Theory and Modern Applications

# Fuzzy ∗-homomorphisms and fuzzy ∗-derivations in induced fuzzy ${C}^{\ast }$-algebras

## Abstract

In this paper, we prove the Ulam-Hyers-Rassias stability of the Cauchy-Jensen additive functional equation

$f\left(\frac{x+y+z}{2}\right)+f\left(\frac{x-y+z}{2}\right)=f\left(x\right)+f\left(z\right)$

in fuzzy Banach spaces.

MSC:39B52, 46S40, 26E50, 46L05, 39B72.

## 1 Introduction

The stability problem of functional equations originated from the question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Th.M. Rassias [3] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (Rassias [3])

Let $f:E\to {E}^{\prime }$ be a mapping from a normed vector space E into a Banach space ${E}^{\prime }$ subject to the inequality $\parallel f\left(x+y\right)-f\left(x\right)-f\left(y\right)\parallel \le ϵ\left({\parallel x\parallel }^{p}+{\parallel y\parallel }^{p}\right)$ for all $x,y\in E$, where ϵ and p are constants with $ϵ>0$ and $0\le p<1$. Then the limit $L\left(x\right)={lim}_{n\to \mathrm{\infty }}\frac{f\left({2}^{n}x\right)}{{2}^{n}}$ exists for all $x\in E$ and $L:E\to {E}^{\prime }$ is the unique additive mapping which satisfies

$\parallel f\left(x\right)-L\left(x\right)\parallel \le \frac{2ϵ}{2-{2}^{p}}{\parallel x\parallel }^{p}$

for all $x\in E$. Also, if for each $x\in E$ the function $f\left(tx\right)$ is continuous in $t\in \mathbb{R}$, then L is linear.

The functional equation $f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$ is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The Ulam-Hyers-Rassias stability of the quadratic functional equation was proved by Skof [4] for mappings $f:X\to Y$, where X is a normed space and Y is a Banach space. Cholewa [5] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [6] proved the Ulam-Hyers-Rassias stability of the quadratic functional equation.

The 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 [721]).

Katsaras [22] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view (see [13, 23, 24]).

In particular, Bag and Samanta [25], following Cheng and Mordeson [26], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [27]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [28].

In this paper we consider a mapping $f:X\to Y$ satisfying the following Cauchy-Jensen functional equation

$f\left(\frac{x+y+z}{2}\right)+f\left(\frac{x-y+z}{2}\right)=f\left(x\right)+f\left(z\right)$
(1.1)

for all $x,y,z\in X$ and establish the fuzzy -homomorphisms and fuzzy -derivations of (1.1) in induced fuzzy ${C}^{\ast }$-algebras.

## 2 Preliminaries

Definition 2.1 Let X be a real vector space. A function $N:X×\mathbb{R}\to \left[0,1\right]$ is called a fuzzy norm on X if for all $x,y\in X$ and all $s,t\in \mathbb{R}$,

(N 1) $N\left(x,t\right)=0$ for $t\le 0$;

(N 2) $x=0$ if and only if $N\left(x,t\right)=1$ for all $t>0$;

(N 3) $N\left(cx,t\right)=N\left(x,\frac{t}{|c|}\right)$ if $c\ne 0$;

(N 4) $N\left(x+y,c+t\right)\ge min\left\{N\left(x,s\right),N\left(y,t\right)\right\}$;

(N 5) $N\left(x,\cdot \right)$ is a non-decreasing function of $\mathbb{R}$ and ${lim}_{t\to \mathrm{\infty }}N\left(x,t\right)=1$;

(N 6) for $x\ne 0$, $N\left(x,\cdot \right)$ is continuous on $\mathbb{R}$.

Example 2.1 Let $\left(X,\parallel \cdot \parallel \right)$ be a normed linear space and $\alpha ,\beta >0$. Then

$N\left(x,t\right)=\left\{\begin{array}{cc}\frac{\alpha t}{\alpha t+\beta \parallel x\parallel },\hfill & t>0,x\in X,\hfill \\ 0,\hfill & t\le 0,x\in X\hfill \end{array}$

is a fuzzy norm on X.

Definition 2.2 Let $\left(X,N\right)$ be a fuzzy normed vector space. A sequence $\left\{{x}_{n}\right\}$ in X is said to be convergent or converge if there exists an $x\in X$ such that ${lim}_{t\to \mathrm{\infty }}N\left({x}_{n}-x,t\right)=1$ for all $t>0$. In this case, x is called the limit of the sequence $\left\{{x}_{n}\right\}$ in X and we denote it by $N\text{-}{lim}_{t\to \mathrm{\infty }}{x}_{n}=x$.

