Uniqueness of positive solutions for several classes of sum operator equations and applications

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R E S E A R C H

Open Access

Uniqueness of positive solutions for several

classes of sum operator equations and

applications

Chen Yang

1

_{, Chengbo Zhai}

2*

_{and Mengru Hao}

2

*_{Correspondence:}

[email protected]; [email protected]

2_{School of Mathematical Sciences,}

Shanxi University, Taiyuan, Shanxi 030006, P.R. China

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

Abstract

In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation

u(t) =

λ

_abG(t,s)f(s,u(s))ds, wherefandGare both nonnegative,

λ

> 0 is a parameter; the other is the elliptic boundary value problem for the Lane-Emden-Fowler equation –

u=

λ

f(x,u),u(x) > 0 in

,u(x) = 0 on

∂

, where

is a bounded domain with smooth boundary inRN₍_N_≥_1),

_λ

_{> 0 and}_f₍_x_,_u_{) is allowed to be singular on}

_∂

_{. The}

new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.

MSC: 47H10; 47H07; 45G15; 35J60; 35J65

Keywords: positive solution; operator equation; normal cone; integral equation; Lane-Emden-Fowler equation

1 Introduction and preliminaries

With the development of nonlinear sciences, nonlinear functional analysis has been an active area of research over the past several decades. As an important branch of nonlinear functional analysis, nonlinear operator theory has attracted much attention and has been widely studied, especially nonlinear operators which arise in the connection with non-linear diﬀerential and integral equations have been extensively studied (see for instance [–]). It is well known that the existence and uniqueness of positive solutions to nonlin-ear operator equations is very important in theory and applications. Many authors have studied this problem; for a small sample of such work, we refer the reader to [, , –]. The operator equation considered in this papers is always of the following form:

Ax=x or A(x,x) =x.

In [], Zhao considered the existence of solutions for the sum operator equation

Ax+Bx=x, (.)

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whereAis increasinge-concave,Bis increasinge-convex andA+Bis a strict set contrac-tion. Motivated by the works [, ], Sanget al.considered the operator equation (.), whereAisϕ-concave,Bisϕ-convex andA+Bis also a strict set contraction. However,

we can see that the conditions of the main results in [, ] are strong and of utmost convenience.

Recently, we considered successively the operator equation (.) and the following op-erator equation:

A(x,x) +Bx=x, (.)

the operators A, B in (.) are increasing, α-concave and sub-homogeneous, respec-tively; the operatorsA,Bin (.) are mixed monotone and increasingα-concave (or sub-homogeneous), respectively. In [], by using the properties of cones and a ﬁxed point theo-rem for increasing generalα-concave operators, we established the existence and unique-ness of positive solutions for the operator equation (.), and we utilized the main results to present the existence and uniqueness of positive solutions for the following two problems; one is a fourth-order two-point boundary value problem for elastic beam equations,

⎧ ⎪ ⎨ ⎪ ⎩

u(t) =f(t,u(t)), ≤t≤,

u() =u() = ,

u() = , u() =g(u()),

wheref ∈C([, ]×R) andg∈C(R) are real functions; and the other is an elliptic value problem for Lane-Emden-Fowler equations

⎧ ⎪ ⎨ ⎪ ⎩

–u=f(x,u) +g(x,u), x∈,

u(x) > , x∈,

u(x) = , x∈∂,

wheref(x,u),g(x,u) are allowed to be singular on∂. In [], by using the properties of cones and a ﬁxed point theorem for mixed monotone operators, we established the exis-tence and uniqueness of positive solutions for the operator equation (.), and we utilized the results obtained to study the existence and uniqueness of positive solutions for a non-linear fractional diﬀerential equation boundary value problem,

Dα

+u(t) =f(t,u(t),u(t)) +g(t,u(t)),  <t< ,

u() =u() =u() =u() = ,

whereDα

+is the Riemann-Liouville fractional derivative of orderα> . These results are

useful and interesting. For completeness, in this paper we will further consider the follow-ing several classes of sum operators:

(i) the sum of increasing operators and decreasing operators; (ii) the sum of increasing operators and mixed monotone operators; (iii) the sum of decreasing operators and mixed monotone operators;

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Motivated by our works [, , ], we will study the above cases (i)-(iv). So this article is a continuation of our papers [, , ], and we will present some interesting results on the existence and uniqueness of positive solutions for the above several classes of sum operator equations. To demonstrate the applicability of our abstract results, we give, in the last section of the paper, some applications to nonlinear integral equations and elliptic boundary value problems for the Lane-Emden-Fowler equations.

In the following two subsections, we state some deﬁnitions, notations, and known re-sults. For convenience of the readers, we refer to [–, –, –] for details.

1.1 Some basic deﬁnitions and notations

Suppose thatEis a real Banach space which is partially ordered by a coneP⊂E,i.e.,x≤y

if and only ify–x∈P. Ifx≤yandx= y, then we denotex<yory>x. Byθwe denote the zero element ofE. Recall that a non-empty closed convex setP⊂Eis a cone if it satisﬁes (i)x∈P,λ≥⇒λx∈P; (ii)x∈P, –x∈P⇒x=θ.

