Fuzzy n-Jordan *-homomorphisms in induced fuzzy C*-algebras

Abstract

Using fixed point method, we prove the fuzzy version of the Hyers-Ulam stability of n-Jordan *-homomorphisms in induced fuzzy C*-algebras associated with the following functional equation

$$f\left(\frac{x+y+z}{3}\right)+f\left(\frac{x-2y+z}{3}\right)+f\left(\frac{x+y-2z}{3}\right)=f\left(x\right).$$

Mathematics Subject Classification (2010): Primary 46S40; 47S40; 39B52; 47H10; 46L05.

1. Introduction and preliminaries

The stability of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms in 1940. More precisely, he proposed the following problem: Given a group , a metric group $\left({\mathcal{G}}^{\prime },d\right)$ and ε > 0, does there exist a δ > 0 such that if a function $f:\mathcal{G}\to {\mathcal{G}}^{\prime }$ satisfies the inequality d(f(xy), f(x)f(y)) < δ for all $x,y\in \mathcal{G}$, then there exists a homomorphism $T:\mathcal{G}\to {\mathcal{G}}^{\prime }$ such that d(f(x), T(x)) < ε for all $x\in \mathcal{G}?$ Hyers [2] gave a partial solution of the Ulam's problem for the case of approximate additive mappings under the assumption that and ${\mathcal{G}}^{\prime }$are Banach spaces. Aoki [3] generalized the Hyers' theorem for approximately additive mappings. Rassias [4] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences.

Let X be a set. A function d: X × X → [0, ∞] is called a generalized metric on X 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 ∈ X;

• (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.

We recall a fundamental result in fixed point theory.

Theorem 1.1. [5, 6]Let (X, d) be a complete generalized metric space and let J: X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either

$$d\left(J^{n}x,J^{n+1}x\right)=\infty$$

for all nonnegative integers n or there exists a positive integer n ₀ such that

• (1) d(Jⁿx, Jⁿ⁺¹x) < ∞, ∀n ≥ n₀;

• (2) the sequence {Jⁿx} converges to a fixed point y* of J;

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

• (4) $d\left(y,y^{*}\right)\le \frac{1}{1-L}d\left(y,Jy\right)$ for all y ∈ Y.

Isac and Rassias [7] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [8–12]).

Katsaras [13] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematics have defined fuzzy normed on a vector space from various points of view [14–20]. In particular, Bag and Samanta [21] following Cheng and Mordeson [22], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [23]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [24].

We use the definition of fuzzy normed spaces given in [16, 17, 21] to investigate a fuzzy version of the Hyers-Ulam stability of n-Jordan *-homomorphisms in induced fuzzy C*- algebras associated with the following functional equation

$$f\left(\frac{x+y+z}{\mathsf{\text{3}}}\right)+f\left(\frac{x-2y+z}{\mathsf{\text{3}}}\right)+f\left(\frac{x+y-2z}{\mathsf{\text{3}}}\right)=f\left(x\right).$$

Definition 1.2. [16–18, 21] Let be a complex vector space. A function $N:\mathcal{X}\times ℝ\to \left[0,1\right]$ is called a fuzzy norm on if for all $x,y\in \mathcal{X}$ and all s, t ∈ ℝ,

N₁: N(x, t) = 0 for t ≤ 0

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

N₃: $N\left(cx,t\right)=N\left(x,\frac{t}{\left|c\right|}\right)$ if c ∈ ℂ-{0}

N₄: N(x + y, s + t) ≥ min{N(x, s), N(y, t)}

N₅: N(x, ·) is a non-decreasing function of ℝ and lim_t→∞N(x, t) = 1

N₆: for x ≠ 0, N(x, .) is continuous on ℝ.

The pair $\left(\mathcal{X},N\right)$ is called a fuzzy normed vector space.

Definition 1.3. [16–18, 21] Let $\left(\mathcal{X},N\right)$ be a fuzzy normed vector space.

• (1) A sequence {x _n } in χ is said to be convergent if there exists an x ∈ χ such that lim_n→∞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- lim _n→∞ x _n = x.

