Existence and multiplicity of positive solutions for a class of fractional differential equations with three point boundary value conditions

R E S E A R C H

Open Access

Existence and multiplicity of positive

solutions for a class of fractional differential

equations with three-point boundary value

conditions

Bingxian Li

_{, Shurong Sun}

_{, Ping Zhao}

_{and Zhenlai Han}

sshrong@163.com

1_{School of Mathematical Sciences,}

University of Jinan, Jinan, Shandong 250022, P.R. China

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

Abstract

In this paper, we consider the nonlinear three-point boundary value problem of fractional differential equations

Dα₀+u(t) +a(t)f

(t,

α

with boundary conditions

u(0) = 0, Dβ₀+u(0) = 0, D β

0+u(1) =bD β

0+u(ξ), 1≤

β

involving Riemann-Liouville fractional derivativesDα₀+andD β

0+, wherea(t) maybe singular att= 0 ort= 1. We use the Banach contraction mapping principle and the Leggett-Williams fixed point theorem to obtain the existence and uniqueness of positive solutions and the existence of multiple positive solutions. We investigate the above fractional differential equations without many preconditions by the fixed point index theory and obtain the existence of a single positive solution. Some examples are given to show the applicability of our main results.

MSC: 34A08; 34B18

Keywords: fractional differential equations; three-point boundary value problem; existence and multiplicity; fixed point theorem; positive solution

1 Introduction

The development of fractional differential equations is accompanied by fractional calcu-lus; see [–]. With wide applications of fractional calculus in the fields of physical, bio-logical, heat conduction, chemical physics, economics,etc., the study of boundary value problems of fractional differential equations and inclusions gradually become hot issues for many mathematicians [–].

In recent years, some researchers focused themselves on the solutions, especially posi-tive solutions of multipoint boundary value problems of fractional differential equations [–].

In , Liet al.[] considered the existence and uniqueness for nonlinear fractional differential equation of the type

Dα_+u(t) +f

t,u(t)= ,  <t< ,  <α≤, subject to the boundary conditions

u() = , Dβ_+u() =aD

β

+u(ξ), ≤β≤,

whereDα

+ is the standard Riemann-Liouville fractional-order derivative. They obtained

the existence and multiplicity results of positive solutions by using some fixed point the-orems.

In , Jianget al.[] discussed the existence of positive solutions for a multipoint boundary value problem of the fractional differential equation

Dα +u(t) +f

t,u(t)= ,  <t< ,  <α≤,

u() = , Dβ_+u() =

m–

i=

biDβ+u(ξi),  <β< ,

whereDα

+ andDβ_+ are the Riemann-Liouville fractional derivatives. By fixed point index

theory they obtained the existence results.

In , Ahmadet al. [] investigated the existence theory for nonlinear fractional differential equations of the type

c_Dq_{x(t) =ft,x(t),  <t<T,n–  <q≤n,n≥,}

subject to the boundary conditions

x() = , x() = , . . . , x(n–)() = , x(T) =

m

i= γi

Iβi_x(η

i) –Iβix(ζi)

,

wherec_Dq_{are the Caputo fractional derivatives. Their results were based on some}

stan-dard tools of fixed point theory, and they obtained the existence theory for nonlinear fractional differential equations of arbitrary order involving nonintersecting finitely many strips of arbitrary length.

Motivated by excellent results and the methods in [, ], in this paper, we investigate the three-point boundary value problem for the fractional differential equation

Dα

+u(t) +a(t)f

t,u(t)= ,  <t< , (.)

u() = , Dβ_+u() = , Dβ_+u() =bDβ_+u(ξ), (.)

whereDα + andD

β

+are the Riemann-Liouville fractional derivatives,  <α≤, ≤β≤,

The main innovation points of this paper are as follows. (i) Compared with [], we pro-pose some techniques to prove the complete continuity of operators and the properties of the Green function that need not add more premise conditions. (ii) Compared with [], the properties of the Green function happen to change with the order of original equa-tions increasing. We use these new properties to obtain more general results. (iii) In [], to prove the existence of the solutions, the function of two variablesf(t,x) was imposed to some preconditions. In this article, we investigate the fractional differential equations without many preconditions by the fixed point index theory and obtain the existence of single positive solutions. In fact, the fractional differential equations can respond better to impersonal law, so it is very important to weaken preconditions. This work is motivated by the []. (iv) Compared with [], we not only obtained the existence of solutions, but also obtained the uniqueness and multiplicity of solutions with three-point boundary value conditions. Our results extend some known results.

