- Research Article
- Open Access
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Stability of a Generalized Euler-Lagrange Type Additive Mapping and Homomorphisms in
-Algebras
Advances in Difference Equations volume 2009, Article number: 273165 (2009)
Abstract
Let be Banach modules over a
-algebra and let
be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital
-algebra:
. We show that if
,
for some
and a mapping
satisfies the functional equation mentioned above then the mapping
is Cauchy additive. As an application, we investigate homomorphisms in unital
-algebras.
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 Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference.
Theorem 1.1 (Th. M. Rassias [4]).
Let be a mapping from a normed vector space
into a Banach space
subject to the inequality

for all , where
and
are constants with
and
. Then the limit

exists for all and
is the unique additive mapping which satisfies

for all . If
, then (1.1) holds for
and (1.3) for
. Also, if for each
the mapping
is continuous in
, then
is
-linear.
Theorem 1.2 (J. M. Rassias [5–7]).
Let be a real normed linear space and
a real Banach space. Assume that
is a mapping for which there exist constants
and
such that
and
satisfies the functional inequality

for all . Then there exists a unique additive mapping
satisfying

for all . If, in addition,
is a mapping such that the transformation
is continuous in
for each fixed
then
is linear.
The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call the generalized Hyers-Ulam stability of functional equations. In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Gvruţa [8], who replaced the bounds
and
by a general control function
.
The functional equation

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [9] for mappings , where
is a normed space and
is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an Abelian group. Czerwik [11] proved the generalized Hyers-Ulam stability of the quadratic functional equation. J. M. Rassias [12, 13] introduced and investigated the stability problem of Ulam for the Euler-Lagrange quadratic mappings (1.6) and

Grabiec [14] has generalized these results mentioned above. In addition, J. M. Rassias [15] generalized the Euler-Lagrange quadratic mapping (1.7) and investigated its stability problem. Thus these Euler-Lagrange type equations (mappings) are called as Euler-Lagrange-Rassias functional equations (mappings).
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 [4–8, 12, 13, 15–55]).
Recently, C. Park and J. Park [45] introduced and investigated the following additive functional equation of Euler-Lagrange type:

whose solution is said to be a generalized additive mapping of Euler-Lagrange type.
In this paper, we introduce the following additive functional equation of Euler-Lagrange type which is somewhat different from (1.8):

where Every solution of the functional equation (1.9) is said to be a generalized Euler-Lagrange type additive mapping.
We investigate the generalized Hyers-Ulam stability of the functional equation (1.9) in Banach modules over a -algebra. These results are applied to investigate
-algebra homomorphisms in unital
-algebras.
Throughout this paper, assume that is a unital
-algebra with norm
and unit
that
is a unital
-algebra with norm
, and that
and
are left Banach modules over a unital
-algebra
with norms
and
respectively. Let
be the group of unitary elements in
and let
For a given mapping
and a given
we define
and
by

for all .
2. Generalized Hyers-Ulam Stability of the Functional Equation (1.9) in Banach Modules Over a
-Algebra
Lemma 2.1.
Let and
be linear spaces and let
be real numbers with
and
for some
Assume that a mapping
satisfies the functional equation (1.9) for all
Then the mapping
is Cauchy additive. Moreover,
for all
and all
Proof.
Since putting
in (1.9), we get
Without loss of generality, we may assume that
Letting
in (1.9), we get

for all Letting
in (2.1), we get

for all Similarly, by putting
in (2.1), we get

for all It follows from (2.1), (2.2) and (2.3) that

for all Replacing
and
by
and
in (2.4), we get

for all Letting
in (2.5), we get that
for all
So the mapping
is odd. Therefore, it follows from (2.5) that the mapping
is additive. Moreover, let
and
Setting
and
for all
in (1.9) and using the oddness of
we get that
Using the same method as in the proof of Lemma 2.1, we have an alternative result of Lemma 2.1 when
Lemma 2.2.
Let and
be linear spaces and let
be real numbers with
for some
Assume that a mapping
with
satisfies the functional equation (1.9) for all
Then the mapping
is Cauchy additive. Moreover,
for all
and all
We investigate the generalized Hyers-Ulam stability of a generalized Euler-Lagrange type additive mapping in Banach spaces.
Throughout this paper, will be real numbers such that
for fixed
Theorem 2.3.
Let be a mapping satisfying
for which there is a function
such that



for all and
Then there exists a unique generalized Euler-Lagrange type additive mapping
such that

for all Moreover,
for all
and all
Proof.
For each with
let
in (2.8), tthen we get the following inequality

