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R E S E A R C H

Open Access

Positive solutions to fractional boundary

value problems with nonlinear boundary

conditions

Wenquan Feng, Shurong Sun

*

, Xinhui Li and Meirong Xu

*Correspondence:

sshrong@163.com

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China

Abstract

We consider the existence of at least one positive solution of the problem –0+u(t) =f(t,u(t)), 0 <t< 1, under the circumstances thatu(0) = 0,

u(1) =H1(

ϕ

(u)) +

EH2(s,u(s))ds, where 1 <

α

< 2,D α

0+is the Riemann-Liouville fractional derivative, andu(1) =H1(

ϕ

(u)) +

EH2(s,u(s))dsrepresents a nonlinear nonlocal boundary condition. By imposing some relatively mild structural conditions onf,H1,H2, and

ϕ

, one positive solution to the problem is ensured. Our results generalize the existing results and an example is given as well.

MSC: 34A08; 34B18

Keywords: fractional differential equation; nonlinear boundary condition; Krasnosel’skiˇi’s fixed point theorem; positive solution

1 Introduction

In this paper we consider the existence of at least one positive solution of the fractional differential equation

+u(t) =ft,u(t),  <t< , ()

subject to the boundary conditions

u() = , u() =H

ϕ(u)+

E H

s,u(s)ds, ()

hereE(, ) is some measurable set,  <α< ,

+is the Riemann-Liouville fractional derivative andϕis a linear functional having the form

ϕ(u) :=

u(t)(t), ()

where the integral appearing in () is taken in the Lebesgue-Stieltjes sense,θis a function of bounded variation.

Let us review briefly some recent results on such problems in order to see our problem ()-() in a more appropriate context.

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So far, in view of their various applications in science and engineering, such as fluid mechanics, control system, viscoelasticity, porous media, edge detection, optical systems, electromagnetism and so forth, see [–], fractional differential equations have attracted great attention of mathematicians.

There are a great number of works on the existence of solutions of various classes of ordinary differential equations and fractional differential equations; readers may refer to [–].

Some of them discussed two-point boundary value problems. For example, Bai and Lü [] studied the following two-point boundary value problem of fractional differential equations:

+u(t) +ft,u(t)= ,  <t< ,

u() =u() = ,

()

where  <α≤,

+is the standard Riemann-Liouville fractional derivative. By means of Guo-Krasnosel’skiˇi’s fixed point theorem and the Leggett-Williams fixed point theorem they obtained the existence of positive solutions.

Some authors discussed multi-point boundary value problems, for instance, by using fixed point index theory, the Krein-Rutman theorem and some other methods, Jiang [] studied the eigenvalue interval of the multi-point boundary value problem

u(t) –Mu(t) =λft,u(t), t[, ],

u() =

n

i= βiu(ξi),

()

where  <α< ,is the Caputo derivative,M,  <ξ

<ξ<· · ·<ξn≤.

There are also results on fractional boundary value problem with integral boundary con-ditions, let us refer to Vong []. He investigated positive solutions of the nonlocal bound-ary value problem for a class of singular fractional differential equations with an integral boundary condition,

cDα

+u(t) +f

t,u(t)= ,

u() =· · ·=u(n–)() = , u() = 

u(s)(s),

()

wheren≥,α∈(n– ,n) andμis a function of bounded variation.

To proceed, Goodrich [] considered the existence of at least one positive solution of the ordinary differential equation

y(t) =ft,y(t),  <t< ,

y() =H

ϕ(y)+

E H

s,y(s)ds, y() = ,

()

in which the boundary condition is more general.

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frac-tional differential equation under the same boundary conditions. Secondly, the bound-ary conditions are more flexible and general than often. Let us take the conditionu() =

H(ϕ(u)) +

EH(s,u(s))dsinto consideration, where H,H,ϕ(u) are defined in the se-quel. IfH(ϕ(u)) +

EH(s,u(s))ds= , the conditions are the standard Dirichlet boundary

conditions. Readers might refer to Bai and Lü []. As far as we are concerned,ϕ(u) varies among many sorts of functionals. IfH(ϕ(u)) =

Fu(t)dt(whereFE(, ) is defined

in the sequel) orH(ϕ(u)) =

[,]u(t)(t), our conditions reduce to integral boundary conditions, while ifH(ϕ(u)) =

n

i=|ai|u(ξi), we have multi-point boundary conditions.

