# On the Positive Almost Periodic Solutions of a Class of Nonlinear Lotka-Volterra Type System with Feedback Control

Authors: Ni Hua; Tian Li-Xin; Yao Hong-Xing
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

With the help of the variable substitution and applying the fixed point theorem, we derive the sufficient conditions which guarantee the existence of the positive almost periodic solutions for a class of Lotka-Volterra type system. The main results improve and generalize the former corresponding results.


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## Body

## 1. Introduction

Denotex(t)=(x1(t),x2(t),…,xn(t)),xt=(x1t,x2t,…,xnt),xit(s)=xi(t+s)  (i=1,2,…,n),s∈[-τ,0],τ is a positive constant or τ=+∞, the norm of a bounded continuous function space C([-τ,0],Rn) is defined as ∥ϕ∥=maxs∈[-τ,0]|ϕ(s)|, where |ϕ|=max{|ϕi|,i=1,2,…,n},C([-τ,0],R+)={ϕ∈C([-τ,0],R):forallθ∈[-τ,0],ϕ(θ)>0}.We call an almost periodic function is positive if and only if each component has its positive infimum. DenoteAP(Rn)={x(t)∈Rn:x(t) is a continuous almost periodic function on Rn},AP(R)={x(t)∈R:x(t) is a continuous almost periodic function on R},Tαf(t)=g(t)referstolimn→∞f(t+αn)=g(t),whereα={αn},{αn} is a sequence of real numbers,T(f,ε)={τ:|f(t+τ)-f(t)|<ε,t∈R},for more almost periodic monographs, see references [1, 2].In [3], Teng first studied the existence of the almost periodic solutions for the following scalar equation:
(1.1)dudt=u(α(t)-β(t))u,
where t∈R,u∈R, and α(t),β(t)∈AP(R), then based on (1.1) and combined Schauder fixed point theorem, he studied the existence of the almost periodic solution for a class of Lotka-Volterra system
(1.2)dxi(t)dt=xi(t)(ai(t)-bi(t)xi(t)-fi(t,xt)),(i=1,2,…,n),
where ai(t),bi(t) are continuous almost periodic functions,fi(t,ϕ) is a continuous almost periodic function in t uniformly with respect to ϕ∈C([-τ,0],Rn), and fi(t,0)≡0(i=1,2,…,n). The author of [4] inherited in the method and ideas in [3], and promoted its conclusions to the following system with feedback control(1.3)dxi(t)dt=xi(t)(ai(t)-bi(t)xi(t)-fi(t,xt,ut)),dui(t)dt=-ci(t)ui(t)+hi(t,xit),
where ui(t) is the control variable, i=1,2,…,n.The authors of [5] used transformation techniques and fixed point theorem and studied a time-delay system with feedback control which is much wider then the system (1.3),(1.4)dxi(t)dt=xi(t)(ai(t)-bi(t)xi(t)-fi(t,xt,ut)),dui(t)dt=-ci(t)ui(t)+hi(t,xt).
In the case of non-Lipschitz condition, they gave a sufficient condition of the existence of the almost periodic solutions for the system (1.4).However, in the real world, the competition between species is not always shown by the linear relationship, while shown by a certain degree of nonlinearity, therefore, studying the following system becomes more realistic and necessary, over this paper, we study the system which is more extensive than the system (1.4) as follows:(1.5)dxi(t)dt=xi(t)(ai(t)-bi(t)xiαi(t)-fi(t,xt,ut)),dui(t)dt=-ci(t)ui(t)+hi(t,xt),
where ui(t) is the control variable, αi is a positive constant, i=1,2,…,n, by using Schauder’s fixed point theorem, we get the sufficient conditions of the existence of the almost periodic solution for the system (1.5), and by using the contraction mapping principle, we give the conclusion of the existence of a unique almost periodic solution for the system (1.5) in the one-dimensional case, some new results are obtained.


## 2. Some Related Lemmas

Lemma 2.1 (see [6]).
