On the Mazur-Ulam Theorem in Non-Archimedean Fuzzy 𝑛-Normed Spaces

Abstract

The motivation of this paper is to present a new notion of non-Archimedean fuzzy 𝑛-normed space over a field with valuation. We obtain a Mazur-Ulam theorem for fuzzy 𝑛-isometry mappings in the strictly convex non-Archimedean fuzzy 𝑛-normed spaces. We also prove that the interior preserving mapping carries the barycenter of a triangle to the barycenter point of the corresponding triangle. And then, using this result, we get a Mazur-Ulam theorem for the interior preserving fuzzy 𝑛-isometry mappings in non-Archimedean fuzzy 𝑛-normed spaces over a linear ordered non-Archimedean field.

1. Introduction

Let 𝕂 be a field. A valuation mapping on 𝕂 is a function |⋅|∶𝕂→ℝ such that for any 𝑟,𝑠∈𝕂 the following conditions are satisfied: (i) |𝑟|≥0 and equality holds if and only if 𝑟=0; (ii) |𝑟𝑠|=|𝑟|⋅|𝑠|; (iii) |𝑟+𝑠|≤|𝑟|+|𝑠|.

A field endowed with a valuation mapping will be called a valued field. The usual absolute values of ℝ and ℂ are examples of valuations. A trivial example of a non-Archimedean valuation is the function |⋅| taking everything except for 0 into 1 and |0|=0. In the following, we will assume that |⋅| is nontrivial; that is, there is an 𝑟₀∈𝕂 such that |𝑟₀|≠0,1.

Throughout this paper, we assume that 𝕂 is a valued field and 𝑛≥2 is a positive integer. We denote the set of all elements of 𝕂 whose norms are 1 by 𝒞; that is, 𝒞={𝑟∈𝕂∣|𝑟|=1}. Moreover, ℕ stands for the set of all positive integers and ℝ (resp., ℂ) denotes the set of all real numbers (resp., complex numbers).

If condition (iii) in the definition of a valuation mapping is replaced with a strong triangle inequality (ultrametric), |𝑟+𝑠|≤max{|𝑟|,|𝑠|}, then the valuation |⋅| is said to be non-Archimedean. In any non-Archimedean field, we have |1|=|−1|=1 and |𝑘|≤1 for all 𝑘∈ℕ.

Let 𝒳 and 𝒴 be metric spaces. A map 𝑓∶𝒳→𝒴 is called a distance preserving mapping (isometry) if 𝑑(𝑓(𝑥),𝑓(𝑦))=𝑑(𝑥,𝑦) for any 𝑥,𝑦∈𝒳. Automatically, an isometry is injective. Two metric spaces 𝒳 and 𝒴 are called isometric if there is an isometry from 𝒳 to 𝒴.

The classical result of Mazur and Ulam states that if 𝒳, 𝒴 are real normed linear spaces and 𝑓∶𝒳→𝒴 is a surjective isometry, then 𝑓 is affine; that is, 𝑓 is a linear mapping up to translation. Numerous generalizations of this fact were presented by many authors (see, e.g., [1–13] and references therein). Unfortunately, the Mazur-Ulam theorem is not true for normed complex vector space. It was a natural step to ask if the theorem holds without the onto assumption. In fact, the onto assumption is essential. Without this assumption, Baker [14] proved that every isometry, not necessary surjective, 𝑓∶𝒳→𝒴, between real normed linear spaces is affine if 𝒴 is strictly convex. Moslehian and Sadeghi presented a non-Archimedean version of this result [11]; they also noted that a Mazur-Ulam theorem generally fails in a non-Archimedean case. Choy et al. [3] 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. Chu et al. [4] studied the Mazur-Ulam theorem in linear 𝑛-normed spaces. Alaca [1] introduced the concepts of 2-isometry, collinearity, and 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. Kubzdela [10] gave some new results for isometries, Mazur-Ulam theorem, and Aleksandrov problem in the framework of non-Archimedean normed spaces. Kang et al. [9] proved that the Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy normed space.

The motivation of this paper is to introduce the notion of non-Archimedean fuzzy 𝑛-normed space over a field with valuation as a generalization of 𝑛-normed space [2, 15, 16], non-Archimedean 2-normed space [3], fuzzy 𝑛-normed space [17], and non-Archimedean fuzzy normed space [9, 18]. We will prove that the Mazur-Ulam theorem holds in the strictly convex non-Archimedean fuzzy 𝑛-normed spaces.

2. Preliminaries

In 1897, Hensel discovered the 𝑝-adic numbers as a number-theoretical analogue of power series in complex analysis. Let 𝑝 be a prime number. For any nonzero rational number 𝑎, there exists a unique integer 𝑟 such that 𝑎=𝑝^𝑟𝑚/𝑘, where 𝑚 and 𝑘 are integers not divisible by 𝑝. Then |𝑎|_𝑝∶=𝑝^−𝑟 defines a non-Archimedean norm on ℚ. The completion of ℚ with respect to the metric 𝑑(𝑎,𝑏)=|𝑎−𝑏|_𝑝 is denoted by ℚ_𝑝 which is called the 𝑝-adic number field. Note that if 𝑝>2, then |2^𝑘|_𝑝=1 for each integer 𝑘 but |2|₂<1.

