R E S E A R C H
Open Access
Ulam type stability problems for alternative
homomorphisms
Sin-Ei Takahasi
1*, Makoto Tsukada
1, Takeshi Miura
2, Hiroyuki Takagi
3and Kotaro Tanahashi
4*Correspondence:
1Department of Information
Sciences, Toho University, Funabashi, 274-8501, Japan Full list of author information is available at the end of the article
Abstract
We introduce an alternative homomorphism with respect to binary operations and investigate the Ulam type stability problem for such a mapping. The obtained results apply to Ulam type stability problems for several important functional equations.
MSC: Primary 39B82; secondary 47H10
Keywords: Ulam type stability; homomorphism; binary operation; fixed point theorem
1 Introduction
In , SM Ulam proposed the following stability problem: Given an approximately ad-ditive mapping, can one find the strictly adad-ditive mapping near it? A year later, DH Hyers gave an affirmative answer to this problem for additive mappings between Banach spaces. Subsequently many mathematicians came to deal with this problem (cf.[–]).
We introduce an alternative homomorphism from a setXwith two binary operations◦ and∗to another setEwith two binary operationsanddefined by
f(x◦y)f(x∗y) =f(x)f(y) (∀x,y∈X),
and we investigate the Ulam type stability problem for such a mapping whenEis a com-plete metric space. In particular, ifst=sfor alls,t∈E, then our results imply the stability results obtained in []. Also the method used in the paper have already applied for some other equations (cf.[–]).
One consequence of Banach’s fixed point theorem
A fixed point theorem has played an important role in the stability problem (cf.[]). The authors used an easy consequence of Banach’s fixed point theorem in []. It will serve again in this paper. Here we review it.
Let Xbe a set and (E,d) a complete metric space. Fix two mappingsf :X→E and
ϕ:X→R+, whereR+denotes the set of all nonnegative real numbers. Denote byf,ϕthe set of all mappingsu:X→Esuch that there exists a finite constantKusatisfying
du(x),f(x)≤Kuϕ(x) (∀x∈X).
For anyu,v∈f,ϕ, we define
ρf,ϕ(u,v) =inf
K≥ :du(x),v(x)≤Kϕ(x) (∀x∈X).
Then (f,ϕ,ρf,ϕ) is a complete metric space which containsf.
Now, fix three mappingsσ:X→X,τ :E→Eandε:X×X→R+. For any mapping
u:X→E, we define the mappingTσ,τu:X→Eby
(Tσ,τu)(x) =τ
u(σx) (x∈X).
Also, we consider three quantities:
ασ,ε=inf
K≥ :ε(σx,σy)≤Kε(x,y) (x,y∈X),
βσ,ϕ=inf
K≥ :ϕ(σx)≤Kϕ(x) (x∈X),
γτ=inf
K≥ :d(τs,τt)≤Kd(s,t) (s,t∈E).
Ifασ,ε<∞,βσ,ϕ<∞andγτ <∞, then we have
ε(σx,σy)≤ασ,εε(x,y) (∀x,y∈X),
ϕ(σx)≤βσ,ϕϕ(x) (∀x∈X),
d(τs,τt)≤γτd(s,t) (∀s,t∈E),
respectively. We will use these inequalities throughout this paper. We now state our fixed point theorem.
Lemma A([, Proposition .]) Let X be a set and(E,d)a complete metric space.Suppose that four mappings f :X→E,ϕ:X→R+,σ:X→X andτ :E→E satisfy
Tσ,τf ∈f,ϕ, βσ,ϕ<∞, γτ <∞ and βσ,ϕγτ < .
Then Tσ,τ(f,ϕ)⊆f,ϕand Tσ,τ has a unique fixed point f∞inf,ϕ.Moreover,
lim
n→∞d
Tσn,τf(x),f∞(x)= and df(x),f∞(x)≤ρf,ϕ(Tσ,τf,f)
–βσ,ϕγτ
ϕ(x)
for all x∈X.
2 A stability of alternative homomorphisms
Let (X,◦,∗) be a setXwith two binary operations◦and∗. Let (E,d,,) be a complete metric space (E,d) with two binary operationsand. Givenf :X→E, we consider the following commutative diagram:
X×X ––––––––(f◦)×(f→∗) E×E
f×f⏐⏐ ⏐⏐
E×E ––––––→
E.
