R E S E A R C H
Open Access
On the regularity of solutions of the
boundary value problem without initial
condition for Schrödinger systems in domain
with conical points
Nguyen Thi Lien
1*and Nguyen Manh Hung
2*Correspondence:
[email protected] 1Department of Mathematics,
Hanoi National University of Education, Hanoi, Vietnam Full list of author information is available at the end of the article
Abstract
In this paper, we deal with the boundary value problems without initial condition for Schrödinger systems in cylinders. We establish several results on the regularity of the solutions.
MSC: 35Q41; 35B65; 35D30
Keywords: regularity of generalized solution; problems without initial condition; cylinders
1 Introduction
The theory of initial boundary value problems for partial differential equations and sys-tems in nonsmooth domains has attracted the attention of many researchers. These prob-lems for Schrödinger systems in domain with conical points are considered in [, ], in which authors consider the existence, the uniqueness and the regularity of solutions of the mentioned problems. While problems without initial conditions arise when describ-ing different nonstationary processes in nature under the hypothesis that the initial con-dition practically has no influence at the present time. These problems are investigated in many works, see [–] for example, with results only about the well-posedness. In [], we studied the boundary value problem without initial condition for Schrödinger systems in domain with conical points. By studying the corresponding problem with initial condition
t=h, then passing to the limitsh→–∞, we obtained the existence and uniqueness of the solution. In the present paper, we continue considering the mentioned problem and our main aim here is to study the regularity of the solution.
There are four sections in this paper. In Section , we set the problem and recall an obvious result about the unique existence of solution. Afterwards, in Sections and , by a similar method to [, ], we give results on the regularity of problems with initial condition
t=hfor Schrödinger systems in domains with conical points. Then, by lettingh→–∞, the smoothness of the generalized solution of our problem is obtained in Section .
2 Setting problem and obvious result
Letbe a bounded domain inRn(n≥) with boundaryS=∂. Assume thatSis an
infinitely differentiable surface everywhere, except for the coordinate origin, in a
borhoodUsatisfying∩Uand coincides with the coneK={x:x/|x| ∈G}, whereG
is a smooth domain on the unit sphereSn–. Fora<b, setb
a=×(a,b),Sba=S×(a,b). If (a,b) =R, we useR to refer to ∞–∞ andSR to refer to S∞–∞. For each multi-index
α= (α, . . . ,αn)∈Nn, set|α|=α+· · ·+αnand∂α=∂xα· · ·∂ αn
xn.
Denoteu(x,t) = (u(x,t), . . . ,us(x,t)),Dαu= (Dαu, . . . ,Dαus),|Dαu|= s
i=|Dαui|and utj= (∂
ju ∂tj , . . . ,
∂jus ∂tj ),|utj|
=s
i=| ∂jui
∂tj|
.
Let us introduce some functional spaces (see []) used in this paper.
We useHk() to denote the space ofs-dimensional vector functions defined inwith the norm
uHk()=
k
|α|=
Dαudx
.
Denote byHk,l(b
a) the space consisting of all vector functionsu:ba−→Cssatisfying
uHk,l(b a)=
ba
k
|α|=
Dαu+ l
j=
|utj|
dx dt
,
andHk,l(–γ,b
a) is the space of vector functions with norm
uHk,l(–γ,b a)=
ba
k
|α|=
Dαu+ l
j=
|utj|
e–γtdx dt
.
In particular,
uHk,(–γ,b a)=
k
|α|=
ba
Dαue–γtdx dt
.
Especially, we setL(–γ,ba) =H,(–γ,ba).
Hβl,k(–γ,ba) is the space of all functionsu(x,t) which have generalized derivativesDαu,
∂ju
∂tj,|α| ≤l, ≤j≤k, satisfying
u
Hlβ,k(–γ,b a)=
ba
l
α=
r(β+|α|–l)Dαu+ k
j=
|utj|
e–γtdx dt<∞.
Hl
β(–γ,ba) is the space of all functionsu(x,t) which have generalized derivativesDαutj,
|α| ≤l, ≤j≤l, satisfying
uHl
β(–γ,
b a)=
ba
l
α+j=
r(β+|α|+j–l)Dαutje–γtdx dt<∞.
