Three-point boundary value problems of fractional functional differential equations with delay

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R E S E A R C H

Open Access

Three-point boundary value problems of

fractional functional differential equations

with delay

Yanan Li, Shurong Sun

*

_{, Dianwu Yang and Zhenlai Han}

*_{Correspondence:}

[email protected]

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

Abstract

In this paper, we study three-point boundary value problems of the following fractional functional diﬀerential equations involving the Caputo fractional derivative:

C_Dα_u₍_t_{) =}_f

₍

_t_,_ut_,C_Dβ_u₍_t_)), _{0 <}_t_{< 1,}

u(0) = 0, u(1) =

λ

u(η),

whereC_Dα_,C_Dβ_{denote Caputo fractional derivatives, 2 <}

_α

_{< 3, 0 <}

_β

_{< 1,}

_η

_∈_{(0, 1),}

1 <

λ

<₂1_η. We use the Green function to reformulate boundary value problems into an abstract operator equation. By means of the Schauder ﬁxed point theorem and the Banach contraction principle, some existence results of solutions are obtained, respectively. As an application, some examples are presented to illustrate the main results.

MSC: 34A08; 34K37

Keywords: fractional functional diﬀerential equation; delay; three-point boundary value problems; ﬁxed point theorem; existence of solutions

1 Introduction

Fractional calculus is a branch of mathematics, it is an emerging field in the area of the applied mathematics that deals with derivatives and integrals of arbitrary orders as well as with their applications. The origins can be traced back to the end of the seventeenth cen-tury. During the history of fractional calculus, it was reported that the pure mathematical formulations of the investigated problems started to be addressed with more applications in various fields. With the help of fractional calculus, we can describe natural phenom-ena and mathematical models more accurately. Therefore, fractional differential equations have received much attention and the theory and its application have been greatly devel-oped; see [–].

Recently, there have been many papers focused on boundary value problems of frac-tional ordinary differential equations [–] and an initial value problem of fracfrac-tional functional differential equations [–]. But the results dealing with the boundary value problems of fractional functional differential equations with delay are relatively scarce [–]. It is well known that in practical problems, the behavior of systems not only depends on the status just at the present, but also on the status in the past. Thus, in many

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cases, we must consider fractional functional diﬀerential equations with delay in order to solve practical problems. Consequently, our aim in this paper is to study the existence of solutions for boundary value problems of fractional functional diﬀerential equations.

In , Rehman [] studied the existence and uniqueness of solutions to nonlinear three-point boundary value problems for the following fractional diﬀerential equation:

C_Dδ

+u(t) =f

t,u(t),CDσ_+u(t)

, t∈[,T],

u() =αu(η), u(T) =βu(η),

where  <δ< ,  <σ< ,α,β∈R,αη( –β) + ( –α)(t–βη)=  andC_Dδ

+,CDσ_+denote Caputo fractional derivatives. By the Banach contraction principle and the Schauder ﬁxed point theorem, they obtained some new existence and uniqueness results.

For  <r< , we denote byCrthe Banach space of all continuous functionsϕ: [–r, ]→ Rendowed with the sup-norm

ϕ[–r,]:=supϕ(s):s∈[–r, ]

.

Ifu: [–r, ]→R, then for anyt∈[, ], we denote byutthe element ofCrdeﬁned by

ut(θ) =u(t+θ), forθ∈[–r, ].

Enlightened by literature [], in this paper we study the following three-point boundary value problem for the fractional functional diﬀerential equation:

C_Dα

u(t) =ft,ut,CDβu(t)

,  <t< , (.)

where  <α < ,  <β <  andC_Dα_,C_Dβ _{denote Caputo fractional derivatives,}_f₍_t_,_u

t, C_Dβ_u₍_t_{)) is a continuous function associated with the boundary conditions}

u() = , u() =λu(η), (.)

andu=ϕ, whereη∈(, ),  <λ<__η andϕis an element of the space

C_r+() :=ψ∈Cr|ψ(s)≥,s∈[–r, ],ψ() = ,CDβψ(s) = 

.

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2 Preliminaries

For the convenience of the reader, we give the following background material from frac-tional calculus theory to facilitate the analysis of boundary value problem (.) and (.). This material can be found in the recent literature; see [, , ].