During the last three decades, 𝑝-adic numbers have gained the interest of physicists for their research, in particular, in problems derived from quantum physics, 𝑝-adic strings, and superstrings (see, e.g., [19]).

Definition 1. Let 𝒳 be a linear space over a field 𝕂 with a non-Archimedean valuation |⋅|. A function ‖⋅‖∶𝒳→[0,∞) is said to be a non-Archimedean norm if it satisfies the following conditions:(i)‖𝑥‖=0 if and only if 𝑥=0, (ii)‖𝑟𝑥‖=|𝑟|‖𝑥‖, 𝑟∈𝕂, 𝑥∈𝒳, (iii)the strong triangle inequality: ‖𝑥+𝑦‖≤max{‖𝑥‖,‖𝑦‖}(𝑥,𝑦∈𝒳).
Then (𝒳,‖⋅‖) is called a non-Archimedean normed space. By a complete non-Archimedean normed space, we mean one in which every Cauchy sequence is convergent.

Definition 2. Let 𝒳 be a linear space over a valued field 𝕂. A function 𝑁∶𝒳^𝑛×ℝ→[0,1] is called a non-Archimedean fuzzy 𝑛-norm on 𝒳 if the following conditions hold for all 𝑥,𝑦,𝑥₁,…,𝑥_𝑛∈𝒳 and all 𝑠,𝑡∈ℝ:(𝑛𝑁₁)𝑁(𝑥₁,…,𝑥_𝑛,𝑡)=0 if 𝑡≤0, (𝑛𝑁₂) for all 𝑡>0, 𝑁(𝑥₁,…,𝑥_𝑛,𝑡)=1 if and only if 𝑥₁,…,𝑥_𝑛 are linearly dependent,(𝑛𝑁₃)𝑁(𝑥₁,…,𝑥_𝑛,𝑡) is invariant under any permutation of 𝑥₁,…,𝑥_𝑛, (𝑛𝑁₄)𝑁(𝑐𝑥₁,…,𝑥_𝑛,𝑡)=𝑁(𝑥₁,…,𝑥_𝑛,𝑡/|𝑐|) for all 𝑡>0 if 𝑐∈𝕂,𝑐≠0, (𝑛𝑁₅)𝑁(𝑥+𝑦,𝑥₂,…,𝑥_𝑛,max{𝑠,𝑡})≥min{𝑁(𝑥,𝑥₂,…,𝑥_𝑛,𝑠),𝑁(𝑦,𝑥₂,…,𝑥_𝑛,𝑡)} for all 𝑠,𝑡>0, (𝑛𝑁₆)lim_𝑡→∞𝑁(𝑥₁,…,𝑥_𝑛,𝑡)=1.
If 𝑁 is a non-Archimedean fuzzy 𝑛-norm on 𝒳, then (𝒳,𝑁) is called a non-Archimedean fuzzy 𝑛-normed space. It should be noticed that from the condition (𝑛𝑁₅) it follows that 𝑁𝑥₁,…,𝑥_𝑛,𝑡≥min𝑁(0,…,0,𝑡),𝑁𝑥₁,…,𝑥_𝑛𝑥,𝑠=𝑁₁,…,𝑥_𝑛,𝑠 (1) for every 𝑡>𝑠>0 and 𝑥₁,…,𝑥_𝑛∈𝒳; that is, 𝑁(𝑥₁,…,𝑥_𝑛,⋅) is nondecreasing for every 𝑥₁,…,𝑥_𝑛∈𝒳. This implies that 𝑁𝑥₁,…,𝑥_𝑛,𝑠+𝑡≥𝑁𝑥₁,…,𝑥_𝑛,max{𝑠,𝑡},𝑠,𝑡>0. (2) If (𝑛𝑁₅) holds, then so is (𝑛𝑁₇)𝑁(𝑥+𝑦,𝑥₂,…,𝑥_𝑛,𝑠+𝑡)≥min{𝑁(𝑥,𝑥₂,…,𝑥_𝑛,𝑠),𝑁(𝑦,𝑥₂,…,𝑥_𝑛,𝑡)}.

Example 3. Let (𝒳,‖⋅,…,⋅‖) be an 𝑛-normed space (see [2]). For each 𝑘∈ℕ, consider 𝑁_𝑘𝑥₁,…,𝑥_𝑛=𝑡,𝑡𝑡+𝑘𝑥₁,…,𝑥_𝑛,𝑡>0,0,𝑡≤0. (3) Then (𝒳,𝑁_𝑘) is a non-Archimedean fuzzy 𝑛-normed space.

Definition 4. Let 𝒳 and 𝒴 be non-Archimedean fuzzy 𝑛-normed spaces, and let 𝑓∶𝒳→𝒴 be a mapping. We call 𝑓 a fuzzy 𝑛-isometry if 𝑁𝑥₁−𝑥₀,…,𝑥_𝑛−𝑥₀𝑓𝑥,𝑡=𝑁₁−𝑓𝑥₀,…,𝑓𝑥_𝑛−𝑓𝑥₀,𝑡 (4) for all 𝑥₀,𝑥₁,…,𝑥_𝑛∈𝒳 and all 𝑡>0.

