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Fuzzy -homomorphisms and fuzzy -derivations in induced fuzzy C -algebras

  • 1Email author,
  • 2,
  • 3,
  • 1 and
  • 4
Advances in Difference Equations20122012:147

https://doi.org/10.1186/1687-1847-2012-147

  • Received: 2 April 2012
  • Accepted: 8 August 2012
  • Published:

Abstract

In this paper, we prove the Ulam-Hyers-Rassias stability of the Cauchy-Jensen additive functional equation

f ( x + y + z 2 ) + f ( x y + z 2 ) = f ( x ) + f ( z )

in fuzzy Banach spaces.

MSC:39B52, 46S40, 26E50, 46L05, 39B72.

Keywords

  • Hyers-Ulam-Rassias stability
  • fixed point method
  • fuzzy Banach -algebra
  • induced fuzzy C -algebra

1 Introduction

The stability problem of functional equations originated from the 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 Th.M. Rassias [3] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (Rassias [3])

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 Ulam-Hyers-Rassias 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 Ulam-Hyers-Rassias 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 [721]).

Katsaras [22] 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 [13, 23, 24]).

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].

In this paper we consider a mapping f : X Y satisfying the following Cauchy-Jensen functional equation
f ( x + y + z 2 ) + f ( x y + z 2 ) = f ( x ) + f ( z )
(1.1)

for all x , y , z X and establish the fuzzy -homomorphisms and fuzzy -derivations of (1.1) in induced fuzzy C -algebras.

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 ,

(N 1) N ( x , t ) = 0 for t 0 ;

(N 2) x = 0 if and only if N ( x , t ) = 1 for all t > 0 ;

(N 3) N ( c x , t ) = N ( x , t | c | ) if c 0 ;

(N 4) N ( x + y , c + t ) min { N ( x , s ) , N ( y , t ) } ;

(N 5) N ( x , ) is a non-decreasing function of R and lim t N ( x , t ) = 1 ;

(N 6) for x 0 , N ( x , ) is continuous on R .

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 ) 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.

     
  2. (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 x a fuzzy norm on X.
  1. (1)

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

     
  2. (2)

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

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

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

     
  2. (2)

    A C -linear mapping D : ( X , N x ) ( X , N x ) 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 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 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 .

     

3 Hyers-Ulam-Rassias stability of CJA functional equation (1.1) in fuzzy Banach -algebras

In this section, using the fixed point alternative approach we prove the Ulam-Hyers-Rassias stability of the functional equation (1.1) in fuzzy Banach spaces. Throughout this paper, assume that X is a vector space and that ( Y , N ) is a fuzzy Banach space.

Theorem 3.1 Let φ : X 3 [ 0 , ) be a function such that there exists an L < 1 2 with φ ( x 2 , y 2 , z 2 ) L φ ( x , y , z ) 2 for all x , y , z X . Let f : X Y be a mapping satisfying
(3.1)
(3.2)
(3.3)
for all x , y , z X and t > 0 . Then there exists a fuzzy -homomorphism H : X Y such that
N ( f ( x ) H ( x ) , t ) ( 2 2 L ) t ( 2 2 L ) t + L φ ( x , 2 x , x )
(3.4)

for all x X and t > 0 .

