Generalized Stability of C ∗-Ternary Quadratic Mappings

We prove the generalized stability of C*-ternary quadratic mappings in C*-ternary rings for the quadratic functional equation f(x


Introduction and preliminaries
A C * -ternary ring 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 [1]).
If a C * -ternary ring (A,[•,•,•]) has an identity, that is, 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 [2]).
Ulam [3] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems, containing the stability problem of homomorphisms.Hyers [4] proved the stability problem of additive mappings in Banach spaces.Rassias [5] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded: let f : E → E be a mapping from a normed vector space E into a Banach space E subject to the inequality for all x, y ∈ E, where and p are constants with > 0 and p < 1. Inequality (1.1) provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept.Gȃvrut ¸a [6] provided a further generalization of Hyers-Ulam theorem (see [7,8]).
A square norm on an inner product space satisfies the important parallelogram equality (1. 2) The functional equation is called the quadratic functional equation whose solution is said to be a quadratic mapping.A generalized stability problem for the quadratic functional equation was proved by Skof [9] for mappings f : , where E 1 is a normed space and E 2 is a Banach space.Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain E 1 is replaced by an Abelian group.Czerwik [11] proved the generalized stability of the quadratic functional equation, and Park [12] proved the generalized stability of the quadratic functional equation in Banach modules over a C * -algebra.Jun and Lee [13] proved the further generalized stability of a Pexiderized quadratic functional equation Recently, a fixed point approach to the stability of Pexiderized quadratic equation was established by Mirzavaziri and Moslehian [14].Throughout this paper, assume that A is a C * -ternary ring with norm • A and that B is a C * -ternary ring with norm • B .
A quadratic mapping Q : ) be a C * -ternary ring derived from a unital commutative C * -algebra A, and let Q : In this paper, we prove the further generalized stability of C * -ternary quadratic mappings in C * -ternary rings.

Stability of C * -ternary quadratic mappings
We prove the further generalized stability of C * -ternary quadratic mappings in C *ternary rings for the quadratic functional equation C. Park and J. Cui 3 Theorem 2.1.Let f : A → B be a mapping for which there exists a function ϕ : for all x, y,z ∈ A. Then there exists a unique C * -ternary quadratic mapping ) for all x, y,z ∈ A.
Proof.If follows from (2.3) that f (0) = 0. Letting y = x in (2.3), we get for all x ∈ A. Hence, for all nonnegative integers m and l with m > l and all x ∈ A. It follows from (2.9) that the sequence for all x ∈ A. Moreover, letting l = 0 and passing the limit m → ∞ in (2.9), we get (2.5).

Abstract and Applied Analysis
It follows from (2.3) that for all x, y ∈ A.
It follows from (2.4) and the continuity of the ternary product that for all x, y,z ∈ A. Now, let T : A → B be another quadratic mapping satisfying (2.5).Then we have which tends to zero as n → ∞ for all x ∈ A. So we can conclude that Q(x) = T(x) for all x ∈ A. This proves the uniqueness of Q.Thus, the mapping Q : A → B is a unique C * -ternary quadratic mapping satisfying (2.5).
Theorem 2.2.Let f : A → B be a mapping for which there exists a function ϕ : for all x, y,z ∈ A. Then there exists a unique C * -ternary quadratic mapping Q : A → B such that The rest of the proof is similar to the proof of Theorem 2.1.
for all nonnegative integers m and l with m > l and all x ∈ A. It follows from(2.19)that the sequence{(1/4 n ) f (2 n x)} is a Cauchy sequence for all x ∈ A. Since B is complete, the sequence {(1/4 n ) f (2 n x)} converges.So one can define the mapping Q : A → B by Q(x) := lim n→∞ 1 4 n f 2 n x (2.20)for all x ∈ A. Moreover, letting l = 0 and passing the limit m → ∞ in (2.19), we get (2.17).It follows from (2.4) and the continuity of the ternary product thatQ [x, y,z] − Q(x),Q(y),Q(z) B = lim n→∞ 1 4 3n f 2 3n [x, y,z] − f 2 n x , f 2 n y , f 2 n z B Remark 2.3.For a Pexiderized quadratic functional equation f (x + y) + g(x − y) = 2h(x) + 2k(y),(2.23)onecan obtain similar results to Theorems 2.1 and 2.2.