Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 734025, 7 pages
http://dx.doi.org/10.1155/2013/734025
Letter to the Editor
Approximately Ternary Homomorphisms on
∗
-Ternary Algebras
Eon Wha Shim,
1
Su Min Kwon,
1
Yun Tark Hyen,
1
Yong Hun Choi,
1
and Abasalt Bodaghi
2
1
Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea
2
Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran
Correspondence should be addressed to Abasalt Bodaghi; abasalt.bodaghi@gmail.com
Received 3 April 2013; Accepted 5 June 2013
Academic Editor: Josip E. Pecaric
Copyright © 2013 Eon Wha Shim et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Gordji et al. established the Hyers-Ulam stability and the superstability of
∗
-ternary homomorphisms and
∗
-ternary derivations
on
∗
-ternary algebras, associated with the following functional equation: ((
2
−
1
)/3)+((
1
−3
3
)/3)+((3
1
+3
3
−
2
)/3) =
(
1
), by the direct method. Under the conditions in the main theorems, we can show that the related mappings must be zero. In
this paper, we correct the conditions and prove the corrected theorems. Furthermore, we prove the Hyers-Ulam stability and the
superstability of
∗
-ternary homomorphisms and
∗
-ternary derivations on
∗
-ternary algebras by using a fxed point approach.
1. Introduction
A
∗
-ternary algebra is a complex Banach space , equipped
with a ternary product (,,) [,,] of
3
into
, which is C-linear in the outer variables, conjugate C-
linear in the middle variable, and associative in the sense
that [,,[,, V]] = [[,,], V] = [[,,],, V], and
satisfes ‖[,,]‖ ≤ ‖‖ ⋅ ‖‖ ⋅ ‖‖ and ‖[,,]‖ = ‖‖
3
.
If a
∗
-ternary algebra ([⋅,⋅,⋅]) 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. A C-linear mapping :
→ between
∗
-ternary algebras is called a
∗
-ternary
homomorphism if
([,,])=[ (), (), ()] (1)
for all ,,∈.A C-linear mapping :→ is called a
∗
-ternary derivation if
([,,])=[ (),,]+[, (),]
+[,,()] (,,∈).
(2)
Ternary structures and their generalization, the so-called
-ary structures, raise certain hopes in view of their applica-
tions in physics (see [1–4]).
Te stability problem of functional equations is originated
from the following question of Ulam [5]: under what con-
dition does there exist an additive mapping near an approxi-
mately additive mapping? In 1941, Hyers [6] gave a partial
afrmative answer to the question of Ulam in the context
of Banach spaces. In 1978, Rassias [7] extended the theorem
of Hyers by considering the unbounded Cauchy diference
‖(+)−()−()‖≤(‖‖
+‖‖
), (>0,∈[0,1)).
Te 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 [8–12]).
Gordji et al. [13] proved the Hyers-Ulam stability and
the superstability of
∗
-ternary homomorphisms and
∗
-
ternary derivations on
∗
-ternary algebras, associated with
the functional equation
(
2
−
1
3
)+(
1
−3
3
3
)+(
3
1
+3
3
−
2
3
)
=(
1
)
(3)