Stability of homomorphisms on fuzzy Lie $C^{\ast }$-algebras via fixed point method

Abstract

In this paper, first, we define fuzzy $C^{\ast }$-algebras and fuzzy Lie $C^{\ast }$-algebras; then, using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy $C^{\ast }$-algebras and fuzzy Lie $C^{\ast }$-algebras for an m-variable additive functional equation.

MSC: 39A10, 39B52, 39B72, 46L05, 47H10, 46B03.

1 Introduction and preliminaries

The stability problem of functional equations originated with a question of Ulam [1] concerning the stability of group homomorphisms: let $(G_1,\ast )$ be a group and let $(G_2,\diamond ,d)$ be a metric group with the metric $d(\cdot ,\cdot )$. Given $ϵ>0$, does there exist a $\delta (ϵ)>0$ such that if a mapping $h:G_1\to G_2$ satisfies the inequality $d(h(x\ast y),h(x)\diamond h(y))<\delta$ for all $x,y\in G_1$, then there is a homomorphism $H:G_1\to G_2$ with $d(h(x),H(x))<ϵ$ for all $x\in G_1$? If the answer is affirmative, we would say that the equation of homomorphism $H(x\ast y)=H(x)\diamond H(y)$ is stable. We recall a fundamental result in fixed-point theory. Let Ω be a set. A function $d:\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\mathrm{\infty }]$ is called a generalized metric on Ω if d satisfies

• (1) $d(x,y)=0$ if and only if $x=y$;

• (2) $d(x,y)=d(y,x)$ for all $x,y\in \mathrm{\Omega }$;

• (3) $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in \mathrm{\Omega }$.

Theorem 1.1 [2]

Let $(\mathrm{\Omega },d)$ be a complete generalized metric space and let $J:\mathrm{\Omega }\to \mathrm{\Omega }$ be a contractive mapping with Lipschitz constant $L<1$. Then for each given element $x\in \mathrm{\Omega }$, 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) $d(J^{n}x,J^{n+1}x)<\mathrm{\infty }$, $\mathrm{\forall }n\ge n_0$;

• (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 $\mathrm{\Gamma }=\{y\in \mathrm{\Omega }\mid d(J^{n_0}x,y)<\mathrm{\infty }\}$;

• (4) $d(y,y^{\ast })\le \frac{1}{1-L}d(y,Jy)$ for all $y\in \mathrm{\Gamma }$.

In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in fuzzy Lie $C^{\ast }$-algebras for the following additive functional equation [3]:

$$\sum _{i=1}^{m}f(m{x}_i+\sum _{j=1,j\ne i}^m{x}_j)+f\left(\sum _{i=1}^m{x}_i\right)=2f\left(\sum _{i=1}^{m}m{x}_i\right)(m\in \mathbb{N},m\ge 2).$$

(1.1)

We use the definition of fuzzy normed spaces given in [4–10] to investigate a fuzzy version of the Hyers-Ulam stability for the Cauchy-Jensen functional equation in the fuzzy normed algebra setting (see also [11–16]).

Definition 1.2 [4]

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 $c\ne 0$;

($N_4$) $N(x+y,s+t)\ge min\{N(x,s),N(y,t)\}$;

($N_5$) $N(x,\cdot )$ is a non-decreasing function of ℝ and ${lim}_{t\to \mathrm{\infty }}N(x,t)=1$;

($N_6$) for $x\ne 0$, $N(x,\cdot )$ is continuous on ℝ.

The pair $(X,N)$ is called a fuzzy normed vector space.

Definition 1.3 [4]

• (1) 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 ${lim}_{n\to \mathrm{\infty }}N(x_n-x,t)=1$ for all $t>0$. In this case, x is called the limit of the sequence $\{x_n\}$ and we denote it by $N\text{-}{lim}_{n\to \mathrm{\infty }}{x}_n=x$.

• (2) Let $(X,N)$ be a fuzzy normed vector space. A sequence $\{x_n\}$ in X is called Cauchy if for each $\epsilon >0$ 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 $N(x_{n+p}-x_n,t)>1-\epsilon$.

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 $x_0\in X$ if for each sequence $\{x_n\}$ converging to $x_0$ in X, then 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 [4, 10]).

Definition 1.4 [12]

A fuzzy normed algebra $(X,\mu ,\ast ,{\ast }^{\prime })$ is a fuzzy normed space $(X,N,\ast )$ with algebraic structure such that

($N_7$) $N(xy,ts)\ge N(x,t)\ast N(y,s)$ for all $x,y\in X$ and all $t,s>0$, in which ∗^′ is a continuous t-norm.

