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# Mazur-Ulam theorem under weaker conditions in the framework of 2-fuzzy 2-normed linear spaces

Journal of Inequalities and Applications20132013:78

https://doi.org/10.1186/1029-242X-2013-78

• Received: 10 September 2012
• Accepted: 6 January 2013
• Published:

## Abstract

The purpose of this paper is to prove that every 2-isometry without any other conditions from a fuzzy 2-normed linear space to another fuzzy 2-normed linear space is affine, and to give a new result of the Mazur-Ulam theorem for 2-isometry in the framework of 2-fuzzy 2-normed linear spaces.

MSC:03E72, 46B20, 51M25, 46B04, 46S40.

## Keywords

• α-2-norm
• 2-fuzzy 2-normed linear spaces
• 2-isometry
• Mazur-Ulam theorem

## 1 Introduction

A satisfactory theory of 2-norm and n-norm on a linear space has been introduced and developed by Gähler in [1, 2]. Freese and Cho  gave some isometry conditions in linear 2-normed spaces. Raja and Vaezpour  introduced the notion of 2-normed hyperset in a hypervector and also constructed some special 2-normed hypersets of strong homomorphisms over hypervector spaces. Different authors introduced the definitions of fuzzy norms on a linear space. For reference, one may see . Following Cheng and Mordeson , Bag and Samanta  introduced the concept of fuzzy norm on a linear space.

Somasundaram and Beaula  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. They gave the notion of α-2-norm on a linear space corresponding to a 2-fuzzy 2-norm with the help of  and also gave some fundamental properties of this space.

Let X and Y be metric spaces. A mapping $f:X\to Y$ is called an isometry if f satisfies ${d}_{Y}\left(f\left(x\right),f\left(y\right)\right)={d}_{X}\left(x,y\right)$ for all $x,y\in X$, where ${d}_{X}\left(\cdot ,\cdot \right)$ and ${d}_{Y}\left(\cdot ,\cdot \right)$ denote the metrics in the spaces X and Y, respectively. Two metric spaces X and Y are defined to be isometric if there exists an isometry of X onto Y. In 1932, Mazur and Ulam  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  showed that an isometry from a real normed linear space into a strictly convex real normed linear space is affine. Also, Jian  investigated the generalizations of the Mazur-Ulam theorem in ${F}^{\ast }$-spaces. Th.M. Rassias and Wagner  described all volume preserving mappings from a real finite dimensional vector space into itself and Väisälä  gave a short and simple proof of the Mazur-Ulam theorem. Chu  proved that the Mazur-Ulam theorem holds when X is a linear 2-normed space. Chu et al.  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 obtained extensions of Th.M. Rassias and Šemrl’s theorem . The Mazur-Ulam theorem has been extensively studied by many authors in different aspects (see [12, 1720]).

Recently, Cho et al.  investigated the Mazur-Ulam theorem on probabilistic 2-normed spaces. Moslehian and Sadeghi  investigated the Mazur-Ulam theorem in non-Archimedean spaces. Choy and Ku  proved that the barycenter of a triangle carries the barycenter of a corresponding triangle. They showed the Mazur-Ulam problem on non-Archimedean 2-normed spaces using the above statement. Chen and Song  introduced the concept of weak n-isometry, and then they got that under some conditions a weak n-isometry is also an n-isometry. Alaca  gave 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 $\mathrm{\Im }\left(X\right)$ is a fuzzy 2-normed linear space. Park and Alaca  introduced 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. They defined the concepts of n-isometry, n-collinearity, n-Lipschitz mapping in this space. Also, they generalized the Mazur-Ulam theorem, that is, when X is a 2-fuzzy n-normed linear space or $\mathrm{\Im }\left(X\right)$ 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. Ren  showed that every generalized area n preserving mapping between real 2-normed linear spaces X and Y which is strictly convex is affine under some conditions.

In the present paper, we give a new version of Mazur-Ulam theorem with a new method when X is a 2-fuzzy 2-normed linear space or $\mathrm{\Im }\left(X\right)$ is a fuzzy 2-normed linear space.