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

Iαf(t) =  (α)

t

t

f(s)

(t–s)–αds, t>t,

where(·) is the gamma function, provided that the right-hand side is point-wise deﬁned on (t, +∞).

Deﬁnition .([]) The Caputo fractional derivative of orderα(n–  <α<n) of a func-tionf: (t, +∞)→Ris given by

C_Dα

f(t) =  (n–α)

t

t

f(n)₍_s₎

(t–s)α+–nds, t>t,

where(·) is the gamma function, provided that the right-hand side is point-wise deﬁned on (t, +∞).

Obviously, the Caputo derivative for every constant function is equal to zero.

From the deﬁnition of the Caputo derivative, we can acquire the following statement.

Lemma .([]) Let f(t)∈L_[_t

,∞).Then C_Dα_Iα_f₍_t₎₌_f₍_t_), _t_>_t

and <α< .

Lemma .([]) Letα> .Then

IαCDαf(t) =f(t) –c–ct–· · ·–cntn–

for some ci∈R,i= , , . . . ,n,where n= [α] + and[α]denotes the integer part ofα. Next, we introduce the Green function of fractional functional diﬀerential equations boundary value problems.

Lemma . Let <α< ,  <η< ,  <λ<__η and h: [, ]→Rbe continuous.Then the boundary value problem

C_Dα

u(t) =h(t),  <t< ,

u() =u() = , u() =λu(η),

(.)

has a unique solution

u(t) =





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where

G(t,s)

=  (α)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(t–s)α–₊(α–)t –λη(( –s)

α–_–_λ₍_η_–_s₎α–_), __≤_s_≤_t_≤_,_s_≤_η_, (t–s)α–₊(α–)t

–λη( –s)

α–_, __≤_s_≤_t_≤_,_η_≤_s_, (α–)t

–λη(( –s)

α–_–_λ₍_η_–_s₎α–_), __≤_t_≤_s_≤_,_s_≤_η_, (α–)t

–λη( –s)

α–_, __≤_t_≤_s_≤_,_η_≤_s_.

(.)

Proof From equation (.), we know

IαCDαu(t) =Iαh(t).

From Lemma ., we have

u(t) –c–ct–ct=  (α)

t



(t–s)α–h(s)ds,

u(t) =  (α)

t



(t–s)α–h(s)ds+c+ct+ct. (.)

According to (.), we know that

c=c= ,

u(t) =α–  (α)

t



(t–s)α–h(s)ds+ ct.

Byu() =λu(η), we have

c=

α–  ( – λη)(α)





( –s)α–h(s)ds–λ

η



(η–s)α–h(s)ds

.

Therefore,

u(t) =  (α)

t



(t–s)α–h(s)ds

+ (α– )t 

( – λη)(α)





η



,

u(t) =α–  (α)

t



(t–s)α–_h₍_s₎_ds

+ t(α– ) ( – λη)(α)





η



.

Now, fort≤η, we have

u(t) =  (α)

t



+ (α– )t 

( – λη)(α)

t

 +

η

t +



η

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–λ

t

 +

η

t

=  (α)

t



(t–s)α–₊(α– )t  – λη

( –s)α–_–_λ₍_η_–_s₎α–_h₍_s₎_ds

+  (α)

η

t

(α– )t

 – λη

( –s)α––λ(η–s)α–h(s)ds

+  (α)



η

(α– )t  – λη( –s)

α–_h₍_s₎_ds_.

Fort≥η, we have

u(t) =  (α)

η

 +

t

η

+ (α– )t 

( – λη)(α)

η

 +

t

η

+



t

η



=  (α)

η



(t–s)α–+(α– )t 

 – λη

( –s)α––λ(η–s)α–h(s)ds

+  (α)

t

η

(t–s)α–+(α– )t 

 – λη( –s)

α–

h(s)ds

+  (α)



t

(α– )t  – λη( –s)

α–_h₍_s₎_ds_.

