Local well-posedness for periodic Benjamin equation with small initial data

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R E S E A R C H

Open Access

Local well-posedness for periodic Benjamin

equation with small initial data

Shaoguang Shi

1*

_{and Junfeng Li}

2,3

*_{Correspondence:}

[email protected]

1_{Department of Mathematics, Linyi}

University, Linyi, 276005, P.R. China Full list of author information is available at the end of the article

Abstract

The local well-posedness of the periodic Benjamin equation with small initial value in

Hs(T),s≥–1/2, is given. It is here shown that –1/2 is the lower endpoint to obtain the bilinear estimates which are the crucial steps to obtain the local well-posedness by the Picard iteration.

MSC: 35Q53; 35Q55

Keywords: Benjamin equation; local well-posedness; bilinear estimate

1 Introduction

The initial value problem (IVP) associated to the periodic Benjamin equation is

⎧ ⎨ ⎩

∂tu–H∂xu–∂xu+∂x(u) = , t∈R;

u(x, ) =ϕ(x), ϕ∈Hs₍_T_). (.)

HereT=R\π,His the Hilbert transform

Hf(x) = p.v.  π

f(x–y)

y dy.

The Benjamin equation physically characterizes the vertical displacement (bounded above and below by rigid horizontal planes) of the interface between a thin layer of ﬂuid atop and a much thicker layer of higher density ﬂuid (cf.[]). The more general form of (.) could be stated as

⎧ ⎨ ⎩

∂tu+αH∂xu+β∂xu+∂x(u) = , α,β,t∈R;

u(x, ) =ϕ(x), ϕ∈Hs₍_T_). (.)

As a KdV type equation, (.) models a large amount of physical quantities. Ifα= , β= , (.) is the IVP of the Benjamin-Ono (BO) equation. The BO equation is a model for one-dimensional long waves in deep stratiﬁed ﬂuids and is completely integrable. It is also a typical dispersive model for which the solution map could not be obtained by a Picard approach inL-based Sobolev spaces. For its well-posedness theory, seee.g.[–] and the references therein. If (α,β) = (, –), (.) agrees with the well-known IVP of KdV

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equation. The history of the well-posedness for KdV equation is quite rich and the litera-ture extensive. For the non-periodic case, seee.g.[–]. The periodic case can be found in []. For the local well-posedness of the dispersion generalized periodic KdV equation, see [–] for example. Some recent results in the context of long waves in deep stratiﬁed ﬂuids can be found in [–].

If α= ,β = , then (.) is the IVP of the Benjamin equation. There is some known work as regards the Benjamin type equations; see [–] for example. The well-posedness of the Benjamin equation in the non-periodic case were fully discussed by Chen

et al.in their recent two papers [, ]. For the periodic case, there were less results for

(.) than the non-periodic case. Only in [], Linares obtained the well-posedness of (.) with initial value inL _{space. In this paper, (.) will be taken as our model to study the}

IVP of the periodic Benjamin equation. We will prove that the local well-posedness result of Linares, valid fors≥, can be extended tos≥–/.

For the scaling argument, the followingγ-periodic IVP for Benjamin equation should be considered:

⎧ ⎨ ⎩

∂tu–H∂

xu–∂xu+∂x(u) = , t∈R; u(x, ) =ϕ(x), ϕ∈Hs(,γ).

(.)

Now, we formulate the main result of this paper as follows.

Theorem . For s≥–/,γandϕ∈Hs_,_{there exists T}₌_T₍_ϕ

Hs) > such that(.)

has a unique solution u∈C([,T],Hs₎_∩_Ys_._{The solution map T}_:_ϕ_→_{u is an analytic}

map on Hs_.

The deﬁnitions ofXs_,_Ys_{, and}_Hs_{will be given in Section . To prove Theorem ., the} following bilinear estimates are crucial to our analysis.

Theorem . For s≥–/and f,g∈Xs_,_{the following two bilinear estimates are true}_:

 γ

k∈ ˙Z\γ

|k|s

R

| ˆwfg(k,λ)|

λ+φ(k) dλ

/

fXsg_Xs (.)

and

 γ

k∈ ˙Z\γ

|k|s R

| ˆwfg(k,λ)|

λ+φ(k) dλ

/

fXsg_Xs, (.)

wherewˆfg(k,λ) =_γikfˆ∗γ gˆ(k,λ),φ(k) =k–|k|k.