## 2 On 2-fuzzy 2-normed linear spaces

In this section, at first we give the concept of linear 2-normed space and later the concept of 2-fuzzy 2-normed linear space and its fundamental properties with help of . For more details, we refer the readers to [7, 8, 28, 29].

Definition 2.1 

Let X be a real vector space of dimension greater than 1 and let $\parallel •,•\parallel$ be a real-valued function on $X×X$ satisfying the following four properties:
1. (1)

$\parallel x,y\parallel =0$ if and only if x and y are linearly dependent,

2. (2)

$\parallel x,y\parallel =\parallel y,x\parallel$,

3. (3)

$\parallel x,\alpha y\parallel =|\alpha |\parallel x,y\parallel$ for any $\alpha \in \mathbb{R}$,

4. (4)

$\parallel x,y+z\parallel \le \parallel x,y\parallel +\parallel x,z\parallel$,

$\parallel •,•\parallel$ is called a 2-norm on X and the pair $\left(X,\parallel •,•\parallel \right)$ is called a linear 2-normed space.

Definition 2.2 

Let X be a linear space over S (a field of real or complex numbers). A fuzzy subset N of $X×\mathbb{R}$ (, the set of real numbers) is called a fuzzy norm on X if and only if:

(N1) For all $t\in \mathbb{R}$ with $t\le 0$, $N\left(x,t\right)=0$,

(N2) For all $t\in \mathbb{R}$ with $t>0$, $N\left(x,t\right)=1$ if and only if $x=0$,

(N3) For all $t\in \mathbb{R}$ with $t>0$, $N\left(\lambda x,t\right)=N\left(x,\frac{t}{|\lambda |}\right)$, if $\lambda \ne 0$, $\lambda \in S$,

(N4) For all $s,t\in \mathbb{R}$, $x,y\in X$, $N\left(x+y,s+t\right)\ge min\left\{N\left(x,s\right),N\left(y,t\right)\right\}$,

(N5) $N\left(x,\cdot \right)$ is a non-decreasing function of $t\in \mathbb{R}$ and ${lim}_{t\to \mathrm{\infty }}N\left(x,t\right)=1$.

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

Theorem 2.1 

Let $\left(X,N\right)$ be an f-NLS. Assume the condition that

(N6) $N\left(x,t\right)>0$ for all $t>0$ implies $x=0$.

Define ${\parallel x\parallel }_{\alpha }=inf\left\{t:N\left(x,t\right)\ge \alpha \right\}$, $\alpha \in \left(0,1\right)$. Then $\left\{{\parallel •\parallel }_{\alpha }:\alpha \in \left(0,1\right)\right\}$ is an ascending family of norms on X. We call these norms α-norms on X corresponding to the fuzzy norm on X.

Definition 2.3 Let X be any non-empty set and $\mathrm{\Im }\left(X\right)$ be the set of all fuzzy sets on X. For $U,V\in \mathrm{\Im }\left(X\right)$ and $\lambda \in S$ the field of real numbers, define
$U+V=\left\{\left(x+y,\nu \wedge \mu \right):\left(x,\nu \right)\in U,\left(y,\mu \right)\in V\right\}$

and $\lambda U=\left\{\left(\lambda x,\nu \right):\left(x,\nu \right)\in U\right\}$.

Definition 2.4 A fuzzy linear space $\stackrel{ˆ}{X}=X×\left(0,1\right]$ over the number field S, where the addition and scalar multiplication operation on X are defined by $\left(x,\nu \right)+\left(y,\mu \right)=\left(x+y,\nu \wedge \mu \right)$, $\lambda \left(x,\nu \right)=\left(\lambda x,\nu \right)$ is a fuzzy normed space if to every $\left(x,\nu \right)\in \stackrel{ˆ}{X}$, there is associated a non-negative real number, $\parallel \left(x,\nu \right)\parallel$, called the fuzzy norm of $\left(x,\nu \right)$, in such a way that
1. (i)

$\parallel \left(x,\nu \right)\parallel =0$ iff $x=0$ the zero element of X, $\nu \in \left(0,1\right]$,

2. (ii)

