SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

An introduction to 2-fuzzy n-normed linear spaces and a new perspective to the Mazur-Ulam problem

Choonkil Park1 and Cihangir Alaca2*

Author Affiliations

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

2 Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, 45140 Manisa, Turkey

For all author emails, please log on.

Journal of Inequalities and Applications 2012, 2012:14  doi:10.1186/1029-242X-2012-14


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/14


Received:24 May 2011
Accepted:19 January 2012
Published:19 January 2012

© 2012 Park and Alaca; 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

The purpose of this article is to introduce the concept of 2-fuzzy n-normed linear space or fuzzy n-normed linear space of the set of all fuzzy sets of a non-empty set. We define the concepts of n-isometry, n-collinearity n-Lipschitz mapping in this space. Also, we generalize the Mazur-Ulam theorem, that is, when X is a 2-fuzzy n-normed linear space or ℑ(X) is a fuzzy n-normed linear space, the Mazur-Ulam theorem holds. Moreover, it is shown that each n-isometry in 2-fuzzy n-normed linear spaces is affine.

Mathematics Subject Classification (2010): 03E72; 46B20; 51M25; 46B04; 46S40.

Keywords:
Mazur-Ulam theorem; α-n-norm; 2-fuzzy n-normed linear spaces; n-isometry; n-Lipschitz mapping.

1. Introduction

A satisfactory theory of 2-norms and n-norms on a linear space has been introduced and developed by Gähler [1,2]. Following Misiak [3], Kim and Cho [4], and Malčeski [5] developed the theory of n-normed space. In [6], Gunawan and Mashadi gave a simple way to derive an (n - 1)-norm from the n-norms and realized that any n-normed space is an (n - 1)-normed space. Different authors introduced the definitions of fuzzy norms on a linear space. Cheng and Mordeson [7] and Bag and Samanta [8] introduced a concept of fuzzy norm on a linear space. The concept of fuzzy n-normed linear spaces has been studied by many authors (see [4,9]).

Recently, Somasundaram and Beaula [10] introduced the concept of 2-fuzzy 2-normed linear space or fuzzy 2-normed linear space of the set of all fuzzy sets of a set. The authors gave the notion of α-2-norm on a linear space corresponding to the 2-fuzzy 2-norm by using some ideas of Bag and Samanta [8] and also gave some fundamental properties of this space.

In 1932, Mazur and Ulam [11] proved the following theorem.

Mazur-Ulam Theorem. Every isometry of a real normed linear space onto a real normed linear space is a linear mapping up to translation.

Baker [12] showed an isometry from a real normed linear space into a strictly convex real normed linear space is affine. Also, Jian [13] investigated the generalizations of the Mazur-Ulam theorem in F*-spaces. Rassias and Wagner [14] described all volume preserving mappings from a real finite dimensional vector space into itself and Väisälä [15] gave a short and simple proof of the Mazur-Ulam theorem. Chu [16] proved that the Mazur-Ulam theorem holds when X is a linear 2-normed space. Chu et al. [17] generalized the Mazur-Ulam theorem when X is a linear n-normed space, that is, the Mazur-Ulam theorem holds, when the n-isometry mapped to a linear n-normed space is affine. They also obtain extensions of Rassias and Šemrl's theorem [18]. Moslehian and Sadeghi [19] investigated the Mazur-Ulam theorem in non-archimedean spaces. Choy et al. [20] proved the Mazur-Ulam theorem for the interior preserving mappings in linear 2-normed spaces. They also proved the theorem on non-Archimedean 2-normed spaces over a linear ordered non-Archimedean field without the strict convexity assumption. Choy and Ku [21] proved that the barycenter of triangle carries the barycenter of corresponding triangle. They showed the Mazur-Ulam problem on non-Archimedean 2-normed spaces using the above statement. Xiaoyun and Meimei [22] introduced the concept of weak n-isometry and then they got under some conditions, a weak n-isometry is also an n-isometry. Cobzaş [23] gave some results of the Mazur-Ulam theorem for the probabilistic normed spaces as defined by Alsina et al. [24]. Cho et al. [25] investigated the Mazur-Ulam theorem on probabilistic 2-normed spaces. Alaca [26] introduced the concepts of 2-isometry, collinearity, 2-Lipschitz mapping in 2-fuzzy 2-normed linear spaces. Also, he gave a new generalization of the Mazur-Ulam theorem when X is a 2-fuzzy 2-normed linear space or ℑ(X) is a fuzzy 2-normed linear space. Kang et al. [27] proved that the Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy normed space. Kubzdela [28] gave some new results for isometries, Mazur-Ulam theorem and Aleksandrov problem in the framework of non-Archimedean normed spaces. The Mazur-Ulam theorem has been extensively studied by many authors (see [29,30]).

