Properties of positive solutions for the operator equation \(Ax=\lambda x\) and applications to fractional differential equations with integral boundary conditions

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R E S E A R C H

Open Access

Properties of positive solutions for the

operator equation

Ax

=

λx

and applications

to fractional differential equations with

integral boundary conditions

Chengbo Zhai

*

_{and Fang Wang}

*_{Correspondence:}

[email protected]

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, P.R. China

Abstract

In this article we present a new ﬁxed point theorem for a class of generalized concave operators and we establish some properties of positive solutions for the operator equationAx=

λ

x. Based on them, the existence and uniqueness of positive solutions for a class of fractional diﬀerential equations with integral boundary conditions is proved. Moreover, we present some properties of positive solutions to the boundary value problem dependent on the parameter.

MSC: 47H10; 47H07; 26A33; 34B18; 34B09

Keywords: positive solution; generalized concave operator; parameter; fractional diﬀerential equation; integral boundary condition

1 Introduction

Owing to various applications of integral boundary value problems in applied fields such as blood flow problems, chemical engineering, thermo-elasticity, underground water flow, and population dynamics, the existence of solutions for fractional differential equations with integral boundary conditions has been extensively studied in recent years (see [–] and the references therein). In these papers, most of the authors have investigated the ex-istence and multiplicity of positive solutions. For example, by means of the monotone iter-ation method, Sun and Zhao [] investigated the existence of positive solutions for a class of Riemann-Liouville fractional differential equations with integral boundary conditions. Zhaoet al.[] used Krasnosel’skii’s fixed point theorem to obtain the existence and nonex-istence of positive solutions for a fractional differential equation with integral boundary conditions. In [], by using the properties of the Green function, au-bounded function,

and ﬁxed point index theory, the authors obtained the existence of positive solutions for a class of nonlinear fractional diﬀerential equations with integral boundary conditions and a parameter. But the uniqueness of positive solutions is not treated in [–]. As far as we know, there are few papers reported on the integral boundary conditions of fractional dif-ferential equations with a parameter. In particular, there are no clear explanations of the relation between positive solutions and the parameter. The reason is that many methods

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used in the literature are independent of the parameters. So we need some properties of positive solutions for the operator equation

Ax=λx, (.)

whereAis a generalized concave operator andλ>  is an eigenvalue. In this article, we first state and prove a new fixed point theorem for a class of generalized concave operators. Further, we establish some properties of positive solutions for the operator equation (.). Here we do not assume the existence of upper-lower solutions for the operatorA, which is usually done in the literature (for example, see [, ]). As applications, we utilize the main results for (.) to the following fractional differential equation with integral boundary conditions:

Dα

+u(t) +λf(t,u(t)) = , t∈(, ),

u() =u() = , u() =β_u(s)ds, (.)

where  <α≤,  <β<α,λ>  is a parameter.Dα

+is the Riemann-Liouville fractional derivative of orderα, which is deﬁned as follows:

Dα_+u(t) =  (n–α)

dn

dtn

t



u(s)

(t–s)α+–nds, n= [α] + ,

here denotes the Euler gamma function and [α] denotes the integer part of number αprovided that the right side is point-wise deﬁned on (, +∞); see []. We establish the existence and uniqueness of positive solutions for problem (.) with any given parameter. Moreover, we present some properties of positive solutions for problem (.) dependent on the parameter.

2 Properties of positive solutions for the operator equationAx=

λ

x

For the discussion of this section, we ﬁrst list some basic notations, concepts in ordered Banach spaces. For the convenience of the reader, we refer to [, , ] for details.

Let (E, · ) be 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. A non-empty closed convex setP⊂Eis a cone if it satisﬁes (i)x∈P,

r≥⇒rx∈P; (ii)x∈P, –x∈P⇒x=θ.

Pis called normal if there is a constantN>  such that, for allx,y∈E,θ≤x≤yimplies

x ≤Ny; in this caseN is the inﬁmum of such constants, it is called the normality constant ofP. We say that an operatorA:E→Eis increasing ifx≤yimpliesAx≤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.

Lemma .(see Theorem . in []) Let P be a normal cone in a real Banach space E and

h>θ.Assume that:

(D) A:P→Pis increasing andAh+x∈Phwithx∈P;

(D) forx∈Pandt∈(, ),there existsϕ(t)∈(t, )such thatA(tx)≥ϕ(t)Ax.

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Now we consider the operator equation

Ax=x. (.)

