Fixed point theorems for monotone operators and applications to nonlinear elliptic problems

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R E S E A R C H

Open Access

Fixed point theorems for monotone

operators and applications to nonlinear

elliptic problems

Jun Wu

1

_{and Yicheng Liu}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics and}

Systems Science, College of Science, National University of Defense Technology, Changsha, 410073, P.R. China

Full list of author information is available at the end of the article

Abstract

In this paper, some fixed point theorems for monotone operators in partially ordered complete metric spaces are proved. Especially, a sufficient and necessary condition for the existence of a fixed point for a class of monotone operators is presented. The main results of this paper are generalizations of the recent results in the literature. Also, the main results can be applied to solve the nonlinear elliptic problems and the delayed hematopoiesis models.

MSC: 47H10; 54H25

Keywords: coupled ﬁxed point; partially ordered metric space; mixed monotone mapping; quadruple ﬁxed point

1 Introduction

In the last decades, the fixed point theorems for the contraction mappings have been im-proved and generalized in different directions. During the extensive applications to the nonlinear integral equations, there were many researchers to investigate the existence of a fixed point for contraction-type mappings in partially ordered metric spaces. In , Bhaskar and Lakshmikantham [] introduced the notion of coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings. Later, Lakshmikan-tham and Ciric presented a coincidence point theorem for a mapping withg-monotone property in []. Also, the concepts of tripled fixed point and quadruple fixed point were introduced by the authors in [] and [], respectively. Meanwhile, they proved the corre-sponding fixed point theorems. More details on the direction of the coupled fixed point theory and its applications can be found in the literature (see,e.g., [–]).

In this manuscript, we give a common method to deal with the existence of a coupled fixed point and the coincidence point for a class of mixed monotone mappings in a par-tially ordered complete metric space. Indeed, we establish some fixed point theorems for the monotone operators in the partially ordered complete metric space. Especially, we present the sufficient and necessary condition for the existence of a fixed point for a class of monotone operators. Our results improve and generalize the main results in the litera-ture [–, ].

In the rest of this section, we recall some basic deﬁnitions.

Let (X,≤) be a partially ordered set, a subsetE⊂Xis said to be a totally ordered subset if eitherx≤yory≤x holds for allx,y∈E. We say the elementsxandyare

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rable if either x≤yory≤xholds. It is said that a triple (X,≤,d) is a partially ordered complete metric space if (X,≤) is a partially ordered set and (X,d) is a complete metric space. Letdenote all the functionsφ: [, +∞)→[, +∞) which satisfy thatφ(r) <rand

lim_t_→_r₊φ(t) <rfor allr> . We should mention that Agarwalet al.[] considered the non-decreasing functionsφ: [, +∞)→[, +∞) satisfyinglim_n_→∞φn(r) =  for allr>  and established some ﬁxed point theorems.

Deﬁnition .(Bhaskar and Lakshmikantham []) Let (X,≤) be a partially ordered set and F:X_→_X_{. The mapping}_F_{is said to have the mixed monotone property if}_F_is

mono-tone non-decreasing in its ﬁrst argument and is monomono-tone non-increasing in its second argument, that is, for anyx,y∈X,

x,x∈X, x≤x ⇒ F(x,y)≤F(x,y) and

y,y∈X, y≤y ⇒ F(x,y)≤F(x,y).

Deﬁnition .(Bhaskar and Lakshmikantham []) An element (x,y)∈X_{is said to be a}

coupled ﬁxed point of the mappingF:X_→_X_if_F₍_x_,_y_{) =}_x_and_F₍_y_,_x_{) =}_y_.

2 Fixed points theorems for monotone operators

Theorem . Let(X,,ρ)be a partially ordered complete metric space and let G:X→X be a monotone non-decreasing operator with respect to the orderon X.Assume that

(i) there is aϕ∈such that

ρG(x˜),G(y˜)≤ϕρ(x˜,˜y) for eachx˜,y˜∈Xwithx˜ ˜y; ()

(ii) there exists anx˜∈Xsuch thatx˜G(x˜);

(iii) either(a)Gis a continuous operator,or(b)if a non-decreasing monotone sequence ˜

xninXtends tox¯,thenx˜n ¯xfor alln. Then the operator G has a ﬁxed point in X.