This means that
f(x◦y)f(x∗y) =f(x)f(y) (∀x,y∈X). ()
In particular, ifst=sfor alls,t∈E, then () and () become
X×X ––––––◦→ X
f×f⏐⏐ ⏐⏐f
E×E ––––––→
E
and
f(x◦y) =f(x)f(y) (∀x,y∈X).
In other words,f is a homomorphism fromXtoE. Thus, if a mappingf :X→E satis-fies (), then we say thatf is analternative homomorphism.
In this section, we establish two general settings, on which we can give an affirmative answer to the Ulam type stability problem for the commutative diagram (). These settings have a property such as duality, that is, each of them works as a complement of the other. Let us describe the first setting. Forε:X×X→R+andδ:X→R+, we consider the following three conditions:
(i) The square operatorx→x◦xis an automorphism ofXwith respect to◦and∗.
We denote byσthe inverse mapping of this automorphism.
(ii) The binary operationsandonEare continuous. The square operator
τ:s→ssis an endomorphism ofEwith respect toand. (iii) α≡ασ,ε<∞,β≡βσ,δ<∞,γ≡γτ <∞andγmax{α,β}< .
Under the above conditions, we show the Ulam type stability for the commutative dia-gram (), as follows.
Theorem Let(X,◦,∗)and(E,d,,)be as above.Suppose that four mappingsσ:X→X,
τ :E→E,ε:X×X→R+andδ:X→R+satisfy(i), (ii),and(iii).If a mapping f :X→E
satisfies
df(x◦y)f(x∗y),f(x)f(y)≤ε(x,y) (∀x,y∈X), ()
df(x)f(σx∗σx),f(x)≤δ(x) (∀x∈X), ()
then there exists a mapping f∞:X→E such that
f∞(x◦y)f∞(x∗y) =f∞(x)f∞(y) (∀x,y∈X), ()
f∞(x)f∞(σx∗σx) =f∞(x) (∀x∈X), ()
df(x),f∞(x)≤ αε(x,x) +δ(x)
–γmax{α,β} (∀x∈X). ()
Moreover,if a mapping g:X→E satisfies(), (),and
∃Kg≥ :d
f(x),g(x)≤Kg
αε(x,x) +δ(x) (∀x∈X), ()
Proof For simplicity, we writeT=Tσ,τ. We note thatα,β, andγ are finite by (iii). Suppose thatf :X→Esatisfies () and (). Putϕ(x) =αε(x,x) +δ(x) for allx∈X. To apply Lemma A tof andϕ, we first observe thatTf ∈f,ϕ. Fixx∈X. Replacingxandyin () byσx, we get
df(σx◦σx)f(σx∗σx),f(σx)f(σx)≤ε(σx,σx).
Since
σx◦σx=σ–(σx) =x,
f(σx)f(σx) =τf(σx)= (Tf)(x),
and
ε(σx,σx)≤αε(x,x),
it follows that
df(x)f(σx∗σx), (Tf)(x)≤αε(x,x).
Using this and (), we have
d(Tf)(x),f(x)≤d(Tf)(x),f(x)f(σx∗σx)+df(x)f(σx∗σx),f(x)
≤αε(x,x) +δ(x)
=ϕ(x).
HenceTf ∈f,ϕandρf,ϕ(Tf,f)≤.
We next estimate the quantityβσ,ϕ. Forx∈X, we have
ϕ(σx) =αε(σx,σx) +δ(σx)
≤αε(x,x) +βδ(x)
≤max{α,β}αε(x,x) +δ(x)
=max{α,β}ϕ(x).
Henceβσ,ϕ≤max{α,β}andβσ,ϕγτ≤γmax{α,β}< by (iii).
Thus we can apply Lemma A. As a consequence,T has a unique fixed pointf∞∈f,ϕ. Moreover,
lim
n→∞d
Tnf(x),f∞(x)= ()
and
df(x),f∞(x)≤ρf,ϕ(Tf,f)
–βσ,ϕγτ
ϕ(x) ()
Here we show (). Ifx,y∈Xandn∈N, then we have
df∞(x◦y)f∞(x∗y),f∞(x)f∞(y)
≤df∞(x◦y)f∞(x∗y),Tnf(x◦y)Tnf(x∗y)
+dTnf(x◦y)Tnf(x∗y),Tnf(x)Tnf(y)
+dTnf(x)Tnf(y),f∞(x)f∞(y). ()
We will see that the right hand side of () tends to asn→ ∞. The first and third terms on the right hand side tend to asn→ ∞, because of () and the continuity ofand in (ii). Moreover, the second term, sayAn(x,y), is estimated as follows: By (i), (ii), and (),
we have
An(x,y) =d
τnfσn(x◦y) τnfσn(x∗y),τnfσnxτnfσny
=dτnfσnx◦σny τnfσnx∗σny,τnfσnxfσny
=dτnfσnx◦σnyfσnx∗σny,τnfσnxfσny
≤γndfσnx◦σnyfσnx∗σny,fσnxfσny ≤γnεσnx,σny
≤γnαnε(x,y),
whereτnandσndenote then-fold compositions of endomorphismsτandσ, respectively. Sinceγ α< by (iii), it follows thatAn(x,y)→ asn→ ∞. Thus the right hand side of ()
tends to , and we obtain ().
