Abstract
The purpose of this article is to introduce the concept of 2fuzzy nnormed linear space or fuzzy nnormed linear space of the set of all fuzzy sets of a nonempty set. We define the concepts of nisometry, ncollinearity nLipschitz mapping in this space. Also, we generalize the MazurUlam theorem, that is, when X is a 2fuzzy nnormed linear space or ℑ(X) is a fuzzy nnormed linear space, the MazurUlam theorem holds. Moreover, it is shown that each nisometry in 2fuzzy nnormed linear spaces is affine.
Mathematics Subject Classification (2010): 03E72; 46B20; 51M25; 46B04; 46S40.
Keywords:
MazurUlam theorem; αnnorm; 2fuzzy nnormed linear spaces; nisometry; nLipschitz mapping.1. Introduction
A satisfactory theory of 2norms and nnorms 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 nnormed space. In [6], Gunawan and Mashadi gave a simple way to derive an (n  1)norm from the nnorms and realized that any nnormed 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 nnormed linear spaces has been studied by many authors (see [4,9]).
Recently, Somasundaram and Beaula [10] introduced the concept of 2fuzzy 2normed linear space or fuzzy 2normed linear space of the set of all fuzzy sets of a set. The authors gave the notion of α2norm on a linear space corresponding to the 2fuzzy 2norm 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.
MazurUlam 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 MazurUlam 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 MazurUlam theorem. Chu [16] proved that the MazurUlam theorem holds when X is a linear 2normed space. Chu et al. [17] generalized the MazurUlam theorem when X is a linear nnormed space, that is, the MazurUlam theorem holds, when the nisometry mapped to a linear nnormed space is affine. They also obtain extensions of Rassias and Šemrl's theorem [18]. Moslehian and Sadeghi [19] investigated the MazurUlam theorem in nonarchimedean spaces. Choy et al. [20] proved the MazurUlam theorem for the interior preserving mappings in linear 2normed spaces. They also proved the theorem on nonArchimedean 2normed spaces over a linear ordered nonArchimedean 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 MazurUlam problem on nonArchimedean 2normed spaces using the above statement. Xiaoyun and Meimei [22] introduced the concept of weak nisometry and then they got under some conditions, a weak nisometry is also an nisometry. Cobzaş [23] gave some results of the MazurUlam theorem for the probabilistic normed spaces as defined by Alsina et al. [24]. Cho et al. [25] investigated the MazurUlam theorem on probabilistic 2normed spaces. Alaca [26] introduced the concepts of 2isometry, collinearity, 2Lipschitz mapping in 2fuzzy 2normed linear spaces. Also, he gave a new generalization of the MazurUlam theorem when X is a 2fuzzy 2normed linear space or ℑ(X) is a fuzzy 2normed linear space. Kang et al. [27] proved that the MazurUlam theorem holds under some conditions in nonArchimedean fuzzy normed space. Kubzdela [28] gave some new results for isometries, MazurUlam theorem and Aleksandrov problem in the framework of nonArchimedean normed spaces. The MazurUlam theorem has been extensively studied by many authors (see [29,30]).
