R E S E A R C H
Open Access
Approximation of homomorphisms and
derivations on non-Archimedean random Lie
C
*
-algebras via fixed point method
Jung Im Kang
1and Reza Saadati
2**Correspondence: [email protected] 2Department of Mathematics, Iran
University Science and Technology, Tehran, Iran
Full list of author information is available at the end of the article
Abstract
In this paper, using fixed point methods, we prove the generalized Hyers-Ulam stability of random homomorphisms in randomC*-algebras and random Lie
C*-algebras and of derivations on non-Archimedean random C*-algebras and
non-Archimedean random Lie C*-algebras for anm-variable additive functional
equation.
MSC: Primary 39A10; 39B52; 39B72; 46L05; 47H10; 46B03
Keywords: additive functional equation; fixed point; non-Archimedean random space; homomorphism inC*-algebras and LieC*-algebras; generalized Hyers-Ulam
stability; derivation onC*-algebras and LieC*-algebras
1 Introduction and preliminaries
By anon-Archimedean fieldwe mean a fieldK equipped with a function (valuation)| · | fromKinto [,∞) such that|r|= if and only ifr= ,|rs|=|r||s|and|r+s| ≤max{|r|,|s|} for allr,s∈K. Clearly,||=|– |= and|n| ≤ for alln∈N. By the trivial valuation we mean the mapping| · |taking everything but into and||= . LetXbe a vector space over a fieldKwith a non-Archimedean non-trivial valuation| · |. A function · :X→ [,∞) is called anon-Archimedean normif it satisfies the following conditions:
(i) x= if and only ifx= ; (ii) for anyr∈K,x∈X,rx=|r|x;
(iii) the strong triangle inequality (ultrametric) holds; namely
x+y ≤maxx,y (x,y∈X).
Then (X, · ) is called a non-Archimedean normed space. From the fact that
xn–xm ≤max
xj+–xj:m≤j≤n–
(n>m)
holds, a sequence{xn}is Cauchy if and only if{xn+–xn}converges to zero in a
non-Archimedean normed space. By a complete non-non-Archimedean normed space, we mean the one in which every Cauchy sequence is convergent.
For any nonzero rational number x, there exists a unique integer nx ∈Z such that
x= abpnx, whereaandbare integers not divisible byp. Then|x|
p:=p–nx defines a
Archimedean norm onQ. The completion ofQwith respect to the metricd(x,y) =|x–y|p
is denoted byQp, which is called thep-adic number field.
A non-Archimedean Banach algebra is a complete non-Archimedean algebraAwhich satisfiesab ≤ abfor alla,b∈A. For more detailed definitions of non-Archimedean Banach algebras, we refer the reader to [, ].
IfUis a non-Archimedean Banach algebra, then an involution onUis a mappingt→t* fromUintoU which satisfies
(i) t**=tfort∈U; (ii) (αs+βt)*=αs*+βt*; (iii) (st)*=t*s*fors,t∈U.
If, in addition,t*t=tfort∈U, thenUis a non-ArchimedeanC*-algebra.
The stability problem of functional equations was originated from a question of Ulam [] concerning the stability of group homomorphisms. Let (G,∗) be a group and let (G,,d)
be a metric group (a metric which is defined on a set with a group property) with the met-ricd(·,·). Given> , does there exist aδ() > such that if a mappingh:G→Gsatisfies
the inequalityd(h(x∗y),h(x)h(y)) <δfor allx,y∈G, then there is a homomorphism
H:G→Gwithd(h(x),H(x)) <for allx∈G? If the answer is affirmative, we would
say that the equation of a homomorphismH(x∗y) =H(x)H(y) is stable (see also [–]). We recall a fundamental result in fixed point theory. Letbe a set. A functiond:× →[,∞] 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 the 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) <∞,∀n≥n ;
() the sequence{Jnx}converges to a fixed pointy*ofJ;
() y*is the unique fixed point ofJin the set={y∈|d(Jnx,y) <∞}; () d(y,y*)≤
–Ld(y,Jy)for ally∈.
In this paper, using the fixed point method, we prove the generalized Hyers-Ulam sta-bility of homomorphisms and derivations in non-Archimedean randomC*-algebras and
non-Archimedean random LieC*-algebras for the following additive functional equation
(see []):
m
i=
f
mxi+ m
j=,j=i
xj
+f
m
i=
xi
= f
m
i=
mxi
(m∈N,m≥). (.)