In the present article, we introduce the concept of 2-fuzzy n-normed linear space or fuzzy n-normed linear space of the set of all fuzzy sets of a non-empty set. We define the concepts of n-isometry, n-collinearity, n-Lipschitz mapping in this space. Also, we generalize the Mazur-Ulam theorem, that is, when X is a 2-fuzzy n-normed linear space or ℑ(X) is a fuzzy n-normed linear space, the Mazur-Ulam theorem holds. It is moreover shown that each n-isometry in 2-fuzzy n-normed linear spaces is affine.

2. Preliminaries

Definition 2.1([31]) Let n ∈ ℕ and let X be a real vector space of dimension d n. (Here we allow d to be infinite.) A real-valued function ∥●, ..., ●∥ on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M1">View MathML</a> satisfying the following properties

(1) ∥x1, x2, ..., xn∥ = 0 if and only if x1, x2, ..., xn are linearly dependent,

(2) ∥x1, x2, ..., xn∥ is invariant under any permutation,

(3) ∥x1, x2, ..., αxn∥ = |α| ∥x1, x2, ..., xn∥ for any α ∈ ℝ,

(4) ∥x1, x2, ..., xn-1, y + z∥ ≤ ∥x1, x2, ..., xn-1, y∥ + ∥x1, x2, ..., xn-1, z∥, is called an n-norm on X and the pair (X, ∥●, ..., ●∥) is called an n-normed linear space.

Definition 2.2 [9] Let X be a linear space over S (field of real or complex numbers). A fuzzy subset N of Xn × ℝ (ℝ, the set of real numbers) is called a fuzzy n-norm on X if and only if:

(N1) For all t ∈ ℝ with t ≤ 0, N(x1, x2, ..., xn, t) = 0,

(N2) For all t ∈ ℝ with t > 0, N(x1, x2, ..., xn, t) = 1 if and only if x1, x2, ..., xn are linearly dependent,

(N3) N(x1, x2, ..., xn, t) is invariant under any permutation of x1, x2, ..., xn,

(N4) For all t ∈ ℝ with t > 0, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M2">View MathML</a>, if λ ≠ 0, λ S,

(N5) For all s, t ∈ ℝ

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M3">View MathML</a>

(N6) N(x1, x2, ..., xn, t) is a non-decreasing function of t ∈ ℝ and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M4">View MathML</a>.

Then (X, N) is called a fuzzy n-normed linear space or in short f-n-NLS.

Theorem 2.1 [9] Let (X, N) be an f-n-NLS. Assume that

(N7) N(x1, x2, ..., xn,t) > 0 for all t > 0 implies that x1, x2, ..., xn are linearly dependent.

Define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M5">View MathML</a>

Then {∥●, ●, ..., ●∥α : α ∈ (0, 1)} is an ascending family of n-norms on X.

We call these n-norms as α-n-norms on X corresponding to the fuzzy n-norm on X.

Definition 2.3 Let X be any non-empty set and ℑ(X) the set of all fuzzy sets on X. For U, V ∈ ℑ(X) and λ S the field of real numbers, define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M6">View MathML</a>

and λU = {(λx, ν): (x, ν) ∈ U}.

Definition 2.4 A fuzzy linear space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M7">View MathML</a> over the number field S, where the addition and scalar multiplication operation on X are defined by (x, ν) + (y, μ) = (x + y, νμ), λ(x, ν) = (λx, ν) is a fuzzy normed space if to every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M8">View MathML</a> there is associated a non-negative real number, ∥(x, ν)∥, called the fuzzy norm of (x, ν), in such away that

(i) ∥(x, ν)∥ = 0 iff x = 0 the zero element of X, ν ∈ (0, 1],

(ii) ∥λ(x, ν)∥ = |λ| ∥(x, ν)∥ for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M8">View MathML</a> and all λ S,

(iii) ∥(x, ν) + (y, μ) || ≤ ∥(x, ν μ)∥ + ∥(y, ν μ)∥ for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M9">View MathML</a>,

(iv) ∥(x, ∨tνt)∥ = ∧t ∥(x, νt)∥ for all νt ∈ (0, 1].