For given points 𝑥,𝑦, and 𝑧 in 𝒳, Δ𝑥𝑦𝑧 denotes the triangle determined by 𝑥,𝑦, and 𝑧. A point (𝑥+𝑦+𝑧)/3 is called a barycenter of Δ𝑥𝑦𝑧. If 𝑝 is a point of a set {𝑡₁𝑥+𝑡₂𝑦+𝑡₃𝑧∣𝑡₁+𝑡₂+𝑡₃=1,𝑡_𝑖∈𝕂,𝑡_𝑖>0,𝑖=1,2,3}, then 𝑝 is called an interior point of Δ𝑥𝑦𝑧. Define a mapping 𝑓 between linear 𝑛-normed spaces to be an interior preserving mapping of the triangle if 𝑓 carries an interior point in a triangle to an interior point in the corresponding triangle.

Remark 5. Let 𝒳 and 𝒴 be non-Archimedean fuzzy 𝑛-normed spaces and 𝑓∶𝒳→𝒴 be a mapping. Then 𝑓 is a fuzzy 𝑛-isometry if and only if 𝑓 satisfies the following property: 𝑁𝑥₁−𝑥₀,…,𝑥_𝑛−𝑥₀,𝑡−𝑁𝑥^′₁−𝑥^′₀,…,𝑥^′_𝑛−𝑥^′₀=𝑁𝑓𝑥,𝑡₁−𝑓𝑥₀,…,𝑓𝑥_𝑛−𝑓𝑥₀𝑓𝑥,𝑡−𝑁^′₁−𝑓𝑥^′₀,…,𝑓𝑥^′_𝑛−𝑓𝑥^′₀,𝑡 (5) for all 𝑥₀,𝑥₁,…,𝑥_𝑛,𝑥^′₀,𝑥^′₁,…,𝑥^′_𝑛∈𝒳 and all 𝑡>0.

Definition 6. Let 𝒳 and 𝒴 be non-Archimedean fuzzy 𝑛-normed spaces and 𝑓∶𝒳→𝒴 be a mapping. Then 𝑓 is called a weak fuzzy 𝑛-isometry if for every 𝜖>0, there exists positive real number 𝛿 such that 𝑁𝑥₁−𝑥₀,…,𝑥_𝑛−𝑥₀,𝑡−𝑁𝑥^′₁−𝑥^′₀,…,𝑥^′_𝑛−𝑥^′₀,𝑡<𝛿implies𝑁𝑓𝑥₁−𝑓𝑥₀,…,𝑓𝑥_𝑛−𝑓𝑥₀𝑓𝑥,𝑡−𝑁^′₁−𝑓𝑥^′₀,…,𝑓𝑥^′_𝑛−𝑓𝑥^′₀,𝑡<𝜖 (6) for all 𝑥₀,𝑥₁,…,𝑥_𝑛,𝑥^′₀,𝑥^′₁,…,𝑥^′_𝑛∈𝒳 and all 𝑡>0.

Definition 7. Let 𝒳 be a non-Archimedean fuzzy 𝑛-normed space. The points 𝑥₀,𝑥₁,…,𝑥_𝑛 are said to be 𝑛-collinear if for every 𝑖∈{0,1,…,𝑛}, {𝑥_𝑗−𝑥_𝑖∣0≤𝑗≠𝑖≤𝑛} is linearly dependent.

Remark 8. Let 𝒳 be a non-Archimedean fuzzy 𝑛-normed space over a valued field 𝕂 and 𝑥₀,𝑥₁, 𝑥₂ mutually disjoint elements of 𝒳. Then 𝑥₀,𝑥₁, and 𝑥₂ are said to be 2-collinear if and only if 𝑥₂−𝑥₀=𝑡(𝑥₁−𝑥₀) for some 𝑡∈𝕂.

Now we define the concept of 𝑛-Lipschitz mapping.

Definition 9. Let 𝒳 and 𝒴 be non-Archimedean fuzzy 𝑛-normed spaces, and let 𝑓∶𝒳→𝒴 be a mapping. Then 𝑓 is called a fuzzy 𝑛-Lipschitz mapping if there is a 𝐿≥0 such that 𝑁𝑓𝑥₁−𝑓𝑥₀,…,𝑓𝑥_𝑛−𝑓𝑥₀𝑥,𝐿𝑡≥𝑁₁−𝑥₀,…,𝑥_𝑛−𝑥₀,𝑡 (7) for all 𝑥₀,𝑥₁,…,𝑥_𝑛∈𝒳 and all 𝑡>0. The smallest such 𝐿 is called the 𝑛-Lipschitz constant.

Definition 10. A non-Archimedean fuzzy 𝑛-normed space 𝒳 over a valued field 𝕂 is called strictly convex, if for each 𝑥₁,…,𝑥_𝑚,𝑤₂,…,𝑤_𝑛∈𝒳 and 𝑠₁,…,𝑠_𝑚>0, 𝑁^𝑚_𝑖=1𝑥_𝑖,𝑤₂,…,𝑤_𝑛,max𝑠₁,…,𝑠_𝑚=min𝑁𝑥₁,𝑤₂,…,𝑤_𝑛,𝑠₁,…,𝑁𝑥_𝑚,𝑤₂,…,𝑤_𝑛,𝑠_𝑚,𝑁𝑥₁,𝑤₂,…,𝑤_𝑛,𝑠₁=⋯=𝑁𝑥_𝑚,𝑤₂,…,𝑤_𝑛,𝑠_𝑚 (8) implies that 𝑥₁=⋯=𝑥_𝑚 and 𝑠₁=⋯=𝑠_𝑚.