Every normed algebra $(X,\parallel \cdot \parallel )$ defines a fuzzy normed algebra $(X,N,min)$, where

$$N(x,t)=\frac{t}{t+\parallel x\parallel }$$

for all $t>0$ iff

$$\parallel xy\parallel \le \parallel x\parallel \parallel y\parallel +s\parallel y\parallel +t\parallel x\parallel (x,y\in X;t,s>0).$$

This space is called the induced fuzzy normed algebra.

Definition 1.5 (1) Let $(X,N,\ast )$ and $(Y,N,\ast )$ be fuzzy normed algebras. An ℝ-linear mapping $f:X\to Y$ is called a homomorphism if $f(xy)=f(x)f(y)$ for all $x,y\in X$.

• (2) An ℝ-linear mapping $f:X\to X$ is called a derivation if $f(xy)=f(x)y+xf(y)$ for all $x,y\in X$.

Definition 1.6 Let $(\mathcal{U},N,\ast ,{\ast }^{\prime })$ be a fuzzy Banach algebra, then an involution on is a mapping $u\to u^{\ast }$ from into which satisfies

• (i) $u^{\ast \ast }=u$ for $u\in \mathcal{U}$;

• (ii) ${(\alpha u+\beta v)}^{\ast }=\overline{\alpha }{u}^{\ast }+\overline{\beta }{v}^{\ast }$;

• (iii) ${(uv)}^{\ast }=v^{\ast }{u}^{\ast }$ for $u,v\in \mathcal{U}$.

If, in addition $N(u^{\ast }u,ts)=N(u,t)\ast N(u,s)$ for $u\in \mathcal{U}$ and $t>0$, then is a fuzzy $C^{\ast }$-algebra.

2 Stability of homomorphisms in fuzzy $C^{\ast }$-algebras

Throughout this section, assume that A is a fuzzy $C^{\ast }$-algebra with norm $N_A$ and that B is a fuzzy $C^{\ast }$-algebra with norm $N_B$.

For a given mapping $f:A\to B$, we define

$$D_{\mu }f(x_1,\dots ,x_m):=\sum _{i=1}^m\mu f(m{x}_i+\sum _{j=1,j\ne i}^m{x}_j)+f\left(\mu \sum _{i=1}^m{x}_i\right)-2f\left(\mu \sum _{i=1}^{m}m{x}_i\right)$$

for all $\mu \in {\mathbb{T}}^1:=\{\nu \in \mathbb{C}:|\nu |=1\}$ and all $x_1,\dots ,x_m\in A$.

Note that a ℂ-linear mapping $H:A\to B$ is called a homomorphism in fuzzy $C^{\ast }$-algebras if H satisfies $H(xy)=H(x)H(y)$ and $H(x^{\ast })=H{(x)}^{\ast }$ for all $x,y\in A$.

We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy $C^{\ast }$-algebras for the functional equation $D_{\mu }f(x_1,\dots ,x_m)=0$.

Theorem 2.1 Let $f:A\to B$ be a mapping for which there are functions $\phi :A^m\times (0,\mathrm{\infty })\to [0,1]$, $\psi :A^2\times (0,\mathrm{\infty })\to [0,1]$ and $\eta :A\times (0,\mathrm{\infty })\to [0,1]$ such that

$$N_B(D_{\mu }f(x_1,\dots ,x_m),t)\ge \phi (x_1,\dots ,x_m,t),$$

(2.1)

$$\underset{j\to \mathrm{\infty }}{lim}\phi (m^j{x}_1,\dots ,m^j{x}_m,m^{j}t)=1,$$

(2.2)

$$N_B(f(xy)-f(x)f(y),t)\ge \psi (x,y,t),$$

(2.3)

$$\underset{j\to \mathrm{\infty }}{lim}\psi (m^{j}x,m^{j}y,m^{2j}t)=1,$$

(2.4)

$$N_B(f\left(x^{\ast }\right)-f{(x)}^{\ast },t)\ge \eta (x,t),$$

(2.5)

$$\underset{j\to \mathrm{\infty }}{lim}\eta (m^{j}x,m^{j}t)=1$$

(2.6)

for all $\mu \in {\mathbb{T}}^1$, all $x_1,\dots ,x_m,x,y\in A$ and $t>0$. If there exists an $L<1$ such that

$$\phi (mx,0,\dots ,0,mLt)\ge \phi (x,0,\dots ,0,t)$$

(2.7)

for all $x\in A$ and $t>0$, then there exists a unique homomorphism $H:A\to B$ such that

$$N_B(f(x)-H(x),t)\ge \phi (x,0,\dots ,0,(m-mL)t)$$

(2.8)

