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An introduction to 2-fuzzy n-normed linear spaces and a new perspective to the Mazur-Ulam problem
Journal of Inequalities and Applications volume 2012, Article number: 14 (2012)
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.
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 satisfying the following properties
-
(1)
∥x1, x2, ..., xn∥ = 0 if and only if x1, x2, ..., x n are linearly dependent,
-
(2)
∥x1, x2, ..., x n ∥ is invariant under any permutation,
-
(3)
∥x1, x2, ..., αx n ∥ = |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, ..., x n , t) = 0,
(N2) For all t ∈ ℝ with t > 0, N(x1, x2, ..., x n , t) = 1 if and only if x1, x2, ..., x n are linearly dependent,
(N3) N(x1, x2, ..., x n , t) is invariant under any permutation of x1, x2, ..., x n ,
(N4) For all t ∈ ℝ with t > 0, , if λ ≠ 0, λ ∈ S,
(N5) For all s, t ∈ ℝ
(N6) N(x1, x2, ..., x n , t) is a non-decreasing function of t ∈ ℝ and .
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, ..., x n ,t) > 0 for all t > 0 implies that x1, x2, ..., x n are linearly dependent.
Define
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
and λU = {(λx, ν): (x, ν) ∈ U}.
Definition 2.4 A fuzzy linear space over the number field S, where the addition and scalar multiplication operation on X are defined by (x, ν) + (y, μ) = (x + y, ν∧μ), λ(x, ν) = (λx, ν) is a fuzzy normed space if to every 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 and all λ ∈ S,
-
(iii)
∥(x, ν) + (y, μ) || ≤ ∥(x, ν ∧ μ)∥ + ∥(y, ν ∧ μ)∥ for all ,
-
(iv)
∥(x, ∨ t ν t )∥ = ∧ t ∥(x, ν t )∥ for all ν t ∈ (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
and
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
-
(ii)
∥λf∥ = |λ| ∥f∥, λ ∈ S. For
-
(iii)
∥f + g∥ ≤ ∥f∥ + ∥g∥ for every f, g ∈ ℑ(X). For
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 satisfies the following properties
-
(1)
∥f1, f2, ..., f n ∥ = 0 if and only if f1, f2, ..., f n are linearly dependent,
-
(2)
∥f1, f2, ..., f n ∥ is invariant under any permutation,
-
(3)
∥f1, f2, ..., λf n ∥ = |λ| ∥f1, f2, ..., f n ∥ 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 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, ..., f n , t) = 0,
(2-N2) for all t ∈ ℝ with t > 0, N(f1, f2, ..., f n , t) = 1 if and only if f1, f2, ..., f n are linearly dependent,
(2-N3) N(f1, f2, ..., f n , t) is invariant under any permutation of f1, f2, ..., f n ,
(2-N4) for all t ∈ ℝ with t > 0, N(f1, f2, ..., λf n , t) = N(f1, f2, ..., f n , t/|λ|), if λ ≠ 0, λ ∈ S,
(2-N5) for all s, t ∈ ℝ,
(2-N6) N(f1, f2, ..., f n , ·): (0, ∞) → [0, 1] is continuous,
(2-N7) .
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, ..., f n , ·) is a non-decreasing function of ℝ for all f1, f2,...,f n ∈ 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, if λ ≠ 0, λ ∈ S,
(2-N5) for all s, t ∈ ℝ,
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
for all . 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, ..., f n , t) = 0.
(2-N2) For all t ∈ ℝ with t > 0,
(2-N3) For all t ∈ ℝ with t > 0,
(2-N4) For all t ∈ ℝ with t > 0 and λ ∈ F, λ ≠ 0,
(2-N5) We have to prove
-
(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
If
Similarly, if
Thus
(2-N6) It is clear that N(f1, f2, ..., f n , ·): (0, ∞) → [0, 1] is continuous.
(2-N7) For all t ∈ ℝ with t > 0,
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, ..., f n , t) > 0 for all t > 0 implies f1, f2, ..., f n are linearly dependent.
Define
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, ..., f n ∥ α = 0. This implies that inf {t : N(f1, ..., f n , t) ≥ α}. Then, N(f1, f2, ..., f n , t) ≥ α > 0, for all t > 0, α ∈ (0, 1), which implies that f1, f2, ..., f n are linearly dependent, by (2-N8).
Conversely, assume f1, f2, ..., f n are linearly dependent. This implies that N(f1, f2, ..., f n , t) = 1 for all t > 0. For all α ∈ (0, 1), inf {t : N(f1, f2, ..., f n , t) ≥ α}, which implies that ∥f1, f2, ..., f n ∥ α = 0.
-
(ii)
Since N(f1, f2, ..., f n , t) is invariant under any permutation, ∥f1, f2, ..., f n ∥ α = 0 under any permutation.
-
(iii)
If λ ≠ 0, then
Let , then
If λ = 0, then
(iv)
Hence
Thus {∥●, ●, ..., ●∥ α : α ∈ (0, 1)} is an α-n-norm on X.
Let 0 < α1 < α2. Then,
As α1 < α2,
implies that
which implies that
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
for all f0, f1, f2,...,f n ∈ ℑ(X) and α, β ∈ (0, 1).
For a mapping Ψ, consider the following condition which is called the n-distance one preserving property (n DOPP).
Then ∥Ψ(f1) - Ψ(f0), ..., Ψ(f n ) - Ψ(f0)∥ β = 1.
Lemma 4.1 Let f1, f2,...,f n ∈ ℑ(X), α ∈ (0, 1) and ħ ∈ ℝ. Then,
for all 1 ≤ i ≠ j ≤ n.
Proof. It is obviously true.