PuttingP˚ ={x∈P|xis an interior point ofP}, a coneP is said to be solid ifP˚ is non-empty. Moreover,Pis called normal if there exists a constantN>  such that, for allx,y∈ E,θ≤x≤yimplies x ≤N y ; in this caseNis called the normality constant ofP. If

x,x∈E, the set [x,x] ={x∈E|x≤x≤x}is called the order interval betweenxandx.

We say that an operator A:E→E is increasing (decreasing) if x≤yimpliesAx≤Ay

(Ax≥Ay).

For all x,y∈E, the notationx∼ymeans that there exist λ>  andμ>  such that

λx≤y≤μx. Clearly,∼is an equivalence relation. Givenh>θ (i.e.,h≥θ andh=θ), we denote byPhthe setPh={x∈E|x∼h}. It is easy to see thatPh⊂P.

Deﬁnition . LetD=PorD=P˚ andαbe a real number with ≤α< . An operator

A:P→Pis said to beα-concave if it satisﬁes

A(tx)≥tαAx, ∀t∈(, ),x∈D.

Notice that the deﬁnition of anα-concave operator mentioned above is diﬀerent from that in [], because we need not require the cone to be solid in general.

Deﬁnition . An operatorA:P→Pis said to be sub-homogeneous if it satisﬁes

A(tx)≥tAx, ∀t∈(, ),x∈P.

Deﬁnition .(See [, , ]) A:P×P→Pis said to be a mixed monotone operator if

A(x,y) is increasing inxand decreasing iny,i.e.,ui,vi(i= , )∈P,u≤u,v≥vimply

A(u,v)≤A(u,v). An elementx∈Pis called a ﬁxed point ofAifA(x,x) =x.

1.2 Some ﬁxed point theorems and properties

In this subsection, we assume thatEis a real Banach space with a partial order introduced by a conePofE. Takeh∈E,h>θ,Phis given as in Section ..

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Theorem . (See Theorem . in []) Let P be a normal cone in E, A:P→P be an increasingα-concave operator and B:P→P be an increasing sub-homogeneous operator.

Assume that

(i) there ish>θ such thatAh∈PhandBh∈Ph;

(ii) there exists a constantδ> such thatAx≥δBx,∀x∈P.

Then the operator equation(.)has a unique solution x∗ in Ph.Moreover,constructing successively the sequence yn=Ayn–+Byn–,n= , , . . .for any initial value y∈Ph,we

have yn→x∗as n→ ∞.

In the paper [], we present the following ﬁxed point theorem for a class of general mixed monotone operators and established some pleasant properties of nonlinear eigen-value problems for mixed monotone operators.

Theorem . (See Lemma . and Theorem . in []) Let P be a normal cone in E.

Assume that A:P×P→P is a mixed monotone operator and satisﬁes: (i) there existsh∈Pwithh=θsuch thatA(h,h)∈Ph;

(ii) for anyu,v∈Pandt∈(, ),there existsϕ(t)∈(t, ]such that

A(tu,t–_v₎_≥_ϕ₍_t₎_A₍_u_,_v₎_.

Then:

() T:Ph×Ph→Ph;

() there existu,v∈Phandr∈(, )such thatrv≤u<v,

u≤A(u,v)≤A(v,u)≤v;

() the operator equation(.)has a unique solutionx∗inPh;

() for any initial valuesx,y∈Ph,constructing successively the sequences

xn=A(xn–,yn–), yn=A(yn–,xn–), n= , , . . . ,

we havexn→x∗andyn→x∗asn→ ∞.

Theorem .(See Theorem . in []) Assume that the operator A satisﬁes the conditions of Theorem..Let xλ(λ> )denote the unique solution of nonlinear eigenvalue equation

A(x,x) =λx in Ph.Then we have the following conclusions:

(R) Ifϕ(t) >t



 _for_t∈_{(, )}_,_then_x_λ_{is strictly decreasing in}_λ_,_{that is,}_{ <}_λ__<_λ__implies xλ>xλ;

(R) If there existsβ∈(, )such thatϕ(t)≥tβfort∈(, ),thenxλis continuous inλ,that is,λ→λ(λ> )implies xλ–xλ →;

(R) If there exists β ∈(,_) such that ϕ(t)≥tβ for t∈(, ), then limλ→∞ xλ = ,

limλ→+ xλ =∞.

Based on Theorem ., in [] we considered the operator equation (.) and established the following conclusions.

Theorem .(See Theorem . in []) Let P be a normal cone in E,α∈(, ).A:P×P→ P is a mixed monotone operator and satisﬁes

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B:P→P is an increasing sub-homogeneous operator.Assume that

(i) there ish∈Phsuch thatA(h,h)∈PhandBh∈Ph;

(ii) there exists a constantδ> such thatA(x,y)≥δBx,∀x,y∈P.

Then:

() A:Ph×Ph→Ph,B:Ph→Ph;

() there existu,v∈Phandr∈(, )such that

rv≤u<v, u≤A(u,v) +Bu≤A(v,u) +Bv≤v;

xn=A(xn–,yn–) +Bxn–, yn=A(yn–,xn–) +Byn–, n= , , . . . ,

Theorem .(See Theorem . in []) Let P be a normal cone in E,α∈(, ).A:P×P→ P is a mixed monotone operator and satisﬁes

Atx,t–y ≥tA(x,y), t∈(, ),x,y∈P.