3. 2-fuzzy n-normed linear spaces

In this section, we define the concepts of 2-fuzzy n-normed linear spaces and α-n-norms on the set of all fuzzy sets of a non-empty set.

Definition 3.1 Let X be a non-empty and ℑ(X) be the set of all fuzzy sets in X. If f ∈ ℑ(X) then f = {(x, μ): x X and μ ∈ (0, 1]}. Clearly f is bounded function for |f(x)| ≤ 1. Let S be the space of real numbers, then ℑ(X) is a linear space over the field S where the addition and scalar multiplication are defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M10">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M11">View MathML</a>

where λ S.

The linear space ℑ(X) is said to be normed linear space if, for every f ∈ ℑ(X), there exists an associated non-negative real number ∥f∥ (called the norm of f) which satisfies

(i) ∥f∥ = 0 if and only if f = 0. For

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M12">View MathML</a>

(ii) ∥λf∥ = |λ| ∥f∥, λ S. For

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M13">View MathML</a>

(iii) ∥f + g∥ ≤ ∥f∥ + ∥g∥ for every f, g ∈ ℑ(X). For

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M14">View MathML</a>

Then (ℑ(X),∥●∥) is a normed linear space.

Definition 3.2 A 2-fuzzy set on X is a fuzzy set on ℑ(X).

Definition 3.3 Let X be a real vector space of dimension d n (n ∈ ℕ) and ℑ(X) be the set of all fuzzy sets in X. Here we allow d to be infinite. Assume that a [0, 1]-valued function ∥●, ..., ●∥ on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M15">View MathML</a> satisfies the following properties

(1) ∥f1, f2, ..., fn∥ = 0 if and only if f1, f2, ..., fn are linearly dependent,

(2) ∥f1, f2, ..., fn∥ is invariant under any permutation,

(3) ∥f1, f2, ..., λfn∥ = |λ| ∥f1, f2, ..., fn∥ for any λ S,

(4) ∥f1, f2, ..., fn-1, y + z∥ ≤ ∥f1, f2, ..., fn-1, y∥ + ∥f1, f2, ..., fn-1, z∥.

Then (ℑ(X),∥●,...,●∥) is an n-normed linear space or (X, ∥●, ..., ●∥) is a 2-n-normed linear space.

Definition 3.4 Let ℑ(X) be a linear space over the real field S. A fuzzy subset N of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M16">View MathML</a> is called a 2-fuzzy n-norm on X (or fuzzy n-norm on ℑ(X)) if and only if

(2-N1) for all t ∈ ℝ with t ≤ 0, N(f1, f2, ..., fn, t) = 0,

(2-N2) for all t ∈ ℝ with t > 0, N(f1, f2, ..., fn, t) = 1 if and only if f1, f2, ..., fn are linearly dependent,

(2-N3) N(f1, f2, ..., fn, t) is invariant under any permutation of f1, f2, ..., fn,

(2-N4) for all t ∈ ℝ with t > 0, N(f1, f2, ..., λfn, t) = N(f1, f2, ..., fn, t/|λ|), if λ ≠ 0, λ S,

(2-N5) for all s, t ∈ ℝ,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M17">View MathML</a>

(2-N6) N(f1, f2, ..., fn, ·): (0, ∞) → [0, 1] is continuous,

(2-N7) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M18">View MathML</a>.

Then ℑ(X), N) is a fuzzy n-normed linear space or (X, N) is a 2-fuzzy n-normed linear space.

Remark 3.1 In a 2-fuzzy n-normed linear space (X, N), N(f1, f2, ..., fn, ·) is a non-decreasing function of ℝ for all f1, f2,...,fn ∈ ℑ(X).

Remark 3.2 From (2-N4) and (2-N5), it follows that in a 2-fuzzy n-normed linear space,

(2-N4) for all t ∈ ℝ with t > 0, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M19">View MathML</a> if λ ≠ 0, λ S,

(2-N5) for all s, t ∈ ℝ,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M20">View MathML</a>

The following example agrees with our notion of 2-fuzzy n-normed linear space.

Example 3.1 Let (ℑ(X),∥●,●,...,●∥) be an n-normed linear space as in Definition 3.3. Define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M21">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M22">View MathML</a>. Then (X, N) is a 2-fuzzy n-normed linear space.