For the equation(2.1)dxdt=a(t)x(t)+b(t),
where a(t),b(t)∈AP(R) and are continuous, if  Rem(a(t))≠0, then (2.1) exists a unique almost periodic solution η(t),mod(η)⊂mod(a,b), and η(t) can be written as follows
(2.2)η(t)={∫-∞te∫sta(τ)dτb(s)ds,Rem(a(t))<0,-∫t+∞e∫sta(τ)dτb(s)ds,Rem(a(t))>0,
where m(a(t))=limT→+∞1/T∫0Ta(t)dt, Rem(a(t)) is the real part of m(a(t)).Lemma 2.2 (see [2]).
Supposef(t),g(t)∈AP(R), then the following conditions are equivalent.(1)
mod(f)⊃mod(g);(2)
for anyɛ>0,∃δ>0,such that T(f,δ)⊂T(g,ɛ);(3)
Tαf exists which implies Tαg exists (any sense);(4)
Tαf=f, implies Tαg=g(any sense);(5)
Tαf=f, implies there is α′⊂α,so that Tα′g=g(any sense). Consider the equation
(2.3)dx(t)dt=x(t)[a(t)-b(t)xα(t)],
where α>0,a(t),b(t)∈AP(R) and are continuous on R.Lemma 2.3.
If (2.3) satisfies one of the following conditions
(1)
m(a(t))>0,m(b(t))>0,b(t)≥0(2)
m(a(t))<0,m(b(t))<0,b(t)≤0,then (2.3) exists a unique positive almost periodic solution ϕ(t), and mod(ϕ)⊂mod(a,b).Proof.
Letx-α(t)=u(t), then (2.3) can be changed as follows:
(2.4)du(t)dt=-αa(t)u(t)+αb(t),
by the condition m(a(t))>0,α>0, we have Rem(-αa(t))<0, also by the condition m(b(t))>0,b(t)≥0, according to Lemma 2.1, it follows that (2.4) exists a unique positive almost periodic solution η(t), and η(t) can be written as follows:
(2.5)η(t)=α∫-∞te-α∫sta(τ)dτb(s)ds.
Next, we prove thatmod(η)⊂mod(a,b). If there is a sequence β′ such that Tβ′a(t)=a(t),Tβ′b(t)=b(t) are convergent uniformly on R, then there exists a sequence β⊂β′ such that Tβη(t)=ζ(t) is convergent uniformly on R, and ζ(t) is also an almost periodic solution of (2.4), by the uniqueness of the almost periodic solution of (2.4), we can get Tβη(t)=η(t), by Lemma 2.2, it follows mod(η)⊂mod(a,b). Since x-α(t)=u(t), (2.3) exists a unique positive almost periodic solution ϕ(t), and it can be written as follows
(2.6)ϕ(t)=[α∫-∞te-α∫sta(τ)dτb(s)ds]-1/α,
by (2.6), we can easily get mod(ϕ)⊂mod(a,b).
If the condition (2) holds, similarly, we can prove that (2.3) exists a unique positive almost periodic solution ϕ(t), it can be written as follows
(2.7)ϕ(t)=[-α∫t+∞e-α∫sta(τ)dτb(s)ds]-1/α,
and mod(ϕ)⊂mod(a,b). This is the end of the proof of Lemma 2.3.Lemma 2.4 (see [2]).
Suppose that an almost periodic sequence{fn(t)} is convergent uniformly on any compact set of R,  f(t) is an almost periodic function, and mod(fn)⊂mod(f)(n=1,2,…), then {fn(t)} is convergent uniformly on R.


## 3. The Conclusion of the N-dimentional System

Consider the following equation(3.1)dxi(t)dt=xi(t)[ai(t)-bi(t)xiαi(t)],
where αi>0,ai(t),bi(t)∈AP(R),i=1,2,…,n.By Lemma2.3, it follows if one of the following conditions holds(A1)
m(ai(t))>0,m(bi(t))>0,bi(t)≥0(A2)
m(ai(t))<0,m(bi(t))<0,bi(t)≤0,then (3.1) exists a unique positive almost periodic solution x0(t)=(x10(t),x20(t),…,xn0(t)), we can easily get if (A1) holds, then
(3.2)xi0(t)=[αi∫-∞te-αi∫stai(τ)dτbi(s)ds]-1/αi,
if (A2) holds, then
(3.3)xi0(t)=[-αi∫t+∞e-αi∫stai(τ)dτbi(s)ds]-1/αi.
