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Mazur-Ulam theorem under weaker conditions in the framework of 2-fuzzy 2-normed linear spaces
Journal of Inequalities and Applications volume 2013, Article number: 78 (2013)
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.
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 is called an isometry if f satisfies for all , where and 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 -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, 17–20]).
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 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 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 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 be a real-valued function on satisfying the following four properties:
if and only if x and y are linearly dependent,
for any ,
is called a 2-norm on X and the pair 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 (ℝ, the set of real numbers) is called a fuzzy norm on X if and only if:
(N1) For all with , ,
(N2) For all with , if and only if ,
(N3) For all with , , if , ,
(N4) For all , , ,
(N5) is a non-decreasing function of and .
Then is called a fuzzy normed linear space or, in short, f-NLS.
Theorem 2.1 
Let be an f-NLS. Assume the condition that
(N6) for all implies .
Define , . Then 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 be the set of all fuzzy sets on X. For and the field of real numbers, define
Definition 2.4 A fuzzy linear space over the number field S, where the addition and scalar multiplication operation on X are defined by , is a fuzzy normed space if to every , there is associated a non-negative real number, , called the fuzzy norm of , in such a way that
iff the zero element of X, ,
for all and all ,
for all ,
for all .
Definition 2.5 
Let X be a non-empty and be the set of all fuzzy sets in X. If , then . Clearly, f is a bounded function for . Let S be the space of real numbers, then is a linear space over the field S where the addition and multiplication are defined by
The linear space is said to be a normed space if for every , there is associated a non-negative real number called the norm of f in such a way that
if and only if . For
, . For
for every . For
Then is a normed linear space.
Definition 2.6 
A 2-fuzzy set on X is a fuzzy set on .
Definition 2.7 
Let be a linear space over the real field S. A fuzzy subset N of (ℝ, a set of real numbers) is called a 2-fuzzy 2-norm on X (or a fuzzy 2-norm on ) if and only if
(2-N1) for all with , ,
(2-N2) for all with , if and only if and are linearly dependent,
(2-N3) is invariant under any permutation of , ,
(2-N4) for all with , , if , ,
(2-N5) for all ,
(2-N6) is continuous,
Then is a fuzzy 2-normed linear space or is a 2-fuzzy 2-normed linear space.
Remark 2.1 In a 2-fuzzy 2-normed linear space , is a non-decreasing function of ℝ for all .
Theorem 2.2 
Let be a fuzzy 2-normed linear space. Assume that
(2-N8) for all implies and are linearly dependent.
Then is an ascending family of 2-norms on . These 2-norms are called α-2-norms on 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 and , is called an area of f, g and h. We call Ψ a 2-isometry if for all and .
A version of the Mazur-Ulam theorem has been obtained in  as follows.
Theorem 3.1 
Assume that and are fuzzy 2-normed linear spaces. If is a 2-isometry and satisfies , and 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 is a fuzzy 2-normed linear space.
Lemma 3.1 
For all , and . Then
Lemma 3.2 Let and . Then is the unique element of satisfying
with and .
Proof From Lemma 3.1, it is obvious that satisfies
with and .
For the uniqueness of v, assume that also satisfies
with and . Let for some . From Lemma 3.1, we have
Since , we have . So, and . □
Theorem 3.2 Let and be fuzzy 2-normed linear spaces. If is a 2-isometry, then Ψ is affine.
Proof Let . Obviously, and Φ is a 2-isometry. Now, we prove that Φ is linear.
Firstly, we show that Φ is additive. For , with , and from Lemma 3.1, we have
So, we get
for some by Definition 2.7. That is,
Thus, from Lemma 3.2,
for all .
Since , we have
It follows that Φ is additive.
Secondly, we show that for every , and . Let and and . Since and Φ is a 2-isometry, we have
So, for some from Definition 2.7. As , there exists a such that . It is easy to see that
So, or . If , then
So, . This is a contradiction since . Thus, for every , and .
Similarly, we can prove for every , and .
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?
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The authors would like to thank the referees and the area editor Professor Yeol Je Cho for their valuable suggestions and comments.
The authors declare that they have no competing interests.
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.
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Park, C., Alaca, C. Mazur-Ulam theorem under weaker conditions in the framework of 2-fuzzy 2-normed linear spaces. J Inequal Appl 2013, 78 (2013). https://doi.org/10.1186/1029-242X-2013-78
- 2-fuzzy 2-normed linear spaces
- Mazur-Ulam theorem