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ISOMETRIES AND ISOMORPHISMS IN QUASI-BANACH ALGEBRAS

ZHIHUA WANG and WANXIONG ZHANG

Abstract. In this paper, we prove the Hyers-Ulam-Rassias stability of isometries and of homomor-phisms for additive functional equations in quasi-Banach algebras. This is applied to investigate iso-morphisms between quasi-Banach algebras.

1. Introduction and preliminaries

Stability is investigated when one concerns whether a small error of parameters causes a large deviation of the solution. Generally speaking, given a function which satisfies a functional equation approximately called anapproximate solution, we ask: Is there a solution of this equation which is close to the approximate solution in some accuracy? An ealier work was done by Hyers [11] in order to answer Ulam’s question ([20]) on approximately additive mappings. Later there have been given lots of results on stability in the Hyers-Ulam sense or some generalized sense (see books and papers [1, 3, 8, 9,12, 17,18] and references therein).

Received January 25, 2011.

2010Mathematics Subject Classification. Primary 46B03, 47B48, 39B72.

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G. Z. Eskandani [7] established the general solution and investigated the Hyers-Ulam-Rassias stability of the following functional equation

m

X

i=1 f



mxi+ m

X

j=1,j6=i xj



+f

m

X

i=1 xi

!

= 2f m

X

i=1 mxi

!

(1.1)

in quasi-Banach spaces, wherem∈N_andm≥2. The stability of isometries in norms spaces and Banach spaces was investigated in several papers [4, 6, 10, 13]. However, C. Park and Th. M. Rassias [15] proved the Hyers-Ulam stability of isometric additive functional equations in quasi-Banach spaces. C. Park [16] studied the Hyers-Ulam stability of homomorphisms in quasi-Banach algebras. Recently, M. S. Moslehian and Gh. Sadeghi [14] have proved the Hyers-Ulam-Rassias stability of linear mappings in quasi-Banach modules associated to the Cauchy functional equation and a generalized Jensen functional equation.

The main purpose of this paper is to study the Hyers-Ulam-Rassias stability of equation (1.1). More precisely, we prove the Hyers-Ulam-Rassias stability of isometric additive functional equations (1.1) in quasi-Banach algebras. Furthermore, we investigate the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to additive functional equations (1.1). This is applied to investigate isomorphisms between quasi-Banach algebras.

We now give some basic facts concerning quasi-Banach spaces and some preliminary results.

Definition 1.1 (cf. [5, 19]). Let X be a real linear space. A quasi-norm is a real-valued function onX satisfying the following:

(1) kxk ≥0 for allx∈X andkxk= 0 if and only ifx= 0. (2) kλxk=|λ| · kxk for allλ∈R_{and for allx∈X.}

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The pair (X,k · k) is called a quasi-normed spaceif k · kis a quasi-norm on X. The smallest possibleK is called the modulus of concavityof k · k. A quasi-Banach spaceis a complete quasi-normed space.

Aquasi-normk · kis called ap-norm(0< p≤1) if

kx+ykp_{≤ k} xkp_+k

ykp

for allx, y∈X. In this case, aquasi-Banach spaceis called ap-Banach space.

Given ap-norm, the formulad(x, y) := kx−ykp _{gives us a translation invariant metric onX.} By the Aoki-Rolewicz theorem [19] (see also [5]), each quasi-norm is equivalent to somep-norm. Since it is much easier to work withp-norms than quasi-norms, henceforth we restrict our attention mainly top-norms.

Definition 1.2 (cf. [2]). Let (X,k · k) be a quasi-normed space. The quasi-normed space (X,k · k) is called a quasi-normed algebra ifX is an algebra and there is a constant C >0 such thatkxyk ≤Ckxkkykfor allx, y∈X.

Aquasi-Banach algebrais a completequasi-normed algebra. If the quasi-normk · kis ap-norm, then thequasi-Banach algebrais called ap-Banach algebra.

