Research Article | Open Access

Choonkil Park, John Michael Rassias, "Stability of the Jensen-Type Functional Equation in -Algebras: A Fixed Point Approach", *Abstract and Applied Analysis*, vol. 2009, Article ID 360432, 17 pages, 2009. https://doi.org/10.1155/2009/360432

# Stability of the Jensen-Type Functional Equation in -Algebras: A Fixed Point Approach

**Academic Editor:**Bruce Calvert

#### Abstract

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in -algebras and Lie -algebras and also of derivations on -algebras and Lie -algebras for the Jensen-type functional equation .

#### 1. Introduction and Preliminaries

The stability problem of functional equations
originated from a question of Ulam [1] concerning the stability of group homomorphisms.
Hyers [2] gave a first
affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by
considering an unbounded Cauchy difference. The paper of Rassias
[4] has provided a
lot of influence in the development of what we call *generalized Hyers-Ulam stability* of
functional equations. A generalization of the Rassias theorem was
obtained by GΔvruΕ£a [5] by replacing the unbounded
Cauchy difference by a general control function in the spirit of Rassias'
approach. In 1982, Rassias [6] followed the innovative approach of the Rassias' theorem [4]
in which he replaced the factor by for with The stability problems of several functional
equations have been extensively investigated by a number of authors and there
are many interesting results concerning this problem (see [4, 7β27]).

We recall a fundamental result in fixed point theory.

Let be a set. A
function is called a *generalized metric* on if satisfies

(1) if and only if (2) for all (3) for all

Theorem 1.1 (see [28, 29]). *Let be a complete
generalized metric space and let be a strictly
contractive mapping with Lipschitz constant Then for each given element either **for all nonnegative integers or there exists
a positive integer such that *

(1)* for all *(2)*the sequence converges to a
fixed point of *(3)* is the unique
fixed point of in the set *(4)* for all *

By the using fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [17, 30β33]).

This paper is organized as follows: in Sections 2 and 3, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in -algebras and of derivations on -algebras for the Jensen-type functional equation.

In Sections 4 and 5, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in Lie -algebras and of derivations on Lie -algebras for the Jensen-type functional equation.

#### 2. Stability of Homomorphisms in -Algebras

Throughout this section, assume that is a -algebra with norm and that is a -algebra with norm

For a given mapping we definefor all and all

Note that a -linear mapping is called a *homomorphism* in -algebras if satisfies and for all

We prove the generalized Hyers-Ulam stability of homomorphisms in -algebras for the functional equation

Theorem 2.1. *Let be a mapping
for which there exists a function such
that **for all and all If there exists an such that for all then there exists a unique -algebra homomorphism such
that **for all **Proof. *It
follows from thatfor all

Consider the setand introduce the *generalized metric* on :It is easy to show that is complete.

Now we consider the linear mapping such thatfor all

By [28, Theorem 3.1],for all

Letting and in (2.2), we getfor all Sofor all Hence

By Theorem 1.1, there exists a mapping such that

(1) is a fixed
point of that is,for all The mapping is a unique
fixed point of in the setThis implies that is a unique
mapping satisfying (2.13) such that there
exists satisfyingfor all (2) as This implies the equalityfor all (3) which implies the inequalityThis implies that the
inequality (2.5) holds.

It follows from(2.2),(2.6), and (2.16) thatfor all Sofor all Letting and in (2.19), we getfor all So the mapping is Cauchy
additive, that is, for all

Letting in (2.2), we getfor all and all By a similar method to above, we getfor all and all Thus one can show that the mapping is -linear.

It follows from
(2.3) thatfor all Sofor all

It follows from
(2.4) thatfor all Sofor all

Thus is a -algebra
homomorphism satisfying (2.5), as desired.Corollary 2.2. *Let and be nonnegative
real numbers, and let be a mapping
such that **for all and all Then there exists a unique -algebra
homomorphism such
that **for all **Proof. *The proof follows from Theorem 2.1
by takingfor all Then and we get the
desired result.