Since Miura’s transform does not work for the Benjamin equation, the way to set up the ill-posedness of KdV equation fails in dealing with that of the Benjamin equation. However, we can provide a counterexample to show that the bilinear estimates (.) and (.) fail fors< –/.

Theorem . For s< –/, (.)and(.)fail.

Here and in the following, for the sake of convenience, the abbreviationsξ = ( +|ξ|₎/

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from line to line. Similarly,ABis used to representA≤C–_B_and_A_∼_B_{stands for}

ABA.

We end this section with the outline of this paper. In Section , some preliminaries which are essential to our proofs are collected. Section  contains the proofs of Theorem . and Theorem .. In Section , Theorem . is proved by constructing a counterexample to verify that the bilinear estimates (.) and (.) fail fors< –/.

2 Preliminaries

We begin this section with some basic notations. The Fourier transform on [,γ] is deﬁned as

ˆ

u(k) =  γ

γ 

u(x)exp(–ikx)dx.

The Fourier inversion formula is denoted by

u(x) =

exp(πikx)uˆ(k)(dk)γ,

where (dk)γ is the normalized counting measure onZ\γ and

f(x)(dk)γ =

 γ

k∈Z\γ f(k).

It is easy to check that the classical properties of the Fourier transform hold for the mea-sure (dk)γ. That is,

uL_([,_γ_])=ˆu_L₍₍_dk₎ γ),

uv(k) =uˆ∗γvˆ(k) =

ˆ

u(k–k)ˆv(k)(dk)γ.

For more information as regards theγ-periodic Fourier transform, see [] for example. Fors∈R, the spatial Sobolev spaces are deﬁned by

H_xs(,γ) :=u∈S(T) :uHxs=·

s_u_ˆ L₍₍_dk₎

γ)<∞

.

LetPdenote the mean operator

Pu:=

 γ

γ 

u dx,

or equivalentlyPu=uˆ(). The solutions of the Benjamin equation are mean-preserving,

and it will be convenient to assume that

ˆ

u() =  γ

γ 

ϕ(x)dx= ,

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u(t,x–P(u)t) –P(u), which was introduced in [].Ysis the completion of functions

that are Schwartz in time andC∞in space with the following norm:

uYs=u_Xs+k suˆ(k,λ)

L₍₍_dk₎ γ)L(dλ),

where

uXs=k sλ+φ(k)/uˆ(k,λ)

L₍₍_dk₎ γdλ).

In fact, the study of the periodic Benjamin equation has been based on iteration in the spaceXs_{, but this space barely fails to control the}_L∞

t Hxsnorm.Ysis a slight modiﬁcation ofXs_{such that}_u

L∞t HxsuYs.

The nonlinear part of the Benjamin equation isu∂xu, which can be written by a Fourier transform in frequency as

k

k∈ ˙Z\γ

λ∈R

ˆ

u(k,λ)uˆ(k–k,λ–λ)dλ. (.)

During the argument, we mainly focus on the iteration of the two functions in (.), which were understood as two free Benjamin ﬂows iterating with each other. The resonance func-tion

R(k,k) =λ+φ(k) –

λ+φ(k)

–λ–λ+φ(k–k)

= –k|k|+k|k|+ (k–k)|k–k|+ kk(k–k) (.)

gives a description of the set where two Benjamin ﬂows interact. The resonance function is a very important concept in the analysis of the nonlinear dispersion equations. The following fundamental estimates for the resonance function can be proved easily.

Lemma . Let|kk(k–k)| = ,k,k∈Z\γ andγ.Then

R(k,k)kk(k–k), (.)

|k|kk(k–k)

max|k|,|k|



. (.)

Proof Equation (.) is obvious. The proof of (.) is a straightforward calculation with

consideration of the following six cases following from []:

G+++={k–k> ,k> ,k> };

G++–={k–k> ,k> ,k< };

G+––={k–k> ,k< ,k< };

G–––={k–k< ,k< ,k< };

G––+={k–k< ,k< ,k> };

G–++={k–k< ,k> ,k> }.

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Lemma . There exists C> such that for anyε> ,α∈R,and <ρ< ,

I(α) =

R

dβ

( +|β|)( +|α–β|)

log( +|α|)

( +|α|) ; (.)

Iρ(α) =

R

dβ

( +|β|)ρ_{( +}|_α_–_β|₎

 +log( +|α|)

( +|α|)ρ ; (.)

Iε(α) =

R

dβ

( +|β|)+ε_{( +}_|_α_–_β_|₎+ε



( +|α|)+ε. (.)

