Stability of functional inequalities in matrix random normed spaces

(1)

R E S E A R C H

Open Access

Stability of functional inequalities in matrix

random normed spaces

Jung Rye Lee

*

*_{Correspondence: [email protected]}

Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea

Abstract

In this paper, we prove the Hyers-Ulam stability of the Cauchy additive functional inequality and the Cauchy-Jensen additive functional inequality in matrix random normed spaces by using the ﬁxed point method.

MSC: 47L25; 46S50; 47S50; 39B52; 54E70; 39B82

Keywords: operator space; matrix random normed space; functional inequality; Hyers-Ulam stability; ﬁxed point method

1 Introduction

The abstract characterization given for linear spaces of bounded Hilbert space operators

in terms ofmatricially normed spaces[] implies that quotients, mapping spaces and

var-ious tensor products of operator spaces may again be regarded as operator spaces. The proof given in [] appealed to the theory of ordered operator spaces []. Effros and Ruan [] showed that one can give a purely metric proof of this important theorem by using a technique of Pisier [] and Haagerup []. The theory of operator spaces has an increasingly significant effect on operator algebra theory (see [, ]).

The stability problem of functional equations originated from a question of Ulam [] concerning the stability of group homomorphisms. The functional equation

f(x+y) =f(x) +f(y)

is called the Cauchy additive functional equation. In particular, every solution of the

Cauchy additive functional equation is said to be anadditive mapping. Hyers [] gave the

first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [] for additive mappings and by Rassias [] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.

In [], Gilányi showed that iff satisﬁes the functional inequality

f(x) + f(y) –fxy–≤f(xy),

thenf satisﬁes the Jordan-von Neumann functional equation

f(x) + f(y) =f(xy) +fxy–.

(2)

See also []. Gilányi [] and Fechner [] proved the Hyers-Ulam stability of the above functional inequality.

Parket al.[] proved the Hyers-Ulam stability of the following functional inequalities:

f(x) +f(y) +f(z)≤f(x+y+z), (.)

f(x) +f(y) + f(z)≤f

x+y

 +z

. (.)

In the sequel, we adopt the usual terminology, notations and conventions of the theory

of random normed spaces, as in [–]. Throughout this paper,+is the space of

dis-tribution functions, that is, the space of all mappingsF:R∪ {–∞,∞} →[, ] such that

Fis left-continuous and non-decreasing onR,F() =  andF(+∞) = .D+_{is a subset of}

+consisting of all functionsF∈+for whichl–_F₍₊_∞_{) = , where}_l–_f₍_x_{) denotes the left}

limit of the functionf at the pointx, that is,l–_f₍_x_{) =}_lim_t→x_–_f₍_t_{). The space}+_{is partially}

ordered by the usual point-wise ordering of functions,i.e.,F≤Gif and only ifF(t)≤G(t)

for alltinR. The maximal element for+ in this order is the distribution functionε

given by

ε(t) = ⎧ ⎨ ⎩

 ift≤,

 ift> .

Deﬁnition .([]) A mappingT: [, ]×[, ]→[, ] is a continuous triangular norm

(brieﬂy, a continuoust-norm) ifTsatisﬁes the following conditions:

(a) T is commutative and associative; (b) T is continuous;

(c) T(a, ) =afor alla∈[, ];

(d) T(a,b)≤T(c,d)whenevera≤candb≤dfor alla,b,c,d∈[, ].

Deﬁnition . ([]) Arandom normed space(brieﬂy, RN-space) is a triple (X,μ,T),

where Xis a vector space,T is a continuoust-norm andμis a mapping from Xinto

D+_{such that the following conditions hold:}

(RN) μx(t) =ε(t)for allt> if and only ifx= ;

(RN) μαx(t) =μx(_|αt|)for allx∈X,α= ;

(RN) μx+y(t+s)≥T(μx(t),μy(s))for allx,y∈Xand allt,s≥.

Every normed space (X, · ) deﬁnes a random normed space (X,μ,TM), where

μx(t) =

t t+ x

for allt> , andTM is the minimum t-norm. This space is called the induced random

normed space.

