# On the Stability of Quadratic Double Centralizers and Quadratic Multipliers: A Fixed Point Approach

- Abasalt Bodaghi
^{1}Email author, - Idham Arif Alias
^{2}and - MadjidEshaghi Gordji
^{3}

**2011**:957541

https://doi.org/10.1155/2011/957541

© Abasalt Bodaghi et al. 2011

**Received: **3 December 2010

**Accepted: **18 January 2011

**Published: **8 February 2011

## Abstract

We prove the superstability of quadratic double centralizers and of quadratic multipliers on Banach algebras by fixed point methods. These results show that we can remove the conditions of being weakly commutative and weakly without order which are used in the work of M. E. Gordji et al. (2011) for Banach algebras.

## Keywords

## 1. Introduction

*under what condition does there exist an additive mapping near an approximately additive mapping?*Hyers [2] answered the problem of Ulam for Banach spaces. He showed that for two Banach spaces and , if and such that

Găvruţa then generalized the Rassias's result in [4].

is called quadratic functional equation. In addition, every solution of functional eqaution (1.6) is said to be a *quadratic mapping*. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings
, where
is a normed space and
is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an abelian group. Indeed, Czerwik in [7] proved the Cauchy-Rassias stability of the quadratic functional equation. Since then, the stability problems of various functional equation have been extensively investigated by a number of authors (e.g, [8–13]).

One should remember that the functional equation is called *stable* if any approximately solution to the functional equation is near to a true solution of that functional equation, and is super*superstable* if every approximately solution is an exact solution of it (see [14]). Recently, the first and third authors in [15] investigated the stability of quadratic double centralizer: the maps which are quadratic and double centralizer. Later, Eshaghi Gordji et al. introduced a new concept of the quadratic double centralizer and the quadratic multipliers in [16], and established the stability of quadratic double centralizer and quadratic multipliers on Banach algebras. They also established the superstability for those which are weakly commutative and weakly without order. In this paper, we show that the hypothesis on Banach algebras being weakly commutative and weakly without order in [16] can be eliminated, and prove the superstability of quadratic double centralizers and quadratic multipliers on a Banach algebra by a method of fixed point.

## 2. Stability of Quadratic Double Centralizers

A linear mapping
is said to be *left centralizer* on
if
, for all
. Similarly, a linear mapping
satisfying
, for all
is called *right centralizer* on
. A *double centralizer* on
is a pair
, where
is a left centralizer,
is a right centralizer and
, for all
. An operator
is said to be a *multiplier* if
, for all
.

Throughout this paper, let be a complex Banach algebra. Recall that a mapping is a quadratic left centralizer if is a quadratic homogeneous mapping, that is is quadratic and , for all and , and , for all . A mapping is a quadratic right centralizer if is a quadratic homogeneous mapping and , for all . Also, a quadratic double centralizer of an algebra is a pair where is a quadratic left centralizer, is a quadratic right centralizer and , for all (see [16] for details).

Before proceeding to the main results, we will state the following theorem which is useful to our purpose.

Theorem 2.1 (The alternative of fixed point [17]).

Suppose that we are given a complete generalized metric space and a strictly contractive mapping with Lipschitz constant . Then for each given , either , for all , or else exists a natural number such that

(2)the sequence is convergent to a fixed point of ,

(3) is the unique fixed point of in the set ,

Theorem 2.2.

Proof.

for all . Hence is a quadratic mapping.

Passing to the limit as , and again from (2.7), we conclude that , for all . Therefore is a quadratic double centralizer on . This completes the proof of this theorem.

Now, we establish the superstability of double quadratic centralizers on Banach algebras as follows

Corollary 2.3.

for all and, for all . Then is a double quadratic centralizer on .

Proof.

## 3. Stability of Quadratic Multipliers

Assume that
is a complex Banach algebra. Recall that a mapping
is a *quadratic multiplier* if
is a quadratic homogeneous mapping, and
, for all
(see [16]). We investigate the stability of quadratic multipliers.

Theorem 3.1.

Proof.

Passing to the limit as , and from (3.4) we conclude that , for all .

Using Theorem 3.1, we establish the superstability of quadratic multipliers on Banach algebras.

Corollary 3.2.

for all and, for all . Then is a quadratic multiplier on .

Proof.

## Authors’ Affiliations

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