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

Proof Consider the set $X:=\{g:A\to B\}$ and introduce the generalized metric on X:

$$d(g,h)=inf\{C\in {\mathbb{R}}_{+}:N_B(g(x)-h(x),Ct)\ge \phi (x,0,\dots ,0,t),\mathrm{\forall }x\in A,t>0\}.$$

It is easy to show that $(X,d)$ is complete. Now, we consider the linear mapping $J:X\to X$ such that $Jg(x):=\frac{1}{m}g(mx)$ for all $x\in A$. By Theorem 3.1 of [17], $d(Jg,Jh)\le Ld(g,h)$ for all $g,h\in X$. Letting $\mu =1$, $x=x_1$ and $x_2=\cdots =x_m=0$ in equation (2.1), we get

$$N_B(f(mx)-mf(x),t)\ge \phi (x,0,\dots ,0,t)$$

(2.9)

for all $x\in A$ and $t>0$. Therefore

$$N_B(f(x)-\frac{1}{m}f(mx),t)\ge \phi (x,0,\dots ,0,mt)$$

for all $x\in A$ and $t>0$. Hence $d(f,Jf)\le \frac{1}{m}$. By Theorem 1.1, there exists a mapping $H:A\to B$ such that

• (1) H is a fixed point of J, i.e.,

  $$H(mx)=mH(x)$$

  (2.10)

for all $x\in A$. The mapping H is a unique fixed point of J in the set

$$Y=\{g\in X:d(f,g)<\mathrm{\infty }\}.$$

This implies that H is a unique mapping satisfying equation (2.10) such that there exists $C\in (0,\mathrm{\infty })$ satisfying

$$N_B(H(x)-f(x),Ct)\ge \phi (x,0,\dots ,0,t)$$

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

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

  $$\underset{n\to \mathrm{\infty }}{lim}\frac{f(m^{n}x)}{m^n}=H(x)$$

  (2.11)

for all $x\in A$.

• (3) $d(f,H)\le \frac{1}{1-L}d(f,Jf)$, which implies the inequality $d(f,H)\le \frac{1}{m-mL}$. This implies that the inequality (2.8) holds.

It follows from equations (2.1), (2.2), and (2.11) that

$$\begin{array}{c}{N}_B(\sum _{i=1}^{m}H(m{x}_i+\sum _{j=1,j\ne i}^m{x}_j)+H\left(\sum _{i=1}^m{x}_i\right)-2H\left(\sum _{i=1}^{m}m{x}_i\right),t)\hfill \\ =\underset{n\to \mathrm{\infty }}{lim}{N}_B(\sum _{i=1}^{m}f(m^{n+1}{x}_i+\sum _{j=1,j\ne i}^m{m}^n{x}_j)+f\left(\sum _{i=1}^m{m}^n{x}_i\right)-2f\left(\sum _{i=1}^m{m}^{n+1}{x}_i\right),m^{n}t)\hfill \\ \le \underset{n\to \mathrm{\infty }}{lim}\phi (m^n{x}_1,\dots ,m^n{x}_m,m^{n}t)=1\hfill \end{array}$$

for all $x_1,\dots ,x_m\in A$ and $t>0$. So

$$\sum _{i=1}^{m}H(m{x}_i+\sum _{j=1,j\ne i}^m{x}_j)+H\left(\sum _{i=1}^m{x}_i\right)=2H\left(\sum _{i=1}^{m}m{x}_i\right)$$

for all $x_1,\dots ,x_m\in A$.

By a similar method to above, we get $\mu H(mx)=H(m\mu x)$ for all $\mu \in {\mathbb{T}}^1$ and all $x\in A$. Thus one can show that the mapping $H:A\to B$ is ℂ-linear.

It follows from equations (2.3), (2.4), and (2.11) that

$$\begin{array}{rcl}{N}_B(H(xy)-H(x)H(y),t)& =& \underset{n\to \mathrm{\infty }}{lim}{N}_B(f\left(m^{n}xy\right)-f\left(m^{n}x\right)f\left(m^{n}y\right),m^{n}t)\\ \le & \underset{n\to \mathrm{\infty }}{lim}\psi (m^{n}x,m^{n}y,m^{2n}t)=1\end{array}$$

for all $x,y\in A$. So $H(xy)=H(x)H(y)$ for all $x,y\in A$. Thus $H:A\to B$ is a homomorphism satisfying equation (2.7), as desired.

Also by equations (2.5), (2.6), (2.11), and by a similar method we have $H(x^{\ast })=H{(x)}^{\ast }$. □

3 Stability of homomorphisms in fuzzy Lie $C^{\ast }$-algebras

A fuzzy $C^{\ast }$-algebra , endowed with the Lie product

$$[x,y]:=\frac{xy-yx}{2}$$

on , is called a fuzzy Lie $C^{\ast }$-algebra (see [18–20]).