The plan of this paper is as follows. In Section , we shall give some definitions and lemmas to prove our main results. In Section , we establish the existence and uniqueness of single positives solutions to the three-point boundary value problem (.)-(.) by the Banach contraction mapping principle and the fixed point index theory, and we investigate the existence of multiple positives solutions for (.) and (.) by the Leggett-Williams fixed point theorem. In Section , we present examples to illustrate the main results.

In order to facilitate to our study, we make the following assumptions:

(H) f : [, ]×[,∞)→[,∞)is a continuous function;

(H) a(·)∈L(, )is nonnegative,a(t)does not vanish identically on any subinterval of

[, ], and <_a(s)( –s)α–β–_sα–_ds<∞.

2 Preliminaries

For the convenience of the reader, we present some necessary definitions and lemmas from the fractional calculus theory.

Definition .([]) The fractional integral of orderα(α> ) of a functionf : (, +∞)→R is given by

I_α+f(t) =



(α)

t

 f(s) (t–s)–αds,

where(·) is the gamma function, provided that the right side is pointwise defined on (, +∞).

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

Dα +f(t) =



(n–α)

d dt

n t



(t–s)n–α–_f(s)ds,

where(·) is the gamma function, provided that the right side is pointwise defined on (, +∞), andn= [α] +  with [α] standing for the largest integer less thanα.

Definition .([]) LetEbe a real Banach space. A nonempty closed convex setK⊂E

() ifx∈Kandλ> , thenλx∈K; () ifx∈Kand–x∈K, thenx= .

Definition .([]) A mapφis said to be a nonnegative continuous concave functional on a cone K of a real Banach space E ifφ:K→[,∞) is continuous and

φλx+ ( –λ)y≥λφ(x) + ( –λ)φ(y) for allx,y∈Kand ≤λ≤.

Lemma .([]) Letα> –,β> ,and t> .Then

Dβ_+tα=

(α+ )

(α–β+ )t

α–β

.

Lemma .([]) Assume that u(t)∈C(, )∩L(, )and Dα

+u∈C(, )∩L(, )with the

Riemann-Liouville fractional derivative of orderα> .Then I_α+Dα_+u(t) =u(t) +ctα–+ctα–+· · ·+cNtα–N,

where ci∈R,i= , , . . . ,N,and N is the smallest integer greater than or equal toα.

Lemma . For Riemann-Liouville fractional derivatives,we have

Dβ_+ t



(t–s)α–f(s)ds= (α)

(α–β)

t



(t–s)α–β–f(s)ds,

where f ∈L_{(, ),andα,βare two constants withα–β– ≥.} Proof From

Dα

+I_α+f(t) =f(t), I_α+I

β +f(t) =I

α+β + f(t)

we get

Dβ_+ t



(t–s)α–f(s)ds=Dβ_+(α)



(α)

t



(t–s)α–f(s)ds

=Dβ_+(α)I_α+f(t) =(α)Dβ_+I_α+f(t)

=(α)Dβ_+I

β +I

α–β

+ f(t) =(α)I

α–β + f(t)

=(α) 

(α–β)

t



(t–s)α–β–_f(s)ds.

The proof is completed.

The following lemma is fundamental in the proofs of our main results.

such that

d(Tx,Ty)≤ρd(x,y).

Then the operator T has a unique fixed point x∗∈X.

Lemma .(Krein-Rutman []) Let K be a reproducing cone in a real Banach space E,and L:E→E be a compact linear operator with L(K)⊆K and spectral radius r(L).If r(L) > ,

then there existsϕ∈K\{}such that Lϕ=r(L)ϕ.

Lemma .(Fixed point index theory []) Assume that E is a Banach space,K⊂E is a cone,and(K)is a bounded open subset in K.Furthermore,assume that T:(K)→K is a completely continuous operator.Then the following conclusions hold:

(i) If there existsu∈K\{}such thatu=Tu+λufor allu∈∂(K)andλ≥,then

the fixed point indexi(T,(K),K) = ;

(ii) If∈(K)andTu=λufor allu∈∂(K)andλ≥,then the fixed point index

i(T,(K),K) = .