Denote byH◦k,l(–γ,b
a) the completion of infinitely differentiable vector functions vanish-ing nearSb
Now we introduce a differential operator of order m
L(x,t,D) = m
|p|,|q|=
(–)|p|Dpapq(x,t)Dq ,
whereapqares×smatrices with the bounded complex-valued components inR,apq=
a∗qp(a∗qpis the transposed conjugate matrix toapq). Set
B(t,u,v) = m
|p|,|q|=
apqDquDpv dx, t∈R.
We assume further that the formB(t,·,·) satisfies the following condition: there exists a constantμ> independent oftandusuch that
B(t,u,u)≥μu(·,t)
Hm() (.)
for allu∈H◦m() and a.e.t∈R, whereH◦m() is the closure inHm() of infinitely differ-entiable complexs-dimensional vector functions with compact support in.
Now we consider the following problem in the cylinderR:
(–)m–iL(x,t,D)u–ut=f(x,t) inR, (.)
∂ju
∂νj
SR= , j= , . . . ,m– , (.)
whereνis the unit vector of outer normal to the surrounding surfaceSR. Letf ∈L(–γ,R), a complex vector-valued functionu∈H◦m,(–γ,
R) is called a gen-eralized solution of problem (.)-(.) if and only if, for anyT> , the equality
(–)m–i
T
–∞B(t,u,η)dt+
T–∞
uηtdx dt=
T–∞
fηdx dt (.)
holds for allη∈H◦m,(γ,
R),η(x,t) = witht≥T.
For anyh∈R, we also consider the following problem with initial condition correspond-ing to problem (.)-(.):
(–)m–iL(x,t,D)v–vt=f(x,t) in∞h , (.)
v|t=h= , x∈, (.)
∂jv
∂νj
S∞h = , j= , . . . ,m– . (.)
A functionv∈H◦m,(–γ,∞
h ) is a generalized solution of problem (.)-(.) if and only if, for anyT> , we have
(–)m–i
T
h
B(t,v,η)dt+
Th
vηtdx dt=
Th
fηdx dt (.)
for allη∈H◦m,(γ,∞
We recall the results about the solvability of problem (.)-(.) proved in [].
Theorem . Assume that:
(i) sup{|∂apq
∂t |: (x,t)∈R, ≤ |p|,|q| ≤m}=μ<∞;
(ii) |∂apq
∂t | ≤μ·eγt,for all(x,t)∈R,≤ |p|,|q| ≤m;
(iii) f,ft∈L(–γ,R),
whereμ,μare positive constants.Then for allγ >γ=mμμ,m =
|α|≤m,there exists a
uniquely generalized solution u∈H◦m,(–γ,
R)of problem(.)-(.)satisfying
u◦
Hm,(–γ,R)≤C
fL(–γ,R)+ftL(–γ,R).
3 The regularity with respect to time variable of the problem with initial condition
The regularity of problem (.)-(.) is implied by analogous property for problem (.)-(.). In [], the regularity with respect to time variable of problem (.)-(.) is studied in the caseh= andftk ∈L∞(,∞;L()). Now, by a similar method, we consider the
problem in the caseh∈Randftk∈L(–γ,∞h ). Moreover, we also show that the constants Cin all prior estimates do not depend onh.
Theorem . Assume that l∈Nand there existsμ> such that
(i) sup{|∂a∂pqt |: (x,t)∈R, ≤ |p|,|q| ≤m,}=μ<∞,|∂k∂atkpq| ≤μ,μ> ,for
≤k≤l+ ;
(ii) ftk∈L(–γ,∞h ),fork≤l+ ,ftk(x,h) = for allx∈,≤k≤l. Then for allγ > (l+ )γ,the solution v∈
◦
Hm,(–γ,
R)of problem(.)-(.)has
deriva-tives with respect to t up to order l satisfying vtk∈Hm,(–γ,∞h ),∀k= ,l,such that
vtk
Hm,(–γ,∞ h)≤C
k+
m=
ftmL(–γ,∞
h), (.)
where the constant C does not depend on v,f,h.
Proof Suppose that{ϕk}∞k= is an orthogonal basis of Hm() which is orthonormal in
L(). For anyN∈N, the approximating solutions can be found in the formvN(x,t) = N
k=CkN(t)ϕk(x) where (CNk(t))Nk=is the solution of the ordinary differential system
(–)m–iBt,vN,ϕk –
vNt ϕkdx=
fϕkdx, (.)
