# Multiple Positive Periodic Solutions for Delay Differential System

Authors: Zhao-Cai Hao; Ti-Jun Xiao; Jin Liang
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We obtain some existence results for multiple positive periodic solutions of some delay differential systems. Examples are presented as applications. For a general positive integerm≥2, main results of this paper do not appear in
former literatures as we know. Comparing with the existing results, our results are new also when m=1.


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## Body

## 1. Introduction

It is known that multiple delay Logistic equations(1.1)y'(t)=-a(t)y(t)+f(t,y(t-τ0(t)),y(t-τ1(t)),…,y(t-τn(t))),y'(t)=a(t)y(t)-f(t,y(t-τ0(t)),y(t-τ1(t)),…,y(t-τn(t))),
are generalizations of many biological models, such as Logistic models of Single-species growth (see [1–3]),
(1.2)y'(t)=a(t)y(t)[1-y(t-τ(t))K(t)],y'(t)=y(t)[a(t)-∑i=1nbi(t)y(t-τi(t))],y'(t)=a(t)y(t)[1-∏i=1ny(t-τi(t))K(t)],
and red blood cell models (see [4–7]),(1.3)y'(t)=-a(t)y(t)+b(t)e-y(t-τ(t)),y'(t)=-a(t)y(t)+b(t)1+yn(t-τ(t)).
For biological models, positive periodic solutions are often important and many results have been achieved in this direction, for instance, [8–10].To the best of our knowledge, few papers concerning the existence of multiple positive solutions of (1.1)   can be found in literature. Furthermore, no papers have yet deal with the more general nonautonomous delay differential systems(1.4)y1′(t)=-a1(t)y1(t)+f1(t,Y1(t),Y2(t),…,Ym(t)),y2′(t)=-a2(t)y2(t)+f2(t,Y1(t),Y2(t),…,Ym(t)),⋮ym′(t)=-am(t)ym(t)+fm(t,Y1(t),Y2(t),…,Ym(t)),(1.5)y1′(t)=a1(t)y1(t)-f1(t,Y1(t),Y2(t),…,Yn(t)),y2′(t)=a2(t)y2(t)-f2(t,Y1(t),Y2(t),…,Yn(t)),⋮yn′(t)=an(t)yn(t)-fm(t,Y1(t),Y2(t),…,Yn(t)),
where m,n are all positive integer and(1.6)ai(·),i∈Λ1:={1,2,…,m},fi(·,…,·),i∈Λ1,τj(·),j∈Λ2:={1,2,…,n}
are given functions and signs Yi, i∈Λ1 are given as follows:(1.7)Yi(t):=(yi(t-τ1(t)),yi(t-τ2(t)),…,yi(t-τn(t))),t∈R,i∈Λ1.
The extension to systems is a natural one; for example, many occurrences in nature involve two or more populations coexisting in an environment, with the model being best described by a system of differential equations (see [11]).The aim of this paper is to investigate systems (1.4) and (1.5). In what follows we only discuss the existence of positive periodic solutions of system (1.4); similar results can be obtained for system (1.5). By using multiple fixed-point theorems (see Lemmas 2.1 and 2.2), which are different from those used in [8–10], we obtain the existence of multiple positive periodic solutions of system (1.4) (see Theorems 3.1, 4.1, and 4.3). Some examples are given also to illustrate our main theorems. Main results of this paper are new also even if m=1 (see Remark 4.5).This paper is organized as follows. In Section2, we make some preliminaries. In Section 3, we derive existence result (see Theorem 3.1) for two positive periodic solutions of system (1.4). Example 3.2 is given below Theorem 3.1. The existence of three positive periodic solutions of system (1.4) is presented in Section 4 (see Theorems 4.1 and   4.3). Applications of Theorems 4.1 and 4.3 may be seen from Examples 4.2 and 4.4.