Next, we show (). Forx∈X, we replacexandyin () byσxto get
f∞(σx◦σx)f∞(σx∗σx) =f∞(σx)f∞(σx).
Sinceσx◦σx=xand
f∞(σx)f∞(σx) =τf∞(σx)= (Tf∞)(x) =f∞(x),
we obtain ().
Finally, we show the last statement. Sincegsatisfies () and (), we have
(Tg)(x) =τg(σx)=g(σx)g(σx)
=g(σx◦σx)g(σx∗σx)
=g(x)g(σx∗σx)
=g(x)
The next corollary is obtained in [].
Corollary ([, Corollary .]) Let X be a set with a binary operation◦such that the square operation x→x◦x is an automorphism of X with respect to◦and E a complete metric space with a continuous binary operationsuch that the square operationτ :s→ ss is an endomorphism of E with respect to. Letε:X×X→R+ and suppose that α≡ασ,ε<∞,γ ≡γτ <∞andγ α< ,whereσdenotes the inverse mapping of the square operation x→x◦x.If a mapping f :X→E satisfies
df(x◦y),f(x)f(y)≤ε(x,y) (∀x,y∈X),
then there exists a unique mapping f∞:X→E such that
f∞(x◦y) =f∞(x)f∞(y) and df(x),f∞(x)≤ α
–αγε(x,x)
for all x,y∈X.
Proof Consider the case that∗=◦andst=sfors,t∈E, in Theorem . In this case,τ is clearly an endomorphism ofEwith respect to. Therefore the corollary follows
immedi-ately from Theorem withδ= .
Now we turn to another setting. Let (X,◦,∗) and (E,d,,) be as in the first part of this section. Forε:X×X→R+andδ:X→R+, we consider the following three conditions:
(iv) The square operatorσ˜ :x→x◦xis an endomorphism ofXwith respect to◦and∗. (v) The binary operationsandonEare continuous. The square operators→ssis an automorphism ofEwith respect toand. We denote byτ˜the inverse mapping of this automorphism.
(vi) α˜≡ασ˜,ε<∞,β˜≡βσ˜,δ<∞,γ˜≡γτ˜<∞, andγ˜max{ ˜α,β˜}< .
Under the above conditions, we show the Ulam type stability for the commutative dia-gram (), as follows.
Theorem Let(X,◦,∗)and(E,d,,)be as above.Suppose that four mappingsσ˜ :X→X, ˜
τ :E→E,ε:X×X→R+andδ:X→R+satisfy(iv), (v),and(vi).If a mapping f:X→E
satisfies()and
df(x◦x)f(x∗x),f(x◦x)≤δ(x) (∀x∈X), ()
then there exists a mapping f∞:X→E satisfying()
f∞(x◦x)f∞(x∗x) =f∞(x◦x) (∀x∈X), ()
df(x),f∞(x)≤γ˜{ε(x,x) +δ(x)}
–γ˜max{ ˜α,β˜} (∀x∈X). ()
Moreover,if a mapping g:X→E satisfies(), (),and
∃Kg≥ :d
f(x),g(x)≤Kgγ˜
ε(x,x) +δ(x) (∀x∈X), ()
Proof For simplicity, we write T˜ =Tσ˜,τ˜, that is, (Tf˜ )(x) =τ˜(f(σ˜x)) forx∈X. We note
that α˜, β˜ and γ˜ are finite by (vi). Suppose that f :X→E satisfies () and (). Put ˜
ϕ(x) =γ˜{ε(x,x) +δ(x)}for allx∈X. To apply Lemma A tof andϕ˜, we first observe that ˜
Tf ∈f,ϕ˜. Fixx∈X. Sinceτ˜(f(x)f(x)) =f(x), it follows from () and () that
d(Tf˜ )(x),f(x)
=dτ˜f(σ˜x),f(x)
=dτ˜f(x◦x),τ˜f(x)f(x)
≤ ˜γdf(x◦x),f(x)f(x)
≤ ˜γdf(x◦x),f(x◦x)f(x∗x)+df(x◦x)f(x∗x),f(x)f(x) ≤ ˜γδ(x) +ε(x,x)
=ϕ˜(x).