Now, consider (1.5), suppose(B1)
ai(t),bi(t),ci(t) are continuous almost periodic functions,fi(t,ϕ,ψ) is continuous for all variables, and is almost periodic in t uniformly with respect to (ϕ,ψ)∈C([-τ,0],Rn)×C([-τ,0],Rn),  hi(t,ϕ) is continuous for all variables, and is almost periodic in t uniformly with respect to ϕ∈C([-τ,0],Rn);(B2)
hi(t,ϕ)>0,  for all ϕi>0,t∈R(i=1,2,…,n);(B¯2)
hi(t,ϕ)<0,  for all ϕi>0,t∈R(i=1,2,…,n);(B3)
m(ai(t))>0,m(bi(t))>0,bi(t)≥0,m(ci(t))>0(i=1,2,…,n);(B¯3)
m(ai(t))<0,m(bi(t))<0,bi(t)≤0,m(ci(t))<0(i=1,2,…,n).Following the front of thex0(t) defined, suppose that (B1),(B2),(B3) or (B1),(B¯2),(B¯3) hold, and denoted the unique positive almost periodic solution of the following equation
(3.4)dui(t)dt=-ci(t)ui(t)+hi(t,xt0),
by u0(t)=(u10(t),u20(t),…,un0(t)),where (3.5)ui0(t)={∫-∞te∫st(-ci(τ))dτhi(s,xs0)ds,Rem(ci(t))>0,-∫t+∞e∫st(-ci(τ))dτhi(s,xs0)ds,Rem(ci(t))<0.
For any (ϕ,ψ)∈C([-τ,0],Rn)×C([-τ,0],Rn), we  define
(3.6)F1(t,ϕ,ψ)=(a1(t)-b1(t)ϕ1α1(0)-f1(t,ϕ,ψ),…,an(t)-bn(t)ϕnαn(0)-fn(t,ϕ,ψ)),H1(t,ϕ,ψ)=(-c1(t)ψ1(0)+h1(t,ϕ),…,-cn(t)ψn(0)+hn(t,ϕ))X1={(ϕ(t),ψ(t))∈AP(R2n):mod⁡(ϕ,ψ)⊂mod⁡(F1,H1)},thenX1 is a Banach space with the model of ∥(ϕ(t),ψ(t))∥=supt∈R(|ϕ(t)|+|ψ(t)|),construct a bounded closed convex set G1 as follows:(3.7)G1={(ϕ,ψ)∈X1:0≤ϕi(t)≤xi0(t),0≤ψi(t)≤ui0(t)}.(B4)
For any(ϕ,ψ)∈G1 and t∈R, 0≤fi(t,ϕt,ψt)≤fi(t,xt0,ut0),0≤hi(t,ϕt)≤hi(t,xt0),m(ai(t)-fi(t,xt0,ut0))>0,(i=1,2,…,n)(B4̅)
For any(ϕ,ψ)∈G1 and t∈R,0≥fi(t,ϕt,ψt)≥fi(t,xt0,ut0),0≥hi(t,ϕt)≥hi(t,xt0),m(ai(t)-fi(t,xt0,ut0))<0,(i=1,2,…,n).Theorem 3.1.
If(B1), (B2), (B3), (B4) or (B1), (B̅2), (B̅3), (B̅4) hold, then (1.5) exists at least a positive almost periodic solution in G1.Proof.
If (B1), (B2), (B3) hold, for any (ϕ,ψ)∈G1, consider the following equation(3.8)dxi(t)dt=xi(t)(ai(t)-fi(t,ϕt,ψt)-bi(t)xiαi(t)),(i=1,2,…,n)
by the condition (B4), it follows m(ai(t)-fi(t,ϕt,ψt))>0, by Lemma 2.3,it follows that (3.8) exists a unique positive almost periodic solutionZϕψ(t)=(Zϕψ1(t),Zϕψ2(t),…,Zϕψn(t)), and it can be written as follows
(3.9)Zϕψi(t)=[αi∫-∞te-αi∫st(ai(τ)-fi(τ,ϕτ,ψτ))dτbi(s)ds]-1/αi,(i=1,2,…,n),
and mod(Zϕψi(t))⊂mod(ai(t)-fi(t,ϕt,ψt),bi(t)), since fi(τ,ϕτ,ψτ)-ai(τ)>-ai(τ), thus by (3.2) and (3.9), we can get 0<Zϕψi(t)≤xi0(t),i=1,2,…,n, on the other hand, since mod(ai(t)-fi(t,ϕt,ψt),bi(t))⊂mod(F1), it follows mod(Zϕψ(t))⊂mod(F1)°.