## 2. Preliminaries

We make the basic assumption throughout this paper that(2.1)T>0is a fixed constant;ai∈C(R,[0,∞)),a(t)≢0,ai(t)=ai(t+T),t∈R,i∈Λ1;fi∈C(R×[0,∞)m×n,[0,∞)),i∈Λ1;fiisT-periodic function in relative tot,i∈Λ1;τj∈C(R,[0,∞)),τj(t+T)=τj(t),t∈R,j∈Λ2.Let us now provide some preparations. LetS be a real Banach space and let P be a cone in S. A map α is said to be a nonnegative continuous concave functional on cone P if α:P→[0,∞) is continuous and(2.2)α(tx+(1-t)y)≥tα(x)+(1-t)α(y)∀x,y∈P,t∈[0,1].
For numbers M, N such that 0<M<N, and a nonnegative continuous concave functional α on cone P, we define(2.3)PM:={x∈P:∥x∥<M},P(α,M):={x∈P:α(x)<M},P(α,M,N):={x∈P:M≤α(x),∥x∥≤N}.
Setting u=(u1,u2,…,un)∈[0,∞)n, we define(2.4)|u|0:=maxj∈Λ2{uj}.
Write(2.5)D:={y(t):y∈C(R,R),y(t+T)=y(t)},E:=D×D×⋯×D︸m,∥y∥0:=supt∈[0,T]|y(t)|fory∈D,∥y∥:=∑i∈Λ1∥yi∥0fory=(y1,y2,…,ym)∈E,P:={y=(y1,y2,…,ym)∈E:yi(t)≥δi∥yi∥0,i∈Λ1},
where(2.6)δi=e-∫0Tai(s)ds,i∈Λ1.
Then (D,∥·∥0) and (E,∥·∥) are all Banach spaces and P is a cone in E. Set(2.7)M1i:=1e∫0Tai(s)ds-1,M2i:=e∫0Tai(s)dse∫0Tai(s)ds-1,i∈Λ1,Gi(t,s):=e∫tsai(ξ)dξe∫0Tai(ξ)dξ-1,(t,s)∈R×[t,t+T],i∈Λ1.
It is easy to see that for any (t,s)∈R×[t,t+T], functions Gi(t,s), i∈Λ1 have properties(2.8)M1i:=Gi(t,t)≤Gi(t,s)≤Gi(t,t+T):=M2i,i∈Λ1,δi=M1iM2i≤Gi(t,s)Gi(t,t+T)≤1,i∈Λ1.Now we define an operatorA:E→E as follows:(2.9)Ay(t):=(A1(t),A2(t),…,Am(t)),t∈R,y=(y1,y2,…,ym)∈E,
where(2.10)Ai(t):=∫tt+TGi(t,s)fi(s,Y1(s),Y2(s),…,Yn(s))ds,t∈R,i∈Λ1,
signs Yi, i∈Λ1 are given in (1.7) and we often use them in the remainder of this paper. It is easy to say that a T-periodic solution of operator equation(2.11)y=Ay,
on P, that is, a fixed point of operator A, is a T-positive periodic solution of system (1.4). So, our main results concerning multiple positive solutions of system (1.4) will arise as application of the following fixed-point theorem.Lemma 2.1 (see [12]).
LetP be a cone in a real Banach space B. Let  α and γ be increasing, nonnegative, continuous functionals on P, and let θ be a nonnegative continuous functional on P with θ(0)=0 such that, for some c>0 and M>0,
(2.12)γ(x)≤θ(x)≤α(x),∥x∥≤Mγ(x),∀x∈P(γ,c)¯.
Suppose there exists a completely continuous operator A:P(γ,c)¯→P and 0<a<b<c such that
(2.13)θ(λx)≤λθ(x),for0≤λ≤1,x∈∂P(θ,b),
and(i)
γ(Ax)>c, for all  x∈∂P(γ,c);(ii)
θ(Ax)<b, for all  x∈∂P(θ,b);(iii)
P(α,a)≠ϕ, and α(Ax)>a, for all x∈∂P(α,a).
