Stability of homogeneity almost everywhere

Acta Math. Hungar., 117 (3) (2007), 219�229.DOI: 10.1007/s10474-007-6092-8

First published online May 26, 2007

STABILITY OF HOMOGENEITY ALMOSTEVERYWHERE

W. JAB�O�SKI

Department of Mathematics, University of Rzeszow, Rejtana 16 A, 35-310 Rzeszów, Polande-mail: [email protected]

(Received May 8, 2006; accepted November 14, 2006)

Abstract. We consider approximately φ-homogeneous mappings almosteverywhere, that is functions F such that the di�erence F (αx)− φ(α)F (x) isin some sense bounded almost everywhere in a product space. We will prove, un-der some assumptions, that then either we have some kind of boundedness ofφand F , or there exist a homomorphism φ̃ and a φ̃-homogeneous function F , whichare almost everywhere equal to φ and F , respectively. From this result we derivethe superstability e�ect for the multiplicativity almost everywhere.

The problem of the stability of functional equations can be formulatedas a question whether for a function, which satis�es the given equation withsome error, we can �nd a �true solution� of this equation which is close to thegiven disturbed solution. The �disturbed� solution of the functional equationis often called the �approximate solution� of the equation.

Originally the problem of stability of functional equations was posed byS. Ulam [11] for homomorphisms. The �rst positive answer to this problemhas been given by D. H. Hyres [6], who proved that every approximatelyadditive mapping between Banach spaces is close to some additive function.The stability problem has been generalized in various ways and many authorsconsidered stability of di�erent functional equations (cf. for example [4, 7]).

Following the famous problem by P. Erd®s (cf. [3]) concerning �additivefunctions almost everywhere�, we are going to consider the stability problem

Key words and phrases: homogeneity equation, almost everywhere, stability.2000 Mathematics Subject Classi�cation: 39B72.

0236�5294/$ 20.00 c© 2007 Akadémiai Kiadó, Budapest

220 W. JAB�O�SKI

for the φ-homogeneity equation

F (αx) = φ(α)F (x)

assumed to hold almost everywhere in some product space. We will showhere some kind of superstability, that is we will prove that essentially un-bounded functions φ and F such that the di�erence F (αx)− φ(α)F (x) issuitably bounded have to be almost everywhere equal to a homomorphismφ̃, and a φ̃-homogeneous function F , respectively. Finally we will considerapproximately multiplicative functions almost everywhere.

We start with some de�nitions and facts which allow us to generalize ho-mogeneity equations. Classical φ-homogeneity equations with a multiplica-tive function φ are considered in vector spaces over some �eld. To cover thisspecial case we replace a �eld with a group with zeroG, and a vector spacewith some G-space.

By a group with zero we mean a structure (G, ·, 0) where G∗ := G \ {0}6= ∅, (

G∗, ·|G∗×G∗)is a group in the classical meaning and α · 0 = 0 · α = 0

for every α ∈ G.In the sequel we will need the notion of aG-space. Assume that (G, ·,0) is

a group with zero and let X be a nonempty set with a �xed element θ whichwill be called zero. Let be given an action of the groupG on the set X, i.e.let · : G×X → X satisfy the following conditions:

(g1g2)x = g1(g2x) for g1, g2 ∈ G, x ∈ X,

1x = x for x ∈ X, gθ = θ for g ∈ G, 0x = θ for x ∈ X.

The structure (X,G) satisfying the above conditions will be called aG-space.A G-space X will be called trivial providedX = {θ}. As it is easy to see, thegroup G is a G-space itself. Moreover a pair (Rn,R) with a multiplicationof vectors by scalars is a classical example of anR-space.