B:P→P is an increasingα-concave operator.Assume that

(i) there ish∈Phsuch thatA(h,h)∈PhandBh∈Ph;

(ii) there exists a constantδ> such thatA(x,y)≤δBx,∀x,y∈P.

Then:

() A:Ph×Ph→Ph,B:Ph→Ph;

() there existu,v∈Phandr∈(, )such that

rv≤u<v, u≤A(u,v) +Bu≤A(v,u) +Bv≤v;

xn=A(xn–,yn–) +Bxn–, yn=A(yn–,xn–) +Byn–, n= , , . . . ,

2 Main results

In this section we consider the existence and uniqueness of positive solutions for several classes of sum operator equations. We always assume thatEis a real Banach space with a partial order induced by a conePofE. Takeh∈E,h>θandPhas given in the Introduction.

2.1 The sum of increasing operators and decreasing operators

Now we ﬁrst consider the following sum operator equations:

Ax+Bx=x, (.)

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Theorem . Let P be a normal cone,A:P→P be an increasing operator and B:P→P be a decreasing operator.Assume that:

(H) for anyx∈Pandt∈(, ),there existϕi(t)∈(t, )(i= , )such that

A(tx)≥ϕ(t)Ax, B(tx)≤



ϕ(t)

Bx; (.)

(H) there existsh∈Phsuch thatAh+Bh∈Ph.

Then:

(i) there existu,v∈Phandr∈(, )such that

rv≤u<v, u≤Au+Bv≤Av+Bu≤v;

(ii) the operator equation(.)has a unique solutionx∗inPh;

(iii) for any initial valuesx,y∈Ph,constructing successively the sequences

xn=Axn–+Byn–, yn=Ayn–+Bxn–, n= , , . . . ,

we havexn→x∗,yn→x∗asn→ ∞.

Proof Firstly, from (.), we have

A



tx

≤ 

ϕ(t)

Ax, B



tx

≥ϕ(t)Bx, ∀x∈P,t∈(, ). (.)

SinceAh+Bh∈Ph, there exist constantsλ,λ>  such that

λh≤Ah+Bh≤λh.

Also fromh∈Ph, there exists a constantt∈(, ) such that

th≤h≤



t

h.

Letϕ(t) =min{ϕ(t),ϕ(t)},t∈(, ). Thenϕ(t)∈(t, ). From (.) and (.), we obtain

Ah+Bh≥A(th) +B



t

h

≥ϕ(t)Ah+ϕ(t)Bh

≥ϕ(t)(Ah+Bh)≥λϕ(t)h,

Ah+Bh≤A



t

h

+B(th)≤



ϕ(t)

Ah+



ϕ(t)

Bh

≤ 

ϕ(t)

(Ah+Bh)≤

λ

ϕ(t)

h.

Note thatλϕ(t),ϕλ(t) > , we can getAh+Bh∈Ph.

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we have

Ttx,t–y =A(tx) +Bt–y ≥ϕ(t)Ax+ϕ(t)By≥ϕ(t)(Ax+By) =ϕ(t)T(x,y).

Hence, the operatorTsatisﬁes the condition (ii) in Theorem .. An application of The-orem . implies that: there are u,v∈Ph andr∈(, ) such thatrv≤u<v,u≤

T(u,v)≤T(v,u)≤v; operator equationT(x) =xhas a unique positivex∗∈Ph; for

any initial valuesx,y∈Ph, constructing successively the sequences

xn=T(xn–,yn–), yn=T(yn–,xn–), n= , , . . . ,

we have xn–x∗ → and yn–x∗ → asn→ ∞. That is: (i) there existu,v∈Phandr∈(, )such that

(ii) the operator equation (.) has a unique solutionx∗inPh;

(iii) for any initial valuesx,y∈Ph, constructing successively the sequences

we havexn→x∗,yn→x∗asn→ ∞.

Note thath∈Ph, and we can easily obtain the following conclusions.

Corollary . Let P be a normal cone,A:Ph→Phbe an increasing operator and B:Ph→ Phbe a decreasing operator.Assume that:

(H) for anyx∈Phandt∈(, ),there existϕi(t)∈(t, )(i= , )such that

A(tx)≥ϕ(t)Ax, B(tx)≤



ϕ(t)

Bx.

Then:

Corollary . Let α,α ∈(, ). Let P be a normal cone, A:P→P be an increasing

α-concave operator and B:P→P be a decreasingα-convex operator.Assume that(H)

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Proof Letϕ(t) =tα,ϕ(t) =tα,t∈(, ). Thenϕ(t),ϕ(t)∈(t, ) fort∈(, ) and

A(tx)≥tα_Ax₌_ϕ

(t)Ax, B(tx)≤t–αBx=



ϕ(t)

Bx, x∈P.

Hence, the conclusions follow from Theorem ..