$\parallel \lambda \left(x,\nu \right)\parallel =|\lambda |\parallel \left(x,\nu \right)\parallel$ for all $\left(x,\nu \right)\in \stackrel{ˆ}{X}$ and all $\lambda \in S$,

3. (iii)

$\parallel \left(x,\nu \right)+\left(y,\mu \right)\parallel \le \parallel \left(x,\nu \wedge \mu \right)\parallel +\parallel \left(y,\nu \wedge \mu \right)\parallel$ for all $\left(x,\nu \right),\left(y,\mu \right)\in \stackrel{ˆ}{X}$,

4. (iv)

$\parallel \left(x,{\bigvee }_{t}{\nu }_{t}\right)\parallel ={\bigwedge }_{t}\parallel \left(x,{\nu }_{t}\right)\parallel$ for all ${\nu }_{t}\in \left(0,1\right]$.

Definition 2.5 

Let X be a non-empty and $\mathrm{\Im }\left(X\right)$ be the set of all fuzzy sets in X. If $f\in \mathrm{\Im }\left(X\right)$, then . Clearly, f is a bounded function for $|f\left(x\right)|\le 1$. Let S be the space of real numbers, then $\mathrm{\Im }\left(X\right)$ is a linear space over the field S where the addition and multiplication are defined by
and
$\lambda f=\left\{\left(\lambda x,\mu \right):\left(x,\mu \right)\in f\right\},$

where $\lambda \in S$.

The linear space $\mathrm{\Im }\left(X\right)$ is said to be a normed space if for every $f\in \mathrm{\Im }\left(X\right)$, there is associated a non-negative real number $\parallel f\parallel$ called the norm of f in such a way that
1. (i)
$\parallel f\parallel =0$ if and only if $f=0$. For

2. (ii)
$\parallel \lambda f\parallel =|\lambda |\parallel f\parallel$, $\lambda \in S$. For
$\begin{array}{rcl}\parallel \lambda f\parallel & =& \left\{\parallel \lambda \left(x,\mu \right)\parallel :\left(x,\mu \right)\in f,\lambda \in S\right\}\\ =& \left\{|\lambda |\parallel \left(x,\mu \right)\parallel :\left(x,\mu \right)\in f\right\}=|\lambda |\parallel f\parallel .\end{array}$

3. (iii)
$\parallel f+g\parallel \le \parallel f\parallel +\parallel g\parallel$ for every $f,g\in \mathrm{\Im }\left(X\right)$. For
$\begin{array}{rcl}\parallel f+g\parallel & =& \left\{\parallel \left(x,\mu \right)+\left(y,\eta \right)\parallel :x,y\in X,\mu ,\eta \in \left(0,1\right]\right\}\\ =& \left\{\parallel \left(x+y\right),\left(\mu \wedge \eta \right)\parallel :x,y\in X,\mu ,\eta \in \left(0,1\right]\right\}\\ =& \left\{\parallel \left(x,\mu \wedge \eta \right)\parallel +\parallel \left(y,\mu \wedge \eta \right)\parallel :\left(x,\mu \right)\in f,\left(y,\eta \right)\in g\right\}\\ =& \parallel f\parallel +\parallel g\parallel .\end{array}$

Then $\left(\mathrm{\Im }\left(X\right),\parallel •\parallel \right)$ is a normed linear space.

Definition 2.6 

A 2-fuzzy set on X is a fuzzy set on $\mathrm{\Im }\left(X\right)$.

Definition 2.7 

Let $\mathrm{\Im }\left(X\right)$ be a linear space over the real field S. A fuzzy subset N of $\mathrm{\Im }\left(X\right)×\mathrm{\Im }\left(X\right)×\mathbb{R}$ (, a set of real numbers) is called a 2-fuzzy 2-norm on X (or a fuzzy 2-norm on $\mathrm{\Im }\left(X\right)$) if and only if

(2-N1) for all $t\in \mathbb{R}$ with $t\le 0$, $N\left({f}_{1},{f}_{2},t\right)=0$,

(2-N2) for all $t\in \mathbb{R}$ with $t>0$, $N\left({f}_{1},{f}_{2},t\right)=1$ if and only if ${f}_{1}$ and ${f}_{2}$ are linearly dependent,

