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*
n\ge 3
*and*
\phi :{X}^{n}\to [0,\mathrm{\infty})
*be a mapping such that there exists*
L<\frac{1}{{(n-m+1)}^{n-2}}
*such that*

\phi (\frac{{x}_{1}}{n-m+1},\dots ,\frac{{x}_{n}}{n-m+1})\le \frac{L\phi ({x}_{1},{x}_{2},\dots ,{x}_{n})}{n-m+1}

*for all* {x}_{1},\dots ,{x}_{n}\in X. *Let* f:X\to Y *with* f(0)=0 *be a mapping satisfying*

N(\mathrm{\Delta}f({x}_{1},\dots ,{x}_{n}),t)\ge \frac{t}{t+\phi ({x}_{1},\dots ,{x}_{n})}

(2.1)

*and*

N(f([abc])-[f(a)f(b)f(c)],t)\ge \frac{t}{t+\phi (a,b,c,0,\dots ,0)}

(2.2)

*for all* \mu \in {\mathbb{T}}_{\frac{1}{{n}_{o}}}^{1}, {x}_{1},\dots ,{x}_{n},a,b,c\in X *and* t>0. *Then there exists a unique ternary homomorphism* H:X\to Y *such that*

N(f(x)-H(x),t)\ge \frac{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t+L\phi (x,\dots ,x)}

(2.3)

*for all* x\in X *and* t>0.

*Proof* Letting \mu =1 and putting {x}_{1}={x}_{2}=\cdots ={x}_{n}=x in (2.1), we have

N(\left(\genfrac{}{}{0ex}{}{n}{m}\right)f((n-m+1)x)-\left(\genfrac{}{}{0ex}{}{n}{m}\right)(n-m+1)f(x),t)\ge \frac{t}{t+\phi (x,\dots ,x)}

(2.4)

for all x\in X and t>0. Set {S}_{0}:=\{h:X\to Y:h(0)=0\} and define a mapping {d}_{0}:{S}_{0}\times {S}_{0}\to [0,\mathrm{\infty}] by

{d}_{0}(f,g)=inf\{\mu \in {\mathbb{R}}^{+}:N(g(x)-h(x),\mu t)\ge \frac{t}{t+\phi (x,\dots ,x)},\phantom{\rule{0.25em}{0ex}}\mathrm{\forall}x\in X,t>0\},

where inf\mathrm{\varnothing}=+\mathrm{\infty}. Also, put S:=\{h\in {S}_{0}:{d}_{0}(h,f)<\mathrm{\infty}\}. Suppose that *d* is the restriction of {d}_{0} on S\times S. By using the same technique in the proof of Theorem 3.2 [50], we can show that (S,d) is a complete metric space. Now, we define a mapping J:S\to S by

Jg(x):=(n-m+1)g\left(\frac{x}{n-m+1}\right)

for all x\in X. It is easy to see that d(Jg,Jh)\le Ld(g,h) for all g,h\in S. This implies that

d(f,Jf)\le \frac{L}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)}.

Thus, by Banach’s fixed point theorem (Theorem 1.2), *J* has a unique fixed point H:X\to Y in *S* satisfying

H\left(\frac{x}{n-m+1}\right)=\frac{H(x)}{n-m+1}

(2.5)

for all x\in X. This implies that *H* is a unique mapping with (2.5) such that there exists \mu \in (0,\mathrm{\infty}) satisfying

N(f(x)-H(x),\mu t)\ge \frac{t}{t+\phi (x,\dots ,x)}

for all x\in X and t>0.

Moreover, we have d({J}^{p}f,H)\to 0 as p\to \mathrm{\infty}, which implies

N\text{-}\underset{p\to \mathrm{\infty}}{lim}\frac{f(\frac{x}{{(n-m+1)}^{p}})}{{(n-m+1)}^{-p}}=H(x)

(2.6)

for all x\in X. Thus it follows from (2.1) and (2.6) that

\underset{{k}_{l}\ne {i}_{j},\phantom{\rule{0.25em}{0ex}}\mathrm{\forall}j\in \{1,\dots ,m\}}{\underset{1\le {k}_{l}\le n}{\sum _{1\le {i}_{1}<\cdots <{i}_{m}\le n}}}H(\frac{{\sum}_{j=1}^{m}\mu {x}_{{i}_{j}}}{m}+\sum _{l=1}^{n-m}\mu {x}_{{k}_{l}})=\frac{(n-m+1)}{n}\left(\genfrac{}{}{0ex}{}{n}{m}\right)\sum _{i=1}^{n}\mu H({x}_{i})

for all \mu \in {\mathbb{T}}_{\frac{1}{{n}_{o}}}^{1} and {x}_{1},\dots ,{x}_{n}\in X. This means that H:X\to Y 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

N(\alpha ,\beta )\ge \frac{t}{t+\phi (\frac{a}{{(n-m+1)}^{p}},\frac{b}{{(n-m+1)}^{p}},\frac{c}{{(n-m+1)}^{p}},0,0,\dots ,0)}

for all a,b,c\in X and t>0, where

Then we have, as p\to +\mathrm{\infty},

for all a,b,c\in X and t>0. So, it follows that

N(H([abc])-[H(a)H(b)H(c)],t)=1

for all a,b,c\in X and t>0. Hence we have H([abc])=[H(a)H(b)H(c)] for all a,b,c\in X. This means that *H* is a ternary homomorphism. This completes the proof. □

