C ∗ -Ternary Biderivations and C ∗ -Ternary Bihomomorphisms

: In this paper, we investigate C ∗ -ternary biderivations and C ∗ -ternary bihomomorphism in C ∗ -ternary algebras, associated with bi-additive s-functional inequalities.


Introduction and Preliminaries
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms.
The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gȃvruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.
A C * -ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z) → [x, y, z] of A 3 into A, which is C-linear in the outer variables, conjugate C-linear in the middle variable, and associative in the sense that [x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v], and satisfies [x, y, z] ≤ x · y · z and [x, x, x] = x 3 (see [21]).
If a C * -ternary algebra (A, [·, ·, ·]) has an identity, i.e., an element e ∈ A such that x = [x, e, e] = [e, e, x] for all x ∈ A, then it is routine to verify that A, endowed with x • y := [x, e, y] and x * := [e, x, e], is a unital C * -algebra. Conversely, if (A, •) is a unital C * -algebra, then [x, y, z] := x • y * • z makes A into a C * -ternary algebra.
In this paper, we prove the Hyers-Ulam stability of C * -ternary bihomomorphisms and C * -ternary bi-derivations in C * -ternary algebras.
This paper is organized as follows: In Sections 2 and 3, we correct and prove the results on C * -ternary bihomomorphisms and C * -ternary derivations in C * -ternary algebras, given in [24]. In Sections 4 and 5, we investigate C * -ternary biderivations and C * -ternary bihomomorphisms in C * -ternary algebras associated with the following bi-additive s-functional inequalities where s is a fixed nonzero complex number with |s| < 1. Throughout this paper, let X be a complex normed space and Y a complex Banach space. Assume that s is a fixed nonzero complex number with |s| < 1.

C * -Ternary Bihomomorphisms in C * -Ternary Algebras
In this section, we correct and prove the results on C * -ternary bihomomorphisms in C * -ternary algebras, given in [24].
Throughout this paper, assume that A and B are C * -ternary algebras.
We prove the Hyers-Ulam stability of C * -ternary bihomomorphisms in C * -ternary algebras.
Theorem 1. Let r < 2 and θ be nonnegative real numbers, and let f : for all λ, µ ∈ T 1 and all x, y, z, w ∈ A. Then there exists a unique C * -ternary bi-homomorphism H : for all x, z ∈ A.
Proof. By the same reasoning as in the proof of ([24] Theorem 2.3), there exists a unique C-bilinear mapping H : for all x, z ∈ A. It follows from (7) that for all x, y, z, w ∈ A, as desired.
Similarly, we can obtain the following.
Theorem 2. Let r > 6 and θ be nonnegative real numbers, and let f : A × A → B be a mapping satisfying f (0, 0) = 0, (6) and (7). Then there exists a unique C * -ternary bihomomorphism H : for all x, z ∈ A.
Proof. By the same reasoning as in the proof of ([24] Theorem 2.5), there exists a unique C-bilinear mapping H : A × A → B satisfying (9). The C-bilinear mapping H : It follows from (7)   for all x, y, z, w ∈ A, as desired.
Theorem 3. Let r < 1 2 and θ be nonnegative real numbers, and let f : ≤ θ · x r · y r · z r · w r for all λ, µ ∈ T 1 and all x, y, z, w ∈ A. Then there exists a unique C * -ternary bihomomorphism H : for all x, z ∈ A.
Proof. By the same reasoning as in the proof of ([24] Theorem 2.6), there exists a unique C-bilinear mapping H : A × A → B satisfying (12). The C-bilinear mapping H : for all x, z ∈ A. The rest of the proof is similar to the proof of Theorem 1.
Proof. By the same reasoning as in the proof of ([24] Theorem 2.7), there exists a unique C-bilinear mapping H : A × A → B satisfying (13). The C-bilinear mapping H : A × A → B is defined by The rest of the proof is similar to the proof of Theorem 1.

C * -Ternary Biderivations on C * -Ternary Algebras
In this section, we correct and prove the results on C * -ternary biderivations on C * -ternary algebras, given in [24].
Throughout this paper, assume that A is a C * -ternary algebra.
Theorem 5. Let r < 2 and θ be nonnegative real numbers, and let f : A × A → A be a mapping satisfying f (0, 0) = 0 and for all λ, µ ∈ T 1 and all x, y, z, w ∈ A. Then there exists a unique C * -ternary biderivation δ : for all x, z ∈ A.
Proof. By the same reasoning as in the proof of ([24] Theorems 2.3 and 3.1), there exists a unique C-bilinear mapping δ : A × A → A satisfying (16). The C-bilinear mapping δ : It follows from (15) that for all x, y, z, w ∈ A, as desired.
Similarly, we can obtain the following.
Proof. By the same reasoning as in the proof of ([24] Theorem 2.5), there exists a unique C-bilinear mapping δ : A × A → A satisfying (17). The C-bilinear mapping δ : It follows from (15) that for all x, y, z, w ∈ A, as desired.
Theorem 7. Let r < 1 2 and θ be nonnegative real numbers, and let f : A × A → A be a mapping satisfying f (0, 0) = 0 and D λ,µ f (x, y, z, w) ≤ θ · x r · y r · z r · w r , ≤ θ · x r · y r · z r · w r for all λ, µ ∈ T 1 and all x, y, z, w ∈ A. Then there exists a unique C * -ternary biderivation δ : for all x, z ∈ A.
Proof. By the same reasoning as in the proof of ([24] Theorem 2.6), there exists a unique C-bilinear mapping δ : A × A → A satisfying (20). The C-bilinear mapping δ : for all x, z ∈ A.
The rest of the proof is similar to the proof of Theorem 5.
Proof. By the same reasoning as in the proof of ([24] Theorem 2.7), there exists a unique C-bilinear mapping δ : A × A → A satisfying (21). The C-bilinear mapping δ : A × A → A is defined by δ(x, z) = lim n→∞ 4 n f x 2 n , z 2 n for all x, z ∈ A. The rest of the proof is similar to the proof of Theorem 5.
It follows from (36)