ThenA has at least two fixed points x1 and x2 belonging to P(γ,c)¯ such that
(2.14)a<α(x1),withθ(x1)<b,b<θ(x2),withγ(x2)<c.Lemma 2.2 (see [13]).
LetP be a cone in a real Banach space E, let A:Pc¯→Pc¯ be completely continuous, and let α be a nonnegative continuous concave functional on P with α(x)≤∥x∥ for all x∈Pc¯. Suppose that there exists 0<d<M<N≤c such that(i)
{x∈P(α,M,N):α(x)>M}≠ϕ and  α(Ax)>M for x∈P(α,M,N);(ii)
∥Ax∥<d for  all ∥x∥≤d;(iii)
α(Ax)>M for x∈P(α,M,c) with ∥Ax∥>N.
ThenA has at least three fixed points x1, x2, x3 satisfying
(2.15)∥x1∥<d,M<α(x2),∥x3∥>d,α(x3)<M.


## 3. Existence of Two Positive Solutions of System (1.4)

In this section, we apply Lemma2.1 to establish Theorem 3.1, the existence result of two positive solutions of system (1.4). Example 3.2 will be given as an application of Theorem 3.1.Theorem 3.1.
Assume that there exist numbers0<a<b<c such that the following three assumptions are satisfied.(H1)
One has(3.1)fk0(t,U1,U2,…,Um)>c+1TM1,t∈R,φ∈[c,cδ-1],
where k0∈Λ1 is fixed and
(3.2)Ui:=(Ui1,Ui2,…,Uin),Uij∈[0,∞),i∈Λ1,j∈Λ2,M1:=min{M11,M12,…,M1m},δ:=min{δ1,δ2,…,δm},φ:=|U1|0+|U2|0+⋯+|Um|0,(H2)
t∈R, Ui∈[0,∞)n, i∈Λ1, and φ∈[b,bδ-1] imply(3.3)∑i=1mfi(t,U1,U2,…,Um)<b(1+δ)TM2,
where
(3.4)M2:=max{M21,M22,…,M2m},(H3)
t∈R, Ui∈[0,∞)n, i∈Λ1, and φ∈[δa,a] imply(3.5)∑i=1mfi(t,U1,U2,…,Um)>a+1TM1.
Then system (1.4) has at least two T-positive periodic solutions.Proof.
We begin by defining(3.6)γ(y):=θ(y):=∑i=1mmint∈[0,T]yi(t),y=(y1,y2,…,ym)∈P,α(y):=∥y∥,y∈P.
Clearly, α and γ are increasing, nonnegative, continuous functionals on P, and θ is nonnegative a continuous functional on P with θ(0)=0. Moreover, we observe that
(3.7)γ(y)=θ(y)≤α(y),∀y∈P,(3.8)∥y∥≤δ-1γ(y),∀y∈P,(3.9)θ(λx)=λθ(x),for0≤λ≤1,x∈∂P(θ,b).
Now, we proceed to show that other conditions of Lemma 2.1 are also satisfied.
Firstly, we will  show that(3.10)A:P(γ,c)¯→Piscompletelycontinuous.
In fact, we have from (2.8), for any y=(y1,y2,…,ym)∈P(γ,c)¯,
(3.11)∥Ai∥0≤M2i∫0Tfi(s,Y1(s),Y2(s),…,Yn(s))ds,i∈Λ1,
which yields
(3.12)Ai(t)≥M1i∫0Tfi(s,Y1(s),Y2(s),…,Yn(s))ds≥M1iM2i∥Ai∥0=δi∥Ai∥0,t∈R,y∈P(γ,c)¯,i∈Λ1.
Hence Ay∈P  for  all  y∈P(γ,c)¯. Furthermore, we know from the continuity of functions fi(·,…,·),  ai(·),  Γi(·,·), i∈Λ1 that the operator A is completely continuous. Hence, we conclude that (3.10) holds.
Secondly, let us prove(3.13)γ(Ay)>c,∀y∈∂P(γ,c).