Definition 3.1 Let A and B be fuzzy Lie $C^{\ast }$-algebras. A ℂ-linear mapping $H:A\to B$ is called a fuzzy Lie $C^{\ast }$-algebra homomorphism if $H([x,y])=[H(x),H(y)]$ for all $x,y\in A$.

Throughout this section, assume that A is a fuzzy Lie $C^{\ast }$-algebra with norm $N_A$ and that B is a fuzzy Lie $C^{\ast }$-algebra with norm $N_B$.

We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy Lie $C^{\ast }$-algebras for the functional equation $D_{\mu }f(x_1,\dots ,x_m)=0$.

Theorem 3.2 Let $f:A\to B$ be a mapping for which there are functions $\phi :A^m\times (0,\mathrm{\infty })\to [0,1]$ and $\psi :A^2\times (0,\mathrm{\infty })\to [0,1]$ such that

$$\underset{j\to \mathrm{\infty }}{lim}\phi (m^j{x}_1,\dots ,m^j{x}_m,m^{j}t)=1,$$

(3.1)

$$N_B(D_{\mu }f(x_1,\dots ,x_m),t)\ge \phi (x_1,\dots ,x_m,t),$$

(3.2)

$$N_B(f([x,y])-[f(x),f(y)],t)\ge \psi (x,y,t),$$

(3.3)

$$\underset{j\to \mathrm{\infty }}{lim}\psi (m^{j}x,m^{j}y,m^{2j}t)=1$$

(3.4)

for all $\mu \in {\mathbb{T}}^1$, all $x_1,\dots ,x_m,x,y\in A$ and $t>0$. If there exists an $L<1$ such that

$$\phi (mx,0,\dots ,0,mlt)\ge \phi (x,0,\dots ,0,t)$$

for all $x\in A$ and $t>0$, then there exists a unique homomorphism $H:A\to B$ such that

$$N_B(f(x)-H(x),t)\ge \phi (x,0,\dots ,0,(m-mL)t)$$

(3.5)

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

Proof By the same reasoning as the proof of Theorem 2.1, we can find that the mapping $H:A\to B$ is given by

$$H(x)=\underset{n\to \mathrm{\infty }}{lim}\frac{f(m^{n}x)}{m^n}$$

for all $x\in A$.

It follows from equation (3.3) that

$$\begin{array}{rcl}{N}_B(H([x,y])-[H(x),H(y)],t)& =& \underset{n\to \mathrm{\infty }}{lim}{N}_B(f(m^{2n}[x,y])-[f\left(m^{n}x\right),f\left(m^{n}y\right)],m^{2n}t)\\ \ge & \underset{n\to \mathrm{\infty }}{lim}\psi (m^{n}x,m^{n}y,m^{2n}t)=1\end{array}$$

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

$$H([x,y])=[H(x),H(y)]$$

for all $x,y\in A$.

Thus $H:A\to B$ is a fuzzy Lie $C^{\ast }$-algebra homomorphism satisfying equation (3.5), as desired. □

Corollary 3.3 Let $0<r<1$ and θ be nonnegative real numbers, and let $f:A\to B$ be a mapping such that

$$N_B(D_{\mu }f(x_1,\dots ,x_m),t)\ge \frac{t}{t+\theta ({\parallel x_1\parallel }_A^r+{\parallel x_2\parallel }_A^r+\cdots +{\parallel x_m\parallel }_A^r)},$$

(3.6)

$$N_B(f([x,y])-[f(x),f(y)],t)\ge \frac{t}{t+\theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r}$$

(3.7)

for all $\mu \in {\mathbb{T}}^1$, all $x_1,\dots ,x_m,x,y\in A$ and $t>0$. Then there exists a unique homomorphism $H:A\to B$ such that

$$N_B(f(x)-H(x),t)\le \frac{t}{t+\frac{\theta }{m-m^r}{\parallel x\parallel }_A^r}$$

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

Proof The proof follows from Theorem 3.2 by taking

$$\begin{array}{c}\phi (x_1,\dots ,x_m,t)=\frac{t}{t+\theta ({\parallel x_1\parallel }_A^r+{\parallel x_2\parallel }_A^r+\cdots +{\parallel x_m\parallel }_A^r)},\hfill \\ \psi (x,y,t):=\frac{t}{t+\theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r}\hfill \end{array}$$

for all $x_1,\dots ,x_m,x,y\in A$ and $t>0$. Putting $L=m^{r-1}$, we get the desired result. □