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R E S E A R C H

Open Access

Stability of homomorphisms on fuzzy Lie

C

-algebras via fixed point method

Javad Vahidi

1

and Sung Jin Lee

2*

*Correspondence: hyper@daejin.ac.kr

2Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea

Full list of author information is available at the end of the article

Abstract

In this paper, first, we define fuzzyC∗-algebras and fuzzy LieC∗-algebras; then, using fixed point methods, we prove the generalized Hyers-Ulam stability of

homomorphisms in fuzzyC∗-algebras and fuzzy LieC∗-algebras for anm-variable additive functional equation.

MSC: Primary 39A10; 39B52; 39B72; 46L05; 47H10; 46B03

Keywords: fuzzy normed spaces; additive functional equation; fixed point;

homomorphism inC∗-algebras and LieC∗-algebras; generalized Hyers-Ulam stability

1 Introduction and preliminaries

The stability problem of functional equations originated with a question of Ulam [] con-cerning the stability of group homomorphisms: let (G,∗) be a group and let (G,,d) be a metric group with the metricd(·,·). Given> , does there exist aδ() >  such that if a mappingh:G→Gsatisfies the inequalityd(h(xy),h(x)h(y)) <δfor allx,yG, then there is a homomorphismH:G→Gwithd(h(x),H(x)) <for allxG? If the answer is affirmative, we would say that the equation of homomorphismH(xy) =H(x)H(y) is stable. We recall a fundamental result in fixed-point theory. Letbe a set. A function

d:×→[,∞] is called ageneralized metriconifdsatisfies () d(x,y) = if and only ifx=y;

() d(x,y) =d(y,x)for allx,y;

() d(x,z)≤d(x,y) +d(y,z)for allx,y,z.

Theorem .[] Let(,d)be a complete generalized metric space and let J:be a contractive mapping with Lipschitz constant L< .Then for each given element x,

either d(Jnx,Jn+x) =for all nonnegative integers n or there exists a positive integer n

such that

() d(Jnx,Jn+x) <,nn ;

() the sequence{Jnx}converges to a fixed pointyofJ;

() yis the unique fixed point ofJin the set={y|d(Jnx,y) <∞};

() d(y,y∗)≤–Ld(y,Jy)for ally.

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functional equation []:

m

i=

f

mxi+ m

j=,j=i

xj

+f

m

i=

xi

= f

m

i=

mxi

(m∈N,m≥). (.)

We use the definition of fuzzy normed spaces given in [–] to investigate a fuzzy ver-sion of the Hyers-Ulam stability for the Cauchy-Jensen functional equation in the fuzzy normed algebra setting (see also [–]).

Definition .[] LetXbe a real vector space. A functionN:X×R→[, ] is called a

fuzzy normonXif for allx,yXand alls,t∈R, (N) N(x,t) = fort≤;

(N) x= if and only ifN(x,t) = for allt> ; (N) N(cx,t) =N(x,|tc|)ifc= ;

(N) N(x+y,s+t)≥min{N(x,s),N(y,t)};

(N) N(x,·)is a non-decreasing function ofRandlimt→∞N(x,t) = ;

(N) forx= ,N(x,·)is continuous onR.

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

Definition .[] () Let (X,N) be a fuzzy normed vector space. A sequence{xn}inXis

said tobe convergentorconvergeif there exists anxXsuch thatlimn→∞N(xnx,t) = 

for allt> . In this case,xis called thelimit of the sequence{xn}and we denote it by

N-limn→∞xn=x.

() Let (X,N) be a fuzzy normed vector space. A sequence{xn}inXis calledCauchyif

for eachε>  and eacht>  there exists ann∈Nsuch that for allnnand allp> , we haveN(xn+pxn,t) >  –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 com-pleteand the fuzzy normed vector space is called afuzzy Banach space.

We say that a mappingf:XYbetween fuzzy normed vector spacesXandYis con-tinuous at a pointx∈Xif for each sequence{xn}converging toxinX, then the sequence

{f(xn)}converges tof(x). Iff :XYis continuous at eachxX, thenf:XY is said to becontinuousonX(see [, ]).

Definition .[] Afuzzy normed algebra(X,μ,∗,∗) is a fuzzy normed space (X,N,∗) with algebraic structure such that

(N) N(xy,ts)≥N(x,t)∗N(y,s)for allx,yXand allt,s> , in which∗is a continuous

t-norm.

