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# Superstability and Stability of the Pexiderized Multiplicative Functional Equation

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 486325 (2010)

## Abstract

We obtain the superstability of the Pexiderized multiplicative functional equation and investigate the stability of this equation in the following form: .

## 1. Introduction

The superstability of the functional equation was studied by Baker et al. [1]. They proved that if is a functional on a real vector space satisfying for some fixed and all , then is either bounded or else for all . This result was genealized with a simplified proof by Baker [2] as follows.

Theorem 1.1 (Baker [2]).

Let , be a semigroup and satisfying

for all . Put Then for all or else for all .

A different generalization of the result of Baker et al. was given by Székelyhidi [3]. It involves an interesting generalization of the class of bounded functions on a group or semigroup and may be stated as follows.

Theorem 1.2 (Székelyhidi [3]).

Let be a commutative group with identity and let be functions such that there exist functions with

for all . Then is bounded or is an exponential and .

In this paper, we prove the superstability of the Pexiderized multiplicative functional equation (PMFE)

That is, we prove that if are functional on a semigroup with identity satisfying and

for all and for a function with some coditions, then is bounded or else is an exponential and . This is a generalization of the result of Székelyhidi. Also we investigate the stability of the Pexiderized multipicative functional equation (1.3) in the sense of Ger [4].

## 2. Superstability of the PMFE

In this section, let be a semigroup with identity and a function with

for all and

for all

Example 2.1.

The following functions satisfy conditions (2.1) and (2.2) above.

(a) for every and

(b), for every and is a functional on .

(c), for every .

(d), for every .

Example 2.2.

Let and also ,

for all and for some . Let . Then satisfy the conditions (1.4), (2.1), (2.2) and

In particular, we know that , and

Theorem 2.3.

Let be a semigroup with identity . If are functions with for some satisfying and condition (1.4), that is,

then

for all and .

Proof.

If we replace by and also by in (1.4), we get

Also we replace by in (1.4), then we have

for all . An induction argument implies that for all ,

Indeed, if inequality (2.9) holds, using inequality (1.4) and (2.8) we have

for all . By (2.9), we have

Thus we can easily show that from as and thus as . By (1.4),

and thus we have

for all . Then, by (2.2),

and so

for all . Thus we have as . Since

as , we can define by

for all . Then

for all .

Corollary 2.4.

Let be a semigroup with identity and functions satisfying the inequality

for all . If , then is bounded or else is exponential and

Theorem 2.5.

Let be a semigroup with identity and functions satisfying condition (1.4), that is,

If satisfies that for some and for some then

for all and .

Proof.

Let and for every and . Then

for all , and where . By Theorem 2.3, we complete the proof.

Corollary 2.6.

Let be a semigroup with identity . If are nonzero functions satisfying condition (1.4), that is,

then either is bounded, or else

for all and .

Proof.

Let for some . If is unbounded, then there exists such that . By Theorem 2.5, we complete the proof.

## 3. Stability of the PMFE

In 1940, Ulam gave a wide-ranging talk in the Mathematical Club of the University of Wisconsin in which he discussed a number of important unsolved problems [5]. One of those was the question concerning the stability of homomorphisms.

*Let*
*be a group and let*
*be a metric group with a metric*
*. Given*
*, does there exist a*
*such that if a mapping*
*satisfies the inequality*
*for all*
*, then there exists a homomorphism*
*with*
*for all*
*?*

In the next year, Hyers [6] answered the Ulam's question for the case of the additive mapping on the Banach spaces . Thereafter, the result of Hyers has been generalized by Rassias [7]. Since then, the stability problems of various functional equations have been investigated by many authors (see [6, 8–18]).

Ger [4] suggested another type of stability for the exponential equation in the following type:

In this section, the stability problem for the Pexiderized multiplicative functional equation in the following form:

will be investigated.

Throughout this section, we denote by a commutative semigroup and by a function such that

for all . Also we let

for all Inequality (3.3) implies that

(a)for all

(b)for all

(c)for all

(d)for all for

because

Example 3.1.

The following functions satisfy condition (3.3) above.

(a) for every and

(b), for every .

Example 3.2.

Let and also ,

for all and for some . Let . Then satisfy condition (3.3) and

In particular, we know that if we let then

Theorem 3.3.

If are functions such that

for all , then there exists a function and there exists a constant such that for all and

for all . Moreover, if is bounded, then

for all and for some constant .

Proof.

If we define functions by

for all , then equality (3.13) may be transformed into

and thus

for all . For the case of , the above inequality implies

and so

for all . Putting instead of and instead of in (3.20), respectively, we get

Letting by and by in (3.20), we have

and also

From (3.21), (3.22) and (3.23),

for all . Now replacing by and by , respectively, we have

for all . Replacing by and by in (3.21), (3.22), and (3.23), respectively, one obtains

for all . Also from (3.22) and (3.23), we have

for all . Thus we have

for all . For arbitrary positive integer , putting instead of in (3.24) and instead of in (3.28), respectively, we see that

for all . By (3.29) with ,

for all . By (3.30), for every positive integer with , we have

as . This proves that is a Cauchy sequence in . Thus we can define a function by

for all . Then, by (3.20) and (3.31), we have

for all . Thus

for all . Now replacing by and then by in (3.20), respectively, we obtain

and so

for all . By (3.22), (3.23), and (3.36), we have

and thus

for all and for fixed . By (3.3) and (3.30),

for all . By (3.20), (3.38) and (3.39), for all with , there exits a constant such that

Now we define a function by

for all . Then

for all . By (3.40), we have

and thus for all

If is bounded, there exist constants such that

and so

and by the same method above, we have

for all . Therefore, we have

for all .

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Lee, Y. Superstability and Stability of the Pexiderized Multiplicative Functional Equation.
*J Inequal Appl* **2010, **486325 (2010). https://doi.org/10.1155/2010/486325

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### Keywords

- Banach Space
- Positive Integer
- Vector Space
- Commutative Group
- Functional Equation