Positive periodic solutions for a nonlinear difference system via a continution theorem

(1)

Positive periodic solutions for a nonlinear difference system via a continution theorem

Genqiang Wang and Sui Sun Cheng

Abstract. Based on a continuation theorem of Mawhin, the existence of a positive periodic solution for a nonlinear difference system is studied.

Keywords: Nonlinear difference system, positive periodic solution, continution theo- rem.

Mathematical subject classification: 39A11.

1 Introduction

In [1], we explained that scalar difference equations of the form

yn+1= ynexp{f(n,yn,yn−1, ...,yn−k)}, n ∈ Z = {0,±1,±2, ...}, (1) where f = f(t,u0,u1, ...,uk) is a real continuous function defined on R^k⁺² such that

f(t+ω,u0, ...,uk)= f(t,u0, ...,uk), (t,u0, ...,uk)∈ R^k⁺²,

andωis a positive integer, are of interest since they include well known equations such as

yn+1=ynexp

μ(1−yn) K

, K >0,

and they are intimately related to delay differential equations with piecewise constant independent arguments [2]:

y⁰(t)= y(t) f ([t ],y([t]) ,y([t −1]) ,y([t−2]) , ...,y([t−k])) , t ∈ R.

We also show that continuation theorems can be used to show existence of periodic solutions of these equations.

Received 13 January 2005.

(2)

Note that in the above equations, only one time dependent variable yt is in- volved. In real problems, multiple time dependent variables may interact, and therefore it is natural to study systems of difference equations.

In this paper, we consider one such system of the form y_i⁽ⁿ⁺¹⁾ =y_i⁽ⁿ⁾exp



r_i⁽ⁿ⁾− Xk

j=1

a_{i j}⁽ⁿ⁾y⁽ⁿ⁾_j − Xk

j=1

b_{i j}⁽ⁿ⁾y

n−τ_{i j}⁽ⁿ⁾

j



,

i ∈ {1, ...,k},n ∈ Z, (2)

where ri =n

r_i⁽ⁿ⁾o

n∈Z, ai j =n a⁽ⁿ⁾_{i j} o

n∈Z, bi j =n b_{i j}⁽ⁿ⁾o

n∈Z and τi j =n τ_{i j}⁽ⁿ⁾o

n∈Z, are realω-periodic sequences such that

r_i⁽ⁿ⁾ = r_i⁽ⁿ⁺^ω), n∈ Z a_{i j}⁽ⁿ⁾ = a_{i j}⁽ⁿ⁺^ω), n ∈ Z b_{i j}⁽ⁿ⁾ = b⁽ⁿ_{i j}⁺^ω), n∈ Z τ_{i j}⁽ⁿ⁾ = τ_{i j}⁽ⁿ⁺^ω), n ∈ Z for i, j∈ {1, ...,k}.We assume further that

a_{i j}⁽ⁿ⁾,b⁽ⁿ⁾_{i j} >0, i, j∈ {1, ...,k};n∈ Z, X

0≤n≤ω−1

r_i⁽ⁿ⁾>0, i ∈ {1, ...,k},

and X

0≤n≤ω−1

a_ii⁽ⁿ⁾+b⁽ⁿ⁾_ii

6=0, i∈ {1, ...,k}. The numberωis a positive integer as before.

A solution of (2) is a real vector sequence of the form y =

y⁽ⁿ⁾ _n_∈_Z where y⁽ⁿ⁾ =

y₁⁽ⁿ⁾,y₂⁽ⁿ⁾, ...,y_k⁽ⁿ⁾†

which renders (2) into an identity after substitution.

As in [1], we are concerned with the existence of positive solutions which are ω-periodic, that is, solutions that satisfy y⁽ⁿ⁺^ω) = y⁽ⁿ⁾for n ∈ Z and y_i⁽ⁿ⁾ >0 for n∈ Z and i ∈ {1, ...,k}.