Lemma . There exists C> such that for anyρ> /,γ,k,k∈Z\γ,andλ,λ∈R,

the following estimates are true:

k=

log( +|λ+φ(k) +φ(k–k)|)

( +|λ+φ(k) +φ(k–k)|)

≤Cγ; (.)

k=

log( +|λ+φ(k) –φ(k–k)|)

( +|λ+φ(k) –φ(k–k)|) ≤

Cγ; (.)

k=

log( +|λ+φ(k) –φ(k–k)|)

( +|λ+φ(k) –φ(k–k)|)ρ ≤

Cγ. (.)

Proof The proof of (.) is given ﬁrst. The fact that, for anyε> ,

log +|a|Cε

 +|a|ε,

gives the following estimate by takingε= /:

k=



( +|λ+φ(k) +φ(k–k)|)/

≤C. (.)

Letβ(k,λ),β(k,λ) be two complex roots ofλ+φ(k) +φ(k–k). In view ofλ+φ(k) +

φ(k–k) being a polynomial of degree  in the variablek, (.) can be rewritten as

k=



( +|k–β(k,λ)||k–β(k,λ)|)/

≤C. (.)

Since

k–βj(k,λ)≥k–Rj(k,λ),

βj(k,λ) can be substituted by its real partRj(k,λ) forj= , . Set

Z=

k:k–βj(k,λ)≤,j= , 

.

ThenZhas no more than γ numbersk, therefore (.) can be controlled by

γ +

|k–βj(k,λ)|>



( +|k–R(k,λ)||k–R(k,λ)|)/

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By the elementary inequality

( +a)( +a)≤( +aa)

forai≥,i= , , it is suﬃcient to estimate

k



( +|k–R(k,λ)|)/( +|k–R(k,λ)|)/

.

And hence (.) follows by the Hölder inequality withkin the above term.

The same conclusion can be drawn for (.) by replacingk withkin (.), while the

proof of (.) can be handled in much the same way as that of (.) for (ρ– /) > .

In the following analysis,ρ= / in (.) is always used. Next, we use the lower bound of the resonance function to recover the derivative on the nonlinear termu∂xu.

Lemma . Let

Qs=| k|s+_|_k

(k–k)|–s

σ(λ,λ,k,k)

and

Qs,r=| k|s+_|_k

(k–k)|–s

σ(–r)₍_λ_,_λ ,k,k)

,

where

σ(λ,λ,k,k) =maxλ+φ(k),λ+φ(k),λ–λ+φ(k–k)

with kk(k–k)= .Then for s≥–/,  <r< /,andγ ,we have

Qs, Qs,r



|n|–r.

Proof It can be conﬁrmed easily by Lemma . that

σ(λ,λ,k,k)kk(k–k).

Putting this lower estimate intoQsandQs,r, we can obtain

Qs≤| k|s+_|_k

(k–k)|–s |kk(k–k)|

|k|s+ |k(k–k)|s+



and

Qs,r≤| k|s+_|_k

(k–k)|–s |kk(k–k)|(–r)

|k|s+ |k|–r_|_k

(k–k)|s+(–r) 

|k|–r,

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3 Proofs of Theorem 1.1 and Theorem 1.2

For a functionf ∈Xs_{, set}

Pf(k,λ) =|k|s

λ+φ(k)/fˆ(k,λ).

Then

fXs=

 γ

k∈ ˙Z/γ

R

Pf(k,λ)



dλ

/

=Pf(k,λ)_L₍₍_dk₎ γdλ),

w_ˆfg(k,λ)  γ|k|

k∈ ˙Z\γ

R

fˆ(k–k,λ–λ)gˆ(k,λ)dλ.

(.)

Sinceuˆ() = , it is convenient to assume thatkk(k–k)= .

We ﬁrst prove Theorem .. The integral in the left hand of (.) can be controlled by

 γ

k

R

|k(k–k)|–s|k|s+PfPgdλ

( +|λ+φ(k)|)/_{( +}|_λ

+φ(k)|)/( +|λ–λ+φ(k–k)|)/

,

wherePf :=Pf(k–k,λ–λ),Pg:=Pg(k,λ). Denote by

Q(k,λ,k,λ) = |

k|s+|k(k–k)|–s λ+φ(k)/_λ

+φ(k)/λ–λ+φ(k–k)/

,

then (.) equals

 γ

k

R

k

RQ(k,

λ,k,λ)PfPgdλ



dλ

/

Pf(k,λ)_L₍_λ_,(_dk₎

γ)Pg(k,λ)L(λ,(dk)γ)

=fXsg_Xs. (.)

