Open Access

Homomorphisms and derivations in induced fuzzy C -algebras

  • Hassan Azadi Kenary1,
  • Mojtaba Ghirati1,
  • Choonkil Park2Email author and
  • Madjid Eshaghi Gordji3
Journal of Inequalities and Applications20132013:88

https://doi.org/10.1186/1029-242X-2013-88

Received: 20 February 2012

Accepted: 25 January 2013

Published: 5 March 2013

Abstract

Using fixed point method, we establish the Hyers-Ulam stability of fuzzy -homomorphisms in fuzzy C -algebras and fuzzy -derivations on fuzzy C -algebras associated to the following ( m , n ) -Cauchy-Jensen additive functional equation:

1 i 1 < < i m n 1 k l ( i j , j { 1 , , m } ) n f ( j = 1 m x i j m + l = 1 n m x k l ) = ( n m + 1 ) n ( n m ) i = 1 n f ( x i ) .

MSC:47S40, 39B52, 46S40, 47H10, 26E50.

Keywords

Hyers-Ulam stabilityfixed point methodfuzzy Banach algebra

1 Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Rassias [3] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (T.M. Rassias)

Let f : E E be a mapping from a normed vector space E into a Banach space E subject to the inequality f ( x + y ) f ( x ) f ( y ) ϵ ( x p + y p ) for all x , y E , where ϵ and p are constants with ϵ > 0 and 0 p < 1 . Then the limit L ( x ) = lim n f ( 2 n x ) 2 n exists for all x E , and L : E E is the unique additive mapping which satisfies
f ( x ) L ( x ) 2 ϵ 2 2 p x p

for all x E . Also, if for each x E , the function f ( t x ) is continuous in t R , then L is -linear.

The functional equation f ( x + y ) + f ( x y ) = 2 f ( x ) + 2 f ( y ) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers-Ulam stability of the quadratic functional equation was proved by Skof [4] for mappings f : X Y , where X is a normed space and Y is a Banach space. Cholewa [5] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [6] proved the Hyers-Ulam stability of the quadratic functional equation.

The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [720]).

Katsaras [21] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view (see [2224]). In particular, Bag and Samanta [25], following Cheng and Mordeson [26], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [27]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [28].

Now, we consider a mapping f : X Y satisfying the following functional equation, which is introduced by the first author:
1 i 1 < < i m n 1 k l ( i j , j { 1 , , m } ) n f ( j = 1 m x i j m + l = 1 n m x k l ) = ( n m + 1 ) n ( n m ) i = 1 n f ( x i )
(1)

for all x 1 , , x n X , where m , n N are fixed integers with n 2 , 1 m n . Especially, we observe that in the case m = 1 , equation (1) yields the Cauchy-type additive equation f ( l = 1 n x k l ) = l = 1 n f ( x i ) . We observe that in the case m = n , equation (1) yields the Jensen-type additive equation f ( j = 1 n x j n ) = 1 n l = 1 n f ( x i ) . Therefore, equation (1) is a generalized form of the Cauchy-Jensen additive equation and thus every solution of equation (1) may be analogously called a general ( m , n ) -Cauchy-Jensen additive. For the case m = 2 , we have established new theorems about the Hyers-Ulam stability in quasi β-normed spaces [29]. Let X and Y be linear spaces. For each m with 1 m n , a mapping f : X Y satisfies equation (1) for all n 2 if and only if f ( x ) f ( 0 ) = A ( x ) is a Cauchy additive, where f ( 0 ) = 0 if m < n . In particular, we have f ( ( n m + 1 ) x ) = ( n m + 1 ) f ( x ) and f ( m x ) = m f ( x ) for all x X .