Our system (2) can be used to describe multispecies ecological competition systems or multi-nation competition models. The analogous problem for differential systems has been treated by Smith [4], Cushing [5], Zanolin [6], Fan and

(3)

Wang [7] and others. In particular, in [7], the authors study differential systems of the form

y_i⁰(t)=yi(t)



ri(t)− Xk

j=1

ai j(t)yj(t)− Xk

j=1

bi j(t)yj(t−τi j)



, i=1,2, ...,k.

As for our system, we can also show that it is related to differential systems with piecewise constant independent arguments of the form

y_i⁰(t)=yi(t)



ri([t])− Xk

j=1

ai j([t])yj(n)− Xk

j=1

bi j([t])yj [t]−τi j([t])



,

i ∈ {1, ...,k},t ∈ R, (3)

where [x] is the greatest-integer function, ri(t) ,ai j(t)and bi j(t)are real contin- uousω-periodic functions defined on R.Indeed, once the existence of a positive ω-periodic solution of (2) can be demonstrated, we may then make immediate statements about the existence of positiveω-periodic solutions of (3). The proof of our assertion is not much different from that of Theorem 1 in [1], and hence is not included here.

As in [1], we will invoke a continuation theorem of Mawhin for obtaining periodic solutions of (2). For the sake of easy reference, we briefly describe this result here. Let X and Y be two Banach spaces and L : DomL ⊂ X →Y is a linear mapping and N : X →Y a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim KerL = codim ImL < +∞, and ImL is closed in Y. If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X and Q:Y →Y such that Im P=KerL and ImL =Ker Q =Im(I −Q).It follows that L_|DomL∩Ker P : (I −P)X → ImL has an inverse which will be denoted by KP. Ifis an open and bounded subset of X , the mapping N will be called L-compact onˉ if Q N ˉ

is bounded and KP(I −Q)N ˉ

is compact. Since Im Q is isomorphic to Ker L there exist an isomorphism J :Im Q→KerL.

Theorem A (Mawhin’s continuation theorem [1]). Let L be a Fredholm mapping of index zero, and let N be L-compact on. Supposeˉ

(i) for eachλ∈(0,1), x ∈∂,L x 6=λN x;and

(ii) for each x ∈∂∩KerL,Q N x 6=0 and deg(J Q N, ∩KerL,0)6=0.

(4)

Then the equation L x =N x has at least one solution inˉ ∩domL.

We recall the useful nonstandard “summation” operation [1] for any real sequence

u⁽ⁿ⁾ _n_∈_Z: Mβ n=γ

u⁽ⁿ⁾ =





 Pβ

n=γu⁽ⁿ⁾, γ ≤β

0, β =γ −1

−Pγ−1

n=β+1u⁽ⁿ⁾, β < γ −1 .

As usual, the forward difference is defined by1u^(k) =u^(k⁺¹⁾−u^(k). We will also employ the following notations for the ‘time’ averages:

ri = 1 ω

X

0≤n≤ω−1

r_i⁽ⁿ⁾,

Ri = 1 ω

X

0≤n≤ω−1

r_i⁽ⁿ⁾

, ai j = 1

ω X

0≤n≤ω−1

a_{i j}⁽ⁿ⁾,

bi j = 1 ω

X

0≤n≤ω−1

b_{i j}⁽ⁿ⁾.

2 Existence Criteria

The main result of our paper is the following.

Theorem 1. Suppose the following set of conditions hold:

(i) for each i ∈ {1, ...,k},ri >0,

(ii) for i, j ∈ {1, ...,k},the inverse of the matrix ai j+bi j

k×kexists and all its components are positive, and

(iii) for each i ∈ {1, ...,k}, ri > X

1≤j≤k,j6=i

ai j+bi j

rj

aj j +bj j

exp 1

2 Rj +rj

ω

.

Then (2) has a positiveω-periodic solution.