Let X be a nonempty set. A nonempty family J (X) ⊂ 2X is called anideal in X provided

(i) A ∈ J (X), B ⊂ A =⇒ B ∈ J (X);(ii) A,B ∈ J (X) =⇒ A ∪B ∈ J (X).An ideal J (X) is called proper, if

(iii) X 6∈ J (X).The ideal J (X) = {∅} is said to be trivial. OtherwiseJ (X) will be called

nontrivial. Let (G, ·, 0) be a group with zero. An ideal J (G) ⊂ 2G is calledlinearly invariant if

(iv) α ∈ G, A ∈ J (G) =⇒ α(A∗)−1 = {αβ−1 : β ∈ A∗} ∈ J (G).Remark 1 [9, Lemma 1, p. 438]. The notion of linearly invariant ideals

has been de�ned for the group in the classical meaning. IfJ (G) is a linearly

Acta Mathematica Hungarica 117, 2007

STABILITY OF HOMOGENEITY ALMOST EVERYWHERE 221

invariant ideal in a group with zero then eitherJ (G) = {∅} or {0} ∈ J (G).Then we have (A∗)−1α, αA,Aα ∈ J (G) for α ∈ G, A ∈ J (G).

Let (G, ·, 0) be a group and letX be a G-space. Now we de�ne the notionof linearly independent ideals inX. We propose here the same name for twodi�erent notions, but this will not cause misunderstanding because in eachcase we will mark whether the set considered is a member of an ideal in agroup G or it belongs to an ideal in a G-space X.

An ideal J (X) in X is said to be linearly invariant provided(v) α ∈ G,U ∈ J (X) =⇒ αU ∈ J (X).Finally let J (X) be an ideal in a nonempty set X. We will say that

a condition W de�ned on a set A ⊂ X holds J (X)-almost everywhere in A(we will write J (X)-a.e. in A), if there exists a set U ∈ J (X) such that forevery x ∈ A \ U we have W (x). Moreover, for a function f : X → R de�ne

J (X) ess infx∈X

f(x) = supU∈J (X)

infx∈X\U

f(x).

Now let (G, ·, 0) be a group with zero and let X be a G-space. Assume thatwe are given proper linearly invariant idealsJ (G) and J (X) in the group Gand in the G-space X, respectively. In the product setG×X we de�ne nowa

(J (G),J (X))-ideal which is, in some sense, conjugate with given ideals

in G and X.Definition 1.By a

(J (G),J (X))-ideal we will mean an idealJ (G×X)

⊂ 2G×X satisfying the following conditions:1) U1 ×X ∈ J (G×X) for U1 ∈ J (G) and G× U2 ∈ J (G×X) for

U2 ∈ J (X);2) if M ∈ J (G×X) then there exist sets U1 ∈ J (G) and U2 ∈ J (X)

such that

Mα :={

x ∈ X : (α, x) ∈ M} ∈ J (X) for α ∈ G \ U1

andMx :=

{α ∈ G : (α, x) ∈ M

} ∈ J (G) for x ∈ X \ U2;

3){

(α, x) ∈ G×X : αx ∈ U} ∈ J (G×X) for every U ∈ J (X).

As one can easily check, from the condition 2) and from the fact that bothideals J (G) and J (X) are proper, we obtain that also the

(J (G),J (X))-

ideal J (G×X) is proper. Some examples of such ideals in someG-spacesare given in [8].

In [8] it has been considered the φ-homogeneity equation almost every-where, that is the equation

(1) F (αx) = φ(α)F (x) J (G×X)-a.e. in G×X,



where (G, ·, 0) and (H, ·, 0) are groups with zero, (X,G) is a G-space, (Y,H)is a nontrivial H-space, J (G) and J (X) are proper linearly invariant ide-als in the group G and in the G-space X, respectively, J (G×X) is a(J (G),J (X)

)-ideal in G×X and φ : G → H, F : X → Y are unknown

functions. We recall here the basic result of [8].Proposition 1 [8, Proposition 1]. Let φ : G → H be a homomorphism.

If a function F : X → Y satis�es the equation (1), then there exists a func-tion F : X → Y such that(2) F (αx) = φ(α)F (x) for (α, x) ∈ G∗ ×X,

andF (x) = F (x) J (X)-a.e. in X.