CkN(h) = , k= , . . . ,N. (.)
From the assumptions it follows that the coefficientsCkN(t), defined uniquely by (.), have derivatives up to order l + andvN(x,h) = . It is very easy to check thatDpvN(x,h) = , ∀≤ |p| ≤m,x∈.
Step : Prove thatDpvN
Taking the derivative (k–) times of both sides of (.) with respect tot, then multiplying both sides by dk
dtk(CkN(t)) and taking the sum with respect tokfrom toN, we arrive at
–i
vNtk
dx+ (–)m m
|p|,|q|=
apqDqvNtk–DpvNtkdx
=i
ftk–vN
tkdx+ (–)
m–
m
|p|,|q|= k–
s=
k–
s
∂k–s–apq ∂tk–s– D
qvN tsDpvN
tk–s–dx. (.)
Because of the fact thatftk(x,h) = for allx∈, ≤k≤l– , using induction with respect
tokfrom (.) we obtainDpvNtk(x,h) = for all ≤k≤l, ≤ |p| ≤m,x∈.
Step :A prioriestimate forvNtk.
Fork=l+ , adding (.) to its complex conjugate, integrating with respect totfromh
toτ, and then integrating by parts, we get
(–)mBτ,vNtl(x,τ),vNtl(x,τ)
= (–)m m
|p|,|q|=
τ
h
∂apq ∂t D
qvN
tlDpvNtl dx dt
+ (–)mRe m
|p|,|q|= l·
τh
∂apq ∂t D
qvN
tlDpvNtl dx dt
+ (–)mRe m
|p|,|q|=
l–
s=
l s τh
∂l–s+apq ∂tl–s+ D
qvN tsDpvN
tldx dt
+ (–)mRe m
|p|,|q|=
l–
s=
l s τh
∂l–sapq ∂tl–s D
qvN
ts+DpvNtldx dt
– (–)mRe m
|p|,|q|=
l– s= l s
∂l–sapq ∂tl–s (x,τ)D
qvN
ts(x,τ)DpvN
tl(x,τ)dx
+ Im
ftl(x,τ)vN
tl(x,τ)dx– Im
τh ftl+vN
tldx dt. (.)
We denote byI,II,III,IV,V, andVIthe terms from the first, second, third, fourth, fifth, sixth, and seventh, respectively, of the right-hand sides of (.). We will give estimations for these terms. Using Cauchy’s inequality, for any> , we have
I≤μ
m
|p|,|q|=
τh
DqvNtl
+DpvNtl
dx dt=μm
τ
h
vNtl(·,t)
Hm()dt,
II≤lμm
τ
h
vNtl(·,t)
Hm()dt,
III≤μml–
τ
h vN
tl(·,t)
Hm()dt+C
l– s= τ h vN
ts(·,t)
Hm()dt,
IV≤μm
l–l–
τ
h
vNtl(·,t)
Hm()dt+C
l–
s=
τ
h
vNts(·,t)
V≤μm l– vNts(·,τ)
Hm()+C
l–
s=
vNts(·,τ)
Hm(),
VI≤Cftl(·,τ)
L()+
τ
h
ftl+(·,t)
L()dt
+
vN tl(·,τ)
Hm()+
τ
h vN
tl(·,t)
Hm()dt
,
whereC=max{μmM,},M=maxs=,...,l–
l s. Denote= ((l+– –l)μm + )
> forl> , then for all <<μ, using above
estimations and (.), we obtain
vNtl(·,τ)
Hm()≤
(l+ )μm + μ–
:=αl τ
h
vNtl(·,t)
Hm()dt+C·
l–
k=
vNtk(·,τ)
Hm()
+C·
l–
k=
τ
h
vNtk(·,t)
Hm()dt
+C·ftl(·,τ)
L()+C·
τ
h
ftl+(·,t)
L()dt, (.)
whereCis a positive constant.
On the other hand, in [] one already has the following estimate:
vN(·,τ)Hm()≤α
τ
h
eα(τ–t)f(·,t)L()+ τ
h
fs(·,s)
L()ds
dt
+Cf(·,τ)L
()+
τ
h
ft(·,t)
L()dt
. (.)