Deﬁnition . Let (X,μ,T) be an RN-space.

(3)

() A sequence{xn}inXis called aCauchy sequenceif, for every> andλ> , there exists a positive integerNsuch thatμxn–xm() >  –λwhenevern≥m≥N.

() An RN-space(X,μ,T)is said to becompleteif and only if every Cauchy sequence in

Xis convergent to a point inX.

Theorem .([]) If(X,μ,T)is an RN-space and{xn}is a sequence such that xn→x,

thenlim_n→∞μxn(t) =μx(t)almost everywhere.

We introduce the concept of matrix random normed space.

Deﬁnition . Let (X,μ,T) be a random normed space. Then

() (X,{μ(n)},T)is called amatrix random normed spaceif for each positive integern, (Mn(X),μ(n),T)is a random normed space andμ(AxBk) (t)≥μ

(n)

x (_{A · B}t )for allt> ,

A∈Mk,n(R),x= [xij]∈Mn(X)andB∈Mn,k(R)with A · B = .

() (X,{μ(n)_}_,_T₎_{is called a}_{matrix random Banach space}_if₍_X_,_μ_,_T₎_{is a random}

Banach space and(X,{μ(n)_}_,_T₎_{is a matrix random normed space.}

LetE,Fbe vector spaces. For a given mappingh:E→Fand a given positive integern,

deﬁnehn:Mn(E)→Mn(F) by

hn

[xij]

=h(xij)

for all [xij]∈Mn(E).

LetXbe a set. A functiond:X×X→[,∞] is called ageneralized metriconXifd

satisﬁes

() d(x,y) = if and only ifx=y; () d(x,y) =d(y,x)for allx,y∈X;

() d(x,z)≤d(x,y) +d(y,z)for allx,y,z∈X.

We recall a fundamental result in ﬁxed point theory.

Theorem .([, ]) Let(X,d)be a complete generalized metric space,and let J:X→ X be a strictly contractive mapping with a Lipschitz constantα< .Then,for each given element x∈X,either

dJnx,Jn+x=∞

for all nonnegative integers n or there exists a positive integer nsuch that

() d(Jn_x_,_Jn+_x_{) <}_∞_,_∀_n_≥_n ;

() the sequence{Jn_x_}_{converges to a ﬁxed point}_y∗_of_J_;

() y∗is the unique ﬁxed point ofJin the setY={y∈X|d(Jn_x_,_y_{) <}∞}_; () d(y,y∗)≤_–α d(y,Jy)for ally∈Y.

The stability problem in a random normed space was considered by Mihet and Radu []; next some authors proved some stability results in random normed spaces by diﬀer-ent methods (see [–]).

(4)

using ﬁxed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [–]).

Throughout this paper, letXbe a normed space and (Y,{μ(n)},T) be a matrix random

Banach space. In Section , we prove the Hyers-Ulam stability of the Cauchy additive func-tional inequality (.) in matrix normed spaces by using the direct method. In Section , we prove the Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality (.) in matrix normed spaces by using the ﬁxed point method.

2 Hyers-Ulam stability of the Cauchy additive functional inequality

In this section, we prove the Hyers-Ulam stability of the Cauchy additive functional in-equality (.) in matrix random normed spaces by using the ﬁxed point method.

Theorem . Letϕ:X_→_[,_∞₎_{be a function such that there exists}_α_{< }_with

ϕ(a,b,c)≤α

ϕ(a, b, c)

for all a,b,c∈X.Let f :X→Y be an odd mapping satisfying

μ(_fn)

n([xij])+fn([yij])+fn([zij])(t)≥min

μ(_fn)

n([xij+yij+zij])

t



, t

t+n_i_,_j_=ϕ(xij,yij,zij)

(.)

for all t> and x= [xij],y= [yij],z= [zij]∈Mn(X).Then A(a) :=liml→∞lf(_al)exists for each a∈X and deﬁnes an additive mapping A:X→Y such that

μfn([xij])–An([xij])(t)≥

( –α)t

( –α)t+n_αn

i,j=ϕ(xij,xij, –xij)

(.)

for all t> and x= [xij]∈Mn(X).