(5)

In order to provide a proof, we proceed in a manner similar to that of Theorem 1 in [1]. However, there are sufficient difference to warrant some details in the following discussions. We first note that if

x =

x₁⁽ⁿ⁾,x₂⁽ⁿ⁾, ...,x_k⁽ⁿ⁾†

n∈Z

is aω-periodic solution of the following system x_i⁽ⁿ⁾ =x_i⁽⁰⁾+

n−1

M

s=0



r_i^(s⁾− Xk

j=1

a^(s)_{i j} exp

x^(s)_j −

Xk j=1

b^(s)_{i j} exp x

s−τ_{i j}^(s)

j



,

i ∈ {1, ...,k},n ∈ Z, (4)

then we can easily check that y =

y₁⁽ⁿ⁾,y₂⁽ⁿ⁾, ...,y_k⁽ⁿ⁾ †

n∈Z

=

e^x¹⁽ⁿ⁾,e^x²⁽ⁿ⁾, ...,e^x^k⁽ⁿ⁾ †

n∈Z

is a positiveω-periodic solution of (1).

We will therefore seek anω-periodic solution of (4). Let Xω be the Banach space of all real vectorω-periodic sequences of the form x = {x⁽ⁿ⁾}n∈Z where x⁽ⁿ⁾ =

x₁⁽ⁿ⁾,x₂⁽ⁿ⁾, ...,x_k⁽ⁿ⁾†

and endowed with the usual linear structure as well as the norm

kxk1= X

1≤i≤k

0≤maxn≤ω−1

x_i⁽ⁿ⁾

2!¹₂ .

Let Yωbe the Banach space of all real sequences of the form y = {y⁽ⁿ⁾}ⁿ∈Z =

nα+h⁽ⁿ⁾ _n_∈_Z

such that y⁽⁰⁾ = 0, whereα = (α1, ..., αk)^† ∈ R^k and {h⁽ⁿ⁾}n∈Z ∈ Xω, and endowed with the usual linear structure as well as the normkyk2= |α| + khk1, here|α| = P

1≤i≤kα_i²¹₂

.Let the zero element of Xω and Yωbe denoted byθ1

andθ2respectively.

Define the mappings L: Xω →Yω and N : Xω →Yωrespectively by

(L x)⁽ⁿ⁾ =x⁽ⁿ⁾−x⁽⁰⁾, n ∈ Z. (5)

(6)

and

(N x)⁽ⁿ⁾_i =

n−1

M

s=0



r_i^(s)− Xk

j=1

a_{i j}^(s)exp x^(s)_j − Xk

j=1

b^(s)_{i j} exp x

s−τ_{i j}^(s)

j



,

i∈ {1, ...,k},n∈ Z. (6) Let

hˉ⁽ⁿ⁾_i =

n−1

M

s=0



r_i^(s)− Xk

j=1

a_{i j}^(s)exp x^(s_j ⁾− Xk

j=1

b^(s)_{i j} exp x

s−τ_{i j}^(s) j

!



− n ω

ω−1

M

s=0



r_i^(s)− Xk

j=1

b_{i j}^(s)exp x

s−τ_{i j}^(s)

j

!

 (7)

for i=1, ...,k and n∈ Z.

Sincehˉ = { ˉh⁽ⁿ⁾}n∈Z ∈ Xω andhˉ⁽⁰⁾ = θ1, N is a well-defined operator from Xω to Yω. On the other hand, direct calculation shows that Ker L = {x ∈ Xω | x⁽ⁿ⁾ =x⁽⁰⁾,n ∈ Z,x⁽⁰⁾∈ R^k}and ImL =Xω∩Yω.Let us define P : Xω → Xω

and Q:Yω →Yωrespectively by

(P x)⁽ⁿ⁾ =x⁽⁰⁾, n ∈ Z, (8)

for x = {x⁽ⁿ⁾}ⁿ∈Z ∈and

(Qy)⁽ⁿ⁾=nα (9)

for y = {nα +h⁽ⁿ⁾}n∈Z ∈ Yω. The operators P and Q are projections and Xω =Ker P⊕KerL,Yω =ImL⊕Im Q. It is easy to see that dim KerL =k = dim Im Q=codimImL,and

ImL = {y ∈ Xω |y(0)=0} ⊂Yω.