2 Random spaces
In the section, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces as in [–]. Throughout this paper,+is the space of
distri-bution functions, that is, the space of all mappingsF:R∪ {–∞,∞} →[, ] such thatF is left-continuous and non-decreasing onR,F() = andF(+∞) = .D+is a subset of+
limit of the functionf at the pointx, that is,l–f(x) =lim
t→x–f(t). The space+is partially
ordered by the usual point-wise ordering of functions,i.e.,F≤Gif and only ifF(t)≤G(t) for alltinR. The maximal element for+ in this order is the distribution functionε
given by
ε(t) = ⎧ ⎨ ⎩
ift≤, ift> .
Definition .[] A mappingT: [, ]×[, ]→[, ] is a continuous triangular norm (briefly, a continuoust-norm) ifTsatisfies the following conditions:
(a) Tis commutative and associative; (b) Tis continuous;
(c) T(a, ) =afor alla∈[, ];
(d) T(a,b)≤T(c,d)whenevera≤candb≤dfor alla,b,c,d∈[, ].
Typical examples of continuous t-norms are TP(a,b) = ab, TM(a,b) =min(a,b) and
TL(a,b) =max(a+b– , ) (the Lukasiewiczt-norm).
Definition .[] Anon-Archimedean random normed space(briefly, NA-RN-space) is
a triple (X,μ,T), whereXis a vector space,Tis a continuoust-norm, andμis a mapping fromXintoD+such that the following conditions hold:
(RN) μx(t) =ε(t)for allt> if and only ifx= ; (RN) μαx(t) =μx(|αt|)for allx∈X,α= ;
(RN) μx+y(t)≥T(μx(t),μy(t))for allx,y∈Xand allt≥.
Every normed space (X, · ) defines a non-Archimedean random normed space (X,μ,TM), where
μx(t) =
t t+x
for allt> , andTM is the minimum t-norm. This space is called the induced random
normed space.
Definition .[] Anon-Archimedean random normed algebra(X,μ,T,T) is a
non-Archimedean random normed space (X,μ,T) with an algebraic structure such that
(RN-) μxy(t)≥T(μx(t),μy(t))for allx,y∈Xand allt> , in whichTis a continuous
t-norm.
Every non-Archimedean normed algebra (X, · ) defines a non-Archimedean random normed algebra (X,μ,TM), where
μx(t) =
t t+x for allt> iff
xy ≤ xy+ty+tx (x,y∈X;t> ).
Definition .
() Let(X,μ,TM)and(Y,μ,TM)be non-Archimedean random normed algebras. An
R-linear mappingf :X→Yis called ahomomorphismiff(xy) =f(x)f(y)for all
x,y∈X.
() AnR-linear mappingf :X→Xis called aderivationiff(xy) =f(x)y+xf(y)for all
x,y∈X.
Definition . Let (U,μ,T,T) be a non-Archimedean random Banach algebra, then an involution onUis a mappingu→u*fromU intoUwhich satisfies
(i) u**=uforu∈U; (ii) (αu+βv)*=αu*+βv*; (iii) (uv)*=v*u*foru,v∈U.
If, in addition,μu*u(t) =T(μu(t),μu(t)) foru∈Uandt> , thenUis a non-Archimedean
randomC*-algebra.
Definition . Let (X,μ,T) be an NA-RN-space.
() A sequence{xn}inXis said to beconvergenttoxinXif, for every> andλ> , there exists a positive integerNsuch thatμxn–x() > –λwhenevern≥N. () A sequence{xn}inXis called aCauchy sequenceif, for every> andλ> , there
exists a positive integerNsuch thatμxn–xn+() > –λwhenevern≥m≥N. () An RN-space(X,μ,T)is said to becompleteif and only if every Cauchy sequence in
Xis convergent to a point inX.
3 Stability of homomorphisms and derivations in non-Archimedean random C*-algebras
Throughout this section, assume thatAis a non-Archimedean randomC*-algebra with
the normμA· and thatBis a non-Archimedean randomC*-algebra with the normμB· . For a given mappingf :A→B, we define
Dλf(x, . . . ,xm) := m
i=
λf
mxi+ m
j=,j=i
xj
+f
λ
m
i=
xi
– f
λ
m
i=
mxi
for allλ∈T:={ν∈C:|ν|= }and allx, . . . ,xm∈A.
Note that aC-linear mappingH:A→Bis called ahomomorphismin non-Archime-dean randomC*-algebras ifHsatisfiesH(xy) =H(x)H(y) andH(x*) =H(x)*for allx,y∈A.
We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archime-dean randomC*-algebras for the functional equationDλf(x
, . . . ,xm) = .
Theorem . Let f :A→Bbe a mapping for which there are functions ϕ:Am→D+,
ψ:A→D+,andη:A→D+such that|m|< is far from zero and
μBD
λf(x,...,xm)(t)≥ϕx,...,xm(t), (.)