Proof Letn= . Then (.) is equivalent to

μf(a)+f(b)+f(c)(t)≥min

μf(a+b+c)

t



, t

t+ϕ(a,b,c)

(.)

for allt>  anda,b,c∈X.

Lettingb=aandc= –ain (.), we get

μf(a)–f(a)(t)≥

t

t+ϕ(a,a, –a), (.)

and so

μf(a)–f(a_)(t)≥ t

t+ϕ(a_,a_, –a)≥

t

t+α_ϕ(a,a, –a) (.)

for allt>  anda∈X. Consider the set

(5)

and introduce the generalized metric onS:

d(g,h) =inf

ν∈R+:μg(a)–h(a)(νt)≥

t

t+ϕ(a,a, –a),∀a∈X,∀t> 

,

where, as usual,infφ= +∞. It is easy to show that (S,d) is complete (see the proof of [,

Lemma .]).

Now we consider the linear mappingJ:S→Ssuch that

Jg(a) := g

a



for alla∈X.

Letg,h∈Sbe given such thatd(g,h) =ε. Then

μg(a)–h(a)(εt)≥ t t+ϕ(a,a)

for alla∈Xandt> . Hence

μJg(a)–Jh(a)(αεt) =μg(a_)–h(a_)(αεt) =μg(a_)–h(a_)

α

εt

≥

αt

 αt

 +ϕ(

a

,

a

, –a) ≥

αt

 αt

 + α

ϕ(a,a, –a)

= t

t+ϕ(a,a, –a)

for alla∈Xandt> . Sod(g,h) =εimplies thatd(Jg,Jh)≤αε. This means that

d(Jg,Jh)≤αd(g,h)

for allg,h∈S.

It follows from (.) thatd(f,Jf)≤α

.

By Theorem ., there exists a mappingA:X→Ysatisfying the following:

() Ais a ﬁxed point ofJ,i.e.,

A

a



=

A(a)

for alla∈X. The mappingAis a unique ﬁxed point ofJin the set

M=g∈S:d(f,g) <∞.

() d(Jlf,A)→asl→ ∞. This implies the equality

lim l→∞

l_f

a

l

=A(a)

(6)

() d(f,A)≤_–α d(f,Jf), which implies the inequality

d(f,A)≤ α

 – α. (.)

By (.),

μ_l_f₍a+b

l )–

l_f₍a

l)–

l_f₍b

l)

lt≥ t

t+ϕ(_al,_bl, –a_+lb)

for alla,b∈Xandt> . So

μ_l_f₍a+b

l )–lf(al)–lf(bl)(t)≥

t

l

t

l + αl

lϕ(a,b, –a–b)

for alla,b∈Xandt> . Sincelim_l→∞

t

l t

l+

αl

lϕ(a,b,–a–b)

=  for alla,b∈Xandt> ,

μA(a+b)–A(a)–A(b)(t) = 

for alla,b∈Xandt> . ThusA(a+b) –A(a) –A(b) = . So the mappingA:X→Y is

additive.

We note thatej∈M,n(R) is thatjth component is  and the others are zero,Eij∈Mn(R)

is that (i,j)-component is  and the others are zero, andEij⊗x∈Mn(X) is that (i,j

)-component isxand the others are zero. Sinceμ(_En)

kl⊗x(t) =μx(t), we have

μ(_[n_x)

ij](t) =μ (n) n

i,j=Eij⊗xij(t)≥min

μ(_En)

ij⊗xij(tij) :i,j= , , . . . ,n

=minμxij(tij) :i,j= , , . . . ,n

,

wheret=n_i_,_j_=tij. Soμ(_[n_x_ij)_](t)≥min{μxij(

t

n) :i,j= , , . . . ,n}.