3. On the Mazur-Ulam Problem

Lemma 11. Let 𝒳 be a non-Archimedean fuzzy 𝑛-normed space over a valued field 𝕂, 𝑥₁,…,𝑥_𝑛∈𝒳 and all 𝑡>0. Then 𝑁𝑥₁,…,𝑥_𝑖,…,𝑥_𝑗,…,𝑥_𝑛𝑥,𝑡=𝑁₁,…,𝑥_𝑖,…,𝑥_𝑗−1,𝑥_𝑗+𝛼𝑥_𝑖,𝑥_𝑗+1,…,𝑥_𝑛,𝑡 (9) for all 𝛼∈𝕂.

Proof. Let 𝑥₁,…,𝑥_𝑛∈𝒳,𝑡>0, and 𝛼∈𝕂, then 𝑁𝑥₁,…,𝑥_𝑖,…,𝑥_𝑗−1,𝑥_𝑗+𝛼𝑥_𝑖,𝑥_𝑗+1,…,𝑥_𝑛𝑁𝑥,𝑡≥min₁,…,𝑥_𝑖,…,𝑥_𝑗,…,𝑥_𝑛,𝑁𝑥,𝑡₁,…,𝑥_𝑖,…,𝑥_𝑗−1,𝑥_𝑖,𝑥_𝑗+1,…,𝑥_𝑛,𝑡|𝛼|≥𝑁𝑥₁,…,𝑥_𝑖,…,𝑥_𝑗,…,𝑥_𝑛,𝑁𝑥,𝑡₁,…,𝑥_𝑖,…,𝑥_𝑗,…,𝑥_𝑛𝑥,𝑡=𝑁₁,…,𝑥_𝑖,…,𝑥_𝑗−1,𝑥_𝑗+𝛼𝑥_𝑖−𝛼𝑥_𝑖,𝑥_𝑗+1,…,𝑥_𝑛𝑁𝑥,𝑡≥min₁,…,𝑥_𝑖,…,𝑥_𝑗−1,𝑥_𝑗+𝛼𝑥_𝑖,𝑥_𝑗+1,…,𝑥_𝑛,𝑁𝑥,𝑡₁,…,𝑥_𝑖,…,𝑥_𝑗−1,−𝛼𝑥_𝑖,𝑥_𝑗+1,…,𝑥_𝑛𝑥,𝑡=𝑁₁,…,𝑥_𝑖,…,𝑥_𝑗−1,𝑥_𝑗+𝛼𝑥_𝑖,𝑥_𝑗+1,…,𝑥_𝑛,𝑡. (10)
It follows that (9) holds.

Lemma 12. Let 𝒳 be a strictly convex non-Archimedean fuzzy 𝑛-normed space over a linear ordered non-Archimedean field 𝕂 with 𝒞={2^𝑛∣𝑛∈ℤ}, and let 𝑥,𝑦∈𝒳 and 𝑡>0. Then 𝑢=(𝑥+𝑦)/2 is the unique element of 𝒳 satisfying 𝑁𝑥−𝑢,𝑥₂−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑦−𝑢,𝑥₂−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛−𝑥,𝑡 (11) for all 𝑥₂,…,𝑥_𝑛∈𝒳 and 𝑥,𝑦, and 𝑢 are 2-collinear.

