On approximate cubic homomorphisms

In this paper, we investigate the generalized Hyers--Ulam--Rassias stability of the system of functional equations $$f(xy)=f(x)f(y), \qquad\qquad. f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x), $$ on Banach algebras. Indeed we establish the superstability of above system by suitable control functions.


Introduction
A definition of stability in the case of homomorphisms between metric groups was suggested by a problem by S. M. Ulam [21] in 1940. Let (G 1 , .) be a group and let (G 2 , * ) be a metric group with the metric d(., .). Given ǫ > 0, does there exist a δ > 0 such that if a mapping h : G 1 −→ G 2 satisfies the inequality d(h(x.y), h(x) * h(y)) < δ for all x, y ∈ G 1 , then there exists a homomorphism H : G 1 −→ G 2 with d(h(x), H(x)) < ǫ for all x ∈ G 1 ? In this case, the equation of homomorphism h(x.y) = h(x) * h(y) is called stable. In the other hand we are looking for situations when the homomorphisms are stable, i.e., 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, D. H. Hyers [8] gave a positive answer to the question of Ulam for Banach spaces. Let f : E 1 −→ E 2 be a mapping between Banach spaces such that for all x, y ∈ E 1 and for some δ ≥ 0. Then there exists a unique additive mapping T : then the mapping T is linear. Th. M. Rassias [20] succeeded in extending the result of Hyers' Theorem by weakening the condition for the Cauchy difference controlled by ( x p + y p ), p ∈ [0, 1) 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 Th. M. Rassias [20], and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Th. M. Rassias is called Hyers-Ulam-Rassias stability. And 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][6][7], [9], [12][13] and [16][17][18]). D.G. Bourgin [4] is the first mathematician dealing with stability of (ring) homomorphism f (xy) = f (x)f (y). The topic of approximate homomorphisms was studied by a number of mathematicians, see [2,3,10,14,15,19] and references therein. Jun and Kim [11] introduced the following functional equation 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 f (x) = cx 3 is a solution of the functional equation (1.1). Thus, it is natural that (1.1) is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic function. Let R be a ring. Then a mapping f : R −→ R is called a cubic homomorphism if f is a cubic function satisfying for all a, b ∈ R. For instance, let R be commutative, then the mapping f : R −→ R defined by f (a) = a 3 (a ∈ R), 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.

Main results
In the following we suppose that A is a normed algebra, B is a Banach algebra and f is a mapping from A into B, and ϕ, ϕ 1 , ϕ 2 are maps from A × A into R + . Also, we put 0 p = 0 for p ≤ 0.
for all x, y ∈ A. Assume that the series converges and that for all x, y ∈ A. Then there exists a unique cubic homomorphism T : for all x ∈ A.
and then dividing by 2 4 in (2.4) to obtain for all x ∈ A. Now by induction we have In order to show that the functions T n (x) = f (2 n x) 2 3n is a convergent sequence, we use the Cauchy convergence criterion. Indeed, replace x by 2 m x and divide by 2 3m in (2.6), where m is an arbitrary positive integer. We find that for all positive integers m,n. Hence by the Cauchy criterion the limit T (x) = lim n→∞ T n (x) exists for each x ∈ A. By taking the limit as n −→ ∞ in (2.3) holds for all x ∈ A. If we replace x by 2 n x and y by 2 n y respectively in (2.2) and divide by 2 3n , we see that Taking the limit as n −→ ∞, we find that T satisfies (1.1) (see Theorem 3.1 of [11]). On the other hand we have for all x, y ∈ A. We find that T satisfies (1.2). To prove the uniqueness property of T , letT : A → A be a functions satisfiesT (2x+y)+T (2x−y) = 2T (x + y) + 2T (x − y) + 12T (x) and T (x) − f (x) ≤ 1 16 Ψ(x, 0) . Since T,T are cubic, then we have for all x ∈ A, hence,

By taking n → ∞ we get, T (x) =T (x).
Corollary 2.2. Let θ 1 and θ 2 be nonnegative real numbers, and let p ∈ (−∞, 3). Suppose that for all x, y ∈ A. Then there exists a unique cubic homomorphism T : A −→ A such that Proof. In Theorem 2.1, let ϕ 1 (x, y) = θ 1 and ϕ 2 (x, y) = θ 2 ( x p + y p ) for all x, y ∈ A. Corollary 2.3. Let θ 1 and θ 2 be nonnegative real numbers. Suppose that for all x, y ∈ A. Then there exists a unique cubic homomorphism T : for all x ∈ A.
Proof. It follows from Corollary 2.2. for all x, y ∈ A. Moreover, Suppose that and that for all x, y ∈ A. Then f is a cubic homomorphism.
Proof. Letting x = y = 0 in (2.7), we get that f (0) = 0. So by y = 0, in (2.7), we get f (2x) = 2 3 f (x) for all x ∈ A. By using induction we have for all x ∈ A and n ∈ N. On the other hand by Theorem 2.1, the mapping T : A → A defined by , is a cubic homomorphism. Therefore it follows from (2.8) that f = T. Hence it is a cubic homomorphism.
Corollary 2.5. Let p, q, θ ≥ 0 and p + q < 3. Let lim n→∞ ϕ(2 n x, 2 n y) for all x, y ∈ A. Moreover, Suppose that and that for all x, y ∈ A. Then f is a cubic homomorphism.
Proof. If q = 0, then by Corollary 2.4 we get the result. If q = 0, it follows from Theorem 2.1, by putting ϕ 1 (x, y) = ϕ(x, y) and ϕ 2 (x, y) = θ( x p y p ) for all x, y ∈ A. for all x, y ∈ A. Moreover, suppose that for all x, y ∈ A. Then f is a cubic homomorphism.
Proof. Let ϕ(x, y) = θ y p . Then by Corollary 2.4, we get the result.
for all x, y ∈ A. Assume that the series converges and that lim n−→∞ 2 6n ϕ 1 ( x 2 n , y 2 n ) = 0, for all x, y ∈ A. Then there exists a unique cubic homomorphism T :

10)
for all x ∈ A.
Proof. Setting y = 0 in (2.9) yields Replacing x by x 2 in (2.11) to get for all x ∈ A. By (2.12) we use iterative methods and induction on n to prove our next relation In order to show that the functions T n (x) = 2 3n f ( x 2 n ) is a convergent sequence, replace x by x 2 m in (2.13), and then multiplying by 2 3m , where m is an arbitrary positive integer. We find that for all positive integers. Hence by the Cauchy criterion the limit T (x) = lim n−→∞ T n (x) exists for each x ∈ A. By taking the limit as n −→ ∞ in (2.13), we see that for all x, y ∈ A. Then f is a cubic homomorphism.
Proof. Letting x = y = 0 in (2.14), we get that f (0) = 0. So by y = 0, in (2.14), we get f (2x) = 2 3 f (x) for all x ∈ A. By using induction we have for all x ∈ A and n ∈ N. On the other hand by Theorem 2.8, the mapping is a cubic homomorphism. Therefore it follows from (2.15) that f = T.
Hence f a cubic homomorphism.

Example. Let
then A is a Banach algebra equipped with the usual matrix-like operations and the following norm:
Hence T is cubic homomorphism. Also from this example it is clear that the superstability of the system of functional equations with the control functions in Corollaries 2.4, 2.5 and 2.6 does not hold.