Thirdly, compared to Goodrich [], we make an adjustment to the Green function and define a functionr(·) instead of a constant which affects the defined cone.

This paper is organized as follows. In Section , we review some preliminaries and lem-mas. In Section , a theorem and five corollaries about the existence of at least one positive solution of problem ()-() are obtained. Lastly, we give an example to illustrate the ob-tained theorem.

2 Preliminaries and lemmas

For the convenience of the readers, we give some background materials from fractional calculus theory to facilitate the analysis of the boundary value problem ()-().

Definition .([]) The Riemann-Liouville fractional integral of orderα>  of a function

y: (, +∞)→Ris given by

+y(t) = 

(α) t

(ts)α–y(s)ds,

provided the right side is pointwise defined on (, +∞).

Definition .([]) The Riemann-Liouville typed fractional derivative of orderα(α> ) of a continuous functionf : (, +∞)→Ris given by

+f(t) = 

(nα)

d dt

n t

(ts)nα+f(s)ds,

wheren= [α] + , [α] denotes the integer part of numberα, provided that the right side is pointwise defined on (, +∞).

Lemma .([]) Letα> .If we assume uC(, )∩L(, ),then the fractional differential equation

+u(t) = 

has u(t) =c–+c–+· · ·+cntαn,ci∈R,i= , , . . . ,n,as a unique solution,where n is the smallest integer greater than or equal toα.

Lemma .([]) Let uC(, )∩L(, )with a fractional derivative of orderα(α> )

that belongs to C(, )∩L(, ).Then

Iα++u(t) =u(t) +c–+c–+· · ·+cntαn, for some ci∈R,i= , , . . . ,n,

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Remark .([]) The Riemann-Liouville type fractional derivative and integral of order

α(α> ) have the following properties:

+Iα+u(t) =u(t), Iα+Iβ+u(t) =I

α+β

+ u(t), α,β> ,uL(, ).

Lemma . Let uC[, ]and <α< .Then the fractional differential equation bound-ary value problem

+u(t) +y(t) = ,  <t< ,

u() = , u() =H

ϕ(u)+

E H

s,u(s)ds,

has a unique solution,

u(t) =–

H

ϕ(u)+

E H

s,u(s)ds

+

G(t,s)y(s)ds,

where

G(t,s) = 

(α) ⎧ ⎨ ⎩

–( –s)α–– (ts)α–, st,

–( –s)α–, ts. ()

Proof We may apply Lemma . to reduce () to an equivalent integral equation,

u(t) = –I+αy(s) +c–+c–, c,c∈R.

Consequently, the general solution of () is

u(t) = – t

(ts)α–

(α) y(s)ds+ct

α–+c

–, c,c∈R.

By (), we have

c=H

ϕ(u)+

E H

s,u(s)ds+ 

( –s)α– (α) y(s)ds,

c= .

Therefore, the unique solution of problem () and () is

u(t) =–

H

ϕ(u)+

E H

s,u(s)ds

t

(ts)α–

(α) y(s)ds+t

α–

( –s)α– (α) y(s)ds

=–

H

ϕ(u)+

E H

s,u(s)ds

+

G(t,s)y(s)ds.

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Lemma .([]) Let(a,b)⊂(, )be an arbitrary but fixed interval.Then the function G(t,s)defined by()satisfies the following conditions:

() G(t,s) > ,fort,s∈(, );

() there exists a positive functionγ(·)∈C(, )such that

min

atbG(t,s)≥γ(s)max≤t≤G(t,s) =γ(s)G(s,s), for each≤s≤. ()

Lemma .([]) Let B be a Banach space,and let KB be a cone.Assume,are

open and bounded subsets of B with∈,¯⊂,and let T:K∩(¯\)→K be a

completely continuous operator such that

(i) Tuu,uK,andTuu,uK;or (ii) Tuu,uK,andTuu,uK.