Corollary . Letα ∈(, )and P be a normal cone. Let A:P→P be an increasing

α-concave operator and A:P→P be an increasing sub-homogeneous operator,B:P→P

be a decreasing operator which satisﬁes(.).Assume that:

(H) there existsh∈Phsuch thatAh+Ah∈Ph; (H) there existsδ> such thatAx≥δAx,x∈P; (H) there existsh∈Phsuch thatBh∈Ph.

Then:

rv≤u<v, u≤Au+Au+Bv≤Av+Av+Bu≤v;

(ii) the following operator equation:

Ax+Ax+Bx=x, (.)

has a unique solutionx∗inPh;

xn=Axn–+Axn–+Byn–,

yn=Ayn–+Ayn–+Bxn–, n= , , . . . ,

Proof Deﬁne an operatorA=A+AbyAx=Ax+Ax. ThenA:P→Pis an increasing

operator andAh∈Ph. Sinceh,h∈Ph, there existt,t∈(, ) such that

th≤h≤



t

h, th≤h≤



t

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Thenh≤_th≤_tt h,h≥th≥tth, and thus

Bh≥B



tt

h

≥ϕ(tt)Bh, Bh≤B(tth)≤



ϕ(tt)

Bh.

Note thatϕ(tt),_ϕ_₍_tt ₎>  andBh∈Ph, we can getBh∈Ph. Hence,Ah+Bh∈Ph.

From the proof of Theorem ., there existsβ(t)∈(α, ) with respect tot, such that

A(tx)≥tβ(t)_Ax_, _∀_t_∈_{(, ),}_x_∈_P_.

Letϕ(t) =tβ(t),t∈(, ). Thenϕ(t)∈(t, ) andA(tx)≥ϕ(t)Ax,x∈P.

Therefore, operatorsA,Bsatisfy all the conditions of Theorem .. So we easily obtain the following conclusions:

(i) there existu,v∈Ph, andr∈(, )such that

rv≤u<v, u≤Au+Au+Bv≤Av+Av+Bu≤v;

(ii) the operator equation (.) has a unique solutionx∗inPh;

xn=Axn–+Axn–+Byn–,

yn=Ayn–+Ayn–+Bxn–, n= , , . . . ,

Corollary . Assume that all the conditions of Theorem.hold.Let xλ(λ> )denote

the unique solution of operator equation(.).Then we have the following conclusions: (i) ifϕi(t) >t



 (_i_{= , })for_t∈_{(, )},then_x_λis strictly decreasing in_λ,that is,

 <λ<λimpliesxλ>xλ;

(ii) if there existsβ∈(, )such thatϕi(t)≥tβ₍_i_{= , }₎_for_t_∈_{(, )}_,_then_x λis continuous inλ,that is,λ→λ(λ> )implies xλ–xλ →;

(iii) if there existsβ∈(,_)such thatϕi(t)≥tβ(i= , )fort∈(, ),then

limλ→∞ xλ = ,limλ→+ xλ =∞.

Proof Deﬁne an operatorT=A+BbyT(x,y) =Ax+By. ThenT:P×P→Pis a mixed monotone operator. From the proof of Theorem ., we haveT(h,h)∈Ph, and

Ttx,t–y ≥ϕ(t)T(x,y), t∈(, ),x,y∈P,

whereϕ(t) =min{ϕ(t),ϕ(t)}. Evidently,ϕ(t)∈(t, ) fort∈(, ). Hence, the conclusions

follow from Theorem ..

Similarly, we can easily obtain the following result.

Corollary . Assume that all the conditions of Corollary.hold.Let xλ(λ> )denote

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(i) ifα,α∈(,_),thenxλis strictly decreasing inλ,that is, <λ<λimplies

xλ>xλ;

(ii) xλis continuous inλ,that is,λ→λ(λ> )implies xλ–xλ →;

(iii) ifα,α∈(,_),thenlimλ→∞ xλ = ,limλ→+ xλ =∞.

2.2 The sum of increasing operators and mixed monotone operators

Next, we consider the following sum operator equations:

Ax+B(x,x) =x, (.)

Ax+B(x,x) =λx, λ> . (.)

Theorem . Let P be a normal cone,A:P→P be an increasing operator and B:P×P→ P be a mixed monotone operator.Assume that:

(H) for anyx∈P,t∈(, ),there existsϕ(t)∈(t, )such that

A(tx)≥ϕ(t)Ax; (.)

(H) for anyx,y∈P,t∈(, ),there existsϕ(t)∈(t, )such that

Btx,t–y ≥ϕ(t)B(x,y); (.)

(H) there existsh∈Phsuch thatAh+B(h,h)∈Ph.

Then:

rv≤u<v, u≤Au+B(u,v)≤Av+B(v,u)≤v;

(ii) the operator equation(.)has a unique solutionx∗inPh;

xn=Axn–+B(xn–,yn–), yn=Ayn–+B(yn–,xn–), n= , , . . . ,

Proof From (.), we obtain

Bt–x,ty ≤  ϕ(t)

B(x,y), x,y∈P,t∈(, ). (.)

SinceAh+B(h,h)∈Ph, there exist constantsλ,λ>  such that

λh≤Ah+B(h,h)≤λh.

