Open Access

Stability of homomorphisms on fuzzy Lie C -algebras via fixed point method

Journal of Inequalities and Applications20142014:33

DOI: 10.1186/1029-242X-2014-33

Received: 20 August 2013

Accepted: 17 December 2013

Published: 24 January 2014

Abstract

In this paper, first, we define fuzzy C -algebras and fuzzy Lie C -algebras; then, using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy C -algebras and fuzzy Lie C -algebras for an m-variable additive functional equation.

MSC: 39A10, 39B52, 39B72, 46L05, 47H10, 46B03.

Keywords

fuzzy normed spaces additive functional equation fixed point homomorphism in C -algebras and Lie C -algebras generalized Hyers-Ulam stability

1 Introduction and preliminaries

The stability problem of functional equations originated with a question of Ulam [1] concerning the stability of group homomorphisms: let ( G 1 , ) be a group and let ( G 2 , , d ) be a metric group with the metric d ( , ) . Given ϵ > 0 , does there exist a δ ( ϵ ) > 0 such that if a mapping h : G 1 G 2 satisfies the inequality d ( h ( x y ) , h ( x ) h ( y ) ) < δ for all x , y G 1 , then there is a homomorphism H : G 1 G 2 with d ( h ( x ) , H ( x ) ) < ϵ for all x G 1 ? If the answer is affirmative, we would say that the equation of homomorphism H ( x y ) = H ( x ) H ( y ) is stable. We recall a fundamental result in fixed-point theory. Let Ω be a set. A function d : Ω × Ω [ 0 , ] is called a generalized metric on Ω if d satisfies
  1. (1)

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

     
  2. (2)

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

     
  3. (3)

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

     

Theorem 1.1 [2]

Let ( Ω , d ) be a complete generalized metric space and let J : Ω Ω be a contractive mapping with Lipschitz constant L < 1 . Then for each given element x Ω , 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 ) < , n 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 Γ = { y Ω d ( J n 0 x , y ) < } ;

     
  4. (4)

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

     
In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in fuzzy Lie C -algebras for the following additive functional equation [3]:
i = 1 m f ( m x i + j = 1 , j i m x j ) + f ( i = 1 m x i ) = 2 f ( i = 1 m m x i ) ( m N , m 2 ) .
(1.1)

We use the definition of fuzzy normed spaces given in [410] to investigate a fuzzy version of the Hyers-Ulam stability for the Cauchy-Jensen functional equation in the fuzzy normed algebra setting (see also [1116]).

Definition 1.2 [4]

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

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

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

The pair ( X , N ) is called a fuzzy normed vector space.

Definition 1.3 [4]
  1. (1)

    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 n N ( x n x , t ) = 1 for all t > 0 . In this case, x is called the limit of the sequence { x n } and we denote it by N - lim n x n = x .

     
  2. (2)

    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 0 X if for each sequence { x n } converging to x 0 in X, then 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 [4, 10]).

Definition 1.4 [12]

A fuzzy normed algebra ( X , μ , , ) is a fuzzy normed space ( X , N , ) with algebraic structure such that

( N 7 ) N ( x y , t s ) N ( x , t ) N ( y , s ) for all x , y X and all t , s > 0 , in which is a continuous t-norm.

Every normed algebra ( X , ) defines a fuzzy normed algebra ( X , N , min ) , where
N ( x , t ) = t t + x
for all t > 0 iff
x y x y + s y + t x ( x , y X ; t , s > 0 ) .

This space is called the induced fuzzy normed algebra.

Definition 1.5 (1) Let ( X , N , ) and ( Y , N , ) be fuzzy normed algebras. An -linear mapping f : X Y is called a homomorphism if f ( x y ) = f ( x ) f ( y ) for all x , y X .
  1. (2)

    An -linear mapping f : X X is called a derivation if f ( x y ) = f ( x ) y + x f ( y ) for all x , y X .

     
Definition 1.6 Let ( U , N , , ) be a fuzzy Banach algebra, then an involution on https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2014-33/MediaObjects/13660_2013_Article_994_IEq87_HTML.gif is a mapping u u from https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2014-33/MediaObjects/13660_2013_Article_994_IEq87_HTML.gif into https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2014-33/MediaObjects/13660_2013_Article_994_IEq87_HTML.gif which satisfies
  1. (i)

    u = u for u U ;

     
  2. (ii)

    ( α u + β v ) = α ¯ u + β ¯ v ;

     
  3. (iii)

    ( u v ) = v u for u , v U .

     

If, in addition N ( u u , t s ) = N ( u , t ) N ( u , s ) for u U and t > 0 , then https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2014-33/MediaObjects/13660_2013_Article_994_IEq87_HTML.gif is a fuzzy C -algebra.