Theorem 2.3. *Let be a mapping
for which there exists a function satisfying (2.2),
(2.3), and (2.4). If there exists
an such that for all then there exists a unique -algebra
homomorphism such
that **for all **Proof. *We
consider the linear mapping such thatfor all

It follows from
(2.11) thatfor all Hence

By Theorem 1.1, there exists a mapping such that

(1) is a fixed
point of that is,for all The mapping is a unique
fixed point of in the setThis implies that is a unique
mapping satisfying (2.35) such that
there exists satisfyingfor all (2) as This implies the equalityfor all (3) which implies the inequalitywhich implies that the inequality (2.32) holds.

The rest of the proof is similar to the proof of
Theorem 2.1.Corollary 2.4. *Let and be nonnegative
real numbers, and let be a mapping
satisfying (2.27), (2.28) and
(2.29). Then there exists a unique -algebra
homomorphism such
that **for all **Proof. *The
proof follows from Theorem 2.3 by takingfor all Then and we get the
desired result.

Theorem 2.5. *Let be an odd
mapping for which there exists a function satisfying (2.2),
(2.3), (2.4) and (2.6). If there exists an such that for all then there exists a unique -algebra
homomorphism such
that **for all **Proof. *Consider
the setand introduce the *generalized metric* on :It is easy to show that is complete.

Now we consider the linear mapping such thatfor all

By [28, Theorem 3.1],for all

Letting and relpacing by in (2.2), we getfor all Sofor all Hence

By Theorem 1.1, there exists a mapping such that

(1) is a fixed
point of that is,for all The mapping is a unique
fixed point of in the setThis implies that is a unique
mapping satisfying (2.49) such that
there exists satisfyingfor all (2) as This implies the equalityfor all (3) which implies the inequalityThis implies that the inequality(2.42) holds.

The rest of the proof is similar to the proof of
Theorem 2.1.Corollary 2.6. *Let and be nonnegative
real numbers, and let be an odd
mapping such that **for all and all Then there exists a unique -algebra
homomorphism such
that **for all **Proof. *The
proof follows from Theorem 2.5 by takingfor all Then and we get the
desired result.

Theorem 2.7. *Let be an odd
mapping for which there exists a function satisfying (2.2),
(2.3) and (2.4). If there exists
an such that for all then there exists a unique -algebra
homomorphism such
that **for all **Proof. *We
consider the linear mapping such thatfor all

It follows from
(2.47) thatfor all Hence

By Theorem 1.1, there exists a mapping such that

(1) is a fixed
point of that is,for all The mapping is a unique
fixed point of in the setThis implies that is a unique
mapping satisfying (2.60) such that
there exists satisfyingfor all (2) as This implies the equalityfor all (3) which implies the inequalitywhich implies that the
inequality (2.57) holds.

The rest of the proof is similar to the proof of
Theorem 2.1.Corollary 2.8. *Let and be nonnegative
real numbers, and let be an odd
mapping satisfying (2.54). Then there exists a unique -algebra
homomorphism such
that **for all **Proof. *The
proof follows from Theorem 2.7 by takingfor all Then and we get the
desired result.

#### 3. Stability of Derivations on -Algebras

Throughout this section, assume that is a -algebra with norm

Note that a -linear mapping is called a *derivation* on if satisfies for all

We prove the generalized Hyers-Ulam stability of derivations on -algebras for the functional equation

Theorem 3.1. *Let be a mapping
for which there exists a function such
that **for all and all If there exists an such that for all Then there exists a unique derivation such
that **for all **Proof. *By the
same reasoning as the proof of Theorem 2.1, there exists a unique involutive -linear mapping satisfying (3.3). The mapping is given
byfor all

It follows from
(3.2) thatfor all Sofor all Thus is a derivation
satisfying (3.3).Corollary 3.2. *Let and be nonnegative
real numbers, and let be a mapping
such that **for all and all Then there exists a unique derivation such
that **for all **Proof. *The
proof follows from Theorem 3.1 by takingfor all Then and we get the
desired result.

Theorem 3.3. *Let be a mapping
for which there exists a function satisfying (3.1) and
(3.2). If there exists an such that for all then there exists a unique derivation such
that **for all **Proof. *The
proof is similar to the proofs of Theorems 2.3 and 3.1.Corollary 3.4. *Let and be nonnegative real
numbers, and let be a mapping
satisfying (3.7) and (3.8). Then there exists a unique derivation such
that **for all **Proof. *The
proof follows from Theorem 3.3 by takingfor all Then and we get the desired result.*Remark 3.5. *For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 2.5 and
2.7 and Corollaries 2.6 and 2.8.