Proof. Let 𝑢∶=(𝑥+𝑦)/2,𝑥₂,…,𝑥_𝑛∈𝒳 and 𝑡>0. By Lemma 11, we have 𝑁𝑥−𝑢,𝑥₂−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑥−𝑥+𝑦2,𝑥₂−𝑥+𝑦2,…,𝑥_𝑛−𝑥+𝑦2,𝑡=𝑁𝑥−𝑥+𝑦2,𝑥₂−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑦−𝑥,𝑥₂−𝑥,…,𝑥_𝑛|2|𝑡,𝑁−𝑥,𝑦−𝑢,𝑥₂−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑦−𝑥+𝑦2,𝑥₂−𝑥+𝑦2,…,𝑥_𝑛−𝑥+𝑦2,𝑡=𝑁𝑦−𝑥+𝑦2,𝑥₂−𝑦,…,𝑥_𝑛−𝑦,𝑡=𝑁𝑦−𝑥,𝑥₂−𝑦,…,𝑥_𝑛−𝑦,|2|𝑡=𝑁𝑦−𝑥,𝑥₂−𝑥,…,𝑥_𝑛−𝑥,|2|𝑡. (12) Hence, the existence holds. For the uniqueness of 𝑢, we may assume that there is another 𝑣∈𝒳, collinear with 𝑥,𝑦 such that 𝑁𝑥−𝑣,𝑥₂−𝑣,…,𝑥_𝑛−𝑣,𝑡=𝑁𝑦−𝑣,𝑥₂−𝑣,…,𝑥_𝑛−𝑣,𝑡=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛−𝑥,𝑡. (13) Since 𝑥,𝑦, and 𝑣 are collinear, 𝑣=𝑠𝑥+(1−𝑠)𝑦 for some 𝑠∈𝕂. We may assume 𝑠≠0 and 𝑠≠1. Then 𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑣,𝑥₂−𝑣,…,𝑥_𝑛−𝑣,𝑡=𝑁𝑥−𝑣,𝑥₂−𝑥,…,𝑥_𝑛()−𝑥,𝑡=𝑁𝑥−𝑠𝑥−1−𝑠𝑦,𝑥₂−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑡−𝑥,||,𝑁1−𝑠𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑦−𝑣,𝑥₂−𝑣,…,𝑥_𝑛−𝑣,𝑡=𝑁𝑦−𝑣,𝑥₂−𝑦,…,𝑥_𝑛()−𝑦,𝑡=𝑁𝑦−𝑠𝑥−1−𝑠𝑦,𝑥₂−𝑦,…,𝑥_𝑛−𝑦,𝑡=𝑁𝑥−𝑦,𝑥₂−𝑦,…,𝑥_𝑛𝑡−𝑦,|𝑠|=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑡−𝑥,|𝑠|. (14) Then 𝑁𝑥−𝑦+𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑡−𝑥,max|𝑠|,𝑡|1−𝑠|=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛1−𝑥,|2|𝑡max|𝑠|,𝑡|1−𝑠|=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑁−𝑥,𝑡=min𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑡−𝑥,|𝑠|,𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑡−𝑥,||,𝑁1−𝑠𝑥−𝑦+𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑡−𝑥,max𝑡,|𝑠|=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛1−𝑥,|2|𝑡max𝑡,|𝑠|=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑁−𝑥,𝑡=min𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛,𝑁−𝑥,𝑡𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑡−𝑥,|𝑠|. (15) By the strict convexity of 𝒳, we have |1−𝑠|=|𝑠|=1. Then there exist two integers 𝑘₁,𝑘₂ such that 1−𝑠=2^𝑘1 and 𝑠=2^𝑘2. Since 2^𝑘1+2^𝑘2=1, we know that 𝑘₁<0,𝑘₂<0. Without loss of generality, we let 1−𝑠=2^−𝑛1 and 𝑠=2^−𝑛2 with 𝑛₁≥𝑛₂. If 𝑛₁>𝑛₂, then 1=2^−𝑛1+2^−𝑛2=2^−𝑛1(1+2^{𝑛1−𝑛2}); that is, 2^𝑛1=1+2^{𝑛1−𝑛2}. This is a contradiction. Thus, 𝑛₁=𝑛₂; that is, 𝑠=1/2. This completes the proof.

Lemma 13. Let 𝒳 and 𝒴 be non-Archimedean fuzzy 𝑛-normed spaces over a valued field 𝕂. If 𝑓∶𝒳→𝒴 is a fuzzy 𝑛-isometry and 𝑥₀,𝑥₁, and 𝑥₂ are 2-collinear, then 𝑓(𝑥₀),𝑓(𝑥₁), and 𝑓(𝑥₂) are 2-collinear.

Proof. Since dim𝒳≥𝑛, for any 𝑥₀∈𝒳, there exist 𝑦₁,…,𝑦_𝑛∈𝒳 such that 𝑦₁−𝑥₀,…,𝑦_𝑛−𝑥₀ are linearly independent. Then 𝑁𝑦₁−𝑥₀,…,𝑦_𝑛−𝑥₀𝑓𝑦,𝑡=𝑁₁−𝑓𝑥₀,…,𝑓𝑦_𝑛−𝑓𝑥₀,𝑡≠1, (16) and hence, the set {𝑓(𝑥)−𝑓(𝑥₀)∶𝑥∈𝒳} contains 𝑛 linearly independent vectors. Assume that 𝑥₀,𝑥₁, and 𝑥₂ are 2-collinear. Then, for any 𝑥₃,…,𝑥_𝑛∈𝒳, 𝑁𝑥₁−𝑥₀,…,𝑥_𝑛−𝑥₀𝑓𝑥,𝑡=𝑁₁−𝑓𝑥₀,…,𝑓𝑥_𝑛−𝑓𝑥₀,𝑡=1. (17) By (𝑛𝑁₂), it follows that 𝑓(𝑥₁)−𝑓(𝑥₀),…,𝑓(𝑥_𝑛)−𝑓(𝑥₀) are linearly dependent. If there exist 𝑥₃,…,𝑥_𝑛−1 such that 𝑓(𝑥₁)−𝑓(𝑥₀),…,𝑓(𝑥_𝑛−1)−𝑓(𝑥₀) are linearly independent, then 𝑓(𝑥)−𝑓𝑥₀𝑓𝑥∶𝑥∈𝒳⊂span₁−𝑓𝑥₀,…,𝑓𝑥_𝑛−1−𝑓𝑥₀ (18) which contradicts the fact that {𝑓(𝑥)−𝑓(𝑥₀)∶𝑥∈𝒳} contains 𝑛 linearly independent vectors. Hence, for any 𝑥₃,…,𝑥_𝑛−1∈𝒳, 𝑓(𝑥₁)−𝑓(𝑥₀),…,𝑓(𝑥_𝑛−1)−𝑓(𝑥₀) are linearly dependent. If there exist 𝑥₃,…,𝑥_𝑛−2 such that 𝑓(𝑥₁)−𝑓(𝑥₀),…,𝑓(𝑥_𝑛−2)−𝑓(𝑥₀) are linearly independent, then 𝑓(𝑥)−𝑓𝑥₀𝑓𝑥∶𝑥∈𝒳⊂span₁−𝑓𝑥₀,…,𝑓𝑥_𝑛−2−𝑓𝑥₀ (19) which contradicts the fact that {𝑓(𝑥)−𝑓(𝑥₀)∶𝑥∈𝒳} contains 𝑛 linearly independent vectors. And thus, 𝑓(𝑥₁)−𝑓(𝑥₀),𝑓(𝑥₂)−𝑓(𝑥₀) are linearly dependent. Therefore, 𝑓(𝑥₀),𝑓(𝑥₁), and 𝑓(𝑥₂) are 2-collinear.