Theorem . Let P be a normal cone in a real Banach space E,h>θ,and let A:P→P be

an increasing operator,satisfying: (i) there ish∈Phsuch thatAh∈Ph;

Remark . We say an operatorAis generalized concave if it satisﬁes the condition (ii)

in Theorem .; see [].

the monotonicity of operatorA, we have

Ah≥A(th)≥ϕ(t)Ah≥ϕ(t)λh,

we can easily get the conclusion ().

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≥ · · · ≥ϕ(t)·ϕ(t)· · ·ϕ(t)·ϕ(t)Ax

SincePis normal, we have

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whereN is the normality constant. So{un}and{vn}are Cauchy sequences in complete

Next we state and prove some properties of positive solutions for the operator equation (.).

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

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whereNis the normality constant. Letλ→λ+ , we havexλ–xλ →. Similarly, let λ→λ– , we also havexλ–xλ →. So the conclusion (ii) holds.

(iii) Letλ= ,λ=λin (.), we havex≥λxλ,∀λ> . Thus, we can easily obtainxλ ≤

N

λx, whereNis the normal constant. Letλ→ ∞, thenxλ →. Similarly, letλ=λ, λ=  in (.), thenxλ≥ _λx,∀ <λ< . Thusxλ ≥_N_λx, whereNis the normality

constant. Letλ→ + , we havexλ → ∞.

Remark . () We do not suppose the condition of upper-lower solutions which is

com-mon in many known results and is diﬃcult to verify. Moreover, we give the iterative forms. The existence of a unique positive solution is proved only in the case where the conePis normal and the operatorsAis generalized concave.

() The eigenvalue problem for generalized concave operators has not been studied in the literature, so Theorem . complements the eigenvalue results for generalized concave operators.

3 Properties of positive solutions for problem (1.2)

Whenλ=  in (.), Cabada and Hamdi [] established the existence of one positive solu-tion for problem (.) under sublinear case or superlinear case. The method used there is by Guo-Krasnosel’skii fixed point theorem. Different from the main result and the method, we will apply Theorem . and present some properties of positive solutions for problem (.) dependent on the parameter, and the method is also different from those in previous works.

Lemma .(see []) Let <α≤andα=β.Assume y∈C[, ],then the following

frac-tional diﬀerential equation with integral boundary conditions:

Dα

+u(t) +y(t) = , t∈(, ),

u() =u() = , u() =β_u(s)ds,

has a unique solution u∈C_{[, ],}_{given by the expression}

u(t) =





G(t,s)y(s)ds, (.)

where

G(t,s) =

⎧ ⎨ ⎩

tα–(–s)α–(α–β+βs)–(α–β)(t–s)α–

(α–β)(α) , ≤s≤t≤,

tα–(–s)α–(α–β+βs)

(α–β)(α) , ≤t≤s≤.

(.)

Lemma . Let <α ≤ and <β <α. The function G(t,s)deﬁned by (.)has the

following properties:

( –s)α–_β_s

(α–β)(α)t

α–_≤_G₍_t_,_s₎_≤( –s)α–(α–β+βs)

(α–β)(α) t

α–_, _t_,_s_∈_{[, ].}

Proof Evidently, the right inequality holds. So we only need to prove the left inequality. If ≤s≤t≤, then we have ≤t–s≤t–ts= ( –s)t, and thus

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Sinceα–β> , we obtain

G(t,s) = t

α–_{( –}_s₎α–_(α_–_β₊_β_s_{) – (α}_–_β₎₍_t_–_s₎α–

(α–β)(α)

≥ 

(α–β)(α)

tα–( –s)α–(α–β+βs) – (α–β)( –s)α–tα–

= ( –s) α–_β_s

(α–β)(α)t α–_.

When ≤t≤s≤, fromα–β> , we have

G(t,s) = t

α–_{( –}_s₎α–_(α_–_β₊_β_s₎

(α–β)(α)

≥ ( –s)α–βs

(α–β)(α)t α–_.

So the left inequality also holds.

In the following considerations we will work in the Banach spaceC[, ], the space of all continuous functions on [, ] with the standard normx=sup{|x(t)|:t∈[, ]}. Evi-dently, this space can be equipped with a partial order given by

x,y∈C[, ], x≤y ⇔ x(t)≤y(t) fort∈[, ].

SetP={x∈C[, ]|x(t)≥,t∈[, ]}, the standard cone. We know thatPis a normal cone inC[, ] and the normality constant is .