for all For convenience, set

for all and all
Letting
in (2.10), we get

for all Similarly, letting
in (2.10), we get

for all It follows from (2.10), (2.12) and (2.13) that

for all Replacing
and
by
and
in (2.14), we get that

for all Putting
in (2.15), we get

for all Replacing
and
by
and
in (2.15), respectively, we get

for all It follows from (2.16) and (2.17) that

for all where

It follows from (2.6) that

for all Replacing
by
in (2.18) and dividing both sides of (2.18) by
we get

for all and all
Therefore, we have

for all and all integers
It follows from (2.20) and (2.22) that the sequence
is Cauchy in
for all
and thus converges by the completeness of
Thus we can define a mapping
by

for all Letting
in (2.22) and taking the limit as
in (2.22), we obtain the desired inequality (2.9).
It follows from (2.7) and (2.8) that

for all Therefore, the mapping
satisfies (1.9) and
Hence by Lemma 2.2,
is a generalized Euler-Lagrange type additive mapping and
for all
and all
To prove the uniqueness, let be another generalized Euler-Lagrange type additive mapping with
satisfying (2.9). By Lemma 2.2, the mapping
is additive. Therefore, it follows from (2.9) and (2.20) that

So for all
Theorem 2.4.
Let be a mapping satisfying
for which there is a function
satisfying (2.6), (2.7) and

for all and all
Then there exists a unique
-linear generalized Euler-Lagrange type additive mapping
satisfying (2.9) for all
Moreover,
for all
and all
Proof.
By Theorem 2.3, there exists a unique generalized Euler-Lagrange type additive mapping satisfying (2.9) and moreover
for all
and all
By the assumption, for each , we get

for all . So

for all and all
Since
for all
and

for all and all
By the same reasoning as in the proofs of [41, 43],

for all and all
Since
for all
the unique generalized Euler-Lagrange type additive mapping
is an
-linear mapping.
Corollary 2.5.
Let and
be real numbers such that
and
for all
where
Assume that a mapping
with
satisfies the inequality

for all and all
. Then there exists a unique
-linear generalized Euler-Lagrange type additive mapping
such that

for all where

Moreover, for all
and all
Proof.
Define and apply Theorem 2.4.
Corollary 2.6.
Let with
Assume that a mapping
with
satisfies the inequality

for all and all
Then there exists a unique
-linear generalized Euler-Lagrange type additive mapping
such that

for all Moreover,
for all
and all
Proof.
Define Applying Theorem 2.4, we obtain the desired result.
Theorem 2.7.
Let be a mapping satisfying
for which there is a function
such that



for all and
Then there exists a unique generalized Euler-Lagrange type additive mapping
such that

for all Moreover,
for all
and all
Proof.
By a similar method to the proof of Theorem 2.3, we have the following inequality

for all where

It follows from (2.36) that

for all Replacing
by
in (2.40) and multiplying both sides of (2.40) by
we get

for all and all
Therefore, we have

for all and all integers
It follows from (2.42) and (2.44) that the sequence
is Cauchy in
for all
and thus converges by the completeness of
Thus we can define a mapping
by

for all Letting
in (2.44) and taking the limit as
in (2.44), we obtain the desired inequality (2.39).
The rest of the proof is similar to the proof of Theorem 2.3.
Theorem 2.8.
Let be a mapping with
for which there is a function
satisfying (2.36), (2.37) and

for all and all
Then there exists a unique
-linear generalized Euler-Lagrange type additive mapping
satisfying (2.39) for all
Moreover,
for all
and all
Proof.
The proof is similar to the proof of Theorem 2.4.
Corollary 2.9.
Let and
be real numbers such that
and
for all
where
Assume that a mapping
with
satisfies the inequality

for all and all
Then there exists a unique
-linear generalized Euler-Lagrange type additive mapping
such that

for all where

Moreover, for all
and all
Proof.
Define Applying Theorem 2.8, we obtain the desired result.
Corollary 2.10.
Let with
Assume that a mapping
with
satisfies the inequality

for all and all
Then there exists a unique
-linear generalized Euler-Lagrange type additive mapping
such that

for all Moreover,
for all
and all
Proof.
Define Applying Theorem 2.8, we obtain the desired result.
Remark 2.11.
In Theorems 2.7 and 2.8 and Corollaries 2.9 and 2.10 one can assume that instead of
For the case in Corollaries 2.5 and 2.9, using an idea from the example of Gajda [56], we have the following counterexample.
Example 2.12.
Let be defined by

