l)]aaaaaaaaaaaaaaaaa×(t-k)-a13(k)∑l=0+∞‍K13(-l)x3(k-l)]},x3(t)=x3(k)exp{-a13(k)∑l=0+∞‍K13(-l)x3(k-l)][-a13(k)∑l=0+∞‍K13(-l)x3(k-l)]r3(k)-a33(k)x3(k)aaaaaaaaaaaaaaaaa-a31(k)∑l=0+∞‍K31(-l)x1(k-l)aaaaaaaaaaaaaaaaa-a32(k)∑l=0+∞‍K32(-l)x2(k-l)]aaaaaaaaaaaaaaaaa×(t-k)-a13(k)∑l=0+∞‍K13(-l)x3(k-l)]}.
Let t→k+1; then (6) reads as
(7)x1(k+1)=x1(k)exp{∑l=0+∞K13(-l)x3(k-l)r1(k)-a11(k)x1(k)aaaaaaaaaaaaaaaaaaa-a12(k)∑l=0+∞‍K12(-l)x2(k-l)aaaaaaaaaaaaaaaaaaa-a13(k)∑l=0+∞K13(-l)x3(k-l)},x2(k+1)=x2(k)exp{∑l=0+∞K13(-l)x3(k-l)r2(k)-a22(k)x2(k)aaaaaaaaaaaaaaaaaaa-a21([t])∑l=0+∞‍K21(-l)x1(k-l)aaaaaaaaaaaaaaaaaaa-a23(k)∑l=0+∞‍K23(-l)x3(k-l)},x3(k+1)=x3(k)exp{∑l=0+∞K13(-l)x3(k-l)r3(k)-a33(k)x3(k)aaaaaaaaaaaaaaaaaaa-a31(k)∑l=0+∞‍K31(-l)x1(k-l)aaaaaaaaaaaaaaaaaaa-a32(k)∑l=0+∞‍K32(-l)x2(k-l)}
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)ri,aij:Z→R+ are positive ω-periodic; that is, ri(k+ω)=ri(k) and aij(k+ω)=aij(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(8)0≤∑l=0+∞‍K12(-l)<+∞,0≤∑l=0+∞‍K13(-l)<+∞,0≤∑l=0+∞‍K21(-l)<+∞,0≤∑l=0+∞‍K23(-l)<+∞,0≤∑l=0+∞‍K31(-l)<+∞,0≤∑l=0+∞‍K32(-l)<+∞.


## 3. Existence of Positive Periodic Solutions

For convenience and simplicity on the following discussion, we always use the notations below throughout the paper:(9)Iω∶={0,1,2,…,ω-1},f¯∶=1ω∑k=0ω-1‍f(k),
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].LetX,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 KP. 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 KP(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).
LetL be a Fredholm mapping of index zero and let N be L-compact on Ω¯. Suppose(a)
for eachλ∈(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 equationLx=Nx has at least one solution lying in DomL⋂‍Ω¯.Lemma 2 (see [17]).
Letg:Z→R be ω periodic; that is, g(k+ω)=g(k); then for any fixed k1,k2∈Iω and any k∈Z, one has
(10)g(k)≤g(k1)+∑s=0ω-1|g(s+1)-g(s)|,g(k)≥g(k2)-∑s=0ω-1|g(s+1)-g(s)|.Lemma 3.
