# Superstability and Stability of the Pexiderized Multiplicative Functional Equation

- YoungWhan Lee
^{1}Email author

**2010**:486325

https://doi.org/10.1155/2010/486325

© YoungWhan Lee. 2010

**Received: **14 August 2009

**Accepted: **28 December 2009

**Published: **12 January 2010

## Abstract

## 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]).

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]).

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

Example 2.1.

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

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

Example 2.2.

In particular, we know that , and

Theorem 2.3.

Proof.

Corollary 2.4.

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

Theorem 2.5.

Proof.

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

Corollary 2.6.

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 Inequality (3.3) implies that

Example 3.1.

The following functions satisfy condition (3.3) above.

Example 3.2.

Theorem 3.3.

for all and for some constant .

Proof.

## Authors’ Affiliations

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