Homomorphisms and derivations in induced fuzzy $C^{\ast }$-algebras

Abstract

Using fixed point method, we establish the Hyers-Ulam stability of fuzzy ∗-homomorphisms in fuzzy $C^{\ast }$-algebras and fuzzy ∗-derivations on fuzzy $C^{\ast }$-algebras associated to the following $(m,n)$-Cauchy-Jensen additive functional equation:

$$\underset{1\le k_l(\ne i_j,\mathrm{\forall }j\in \{1,\dots ,m\})\le n}{\sum _{1\le i_1<\cdots <i_m\le n}}f(\frac{{\sum }_{j=1}^m{x}_{i_j}}{m}+\sum _{l=1}^{n-m}{x}_{k_l})=\frac{(n-m+1)}{n}\left(\genfrac{}{}{0ex}{}{n}{m}\right)\sum _{i=1}^{n}f(x_i).$$

MSC:47S40, 39B52, 46S40, 47H10, 26E50.

1 Introduction

The stability problem of functional equations originated from a 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 Rassias [3] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (T.M. Rassias)

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 ℝ-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 Hyers-Ulam 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 Hyers-Ulam 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 [7–20]).

Katsaras [21] 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 [22–24]). 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].

Now, we consider a mapping $f:X\to Y$ satisfying the following functional equation, which is introduced by the first author:

$$\underset{1\le k_l(\ne i_j,\mathrm{\forall }j\in \{1,\dots ,m\})\le n}{\sum _{1\le i_1<\cdots <i_m\le n}}f(\frac{{\sum }_{j=1}^m{x}_{i_j}}{m}+\sum _{l=1}^{n-m}{x}_{k_l})=\frac{(n-m+1)}{n}\left(\genfrac{}{}{0ex}{}{n}{m}\right)\sum _{i=1}^{n}f(x_i)$$

(1)

for all $x_1,\dots ,x_n\in X$, where $m,n\in \mathbb{N}$ are fixed integers with $n\ge 2$, $1\le m\le n$. Especially, we observe that in the case $m=1$, equation (1) yields the Cauchy-type additive equation $f({\sum }_{l=1}^n{x}_{k_l})={\sum }_{l=1}^{n}f(x_i)$. We observe that in the case $m=n$, equation (1) yields the Jensen-type additive equation $f(\frac{{\sum }_{j=1}^n{x}_j}{n})=\frac{1}{n}{\sum }_{l=1}^{n}f(x_i)$. Therefore, equation (1) is a generalized form of the Cauchy-Jensen additive equation and thus every solution of equation (1) may be analogously called a general $(m,n)$-Cauchy-Jensen additive. For the case $m=2$, we have established new theorems about the Hyers-Ulam stability in quasi β-normed spaces [29]. Let X and Y be linear spaces. For each m with $1\le m\le n$, a mapping $f:X\to Y$ satisfies equation (1) for all $n\ge 2$ if and only if $f(x)-f(0)=A(x)$ is a Cauchy additive, where $f(0)=0$ if $m<n$. In particular, we have $f((n-m+1)x)=(n-m+1)f(x)$ and $f(mx)=mf(x)$ for all $x\in X$.

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 $s,t\in \mathbb{R}$,

(N1) $N(x,t)=0$ for $t\le 0$;

(N2) $x=0$ if and only if $N(x,t)=1$ for all $t>0$;

(N3) $N(cx,t)=N(x,\frac{t}{|c|})$ if $c\ne 0$;

(N4) $N(x+y,c+t)\ge min\{N(x,s),N(y,t)\}$;

(N5) $N(x,\cdot )$ is a non-decreasing function of ℝ and ${lim}_{t\to \mathrm{\infty }}N(x,t)=1$;

(N6) for $x\ne 0$, $N(x,\cdot )$ is continuous on 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 $(X,N)$ be 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 be a fuzzy norm on X.

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

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

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

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

• (2) A ℂ-linear mapping $D:(X,N)\to (X,N)$ 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 nonempty 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 a 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 $y\in Y$.

Throughout this paper, assume that X, Y are unital fuzzy Banach ∗-algebras.