HenceTf˜ ∈f,ϕ˜andρf,ϕ˜(Tf˜ ,f)≤.
We next estimate the quantityβσ˜,ϕ˜. Forx∈X, we have
˜
ϕ(σ˜x) =γ˜ε(σ˜x,σ˜x) +δ(σ˜x)
≤ ˜γαε˜ (x,x) +βδ˜ (x) ≤ ˜γmax{ ˜α,β˜}ε(x,x) +δ(x)
=max{ ˜α,β˜} ˜ϕ(x).
Henceβσ˜,ϕ˜≤max{ ˜α,β˜}andβσ˜,ϕ˜γτ˜≤ ˜γmax{ ˜α,β˜}< by (vi).
Thus we can apply Lemma A. As a consequence,T˜ has a unique fixed pointf∞∈f,ϕ˜. Moreover,
lim
n→∞d
˜
Tnf(x),f∞(x)= ()
and
df(x),f∞(x)≤ρf,ϕ˜(
˜ Tf,f) –βσ˜,ϕ˜γτ˜ ˜
ϕ(x) ()
for allx∈X. Sinceρf,ϕ˜(Tf˜ ,f)≤ andβσ˜,ϕ˜γτ˜≤ ˜γmax{ ˜α,β˜}< , () implies (). Here we show (). Ifx,y∈Xandn∈N, then we have
df∞(x◦y)f∞(x∗y),f∞(x)f∞(y)
≤df∞(x◦y)f∞(x∗y),T˜nf(x◦y)T˜nf(x∗y)
+dT˜nf(x◦y)T˜nf(x∗y),T˜nf(x)T˜nf(y)
+dT˜nf(x)T˜nf(y),f
∞(x)f∞(y).
as follows: By (iv), (v), and (),
˜
An(x,y) =d
˜
τnfσ˜n(x◦y)τ˜nfσ˜n(x∗y),τ˜nfσ˜nx ˜τnfσ˜ny
=dτ˜nfσ˜nx◦ ˜σnyτ˜nfσ˜nx∗ ˜σny,τ˜nfσ˜nxfσ˜ny
=dτ˜nfσ˜nx◦ ˜σnyfσ˜nx∗ ˜σny,τ˜nfσ˜nxfσ˜ny
≤ ˜γndfσ˜nx◦ ˜σnyfσ˜nx∗ ˜σny,fσ˜nxfσ˜ny ≤ ˜γnεσ˜nx,σ˜ny
≤ ˜γnα˜nε(x,y),
whereτ˜nandσ˜ndenote then-fold compositions of endomorphismsτ˜andσ˜, respectively.
Sinceγ˜α˜< by (vi), it follows thatA˜n(x,y)→ asn→ ∞. Thus we obtain ().
Next, we show (). Replacingyin () byx, we have
f∞(x◦x)f∞(x∗x) =f∞(x)f∞(x). ()
Also since
˜
τf∞(x◦x)=τ˜f∞(σ˜x)= (Tf˜ ∞)(x) =f∞(x) =τ˜f∞(x)f∞(x),
it follows that
f∞(x◦x) =f∞(x)f∞(x).
Combining with (), we obtain ().
Finally, we show the last statement. Sincegsatisfies () and (), we have
g(σ˜x) =g(x◦x) =g(x◦x)g(x∗x) =g(x)g(x) =τ˜–g(x),
that is, (Tg)(x) =˜ g(x) for allx∈X. This says thatgis a fixed point ofT. Also, by (), we˜ haveg∈f,ϕ˜. Hence the uniqueness of a fixed point ofT˜ inf,ϕ˜ implies thatg=f∞.
The next corollary is obtained in [].