Solution. (2-N1) For all t ∈ ℝ with t ≤ 0, by definition, we have N(f1, f2, ..., fn, t) = 0.

(2-N2) For all t ∈ ℝ with t > 0,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M23">View MathML</a>

(2-N3) For all t ∈ ℝ with t > 0,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M24">View MathML</a>

(2-N4) For all t ∈ ℝ with t > 0 and λ F, λ ≠ 0,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M25">View MathML</a>

(2-N5) We have to prove

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M26">View MathML</a>

(i) s + t < 0,

(ii) s = t = 0,

(iii) s + t > 0; s > 0, t < 0; s < 0, t > 0, then the above relation is obvious. If

(iv) s > 0, t > 0, s + t > 0, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M27">View MathML</a>

If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M28">View MathML</a>

Similarly, if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M29">View MathML</a>

Thus

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M30">View MathML</a>

(2-N6) It is clear that N(f1, f2, ..., fn, ·): (0, ∞) → [0, 1] is continuous.

(2-N7) For all t ∈ ℝ with t > 0,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M31">View MathML</a>

as desired.

As a consequence of Theorem 3.2 in [10], we introduce an interesting notion of ascending family of α-n-norms corresponding to the fuzzy n-norms in the following theorem.

Theorem 3.1 Let (ℑ(X), N) is a fuzzy n-normed linear space. Assume that

(2-N8) N(f1, f2, ..., fn, t) > 0 for all t > 0 implies f1, f2, ..., fn are linearly dependent.

Define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M32">View MathML</a>

Then {∥●, ●, ..., ●∥α : α ∈ (0, 1)} is an ascending family of n-norms on ℑ(X).

These n-norms are called α-n-norms on ℑ(X) corresponding to the 2-fuzzy n-norm on X.

Proof. (i) Let ∥f1, ..., fnα = 0. This implies that inf {t : N(f1, ..., fn, t) ≥ α}. Then, N(f1, f2, ..., fn, t) ≥ α > 0, for all t > 0, α ∈ (0, 1), which implies that f1, f2, ..., fn are linearly dependent, by (2-N8).

Conversely, assume f1, f2, ..., fn are linearly dependent. This implies that N(f1, f2, ..., fn, t) = 1 for all t > 0. For all α ∈ (0, 1), inf {t : N(f1, f2, ..., fn, t) ≥ α}, which implies that ∥f1, f2, ..., fnα = 0.

(ii) Since N(f1, f2, ..., fn, t) is invariant under any permutation, ∥f1, f2, ..., fnα = 0 under any permutation.

(iii) If λ ≠ 0, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M33">View MathML</a>

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M34">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M35">View MathML</a>

If λ = 0, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M36">View MathML</a>

(iv)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M37">View MathML</a>

Hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M38">View MathML</a>

Thus {∥●, ●, ..., ●∥α : α ∈ (0, 1)} is an α-n-norm on X.

Let 0 < α1 < α2. Then,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M39">View MathML</a>

As α1 < α2,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M40">View MathML</a>

implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M41">View MathML</a>

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M42">View MathML</a>

Hence {∥●, ●, ..., ●∥α : α ∈ (0, 1)} is an ascending family of α-n-norms on x corresponding to the 2-fuzzy n-norm on X.

4. On the Mazur-Ulam problem

In this section, we give a new generalization of the Mazur-Ulam theorem when X is a 2-fuzzy n-normed linear space or ℑ(X) is a fuzzy n-normed linear space. Hereafter, we use the notion of fuzzy n-normed linear space on ℑ(X) instead of 2-fuzzy n-normed linear space on X.

Definition 4.1 Let ℑ(X) and ℑ(X) be fuzzy n-normed linear spaces and Ψ : ℑ(X) → ℑ(Y) a mapping. We call Ψ an n-isometry if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M43">View MathML</a>

for all f0, f1, f2,...,fn ∈ ℑ(X) and α, β ∈ (0, 1).

For a mapping Ψ, consider the following condition which is called the n-distance one preserving property (nDOPP).

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M44">View MathML</a>

Then ∥Ψ(f1) - Ψ(f0), ..., Ψ(fn) - Ψ(f0)∥β = 1.

Lemma 4.1 Let f1, f2,...,fn ∈ ℑ(X), α ∈ (0, 1) and ħ ∈ ℝ. Then,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M45">View MathML</a>

for all 1 ≤ i j n.