The function F is unique with respect to the ideal J (X).Now we give some facts which will be useful in the sequel.Lemma 1 [8, Lemma 1]. Let (G, ·, 0) and (H, ·, 0) be groups with zero

and assume that φ : G → H is a homomorphism, i.e. the equationφ(αβ) = φ(α)φ(β) for α, β ∈ G

is satis�ed. If φ 6= 0 then φ(1) = 1 and φ(G∗) ⊂ H∗.Lemma 2 [8, Lemma 2]. Let (G, ·, 0) be a group with zero and let X

be a G-space. Assume that J (G) and J (X) are proper linearly invariantideals in G and X, respectively. Let moreover J (G×G) and J (G×X) be(J (G),J (G)

)-ideal in G×G and

(J (G),J (X))-ideal in G×X, respec-

tively. Assume that a property W , de�ned on G×X, satis�es the conditionW (α, x) for (α, x) ∈ (G×X) \M,

with some M ∈ J (G×X). Then there exists a set N ∈ J (G×G) such thatfor every (α,β) ∈ (G×G∗)\N there exists a set U0 ∈ J (X) with the property

W (β, x) ∧W (α, βx) ∧W (αβ, x) for each x ∈ X \ U0.

Lemma 3 [5, 8]. Let (G, ·, 0) and (H, ·, 0) be groups with zero. LetJ (G) be a proper linearly invariant ideal inG and assume that J (G×G) isa

(J (G),J (G))-ideal in G×G. Assume that a function φ : G → H satis�es

the conditionφ(αβ) = φ(α)φ(β) J (G×G)-a.e. in G×G.

Then there exists a homomorphism φ̃ : G → H such hat

φ(α) = φ̃(α) J (G)-a.e. in G.



Superstability of φ-homogeneity

Now we are going to study the superstability of theφ-homogeneity equa-tion. We will assume that (G, ·, 0) is a group with zero, X is a G-space,J (G) and J (X) are proper linearly invariant ideals in G and X, respec-tively. Let J (G×G) and J (G×X) be

(J (G),J (G))- and

(J (G),J (X))-

ideals in G×G and G×X, respectively. Assume thatY is a linear space overK ∈ {R,C}. A function φ : G →K will be called a homomorphism provided

φ(αβ) = φ(α)φ(β) for every α, β ∈ G.

We begin withLemma 4. Let φ : G → K be a homomorphism. Assume thatB : G×X

→ 2Y and let a function f : X → Y satisfy the condition

(3) f(αx)− φ(α)f(x) ∈ B(α, x) J (G×X)-a.e. in G×X.

Then there exists a set N ∈ J (G×X) such that for every (α, x) ∈ (G∗ ×X)\N there exists a set A ∈ J (G) such that

φ(β)(f(αx)− φ(α)f(x)

) ∈ B(β, αx)−B(βα, x) for β ∈ G \A.

Proof. From (3) it follows that there exists a setM ∈ J (G×X) suchthat

(4) f(αx)− φ(α)f(x) ∈ B(α, x) for (α, x) ∈ (G×X) \M.

Since M ∈ J (G×X), we �nd a set U1 ∈ J (X) such that Mx ∈ J (G) foreach x ∈ X \ U1. Let

N := (G× U1) ∪{

(α, x) ∈ G×X : αx ∈ U1

}.

Then N ∈ J (G×X). Fix (α,x) ∈ (G∗×X) \N arbitrarily. Then (α,x) 6∈ G× U1, so x 6∈ U1. Hence Mx ∈ J (G). Moreover we have αx 6∈ U1 and thenMαx ∈ J (G). Put A := Mxα−1 ∪Mαx. Clearly A ∈ J (G). Fix β ∈ G \A.Then β 6∈ Mxα−1, so (βα, x) 6∈ M . From (3) we obtain

f(βαx)− φ(βα)f(x) ∈ B(βα, x).