(2-N3) $N\left({f}_{1},{f}_{2},t\right)$ is invariant under any permutation of ${f}_{1}$, ${f}_{2}$,

(2-N4) for all $t\in \mathbb{R}$ with $t>0$, $N\left({f}_{1},\lambda {f}_{2},t\right)=N\left({f}_{1},{f}_{2},\frac{t}{|\lambda |}\right)$, if $\lambda \ne 0$, $\lambda \in S$,

(2-N5) for all $s,t\in \mathbb{R}$,
$N\left({f}_{1},{f}_{2}+{f}_{3},s+t\right)\ge min\left\{N\left({f}_{1},{f}_{2},s\right),N\left({f}_{1},{f}_{3},t\right)\right\},$

(2-N6) $N\left({f}_{1},{f}_{2},\cdot \right):\left(0,\mathrm{\infty }\right)\to \left[0,1\right]$ is continuous,

(2-N7) ${lim}_{t\to \mathrm{\infty }}N\left({f}_{1},{f}_{2},t\right)=1$.

Then $\left(\mathrm{\Im }\left(X\right),N\right)$ is a fuzzy 2-normed linear space or $\left(X,N\right)$ is a 2-fuzzy 2-normed linear space.

Remark 2.1 In a 2-fuzzy 2-normed linear space $\left(X,N\right)$, $N\left({f}_{1},{f}_{2},\cdot \right)$ is a non-decreasing function of for all ${f}_{1},{f}_{2}\in \mathrm{\Im }\left(X\right)$.

Theorem 2.2 

Let $\left(\mathrm{\Im }\left(X\right),N\right)$ be a fuzzy 2-normed linear space. Assume that

(2-N8) $N\left({f}_{1},{f}_{2},t\right)>0$ for all $t>0$ implies ${f}_{1}$ and ${f}_{2}$ are linearly dependent.

Define ${\parallel {f}_{1},{f}_{2}\parallel }_{\alpha }=inf\left\{t:N\left({f}_{1},{f}_{2}t\right)\ge \alpha ,\alpha \in \left(0,1\right)\right\}$.

Then $\left\{{\parallel •,•\parallel }_{\alpha }:\alpha \in \left(0,1\right)\right\}$ is an ascending family of 2-norms on $\mathrm{\Im }\left(X\right)$. These 2-norms are called α-2-norms on $\mathrm{\Im }\left(X\right)$ corresponding to the 2-fuzzy 2-norm on X.

## 3 On the Mazur-Ulam theorem

Recently, Alaca  introduced the concept of 2-isometry which is suitable to represent the notion of area-preserving mappings in fuzzy 2-normed linear spaces as follows.

For $f,g,h\in \mathrm{\Im }\left(X\right)$ and $\alpha ,\beta \in \left(0,1\right)$, ${\parallel f-h,g-h\parallel }_{\alpha }$ is called an area of f, g and h. We call Ψ a 2-isometry if ${\parallel f-h,g-h\parallel }_{\alpha }={\parallel \mathrm{\Psi }\left(f\right)-\mathrm{\Psi }\left(h\right),\mathrm{\Psi }\left(g\right)-\mathrm{\Psi }\left(h\right)\parallel }_{\beta }$ for all $f,g,h\in \mathrm{\Im }\left(X\right)$ and $\alpha ,\beta \in \left(0,1\right)$.

A version of the Mazur-Ulam theorem has been obtained in  as follows.

Theorem 3.1 

Assume that $\mathrm{\Im }\left(X\right)$ and $\mathrm{\Im }\left(Y\right)$ are fuzzy 2-normed linear spaces. If $\mathrm{\Psi }:\mathrm{\Im }\left(X\right)\to \mathrm{\Im }\left(Y\right)$ is a 2-isometry and satisfies $\mathrm{\Psi }\left(f\right)$, $\mathrm{\Psi }\left(g\right)$ and $\mathrm{\Psi }\left(h\right)$ are collinear when f, g and h are collinear, then Ψ is affine.

A natural question is whether the 2-isometry in the fuzzy 2-normed linear spaces is also affine without the condition of preserving collinearity. In this section, we find a reply to this question when X is a 2-fuzzy 2-normed linear space or $\mathrm{\Im }\left(X\right)$ is a fuzzy 2-normed linear space.