For any y=(y1,y2,…,ym)∈∂P(γ,c), so that  γ(y)=c, we get, in view of (1.7), (2.4) and (3.8),
(3.14)c=∑i=1mmint∈[0,T]yi(t)≤∑i=1m|Yi(t)|0,t∈R,∑i=1m|Yi(t)|0≤∑i=1m∥yi∥0≤δ-1γ(y)=cδ-1,t∈R.
Consequently, for any y∈∂P(γ,c), condition (H1) and (3.14) imply that
(3.15)γ(Ay)=∑i=1mmint∈[0,T]∫tt+TGi(t,s)fi(s,Y1(s),Y2(s),…,Ym(s))ds≥mint∈[0,T]∫tt+TGk0(t,s)fk0(s,Y1(s),Y2(s),…,Ym(s))ds≥mint∈[0,T]∫tt+TM1k0c+1TM1ds>c,
which gives (3.13).
Thirdly, we verify(3.16)θ(Ay)<b,∀y∈∂P(θ,b).
As before, θ(y)=b and (1.7), (2.4), and (3.8) also tell us that
(3.17)b≤∑i=1m|Yi(t)|0≤bδ-1,t∈R.
Then condition (H2), (3.17), and the fact that the function min is concave imply
(3.18)θ(Ay)=∑i=1mmint∈[0,T]∫tt+TGi(t,s)fi(s,Y1(s),Y2(s),…,Ym(s))ds≤mint∈[0,T]∫tt+T∑i=1mM2ifi(s,Y1(s),Y2(s),…,Ym(s))ds≤mint∈[0,T]∫tt+TM2b(1+δ)TM2ds<b.
Thus (3.16) holds.
Finally, let us prove(3.19)P(α,a)≠ϕ,α(Ay)>a,∀y∈∂P(α,a).
Obviously,
(3.20)y=(y1,y2,…,ym)=(am+1,am+1,…,am+1)∈P(α,a).
In addition, for any y∈∂P(α,a), we get
(3.21)δa≤∑i=1m|Yi(t)|0≤a,t∈R
since ∑i=1m∥yi∥0=a and y=(y1,y2,…,ym)∈P. So we have from condition (H3) that
(3.22)α(Ay)=∑i=1mmaxt∈[0,T]∫tt+TGi(t,s)fi(s,Y1(s),Y2(s),…,Ym(s))ds≥maxt∈[0,T]∫tt+T∑i=1mM1ifi(s,Y1(s),Y2(s),…,Ym(s))ds≥maxt∈[0,T]∫tt+TM1a+1TM1ds>a.
Hence (3.19) holds.
To sum up, (3.6)–(3.10), (3.13), (3.16), and (3.19) tell us that conditions of Lemma 2.1 all hold here. Consequently, system (1.4) has at least two T-positive periodic solutions y1=(y11,y21,…,ym1) and y2=(y12,y22,…,ym2) belonging to P(γ,c)¯ such that
(3.23)a<∥y1∥,with∑i=1mmint∈[0,T]yi1<b,b<∑i=1mmint∈[0,T]yi2<c.As an application of Theorem3.1, we provide the following example. For convenience, all examples in this paper are given when n=1,m=2.Example 3.2.
Assume thatτ1>0 is a fixed constant. Consider the following system:
(3.24)y1′(t)=-(ln2)|sint|y1(t)+14πe|cost|-1g1(t,y1(t-τ1),y2(t-τ1)),y2′(t)=-12(ln3)|cost|y2(t)+14πe|sint|-1g2(t,y1(t-τ1),y2(t-τ1)),
where
(3.25)g1(t,u1,u2):={12e,t∈R,u1+u2∈[0,160],2401e(u1+u2-160)+12e(199-u1-u2)39,t∈R,u1+u2∈[160,199],2401e,t∈R,u1+u2∈[199,∞).g2(t,u1,u2):={4223,t∈R,u1+u2∈[0,160],(u1+u2-160)+42(2/3)(199-u1-u2)39,t∈R,u1+u2∈[160,199],1,t∈R,u1+u2∈[199,∞).