By (.),

μ(_fn)

n([xij])–An([xij])(t)≥min

μf(xij)–A(xij)

t n

:i,j= , , . . . ,n

≥min

( –α)t

( –α)t+n_αϕ₍_x

ij,xij, –xij)

:i,j= , , . . . ,n

≥ ( –α)t

( –α)t+n_αn

for allx= [xij]∈Mn(X). ThusA:X→Y is a unique additive mapping satisfying (.), as

desired.

Corollary . Let r,θbe positive real numbers with r> .Let f :X→Y be an odd mapping satisfying

μ(_fn)

n([xij])+fn([yij])+fn([zij])(t)

≥min

μ(_fn)

n([xij+yij+zij])

t



, t

t+n_i_,_j_=θ( xij r+ yij r+ zij r)

(7)

(r_{– )}_t (r_{– )}_t₊_n_{( + }r₎n

i,j=θ xij r

Proof The proof follows from Theorem . by takingϕ(a,b,c) =θ( a r₊ _b r₊ _c r_{) for}

alla,b,c∈X. Then we can chooseα= –r_{and we get the desired result.}

Theorem . Let f :X→Y be an odd mapping satisfying(.)for which there exists a functionϕ:X_→_[,_∞₎_{such that there exists}_α_{< }_with

ϕ(a,b,c)≤αϕ

a

,

b

,

c



for all a,b,c∈X.Then A(a) :=lim_l→∞ 

lf(la)exists for each a∈X and deﬁnes an additive mapping A:X→Y such that

( –α)t

( –α)t+nn

Proof Let (S,d) be the generalized metric space deﬁned in the proof of Theorem ..

Jg(a) := g

a



for alla∈X.

It follows from (.) thatd(f,Jf)≤_. So

d(f,A)≤   – α.

The rest of the proof is similar to the proof of Theorem ..

Corollary . Let r,θ be positive real numbers with r< .Let f :X→Y be a mapping satisfying(.).Then A(a) :=lim_l→∞l_f₍a

l)exists for each a∈X and deﬁnes an additive mapping A:X→Y such that

( – r₎_t ( – r₎_t₊_n_{( + }r₎n

i,j=θ xij r

Proof The proof follows from Theorem . by takingϕ(a,b,c) =θ( a r₊ _b r₊ _c r_{) for}

(8)

3 Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality

In this section, we prove the Hyers-Ulam stability of Cauchy-Jensen additive functional inequality (.) in matrix random normed spaces by using the ﬁxed point method.

Theorem . Letϕ:X_→_[,_∞₎_{be a function such that there exists}_α_{< }_with

ϕ(a,b,c)≤α

ϕ(a, b, c)

for all a,b,c∈X.Let f :X→Y be an odd mapping satisfying

μ(_fn)

n([xij])+fn([yij])+fn([zij])(t)

≥min

μ(n)

fn([xij+yij+zij])

t



, t

t+n_i_,_j_=ϕ(xij,yij,zij)

(.)

( –α)t

( –α)t+n_αn

i,j=ϕ(xij,xij, –xij)

(.)

Proof Letn= . Then (.) is equivalent to

μf(a)+f(b)+f(c)(t)≥min

μ__f₍a+b

 +c)

t



, t

t+ϕ(a,b,c)

(.)

for allt>  anda,b,c∈X.

Lettingb=aandc= –ain (.), we get

μf(a)–f(a)(t)≥ t

t+ϕ(a,a, –a), (.)

and so

μf(a)–f(a_)(t)≥ t

t+ϕ(a_,a_, –a_)≥

t

t+α_ϕ(a,a, –a) (.)

for allt>  anda∈X.

Let (S,d) be the generalized metric space deﬁned in the proof of Theorem ..

Jg(a) := g

a



for alla∈X.