• (2) A sequence {x _n } in χ is called Cauchy if for each ε > 0 and each t > 0 there exists an n₀ ∈ ℕ such that for all n ≥ n₀ 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 $f:\mathcal{X}\to \mathcal{Y}$ between fuzzy normed vector space $\mathcal{X},\mathcal{Y}$ is continuous at point $x_0\in \mathcal{X}$ if for each sequence {x _n } converging to x₀ in , then the sequence {f(x _n )} converges to f (x₀). If $f:\mathcal{X}\to \mathcal{Y}$ is continuous at each $x\in \mathcal{X}$, then $f:\mathcal{X}\to \mathcal{Y}$ is said to be continuous on (see [24]).

Definition 1.4. Let be a *-algebra and $\left(\mathcal{X},N\right)$ a fuzzy normed space.

• (1) The fuzzy normed space $\left(\mathcal{X},N\right)$ is called a fuzzy normed *-algebra if

  $$N\left(xy,st\right)\ge N\left(x,s\right)\cdot N\left(y,t\right)\mathsf{\text{\&}}N\left(x^{*},t\right)=N\left(x,t\right)$$

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

Example 1.5. Let $\left(\mathcal{X}\mathsf{\text{, }}∥\cdot ∥\right)$ be a normed *-algebra. let

$$N\left(x,t\right)=\left\{\begin{array}{ccc}\hfill \frac{t}{t+∥x∥},\hfill & \hfill t\mathsf{\text{ > 0,}}\hfill & \hfill x\in \mathcal{X}\hfill \\ \hfill 0,\hfill & \hfill t\le \mathsf{\text{0,}}\hfill & \hfill x\in \mathcal{X}\mathsf{\text{.}}\hfill \end{array}\right.$$

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

Definition 1.6. Let $\left(\mathcal{X}\mathsf{\text{, }}∥\cdot ∥\right)$ be a C*-algebra and $N_{\mathcal{X}}$ a fuzzy norm on .

• (1) The fuzzy normed *-algebra $\left(\mathcal{X},N_{\mathcal{X}}\right)$ is called an induced fuzzy normed *-algebra

• (2) The fuzzy Banach *-algebra $\left(\mathcal{X},N_{\mathcal{X}}\right)$ is called an induced fuzzy C*-algebra.

Definition 1.7. Let $\left(\mathcal{X},N_{\mathcal{X}}\right)$ and $\left(\mathcal{Y},N\right)$ be induced fuzzy normed *-algebras. Then a ℂ-linear mapping $H:\left(\mathcal{X},N_{\mathcal{X}}\right)\to \left(\mathcal{Y}\mathsf{\text{, }}N\right)$ is called a fuzzy n-Jordan *-homomorphism if

$$H\left(x^n\right)=H{\left(x\right)}^n\mathsf{\text{\&}}H\left(x^{*}\right)=H{\left(x\right)}^{*}$$

for all $x\in \mathcal{X}$.

Throughout this article, assume that $\left(\mathcal{X},N\right)$ is an induced fuzzy normed *-algebra and that $\left(\mathcal{Y},N\right)$ is an induced fuzzy C*-algebra.

2. Main results

Lemma 2.1. Let $\mathsf{\text{(}}\mathcal{Z}\mathsf{\text{, }}N)$ be a fuzzy normed vector space and let $f:\mathcal{X}\to \mathcal{Z}$ be a mapping such that

$$N\left(f\left(\frac{x+y+z}{\mathsf{\text{3}}}\right)+f\left(\frac{x-2y+z}{3}\right)+f\left(\frac{x+y-2z}{3}\right),t\right)\ge N\left(f\left(x\right),\frac{t}{2}\right)$$

(2.1)

for all $x,y,z\in \mathcal{X}$ and all t > 0. Then f is additive, i.e., f(x + y) = f(x) + f(y) for all $x,y\in \mathcal{X}$.

Proof. Letting x = y = z = 0 in (2.1), we get

$$N\left(3f\left(0\right),t\right)=N\left(f\left(0\right),\frac{t}{3}\right)\ge N\left(f\left(0\right),\frac{t}{2}\right)$$

for all t > 0. By N₅ and N₆, N(f(0), t) = 1 for all t > 0. It follows from N₂ that f(0) = 0.

Letting z = -x, y = x, x = 0 in (2.1), we get

$$N\left(f\left(0\right)+f\left(-x\right)+f\left(x\right),t\right)\ge N\left(f\left(0\right),\frac{t}{2}\right)=1$$

for all t > 0. It follows from N₂ that f(-x) + f(x) = 0 for all $x\in \mathcal{X}$. So

$$f\left(-x\right)=-f\left(x\right)$$

for all $x\in \mathcal{X}$.