Definition 2.3 Let $\left(X,N\right)$ be a fuzzy normed vector space. A sequence $\left\{{x}_{n}\right\}$ in X is called Cauchy if for each $ϵ>0$ and each $t>0$ there exists an ${n}_{0}\in \mathbb{N}$ such that for all $n\ge {n}_{0}$ and all $p>0$, we have $N\left({x}_{n+p}-{x}_{n},t\right)>1-ϵ$.

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping $f:X\to Y$ between fuzzy normed vector spaces X and Y is continuous at a point $x\in X$ if for each sequence $\left\{{x}_{n}\right\}$ converging to ${x}_{0}\in X$ the sequence $\left\{f\left({x}_{n}\right)\right\}$ converges to $f\left({x}_{0}\right)$. If $f:X\to Y$ is continuous at each $x\in X$, then $f:X\to Y$ is said to be continuous on X (see [28]).

Definition 2.4 Let X be a -algebra and $\left(X,N\right)$ a fuzzy normed space.

1. (1)

The fuzzy normed space $\left(X,N\right)$ is called a fuzzy normed -algebra if

$N\left(xy,st\right)\ge N\left(x,s\right)\cdot N\left(y,t\right),\phantom{\rule{2em}{0ex}}N\left({x}^{\ast },t\right)=N\left(x,t\right)$

for all $x,y\in X$ and all positive real numbers s and t.

2. (2)

A complete fuzzy normed -algebra is called a fuzzy Banach -algebra.

Example 2.2 Let $\left(X,\parallel \cdot \parallel \right)$ be a normed -algebra. Let

$N\left(x,t\right)=\left\{\begin{array}{cc}\frac{t}{t+\parallel x\parallel },\hfill & t>0,x\in X,\hfill \\ 0,\hfill & t\le 0,x\in X.\hfill \end{array}$

Then $N\left(x,t\right)$ is a fuzzy norm on X and $\left(X,N\right)$ is a fuzzy normed -algebra.

Definition 2.5 Let $\left(X,\parallel \cdot \parallel \right)$ be a normed ${C}^{\ast }$-algebra and ${N}_{x}$ a fuzzy norm on X.

1. (1)

The fuzzy normed -algebra $\left(X,{N}_{x}\right)$ is called an induced fuzzy normed -algebra.

2. (2)

The fuzzy Banach -algebra $\left(X,{N}_{x}\right)$ is called an induced fuzzy ${C}^{\ast }$-algebra.

Definition 2.6 Let $\left(X,{N}_{x}\right)$ and $\left(Y,N\right)$ be induced fuzzy normed -algebras.

1. (1)

A multiplicative $\mathbb{C}$-linear mapping $H:\left(X,{N}_{x}\right)\to \left(Y,N\right)$ is called a fuzzy -homomorphism if $H\left({x}^{\ast }\right)=H{\left(x\right)}^{\ast }$ for all $x\in X$.

2. (2)

A $\mathbb{C}$-linear mapping $D:\left(X,{N}_{x}\right)\to \left(X,{N}_{x}\right)$ is called a fuzzy -derivation if $D\left(xy\right)=D\left(x\right)y+xD\left(y\right)$ and $D\left({x}^{\ast }\right)=D{\left(x\right)}^{\ast }$ for all $x,y\in X$.

Definition 2.7 Let X be a set. A function $d:X×X\to \left[0,\mathrm{\infty }\right]$ is called a generalized metric on X if d satisfies the following conditions:

1. (1)

$d\left(x,y\right)=0$ if and only if $x=y$ for all $x,y\in X$;

2. (2)

$d\left(x,y\right)=d\left(y,x\right)$ for all $x,y\in X$;

3. (3)

$d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)$ for all $x,y,z\in X$.

Theorem 2.1 Let (X,d) be a complete generalized metric space and $J:X\to X$ be a strictly contractive mapping with Lipschitz constant $L<1$. Then, for all $x\in X$, either $d\left({J}^{n}x,{J}^{n+1}x\right)=\mathrm{\infty }$ for all nonnegative integers n or there exists a positive integer ${n}_{0}$ such that

1. (1)

$d\left({J}^{n}x,{J}^{n+1}x\right)<\mathrm{\infty }$ for all ${n}_{0}\ge {n}_{0}$;

2. (2)

the sequence $\left\{{J}^{n}x\right\}$ converges to a fixed point ${y}^{\ast }$ of J;

3. (3)

${y}^{\ast }$ is the unique fixed point of J in the set $Y=\left\{y\in X:d\left({J}^{{n}_{0}}x,y\right)<\mathrm{\infty }\right\}$;

4. (4)

$d\left(y,{y}^{\ast }\right)\le \frac{1}{1-L}d\left(y,Jy\right)$ for all $y\in Y$.