In the present article, we introduce the concept of 2fuzzy nnormed linear space or fuzzy nnormed linear space of the set of all fuzzy sets of a nonempty set. We define the concepts of nisometry, ncollinearity, nLipschitz mapping in this space. Also, we generalize the MazurUlam theorem, that is, when X is a 2fuzzy nnormed linear space or ℑ(X) is a fuzzy nnormed linear space, the MazurUlam theorem holds. It is moreover shown that each nisometry in 2fuzzy nnormed 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 realvalued function ∥●, ..., ●∥ on satisfying the following properties
(1) ∥x_{1}, x_{2}, ..., x_{n}∥ = 0 if and only if x_{1}, x_{2}, ..., x_{n }are linearly dependent,
(2) ∥x_{1}, x_{2}, ..., x_{n}∥ is invariant under any permutation,
(3) ∥x_{1}, x_{2}, ..., αx_{n}∥ = α ∥x_{1}, x_{2}, ..., x_{n}∥ for any α ∈ ℝ,
(4) ∥x_{1}, x_{2}, ..., x_{n1}, y + z∥ ≤ ∥x_{1}, x_{2}, ..., x_{n1}, y∥ + ∥x_{1}, x_{2}, ..., x_{n1}, z∥, is called an nnorm on X and the pair (X, ∥●, ..., ●∥) is called an nnormed 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 X^{n }× ℝ (ℝ, the set of real numbers) is called a fuzzy nnorm on X if and only if:
(N1) For all t ∈ ℝ with t ≤ 0, N(x_{1}, x_{2}, ..., x_{n}, t) = 0,
(N2) For all t ∈ ℝ with t > 0, N(x_{1}, x_{2}, ..., x_{n}, t) = 1 if and only if x_{1}, x_{2}, ..., x_{n }are linearly dependent,
(N3) N(x_{1}, x_{2}, ..., x_{n}, t) is invariant under any permutation of x_{1}, x_{2}, ..., x_{n},
(N4) For all t ∈ ℝ with t > 0, , if λ ≠ 0, λ ∈ S,
(N5) For all s, t ∈ ℝ
(N6) N(x_{1}, x_{2}, ..., x_{n}, t) is a nondecreasing function of t ∈ ℝ and .
Then (X, N) is called a fuzzy nnormed linear space or in short fnNLS.
Theorem 2.1 [9] Let (X, N) be an fnNLS. Assume that
(N7) N(x_{1}, x_{2}, ..., x_{n},t) > 0 for all t > 0 implies that x_{1}, x_{2}, ..., x_{n }are linearly dependent.
Define
Then {∥●, ●, ..., ●∥_{α }: α ∈ (0, 1)} is an ascending family of nnorms on X.
We call these nnorms as αnnorms on X corresponding to the fuzzy nnorm on X.
Definition 2.3 Let X be any nonempty 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 nonnegative 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. 2fuzzy nnormed linear spaces
In this section, we define the concepts of 2fuzzy nnormed linear spaces and αnnorms on the set of all fuzzy sets of a nonempty set.
Definition 3.1 Let X be a nonempty 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 nonnegative 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 2fuzzy 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) ∥f_{1}, f_{2}, ..., f_{n}∥ = 0 if and only if f_{1}, f_{2}, ..., f_{n }are linearly dependent,
(2) ∥f_{1}, f_{2}, ..., f_{n}∥ is invariant under any permutation,
(3) ∥f_{1}, f_{2}, ..., λf_{n}∥ = λ ∥f_{1}, f_{2}, ..., f_{n}∥ for any λ ∈ S,
(4) ∥f_{1}, f_{2}, ..., f_{n1}, y + z∥ ≤ ∥f_{1}, f_{2}, ..., f_{n1}, y∥ + ∥f_{1}, f_{2}, ..., f_{n1}, z∥.
Then (ℑ(X),∥●,...,●∥) is an nnormed linear space or (X, ∥●, ..., ●∥) is a 2nnormed 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 2fuzzy nnorm on X (or fuzzy nnorm on ℑ(X)) if and only if
(2N1) for all t ∈ ℝ with t ≤ 0, N(f_{1}, f_{2}, ..., f_{n}, t) = 0,
(2N2) for all t ∈ ℝ with t > 0, N(f_{1}, f_{2}, ..., f_{n}, t) = 1 if and only if f_{1}, f_{2}, ..., f_{n }are linearly dependent,
(2N3) N(f_{1}, f_{2}, ..., f_{n}, t) is invariant under any permutation of f_{1}, f_{2}, ..., f_{n},
(2N4) for all t ∈ ℝ with t > 0, N(f_{1}, f_{2}, ..., λf_{n}, t) = N(f_{1}, f_{2}, ..., f_{n}, t/λ), if λ ≠ 0, λ ∈ S,
(2N5) for all s, t ∈ ℝ,
(2N6) N(f_{1}, f_{2}, ..., f_{n}, ·): (0, ∞) → [0, 1] is continuous,
Then ℑ(X), N) is a fuzzy nnormed linear space or (X, N) is a 2fuzzy nnormed linear space.