2 Preliminaries

Definition 2.1 Let X be a real vector space. A function N : X × R [ 0 , 1 ] is called a fuzzy norm on X if for all x , y X and all s , t R ,

(N1) N ( x , t ) = 0 for t 0 ;

(N2) x = 0 if and only if N ( x , t ) = 1 for all t > 0 ;

(N3) N ( c x , t ) = N ( x , t | c | ) if c 0 ;

(N4) N ( x + y , c + t ) min { N ( x , s ) , N ( y , t ) } ;

(N5) N ( x , ) is a non-decreasing function of and lim t N ( x , t ) = 1 ;

(N6) for x 0 , N ( x , ) is continuous on .

Example 2.1 Let ( X , ) be a normed linear space and α , β > 0 . Then
N ( x , t ) = { α t α t + β x , t > 0 , x X , 0 , t 0 , x X

is a fuzzy norm on X.

Definition 2.2 Let ( X , N ) be a fuzzy normed vector space. A sequence { x n } in X is said to be convergent or converge if there exists an x X such that lim t N ( x n x , t ) = 1 for all t > 0 . In this case, x is called the limit of the sequence { x n } in X and we denote it by N - lim t x n = x .

Definition 2.3 Let ( X , N ) be a fuzzy normed vector space. A sequence { x n } in X is called Cauchy if for each ϵ > 0 and each t > 0 , there exists an n 0 N such that for all n n 0 and all p > 0 , we have N ( x n + p x n , t ) > 1 ϵ .

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping f : X Y between fuzzy normed vector spaces X and Y is continuous at a point x X if for each sequence { x n } converging to x 0 X , the sequence { f ( x n ) } converges to f ( x 0 ) . If f : X Y is continuous at each x X , then f : X Y is said to be continuous on X (see [28]).

Definition 2.4 Let X be a -algebra and ( X , N ) be a fuzzy normed space.
  1. (1)
    The fuzzy normed space ( X , N ) is called a fuzzy normed -algebra if
    N ( x y , s t ) N ( x , s ) N ( y , t ) , N ( x , t ) = N ( x , t )
     
for all x , y X and all positive real numbers s and t.
  1. (2)

    A complete fuzzy normed -algebra is called a fuzzy Banach -algebra.

     
Example 2.2 Let ( X , ) be a normed -algebra. Let
N ( x , t ) = { t t + x , t > 0 , x X , 0 , t 0 , x X .

Then N ( x , t ) is a fuzzy norm on X and ( X , N ) is a fuzzy normed -algebra.

Definition 2.5 Let ( X , ) be a normed C -algebra and N be a fuzzy norm on X.
  1. (1)

    The fuzzy normed -algebra ( X , N ) is called an induced fuzzy normed -algebra.

     
  2. (2)

    The fuzzy Banach -algebra ( X , N ) is called an induced fuzzy C -algebra.

     
Definition 2.6 Let ( X , N ) and ( Y , N ) be induced fuzzy normed -algebras.
  1. (1)

    A multiplicative -linear mapping H : ( X , N ) ( Y , N ) is called a fuzzy -homomorphism if H ( x ) = H ( x ) for all x X .

     
  2. (2)

    A -linear mapping D : ( X , N ) ( X , N ) is called a fuzzy -derivation if D ( x y ) = D ( x ) y + x D ( y ) and D ( x ) = D ( x ) for all x , y X .

     
Definition 2.7 Let X be a nonempty set. A function d : X × X [ 0 , ] is called a generalized metric on X if d satisfies the following conditions:
  1. (1)

    d ( x , y ) = 0 if and only if x = y for all x , y X ;

     
  2. (2)

    d ( x , y ) = d ( y , x ) for all x , y X ;

     
  3. (3)

    d ( x , z ) d ( x , y ) + d ( y , z ) for all x , y , z X .

     
Theorem 2.1 Let ( X , d ) be a complete generalized metric space and J : X X be a strictly contractive mapping with a Lipschitz constant L < 1 . Then, for all x X , either d ( J n x , J n + 1 x ) = for all nonnegative integers n or there exists a positive integer n 0 such that
  1. (1)

    d ( J n x , J n + 1 x ) < for all n 0 n 0 ;

     
  2. (2)

    the sequence { J n x } converges to a fixed point y of J;

     
  3. (3)

    y is the unique fixed point of J in the set Y = { y X : d ( J n 0 x , y ) < } ;

     
  4. (4)

    d ( y , y ) 1 1 L d ( y , J y ) for all y Y .

