On Approximate Cubic Homomorphisms

Abstract

We investigate the generalized Hyers-Ulam-Rassias stability of the system of functional equations: , , on Banach algebras. Indeed we establish the superstability of this system by suitable control functions.

1. Introduction

A definition of stability in the case of homomorphisms between metric groups was suggested by a problem by Ulam [2] in 1940. Let be a group and let be a metric group with the 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 this case, the equation of homomorphism is called stable. On the other hand, we are looking for situations when the homomorphisms are stable, that is, if a mapping is an approximate homomorphism, then there exists an exact homomorphism near it. 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. In 1941, Hyers [3] gave a positive answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that

(1.1)

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

(1.2)

for all . Moreover, if is continuous in for each fixed , then the mapping is linear. Rassias [4] succeeded in extending the result of Hyers' theorem by weakening the condition for the Cauchy difference controlled by , to be unbounded. This condition has been assumed further till now, through the complete Hyers direct method, in order to prove linearity for generalized Hyers-Ulam stability problem forms. A number of mathematicians were attracted to the pertinent stability results of Rassias [4], and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Rassias is called Hyers-Ulam-Rassias stability. Then the stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem, see [5–13].

Bourgin [14] is the first mathematician dealing with stability of (ring) homomorphism . The topic of approximate homomorphisms was studied by a number of mathematicians, see [15–22] and references therein.

Jun and Kim [1] introduced the following functional equation:

(1.3)

and they established the general solution and generalized Hyers-Ulam-Rassias stability problem for this functional equation. It is easy to see that the function is a solution of the functional equation (1.3) Thus, it is natural that (1.3) is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic function.

Let be a ring. Then a mapping is called a cubic homomorphism if is a cubic function satisfying

(1.4)

for all For instance, let be commutative, then the mapping defined by is a cubic homomorphism. It is easy to see that a cubic homomorphism is a ring homomorphism if and only if it is zero function. In this paper, we study the stability of cubic homomorphisms on Banach algebras. Indeed, we investigate the generalized Hyers-Ulam-Rassias stability of the system of functional equations:

(1.5)

on Banach algebras. To this end, we need two control functions for our stability. One control function for (1.3) and an other control function for (1.4). So this is the main difference between our hypothesis (where two-degree freedom appears in the election for two control functions and in Theorem 2.1 in what follows), and the conditions (with one control function) that appear, for example, in [1, Theorem  3.1].

2. Main Results

In the following we suppose that is a normed algebra, is a Banach algebra, and is a mapping from into , and are maps from into . Also, we put for

Theorem 2.1.

Let

(2.1)

(2.2)

for all Assume that the series

(2.3)

converges, and that

(2.4)

for all . Then there exists a unique cubic homomorphism such that

(2.5)

for all .

Proof.

Setting in (2.2) yields

(2.6)

and then dividing by in (2.6), we obtain

(2.7)

for all . Now by induction we have

(2.8)

In order to show that the functions are a convergent sequence, we use the Cauchy convergence criterion. Indeed, replace by and divide by in (2.8), where is an arbitrary positive integer. We find that

(2.9)

for all positive integers . Hence by the Cauchy criterion, the limit exists for each . By taking the limit as in (2.8), we see that and (2.5) holds for all . If we replace by and by respectively, in (2.2) and divide by , we see that

(2.10)

Taking the limit as , we find that satisfies (1.3) [1, Theorem  3.1]. On the other hand we have

(2.11)

for all We find that satisfies (1.4). To prove the uniqueness property of , let be a function satisfing and Since are cubic, then we have

(2.12)

for all , hence,

(2.13)

By taking we get

Corollary 2.2.

Let and be nonnegative real numbers, and let . Suppose that

(2.14)

for all . Then there exists a unique cubic homomorphism such that

(2.15)

for all .

Proof.

In Theorem 2.1, let and for all

Corollary 2.3.

Let and be nonnegative real numbers. Suppose that

(2.16)

for all . Then there exists a unique cubic homomorphism such that

(2.17)

for all .

Proof.

The proof follows from Corollary 2.2.

Corollary 2.4.

Let and let be a positive real number. Suppose that

(2.18)

for all Moreover, suppose that

(2.19)

and that

(2.20)

for all Then is a cubic homomorphism.

Proof.

Letting in (2.20), we get that So by , in (2.20) we get for all By using induction we have

(2.21)

for all and On the other hand, by Theorem 2.1, the mapping defined by

(2.22)

is a cubic homomorphism. Therefore it follows from (2.21) that Hence it is a cubic homomorphism.

Corollary 2.5.

Let and . Let

(2.23)

for all Moreover, suppose that

(2.24)

and that

(2.25)

for all Then is a cubic homomorphism.

Proof.

If , then by Corollary 2.4 we get the result. If the following results from Theorem 2.1, by putting and for all

Corollary 2.6.

Let and be a positive real number. Let

(2.26)

for all Then is a cubic homomorphism.

Proof.

Let Then by Corollary 2.4, we get the result.

Theorem 2.7.

Let

(2.27)

(2.28)

for all . Assume that the series

(2.29)

converges and that

(2.30)

for all . Then there exists a unique cubic homomorphism such that

(2.31)

for all .

Proof.

Setting in (2.28) yields

(2.32)

Replacing by in (2.32), we get

(2.33)

for all . By (2.33) we use iterative methods and induction on to prove the following relation

(2.34)

In order to show that the functions are a convergent sequence, replace by in (2.34), and then multiply by , where is an arbitrary positive integer. We find that

(2.35)

for all positive integers. Hence by the Cauchy criterion the limit exists for each . By taking the limit as in (2.34), we see that and (2.31) holds for all . The rest of proof is similar to the proof of Theorem 2.1.

Corollary 2.8.

Let and be a positive real number. Let

(2.36)

for all Moreover, suppose that

(2.37)

(2.38)

for all Then is a cubic homomorphism.

Proof.

Letting in (2.38), we get that So by , in (2.38) we get for all By using induction, we have

(2.39)

for all and On the other hand, by Theorem  2.8, the mapping defined by

(2.40)

is a cubic homomorphism. Therefore, it follows from (2.39) that Hence is a cubic homomorphism.

Example 2.9.

Let

(2.41)

then is a Banach algebra equipped with the usual matrix-like operations and the following norm:

(2.42)

Let

(2.43)

and we define by and

(2.44)

for all Then we have

(2.45)

Thus the limit exists. Also,

(2.46)

Furthermore,

(2.47)

Hence is cubic homomorphism.

Also from this example, it is clear that the superstability of the system of functional equations

(2.48)

with the control functions in Corollaries 2.4, 2.5 and 2.6 does not hold.