Lemma .(Leggett-Williams fixed point theorem []) Let K be a cone in a real Banach space E,Kc={x∈K:x<c},φbe a nonnegative continuous concave functional on K such

thatφ(x)≤ xfor all x∈Kc,and K(φ,b,d) ={x∈K:b≤φ(x),x ≤d}.Suppose that T:Kc→Kcis completely continuous and there exist positive constants <a<b<d≤c such that

(i) {x∈K(φ,b,d) :φ(x) >b} =∅andφ(Tx) >bforx∈K(φ,b,d),

(ii) Tx<aforx∈Ka,

(iii) φ(Tx) >bforx∈K(φ,b,c)withTx>d.

Then T has at least three fixed points x,x,and xwithx<a,b<φ(x),and a<x withφ(x) <b.

Remark . Ifd=c, then condition (i) implies conditions (iii).

3 Main results

LetE=C[, ] be a Banach space of all continuous real functions on [, ] endowed with normu=sup_{≤t≤}|u(t)|, andKbe the cone

K= u∈E:u(t)≥,t∈[, ]. Obviously,Kis a reproducing cone ofE.

Let the nonnegative continuous concave functionalφon the coneKbe defined by

φ(u) = min ξ≤t≤u(t).

Lemma . Let g(t)∈C(, )∩L_{(, ).Then the boundary value problem of the fractional} differential equation

Dα_+u(t) +g(t) = ,  <t< ,  <α≤, (.)

has the unique solution

Proof In view of Definition . and Lemma ., it is clear that equation (.) is equivalent to the integral form

Then the boundary value problem has the unique solution

u(t) = –

Define the operatorT:K→Eand the linear operatorL:K→Kas follows:

Tu(t) = –

t





(α)a(s)(t–s)

α–_fs,u(s)ds+ t α–

A(α)





a(s)( –s)α–β–_fs,u(s)ds

– bt

α–

A(α)

ξ



a(s)(ξ–s)α–β–fs,u(s)ds

=





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

Lu(t) =





G(t,s)a(s)u(s)ds.

Setg(t) =a(t)f(t,u(t)) in Lemma .. We deduce thatuis a solution of the boundary value problem (.)-(.) if and only if it is a fixed point of the operatorT.

Remark . a(t) may be singular att=  ort= .

Lemma . The function G(t,s)in Lemma.satisfies the following properties:

(i) G(t,s)is continuous on[, ]×[, ]; (ii) G(t,s) > for anyt,s∈(, ); (iii) G(t,s)≤G(,s),for anyt,s∈(, );

(iv) there exists a positive functionγ(s)∈C(, )such that

min

ξ≤t≤G(t,s)≥γ(s)max≤t≤G(t,s) =γ(s)G(,s),  <s< . Proof It is easy to see that (i) holds. So we prove that the rest are true. Let

g(t,s) =

tα–_{( –s)}α–β–_–btα–_(ξ–s)α–β–_–A(t–s)α–

A(α) , ≤s≤min{t,ξ}< ,

g(t,s) =

tα–_{( –s)}α–β–_–A(t–s)α–

A(α) ,  <ξ≤s≤t≤,

g(t,s) =

tα–_{( –s)}α–β–_–btα–_(ξ–s)α–β–

A(α) , ≤t≤s≤ξ< ,

g(t,s) =

tα–_{( –s)}α–β–

A(α) , max{t,ξ} ≤s≤. We will first show that

g(t,s) > , ≤s≤min{t,ξ}< .

Since

g(t,s) =



A(α)

tα–( –s)α–β––btα–(ξ–s)α–β––A(t–s)α–

= 

A(α)

tα– –bξα–β–+bξα–β–( –s)α–β––btα–ξα–β–

 – s

ξ

α–β–

–Atα–

 –s

t

= 

By a similar argument we can conclude that

>(α– )t

α–

A

A

( –s)α––

 –s

t

α–

+bξα–β–

( –s)α–β–_–

 – s

ξ

α–β–

>(α– )t

α–

A

A( –s)α–– ( –s)α– +bξα–β–( –s)α–β–– ( –s)α–β– = .

So we have ∂h(t,s)

∂t > , and thusg(t,s) is increasing with respect toton [s, ].

Next, we show thatg(t,s) is increasing with respect toton [s, ].