Every normed algebra (X, · ) defines a fuzzy normed algebra (X,N,min), where

N(x,t) = t

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for allt>  iff

xyxy+sy+tx (x,yX;t,s> ).

This space is called the induced fuzzy normed algebra.

Definition . () Let (X,N,∗) and (Y,N,∗) be fuzzy normed algebras. AnR-linear map-pingf :XYis called ahomomorphismiff(xy) =f(x)f(y) for allx,yX.

() AnR-linear mappingf :XXis called aderivationiff(xy) =f(x)y+xf(y) for all

x,yX.

Definition . Let (U,N,∗,∗) be a fuzzy Banach algebra, then an involution onU is a mappinguu∗fromUintoUwhich satisfies

(i) u∗∗=uforuU; (ii) (αu+βv)∗=αu∗+βv∗; (iii) (uv)∗=vu∗foru,vU.

If, in additionN(uu,ts) =N(u,t)∗N(u,s) foruU andt> , thenU is a fuzzyC∗ -algebra.

2 Stability of homomorphisms in fuzzyC∗-algebras

Throughout this section, assume thatAis a fuzzyC∗-algebra with normNAand thatBis

a fuzzyC∗-algebra with normNB.

For a given mappingf :AB, we define

Dμf(x, . . . ,xm) := m

i=

μf

mxi+ m

j=,j=i

xj

+f

μ

m

i=

xi

– f

μ

m

i=

mxi

for allμ∈T:={νC:|ν|= }and allx

, . . . ,xmA.

Note that aC-linear mappingH:ABis called ahomomorphismin fuzzyC∗-algebras ifHsatisfiesH(xy) =H(x)H(y) andH(x∗) =H(x)∗for allx,yA.

We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzyC∗-algebras for the functional equationDμf(x, . . . ,xm) = .

Theorem . Let f :AB be a mapping for which there are functionsϕ:Am×(,)

[, ],ψ:A×(,∞)→[, ]andη:A×(,∞)→[, ]such that

NB

Dμf(x, . . . ,xm),t

ϕ(x, . . . ,xm,t), (.)

lim

j→∞ϕ

mjx, . . . ,mjxm,mjt

= , (.)

NB

f(xy) –f(x)f(y),tψ(x,y,t), (.)

lim

j→∞ψ

mjx,mjy,mjt= , (.)

NB

fx∗–f(x)∗,tη(x,t), (.)

lim

j→∞η

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for allμ∈T,all x

, . . . ,xm,x,yA and t> .If there exists an L< such that

ϕ(mx, , . . . , ,mLt)≥ϕ(x, , . . . , ,t) (.)

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

NB

f(x) –H(x),tϕx, , . . . , , (mmL)t (.)

for all xA and t> .

Proof Consider the setX:={g:AB}and introduce thegeneralized metriconX:

d(g,h) =infC∈R+:NB

g(x) –h(x),Ctϕ(x, , . . . , ,t),∀xA,t> .

It is easy to show that (X,d) is complete. Now, we consider the linear mappingJ:XX

such thatJg(x) := mg(mx) for allxA. By Theorem . of [],d(Jg,Jh)≤Ld(g,h) for all

g,hX. Lettingμ= ,x=xandx=· · ·=xm=  in equation (.), we get

NB

f(mx) –mf(x),tϕ(x, , . . . , ,t) (.)

for allxAandt> . Therefore

NB f(x) –

mf(mx),t

ϕ(x, , . . . , ,mt)

for allxAandt> . Henced(f,Jf)≤m. By Theorem ., there exists a mappingH:ABsuch that

()His a fixed point ofJ,i.e.,

H(mx) =mH(x) (.)

for allxA. The mappingHis a unique fixed point ofJin the set

Y=gX:d(f,g) <∞.

This implies thatHis a unique mapping satisfying equation (.) such that there exists

C∈(,∞) satisfying

NB

H(x) –f(x),Ctϕ(x, , . . . , ,t)

for allxAandt> .

()d(Jnf,H) asn→ ∞. This implies the equality

lim

n→∞

f(mnx)

mn =H(x) (.)

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()d(f,H)≤ –Ld(f,Jf), which implies the inequalityd(f,H)≤mmL. This implies that the inequality (.) holds.