Then T has a fixed point in K∩(¯\).

In order to get the main results, we first need some structure onH,H,ϕ, andf ap-pearing in problem ()-().

LetBbe the Banach space onC([, ]) equipped with the usual supremum norm · . Then define the coneKBby

K=

uB|u(t)≥, min

atbu(t)≥γu,ϕ

(u),ϕ(u)≥

,

whereγ∗=min{mint∈[a,b]–,mins∈[a,b]γ(s)}. Define the operatorT:C[, ]→C[, ] by

Tu(t) =–

H

ϕ(u)+

E H

s,u(s)ds

+

G(t,s)fs,u(s)ds. ()

Here we come to the nine significant assumptions.

(H) LetH: [, +∞)→[, +∞)andH: [, ]×[, +∞)→[, +∞)be real-valued con-tinuous functions.

(H) The functionalϕ(u) :=

[,]u(t)(t)can be written in the form

ϕ(u) =ϕ(u) +ϕ(u) :=

[,]

u(t)(t) +

[,]

u(t)(t), ()

whereθ,θ,θ: [, ]→Rsatisfyθ,θ,θ∈BV([, ]), andϕ,ϕare continuous linear functionals.

(H) There is a constantC∈[, )such that the functionalϕin()satisfies the inequality

ϕ(u)≤Cu ()

for alluC([, ]). Furthermore, there is a constantC> such that the functional ϕin()satisfies|ϕ(u)| ≥CuwheneveruK.

(H) For each givenε> , there areC> and> wheneverzand we have

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(H) There exists a functionF: [, +∞)→[, +∞)satisfying the growth condition

F(u) <Cu ()

for someC≥, having the property that for each givenε> , there is> such

that

H(x,u) –F(u)<εF(u) ()

for allx∈[, ], wheneveru.

(H) Assume that the nonlinearityf(t,u)splits in the sense thatf(t,u) =a(t)g(u), for con-tinuous functionsa: [, ]→[, +∞)andg:R→[, +∞)such thatais not identi-cally zero on any subinterval of[, ].

(H) Supposelimu→+ g(u)

u = +∞.

(H) Supposelimu→+∞ g(uu) = .

(H) For eachi= , both

[,]

–dθi(t)≥ ()

and

[,]

G(t,s)dθi(t)≥ ()

hold, where()holds for eachs∈[, ].

3 Main results

In this section we state and prove the existence theorem of problem ()-().

Lemma . Let T be the operator defined in().Assume conditions(H)-(H)hold.Then

T:KK,and the operator T is completely continuous.

Proof Now we divide the proof into two steps; in the first step we prove thatT:KK, then in the next, the conclusion that the operatorTis completely continuous is treated.

Step . Here we are going to show thatT :KK. In fact, sinceH,H,a, andg are all nonnegative, it is easy to find that wheneveruK, it follows that (Tu)(t)≥, for each

t∈[, ], where we use the factϕ(u)≥, followingH(ϕ(u))≥. On the other hand, provideduKwe get

min

t∈[a,b]Tu(t)≥tmin∈[a,b]t

α–

H

ϕ(u)+

E H

s,u(s)ds

+ min

t∈[a,b]

G(t,s)fs,u(s)ds

γ

H

ϕ(u)+

E H

s,u(s)ds

+max

t∈[,]

γ(s)G(t,s)fs,u(s)ds

γ

H

ϕ(u)+

E H

s,u(s)ds

+max

t∈[,]

b

a

γ(s)G(t,s)fs,u(s)ds

γ

H

ϕ(u)+

E H

s,u(s)ds

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+ min

t∈[a,b]γ(t)tmax∈[,]

b

a

G(t,s)fs,u(s)ds

γTu, ()

whereγ=mint∈[a,b]–.

