- M. Eshaghi Gordji
^{1}Email author, - T. Karimi
^{2}and - S. Kaboli Gharetapeh
^{3}

**2009**:870843

https://doi.org/10.1155/2009/870843

© M. Eshaghi Gordji et al. 2009

**Received: **29 November 2008

**Accepted: **14 January 2009

**Published: **27 January 2009

## Abstract

Let , and let be two rings. An additive map is called -Jordan homomorphism if for all . In this paper, we establish the Hyers-Ulam-Rassias stability of -Jordan homomorphisms on Banach algebras. Also we show that (a) to each approximate 3-Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique 3-ring homomorphism near to , (b) to each approximate -Jordan homomorphism between two commutative Banach algebras there corresponds a unique -ring homomorphism near to for all .

## 1. Introduction and Preliminaries

Let
be two rings (algebras). An additive map
is called *n*-Jordan homomorphism (*n*-ring homomorphism) if
for all
for all
If
is a linear *n*-ring homomorphism, we say that
is *n*-homomorphism. The concept of *n*-homomorphisms was studied for complex algebras by Hejazian et al. [1] (see also [2, 3]). A 2-Jordan homomorphism is a Jordan homomorphism, in the usual sense, between rings. Every Jordan homomorphism is an *n*-Jordan homomorphism, for all
(e.g., [4, Lemma 6.3.2]), but the converse is false, in general. For instance, let
be an algebra over
and let
be a nonzero Jordan homomorphism on
. Then,
is a 3-Jordan homomorphism. It is easy to check that
is not 2-Jordan homomorphism or 4-Jordan homomorphism. The concept of *n*-Jordan homomorphisms was studied by the first author [5]. A classical question in the theory of functional equations is that "when is it true that a mapping which approximately satisfies a functional equation
must be somehow close to an exact solution of
?" Such a problem was formulated by Ulam [6] in 1940 and solved in the next year for the Cauchy functional equation by Hyers [7]. It gave rise to the *stability theory* for functional equations. Subsequently, various approaches to the problem have been introduced by several authors. For the history and various aspects of this theory we refer the reader to monographs [8–12]. Applying a theorem of Hyers [7], Rassias [13], and Gajda [14], Bourgin [15] proved the stability problem of ring homomorphisms between unital Banach algebras. Badora [16] proved the Hyers-Ulam-Rassias stability of ring homomorphisms, which generalizes the result of Bourgin. Recently, Miura et al. [17] proved the Hyers-Ulam-Rassias stability of Jordan homomorphisms. The stability problem of *n*-homomorphisms between Banach algebras, has been proved by the first author [18]. In this paper, we consider the stability, in the sense of Hyers-Ulam-Rassias, of *n*-Jordan homomorphisms on Banach algebras.

## 2. Main Result

By a following similar way as in [17], we obtain the next theorem.

Theorem 2.1.

Proof.

*n*-Jordan homomorphism. Since it follows from (2.2) that

for all
In other words,
is *n*-Jordan homomorphism. The uniqueness property of
follows from [13, 14].

Theorem 2.2.

*n*-Jordan homomorphism such that

Proof.

which proves , whenever This completes the proof.

By [17, Theorem 1.1] and [5, Theorem 2.5], we have the following theorem.

Theorem 2.3.

Let
be fixed. Suppose
is a Banach algebra, which needs not to be commutative, and suppose
is a semisimple commutative Banach algebra. Then, each *n*-Jordan homomorphism
is a *n*-ring homomorphism.

Let
be fixed. As a direct corollary, we show that to each approximate *n*-Jordan homomorphism
from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique *n*-ring homomorphism near to
.

Corollary 2.4.

Proof.

It follows from Theorems 2.1, 2.2, and 2.3.

Theorem 2.5.

Let
be fixed,
be two commutative algebras, and let
be a *n*-Jordan homomorphism. Then,
is *n*-ring homomorphism.

Proof.

Hence, is 5-ring homomorphism.

Corollary 2.6.

Proof.

It follows from Theorems 2.1, 2.2, and 2.5.

## Authors’ Affiliations

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## Copyright

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