We set
(3.26)a=1,b=40,c=199,a1(t):=(ln2)|sint|,a2(t):=12(ln3)|cost|,t∈R,f1(t,y1(t-τ),y2(t-τ))=14πe|cost|-1g1(t,y1(t-τ),y2(t-τ)),t∈R,f2(t,y1(t-τ),y2(t-τ))=14πe|sint|-1g2(t,y1(t-τ),y2(t-τ)),t∈R
Then
(3.27)T=π,δ=14,M1=13,M2=32.
We may verify that conditions (H1), (H2), and (H3) are all satisfied. Hence, Theorem 3.1 tells us that system (3.24) has at least two π-positive periodic solutions y1=(y11,y21) and y2=(y12,y22) such  that
(3.28)1<maxt∈[0,π]y11+maxt∈[0,π]y21,mint∈[0,π]y11+mint∈[0,π]y21<40,40<mint∈[0,π]y12+mint∈[0,π]y22<199.


## 4. Existence of Three Positive Solutions of System (1.4)

For the sake of convenience we list the assumptions to be used in this section as follows.(H4)
There exists a numberC1>0 such that(4.1)fi(t,U1,U2,…,Um)<C1mTM2,t∈R,Ui∈[0,∞)n,φ≤C1,i∈Λ1,
where φ and Ui, i∈Λ1 are given in (H1).(H5)
There exist numbersC2>C1 and i0∈Λ1 such that(4.2)fi0(t,U1,U2,…,Um)>C3TM1,t∈R,Ui∈[0,∞)n,C2≤φ≤C3,
where(4.3)C3:=C2(1+δ)δ.(H6)
One has(4.4)limφ→∞maxt∈[0,T]fi(t,U1,U2,…,Um)φ<1mTM2,Ui∈[0,∞)n,i∈Λ1.(H7)
There exists a numberC4>C3 such that,(4.5)fi(t,U1,U2,…,Um)≤C4mTM2,t∈R,Ui∈[0,∞)n,φ≤C4,i∈Λ1.Let us now state the first existence result of three positive solutions of system (1.4).Theorem 4.1.
Assume that conditions(H4), (H5), and (H6) hold. Then system (1.4) has at least three T-positive periodic solutions.Proof.
Firstly, we set(4.6)β(y):=∑i=1mmint∈[0,T]yi(t),y=(y1,y2,…,ym)∈P.
Obviously, β is a nonnegative continuous concave functional on P and
(4.7)β(y)≤∥y∥,y=(y1,y2,…,ym)∈PM¯foranyM>0.
Secondly, condition(H6) implies that there exists a number C5≥C3 such that
(4.8)A:PC5¯→PC5¯is completely continuous.
In fact, we know from condition (H6) that there exist numbers
(4.9)τi>0,0<σi<1mTM2,i∈Λ1
satisfying
(4.10)maxt∈[0,T]fi(t,Y1(t),Y2(t),…,Ym(t))∥y∥≤σi,∥y∥≥τi,i∈Λ1.
So
(4.11)fi(t,Y1(t),Y2(t),…,Ym(t))≤σi∥y∥,t∈[0,T],∥y∥≥τi,i∈Λ1.
Set
(4.12)C6i:=maxt∈[0,T]{fi(t,Y1(t),Y2(t),…,Ym(t))},t∈[0,T],∥y∥≤τi,i∈Λ1.
Then
(4.13)fi(t,Y1(t),Y2(t),…,Ym(t))≤C6i+σi∥y∥,t∈[0,T],y∈P,i∈Λ1.
Let us choose
(4.14)C5≥max{C3,mC61TM211-mTM21σ1,mC62TM221-mTM22σ2,…,mC6mTM2m1-mTM2mσm}.