Also, fromh∈Ph, there exists a small constantt∈(, ) such that

th≤h≤



t

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Letϕ(t) =min{ϕ(t),ϕ(t)}. Thenϕ(t)∈(t, ) fort∈(, ). From (.)-(.),

Ah+B(h,h)≥A(th) +B

th,t–h ≥ϕ(t)Ah+ϕ(t)B(h,h) ≥ϕ(t)

Ah+B(h,h)

≥λϕ(t)h,

Ah+B(h,h)≤At_–h +B

t_–h,th ≤



ϕ(t)

Ah+



ϕ(t)

B(h,h)

≤ 

ϕ(t)

Ah+B(h,h)

≤ λ

ϕ(t)

h.

Note thatλϕ(t),_ϕλ₍_t₎ > , we can getAh+B(h,h)∈Ph.

Next, we deﬁne an operatorT=A+BbyT(x,y) =Ax+B(x,y). ThenT:P×P→Pis a mixed monotone operator andT(h,h) =Ah+B(h,h)∈Ph. Moreover, for anyx,y∈Pand t∈(, ), we have

Ttx,t–y =A(tx) +Btx,t–y ≥ϕ(t)Ax+ϕ(t)B(x,y) ≥ϕ(t)Ax+B(x,y)=ϕ(t)T(x,y).

Hence, all the conditions of Theorem . are satisﬁed. An application of Theorem . im-plies that: there areu,v∈Ph andr∈(, ) such thatrv≤u<v,u≤T(u,v)≤

T(v,u)≤v; operator equationT(x,x) =xhas a unique solutionx∗∈Ph; for any initial

valuesx,y∈Ph, constructing successively the sequences

we have xn–x∗ → and yn–x∗ → asn→ ∞. That is: (i) there existu,v∈Phandr∈(, )such that

(ii) the operator equation (.) has a unique solutionx∗inPh;

Corollary . Let P be a normal cone, A:Ph→Ph be an increasing operator and B:

Ph×Ph→Phbe a mixed monotone operator.Assume that:

(H) for anyx∈Ph,t∈(, ),there existsϕ(t)∈(t, )such that

A(tx)≥ϕ(t)Ax;

(H) for anyx,y∈Ph,t∈(, ),there existsϕ(t)∈(t, )such that

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Then:

Corollary . Letα∈(, )and P be a normal cone.Let A:P→P be an increasing op-erator which satisﬁes(H),B:P→P be an increasing sub-homogeneous operator and

B:P×P→P be a mixed monotone operator which satisﬁes

B

tx,t–y ≥tαB(x,y), t∈(, ),x,y∈P. (.)

Assume that:

(H) there existh,h∈Phsuch that

Ah∈Ph, Bh+B(h,h)∈Ph;

(H) there exists a constantδ> such that

B(x,y)≥δBx, x,y∈P.

Then:

rv≤u<v,

u≤Au+Bu+B(u,v)≤Av+Bv+B(v,u)≤v;

(ii) the following operator equation:

Ax+Bx+B(x,x) =x (.)

has a unique solutionx∗inPh;

xn=Axn–+Bxn–+B(xn–,yn–),

yn=Ayn–+Byn–+B(yn–,xn–), n= , , . . . ,

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Proof Deﬁne an operatorB=B+BbyB(x,y) =Bx+B(x,y). ThenB:P×P→Pis a

mixed monotone operator andB(h,h) =Bh+B(h,h)∈Ph. Sinceh,h∈Ph, there

existt,t∈(, ) such that

th≤h≤



t

h, th≤h≤



t

h.

Thenh≤_th≤_tt h,h≥th≥tth, and thus

Ah≥A(tth)≥ϕ(tt)Ah, Ah≤A



tt

h

≤ 

ϕ(tt)

Ah.

Note thatϕ(tt),_ϕ_₍_tt ₎>  andAh∈Ph, we can getAh∈Ph. Hence,Ah+B(h,h)∈Ph.

Note that (H) and from the proof of Theorem ., there existsβ(t)∈(α, ) with respect

totsuch that

Btx,t–y ≥tβ(t)_B₍_x_,_y_), ∀_t∈_{(, ),}_x_,_y∈_P_.

Letϕ(t) =tβ(t),t∈(, ). Thenϕ(t)∈(t, ) and

Btx,t–y ≥ϕ(t)B(x,y), ∀t∈(, ),x,y∈P.

Therefore, the operatorsA,Bsatisfy all the conditions of Theorem .. So we easily obtain the following conclusions:

(i) there existu,v∈Phandr∈(, )such that

rv≤u<v,

(ii) the operator equation (.) has a unique solutionx∗inPh;

yn=Ayn–+Byn–+B(yn–,xn–), n= , , . . . ,

Corollary . Letα∈(, ) and P be a normal cone.Let A:P→P be an increasing operator which satisﬁes(H),B :P→P be an increasingα-concave operator and B :

P×P→P be a mixed monotone operator which satisﬁes

B

tx,t–y ≥tB(x,y), t∈(, ),x,y∈P. (.)