(
x
^
1
(
k
)
,
x
^
2
(
k
)
,
x
^
3
(
k
)
) is an ω periodic solution of (7) with strictly positive components if and only if (ln{x^1(k)},ln{x^2(k)},ln{x^3(k)}) is an ω periodic solution of
(11)x1(k+1)-x1(k)=r1(k)-a11(k)exp(x1(k))aaaaaaaaaaaaaaaaa-a12(k)∑l=0+∞K12(-l)exp(x2(k-l))aaaaaaaaaaaaaaaaa-a13(k)∑l=0+∞‍K13(-l)exp(x3(k-l)),x2(k+1)-x2(k)=r2(k)-a22(k)exp(x2(k))aaaaaaaaaaaaaaaaa-a21(k)∑l=0+∞‍K21(-l)exp(x1(k-l))aaaaaaaaaaaaaaaaa-a23(k)∑l=0+∞‍K23(-l)exp(x3(k-l)),x3(k+1)-x3(k)=r3(k)-a33(k)exp(x3(k))aaaaaaaaaaaaaaaaa-a31(k)∑l=0+∞‍K31(-l)exp(x1(k-l))aaaaaaaaaaaaaaaaa-a32(k)∑l=0+∞‍K32(-l)exp(x2(k-l)).The proofs of Lemma3 are trivial, so we omitted the details here.Define(12)l3={z={z(k)}:z(k)∈R3,k∈Z}.Fora=(a1,a2,a3)T∈R3, define |a|=max{|a1|,|a2|,|a3|}. Let lω⊂l3 denote the subspace of all ω periodic sequences equipped with the usual supremum norm ∥·∥; that is, ∥z∥=maxk∈Iω|z(k)|, for any z={z(k):k∈Z}∈lω. It is easy to show that lω is a finite-dimensional Banach space.Let(13)l0ω={z={z(k)}∈lω:∑k=0ω-1‍z(k)=0},lcω={z={z(k)}∈lω:z(k)=h∈R3,k∈Z},
then it follows that l0ω and lcω are both closed linear subspaces of lω and
(14)lω=l0ω+lcω,dimlcω=3.
Next, we will be ready to establish our result.Theorem 4.
LetB3, 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
(15)θ1=a¯11exp(B3)+a¯12∑l=1∞‍K12(-l)exp(B6),θ2=a¯11exp(B3)+a¯13∑l=1∞‍K13(-l)exp(B11),θ3=a¯11exp(B17)+a¯12∑l=1∞‍K12(-l)exp(B15),θ4=a¯12∑l=1∞‍K12(-l)exp(B15)+a¯13∑l=1∞‍K13(-l)exp(B22),θ5=a¯11exp(B35)+a¯13∑l=1∞‍K13(-l)exp(B33),θ6=a¯13∑l=1∞‍K13(-l)exp(B39),θ7=a¯12∑l=1∞‍K12(-l)exp(B39)+a¯13∑l=1∞‍K13(-l)exp(B41).
Suppose that (H1), (H2) and (H3)  r¯1>{θ1,θ2,θ3,θ4,θ5,θ6,θ7} hold, then system (7) has at least an ω periodic solution with positive components.Proof.
LetX=Y=lω,
(16)(Lz)(k)=z(k+1)-z(k),(Nz)(k)=(f1(k)f2(k)f3(k)),
where z∈X,  k∈Z, and
(17)f1(k)=r1(k)-a11(k)exp(x1(k))-a12(k)∑l=0+∞‍K12(-l)exp(x2(k-l))-a13(k)∑l=0+∞‍K13(-l)exp(x3(k-l)),f2(k)=r2(k)-a22(k)exp(x2(k))-a21(k)∑l=0+∞‍K21(-l)exp(x1(k-l))-a23(k)∑l=0+∞‍K23(-l)exp(x3(k-l)),f3(k)=r3(k)-a33(k)exp(x3(k))-a31(k)∑l=0+∞‍K31(-l)exp(x1(k-l))-a32(k)∑l=0+∞‍K32(-l)exp(x2(k-l)).
Then it is trivial to see that L is a bounded linear operator and
(18)KerL=lcω,ImL=l0ω,dimKerL=3=codimImL,
then it follows that L is a Fredholm mapping of index zero. Define
(19)Py=1ω∑s=0ω-1‍y(s),y∈X,Qz=1ω∑s=0ω-1‍z(s),z∈Y.
It is not difficult to show that P and Q are continuous projectors such that
(20)ImP=KerL,ImL=KerQ=Im(I-Q).
Furthermore, the generalized inverse (to L) KP:ImL→KerP⋂‍DomL exists and is given by
(21)KP(z)=∑s=0ω-1‍z(s)-1ω∑s=0ω-1‍(ω-s)z(s).