Then for any  y=(y1,y2,…,ym)∈PC5¯, we have
(4.15)∥Ay∥=∑i=1mmaxt∈[0,T]∫tt+TGi(t,s)fi(t,Y1(s),Y2(s)),…,Ym(s))ds≤∑i=1mmaxt∈[0,T]∫tt+TGi(t,t+T)[σi∥y∥+C6i]ds≤∑i=1m(σiC5+C6i)TM2i≤C5,
which implies A(y)∈PC5¯ for all y∈PC5¯. Moreover, we know from the proof of (3.10) that A:PC5¯→PC5¯ is completely continuous.
Thirdly, let us show that numbers(4.16)0<C1<C2<C3≤C5
satisfy conditions (i),  (ii), and  (iii) of Lemma 2.2.
Step 1.
We prove that(4.17){y∈P(β,C2,C3):β(y)>C2}≠ϕ,β(Ay)>C2,y∈P(β,C2,C3).
Clearly,  y=(y1,y2,…,ym)=(C3/m,C3/m,…,C3/m)∈{P(β,C2,C3):β(y)>C2}. Moreover, for any y=(y1,y2,…,ym)∈P(β,C2,C3), we have
(4.18)C2≤∑i=1mmint∈[0,T]yi(t)≤∑i=1m|Yi(t)|0≤∥y∥≤C3.
Then condition (H5), (4.6), and (4.18) imply that
(4.19)β(Ay)=∑i=1mmint∈[0,T]∫tt+TGi(t,s)fi(s,Y1(s),Y2(s),…,Ym(s))ds≥mint∈[0,T]∫tt+TGi0(t,s)fi0(s,Y1(s),Y2(s),…,Ym(s))ds≥mint∈[0,T]∫tt+TM1i0C3TM1ds≥C3>C2,
which gives  β(Ay)>C2 for y∈P(β,C2,C3). And then we arrive at (4.17).Step 2.
Condition(H4) implies
(4.20)∥Ay∥<C1for∥y∥≤C1.
In fact, for any y=(y1,y2,…,ym)∈PC1¯, that is, ∥y∥≤C1, from
(4.21)|Yi(t)|0≤∥yi∥0,t∈[0,T],i∈Λ1,
and condition (H4) we have
(4.22)∥Ay∥=∑i=1mmaxt∈[0,T]∫tt+TGi(t,s)fi(s,Y1(s),Y2(s),…,Ym(s))ds≤∑i=1mmaxt∈[0,T]∫tt+TM2ifi(s,Y1(s),Y2(s),…,Ym(s))ds<mmaxt∈[0,T]∫tt+TM2C1mTM2ds=C1,
which yields (4.20).Step 3.
β(Ay)>C2 for y∈P(β,C2,C5) with ∥Ay∥>C3. This is the case because Ay∈P implies
(4.23)β(Ay)=∑i=1mmint∈[0,T]Ai(t)≥δ∑i=1m∥Ai∥0=δ∥Ay∥>δC3>C2.
At present, we may say that hypotheses of Lemma2.2 (the Leggett-Willaims theorem) are satisfied. Hence system (1.4) has at least three T-positive periodic solutions:
(4.24)u=(u1,u2,…,um),v=(v1,v2,…,vm),w=(w1,w2,…,wm),
such that
(4.25)∥u∥<C1,C2<∑i=1mmint∈[0,T]vi,∥w∥>C1,∑i=1mmint∈[0,T]wi<C2.
We give the following example to illustrate Theorem4.1.Example 4.2.
Consider the following system:(4.26)y1′(t)=-14π(ln3)(2+cost)y1(t)+14πe(sint)-1g1(t,y1(t-τ2),y2(t-τ2)),y2′(t)=-14π(ln2)(2+sint)y2(t)+14πe(cost)-1g2(t,y1(t-τ2),y2(t-τ2)),
where τ2>0 is a fixed constant and
(4.27)g1(t,u1,u2):={25,t∈R,u1+u2∈[0,1],(3-u1-u2)5+(36e2+1)(u1+u2-1),t∈R,u1+u2∈[1,3],72e2+2,t∈R,u1+u2∈[3,12],512(u1+u2)-3+72e2,t∈R,u1+u2∈[12,∞),g2(t,u1,u2):=constant<715,(t,u1,u2)∈R×[0,∞)×[0,∞).