It follows that ImL is closed in Yω. Thus the following Lemma is true.

Lemma 1. The mapping L defined by (5) is a Fredholm mapping of index zero.

Lemma 2. Let L and N defined by (5) and (6) respectively. Supposeis an open and bounded subset of Xω. Then N is L-compact on.

(7)

Proof. From (6), (7) and (9), we see that for any x = {x⁽ⁿ⁾}ⁿ∈Z ∈ Xω, (Q N x)_i⁽ⁿ⁾ = n

ω

ω−1

M

s=0



r_i^(s)− Xk

j=1

b_{i j}^(s)exp x

s−τ_{i j}^(s)

j

!

,

i∈ {1,2, ..,k},n∈ Z. (10) We denote the inverse of the mapping L|^DomL∩Ker P:(I −P)X →ImL by KP. Direct calculation leads to

(KP(I −Q)N x)_i⁽ⁿ⁾ =

n−1

M

s=0



r_i^(s)− Xk

j=1

b_{i j}^(s)exp x

s−τ_{i j}^(s) j





−n ω

ω−1

M

s=0



r_i^(s)− Xk

j=1

b^(s)_{i j} exp x

s−τ_{i j}^(s)

j



. (11) It is easy to see that Q N and KP(I −Q)N are continuous on Xω and takes bounded sets into bounded sets respectively. Since the Banach space Xωis finite dimensional, N is L-compact on. The proof is complete.

Let lω, whereω > 2 is positive number, be the space of all realω-periodic sequences of the form u=

u⁽ⁿ⁾ _n_∈_Z.

Lemma 3. If u =

u⁽ⁿ⁾ _n_∈_Z ∈lω, then

0≤maxs,i≤ω−1

u^(s)−u⁽ⁱ⁾≤ 1 2

ω−1

X

k=0

1u^(k), (12)

where the constant factor 1/2 is the best possible.

Proof. Let u=

u⁽ⁿ⁾ _n_∈_Z ∈lωand s,i ∈ {0,1, ..., ω−1}.Without loss of any generality, let s ∈ {i+1, ...,i+ω−1},we have

u^(s) =u⁽ⁱ⁾+

s−1

X

k=i

1u^(k) (13)

and

u⁽ⁱ⁾ =u⁽ⁱ⁺^ω)=u^(s)+

i+Xω−1 k=s

1u^(k). (14)

(8)

From (13) and (14), we see that for any s∈ {i,i+1, . . . ,i+ω−1}, 2u^(s)=2u⁽ⁱ⁾+

s−1

X

k=i

1u^(k)−

i+Xω−1 k=s

1u^(k), (15)

that is

u^(s) =u⁽ⁱ⁾+1 2

(_s₋₁ X

k=i

1u^(k)−

i+Xω−1 k=s

1u^(k) )

. (16)

Thus for any s∈ {i,i+1, . . . ,i+ω−1}, u^(s)−u⁽ⁱ⁾≤ 1

2

i+Xω−1 k=i

1u^(k)= 1 2

ω−1

X

k=0

1u^(k), (17)

so that

0≤s,imax≤ω−1

u^(s)−u⁽ⁱ⁾≤ 1 2

ω−1

X

k=0

1u^(k). (18)

Now we assert that ifβis a constant andβ <1/2,then there are u =

u⁽ⁿ⁾ _n_∈_Z ∈ lω and such that

max

0≤s,i≤ω−1

u^(s)−u⁽ⁱ⁾> β

ω−1

X

k=0

1u^(k). (19)

Indeed, if we let u⁽ⁿ⁾ = j for n =kω+ j,k ∈ Z and j =0,1, ..., ω−1,then max0≤s,i≤ω−1

u^(s)−u⁽ⁱ⁾=ω−1 and 1u⁽ⁿ⁾ =

1, n=0,1, .., ω−2

−(ω−1) , n=ω−1 , (20) and

β

ω−1

X

k=0

1u^(k)=2β (ω−1) < max

0≤s,i≤ω−1

u^(s)−u⁽ⁱ⁾

as required. This shows that the constant 1/2 in (12) is the best possible. The

proof is complete.