     

Throughout this paper, assume that X, Y are unital fuzzy Banach -algebras.

3 Approximate homomorphisms in fuzzy Banach -algebras

In this section, using fixed point method, we prove the Hyers-Ulam stability of homomorphisms in fuzzy Banach -algebras related to functional equation (1).

Theorem 3.1 Let φ : X n [ 0 , ) be a function such that there exists an L < 1 ( n m + 1 ) n 2 with
φ ( x 1 n m + 1 , , x n n m + 1 ) L φ ( x 1 , x 2 , , x n ) n m + 1
for all x 1 , , x n X . Let f : X Y with f ( 0 ) = 0 be a mapping satisfying
(2)
(3)
(4)
for all x 1 , , x n X and all t > 0 . Then there exists a fuzzy -homomorphism H : X Y such that
N ( f ( x ) H ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + L φ ( x , , x )
(5)

for all x X and all t > 0 .

Proof Letting μ = 1 and replacing ( x 1 , , x n ) by ( x , , x ) in (2), we have
N ( ( n m ) f ( ( n m + 1 ) x ) ( n m ) ( n m + 1 ) f ( x ) , t ) t t + φ ( x , , x )
(6)
for all x X and t > 0 . Consider the set S : = { g : X Y ; g ( 0 ) = 0 } and the generalized metric d in S defined by
d ( f , g ) = inf { μ R + : N ( g ( x ) h ( x ) , μ t ) t t + φ ( x , , x ) , x X , t > 0 } ,
where inf = + . It is easy to show that ( S , d ) is complete (see [30]). Now, we consider a linear mapping J : S S such that J g ( x ) : = ( n m + 1 ) g ( x n m + 1 ) for all x X . Let g , h S be such that d ( g , h ) = ϵ . Then N ( g ( x ) h ( x ) , ϵ t ) t t + φ ( x , , x ) for all x X and t > 0 . Hence,
for all x X and t > 0 . Thus, d ( g , h ) = ϵ implies that d ( J g , J h ) L ϵ . This means that d ( J g , J h ) L d ( g , h ) for all g , h S . It follows from (6) that
N ( f ( x n m + 1 ) ( n m + 1 ) 1 f ( x ) , t ( n m ) ) t t + φ ( x n m + 1 , , x n m + 1 ) t t + L φ ( x , , x ) n m + 1
for all x X and all t > 0 . So,
N ( f ( x n m + 1 ) ( n m + 1 ) 1 f ( x ) , L t ( n m + 1 ) ( n m ) ) t t + φ ( x , , x ) .
This implies that d ( f , J f ) L ( n m + 1 ) ( n m ) . By Theorem 2.1, there exists a mapping A : X Y satisfying the following:
  1. (1)
    A is a fixed point of J, that is,
    H ( x n m + 1 ) = H ( x ) n m + 1
    (7)
     
for all x X . The mapping H is a unique fixed point of J in the set Ω = { h S : d ( g , h ) < } . This implies that H is a unique mapping satisfying (7) such that there exists μ ( 0 , ) satisfying N ( f ( x ) H ( x ) , μ t ) t t + φ ( x , , x ) for all x X and t > 0 .
  1. (2)
    d ( J p f , H ) 0 as p . This implies the equality
    N - lim p f ( x ( n m + 1 ) p ) ( n m + 1 ) p = H ( x )
    (8)
     
for all x X .
  1. (3)
    d ( f , H ) d ( f , J f ) 1 L with f Ω , which implies the inequality
    d ( f , H ) L ( n m + 1 ) ( n m ) ( n m + 1 ) ( n m ) L .
     