Letg,h∈Sbe given such thatd(g,h) =ε. Then

(9)

for alla∈Xandt> . Hence

μJg(a)–Jh(a)(αεt) =μg(a_)–h(a_)(αεt) =μg(a_)–h(a_)

α

εt

≥

αt

 αt

 +ϕ(

a

,

a

, –

a

) ≥

αt

 αt

 + α

ϕ(a,a, –a)

= t

t+ϕ(a,a, –a)

for alla∈Xandt> . Sod(g,h) =εimplies thatd(Jg,Jh)≤αε. This means that

d(Jg,Jh)≤αd(g,h) for allg,h∈S.

It follows from (.) thatd(f,Jf)≤α_.

By Theorem ., there exists a mappingA:X→Ysatisfying the following:

() Ais a ﬁxed point ofJ,i.e.,

A

a



=

A(a)

for alla∈X. The mappingAis a unique ﬁxed point ofJin the set

M=g∈S:d(f,g) <∞.

() d(Jl_f_,_A₎_→__as_l_{→ ∞}_{. This implies the equality}

lim l→∞

l_f

a

l

=A(a)

for alla∈X.

() d(f,A)≤_–α d(f,Jf), which implies the inequality

d(f,A)≤ α

 – α. (.)

By (.),

μ_l_f₍a+b

l )–

l_f₍a

l)–

l_f₍b

l)

lt≥ t

t+ϕ(_al,_bl, –a_l++b)

for alla,b∈Xandt> . So

μ_l_f₍a+b

l )–

l_f₍a

l)–

l_f₍b

l)

(t)≥

t

l

t

l + αl

lϕ(a,b, –a+_b)

for alla,b∈Xandt> . Sinceliml→∞

t

l t

l+

αl

lϕ(a,b,–a–b)

=  for alla,b∈Xandt> ,

(10)

for alla,b∈Xandt> . ThusA(a+b) –A(a) –A(b) = . So the mappingA:X→Y is additive.

By (.),

μ(_fn)

n([xij])–An([xij])(t)≥min

μf(xij)–A(xij)

t n

:i,j= , , . . . ,n

≥min

( –α)t

( –α)t+n_αϕ₍_x

ij,xij, –xij)

:i,j= , , . . . ,n

≥ ( –α)t

( –α)t+n_αn

for allx= [xij]∈Mn(X). ThusA:X→Y is a unique additive mapping satisfying (.), as

desired.

Corollary . Let r,θbe positive real numbers with r> .Let f :X→Y be an odd mapping satisfying

μ(_fn)

n([xij])+fn([yij])+fn([zij])(t)

≥min

μ(n)

fn([xij+_yij+zij])

t



, t

t+n_i_,_j_=θ( xij r+ yij r+ zij r)

(.)

(r_{– )}_t

(r_{– )}_t_{+ }_nn

i,j=θ xij r

Proof The proof follows from Theorem . by takingϕ(a,b,c) =θ( a r+ b r+ c r) for

alla,b,c∈X. Then we can chooseα= –r_{and we get the desired result.}

Theorem . Let f :X→Y be an odd mapping satisfying(.)for which there exists a functionϕ:X_→_[,_∞₎_{such that there exists}_α_{< }_with

ϕ(a,b,c)≤αϕ

a

,

b

,

c



for all a,b,c∈X.Then A(a) :=liml→∞_lf(la)exists for each a∈X and deﬁnes an additive mapping A:X→Y such that

( –α)t

( –α)t+nn

(11)

Jg(a) := g

a



for alla∈X.

It follows from (.) thatd(f,Jf)≤_. So

d(f,A)≤   – α.

The rest of the proof is similar to the proof of Theorem ..

Corollary . Let r,θ be positive real numbers with r< .Let f :X→Y be a mapping satisfying(.).Then A(a) :=lim_l→∞l_f₍a

l)exists for each a∈X and deﬁnes an additive mapping A:X→Y such that

( – r₎_t ( – r₎_t_{+ }_nn

i,j=θ xij r

Proof The proof follows from Theorem . by takingϕ(a,b,c) =θ( a r₊ _b r₊ _c r_{) for}

alla,b,c∈X. Then we can chooseα= r–and we get the desired result.

Competing interests

The author declares that she has no competing interests.