Theorem 14. Let 𝒳 and 𝒴 be non-Archimedean fuzzy 𝑛-normed spaces over a linear ordered non-Archimedean field 𝕂 with 𝒞={2^𝑛∣𝑛∈ℤ} such that 𝒴 is strictly convex. If 𝑓∶𝒳→𝒴 is a fuzzy 𝑛-isometry, then 𝑓(𝑥)−𝑓(0) is additive.

Proof. Let 𝑔(𝑥)=𝑓(𝑥)−𝑓(0) for 𝑥∈𝒳. Then 𝑔 is a fuzzy 𝑛-isometry and 𝑔(0)=0. For each 𝑥,𝑦,𝑥₂,…,𝑥_𝑛∈𝒳. Since 𝑔 is a fuzzy 𝑛-isometry, we have 𝑁𝑔(𝑥)−𝑔𝑥+𝑦2,𝑔𝑥₂−𝑔𝑥+𝑦2,…,𝑔𝑥_𝑛−𝑔𝑥+𝑦2,𝑡=𝑁𝑥−𝑥+𝑦2,𝑥₂−𝑥+𝑦2,…,𝑥_𝑛−𝑥+𝑦2,𝑡=𝑁𝑥−𝑥+𝑦2,𝑥₂−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑔(𝑥)𝑦𝑥−𝑥,𝑡=𝑁−𝑔,𝑔₂−𝑔(𝑥),…,𝑔𝑥_𝑛(𝑥),𝑁𝑔𝑦−𝑔,𝑡−𝑔𝑥+𝑦2,𝑔𝑥₂−𝑔𝑥+𝑦2,…,𝑔𝑥_𝑛−𝑔𝑥+𝑦2,𝑡=𝑁𝑦−𝑥+𝑦2,𝑥₂−𝑥+𝑦2,…,𝑥_𝑛−𝑥+𝑦2,𝑡=𝑁𝑦−𝑥+𝑦2,𝑥₂−𝑦,…,𝑥_𝑛−𝑦,𝑡=𝑁𝑥−𝑦,𝑥₂−𝑦,…,𝑥_𝑛−𝑦,𝑡=𝑁𝑥−𝑦,𝑥₂−𝑥,…,𝑥_𝑛𝑔(𝑥)𝑦𝑥−𝑥,𝑡=𝑁−𝑔,𝑔₂−𝑔(𝑥),…,𝑔𝑥_𝑛−𝑔(𝑥),𝑡. (20) Since (𝑥+𝑦)/2, 𝑥, and 𝑦 are collinear, by Lemma 13, 𝑔((𝑥+𝑦)/2), 𝑔(𝑥), and 𝑔(𝑦) are also collinear. It follows from Lemma 12 that 𝑔𝑥+𝑦2=𝑔(𝑥)+𝑔𝑦2, (21) for all 𝑥,𝑦∈𝒳. Hence, 𝑔(𝑥)=𝑓(𝑥)−𝑓(0) is additive since 𝑔(0)=0.

In the following, we prove that the interior preserving mapping carries the barycenter of a triangle to the barycenter point of the corresponding triangle. And then, using this result, we get a Mazur-Ulam theorem on non-Archimedean fuzzy 𝑛-normed spaces over a linear ordered non-Archimedean field 𝕂 with 𝒞={3^𝑛∣𝑛∈ℤ}.

Lemma 15. Let 𝒳 be a strictly convex non-Archimedean fuzzy 𝑛-normed space over a linear ordered non-Archimedean field 𝕂 with 𝒞={3^𝑛∣𝑛∈ℤ}, and let 𝑥,𝑦,𝑧∈𝒳 and 𝑡>0. Then 𝑢∶=(𝑥+𝑦+𝑧)/3 is the unique element of 𝒳 satisfying 𝑁𝑥−𝑢,𝑦−𝑢,𝑥₃−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑦−𝑢,𝑧−𝑢,𝑥₃−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑥−𝑢,𝑧−𝑢,𝑥₃−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑥−𝑦,𝑥−𝑧,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡 (22) for all 𝑥₃,…,𝑥_𝑛∈𝒳 and 𝑢∈{𝑡₁𝑥+𝑡₂𝑦+𝑡₃𝑧∣𝑡₁+𝑡₂+𝑡₃=1,𝑡_𝑖∈𝕂,𝑡_𝑖>0,𝑖=1,2,3}.