## 3 Hyers-Ulam-Rassias stability of CJA functional equation (1.1) in fuzzy Banach ∗-algebras

In this section, using the fixed point alternative approach we prove the Ulam-Hyers-Rassias stability of the functional equation (1.1) in fuzzy Banach spaces. Throughout this paper, assume that X is a vector space and that $\left(Y,N\right)$ is a fuzzy Banach space.

Theorem 3.1 Let $\phi :{X}^{3}\to \left[0,\mathrm{\infty }\right)$ be a function such that there exists an $L<\frac{1}{2}$ with $\phi \left(\frac{x}{2},\frac{y}{2},\frac{z}{2}\right)\le \frac{L\phi \left(x,y,z\right)}{2}$ for all $x,y,z\in X$. Let $f:X\to Y$ be a mapping satisfying

(3.1)
(3.2)
(3.3)

for all $x,y,z\in X$ and $t>0$. Then there exists a fuzzy -homomorphism $H:X\to Y$ such that

$N\left(f\left(x\right)-H\left(x\right),t\right)\ge \frac{\left(2-2L\right)t}{\left(2-2L\right)t+L\phi \left(x,2x,x\right)}$
(3.4)

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

Proof Letting $\mu =1$ and replacing $\left(x,y,z\right)$ by $\left(x,2x,x\right)$ in (3.1), we have

$N\left(f\left(2x\right)-2f\left(x\right),t\right)\ge \frac{t}{t+\phi \left(x,2x,x\right)}$
(3.5)

for all $x\in X$ and $t>0$. Replacing x by $\frac{x}{2}$ in (3.5), we obtain

$N\left(f\left(x\right)-2f\left(\frac{x}{2}\right),t\right)\ge \frac{t}{t+\phi \left(\frac{x}{2},x,\frac{x}{2}\right)}\ge \frac{t}{t+\frac{L}{2}\phi \left(x,2x,x\right)}.$
(3.6)

Consider the set $S:=\left\{g:X\to Y\right\}$ and the generalized metric d in S defined by

$d\left(f,g\right)=inf\left\{\mu \in {\mathbb{R}}^{+}:N\left(g\left(x\right)-h\left(x\right),\mu t\right)\ge \frac{t}{t+\phi \left(x,2x,x\right)},\mathrm{\forall }x\in X,t>0\right\},$

where $inf\mathrm{\varnothing }=+\mathrm{\infty }$. It is easy to show that $\left(S,d\right)$ is complete (see [29]). Now, we consider a linear mapping $J:S\to S$ such that $Jg\left(x\right):=2g\left(\frac{x}{2}\right)$ for all $x\in X$. Let $g,h\in S$ be such that $d\left(g,h\right)=ϵ$. Then $N\left(g\left(x\right)-h\left(x\right),ϵt\right)\ge \frac{t}{t+\phi \left(x,2x,x\right)}$ for all $x\in X$ and $t>0$. Hence

$\begin{array}{rcl}N\left(Jg\left(x\right)-Jh\left(x\right),Lϵt\right)& =& N\left(2g\left(\frac{x}{2}\right)-2h\left(\frac{x}{2}\right),Lϵt\right)=N\left(g\left(\frac{x}{2}\right)-h\left(\frac{x}{2}\right),\frac{Lϵt}{2}\right)\\ \ge & \frac{\frac{Lt}{2}}{\frac{Lt}{2}+\phi \left(\frac{x}{2},x,\frac{x}{2}\right)}\ge \frac{\frac{Lt}{2}}{\frac{Lt}{2}+\frac{L\phi \left(x,2x,x\right)}{2}}=\frac{t}{t+\phi \left(x,2x,x\right)}\end{array}$

for all $x\in X$ and $t>0$. Thus $d\left(g,h\right)=ϵ$ implies that $d\left(Jg,Jh\right)\le Lϵ$. This means that $d\left(Jg,Jh\right)\le Ld\left(g,h\right)$ for all $g,h\in S$. It follows from (3.6) that

$N\left(2f\left(\frac{x}{2}\right)-f\left(x\right),\frac{Lt}{2}\right)\ge \frac{t}{t+\phi \left(x,2x,x\right)}$

for all $x\in X$ and all $t>0$. This implies that $d\left(f,Jf\right)\le \frac{L}{2}$. By Theorem 2.1, there exists a mapping $H:X\to Y$ satisfying the following:

1. (1)

H is a fixed point of J, that is,

$H\left(\frac{x}{2}\right)=\frac{H\left(x\right)}{2}$
(3.7)

for all $x\in X$. The mapping H is a unique fixed point of J in the set $\mathrm{\Omega }=\left\{h\in S:d\left(g,h\right)<\mathrm{\infty }\right\}$. This implies that H is a unique mapping satisfying (3.7) such that there exists $\mu \in \left(0,\mathrm{\infty }\right)$ satisfying $N\left(f\left(x\right)-H\left(x\right),\mu t\right)\ge \frac{t}{t+\phi \left(x,2x,x\right)}$ for all $x\in X$ and $t>0$.