3 Approximate homomorphisms in fuzzy Banach ∗-algebras

In this section, using fixed point method, we prove the Hyers-Ulam stability of homomorphisms in fuzzy Banach ∗-algebras related to functional equation (1).

Theorem 3.1 Let $\phi :X^n\to [0,\mathrm{\infty })$ be a function such that there exists an $L<\frac{1}{{(n-m+1)}^{n-2}}$ 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 $f:X\to Y$ with $f(0)=0$ be a mapping satisfying

(2)

(3)

(4)

for all $x_1,\dots ,x_n\in X$ and all $t>0$. Then there exists a fuzzy ∗-homomorphism $H:X\to Y$ such that

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

(5)

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

Proof Letting $\mu =1$ and replacing $(x_1,\dots ,x_n)$ by $(x,\dots ,x)$ in (2), we have

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

(6)

for all $x\in X$ and $t>0$. Consider the set $S:=\{g:X\to Y;g(0)=0\}$ and the generalized metric d in S defined by

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

where $inf\mathrm{\varnothing }=+\mathrm{\infty }$. It is easy to show that $(S,d)$ is complete (see [30]). Now, we consider a linear mapping $J:S\to S$ such that $Jg(x):=(n-m+1)g(\frac{x}{n-m+1})$ for all $x\in X$. Let $g,h\in S$ be such that $d(g,h)=ϵ$. Then $N(g(x)-h(x),ϵt)\ge \frac{t}{t+\phi (x,\dots ,x)}$ for all $x\in X$ and $t>0$. Hence,

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 (6) that

$$\begin{array}{rcl}N(\frac{f(\frac{x}{n-m+1})}{{(n-m+1)}^{-1}}-f(x),\frac{t}{\left(\genfrac{}{}{0ex}{}{n}{m}\right)})& \ge & \frac{t}{t+\phi (\frac{x}{n-m+1},\dots ,\frac{x}{n-m+1})}\\ \ge & \frac{t}{t+\frac{L\phi (x,\dots ,x)}{n-m+1}}\end{array}$$

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

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

This implies that $d(f,Jf)\le \frac{L}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)}$. By Theorem 2.1, there exists a mapping $A:X\to Y$ satisfying the following:

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

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

  (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 (7) 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 $t>0$.

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

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

  (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

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

This implies that the inequality (5) holds. Furthermore, it follows from (2) and (8) that

for all $x_1,\dots ,x_n\in X$, all $t>0$ and all $\mu \in \mathbb{C}$. Hence,

$$\underset{1\le k_l(\ne i_j,\mathrm{\forall }j\in \{1,\dots ,m\})\le n}{\sum _{1\le i_1<\cdots <i_m\le n}}H(\frac{{\sum }_{j=1}^m\mu x_{i_j}}{m}+\sum _{l=1}^{n-m}\mu x_{k_l})=\frac{(n-m+1)}{n}\left(\genfrac{}{}{0ex}{}{n}{m}\right)\sum _{i=1}^{n}H(\mu x_i)$$

for all $x_1,\dots ,x_n\in X$. So, the mapping $H:X\to Y$ is additive and ℂ-linear. By (3)

(9)

for all $x_1,\dots ,x_{n-1}\in X$ and all $t>0$. Then

for all $x_1,\dots ,x_{n-1}\in X$ and all $t>0$. So,

$$N(H(x_1\cdots x_{n-1})-H(x_1)\cdots H(x_{n-1}),t)=1$$

for all $x_1,\dots ,x_{n-1}\in X$ and all $t>0$. Thus, $H(x_1\cdots x_{n-1})=H(x_1)\cdots H(x_{n-1})$.