Received: 15 July 2013 Accepted: 29 October 2013 Published:27 Nov 2013

References

1. Ruan, Z-J: Subspaces ofC∗-algebras. J. Funct. Anal.76, 217-230 (1988)

2. Choi, M-D, Eﬀros, E: Injectivity and operator spaces. J. Funct. Anal.24, 156-209 (1977)

3. Eﬀros, E, Ruan, Z-J: On the abstract characterization of operator spaces. Proc. Am. Math. Soc.119, 579-584 (1993) 4. Pisier, G: Grothendieck’s Theorem for non-commutativeC∗-algebras with an appendix on Grothendieck’s constants.

J. Funct. Anal.29, 397-415 (1978)

5. Haagerup, U: Decomp. of completely bounded maps. Unpublished manuscript

6. Eﬀros, E: On Multilinear Completely Bounded Module Maps. Contemp. Math., vol. 62, pp. 479-501. Am. Math. Soc., Providence (1987)

7. Eﬀros, E, Ruan, Z-J: On approximation properties for operator spaces. Int. J. Math.1, 163-187 (1990) 8. Ulam, SM: A Collection of the Mathematical Problems. Interscience, New York (1960)

9. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA27, 222-224 (1941) 10. Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn.2, 64-66 (1950) 11. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.72, 297-300 (1978) 12. G˘avruta, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal.

Appl.184, 431-436 (1994)

13. Gilányi, A: Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequ. Math.62, 303-309 (2001) 14. Rätz, J: On inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math.66, 191-200

(2003)

15. Gilányi, A: On a problem by K. Nikodem. Math. Inequal. Appl.5, 707-710 (2002)

16. Fechner, W: Stability of a functional inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math.71, 149-161 (2006)

17. Park, C, Cho, Y, Han, M: Functional inequalities associated with Jordan-von Neumann type additive functional equations. J. Inequal. Appl.2007, Article ID 41820 (2007)

18. Chang, SS, Cho, Y, Kang, S: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Publ., New York (2001) 19. Cho, Y, Rassias, TM, Saadati, R: Stability of Functional Equations in Random Normed Spaces. Springer Optimization

and Its Applications, vol. 86. Springer, Berlin (2013)

20. Schweizer, B, Sklar, A: Probabilistic Metric Spaces. North-Holand, New York (1983)

(12)

22. C˘adariu, L, Radu, V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math.4(1), Article ID 4 (2003)

23. Diaz, J, Margolis, B: A ﬁxed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc.74, 305-309 (1968)

24. Mihe¸t, D, Radu, V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl.343, 567-572 (2008)

25. Cho, Y, Kang, S, Saadati, R: Nonlinear random stability via ﬁxed-point method. J. Appl. Math.2012, Article ID 902931 (2012)

26. Mihe¸t, D, Saadati, R: On the stability of the additive Cauchy functional equation in random normed spaces. Appl. Math. Lett.24, 2005-2009 (2011)

27. Mihe¸t, D, Saadati, R, Vaezpour, SM: The stability of the quartic functional equation in random normed spaces. Acta Appl. Math.110, 797-803 (2010)

28. Isac, G, Rassias, TM: Stability ofψ-additive mappings: applications to nonlinear analysis. Int. J. Math. Math. Sci.19, 219-228 (1996)

29. C˘adariu, L, Radu, V: On the stability of the Cauchy functional equation: a ﬁxed point approach. Grazer Math. Ber.346, 43-52 (2004)

30. C˘adariu, L, Radu, V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl.2008, Article ID 749392 (2008)

31. Mirzavaziri, M, Moslehian, MS: A ﬁxed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc.37, 361-376 (2006)

32. Park, C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a ﬁxed point approach. Fixed Point Theory Appl.2008, Article ID 493751 (2008)

33. Park, C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl.2007, Article ID 50175 (2007)

34. Radu, V: The ﬁxed point alternative and the stability of functional equations. Fixed Point Theory4, 91-96 (2003)

10.1186/1029-242X-2013-569