2. (2)

$d\left({J}^{n}f,H\right)\to 0$ as $n\to \mathrm{\infty }$. This implies the equality

$N\text{-}\underset{n\to \mathrm{\infty }}{lim}{2}^{n}f\left(\frac{x}{{2}^{n}}\right)=H\left(x\right)$
(3.8)

for all $x\in X$.

3. (3)

$d\left(f,H\right)\le \frac{d\left(f,Jf\right)}{1-L}$ with $f\in \mathrm{\Omega }$, which implies the inequality $d\left(f,H\right)\le \frac{L}{2-2L}$. This implies that the inequality (3.4) holds. Furthermore, it follows from (3.1) and (3.8) that

$\begin{array}{c}N\left(\mu H\left(\frac{x+y+z}{2}\right)+\mu H\left(\frac{x-y+z}{2}\right)-H\left(\mu x\right)-H\left(\mu z\right),t\right)\hfill \\ \phantom{\rule{1em}{0ex}}=N\text{-}\underset{n\to \mathrm{\infty }}{lim}\left({2}^{n}\mu f\left(\frac{x+y+z}{{2}^{n+1}}\right)+{2}^{n}\mu f\left(\frac{x-y+z}{{2}^{n+1}}\right)-{2}^{n}f\left(\frac{\mu x}{{2}^{n}}\right)-{2}^{n}f\left(\frac{\mu z}{{2}^{n}}\right),t\right)\hfill \\ \phantom{\rule{1em}{0ex}}\ge \underset{n\to \mathrm{\infty }}{lim}\frac{\frac{t}{{2}^{n}}}{\frac{t}{{2}^{n}}+\phi \left(\frac{x}{{2}^{n}},\frac{y}{{2}^{n}},\frac{z}{{2}^{n}}\right)}\ge \underset{n\to \mathrm{\infty }}{lim}\frac{\frac{t}{{2}^{n}}}{\frac{t}{{2}^{n}}+\frac{{L}^{n}}{{2}^{n}}\phi \left(x,y,z\right)}\to 1\hfill \end{array}$

for all $x,y,z\in X$, all $t>0$ and all $\mu \in \mathbb{C}$. Hence

$\mu H\left(\frac{x+y+z}{2}\right)+\mu H\left(\frac{x-y+z}{2}\right)-H\left(\mu x\right)-H\left(\mu z\right)=0$

for all $x,y,z\in X$. So the mapping $H:X\to Y$ is additive and $\mathbb{C}$-linear. By (3.2),

$N\left({4}^{n}f\left(\frac{xy}{{4}^{n}}\right)-{2}^{n}f\left(\frac{x}{{2}^{n}}\right)\cdot {2}^{n}f\left(\frac{y}{{2}^{n}}\right),{4}^{n}t\right)\ge \frac{t}{t+\phi \left(\frac{x}{{2}^{n}},\frac{y}{{2}^{n}},0\right)}$

for all $x,y\in X$ and all $t>0$. Then

for all $x,y\in X$ and all $t>0$. So $N\left(H\left(xy\right)-H\left(x\right)H\left(y\right),t\right)=1$ for all $x,y\in X$ and all $t>0$. By (3.3)

$N\left({2}^{n}f\left(\frac{{x}^{\ast }}{{2}^{n}}\right)-{2}^{n}f{\left(\frac{x}{{2}^{n}}\right)}^{\ast },{2}^{n}t\right)\ge \frac{t}{t+\phi \left(\frac{x}{{2}^{n}},0,0\right)}$

for all $x\in X$ and all $t>0$. So

$N\left({2}^{n}f\left(\frac{{x}^{\ast }}{{2}^{n}}\right)-{2}^{n}f{\left(\frac{x}{{2}^{n}}\right)}^{\ast },t\right)\ge \frac{\frac{t}{{2}^{n}}}{\frac{t}{{2}^{n}}+\phi \left(\frac{x}{{2}^{n}},0,0\right)}\ge \frac{\frac{t}{{2}^{n}}}{\frac{t}{{2}^{n}}+\frac{{L}^{n}}{{2}^{n}}\phi \left(x,0,0\right)}$

for all $x\in X$ and all $t>0$. Since ${lim}_{n\to +\mathrm{\infty }}\frac{\frac{t}{{2}^{n}}}{\frac{t}{{2}^{n}}+\frac{{L}^{n}}{{2}^{n}}\phi \left(x,0,0\right)}=1$, for all $x\in X$ and $t>0$, we get $N\left(H\left({x}^{\ast }\right)-H{\left(x\right)}^{\ast },t\right)=1$ for all $x\in X$ and all $t>0$. Thus $H\left({x}^{\ast }\right)=H{\left(x\right)}^{\ast }$ for all $x\in X$. □

