R E S E A R C H
Open Access
Stability of additivity and fixed point methods
Janusz Brzd ˛ek
**Correspondence:
jbrzdek@up.krakow.pl Department of Mathematics, Pedagogical University, Podchor ˛a˙zych 2, Kraków, 30-084, Poland
Abstract
We show that the fixed point methods allow to investigate Ulam’s type stability of additivity quite efficiently and precisely. Using them we generalize, extend and complement some earlier classical results concerning the stability of the additive Cauchy equation.
MSC: 39B82; 47H10
Keywords: additivity; fixed point; Ulam stability
Introduction
In applications quite often we have to do with functions that satisfy some equations only approximately. One of possible ways to deal with them is just to replace such functions by corresponding (in suitable ways) exact solutions to those equations. But there of course arises the issue of errors which we commit in this way. Some tools to evaluate such errors are provided within the theory of Ulam’s type stability. For instance, we can introduce the following definition, which somehow describes the main ideas of such stability notion for the Cauchy equation
f(x+y) =f(x) +f(y). ()
Definition Let (A, +) and (X, +) be semigroups,dbe a metric inX,E⊂C⊂RA
+ be
nonempty, andT be an operator mappingCintoRA
+(R+stands for the set of nonnegative
reals). We say that Cauchy equation () is (E,T)-stable provided for everyε∈Eandϕ∈
XAwith
dϕ(x+y),ϕ(x) +ϕ(y)
≤ε(x,y), x,y∈A, ()
there exists a solutionϕ∈XAof equation () such that dϕ(x),ϕ(x)
≤Tε(x), x∈A. ()
(As usual,CDdenotes the family of all functions mapping a setD=∅into a setC=∅.) Roughly speaking, (E,T)-stability of equation () means that every approximate (in the sense of ()) solution of () is always close (in the sense of ()) to an exact solution to (). Let us mention that this type of stability has been a very popular subject of investigations for the last nearly fifty years (see,e.g., [–]). The main motivation for it was given by S.M. Ulam (cf.[, ]) in in his talk at the University of Wisconsin, where he presented, in particular, the following problem.
Let Gbe a group and(G,d)be a metric group. Givenε> , does there existδ> such
that if f :G→Gsatisfies
df(xy),f(x)f(y)<δ, x,y∈G,
then a homomorphism T:G→Gexists with d(f(x),T(x)) <εfor x∈G?
Hyers [] published a partial answer to it, which can be stated as follows. Let E and Y be Banach spaces andε> . Then, for every g:E→Y with
sup x,y∈E
g(x+y) –g(x) –g(y)≤ε,
there is a unique f :E→Y that is additive (i.e., satisfies equation()) and such that
sup x∈E
g(x) –f(x)≤ε.
Quite often we describe that result of Hyers simply saying thatCauchy functional equa-tion()is Hyers-Ulam stable (or has the Hyers-Ulam stability).
In the next few years, Hyers and Ulam published some further stability results for poly-nomial functions, isometries and convex functions in [–]. Let us mention yet that now we are aware of an earlier (than that of Hyers) result concerning such stability that is due to Pólya and Szegö [, Teil I, Aufgabe ] (see also [, Part I, Ch. , Problem ]) and reads as follows (Nstands for the set of positive integers).
For every real sequence(an)n∈Nwithsupn,m∈N|an+m–an–am| ≤, there is a real number ωsuch thatsupn∈N|an–ωn| ≤. Moreover,ω=limn→∞an/n.
The next theorem is considered to be one of the most classical results.
Theorem Let Eand Ebe two normed spaces,Ebe complete,c≥and p= be fixed
real numbers.Let f :E→Ebe a mapping such that
f(x+y) –f(x) –f(y)≤cxp+yp, x,y∈E\ {}. ()
Then there exists a unique additive function T:E→Ewith f(x) –T(x)≤ cx
p
|p–– |, x∈E\ {}. ()
It was motivated by Th.M. Rassias (see [–]) and is composed of the outcomes in [, , ]. Note that Theorem withp= yields the result of Hyers and it is known (see []; cf.also [, ]) that forp= an analogous result is not valid. Moreover, it was shown in [] that estimation () is optimum forp≥ in the general case.
Theorem has a very nice simple form. However, recently, it was shown in [] that it can be significantly improved; namely, in the casep< , eachf :E→E satisfying ()
must actually be additive and the assumption of completeness ofE is not necessary in
Theorem Let Eand Ebe two normed spaces and c≥and p= be fixed real numbers.
Let f :E→Ebe a mapping satisfying().If p≥and Eis complete,then there exists a
unique additive function T:E→Esuch that()holds.If p< ,then f is additive.
The second statement of Theorem , forp< , can be described as theϕ-hyperstability of the additive Cauchy equation forϕ(x,y)≡c(xp+yp). Unfortunately, such result does not remain valid if we restrict the domain off, as the following remark shows it.
