On the Mazur-Ulam problem in non-Archimedean fuzzy 2-normed spaces

Abstract

We study the notion of non-Archimedean fuzzy 2-normed space over a non-Archimedean field and prove that the Mazur-Ulam theorem holds under some conditions in the non-Archimedean fuzzy 2-normed space.

MSC:46S10, 47S10, 26E30, 12J25.

1 Introduction

A mapping $f:X⟶Y$ is called an isometry if f satisfies

$$d_Y(f(x),f(y))=d_X(x,y)$$

for all $x,y\in X$, where $d_X(\cdot ,\cdot )$ and $d_Y(\cdot ,\cdot )$ denote the metrics in the spaces X and Y, respectively.

The theory of isometric mappings originated in the classical paper [1] by Mazur and Ulam in 1932.

Mazur-Ulam theorem Every isometry f of a normed real linear space X onto a normed real linear space is a linear mapping up to translation, that is, $x\mapsto f(x)-f(0)$ is linear, which amounts to the definition that f is affine.

The Mazur-Ulam theorem is not true for a normed complex vector space. In addition, the onto assumption is also essential. Without this assumption, Baker [2] proved that an isometry from a normed real linear space into a strictly convex normed real linear space is affine.

Gähler [3, 4] introduced a new approach for a theory of 2-norm and n-norm on a linear space. Chu [5] studied the Mazur-Ulam theorem in linear 2-normed spaces. Recently, Moslehian and Sadeghi [6] introduced the Mazur-Ulam theorem in the non-Archimedean strictly convex normed spaces. Moreover, Mirmostafaee and Moslehian [7] introduced a non-Archimedean fuzzy norm on a linear space over a non-Archimedean field. In particular, Amyari and Sadeghi [8] proved Mazur-Ulam theorem under the condition of strict convexity in non-Archimedean 2-normed spaces.

In 1984, Katsaras [9] and Wu and Fang [10] introduced the notion of fuzzy norm, and also Wu and Fang gave the generalization of the Kolmogoroff normalized theorem for a fuzzy topological linear space. In addition, fuzzy n-normed linear spaces were studied by Narayanan and Vijayabalaji; see [11].

In this paper, we investigate the notion of non-Archimedean fuzzy 2-normed space over a linear ordered non-Archimedean field and prove that Mazur-Ulam theorem holds under some conditions in the non-Archimedean fuzzy 2-normed space.

Definition 1.1 A non-Archimedean field is a field $\mathcal{K}$ equipped with a (valuation) function from $\mathcal{K}$ into $[0,\mathrm{\infty })$ satisfying the following properties:

• (1) $|a|\ge 0$ and equality holds if and only if $a=0$,

• (2) $|ab|=|a||b|$,

• (3) $|a+b|\le max\{|a|,|b|\}$

for all $a,b\in \mathcal{K}$.

Clearly, $|1|=|-1|=1$ and $|n|\le 1$ for all $n\in \mathbb{N}$. An example of a non-Archimedean valuation is the function $|\cdot |$ taking everything except 0 into 1 and $|0|=0$; see [12]. We call it a non-Archimedean trivial valuation. Also, the most important examples of non-Archimedean spaces are p-adic numbers; see [7].

Definition 1.2 Let X be a linear space over a field $\mathcal{K}$ with a non-Archimedean valuation $|\cdot |$. A function $\parallel \cdot \parallel :X\times X⟶[0,\mathrm{\infty })$ is said to be a non-Archimedean 2-norm if it satisfies the following properties:

• (1) $\parallel x,y\parallel =0$ if and only if x, y are linearly dependent,

• (2) $\parallel x,y\parallel =\parallel y,x\parallel$,

• (3) $\parallel cx,y\parallel =|c|\parallel x,y\parallel$,

• (4) $\parallel x,y+z\parallel \le max\{\parallel x,y\parallel ,\parallel x,z\parallel \}$

for all $x,y,z\in X$ and $c\in \mathcal{K}$. Then $(X,\parallel \cdot \parallel )$ is called a non-Archimedean 2-normed space.

Definition 1.3 Let X be a linear space over a field $\mathcal{K}$ with a non-Archimedean valuation $|\cdot |$. A function $N:X^2\times \mathbb{R}⟶[0,1]$ is said to be a non-Archimedean fuzzy 2-norm on X if for all $x,y\in X$ and all $s,t\in \mathbb{R}$,

(N1) $N(x,y,t)=0$ for $t\le 0$,

(N2) for $t>0$, $N(x,y,t)=1$ if and only if x and y are linearly dependent,

(N3) $N(x,y,t)=N(y,x,t)$,

(N4) $N(x,cy,t)=N(y,x,\frac{t}{|c|})$ for $c\ne 0$,

(N5) $N(x,y+z,max\{s,t\})\ge min\{N(x,y,s),N(x,z,t)\}$,

(N6) $N(x,y,\ast )$ is a nondecreasing function of ℝ and ${lim}_{t\to \mathrm{\infty }}N(x,y,t)=1$.