It follows from equations (.), (.), and (.) that

NB m i= H

mxi+ m

j=,j=i

xj +H m i= xi

– H

m i= mxi ,t = lim

n→∞NB m

i=

f

mn+xi+ m

j=,j=i

mnxj

+f

m

i=

mnxi

– f

m

i=

mn+xi

,mnt

≤ lim

n→∞ϕ

mnx, . . . ,mnxm,mnt

= 

for allx, . . . ,xmAandt> . So m

i=

H

mxi+ m

j=,j=i

xj +H m i= xi

= H

m

i=

mxi

for allx, . . . ,xmA.

By a similar method to above, we getμH(mx) =H(mμx) for allμ∈T and allxA. Thus one can show that the mappingH:ABisC-linear.

It follows from equations (.), (.), and (.) that

NB

H(xy) –H(x)H(y),t= lim

n→∞NB

fmnxyfmnxfmny,mnt

≤ lim

n→∞ψ

mnx,mny,mnt= 

for allx,yA. SoH(xy) =H(x)H(y) for allx,yA. ThusH:ABis a homomorphism satisfying equation (.), as desired.

Also by equations (.), (.), (.), and by a similar method we haveH(x∗) =H(x)∗.

3 Stability of homomorphisms in fuzzy LieC∗-algebras A fuzzyC∗-algebraC, endowed with the Lie product

[x,y] :=xyyx

onC, is called afuzzy Lie C∗-algebra(see [–]).

Definition . LetAandBbe fuzzy LieC∗-algebras. AC-linear mappingH:ABis called afuzzy Lie C∗-algebra homomorphismifH([x,y]) = [H(x),H(y)] for allx,yA.

Throughout this section, assume thatAis a fuzzy LieC∗-algebra with normNAand that

Bis a fuzzy LieC∗-algebra with normNB.

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Theorem . Let f :AB be a mapping for which there are functionsϕ:Am×(,)

[, ]andψ:A×(,)[, ]such that lim

j→∞ϕ

mjx, . . . ,mjxm,mjt

= , (.)

NB

Dμf(x, . . . ,xm),t

ϕ(x, . . . ,xm,t), (.)

NB

f[x,y]–f(x),f(y),tψ(x,y,t), (.)

lim

j→∞ψ

mjx,mjy,mjt=  (.)

for allμ∈T,all x, . . . ,xm,x,yA and t> .If there exists an L< such that ϕ(mx, , . . . , ,mlt)≥ϕ(x, , . . . , ,t)

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

f(x) –H(x),tϕx, , . . . , , (mmL)t (.)

for all xA and t> .

Proof By the same reasoning as the proof of Theorem ., we can find that the mapping

H:ABis given by

H(x) = lim

n→∞

f(mnx)

mn

for allxA.

It follows from equation (.) that

NB

H[x,y]–H(x),H(y),t= lim

n→∞NB

fmn[x,y]–fmnx,fmny,mnt

≥ lim

n→∞ψ

mnx,mny,mnt=  for allx,yAandt> . So

H[x,y]=H(x),H(y) for allx,yA.

ThusH:ABis a fuzzy LieC∗-algebra homomorphism satisfying equation (.), as

desired.

Corollary . Let <r< andθ be nonnegative real numbers,and let f :AB be a mapping such that

NB

Dμf(x, . . . ,xm),t

t

t+θ(xrA+xrA+· · ·+xmrA)

, (.)

NB

f[x,y]–f(x),f(y),tt t+θ· xr

A· yrA

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for allμ∈T,all x

, . . . ,xm,x,yA and t> .Then there exists a unique homomorphism

H:AB such that NB

f(x) –H(x),tt t+mθmrxrA

for all xA and t> .

Proof The proof follows from Theorem . by taking

ϕ(x, . . . ,xm,t) =

t

t+θ(xrA+xrA+· · ·+xmrA)

,

ψ(x,y,t) := t

t+θ· xr A· yrA

for allx, . . . ,xm,x,yAandt> . PuttingL=mr–, we get the desired result.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.2Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea.

Received: 20 August 2013 Accepted: 17 December 2013 Published:24 Jan 2014

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10.1186/1029-242X-2014-33

Cite this article as:Vahidi and Lee:Stability of homomorphisms on fuzzy LieC∗-algebras via fixed point method.

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