Remark Letp< ,a≥,I= (a,∞), andf,T:I→Rbe given byT(x) = andf(x) =xp forx∈I. Then, clearly,
f(x) –T(x)=xp, x∈I,
and (cf.Example )
f(x+y) –f(x) –f(y)≤xp+yp, x,y∈I.
1 The main result
In this paper we prove the following complement to Theorem , which covers also the situation described in Remark (see Remark ).
Theorem Let(X, +)be a commutative semigroup, (E, +)be a commutative group,d be a complete metric in E which is invariant(i.e.,d(x+z,y+z) =d(x,y)for x,y,z∈X),and h:X→R+be a function such that
M:=
n∈N:s(n) +s(n+ ) < =∅,
where s(n) :=inf{t∈R+:h(nx)≤th(x)for all x∈X} for n∈N.Assume that f :X→E satisfies the inequality
df(x+y),f(x) +f(y)≤h(x) +h(y), x,y∈X. ()
Then there exists a unique additive T:X→E such that
df(x),T(x)≤sh(x), x∈X, ()
with
s:=inf
+s(n)
–s(n) –s(n+ ):n∈M .
It is easily seen that Theorem yields the subsequent corollary.
Corollary Let X,E and d be as in Theoremand h:X→(,∞)be such that
lim inf n→∞ supx∈X
h(nx) +h(( +n)x)
Assume that f :X→E satisfies().Then there is a unique additive T:X→E with
df(x),T(x)≤h(x), x∈X. ()
Remark IfE andEare normed spaces andXis a subsemigroup of the group (E, +)
such that /∈X, then it is easily seen that the functionh:X→E, given byh(x) =cxpfor
x∈X, with some realp< andc> , fulfils condition (). This shows that Corollary (and therefore Theorem , as well) complements Theorem and in particular also Theorem . Note that for suchh, () takes the form
f(x) –T(x)≤cxp, x∈X, ()
which is sharper than () forp< .
A bit more involved example ofh:X→Esatisfying () we obtain taking
h(x) =γ(x)A(x)p, x∈X
for any realp< , any bounded functionγ :X→Rwithinfγ(X) > , and anyA:X→ (,∞) such thatA(nx)≥nA(x) forx∈X,∈N(for instance, we can takeA(x) :=α(x)for x∈X, with some additiveα:E→E).
In some cases, estimation () provided in Corollary is optimum as the subsequent example shows. Unfortunately, this is not always the case, because the possibly sharpest such estimation we have in Theorem forp< .
Example Letp< ,a≥,I= [a,∞), andA:I→Rbe additive and such thatX:={x∈I: A(x) > } =∅. Writeh(x) =A(x)pforx∈X. Then it is easily seen thatXis a subsemigroup of the semigroup (I, +) and () is valid. Definef,T:X→RbyT(x) = andf(x) =A(x)p forx∈X. Then
f(x) –T(x)=A(x)p=h(x), x∈X.
We show that
f(x+y) –f(x) –f(y)≤h(x) +h(y), x,y∈X.
Actually, the calculations are very elementary, but for the convenience of readers, we pro-vide them.
So, fixx,y∈X. Suppose, for instance, thatA(x)≤A(y). Then
A(x) +A(y)≥A(x),
which means that
and consequently
f(x+y) –f(x) –f(y)=A(x)p+A(y)p–A(x+y)p
≤A(x)p+A(y)p=h(x) +h(y).
2 Auxiliary result
The proof of Theorem is based on a fixed point result that can be easily derived from [, Theorem ] (cf.[, Theorem ] and []). Let us mention that [, Theorem ] was already used, in a similar way as here, for the first time in [] for proving some stability results for the functional equation of p-Wright affine functions, next in [, ] in proving hyperstability of the Cauchy equation, and (very recently) also for investigations of stability and hyperstability of some other equations in [–] (the Jensen equation, the general linear equation, and the Drygas functional equation, respectively).
The fixed point approach to Ulam’s type stability was proposed for the first time in [] (cf.[] for a generalization; see also []) and later applied in numerous papers; for a survey on this subject, we refer to [].
We need to introduce the following hypotheses.
(H) Xis a nonempty set and(Y,d)is a complete metric space. (H) f,f:X→Xare given maps.
(H) T :YX→YXis an operator satisfying the inequality dTξ(x),Tμ(x)≤dξf(x)
,μf(x)
+dξf(x)
,μf(x)
, ξ,μ∈YX,x∈X. (H) :RX
+→RX+ is an operator defined by
δ(x) :=δf(x)
+δf(x)
, δ∈RX+,x∈X.
Now we are in a position to present the above mentioned fixed point result following from [, Theorem ].
Theorem Assume that hypotheses(H)-(H)are valid.Suppose that there exist func-tionsε:X→R+andϕ:X→Y such that
Tϕ(x) –ϕ(x)≤ε(x), ε∗(x) :=
∞
n=
n
ε(x) <∞, x∈X.
Then there exists a unique fixed pointψofT with
ϕ(x) –ψ(x)≤ε∗(x), x∈X.