Proof Letting μ = 1 and replacing ( x , y , z ) by ( x , 2 x , x ) in (3.1), we have
N ( f ( 2 x ) 2 f ( x ) , t ) t t + φ ( x , 2 x , x )
(3.5)
for all x X and t > 0 . Replacing x by x 2 in (3.5), we obtain
N ( f ( x ) 2 f ( x 2 ) , t ) t t + φ ( x 2 , x , x 2 ) t t + L 2 φ ( x , 2 x , x ) .
(3.6)
Consider the set S : = { g : X Y } 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 , 2 x , x ) , x X , t > 0 } ,
where inf = + . It is easy to show that ( S , d ) is complete (see [29]). Now, we consider a linear mapping J : S S such that J g ( x ) : = 2 g ( x 2 ) 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 , 2 x , x ) for all x X and t > 0 . Hence
N ( J g ( x ) J h ( x ) , L ϵ t ) = N ( 2 g ( x 2 ) 2 h ( x 2 ) , L ϵ t ) = N ( g ( x 2 ) h ( x 2 ) , L ϵ t 2 ) L t 2 L t 2 + φ ( x 2 , x , x 2 ) L t 2 L t 2 + L φ ( x , 2 x , x ) 2 = t t + φ ( x , 2 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 (3.6) that
N ( 2 f ( x 2 ) f ( x ) , L t 2 ) t t + φ ( x , 2 x , x )
for all x X and all t > 0 . This implies that d ( f , J f ) L 2 . By Theorem 2.1, there exists a mapping H : X Y satisfying the following:
  1. (1)
    H is a fixed point of J, that is,
    H ( x 2 ) = H ( x ) 2
    (3.7)

    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 (3.7) such that there exists μ ( 0 , ) satisfying N ( f ( x ) H ( x ) , μ t ) t t + φ ( x , 2 x , x ) for all x X and t > 0 .

     
  2. (2)
    d ( J n f , H ) 0 as n . This implies the equality
    N - lim n 2 n f ( x 2 n ) = H ( x )
    (3.8)

    for all x X .

     
  3. (3)
    d ( f , H ) d ( f , J f ) 1 L with f Ω , which implies the inequality d ( f , H ) L 2 2 L . This implies that the inequality (3.4) holds. Furthermore, it follows from (3.1) and (3.8) that
    N ( μ H ( x + y + z 2 ) + μ H ( x y + z 2 ) H ( μ x ) H ( μ z ) , t ) = N - lim n ( 2 n μ f ( x + y + z 2 n + 1 ) + 2 n μ f ( x y + z 2 n + 1 ) 2 n f ( μ x 2 n ) 2 n f ( μ z 2 n ) , t ) lim n t 2 n t 2 n + φ ( x 2 n , y 2 n , z 2 n ) lim n t 2 n t 2 n + L n 2 n φ ( x , y , z ) 1
     
for all x , y , z X , all t > 0 and all μ C . Hence
μ H ( x + y + z 2 ) + μ H ( x y + z 2 ) H ( μ x ) H ( μ z ) = 0
for all x , y , z X . So the mapping H : X Y is additive and C -linear. By (3.2),
N ( 4 n f ( x y 4 n ) 2 n f ( x 2 n ) 2 n f ( y 2 n ) , 4 n t ) t t + φ ( x 2 n , y 2 n , 0 )
for all x , y X and all t > 0 . Then
N ( 4 n f ( x y 4 n ) 2 n f ( x 2 n ) 2 n f ( y 2 n ) , t ) t 4 n t 4 n + φ ( x 2 n , y 2 n , 0 ) t 4 n t 4 n + L n φ ( x , y , 0 ) 2 n 1 when  n +
for all x , y X and all t > 0 . So N ( H ( x y ) H ( x ) H ( y ) , t ) = 1 for all x , y X and all t > 0 . By (3.3)
N ( 2 n f ( x 2 n ) 2 n f ( x 2 n ) , 2 n t ) t t + φ ( x 2 n , 0 , 0 )
for all x X and all t > 0 . So
N ( 2 n f ( x 2 n ) 2 n f ( x 2 n ) , t ) t 2 n t 2 n + φ ( x 2 n , 0 , 0 ) t 2 n t 2 n + L n 2 n φ ( x , 0 , 0 )

for all x X and all t > 0 . Since lim n + t 2 n t 2 n + L n 2 n φ ( x , 0 , 0 ) = 1 , for all x X and t > 0 , we get N ( H ( x ) H ( x ) , t ) = 1 for all x X and all t > 0 . Thus H ( x ) = H ( x ) for all x X . □