Write

A=(k,λ,k,λ) :λ–λ+φ(k–k)≤λ+φ(k),

then by symmetry, the proof of (.) is reduced to the estimate

 γ

k

R

k

R(χAQ)(k,λ,k,λ)PfPgdλ



dλ

/

, (.)

which can be considered by two cases.

Case I.|λ+φ(k)| ≤ |λ+φ(k)|. In this case, setAis replaced by

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Cauchy-Schwarz’s inequality allows us to control (.) by

_γ

k

R(χAIQ)

₍_k_,_λ_,_k

,λ)dλ

 _

γ

k

RP



fPgdλ

 

L₍₍_dk₎ γ)dλ

.

We claim that there existsC>  such that, fors≥–/,

S_I =sup

k,λ

 γ

k

R(χAIQ)

₍_k_,_λ_,_k

,λ)dλC, (.)

which gives

(.)≤SIPf(k,λ)_L₍₍_dk₎

γ)dλPg(k,λ)L((dk)γ)dλfXsgXs. Indeed, the fact

λ+φ(k)≥maxλ+φ(k)

,λ–λ+φ(k–k)

allows us to writeQ_as

Q=|k|

s+_|_k

(k–k)|–s λ+φ(k)



λ+φ(k) λ–λ+φ(k–k)

≈Qs



λ+φ(k) λ–λ+φ(k–k)

.

Lemma . withs≥–/, (k,λ,k,λ)∈AIgives

S_I  γ sup_k_,λ

k

R

dλ

λ+φ(k) λ–λ+φ(k–k)

.

Let

β=λ+φ(k), α=λ+φ(k) +φ(k–k) =τ–R(k,k), τ =λ+φ(k).

Then (.) can be obtained by (.) and (.).

Case II.|λ+φ(k)|<|λ+φ(k)|. In this case, setAis

AII=λ–λ+φ(k–k)<λ+φ(k),λ+φ(k)<λ+φ(k).

That is, we need to estimate the following term:

_γ

k

R(χAIIQ)(k,λ,k,λ)PfPgdλ

L₍₍_dk₎ γdλ)

. (.)

By duality, (.) equals

sup d_L_((_dk_)γ_d_λ)=

 γ

k,k

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Fubini’s theorem and the Cauchy-Schwarz inequality can be used to control (.) by

sup d_L_((_dk_)γ_d_λ)=

 γ

k

R _k

R(χAIIQ)

₍_k_,_λ_,_k ,λ)dλ

×

k

Rd

₍_k_,_λ₎_P

fdλ

dλ

/ gXs.

Write

S_II=  γ ksup,λ

k

R(χAIIQ)

₍_k_,_λ_,_k ,λ)dλ.

Next, one only needs to show that, fors≥–/,

S_II<∞, (.)

which is similar to the previous analysis of (.). For (k,λ,k,λ)∈AII,Qcould be written

as

Q=|k|

s+_|_k

(k–k)|–s λ+φ(k)



λ+φ(k) λ–λ+φ(k–k)

≈Qs



.

Using Lemma .,S

IIcan be estimated by

S_II  γ ksup,λ

k

λ

dλ

.

Let

β=λ+φ(k), α=λ+φ(k) –φ(k–k) =τ–R(k,k), τ= –λ–φ(k).

Then (.) and (.) can be used to obtain (.). The proof of (.) is completed. Next, we proceed the proof of Theorem . by showing (.). We ﬁrst rewrite (.) as

_γ_R

k

R(χAQ)(·,λ,k,λ)PfPgdλdλ

L₍₍_dk₎ γ)

fXsg_Xs,

where

Q(k,λ,k,λ) = |

k|s+_|_k

(k–k)|–s

λ+φ(k) λ+φ(k)/λ–λ+φ(k–k) /

.

As in the proof of (.), (.) can be considered by two cases.

