Open Access Research

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

Choonkil Park1, Shahram Ghaffary Ghaleh2, Khatereh Ghasemi3 and Sun Young Jang4*

Author Affiliations

1 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea

2 Department of Mathematics, Payame Noor University of Zahedan, Zahedan, Iran

3 Department of Mathematics, Payame Noor University of Khash, Khash, Iran

4 Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea

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Advances in Difference Equations 2012, 2012:42  doi:10.1186/1687-1847-2012-42


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2012/1/42


Received:27 January 2012
Accepted:5 April 2012
Published:5 April 2012

© 2012 Park et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M1">View MathML</a>

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

Keywords:
fuzzy n-Jordan *-homomorphism; induced fuzzy C*-algebra; Hyers-Ulam stability.

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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M3">View MathML</a> and ε > 0, does there exist a δ > 0 such that if a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M4">View MathML</a> satisfies the inequality d(f(xy), f(x)f(y)) < δ for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M100">View MathML</a>, then there exists a homomorphism <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M5">View MathML</a> such that d(f(x), T(x)) < ε for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M6">View MathML</a> Hyers [2] gave a partial solution of the Ulam's problem for the case of approximate additive mappings under the assumption that and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M7">View MathML</a>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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M8">View MathML</a>

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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M9">View MathML</a>;

(4) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M10">View MathML</a>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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M11">View MathML</a>

Definition 1.2. [16-18,21] Let be a complex vector space. A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M13">View MathML</a> is called a fuzzy norm on if for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M14">View MathML</a> 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: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M15">View MathML</a> 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 limt→∞ N(x, t) = 1

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

The pair <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16">View MathML</a> is called a fuzzy normed vector space.

Definition 1.3. [16-18,21] Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16">View MathML</a> be a fuzzy normed vector space.

(1) A sequence {xn} in χ is said to be convergent if there exists an x χ such that limn→∞ N(xn - x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn} and we denote it by N-limn→∞ xn = x.

(2) A sequence {xn} 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(xn+p-xn, 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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17">View MathML</a> between fuzzy normed vector space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M18">View MathML</a> is continuous at point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M19">View MathML</a> if for each sequence {xn} converging to x0 in , then the sequence {f(xn)} converges to f (x0). If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17">View MathML</a> is continuous at each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17">View MathML</a> is said to be continuous on (see [24]).

Definition 1.4. Let be a *-algebra and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16">View MathML</a> a fuzzy normed space.

(1) The fuzzy normed space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16">View MathML</a> is called a fuzzy normed *-algebra if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M21">View MathML</a>

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

Example 1.5. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M22">View MathML</a> be a normed *-algebra. let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M23">View MathML</a>

Then N(x, t) is a fuzzy norm on and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M24">View MathML</a> is a fuzzy normed *-algebra.

Definition 1.6. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M22">View MathML</a> be a C*-algebra and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M25">View MathML</a> a fuzzy norm on .

(1) The fuzzy normed *-algebra <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M26">View MathML</a> is called an induced fuzzy normed *-algebra

(2) The fuzzy Banach *-algebra <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M26">View MathML</a> is called an induced fuzzy C*-algebra.

Definition 1.7. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M26">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M27">View MathML</a> be induced fuzzy normed *-algebras. Then a ℂ-linear mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M28">View MathML</a> is called a fuzzy n-Jordan *-homomorphism if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M29">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M30">View MathML</a>.

Throughout this article, assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M16">View MathML</a> is an induced fuzzy normed *-algebra and that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M27">View MathML</a> is an induced fuzzy C*-algebra.

2. Main results

Lemma 2.1. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M31">View MathML</a>be a fuzzy normed vector space and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M32">View MathML</a> be a mapping such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M33">View MathML</a>

(2.1)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>and all t > 0. Then f is additive, i.e., f(x + y) = f(x) + f(y) for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M35">View MathML</a>.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M36">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M37">View MathML</a>

for all t > 0. It follows from N2 that f(-x) + f(x) = 0 for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M38">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M39">View MathML</a>

for all t > 0. It follows from N2 that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M40">View MathML</a>

(2.2)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M41">View MathML</a>. Let t = 2y-z and s = 2z-y in (2.2), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M42">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M43">View MathML</a>, 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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M44">View MathML</a>be a function such that there exists an <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M45">View MathML</a> with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M46">View MathML</a>

(2.3)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17">View MathML</a>be a mapping such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M47">View MathML</a>

(2.4)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M48">View MathML</a>

(2.5)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M49">View MathML</a>

(2.6)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>, all t > 0 and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M50">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M51">View MathML</a>exists for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>and defines a fuzzy n-Jordan *-homomorphism <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M52">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M53">View MathML</a>

(2.7)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>and all t > 0.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M54">View MathML</a>

(2.8)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>.

Consider the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M55">View MathML</a>

and introduce the generalized metric on S:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M56">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M57">View MathML</a>

for all x X.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M58">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M59">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ . This means that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M60">View MathML</a>

for all g, h S.

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

By Theorem 1.1, there exists a mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61">View MathML</a> satisfying the following:

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M62">View MathML</a>

(2.9)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>. The mapping H is a unique fixed point of J in the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M63">View MathML</a>

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M64">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>;

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M65">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>;

(3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M66">View MathML</a>, which implies the inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M67">View MathML</a>

This implies that the inequality (2.7) holds.

It follows from (2.3) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M68">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>.

By (2.4),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M69">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>, all t > 0 and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M71">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>, all t > 0 and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M72">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a> and all t > 0,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M73">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>, all t > 0 and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70">View MathML</a>. Thus

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M74">View MathML</a>

(2.10)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>, all t > 0 and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70">View MathML</a>. 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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M75">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70">View MathML</a>. By [[26], Theorem 2.1], the mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61">View MathML</a> is ℂ-linear.

By (2.5),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M76">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M77">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M78">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M79">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. Thus, H(xn) - H(x)n = 0 for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>.

By (2.6),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M80">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M81">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M82">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M83">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. Thus, H(x*) - H(x)* = 0 for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>.

Therefore, the mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61">View MathML</a> 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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17">View MathML</a>be a mapping satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M84">View MathML</a>

(2.11)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M85">View MathML</a>

(2.12)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M86">View MathML</a>

(2.13)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>, all t > 0 and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M70">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M87">View MathML</a>exists for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>and defines a fuzzy n-Jordan *-homomorphism <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M88">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>and all t > 0.

Proof. The proof follows from Theorem 2.2 by taking

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M89">View MathML</a>

and L = 3l-p.

Theorem 2.4. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M90">View MathML</a>be a function such that there exists an L < 1 with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M91">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M34">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17">View MathML</a>be a mapping satisfying (2.4), (2.5), and (2.6). Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M92">View MathML</a>exists for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>and defines a fuzzy n-Jordan *-homomorphism <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M93">View MathML</a>

(2.14)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M94">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>.

It follows from (2.8) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M95">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a> and all t > 0. So d(f, Jf) ≤ L. Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M96">View MathML</a>

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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M17">View MathML</a>be a mapping satisfying (2.11), (2.12), and (2.13). Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M97">View MathML</a>exists for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>and defines a fuzzy n-Jordan *-homomorphism <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M61">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M98">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M20">View MathML</a>and all t > 0.

Proof. The proof follows from Theorem 2.4 by taking

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/42/mathml/M99">View MathML</a>

and L = 3p-l. □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

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.

Acknowledgements

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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