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MazurUlam theorem under weaker conditions in the framework of 2fuzzy 2normed linear spaces
Journal of Inequalities and Applications volume 2013, Article number: 78 (2013)
Abstract
The purpose of this paper is to prove that every 2isometry without any other conditions from a fuzzy 2normed linear space to another fuzzy 2normed linear space is affine, and to give a new result of the MazurUlam theorem for 2isometry in the framework of 2fuzzy 2normed linear spaces.
MSC:03E72, 46B20, 51M25, 46B04, 46S40.
1 Introduction
A satisfactory theory of 2norm and nnorm on a linear space has been introduced and developed by Gähler in [1, 2]. Freese and Cho [3] gave some isometry conditions in linear 2normed spaces. Raja and Vaezpour [4] introduced the notion of 2normed hyperset in a hypervector and also constructed some special 2normed 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 2fuzzy 2normed linear space or fuzzy 2normed linear space of the set of all fuzzy sets of a set. They gave the notion of α2norm on a linear space corresponding to a 2fuzzy 2norm 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.
MazurUlam 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 MazurUlam 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 MazurUlam theorem. Chu [14] proved that the MazurUlam theorem holds when X is a linear 2normed space. Chu et al. [15] generalized the MazurUlam theorem when X is a linear nnormed space, that is, the MazurUlam theorem holds, when the nisometry mapped to a linear nnormed space is affine. They also obtained extensions of Th.M. Rassias and Šemrl’s theorem [16]. The MazurUlam theorem has been extensively studied by many authors in different aspects (see [12, 17–20]).
Recently, Cho et al. [21] investigated the MazurUlam theorem on probabilistic 2normed spaces. Moslehian and Sadeghi [22] investigated the MazurUlam theorem in nonArchimedean spaces. Choy and Ku [23] proved that the barycenter of a triangle carries the barycenter of a corresponding triangle. They showed the MazurUlam problem on nonArchimedean 2normed spaces using the above statement. Chen and Song [24] introduced the concept of weak nisometry, and then they got that under some conditions a weak nisometry is also an nisometry. Alaca [25] gave the concepts of 2isometry, collinearity, 2Lipschitz mapping in 2fuzzy 2normed linear spaces. Also, he gave a new generalization of the MazurUlam theorem when X is a 2fuzzy 2normed linear space or \mathrm{\Im}(X) is a fuzzy 2normed linear space. Park and Alaca [26] introduced the concept of 2fuzzy nnormed linear space or fuzzy nnormed linear space of the set of all fuzzy sets of a nonempty set. They defined the concepts of nisometry, ncollinearity, nLipschitz mapping in this space. Also, they generalized the MazurUlam theorem, that is, when X is a 2fuzzy nnormed linear space or \mathrm{\Im}(X) is a fuzzy nnormed linear space, the MazurUlam theorem holds. Moreover, it is shown that each nisometry in 2fuzzy nnormed linear spaces is affine. Ren [27] showed that every generalized area n preserving mapping between real 2normed linear spaces X and Y which is strictly convex is affine under some conditions.
In the present paper, we give a new version of MazurUlam theorem with a new method when X is a 2fuzzy 2normed linear space or \mathrm{\Im}(X) is a fuzzy 2normed linear space.
2 On 2fuzzy 2normed linear spaces
In this section, at first we give the concept of linear 2normed space and later the concept of 2fuzzy 2normed 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]
Let X be a real vector space of dimension greater than 1 and let \parallel \u2022,\u2022\parallel be a realvalued function on X\times X satisfying the following four properties:

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

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

(3)
\parallel x,\alpha y\parallel =\alpha \parallel x,y\parallel for any \alpha \in \mathbb{R},

(4)
\parallel x,y+z\parallel \le \parallel x,y\parallel +\parallel x,z\parallel,
\parallel \u2022,\u2022\parallel is called a 2norm on X and the pair (X,\parallel \u2022,\u2022\parallel ) is called a linear 2normed 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 nondecreasing 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, fNLS.
Theorem 2.1 [7]
Let (X,N) be an fNLS. 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 \u2022\parallel}_{\alpha}:\alpha \in (0,1)\} is an ascending family of norms on X. We call these norms αnorms on X corresponding to the fuzzy norm on X.
Definition 2.3 Let X be any nonempty set and \mathrm{\Im}(X) be the set of all fuzzy sets on X. For U,V\in \mathrm{\Im}(X) and \lambda \in S the field of real numbers, define
and \lambda U=\{(\lambda x,\nu ):(x,\nu )\in U\}.
Definition 2.4 A fuzzy linear space \stackrel{\u02c6}{X}=X\times (0,1] over the number field S, where the addition and scalar multiplication operation on X are defined by (x,\nu )+(y,\mu )=(x+y,\nu \wedge \mu ), \lambda (x,\nu )=(\lambda x,\nu ) is a fuzzy normed space if to every (x,\nu )\in \stackrel{\u02c6}{X}, there is associated a nonnegative real number, \parallel (x,\nu )\parallel, called the fuzzy norm of (x,\nu ), in such a way that