Proof Deﬁnite a sequence{˜xn}inXby

˜

xn=G(x˜n–) forn= , , . . . . ()

Considering the operatorF˜ is non-decreasing monotone for the orderandx˜G(x˜),

we have

˜

x ˜x ˜x · · · ˜xn · · ·.

If there existsnsuch thatxñ=xñ+, thenxñ=G(xñ) andxñis a fixed point ofG. Then the result of Theorem . trivially holds.

Suppose now thatxñ=xñ+for alln. Letan=ρ(xñ+,x˜n), noting that the sequence{˜xn} is a non-decreasing sequence inX_{, we conclude that}

an+ =ρ(x˜n+,x˜n+) =ρ

G(x˜n+),G(x˜n)

≤ϕρ(x˜n+,x˜n)

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Thus we obtain that

 <an+≤ϕ(an+) <an+≤ϕ(an) <an.

This implies that both sequences{an}and{ϕ(an)}are convergent. Setlimn→∞an=a. If a> , notinglimt→r+ϕ(t) <rfor allr> , we have

a= lim

n→∞an+≤nlim→∞ϕ(an) =rlim→a+ϕ(r) <a.

This is a contradiction. Thuslim_n_→∞ρ(x˜n+,x˜n) =limn→∞an= .

Now, we shall prove that{˜xn} is a Cauchy sequence inX. In fact, byϕ∈, we can choose a positive sequence{εm} withlimm→∞εm=  andεm∗ =sup≤t≤εmϕ(t) <εm. For

a ﬁxedm, there exists a large enough positive number N satisfyingaN ≤εm–εm∗. Let

˜

x∈:={˜x∈X_:_ρ₍_x_˜_,_x_˜

N)≤εm,x˜N ˜x}, then by the triangle inequality

ρG(x˜),x˜N

≤ρG(x˜),G(x˜N)

+ρG(x˜N),x˜N

≤ϕρ(x˜,x˜N)

+aN<ε∗m+aN

≤ε∗_m+εm–εm∗ =εm.

Also, xÑ G(xÑ)G(x˜). This means that the set is invariant for the operator G. Clearly, xÑ ∈. Thus xÑ+p∈ for all p∈Z+. So, the sequence{˜xn} is a Cauchy se-quence inX_{. Since (}_X_,_ρ_{) is a complete metric space, there exists a point}_x_¯_∈_X _such

thatlim_n_→∞x˜n=x¯.

Suppose thatGis a continuous operator. Then, by deﬁnition of{˜xn}, we have

¯

x= lim

n→∞x˜n=nlim→∞G(x˜n–) =G(x¯).

Let us assume that the assumption (b) holds, thenx˜n ¯xfor alln∈Z+. Thus from the assumption (i), we have

ρx¯,G(x¯)≤ρx¯,G(x˜n)+ρG(x˜n),G(x¯)

≤ρ(x¯,x˜n+) +ϕ

ρ(x˜n,x¯)→ asn→ ∞.

So,ρ(x¯,G(x¯)) = . The proof of Theorem . is complete. LetF:X_→_X_{be a mapping having the mixed monotone property on}_X_{and deﬁne the}

operatorG:X_→_X_by

G(x˜) =F(x,y),F(y,x) for allx˜= (x,y)∈X.

It is easy to see that the coupled ﬁxed points ofFis the ﬁxed points ofGinX_{. Also, for} ˜

t= (x,y),s˜= (u,v)∈X_{, we introduce a partial order}_in_X_{given by}

˜

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Thus, ifFhas the mixed monotone property onX, then the operatorGis non-decreasing monotone for the order. For˜t= (x,y),˜s= (u,v)∈X_{, let}_ρ₍_t˜_,_˜_s_{) :=}_d₍_x_,_u_{) +}_d₍_y_,_v_{), then}

(X,ρ) is a complete metric space provided (X,d) is a complete metric space. Then, as a consequence of Theorem ., we achieve the following corollary.

Corollary . Let(X,≤,d)be a partially ordered complete metric space and let F:X→X be a mapping having the mixed monotone property on X.Assume that

(i) there is aϕ∈such thatG:X_→_X_satisfying

ρG(x˜),G(y˜)≤ϕρ(x˜,˜y) for eachx˜,y˜∈Xwithx˜ ˜y;

(ii) there exists anx˜∈Xsuch thatx˜G(x˜); (iii) one of(a)and(b)holds:

(a) Gis a continuous operator;

(b) if a non-decreasing monotone sequencex˜ninXtends tox¯,thenx˜n ¯xfor alln. Then the operator G has a ﬁxed point in X_,_{that is}_,_{there exist x}_,_y_∈_{X such that}

x=F(x,y) and y=F(y,x).