On the other hand, by (4)

$$N(\frac{f(\frac{x_1^{\ast }}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}-{\frac{f(\frac{x_1}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}}^{\ast },\frac{t}{{(n-m+1)}^{-p}})\ge \frac{t}{t+\phi (\frac{x_1}{{(n-m+1)}^p},0,\dots ,0)}$$

for all $x_1\in X$ and all $t>0$. So,

$$\begin{array}{rcl}N(\frac{f(\frac{x_1^{\ast }}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}-{\frac{f(\frac{x_1}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}}^{\ast },t)& \ge & \frac{\frac{t}{{(n-m+1)}^p}}{\frac{t}{{(n-m+1)}^p}+\phi (\frac{x_1}{{(n-m+1)}^p},0,\dots ,0)}\\ \ge & \frac{\frac{t}{{(n-m+1)}^p}}{\frac{t}{{(n-m+1)}^p}+\frac{L^p}{{(n-m+1)}^p}\phi (x_1,0,\dots ,0)}\end{array}$$

for all $x_1\in X$ and all $t>0$. Since ${lim}_{p\to +\mathrm{\infty }}\frac{\frac{t}{{(n-m+1)}^p}}{\frac{t}{{(n-m+1)}^p}+\frac{L^p}{{(n-m+1)}^p}\phi (x_1,0,\dots ,0)}=1$ for all $x_1\in X$ and $t>0$, we get

$$N(H\left(x_1^{\ast }\right)-H{(x_1)}^{\ast },t)=1$$

for all $x_1\in X$ and all $t>0$. Thus, $H(x_1^{\ast })=H{(x_1)}^{\ast }$ for all $x_1\in X$. This completes the proof. □

Theorem 3.2 Let $\phi :X^n\to [0,\mathrm{\infty })$ be a function such that there exists an $L<1$ 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 satisfying $f(0)=0$, (2)-(4). Then the limit $A(x):=N\text{-}{lim}_{p\to \mathrm{\infty }}\frac{f({(n-m+1)}^{p}x)}{{(n-m+1)}^p}$ exists for each $x\in X$ and defines a fuzzy ∗-homomorphism $H:X\to Y$ such that

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

(10)

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 $Jg(x):=\frac{g((n-m+1)x)}{n-m+1}$ for all $x\in X$. Let $g,h\in S$ be such that $d(g,h)=ϵ$. Then $N(g(x)-h(x),ϵt)\ge \frac{t}{t+\phi (x,\dots ,x)}$ for all $x\in X$ and $t>0$. Hence,

$$\begin{array}{rcl}N(Jg(x)-Jh(x),Lϵt)& =& N(\frac{g((n-m+1)x)}{n-m+1}-\frac{h((n-m+1)x)}{n-m+1},Lϵt)\\ =& N(g((n-m+1)x)-h((n-m+1)x),(n-m+1)Lϵt)\\ \ge & \frac{(n-m+1)Lt}{(n-m+1)Lt+\phi ((n-m+1)x,\dots ,(n-m+1)x)}\\ \ge & \frac{(n-m+1)Lt}{(n-m+1)Lt+(n-m+1)L\phi (x,\dots ,x)}\\ =& \frac{t}{t+\phi (x,\dots ,x)}\end{array}$$

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 (6) that

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

(11)

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

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

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

  (12)

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 (12) 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 $t>0$.

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

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

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

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

This implies that the inequality (10) holds.

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

From now on, we assume that X has a unit e and a unitary group $\mathcal{U}(X):=\{u\in X:u^{\ast }u=u{u}^{\ast }=e\}$.

Theorem 3.3 Let $\phi :X^n\to [0,\mathrm{\infty })$ be a function such that there exists an $L<\frac{1}{{(n-m+1)}^{n-2}}$ 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 $f:X\to Y$ be a mapping satisfying $f(0)=0$, (2) and

(13)

(14)

for all $u_1,\dots ,u_n\in \mathcal{U}(X)$ and all $t>0$. Then there exists a fuzzy ∗-homomorphism $H:X\to Y$ satisfying (5).