Multiplying both sides of (.) bye–γ τ, then integrating with respect toτ fromhto∞
we obtain
vNHm,(–γ,∞ h)≤C
fL(–γ,∞
h)+ft
L(–γ,∞h) , (.)
where the constantCdoes not depend onh. Now using (.) and (.) forl= yields
vNt Hm,(–γ,∞ h)
≤CftL(–γ,∞
h)+ft
L(–γ,∞
h)+
vNHm,(–γ,∞ h)
+C·
∞
h
e–γ τ
τ
h
eα(τ–t)vN(·,t)Hm()+ft(·,t)L() dt dτ
+C·
∞
h
e–γ τ
τ
h
eα(τ–t)
t
h
vN(·,s)
Hm()+ft(·,s)
L() ds dt dτ. (.)
Remember that
inf
μ>>α=μ>>inf
μm +
μ–
So we can choosesmall enough such thatα<γ. By (.) and using the Fubini theorem
we obtain
vNt Hm,(–γ,∞ h)≤C
k=
ftkL(–γ,∞
h)+
vNHm,(–γ,∞ h)
,
which implies
vNt Hm,(–γ,∞ h)≤C
k=
ftkL
(–γ,∞h). (.)
Repeating the above arguments, from (.) one can prove for anyl, ifftk∈L(–γ,∞h ), k= ,l+ , thatvN has derivative with respect totup to thelth and for anyk= ,l
vNtk
Hm,(–γ,∞ h)≤C
k+
m=
ftmL(–γ,∞
h). (.)
The inequality (.) implies that{vNtk}is uniformly bounded inHm,(–γ,∞h ). By a stan-dard weakly convergence argument, we can conclude that the sequence{vNtk}∞N=possesses
a subsequence convergent to a vector functionvtk inHm,(–γ,∞h ). Moreover, it follows
from (.) that (.) holds.
4 Further results on the regularity of solution of problem with initial condition
Assume thatωis a local coordinate system onSn–. Moreover, assume that the principal
part of the operatorL(x,t,D) at origin can be written in the form
L(,t,D) =r–mQ(ω,t,rDr,Dω), Dr=
i∂ ∂r,
where Qis a linear operator with smooth coefficients. Consider the following spectral
problem:
Q(ω,t,λ,Dω)v(ω) = , ω∈G; (.)
Djωv(ω) = , ω∈∂G,j= , . . . ,m– . (.)
Proceeding similarly to Lemma . in [], the following lemma holds.
Lemma . Let f,ft,ftt∈L(–γ,Kh∞),γ > (m+ )γand f(x,h) =ft(x,h) = .If v(x,t)is
a generalized solution of problem(.)-(.)in the spaceH◦m,(–γ,K∞
h )such that u≡
whenever|x|>R,R=const.,then v∈Hmm,(–γ,Kh∞)and the following estimate holds:
v
Hmm,(–γ,Kh∞)≤
CfL(–γ,K∞
h )+ft
L(–γ,Kh∞)+fttL(–γ,Kh∞)
, (.)
where the constant C is independent of v,f,h.
strip
m–n
≤Imλ≤m–
n
does not contain points of the spectrum of problem(.)-(.)for every t∈[h,∞).Then v∈Hm
(–γ,Kh∞)and the following estimate holds:
vHm
(–γ,Kh∞)≤C m+
k=
ftkL(–γ,K∞
h), (.)
where the constant C is independent of v,f,h.
Proof First, we prove that
vts
Hm,(–γ,Kh∞)≤C m+
k=
ftkL
(–γ,Kh∞), (.)
whereCdoes not depend onh,s≤m.
Doing the same in [, Proposition .] we have
vHm (K)≤C
fL(K)+vtL(K)+vHm m (K)
,
for a.e.t∈[h,∞), whereC= constant.
Multiplying both sides of this inequality bye–γt, then integrating with respect totfrom
hto∞, we get
v
Hm,(–γ,Kh∞)≤
CfL(–γ,K∞
h )+vt
L(–γ,Kh∞)+v
Hm,(–γ,Kh∞)
.
By (.)
vtL(–γ,K∞
h )≤C
k=
ftkL (–γ,Kh∞),
and by Lemma .
v
Hm,(–γ,Kh∞)≤ v
Hmm,(–γ,Kh∞)≤ C
k=
ftkL(–γ,K∞
h ),
from which it follows that
v
Hm,(–γ,Kh∞)≤ C
k=
ftkL(–γ,K∞
h ).