Letting x = 0 and replacing y, z by 3y, 3z, respectively, in (2.1), we get

$$N\left(f\left(y+z\right)+f\left(-2y+z\right)+f\left(y-2z\right),t\right)\ge N\left(f\left(0\right),\frac{t}{2}\right)=1$$

for all t > 0. It follows from N₂ that

$$f\left(y+z\right)+f\left(-\mathsf{\text{2}}y+z\right)+f\left(y-\mathsf{\text{2}}z\right)=0$$

(2.2)

for all $y,z\in \mathcal{X}$. Let t = 2y-z and s = 2z-y in (2.2), we obtain

$$f\left(t+s\right)=f\left(t\right)+f\left(s\right)$$

for all $t,s\in \mathcal{X}$, as desired. □

Using fixed point method, we prove the Hyers-Ulam stability of fuzzy n-Jordan *-homomorphisms in induced fuzzy C*-algebras.

Theorem 2.2. Let $\phi :{\mathcal{X}}^3\to \left[0,\infty \right)$ be a function such that there exists an $L<\frac{3}{3^n}$ with

$$\phi \left(\frac{x}{3},\frac{y}{3},\frac{z}{3}\right)\le \frac{L}{3}\phi \left(x,y,z\right)$$

(2.3)

for all $x,y,z\in \mathcal{X}$. Let $f:\mathcal{X}\to \mathcal{Y}$ be a mapping such that

$$\begin{array}{c}N\left(f\left(\frac{\mu x+\mu y+\mu z}{3}\right)+f\left(\frac{\mu x-\mathsf{\text{2}}\mu y+\mu z}{3}\right)+f\left(\frac{\mu x+\mu y-\mathsf{\text{2}}\mu z}{3}\right)-\mu f\left(x\right),t\right)\hfill \\ \ge \frac{t}{t+\phi \left(x,y,z\right)},\hfill \end{array}$$

(2.4)

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

(2.5)

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

(2.6)

for all $x,y,z\in \mathcal{X}$, all t > 0 and all $\mu \in {\mathbb{T}}^{\mathsf{\text{1}}}:=\left\{\lambda \in ℂ:\left|\lambda \right|=\mathsf{\text{1}}\right\}$. Then $H\left(x\right)=N-\mathsf{\text{li}}{\mathsf{\text{m}}}_{n\to \infty }{\mathsf{\text{3}}}^{n}f\left(\frac{x}{3^n}\right)$ exists for each $x\in \mathcal{X}$ and defines a fuzzy n-Jordan *-homomorphism $H:\mathcal{X}\to \mathcal{Y}$ such that

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

(2.7)

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

Proof. Letting μ = 1 and y = z = 0 in (2.4), we get

$$N\left(3f\left(\frac{x}{3}\right)-f\left(x\right),t\right)\ge \frac{t}{t+\phi \left(x,0,0\right)}$$

(2.8)

for all $x\in \mathcal{X}$.

Consider the set

$$S:=\left\{g:\mathcal{X}\to \mathcal{Y}\right\}$$

and introduce the generalized metric on S:

$$d\left(g,h\right)=\mathsf{\text{inf}}\left\{\alpha \in {ℝ}_{+}:N\left(g\left(x\right)-h\left(x\right),\alpha t\right)\ge \frac{t}{t+\phi \left(x,0,0\right)},\forall x\in \mathcal{X},\forall t>0\right\},$$

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see the proof of [[25], Lemma 2.1]).

Now we consider the linear mapping J: S → S such that

$$Jg\left(x\right):=\mathsf{\text{3}}g\left(\frac{x}{3}\right)$$

for all x ∈ X.

Let g, h ∈ S be given such that d(g, h) = ε. Then

$$N\left(g\left(x\right)-h\left(x\right),\epsilon t\right)\ge \frac{t}{t+\phi \left(x,0,0\right)}$$

for all $x\in \mathcal{X}$ and all t > 0. Hence

$$\begin{array}{cc}\hfill N\left(Jg\left(x\right)-Jh\left(x\right),L\epsilon t\right)& =N\left(3g\left(\frac{x}{3}\right)-3h\left(\frac{x}{3}\right),L\epsilon t\right)=N\left(g\left(\frac{x}{3}\right)-h\left(\frac{x}{3}\right),\frac{L}{3}\epsilon t\right)\hfill \\ \ge \frac{\frac{Lt}{3}}{\frac{Lt}{3}+\phi \left(\frac{x}{3},0,0\right)}\ge \frac{\frac{Lt}{3}}{\frac{Lt}{3}+\frac{L}{3}\phi \left(x,0,0\right)}\hfill \\ =\frac{t}{t+\phi \left(x,0,0\right)}\hfill \end{array}$$

for all $x\in \mathcal{X}$ and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that

$$d\left(Jg,Jh\right)\le Ld\left(g,h\right)$$

for all g, h ∈ S.