This implies that the inequality (5) holds. Furthermore, it follows from (2) and (8) that
for all x 1 , , x n X , all t > 0 and all μ C . Hence,
1 i 1 < < i m n 1 k l ( i j , j { 1 , , m } ) n H ( j = 1 m μ x i j m + l = 1 n m μ x k l ) = ( n m + 1 ) n ( n m ) i = 1 n H ( μ x i )
for all x 1 , , x n X . So, the mapping H : X Y is additive and -linear. By (3)
(9)
for all x 1 , , x n 1 X and all t > 0 . Then
for all x 1 , , x n 1 X and all t > 0 . So,
N ( H ( x 1 x n 1 ) H ( x 1 ) H ( x n 1 ) , t ) = 1

for all x 1 , , x n 1 X and all t > 0 . Thus, H ( x 1 x n 1 ) = H ( x 1 ) H ( x n 1 ) .

On the other hand, by (4)
N ( f ( x 1 ( n m + 1 ) p ) ( n m + 1 ) p f ( x 1 ( n m + 1 ) p ) ( n m + 1 ) p , t ( n m + 1 ) p ) t t + φ ( x 1 ( n m + 1 ) p , 0 , , 0 )
for all x 1 X and all t > 0 . So,
N ( f ( x 1 ( n m + 1 ) p ) ( n m + 1 ) p f ( x 1 ( n m + 1 ) p ) ( n m + 1 ) p , t ) t ( n m + 1 ) p t ( n m + 1 ) p + φ ( x 1 ( n m + 1 ) p , 0 , , 0 ) t ( n m + 1 ) p t ( n m + 1 ) p + L p ( n m + 1 ) p φ ( x 1 , 0 , , 0 )
for all x 1 X and all t > 0 . Since lim p + t ( n m + 1 ) p t ( n m + 1 ) p + L p ( n m + 1 ) p φ ( x 1 , 0 , , 0 ) = 1 for all x 1 X and t > 0 , we get
N ( H ( x 1 ) H ( x 1 ) , t ) = 1

for all x 1 X and all t > 0 . Thus, H ( x 1 ) = H ( x 1 ) for all x 1 X . This completes the proof. □

Theorem 3.2 Let φ : X n [ 0 , ) be a function such that there exists an L < 1 with
φ ( x 1 , , x n ) ( n m + 1 ) L φ ( x 1 n m + 1 , , x n n m + 1 )
for all x 1 , x 2 , , x n X . Let f : X Y be a mapping satisfying f ( 0 ) = 0 , (2)-(4). Then the limit A ( x ) : = N - lim p f ( ( n m + 1 ) p x ) ( n m + 1 ) p exists for each x X and defines a fuzzy -homomorphism H : X Y such that
N ( f ( x ) H ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + φ ( x , , x )
(10)

for all x X and all t > 0 .

Proof Let ( S , d ) be a generalized metric space defined as in the proof of Theorem 3.1. Consider the linear mapping J : S S such that J g ( x ) : = g ( ( n m + 1 ) x ) n m + 1 for all x X . Let g , h S be such that d ( g , h ) = ϵ . Then N ( g ( x ) h ( x ) , ϵ t ) t t + φ ( x , , x ) for all x X and t > 0 . Hence,
N ( J g ( x ) J h ( x ) , L ϵ t ) = N ( g ( ( n m + 1 ) x ) n m + 1 h ( ( n m + 1 ) x ) n m + 1 , L ϵ t ) = N ( g ( ( n m + 1 ) x ) h ( ( n m + 1 ) x ) , ( n m + 1 ) L ϵ t ) ( n m + 1 ) L t ( n m + 1 ) L t + φ ( ( n m + 1 ) x , , ( n m + 1 ) x ) ( n m + 1 ) L t ( n m + 1 ) L t + ( n m + 1 ) L φ ( x , , x ) = t t + φ ( x , , x )
for all x X and t > 0 . Thus, d ( g , h ) = ϵ implies that d ( J g , J h ) L ϵ . This means that d ( J g , J h ) L d ( g , h ) for all g , h S . It follows from (6) that
N ( f ( x ) f ( ( n m + 1 ) x ) n m + 1 , t ( n m + 1 ) ( n m ) ) t t + φ ( x , , x )
(11)
for all x X and t > 0 . So, d ( f , J f ) 1 ( n m + 1 ) ( n m ) . By Theorem 2.1, there exists a mapping H : X Y satisfying the following:
  1. (1)
    A is a fixed point of J, that is,
    ( n m + 1 ) H ( x ) = H ( ( n m + 1 ) x )
    (12)
     