Consider the following equation(3.10)dui(t)dt=-ci(t)ui(t)+hi(t,ϕt),
by Lemma 2.1, (3.10) exists a unique positive almost periodic solution
(3.11)Vϕ(t)=(Vϕ1(t),Vϕ2(t),…,Vϕn(t)),
and it can be written as follows:
(3.12)Vϕi(t)=∫-∞te∫st(-ci(τ))dτhi(s,ϕs)ds,Rem(ci(t))>0,
also by the condition 0≤hi(t,ϕt)≤hi(t,xt0),(i=1,2,…,n), the first formula of (3.5) and (3.12), we have 0<Vϕi(t)≤ui0(t), and mod(Vϕi(t))⊂mod(ci(t),hi(t,ϕt)), also mod(ci(t),hi(t,ϕt))⊂mod(H1), thus mod(Vϕi(t))⊂mod(H1), therefore, we can get mod(Zϕψ,Vϕ)⊂mod(F1,H1), hence we can define
(3.13)P:G1⟶G1,P(ϕ,ψ)=(Zϕψ,Vϕ).
Now we are committed to prove the continuity ofP in G1. Suppose {(ϕk,ψk)⊂G1}, and when k→∞,(ϕk,ψk)→(ϕ,ψ)∈G1, define S={ϕk,ψk}∪{(ϕ,ψ)}, then S is a compact set of X1, since m(ai(t)-fi(τ,xτ0,uτ0))>0,i=1,2,…,n, there exist the positive numbers λandH, such that when t≥s,e-αi∫st(ai(τ)-fi(τ,xτ0,uτ0))dτ≤Heλ(s-t),i=1,2,…,n,similarly, since m(ci(t))>0,i=1,2,…,n, there exist the positive numbers μ and I, such that when t≥s,e∫st(-ci(τ))dτ≤Ieu(s-t),  for any (ϕ,ψ)∈G1,itfollows0≤fi(t,ϕt,ψt)≤fi(t,xt0,ut0).
Therefore, by the mean value theorem(3.14)|e-αi∫st(ai(τ)-fi(τ,ϕτk,ψτk))dτ-e-αi∫st(ai(τ)-fi(τ,ϕτ,ψτ))dτ|=|αi∫st(fi(τ,ϕτk,ψτk)-fi(τ,ϕτ,ψτ))dτ|eξ≤|αi∫st(fi(τ,ϕτk,ψτk)-fi(τ,ϕτ,ψτ))dτ|e-αi∫st(ai(τ)-fi(τ,xτ0,uτ0))dτ≤(t-s)‖αi(fi(t,ϕtk,ψtk)-fi(t,ϕt,ψt))‖Heλ(s-t),t≥s,i=1,2,⋯,n.
Where,(3.15)ξ≤max⁡{-αi∫st(ai(τ)-fi(τ,ϕτk,ψτk))dτ,-αi∫st(ai(τ)-fi(τ,ϕτ,ψτ))dτ}≤-αi∫st(ai(τ)-fi(τ,xτ0,uτ0))dτ.
Also by the mean value theorem, we can obtain the following:
(3.16)‖Pi(ϕk,ψk)-Pi(ϕ,ψ)‖=‖(Zϕkψki(t)-Zϕψi(t),Vϕki(t)-Vϕi(t))‖=‖(-[Zϕkψki(t)]-αi-[Zϕψi(t)]-αiαiζi-αi-1,Vϕki(t)-Vϕi(t))‖,
where, ζi is between the positive almost periodic function Zϕkψki(t) and Zϕψi(t), thus
(3.17)0<ζi≤xi0(t).