Theorem 3.2 Let φ : X 3 [ 0 , ) be a function such that there exists an L < 1 with φ ( x , y , z ) 2 L φ ( x 2 , y 2 , z 2 ) for all x , y , z X . Let f : X Y be a mapping satisfying (3.1)-(3.3). Then the limit H ( x ) : = N - lim n f ( 2 n x ) 2 n exists for each x X and defines a fuzzy -homomorphism H : X Y such that
N ( f ( x ) H ( x ) , t ) ( 2 2 L ) t ( 2 2 L ) t + φ ( x , 2 x , x )
(3.9)

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 ( 2 x ) 2 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 , 2 x , x ) for all x X and t > 0 . Hence
N ( J g ( x ) J h ( x ) , L ϵ t ) = N ( g ( 2 x ) 2 h ( 2 x ) 2 , L ϵ t ) = N ( g ( 2 x ) h ( 2 x ) , 2 L ϵ t ) 2 L t 2 L t + φ ( 2 x , , 4 x , 2 x ) 2 L t 2 L t + 2 L φ ( x , , 2 x , x ) = t t + φ ( x , 2 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 (3.5) that
N ( f ( 2 x ) 2 f ( x ) , t 2 ) t t + φ ( x , 2 x , x )
(3.10)
for all x X and t > 0 . So d ( f , J f ) 1 2 . By Theorem 2.1, there exists a mapping H : X Y satisfying the following:
  1. (1)
    H is a fixed point of J, that is,
    2 H ( x ) = H ( 2 x )
    (3.11)

    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 (3.11) such that there exists μ ( 0 , ) satisfying N ( f ( x ) H ( x ) , μ t ) t t + φ ( x , 2 x , x ) for all x X and t > 0 .

     
  2. (2)

    d ( J n f , H ) 0 as n . This implies the equality H ( x ) = N - lim n f ( 2 n x ) 2 n for all x X .

     
  3. (3)

    d ( f , H ) d ( f , J f ) 1 L with f Ω , which implies the inequality d ( f , H ) 1 2 2 L . This implies that the inequality (3.9) holds. The rest of the proof is similar to that of the proof of Theorem 3.1. □

     

4 Hyers-Ulam-Rassias stability of CJA functional equation (1.1) in induced fuzzy C -algebras

Throughout this section, assume that X is a unital C -algebra with unit e and unitary group U ( X ) : = { u X : u u = u u = e } and that Y is a unital C -algebra.

Using the fixed point method, we prove the Hyers-Ulam-Rassias stability of the Cauchy-Jensen additive functional equation (1.1) in induced fuzzy C -algebras.

Theorem 4.1 Let φ : X 3 [ 0 , ) be a function such that there exists an L < 1 2 with φ ( x 2 , y 2 , z 2 ) L φ ( x , y , z ) 2 for all x , y , z X . Let f : X Y be a mapping satisfying (3.1) and
(4.1)
(4.2)

for all u , v U ( X ) and all t > 0 . Then there exists a fuzzy -homomorphism H : X Y satisfying (3.4).

