1 Introduction

Let T＞3 be an integer.In this paper,we are concerned with the existence and multiplicity of positive T-periodic solutions of the following singular discrete systems

and

where u=(u1,···,un)∈ Rn,ai,bi:Z → [0,∞)are T-periodic functions with

gi ∈ C(,[0,∞))and fi:{0} → [0,∞)are continuous,i=1,2,···,n; τ:Z → Z is a T-periodic function and λ is a positive parameter.

In the past few years,there has been considerable interest in the existence of periodic solutions of equations

Let Tλ:X→X be a mapping with components(,···,):

where a,b∈ C(R,[0,∞))are T-periodic functions with

and τ is a continuous T-periodic function.Equations(1.3)and(1.4)have been proposed as models for a variety of physiological processes and conditions including production of blood cells,respiration,and cardiac arrhythmias.See for example,[1-8,12]and the references therein.On the other hand,many authors paid their attention to the existence of positive periodic solutions of singular systems of the first-order and second-order differential equations,see Chu[9],Jiang[10],Wang[11,12]and the references therein.It has been shown that many results of nonsingular systems still valid for singular cases.

Let

and for any u=(u1,···,un)∈ ,

Recently,Wang[12]studied the existence and multiplicity of positive periodic solutions of the following singular non-autonomous n-dimensional system

under assumptions

Thus by the similar manners as in the proof of Lemmas 2.3 and 2.5,we can easily obtain the following lemmas.

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(H2)fi:{0} → (0,∞)are continuous,i=1,···,n.

By using Krasnoselskii fixed point theorem in a cone,the author established the existence and multiplicity of positive periodic solutions of(1.5)with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter.

And then Lemma 2.3 shows

We make the following assumptions:

(C1)ai,bi:Z→[0,∞)are T-periodic functions withi=1,2,···,n;τ:Z → Z is a T-periodic function.

(C2)gi∈C(,[0,∞))satisfies 0＜li≤gi(u)≤Li＜∞,fi:{0}→(0,∞)is continuous,i=1,2,···,n.

(C3)0 ≤ liai(t)≤ Liai(t)＜ 1,t∈ T:={0,1,···,T −1},i=1,2,···,n.

It follows from Theorem A that

Theorem 1.1 Let(C1)-(C3)hold.Suppose= ∞ f or some i=1,2,···,n,then:

(i)If=0,i=1,2,···,n,then for all λ ＞ 0 ,(1.1)admits a positive periodic solution.

(ii)If= ∞ ,i=1,2,···,n,then(1.1)admits two positive periodic solutions for λ ＞ 0 sufficiently small.

(iii)There exists a λ0 ＞ 0 such that(1.1)admits a positive periodic solution for 0＜λ＜λ0.

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Remark 1.1 Theorem 1.1,which improves the corresponding ones established for single difference equations in[17-21],is the discrete analogues of[12,Theorem 1.1]when gi ≡ 1,i=1,2,···,n and τ≡ 0.For more details on the periodic solutions of systems(1.1)and(1.2),we refer the readers to[13-16].

The following well-known theorem plays a key role in proving our main results.

Theorem A[22,23]Let E be a Banach space and P be a cone in E.For r＞0,define Pr={u ∈ P :‖u‖ ＜ r }.Assume T:→P is completely continuous such that Tu≠u for u ∈ ∂Pr={u ∈ P:‖u‖ =r}.

(i)If ‖Tu‖＞ ‖u‖ for u ∈ ∂Pr,then i(T,Pr,P)=0.

(ii)If ‖Tu‖＜ ‖u‖ for u∈ ∂Pr,then i(T,Pr,P)=1.

2 Preliminaries

Set

For r＞0,de fine

Let E={u:Z→R|u(t+T)=u(t),t∈Z}be a Banach space with the normand X be a Banach space defined by

which is equipped with the norm

Define

It is not difficult to check that K is a cone in X.For r＞ 0,let

then ∂Ωr={u ∈ K:‖u‖=r}.

and

where

It follows from(C3)that

Moreover,we can easily get

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Lemma 2.1 Let(C1)-(C3)hold.Then Tλ(K)⊂K and Tλ:K→K is compact and continuous.