#### 4. Stability of Homomorphisms in Lie -Algebras

A -algebra endowed with the Lie product on is called a *Lie *-*algebra* (see [13β15]).

*Definition 4.1. *Let and be Lie -algebras. A -linear mapping is called
a Lie -algebra
homomorphism if for all

Throughout this section, assume that is a Lie -algebra with norm and that is a Lie -algebra with norm

We prove the generalized Hyers-Ulam stability of homomorphisms in Lie -algebras for the functional equation

Theorem 4.2. *Let be a mapping
for which there exists a function satisfying (2.2) such that **for all If there exists an such that for all then there exists a unique Lie -algebra
homomorphism satisfying (2.5).**Proof. *By the same reasoning as the proof
of Theorem 2.1, there exists a unique -linear mapping satisfying (2.5). The mapping is given
byfor all

It follows from
(4.1) thatfor all Sofor all

Thus is a Lie -algebra
homomorphism satisfying (2.5), as
desired.Corollary 4.3. *Let and be nonnegative
real numbers, and let be a mapping
satisfying (2.27) such
that **for all Then there exists a unique Lie -algebra
homomorphism satisfying (2.30).**Proof. *The
proof follows from Theorem 4.2 by takingfor all Then and we get the
desired result.

Theorem 4.4. *Let be a mapping
for which there exists a function satisfying (2.2) and
(4.1). If there exists an such that for all then there exists a unique Lie -algebra
homomorphism satisfying (2.32).**Proof. *The proof is similar to the proofs
of Theorems 2.3 and 4.2.Corollary 4.5. *Let and be nonnegative
real numbers, and let be a mapping
satisfying (2.27) and (4.5). Then there exists a unique Lie -algebra
homomorphism satisfying (2.40).**Proof. *The
proof follows from Theorem 4.4 by takingfor all Then and we get the
desired result.*Remark 4.6. *For inequalities controlled by the
product of powers of norms, one can obtain similar results to Theorems 2.5 and 2.7 and Corollaries 2.6 and 2.8.

#### 5. Stability of Lie Derivations on -Algebras

*Definition 5.1. *Let be a Lie -algebra. A -linear mapping is called
a Lie derivation if for all

Throughout this section, assume that is a Lie -algebra with norm

We prove the generalized Hyers-Ulam stability of derivations on Lie -algebras for the functional equation

Theorem 5.2. *Let be a mapping
for which there exists a function satisfying (3.1) such that **for all If there exists an such that for all Then there exists a unique Lie derivation satisfying (3.3).**Proof. *By the same reasoning as the proof
of Theorem 2.1, there exists a unique involutive -linear mapping satisfying (3.3). The mapping is given
byfor all

It follows from
(5.1) thatfor all Sofor all Thus is a derivation
satisfying (3.3).Corollary 5.3. *Let and be nonnegative
real numbers, and let be a mapping
satisfying (3.7) such
that **for all Then there exists a unique Lie derivation satisfying (3.9).**Proof. *The
proof follows from Theorem 5.2 by takingfor all Then and we get the
desired result.

Theorem 5.4. *Let be a mapping
for which there exists a function satisfying (3.1) and
(5.1). If there exists an such that for all then there exists a unique Lie derivation satisfying (3.11).**Proof. *The
proof is similar to the proofs of Theorems 2.3 and 5.2.Corollary 5.5. *Let and be nonnegative
real numbers, and let be a mapping
satisfying (3.7) and (5.5). Then there exists a unique Lie
derivation satisfying (3.12).**Proof. *The
proof follows from Theorem 5.4 by takingfor all Then and we get the
desired result.*Remark 5.6. *For inequalities controlled by the
product of powers of norms, one can obtain similar results to Theorems 2.5 and 2.7 and Corollaries 2.6 and 2.8.

#### Acknowledgment

The first author was supported by the R & E program of KOSEF in 2008.

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#### Copyright

Copyright © 2009 Choonkil Park and John Michael Rassias. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.