Obviously, QN and KP(I-Q)N are continuous. Since X is a finite-dimensional Banach space, it is not difficult to show that KP(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 subsetΩ for the application of the continuation theorem. Corresponding to the operator equation Lz=λNz, λ∈(0,1), we have
(22)x1(k+1)-x1(k)=λ[∑l=0+∞K13(-l)exp(x3(k-l))r1(k)-a11(k)exp(x1(k))a-a12(k)∑l=0+∞‍K12(-l)exp(x2(k-l))a-a13(k)∑l=0+∞K13(-l)exp(x3(k-l))],x2(k+1)-x2(k)=λ[∑l=0+∞K13(-l)exp(x3(k-l))r2(k)-a22(k)exp(x2(k))a-a21(k)∑l=0+∞‍K21(-l)exp(x1(k-l))a-a23(k)∑l=0+∞‍K23(-l)exp(x3(k-l))],x3(k+1)-x3(k)=λ[∑l=0+∞K13(-l)exp(x3(k-l))r3(k)-a33(k)exp(x3(k))a-a31(k)∑l=0+∞‍K31(-l)exp(x1(k-l))a-a32(k)∑l=0+∞‍K32(-l)exp(x2(k-l))].
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
(23)∑k=0ω-1‍[a11(k)exp(x1(k))+a12(k)∑l=0+∞‍K12(-l)exp(x2(k-l))+a13(k)∑l=0+∞‍K13(-l)exp(x3(k-l))]=r¯1ω,(24)∑k=0ω-1‍[a22(k)exp(x2(k))+a21(k)∑l=0+∞‍K21(-l)exp(x1(k-l))+a23(k)∑l=0+∞‍K23(-l)exp(x3(k-l))]=r¯2ω,(25)∑k=0ω-1‍[a33(k)exp(x3(k))+a31(k)∑l=0+∞‍K31(-l)exp(x1(k-l))+a32(k)∑l=0+∞‍K32(-l)exp(x2(k-l))]=r¯3ω.
It follows from (22), (23), (24), and (25) that
(26)∑k=0ω-1|x1(k+1)-x1(k)|≤2r¯1ω,∑k=0ω-1‍|x2(k+1)-x2(k)|≤2r¯2ω,∑k=0ω-1‍|x3(k+1)-x3(k)|≤2r¯3ω.
In view of the hypothesis that z={z(k)}∈X, there exist ξi,ηi∈Iω such that
(27)xi(ξi)=mink∈Iω{xi(k)},xi(ηi)=maxk∈Iω{xi(k)}(i=1,2,3).
By (23), (24), and (25), we have
(28)a¯11(k)exp(x1(ξ1))≤∑k=0ω-1‍a11(k)exp(x1(k))<r¯1ω,a¯22(k)exp(x2(ξ2))≤∑k=0ω-1‍a22(k)exp(x2(k))<r¯2ω,a¯33(k)exp(x3(ξ3))≤∑k=0ω-1‍a33(k)exp(x3(k))<r¯3ω.
Thus
(29)x1(ξ1)<ln[r¯1a¯11],x2(ξ2)<ln[r¯2a¯22],x3(ξ3)<ln[r¯3a¯33].
In the sequel, we consider six cases.
(a) Ifx1(η1)≥x2(η2)≥x3(η3), then it follows from (23) that
(30)[a¯11+a¯12∑l=0+∞‍K12(-l)+a¯13∑l=0+∞‍K13(-l)]×exp(x1(η1))ω≥r¯1ω
which leads to
(31)x1(η1)>ln[r¯1a¯11+a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)]∶=M1,
It follows from (29), (31), and Lemma 2 that
(32)x1(k)≤x1(ξ1)+∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≤ln[r¯1a¯11]+2r¯1ω∶=B1,x1(k)≥x1(η1)-∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≥M1-2r¯1ω∶=B2.
By (32), we derive
(33)maxk∈Iω{x1(k)}≤max{|B1|,|B2|}∶=B3.
From (23) and (33), we obtain that
(34)a¯11ωexp(B3)+[a¯12∑l=0+∞‍K12(-l)+a¯13∑l=0+∞‍K13(-l)]×ωexp(x2(η2))≥r¯1ω.
Then
(35)x2(η2)≥ln[r¯1-a¯11exp(B3)a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)].
Thus by (29), (35), and Lemma 2, we get
(36)x2(k)≤x2(ξ2)+∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≤ln[r¯2a¯22]+2r¯2ω∶=B4,x2(k)≥x2(η2)-∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≥ln[r¯1-a¯11exp(B3)a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)]aaaaa-2r¯2ω∶=B5.