Remark 3.1 In a 2fuzzy nnormed linear space (X, N), N(f_{1}, f_{2}, ..., f_{n}, ·) is a nondecreasing function of ℝ for all f_{1}, f_{2},...,f_{n} ∈ ℑ(X).
Remark 3.2 From (2N4) and (2N5), it follows that in a 2fuzzy nnormed linear space,
(2N4) for all t ∈ ℝ with t > 0, if λ ≠ 0, λ ∈ S,
(2N5) for all s, t ∈ ℝ,
The following example agrees with our notion of 2fuzzy nnormed linear space.
Example 3.1 Let (ℑ(X),∥●,●,...,●∥) be an nnormed linear space as in Definition 3.3. Define
for all . Then (X, N) is a 2fuzzy nnormed linear space.
Solution. (2N1) For all t ∈ ℝ with t ≤ 0, by definition, we have N(f_{1}, f_{2}, ..., f_{n}, t) = 0.
(2N2) For all t ∈ ℝ with t > 0,
(2N3) For all t ∈ ℝ with t > 0,
(2N4) For all t ∈ ℝ with t > 0 and λ ∈ F, λ ≠ 0,
(2N5) 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
(2N6) It is clear that N(f_{1}, f_{2}, ..., f_{n}, ·): (0, ∞) → [0, 1] is continuous.
(2N7) 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 αnnorms corresponding to the fuzzy nnorms in the following theorem.
Theorem 3.1 Let (ℑ(X), N) is a fuzzy nnormed linear space. Assume that
(2N8) N(f_{1}, f_{2}, ..., f_{n}, t) > 0 for all t > 0 implies f_{1}, f_{2}, ..., f_{n }are linearly dependent.
Define
Then {∥●, ●, ..., ●∥_{α }: α ∈ (0, 1)} is an ascending family of nnorms on ℑ(X).
These nnorms are called αnnorms on ℑ(X) corresponding to the 2fuzzy nnorm on X.
Proof. (i) Let ∥f_{1}, ..., f_{n}∥_{α }= 0. This implies that inf {t : N(f_{1}, ..., f_{n}, t) ≥ α}. Then, N(f_{1}, f_{2}, ..., f_{n}, t) ≥ α > 0, for all t > 0, α ∈ (0, 1), which implies that f_{1}, f_{2}, ..., f_{n }are linearly dependent, by (2N8).
Conversely, assume f_{1}, f_{2}, ..., f_{n }are linearly dependent. This implies that N(f_{1}, f_{2}, ..., f_{n}, t) = 1 for all t > 0. For all α ∈ (0, 1), inf {t : N(f_{1}, f_{2}, ..., f_{n}, t) ≥ α}, which implies that ∥f_{1}, f_{2}, ..., f_{n}∥_{α }= 0.
(ii) Since N(f_{1}, f_{2}, ..., f_{n}, t) is invariant under any permutation, ∥f_{1}, f_{2}, ..., f_{n}∥_{α }= 0 under any permutation.
(iii) If λ ≠ 0, then
If λ = 0, then
(iv)
Hence
Thus {∥●, ●, ..., ●∥_{α }: α ∈ (0, 1)} is an αnnorm on X.
Let 0 < α_{1 }< α_{2}. Then,
As α_{1 }< α_{2},
implies that
which implies that
Hence {∥●, ●, ..., ●∥_{α }: α ∈ (0, 1)} is an ascending family of αnnorms on x corresponding to the 2fuzzy nnorm on X.
4. On the MazurUlam problem
In this section, we give a new generalization of the MazurUlam theorem when X is a 2fuzzy nnormed linear space or ℑ(X) is a fuzzy nnormed linear space. Hereafter, we use the notion of fuzzy nnormed linear space on ℑ(X) instead of 2fuzzy nnormed linear space on X.