LetD={˜x∈X_:_x_˜_and_G₍_x_˜_{) are comparable}_}_{, then we have the following theorem.}

Theorem . Let(X,≤,d)be a partially ordered complete metric space and let F:X_→_X

be a mapping having the mixed monotone property on X.Assume that(i)in Theorem. and one of following conditions holds:

(a) Gis a continuous operator;

(b) if a monotone sequencex˜ninXtends tox¯,thenx˜nandx¯are comparable for alln. Then the operator G has a ﬁxed point in Xif and only if D=φ.Furthermore,if D is a totally ordered nonempty subset,then the operator G has a unique ﬁxed point in X_.

Proof It is easy to see that all the ﬁxed points ofGfall in the setD. Thus if the operatorG has a ﬁxed point inX_{, then}_D₌_φ_.

We supposeD=φ. If the condition (a) holds andx˜∈D, then there are two cases:x˜

G(x˜) orG(x˜) ˜x. For the ﬁrst case, following Theorem ., we claim that the operatorG

has a ﬁxed point inX_{. For the other case:}_G₍_x_˜

) ˜x, noting the symmetry of the metric,

we see that the formula () holds fory˜ ˜x. Thus

ρG(x˜),G(y˜)≤ϕρ(x˜,˜y) for eachx˜,y˜∈Xsatisfyingx˜is comparable withy˜.

Constructing the same sequence{˜xn}inXbyx˜n=G(x˜n–), forn= , , . . . , we have

· · · ˜xn · · · ˜x ˜x ˜x.

For a mini-revise to the proof of Theorem . and resetting:={˜x∈X_:_ρ₍_x_˜_,_x_˜

N)≤εm,x˜

˜

xN}, we conclude that the sequence{˜xn}tends to a ﬁxed point ofG.

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withx˜nfor alln∈Z+. Then we have

ρx¯,G(x¯)≤ρx¯,G(x˜n)

+ρG(x˜n),G(x¯)

≤ρ(x¯,x˜n+) +ϕ

ρ(x˜n,x¯)→ asn→ ∞.

Thus the operatorGhas a ﬁxed pointx¯inX_.

Next, we suppose thatDis a totally ordered nonempty subset. It is sufficient to prove the uniqueness of a fixed point ofG. Letxãndy˜be two fixed points ofG, thenx˜is comparable withy˜,G(x˜) =xãndG(y˜) =y˜. Following the assumption (i), we have

ρ(x˜,y˜) =ρG(x˜),G(y˜)≤ϕρ(x˜,y˜)<ρ(x˜,y˜).

Thusρ(x˜,y˜) = , that is,x˜=y˜. The proof of Theorem . is complete. Following Theorem ., we have the next two corollaries.

Corollary .([], Theorem .) Let(X,≤)be a partially ordered set and suppose there is a metric d on X such that(X,d)is a complete metric space.Let F:X_→_{X be a continuous}

mapping having the mixed monotone property on X.Assume that there exists a k∈[, ) with

dF(x,y),F(u,v)≤k 

d(x,u) +d(y,v) for each x≤u and v≤y.

If there exist(x,y)∈X such that x≤F(x,y)and F(y,x)≤y,then there exist(x,y)∈

X such that

Proof Takingϕ(r) =krforr≥,t˜= (x,y),˜s= (u,v), if˜t ˜sthen

ρG(˜t),G(˜s)=dF(x,y),F(u,v)+dF(y,x),F(v,u)

≤kd(x,u) +d(y,v)=ϕρ(˜s,˜t).

Thus Corollary . is an immediate consequence of Theorem ..

Corollary .([], Theorem .) Let(X,≤)be a partially ordered set and suppose there is a metric d on X such that(X,d)is a complete metric space.Assume that X has the following property:

(i) if a non-decreasing sequencexn→x,thenxn≤xfor alln;

(ii) if a non-increasing sequenceyn→y,theny≤ynfor alln.

Let F:X_→_{X be a mapping having the mixed monotone property on X}_._{Assume that there}

exists a k∈[, )with

dF(x,y),F(u,v)≤k 

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If there exist(x,y)∈X such that x≤F(x,y)and F(y,x)≤y,then there exist(x,y)∈

X such that

Proof It follows from Theorem . immediately.