It follows from (2.8) that d(f, Jf) ≤ 1.

By Theorem 1.1, there exists a mapping $H:\mathcal{X}\to \mathcal{Y}$ satisfying the following:

• (1) H is a fixed point of J, i.e.,

  $$H\left(\frac{x}{3}\right)=\frac{1}{3}H\left(x\right)$$

  (2.9)

for all $x\in \mathcal{X}$. The mapping H is a unique fixed point of J in the set

$$M=\left\{g\in S:d\left(f,g\right)<\infty \right\}.$$

This implies that H is a unique mapping satisfying (2.9) such that there exists a α ∈ (0, ∞) satisfying

$$N\left(f\left(x\right)-H\left(x\right),\alpha t\right)\ge \frac{t}{t\mathsf{\text{ + }}\phi \mathsf{\text{ (}}x\mathsf{\text{, 0, 0)}}}$$

for all $x\in \mathcal{X}$;

• (2) d(J^k f, H) → 0 as k → ∞. This implies the equality

  $$N-\underset{k\to \infty }{\mathsf{\text{lim}}}{3}^{k}f\left(\frac{x}{3^k}\right)=H\left(x\right)$$

for all $x\in \mathcal{X}$;

• (3) $d\left(f,H\right)\le \frac{1}{1-L}d\left(f,Jf\right)$, which implies the inequality

  $$d\left(f,H\right)\le \frac{1}{1-L}.$$

This implies that the inequality (2.7) holds.

It follows from (2.3) that

$$\sum _{k=0}^{\infty }{3}^k\phi \left(\frac{x}{3^k},\frac{y}{3^k},\frac{z}{3^k}\right)<\infty$$

for all $x,y,z\in \mathcal{X}$.

By (2.4),

$$\begin{array}{c}N\left(3^{k}f\left(\frac{\mu x+\mu y+\mu z}{3^{k+1}}\right)+3^{k}f\left(\frac{\mu x-2\mu y+\mu z}{3^{k+1}}\right)+3^{k}f\left(\frac{\mu x+\mu y-2\mu z}{3^{k+1}}\right)-\mu 3^{k}f\left(\frac{x}{3^k}\right),3^{k}t\right)\hfill \\ \ge \frac{t}{t+\phi \left(\frac{x}{3^k},\frac{y}{3^k},\frac{z}{3^k}\right)}\hfill \end{array}$$

for all $x,y,z\in \mathcal{X}$, all t > 0 and all $\mu \in {\mathbb{T}}^{\mathsf{\text{1}}}$. So

$$\begin{array}{c}N\left(3^{k}f\left(\frac{\mu x+\mu y+\mu z}{3^{k+1}}\right)+3^{k}f\left(\frac{\mu x-2\mu y+\mu z}{3^{k+1}}\right)+3^{k}f\left(\frac{\mu x+\mu y-2\mu z}{3^{k+1}}\right)-\mu 3^{k}f\left(\frac{x}{3^k}\right),t\right)\hfill \\ \ge \frac{\frac{t}{3^k}}{\frac{t}{3^k}+\phi \left(\frac{x}{3^k},\frac{y}{3^k},\frac{z}{3^k}\right)}=\frac{t}{t+3^k\phi \left(\frac{x}{3^k},\frac{y}{3^k},\frac{z}{3^k}\right)}\hfill \end{array}$$

for all $x,y,z\in \mathcal{X}$, all t > 0 and all $\mu \in {\mathbb{T}}^{\mathsf{\text{1}}}$. Since ${\mathsf{\text{lim}}}_{k\to \infty }\frac{t}{t+3^k\phi \left(\frac{x}{3^k},\frac{y}{3^k},\frac{z}{3^k}\right)}=1$ for all $x,y,z\in \mathcal{X}$ and all t > 0,