Lemma 3.1 

For all $f,g\in \mathrm{\Im }\left(X\right)$, $\alpha \in \left(0,1\right)$ and $\lambda \in \mathbb{R}$. Then
${\parallel f,g\parallel }_{\alpha }={\parallel f,g+\lambda f\parallel }_{\alpha }.$
Lemma 3.2 Let $f,g,h\in \mathrm{\Im }\left(X\right)$ and $\alpha \in \left(0,1\right)$. Then $v=\frac{f+g}{2}$ is the unique element of $\mathrm{\Im }\left(X\right)$ satisfying
${\parallel f-h,f-v\parallel }_{\alpha }={\parallel g-v,g-h\parallel }_{\alpha }=\frac{1}{2}{\parallel f-h,g-h\parallel }_{\alpha }$

with ${\parallel f-h,g-h\parallel }_{\alpha }\ne 0$ and $v\in \left\{kf+\left(1-k\right)g:k\in \mathbb{R}\right\}$.

Proof From Lemma 3.1, it is obvious that $v=\frac{f+g}{2}$ satisfies
${\parallel f-h,f-v\parallel }_{\alpha }={\parallel g-v,g-h\parallel }_{\alpha }=\frac{1}{2}{\parallel f-h,g-h\parallel }_{\alpha }$

with ${\parallel f-h,g-h\parallel }_{\alpha }\ne 0$ and $v\in \left\{kf+\left(1-k\right)g:k\in \mathbb{R}\right\}$.

For the uniqueness of v, assume that $u\in \mathrm{\Im }\left(X\right)$ also satisfies
${\parallel f-h,f-u\parallel }_{\alpha }={\parallel g-u,g-h\parallel }_{\alpha }=\frac{1}{2}{\parallel f-h,g-h\parallel }_{\alpha }$
with ${\parallel f-h,g-h\parallel }_{\alpha }\ne 0$ and $u\in \left\{kf+\left(1-k\right)g:k\in \mathbb{R}\right\}$. Let $u=kf+\left(1-k\right)g$ for some $k\in \mathbb{R}$. From Lemma 3.1, we have
$\begin{array}{rcl}{\parallel f-h,g-h\parallel }_{\alpha }& =& 2{\parallel f-h,f-u\parallel }_{\alpha }\\ =& 2{\parallel f-h,f-\left(kf+\left(1-k\right)g\right)\parallel }_{\alpha }\\ =& 2|1-k|{\parallel f-h,f-g\parallel }_{\alpha }\\ =& 2|1-k|{\parallel f-h,g-h\parallel }_{\alpha }\end{array}$
and
$\begin{array}{rcl}{\parallel f-h,g-h\parallel }_{\alpha }& =& 2{\parallel g-h,g-u\parallel }_{\alpha }\\ =& 2{\parallel g-h,g-\left(kf+\left(1-k\right)g\right)\parallel }_{\alpha }\\ =& 2|k|{\parallel g-h,g-f\parallel }_{\alpha }\\ =& 2|k|{\parallel f-h,g-h\parallel }_{\alpha }.\end{array}$

Since ${\parallel f-h,g-h\parallel }_{\alpha }\ne 0$, we have $1=2|1-k|=2|k|$. So, $k=\frac{1}{2}$ and $u=v=\frac{f+g}{2}$. □

Theorem 3.2 Let $\mathrm{\Im }\left(X\right)$ and $\mathrm{\Im }\left(Y\right)$ be fuzzy 2-normed linear spaces. If $\mathrm{\Psi }:\mathrm{\Im }\left(X\right)\to \mathrm{\Im }\left(Y\right)$ is a 2-isometry, then Ψ is affine.

Proof Let $\mathrm{\Phi }\left(f\right)=\mathrm{\Psi }\left(f\right)-\mathrm{\Psi }\left(0\right)$. Obviously, $\mathrm{\Phi }\left(0\right)=0$ and Φ is a 2-isometry. Now, we prove that Φ is linear.