Assume that(H)holds and

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Then:

rv≤u<v, u≤Au+Bu+B(u,v)≤Av+Bv+B(v,u)≤v;

(ii) the operator equation(.)has a unique solutionx∗inPh;

yn=Ayn–+Byn–+B(yn–,xn–), n= , , . . . ,

Proof Consider the same operatorBdeﬁned by the proof of Corollary ., we have B:

P×P→Pis a mixed monotone operator andB(h,h)∈Ph. From Deﬁnition ., we have A(tx)≤ 

tαAx,t∈(, ),x∈P. Sinceh,h∈Ph, there existt,t∈(, ) such that

th≤h≤



t

h, th≤h≤



t

h.

Thentth≤h≤_tt h, and thus

Ah≥A(tth)≥(tt)αAh, Ah≤A



tt

h

≤ 

(tt)α

Ah.

Note that (tt)α,₍_tt₎α>  andAh∈Ph, we can getAh∈Ph. Hence,Ah+B(h,h)∈Ph.

Note that (H) and from the proof of Theorem ., we know that there existsβ(t)∈

(α, ) with respect totsuch that

Btx,t–y ≥tβ(t)_B₍_x_,_y_), ∀_t∈_{(, ),}_x_,_y∈_P_.

Letϕ(t) =tβ(t),t∈(, ). Thenϕ(t)∈(t, ) andB(tx,t–y)≥ϕ(t)B(x,y),x,y∈P.

Therefore, the operatorsA, Bsatisfy all the conditions of Theorem .. So we easily obtain the following conclusions:

(i) there existu,v∈Phandr∈(, )such that

rv≤u<v,

(ii) the operator equation (.) has a unique solutionx∗inPh;

yn=Ayn–+Byn–+B(yn–,xn–), n= , , . . . ,

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Corollary . Assume that all the conditions of Theorem.hold.Let xλ(λ> )denote

the unique solution of operator equation(.).Then we have the following conclusions: (i) ifϕi(t) >t



 ₍_i_{= , }₎_for_t∈_{(, )}_,_then_x_λ_{is strictly decreasing in}_λ_,_{that is,}

(ii) if there existsβ∈(, )such thatϕi(t)≥tβ₍_i_{= , }₎_for_t_∈_{(, )}_,_then_x λis continuous inλ,that is,λ→λ(λ> )implies xλ–xλ →;

(iii) if there existsβ∈(,

)such thatϕi(t)≥t

β₍_i_{= , }₎_for_t_∈_{(, )}_,_then

2.3 The sum of decreasing operators and mixed monotone operators

In the following we also consider the operator equations (.) and (.).

Theorem . Let P be a normal cone,A:P→P be a decreasing operator and B:P×P→ P be a mixed monotone operator.Assume that(H)and(H)hold and

(H) for anyx∈Pandt∈(, ),there existsϕ(t)∈(t, )such that

A(tx)≤ 

ϕ(t)

Ax. (.)

Then:

rv≤u<v, u≤Av+B(u,v)≤Au+B(v,u)≤v;

xn=Ayn–+B(xn–,yn–), yn=Axn–+B(yn–,xn–), n= , , . . . ,

Proof From (.), we have

A



tx

≥ϕ(t)Ax, t∈(, ),x∈P. (.)

SinceAh+B(h,h)∈Ph, there exist constantsλ,λ>  such that

λh≤Ah+B(h,h)≤λh.

Also fromh∈Ph, there exists a small constantt∈(, ) such that

th≤h≤



t

h.

Letϕ(t) =min{ϕ(t),ϕ(t)}. Thenϕ(t)∈(t, ) fort∈(, ). From (H) and (.), (.),

Ah+B(h,h)≥A



t

h

+Bth,t–h ≥ϕ(t)Ah+ϕ(t)B(h,h)

≥ϕ(t)

Ah+B(h,h)

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Ah+B(h,h)≤A(th) +B

t_–h,th ≤



ϕ(t)

Ah+



ϕ(t)

B(h,h)

≤ 

ϕ(t)

Ah+B(h,h)

≤ λ

ϕ(t)

h.

Note thatλϕ(t),ϕλ(t) > , we can getAh+B(h,h)∈Ph.

Next, we deﬁne an operatorT=A+BbyT(x,y) =Ay+B(x,y). ThenT:P×P→Pis a mixed monotone operator andT(h,h) =Ah+B(h,h)∈Ph.

Moreover, for anyx,y∈Pandt∈(, ), we have

Ttx,t–y =At–y +Btx,t–y ≥ϕ(t)Ay+ϕ(t)B(x,y) ≥ϕ(t)Ay+B(x,y)=ϕ(t)T(x,y).

Hence, all the conditions of Theorem . are satisﬁed. Application of Theorem . implies that: there areu,v∈Phandr∈(, ) such thatrv≤u<v,u≤T(u,v)≤T(v,u)≤

v; operator equationT(x,x) =xhas a unique solutionx∗∈Ph; for any initial valuesx,y∈

Ph, constructing successively the sequences

we have xn–x∗ → and yn–x∗ → asn→ ∞. That is, (i) there existu,v∈Phandr∈(, )such that

rv≤u<v, u≤Av+B(u,v)≤Au+B(v,u)≤v;

(ii) the operator equation (.) has a unique solutionx∗inPh;

Corollary . Let P be a normal cone,A:Ph →Ph be a decreasing operator and B:

Ph×Ph→Phbe a mixed monotone operator.Assume that:

(H) for anyx∈Phandt∈(, ),there existsϕ(t)∈(t, )such that

A(tx)≤ 

ϕ(t)

Ax;

(H) for anyx,y∈Ph,t∈(, ),there existsϕ(t)∈(t, )such that

Btx,t–y ≥ϕ(t)B(x,y).