Definition 1.3 (cf. [15]). Let X and Y be quasi-Banach algebras with norms k · kX and

k · kY, respectively. An additive mappingA:X →Y is called an isometric additive mapping if the additive mappingA:X →Y satisfies

kA(x)−A(y)kY =kx−ykX

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2. Stability of isometric additive mappings in quasi-Banach algebras

Throughout this section and Section 3, assume thatX is a quasi-normed algebra with quasi-norm

k · kX and thatY is ap-Banach algebra withp-norm k · kY. LetK be the modulus of concavity ofk · kY. For convenience, we use the following abbreviation for a given mappingf:X →Y:

Df(x1,· · ·, xm) = m

X

i=1 f



mxi+ m

X

j=1,j6=i xj



+f

m

X

i=1 xi

!

−2f m

X

i=1 mxi

!

for allxj ∈X (1≤j ≤m). We prove the Hyers-Ulam-Rassias stability of the isometric additive functional equation (1.1) in quasi-Banach algebras.

Theorem 2.1. Let ϕ:Xm_{→[0,∞) be a mapping such that}

lim n→∞

1 mnϕ(m

n_{x1,· · ·, m}n_x m) = 0 (2.1)

˜ ϕ(x) :=

∞ X

i=0 1 mip(ϕ(m

i

x,0,· · ·,0))p <∞

(2.2)

for allx, xj∈X (1≤j≤m). Suppose that a mappingf:X →Y satisfies

kDf(x1,· · ·, xm)kY ≤ϕ(x1,· · ·, xm) (2.3)

| kf(x)kY − kxkX |≤ϕ(x,· · ·, x

| {z }

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for allx, xj ∈X (1≤j≤m). Then there exists a unique isometric additive mappingA:X →Y

such that

kf(x)−A(x)kY ≤ 1 m[ ˜ϕ(x)]

1

p

(2.5)

for allx∈X.

Proof. By the Eskandani’s theorem [7, Theorem 2.2], it follows from (2.1), (2.2) and (2.3) that there exists a unique additive mappingA:X →Y satisfying (2.5). The additive mapping A: X→Y is given by

A(x) := lim n→∞

1 mnf(m

n x) (2.6)

for allx∈X.

It follows from (2.4) that

| k 1

mnf(m n

x)kY − kxkX | ≤ 1

mn | kf(m n

x)kY − kmnxkX|

≤ 1

mnϕ(m n

x,· · ·, mnx

| {z }

m−times )

which tends to zero asn→ ∞for allx∈X. So

kA(x)kY = lim n→∞k

1 mnf(m

n_x)k

Y =kxkX

for allx∈X. SinceA:X →Y is additive,

kA(x)−A(y)kY =kA(x−y)kY =kx−ykX

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Theorem 2.2. Let φ:Xm_{→[0,∞)be a mapping such that}

lim n→∞m

n_φ(x1 mn,· · ·,

xm mn) = 0 (2.7)

˜ φ(x) :=

∞ X

i=1

mip(φ( x

mi,0,· · · ,0)) p

<∞

(2.8)

for allx, xj∈X (1≤j≤m). Suppose that a mappingf:X →Y satisfies

kDf(x1,· · ·, xm)kY ≤φ(x1,· · ·, xm) (2.9)

| kf(x)kY − kxkX|≤φ(x,· · ·, x

| {z }

m−times ) (2.10)

for allx, xj ∈X (1≤j≤m). Then there exists a unique isometric additive mappingA:X →Y

such that

kf(x)−A(x)kY ≤ 1 m[ ˜φ(x)]

1

p

(2.11)

for allx∈X.

Proof. By the Eskandani’s theorem [7, Theorem 2.3], it follows from (2.7), (2.8) and (2.9) that there exists a unique additive mappingA:X →Y satisfying (2.11). The additive mapping A: X→Y is given by

A(x) := lim n→∞m

n_f( x mn) (2.12)

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By (2.10), we have

| kmnf( x

mn)kY − kxkX| ≤m n _{| k}

f( x

mn)kY − k x mnkX|

≤mnϕ( x mn,· · ·,

x mn

| {z }

m−times )

which tends to zero asn→ ∞for allx∈X. By (2.12), we obtain

kA(x)kY = lim n→∞km

n_f( x

mn)kY =kxkX for allx∈X. Hence

kA(x)−A(y)kY =kA(x−y)kY =kx−ykX

for allx∈X. So the additive mappingA:X →Y is an isometry. This completes the proof of the

theorem.