Proof By the same reasoning as in the proof of Theorem 3.1, there is a C -linear mapping H : X Y satisfying (3.4). The mapping H : X Y is given by
N - lim p 2 n f ( x 2 n ) = H ( x )
for all x X . By (4.1),
N ( 4 n f ( u v 4 n ) 2 n f ( u 2 n ) 2 n f ( v 2 n ) , 4 n t ) t t + φ ( u 2 n , v 2 n , 0 )
for all u , v U ( X ) and all t > 0 . Then
N ( 4 n f ( u v 4 n ) 2 n f ( u 2 n ) 2 n f ( v 2 n ) , t ) t 4 n t 4 n + φ ( u 2 n , v 2 n , 0 ) t 4 n t 4 n + L n φ ( u , v , 0 ) 2 n 1 when  n +
for all x , y U ( X ) and all t > 0 . So N ( H ( u v ) H ( u ) H ( v ) , t ) = 1 for all u , v U ( X ) and all t > 0 . Therefore
H ( u v ) = H ( u ) H ( v ) ,
(4.3)
for all u , v U ( X ) . Since H is C -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 (4.3) 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 . By (4.2)
N ( 2 n f ( u 2 n ) 2 n f ( u 2 n ) , 2 n t ) t t + φ ( u 2 n , 0 , 0 )
for all u U ( X ) and all t > 0 . So
N ( 2 n f ( u 2 n ) 2 n f ( u 2 n ) , t ) t 2 n t 2 n + φ ( u 2 n , 0 , 0 ) t 2 n t 2 n + L n 2 n φ ( u , 0 , 0 )
for all u U ( X ) and all t > 0 . Since lim n + t 2 n t 2 n + L n 2 n φ ( u , 0 , 0 ) = 1 , for all u U ( X ) and t > 0 , we get N ( H ( u ) H ( u ) , t ) = 1 for all u U ( X ) and all t > 0 . Thus
H ( u ) = H ( u )
(4.4)
for all u U ( X ) . Since H is C -linear, i.e., 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 (4.4) 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 4.2 Let φ : X 3 [ 0 , ) be a function such that there exists an L < 1 with φ ( x , y , z ) 2 L φ ( x 2 , y 2 , z 2 ) for all x , y , z X . Let f : X Y be a mapping satisfying (3.1), (4.1) and (4.2). Then the limit H ( x ) : = N - lim n f ( 2 n x ) 2 n exists for each x X and defines a fuzzy -homomorphism H : X Y such that
N ( f ( x ) H ( x ) , t ) ( 2 2 L ) t ( 2 2 L ) t + φ ( x , 2 x , x )
(4.5)

for all x X and all t > 0 .

5 Hyers-Ulam-Rassias stability of fuzzy -derivations in fuzzy Banach -algebras and in induced fuzzy C -algebras

In this section, assume that ( X , N X ) is a fuzzy Banach -algebra. Using the fixed point method, we prove the Hyers-Ulam-Rassias stability of fuzzy -derivations in fuzzy Banach -algebras.

Theorem 5.1 Let φ : X 2 [ 0 , ) be a function such that there exists an L < 1 2 with φ ( x 2 , y 2 , z 2 ) L φ ( x , y , z ) 2 for all x , y , z X . Let f : X X be a mapping satisfying (3.1), (3.3) and
N X ( f ( x y ) x f ( y ) y f ( x ) , t ) t t + φ ( x , y , 0 )
(5.1)
for all x , y X and all t > 0 . Then δ ( x ) : = N - lim n 2 n f ( x 2 n ) exists for each x X and defines a fuzzy -derivation δ : X X such that
N ( f ( x ) δ ( x ) , t ) ( 2 2 L ) t ( 2 2 L ) t + L φ ( x , 2 x , x )
(5.2)

for all x X and all t > 0 .

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

Theorem 5.2 Let φ : X 2 [ 0 , ) be a function such that there exists an L < 1 with φ ( x , y , z ) 2 L φ ( x 2 , y 2 , z 2 ) for all x , y , z X . Let f : X Y be a mapping satisfying (3.1) and (5.1). Then the limit δ ( x ) : = N - lim p f ( 2 n x ) 2 n exists for each x X and defines a fuzzy -derivation δ : X Y such that
N ( f ( x ) δ ( x ) , t ) ( 2 2 L ) t ( 2 2 L ) t + φ ( x , 2 x , x )
(5.3)

for all x X and all t > 0 .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, College of Sciences, Yasouj University, Yasouj, 75914-353, Iran
(2)
Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, 73711-13119, Iran
(3)
Department of Mathematics, Semnan University, Semnan, P.O. Box 35195-363, Iran
(4)
Department of Mathematics, Hallym University, Chunchen, 200-702, South Korea

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