By (3.9), (3.12), and (3.14), it follows that
(3.18)|-[Zϕkψki(t)]-αi-[Zϕψi(t)]-αiαiζi-αi-1|=|1ζi-αi-1∫-∞t(e-αi∫st(ai(τ)-fi(τ,ϕτk,ψτk))dτ-e-αi∫st(ai(τ)-fi(τ,ϕτ,ψτ))dτ)bi(s)ds|≤1ζi-αi-1∫-∞t|e-αi∫st(ai(τ)-fi(τ,ϕτk,ψτk))dτ-e-αi∫st(ai(τ)-fi(τ,ϕτ,ψτ))dτ||bi(s)|ds≤H(xi0(t))-αi-1‖αi‖‖bi(t)‖‖fi(t,ϕtk,ψtk)-fi(t,ϕτ,ψτ)‖∫-∞t(t-s)eλ(s-t)ds≤Hλ2‖(xi0(t))αi+1‖‖αi‖‖bi(t)‖‖fi(t,ϕtk,ψtk)-fi(t,ϕt,ψt)‖,|Vϕki(t)-Vϕi(t)|=|∫-∞te∫st(-ci(τ))dτhi(s,ϕsk)ds-∫-∞te∫st(-ci(τ))dτhi(s,ϕs)ds|≤∫-∞te∫st(-ci(τ))dτ|hi(s,ϕsk)-hi(s,ϕs)|ds≤Iμ‖hi(t,ϕtk)-hi(t,ϕt)‖,thus we have
(3.19)‖Pi(ϕk,ψk)-Pi(ϕ,ψ)‖≤Hλ2‖(xi0(t))αi+1‖‖αi‖‖bi(t)‖‖fi(t,ϕtk,ψtk)-fi(t,ϕt,ψt)‖+Iμ‖hi(t,ϕtk)-hi(t,ϕt)‖.
Hence
(3.20)‖P(ϕk,ψk)-P(ϕ,ψ)‖≤Hλ2‖(x0(t))α+1‖‖α‖‖b(t)‖‖f(t,ϕtk,ψtk)-f(t,ϕt,ψt)‖+Iμ‖h(t,ϕtk)-h(t,ϕt)‖.
In addition, taking into account that f(t,ϕ,ψ),h(t,ϕ) are continuous uniformly on R×S, when k→∞,  it follows ∥f(t,ϕtk,ψtk)-f(t,ϕt,ψt)∥→0,∥h(t,ϕtk)-h(t,ϕt)∥→0, thus when k→∞, we have
(3.21)‖P(ϕk,ψk)-P(ϕ,ψ)‖⟶0,
therefore, P is continuous.Then to proveP is relatively compact in G1. By the boundedness ofG1 and (3.8), (3.10), we can obtain that there exists a positive number A such that ∥d(P(ϕ,ψ))/dt∥≤A,forall(ϕ,ψ)∈G1, thus P(G1) is uniformly bounded and equicontinuous on R, by the theorem of Ascoli-arzela, for any sequence {(ξk,ζk)}in G1, there exist a subsequences (also denoted by {(ξk,ζk)}) such that {(ξk,ζk)} is convergent uniformly in any compact set of R, also combined with Lemma 2.4,{(ξk,ζk)}is convergent uniformly on R, that is to say Pis relatively compact in G1. According to Schauder's fixed point theorem, P exists as a fixed point in G1, that is to say (1.5) exists at least a positive almost periodic solution in G1. Similarly, when (B1), (B2), (B̅3), (B̅4) hold, we can prove that (1.5) exists at least a positive almost periodic solution in G1.Remark 3.2.
Whenαi≡1, then (1.5) turns into (1.4), obviously, Theorem 3.1 is the generalization of Theorems 2.1 and 2.2 in [5];Remark 3.3.
In Theorem 2.2 of [5], it requires the conditionfi(t,0,0)≡0, in this paper, we do not require fi(t,0,0)≡0, this can be seen in the weak conditions we get the similar results, thus Theorem 3.1 extends the results of Theorem 2.2  of the paper [5].


