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

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.

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(xy),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(xy)=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,yΩ;

  3. (3)

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

Theorem 1.1 [2]

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

  4. (4)

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

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 ) =2f ( i = 1 m m x i ) (mN,m2).
(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,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,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 x0, 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 xX 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)>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 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:XY is continuous at each xX, then f:XY 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(xy,ts)N(x,t)N(y,s) for all x,yX 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

xyxy+sy+tx(x,yX;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:XY is called a homomorphism if f(xy)=f(x)f(y) for all x,yX.

  1. (2)

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

Definition 1.6 Let (U,N,, ) be a fuzzy Banach algebra, then an involution on is a mapping u u from into which satisfies

  1. (i)

    u =u for uU;

  2. (ii)

    ( α u + β v ) = α ¯ u + β ¯ v ;

  3. (iii)

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

If, in addition N( u u,ts)=N(u,t)N(u,s) for uU and t>0, then 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:AB, we define

D μ f( x 1 ,, x m ):= i = 1 m μf ( m x i + j = 1 , j i m x j ) +f ( μ i = 1 m x i ) 2f ( μ i = 1 m m x i )

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

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

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:AB 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,yA and t>0. If there exists an L<1 such that

φ(mx,0,,0,mLt)φ(x,0,,0,t)
(2.7)

for all xA and t>0, then there exists a unique homomorphism H:AB such that

N B ( f ( x ) H ( x ) , t ) φ ( x , 0 , , 0 , ( m m L ) t )
(2.8)

for all xA and t>0.

Proof Consider the set X:={g:AB} 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:XX such that Jg(x):= 1 m g(mx) for all xA. By Theorem 3.1 of [17], d(Jg,Jh)Ld(g,h) for all g,hX. 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 xA and t>0. Therefore

N B ( f ( x ) 1 m f ( m x ) , t ) φ(x,0,,0,mt)

for all xA and t>0. Hence d(f,Jf) 1 m . By Theorem 1.1, there exists a mapping H:AB such that

  1. (1)

    H is a fixed point of J, i.e.,

    H(mx)=mH(x)
    (2.10)

for all xA. 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 xA 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 xA.

  1. (3)

    d(f,H) 1 1 L d(f,Jf), 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 ) =2H ( i = 1 m m x i )

for all x 1 ,, x m A.

By a similar method to above, we get μH(mx)=H(mμx) for all μ T 1 and all xA. Thus one can show that the mapping H:AB 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,yA. So H(xy)=H(x)H(y) for all x,yA. Thus H:AB 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 , endowed with the Lie product

[x,y]:= x y y x 2

on , 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:AB is called a fuzzy Lie C -algebra homomorphism if H([x,y])=[H(x),H(y)] for all x,yA.

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:AB 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,yA and t>0. If there exists an L<1 such that

φ(mx,0,,0,mlt)φ(x,0,,0,t)

for all xA and t>0, then there exists a unique homomorphism H:AB such that

N B ( f ( x ) H ( x ) , t ) φ ( x , 0 , , 0 , ( m m L ) t )
(3.5)

for all xA and t>0.

Proof By the same reasoning as the proof of Theorem 2.1, we can find that the mapping H:AB is given by

H(x)= lim n f ( m n x ) m n

for all xA.

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,yA and t>0. So

H ( [ x , y ] ) = [ H ( x ) , H ( y ) ]

for all x,yA.

Thus H:AB 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:AB 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,yA and t>0. Then there exists a unique homomorphism H:AB such that

N B ( f ( x ) H ( x ) , t ) t t + θ m m r x A r

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

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Correspondence to Sung Jin Lee.

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Vahidi, J., Lee, S.J. Stability of homomorphisms on fuzzy Lie C -algebras via fixed point method. J Inequal Appl 2014, 33 (2014). https://doi.org/10.1186/1029-242X-2014-33

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Keywords

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