The pair $(X,N)$ is called a non-Archimedean fuzzy 2-normed space.

The property (N4) implies that $N(-x,y,t)=N(x,y,t)$ for all $x,y\in X$ and $t>0$. It is easy to show that (N5) is equivalent to the following condition:

$$N(x,y+z,t)\ge min\{N(x,y,t),N(x,z,t)\}\text{for all }x,y,z\in X\text{ and }t\in \mathbb{R}.$$

Example 1.4 Let $(X,\parallel \cdot ,\cdot \parallel )$ be a non-Archimedean 2-normed space. Define

$$N(x,y,t)=\{\begin{array}{cc}\frac{t}{t+\parallel x,y\parallel }\hfill & \text{when }t>0,t\in \mathbb{R},\hfill \\ 0\hfill & \text{when }t\le 0,\hfill \end{array}$$

where $x,y\in X$. Then $(X,N)$ is a non-Archimedean fuzzy 2-normed space.

Definition 1.5 A non-Archimedean fuzzy 2-normed space is said to be strictly convex if $N(x,y+z,max\{s,t\})=min\{N(x,y,s),N(x,z,t)\}$ and $N(x,y,s)=N(x,z,t)$ imply $y=z$ and $s=t$.

Definition 1.6 Let $(X,N)$ and $(Y,N)$ be two non-Archimedean fuzzy 2-normed spaces. We call $f:(X,N)⟶(Y,N)$ a fuzzy 2-isometry if $N(a-c,b-c,t)=N(f(a)-f(c),f(b)-f(c),t)$ for all $a,b,c\in X$ and $t>0$.

Definition 1.7 Let X be a non-Archimedean fuzzy 2-normed space, and let a, b, c be mutually disjoint elements of X. Then a, b and c are said to be collinear if $b-c=r(a-c)$ for some real number r.

We denote the set of all elements of $\mathcal{K}$ whose norms are 1 by $\mathcal{C}$, that is,

$$\mathcal{C}=\{r\in \mathcal{K}||r|=1\}.$$

2 Main results

Lemma 2.1 Let $(X,N)$ be a non-Archimedean fuzzy 2-normed space over a linear ordered non-Archimedean field $\mathcal{K}$. Then

$$N(x,y,t)=N(x,y+rx,t)\mathit{\text{for all }}r\in \mathcal{K}.$$

Proof Let $x,y\in X$ and let $r\in \mathcal{K}$. Without loss of generality, we may assume $t>0$. Then

$$N(x,y+rx,t)\ge min\{N(x,y,t),N(x,rx,t)\}=N(x,y,t).$$

Conversely,

$$\begin{array}{rcl}N(x,y,t)& =& N(x,y+rx-rx,t)\ge min\{N(x,y+rx,t),N(x,rx,t)\}\\ =& N(x,y+rx,t).\end{array}$$

Thus $N(x,y,t)=N(x,y+rx,t)$ for all $r\in \mathcal{K}$. □

Lemma 2.2 Let $(X,N)$ be a non-Archimedean fuzzy 2-normed space over a linear ordered non-Archimedean field $\mathcal{K}$ with $\mathcal{C}=\{2^n|n\in \mathbb{Z}\}$, and let $a,b,c\in X$ and $t>0$. Suppose that X is strictly convex. Then $\alpha =\frac{a+b}{2}$ is the unique element of X such that

$$N(a-c,a-\alpha ,t)=N(b-\alpha ,b-c,t)=N(a-c,b-c,t),$$

where a, b and α are collinear.

Proof Let $\alpha =\frac{a+b}{2}\in X$ and $t>0$. By Lemma 2.1, we have

$$\begin{array}{rcl}N(a-c,a-\alpha ,t)& =& N(a-c,a-\frac{a+b}{2},t)\\ =& N(a-c,\frac{a-b}{2},t)\\ =& N(a-c,a-b,|2|t)\\ =& N(a-c,a-b,t)\\ =& N(a-c,b-c,t).\end{array}$$

Similarly,

$$\begin{array}{rcl}N(b-\alpha ,b-c,t)& =& N(b-\frac{a+b}{2},b-c,t)=N(b-a,b-c,t)\\ =& N(a-c,b-c,t).\end{array}$$

Hence we have $N(a-c,a-\alpha ,t)=N(a-c,b-c,t)=N(b-\alpha ,b-c,t)$, that is, the existence part holds. To show the uniqueness part, assume that β is an element of X such that

$$N(a-c,a-\beta ,t)=N(b-\beta ,b-c,t)=N(a-c,b-c,t),$$

where a, b and β are collinear. Since a, b and β are collinear, there exists a real number s such that

$$\beta =sa+(1-s)b.$$

We may assume $s\ne 0$ and $s\ne 1$.