## 4. The Conclusions Of The One-dimentional  System

Here we will discuss the system (1.5) in the one-dimensional case(4.1)dx(t)dt=x(t)(a(t)-b(t)xα(t)-f(t,xt,ut)),du(t)dt=-c(t)u(t)+h(t,xt),
where α>0,t∈R,a(t),b(t),c(t)are continuous almost periodic functions, f(t,ϕ,ψ) is almost periodic in t uniformly with respect to (ϕ,ψ)∈C([-τ,0],R+)×C([-τ,0],R+), and f(t,0,0)≡0,h(t,ϕ) is almost periodic in t uniformly with respect to ϕ∈C([-τ,0],R+), and h(t,0)≡0. For convenience, we introduce the following notations. For a continuous bounded function f(t), denote fL=inft∈Rf(t),fM=supt∈Rf(t),(C1)
aL>0,bL>0,cL>0;(C2)
f(t,ϕ,ψ)≥0,h(t,ϕ)≥0forϕ≥0,ψ≥0andt∈R;(C3)
|f(t,x1,y1)-f(t,x2,y2)|≤L1(|x1-x2|+|y1-y2|),|h(t,x1)-h(t,x2)|≤L2|x1-x2|fort∈R,x1,x2,y1,y2∈R+,L1,L2 are positive constants; (C4)
L1γ1/αaLβ1/α+1+L1/aLβ1/α+L1γ2/aL+L2/αcLβ1/α+1+L1γ1/aL<1,  where,β=min{bL/aM(α2+α+1),[bL/aM(α2+α+1)]α/2α+1},γ1=(bM+αaL(β+αβ2+(1/α)))/aL,γ2=αβ.In the next paper, we suppose that initial values(x(θ),u(θ)) of system (4.1) satisfy x(θ)≥0,u(θ)≥0,θ∈[-τ,0], for the solution (x(t),u(t)) of the system (4.1), we only consider its positive initial value, that is to say
(4.2)x(0)>0,u(0)>0.
It is not difficult for us to see the solution (x(t),u(t)) of (4.1) with the positive initial value is always positive.Theorem 4.1.
If(C1),(C2),(C3),(C4) hold, then (4.1) exists a unique positive almost periodic solution.Proof.
Lety(t)=1/xα(t), then (4.1) can be changed into the following equation:
(4.3)dy(t)dt=-αa(t)y(t)+αb(t)+αy(t)f(t,yt-1/α,ut),du(t)dt=-c(t)u(t)+h(t,yt-1/α).
For(ϕ,ψ)∈C([-τ,0],R+)×C([-τ,0],R+), define
(4.4)F(t,ϕ,ψ)=-a(t)ϕ(0)+b(t)+f(t,ϕ-1/α,ψ)ϕ(0),H(t,ϕ,ψ)=-c(t)ψ(0)+h(t,ϕ-1/α),X={(ϕ,ψ)∈AP(R2):β≤ϕ≤γ1,0<ψ≤γ2,mod⁡(ϕ,ψ)⊂mod⁡(F,H)},
then X is a Banach space with the model of ∥ϕ,ψ∥=supt∈R(|ψ(t)|+|ψ(t)|), for all(ϕ,ψ)∈X, consider the following equation:
(4.5)dy(t)dt=-αa(t)y(t)+αb(t)+αϕ(t)f(t,ϕt-1/α,ψt),
by Lemma 2.1, it follows that (4.5) exists a unique positive almost periodic solution Zϕψ(t), it can be written as follows:(4.6)Zϕψ(t)=α∫-∞te-α∫sta(τ)dτ[b(s)+ϕ(s)f(s,ϕs-1/α,ψs)]ds,
similar to the proof of Lemma 2.3 we can obtain
(4.7)mod⁡(Zϕψ)⊂mod⁡(a(t),b(t)+f(t,ϕt-1/α,ψ)ϕ(t))⊂mod⁡(F),
in addition,
(4.8)Zϕψ(t)≥α∫-∞te-α∫sta(τ)dτ(bL-|ϕ(s)||f(s,ϕs-1/α,ψs)|)ds=α∫-∞te-α∫sta(τ)dτ(bL-|ϕ(s)||f(s,ϕs-1/α,ψs)-f(s,0,0)|)ds≥α∫-∞te-α∫sta(τ)dτ(bL-L1|ϕ(s)|(|ϕs-1/α|+|ψs|))ds≥bL-γ1L1(β-1/α+γ2)aM.