Consequently, for eachi= ,  we deduce that

ϕi(Tu) =

[,]

–

H

ϕ(u)+

E H

s,u(s)ds

dθi(t)

+

[,]

G(t,s)a(s)gu(s)ds dθi(t)

=

H

ϕ(u)+

E H

s,u(s)ds

[,]

–dθ

i(t)

+ 

[,]

G(t,s)dθi(t)

a(s)gu(s)ds

≥, ()

where the inequality follows from assumption (H). Thus,ϕi(Tu)≥, therefore as desired

we concludeT(K)⊂K.

Step . In this part we turn to the proof thatT() in the sequel is bounded as well as equicontinuous with the help of the Arzelà-Ascoli theorem.

With the continuity ofH,H,a, andg, it is easy to findTis continuous.

LetKbe bounded, namely, there exists a numberM> , such that for eachu

we haveuM. By the continuity ofH,H, andϕ, we findH,Hare bounded, so there exist constantsP>  andQ>  such that|H(ϕ())| ≤Pand|H(t,u(t))| ≤Q.E(, ) is a measurable set, takeL=maxt∈[,],u∈[,M]|f(t,u)|+ , then

(Tu)(t)=–

H

ϕ(u)+

E H

s,u(s)ds

+

G(t,s)fs,u(s)ds

H

ϕ(u)+

E

H

s,u(s)ds+ 

G(t,s)fs,u(s)ds

P+Qm(E) +L

G(t,s)ds

P+Qm(E) +L

G(s,s)ds

=P+Qm(E) + L

(α+ ). ()

HenceT() is bounded.

For eachu,t,t∈[, ],t<t, and assuming there is aδ> , such thatt–t<δ, for eachε>  put

δ=min

ε,

εNα– , ε

(α– )N

α– ,

ε

α

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whereN=P+Qm(E) +(αL+), we find|(Tu)(t) – (Tu)(t)|<ε, so thatT() is equicontin-uous.

Indeed,

(Tu)(t) – (Tu)(t)

=tα–

H

ϕ(u)+

E H

s,u(s)ds

+

G(t,s)f

s,u(s)ds

–

H

ϕ(u)+

E H

s,u(s)ds

G(t,s)f

s,u(s)ds

=tα–––H

ϕ(u)+

E H

s,u(s)ds

+

G(t,s) –G(t,s)

fs,u(s)ds

––tα–H

ϕ(u)+

E

H

s,u(s)ds

+ 

G(t,s) –G(t,s)

fs,u(s)ds. ()

By the continuity ofH,H, andϕ, we findH,H are bounded, so there exist constants

P>  andQ>  such that|H(ϕ())| ≤Pand|H(t,u(t))| ≤Q.E(, ) is a measurable set, takeL=maxt∈[,],u∈[,M]|f(t,u)|+ , then

(Tu)(t) – (Tu)(t)

––tα–P+Qm(E)+

G(t,s) –G(t,s)

fs,u(s)ds

––tα–P+Qm(E)

+ L

(α) t

tα–( –s)α–– (t–s)α––tα–( –s)

α–+ (t

–s)α–ds

+ t

t

tα–( –s)α–– (t–s)α–––( –s)

α–ds

+ 

t

–( –s)α––tα–( –s)α–ds

––tα–P+Qm(E)+ L

(α)

tα––tα–

t

( –s)α–ds

+ t

(t–s)α–– (t–s)α–

ds+––tα–

t

t

( –s)α–ds

+ t

t

(t–s)α–ds+––t

α– 

t

( –s)α–ds

=tα––tα–P+Qm(E) + L

(α+ )

+ L

(α+ )

tα. ()

Next we discuss the following three cases. Case .δt<t< .

tα–––≤α– 

δ–α(t–t)≤(α– )δ

α– εN,

tαα

δ–α(t–t)≤αδ α ε

N.

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Case . ≤t<δ<t< δ.

tα––tα–≤tα–≤(δ)α–≤ ε

N, ()

tαtα≤(δ)αεN.

Case . ≤t<t<δ.

–

 –tα–≤tα–≤δα–≤ ε

N, ()

tαtαδαε

N.