Hence, we can conclude (.) holds, where

G(t,s) =  (α)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(t–s)α–₊(α–)t –λη(( –s)

α–_–_λ₍_η_–_s₎α–_), __≤_s_≤_t_≤_,_s_≤_η_, (t–s)α–₊(α–)t

–λη( –s)

α–_, __≤_s_≤_t_≤_,_η_≤_s_, (α–)t

–λη(( –s)

α–_–_λ₍_η_–_s₎α–_), __≤_t_≤_s_≤_,_s_≤_η_, (α–)t

–λη( –s)

α–_, __≤_t_≤_s_≤_,_η_≤_s_.

The proof is completed.

Lemma .([] Schauder ﬁxed point theorem) Let(D,d)be a complete metric space,U be a closed convex subset of D,and T:D→D be the map such that the set Tu:u∈U is relatively compact in D.Then the operator T has at least one ﬁxed point u*∈U:

Tu*=u*.

3 Main results

In this section, we discuss the existence and uniqueness of solutions for boundary value problem (.) and (.) by the Schauder ﬁxed point theorem and the Banach contraction principle.

For convenience, we deﬁne the Banach spaceX={u|u∈C[–r, ],C_Dβ_u_∈_C_[–_r_{, ],  <}

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C(I), we deﬁne

uI=max t∈I

u(t)+max t∈I

C_Dβ

u(t). (.)

Foru=ϕ, in view of the deﬁnitions ofutandϕ, we have

u=u(θ) =ϕ(θ), forθ∈[–r, ]. Thus, we have

u(t) =ϕ(t), fort∈[–r, ].

Since f : [, ]×Cr ×R→R is a continuous function, set f(t,ut,CDβu(t)) :=h(t) in Lemma .. We have by Lemma . that a functionuis a solution of boundary value prob-lem (.) and (.) if and only if it satisﬁes

u(t) =

⎧ ⎨ ⎩



G(t,s)f(s,us,

C_Dβ_u₍_s₎₎_ds_, _t_∈_{(, ),}

ϕ(t), t∈[–r, ]. We deﬁne an operatorT:X→Xas follows:

Tu(t) =u(t) =

⎧ ⎨ ⎩



G(t,s)f(s,us,

C_Dβ_u₍_s₎₎_ds_, _t_∈_{(, ),}

ϕ(t), t∈[–r, ], (.)

and

l=max ≤t≤





G(t,s)g(s)ds

,

l*=max ≤t≤





_∂∂_tG(t,s)g(s)ds

,

Q= 

( –β)+

 +ληα– ( –β)( –λη)+

 α +

ληα–+   – λη .

Theorem . Assume the following:

(H) There exists a nonnegative functiong∈L[, ]such that

f(t,v,w)≤g(t) +a|v|k+b|w|k

for eachv∈Cr,w∈R,wherea,b∈Rare nonnegative constants and <k,k< ;or

(H) There exists a nonnegative functiong∈L[, ]such that

f(t,v,w)≤g(t) +a|v|k+b|w|k

for eachv∈Cr,w∈R,wherea,b∈Rare nonnegative constants andk,k> .

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Proof Suppose (H) holds. Choose

ω≥max



l+ l

*

( –β)

, (aQ)–k__{, (}_bQ₎–k_

(.)

and deﬁne the coneU={u∈X|u ≤ω,ω> }. For anyu∈U, we have

Tu(t)=





G(t,s)fs,us,CDβu(s)

ds

≤





G(t,s)g(s)ds+a|ω|k+b|ω|k t



(t–s)α– (α) ds

+(α– )t 

 – λη





( –s)α– (α) ds+

(α– )λt  – λη

η



(η–s)α– (α) ds

≤l+a|ω|k+b|ω|k t

α

α(α)+

t

( – λη)(α)+

λtηα– ( – λη)(α)

≤l+a|ω|

k₊_b_|_ω_|k (α)

 α+

ληα–_{+ }  – λη

. (.)

Also,

Tu(t)≤





_∂∂_tG(t,s)fs,us,CDβu(s)ds

≤





_∂∂_tG(t,s)g(s)ds+a|ω|k+b|ω|k t



(t–s)α– (α– )ds

+(α– )t  – λη





( –s)α– (α) ds+

(α– )λt

 – λη

η



(η–s)α– (α) ds

≤l*+a|ω|k+b|ω|kt

α– (α)+

t

( –λη)(α)+

λtηα– ( –λη)(α)

≤l*+a|ω|

k₊_b_|_ω_|k (α)

 + +λη

α–  –λη

.

