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

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 [3] gave some isometry conditions in linear 2-normed spaces. Raja and Vaezpour [4] 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 [5]. Following Cheng and Mordeson [6], Bag and Samanta [7] introduced the concept of fuzzy norm on a linear space.

Somasundaram and Beaula [8] 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 [7] 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 $d_Y(f(x),f(y))=d_X(x,y)$ 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. 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 [9] 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 [10] showed that an isometry from a real normed linear space into a strictly convex real normed linear space is affine. Also, Jian [11] investigated the generalizations of the Mazur-Ulam theorem in $F^{\ast }$-spaces. Th.M. Rassias and Wagner [12] described all volume preserving mappings from a real finite dimensional vector space into itself and Väisälä [13] gave a short and simple proof of the Mazur-Ulam theorem. Chu [14] proved that the Mazur-Ulam theorem holds when X is a linear 2-normed space. Chu et al. [15] 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 [16]. The Mazur-Ulam theorem has been extensively studied by many authors in different aspects (see [12, 17–20]).

Recently, Cho et al. [21] investigated the Mazur-Ulam theorem on probabilistic 2-normed spaces. Moslehian and Sadeghi [22] investigated the Mazur-Ulam theorem in non-Archimedean spaces. Choy and Ku [23] 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 [24] 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 [25] 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 $\mathrm{\Im }(X)$ is a fuzzy 2-normed linear space. Park and Alaca [26] 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 $\mathrm{\Im }(X)$ 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. Ren [27] 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 $\mathrm{\Im }(X)$ is a fuzzy 2-normed linear space.

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 [8]. For more details, we refer the readers to [7, 8, 28, 29].

Definition 2.1 [28]

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

Definition 2.2 [7]

Let X be a linear space over S (a field of real or complex numbers). A fuzzy subset N of $X\times \mathbb{R}$ (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 $t\le 0$, $N(x,t)=0$,

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

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

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

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

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

Theorem 2.1 [7]

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

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

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

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

Definition 2.5 [8]

where $\lambda \in S$.

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

Definition 2.6 [8]

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

Definition 2.7 [8]

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

(2-N1) for all $t\in \mathbb{R}$ with $t\le 0$, $N(f_1,f_2,t)=0$,

(2-N2) for all $t\in \mathbb{R}$ with $t>0$, $N(f_1,f_2,t)=1$ if and only if $f_1$ and $f_2$ are linearly dependent,

(2-N3) $N(f_1,f_2,t)$ is invariant under any permutation of $f_1$, $f_2$,

(2-N4) for all $t\in \mathbb{R}$ with $t>0$, $N(f_1,\lambda f_2,t)=N(f_1,f_2,\frac{t}{|\lambda |})$, if $\lambda \ne 0$, $\lambda \in S$,

(2-N6) $N(f_1,f_2,\cdot ):(0,\mathrm{\infty })\to [0,1]$ is continuous,

(2-N7) ${lim}_{t\to \mathrm{\infty }}N(f_1,f_2,t)=1$.

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

Remark 2.1 In a 2-fuzzy 2-normed linear space $(X,N)$, $N(f_1,f_2,\cdot )$ is a non-decreasing function of ℝ for all $f_1,f_2\in \mathrm{\Im }(X)$.

Theorem 2.2 [8]

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

(2-N8) $N(f_1,f_2,t)>0$ for all $t>0$ implies $f_1$ and $f_2$ are linearly dependent.

Define ${\parallel f_1,f_2\parallel }_{\alpha }=inf\{t:N(f_1,f_{2}t)\ge \alpha ,\alpha \in (0,1)\}$.

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

Recently, Alaca [25] 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 $f,g,h\in \mathrm{\Im }(X)$ and $\alpha ,\beta \in (0,1)$, ${\parallel f-h,g-h\parallel }_{\alpha }$ is called an area of f, g and h. We call Ψ a 2-isometry if ${\parallel f-h,g-h\parallel }_{\alpha }={\parallel \mathrm{\Psi }(f)-\mathrm{\Psi }(h),\mathrm{\Psi }(g)-\mathrm{\Psi }(h)\parallel }_{\beta }$ for all $f,g,h\in \mathrm{\Im }(X)$ and $\alpha ,\beta \in (0,1)$.

A version of the Mazur-Ulam theorem has been obtained in [25] as follows.

Theorem 3.1 [25]

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

Lemma 3.1 [25]

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

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

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

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

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

It follows that Φ is additive.

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

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

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 [25] 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?