Hence|(Tu)(t) – (Tu)(t)| ≤ε.

From the Arzelà-Ascoli theorem,T is completely continuous. The proof is complete.

With Lemma . in hand, we are now ready to present the first existence theorem for problem () and ().

Theorem . Assume that conditions(H)-(H)hold.Suppose that

CC+Cm(E) <  ()

and that E(, ).Then problem()-()has at least one positive solution.

Proof Begin by selecting a numberηsuch that

η

b

a

γG

,s a(s)ds≥. ()

From (H), there exists a numberr>  such thatg(u)≥ηuwhenever  <ur. Then set

r:=

uB:u<r

, ()

foruK∂rwe find

(Tu)

 ≥

G

,s a(s)g

u(s)ds

b

a

γG

,s a(s)ds

u, ()

from the definition of norm · , we getTuu, and so,T is a cone expansion on

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From condition (), we choose a positive numberεsmall enough, so that we may as-sume

CCε+CC+ε+ (C+Cε)m(E)≤. ()

Sinceϕ(u)≥, it follows thatϕ(u)≥ϕ(u)≥Cu, ifuMCε, thenϕ(u) >, and

then by condition (H), we have

H

ϕ(u)–Cϕ(u)<εCϕ(u). ()

ForE(, ), we may select  <a<b<  such thatE⊆(a,b). By the definition of the cone, we have

min

tEu(t)≥tmin∈[a,b]u(t)≥r

u. ()

Ifu

r∗, then

min

tEu(t)≥tmin∈[a,b]u(t)≥r

ur

r∗ =. ()

Hence, by condition (H) we get

H

x,u(s)–Fu(s)<εFu(s). ()

Provided that

u ≥max

C

,

r

, ()

both () and () hold.

Next we are going to discuss these two cases:gis bounded and unbounded on [, +∞), respectively.

Now ifgis bounded on [, +∞), then there existsr>  sufficiently large such that

g(u)≤r, for anyu≥. ()

Indeed, we might assume without loss of generality that

g(u)≤ rεG(s,s)a(s)ds

, ()

whereεis selected sufficiently small such that both () and () hold. We define a number

r:=max

r

r∗,r, C

,

r

.

Set

r:=

(11)

Then for eachuK∂r∗we find that

TuH

ϕ(u)+

E H

s,u(s)ds+

G(s,s)a(s)gu(s)ds

H

ϕ(u)–Cϕ(u)+Cϕ(u)+

E

H

s,u(s)–Fu(s)ds

+

E

Fu(s)ds+ 

rεds

εCCu+CCu+m(E)C( +ε)u+εu

=CCε+CC+m(E)C( +ε) +ε

u

u, ()

whenceT is a cone compression onK∂r.

On the other hand, assumegis unbounded on [, +∞). From condition (H) there is a numberr>  such thatg(u)≤ηuwheneveru>r, whereηis picked with

η 

G(s,s)a(s)dsε. ()

Sincegis unbounded on [, +∞), we can find a numberrsatisfying

r>max

r

r∗ ,r, C

,

r

,

such thatg(u)≤g(r∗) for anyu∈[,r∗]. Take

r:=

uB:u<r∗. ()

Then for eachuK∂r∗we find that

TuH

ϕ(u)+

E H

s,u(s)ds+

G(s,s)a(s)gu(s)ds

H

ϕ(u)–Cϕ(u)+Cϕ(u)+

E

H

s,u(s)–Fu(s)ds

+

E

F

u(s)ds+ 

G(s,s)a(s)grds

Cεϕ(u) +CCu+

E

( +ε)Fu(s)ds+ 

G(s,s)a(s)grds

εCCu+CCu+m(E)C( +ε)u+εu

=CCε+CC+m(E)C( +ε) +ε

u

u, ()

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Therefore, in either case, definer∗=max{r,r∗}, we findTuuwheneveruK

r

. From Lemma ., we claim that problem ()-() has at least one positive solution;

the proof is complete.

