- Research Article
- Open Access

# On Approximate -Ternary -Homomorphisms: A Fixed Point Approach

- MEshaghi Gordji
^{1, 2}, - Z Alizadeh
^{1, 2}, - YJ Cho
^{3}Email author and - H Khodaei
^{1, 2}

**2011**:454093

https://doi.org/10.1155/2011/454093

© M. Eshaghi Gordji et al. 2011

**Received:**21 November 2010**Accepted:**6 March 2011**Published:**14 March 2011

## Abstract

Using fixed point methods, we prove the stability and superstability of -ternary additive, quadratic, cubic, and quartic homomorphisms in -ternary rings for the functional equation , for each .

## Keywords

- Functional Equation
- Banach Algebra
- Generalize Stability
- Lipschitz Constant
- Unique Fixed Point

## 1. Introduction

Following the terminology of [1], a nonempty set
with a ternary operation
is called a *ternary groupoid*, which is denoted by
. The ternary groupoid
is said to be *commutative* if
for all
and all permutations
of
. If a binary operation
is defined on
such that
for all
, then we say that
is derived from
. We say that
is a *ternary semigroup* if the operation
is associative, that is, if
holds for all
(see [2]). Since it is extensively discussed in [3], the full description of a physical system
implies the knowledge of three basis ingredients: the set of the observables, the set of the states, and the dynamics that describes the time evolution of the system by means of the time dependence of the expectation value of a given observable on a given statue. Originally, the set of the observable was considered to be a
-algebra [4]. In many applications, however, it was shown not to be the most convenient choice and the
-algebra was replaced by a von Neumann algebra because the role of the representation turns out to be crucial mainly when long-range interactions are involved (see [5] and references therein). Here we used a different algebraic structure.

A
-*ternary ring* is a complex Banach space
, equipped with a ternary product
of
into
, which is
-linear in the outer variables, conjugate
-linear in the middle variable and associative in the sense that
and satisfies
and
.

If a -ternary ring has an identity, that is, an element such that for all , then it is routine to verify that , endowed with and , is a unital -algebra. Conversely, if is a unital -algebra, then makes into a -ternary algebra.

Consider the functional equation
in a certain general setting. A function
is an approximate solution of
if
and
are close in some sense. The Ulam stability problem asks whether or not there exists a true solution of
near
. A functional equation is said to be *superstable* if every approximate solution of the equation is an exact solution of the functional equation. The problem of stability of functional equations originated from a question of Ulam [6] concerning the stability of group homomorphisms.

for all , then there exists a homomorphism with for all ?

If the answer is affirmative, we say that the equation of homomorphism
is *stable*. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation?

In 1941, Hyers [7] gave a first affirmative answer to the question of Ulam for Banach spaces.

for all and some . Then there exists a unique additive mapping such that for all .

for all , where and . This result was later extended to all and generalized by Gajda [11], Th. M. Rassias and Šemrl [12], and Isac and Th. M. Rassias [13].

In 2000, Lee and Jun [14] have improved the stability problem for approximately additive mappings. The problem when is not true. Counter examples for the corresponding assertion in the case were constructed by Gadja [11], Th. M. Rassias and Šemrl [12].

and by a general control function . In 1949 and 1951, Bourgin [19, 20] is the first mathematician dealing with stability of (ring) homomorphism . The topic of approximation of functional equations on Banach algebras was studied by a number of mathematicians (see [21–33]).

is related to a symmetric biadditive mapping [34, 35]. It is natural that this equation is called a *quadratic functional equation*. For more details about various results concerning such problems, the readers refer to [36–43].

*cubic functional equation*. In 2005, Lee et al. [45] considered the following functional equation

It is easy to see that the mapping
is a solution of the functional equation (1.7), which is called the *quartic functional equation*.

## 2. Preliminaries

*ternary quadratic mapping*if is a quadratic mapping satisfies

for all . Let be a -ternary ring derived from a unital commutative -algebra and let satisfy for all . It is easy to show that the mapping is a -ternary quadratic mapping.

and they have obtained the stability of the functional equations (2.2) and (2.3).