Proof. It is obviously true.

Lemma 4.2 For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M46">View MathML</a>, if f0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M47">View MathML</a> are linearly dependent with some direction, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M48">View MathML</a> for some t > 0, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M49">View MathML</a>

for all f1, f2,...,fn ∈ ℑ(X) and α ∈ (0, 1).

Proof. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M48">View MathML</a> for some t > 0. Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M50">View MathML</a>

for all f1, f2,...,fn ∈ ℑ(X) and α ∈ (0, 1).

Definition 4.2 The elements f0, f1, f2, ..., fn of ℑ(X) are said to be n-collinear if for every i, {fj - fi : 0 ≤ j i n} is linearly dependent.

Remark 4.1 The elements f0, f1, and f2 are said to be 2-collinear if and only if f2 - f0 = r(f1 - f0) for some real number r.

Now we define the concept of n-Lipschitz mapping.

Definition 4.3 We call Ψ an n-Lipschitz mapping if there is a κ ≥ 0 such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M51">View MathML</a>

for all f0, f1, f2,...,fn ∈ ℑ(X) and α, β ∈ (0, 1). The smallest such κ is called the n-Lipschitz constant.

Lemma 4.3 Assume that if f0, f1, and f2 are 2 -collinear then Ψ(f0), Ψ(f1) and Ψ(f2) are 2-collinear, and that Ψ satisfies (nDOPP). Then Ψ preserves the n-distance k for each k ∈ ℕ.

Proof. Suppose that there exist f0, f1 ∈ ℑ(X) with f0 f1 such that Ψ(f0) = Ψ(f1). Since dimℑ(X) ≥ n, there are f2,...,fn ∈ ℑ(X) such that f1 - f0, f2 - f0, ..., fn - f0 are linearly independent. Since ∥f1 - f0, f2 - f0, ..., fn - f0α ≠ 0, we can set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M52">View MathML</a>

Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M53">View MathML</a>

Since Ψ preserves the unit n-distance,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M54">View MathML</a>

But it follows from Ψ (f0) = Ψ (f1) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M55">View MathML</a>

which is a contradiction. Hence, Ψ is injective.

Let f0, f1, f2, ..., fn be elements of ℑ(X), k ∈ ℕ and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M56">View MathML</a>

We put

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M57">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M58">View MathML</a>

for all i = 0, 1, ..., k - 1. Since Ψ satisfies (nDOPP),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M59">View MathML</a>

(4.1)

for all i = 0, 1, ..., k - 1. Since g0, g1, and g2 are 2-collinear, Ψ (g0), Ψ(g1) and Ψ(g2) are also 2-collinear. Thus there is a real number r0 such that Ψ(g2) - Ψ(g1) = r0 (Ψ(g1) - Ψ(g0)). It follows from (4.1) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M60">View MathML</a>

Thus, we have r0 = 1 or -1. If r0 = -1, Ψ(g2) - Ψ(g1) = -Ψ(g1) + Ψ(g0), that is, Ψ(g2) = Ψ(g0). Since Ψ is injective, g2 = g0, which is a contradiction. Thus r0 = 1. Then we have Ψ(g2) - Ψ(g1) = Ψ(g1) - Ψ(g0). Similarly, one can obtain that Ψ(gi+1) - Ψ(gi) = Ψ(gi) - Ψ(gi-1) for all i = 0, 1, ..., k - 1. Thus Ψ(gi+1) - Ψ(gi) = Ψ(g1) - Ψ(g0) for all i = 0, 1, ..., k - 1. Hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M61">View MathML</a>

Hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M62">View MathML</a>

This completes the proof.

Lemma 4.4 Let h, f0, f1, ..., fn be elements of ℑ(X) and let h, f0, f1 be 2-collinear. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M63">View MathML</a>

Proof. Since h, f0, f1 are 2-collinear, there exists a real number r such that f1 - h = r(f0 - h). It follows from Lemma 4.1 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M64">View MathML</a>

This completes the proof.

Theorem 4.1 Let Ψ be an n-Lipschitz mapping with the n-Lipschitz constant κ ≤ 1. Assume that if f0, f1, ..., fn are m-collinear then Ψ(f0), Ψ(f1), ..., Ψ(fm) are m-collinear, m = 2, n, and that Ψ satisfies (nDOPP), then Ψ is an n-isometry.