Proof. Let 𝑢∶=(𝑥+𝑦+𝑧)/3,𝑥₃,…,𝑥_𝑛∈𝒳 and 𝑡>0. By Lemma 11, we have 𝑁𝑥−𝑢,𝑦−𝑢,𝑥₃−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑥−𝑢,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑥+𝑦+𝑧3,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁2𝑥−𝑦−𝑧,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,|3|𝑡=𝑁𝑦−𝑧,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑦−𝑥,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑦,𝑥−𝑧,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡. (23)
Similarly, we have 𝑁𝑦−𝑢,𝑧−𝑢,𝑥₃−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑥−𝑦,𝑥−𝑧,𝑥₃−𝑥,…,𝑥_𝑛,𝑁−𝑥,𝑡𝑥−𝑢,𝑧−𝑢,𝑥₃−𝑢,…,𝑥_𝑛−𝑢,𝑡=𝑁𝑥−𝑦,𝑥−𝑧,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡. (24)
Hence, the existence holds. For the uniqueness of 𝑢, we may assume that there is another 𝑣∈𝒳 satisfying (22). Since 𝑣∈{𝑡₁𝑥+𝑡₂𝑦+𝑡₃𝑧∣𝑡₁+𝑡₂+𝑡₃=1,𝑡_𝑖∈𝕂,𝑡_𝑖>0,𝑖=1,2,3}, 𝑣=𝑠₁𝑥+𝑠₂𝑦+𝑠₃𝑧, 𝑠₁+𝑠₂+𝑠₃=1 for some 𝑠_𝑖∈𝕂,𝑠_𝑖>0,𝑖=1,2,3. Then 𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑣,𝑦−𝑣,𝑥₃−𝑣,…,𝑥_𝑛−𝑣,𝑡=𝑁𝑥−𝑣,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁1−𝑠₁𝑥−𝑠₂𝑦−1−𝑠₁−𝑠₂𝑧,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁1−𝑠₁−𝑠₂𝑥−1−𝑠₁−𝑠₂𝑧,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑧,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,1−𝑠₁−𝑠₂=𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,1−𝑠₁−𝑠₂,𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑣,𝑧−𝑣,𝑥₃−𝑣,…,𝑥_𝑛−𝑣,𝑡=𝑁𝑥−𝑣,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁1−𝑠₁𝑥−𝑠₂𝑦−1−𝑠₁−𝑠₂𝑧,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑠−𝑥,𝑡=𝑁₂𝑥−𝑠₂𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑧,𝑦−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,𝑠₂,𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑦−𝑣,𝑧−𝑣,𝑥₃−𝑣,…,𝑥_𝑛−𝑣,𝑡=𝑁𝑦−𝑣,𝑧−𝑦,𝑥₃−𝑦,…,𝑥_𝑛−𝑦,𝑡=𝑁1−𝑠₂𝑦−𝑠₁𝑥−1−𝑠₁−𝑠₂𝑧,𝑧−𝑦,𝑥₃−𝑦,…,𝑥_𝑛−𝑦,𝑡=𝑁−𝑠₁𝑥+𝑠₁𝑧,𝑧−𝑦,𝑥₃−𝑦,…,𝑥_𝑛−𝑦,𝑡=𝑁𝑥−𝑧,𝑧−𝑦,𝑥₃−𝑦,…,𝑥_𝑛𝑡−𝑦,𝑠₁=𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,𝑠₁. (25)
Hence,𝑁𝑥−𝑦+𝑥−𝑦+𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,max𝑠₁,𝑡𝑠₂,𝑡1−𝑠₁−𝑠₂=𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛1−𝑥,|3|𝑡max𝑠₁,𝑡𝑠₂,𝑡1−𝑠₁−𝑠₂=𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑁−𝑥,𝑡=min𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,𝑠₁,𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,𝑠₂,𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,1−𝑠₁−𝑠₂,𝑁𝑥−𝑦+𝑥−𝑦+𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,max𝑡,𝑠₁,𝑡𝑠₂=𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛1−𝑥,|3|𝑡max𝑡,𝑠₁,𝑡𝑠₂=𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑁−𝑥,𝑡=min𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛,𝑁−𝑥,𝑡𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,𝑠₁,𝑁𝑥−𝑦,𝑧−𝑥,𝑥₃−𝑥,…,𝑥_𝑛𝑡−𝑥,𝑠₂. (26)By the strict convexity, we have |𝑠₁|=|𝑠₂|=|𝑠₃|=1. Then there exist integers 𝑘₁, 𝑘₂, and 𝑘₃ such that 𝑠₁=3^𝑘1, 𝑠₂=3^𝑘2, and 𝑠₃=3^𝑘3. Since 3^𝑘1+3^𝑘2+3^𝑘3=1, we know that 𝑘₁<0, 𝑘₂<0, 𝑘₃<0. Without loss of generality, we let 𝑠₁=3^−𝑛1, 𝑠₂=3^−𝑛2, and 𝑠₃=3^−𝑛3 with 𝑛₁≥𝑛₂≥𝑛₃. Assume that the above one of the inequalities holds. Then 1=𝑠₁+𝑠₂+𝑠₃=3^−𝑛1(1+3^{𝑛1−𝑛2}+3^{𝑛1−𝑛3}). So 3^𝑛1=1+3^{𝑛1−𝑛2}+3^{𝑛1−𝑛3}. This is a contradiction. Thus, 𝑠₁=𝑠₂=𝑠₃=1/3 which means 𝑢=𝑣. This completes the proof.