Corollary 2.1. Let θ, rj (1 ≤ j ≤ m) be non-negative real numbers such that rj > 1 or 0< rj<1. Suppose that a mapping f:X →Y satisfies

kDf(x1,· · ·, xm)kY ≤θ m

X

i=1

kxikrXi

| kf(x)kY − kxkX |≤θ m

X

i=1

kxkri

X

for allx, xj ∈X (1≤j≤m). Then there exists a unique isometric additive mappingA:X →Y

such that

kf(x)−A(x)kY ≤

θ

|mp_−mpr_1|1p kxkr1

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for allx∈X.

Proof. The result follows from the proofs of Theorems2.1and2.2.

3. Stability of homomorphisms in quasi-Banach algebras

We prove the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, associated to the additive functional equation (1.1).

Theorem 3.1. Suppose that a mappingf:X →Y satisfies

kDf(x1,· · · , xm)kY ≤ϕ(x1,· · · , xm) (3.1)

kf(xy)−f(x)f(y)kY ≤ψ(x, y) (3.2)

for allx, y, xj ∈X (1≤j≤m), whereϕ:Xm→[0,∞)satisfies (2.1)and (2.2), andψ:X×X → [0,∞)satisfies the following

lim n→∞

1 mnψ(m

n

x, mny) = 0 (3.3)

for all x, y∈X. Iff(tx) is continuous int ∈R for each fixed x∈X, then there exists a unique homomorphismH:X →Y such that

kf(x)−H(x)kY ≤ 1 m[ ˜ϕ(x)]

1

p

(3.4)

for allx∈X.

Proof. By Theorem 2.1, there exists a unique additive mapping H: X → Y satisfying (3.4). The additive mappingH:X →Y is given by

H(x) := lim n→∞

1 mnf(m

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for allx∈X. By the same reasoning as in the proof of Theorem of [17], the mappingH:X →Y isR_-linear.

It follows from (3.2) that

kH(xy)−H(x)H(y)kY = lim n→∞

1 m2nkf(m

2n_xy)−f(mn_x)f(mn_y)k Y

≤ lim n→∞

1 m2nψ(m

n_{x, m}n_{y) = 0}

for allx, y∈X. Hence, we get

H(xy) =H(x)H(y)

for allx, y∈X. Thus the mapping H:X →Y is a unique homomorphism satisfying (3.4). This

completes the proof of the theorem.

Theorem 3.2. Suppose that a mappingf:X →Y satisfies

kDf(x1,· · ·, xm)kY ≤φ(x1,· · ·, xm) (3.6)

kf(xy)−f(x)f(y)kY ≤Ψ(x, y) (3.7)

for allx, y, xj ∈X (1≤j≤m), whereφ:Xm→[0,∞)satisfies (2.7)and (2.8), andΨ :X×X → [0,∞)satisfies the following

lim n→∞m

n_Ψ( x mn,

y mn) = 0 (3.8)

for all x, y∈X. Iff(tx) is continuous int ∈R _{for each fixed x∈X, then there exists a unique}

homomorphismH:X →Y such that

kf(x)−H(x)kY ≤ 1 m[ ˜φ(x)]

1

p

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for allx∈X.

Proof. By Theorem 2.2, there exists a unique additive mapping H: X → Y satisfying (3.9). The additive mappingH:X →Y is given by

H(x) := lim n→∞m

n f( x

mn) (3.10)

for allx∈X. By the same reasoning as in the proof of Theorem of [17], the mappingH:X →Y isR_-linear.

It follows from (3.8) that

kH(xy)−H(x)H(y)kY = lim n→∞m

2n_kf( xy

mn_·mn)−f( x mn)f(

y mn)kY

≤ lim n→∞m

2n_Ψ( x mn,

y mn) = 0 for allx, y∈X. Hence, we get

H(xy) =H(x)H(y)

for allx, y∈X. Thus the mapping H:X →Y is a unique homomorphism satisfying (3.9). This

completes the proof of the theorem.