μBf(xy)–f(x)f(y)(t)≥ψx,y(t), (.)
for allλ∈T,all x
, . . . ,xm,x,y∈A and t> .If there exists an L< such that
ϕmx,...,mxm
|m|Lt≥ϕx,...,xm(t), (.)
ψmx,my
|m|Lt≥ψx,y(t), (.)
ηmx
|m|Lt≥ηx(t), (.)
for all x,y,x, . . . ,xm∈A and t> ,then there exists a unique random homomorphism
H:A→Bsuch that
μBf(x)–H(x)(t)≥ϕx,,...,
|m|–|m|Lt (.)
for all x∈Aand t> .
Proof It follows from (.), (.), (.), andL< that
lim
n→∞ϕmnx,...,mnxm
|m|nt= , (.)
lim n→∞ψm
nx,mny|m|nt= , (.)
lim n→∞ηmnx
|m|nt= , (.)
for allx,y,x, . . . ,xm∈Aandt> .
Let us defineto be the set of all mappingsg:A–→Band introduce a generalized metric onas follows:
d(g,h) =infk∈(,∞) :μg(x)–h(x)B (kt) >φx,,...,(t),∀x∈A,t>
.
It is easy to show that (,d) is a generalized complete metric space (see []).
Now, we consider the functionJ:–→defined byJg(x) =mg(mx) for allx∈Aand g∈. Note that for allg,h∈, we have
d(g,h) <k =⇒ μBg(x)–h(x)(kt) >φx,,...,(t)
=⇒ μB
mg(mx)–mh(mx)
(kt) >|m|φmx,,...,
|m|t =⇒ μB
mg(mx)–mh(mx)(kLt) >φmx,,...,(t)
=⇒ d(Jg,Jh) <kL.
From this it is easy to see thatd(Jg,Jk)≤Ld(g,h) for allg,h∈, that is,Jis a self-function ofwith the Lipschitz constantL.
Puttingμ= ,x=xandx=x=· · ·=xm= in (.), we have
μBf(mx)–mf(x)(t)≥φx,,...,(t)
for allx∈Aandt> . Then μB
f(x)–mf(mx)(t)≥φx,,...,
for allx∈Aandt> , that is,d(Jf,f)≤|m|<∞. Now, from the fixed point alternative, it follows that there exists a fixed pointHofJinsuch that
H(x) = lim n→∞
|m|nf
mnx (.)
for allx∈Asincelimn→∞d(Jnf,H) = .
On the other hand, it follows from (.), (.), and (.) that
μBD
λH(x,...,xm)(t) =nlim→∞μB
mnDf(mnx,...,mnxm)
(t)
≥ lim n→∞φm
nx,...,mnx m
|m|nt= .
By a similar method to the above, we getλH(mx) =H(mλx) for allλ∈Tand allx∈A. Thus, one can show that the mappingH:A→BisC-linear.
It follows from (.), (.), and (.) that
μBH(xy)–H(x)H(y)(t) = lim n→∞μ
B
f(mnxy)–f(mnx)f(mny)
|m|nt
≥ lim n→∞ψm
nx,mny|m|nt=
for allx,y∈A. So,H(xy) =H(x)H(y) for allx,y∈A. Thus,H:A→Bis a homomorphism satisfying (.) as desired.
Also by (.), (.), (.) and by a similar method, we haveH(x*) =H(x)*.
Corollary . Let r> andθ be nonnegative real numbers,and let f :A→Bbe a map-ping such that
μBD
λf(x,...,xm)(t)≥
t
t+θ·(xrA+xrA+· · ·+xmrA)
,
μBf(xy)–f(x)f(y)(t)≥ t
t+θ·(xrA· yrA), μBf(x*)–f(x)*(t)≥
t t+θ· xrA
for allλ∈T,all x, . . . ,xm,x,y∈Aand t> .Then there exists a unique homomorphism
H:A→Bsuch that
μBf(x)–H(x)(t)≥ (|m|–|m|
r)t
(|m|–|m|r)t+θ|m|–|m|rxr
A
for all x∈Aand t> .
Proof The proof follows from Theorem .. By taking
ϕx,...,xm(t) =
t
t+θ·(xrA+xrA+· · ·+xmrA)
,
ψx,y(t) :=
t
ηx(t) =
t t+θ· xr
A
for allx, . . . ,xm,x,y∈A,L=|m|r–andt> , we get the desired result.
We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean ran-domC*-algebras for the functional equationDλf(x, . . . ,xm) = .