2 Stability of homomorphisms in fuzzy C -algebras

Throughout this section, assume that A is a fuzzy C -algebra with norm N A and that B is a fuzzy C -algebra with norm N B .

For a given mapping f : A B , we define
D μ f ( x 1 , , x m ) : = i = 1 m μ f ( m x i + j = 1 , j i m x j ) + f ( μ i = 1 m x i ) 2 f ( μ i = 1 m m x i )

for all μ T 1 : = { ν C : | ν | = 1 } and all x 1 , , x m A .

Note that a -linear mapping H : A B is called a homomorphism in fuzzy C -algebras if H satisfies H ( x y ) = H ( x ) H ( y ) and H ( x ) = H ( x ) for all x , y A .

We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy C -algebras for the functional equation D μ f ( x 1 , , x m ) = 0 .

Theorem 2.1 Let f : A B be a mapping for which there are functions φ : A m × ( 0 , ) [ 0 , 1 ] , ψ : A 2 × ( 0 , ) [ 0 , 1 ] and η : A × ( 0 , ) [ 0 , 1 ] such that
N B ( D μ f ( x 1 , , x m ) , t ) φ ( x 1 , , x m , t ) ,
(2.1)
lim j φ ( m j x 1 , , m j x m , m j t ) = 1 ,
(2.2)
N B ( f ( x y ) f ( x ) f ( y ) , t ) ψ ( x , y , t ) ,
(2.3)
lim j ψ ( m j x , m j y , m 2 j t ) = 1 ,
(2.4)
N B ( f ( x ) f ( x ) , t ) η ( x , t ) ,
(2.5)
lim j η ( m j x , m j t ) = 1
(2.6)
for all μ T 1 , all x 1 , , x m , x , y A and t > 0 . If there exists an L < 1 such that
φ ( m x , 0 , , 0 , m L t ) φ ( x , 0 , , 0 , t )
(2.7)
for all x A and t > 0 , then there exists a unique homomorphism H : A B such that
N B ( f ( x ) H ( x ) , t ) φ ( x , 0 , , 0 , ( m m L ) t )
(2.8)

for all x A and t > 0 .

Proof Consider the set X : = { g : A B } and introduce the generalized metric on X:
d ( g , h ) = inf { C R + : N B ( g ( x ) h ( x ) , C t ) φ ( x , 0 , , 0 , t ) , x A , t > 0 } .
It is easy to show that ( X , d ) is complete. Now, we consider the linear mapping J : X X such that J g ( x ) : = 1 m g ( m x ) for all x A . By Theorem 3.1 of [17], d ( J g , J h ) L d ( g , h ) for all g , h X . Letting μ = 1 , x = x 1 and x 2 = = x m = 0 in equation (2.1), we get
N B ( f ( m x ) m f ( x ) , t ) φ ( x , 0 , , 0 , t )
(2.9)
for all x A and t > 0 . Therefore
N B ( f ( x ) 1 m f ( m x ) , t ) φ ( x , 0 , , 0 , m t )
for all x A and t > 0 . Hence d ( f , J f ) 1 m . By Theorem 1.1, there exists a mapping H : A B such that
  1. (1)
    H is a fixed point of J, i.e.,
    H ( m x ) = m H ( x )
    (2.10)
     
for all x A . The mapping H is a unique fixed point of J in the set
Y = { g X : d ( f , g ) < } .
This implies that H is a unique mapping satisfying equation (2.10) such that there exists C ( 0 , ) satisfying
N B ( H ( x ) f ( x ) , C t ) φ ( x , 0 , , 0 , t )
for all x A and t > 0 .
  1. (2)
    d ( J n f , H ) 0 as n . This implies the equality
    lim n f ( m n x ) m n = H ( x )
    (2.11)
     
for all x A .
  1. (3)

    d ( f , H ) 1 1 L d ( f , J f ) , which implies the inequality d ( f , H ) 1 m m L . This implies that the inequality (2.8) holds.

     
It follows from equations (2.1), (2.2), and (2.11) that
N B ( i = 1 m H ( m x i + j = 1 , j i m x j ) + H ( i = 1 m x i ) 2 H ( i = 1 m m x i ) , t ) = lim n N B ( i = 1 m f ( m n + 1 x i + j = 1 , j i m m n x j ) + f ( i = 1 m m n x i ) 2 f ( i = 1 m m n + 1 x i ) , m n t ) lim n φ ( m n x 1 , , m n x m , m n t ) = 1
for all x 1 , , x m A and t > 0 . So
i = 1 m H ( m x i + j = 1 , j i m x j ) + H ( i = 1 m x i ) = 2 H ( i = 1 m m x i )

for all x 1 , , x m A .