It follows from (36) that
(37)maxk∈Iω{x2(k)}≤max{|B4|,|B5|}∶=B6.
In view of (33), (37), and (23), we get
(38)a¯11ωexp(B3)+a¯12ω∑l=0+∞‍K12(-l)exp(B6)+a¯13ω∑l=0+∞‍K13(-l)exp(x3(η3))≥r¯1ω.
Then
(39)x3(η3)≥ln[r¯1-a¯11exp(B3)-a¯12∑l=0+∞K12(-l)exp(B6)a¯13∑l=0+∞K13(-l)].
Thus by (29), (39) and Lemma 2, we get
(40)x3(k)≤x3(ξ3)+∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≤ln[r¯3a¯33]+2r¯3ω∶=B7,x3(k)≥x3(η3)-∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≥ln[r¯1-a¯11exp(B3)-a¯12∑l=0+∞K12(-l)exp(B6)a¯13∑l=0+∞K13(-l)]aaaaa-2r¯3ω∶=B8.
It follows from (40) that
(41)maxk∈Iω{x3(k)}≤max{|B7|,|B8|}∶=B9.
(b) Ifx1(η1)≥x3(η3)≥x2(η2), then it follows from (23) that
(42)[a¯11+a¯12∑l=0+∞‍K12(-l)+a¯13∑l=0+∞‍K13(-l)]×exp(x1(η1))ω≥r¯1ω
which leads to
(43)x1(η1)>ln[r¯1a¯11+a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)]∶=M1.
It follows from (29), (43), and Lemma 2 that
(44)x1(k)≤x1(ξ1)+∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≤ln[r¯1a¯11]+2r¯1ω∶=B1,x1(k)≥x1(η1)-∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≥M1-2r¯1ω∶=B2.
By (44), we derive
(45)maxk∈Iω{x1(k)}<max{|B1|,|B2|}∶=B3.
From (23) and (45), we obtain that
(46)a¯11ωexp(B3)+[a¯12∑l=0+∞‍K12(-l)+a¯13∑l=0+∞‍K13(-l)]×ωexp(x3(η3))≥r¯1ω.
Then
(47)x3(η3)≥ln[r¯1-a¯11exp(B3)a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)],
Thus by (29), (47), and Lemma 2, we get
(48)x3(k)≤x3(ξ3)+∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≤ln[r¯3a¯33]+2r¯3ω∶=B7,x3(k)≥x3(η3)-∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≥ln[r¯1-a¯11exp(B3)a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)]aaaaa-2r¯3ω∶=B10.
It follows from (48) that
(49)maxk∈Iω{x3(k)}≤max{|B7|,|B10|}∶=B11.
In view of (45), (49), and (23), we get
(50)a¯11ωexp(B3)+a¯12ω∑l=0+∞‍K12(-l)exp(x2(η2))+a¯13ω∑l=0+∞‍K13(-l)exp(B11)≥r¯1ω.
Then
(51)x2(η2)≥ln[r¯1-a¯11exp(B3)-a¯13∑l=0+∞K13(-l)exp(B11)a¯12∑l=0+∞K12(-l)].
Thus by (29), (51), and Lemma 2, we get
(52)x3(k)≤x3(ξ3)+∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≤ln[r¯3a¯33]+2r¯3ω∶=B7,x3(k)≥x3(η3)-∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≥ln[r¯1-a¯11exp(B3)-a¯13∑l=0+∞K13(-l)exp(B11)a¯12∑l=0+∞K12(-l)]aaaaa-2r¯3ω:=B12.
It follows from (52) that
(53)maxk∈Iω{x3(k)}≤max{|B7|,|B12|}∶=B13.
(c) Ifx2(η2)≥x1(η1)≥x3(η3), then it follows from (23) that
(54)[a¯11+a¯12∑l=0+∞‍K12(-l)+a¯13∑l=0+∞‍K13(-l)]×exp(x2(η2))ω≥r¯1ω
which leads to
(55)x2(η2)>ln[r¯1a¯11+a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)]∶=M1.
It follows from (29), (55), and Lemma 2 that
(56)x2(k)≤x2(ξ2)+∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≤ln[r¯2a¯22]+2r¯2ω∶=B4,x2(k)≥x2(η2)-∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≥M1-2r¯2ω∶=B14.