Since φ is a homomorphism we get

f(βαx)− φ(β)φ(α)f(x) ∈ B(βα, x).



Next, β 6∈ Mαx and then (β, αx) 6∈ M . Thus from (3) we have

f(βαx)− φ(β)f(αx) ∈ B(β, αx).

From the above conditions we obtainφ(β)

(f(αx)− φ(α)f(x)

)= φ(β)f(αx)− f(βαx) + f(βαx)− φ(β)φ(α)f(x)

∈ (B(βα, x)−B(β, αx)

). ¤

Lemma 5. Let φ : G→K, B : G×X → 2Y and assume that the functionf : X → Y satis�es condition (3). Then there exists a set N ∈ J (G×G)such that for every (α, β) ∈ (G×G∗) \N there exists a set U ∈ J (X) with

(5)(φ(αβ)− φ(α)φ(β)

)f(x) ∈ B(α, βx)−B(αβ, x) + φ(α)B(β, x)

for each x ∈ X \ U .Proof. It follows from (3) that there exists a set M ∈ J (G×X) such

that (4) holds. Let us consider the condition (see Lemma 2)

W (α, x) :=(F (αx)− φ(α)F (x) ∈ B(α, x)

).

Then, on account of Lemma 2, there exists a setN ∈ J (G×G) such that forevery (α, β) ∈ (G×G∗) \N there exists a set U ∈ J (X) such that for eachx ∈ X \ U

f(βx)− φ(β)f(x) ∈ B(β, x), f(αβx)− φ(α)f(βx) ∈ B(α, βx),

f(αβx)− φ(αβ)f(x) ∈ B(αβ, x)

hold. Thus(φ(αβ)− φ(α)φ(β)

)f(x) = φ(αβ)f(x)− f(αβx)

+ f(αβx)− φ(α)f(βx) + φ(α)f(βx)− φ(α)φ(β)f(x)

∈ B(α, βx)−B(αβ, x) + φ(α)B(β, x). ¤

Now we prove our main results. We will assume thatY is a locally convexlinear topological space overK. By B(Y ) we denote the family of all boundedsubsets of the space Y . Let δ : G→K and V ∈ B(Y ). Assume moreover thatψ : G → [0,∞) is a homomorphism and let the functionK : X → K satisfythe inequality

(6)∣∣K(αx)

∣∣ 5 ψ(α)∣∣K(x)

∣∣ for α ∈ G, x ∈ X.

We are going to prove the following theorem.



Theorem 1. Assume that φ : G → K is a nonzero homomorphism, thatis (cf. Lemma 1) φ(G∗) ⊂ K∗, and let the function f : X → Y satisfy thecondition

(7) f(αx)− φ(α)f(x) ∈ δ(α)K(x)V J (G×X)-a.e. in G×X.

If for each α ∈ G∗

(8) J (G) ess infβ∈G∗

∣∣δ(β)∣∣ψ(α) +

∣∣δ(αβ)∣∣

∣∣φ(β)∣∣ = 0,

then there exists a function F : X → Y such that

F (αx) = φ(α)F (x) for (α, x) ∈ G∗ ×X

andf(x) = F (x) J (X)-a.e. in X.

Proof. In virtue of (7) there exists a setM ∈ J (G×X) with

f(αx)− φ(α)f(x) ∈ δ(α)K(x)V for (α, x) ∈ (G×X) \M.

Put B(α,x) := δ(α)K(x)V and let N ∈ J (G×X) be the set from Lemma 4.Suppose that for some (α, x) ∈ (G∗ ×X) \N we have f(αx) 6= φ(α)f(x).Then there exists a set A ∈ J (G) such that for every β ∈ G \A we obtain

φ(β)(f(αx)− φ(α)f(x)

) ∈ (δ(βα)K(x)V − δ(β)K(αx)V

)

⊂ ∣∣δ(βα)∣∣ ∣∣K(x)

∣∣ aconv V +∣∣δ(β)

∣∣ ∣∣K(αx)∣∣ aconv V

⊂ (∣∣δ(βα)

∣∣ +∣∣δ(β)

∣∣ψ(α))∣∣K(x)

∣∣ aconv V.