Leth(t,s) =g(t,s)(α). Then we have

h(t,s) =

tα–_{( –s)}α–β–_–A(t–s)α–

A ,  <ξ≤s≤t≤,

and

∂h(t,s) ∂t =

(α– )tα– A

( –s)α–β––A

 –s

t

α–

≥(α– )tα–

A

( –s)α–β––A( –s)α–

=(α– )t

α–_{( –s)}α– A

( –s)–β–A

=(α– )(t( –s))

α–

A

( –s)–β+bξα–β–– 

≥(α– )(t( –s))α–

A

( –s)–β– ≥.

So we have ∂h(t,s)

∂t ≥, and thusg(t,s) is increasing with respect toton [s, ].

Then we can conclude thatG(t,s) is increasing with respect totfort∈[, ]. Hence,

G(t,s)≤G(,s) fors,t∈[, ]. On the other hand, we know that

min

ξ≤t≤G(t,s) =

⎧ ⎨ ⎩

minξ≤t≤{g(t,s),g(t,s)}, ≤s≤ξ, minξ≤t≤{g(t,s),g(t,s)}, ξ≤s≤

=

⎧ ⎨ ⎩

g(ξ,s), ≤s≤ξ, g(ξ,s), ξ≤s≤.

Let

γ(s)≤

⎧ ⎨ ⎩

g(ξ,s)

G(,s),  <s≤ξ,

g(ξ,s)

where

The proof is completed.

≤M(t–t) (α)

t



a(s)ds+ M

(α)

_tt



a(s)(t–s)α–ds

+M( +b)(t

α–  –t

α–  ) A(α)



 a(s)ds.

Sincea(t)∈L_{(, ), by the integration of Cauchy’s test for convergence and the uniform}

continuity oftα_,tα–_{on [, ], we can get that{Tu,u∈}is equicontinuous. By the}

Arzela-Ascoli theorem we conclude thatT:K→Kis a completely continuous operator. By the same method we can get thatL:K→Kis a completely continuous operator. The

proof is completed.

Now we use the Banach contraction mapping principle to prove that the boundary value problem (.)-(.) has a unique solution on [, ].

Theorem . Assume that(H)and(H)hold and there exists a constantsuch that

f(t,x) –f(t,y)≤|x–y|

for almost every t∈[, ]and all x,y∈[,∞).

If

 <





G(,s)a(s)ds< ,

then the boundary value problem(.)-(.)has a unique positive solution on[, ].

Proof By Lemma .,T:K→K. Moreover, for anyu,v∈K, we deduce

Tu(t) –Tv(t)=





G(t,s)a(s)fs,u(s)–fs,v(s)ds

≤ 



G(t,s)a(s)fs,u(s)–fs,v(s)ds

≤





G(,s)a(s)u(s) –v(s)ds

≤





G(,s)a(s)dsu–v.

This implies that

Tu–Tv ≤αu–v,

whereα=_G(,s)a(s)ds∈(, ). SoTis a contraction mapping. Hence, by the Banach contraction mapping principle the boundary value problem (.)–(.) has a unique posi-tive solution on [, ]. The proof is completed.

Remark . For a∈C[, ], the integral condition reduces to σa< , where σ =



Now we study the existence of solutions for the boundary value problem (.)-(.) by the fixed point index theory.

Lemma . Assume that(H)-(H)hold.Then the spectral radius of the operator L is

Define

f=lim inf

u→+ _tmin_∈[,]

f(t,u)

u , f∞=lim sup

u→∞

max

t∈[,] f(t,u)

u , Kc= x∈K:x<c

,

r(L) = 

μ, μ∈R +_.

Lemma . Assume that(H)-(H)hold andμ<f≤ ∞.Then there existsρ> such that forρ∈(,ρ],if u=Tu,u∈∂Kρ,then i(T,Kρ,K) = .

Proof It follows fromμ<fthat there existε>  andρ>  such that fort∈[, ] and

≤u≤ρ,

f(t,u)≥(μ+ε)u. (.)

For  <ρ<ρ, assume thatu=Tu,u∈∂Kρ. By Lemma . and Lemma .(i) we need

only to prove that

u=Tu+λϕ, λ> ,

whereϕ∈K\ {}withLϕ=r(L)ϕ.

Otherwise, there existu∈∂Kρandλ>  such that

u=Tu+λϕ. (.)

Then,u≥Tuandu≥λϕ. By (.) we get that

Tu(t) =





G(t,s)a(s)fs,u(s)

ds≥(μ+ε)Lu(t). (.)