Consider the function by the formula

It is clear that is continuous and bounded by 2 on
. We prove that

for all and all
If
or
then

Now suppose that Then there exists a nonnegative integer
such that

Therefore

Hence

for all From the definition of
and (2.56), we have

Therefore satisfies (2.54). Let
be an additive mapping such that

for all Then there exists a constant
such that
for all rational numbers
So we have

for all rational numbers Let
with
If
is a rational number in
, then
for all
So

which contradicts with (2.61).
3. Homomorphisms in Unital
-Algebras
In this section, we investigate -algebra homomorphisms in unital
-algebras.
We will use the following lemma in the proof of the next theorem.
Lemma 3.1 (see [43]).
Let be an additive mapping such that
for all
and all
Then the mapping
is
-linear.
Theorem 3.2.
Let and
be real numbers such that
for all
where
and
Let
be a mapping with
for which there is a function
satisfying (2.7) and



for all for all
all
and all
Then the mapping
is a
-algebra homomorphism.
Proof.
Since letting
and
for all
in (3.1), we get

for all By the same reasoning as in the proof of Lemma 2.1, the mapping
is additive and
for all
and
So by letting
and
for all
in (3.1), we get that
for all
and all
. Therefore, by Lemma 3.1, the mapping
is
-linear. Hence it follows from (2.7), (3.2) and (3.3) that

for all and all
So
and
for all
and all
Since
is
-linear and each
is a finite linear combination of unitary elements (see [57]), that is,
where
and
for all
we have

for all Therefore, the mapping
is a
-algebra homomorphism, as desired.
The following theorem is an alternative result of Theorem 3.2.
Theorem 3.3.
Let and
be real numbers such that
for all
where
and
Let
be a mapping with
for which there is a function
satisfying (2.37) and

for all for all
all
and all
. Then the mapping
is a
-algebra homomorphism.
Remark 3.4.
In Theorems 3.2 and 3.3, one can assume that instead of
Theorem 3.5.
Let be a mapping with
for which there is a function
satisfying (2.6), (2.7), (3.2), (3.3) and

for all and all
. Assume that
is invertible. Then the mapping
is a
-algebra homomorphism.
Proof.
Consider the -algebras
and
as left Banach modules over the unital
-algebra
By Theorem 2.4, there exists a unique
-linear generalized Euler-Lagrange type additive mapping
defined by

for all Therefore, by (2.7), (3.2) and (3.3), we get

for all and for all
So
and
for all
and all
Therefore, by the additivity of
we have

for all and all
Since
is
-linear and each
is a finite linear combination of unitary elements, that is,
where
and
for all
it follows from (3.11) that

for all Since
is invertible and

for all for all
therefore, the mapping
is a
-algebra homomorphism.
The following theorem is an alternative result of Theorem 3.5.
Theorem 3.6.
Let be a mapping with
for which there is a function
satisfying (2.36), (2.37), (3.7) and

for all and all
. Assume that
is invertible. Then the mapping
is a
-algebra homomorphism.
Corollary 3.7.
Let and
be real numbers such that
and
for all
where
Assume that a mapping
with
satisfies the inequalities

for all all
all
and all
. Assume that
is invertible. Then the mapping
is a
-algebra homomorphism.
Proof.
The result follows from Theorem 3.6 (resp., Theorem 3.5).
Remark 3.8.
In Theorem 3.6 and Corollary 3.7, one can assume that instead of
Theorem 3.9.
Let be a mapping with
for which there is a function
satisfying (2.6), (2.7), (3.2), (3.3) and

for and all
. Assume that
is invertible and for each fixed
the mapping
is continuous in
. Then the mapping
is a
-algebra homomorphism.
Proof.
Put in (3.16). By the same reasoning as in the proof of Theorem 2.3, there exists a unique generalized Euler-Lagrange type additive mapping
defined by

for all By the same reasoning as in the proof of [4], the generalized Euler-Lagrange type additive mapping
is
-linear.
By the same method as in the proof of Theorem 2.4, we have

for all So

for all Since
for all
and

for and for all
For each element we have
where
. Thus

for all and all
So

for all and all
Hence the generalized Euler-Lagrange type additive mapping
is
-linear. The rest of the proof is the same as in the proof of Theorem 3.5.
The following theorem is an alternative result of Theorem 3.9.
Theorem 3.10.
Let be a mapping with
for which there is a function
satisfying (2.36), (2.37), (3.7) and

for and all
. Assume that
is invertible and for each fixed
the mapping
is continuous in
. Then the mapping
is a
-algebra homomorphism.
Remark 3.11.
In Theorem 3.10, one can assume that instead of
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Acknowledgments
The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper. C. Park author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).
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Najati, A., Park, C. Stability of a Generalized Euler-Lagrange Type Additive Mapping and Homomorphisms in -Algebras.
Adv Differ Equ 2009, 273165 (2009). https://doi.org/10.1155/2009/273165
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DOI: https://doi.org/10.1155/2009/273165
Keywords
- Banach Space
- Functional Equation
- Stability Problem
- Additive Mapping
- Alternative Result