Thus

f(αx)− φ(α)f(x) ∈⋂

β∈G∗\A

∣∣δ(βα)∣∣ +

∣∣δ(β)∣∣ψ(α)∣∣φ(β)

∣∣∣∣K(x)

∣∣ aconv V,

which jointly with (8) and the boundedness of the setaconvV leads to a con-tradiction. Thus f(αx) = φ(α)f(x) for (α, x) ∈ (G∗ ×X) \N . Then Propo-sition 1 �nishes the proof. ¤

Now we will prove that the assumption that φ is a homomorphismcan be replaced in Theorem 1 with the one on the J (X)-essentially K-unboundedness of the function f .



Definition 2. A function f : X → Y is called J (X)-essentially K-bounded provided there exist sets U ∈ J (X) and W ∈ B(Y ) with the prop-erty

f(x) ∈ K(x)W for x ∈ X \ U.

Otherwise f is said to be J (X)-essentially K-unbounded. In the caseK = 1we will simply say that a function f is (or is not) J (X)-essentially bounded.

For a function φ : G → K denote suppφ :={

α ∈ G : φ(α) 6= 0}. We

prove the following theorem.Theorem 2. Assume that the functions φ : G → K and f : X → Y sat-

isfy condition (7). If f is J (X)-essentially K-unbounded then there exists ahomomorphism φ̃ : G → K such that

φ(β) = φ̃(β) J (G)-e.a. in G.

Furthermore, if for every A ∈ J (G) there exists an α ∈ G \A with φ(α) 6= 0and for each α ∈ G∗

(9) J (G) ess infβ∈supp φ

∣∣δ(β)∣∣ψ(α) +

∣∣δ(αβ)∣∣

∣∣φ(β)∣∣ = 0

holds, then there exists a function F : X → Y such that

F (αx) = φ̃(α)F (x) for (α, x) ∈ G∗ ×X

andf(x) = F (x) J (X)-a.e. in X.

Proof. Assume that f is a J (X)-essentially K-unbounded function. Wewill show that then φ satis�es the equation of homomorphismJ (G×G)-a.e.in G×G. Indeed, on account of Lemma 5, for B(α, x) = δ(α)K(x)V thereexists a set N ∈ J (G×G) such that for every (α, β) ∈ (G×G∗) \N thereexists a set U ∈ J (X) with(φ(αβ)− φ(α)φ(β)

)f(x) ∈ δ(α)K(βx)V − δ(αβ)K(x)V + φ(α)δ(β)K(x)V

⊂ (δ(α)ψ(β) + δ(αβ) + φ(α)δ(β))K(x) aconv V

for x ∈ X \ U . Suppose that for some (α, β) ∈ (G×G∗) \N ,

φ(αβ) 6= φ(α)φ(β)



holds. Then we get

f(x) ∈ δ(αβ) + δ(β)ψ(α) + δ(α)φ(β)∣∣φ(αβ)− φ(α)φ(β)∣∣ K(x) aconv V

for each x ∈ X \U . Thus we have obtained a contradiction from the assump-tion of J (X)-essentially K-unboundedness of the function f . Therefore wehave

φ(αβ) = φ(α)φ(β) for (α, β) ∈ (G×G) \N1,

where N1 := N ∪ (G× {0}) ∈ J (G×G). Hence φ satis�es the equation of

homomorphism J (G×G)-a.e. in G×G. In virtue of Lemma 3, there existsa homomorphism φ̃ : G →K such that φ(α) = φ̃(α) for α ∈ G \A with someA ∈ J (G). From (7) we then obtain

f(αx)− φ̃(α)f(x) ∈ δ(α)K(x)V for (α, x) ∈ (G×X) \M1

where M1 := (A×X) ∪M ∈ J (G×X).Now assume that for every A′ ∈ J (G) there exists an α ∈ G \A′ with

φ(α) 6= 0. Then, since φ = φ̃ almost everywhere in G, so (cf. Lemma 1) φ̃ isa nonzero homomorphism andG \ suppφ ∈ J (G). Hence from (9) we obtain

J (G) ess infβ∈G∗

∣∣δ(β)∣∣ψ(α) +

∣∣δ(αβ)∣∣

∣∣ φ̃(β)∣∣ = 0.