Moreover,ψ(x) :=limn→∞(Tnϕ)(x)for x∈X. 3 Proof of Theorem 3
Note that () withy=mxgives
Define operatorsTm:EX→EXand m:RX+→RX+ by Tmξ(x) :=ξ
( +m)x–ξ(mx), x∈X,ξ∈EX,m∈N, mδ(x) :=δ
( +m)x+δ(mx), x∈X,δ∈RX+,m∈N.
()
Then it is easily seen that, for eachm∈N, := m has the form described in (H) with f(x) =mxandf(x) = ( +m)x. Moreover, sincedis invariant, () can be written in the
form
dTmf(x),f(x)≤ +s(m)h(x) =:εm(x), x∈X,m∈N, ()
and
dTmξ(x),Tmμ(x)
=dξ( +m)x–ξ(mx),μ( +m)x–μ(mx)
≤dξ( +m)x,μ( +m)x+dξ(mx),μ(mx) ()
for everyξ,μ∈EX,x∈X,m∈N. Consequently, for eachm∈N, also (H) is valid with Y:=EandT :=Tm.
It is easy to show by induction onnthat
n mεm(x)≤
+s(m)h(x)s(m) +s( +m)n
forx∈X,n∈N(nonnegative integers) andm∈M. Hence
ε∗m(x) :=
∞
n=
n
mεm
(x)
≤ +s(m)h(x)
∞
n=
s(m) +s( +m)n
= ( +s(m))h(x)
–s(m) –s( +m), x∈X,m∈M.
Now, we can use Theorem withY=Eandϕ=f. According to it, the limit
Tm(x) := lim n→∞
Tn
mf
(x)
exists for eachx∈Xandm∈M,
df(x),Tm(x)
≤ ( +s(m))h(x)
–s(m) –s( +m)), x∈X,m∈M, () and the functionTm:X→E, defined in this way, is a solution of the equation
Now we show that
dTmnf(x+y),Tn
mf(x) +Tmnf(y)
≤s(m) +s( +m)nh(x) +h(y) ()
for everyx,y∈X,n∈Nandm∈M. Since the casen= is just (), takek∈Nand
assume that () holds forn=kand everyx,y∈X,m∈M. Then
dTmk+f(x+y),Tmk+f(x) +Tmk+f(y)
=dTmkf( +m)(x+y)–Tmkfm(x+y),
Tk mf
( +m)x–Tmkf(mx) +Tmkf( +m)y–Tmkf(my)
≤dTmkf( +m)x+ ( +m)y,Tmkf( +m)x+Tmkf( +m)y +dTmkf(mx+my),Tmkf(mx) +Tmkf(my)
≤s(m) +s( +m)kh( +m)x+h( +m)y +s(m) +s( +m)kh(mx) +h(my)
≤s(m) +s( +m)k+h(x) +h(y), x,y∈X,m∈M.
Thus, by induction we have shown that () holds for everyx,y∈X,n∈N, andm∈M.
Lettingn→ ∞in (), we obtain the equality
Tm(x+y) =Tm(x) +Tm(y), x,y∈X,m∈M. ()
Next, we prove that each additive functionT:X→Ysatisfying the inequality
df(x),T(x)≤Lh(x), x∈X, ()
with someL> , is equal toTmfor eachm∈M. To this end, fixm∈Mand an additive
T:X→Ysatisfying (). Note that by (),
dT(x),Tm(x)≤dT(x),f(x)+df(x),Tm(x) =Lh(x)
∞
n=
s(m) +s( +m) n
, x∈X ()
for someL> (the caseh(x)≡ is trivial, so we exclude it here). Observe yet thatTand
Tmare solutions to equation () for allm∈M.
We show that for eachj∈N,
dT(x),Tm(x)≤Lh(x) ∞
n=j
s(m) +s( +m) n
The casej= is exactly (). So, fixl∈Nand assume that () holds forj=l. Then, in view of (),
dT(x),Tm(x)=dT( +m)x
–T(mx),Tm
( +m)x
–Tm(mx)
≤dT( +m)x
,Tm
( +m)x
+dT(mx),Tm(mx)
≤L
h( +m)x
+h(mx)
∞
n=l
s(m) +s( +m) n
≤Lh(x) ∞
n=l+
s(m) +s( +m) n
, x∈X.
Thus we have shown (). Now, lettingj→ ∞in (), we get
T=Tm. ()
Thus we have also proved thatTm=Tmfor eachm∈M, which (in view of ()) yields
df(x),Tm(x)
≤ ( +s(m))h(x)
–s(m) –s( +m), x∈X,m∈M. () This implies () withT:=Tm; clearly, equality () means the uniqueness ofTas well.
Remark Note that from the above proof we can derive a much stronger statement on the uniqueness ofTthan the one formulated in Theorem . Namely, it is easy to see that T:=Tmis the unique additive mapping such that () holds with someL> .
Competing interests
The author declares that he has no competing interests.
Received: 30 July 2013 Accepted: 4 October 2013 Published:08 Nov 2013
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