Lemma 4.2 For , if f0 and are linearly dependent with some direction, that is, for some t > 0, then
for all f1, f2,...,f n ∈ ℑ(X) and α ∈ (0, 1).
Proof. Let for some t > 0. Then we have
for all f1, f2,...,f n ∈ ℑ(X) and α ∈ (0, 1).
Definition 4.2 The elements f0, f1, f2, ..., f n of ℑ(X) are said to be n-collinear if for every i, {f j - f i : 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
for all f0, f1, f2,...,f n ∈ 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 (n DOPP). 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,...,f n ∈ ℑ(X) such that f1 - f0, f2 - f0, ..., f n - f0 are linearly independent. Since ∥f1 - f0, f2 - f0, ..., f n - f0∥ α ≠ 0, we can set
Then we have
Since Ψ preserves the unit n-distance,
But it follows from Ψ (f0) = Ψ (f1) that
which is a contradiction. Hence, Ψ is injective.
Let f0, f1, f2, ..., f n be elements of ℑ(X), k ∈ ℕ and
We put
Then
for all i = 0, 1, ..., k - 1. Since Ψ satisfies (n DOPP),
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
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(g i ) = Ψ(g i ) - Ψ(gi-1) for all i = 0, 1, ..., k - 1. Thus Ψ(gi+1) - Ψ(g i ) = Ψ(g1) - Ψ(g0) for all i = 0, 1, ..., k - 1. Hence
Hence
This completes the proof.
Lemma 4.4 Let h, f0, f1, ..., f n be elements of ℑ(X) and let h, f0, f1 be 2-collinear. Then
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
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, ..., f n are m-collinear then Ψ(f0), Ψ(f1), ..., Ψ(f m ) are m-collinear, m = 2, n, and that Ψ satisfies (n DOPP), 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, ..., f n ∈ X, there are two cases depending upon whether ∥f1 - f0, ..., f n - f0∥ α = 0 or not. In the case ∥f1 - f0, ..., f n - f0∥ α = 0, f1 - f0, ..., f n - f0 are linearly dependent, that is, n-collinear. Thus f1 - f0, ..., f n - f0 are linearly dependent. Thus ∥Ψ(f1) - Ψ(f0), ..., Ψ(f n ) - Ψ(f0)∥ β = 0.
In the case ∥f1 - f0, ..., f n - f0∥ α > 0, there exists an n0 ∈ ℕ such that
Assume that
We can set
Then we get
It follows from Lemma 4.3 that
By the definition of h,
Since
h - f1 and f1 - f0 have the same direction. It follows from Lemma 4.2 that
Since Ψ(h), Ψ(f1), Ψ(f2) are 2-collinear, we have
by Lemma 4.4. By the assumption,
which is a contradiction. Hence Ψ is an n-isometry.
Lemma 4.5 Let g0, g1 be elements of ℑ(X). Then is the unique element of ℑ(X) satisfying
for some g2,...,g n ∈ ℑ(X) with ∥g0 - g n , g1 - g n , g2 - g n , ..., gn-1- g n ∥ α ≠ 0 and v, g0, g1 2-collinear.
Proof. Let ∥g0 - g n , g1 - g n , g2 - g n , ..., gn-1- g n ∥ α ≠ 0 and .
Then v, g0, g1 are 2-collinear. It follows from Lemma 4.1 and g n - g 0 = g1 - g0 - (g1 - g n ) that
and similarly
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
and
Since ∥g0 - g n , g1 - g n , g2 - g n , ..., gn-1- g n ∥ α ≠ 0, we have . Therefore, we get 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,...,g n ∈ ℑ(X) such that g1 - f0, ..., g n - f0 are linearly independent. Then
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,...,f n ∈ ℑ(X),
i.e. Ψ(f1) - Ψ(f0), ..., Ψ(f n ) - Ψ(f0) are linearly dependent.
If there exist f3, ..., fn-1such that Ψ(f1) - Ψ(f0), ..., Ψ(fn- 1) - Ψ(f0) are linearly independent, then
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-2such that Ψ(f1) - Ψ(f0), ..., Ψ(fn-2) - Ψ(f0) are linearly independent, then
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,...,f n ∈ ℑ(X) such that
Since Ψ is an n-isometry, we have
It follows from Lemma 4.1 that
And we get
By Lemma 4.6, we obtain that , Ψ(f0), and Ψ(f1) are 2-collinear. By Lemma 4.5, we get 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 -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) = k Ψ(f). Since dimℑ(X) ≥ n, there exist f1, ..., fn-1 ∈ ℑ(X) such that ∥f, f1, f2, ..., fn-1∥ α ≠ 0. Since Ψ(0) = 0, for every f0, f1, f2, ..., fn-1 ∈ ℑ(X),
Thus we have
Since ∥f, f1, f2, ..., fn-1∥ α ≠ 0, |k| = r. Then Ψ(rf) = r Ψ(f) or Ψ(rf) = -r Ψ(f). First of all, assume that k = -r, that is, Ψ(rf) = -r Ψ(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
Then we have
And also we have
Since ∥rf - q2f, h1 - q2f, h2 - q2f, ..., hn-1- q2f∥ α ≠ 0,
Thus we have r + q2 < q2 - q1, which is a contradiction. Hence k = r, that is, Ψ(rf) = r Ψ(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 (n DOPP), 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.
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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.
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Park, C., Alaca, C. An introduction to 2-fuzzy n-normed linear spaces and a new perspective to the Mazur-Ulam problem. J Inequal Appl 2012, 14 (2012). https://doi.org/10.1186/1029-242X-2012-14
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DOI: https://doi.org/10.1186/1029-242X-2012-14