Consideringu≥λϕ, we have

Lu≥λLϕ.

ForLϕ=r(L)ϕ, (μ+ε)r(L) > , so that (μ+ε)r(L)ϕ>ϕ. So we can conclude

Tu≥(μ+ε)λLϕ>λϕ.

Together with (.), we haveu≥λϕ. By (.) we haveTu≥λϕ. Sou≥λϕ.

Re-peating this process, we get thatu≥nλϕ, so that we haveu ≥nλϕ → ∞,n→ ∞.

This is a contradiction.

It follows from Lemma .(i) thati(T,Kρ,K) = ,ρ∈(,ρ]. The proof is completed.

Lemma . Assume that(H)-(H)hold and≤f∞<μ.Then there existsτ> such that for eachτ>τ,ifλu=Tu,u∈∂Kτ,then i(T,Kτ,K) = .

Proof Let>  satisfyf∞<μ–. Then there existsτ>  such that foru>τandt∈[, ], f(t,u)≤(μ–)u. (.) Set(t) =maxu∈[,τ]f(t,u). Then, for allu∈R+andt∈[, ], we have

f(t,u)≤(μ–)u+(t). (.) Let

F=





a(s)G(t,s)(s)ds, τ=

_μF_–μI_– –L

–

.

Takeτ>τ. We will show thatλu=Tufor allu∈∂Kτ andλ≥.

Otherwise, there existu∈∂Kτ andλ≥ such that

Tu=λu. (.)

Together with (.), we have

u≤λu=Tu≤(μ–)Lu+F.

Then (_μI_– –L)u(t)≤ μF– fort∈[, ]. So we deduce

F

μ– – (

I

μ– –L)u∈K. It follows

fromL(K)⊂Kthatu(t)≤μF–(

I

μ– –L)–fort∈[, ]. Therefore, we haveu ≤τ<τ.

This is a contradiction. So we can conclude that

Tu=λu

for allu∈∂Kτ andλ≥.

By Lemma .(ii) we get thati(T,Kτ,K) =  for eachτ<τ. The proof is completed.

Theorem . Suppose that(H)-(H)hold,μ<f≤ ∞,and≤f∞<μ.Then the bound-ary value problem(.)-(.)has at least one positive solution on[, ].

Proof It follows fromμ<f≤ ∞and Lemma . that there exists  <ρ<τ such that

either there existsu∈∂Kρwithu=Tuori(T,Kρ,K) = . By ≤f∞<μand Lemma .

there existsτ >  such thati(T,Kτ,K) = . So we can conclude thatT has a fixed point u∈K withρ<u<τ by the properties of index. Hence, the boundary value problem (.)-(.) has at least one positive solution on [, ]. The proof is completed. Now we study the multiplicity of solutions for the boundary value problem (.)-(.) by the Leggett-Williams fixed point theorem.

(i) f(t,u) <Ma,(t,u)∈[, ]×[,a], (ii) f(t,u)≤Mc,(t,u)∈[, ]×[,c], (iii) f(t,u)≥Nb,(t,u)∈[ξ, ]×[b,c],

where γ(s)∈(, ), M= (_a(s)G(,s)ds)–_{, and N = (}

ξ a(s)γ(s)G(,s)ds)

–_{. Then the} boundary value problem(.)-(.)has at least three positive solutions u,u,and uwith

u<a, b<φ(u) <u ≤c, and a<u, φ(u) <b.

Proof Ifu∈Kc, thenu ≤c. So ≤u(t)≤c,t∈[, ]. By condition (ii) we have

_Tu(t)=  

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

≤ 



G(,s)a(s)Mc ds

=Mc





a(s)G(,s)ds=c,

which implies thatTu ≤c,u∈Kc. Hence,T:Kc→Kc. In view of Lemma .,T:Kc→

Kcis completely continuous.

Next, by using the analogous argument it follows from condition (i) that ifu∈Ka, then

Tu<a.

Chooseu(t) = b+c

 ,t∈[, ]. It is easy to see thatu(t)∈K(φ,b,c),φ(

b+c

 ) =

b+c

 >b.

Therefore,{u∈K(φ,b,c)|φ(u) >b} =∅.

On the other hand, ifu∈K(φ,b,c), thenb≤u(t)≤c,t≤[ξ, ]. By condition (iii) we

have

ft,u(t)≥Nb.

