Stability of a Generalized Euler-Lagrange Type Additive Mapping and Homomorphisms in -Algebras

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

(1.1)

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

(1.2)

exists for all and is the unique additive mapping which satisfies

(1.3)

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

(1.4)

for all . Then there exists a unique additive mapping satisfying

(1.5)

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

(1.6)

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

(1.7)

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:

(1.8)

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):

(1.9)

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

(1.10)

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

(2.1)

for all Letting in (2.1), we get

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

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

(2.6)

(2.7)

(2.8)

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

(2.9)

for all Moreover, for all and all

Proof.

For each with let in (2.8), tthen we get the following inequality

(2.10)

for all For convenience, set

(2.11)

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

(2.12)

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

(2.13)

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

(2.14)

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

(2.15)

for all Putting in (2.15), we get

(2.16)

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

(2.17)

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

(2.18)

for all where

(2.19)

It follows from (2.6) that

(2.20)

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

(2.21)

for all and all Therefore, we have

(2.22)

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

(2.23)

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

(2.24)

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

(2.25)

So for all

Theorem 2.4.

Let be a mapping satisfying for which there is a function satisfying (2.6), (2.7) and

(2.26)

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

(2.27)

for all . So

(2.28)

for all and all Since for all and

(2.29)

for all and all

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

(2.30)

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

(2.31)

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

(2.32)

for all where

(2.33)

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

(2.34)

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

(2.35)

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

(2.36)

(2.37)

(2.38)

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

(2.39)

for all Moreover, for all and all

Proof.

By a similar method to the proof of Theorem 2.3, we have the following inequality

(2.40)

for all where

(2.41)

It follows from (2.36) that

(2.42)

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

(2.43)

for all and all Therefore, we have

(2.44)

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

(2.45)

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

(2.46)

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

(2.47)

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

(2.48)

for all where

(2.49)

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

(2.50)

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

(2.51)

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

(2.52)

Consider the function by the formula

(2.53)

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

(2.54)

for all and all If or then

(2.55)

Now suppose that Then there exists a nonnegative integer such that

(2.56)

Therefore

(2.57)

Hence

(2.58)

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

(2.59)

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

(2.60)

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

(2.61)

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

(2.62)

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

(3.1)

(3.2)

(3.3)

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

(3.4)

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

(3.5)

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

(3.6)

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

(3.7)

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

(3.8)

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

(3.9)

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

(3.10)

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

(3.11)

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

(3.12)

for all Since is invertible and

(3.13)

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

(3.14)

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

(3.15)

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

(3.16)

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

(3.17)

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

(3.18)

for all So

(3.19)

for all Since for all and

(3.20)

for and for all

For each element we have where . Thus

(3.21)

for all and all So

(3.22)

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

(3.23)

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