Theorem 3.2 Let $\phi :{X}^{3}\to \left[0,\mathrm{\infty }\right)$ be a function such that there exists an $L<1$ with $\phi \left(x,y,z\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2},\frac{z}{2}\right)$ for all $x,y,z\in X$. Let $f:X\to Y$ be a mapping satisfying (3.1)-(3.3). Then the limit $H\left(x\right):=N\text{-}{lim}_{n\to \mathrm{\infty }}\frac{f\left({2}^{n}x\right)}{{2}^{n}}$ exists for each $x\in X$ and defines a fuzzy -homomorphism $H:X\to Y$ such that

$N\left(f\left(x\right)-H\left(x\right),t\right)\ge \frac{\left(2-2L\right)t}{\left(2-2L\right)t+\phi \left(x,2x,x\right)}$
(3.9)

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

Proof Let $\left(S,d\right)$ be a generalized metric space defined as in the proof of Theorem 3.1. Consider the linear mapping $J:S\to S$ such that $Jg\left(x\right):=\frac{g\left(2x\right)}{2}$ for all $x\in X$. Let $g,h\in S$ be such that $d\left(g,h\right)=ϵ$. Then $N\left(g\left(x\right)-h\left(x\right),ϵt\right)\ge \frac{t}{t+\phi \left(x,2x,x\right)}$ for all $x\in X$ and $t>0$. Hence

$\begin{array}{rcl}N\left(Jg\left(x\right)-Jh\left(x\right),Lϵt\right)& =& N\left(\frac{g\left(2x\right)}{2}-\frac{h\left(2x\right)}{2},Lϵt\right)=N\left(g\left(2x\right)-h\left(2x\right),2Lϵt\right)\\ \ge & \frac{2Lt}{2Lt+\phi \left(2x,,4x,2x\right)}\ge \frac{2Lt}{2Lt+2L\phi \left(x,,2x,x\right)}\\ =& \frac{t}{t+\phi \left(x,2x,x\right)}\end{array}$

for all $x\in X$ and $t>0$. Thus $d\left(g,h\right)=ϵ$ implies that $d\left(Jg,Jh\right)\le Lϵ$. This means that $d\left(Jg,Jh\right)\le Ld\left(g,h\right)$ for all $g,h\in S$. It follows from (3.5) that

$N\left(\frac{f\left(2x\right)}{2}-f\left(x\right),\frac{t}{2}\right)\ge \frac{t}{t+\phi \left(x,2x,x\right)}$
(3.10)

for all $x\in X$ and $t>0$. So $d\left(f,Jf\right)\le \frac{1}{2}$. By Theorem 2.1, there exists a mapping $H:X\to Y$ satisfying the following:

1. (1)

H is a fixed point of J, that is,

$2H\left(x\right)=H\left(2x\right)$
(3.11)

for all $x\in X$. The mapping H is a unique fixed point of J in the set $\mathrm{\Omega }=\left\{h\in S:d\left(g,h\right)<\mathrm{\infty }\right\}$. This implies that H is a unique mapping satisfying (3.11) such that there exists $\mu \in \left(0,\mathrm{\infty }\right)$ satisfying $N\left(f\left(x\right)-H\left(x\right),\mu t\right)\ge \frac{t}{t+\phi \left(x,2x,x\right)}$ for all $x\in X$ and $t>0$.

2. (2)

$d\left({J}^{n}f,H\right)\to 0$ as $n\to \mathrm{\infty }$. This implies the equality $H\left(x\right)=N\text{-}{lim}_{n\to \mathrm{\infty }}\frac{f\left({2}^{n}x\right)}{{2}^{n}}$ for all $x\in X$.

3. (3)

$d\left(f,H\right)\le \frac{d\left(f,Jf\right)}{1-L}$ with $f\in \mathrm{\Omega }$, which implies the inequality $d\left(f,H\right)\le \frac{1}{2-2L}$. This implies that the inequality (3.9) holds. The rest of the proof is similar to that of the proof of Theorem 3.1. □

## 4 Hyers-Ulam-Rassias stability of CJA functional equation (1.1) in induced fuzzy ${C}^{\ast }$-algebras

Throughout this section, assume that X is a unital ${C}^{\ast }$-algebra with unit e and unitary group $\mathcal{U}\left(X\right):=\left\{u\in X:{u}^{\ast }u=u{u}^{\ast }=e\right\}$ and that Y is a unital ${C}^{\ast }$-algebra.