Hence,

φ(Tu) = min ξ≤t≤

_Tu(t)= min ξ≤t≤





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

≥ min

ξ≤t≤





G(t,s)a(s)Nbds

>



ξ

γ(s)G(,s)a(s)Nbds

=Nb



ξ

a(s)γ(s)G(,s)ds=b,

which implies thatφ(Tu) >bforu∈K(φ,b,c).

In conclusion, by Lemma . and Remark . the boundary value problem (.)-(.) has at least three positive solutionu,u, anduwith

u<a, b<φ(u) <u ≤c, and a<u, φ(u) <b.

4 Examples

In this section, we present some examples to illustrate our main results.

Example . Consider the following boundary value problem:

D

 

+u(t) +a(t)

et

( +et_{)( +u)}+sin

_{t+ }

= ,  <t< , (.)

u() = , D

 

+u() = , D  

+u() =bD  

+u(ξ), (.)

whereb=_ andξ=_. Here

α=

, β= 

, a(t) = ( –t)

t

, f(t,u) = e

t

( +et_{)( +u)}+sin

_{t+ ,}

(t,u)∈[, ]×[,∞],=  .

It is clear that|f(t,u) –f(t,v)| ≤|u–v|for (t,u), (t,v)∈[, ]×[,∞]. Sincea(t) is singular att= , by simple calculation we have

 <





G(,s)a(s)ds≤





( –s)( –s)α–β–– ( –bξα–β–)( –s)α–( –s)

s( –bξα–β–)(_α) ≈. < .

By Theorem . we see that the boundary value problem (.)-(.) has a unique solution.

Example . Consider the following boundary value problem:

D

 

+u(t) +a(t)f(t,u) = ,  <t< , (.)

u() = , D

 

+u() = , D  

+u() =bD  

+u(ξ), (.)

where

f(t,u) =

u_+t_{, u}∈_{[, ],t}∈_{[, ],}

u+t, u∈(,∞),t∈[, ],

andb=  ,ξ=

 .

Hereα=_,β=_,a(t) =_–_t. It is easy to see that (H) and (H) are satisfied. By simple

calculation we can conclude that

f=∞, f∞= .

Example . Consider the following boundary value problem:

D

 

+u(t) +a(t)f(t,u) = ,  <t< , (.)

u() = , D

 

+u() = , D  

+u() =bD  

+u(ξ), (.)

where

f(t,u) =

u₊ t

, u∈[, ],t∈[, ],

, u> ,t∈[, ],

andb= ,ξ=

 .

Hereα=_,β=_,a(t) = . By simple calculation we have

A=  –bξα–β–= .,





a(s)G(,s)ds=





( –s)α–β–_{– ( –bξ}α–β–_{)( –s)}α–

( –bξα–β–_)(α) ds≈.,

and



ξ

γ(s)a(s)G(,s)ds=



 

g(ξ,s)ds≈..

Hence,M= (_a(s)G(,s)ds)–_{≈. andN= (}

ξγ(s)a(s)G(,s)ds)

–_{≈..}

Choosinga=_,b= , andc= , we have

f(t,u) = u+ t

≤. <Ma= ., (t,u)∈[, ]×

, 

,

f(t,u) = u₊ t

≤. <Mc= ., (t,u)∈[, ]×[, ],

f(t,u) = ≤Mc= ., (t,u)∈[, ]×(, ],

f(t,u) = ≥Nb= ., (t,u)∈

 , 

×(, ].

Hence, by Theorem . we obtain that the boundary value problem (.)-(.) has at least three positive solutionsu,u, andusuch that

u<



,  <φ(u) <u ≤, and 

<u, φ(u) < .

5 Conclusion

and we investigate the existence of multiple positive solutions for (.)-(.) by the Leggett-Williams fixed point theorem. As applications, examples are presented to illustrate the main results.

We propose some techniques to prove the completely continuity of operators and the properties of the Green function that need not adding more preconditions. With the ele-vation of order, the properties of the Green function happen to change. We use these new properties to obtain more general results. We investigate the fractional differential equa-tions without many precondiequa-tions by the fixed point index theory and obtain the existence of single positive solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China.}2_{School of Control Science}

and Engineering, University of Jinan, Jinan, Shandong 250022, P.R. China.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (61374074, 11571202) and by the Shandong Provincial Natural Science Foundation (ZR2013AL003).

Received: 15 May 2015 Accepted: 28 November 2015 References

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