Then:

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Corollary . Assume that all the conditions of Theorem.hold.Let xλ(λ> )denote

the unique solution of operator equation(.).Then we have the following conclusions: (i) ifϕi(t) >t



 ₍_i_{= , }₎_for_t∈_{(, )}_,_then_x_λ_{is strictly decreasing in}_λ_,_{that is,}

(ii) if there existsβ∈(, )such thatϕi(t)≥tβ(i= , )fort∈(, ),thenxλis continuous inλ,that is,λ→λ(λ> )implies xλ–xλ →;

(iii) if there existsβ∈(,_)such thatϕi(t)≥tβ₍_i_{= , }₎_for_t_∈_{(, )}_,_then

2.4 The sum of increasing operators, decreasing operators, and mixed monotone operators

From the above results, we can easily obtain the following results on operator equa-tions:

Ax+Bx+B(x,x) =x, (.)

Ax+Ax+Bx+B(x,x) =x. (.)

By Theorem . and Corollary ., Corollary ., we have the following conclu-sions.

Theorem . Letα∈(, )and P be a normal cone.Let A:P→P be a decreasing op-erator which satisﬁes(H),operators B,Bbe the same as for Corollary..Assume that

(H), (H)hold.Then:

rv≤u<v,

u≤Av+Bu+B(u,v)≤Au+Bv+B(v,u)≤v;

(ii) the operator equation(.)has a unique solutionx∗inPh;

xn=Ayn–+Bxn–+B(xn–,yn–),

yn=Axn–+Byn–+B(yn–,xn–), n= , , . . . ,

Theorem . Letα∈(, )and P be a normal cone.Let A:P→P be a decreasing oper-ator which satisﬁes(H),operators B,Bbe the same as for Corollary..Assume that

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rv≤u<v,

u≤Av+Bu+B(u,v)≤Au+Bv+B(v,u)≤v;

(ii) the operator equation(.)has a unique solutionx∗inPh;

xn=Ayn–+Bxn–+B(xn–,yn–),

yn=Axn–+Byn–+B(yn–,xn–), n= , , . . . ,

From Corollary ., Corollary ., and Corollary ., we can easily obtain the following results.

Theorem . Letα,α∈(, )and P be a normal cone,operators A,Asatisfy the

con-ditions of Corollary.,where Aisα-concave,operators B,Bsatisfy the conditions of

Corollary.,where Bsatisﬁes(.)withαreplaced byα.Then: (i) there existu,v∈Phandr∈(, )such that

rv≤u<v,

u≤Au+Au+Bu+B(u,v)≤Av+Av+Bv+B(v,u)≤v;

(ii) the operator equation(.)has a unique solutionx∗inPh;

xn=Axn–+Axn–+Bxn–+B(xn–,yn–),

yn=Ayn–+Ayn–+Byn–+B(yn–,xn–),

wheren= , , . . .,we havexn→x∗,yn→x∗asn→ ∞.

Theorem . Letα,α∈(, )and P be a normal cone,operator A:P→P isα-concave,

operators B,Bsatisfy the conditions of Corollary.,where Bisα-concave.Then: (i) there existu,v∈Phandr∈(, )such that

rv≤u<v,

u≤Au+Au+Bu+B(u,v)≤Av+Av+Bv+B(v,u)≤v;

(ii) the operator equation(.)has a unique solutionx∗inPh;

xn=Axn–+Axn–+Bxn–+B(xn–,yn–),

yn=Ayn–+Ayn–+Byn–+B(yn–,xn–),

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3 Some applications

In this section, we will apply the main results to study nonlinear problems which in-clude nonlinear integral equations and nonlinear elliptic boundary value problems for the Lane-Emden-Fowler equations. And then we will obtain new results on the existence and uniqueness of positive solutions for these problems, which are not consequences of the corresponding ﬁxed point theorems in the literature.

3.1 Applications to nonlinear integral equations

A standard approach, in studying the existence of positive solutions of boundary value problems (BVPs for short) for ordinary diﬀerential equations, is to rewrite the problem as an equivalent positive-solution problem for a Hammerstein integral equation of the form

u(t) =λ b

a

G(t,s)fs,u(s) ds, (.)

in the spaceE=C[a,b], where the nonlinearityf and the kernelG(the Green function of the problem) are both nonnegative, λ>  is a parameter. One seeks ﬁxed points of a Hammerstein integral operator in a suitable cone of positive functions.