**Theorem 2.2**
*Let*
\phi :{X}^{n}\to [0,\mathrm{\infty})
*be a mapping such that there exists*
L<1
*with*

\phi ({x}_{1},\dots ,{x}_{n})\le (n-m+1)L\phi (\frac{{x}_{1}}{n-m+1},\dots ,\frac{{x}_{n}}{n-m+1})

*for all* {x}_{1},{x}_{2},\dots ,{x}_{n}\in X. *Let* f:X\to Y *be a mapping with* f(0)=0 *satisfying* (2.1). *Then the limit* H(x):=N\text{-}{lim}_{p\to \mathrm{\infty}}\frac{f({(n-m+1)}^{p}x)}{{(n-m+1)}^{p}} *exists for all* x\in X *and* H:X\to Y *is defined as a ternary homomorphism such that*

N(f(x)-H(x),t)\ge \frac{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t+\phi (x,\dots ,x)}

(2.7)

*for all* x\in X *and* t>0.

*Proof* Let (S,d) be the metric space defined as in the proof of Theorem 2.1. Consider the mapping T:S\to S defined by Tg(x):=\frac{g((n-m+1)x)}{n-m+1} for all x\in X. One can show that d(g,h)=\u03f5 implies that d(Tg,Th)\le L\u03f5 for all positive real numbers *ϵ*. This means that *T* is a contraction on (S,d). The mapping

H(x):=N\text{-}\underset{p\to \mathrm{\infty}}{lim}\frac{f({(n-m+1)}^{p}x)}{{(n-m+1)}^{p}}

is the unique fixed point of *T* in *S* and *H* has the following property:

(n-m+1)H(x)=H((n-m+1)x)

(2.8)

for all x\in X. This implies that *H* is a unique mapping satisfying (2.8) such that there exists \mu \in (0,\mathrm{\infty}) satisfying N(f(x)-H(x),\mu t)\ge \frac{t}{t+\phi (x,\dots ,x)} for all x\in X and t>0.

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* \phi :{X}^{n}\to [0,\mathrm{\infty}) *be a function such that there exists* L<\frac{1}{{(n-m+1)}^{n-2}} *with*

\phi (\frac{{x}_{1}}{n-m+1},\dots ,\frac{{x}_{n}}{n-m+1})\le \frac{L\phi ({x}_{1},{x}_{2},\dots ,{x}_{n})}{n-m+1}

*for all* {x}_{1},\dots ,{x}_{n}\in X. *Let* f:X\to X *be a mapping with* f(0)=0 *satisfying* (2.1) *and*

N(f([abc])-[f(a)bc]-[af(b)c]-[abf(c)],t)\ge \frac{t}{t+\phi (a,b,c,0,0,\dots ,0)}

(2.9)

*for all* a,b,c\in X *and* t>0. *Then* D(x):=N\text{-}{lim}_{p\to \mathrm{\infty}}\frac{f(\frac{x}{{(n-m+1)}^{p}})}{{(n-m+1)}^{-p}} *exists for all* x\in X *and* D:X\to X *is defined as a unique ternary derivation such that*

N(f(x)-D(x),t)\ge \frac{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t+L\phi (x,\dots ,x)}

(2.10)

*for all* x\in X *and* t>0.

*Proof* By the same reasoning as that in the proof of Theorem 2.1, the mapping D:X\to X 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 a,b,c\in X and t>0. Then we have

for all a,b,c\in X and t>0. So, we have

N(D([abc])-[D(a)bc]-[aD(b)c]-[abH(c)],t)=1

for all a,b,c\in X and t>0. Hence we have D([abc])=[D(a)bc]+[aD(b)c]+[abD(c)] for all a,b,c\in X . This means that *D* is a ternary derivation. This completes the proof. □

**Theorem 2.4** *Let* *X* *be a fuzzy Banach algebra*. *Let* \phi :{X}^{n}\to [0,\mathrm{\infty}) *be a function such that there exists* L<1 *with*

\phi ({x}_{1},\dots ,{x}_{n})\le (n-m+1)L\phi (\frac{{x}_{1}}{n-m+1},\dots ,\frac{{x}_{n}}{n-m+1})

*for all* {x}_{1},{x}_{2},\dots ,{x}_{n}\in X. *Let* f:X\to X *be a mapping with* f(0)=0 *satisfying* (2.1) *and* (2.9). *Then the limit* D(x):=N\text{-}{lim}_{p\to \mathrm{\infty}}\frac{f({(n-m+1)}^{p}x)}{{(n-m+1)}^{p}} *exists for all* x\in X *and* D:X\to X *is defined as a ternary derivation such that*

N(f(x)-D(x),t)\ge \frac{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t+\phi (x,\dots ,x)}

(2.12)

*for all* x\in X *and* t>0.