On Approximate -Ternary -Homomorphisms: A Fixed Point Approach

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 .

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.

Let be a group and be a metric group with fthe metric . Given , does there exist a such that, if a mapping satisfies the inequality

(1.1)

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.

Let and be Banach spaces. Assume that satisfies

(1.2)

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

A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [8] in 1950 (see also [9]). In 1978, a generalized solution for approximately linear mappings was given by Th. M. Rassias [10]. He considered a mapping satisfying the condition

(1.3)

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].

On the other hand, J. M. Rassias [15–17] considered the Cauchy difference controlled by a product of different powers of norm. Furthermore, a generalization of Th. M. Rassias theorems was obtained by Găvruţa [18], who replaced

(1.4)

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]).

The functional equation:

(1.5)

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].

In 2002, Jun and Kim [44] introduced the following cubic functional equation:

(1.6)

and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.6). Obviously, the mapping satisfies the functional equation (1.6), which is called the cubic functional equation. In 2005, Lee et al. [45] considered the following functional equation

(1.7)

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

In 2007, Park and Cui [46] investigated the generalized stability of a quadratic mapping , which is called a -ternary quadratic mapping if is a quadratic mapping satisfies

(2.1)

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.

Recently, in 2010, Bae and Park [47] investigated the following functional equations

(2.2)

for each , and

(2.3)

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

We can rewrite the functional equations (2.2) and (2.3) by

(2.4)

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 homomorphismbriefly, -ternary 1-homomorphism if is an additive mapping satisfying (2.1) for all .

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

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

(4)A mapping is called a -ternary quartic homomorphismbriefly, -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.

Suppose that is a complete generalized metric space and is a strictly contractive mapping with the Lipschitz constant . Then, for any , either

(2.5)

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).

Throughout this section, we suppose that and are two -ternary algebras. For convenience, we use the following abbreviation: for any function ,

(3.1)

for all .

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

Theorem 3.1.

Let be a mapping for which there exist functions and such that

(3.2)

(3.3)

for all . If there exists a constant such that

(3.4)

for all , then there exists a unique -ternary -homomorphism such that

(3.5)

for all .

Proof.

It follows from (3.4) that

(3.6)

(3.7)

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

Let . We introduce a generalized metric on as follows:

(3.8)

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

Now, we consider the mapping defined by for all and . Note that, for all and ,

(3.9)

Hence we see that

(3.10)

for all , that is, is a strictly self-mapping of with the Lipschitz constant . Putting in (3.2), we have

(3.11)

for all and so

(3.12)

for all , that is, .

Now, from Theorem 2.2, it follows that there exists a fixed point of in such that

(3.13)

for all since .

On the other hand, it follows from (3.2), (3.6), and (3.13) that

(3.14)

for all and so . By the result in [44, 45, 47], is -mapping and so it follows from the definition of , (3.3) and (3.7) that

(3.15)

for all and so .

According to Theorem 2.2, since is the unique fixed point of in the set ,   is the unique mapping such that

(3.16)

for all and . Again, using Theorem 2.2, we have

(3.17)

and so

(3.18)

for all . This completes the proof.

Corollary 3.2.

Let be nonnegative real numbers with and . Suppose that is a mapping such that

(3.19)

(3.20)

for all . Then there exists a unique -ternary -homomorphism satisfying

(3.21)

for all .

Proof.

The proof follows from Theorem 3.1 by taking

(3.22)

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

Remark 3.3.

Let be a mapping with such that there exist functions and satisfying (3.2) and (3.3). Let be a constant such that

(3.23)

for all . By the similar method as in the proof of Theorem 3.1, one can show that there exists a unique -ternary -homomorphism satisfying

(3.24)

for all . For the case

(3.25)

where , are nonnegative real numbers and , and , there exists a unique -ternary -homomorphism satisfying

(3.26)

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.

Suppose that there exist functions , and a constant such that

(3.27)

for all . Moreover, if is a mapping such that

(3.28)

(3.29)

for all , then is a -ternary -homomorphism.

Proof.

It follows from (3.27) that

(3.30)

(3.31)

for all . We have since . Letting in (3.28), we get for all . By using induction, we obtain

(3.32)

for all and and so

(3.33)

for all and . It follows from (3.29) and (3.33) that

(3.34)

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

On the other hand, we have

(3.35)

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.

Let , , be nonnegative real numbers with and . If is a function such that

(3.36)

for all , then is a -ternary -homomorphism.

Remark 3.6.

Let be nonnegative real numbers with . Suppose that there exists a function and a constant such that

(3.37)

for all . Moreover, if is a mapping such that

(3.38)

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.

By Corollary 3.2, there exists a unique -ternary -homomorphism such that

(3.39)

for all . It follows from (3.39) that

(3.40)

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

Let and . It follows from (3.20) that

(3.41)

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.

Let be positive real numbers with and and . Suppose that is a mapping satisfying (3.19) and

(3.42)

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

Proof.

By Theorem 3.1 there exists a unique -ternary -homomorphism such that

(3.43)

for all . It follows from (3.43) that

(3.44)

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

Let and . It follows from (3.20) that

(3.45)

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.