Definition 4.1 Let ℑ(X) and ℑ(X) be fuzzy nnormed linear spaces and Ψ : ℑ(X) → ℑ(Y) a mapping. We call Ψ an nisometry if
for all f_{0}, f_{1}, f_{2},...,f_{n} ∈ ℑ(X) and α, β ∈ (0, 1).
For a mapping Ψ, consider the following condition which is called the ndistance one preserving property (nDOPP).
Then ∥Ψ(f_{1})  Ψ(f_{0}), ..., Ψ(f_{n})  Ψ(f_{0})∥_{β }= 1.
Lemma 4.1 Let f_{1}, f_{2},...,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 f_{0 }and are linearly dependent with some direction, that is, for some t > 0, then
for all f_{1}, f_{2},...,f_{n} ∈ ℑ(X) and α ∈ (0, 1).
Proof. Let for some t > 0. Then we have
for all f_{1}, f_{2},...,f_{n} ∈ ℑ(X) and α ∈ (0, 1).
Definition 4.2 The elements f_{0}, f_{1}, f_{2}, ..., f_{n }of ℑ(X) are said to be ncollinear if for every i, {f_{j } f_{i }: 0 ≤ j ≠ i ≤ n} is linearly dependent.
Remark 4.1 The elements f_{0}, f_{1}, and f_{2 }are said to be 2collinear if and only if f_{2 } f_{0 }= r(f_{1 } f_{0}) for some real number r.
Now we define the concept of nLipschitz mapping.
Definition 4.3 We call Ψ an nLipschitz mapping if there is a κ ≥ 0 such that
for all f_{0}, f_{1}, f_{2},...,f_{n} ∈ ℑ(X) and α, β ∈ (0, 1). The smallest such κ is called the nLipschitz constant.
Lemma 4.3 Assume that if f_{0}, f_{1}, and f_{2 }are 2 collinear then Ψ(f_{0}), Ψ(f_{1}) and Ψ(f_{2}) are 2collinear, and that Ψ satisfies (nDOPP). Then Ψ preserves the ndistance k for each k ∈ ℕ.
Proof. Suppose that there exist f_{0}, f_{1} ∈ ℑ(X) with f_{0 }≠ f_{1 }such that Ψ(f_{0}) = Ψ(f_{1}). Since dimℑ(X) ≥ n, there are f_{2},...,f_{n} ∈ ℑ(X) such that f_{1 } f_{0}, f_{2 } f_{0}, ..., f_{n } f_{0 }are linearly independent. Since ∥f_{1 } f_{0}, f_{2 } f_{0}, ..., f_{n } f_{0}∥_{α }≠ 0, we can set
Then we have
Since Ψ preserves the unit ndistance,
But it follows from Ψ (f_{0}) = Ψ (f_{1}) that
which is a contradiction. Hence, Ψ is injective.
Let f_{0}, f_{1}, f_{2}, ..., f_{n }be elements of ℑ(X), k ∈ ℕ and
We put
Then
for all i = 0, 1, ..., k  1. Since Ψ satisfies (nDOPP),
for all i = 0, 1, ..., k  1. Since g_{0}, g_{1}, and g_{2 }are 2collinear, Ψ (g_{0}), Ψ(g_{1}) and Ψ(g_{2}) are also 2collinear. Thus there is a real number r_{0 }such that Ψ(g_{2})  Ψ(g_{1}) = r_{0 }(Ψ(g_{1})  Ψ(g_{0})). It follows from (4.1) that
Thus, we have r_{0 }= 1 or 1. If r_{0 }= 1, Ψ(g_{2})  Ψ(g_{1}) = Ψ(g_{1}) + Ψ(g_{0}), that is, Ψ(g_{2}) = Ψ(g_{0}). Since Ψ is injective, g_{2 }= g_{0}, which is a contradiction. Thus r_{0 }= 1. Then we have Ψ(g_{2})  Ψ(g_{1}) = Ψ(g_{1})  Ψ(g_{0}). Similarly, one can obtain that Ψ(g_{i+1})  Ψ(g_{i}) = Ψ(g_{i})  Ψ(g_{i1}) for all i = 0, 1, ..., k  1. Thus Ψ(g_{i+1})  Ψ(g_{i}) = Ψ(g_{1})  Ψ(g_{0}) for all i = 0, 1, ..., k  1. Hence
Hence
This completes the proof.