Firstly, we show that Φ is additive. For $f,g,h\in \mathrm{\Im }\left(X\right)$, $\alpha ,\beta \in \left(0,1\right)$ with ${\parallel f-h,g-h\parallel }_{\alpha }\ne 0$, ${\parallel \mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(h\right),\mathrm{\Phi }\left(g\right)-\mathrm{\Phi }\left(h\right)\parallel }_{\beta }\ne 0$ and from Lemma 3.1, we have
$\begin{array}{rcl}{\parallel \mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(h\right),\mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(\frac{f+g}{2}\right)\parallel }_{\beta }& =& {\parallel f-h,f-\frac{f+g}{2}\parallel }_{\alpha }\\ =& {\parallel f-h,\frac{f-g}{2}\parallel }_{\alpha }\\ =& \frac{1}{2}{\parallel f-h,f-g\parallel }_{\alpha }\\ =& \frac{1}{2}{\parallel f-h,g-h\parallel }_{\alpha }\\ =& \frac{1}{2}{\parallel \mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(h\right),\mathrm{\Phi }\left(g\right)-\mathrm{\Phi }\left(h\right)\parallel }_{\beta }.\end{array}$
Similarly,
${\parallel \mathrm{\Phi }\left(g\right)-\mathrm{\Phi }\left(h\right),\mathrm{\Phi }\left(g\right)-\mathrm{\Phi }\left(\frac{f+g}{2}\right)\parallel }_{\beta }=\frac{1}{2}{\parallel \mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(h\right),\mathrm{\Phi }\left(g\right)-\mathrm{\Phi }\left(h\right)\parallel }_{\beta }.$
And
$\begin{array}{rcl}{\parallel \mathrm{\Phi }\left(\frac{f+g}{2}\right)-\mathrm{\Phi }\left(g\right),\mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(g\right)\parallel }_{\beta }& =& {\parallel \frac{f+g}{2}-g,f-g\parallel }_{\alpha }\\ =& \frac{1}{2}{\parallel f-g,f-g\parallel }_{\alpha }=0.\end{array}$
So, we get
$\mathrm{\Phi }\left(\frac{f+g}{2}\right)-\mathrm{\Phi }\left(g\right)=k\left(\mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(g\right)\right)$
for some $k\in \mathbb{R}$ by Definition 2.7. That is,
$\mathrm{\Phi }\left(\frac{f+g}{2}\right)=k\mathrm{\Phi }\left(f\right)+\left(1-k\right)\mathrm{\Phi }\left(g\right).$
Thus, from Lemma 3.2,
$\mathrm{\Phi }\left(\frac{f+g}{2}\right)=\frac{\mathrm{\Phi }\left(f\right)+\mathrm{\Phi }\left(g\right)}{2}$

for all $f,g\in \mathrm{\Im }\left(X\right)$.

Since $\mathrm{\Phi }\left(0\right)=0$, we have
$\mathrm{\Phi }\left(\frac{f}{2}\right)=\mathrm{\Phi }\left(\frac{f+0}{2}\right)=\frac{\mathrm{\Phi }\left(f\right)+\mathrm{\Phi }\left(0\right)}{2}=\frac{\mathrm{\Phi }\left(f\right)}{2}$
and
$\begin{array}{rcl}\mathrm{\Phi }\left(f+g\right)& =& \mathrm{\Phi }\left(\frac{2f+2g}{2}\right)=\frac{\mathrm{\Phi }\left(2f\right)+\mathrm{\Phi }\left(2g\right)}{2}=\frac{\mathrm{\Phi }\left(2f\right)}{2}+\frac{\mathrm{\Phi }\left(2g\right)}{2}\\ =& \mathrm{\Phi }\left(f\right)+\mathrm{\Phi }\left(g\right).\end{array}$

It follows that Φ is additive.