By (56), we derive
(57)maxk∈Iω{x2(k)}≤max{|B4|,|B14|}∶=B15.
From (23) and (57), we obtain that
(58)[a¯11+a¯13∑l=0+∞‍K13(-l)]ωexp(x1(η1))+[a¯12∑l=0+∞‍K12(-l)]ωexp(B15)≥r¯1ω.
Then
(59)x1(η1)≥ln[r¯1-a¯12∑l=0+∞K12(-l)exp(B15)a¯11+a¯13∑l=0+∞K13(-l)].
Thus by (29), (59), and Lemma 2, we get
(60)x1(k)≤x1(ξ1)+∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≤ln[r¯1a¯11]+2r¯1ω∶=B1,x1(k)≥x1(η1)-∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≥ln[r¯1-a¯12∑l=0+∞K12(-l)exp(B15)a¯11+a¯13∑l=0+∞K13(-l)]-2r¯1ω∶=B16.
It follows from (60) that
(61)maxk∈Iω{x1(k)}≤max{|B1|,|B16|}∶=B17.
In view of (57), (61), and (23), we get
(62)a¯11ωexp(B17)+a¯12ω∑l=0+∞‍K12(-l)exp(B15)+a¯13ω∑l=0+∞‍K13(-l)exp(x3(η3))≥r¯1ω.
Then
(63)x3(η3)≥ln[r¯1-a¯11exp(B17)-a¯12∑l=0+∞K12(-l)exp(B15)a¯13∑l=0+∞K13(-l)].
Thus by (29), (63), and Lemma 2, we get
(64)x3(k)≤x3(ξ3)+∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≤ln[2r¯3a¯33]+2r¯3ω∶=B7,x3(k)≥y(η3)-∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≥ln[r¯1-a¯11exp(B17)a¯13∑l=0+∞K13(-l)aaaaa-a¯12∑l=0+∞K12(-l)exp(B15)a¯13∑l=0+∞K13(-l)]aaaaa-2r¯3ω∶=B18.
It follows from (64) that
(65)maxk∈Iω{x3(k)}≤max{|B7|,|B18|}∶=B19.
(d) Ifx2(η2)≥x3(η3)≥x1(η1), then it follows from (23) that
(66)[a¯11+a¯12∑l=0+∞‍K12(-l)+a¯13∑l=0+∞‍K13(-l)]×exp(x2(η2))ω≥r¯1ω
which leads to
(67)x2(η2)>ln[r¯1a¯11+a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)]∶=M1.
It follows from (29), (67), and Lemma 2 that
(68)x2(k)≤x2(ξ2)+∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≤ln[r¯2a¯22]+2r¯2ω∶=B4,x2(k)≥x2(η2)-∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≥M1-2r¯2ω∶=B14.
By (56), we derive
(69)maxk∈Iω{x2(k)}≤max{|B4|,|B14|}∶=B20.
From (23) and (57), we obtain that
(70)[a¯11+a¯13∑l=0+∞‍K13(-l)]ωexp(x3(η3))+[a¯12∑l=0+∞‍K12(-l)]ωexp(B20)≥r¯1ω.
Then
(71)x3(η3)≥ln[r¯1-a¯12∑l=0+∞K12(-l)exp(B20)a¯11+a¯13∑l=0+∞K13(-l)].
Thus by (29), (71), and Lemma 2, we get
(72)x3(k)≤x3(ξ3)+∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≤ln[r¯3a¯33]+2r¯3ω∶=B7,x3(k)≥x3(η3)-∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≥ln[r¯1-a¯12∑l=0+∞K12(-l)exp(B20)a¯11+a¯13∑l=0+∞K13(-l)]-2r¯3ω∶=B21.
It follows from (72) that
(73)maxk∈Iω{x3(k)}≤max{|B7|,|B20|}∶=B22.
In view of (69), (73), and (23), we get
(74)a¯11ωexp(x1(η1))+a¯12ω∑l=0+∞‍K12(-l)exp(B20)+a¯13ω∑l=0+∞‍K13(-l)exp(B22)≥r¯1ω.
Then
(75)x1(η1)≥ln[r¯1-a¯12∑l=0+∞K12(-l)exp(B20)a¯13∑l=0+∞K13(-l)a-a¯13∑l=0+∞K13(-l)exp(B22)a¯13∑l=0+∞K13(-l)].