Now assume that (.) is true fors– . By differentiating the systems (.)stimes with respect totand by puttingw=vts, we obtain
(–)m–Lw= –i(w
t+fts) + (–)m
s
k=
s k
Ltkvts–k,
whereLtk=m|p|,|q|=Dp( ∂ka
pq ∂tk Dq).
From the inductive hypothesis and repeating the arguments of the proof in the cases= , the inequality (.) holds fors.
Since
v
Hm(–γ,Kh∞)≤ C
m–
s=
vts
Hm,(–γ,Kh∞)
+vtm
H,(–γ,Kh∞)
,
from (.) and Theorem ., (.) is true. The lemma is proved.
Theorem . Let l be a nonnegative integer.Assume that v(x,t)is a weak solution in the space Hm,(–γ,∞
h )withγ > ((m+l)+)γof problem(.)-(.)and ftk∈Hl,(–γ,∞h ), for k≤m+l+ ,ftk(x,h) = for k≤m+l.In addition,suppose that in the strip
m–n
≤Imλ≤m+l–
n
there is no point from the spectrum of (.)-(.).Then we have v∈Hm+l
(–γ,∞h )and the
following estimate holds:
v
Hm+l(–γ,∞h)≤C m+l+
k=
ftk
Hl,(–γ,∞h). (.)
Proof . Firstly, we study the caseu≡ outsideU.
We use the induction byl. Forl= , this theorem is proved by Lemma . with noting thatH,(–γ,Kh∞) =L(–γ,Kh∞). Assume that the theorem’s assertion holds up tol– , we need to prove that this holds up tol.
Firstly we will prove the following inequality:
vtj
Hm+l–j
–γ,Kh∞ ≤C m+l+
k=
ftk
Hl,(–γ,Kh∞) (.)
for allj=l,l– , . . . , , where the constantCis independent ofh.
Sinceftk ∈Hl(–γ,Kh∞),ftk ∈L(–γ,Kh∞). So using similar arguments in the proof of
Theorem . we getvtl∈Hm,(–γ,Kh∞). By Lemma ., one obtainsvtl ∈Hm(–γ,Kh∞).
This means that (.)holds forj=l.
Assume that (.) holds forj=l,l– , . . . ,s+ .
By the assumption of the induction of l,v∈Hm+l–
(–γ,Kh∞), put w=vts then w∈ Hm+l–s–(–γ,Kh∞). Differentiating (.)s-times with respect tot, we have
Lw=F= –i(wt+fts) +
s
p=
s p
where Ltk =
m
|p|,|q|=(–)|p|Dp( ∂kapq
∂tk Dq). Following the assumption of the induction of j
and the hypothesis of the functionf one haswt∈Hm+l–s–(–γ,Kh∞),fts∈Hl–s(–γ,K∞
h ). It follows thatF∈Hl–s(–γ,Kh∞)⊂H–l–s–(–γ,Kh∞). Because the strip m+l–s– –n ≤ Imλ≤m+l–s–n does not contain spectral points of problem (.)-(.), then using
results in [], one getsw∈H–m+l–s–(K), a.e.t∈(h,∞). Sow∈H–m+l–s–,(–γ,Kh∞). Note thatF∈Hl–s
(–γ,Kh∞), then by using similar method to the one used in Lemma . in [] we havew∈H–m+l–s–,(–γ,Kh∞) and the following estimate holds:
w
H–m+l–s–,(–γ,Kh∞)≤C
F
Hl–s,+w
H–l–s+,(–γ,Kh∞)
, (.)
whereCis a constant independent ofh.
So using the inductive assumption and (.), we obtain
vtj
Hm+l–s
–γ,Kh∞ ≤C m+l+
k=
ftk
Hl,(–γ,Kh∞). (.)
It means that (.) is proved and our theorem is completed by fixingj= in (.). . The general case: Consider a functionϕ∈C∞ (U) such thatϕ≡ in some
neigh-borhood of . Definitev=ϕv, which satisfies the system
(–)m–iL(x,t,D)v– (v)t=ϕf +L(x,t,D)v,
whereL(x,t,D) is a linear operator having order less than mand the coefficients of this operator is equal to outsideU. So the first case of this theorem implies that
vHm+l
(–γ,∞h)≤ C
m+l+
k=
ftk
Hl,(–γ,∞h). (.)