By a similar method to above, we get μ H ( m x ) = H ( m μ x ) for all μ T 1 and all x A . Thus one can show that the mapping H : A B is -linear.

It follows from equations (2.3), (2.4), and (2.11) that
N B ( H ( x y ) H ( x ) H ( y ) , t ) = lim n N B ( f ( m n x y ) f ( m n x ) f ( m n y ) , m n t ) lim n ψ ( m n x , m n y , m 2 n t ) = 1

for all x , y A . So H ( x y ) = H ( x ) H ( y ) for all x , y A . Thus H : A B is a homomorphism satisfying equation (2.7), as desired.

Also by equations (2.5), (2.6), (2.11), and by a similar method we have H ( x ) = H ( x ) . □

3 Stability of homomorphisms in fuzzy Lie C -algebras

A fuzzy C -algebra https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2014-33/MediaObjects/13660_2013_Article_994_IEq132_HTML.gif , endowed with the Lie product
[ x , y ] : = x y y x 2

on https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2014-33/MediaObjects/13660_2013_Article_994_IEq132_HTML.gif , is called a fuzzy Lie C -algebra (see [1820]).

Definition 3.1 Let A and B be fuzzy Lie C -algebras. A -linear mapping H : A B is called a fuzzy Lie C -algebra homomorphism if H ( [ x , y ] ) = [ H ( x ) , H ( y ) ] for all x , y A .

Throughout this section, assume that A is a fuzzy Lie C -algebra with norm N A and that B is a fuzzy Lie C -algebra with norm N B .

We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy Lie C -algebras for the functional equation D μ f ( x 1 , , x m ) = 0 .

Theorem 3.2 Let f : A B be a mapping for which there are functions φ : A m × ( 0 , ) [ 0 , 1 ] and ψ : A 2 × ( 0 , ) [ 0 , 1 ] such that
lim j φ ( m j x 1 , , m j x m , m j t ) = 1 ,
(3.1)
N B ( D μ f ( x 1 , , x m ) , t ) φ ( x 1 , , x m , t ) ,
(3.2)
N B ( f ( [ x , y ] ) [ f ( x ) , f ( y ) ] , t ) ψ ( x , y , t ) ,
(3.3)
lim j ψ ( m j x , m j y , m 2 j t ) = 1
(3.4)
for all μ T 1 , all x 1 , , x m , x , y A and t > 0 . If there exists an L < 1 such that
φ ( m x , 0 , , 0 , m l t ) φ ( x , 0 , , 0 , t )
for all x A and t > 0 , then there exists a unique homomorphism H : A B such that
N B ( f ( x ) H ( x ) , t ) φ ( x , 0 , , 0 , ( m m L ) t )
(3.5)

for all x A and t > 0 .

Proof By the same reasoning as the proof of Theorem 2.1, we can find that the mapping H : A B is given by
H ( x ) = lim n f ( m n x ) m n

for all x A .

It follows from equation (3.3) that
N B ( H ( [ x , y ] ) [ H ( x ) , H ( y ) ] , t ) = lim n N B ( f ( m 2 n [ x , y ] ) [ f ( m n x ) , f ( m n y ) ] , m 2 n t ) lim n ψ ( m n x , m n y , m 2 n t ) = 1
for all x , y A and t > 0 . So
H ( [ x , y ] ) = [ H ( x ) , H ( y ) ]

for all x , y A .

Thus H : A B is a fuzzy Lie C -algebra homomorphism satisfying equation (3.5), as desired. □

Corollary 3.3 Let 0 < r < 1 and θ be nonnegative real numbers, and let f : A B be a mapping such that
N B ( D μ f ( x 1 , , x m ) , t ) t t + θ ( x 1 A r + x 2 A r + + x m A r ) ,
(3.6)
N B ( f ( [ x , y ] ) [ f ( x ) , f ( y ) ] , t ) t t + θ x A r y A r
(3.7)
for all μ T 1 , all x 1 , , x m , x , y A and t > 0 . Then there exists a unique homomorphism H : A B such that
N B ( f ( x ) H ( x ) , t ) t t + θ m m r x A r

for all x A and t > 0 .

Proof The proof follows from Theorem 3.2 by taking
φ ( x 1 , , x m , t ) = t t + θ ( x 1 A r + x 2 A r + + x m A r ) , ψ ( x , y , t ) : = t t + θ x A r y A r

for all x 1 , , x m , x , y A and t > 0 . Putting L = m r 1 , we get the desired result. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Iran University of Science and Technology
(2)
Department of Mathematics, Daejin University

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© Vahidi and Lee; licensee Springer. 2014

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