Hence,

C_Dβ_Tu₍_t₎

≤ 

( –β)

t



(t–s)–βTu(s)ds

≤l* t

–β

( –β)( –β)+

(a|ω|k₊_b_|_ω_|k₎_t–β

( –β)

tα– (α)+

t

( –λη)(α)+

λtηα– ( –λη)(α)

≤ l*

( –β)+

a|ω|k₊_b_|_ω_|k ( –β)(α)

 + +λη

α–  –λη

.

In view of (.) and (.), we obtain

Tu(t)≤l+ l *

( –β)

+a|ω|

k₊_b_|_ω_|k (α)

 ( –β)+

 +ληα– ( –β)( –λη)+

 α +

ληα–_{+ }  – λη

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≤ ω

 +

a|ω|k+b|ω|kQ

≤ ω  + ω  + ω 

≤ω, (.)

which implies thatT:U→U. The continuity of the operatorT follows from the conti-nuity off andG.

Now, if (H) holds, we choose

 <ω≤min



l+ l

*

( –β)

,  aQ 

–k_

,

 bQ



–k_

(.)

and by the same process as above, we obtain

Tu(t)≤l+ l *

( –β)

+a|ω|

k₊_b_|_ω_|k (α)

 ( –β)+

 +ληα– ( –β)( –λη)+

 α +

ληα–_{+ }  – λη

≤ ω

 +

a|ω|k+b|ω|kQ

≤ ω  + ω  + ω 

≤ω,

which implies thatT:U→U.

Now, we show thatTis a completely continuous operator.

LetL=max≤t≤|f(t,ut,CDβu(t))|+ . Then foru∈Uandt,t∈[–r, ] witht<t, in view of Lemma ., if ≤t<t≤, then

Tu(t) –Tu(t)

=





G(t,s)f

s,us,CDβu(s)

ds–





G(t,s)f

s,us,CDβu(s)

ds

≤

t



G(t,s) –G(t,s)L ds+



t

G(t,s) –G(t,s)L ds

+

t

t

G(t,s) –G(t,s)L ds

< L (α)

_t(t–s)α–– (t–s)α–

+(α– )( –s)

α–₍_t –t)  – λη –

λ(η–s)α–₍_α_{– )(}_t –t)  – λη

ds

+



t

(α– )(t –t)  – λη

( –s)α––λ(η–s)α–ds

+

t

t

(α– )(t –t)  – λη

( –s)α–_–_λ₍_η_–_s₎α–_{– (}_t

–s)α–ds

≤ L

(α)





(α– )(t

–t)( –s)α–  – λη ds–

η



(α– )(t

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+

t



(t–s)α–ds–

t



(t–s)α–ds

≤ L

(α)

tα–tα

α +

(t

–t)( –ληα–)  – λη

.

If –r≤t<t≤, then

Tu(t) –Tu(t)=ϕ(t) –ϕ(t).

If –r≤t<  <t≤, then

Tu(t) –Tu(t)≤Tu(t) –Tu()+Tu() –Tu(t)

≤





G(t,s) –G(,s)f

s,us,CDβu(s)ds+ϕ() –ϕ(t)

≤ L

(α)

tα

α +

t

( –ληα–)  – λη

+ϕ(t).

Hence, if ≤t<t≤, we have

C_Dβ

Tu(t) –CDβTu(t)

= 

( –β)

_t(t–s)–βTu(s)ds–

t



(t–s)–βTu(s)ds

≤ 

( –β)

t



(t–s)–βTu(s)ds–

t



(t–s)–βTu(s)ds

+ 

( –β)

_t(t–s)–βTu(s)ds–

t



(t–s)–βTu(s)ds

≤ 

( –β)

_tt(t–s)–βTu(s)ds+

t



(t–s)–β– (t–s)–βTu(s)ds

≤ 

( –β)

_tt(t–s)–β





_∂∂_sG(s,z)fz,uz,CDβu(z)dz

ds

+

t



(t–s)–β– (t–s)–β





_∂∂_sG(s,z)fz,uz,CDβu(z)dz

ds

≤ L( –λη+ληα–)

( –β)( –λη)(α)

t

t

(t–s)–βds+

t



(t–s)–β– (t–s)–β

ds

≤ L( –λη+ληα–)

( –λη)( –β)α)t –β

 –t –β

 .