Case I.|λ+φ(k)| ≤ |λ+φ(k)|. In this case, the setAis replaced by

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By duality ofL₍₍_dk₎

γ), one only needs to estimate

sup d_L_((_dk₎

γ)=  γ

k d(k)

R k R(

χAIQ)(k,λ,k,λ)PfPgdλdλ,

which can be bounded by

sup d_L_((_dk₎

γ)=  γ/ k R k R(

χAIQ)

₍_k_,_λ_,_k ,λ)dλ

×

k

Rd

₍_k₎_P

f dλ

dλ

/ gXs

by Cauchy-Schwarz’s inequality. Therefore, it is suﬃcient to show that fors≥–/,

S_I =sup

k  γ k R

R(χAIQ)

₍_k_,_λ_,_k

,λ)dλdλC.

For some  <r< /,

S I =sup

k  γ k R R 

λ+φ(k)r(χAIQr)

₍_k_,_λ_,_k

,λ)dλdλ,

where

Q_r=|k|

s+_|_k

(k–k)|–s λ+φ(k)(–r) ·



λ+φ(k) λ–λ+φ(k–k)

=Qs,r



λ+φ(k) λ–λ+φ(k–k)

.

By Lemma .,S

I could be controlled by

sup k  γ k R R 

|k|–r ·

dλdλ

λ+φ(k)+rλ–λ+φ(k–k)+r

.

Applying (.) withα=λ+φ(k) +φ(k–k), one has

R

dλ

λ+φ(k)+rλ–λ+φ(k–k)+r



( +|λ+φ(k) +φ(k–k)|)+r

;

hence

S_I sup

k  γ k 

|k|–r

R



( +|λ+φ(k) +φ(k–k)|)+r

dλ,

and ﬁnallyS_I <∞forr< /.

Case II.|λ+φ(k)|<|λ+φ(k)|. In this case,Ais replaced by

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Write

 +λ+φ(k)= +λ+φ(k)r +λ+φ(k)–r, where / <r< . As in Case I, let

Qr= |

k|s+_|_k

(k–k)|–s

λ+φ(k)/λ+φ(k)–rλ–λ+φ((k–k))/

.

Then it suﬃces to estimate

 γ k R  γ k R 

λ+φ(k)r(χAIIQr)(k,λ,k,λ)PfPgdλdλ

/

. (.)

Applying the Cauchy-Schwarz inequality inλ, (.) can be bounded by

 γ k R dλ

λ+φ(k)r

R

PfPg γ

k

R(χAIIQr)(k,λ,k,λ)dλ



dλ

/

.

Since r> , (.) is dominated by

 γ k R  γ k R(

χAIIQr)(k,λ,k,λ)PfPgdλ



dλ

/

.

By the deﬁnition of theL₍₍_dk₎_γ_d_λ₎-norm, (.) can be further controlled by

sup d_L₍₍_dk₎

γdλ)=  γ

k

Rd(k,λ)

 γ

k

R(χAIIQr)(k,λ,k,λ)PfPgdλdλ. (.)

By the Cauchy-Schwarz inequality, we can obtain the following estimate for (.):

sup d

L((dk)γdλ)=  γ k R  γ k R(

χAIQr)

₍_k_,_λ_,_k ,λ)dλ

×  γ k Rd

₍_k_,_λ₎_P

fdλ

dλ

/ gXs.

Fors≥–/, we claim that

S_II=sup

k,λ  γ k R(

χAIQr)

₍_k_,_λ_,_k

,λ)dλ<∞.

In fact, by Lemma ., we have

S_II=sup

k,λ  γ k R dλ

λ+φ(k)(–r)_λ_–_λ

+φ(k–k)

.

Applying (.) withρ= ( –r),β=λ+φ(k), andα=λ+φ(k) –φ(k–k) for some / <

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Having disposed of the proof for Theorem ., we can now prove Theorem .. Taking Fourier transform with respect toxin (.) gives

∂tuˆ(k,t) –ik|k|ˆu(k,t) +ikuˆ(k,t) = –wˆ(k,t), uˆ(k, ) =ϕˆ(k), (.)

wherewˆ(k,t) =u∂xu(k,t). TheYsnorm cannot be calculated directly on the equation since the function has no integrability in time. A cutting oﬀ according to time is necessary. Let ψ(t) be a bump function with support in{|t|< }andψ(t) =  for|t|< /. Then the fol-lowing can be shown:

ψ(t)u(x,t) =  γψ(t)

k∈ ˙Z/γ ˆ

ϕ(k)ei(kx–tφ(k))

+ i γ

∞

n=

intn n! ψ(t)

k∈ ˙Z/γ

ei(kx–tφ(k))

×

∞ –∞

ψλ+φ(k)λ+φ(k)n–wˆ(k,λ)dλ

+ i

(γ)ψ(t)

k∈ ˙Z/γ

ei(kx)