Thus by (29), (75), and Lemma 2, we get
(76)x1(k)≤x1(ξ1)+∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≤ln[r¯1a¯11]+2r¯1ω∶=B1,x1(k)≥x1(η1)-∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≥ln[r¯1-a¯12∑l=0+∞K12(-l)exp(B20)a¯13∑l=0+∞K13(-l)aaaaaa-a¯13∑l=0+∞K13(-l)exp(B22)a¯13∑l=0+∞K13(-l)]aaaaa-2r¯1ω∶=B23.
It follows from (76) that
(77)maxk∈Iω{x3(k)}≤max{|B1|,|B23|}∶=B24.
(e) Ifx3(η3)≥x1(η1)≥x2(η2), then it follows from (23) that
(78)[a¯11+a¯12∑l=0+∞‍K12(-l)+a¯13∑l=0+∞‍K13(-l)]×exp(x3(η3))ω≥r¯1ω
which leads to
(79)x3(η3)>ln[r¯1a¯11+a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)].
It follows from (29), (79), and Lemma 2 that
(80)x3(k)≤x3(ξ3)+∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≤ln[r¯3a¯33]+2r¯3ω∶=B7,x3(k)≥x3(η3)-∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≥ln[r¯1a¯11+a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)]aaaaa-2r¯3ω∶=B32.
By (80), we derive
(81)maxk∈Iω{x3(k)}≤max{|B7|,|B32|}∶=B33.
From (23) and (33), we obtain that
(82)[a¯11+a¯12∑l=0+∞‍K12(-l)]ωexp(x1(η1))+a¯13∑l=0+∞‍K13(-l)ωexp(B33)≥r¯1ω.
Then
(83)x1(η1)≥ln[r¯1-a¯13∑l=0+∞K13(-l)exp(B33)a¯11+a¯12∑l=0+∞K12(-l)].
Thus by (29), (83), and Lemma 2, we get
(84)x1(k)≤x1(ξ1)+∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≤ln[r¯1a¯11]+2r¯1ω∶=B1,(85)x1(k)≥x1(η1)-∑s=0ω-1|x1(s+1)-x1(s)|aaaaa≥ln[r¯1-a¯13∑l=0+∞K13(-l)exp(B33)a¯11+a¯12∑l=0+∞K12(-l)]aaaaa-2r¯1ω∶=B34.
It follows from (83) and (84) that
(86)maxk∈Iω{x1(k)}≤max{|B1|,|B34|}∶=B35.
In view of (83), (84), and (23), we get
(87)a¯11ωexp(B35)+a¯12ω∑l=0+∞‍K12(-l)exp(x2(η2))+a¯13ω∑l=0+∞‍K13(-l)exp(B33)≥r¯1ω.
Then
(88)x2(η2)≥ln[r¯1-a¯11exp(B1)-a¯13∑l=0+∞K13(-l)exp(B33)a¯13∑l=0+∞K13(-l)].
Thus by (29), (88), and Lemma 2, we get
(89)x2(k)≤x2(ξ2)+∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≤ln[r¯2a¯22]+2r¯2ω∶=B4,x2(k)≥x2(η3)-∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≥ln[r¯1-a¯11exp(B4)-a¯13∑l=0+∞K13(-l)exp(B33)a¯13∑l=0+∞K13(-l)]aaaaa-2r¯1ω∶=B36.
It follows from (89) that
(90)maxk∈Iω{x2(k)}≤max{|B4|,|B36|}∶=B37.
(f) Ifx3(η3)≥x2(η2)≥x1(η1), then it follows from (23) that
(91)[a¯11+a¯12∑l=0+∞‍K12(-l)+a¯13∑l=0+∞‍K13(-l)]×exp(x3(η3))ω≥r¯1ω
which leads to
(92)x3(η3)>ln[r¯1a¯11+a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)].
It follows from (29), (92), and Lemma 2 that
(93)x3(k)≤x3(ξ3)+∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≤ln[r¯3a¯33]+2r¯3ω∶=B7,x3(k)≥x3(η3)-∑s=0ω-1‍|x3(s+1)-x3(s)|aaaaa≥ln[r¯1a¯11+a¯12∑l=0+∞K12(-l)+a¯13∑l=0+∞K13(-l)]aaaaa-2r¯3ω∶=B38.