Proof. It follows from Lemma 4.3 that Ψ preserves n-distance k for all k ∈ ℕ. For f0, f1, ..., fn X, there are two cases depending upon whether ∥f1 - f0, ..., fn - f0α = 0 or not. In the case ∥f1 - f0, ..., fn - f0α = 0, f1 - f0, ..., fn - f0 are linearly dependent, that is, n-collinear. Thus f1 - f0, ..., fn - f0 are linearly dependent. Thus ∥Ψ(f1) - Ψ(f0), ..., Ψ(fn) - Ψ(f0)∥β = 0.

In the case ∥f1 - f0, ..., fn - f0α > 0, there exists an n0 ∈ ℕ such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M65">View MathML</a>

Assume that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M66">View MathML</a>

We can set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M67">View MathML</a>

Then we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M68">View MathML</a>

It follows from Lemma 4.3 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M69">View MathML</a>

By the definition of h,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M70">View MathML</a>

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M71">View MathML</a>

h - f1 and f1 - f0 have the same direction. It follows from Lemma 4.2 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M72">View MathML</a>

Since Ψ(h), Ψ(f1), Ψ(f2) are 2-collinear, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M73">View MathML</a>

by Lemma 4.4. By the assumption,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M74">View MathML</a>

which is a contradiction. Hence Ψ is an n-isometry.

Lemma 4.5 Let g0, g1 be elements of ℑ(X). Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M75">View MathML</a> is the unique element of ℑ(X) satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M76">View MathML</a>

for some g2,...,gn ∈ ℑ(X) with ∥g0 - gn, g1 - gn, g2 - gn, ..., gn-1 - gnα ≠ 0 and v, g0, g1 2-collinear.

Proof. Let ∥g0 - gn, g1 - gn, g2 - gn, ..., gn-1 - gnα ≠ 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M87">View MathML</a>.

Then v, g0, g1 are 2-collinear. It follows from Lemma 4.1 and gn - g0 = g1 - g0 - (g1 - gn) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M77">View MathML</a>

and similarly

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M78">View MathML</a>

Now we prove the uniqueness.

Let u be an element of ℑ(X) satisfying the above properties. Since u, g0, g1 are 2-collinear, there exists a real number t such that u = tg0 + (1 - t)g1. It follows from Lemma 4.1 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M79">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M80">View MathML</a>

Since ∥g0 - gn, g1 - gn, g2 - gn, ..., gn-1 - gnα ≠ 0, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M81">View MathML</a>. Therefore, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M82">View MathML</a> and hence v = u.

Lemma 4.6 If Ψ is an n-isometry and f0, f1, f2 are 2-collinear then Ψ(f0), Ψ(f1), Ψ(f2) are 2-collinear.

Proof. Since dimℑ(X) ≥ n, for any f0 ∈ ℑ(X), there exist g1,...,gn ∈ ℑ(X) such that g1 - f0, ..., gn - f0 are linearly independent. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M83">View MathML</a>

and hence, the set A = {Ψ(f) -Ψ(f0) : f ∈ ℑ(X)} contains n linearly independent vectors.

Assume that f0, f1, f2 are 2-collinear. Then, for any f3,...,fn ∈ ℑ(X),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M84">View MathML</a>

i.e. Ψ(f1) - Ψ(f0), ..., Ψ(fn) - Ψ(f0) are linearly dependent.

If there exist f3, ..., fn-1 such that Ψ(f1) - Ψ(f0), ..., Ψ(fn-1) - Ψ(f0) are linearly independent, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M85">View MathML</a>

which contradicts the fact that A contains n linearly independent vectors.

Then, for any f3, ..., fn-1, Ψ(f1) - Ψ(f0), ..., Ψ(fn-1) - Ψ(f0) are linearly dependent.

If there exist f3, ..., fn-2 such that Ψ(f1) - Ψ(f0), ..., Ψ(fn-2) - Ψ(f0) are linearly independent, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M86">View MathML</a>

which contradicts the fact that A contains n linearly independent vectors.

And so on, Ψ(f1) - Ψ(f0), Ψ(f2) - Ψ(f0) are linearly dependent. Thus Ψ(f0), Ψ(f1), and Ψ(f2) are 2-collinear.

Theorem 4.2 Every n-isometry mapping is affine.