If –r≤t<t≤, in view of the deﬁnition ofϕ, we have

C_Dβ

Tu(t) –CDβTu(t)=CDβϕ(t) –CDβϕ(t)= .

If –r≤t<  <t≤, then

C_Dβ

Tu(t) –CDβTu(t)

= 

( –β)

t



(t–s)–βTu(s)ds

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≤ 

( –β)

t



(t–s)–β





∂

∂sG(s,z)

fz,uz,CDβu(z)dz

ds

≤ Lt

–β

 ( –β).

Hence, if ≤t<t≤, we have

Tu(t) –Tu(t)

≤ L

(α)

tα

 –tα

α +

(t

–t)( –ληα–)  – λη

+ L( –λη+λη

α–₎ ( –λη)( –β)(α)

t–_ β–t_–β

≤ L

(α)

tα–tα

α +

(t

–t)( –ληα–)  – λη

+  –λη+λη

α– ( –λη)( –β)

t_–β–t–_ β

.

If –r≤t<t≤, we have

Tu(t) –Tu(t)=ϕ(t) –ϕ(t).

If –r≤t<  <t≤, then

Tu(t) –Tu(t)≤  (α)

tα

α +

t

( –ληα–)  – λη

+ϕ(t)+

Lt–_ β

( –β).

In any case, it implies thatTu(t) –Tu(t) → ast→t,i.e., for any> , there exists δ> , independent oft,t andu, such that|Tu(t) –Tu(t)| ≤, whenever|t–t|<δ. ThereforeT:X→Xis completely continuous. The proof is completed.

For convenience, we denote

M= 

(α)

 α+

  – λη+

ληα–  – λη

,

N= 

( –β)(α)

 + +λη

α–  –λη

.

Theorem . Assume that

(H) There exists a constantp> such that|f(t,μ,ν) –f(t,μ¯,ν¯)| ≤p(|μ–μ¯|+|ν–ν¯|)for

eachμ,μ¯∈Cr,ν,ν¯∈R.If

p< (M+N)–,

then boundary value problem(.)and(.)has a unique solution.

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contraction. For eacht∈[, ],

Tu(t) –Tu¯(t)≤





G(t,s)fs,us,CDβu(s)

–fs,u¯s,CDβu¯(s)ds

≤ pu–u¯

(α)

t



(t–s)α–ds+(α– )t 

 – λη





( –s)α–ds

+λ(α– )t 

 – λη

η



(η–s)α–ds

≤ pu–u¯

(α)

tα

α +

t  – λη+

t_ληα–  – λη

≤ pu–u¯

(α)

 α+

  – λη+

ληα–  – λη

≤pu–u¯M. (.)

By a similar method, we get

C_Dβ_Tu₍_t_{) –}C_Dβ_T_u_¯₍_t₎₌ 

( –β)

t



(t–s)–β_Tu₍_s_{) –}_T_u_¯₍_s₎_ds

≤ 

( –β)

t



(t–s)–β





_∂∂_sG(s,z)fz,uz,CDβu(z)

–fz,u¯z,CDβu¯(z)dz

ds

≤ pu–u¯

( –β)

t



(t–s)–β





∂

∂sG(s,z)

dz

ds. (.)

In view of the deﬁnition ofG(t,s), we obtain





∂G_∂(t_t,s)ds

≤

t



(t₍–_αs_{– )})α–ds+(α– )t  – λη





( –₍s_α)α₎–ds+λ(α– )t  – λη

η



(η–₍s_α)α₎–ds

≤ tα–

(α)+

t

( –λη)(α)+

tληα– ( –λη)(α)

≤ 

(α)

 + +λη

α–  –λη

.