Theorem . Assume that:

(H) f : [, ]×[, +∞)→[, +∞)is continuous withf(t, )≡;

(H) f(t,x)is increasing inxfor eacht∈[, ];

(H) for anyr∈(, ),there existsϕ(r)∈(r, )such that

f(t,rx)≥ϕ(r)f(t,x), ∀t∈[, ],x∈[, +∞).

Then the following conclusions hold:

() For any givenλ> ,problem(.)has a unique positive solutionu∗λinPh,where

h(t) =tα–_,_t_∈_{[, ]}_._Moreover_,_{for any initial value}_u

∈Ph,constructing successively

the sequence

un(t) =λ





G(t,s)fs,un–(s)

ds, n= , , . . . ,

we haveun(t)→u∗λ(t)asn→+∞,whereG(t,s)is given as in Lemma.. () u∗_λis strictly increasing inλ,that is, <λ<λimpliesu∗λ<u

∗ λ. () If there existsγ∈(, )such thatϕ(t)≥tγ _for_t_∈_{(, )}_,_then_u∗

λis continuous inλ, that is,λ→λ(λ> )impliesu∗λ–u∗λ →.

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Proof For anyu∈P, we deﬁne an operatorAby

sequel we check thatAsatisﬁes all assumptions of Theorem ..

First of all, we show thatAsatisﬁes the second condition of Theorem .. From (H), for

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It is easy to check thatu∗λis a unique positive solution of problem (.) for givenλ> . From Theorem ., u∗λis strictly increasing in λ, that is,  <λ<λ impliesu∗λ≤u

∗ λ,

u∗_λ_=u∗_λ_. Further,limλ→+u∗_λ= ,lim_λ_→∞u∗_λ=∞. Moreover, if there existsγ∈(, ) such thatϕ(t)≥tγ _for_t_∈_{(, ), Theorem . means that}_u∗

λis continuous inλ, that is, λ→λ(λ> ) impliesu∗λ–u∗λ →.

LetAλ=λA, thenAλalso satisﬁes all the conditions of Theorem .. By Theorem ., for any initial valueu∈Ph, constructing successively the sequenceun=Aλun–,n= , , . . . ,

we haveun→u∗λasn→ ∞. That is,

un(t) =λ





ds, n= , , . . . ,t∈[, ],

andun(t)→uλ∗(t) asn→+∞.

In Theorem ., letλ= , we can easily obtain the following conclusions.

Corollary . Assume(H)-(H)hold. Then the following Riemann-Liouville fractional

diﬀerential equation with integral boundary conditions

Dα

+u(t) +f(t,u(t)) = , t∈(, ),

u() =u() = , u() =β_u(s)ds,

where <α≤,  <β<α,has a unique positive solution u∗in Ph,where h(t) =tα–,t∈

[, ].Moreover,for any initial value u∈Ph,constructing successively the sequence

un(t) =





ds, n= , , . . . ,

we have un(t)→u∗(t)as n→+∞,where G(t,s)is given as in Lemma..

Remark . Comparing Theorem . and Corollary . with many main results in the

literature, here we present an alternative approach to the study of similar problems under diﬀerent conditions. Our results cannot only guarantee the existence of a unique positive solution for any given parameter, but they also help to construct an iterative scheme for approximating it. Moreover, we can show that the positive solution with respect to the parameter has some deﬁnite properties. So our results are seldom seen in the literature.

Remark . () We can see that there is a large number of functions which satisfy the

conditions of Theorem . or Corollary .. For example, letf(t,x) =a(t)[x +b], where

a: [, ]→[, +∞) is continuous witha(t)≡,b> . Takeϕ(r) =r_{. Also let}_f₍_t_,_x_{) =}

g(t) +x ₊_x ₊· · ·₊_xn₊_b_{, where}_g_{: [, ]}→_{[, +}∞_{) is continuous,}_n≥_,_b_{> . Take}

ϕ(r) =r_{. We can easily prove that (H}__)-(H__{) in Theorem . hold.}

() If the Green functions satisfy some properties similar to Lemma ., then Theo-rems . and . can be applied to many fractional diﬀerential equations boundary value problems.

Competing interests

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Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibilities. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The research was supported by the Youth Science Foundation of China (11201272) and Shanxi Province Science Foundation (2015011005), 131 Talents Project of Shanxi Province (2015).

Received: 14 July 2015 Accepted: 18 November 2015

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