Using the fixed point method, we prove the Hyers-Ulam-Rassias stability of the Cauchy-Jensen additive functional equation (1.1) in induced fuzzy ${C}^{\ast }$-algebras.

Theorem 4.1 Let $\phi :{X}^{3}\to \left[0,\mathrm{\infty }\right)$ be a function such that there exists an $L<\frac{1}{2}$ with $\phi \left(\frac{x}{2},\frac{y}{2},\frac{z}{2}\right)\le \frac{L\phi \left(x,y,z\right)}{2}$ for all $x,y,z\in X$. Let $f:X\to Y$ be a mapping satisfying (3.1) and

(4.1)
(4.2)

for all $u,v\in \mathcal{U}\left(X\right)$ and all $t>0$. Then there exists a fuzzy -homomorphism $H:X\to Y$ satisfying (3.4).

Proof By the same reasoning as in the proof of Theorem 3.1, there is a $\mathbb{C}$-linear mapping $H:X\to Y$ satisfying (3.4). The mapping $H:X\to Y$ is given by

$N\text{-}\underset{p\to \mathrm{\infty }}{lim}{2}^{n}f\left(\frac{x}{{2}^{n}}\right)=H\left(x\right)$

for all $x\in X$. By (4.1),

$N\left({4}^{n}f\left(\frac{uv}{{4}^{n}}\right)-{2}^{n}f\left(\frac{u}{{2}^{n}}\right)\cdot {2}^{n}f\left(\frac{v}{{2}^{n}}\right),{4}^{n}t\right)\ge \frac{t}{t+\phi \left(\frac{u}{{2}^{n}},\frac{v}{{2}^{n}},0\right)}$

for all $u,v\in \mathcal{U}\left(X\right)$ and all $t>0$. Then

for all $x,y\in \mathcal{U}\left(X\right)$ and all $t>0$. So $N\left(H\left(uv\right)-H\left(u\right)H\left(v\right),t\right)=1$ for all $u,v\in \mathcal{U}\left(X\right)$ and all $t>0$. Therefore

$H\left(uv\right)=H\left(u\right)H\left(v\right),$
(4.3)

for all $u,v\in \mathcal{U}\left(X\right)$. Since H is $\mathbb{C}$-linear and each $x\in X$ is a finite linear combination of unitary elements, i.e.,

$x=\sum _{j=1}^{m}{\lambda }_{j}{u}_{j}\left({\lambda }_{j}\in \mathbb{C},{u}_{j}\in U\left(X\right)\right),$

it follows from (4.3) that

$H\left(xv\right)=H\left(\sum _{j=1}^{m}{\lambda }_{j}{u}_{j}v\right)=\sum _{j=1}^{n}{\lambda }_{j}H\left({u}_{j}v\right)=\sum _{j=1}^{n}{\lambda }_{j}H\left({u}_{j}\right)H\left(v\right)=H\left(\sum _{j=1}^{m}{\lambda }_{j}{u}_{j}\right)H\left(v\right)$

for all $v\in \mathcal{U}\left(X\right)$. So $H\left(xv\right)=H\left(x\right)H\left(v\right)$. Similarly, one can obtain that $H\left(xy\right)=H\left(x\right)H\left(y\right)$ for all $x,y\in X$. By (4.2)

$N\left({2}^{n}f\left(\frac{{u}^{\ast }}{{2}^{n}}\right)-{2}^{n}f{\left(\frac{u}{{2}^{n}}\right)}^{\ast },{2}^{n}t\right)\ge \frac{t}{t+\phi \left(\frac{u}{{2}^{n}},0,0\right)}$

for all $u\in \mathcal{U}\left(X\right)$ and all $t>0$. So

$N\left({2}^{n}f\left(\frac{{u}^{\ast }}{{2}^{n}}\right)-{2}^{n}f{\left(\frac{u}{{2}^{n}}\right)}^{\ast },t\right)\ge \frac{\frac{t}{{2}^{n}}}{\frac{t}{{2}^{n}}+\phi \left(\frac{u}{{2}^{n}},0,0\right)}\ge \frac{\frac{t}{{2}^{n}}}{\frac{t}{{2}^{n}}+\frac{{L}^{n}}{{2}^{n}}\phi \left(u,0,0\right)}$

for all $u\in \mathcal{U}\left(X\right)$ and all $t>0$. Since ${lim}_{n\to +\mathrm{\infty }}\frac{\frac{t}{{2}^{n}}}{\frac{t}{{2}^{n}}+\frac{{L}^{n}}{{2}^{n}}\phi \left(u,0,0\right)}=1$, for all $u\in \mathcal{U}\left(X\right)$ and $t>0$ , we get $N\left(H\left({u}^{\ast }\right)-H{\left(u\right)}^{\ast },t\right)=1$ for all $u\in \mathcal{U}\left(X\right)$ and all $t>0$. Thus