Corollary 3.1. Let θ, δ be non-negative real numbers and let rj (1 ≤ j ≤ m), s1, s2 be

non-negative real numbers such that rj > 1, s1, s2 > 2 or 0 < rj < 1, s1, s2 < 2. Suppose that a

mappingf:X→Y satisfies

kDf(x1,· · · , xm)kY ≤θ m

X

i=1

kxikrXi (3.11)

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for allx, y, xj∈X (1≤j ≤m). Iff(tx)is continuous int∈Rfor each fixed x∈X, then there

exists a unique homomorphismH:X →Y such that

kf(x)−H(x)kY ≤

θ

|mp_−mpr_1|p1 kxkr1

X

for allx∈X.

Proof. The result follows from the proofs of Theorems3.1and3.2.

Corollary 3.2. Let θ, δ be non-negative real numbers and let rj (1 ≤ j ≤ m), s1, s2 be

non-negative real numbers such that

m

P

i=1

ri >1,s1+s2 >2 or m

P

i=1

ri <1,s1+s2 <2 andrj 6= 0 for

somej(2≤j≤m). Suppose that a mappingf:X →Y satisfies

kDf(x1,· · · , xm)kY ≤θ m

Y

i=1

kxikr_Xi (3.13)

kf(xy)−f(x)f(y)kY ≤δkxksX1kyk s2

X (3.14)

for allx, y, xj ∈X (1≤j ≤m). Iff(tx) is continuous in t ∈R for each fixedx∈X, then the

mappingf:X→Y is a homomorphism.

Proof. The result follows from the proofs of Theorems3.1and3.2.

4. Isomorphisms between quasi-Banach algebras

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We investigate isomorphisms between quasi-Banach algebras associated to the additive func-tional equation (1.1).

Theorem 4.1. Suppose thatf:X →Y is a bijective mapping satisfying (3.1)such that

f(xy) =f(x)f(y) (4.1)

for all x, y∈X. If f(tx) is continuous in t∈R for each fixedx∈X and lim n→∞

1 mnf(m

n_{e) =e}′_, then the mappingf:X→Y is an isomorphism.

Proof. By Theorem3.1, there exists a homomorphismH:X →Y satisfying (3.4). The mapping H:X →Y is given by

H(x) := lim n→∞

1 mnf(m

n_x) (4.2)

for allx∈X. By (4.1), we have

H(x) =H(ex) = lim n→∞

1 mnf(m

n_{ex) = lim} n→∞

1 mnf(m

n e·x)

= lim n→∞

1 mnf(m

n

e)f(x) =e′f(x) =f(x)

for allx∈X. So the bijective mappingf:X →Y is an isomorphism. This completes the proof

of the theorem.

Theorem 4.2. Suppose that f: X → Y is a bijective mapping satisfying (3.6) and (4.1). If

f(tx) is continuous in t ∈ R for each fixed x ∈ X and lim n→∞m

n_f( e

mn) = e′, then the mapping

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Proof. By Theorem3.2, there exists a homomorphismH:X →Y satisfying (3.9). The mapping H:X →Y is given by

H(x) := lim n→∞m

n f( x

mn) (4.3)

for allx∈X.

The rest of the proof is similar to the proof of Theorem4.1. This completes the proof of the

theorem.

Corollary 4.1. Let θ, rj (1 ≤ j ≤ m) be non-negative real numbers such that rj > 1 or 0 < rj < 1. Suppose that a bijective mapping f:X →Y satisfies (3.11) and (4.1). Iff(tx) is

continuous int∈R _{for each fixed x∈X and lim} n→∞m

n_f( e

mn) =e′ or lim

n→∞

1 mnf(m

n_{e) =e}′_{, then} the mappingf:X →Y is an isomorphism.

Proof. The result follows from the proofs of Theorems4.1and4.2.

Acknowledgment. The authors are very grateful to the referees for their helpful comments and suggestions.

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