Lemma 4.4 Let h, f_{0}, f_{1}, ..., f_{n }be elements of ℑ(X) and let h, f_{0}, f_{1 }be 2collinear. Then
Proof. Since h, f_{0}, f_{1 }are 2collinear, there exists a real number r such that f_{1 } h = r(f_{0 } h). It follows from Lemma 4.1 that
This completes the proof.
Theorem 4.1 Let Ψ be an nLipschitz mapping with the nLipschitz constant κ ≤ 1. Assume that if f_{0}, f_{1}, ..., f_{n }are mcollinear then Ψ(f_{0}), Ψ(f_{1}), ..., Ψ(f_{m}) are mcollinear, m = 2, n, and that Ψ satisfies (nDOPP), then Ψ is an nisometry.
Proof. It follows from Lemma 4.3 that Ψ preserves ndistance k for all k ∈ ℕ. For f_{0}, f_{1}, ..., f_{n }∈ X, there are two cases depending upon whether ∥f_{1 } f_{0}, ..., f_{n } f_{0}∥_{α }= 0 or not. In the case ∥f_{1 } f_{0}, ..., f_{n } f_{0}∥_{α }= 0, f_{1 } f_{0}, ..., f_{n } f_{0 }are linearly dependent, that is, ncollinear. Thus f_{1 } f_{0}, ..., f_{n } f_{0 }are linearly dependent. Thus ∥Ψ(f_{1})  Ψ(f_{0}), ..., Ψ(f_{n})  Ψ(f_{0})∥_{β }= 0.
In the case ∥f_{1 } f_{0}, ..., f_{n } f_{0}∥_{α }> 0, there exists an n_{0 }∈ 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  f_{1 }and f_{1 } f_{0 }have the same direction. It follows from Lemma 4.2 that
Since Ψ(h), Ψ(f_{1}), Ψ(f_{2}) are 2collinear, we have
by Lemma 4.4. By the assumption,
which is a contradiction. Hence Ψ is an nisometry.
Lemma 4.5 Let g_{0}, g_{1 }be elements of ℑ(X). Then is the unique element of ℑ(X) satisfying
for some g_{2},...,g_{n} ∈ ℑ(X) with ∥g_{0 } g_{n}, g_{1 } g_{n}, g_{2 } g_{n}, ..., g_{n1 } g_{n}∥_{α }≠ 0 and v, g_{0}, g_{1 }2collinear.
Proof. Let ∥g_{0 } g_{n}, g_{1 } g_{n}, g_{2 } g_{n}, ..., g_{n1 } g_{n}∥_{α }≠ 0 and .
Then v, g_{0}, g_{1 }are 2collinear. It follows from Lemma 4.1 and g_{n } g_{0 }= g_{1 } g_{0 } (g_{1 } g_{n}) that
and similarly
Now we prove the uniqueness.
Let u be an element of ℑ(X) satisfying the above properties. Since u, g_{0}, g_{1 }are 2collinear, there exists a real number t such that u = tg_{0 }+ (1  t)g_{1}. It follows from Lemma 4.1 that
and
Since ∥g_{0 } g_{n}, g_{1 } g_{n}, g_{2 } g_{n}, ..., g_{n1 } g_{n}∥_{α }≠ 0, we have . Therefore, we get and hence v = u.
Lemma 4.6 If Ψ is an nisometry and f_{0}, f_{1}, f_{2 }are 2collinear then Ψ(f_{0}), Ψ(f_{1}), Ψ(f_{2}) are 2collinear.