Theorem . Let f :A→Abe a mapping for which there are functionsϕ:Am→D+,
ψ:A→D+,andη:A→D+such that|m|< is far from zero and
μAD
λf(x,...,xm)(t)≥ϕx,...,xm(t),
μAf(xy)–f(x)y–xf(y)(t)≥ψx,y(t),
μAf(x*)–f(x)*(t)≥ηx(t),
for allλ∈Tand all x
, . . . ,xm,x,y∈A and t> .If there exists an L< such that(.),
(.),and(.)hold,then there exists a unique derivationδ:A→Asuch that μAf(x)–δ(x)(t)≥ϕx,,...,
|m|–|m|Lt
for all x∈Aand t> .
4 Stability of homomorphisms and derivations in non-Archimedean Lie C*-algebras
A non-Archimedean randomC*-algebraC, endowed with the Lie product
[x,y] :=xy–yx
onC, is called aLie non-Archimedean random C*-algebra.
Definition . LetAandBbe random LieC*-algebras. AC-linear mappingH:A→B
is called anon-Archimedean Lie C*-algebra homomorphismifH([x,y]) = [H(x),H(y)] for
allx,y∈A.
Throughout this section, assume thatAis a non-Archimedean random LieC*-algebra
with the normμAand thatBis a non-Archimedean random LieC*-algebra with the norm
μB.
We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archime-dean random LieC*-algebras for the functional equationDλf(x
, . . . ,xm) = .
Theorem . Let f :A→Bbe a mapping for which there are functionsϕ:Am→D+and
ψ:A→D+such that(.)and(.)hold and
μBf([x,y])–[f(x),f(y)](t)≥ψx,y(t) (.)
Proof By the same reasoning as in the proof of Theorem ., we can find the mapping H:A→Bgiven by
H(x) = lim n→∞
f(mnx)
|m|n (.)
for allx∈A. It follows from (.) and (.) that
μBH([x,y])–[H(x),H(y)](t) = lim n→∞μ
B
f(mn[x,y])–[f(mnx),f(mny)]
|m|nt
≥ lim n→∞ψm
nx,mny|m|nt=
for allx,y∈Aandt> , then
H[x,y]=H(x),H(y)
for allx,y∈A. Thus,H:A→Bis a LieC*-algebra homomorphism satisfying (.), as
desired.
Corollary . Let r> andθ be nonnegative real numbers,and let f :A→Bbe a map-ping such that
μBD
λf(x,...,xm)(t)≥
t
t+θ(xrA+xrA+· · ·+xmrA)
,
μBf([x,y])–[f(x),f(y)](t)≥ t t+θ· xr
A· yrA
,
μBf(x*)–f(x)*(t)≥
t t+θ· xr
A
for allλ∈T,all x
, . . . ,xm,x,y∈Aand t> .Then there exists a unique homomorphism
H:A→Bsuch that
μBf(x)–H(x)(t)≥ (|m|–|m|
r)t
(|m|–|m|r)t+θxr
A
for all x∈Aand t> .
Proof The proof follows from Theorem . and a method similar to Corollary ..
Definition . LetAbe a non-Archimedean random LieC*-algebra. AC-linear mapping
δ:A→Ais called aLie derivationifδ([x,y]) = [δ(x),y] + [x,δ(y)] for allx,y∈A.
We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean ran-dom LieC*-algebras for the functional equationDλf(x
, . . . ,xm) = .
Theorem . Let f :A→Abe a mapping for which there are functionsϕ:Am→D+and
ψ:A→D+such that(.)and(.)hold and
for all x,y∈A.If there exists an L< and(.), (.),and(.)hold,then there exists a unique Lie derivationδ:A→Asuch that(.)holds.
Proof By the same reasoning as the proof of Theorem ., there exists a uniqueC-linear mappingδ:A→Asatisfying (.); the mappingδ:A→Ais given by
δ(x) = lim n→∞
f(mnx)
|m|n (.)
for allx∈A.
It follows from (.) and (.) that
μAδ([x,y])–[δ(x),y]–[x,δ(y)](t)
= lim n→∞μ
A
f(mn[x,y])–[f(mnx),·mny]–[mnx,f(mny)]
|m|nt
≥ lim n→∞ψm
nx,mny|m|nt=
for allx,y∈Aandt> , then
δ[x,y]=δ(x),y+x,δ(y)
for allx,y∈A. Thus,δ:A→Ais a Lie derivation satisfying (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
Author details
1National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, 463-1 Jeonmin-Dong, Yuseong-Gu, Daejeon,
305-811, Korea.2Department of Mathematics, Iran University Science and Technology, Tehran, Iran.
Received: 18 April 2012 Accepted: 15 October 2012 Published: 29 October 2012 References
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