$H\left({u}^{\ast }\right)=H{\left(u\right)}^{\ast }$
(4.4)

for all $u\in \mathcal{U}\left(X\right)$. Since H is $\mathbb{C}$-linear, i.e., $x\in X$ is a finite linear combination of unitary elements, i.e., $x={\sum }_{j=1}^{m}{\lambda }_{j}{u}_{j}$ (${\lambda }_{j}\in \mathbb{C}$, ${u}_{j}\in \mathcal{U}\left(X\right)$), it follows from (4.4) that

$H\left({x}^{\ast }\right)=H\left(\sum _{j=1}^{m}\overline{{\lambda }_{j}}{u}_{j}^{\ast }\right)=\sum _{j=1}^{n}\overline{{\lambda }_{j}}H\left({u}_{j}^{\ast }\right)=\sum _{j=1}^{n}\overline{{\lambda }_{j}}H{\left({u}_{j}\right)}^{\ast }=H{\left(\sum _{j=1}^{m}{\lambda }_{j}{u}_{j}\right)}^{\ast }=H{\left(x\right)}^{\ast }$

for all $x\in X$. So $H\left({x}^{\ast }\right)=H{\left(x\right)}^{\ast }$ for all $x\in X$. Therefore, the mapping $H:X\to Y$ is a -homomorphism. □

Similarly, we have the following. We will omit the proof.

Theorem 4.2 Let $\phi :{X}^{3}\to \left[0,\mathrm{\infty }\right)$ be a function such that there exists an $L<1$ with $\phi \left(x,y,z\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2},\frac{z}{2}\right)$ for all $x,y,z\in X$. Let $f:X\to Y$ be a mapping satisfying (3.1), (4.1) and (4.2). Then the limit $H\left(x\right):=N\text{-}{lim}_{n\to \mathrm{\infty }}\frac{f\left({2}^{n}x\right)}{{2}^{n}}$ exists for each $x\in X$ and defines a fuzzy -homomorphism $H:X\to Y$ such that

$N\left(f\left(x\right)-H\left(x\right),t\right)\ge \frac{\left(2-2L\right)t}{\left(2-2L\right)t+\phi \left(x,2x,x\right)}$
(4.5)

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

## 5 Hyers-Ulam-Rassias stability of fuzzy ∗-derivations in fuzzy Banach ∗-algebras and in induced fuzzy ${C}^{\ast }$-algebras

In this section, assume that $\left(X,{N}_{X}\right)$ is a fuzzy Banach -algebra. Using the fixed point method, we prove the Hyers-Ulam-Rassias stability of fuzzy -derivations in fuzzy Banach -algebras.

Theorem 5.1 Let $\phi :{X}^{2}\to \left[0,\mathrm{\infty }\right)$ be a function such that there exists an $L<\frac{1}{2}$ with $\phi \left(\frac{x}{2},\frac{y}{2},\frac{z}{2}\right)\le \frac{L\phi \left(x,y,z\right)}{2}$ for all $x,y,z\in X$. Let $f:X\to X$ be a mapping satisfying (3.1), (3.3) and

${N}_{X}\left(f\left(xy\right)-xf\left(y\right)-yf\left(x\right),t\right)\ge \frac{t}{t+\phi \left(x,y,0\right)}$
(5.1)

for all $x,y\in X$ and all $t>0$. Then $\delta \left(x\right):=N\text{-}{lim}_{n\to \mathrm{\infty }}{2}^{n}f\left(\frac{x}{{2}^{n}}\right)$ exists for each $x\in X$ and defines a fuzzy -derivation $\delta :X\to X$ such that

$N\left(f\left(x\right)-\delta \left(x\right),t\right)\ge \frac{\left(2-2L\right)t}{\left(2-2L\right)t+L\phi \left(x,2x,x\right)}$
(5.2)

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

Proof The proof is similar to the proof of Theorem 3.1. □

Theorem 5.2 Let $\phi :{X}^{2}\to \left[0,\mathrm{\infty }\right)$ be a function such that there exists an $L<1$ with $\phi \left(x,y,z\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2},\frac{z}{2}\right)$ for all $x,y,z\in X$. Let $f:X\to Y$ be a mapping satisfying (3.1) and (5.1). Then the limit $\delta \left(x\right):=N\text{-}{lim}_{p\to \mathrm{\infty }}\frac{f\left({2}^{n}x\right)}{{2}^{n}}$ exists for each $x\in X$ and defines a fuzzy -derivation $\delta :X\to Y$ such that

$N\left(f\left(x\right)-\delta \left(x\right),t\right)\ge \frac{\left(2-2L\right)t}{\left(2-2L\right)t+\phi \left(x,2x,x\right)}$
(5.3)

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

## References

1. Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.

2. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

3. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

4. Skof F: Local properties and approximation of operators. Rend. Semin. Mat. Fis. Milano 1983, 53: 113–129. 10.1007/BF02924890

5. Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76–86. 10.1007/BF02192660

6. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 239–248.

7. Eshaghi Gordji M, Savadkouhi MB: Stability of mixed type cubic and quartic functional equations in random normed spaces. J. Inequal. Appl. 2009., 2009: Article ID 527462

8. Eshaghi Gordji M, Savadkouhi MB, Park C: Quadratic-quartic functional equations in RN-spaces. J. Inequal. Appl. 2009., 2009: Article ID 868423

9. Eshaghi Gordji M, Khodaei H: Stability of Functional Equations. Lap Lambert Academic Publishing, Saarbrücken; 2010.

10. Eshaghi Gordji M, Zolfaghari S, Rassias JM, Savadkouhi MB: Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces. Abstr. Appl. Anal. 2009., 2009: Article ID 417473

11. Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211

12. Park C: On the stability of the linear mapping in Banach modules. J. Math. Anal. Appl. 2002, 275: 711–720. 10.1016/S0022-247X(02)00386-4

13. Park C: Fuzzy stability of a functional equation associated with inner product spaces. Fuzzy Sets Syst. 2009, 160: 1632–1642. 10.1016/j.fss.2008.11.027

14. Park C: Generalized Hyers-Ulam-Rassias stability of n -sesquilinear-quadratic mappings on Banach modules over ${C}^{\ast }$ -algebras. J. Comput. Appl. Math. 2005, 180: 279–291. 10.1016/j.cam.2004.11.001

15. Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007., 2007: Article ID 50175

16. Park C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008., 2008: Article ID 493751

17. Rassias TM: On the stability of the quadratic functional equation and its application. Stud. Univ. Babeş-Bolyai, Math. 1998, XLIII: 89–124.

18. Rassias TM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 2000, 251: 264–284. 10.1006/jmaa.2000.7046

19. Rassias TM, Šemrl P: On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 1993, 173: 325–338. 10.1006/jmaa.1993.1070

20. Saadati R, Vaezpour M, Cho YJ: A note to paper ’On the stability of cubic mappings and quartic mappings in random normed spaces’. J. Inequal. Appl. 2009., 2009: Article ID 214530. doi:10.1155/2009/214530

21. Saadati R, Zohdi MM, Vaezpour SM: Nonlinear L -random stability of an ACQ functional equation. J. Inequal. Appl. 2011., 2011: Article ID 194394. doi:10.1155/2011/194394

22. Katsaras AK: Fuzzy topological vector spaces. Fuzzy Sets Syst. 1984, 12: 143–154. 10.1016/0165-0114(84)90034-4

23. Felbin C: Finite-dimensional fuzzy normed linear space. Fuzzy Sets Syst. 1992, 48: 239–248. 10.1016/0165-0114(92)90338-5

24. Krishna SV, Sarma KKM: Separation of fuzzy normed linear spaces. Fuzzy Sets Syst. 1994, 63: 207–217. 10.1016/0165-0114(94)90351-4

25. Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 2003, 11: 687–705.

26. Cheng SC, Mordeson JN: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 1994, 86: 429–436.

27. Karmosil I, Michalek J: Fuzzy metric and statistical metric spaces. Kybernetika 1975, 11: 326–334.

28. Bag T, Samanta SK: Fuzzy bounded linear operators. Fuzzy Sets Syst. 2005, 151: 513–547. 10.1016/j.fss.2004.05.004

29. Mihet D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 2008, 343: 567–572. 10.1016/j.jmaa.2008.01.100

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Azadi Kenary, H., Zohdi, A., Eshaghi Gordji, M. et al. Fuzzy -homomorphisms and fuzzy -derivations in induced fuzzy ${C}^{\ast }$-algebras. Adv Differ Equ 2012, 147 (2012). https://doi.org/10.1186/1687-1847-2012-147

• induced fuzzy ${C}^{\ast }$-algebra