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

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

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

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 and ε > 0, does there exist a δ > 0 such that if a function satisfies the inequality d(f(xy), f(x)f(y)) < δ for all , then there exists a homomorphism such that d(f(x), T(x)) < ε for all Hyers [2] gave a partial solution of the Ulam's problem for the case of approximate additive mappings under the assumption that and 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

for all nonnegative integers n or there exists a positive integer n₀ 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 ;

(4) 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

Definition 1.2. [16-18,21] Let be a complex vector space. A function is called a fuzzy norm on if for all 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: 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 is called a fuzzy normed vector space.

Definition 1.3. [16-18,21] Let 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 between fuzzy normed vector space is continuous at point if for each sequence {x_n} converging to x₀ in , then the sequence {f(x_n)} converges to f (x₀). If is continuous at each , then is said to be continuous on (see [24]).

Definition 1.4. Let be a *-algebra and a fuzzy normed space.

(1) The fuzzy normed space is called a fuzzy normed *-algebra if

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

Example 1.5. Let be a normed *-algebra. let

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

Definition 1.6. Let be a C*-algebra and a fuzzy norm on .

(1) The fuzzy normed *-algebra is called an induced fuzzy normed *-algebra

(2) The fuzzy Banach *-algebra is called an induced fuzzy C*-algebra.

Definition 1.7. Let and be induced fuzzy normed *-algebras. Then a ℂ-linear mapping is called a fuzzy n-Jordan *-homomorphism if

for all .

Throughout this article, assume that is an induced fuzzy normed *-algebra and that is an induced fuzzy C*-algebra.

Lemma 2.1. Let be a fuzzy normed vector space and let be a mapping such that

for all and all t > 0. Then f is additive, i.e., f(x + y) = f(x) + f(y) for all .

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

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

for all t > 0. It follows from N₂ that f(-x) + f(x) = 0 for all . So

for all .

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

for all t > 0. It follows from N₂ that

for all . Let t = 2y-z and s = 2z-y in (2.2), we obtain

for all , 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 be a function such that there exists an with

for all . Let be a mapping such that

for all , all t > 0 and all . Then exists for each and defines a fuzzy n-Jordan *-homomorphism such that

for all and all t > 0.

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

for all .

Consider the set

and introduce the generalized metric on S:

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

for all x ∈ X.

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

for all and all t > 0. Hence

for all and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that

for all g, h ∈ S.

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

By Theorem 1.1, there exists a mapping satisfying the following:

(1) H is a fixed point of J, i.e.,

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

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

for all ;

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

for all ;

(3) , which implies the inequality

This implies that the inequality (2.7) holds.

It follows from (2.3) that

for all .

By (2.4),

for all , all t > 0 and all . So

for all , all t > 0 and all . Since for all and all t > 0,

for all , all t > 0 and all . Thus

for all , all t > 0 and all . 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

for all and all . By [[26], Theorem 2.1], the mapping is ℂ-linear.

By (2.5),

for all and all t > 0. So

for all and all t > 0. Since for all and all t > 0,

for all and all t > 0. Thus, H(xⁿ) - H(x)ⁿ = 0 for all .

By (2.6),

for all and all t > 0. So

for all and all t > 0. Since for all and all t > 0,

for all and all t > 0. Thus, H(x*) - H(x)* = 0 for all .

Therefore, the mapping 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 be a mapping satisfying

for all , all t > 0 and all . Then exists for each and defines a fuzzy n-Jordan *-homomorphism such that

for all and all t > 0.

Proof. The proof follows from Theorem 2.2 by taking

and L = 3^l-p.

Theorem 2.4. Let be a function such that there exists an L < 1 with

for all . Let be a mapping satisfying (2.4), (2.5), and (2.6). Then exists for each and defines a fuzzy n-Jordan *-homomorphism such that

for all 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

for all .

It follows from (2.8) that

for all and all t > 0. So d(f, Jf) ≤ L. Hence

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 be a mapping satisfying (2.11), (2.12), and (2.13). Then exists for each and defines a fuzzy n-Jordan *-homomorphism such that

for all and all t > 0.

Proof. The proof follows from Theorem 2.4 by taking

and L = 3^p-l. □

The authors declare that they have no competing interests.

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.

C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). S.Y. Jang was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2011-0004872).

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