By (93), we derive
(94)maxk∈Iω{x3(k)}<max{|B7|,|B38|}∶=B39.
From (23) and (94), we obtain that
(95)[a¯11+a¯12∑l=0+∞‍K12(-l)]ωexp(x2(η2))+a¯13∑l=0+∞‍K13(-l)ωexp(B39)≥r¯1ω.
Then
(96)x2(η2)≥ln[r¯1-a¯13∑l=0+∞K13(-l)exp(B39)a¯11+a¯12∑l=0+∞K12(-l)].
Thus by (29), (96), and Lemma 2, we get
(97)x2(k)≤x2(ξ2)+∑s=0ω-1‍|x2(s+1)-x2(s)|aaaaa≤ln[r¯2a¯22]+2r¯2ω∶=B4,x2(k)≥x2(η2)-∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≥ln[r¯1-a¯13∑l=0+∞K13(-l)exp(B39)a¯11+a¯12∑l=0+∞K12(-l)]aaaaa-2r¯2ω∶=B40.
It follows from (97) that
(98)maxk∈Iω{x2(k)}≤max{|B4|,|B40|}∶=B41.
In view of (94), (98), and (23), we get
(99)a¯11ωexp(x1(η1))+a¯12ω∑l=0+∞‍K12(-l)exp(B39)+a¯13ω∑l=0+∞‍K13(-l)exp(B41)≥r¯1ω.
Then
(100)x1(η1)≥ln[r¯1-a¯12∑l=0+∞K12(-l)exp(B39)a¯11a-a¯13∑l=0+∞K13(-l)exp(B41)a¯11].
Thus by (29), (100), and Lemma 2, we get
(101)x1(k)≤x1(ξ1)+∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≤ln[r¯1a¯11]+2r¯1ω∶=B1,x1(k)≥x1(η1)-∑s=0ω-1‍|x1(s+1)-x1(s)|aaaaa≥ln[r¯1-a¯12∑l=0+∞K12(-l)exp(B39)a¯11aaaaaa-a¯13∑l=0+∞K13(-l)exp(B41)a¯11]aaaaa-2r¯1ω∶=B42.
It follows from (101) that
(102)maxk∈Iω{x2(k)}≤max{|B1|,|B42|}∶=B43.
Obviously, Bi(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{x1*}|,|ln{x2*}|,|ln{x3*}|}<B0, where (x1*,x2*,x3*)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.
LetΩ:={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. Then(103)QNz=(r¯1-a¯11exp(x1)-a¯12∑l=0+∞‍K12(-l)exp(x2)-a¯13∑l=0+∞‍K13(-l)exp(x3)r¯2-a¯22exp(x2)-a¯21∑l=0+∞‍K21(-l)exp(x1)-a¯23∑l=0+∞‍K23(-l)exp(x3)r¯3-a¯33exp(x3)-a¯31∑l=0+∞‍K31(-l)exp(x1)-a¯32∑l=0+∞‍K32(-l)exp(x2))≠0.Now let us consider homotopic ϕ(x1,x2,x3μ)=μQNz+(1-μ)Gz, μ∈[0,1], where
(104)Gz=(r¯1-a¯11exp(x1)r¯2-a¯22exp(x2)r¯3-a¯33exp(x3)).
Letting J be the identity mapping and by direct calculation, we get(105)deg{JQN(x1,x2,x3)T;Ω⋂‍kerL;0}=deg{QN(x1,x2,x3)T;Ω⋂‍kerL;0}=deg{ϕ(x1,x2,x3,1);Ω⋂‍kerL;0}=deg{ϕ(x1,x2,x3,0);Ω⋂‍kerL;0}=sign{det(-a¯11exp(x1*)000-a¯22exp(x2*)000-a¯33exp(x3*))}=sign{-a¯11a¯22a¯33exp(x1*+x2*+x3*)}=-1≠0.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)}={(x1*(k),x2*(k),x3*(k))T}. Let x¯1*(k)=exp{x1*(k)}, x¯2*(k)=exp{x2*(k)}, and x¯3*(k)=exp{x3*(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.



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*Source: 963046-2013-03-11.xml*