Corollary ([, Corollary .]) Let X be a set with a binary operation◦such that the square operationσ˜ :x→x◦x is an endomorphism of X with respect to◦and E a complete metric space with a continuous binary operation such that the square operation s→ ss is an automorphism of E with respect to.Letε:X×X→R+ and suppose that
˜
α≡ασ˜,ε<∞,γ˜≡γτ˜<∞andγ˜α˜< ,whereτ˜denotes the inverse mapping of the square operation s→ss.If a mapping f :X→E satisfies
df(x◦y),f(x)f(y)≤ε(x,y) (∀x,y∈X),
then there exists a unique mapping f∞:X→E such that
f∞(x◦y) =f∞(x)f∞(y) and df(x),f∞(x)≤ γ˜
–α˜γ˜ε(x,x)
Proof Consider the case that∗=◦andst=sfors,t∈E, in Theorem . Thenτ˜is clearly an endomorphism ofEwith respect to. Therefore the corollary follows immediately from
Theorem withδ= .
3 Application I
The Ulam type stability problem for Euler-Lagrange type additive mappings has been in-vestigated in []. Here we take up the following Euler-Lagrange type mappingf :X→E satisfying
f(ax+by) +f(bx+ay) + (a+b)f(–x) +f(–y)= (∀x,y∈X), ()
whereXis a complex normed space,Ea complex Banach space anda,b∈Cwitha+b= . The following is an Ulam type stability result for this mapping.
Corollary (cf.[, Theorem .]) Letε:X×X→R+and suppose that
(vii) ∃K≥ :|a+b|K< andε(x,y)≤Kε(–(a+b)x, –(a+b)y)(∀x,y∈X). If a mapping f :X→E satisfies
f(ax+by) +f(bx+ay) + (a+b)f(–x) +f(–y)≤ε(x,y) (∀x,y∈X), ()
then there exists a unique mapping f∞:X→E satisfying()and
f(x) –f∞(x)≤ K
( –|a+b|K)ε(–x, –x) (∀x∈X). ()
Proof Putu= –x,v= –yfor eachx,y∈X. Under these transformations, () changes into the following estimate:
f(–au–bv) +f(–bu–av)+a+b
f(u) +f(v)≤ε(u,v) (∀u,v∈X), ()
whereε(u,v) =ε(–u, –v) (∀u,v∈X).
Now we defineu◦v= –au–bv,u∗v= –bu–avfor eachu,v∈X. In this case, we can easily see that the square operatoru→u◦uis an endomorphism ofXwith respect to◦
and∗. Also sincea+b= , this endomorphism is bijective and so automorphic. We denote
byσthe inverse mapping of this automorphism. Moreover, we definest= –
(a+b)(s+t),
st=(s+t) for eachs,t∈E. Then we can also see that the binary operationsand
onEare continuous and the square operatorτ :s→ssis an automorphism ofEwith respect toand. Note that () changes into the following:
f(u◦v)f(u∗v) –f(u)f(v)≤ε(u,v) (∀u,v∈X). ()
second condition of (vii) and henceγτασ,ε≤ |a+b|K< from the first condition of (vii). Therefore, by Theorem , there exists a unique mappingf∞:X→Esuch that
f∞(u◦v)f∞(u∗v) =f∞(u)f∞(v) (∀u,v∈X),
namely, () holds and
f(u) –f∞(u)≤ ασ,εε(u,u)
–γτmax{ασ,ε,βσ,δ}
≤ K
( –|a+b|K)ε(–u, –u) (∀u∈X),
and so () holds.
The following is also an Ulam type stability result for the mapping satisfying ().
Corollary (cf.[, Theorem .]) Letε:X×X→R+and suppose that (viii) ∃K≥ :K<|a+b|andε(–(a+b)x, –(a+b)y)≤Kε(x,y)(∀x,y∈X).
If a mapping f :X→E satisfies(),then there exists a unique mapping f∞:X→E
satis-fying()and
f(x) –f∞(x)≤
(|a+b|–K)ε(–x, –x) (∀x∈X). ()
Proof As observed in the proof of Corollary , () changes into (). Now we define u◦v= –au–bv,u∗v= –bu–avfor eachu,v∈X. In this case, we can easily see that the square operatorσ˜ :u→u◦uis an endomorphism ofXwith respect to◦and∗. Moreover, we definest= –
(a+b)(s+t),st=
(s+t) for eachs,t∈E. Then we can also see that
the binary operationsandonEare continuous and the square operators→ssis an endomorphism ofEwith respect toand. Also sincea+b= , this endomorphism is bijective and so automorphic. We denote byτ˜the inverse mapping of this automorphism. Note that () changes into (). Sincex◦x=x∗x(∀x∈X) andss=s(∀s∈E), it follows thatf(x◦x)f(x∗x) =f(x◦x) for allx∈Xand then () holds withδ= .
