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

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.

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,yE, where ϵ and p are constants with ϵ>0 and 0p<1. Then the limit L(x)= lim n f ( 2 n x ) 2 n exists for all xE and L:E E is the unique additive mapping which satisfies

f ( x ) L ( x ) 2 ϵ 2 2 p x p

for all xE. Also, if for each xE the function f(tx) is continuous in tR, then L is linear.

The functional equation f(x+y)+f(xy)=2f(x)+2f(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:XY, 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:XY 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,zX 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,yX and all s,tR,

(N 1) N(x,t)=0 for t0;

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

(N 3) N(cx,t)=N(x, t | c | ) if c0;

(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 x0, 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 xX 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)>1ϵ.

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:XY between fuzzy normed vector spaces X and Y is continuous at a point xX if for each sequence { x n } converging to x 0 X the sequence {f( x n )} converges to f( x 0 ). If f:XY is continuous at each xX, then f:XY 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(xy,st)N(x,s)N(y,t),N ( x , t ) =N(x,t)

    for all x,yX 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 xX.

  2. (2)

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

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,yX;

  2. (2)

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

  3. (3)

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

Theorem 2.1 Let (X,d) be a complete generalized metric space and J:XX be a strictly contractive mapping with Lipschitz constant L<1. Then, for all xX, 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={yX:d( J n 0 x,y)<};

  4. (4)

    d(y, y ) 1 1 L d(y,Jy) for all yY.

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,zX. Let f:XY be a mapping satisfying

(3.1)
(3.2)
(3.3)

for all x,y,zX and t>0. Then there exists a fuzzy -homomorphism H:XY 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 xX and t>0.

Proof Letting μ=1 and replacing (x,y,z) by (x,2x,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 xX 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:XY} 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:SS such that Jg(x):=2g( x 2 ) for all xX. Let g,hS be such that d(g,h)=ϵ. Then N(g(x)h(x),ϵt) t t + φ ( x , 2 x , x ) for all xX 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 xX and t>0. Thus d(g,h)=ϵ implies that d(Jg,Jh)Lϵ. This means that d(Jg,Jh)Ld(g,h) for all g,hS. 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 xX and all t>0. This implies that d(f,Jf) L 2 . By Theorem 2.1, there exists a mapping H:XY satisfying the following:

  1. (1)

    H is a fixed point of J, that is,

    H ( x 2 ) = H ( x ) 2
    (3.7)

    for all xX. The mapping H is a unique fixed point of J in the set Ω={hS: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),μt) t t + φ ( x , 2 x , x ) for all xX 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 xX.

  3. (3)

    d(f,H) d ( f , J f ) 1 L with fΩ, 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,zX, all t>0 and all μC. Hence

μH ( x + y + z 2 ) +μH ( x y + z 2 ) H(μx)H(μz)=0

for all x,y,zX. So the mapping H:XY 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,yX 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,yX and all t>0. So N(H(xy)H(x)H(y),t)=1 for all x,yX 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 xX 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 xX and all t>0. Since lim n + t 2 n t 2 n + L n 2 n φ ( x , 0 , 0 ) =1, for all xX and t>0, we get N(H( x )H ( x ) ,t)=1 for all xX and all t>0. Thus H( x )=H ( x ) for all xX. □

Theorem 3.2 Let φ: X 3 [0,) be a function such that there exists an L<1 with φ(x,y,z)2Lφ( x 2 , y 2 , z 2 ) for all x,y,zX. Let f:XY 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 xX and defines a fuzzy -homomorphism H:XY 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 xX 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:SS such that Jg(x):= g ( 2 x ) 2 for all xX. Let g,hS be such that d(g,h)=ϵ. Then N(g(x)h(x),ϵt) t t + φ ( x , 2 x , x ) for all xX 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 xX and t>0. Thus d(g,h)=ϵ implies that d(Jg,Jh)Lϵ. This means that d(Jg,Jh)Ld(g,h) for all g,hS. 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 xX and t>0. So d(f,Jf) 1 2 . By Theorem 2.1, there exists a mapping H:XY satisfying the following:

  1. (1)

    H is a fixed point of J, that is,

    2H(x)=H(2x)
    (3.11)

    for all xX. The mapping H is a unique fixed point of J in the set Ω={hS: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),μt) t t + φ ( x , 2 x , x ) for all xX 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 xX.

  3. (3)

    d(f,H) d ( f , J f ) 1 L with fΩ, 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):={uX: 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,zX. Let f:XY be a mapping satisfying (3.1) and

(4.1)
(4.2)

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

Proof By the same reasoning as in the proof of Theorem 3.1, there is a C-linear mapping H:XY satisfying (3.4). The mapping H:XY is given by

N- lim p 2 n f ( x 2 n ) =H(x)

for all xX. 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,vU(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,yU(X) and all t>0. So N(H(uv)H(u)H(v),t)=1 for all u,vU(X) and all t>0. Therefore

H(uv)=H(u)H(v),
(4.3)

for all u,vU(X). Since H is C-linear and each xX 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(xv)=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 vU(X). So H(xv)=H(x)H(v). Similarly, one can obtain that H(xy)=H(x)H(y) for all x,yX. 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 uU(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 uU(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 uU(X) and t>0 , we get N(H( u )H ( u ) ,t)=1 for all uU(X) and all t>0. Thus

H ( u ) =H ( u )
(4.4)

for all uU(X). Since H is C-linear, i.e., xX 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 xX. So H( x )=H ( x ) for all xX. Therefore, the mapping H:XY 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)2Lφ( x 2 , y 2 , z 2 ) for all x,y,zX. Let f:XY 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 xX and defines a fuzzy -homomorphism H:XY 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 xX 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,zX. Let f:XX 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,yX and all t>0. Then δ(x):=N- lim n 2 n f( x 2 n ) exists for each xX and defines a fuzzy -derivation δ:XX 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 xX 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)2Lφ( x 2 , y 2 , z 2 ) for all x,y,zX. Let f:XY 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 xX and defines a fuzzy -derivation δ:XY 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 xX and all t>0.

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Correspondence to H Azadi Kenary.

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The authors declare that they have no competing interests.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Azadi Kenary, H., Zohdi, A., Eshaghi Gordji, M. et al. Fuzzy -homomorphisms and fuzzy -derivations in induced fuzzy C -algebras. Adv Differ Equ 2012, 147 (2012). https://doi.org/10.1186/1687-1847-2012-147

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Keywords

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