$$N\left(H\left(\frac{\mu x+\mu y+\mu z}{3}\right)+H\left(\frac{\mu x-2\mu y+\mu z}{3}\right)+H\left(\frac{\mu x+\mu y-2\mu z}{3}\right)-\mu H\left(x\right),t\right)=1$$

for all $x,y,z\in \mathcal{X}$, all t > 0 and all $\mu \in {\mathbb{T}}^{\mathsf{\text{1}}}$. Thus

$$H\left(\frac{\mu x+\mu y+\mu z}{3}\right)+H\left(\frac{\mu x-2\mu y+\mu z}{3}\right)+H\left(\frac{\mu x+\mu y-2\mu z}{3}\right)=\mu H\left(x\right)$$

(2.10)

for all $x,y,z\in \mathcal{X}$, all t > 0 and all $\mu \in {\mathbb{T}}^{\mathsf{\text{1}}}$. Letting x = y = z = 0 in (2.10), we get H(0) = 0. Let μ = 1 and x = 0 in (2.10). By the same reasoning as in the proof of Lemma 2.1, one can easily show that H is additive. Letting y = z = 0 in (2.10), we get

$$H\left(\mu x\right)=3H\left(\frac{\mu x}{3}\right)=\mu H\left(x\right)$$

for all $x\in \mathcal{X}$ and all $\mu \in {\mathbb{T}}^{\mathsf{\text{1}}}$. By [[26], Theorem 2.1], the mapping $H:\mathcal{X}\to \mathcal{Y}$ is ℂ-linear.

By (2.5),

$$N\left(3^{nk}f\left(\frac{x^k}{3^{nk}}\right)-3^{nk}f{\left(\frac{x}{3^k}\right)}^n,3^{nk}t\right)\ge \frac{t}{t+\phi \left(\frac{x}{3^k},0,0\right)}$$

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

$$N\left(3^{nk}f\left(\frac{x^k}{3^{nk}}\right)-3^{nk}f{\left(\frac{x}{3^k}\right)}^n,t\right)\ge \frac{\frac{t}{3^{nk}}}{\frac{t}{3^{nk}}+\phi \left(\frac{x}{3^k},0,0\right)}=\frac{t}{t+{\left(3^{n-1}L\right)}^k\phi \left(x,0,0\right)}$$

for all $x\in \mathcal{X}$ and all t > 0. Since ${\mathsf{\text{lim}}}_{k\to \infty }\frac{t}{t+{\left(3^{n-1}L\right)}^k\phi \left(x,0,0\right)}=1$ for all $x\in \mathcal{X}$ and all t > 0,

$$N\left(H\left(x^n\right)-H{\left(x\right)}^n,t\right)=\mathsf{\text{1}}$$

for all $x\in \mathcal{X}$ and all t > 0. Thus, H(xⁿ) - H(x)ⁿ = 0 for all $x\in \mathcal{X}$.

By (2.6),

$$N\left(3^{k}f\left(\frac{x^{*}}{3^k}\right)-3^{k}f{\left(\frac{x}{3^k}\right)}^{*},3^{k}t\right)\ge \frac{t}{t+\phi \left(\frac{x}{3^k},0,0\right)}$$

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

$$N\left(3^{k}f\left(\frac{x^{*}}{3^k}\right)-3^{k}f{\left(\frac{x}{3^k}\right)}^{*},t\right)\ge \frac{\frac{t}{3^k}}{\frac{t}{3^k}+\phi \left(\frac{t}{3^k},0,0\right)}=\frac{t}{t+3^k\phi \left(\frac{t}{3^k},0,0\right)}$$

for all $x\in \mathcal{X}$ and all t > 0. Since ${\mathsf{\text{lim}}}_{k\to \infty }\frac{t}{t+3^k\phi \left(\frac{t}{3^k},0,0\right)}=1$ for all $x\in \mathcal{X}$ and all t > 0,

$$N\left(H\left(x^{*}\right)-H{\left(x\right)}^{*},t\right)=\mathsf{\text{1}}$$

for all $x\in \mathcal{X}$ and all t > 0. Thus, H(x*) - H(x)* = 0 for all $x\in \mathcal{X}$.