Hence,

C_Dβ

Tu(t) –CDβTu¯(t)≤ pu–u¯ ( –β)

t



(t–s)–β

(α)

 + +λη

α–  –λη

ds

≤ pu–u¯

( –β)  (α)

  –β

 + +λη

α–  –λη

≤ pu–u¯

( –β)(α)

 + +λη

α–  –λη

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Clearly, for eacht∈[–r, ], we have|Tu(t) –Tu¯(t)|= . Therefore, by (.) and (.), we get

Tu–Tu¯ ≤pu–u¯M+pu–u¯N≤pu–u¯(N+M)≤ u–u¯

andT is a contraction. As a consequence of the Banach contraction principle, we get that

T has a ﬁxed point which is a solution of boundary value problem (.) and (.).

4 Example

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

Example . Consider boundary value problems of the following fractional functional diﬀerential equations:

C_Dα

u(t) = e t_{– }  +

e–t |ut|

k₊ e–t 

C_Dβ

u(t)k, (.)

u() =u() = , u() =  u





, (.)

whereC_Dα_,C_Dβ_{denote Caputo fractional derivatives,  <}_α_{< ,  <}_β_{< ,}_t_∈_{(, ).}

Chooseλ=_,η=_,φ(t) =e_t–,a=_e–t,b=_e–t and

ft,ut,CDβu(t)

=e t_{– }

 +

e–t |ut|

k₊ e–t 

C_Dβ

u(t)k.

Then, fort∈(, ), we have

ft,ut,CDβu(t)≤φ(t) +a|ut|k+bCDβu(t)k.

For  <k,k< , (H) is satisﬁed and fork,k> , (H) is satisﬁed. Therefore, by Theo-rem ., boundary value problem (.) and (.) has a solution.

Example . Consider boundary value problems of the following fractional functional diﬀerential equations:

C_Dα_u₍_t_{) =} |ut|+|CD β_u₍_t₎_|

( + et_{)( +}|_u

t|+|CDβu(t)|)

, (.)

u() =u() = , u() =  u





, (.)

whereC_Dα_,C_Dβ_{denote Caputo fractional derivatives,  <}_α_{< ,  <}_β_{< ,}_t_∈_{(, ).}

Chooseλ=_,η=_and

ft,ut,CDβu(t)

= |ut|+|

C_Dβ_u₍_t₎_|

( + et_{)( +}_|_u

t|+|CDβu(t)|) .

Set

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Letμ,μ¯∈Cr,ν,ν¯∈R. Then for eacht∈[, ],

f(t,μ,ν) –f(t,μ¯,ν¯)=   + et

|μ|+|ν|  +|μ|+|ν|–

| ¯μ|+|¯ν|  +| ¯μ|+|¯ν|

= |μ+μ¯|–|ν–ν¯|

( + et_{)( +}_{| ¯}_μ_|₊_|¯_ν_|_{)( +}_|_μ_|₊_|_ν_|₎

≤ 

 + et

|μ+μ¯|–|ν–ν¯|

≤ 



|μ+μ¯|–|ν–ν¯|. For eacht∈[–, ],

f(t,μ,ν) –f(t,μ¯,ν¯)=   + et

|μ|+|ν|  +|μ|+|ν|–

| ¯μ|+|¯ν|  +| ¯μ|+|¯ν|

= |μ+μ¯|–|ν–ν¯|

( + et_{)( +}_{| ¯}_μ_|₊_|¯_ν_|_{)( +}_|_μ_|₊_|_ν_|₎

≤ 

 + et

|μ+μ¯|–|ν–ν¯|

≤ 



|μ+μ¯|–|ν–ν¯|.

Thus the condition (H) holds withp=_. Forλ=_,η=_, we have

M= 

(α)

 α+



 – ×_×_ +  (

 )

α–

 – ×_×_

=  (α)

 α+

 +

 

 

α– ,

N= 

( –β)(α)

 + +  (

 )

α–

 –_×_

= 

( –β)(α)

 + +  (

 )

α–  

.

By  <α< ,  <β< , we have

 ≤

 α <

 

and

M>  (α)



, N>

 ( –β)(α).

It implies that

p= 

< . < (M+N) –_.

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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.

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 (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

Received: 6 December 2012 Accepted: 7 February 2013 Published: 22 February 2013 References

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doi:10.1186/1687-2770-2013-38