Proof. Since dimℑ(X) ≥ n, for any f_{0} ∈ ℑ(X), there exist g_{1},...,g_{n} ∈ ℑ(X) such that g_{1 } f_{0}, ..., g_{n } f_{0 }are linearly independent. Then
and hence, the set A = {Ψ(f) Ψ(f_{0}) : f ∈ ℑ(X)} contains n linearly independent vectors.
Assume that f_{0}, f_{1}, f_{2 }are 2collinear. Then, for any f_{3},...,f_{n} ∈ ℑ(X),
i.e. Ψ(f_{1})  Ψ(f_{0}), ..., Ψ(f_{n})  Ψ(f_{0}) are linearly dependent.
If there exist f_{3}, ..., f_{n1 }such that Ψ(f_{1})  Ψ(f_{0}), ..., Ψ(f_{n1})  Ψ(f_{0}) are linearly independent, then
which contradicts the fact that A contains n linearly independent vectors.
Then, for any f_{3}, ..., f_{n1}, Ψ(f_{1})  Ψ(f_{0}), ..., Ψ(f_{n1})  Ψ(f_{0}) are linearly dependent.
If there exist f_{3}, ..., f_{n2 }such that Ψ(f_{1})  Ψ(f_{0}), ..., Ψ(f_{n2})  Ψ(f_{0}) are linearly independent, then
which contradicts the fact that A contains n linearly independent vectors.
And so on, Ψ(f_{1})  Ψ(f_{0}), Ψ(f_{2})  Ψ(f_{0}) are linearly dependent. Thus Ψ(f_{0}), Ψ(f_{1}), and Ψ(f_{2}) are 2collinear.
Theorem 4.2 Every nisometry mapping is affine.
Proof. Let Ψ be an nisometry and Φ(f) = Ψ(f)  Ψ(0). Then Φ is an nisometry and Φ(0) = 0. Thus we may assume that Ψ(0) = 0. Hence it suffices to show that Ψ is linear.
Let f_{0}, f_{1} ∈ ℑ(X) with f_{0 }≠ f_{1}. Since dimℑ(X) ≥ n, there exist f_{2},...,f_{n} ∈ ℑ(X) such that
Since Ψ is an nisometry, we have
It follows from Lemma 4.1 that
And we get
By Lemma 4.6, we obtain that , Ψ(f_{0}), and Ψ(f_{1}) are 2collinear. 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 2collinear. 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 f_{1}, ..., f_{n1} ∈ ℑ(X) such that ∥f, f_{1}, f_{2}, ..., f_{n1}∥_{α }≠ 0. Since Ψ(0) = 0, for every f_{0}, f_{1}, f_{2}, ..., f_{n1} ∈ ℑ(X),
Thus we have
Since ∥f, f_{1}, f_{2}, ..., f_{n1}∥_{α }≠ 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 q_{1}, q_{2 }such that o < q_{1 }< r < q_{2}. Since dimℑ(X) ≥ n, there exist h_{1}, ..., h_{n1} ∈ ℑ(X) such that
Then we have
And also we have
Since ∥rf  q_{2}f, h_{1 } q_{2}f, h_{2 } q_{2}f, ..., h_{n1 } q_{2}f∥_{α }≠ 0,
Thus we have r + q_{2 }< q_{2 } q_{1}, which is a contradiction. Hence k = r, that is, Ψ(rf) = rΨ(f) for all positive real numbers r. Therefore Ψ is ℝlinear, as desired.
We get the following corollary from Theorems 4.1 and 4.2.
Corollary 4.1 Let Ψ be an nLipschitz mapping with the nLipschitz constant κ ≤ 1. Suppose that if f, g, h are 2collinear, then Ψ(f), Ψ(g), Ψ(h) are 2collinear. If Ψ satisfies (nDOPP), then Ψ is an affine nisometry.
5. Conclusion
In this article, the concept of 2fuzzy nnormed linear space is defined and the concepts of nisometry, ncollinearity, nLipschitz mapping are given. Also, the MazurUlam theorem is generalized into 2fuzzy nnormed 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. ElGebeily for giving useful suggestions and comments for the improvement of this article.
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