We set(4.28)a1(t):=14π(ln3)(2+cost),a2(t):=14π(ln2)(2+sint),t∈R,f1(t,y1(t-τ),y2(t-τ)):=14πe(sint)-1g1(t,y1(t-τ),y2(t-τ)),t∈R,f2(t,y1(t-τ),y2(t-τ)):=14πe(cost)-1g2(t,y1(t-τ),y2(t-τ)),t∈R,C1:=1,C2:=3.
Then
(4.29)T=2π,δ=13,M1=13,M2=2.
We may verify also that conditions (H4), (H5), and (H6) hold. Hence, Theorem 4.1 tells us that system (4.26) has at least three 2π-positive periodic solutions:
(4.30)y1=(y11,y21),y2=(y12,y22),y3=(y13,y23)
such that
(4.31)∥y1∥<1,3<mint∈[0,2π]y12+mint∈[0,2π]y22,∥y3∥>1,mint∈[0,2π]y13+mint∈[0,2π]y23<3.
The second existence result of three positive solutions of system (1.4) is as follows.Theorem 4.3.
Assume that conditions(H4),  (H5), and (H7) hold. Then system (1.4) has at least three T-positive periodic solutions.Proof.
If we can get (4.8) with C5 replaced by C4 in this case, then the proof is complete. In fact, for any y=(y1,y2,…,ym)∈PC4¯, condition (H7) implies
(4.32)∥Ay∥=∑i=1mmaxt∈[0,T]∫tt+TGi(t,s)fi(s,Y1(s),Y2(s),…,Ym(s))ds≤∑i=1mmaxt∈[0,T]∫tt+TGi(t,t+T)fi(s,Y1(s),Y2(s),…,Ym(s))ds≤mmaxt∈[0,T]∫tt+TM2C4mTM2ds≤C4.
Then
(4.33)A:PC4¯→PC4¯,
as desired. This ends the proof.The following example is an application of Theorem4.3.Example 4.4.
Consider system(4.34)y1′(t)=-14π(ln2)(2+cost)y1(t)+14πe(cost)-1g1(t,y1(t-τ3),y2(t-τ3)),y2′(t)=-14π(ln2)(2+sint)y2(t)+14πe(cost)-1g2(t,y1(t-τ3),y2(t-τ3)),
where τ3>0 is a fixed constant and
(4.35)g1(t,u1,u2):={2(1+u2)3(u1+u2+1),t∈R,u1+u2∈[0,2],(1+u2)(4-u1-u2)3(u1+u2+1)+12(24e2+1)(u1+u2-2),t∈R,u1+u2∈[2,4],24e2+1,t∈R,u1+u2∈[4,∞),g2(t,u1,u2):=constant<1,(t,u1,u2)∈R×[0,∞)×[0,∞).
If we set(4.36)a1(t):=14π(ln2)(2+cost),a2(t):=14π(ln2)(2+sint),t∈R,fi(t,y1(t-τ),y2(t-τ))=14πe(cost)-1gi(t,y1(t-τ),y2(t-τ)),t∈R,i=1,2,
then
(4.37)T=2π,δ=12,M1=1,M2=2.
We choose
(4.38)C1:=2,C2:=4,C4:=50e2.
Then assumptions (H4), (H5), and (H7) hold. Hence we know from Theorem 4.3 that system (4.34) has at least three 2π-positive periodic solutions:
(4.39)y1=(y11,y21),y2=(y12,y22),y3=(y13,y23)
such that
(4.40)∥y1∥<2,4<mint∈[0,2π]y12+mint∈[0,2π]y22,∥y3∥>2,mint∈[0,2π]y13+mint∈[0,2π]y23<4.We end this paper by the following remark.Remark 4.5.
For a general positive integerm≥2, main results of this paper do not appear in former literatures as we know. Comparing with the existing results, our Theorems 3.1, 4.1, and 4.3 are new also when m=1.



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*Source: 239209-2010-02-08.xml*