Therefore, the mapping $H:\mathcal{X}\to \mathcal{Y}$ is a fuzzy n-Jordan *-homomorphism. □

Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > n. Let be a normed vector space with norm || · ||. Let $f:\mathcal{X}\to \mathcal{Y}$ be a mapping satisfying

$$\begin{array}{c}N\left(f\left(\frac{\mu x+\mu y+\mu z}{3}\right)+f\left(\frac{\mu x-2\mu y+\mu z}{3}\right)+f\left(\frac{\mu x+\mu y-2\mu z}{3}\right)-\mu f\left(x\right),t\right)\hfill \\ \ge \frac{t}{t+\theta \left({∥x∥}^p+{∥y∥}^p+{∥z∥}^p\right)},\hfill \end{array}$$

(2.11)

$$N\left(f\left(x^n\right)-f{\left(x\right)}^n,t\right)\ge \frac{t}{t+\theta {∥x∥}^p},$$

(2.12)

$$N\left(f\left(x^{*}\right)-f{\left(x\right)}^{*},t\right)\ge \frac{t}{t+\theta {∥x∥}^p}$$

(2.13)

for all $x,y,z\in \mathcal{X}$, all t > 0 and all $\mu \in {\mathbb{T}}^{\mathsf{\text{1}}}$. Then $H\left(x\right)=N-{\mathsf{\text{lim}}}_{n\to \infty }{3}^{n}f\left(\frac{x}{3^n}\right)$ exists for each $x\in \mathcal{X}$ and defines a fuzzy n-Jordan *-homomorphism $H:\mathcal{X}\to \mathcal{Y}$ such that

$$N\left(f\left(x\right)-H\left(x\right),t\right)\ge \frac{\left(3^p-3\right)t}{\left(3^p-3\right)t+3^p\theta {∥x∥}^p}$$

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

Proof. The proof follows from Theorem 2.2 by taking

$$\phi \left(x,y,z\right)=\theta \left({∥x∥}^p+{∥y∥}^p+{∥z∥}^p\right)$$

and L = 3^l-p.

Theorem 2.4. Let $\phi :{\mathcal{X}}^{\mathsf{\text{3}}}\to \mathsf{\text{[0,}}\infty \mathsf{\text{)}}$ be a function such that there exists an L < 1 with

$$\phi \left(x,y,z\right)\le 3L\phi \left(\frac{x}{3},\frac{y}{3},\frac{z}{3}\right)$$

for all $x,y,z\in \mathcal{X}$. Let $f:\mathcal{X}\to \mathcal{Y}$ be a mapping satisfying (2.4), (2.5), and (2.6). Then $H\left(x\right)=N-{\mathsf{\text{lim}}}_{n\to \infty }\frac{1}{3^n}f\left(3^{n}x\right)$ exists for each $x\in \mathcal{X}$ and defines a fuzzy n-Jordan *-homomorphism $H:\mathcal{X}\to \mathcal{Y}$ such that

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

(2.14)

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

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2.

Consider the linear mapping J: S → S such that

$$Jg\left(x\right):=\frac{1}{3}g\left(3x\right)$$

for all $x\in \mathcal{X}$.

It follows from (2.8) that

$$N\left(f\left(x\right)-\frac{1}{3}f\left(3x\right),\frac{1}{3}t\right)\ge \frac{t}{t+\phi \left(3x,0,0\right)}\ge \frac{t}{t+3L\phi \left(x,0,0\right)}$$

for all $x\in \mathcal{X}$ and all t > 0. So d(f, Jf) ≤ L. Hence

$$d\left(f,H\right)\le \frac{L}{1-L},$$

which implies that the inequality (2.14) holds.

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.5. Let θ ≥ 0 and let p be a positive real number with p < 1. Let be a normed vector space with norm || · || Let $f:\mathcal{X}\to \mathcal{Y}$ be a mapping satisfying (2.11), (2.12), and (2.13). Then $H\left(x\right)=N-{\mathsf{\text{lim}}}_{n\to \infty }\frac{1}{3^n}f\left(3^{n}x\right)$ exists for each $x\in \mathcal{X}$ and defines a fuzzy n-Jordan *-homomorphism $H:\mathcal{X}\to \mathcal{Y}$ such that

$$N\left(f\left(x\right)-H\left(x\right),t\right)\ge \frac{\left(3-3^p\right)t}{\left(3-3^p\right)t+3^p\theta {∥x∥}^p}$$

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

Proof. The proof follows from Theorem 2.4 by taking

$$\phi \left(x,y,z\right)=\theta \left({∥x∥}^p+{∥y∥}^p+{∥z∥}^p\right)$$

and L = 3^p-l. □