Theorem 16. Let 𝒳 and 𝒴 be non-Archimedean fuzzy 𝑛-normed spaces over a linear ordered non-Archimedean field 𝕂 with 𝒞={3^𝑛∣𝑛∈ℤ} such that 𝒴 is strictly convex. If 𝑓∶𝒳→𝒴 is an interior preserving fuzzy 𝑛-isometry, then 𝑓(𝑥)−𝑓(0) is additive.

Proof. Let 𝑔(𝑥)=𝑓(𝑥)−𝑓(0) for 𝑥∈𝒳. Then 𝑔 is a fuzzy 𝑛-isometry and 𝑔(0)=0. For 𝑎,𝑏,𝑐∈𝒳, let Δ𝑎𝑏𝑐 be a triangle determined by the points 𝑎, 𝑏, 𝑐, and 𝑥, an interior point of Δ𝑎𝑏𝑐. Since 𝑓 is an interior preserving map, there exist 𝑠_𝑖∈𝕂,𝑠_𝑖>0(𝑖=1,2,3) with 𝑠₁+𝑠₂+𝑠₃=1 such that 𝑓(𝑥)=𝑠₁𝑓(𝑎)+𝑠₂𝑓(𝑏)+𝑠₃𝑓(𝑐). Then 𝑔(𝑥)=𝑠₁𝑓(𝑎)+𝑠₂𝑓(𝑏)+𝑠₃𝑓(𝑐)−𝑓(0)=𝑠₁𝑓(𝑎)−𝑓(0)+𝑠₂𝑓(𝑏)−𝑓(0)+𝑠₃𝑓(𝑐)−𝑓(0)=𝑠₁𝑔(𝑎)+𝑠₂𝑔(𝑏)+𝑠₃𝑔(𝑐), (27) and hence, 𝑔(𝑥) is an interior point of Δ𝑔(𝑎)𝑔(𝑏)𝑔(𝑐). Therefore, 𝑔 is also an interior preserving mapping.
Now let 𝑥,𝑦,𝑧,𝑥₃,…,𝑥_𝑛∈𝒳. Since 𝑔 is a fuzzy 𝑛-isometry, we have 𝑁𝑔(𝑥)−𝑔𝑥+𝑦+𝑧3,𝑔𝑦−𝑔𝑥+𝑦+𝑧3,𝑔𝑥₃−𝑔𝑥+𝑦+𝑧3,…,𝑔𝑥_𝑛−𝑔𝑥+𝑦+𝑧3,𝑡=𝑁𝑥−𝑥+𝑦+𝑧3,𝑦−𝑥+𝑦+𝑧3,𝑥₃−𝑥+𝑦+𝑧3,…,𝑥_𝑛−𝑥+𝑦+𝑧3,𝑡=𝑁𝑥−𝑦,𝑥−𝑥+𝑦+𝑧3,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑦,2𝑥−𝑦−𝑧,𝑥₃−𝑥,…,𝑥_𝑛−𝑥,𝑡=𝑁𝑥−𝑦,𝑥−𝑧,𝑥₃−𝑥,…,𝑥_𝑛𝑔(𝑥)𝑦(𝑥)(𝑧),𝑔𝑥−𝑥,𝑡=𝑁−𝑔,𝑔−𝑔₃−𝑔(𝑥),…,𝑔𝑥_𝑛−𝑔(𝑥),𝑡, (28) and similarly, we can obtain𝑁𝑔𝑦−𝑔𝑥+𝑦+𝑧3,𝑔(𝑧)−𝑔𝑥+𝑦+𝑧3,𝑔𝑥₃−𝑔𝑥+𝑦+𝑧3,…,𝑔𝑥_𝑛−𝑔𝑥+𝑦+𝑧3𝑔(𝑥)𝑦(𝑥)(𝑧)𝑥,𝑡=𝑁−𝑔,𝑔−𝑔,𝑔₃−𝑔(𝑥),…,𝑔𝑥_𝑛−𝑔(𝑥),𝑡,𝑁𝑔(𝑥)−𝑔𝑥+𝑦+𝑧3,𝑔(𝑧)−𝑔𝑥+𝑦+𝑧3,𝑔𝑥₃−𝑔𝑥+𝑦+𝑧3,…,𝑔𝑥_𝑛−𝑔𝑥+𝑦+𝑧3𝑔(𝑥)𝑦(𝑥)(𝑧),𝑔𝑥,𝑡=𝑁−𝑔,𝑔−𝑔₃−𝑔(𝑥),…,𝑔𝑥_𝑛−𝑔(𝑥),𝑡. (29)(30)Since (𝑥+𝑦+𝑧)/3 is an interior point of the triangle Δ𝑥𝑦𝑧 and 𝑔 is an interior preserving mapping, 𝑔((𝑥+𝑦+𝑧)/3)∈{𝑡₁𝑔(𝑥)+𝑡₂𝑔(𝑦)+𝑡₃𝑔(𝑧)∣𝑡₁+𝑡₂+𝑡₃=1,𝑡_𝑖∈𝕂,𝑡_𝑖>0,𝑖=1,2,3}. By Lemma 15, 𝑔𝑥+𝑦+𝑧3=𝑔(𝑥)+𝑔𝑦+𝑔(𝑧)3. (31) Hence, 𝑔(𝑥)=𝑓(𝑥)−𝑓(0) is additive since 𝑔(0)=0. This completes the proof.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (NNSFC) (Grant no. 11171022).