Secondly, we show that $\mathrm{\Phi }\left(rf\right)=r\mathrm{\Phi }\left(f\right)$ for every $r\in \mathbb{R}$, $f\in \mathrm{\Im }\left(X\right)$ and $\alpha ,\beta \in \left(0,1\right)$. Let $r\in {\mathbb{R}}^{+}$ and $f\in \mathrm{\Im }\left(X\right)$ and $\alpha ,\beta \in \left(0,1\right)$. Since $\mathrm{\Phi }\left(0\right)=0$ and Φ is a 2-isometry, we have
$\begin{array}{rcl}{\parallel \mathrm{\Phi }\left(rf\right),\mathrm{\Phi }\left(f\right)\parallel }_{\beta }& =& {\parallel \mathrm{\Phi }\left(rf\right)-\mathrm{\Phi }\left(0\right),\mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(0\right)\parallel }_{\beta }\\ =& {\parallel rf-0,f-0\parallel }_{\alpha }\\ =& {\parallel rf,f\parallel }_{\alpha }\\ =& 0.\end{array}$
So, $\mathrm{\Phi }\left(rf\right)=s\mathrm{\Phi }\left(f\right)$ for some $s\in \mathbb{R}$ from Definition 2.7. As $dim\mathrm{\Im }\left(X\right)>1$, there exists a $g\in \mathrm{\Im }\left(X\right)$ such that ${\parallel f,g\parallel }_{\alpha }\ne 0$. It is easy to see that
$\begin{array}{rcl}r{\parallel f,g\parallel }_{\alpha }& =& {\parallel rf,g\parallel }_{\alpha }={\parallel \mathrm{\Phi }\left(rf\right),\mathrm{\Phi }\left(g\right)\parallel }_{\beta }={\parallel s\mathrm{\Phi }\left(f\right),\mathrm{\Phi }\left(g\right)\parallel }_{\beta }\\ =& |s|{\parallel \mathrm{\Phi }\left(f\right),\mathrm{\Phi }\left(g\right)\parallel }_{\beta }=|s|{\parallel f,g\parallel }_{\alpha }.\end{array}$
So, $s=r$ or $s=-r$. If $s=-r$, then
$\begin{array}{rcl}|r-1|{\parallel f,g\parallel }_{\alpha }& =& {\parallel \left(r-1\right)f,g\parallel }_{\alpha }={\parallel rf-f,g-0\parallel }_{\alpha }\\ =& {\parallel \mathrm{\Phi }\left(rf\right)-\mathrm{\Phi }\left(f\right),\mathrm{\Phi }\left(g\right)-\mathrm{\Phi }\left(0\right)\parallel }_{\beta }\\ =& {\parallel -r\mathrm{\Phi }\left(f\right)-\mathrm{\Phi }\left(f\right),\mathrm{\Phi }\left(g\right)\parallel }_{\beta }\\ =& \left(r+1\right){\parallel \mathrm{\Phi }\left(f\right),\mathrm{\Phi }\left(g\right)\parallel }_{\beta }\\ =& \left(r+1\right){\parallel f,g\parallel }_{\alpha }.\end{array}$

So, $|r-1|=r+1$. This is a contradiction since $r\in {\mathbb{R}}^{+}$. Thus, $\mathrm{\Phi }\left(rf\right)=r\mathrm{\Phi }\left(f\right)$ for every $r\in {\mathbb{R}}^{+}$, $f\in \mathrm{\Im }\left(X\right)$ and $\alpha ,\beta \in \left(0,1\right)$.

Similarly, we can prove $\mathrm{\Phi }\left(rf\right)=r\mathrm{\Phi }\left(f\right)$ for every $r\in {\mathbb{R}}^{-}$, $f\in \mathrm{\Im }\left(X\right)$ and $\alpha ,\beta \in \left(0,1\right)$.

Hence, we prove that Φ is linear and Ψ is affine. □

Remark 3.1 Theorem 3.1 has been substantially improved by Theorem 3.2.

Remark 3.2 It is clear that the Mazur-Ulam theorem has been proved under much weaker conditions than the main result of Alaca  in the framework of 2-fuzzy 2-normed linear spaces.

Open problem How can obtain some results for the Aleksandrov problem in fuzzy 2-normed linear spaces with the help of this technique?

## Declarations

### Acknowledgements

The authors would like to thank the referees and the area editor Professor Yeol Je Cho for their valuable suggestions and comments.

## Authors’ Affiliations

(1)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, South Korea
(2)
Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, Manisa, 45140, Turkey

## References 