for all x X . The mapping H is a unique fixed point of J in the set Ω = { h S : d ( g , h ) < } . This implies that H is a unique mapping satisfying (12) such that there exists μ ( 0 , ) satisfying N ( f ( x ) H ( x ) , μ t ) t t + φ ( x , , x ) for all x X and t > 0 .
  1. (2)
    d ( J p f , H ) 0 as p . This implies the equality
    H ( x ) = N - lim p f ( ( n m + 1 ) p x ) ( n m + 1 ) p
     
for all x X .
  1. (3)
    d ( f , H ) d ( f , J f ) 1 L with f Ω , which implies the inequality
    d ( f , H ) 1 ( n m + 1 ) ( n m ) ( n m + 1 ) ( n m ) L .
     

This implies that the inequality (10) holds.

The rest of the proof is similar to the proof of Theorem 3.1. □

From now on, we assume that X has a unit e and a unitary group U ( X ) : = { u X : u u = u u = e } .

Theorem 3.3 Let φ : X n [ 0 , ) be a function such that there exists an L < 1 ( n m + 1 ) n 2 with
φ ( x 1 n m + 1 , , x n n m + 1 ) L φ ( x 1 , x 2 , , x n ) n m + 1
for all x 1 , , x n X . Let f : X Y be a mapping satisfying f ( 0 ) = 0 , (2) and
(13)
(14)

for all u 1 , , u n U ( X ) and all t > 0 . Then there exists a fuzzy -homomorphism H : X Y satisfying (5).

Proof By the same reasoning as in the proof of Theorem 3.1, there is a -linear mapping H : X Y satisfying (5). The mapping H : X Y is given by
N - lim p f ( x ( n m + 1 ) p ) ( n m + 1 ) p = H ( x )
for all x X . By (13)
for all u 1 , , u n 1 U ( X ) and all t > 0 . Then
as p + for all u 1 , , u n 1 U ( X ) and all t > 0 . So,
N ( H ( u 1 u n 1 ) H ( u 1 ) H ( u n 1 ) , t ) = 1
for all u 1 , , u n 1 U ( X ) and all t > 0 . Thus,
H ( u 1 u n 1 ) = H ( u 1 ) H ( u n 1 ) .
(15)
Since H is -linear and each x X is a finite linear combination of unitary elements, i.e.,
x = j = 1 m λ j u j ( λ j C , u j U ( X ) ) ,
it follows from (15) that
H ( x v ) = H ( j = 1 m λ j u j v ) = j = 1 n λ j H ( u j v ) = j = 1 n λ j H ( u j ) H ( v ) = H ( j = 1 m λ j u j ) H ( v )
for all v U ( X ) . So, H ( x v ) = H ( x ) H ( v ) . Similarly, one can obtain that H ( x y ) = H ( x ) H ( y ) for all x , y X . Thus by induction, one can easily show that H ( x 1 x n 1 ) = H ( x 1 ) H ( x n 1 ) . By (4)
N ( f ( u 1 ( n m + 1 ) p ) ( n m + 1 ) p f ( u 1 ( n m + 1 ) p ) ( n m + 1 ) p , t ( n m + 1 ) p ) t t + φ ( u 1 ( n m + 1 ) p , 0 , , 0 )
for all u 1 U ( X ) and all t > 0 . So,
N ( f ( u 1 ( n m + 1 ) p ) ( n m + 1 ) p f ( u 1 ( n m + 1 ) p ) ( n m + 1 ) p , t ) t ( n m + 1 ) p t ( n m + 1 ) p + φ ( u 1 ( n m + 1 ) p , 0 , , 0 ) t ( n m + 1 ) p t ( n m + 1 ) p + L p φ ( u 1 , 0 , , 0 ) ( n m + 1 ) p
for all u 1 U ( X ) and all t > 0 . Since lim p + t ( n m + 1 ) p t ( n m + 1 ) p + L p φ ( u 1 , 0 , , 0 ) ( n m + 1 ) p = 1 for all u 1 U ( X ) and all t > 0 , we get
N ( H ( u 1 ) H ( u 1 ) , t ) = 1
for all u 1 U ( X ) and all t > 0 . Thus,
H ( u 1 ) = H ( u 1 )
(16)
for all u 1 U ( X ) . Since H is -linear and each x X is a finite linear combination of unitary elements, i.e., x = j = 1 m λ j u j ( λ j C , u j U ( X ) ), it follows from (16) that
H ( x ) = H ( j = 1 m λ j ¯ u j ) = j = 1 n λ j ¯ H ( u j ) = j = 1 n λ j ¯ H ( u j ) = H ( j = 1 m λ j u j ) = H ( x )

