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

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

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

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

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

The functional equation $f(x+y)+f(x-y)=2f(x)+2f(y)$ 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 [7–21]).

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

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

(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 $N:X\times \mathbb{R}\to [0,1]$ is called a fuzzy norm on X if for all $x,y\in X$ and all $s,t\in \mathbb{R}$,

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

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

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

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

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

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

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

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

is a fuzzy norm on X.

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

Definition 2.3 Let $(X,N)$ be a fuzzy normed vector space. A sequence $\{x_n\}$ 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 $N(x_{n+p}-x_n,t)>1-ϵ$.

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 $\{x_n\}$ converging to $x_0\in X$ the sequence $\{f(x_n)\}$ converges to $f(x_0)$. 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 $(X,N)$ a fuzzy normed space.

• (1) The fuzzy normed space $(X,N)$ is called a fuzzy normed ∗-algebra if

  $$N(xy,st)\ge N(x,s)\cdot N(y,t),N(x^{\ast },t)=N(x,t)$$

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

• (2) A complete fuzzy normed ∗-algebra is called a fuzzy Banach ∗-algebra.

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

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

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

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

• (1) The fuzzy normed ∗-algebra $(X,N_x)$ is called an induced fuzzy normed ∗-algebra.

• (2) The fuzzy Banach ∗-algebra $(X,N_x)$ is called an induced fuzzy $C^{\ast }$-algebra.

Definition 2.6 Let $(X,N_x)$ and $(Y,N)$ be induced fuzzy normed ∗-algebras.

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

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

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

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

• (2) $d(x,y)=d(y,x)$ for all $x,y\in X$;

• (3) $d(x,z)\le d(x,y)+d(y,z)$ 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 $d(J^{n}x,J^{n+1}x)=\mathrm{\infty }$ for all nonnegative integers n or there exists a positive integer $n_0$ such that

• (1) $d(J^{n}x,J^{n+1}x)<\mathrm{\infty }$ for all $n_0\ge n_0$;

• (2) the sequence $\{J^{n}x\}$ converges to a fixed point $y^{\ast }$ of J;

• (3) $y^{\ast }$ is the unique fixed point of J in the set $Y=\{y\in X:d(J^{n_0}x,y)<\mathrm{\infty }\}$;

• (4) $d(y,y^{\ast })\le \frac{1}{1-L}d(y,Jy)$ 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 $(Y,N)$ is a fuzzy Banach space.

Theorem 3.1 Let $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $L<\frac{1}{2}$ with $\phi (\frac{x}{2},\frac{y}{2},\frac{z}{2})\le \frac{L\phi (x,y,z)}{2}$ 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

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

(3.4)

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

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

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

(3.5)

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

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

(3.6)

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

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

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

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

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

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

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

• (1) H is a fixed point of J, that is,

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

  (3.7)

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

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

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

  (3.8)

  for all $x\in X$.

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

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

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

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

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

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

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

$$\begin{array}{rcl}N(4^{n}f\left(\frac{xy}{4^n}\right)-2^{n}f\left(\frac{x}{2^n}\right)\cdot 2^{n}f\left(\frac{y}{2^n}\right),t)& \ge & \frac{\frac{t}{4^n}}{\frac{t}{4^n}+\phi (\frac{x}{2^n},\frac{y}{2^n},0)}\\ \ge & \frac{\frac{t}{4^n}}{\frac{t}{4^n}+\frac{L^n\phi (x,y,0)}{2^n}}\to 1\text{when }n\to +\mathrm{\infty }\end{array}$$

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

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

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

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

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

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

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

(3.9)

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

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

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

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

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

(3.10)

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

• (1) H is a fixed point of J, that is,

  $$2H(x)=H(2x)$$

  (3.11)

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

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

• (3) $d(f,H)\le \frac{d(f,Jf)}{1-L}$ with $f\in \mathrm{\Omega }$, which implies the inequality $d(f,H)\le \frac{1}{2-2L}$. 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 $\mathcal{U}(X):=\{u\in X:u^{\ast }u=u{u}^{\ast }=e\}$ 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 $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $L<\frac{1}{2}$ with $\phi (\frac{x}{2},\frac{y}{2},\frac{z}{2})\le \frac{L\phi (x,y,z)}{2}$ 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 $u,v\in \mathcal{U}(X)$ 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

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

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

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

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

$$\begin{array}{rcl}N(4^{n}f\left(\frac{uv}{4^n}\right)-2^{n}f\left(\frac{u}{2^n}\right)\cdot 2^{n}f\left(\frac{v}{2^n}\right),t)& \ge & \frac{\frac{t}{4^n}}{\frac{t}{4^n}+\phi (\frac{u}{2^n},\frac{v}{2^n},0)}\\ \ge & \frac{\frac{t}{4^n}}{\frac{t}{4^n}+\frac{L^n\phi (u,v,0)}{2^n}}\to 1\text{when }n\to +\mathrm{\infty }\end{array}$$

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

$$H(uv)=H(u)H(v),$$

(4.3)

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

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

it follows from (4.3) that

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

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

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

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

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

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

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

(4.4)

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

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

for all $x\in X$. So $H(x^{\ast })=H{(x)}^{\ast }$ 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 $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $L<1$ with $\phi (x,y,z)\le 2L\phi (\frac{x}{2},\frac{y}{2},\frac{z}{2})$ 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 $H(x):=N\text{-}{lim}_{n\to \mathrm{\infty }}\frac{f(2^{n}x)}{2^n}$ exists for each $x\in X$ and defines a fuzzy ∗-homomorphism $H:X\to Y$ such that

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

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

$$N_X(f(xy)-xf(y)-yf(x),t)\ge \frac{t}{t+\phi (x,y,0)}$$

(5.1)

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

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

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

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

(5.3)

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