Proof In view of the definition of K,for u ∈ K and i=1,2,···,n,

Indeed,since aiis T-periodic and u∈K,we get

and thus Tλu∈X.One can show that,for u∈K and t∈T,

Therefore Tλ(K)⊂ K and Tλ:K → K is compact and continuous.The proof is completed.

Using the similar methods as in the proof of[12,Lemma 2.2]with obvious changes,we can obtain the following lemma.

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Lemma 2.2 Let(C1)-(C3)hold.Then u∈K{0}is a positive periodic solution of system(1.1)if and only if u is a fixed point of Tλin K{0}.

Lemma 2.3 Let(C1)-(C3)hold.For any η＞0 and u∈K{0},if there exists a fisuch that for t∈ T ,then ‖Tλu‖ ≥ λ Γη‖u‖.

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which implies‖Tλu‖≥ λΓη‖u‖.The proof is completed.

Let:[1,∞)→ R+be a function defined by

Thenis nondecreasing on[1,∞).

Lemma 2.4[11,12]exists(which can be infinity),thenexists and

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Lemma 2.5 Suppose(C1)-(C3)hold and r＞.If there exists an ε＞ 0 such that(r)≤ εr,i=1,2,···,n,then ‖Tλu‖≤ λεΛ‖u‖ for u ∈ ∂Ωr.

Proof For u∈∂Ωr,we have

and the proof is completed.

When u∈∂Ωr,r＞0,the definitions of M(r)and m(r)yield

(H1)ai,bi ∈ C(R,[0,∞))are ω-periodic functions such that

Lemma 2.6 Let(C1)-(C3)hold.If u ∈ ∂ Ωrand r＞ 0 ,then‖Tλu‖ ≥ m(r).

Lemma 2.7 Let(C1)-(C3)hold.If u∈∂Ωrand r＞0,then‖Tλu‖≤ λΛM(r).

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3 Proof of Theorem 1.1

(i)It follows from the assumption that there exists an r1＞0 such that

for u ∈ with 0 ＜ ‖u‖≤ r1,where η ＞ 0 is chosen satisfying λΓη ＞ 1.If u∈∂Ωr1,then

Lemma 2.3 implies‖Tλu‖≥ λΓη‖u‖＞ ‖u‖,for u ∈ ∂Ωr1.

On the other hand,since=0,i=1,2,···,n,Lemma 2.4 yields=0,i=1,2,···,n.Therefore there exists an r2 ＞ such that

Proof Since u∈K{0}and for t∈ T ,we have

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where ε＞ 0 satisfies λΛε＜ 1.And then by Lemma 2.5,we get

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Our main results can be stated as below.

consequently i(Tλ,Ωr1,K)=1.Hence,Tλ has a fixed point u inΩr1,which is just a positive periodic solution of system(1.1).

(ii)Let r1 ＞ 0 be fixed.By Lemma 2.7,there exists a λ0＞ 0 such that

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In view of= ∞ for some i=1,2,···,n,there is a positive number r2＜ r1such that fi(u)≥ η‖u‖for u ∈ with 0 ＜ ‖u‖ ≤ r2,where η ＞ 0 is chosen so that λΓη ＞ 1.Then for u ∈ ∂Ωr2,we get

Lemma 2.3 implies‖Tλu‖≥ λΓη‖u‖＞ ‖u‖,for u ∈ ∂Ωr2.

It follows from= ∞,i=1,2,···,n that there exists an^H ＞ 0 such that

for u ∈ with ‖u‖ ≥ ,where η ＞ 0 is chosen so that λΓη ＞ 1.Let r3=max{2r1,}.If u∈∂Ωr,then

3

which yields

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However,to the best of our knowledge,the existence results of positive periodic solutions for first-order discrete systems(1.1)and(1.2)with singular nonlinearities are relatively little.Motivated by the above considerations,in this paper,we study the existence and multiplicity of positive T-periodic solutions of singular discrete systems(1.1)and(1.2).Obviously,(1.1)is a discrete analogue of system(1.5)when gi ≡ 1,i=1,2,···,n and τ≡ 0,and we are interested in establishing the similar results as[12,Theorem 1.1]for systems(1.1)and(1.2).