Note that the condition (C4) holds, thus we have L1<αaLβ1+(1/α)/γ1, hence
(4.9)Zϕψ(t)≥bL-γ1αaLβ1+1/α/γ1(β-1/α+αβ)aM≥bL-αaLβ1+1/α(β-1/α+αβ)aM=bL-αaLβ-α2aLβ2+(1/α)aM.
Note that β=min{bL/aM(α2+α+1),[bL/aM(α2+α+1)]α/(2α+1)}, if 0<bL/(aM(α2+α+1))≤1, then β=bL/aM(α2+α+1), thus we can get
(4.10)Zϕψ(t)≥bL-αaLβ-α2aLβ2+(1/α)aM≥bL-αaMβ-α2aMβaM=β,
if bL/aM(α2+α+1)>1, then β=[bL/aM(α2+α+1)]α/(2α+1), therefore
(4.11)Zϕψ(t)≥bL-αaLβ-α2aLβ2+(1/α)aM≥bL-αaMβ2+(1/α)-α2aMβ2+(1/α)aM=bLaM(α2+α+1)≥β.
No matter what 0<bL/aM(α2+α+1)≤1 or bL/aM(α2+α+1)>1, we always have
(4.12)Zϕψ(t)≥β.
Also we have
(4.13)Zϕψ(t)≤α∫-∞te-α∫sta(τ)dτ(|b(s)|+|ϕ(s)||f(s,ϕs-1/α,ψs)|)ds=α∫-∞te-α∫sta(τ)dτ(|b(s)|+|ϕ(s)||f(s,ϕs-1/α,ψs)-f(s,0,0)|)ds≤α∫-∞te-α∫sta(τ)dτ(|b(s)|+|ϕ(s)|L1(|ϕs-1/α|+|ψs|))ds≤bM+‖ϕ(t)‖L1(‖ϕt-1/α‖+‖ψt‖)aL≤bM+γ1L1(β-1/α+γ2)aL≤bM+γ1αaLβ1+1/α/γ1(β-1/α+γ2)aL=bM+αaLβ1+1/α(β-1/α+αβ)aL=γ1.
Consider the following equation:
(4.14)du(t)dt=-c(t)u(t)+h(t,ϕt-1/α).
By Lemma 2.1 we can see that (4.14) has a unique positive almost periodic solution Vϕ(t), it can be written as follows
(4.15)Vϕ(t)=∫-∞te-∫stc(τ)dτh(s,ϕs-1/α)ds.
Similar to the proof of Lemma 2.3, we can get
(4.16)mod⁡(Vϕ(t))⊂mod⁡(c(t),h(t,ϕt-1/α))⊂mod⁡(H),Vϕ(t)≤∫-∞te-∫stc(τ)dτ|h(s,ϕs-1/α)|ds=∫-∞te-∫stc(τ)dτ|h(s,ϕs-1/α)-h(s,0)|ds≤L2∫-∞te-∫stc(τ)dτ|ϕs-1/α|ds≤L2β-1/αcL.
Note that the condition (C4) holds, it follows thatL2<αcLβ1+(1/α), hence
(4.17)Vϕ(t)≤αcLβ1+1/αβ-1/αcL=αβ=γ2.
Now, define a mapping
(4.18)T(ϕ,ψ)(t)=(T1(ϕ,ψ)(t),T2(ϕ,ψ)(t))=(Zϕψ(t),Vϕ(t)).