Let (X, · ) be a real Banach space and letKbe a cone. The relationx≤yholds if and only ify–x∈K. DenoteK+=K\ {θ}andKf ={x∈K+:λf ≤x≤μf for some positive real numbersλandμ} for a given f ∈K+_{. Let} _M₍_x_,_y_{) =}_inf_{_λ _:_x_≤_λ_y_} _and _d₍_x_,_y_{) =}

log(max{M(x,y),M(y,x)}) forx,y∈Kf. Thenddeﬁnes a metric on Kf which is known as the Thompson metric []. More details about the Thompson metric can be found in the references [–]

At this stage, we state our main results in the real Banach space.

Theorem . Let(X, · )be a real Banach space,let K⊂X a cone and f ∈K+_._Suppose

that Ai:Kf ×Kf →Kf are two mixed monotone maps satisfying Ai(tx,t–y)≥tiAi(x,y) and Ai(f,f)∈Kf for t∈(, ),i= ,p with <p< .Let A=A+Apand assume that there exists a point(x,y)∈Kf×Kf such that

x≤A(x,y)≤A(y,x)≤y.

Then A has a unique ﬁxed point in Kf,that is,there exists a unique point x∈Kf such that A(x,x) =x.

In order to prove this result, we need some technique lemmas.

Lemma .([], Lemma .) Under the assumptions of Theorem.,there existsδx,y∈ (p, )such that

Atx,t–y≥tδx,y_A₍_x_,_y₎ _{for all t}∈_{(, )}_{and x}_,_y∈_K

f,

where

δx,y=

x,y+p

x,y+ 

and x,y=M

A(x,y),Ap(x,y)

.

Lemma . Under the assumptions of Theorem.,then

dA(x,y),A(u,v)≤δu,vmax

d(x,u),d(y,v).

Proof Noting thate–d(x,y)_y_≤_x_≤_ed(x,y)_y_{for all}_x_,_y_∈_K

f, we have

A(x,y)≥Ae–d(x,u)u,ed(y,v)v

≥Ae–max{d(x,u),d(y,v)}u,emax{d(x,u),d(y,v)}v

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On the other hand, sinceA(tx,t–_y₎_≤_tδx,y_A₍_x_,_y_{) for}_t_{> , then we have}

A(x,y)≤Aed(x,u)u,e–d(y,v)v≤Aemax{d(x,u),d(y,v)}u,e–max{d(x,u),d(y,v)}v

≤eδu,vmax{d(x,u),d(y,v)}_A₍_u_,_v_).

Thusd(A(x,y),A(u,v))≤δu,vmax{d(x,u),d(y,v)}.

Lemma . Under the assumptions of Theorem.,the successive sequences{xn}and{yn} are Cauchy sequences,where

xn+=A(xn,yn), yn+=A(yn,xn), n= , , . . . .

Proof Sincex≤A(x,y)≤A(y,x)≤y, it follows by an induction argument that

x≤x≤ · · · ≤xn≤ · · · ≤yn≤ · · · ≤y≤y.

Noting that

A(yn,xn)≤A(y,x)≤M

A(y,x),Ap(x,y)

Ap(x,y)

≤MA(y,x),Ap(x,y)

Ap(yn,xn),

we have, for alln,

yn,xn=M

A(yn,xn),Ap(yn,xn)

≤MA(y,x),Ap(x,y)

:=.

Thus

δyn,xn=

yn,xn+p

yn,xn+ 

≤+p

+ 

:=δ.

Next, we claim that

xn≥e–d(x,y)δ

n

_y

n, n= , , , . . . . ()

In fact, it holds forn= . For arbitraryn, by induction argument, we have

xn+=A(xn,yn)≥A

e–d(x,y)δn_y_n,_ed(x,y)δn_x_n

≥e–d(x,y)δnδyn,xn_A₍_y_n,_x_n)≥_e–d(x,y)δn+_y_n_+_.

Thus () holds for alln. On the other hand, since

A(xn,yn)≤A(yn,xn)≤yn,xnAp(yn,xn)

≤Ap(yn,xn)≤Ap

ed(x,y)δn_x_n,_e–d(x,y)δn_y n

≤epd(x,y)δ n

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we obtain that, for alln,

xn,yn=M

A(xn,yn),Ap(xn,yn)

≤epd(x,y):=.