Proof. Let Ψ be an n-isometry and Φ(f) = Ψ(f) - Ψ(0). Then Φ is an n-isometry and Φ(0) = 0. Thus we may assume that Ψ(0) = 0. Hence it suffices to show that Ψ is linear.

Let f0, f1 ∈ ℑ(X) with f0 f1. Since dimℑ(X) ≥ n, there exist f2,...,fn ∈ ℑ(X) such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M87">View MathML</a>

Since Ψ is an n-isometry, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M88">View MathML</a>

It follows from Lemma 4.1 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M89">View MathML</a>

And we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M90">View MathML</a>

By Lemma 4.6, we obtain that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M91">View MathML</a>, Ψ(f0), and Ψ(f1) are 2-collinear. By Lemma 4.5, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M92">View MathML</a> for all f, g ∈ ℑ(X) and α, β ∈ (0, 1). Since Ψ(0) = 0, we can easily show that Ψ is additive. It follows that Ψ is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M93">View MathML</a>-linear.

Let r ∈ ℝ+ with r ≠ 1 and f ∈ ℑ(X). By Lemma 4.6, Ψ(0), Ψ(f) and Ψ(rf) are also 2-collinear. It follows from Ψ(0) = 0 that there exists a real number k such that Ψ(rf) = kΨ(f). Since dimℑ(X) ≥ n, there exist f1, ..., fn-1 ∈ ℑ(X) such that ∥f, f1, f2, ..., fn-1α ≠ 0. Since Ψ(0) = 0, for every f0, f1, f2, ..., fn-1 ∈ ℑ(X),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M94">View MathML</a>

Thus we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M95">View MathML</a>

Since ∥f, f1, f2, ..., fn-1α ≠ 0, |k| = r. Then Ψ(rf) = rΨ(f) or Ψ(rf) = -rΨ(f). First of all, assume that k = -r, that is, Ψ(rf) = -rΨ(f). Then there exist positive rational numbers q1, q2 such that o < q1 < r < q2. Since dimℑ(X) ≥ n, there exist h1, ..., hn-1 ∈ ℑ(X) such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M96">View MathML</a>

Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M97">View MathML</a>

And also we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M98">View MathML</a>

Since ∥rf - q2f, h1 - q2f, h2 - q2f, ..., hn-1 - q2fα ≠ 0,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/14/mathml/M99">View MathML</a>

Thus we have r + q2 < q2 - q1, which is a contradiction. Hence k = r, that is, Ψ(rf) = rΨ(f) for all positive real numbers r. Therefore Ψ is ℝ-linear, as desired.

We get the following corollary from Theorems 4.1 and 4.2.

Corollary 4.1 Let Ψ be an n-Lipschitz mapping with the n-Lipschitz constant κ ≤ 1. Suppose that if f, g, h are 2-collinear, then Ψ(f), Ψ(g), Ψ(h) are 2-collinear. If Ψ satisfies (nDOPP), then Ψ is an affine n-isometry.

5. Conclusion

In this article, the concept of 2-fuzzy n-normed linear space is defined and the concepts of n-isometry, n-collinearity, n-Lipschitz mapping are given. Also, the Mazur-Ulam theorem is generalized into 2-fuzzy n-normed linear spaces.

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

The authors would like to thank the referees and area editor Professor Mohamed A. El-Gebeily for giving useful suggestions and comments for the improvement of this article.

References

  1. Gähler, S: Lineare 2-normierte raume. Math Nachr. 28, 1–43 (1964). Publisher Full Text OpenURL

  2. Gähler, S: Untersuchungen über verallgemeinerte m-metrische räume. I Math Nachr. 40, 165–189 (1969). Publisher Full Text OpenURL

  3. Misiak, A: n-inner product spaces. Math Nachr. 140, 299–319 (1989). Publisher Full Text OpenURL

  4. Kim, SS, Cho, YJ: Strict convexity in linear n-normed spaces. Demonstratio Math. 29, 739–744 (1996)

  5. Malčeski, R: Strong n-convex n-normed spaces. Mat Bilten. 21, 81–102 (1997)

  6. Gunawan, H, Mashadi, M: On n-normed spaces. Intern J Math Math Sci. 27, 631–639 (2001). Publisher Full Text OpenURL

  7. Cheng, SC, Mordeson, JN: Fuzzy linear operators and fuzzy normed linear spaces. Bull Calcutta Math Soc. 86, 429–436 (1994)