Thus Theorem 1 �nishes our proof. ¤As a simple consequence of Theorem 2 we obtainCorollary 1. Let

(Y, ‖ · ‖) be a normed space over K and let v : G

→ [0,∞) be given. Assume that the functions φ : G → K and f : X → Ysatisfy the inequality

∥∥f(αx)− φ(α)f(x)∥∥ 5 v(α) J (G×X)-a.e. in G×X.

If f is J (X)-essentially unbounded then there exists a homomorphism φ̃ : G→ K such that

φ̃(β) = φ(β) J (G)-e.a. in G.

Furthermore, if for every A ∈ J (G) there exists an α ∈ G \A with φ(α) 6= 0and for each α ∈ G∗

J (G) ess infβ∈supp φ

v(β) + v(αβ)∣∣φ(β)∣∣ = 0



holds, then there exists a function F : X → Y such that

F (αx) = φ̃(α)F (x) for (α, x) ∈ G∗ ×X

andF (x) = f(x) J (X)-a.e. in X.

Superstability of multiplicativity

Finally, using Corollary 1 we will show the superstability phenomenon ofthe multiplicativity (see for example [1, 2, 10]). Let J (G) be a nontrivialproper linearly invariant ideal in a group with zero(G, ·, 0) and let J (G×G)be a

(J (G),J (G))-ideal in G×G. A simple consequence of Corollary 1 is

the following result, which we give without a proof.Corollary 2. Let v : G → [0,∞) be given. If a function f : G → K

satis�es the inequality∣∣f(xy)− f(x)f(y)

∣∣ 5 v(x) J (G×G)-a.e. in G×G,

then either f is J (G)-essentially bounded or it is J (G)-a.e. in G equal to amultiplicative function f̃ : G → K.

References[1] J. A. Baker, The stability of the cosine equation,Proc. Amer. Math. Soc., 80 (1980),

411�416.[2] J. A. Baker, J. Lawrence and F. Zorzitto, The stability of the equationf(x + y) =

f(x)f(y), Proc. Amer. Math. Soc., 74 (1979), 242�246.[3] Paul Erd®s, P 310, Colloq. Math., 7 (1963), 267�269.[4] G. L. Forti, Hyers-Ulam stability of functional equations in several variables,Aequa-

tiones Math., 50 (1995), 143�190.[5] R. Ger, On a system of functional equations occurring in projective geometry,Rad.

Mat., 8 (1992), 189�200.[6] D. H. Hyers, On the stability of the linear functional equation,Proc. Nat. Acad. Sci.

USA, 27 (1941), 222�224.[7] D. H. Hyers, G. Isac and Th. M. Rasias, Stability of Functional Equations in Several

Variables, vol. 34, in: Progress in Nonlinear Di�erential Equations and TheirApplications, Birkhäuser Verlag (Boston � Basel � Berlin, 1998).

[8] W. Jabªo«ski, Homogeneity almost everywhere,Acta Math. Hungar., 113 (2006), 73�83.

[9] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities.Cauchy's Equation and Jensen's Inequality, P.W.N. (Warszawa � Kraków �Katowice, 1985).



[10] L. Székelyhidi, On a theorem of Baker, Lawrence and Zorzitto,Proc. Amer. Math.Soc., 84 (1982), 95�96.

[11] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ. (New York�London, 1960).


Stability of homogeneity almost everywhere

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