Thus

δxn,yn=

xn,yn+p

xn,yn+ 

≤+p

+ 

:=δ.

Then following Lemma ., we have

d(xn+,xn) =d

A(xn,yn),A(xn–,yn–)

≤δxn–,yn–max

d(xn,xn–),d(yn,yn–)

≤δmax

.

Similarly, we have

d(yn+,yn) =d

A(yn,xn),A(yn–,xn–)

≤δyn–,xn–max

d(yn,yn–),d(xn,xn–)

≤δmax

.

Letδ=max{δ,δ}, thenδ<  and

maxd(xn+,xn),d(yn+,yn)

≤δmaxd(xn,xn–),d(yn,yn–)

.

Thus, for alln,

maxd(xn+,xn),d(yn+,yn)

≤δnmaxd(x,x),d(y,y)

.

Furthermore, for anyk∈Z+, we have

maxd(xn+k,xn),d(yn+k,yn)

≤ δn

 –δmax

d(x,x),d(y,y)

.

This shows that both successive sequences{xn}and{yn}are Cauchy sequences.

Proof of Theorem . By Lemma ., there are a,b∈K such that lim_n_→∞xn=aand

limn→∞yn=b. Obviously,xn≤aandb≤ynfor alln∈Z+. Thusa,b∈Kf. Next, noting that

dxn+,A(a,b)

=dA(xn,yn),A(a,b)≤δa,bmax

d(xn,a),d(yn,b)

and

dyn+,A(b,a)

=dA(yn,xn),A(b,a)≤δb,amax

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and lettingngo to inﬁnity, we get thata=A(a,b) andb=A(b,a). It follows from

d(a,b) =dA(a,b),A(b,a)≤δb,amax

d(b,a),d(a,b)=δb,ad(a,b)

thatd(a,b) =  anda=b. The uniqueness is obvious. ThusAhas a unique ﬁxed point in Kf, that is, there exists a unique pointa∈Kf such thatA(a,a) =a. The proof is complete.

Remark . Our result in Theorem . improved the corresponding result in [] (The-orem .) and removed some restriction conditions: the successive sequences have con-vergent subsequences.

3 Application to the nonlinear elliptic problems

Letbe the open unit ball inRn_,_n_≥_{, with center at the origin. We consider positive} solutions of the Dirichlet problem

⎧ ⎨ ⎩

∇(a(|x|)∇(u)) +b(|x|)(up+_+c_uq) =  in,

u=  on∂. ()

Whena(|x|) =b(|x|) = ,n≥ andc= , it is well known that () has no positive solution ifp>n_n_–+, and that the positive solution of () is unique ifp> , see [] and []. Also, in this case when  <p< , () has a unique positive radial solution [].

In this section, we assume thatp∈(, ],q∈(, ) andc>  are constants,a(r) andb(r) are positive and continuous for ≤r≤. Our result is as follows.

Theorem . Problem()has a unique positive radial solution if bmax

namin< ,where amin:=

min{a(r) :r∈[, ]}and bmax:=max{b(r) :r∈[, ]}.

To this end, we should establish a technique lemma.

Lemma . The function u is a positive radial solution of problem()if and only if u is a positive solution of the integral equation

u(r) =





G(r,t)b(t)

up(t) + c  +uq₍_t₎

dt,

where

G(r,t) =

⎧ ⎨ ⎩



t



a(s)(

t s)

n–_ds_, __≤_r_<_t_≤_,



r



a(s)(

t

s)n–ds, ≤t≤r≤.

Proof Assuming solutions to be functions ofr, the radial distance from the origin, () reduces to

⎧ ⎨ ⎩

(a(r)u(r))+n–_r a(r)u(r) +b(r)[up₍_r_{) +} c

+uq(r)] = ,  <r< ,

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whereu(r) =_drdu(r). Then the Green function for problem () is

G(r,t) =

⎧ ⎨ ⎩  t 

a(s)(

t s)

n–_ds_, __≤_r_<_t_≤_,



r



a(s)(

t

s)n–ds, ≤t≤r≤,

which is positive on [, )×[, ). Thus the functionuis a positive radial solution of prob-lem () if and only ifuis a positive solution of the integral equation

u(r) =





G(r,t)b(t)

up(t) + c  +uq₍_t₎

dt.