$$\begin{array}{rcl}N(a-c,b-c,t)& =& N(a-c,a-\beta ,t)=N(a-c,a-(sa+(1-s)b),t)\\ =& N(a-c,a-b,\frac{t}{|1-s|})\\ =& N(a-c,b-c,\frac{t}{|1-s|}).\end{array}$$

Similarly, we have

$$N(a-c,b-c,t)=N(a-c,b-c,\frac{t}{|s|}),$$

that is,

$$N(a-c,b-c,t)=N(a-c,b-c,\frac{t}{|1-s|})=N(a-c,b-c,\frac{t}{|s|}).$$

We note that

$$\begin{array}{r}N(a-c+a-c,b-c,max\{\frac{t}{|s|},\frac{t}{|1-s|}\})\\ \ge min\{N(a-c,b-c,\frac{t}{|s|}),N(a-c,b-c,\frac{t}{|1-s|})\}\\ =N(a-c,b-c,\frac{t}{|s|})=N(a-c,b-c,\frac{t}{|1-s|}),\end{array}$$

and

$$\begin{array}{r}N(a-c+a-c,b-c,max\{\frac{t}{|s|},\frac{t}{|1-s|}\})\\ =N(2(a-c),b-c,max\{\frac{t}{|s|},\frac{t}{|1-s|}\})\\ =N(a-c,b-c,max\{\frac{t}{|s|},\frac{t}{|1-s|}\}).\end{array}$$

The previous note implies that

$$N(a-c,b-c,t)=N(a-c,b-c,\frac{t}{|s|})=N(a-c,b-c,\frac{t}{|1-s|}).$$

The strict convexity of X implies that $|s|=|1-s|=1$. Then there exist elements $t_1$ and $t_2$ in ℤ such that $1-s=2^{t_1}$ and $s=2^{t_2}$. Since $2^{t_1}+2^{t_2}=1$, we know that $t_1,t_2<0$. Without loss of generality, we let $1-s=2^{-n_1}$ and $s=2^{-n_2}$ with $n_1\ge n_2$. If $n_1⪈n_2$, then

$$1=2^{-n_1}+2^{-n_2}=2^{-n_1}(1+2^{n_1-n_2}).$$

Hence $2^{n_1}=1+2^{n_1-n_2}$. This is a contradiction. Thus $n_1=n_2$, that is, $s=\frac{1}{2}$. This implies that $\beta =\frac{a+b}{2}=\alpha$. Therefore the proof is completed. □

Theorem 2.3 Let X and Y be non-Archimedean fuzzy 2-normed spaces over a linear ordered non-Archimedean field $\mathcal{K}$ with $\mathcal{C}=\{2^n|n\in \mathbb{Z}\}$. Let X and Y be strict convexities. Suppose that $f:X⟶Y$ is a fuzzy 2-isometry satisfying that $f(a)$, $f(b)$ and $f(c)$ are collinear when a, b and c are collinear. Then $f(x)-f(0)$ is additive.

Proof Let $g(x)=f(x)-f(0)$. Since f is a fuzzy 2-isometry, so is g. It is easy to show that if a, b and c are collinear, then $g(a)$, $g(b)$ and $g(c)$ are collinear. Since $g:X⟶Y$ is a fuzzy 2-isometry, we have

$$\begin{array}{rl}N(g(a)-g(c),g(a)-g\left(\frac{a+b}{2}\right),t)& =N(a-c,a-\frac{a+b}{2},t)\\ =N(a-c,a-b,t)=N(a-c,b-c,t)\\ =N(g(a)-g(c),g(b)-g(c),t).\end{array}$$

Similarly, we get $N(g(b)-g(\frac{a+b}{2}),g(b)-g(c),t)=N(g(a)-g(c),g(b)-g(c),t)$. Hence

$$\begin{array}{rcl}N(g(a)-g(c),g(a)-g\left(\frac{a+b}{2}\right),t)& =& N(g(b)-g\left(\frac{a+b}{2}\right),g(b)-g(c),t)\\ =& N(g(a)-g(c),g(b)-g(c),t).\end{array}$$

By the uniqueness of Lemma 2.2, we have $g(\frac{a+b}{2})=\frac{g(a)+g(b)}{2}$ for all $a,b\in X$. Thus $f(x)-f(0)$ is additive, as desired. □

Example 2.4 Let $\mathcal{K}={\mathbb{Z}}_3$, where ${\mathbb{Z}}_3=\{0,1,2\}$. Suppose that the field $\mathcal{K}$ has a non-Archimedean trivial valuation $|\cdot |$. Then $|2|=1$, that is, $\mathcal{C}=\{2^n|n\in \mathbb{Z}\}$.