Moreover,βσ˜,δ= must hold withδ= . It is also obvious thatγτ˜ =|a+b|–from the definition ofτ˜. We also note thatασ˜,ε≤Kfrom the second condition of (viii) and hence
γτ˜ασ˜,ε≤ |a+b|–K< from the first condition of (viii).
Therefore, by Theorem , there exists a unique mappingf∞:X→Esuch that
f∞(u◦v)f∞(u∗v) =f∞(u)f∞(v) (∀u,v∈X),
namely, () holds and
f(u) –f∞(u)≤ γτ˜ε(u,u)
–γτ˜max{ασ˜,ε,βσ˜,δ}
≤ |a+b|–
( –|a+b|–K)ε(–u, –u)
=
(|a+b|–K)ε(–u, –u) (∀u∈X),
Corollary (cf.[, Corollary .]) Suppose that|a+b| = ,δ,p,q≥and p+q= .If a mapping f :X→E satisfies
f(ax+by) +f(bx+ay) + (a+b)f(–x) +f(–y)≤δxpyq
for all x,y∈X,then there exists a unique mapping f∞:X→E satisfying()and
f(x) –f∞(x)≤ δ
(||a+b|p+q–|a+b||x
p+q (∀x∈X).
Proof Putε(x,y) =δxpyqfor eachx,y∈X. (a) The case where either
|a+b|> , p+q> ,
or
|a+b|< , p+q< .
PutK=|a+b|–(p+q). ThenKsatisfies (vii). Note also that
K
( –|a+b|K)ε(–x, –x) =
δ
(|a+b|p+q–|a+b|)x p+q
for allx∈X. Then the desired result follows from Corollary . (b) The case where either
|a+b|> , p+q< ,
or
|a+b|< , p+q> .
PutK=|a+b|p+q. ThenKsatisfies (viii). Note also that
(|a+b|–K)ε(–x, –x) =
δ
(|a+b|–|a+b|p+q)x p+q
for allx∈X. Then the desired result follows from Corollary .
4 Application II
Lemma B([, Théorème )] A mapping f :X→Rsatisfies
maxf(x+y),f(x–y)=f(x) +f(y) (∀x,y∈X), ()
if and only if f(x) =|π(x)|(∀x∈X)for some additive functionπ:X→R.
In this section, we deal with the Ulam type stability problem for Equation (). Putx◦y= x+yandx∗y=x–yfor eachx,y∈X. Moreover, putst=s+tandst=max{s,t}for eachs,t∈R. Then () changes into (). Also we can easily see that the square operation
˜
σ :x→x◦xis endomorphic with respect to◦and∗and that the square operators→ ssis automorphic with respect toand. Denote by τ˜ the inverse mapping of this automorphism. In this case, it is obvious thatτ˜(s) = sfor eachs∈Rand henceγτ˜= /.
Now letεbe a nonnegative constant and suppose thatf :X→Rsatisfies
maxf(x+y),f(x–y)–f(x) +f(y)≤ε (∀x,y∈X). ()
Puttingx=y= in (), we obtain
f()≤ε. ()
Also, puttingx=yin (), we obtain
–ε+f()≤–ε+maxf(x+x),f()≤f(x) (∀x∈X). ()
Combining () and (), we obtain
–ε≤f(x) (∀x∈X). ()
Putδ= ε. By () and (), we obtain
≤maxf(x+x),f()–f(x+x)≤ε+ε=δ (∀x∈X),
and hence () holds. Moreover, note thatασ˜,ε=βσ˜,δ= sinceεandδare constant. Then Lemma B and Theorem easily imply the following.
Corollary Let X be an Abelian group andεa nonnegative constant.If f:X→Rsatisfies (),then there exists an additive mappingπ:X→Rsuch that
f(x) –π(x)≤ε (∀x∈X).
For the related results, see [, ].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Department of Information Sciences, Toho University, Funabashi, 274-8501, Japan.2Department of Mathematics, Niigata
University, Niigata, 950-2181, Japan. 3Department of Mathematical Sciences, Faculty of Science, Shinshu University,
Matsumoto, 390-8621, Japan. 4Department of Mathematics, Tohoku Pharmaceutical University, Sendai, 981-8558, Japan.
Acknowledgements
The authors are deeply grateful to the referees for careful reading of the paper and for the helpful suggestions and comments. All authors are partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.
Received: 29 January 2014 Accepted: 27 May 2014 Published:04 Jun 2014 References
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