Proof of Theorem. LetKdenote the cone of nonnegative functions inC[, ], the rela-tionx≤yholds if and only ifx(t)≤y(t) for allt∈[, ],K+₌_K_{\ {}__}_and_f₍_r_{) =  –}_r

for r∈ [, ], then f ∈ K+ and ≤f(r)≤fmax = . Denote Kf ={x∈ K+: λf ≤x≤

μf for some positive numbersλ,μ∈R}.

Now we introduce the mapsA,Aq:Kf×Kf→Kf deﬁned by

A(x,y)(r) =





G(r,t)b(t)xp(t)dt,

Aq(x,y)(r) =





G(r,t)b(t) c  +yq₍_t₎dt.

Forg,h∈Kf, then there existλg,λh,μg,μhsuch thatμgf ≤g≤λgf andμhf ≤h≤λhf. By direct computation, we have

bminμpg n(n+ )amax

f(r)≤A(g,h)(r)≤





G(r,t)b(t)λp_gdt≤bmaxλ p g namin

f(r),

cbmin

namax( +hqmax)

f(r)≤Aq(g,h)(r)≤





G(r,t)b(t)c dt≤

cbmax

namin

f(r).

Thus the mapAiis well deﬁned andAi(f,f)∈Kf fori= ,q. Also,A(f,f) +Aq(f,f)∈Kf. Obviously,Aiis a mixed monotone map inKfandA(tx,t–y)≥tA(x,y) andAq(tx,t–y)≥ tq_A_q(_x_,_y_).

Since bmax

namin< , we choose a positive numberk(large enough) satisfying

bmax

namin

k+ c 

≤k and bmin

namax

c  +k ≤.

LetA=A+Aq,x(r) =__nabmin_max_+c_k( –r),y(r) =k( –r), thenx,y∈Kf and

A(x,y)(r) =





G(r,t)b(t)

xp_(t) + c  +yq_(t)

dt

≥ cbmin

 +k





G(r,t)dt≥ cbmin namax( +k)

 –r

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and

A(y,x)(r) =





G(r,t)b(t)

yp_(t) + c  +xq_(t)

dt

≤





G(r,t)bmax

c +kf(t)

dt

≤

c +k

bmax

namin

f(r)

≤kf(r) =y(r).

Thus

x≤A(x,y)≤A(y,x)≤y.

Applying Theorem . to the operatorA, we conclude that there is a unique pointuin Kf such thatu(r) =A(u,u)(r). On the other hand, for allg,h∈K+, we have

f(r)

gp_max+ c 

bmax

namin ≥





G(r,t)b(t)

gp(t) + c  +hq₍_t₎

dt

≥ cbmin

namax( +hqmax)

f(r).

This means thatA(g,h)∈Kf. Thus problem () has a unique positive radial solution.

4 Application to the delayed hematopoiesis models

In this section, we consider the positive periodic solution of the following hematopoiesis model with delays:

x(t) = –a(t)x(t) + n

i=

bi(t)

 +xq₍_t_–_τ_i(_t₎₎, ()

wherea,bi,τi∈C(R,R) are positiveT-periodic functions and ≤τi(t)≤tfor allt∈[,T], qis a nonnegative constant (i= , , . . . ,n). In the case whenq≥, Wu [] proved that () had a unique positiveT-periodic solution.

Here we assume thatq∈(, ) and our result is as follows.

Theorem . Problem()has a unique positive T -periodic solution.

Proof LetKdenote the cone of nonnegativeT-periodic functions inC(R,R), the relation x≤yholds if and only ifx(t)≤y(t) for allt∈[,T],K+₌_K_{\ {}__}_and_f₍_r₎_≡_{ for}_r_∈_[,_T_].

DenoteKf={x∈K+:λf ≤x≤μf for some positive numbersλ,μ∈R}. It is easy to show that the functionxis a positive T-periodic solution of problem () if and only ifxis a positive solution of the integral equation

x(r) = e βT

eβT_{– }

r

r–T e–β(r–s)

β–a(s)x(s) + n

i=

bi(s)  +xq₍_s_–_τ

i(s))

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whereβ>amaxis a constant. Deﬁne the mapsA,Aq:Kf ×Kf→Kf by

A(x,y)(t) =

eβT eβT_{– }

t

t–T

e–β(t–s)_β_–_a₍_s₎_x₍_s₎_ds_,

Aq(x,y)(t) = e βT

eβT_{– }

t

t–T e–β(t–s)

n

i=

bi(s)  +yq₍_s_–_τ

i(s)) ds.