∞ –∞

 –ψ(λ+φ(k)) λ+φ(k) e

iλt_w_ˆ₍_k_,_λ₎_d_λ

– i

(γ)ψ(t)

k∈ ˙Z/γ

ei(kx–tφ(k))

∞ –∞

 –ψ(λ+φ(k))

λ+φ(k) wˆ(k,λ)dλ,

wherewˆ(k,λ) = _γikuˆ∗ ˆu(k,λ). We adopt the idea in [] dealing with the KdV equation here to prove Theorem .. LetT denote the map deﬁned by the above expressions of ψ(t)u(x,t). Then the task is to show that the mapu→Tuis contracted in the spaceYs. To do this, we need the following estimates.

Theorem . Let s≥–/.Then there exist constants C=C(ψ) > such that for all u,v∈ Ys_,

TuYs≤Cu

Ys+ϕHs, (.)

Tu–TvYs≤Cu+v_Ysu–v_Ys. (.)

A straightforward application of Theorem . can produce the following proposition.

Proposition . Let C=C(ψ) be the constant appearing in Theorem .. If ϕHs ≤

/(C_),_{then T is a contraction in the closed ball B}_{(, /(}_C₎₎_{of Y}s_.

Therefore, the proof of Theorem . for suﬃciently small data (ϕHs≤/(c)) follows

from Proposition ..

Next, we give the sketch of the proof for Theorem .. Computing theYs_{-norm of the} terms in (.), the following estimate can be obtained:

TuYsϕ_Hs+

w Xs,–  +

 γ

k∈ ˙Z/γ |k|s

R

| ˆw(k,λ)|

λ+ (k) dλ

/

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Then (.) follows from (.) and the bilinear estimates in Theorem . withf =g=u. Noticing thatTu–Tvis a dispersion Benjamin equation with nonlinear termu∂xu–v∂xv with initial dataϕ= , (.) can be proved by the bilinear estimates withf =u+vand

g=u–v.

4 Proof of Theorem 1.3

As explained in the above sections, the method used to prove the local well-posedness of the Benjamin equation is based on the bilinear estimates (.) and (.). In this section, a counterexample will be given to show that (.) and (.) are not true fors< –/ inspired by some ideas from [].

For anyF=Pf(k,λ)∈L((dk)γdλ), the bilinear formBs(F,F) can be deﬁned by

ks+ λ+φ(k)/

k=,k=k

∞ –∞

(k(k–k))–sF(k,λ)F(k–k,λ–λ)dλ λ+φ(k)/λ–λ+φ(k–k) /

.

Then (.) equals

Bs(F,F)_L₍₍_dk₎

γdλ)cF



L₍₍_dk₎

γdλ), (.)

wheres≥–/. A counterexample will be given next to show that (.) fails fors< –/. Choosing

F=akχ

λ+φ(k),

where

χb

λ+φ(k)=

, |λ+φ(k)| ≤b; , elsewhere

and

ak=

, k=N,  –N; , elsewhere,

thenBs(F,F)(k,λ) atk=  can be evaluated as

Bs(F,F)(,λ)≈



( +|λ+φ(k)|)/ ·



Ns.

For thoseλwhich are in a unit size interval ofλ, the following can be shown:

λ+φ(N)≤, λ–λ+φ( –N)≤.

Moreover, the lower bound estimate on the resonance function implies

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Hence, for anyN∈N,



N ·



Ns≤C.

Therefore,s≥–/. The same function can be used to show that (.) fails fors< –/. This completes the proof of Theorem ..

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.

Author details

1_{Department of Mathematics, Linyi University, Linyi, 276005, P.R. China.}2_{Laboratory of Mathematics and Complex}

Systems, Ministry of Education, Beijing, 100875, P.R. China.3_{School of Mathematical Sciences, Beijing Normal University,}

Beijing, 100875, P.R. China.

Acknowledgements

The authors are thankful to the honorable reviewers for their valuable suggestions and comments, which improved the paper. This work was partially supported by National Natural Science Foundation of China (Grant Nos. 11171026, 11271175, 11301249, 11301250), National Natural Science Foundation of Shandong Province (Grant No. ZR2012AQ026), the Applied Mathematics Enhancement Program of Linyi University, and also the Program for Changjiang Scholars and Innovative Research Team in University.

Received: 14 January 2015 Accepted: 27 March 2015

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