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Homomorphisms and derivations in induced fuzzy C -algebras

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.

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,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 Hyers-Ulam 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 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:XY 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,nN are fixed integers with n2, 1mn. 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 1mn, a mapping f:XY satisfies equation (1) for all n2 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((nm+1)x)=(nm+1)f(x) and f(mx)=mf(x) for all xX.

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,

(N1) N(x,t)=0 for t0;

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

(N3) N(cx,t)=N(x, t | c | ) if c0;

(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 x0, 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 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) be 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.

  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 xX.

  2. (2)

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

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:XY 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:XY 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 xX 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 xX and t>0. Consider the set S:={g:XY;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:SS such that Jg(x):=(nm+1)g( x n m + 1 ) for all xX. Let g,hS be such that d(g,h)=ϵ. Then N(g(x)h(x),ϵt) t t + φ ( x , , x ) for all xX and t>0. Hence,

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 (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 xX 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,Jf) L ( n m + 1 ) ( n m ) . By Theorem 2.1, there exists a mapping A:XY 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 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 (7) such that there exists μ(0,) satisfying N(f(x)H(x),μt) t t + φ ( x , , x ) for all xX 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 xX.

  1. (3)

    d(f,H) d ( f , J f ) 1 L with fΩ, 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 μC. 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:XY 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 )(nm+1)Lφ ( x 1 n m + 1 , , x n n m + 1 )

for all x 1 , x 2 ,, x n X. Let f:XY 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 xX and defines a fuzzy -homomorphism H:XY 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 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 ( ( n m + 1 ) x ) n m + 1 for all xX. Let g,hS be such that d(g,h)=ϵ. Then N(g(x)h(x),ϵt) t t + φ ( x , , x ) for all xX 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 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 (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 xX and t>0. So, d(f,Jf) 1 ( n m + 1 ) ( n m ) . By Theorem 2.1, there exists a mapping H:XY satisfying the following:

  1. (1)

    A is a fixed point of J, that is,

    (nm+1)H(x)=H ( ( n m + 1 ) x )
    (12)

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 (12) such that there exists μ(0,) satisfying N(f(x)H(x),μt) t t + φ ( x , , x ) for all xX 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 xX.

  1. (3)

    d(f,H) d ( f , J f ) 1 L with fΩ, 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):={uX: 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:XY 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:XY satisfying (5).

Proof By the same reasoning as in the proof of Theorem 3.1, there is a -linear mapping H:XY satisfying (5). The mapping H:XY is given by

N- lim p f ( x ( n m + 1 ) p ) ( n m + 1 ) p =H(x)

for all xX. 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 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 (15) 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. 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 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 (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 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 3.4 Let φ: X n [0,) be a function such that there exists an L<1 with

φ( x 1 ,, x n )(nm+1)Lφ ( x 1 n m + 1 , , x n n m + 1 )

for all x 1 , x 2 ,, x n X. Let f:XY 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 xX and defines a fuzzy -homomorphism H:XY 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 xX 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:XX 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 xX and defines a fuzzy -derivation D:XX 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 xX 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 )(nm+1)Lφ ( x 1 n m + 1 , , x n n m + 1 )

for all x 1 , x 2 ,, x n X. Let f:XX 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 xX and defines a fuzzy -derivation D:XX 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 xX and all t>0.

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Correspondence to Choonkil Park.

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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., Ghirati, M., Park, C. et al. Homomorphisms and derivations in induced fuzzy C -algebras. J Inequal Appl 2013, 88 (2013). https://doi.org/10.1186/1029-242X-2013-88

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Keywords

  • Hyers-Ulam stability
  • fixed point method
  • fuzzy Banach algebra