Now, we consider the following system x_i⁽ⁿ⁾−x_i⁽⁰⁾=λ

n−1

M

s=0



r_i^(s)− Xk

j=1

b_{i j}^(s)exp x

s−τ_{i j}^(s)

j

!

, i ∈ {1, ...,k},n ∈ Z, (21) whereλ∈(0,1).

(9)

Lemma 4. Suppose the condition (iii) in Theorem 1 holds. Then there ex- ist positive constants H1, ...,Hk such that for any solution x = {x⁽ⁿ⁾}ⁿ∈Z =

x₁⁽ⁿ⁾, ...,x_k⁽ⁿ⁾†

n∈Z

∈ Xω of (21), we have the following inequalities

0≤maxn≤ω−1

x_i⁽ⁿ⁾

≤ Hi, i ∈ {1, ...,k}. (22) Proof. Let x= {x⁽ⁿ⁾}ⁿ∈Z be aω-periodic solution of (21). Then

ω−1

M

s=0



r_i^(s⁾− Xk

j=1

a_{i j}^(s)exp x^(s_j ⁾− Xk

j=1

b_{i j}^(s)exp x

s−τ_{i j}^(s)

j



=0, i ∈ {1, ...,k}. (23)

It leads to

ω−1

M

s=0



 Xk

j=1

a_{i j}^(s)exp x^(s)_j + Xk

j=1

b_{i j}^(s)exp x

s−τ_{i j}^(s)

j



=ωri. (24)

From (21), we have 1x_i⁽ⁿ⁾ =λ



r_i⁽ⁿ⁾− Xk

j=1

a⁽ⁿ⁾_{i j} exp x⁽ⁿ⁾_j − Xk

j=1

b_{i j}⁽ⁿ⁾exp x

n−τ_{i j}⁽ⁿ⁾

j

!

,

i∈ {1, ...,k},n∈ Z. (25) By (24) and (25), we see that

ω−1

M

s=0

1x_i^(s)

≤

ω−1

M

s=0



r_i^(s) +

Xk j=1

a_{i j}^(s)exp x^(s)_j + Xk

j=1

b_{i j}^(s)exp x

s−τ_{i j}^(s)

j





=

ω−1

M

s=0

r_i^(s)

+

ω−1

M

s=0



 Xk

j=1

a_{i j}^(s)exp x^(s)_j

+ Xk

j=1

b_{i j}^(s)exp x

s−τ_{i j}^(s)

j





= Ri +ri

ω.

(26)

(10)

Let x_i^(μⁱ⁾ =max0≤n≤ω−1x_i⁽ⁿ⁾ and x_i^(νⁱ⁾ =min0≤n≤ω−1x_i⁽ⁿ⁾, where 0 ≤ μi, νi ≤ ω−1.From (24), we have

ωri >

ω−1

M

s=0



 Xk

j=1

a_{i j}^(s)exp x^(ν_j ^j⁾+ Xk

j=1

b_{i j}^(s)exp x^(ν_j ^j⁾





= Xk

j=1

ai j+bi j

ωexp x^(ν_j ^j⁾

> aii +b

ωexp x_i^(νⁱ⁾,

(27)

that is,

x_i^(νⁱ⁾ ≤ln ri

aii +bii

. (28)

In view of Lemma 3, (26) and (28), we see that for any n=0,1, ..., ω−1, x_i⁽ⁿ⁾ ≤x_i^(νⁱ⁾+1

2

ω−1

X

k=0

1x_i^(k)

≤ Bi, (29) where

Bi =ln ri

aii+bii

+1

2 Ri+ri

ω. (30) Furthermore, from (24), we have

ωri ≤

ω−1

M

s=0



 Xk

j=1

a_{i j}^(s)exp x^(μ_j ^j⁾+ Xk

j=1

b^(s)_{i j} exp x^(μ_j ^j⁾





= Xk

j=1

ai j +bi j

ωexp x^(μ_j ^j⁾.