Writev=v+vwherev= ( –ϕ)v. The functionvis equal to inU, so we can apply
the famous results on the smoothness of solutions of Schrödinger systems in a smooth domain to this function and obtain
v
Hm+l(–γ,∞h)≤ C
m+l+
k=
ftk
Hl,(–γ,∞h)
. (.)
From inequalities (.) and (.) it follows that
vHm+l
(–γ,∞h)≤ C
m+l+
k=
ftk
Hl,(–γ,∞h).
The theorem is proved.
5 The regularity of solution of problem (2.2)-(2.3)
It is well known that there is a smooth functionχ(t) which is equal to on [,∞), is equal to on (–∞, ], and assumes value in [, ] on [; ] (see [, Th. .] for more details). Moreover, we can suppose that all derivatives ofχ(t) are bounded.
Leth∈(–∞, ] be an integer. We setfh(x,t) =χ(t–h)f(x,t) and we consider a gen-eralized solutionuh andujof problems (.)-(.) in cylinders∞
h and∞j withf(x,t) replaced byfh(x,t) andfj(x,t), respectively. Ifh>j,uh can be defined in H◦m,(–γ,∞
j ) withuh(x,t) = ,∀j≤t≤h, then we putujh=uj(x,t) –uh(x,t), andujhis the generalized solution of problem (.)-(.) in whichf(x,t) is replaced byfjh=fj(x,t) –fh(x,t). Accord-ing to (.),
ujh
tk
Hm+l(–γ,∞j )=u
jh tk
Hm+l(–γ,R)≤C m+l+
m=
ftjhm
Hl,(–γ,∞j ).
Because
fjhHl,(–γ,∞ j )=
h+
j
e–γtfh–fjHl ()dt
= h+
j
e–γtχ(t–h) –χ(t–j)f
Hl()dt
≤ h+
j
e–γtf
Hl()dt.
Since f ∈ Hl,(–γ,R), lim
h+
j e
–γtf
Hl()dt = when h,j → –∞. So
limfjh
Hl,(–γ,∞j ) = when h,j → –∞. Repeating this argument, we discover
limftjhm
Hl,(–γ,∞j )
= whenh,j→–∞, for allm= , . . . ,k+ . It follows that{uhtk}–∞h=is a
Cauchy sequence anduhtkis convergent toutkinHm+l(–γ,R). Moreover,Hm+l(–γ,R)
is continuously embedded inHm,(–γ,R), since
u
Hm+l(–γ,R)≤ m+l
|α|=
R
e–γtr(|α|–m–l)Dαu≤CuHm,(–γ,R).
So it is very easy to verify thatu∈Hm,(–γ,
R) is a generalized solution of problem (.)-(.); see []. We obtain the following main results.
Theorem . Assume that l∈Nand there existsμ> such that
(i) sup{|∂a∂pqt |: (x,t)∈R, ≤ |p|,|q| ≤m,}=μ<∞,|∂k∂atkpq| ≤μ,μ> ,for
≤k≤l+ ;
(ii) |∂a∂pqt | ≤μ·eγt,for all(x,t)∈R,≤ |p|,|q| ≤m; (iii) ftk∈L(–γ,R),for allx∈,≤k≤l+ . Then for allγ> (l+ )γ,the solution u∈
◦
Hm,(–γ,
R)of problem(.)-(.)has
deriva-tives with respect to t up to order l satisfying utk∈Hm,(–γ,R),∀k= ,l,such that
utkHm,(–γ,R)≤C
k+
m=
ftmL(–γ,R),
Theorem . Let l be a nonnegative integer. Assume that u(x,t)is a weak solution in the space Hm,(–γ,
R) withγ > ((m+l) + )γ of the problem(.)-(.)and ftk ∈ Hl,(–γ,R),for k≤m+l+ .In addition,suppose that in the strip
m–n
≤Imλ≤m+l–
n
there is no point from the spectrum of (.)-(.).Then we have u∈Hm+l
(–γ,R)and the following estimate holds:
u
Hm+l(–γ,R)≤C m+l+
k=
ftk
Hl,(–γ,R)
.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam.2National Institute of Education
Management, Hanoi, Vietnam.
Acknowledgements
We would like to thank the reviewers for a careful reading and valuable comments on the original manuscript.
Received: 25 March 2014 Accepted: 4 July 2014
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doi:10.1186/s13661-014-0181-8