For arbitraryx,y∈Kf, there are positive numbersλx,λy,μx,μysuch thatλx≤x(t)≤μx andλy≤y(t)≤μyfor allt∈[,T]. Furthermore, we deduce that

A(x,y)(t) =

eβT eβT_{– }

t

t–T

e–β(t–s)β–a(s)x(s)ds

≥ eβT

eβT_{– }

t

t–T

e–β(t–s)ds(β–amax)λx

=

 –amax

β

λx

and

A(x,y)(t)≤

 –amin

β

μx.

Similarly, we have



β

n

i=

bimin

 +μqy

≤Aq(x,y)(t)≤ 

β

n

i=

bimax

 +λqy

for allt∈[,T].

This means A(x,y) ∈ Kf and Aq(x,y) ∈ Kf. Thus A and Aq are well deﬁned and A(f,f),Aq(f,f)∈Kf. Also,A(f,f) +Aq(f,f)∈Kf. Obviously,AandAqare mixed mono-tone maps inKf andA(tx,t–y)≥tA(x,y) andAq(tx,t–y)≥tqAq(x,y).

Sinceβ>amax, we can choose a constantk>  satisfying

 –amin

β

k+ 

β

n

i=

bimax≤k and



β

n

i=

bimin

 +kq < .

LetA=A+Aq,x(t)≡ β

n i=

bimin

+kq andy(t)≡kfor allt∈[,T], then we have

A(x,y)(t) =

eβT eβT_{– }

t

t–T e–β(t–s)

β–a(s)x(s) +

n

i=

bi(s)  +yq_(s–τi(s))

ds

≥ eβT

eβT_{– }

t

t–T e–β(t–s)

n

i=

bimin

 +kqds

= 

β

n

i=

bimin

(13)

and

A(y,x)(t) =

eβT eβT_{– }

t

t–T e–β(t–s)

β–a(s)y(s) +

n

i=

bi(s)  +xq_(s–τi(s))

ds

≤ eβT

eβT_{– }

t

t–T e–β(t–s)

(β–amin)k+

n

i=

bimax

ds

= 

β

(β–amin)k+

n

i=

bimax

≤k=y(t).

Thus

x≤A(x,y)≤A(y,x)≤y.

Applying Theorem . to the operatorA, we conclude that there is a unique pointxin Kf such thatx(t) =A(x,x)(t). On the other hand, for allg,h∈K+, we have

f(t)

n i=bimax

β +

 –amin

β

gmax

≥A(g,h)(t)≥ 

β

n

i=

bimin

 +hqmax

f(t).

This means thatA(g,h)∈Kf. Thus problem () has a unique positiveT-periodic

solu-tion.

Remark . Using similar ideas, it is possible to extend our results to investigate the ex-istence and uniqueness of nonlinear singular boundary value problems and fractional dif-ferential equation boundary value problems, which are mentioned extensively in the liter-ature [, , ].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and read and approved the ﬁnal manuscript.

Author details

1_{College of Mathematics and Computer Science, Changsha University of Science Technology, Changsha, 410114,}

P.R. China.2_{Department of Mathematics and Systems Science, College of Science, National University of Defense}

Technology, Changsha, 410073, P.R. China.

Acknowledgements

The authors are grateful to the reviewers for their valuable comments and suggestions. This work was partly supported by the National Natural Science Foundation of China (11201481) and Hunan Provincial Education Department Foundation (12B007).

Received: 12 February 2013 Accepted: 8 May 2013 Published: 27 May 2013 References

1. Bhaskar, TG, Lakshmikantham, V: Fixed point theory in partially ordered metric spaces and applications. Nonlinear Anal.65, 1379-1393 (2006)

2. Lakshmikantham, V, Ciric, L: Couple ﬁxed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.70, 4341-4349 (2009)

3. Berinde, V, Borcut, M: Tripled ﬁxed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal.74(15), 4889-4897 (2011)

(14)

5. Abbas, M, Khan, MA, Radenovic, S: Common coupled ﬁxed point theorems in cone metric space forw-compatible mappings. Appl. Math. Comput.217, 195-203 (2010)