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By Theorem A,we can easily obtain

consequently

Hence Tλhas two fixed points lying in which are positive periodic solutions of(1.1).

(iii)For a fixed number r1 ＞ 0,Lemma 2.7 implies there exists a λ0 ＞ 0 such that

On the other hand,since= ∞ f or some i=1,2,···,n,there is a positive number r2＜r1such that

for u ∈ with 0＜ ‖u‖≤ r2,where η＞ 0 is chosen so that λΓη＞ 1.If u ∈ ∂Ωr2,then

It follows from Lemma 2.3 that‖Tλu‖≥ λΓη‖u‖＞ ‖u‖,for u ∈ ∂Ωr2.

Using Theorem A again,we can get

so i(Tλ,Ωr2,K)=1.Hence,Tλhas a fixed point u inΩr1Ωr2for 0＜ λ ＜ λ 0,which is a positive periodic solution of system(1.1).The proof is completed.

4 Positive Periodic Solutions of System(1.2)

In this Section,we shall establish the existence and multiplicity of positive T-periodic solutions of singular discrete system(1.2),that is,

where λ, τ,ai,bi,fi(u),gi(u)satisfy the same assumptions stated for system(1.1).In view of(1.2),we can de fine an operator Tλ:X →X with components(,···,):

where

Clearly,(C1)and(C2)imply for all t∈ T and i=1,2,···,n,

and 0 ＜ 1.Here

De fine a cone in X by

By the similar arguments as in Sections 2 and 3,we can establish the following theorems.

Theorem 4.1 Let(C1)and(C2)hold.Assume= ∞ f or some i=1,2,···,n.

(i)If=0,i=1,2,···,n,then for all λ ＞ 0 ,(1.2)admits a positive periodic solution.

(ii)If= ∞ ,i=1,2,···,n,then(1.2)admits two positive periodic solutions for λ ＞ 0 sufficiently small.

(iii)There exists a λ0 ＞ 0 such that(1.2)admits a positive periodic solution for 0＜λ＜λ0.

Finally,consider discrete systems(1.1)and(1.2)without singularities,that is,we replace(C2)with the following condition.

()gi∈ C(,[0,∞))satisfies 0＜ li≤ gi(u)≤ Li＜ ∞,fi:→ [0,∞)is continuous and fi(u) ＞ 0 for u ∈with u≠0,i=1,2,···,n.

Then the following two theorems can be established by the similar methods adopted in Sections 2 and 3.

Theorem 4.2 Let(C1),()and(C3)hold.Assume=0 for i=1,2,···,n.

(i)If = ∞,i=1,2,···,n,then for all λ ＞ 0,(1.1)admits a positive periodic solution.

(ii)If=0,i=1,2,···,n,then(1.1)admits two positive periodic solutions for λ ＞ 0 sufficiently large.

(iii)There exists a λ0 ＞ 0 such that(1.1)admits a positive periodic solution for λ＞λ0.

Theorem 4.3 Let(C1)and()hold.Assume=0 for i=1,2,···,n.

(i)If= ∞ ,i=1,2,···,n,then for all λ ＞ 0 ,(1.2)admits a positive periodic solution.

(ii)If=0,i=1,2,···,n,then(1.2)admits two positive periodic solutions for λ ＞ 0 sufficiently large.

(iii)There exists a λ0 ＞ 0 such that(1.2)admits a positive periodic solution for λ＞λ0.

Remark 4.1 Note that Theorems 4.1-4.3 enrich and complement Theorem 1.1.And obviously,Lemma 2.6 is crucial to prove Theorems 4.2-4.3.

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