From the above discussion we can see T(ϕ,ψ)(t)∈X, thus T:X→X, for given any (ϕ1,ψ1),(ϕ2,ψ2)∈X, we have
(4.19)‖T(ϕ1,ψ1)-T(ϕ2,ψ2)‖=‖(T1(ϕ1,ψ1)-T1(ϕ2,ψ2),T2(ϕ1,ψ1)-T2(ϕ2,ψ2))‖=‖(Zϕ1ψ1(t)-Zϕ2ψ2(t),Vϕ1(t)-Vϕ2(t))‖=sup⁡t∈R(|α∫-∞te-α∫sta(τ)dτ[ϕ1(s)f(s,ϕ1s-1/α,ψ1s)-ϕ2(s)f(s,ϕ2s-1/α,ψ2s)]ds|+|∫-∞te-∫stc(τ)dτ[h(s,ϕ1s-1/α)-h(s,ϕ2s-1/α)]ds|)≤sup⁡t∈R(α∫-∞te-α∫sta(τ)dτ(|ϕ1(s)||f(s,ϕ1s-1/α,ψ1s)-f(s,ϕ2s-1/α,ψ2s)|+|f(s,ϕ2s-1/α,ψ2s)||ϕ1(s)-ϕ2(s)|)ds+∫-∞te-∫stc(τ)dτ|h(s,ϕ1s-1/α)-h(s,ϕ2s-1/α)|dsα∫-∞te-α∫sta(τ)dτ)≤sup⁡t∈R(α∫-∞te-α∫sta(τ)dτ(L1|ϕ1(s)|[|ϕ1s-1/α-ϕ2s-1/α|+|ψ1s-ψ2s|]+L1(|ϕ2s-1/α|+|ψ2s|)|ϕ1(s)-ϕ2(s)|)ds+L2∫-∞te-∫stc(τ)dτ|ϕ1s-1/α-ϕ2s-1/α|ds).
By the mean value theorem (where ξ of the following formula is between ϕ1s and ϕ2s), the above formula can be changed as follows:
(4.20)‖T(ϕ1,ψ1)-T(ϕ2,ψ2)‖≤sup⁡t∈R(α∫-∞te-α∫sta(τ)dτ(L1|ϕ1(s)|[|1αξ(-1/α)-1||ϕ1s-ϕ2s|+|ψ1s-ψ2s|]+L1(|ϕ2s-1/α|+|ψ2s|)|ϕ1(s)-ϕ2(s)|1α)+L2∫-∞te-∫stc(τ)dτ|1αξ(-1/α)-1||ϕ1s-ϕ2s|ds∫)≤sup⁡t∈R(α∫-∞te-α∫sta(τ)dτ(L1γ1[1αβ-1/α-1|ϕ1s-ϕ2s|+|ψ1s-ψ2s|]+L1(β(-1/α)+γ2)|ϕ1(s)-ϕ2(s)|1α)ds+L2∫-∞te-∫stc(τ)dτ1αβ(-1/α)-1|ϕ1s-ϕ2s|ds∫)≤1aL(L1γ11αβ(-1/α)-1‖ϕ1t-ϕ2t‖+L1γ1‖ψ1t-ψ2t‖+L1(β(-1/α)+γ2)‖ϕ1(t)-ϕ2(t)‖L1γ11α)+1cLL21αβ(-1/α)-1‖ϕ1t-ϕ2t‖=(1aLL1γ11αβ-1/α-1+1aLL1β-1/α+1aLL1γ2+1cLL21αβ(-1/α)-1)‖ϕ1-ϕ2‖+1aLL1γ1‖ψ1-ψ2‖≤(1aLL1γ11αβ(-1/α)-1+1aLL1β-1/α+1aLL1γ2+1cLL21αβ(-1/α)-1+1aLL1γ1)×‖(ϕ1,ψ1)-(ϕ2,ψ2)‖
notice that the condition (C4) holds, it follows
(4.21)‖T(ϕ1,ψ1)-T(ϕ2,ψ2)‖<‖(ϕ1,ψ1)-(ϕ2,ψ2)‖,
by (4.21), T is a contraction mapping, thus T exists a unique fixed point in X, this fixed point is the only positive almost periodic solution(φ*(t),u*(t)) of (4.3), and β≤φ*(t)≤γ1,0<u*(t)≤γ2. note that y(t)=1/xα(t),  x(t)=u-1/α(t), thus (4.1) exists a positive almost periodic solution (ψ*(t),u*(t)), and 1/(γ11/α)≤ψ*(t)≤1/β1/α,0<u*(t)≤γ2. This completes the proof of Theorem 4.1.



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*Source: 135075-2012-03-14.xml*