Proof By the same reasoning as in the proof of Theorem 3.1, there is a ℂ-linear mapping $H:X\to Y$ satisfying (5). The mapping $H:X\to Y$ is given by

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

for all $x\in X$. By (13)

for all $u_1,\dots ,u_{n-1}\in \mathcal{U}(X)$ and all $t>0$. Then

as $p\to +\mathrm{\infty }$ for all $u_1,\dots ,u_{n-1}\in \mathcal{U}(X)$ and all $t>0$. So,

$$N(H(u_1\cdots u_{n-1})-H(u_1)\cdots H(u_{n-1}),t)=1$$

for all $u_1,\dots ,u_{n-1}\in \mathcal{U}(X)$ and all $t>0$. Thus,

$$H(u_1\cdots u_{n-1})=H(u_1)\cdots H(u_{n-1}).$$

(15)

Since H is 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 (15) 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 $x,y\in X$. Thus by induction, one can easily show that $H(x_1\cdots x_{n-1})=H(x_1)\cdots H(x_{n-1})$. By (4)

$$N(\frac{f(\frac{u_1^{\ast }}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}-{\frac{f(\frac{u_1}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}}^{\ast },\frac{t}{{(n-m+1)}^{-p}})\ge \frac{t}{t+\phi (\frac{u_1}{{(n-m+1)}^p},0,\dots ,0)}$$

for all $u_1\in \mathcal{U}(X)$ and all $t>0$. So,

$$\begin{array}{rcl}N(\frac{f(\frac{u_1^{\ast }}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}-{\frac{f(\frac{u_1}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}}^{\ast },t)& \ge & \frac{\frac{t}{{(n-m+1)}^p}}{\frac{t}{{(n-m+1)}^p}+\phi (\frac{u_1}{{(n-m+1)}^p},0,\dots ,0)}\\ \ge & \frac{\frac{t}{{(n-m+1)}^p}}{\frac{t}{{(n-m+1)}^p}+\frac{L^p\phi (u_1,0,\dots ,0)}{{(n-m+1)}^p}}\end{array}$$

for all $u_1\in \mathcal{U}(X)$ and all $t>0$. Since ${lim}_{p\to +\mathrm{\infty }}\frac{\frac{t}{{(n-m+1)}^p}}{\frac{t}{{(n-m+1)}^p}+\frac{L^p\phi (u_1,0,\dots ,0)}{{(n-m+1)}^p}}=1$ for all $u_1\in \mathcal{U}(X)$ and all $t>0$, we get

$$N(H\left(u_1^{\ast }\right)-H{(u_1)}^{\ast },t)=1$$

for all $u_1\in \mathcal{U}(X)$ and all $t>0$. Thus,

$$H\left(u_1^{\ast }\right)=H{(u_1)}^{\ast }$$

(16)

for all $u_1\in \mathcal{U}(X)$ . Since H is ℂ-linear and each $x\in X$ is a finite linear combination of unitary elements, i.e., $x={\sum }_{j=1}^m{\lambda }_j{u}_j$ (${\lambda }_j\in \mathbb{C}$, $u_j\in \mathcal{U}(X)$), it follows from (16) 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 3.4 Let $\phi :X^n\to [0,\mathrm{\infty })$ be a function such that there exists an $L<1$ 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 satisfying $f(0)=0$, (2), (13) and (14). Then the limit $A(x):=N\text{-}{lim}_{p\to \mathrm{\infty }}\frac{f({(n-m+1)}^{p}x)}{{(n-m+1)}^p}$ exists for each $x\in X$ and defines a fuzzy ∗-homomorphism $H:X\to Y$ such that

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

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

4 Approximate derivations on fuzzy Banach ∗-algebras

In this section, we assume that $(X,N)$ is a fuzzy Banach ∗-algebra. Using fixed point method, we prove the Hyers-Ulam stability of derivations on fuzzy Banach ∗-algebras related to functional equation (1).

Theorem 4.1 Let $\phi :X^n\to [0,\mathrm{\infty })$ be a function such that there exists an $L<\frac{1}{{(n-m+1)}^{n-2}}$ 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 satisfying $f(0)=0$,

(17)

(18)

(19)

for all $x_1,\dots ,x_{n-1}\in X$ and all $t>0$. 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 defines a fuzzy ∗-derivation $D:X\to X$ such that

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

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

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

Theorem 4.2 Let $\phi :X^n\to [0,\mathrm{\infty })$ be a function such that there exists an $L<1$ 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 satisfying $f(0)=0$, (17), (18) and (19). 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 defines a fuzzy ∗-derivation $D:X\to X$ such that

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

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