6. Amini-Harandi, A, Emami, H: A ﬁxed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary diﬀerential equations. Nonlinear Anal.72, 2238-2242 (2010)

7. Chen, YZ: Existence theorems of coupled ﬁxed points. J. Math. Anal. Appl.154, 142-150 (1991)

8. Choudhury, BS, Kundu, A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal.73, 2524-2531 (2010)

9. Haghi, RH, Rezapour, S, Shahzad, N: Some ﬁxed point generalizations are not real generalizations. Nonlinear Anal.74, 1799-1803 (2011)

10. Huang, C-Y: Fixed point theorems for a class of positive mixed monotone operators. Math. Nachr.285, 659-669 (2012) 11. Huang, MJ, Huang, CY, Tsai, TM: Hilbert’s projective metric and nonlinear elliptic problems. Nonlinear Anal.71,

2576-2584 (2009)

12. Figueroa, R, López Pouso, R: Coupled ﬁxed points of multivalued operators and ﬁrst-order ODEs with state-dependent deviating arguments. Nonlinear Anal.74, 6876-6889 (2011)

13. Karapinar, E, Berinde, V: Quadruple ﬁxed point theorems for nonlinear contractions in partially ordered metric spaces. Banach J. Math. Anal.6, 74-79 (2012)

14. Karapinar, E: Couple ﬁxed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl.59, 3656-3668 (2010)

15. Thompson, AC: On certain contraction mappings in a partially ordered vector space. Proc. Am. Math. Soc.14, 438-443 (1963)

16. Samet, B, Vetro, C, Vetro, P: Fixed point theorems forϕ-contractive type mappings. Nonlinear Anal.75, 2154-2165 (2012)

17. Shatanawi, W: Partially ordered cone metric spaces and coupled ﬁxed point results. Comput. Math. Appl.60, 2508-2515 (2010)

18. Shatanawi, W, Samet, B: On (ψ,φ)-weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl.62, 3204-3214 (2011)

19. Shatanawi, W: Onw-compatible mappings and common coupled coincidence point in cone metric spaces. Appl. Math. Lett.25, 925-931 (2012)

20. Xiao, JZ, Zhu, XH, Cao, YF: Common coupled ﬁxed point results for probabilisticϕ-contractions in Menger spaces. Nonlinear Anal.74, 4589-4600 (2011)

21. Liu, YC, Li, ZX: Krasnoselskii type ﬁxed point theorems and applications. Proc. Am. Math. Soc.136, 1213-1220 (2008) 22. Gidas, B, Ni, W-M, Nirenberg, L: Symmetry and related properties via the maximum principle. Commun. Math. Phys.

68, 209-243 (1979)

23. Ni, W-M, Nussbaum, RD: Uniqueness and nonuniqueness for positive radial solutions ofu+f(u,r) = 0. Commun. Pure Appl. Math.38, 67-108 (1985)

24. Bushell, PJ: The Cayley-Hilbert metric and positive operators. Linear Algebra Appl.84, 271-280 (1986)

25. Zhai, C, Hao, M: Fixed point theorems for mixed monotone operators with perturbation and applications to fractional diﬀerential equation boundary value problems. Nonlinear Anal.75, 2542-2551 (2012)

26. Wu, J, Liu, Y: Fixed point theorems and uniqueness of the periodic solution for the hematopoiesis models. Abstr. Appl. Anal.2012, Article ID 387193 (2012). doi:10.1155/2012/387193

27. Yang, XT: Existence and global attractivity of unique positive almost periodic solution for a model of hematopoiesis. Appl. Math. J. Chin. Univ. Ser. A25, 25-34 (2010)

28. Agarwal, RP, El-Gebelly, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.87, 109-116 (2008)

29. Guo, DJ, Lakshmikantham, V: Coupled ﬁxed points of nonlinear operators with applications. Nonlinear Anal.11, 623-632 (1987)

30. Guo, DJ: Fixed points of mixed monotone operators with applications. Appl. Anal.31, 215-224 (1988)

31. Guo, DJ: Existence and uniqueness of positive ﬁxed points for mixed monotone operators and applications. Appl. Anal.46, 91-100 (1992)

32. Guo, DJ, Cho, YJ, Zhu, J: Partial Ordering Methods in Nonlinear Problems. Nova Science Publishers, Hauppauge (2004)

doi:10.1186/1687-1812-2013-134