(31)

By (29) and (31), we see that ai i +bii

exp x_i^(μⁱ⁾

> ri − X

1≤j≤k,j6=i

ai j +bi j

exp x^(μ_j ^j⁾

> ri − X

1≤j≤k,j6=i

ai j +bi j

rj

aj j+bj j

exp 1

2 Rj +rj

ω

,

(32)

(11)

that is

x_i^(μⁱ⁾ >Ci, (33)

where Ci =ln





ri −P

1≤j≤k,j6=i ai j +bi j

rj

aj j+bj j exp ¹₂ Rj +rj

ω aii+bii



. (34) In view of Lemma 3, (26) and (33), we see that for any n=0,1, ..., ω−1,

x_i⁽ⁿ⁾>x_i^(μⁱ⁾−1 2

ω−1

X

k=0

1x_i^(k)

>Ci −1

2 Ri +ri

ω. (35)

From (29) and (35), we have

0≤maxn≤ω−1

x_i⁽ⁿ⁾

≤ Hi, (36) where Hi =max

|Bi|,C_i− ¹2 Ri +ri

ω

+1.The proof is complete.

Proof of Theorem 1. Let L,N,P and Q be defined by (5), (6), (8) and (9) respectively. By conditions (i) and (ii), we know that the linear system of the form

ri− Xk

j=1,

ai j +bi j

vj =0, i∈ {1, ...,k}, (37)

has the unique solutionv^∗ = v₁^∗, v^∗₂, ..., v_k^∗†

andv_i^∗>0 for i ∈ {1, ...,k}.Pick M such that

Xk i=1

lnv_i^∗2

!¹2

< M. (38)

From Lemma 4, we know there exist positive constants H1, ...,Hk such that for any solution x = {x⁽ⁿ⁾}n∈Z =

x₁⁽ⁿ⁾, ...,x_k⁽ⁿ⁾ †

n∈Z

∈ Xωof (21), we have the following inequalities

0≤maxn≤ω−1

x_i⁽ⁿ⁾

≤ Hi, i =1, ...,k. (39) Let H =Pk

i=1H_i²₂¹

+M.Thenkxk1< H.Set

= {x ∈ Xω| kxk1<H}.

(12)

It is easy to see thatis an open and bounded subset of Xωand for eachλ∈(0,1) and x ∈∂, L x 6= λN x . Furthermore, in view of Lemma 1 and Lemma 2, L is a Fredholm mapping of index zero and N is L-compact on. Noting that H >Pk

i=1H_i² ¹₂

,by Lemma 4, for eachλ∈(0,1)and x ∈∂, L x 6=λN x.

Next note that a vector sequence x= {x⁽ⁿ⁾}n∈Z ∈∂∩KerL must be a constant vector andkxk1=H > M.Hence

kQ N xk2=





 Xk

i=1







r_i − Xk

j=1

a_{i j} +bi j

exp xj









2





1 2

6=0.

So

Q N x 6=θ2.