for all x X . So, H ( x ) = H ( x ) for all x X . Therefore, the mapping H : X Y is a -homomorphism. □

Similarly, we have the following. We will omit the proof.

Theorem 3.4 Let φ : X n [ 0 , ) be a function such that there exists an L < 1 with
φ ( x 1 , , x n ) ( n m + 1 ) L φ ( x 1 n m + 1 , , x n n m + 1 )
for all x 1 , x 2 , , x n X . Let f : X Y be a mapping satisfying f ( 0 ) = 0 , (2), (13) and (14). Then the limit A ( x ) : = N - lim p f ( ( n m + 1 ) p x ) ( n m + 1 ) p exists for each x X and defines a fuzzy -homomorphism H : X Y such that
N ( f ( x ) H ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + φ ( x , , x )

for all x X and all t > 0 .

4 Approximate derivations on fuzzy Banach -algebras

In this section, we assume that ( X , N ) is a fuzzy Banach -algebra. Using fixed point method, we prove the Hyers-Ulam stability of derivations on fuzzy Banach -algebras related to functional equation (1).

Theorem 4.1 Let φ : X n [ 0 , ) be a function such that there exists an L < 1 ( n m + 1 ) n 2 with
φ ( x 1 n m + 1 , , x n n m + 1 ) L φ ( x 1 , x 2 , , x n ) n m + 1
for all x 1 , , x n X . Let f : X X be a mapping satisfying f ( 0 ) = 0 ,
(17)
(18)
(19)
for all x 1 , , x n 1 X and all t > 0 . Then D ( x ) : = N - lim p f ( x ( n m + 1 ) p ) ( n m + 1 ) p exists for all x X and defines a fuzzy -derivation D : X X such that
N ( f ( x ) D ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + L φ ( x , , x )

for all x X and all t > 0 .

Proof The proof is similar to the proof of Theorem 3.1. □

Theorem 4.2 Let φ : X n [ 0 , ) be a function such that there exists an L < 1 with
φ ( x 1 , , x n ) ( n m + 1 ) L φ ( x 1 n m + 1 , , x n n m + 1 )
for all x 1 , x 2 , , x n X . Let f : X X be a mapping satisfying f ( 0 ) = 0 , (17), (18) and (19). Then the limit D ( x ) : = N - lim p f ( ( n m + 1 ) p x ) ( n m + 1 ) p exists for all x X and defines a fuzzy -derivation D : X X such that
N ( f ( x ) D ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + φ ( x , , x )

for all x X and all t > 0 .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, College of Sciences, Yasouj University
(2)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University
(3)
Department of Mathematics, Semnan University

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© Azadi Kenary et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.