Next we are going to give some corollaries sinceϕ(u) admits a wide variety of function-als. First we assumeH(ϕ(u)) = , then, respectively, thatH(ϕ(u)) =

Fu(t)dt,H(ϕ(u)) =

n

i=|ai|u(ξi) andH(ϕ(u)) =

[,]u(t)(t), whereFE(, ) is not Lebesgue null. In addition, we knowu() =  is also well defined and ifu() =  the problem as well as the boundary conditions are similar to [].

Corollary . Assume that conditions(H), (H)-(H)hold,then the problem with

Dirich-let conditions

+u(t) =ft,u(t),  <t< ,

u() = , u() = ,

()

has at least one positive solution.

Corollary . Assume that conditions(H)-(H)hold.Suppose,in addition,

CC+Cm(E) < ,

and FE(, ) is not Lebesgue null,and m(F)≤C,then the problem with integral

conditions

+u(t) =ft,u(t),  <t< ,

u() =

F

u(t)dt+

E H

s,u(s)ds, u() = ,

()

has at least one positive solution.

Corollary . Assume that conditions(H)-(H)hold.Suppose,in addition,

CC+Cm(E) < ,

and E(, )is not Lebesgue null,andni=|ai| ≤C,then the problem with multi-point

conditions

+u(t) =ft,u(t),  <t< ,

u() =

n

i=

|ai|u(ξi) +

E H

s,u(s)ds, u() = ,

()

has at least one positive solution.

Corollary . Assume that conditions(H)-(H)hold.Suppose,in addition,

(13)

and E(, )is not Lebesgue null,and the total variation ofθover[, ]satisfies V[,](θ)≤

C,then the problem with the Lebesgue-Stieltjes integral conditions

+u(t) =ft,u(t),  <t< ,

u() =

[,]

u(t)(t) +

E H

s,u(s)ds, u() = ,

()

has at least one positive solution.

Specially, takeθ(t) =tt, for anyx,y[, ] we have

θ(x) –θ(y)=(xy)(x+y– )≤ |xy|,

then

V[,](θ) =

n

i=

θ(xi) –θ(xi–)≤

n

i=

|xixi–|= n

i=

(xixi–) = ,

thus we have the following corollary.

Corollary . Assume that conditions(H)-(H)and(H)-(H)hold.Suppose,in

addi-tion,

CC< ,

then the problem

+u(t) =f

t,u(t),  <t< ,

u() =

[,]

u(t)dt–t, u() = ,

()

has at least one positive solution.

4 Example

In this part we give an example of Theorem .. Defineϕ(u) in the first place by

ϕ(u) =  u

 –

 u

 +

[,]

u(t)dt, ()

where

ϕ(u) =  u

 –

 u

 , ϕ(u) =

[,]

u(t)dt. ()

Then defineH,Hby

(14)

and

H(x,u) = u+ x+ex

u. ()

It is clear that

lim

z→∞ln(z+ ) +z

z= , ()

and moreover

lim u→∞

|(u+ x+exu) – u|

u = , ()

so conditions (H) and (H) hold, and we seeC= ,C= , andF(u) = u. Now we consider the boundary value problem

+u(t) =ft,u(t),  <t< ,

u() =lnϕ(u) + +ϕ(u) +

 

u(s) + s+esu(s)ds, u() = ,

()

wheref(t,u(t)) is a given function with conditions (H) and (H) satisfied. HereE= [,] is chosen such thatm(E) = andE⊂(, ).

What is more, for eachuK

ϕ(u)≤ u+

 u+

 –

  u=



u ()

and

ϕ(u)≥  r

u. ()

Then we find that C = ∈[, ] andC= r∗> , so condition (H) is met as well. Finally, after straightforward numerical calculations, condition (H) can also be achieved, since

CC+Cm(E) =  · + ·

 < .

As a consequence, each of conditions (H)-(H) is satisfied. From Theorem ., problem () has at least one positive solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments, which have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003), also supported by Graduate Innovation Foundation of University of Jinan (YCX13013).

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doi:10.1186/s13661-014-0225-0

References

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