SetP={u∈C[a,b]|u(t)≥,t∈[a,b]}, the standard cone. It is easy to see thatPis a nor-mal cone of which the nornor-mality constant is . ThenPh={x∈P|there areλ(x)≥λ(x) >

 such thatλ(x)h(t)≤x(t)≤λ(x)h(t),t∈[a,b]}. Assume thatG(t,s) : [a,b]×[a,b]→

[, +∞) is continuous with G(t,s)≡ and there exist h,m,n∈C([a,b], [, +∞)) with

h(t),m(t),n(t)≡, such that

m(s)h(t)≤G(t,s)≤n(s)h(t) for allt,s∈[a,b]. (.)

Theorem . Assume that f(t,x) =f(t,x) +f(t,x)≡and

(H) fi: [a,b]×[, +∞)→[, +∞)is continuous (i= , ),f(t,x)is increasing in x∈

[, +∞)for ﬁxedt∈[a,b]andf(t,x)is decreasing inx∈[, +∞)for ﬁxedt∈[a,b]; (H) forη∈(, ),there existϕi(η)∈(η, )(i= , )such that

f(t,ηx)≥ϕ(η)f(t,x), f(t,ηx)≤



ϕ(η)

f(t,x), ∀t∈[a,b],x∈[, +∞).

Then,for any givenλ> ,the integral equation(.)has a unique positive solution u∗_λin Ph. Moreover,for any initial values x,y∈Ph,constructing successively the sequences:

xn=λ b

a

G(t,s)f

s,xn–(s) +f

s,yn–(s) ds,

yn=λ b

a

G(t,s)f

s,yn–(s) +f

s,xn–(s) ds, n= , , . . . ,

we have xn→u∗_λ,yn→u∗_λas n→+∞.Further, (i)ifϕi(t) >t (i= , )for t∈(, ),then u∗_λis strictly increasing inλ,that is,  <λ<λimplies u∗λ<u

∗

λ; (ii)if there existsβ∈(, ) such thatϕi(t)≥tβ(i= , )for t∈(, ),then u∗λis continuous inλ,that is,λ→λ(λ>

)implies u∗λ–u∗λ →; (iii)if there existsβ∈(,



)such thatϕi(t)≥t

β ₍_i_{= , )}_for

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Proof Deﬁne two operatorsA:P→EandB:P→Eby

Au(t) =

b

a

G(t,s)f

s,u(s) ds, Bu(t) =

b

a

G(t,s)f

s,u(s) ds.

It is easy to see thatuis the solution of (.) if and only ifu=λ(Au+Bu). From (H), we

know thatA:P→Pis increasing andB:P→Pis decreasing. Further, from (H), we can

prove thatA,Bsatisfy (H). Next we prove thatAh+Bh∈Ph. Sethmax=maxt∈[a,b]h(t),

hmin=mint∈[a,b]h(t). Thenhmax≥hmin> .

For anyt∈[a,b], from (H) and (.), we have

Ah(t) +Bh(t) =

b

a

G(t,s)f

s,h(s) +f

s,h(s) ds

≥ b

a

h(t)m(s)f(s,hmin) +f(s,hmax)

ds

=h(t)

b

a

m(s)f(s,hmin) +f(s,hmax)

ds,

Ah(t) +Bh(t)≤

b

a

h(t)n(s)f(s,hmax) +f(s,hmin)

ds

=h(t)

b

a

n(s)f(s,hmax) +f(s,hmin)

ds.

Letr=

b

a m(s)[f(s,hmin) +f(s,hmax)]ds,r= b

a n(s)[f(s,hmax) +f(s,hmin)]ds. Note that f =f+f≥ is continuous withf ≡ and from (.), we get  <r≤r and in

con-sequence,rh≤Ah+Bh≤rh. That is,Ah+Bh∈Ph. Hence, all the conditions of

The-orem . are satisﬁed. It follows from TheThe-orem . and Corollary . that the operator equationAu+Bu=_λuhas a unique solutionu∗_λinPh, that is,λ(Au∗_λ+Bu∗_λ) =u∗_λ. Sou∗_λis a unique positive solution of the integral equation (.) inPhfor givenλ> . From

Corol-lary ., we have (i) ifϕi(t) >t 

 ₍_i_{= , ) for}_t∈_{(, ), then}_u∗

λis strictly increasing inλ, that is,  <λ<λimpliesu∗λ<u

∗

λ; (ii) if there existsβ∈(, ) such thatϕi(t)≥t

β₍_i_{= , ) for}

t∈(, ), thenu∗_λis continuous inλ, that is,λ→λ(λ> ) implies u∗λ–u∗λ →; (iii) if

there exists β∈(,_) such thatϕi(t)≥tβ (i= , ) fort∈(, ), thenlimλ→+ u∗λ = ,

limλ→+∞ u∗λ = +∞.

LetAλ=λA,Bλ=λB. ThenAλ,Bλalso satisfy the conditions of Theorem .. By Theo-rem ., for any initial valuesx,y∈Ph, constructing successively the sequences

xn=Aλxn–+Bλyn–, yn=Aλyn–+Bλxn–, n= , , . . . ,

we havexn→u∗_λ,yn→u∗_λasn→+∞. That is,

xn=λ b

a

G(t,s)f

s,xn–(s) +f

s,yn–(s) ds→u∗λ(t),

yn=λ b

a

G(t,s)f

s,yn–(s) +f

s,xn–(s) ds→u∗λ(t)