The isomorphism J: Im Q → KerL is defined by J Qy = α for y = {nα + h⁽ⁿ⁾}ⁿ∈Z ∈Yω.Then

(J Q N x)⁽ⁿ⁾_i = 1 ω

ω−1

M

s=0



r_i^(s)− Xk

j=1

b^(s)_{i j} exp x

s−τ_{i j}^(s) j





=ri − Xk

j=1

a_{i j} +bi j

exp xj,

(40)

for n ∈ Z and i ∈ {1, ...,k}. Since (37) has the unique solution v^∗ = v₁^∗, v^∗₂, ..., v^∗_k†

with positive components and such that (38) is satisfied, we see that the system

ri − Xk

j=1

a_{i j} +bi j

exp xj

=0,i∈ {1, ...,k} (41)

has a unique solution x = x₁^∗,x₂^∗, ...,x_k^∗†

in ∩ Ker L, so that from the condition (ii) we have

deg(J Q N x, ∩Ker L, θ1)= sign detϒJ Q N (x)6=0.

whereϒJ Q N(x)is the Jacobi matrix of J Q N at x.By Theorem A, we see that equation L x =N x has at least one solution in∩DomL. In other words, (4) has aω-periodic solution x = {x⁽ⁿ⁾}ⁿ∈Z,and hencen

e^x¹⁽ⁿ⁾, ...,e^x^k⁽ⁿ⁾o

n∈Z is a positiveω-periodic solution of (2). The proof is complete.

(13)

We remark that by the relationship that exists between (2) and (3), under the same assumption of Theorem 1, system (3) has a positiveω-periodic solution.

We now illustrate our main result by considering the following system











y₁⁽ⁿ⁺¹⁾ =y₁⁽ⁿ⁾exp r₁⁽ⁿ⁾−a₁₁⁽ⁿ⁾y₁⁽ⁿ⁾−b⁽ⁿ⁾₁₂ y

n−τ₁₂⁽ⁿ⁾

2

! ,

y₂⁽ⁿ⁺¹⁾ =y₂⁽ⁿ⁾exp r₂⁽ⁿ⁾−a₂₂⁽ⁿ⁾y₂⁽ⁿ⁾−b⁽ⁿ⁾₂₁ y

n−τ₂₁⁽ⁿ⁾

1

! ,

where ri,bi j,aii andτi j for i, j∈ {1,2}are 2-periodic sequences and

r₁⁽⁰⁾ =0, r₁⁽¹⁾ =1, r₂⁽⁰⁾ =1, r₂⁽¹⁾=0, a₁₁⁽⁰⁾ =1/3, a₁₁⁽¹⁾ =2/3, a₂₂⁽⁰⁾ =2/3, a⁽¹⁾₂₂ =1/3, b⁽⁰⁾₁₂ =1/6e, b⁽¹⁾₁₂ =1/4e, b⁽⁰⁾₂₁ =1/5e, b₂₁⁽¹⁾=1/7e.

It is easily verified from Theorem 1 that it has a positive 2-periodic solution.

References

[1] G. Q. Wang and S. S. Cheng, Positive periodic solutions for a nonlinear difference equation via a continuation theorem, Adv. Difference Eq., 4 (2004), 311–320.

[2] K. L. Cooke and J. Wiener, A survey of differential equations with piecewise continuous arguments, Delay differential equations and dynamical systems (Claremont, CA, 1990), 1–15, Lecture Notes in Math., 1475, Springer, Berlin, (1991).

[3] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math, Springer-Verlag, 586 (1977).

[4] H. L. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal., 17(6) (1986), 1289–1318.

[5] J. M. Cushing, Two species competition in a periodic enviroment, J. Math. Biol., 10 (1980), 385–400.

[6] F. Zanolin, Coexistence states for periodic Kolmogorov systems, In “The first 60 years of Nonlinear Analysis of Jean Mawhin”, World Scientific Publ., (2004), pp.

233–246.

[7] M. Fan and K. Wang, Existence and global attractivity of positive periodic solution of multispecies ecological competition system, Acta Math. Sinica, 43(1) (2000), 77–82.

(14)

Genqiang Wang

Department of Computer Science

Guangdong Polytechnic Normal University Guangzhou, Guangdong 510665

P. R. CHINA

E-mail: [email protected]

Sui Sun Cheng

Department of Mathematics Tsing Hua University Hsinchu, Taiwan 30043 R. O. CHINA

E-mail: [email protected]