In this section, by using the idea of Gavruta and Gavruta [14], we prove the generalized Hyers-Ulam-Rassias stability of ternary homomorphisms related to functional equation (1.1) on ternary fuzzy Banach algebras (see also [50]).
Theorem 2.1
Let
and
be a mapping such that there exists
such that
for all . Let with be a mapping satisfying
(2.1)
and
(2.2)
for all , and . Then there exists a unique ternary homomorphism such that
(2.3)
for all and .
Proof Letting and putting in (2.1), we have
(2.4)
for all and . Set and define a mapping by
where . Also, put . Suppose that d is the restriction of on . By using the same technique in the proof of Theorem 3.2 [50], we can show that is a complete metric space. Now, we define a mapping by
for all . It is easy to see that for all . This implies that
Thus, by Banach’s fixed point theorem (Theorem 1.2), J has a unique fixed point in S satisfying
(2.5)
for all . This implies that H is a unique mapping with (2.5) such that there exists satisfying
for all and .
Moreover, we have as , which implies
(2.6)
for all . Thus it follows from (2.1) and (2.6) that
for all and . This means that is additive. By using the same technique as in the proof of Theorem 2.1 [51], we can show that H is ℂ-linear. On the other hand, by (2.2), we have
for all and , where
Then we have, as ,
for all and . So, it follows that
for all and . Hence we have for all . This means that H is a ternary homomorphism. This completes the proof. □
Theorem 2.2
Let
be a mapping such that there exists
with
for all . Let be a mapping with satisfying (2.1). Then the limit exists for all and is defined as a ternary homomorphism such that
(2.7)
for all and .
Proof Let be the metric space defined as in the proof of Theorem 2.1. Consider the mapping defined by for all . One can show that implies that for all positive real numbers ϵ. This means that T is a contraction on . The mapping
is the unique fixed point of T in S and H has the following property:
(2.8)
for all . This implies that H is a unique mapping satisfying (2.8) such that there exists satisfying for all and .
The rest of the proof is similar to the proof of Theorem 2.1. This completes the proof. □
Now, we investigate the Hyers-Ulam-Rassias stability of ternary derivations in ternary fuzzy Banach algebras.
Theorem 2.3 Let X be a fuzzy Banach algebra. Let be a function such that there exists with
for all . Let be a mapping with satisfying (2.1) and
(2.9)
for all and . Then exists for all and is defined as a unique ternary derivation such that
(2.10)
for all and .
Proof By the same reasoning as that in the proof of Theorem 2.1, the mapping is a unique ℂ-linear mapping which satisfies (2.10).
Now, we show that D is a ternary derivation. By (2.9), we have
for all and . Then we have
for all and . So, we have
for all and . Hence we have for all . This means that D is a ternary derivation. This completes the proof. □
Theorem 2.4 Let X be a fuzzy Banach algebra. Let be a function such that there exists with
for all . Let be a mapping with satisfying (2.1) and (2.9). Then the limit exists for all and is defined as a ternary derivation such that
(2.12)
for all and .