Obviously, the monomial is a solution of the functional equation (2.4) for each .

For , Bae and Park [47, 48] showed that the functional equation (2.4) is equivalent to the additive equation and quadratic equation, respectively.

If , the functional equation (2.4) is equivalent to the cubic equation [44]. Moreover, Lee et al. [45] solved the solution of the functional equation (2.4) for .

In this paper, using the idea of Park and Cui [46], we study the further generalized stability of -ternary additive, quadratic, cubic, and quartic mappings over -ternary algebra via fixed point method for the functional equation (2.4). Moreover, we establish the superstability of this functional equation by suitable control functions.

Definition 2.1.

Let and be two -ternary algebras.

(1)A mapping
is called a
*-ternary additive homomorphism*
briefly,
-ternary 1-homomorphism
if
is an additive mapping satisfying (2.1) for all
.

(2)A mapping
is called a
*-ternary quadratic mapping*
briefly,
-ternary 2-homomorphism
if
is a quadratic mapping satisfying (2.1) for all
.

(3)A mapping
is called a
*-ternary cubic mapping*
briefly,
-ternary 3-homomorphism
if
is a cubic mapping satisfying (2.1) for all
.

(4)A mapping
is called a
*-ternary quartic homomorphism*
briefly,
-ternary 4-homomorphism
if
is a quartic mapping satisfying (2.1) for all
.

Now, we state the following notion of fixed point theorem. For the proof, refer to [49] (see also Chapter 5 in [50] and [51, 52]). In 2003, Radu [53] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also [54–57]).

Let
be a generalized metric space. We say that a mapping
satisfies a Lipschitz condition if there exists a constant
such that
for all
, where the number
is called the Lipschitz constant. If the Lipschitz constant
is less than 1, then the mapping
is called a *strictly contractive mapping*. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity.

The following theorem was proved by Diaz and Margolis [49] and Radu [53].

Theorem 2.2.

or there exists a natural number such that

(1) for all ;

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

(3) is the unique fixed point of in ;

(4) for all .

## 3. Approximation of -Ternary -Homomorphisms between -Ternary Algebras

In this section, we investigate the generalized stability of -ternary -homomorphism between -ternary algebras for the functional equation (2.4).

for all .

From now on, let be a positive integer less than 5.

Theorem 3.1.

for all .

Proof.

for all . By (3.6), and so . Letting in (3.2), we get and so .

It is easy to show that is a generalized complete metric space [55].

for all , that is, .

for all since .

for all and so .

for all . This completes the proof.

Corollary 3.2.

for all .

Proof.

for all . Then we can choose and so the desired conclusion follows.

Remark 3.3.

for all .

In the following, we formulate and prove a theorem in superstability of -ternary -homomorphism in -ternary rings for the functional equation (2.4).

Theorem 3.4.

for all , then is a -ternary -homomorphism.

Proof.

for all , and . Hence, letting in (3.34) and using (3.31), we have for all .

for all and . Thus, letting in (3.35) and using (3.30), we have for all . Therefore, is a -ternary -homomorphism. This completes the proof.

Corollary 3.5.

for all , then is a -ternary -homomorphism.

Remark 3.6.

for all , then is a -ternary -homomorphism.

In the rest of this section, assume that is a unital -ternary algebra with the unit and is a -ternary algebra with the unit .

Theorem 3.7.

Let , , be positive real numbers with , and . Suppose that is a mapping satisfying (3.19) and (3.20). If there exist a real number and such that , then the mapping is a -ternary -homomorphism.

Proof.

for all and . Therefore, by the assumption, we get that .

for all and so for all . Letting in the last equality, we get for all .

Similarly, one can show that for all when and . Therefore, the mapping is a -ternary -homomorphism. This completes the proof.

Theorem 3.8.

for all . If there exist a real number and such that , then the mapping is a -ternary -homomorphism.

Proof.

for all and . Therefore, by the assumption, we get that .

for all and so for all . Letting in the last equality, we get for all .

Similarly, one can show that for all when and . Therefore, the mapping is a -ternary -homomorphism. This completes the proof.

## Declarations

### Acknowledgment

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

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