  8. Bag, T, Samanta, SK: Finite dimensional fuzzy normed linear spaces. J Fuzzy Math. 11, 687–705 (2003)

  9. Narayanan, AL, Vijayabalaji, S: Fuzzy n-normed linear space. Intern J Math Math Sci. 24, 3963–3977 (2005)

  10. Somasundaram, RM, Beaula, T: Some Aspects of 2-fuzzy 2-normed linear spaces. Bull Malays Math Sci Soc. 32, 211–221 (2009)

  11. Mazur, S, Ulam, S: Sur les transformationes isométriques d'espaces vectoriels normés. CR Acad Sci Paris. 194, 946–948 (1932)

  12. Baker, JA: Isometries in normed spaces. Am Math Monthly. 78, 655–658 (1971). Publisher Full Text OpenURL

  13. Jian, W: On the generations of the Mazur-Ulam isometric theorem. J Math Anal Appl. 263, 510–521 (2001). Publisher Full Text OpenURL

  14. Rassias, ThM, Wagner, P: Volume preserving mappings in the spirit of the Mazur-Ulam theorem. Aequationes Math. 66, 85–89 (2003). Publisher Full Text OpenURL

  15. Väisälä, J: A proof of the Mazur-Ulam theorem. Am Math Monthly. 110, 633–635 (2003). Publisher Full Text OpenURL

  16. Chu, H: On the Mazur-Ulam problem in linear 2-normed spaces. J Math Anal Appl. 327, 1041–1045 (2007). Publisher Full Text OpenURL

  17. Chu, H, Choi, S, Kang, D: Mappings of conservative distances in linear n-normed spaces. Nonlinear Anal. 70, 1168–1174 (2009). Publisher Full Text OpenURL

  18. Rassias, ThM, Šemrl, P: On the Mazur-Ulam problem and the Aleksandrov problem for unit distance preserving mappings. Proc Amer Math Soc. 118, 919–925 (1993). Publisher Full Text OpenURL

  19. Moslehian, MS, Sadeghi, Gh: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal. 69, 3405–3408 (2008). Publisher Full Text OpenURL

  20. Choy, J, Chu, H, Ku, S: Characterizations on Mazur-Ulam theorem. Nonlinear Anal. 72, 1291–1297 (2010). Publisher Full Text OpenURL

  21. Choy, J, Ku, S: Characterization on 2-isometries in non-Archimedean 2-normed spaces. J Chungcheong Math Soc. 22, 65–71 (2009)

  22. Chen, XY, Song, MM: Characterizations on isometries in linear n-normed spaces. Nonlinear Anal. 72, 1895–1901 (2010). Publisher Full Text OpenURL

  23. Cobzaş, S: A Mazur-Ulam theorem for probabilistic normed spaces. Aequa-tiones Math. 77, 197–205 (2009). Publisher Full Text OpenURL

  24. Alsina, C, Schweizer, B, Sklar, A: On the definition of a probabilistic normed space. Aequationes Math. 46, 91–98 (1993). Publisher Full Text OpenURL

  25. Cho, YJ, Rahbarnia, F, Saadati, R, Sadeghi, Gh: Isometries in probabilistic 2-normed spaces. J Chungcheong Math Soc. 22, 623–634 (2009)

  26. Alaca, C: A new perspective to the Mazur-Ulam problem in 2-fuzzy 2-normed linear spaces. Iranian J Fuzzy Syst. 7, 109–119 (2010)

  27. Kang, D, Koh, H, Cho, IG: On the Mazur-Ulam theorem in non-Archimedean fuzzy normed spaces. App Math Lett. 25, 301–304 (2012). Publisher Full Text OpenURL

  28. Kubzdela, K: Isometries, Mazur-Ulam theorem and Aleksandrov problem for non-Archimedean normed spaces. In: Nonlinear Anal

  29. Rassias, ThM: On the A.D. Aleksandrov problem of conservative distances and the Mazur-Ulam theorem. Nonlinear Anal. 47, 2597–2608 (2001). Publisher Full Text OpenURL

  30. Xiang, S: Mappings of conservative distances and the Mazur-Ulam theorem. J Math Anal Appl. 254, 262–274 (2001). Publisher Full Text OpenURL

  31. Cho, YJ, Lin, PCS, Kim, SS, Misiak, A: Theory of 2-Inner Product Spaces. Nova Science Publishers, New York (2001)