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

(ii)
\parallel \lambda (x,\nu )\parallel =\lambda \parallel (x,\nu )\parallel for all (x,\nu )\in \stackrel{\u02c6}{X} and all \lambda \in S,

(iii)
\parallel (x,\nu )+(y,\mu )\parallel \le \parallel (x,\nu \wedge \mu )\parallel +\parallel (y,\nu \wedge \mu )\parallel for all (x,\nu ),(y,\mu )\in \stackrel{\u02c6}{X},

(iv)
\parallel (x,{\bigvee}_{t}{\nu}_{t})\parallel ={\bigwedge}_{t}\parallel (x,{\nu}_{t})\parallel for all {\nu}_{t}\in (0,1].
Definition 2.5 [8]
Let X be a nonempty and \mathrm{\Im}(X) be the set of all fuzzy sets in X. If f\in \mathrm{\Im}(X), then f=\{(x,\mu ):x\in X\text{and}\mu \in (0,1]\}. Clearly, f is a bounded function for f(x)\le 1. Let S be the space of real numbers, then \mathrm{\Im}(X) is a linear space over the field S where the addition and multiplication are defined by
and
where \lambda \in S.
The linear space \mathrm{\Im}(X) is said to be a normed space if for every f\in \mathrm{\Im}(X), there is associated a nonnegative real number \parallel f\parallel called the norm of f in such a way that

(i)
\parallel f\parallel =0 if and only if f=0. For

(ii)
\parallel \lambda f\parallel =\lambda \parallel f\parallel, \lambda \in S. For
\begin{array}{rcl}\parallel \lambda f\parallel & =& \{\parallel \lambda (x,\mu )\parallel :(x,\mu )\in f,\lambda \in S\}\\ =& \{\lambda \parallel (x,\mu )\parallel :(x,\mu )\in f\}=\lambda \parallel f\parallel .\end{array} 
(iii)
\parallel f+g\parallel \le \parallel f\parallel +\parallel g\parallel for every f,g\in \mathrm{\Im}(X). For
\begin{array}{rcl}\parallel f+g\parallel & =& \{\parallel (x,\mu )+(y,\eta )\parallel :x,y\in X,\mu ,\eta \in (0,1]\}\\ =& \{\parallel (x+y),(\mu \wedge \eta )\parallel :x,y\in X,\mu ,\eta \in (0,1]\}\\ =& \{\parallel (x,\mu \wedge \eta )\parallel +\parallel (y,\mu \wedge \eta )\parallel :(x,\mu )\in f,(y,\eta )\in g\}\\ =& \parallel f\parallel +\parallel g\parallel .\end{array}
Then (\mathrm{\Im}(X),\parallel \u2022\parallel ) is a normed linear space.
Definition 2.6 [8]
A 2fuzzy 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 2fuzzy 2norm on X (or a fuzzy 2norm on \mathrm{\Im}(X)) if and only if
(2N1) for all t\in \mathbb{R} with t\le 0, N({f}_{1},{f}_{2},t)=0,
(2N2) 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,
(2N3) N({f}_{1},{f}_{2},t) is invariant under any permutation of {f}_{1}, {f}_{2},
(2N4) 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,
(2N5) for all s,t\in \mathbb{R},
(2N6) N({f}_{1},{f}_{2},\cdot ):(0,\mathrm{\infty})\to [0,1] is continuous,
(2N7) {lim}_{t\to \mathrm{\infty}}N({f}_{1},{f}_{2},t)=1.
Then (\mathrm{\Im}(X),N) is a fuzzy 2normed linear space or (X,N) is a 2fuzzy 2normed linear space.
Remark 2.1 In a 2fuzzy 2normed linear space (X,N), N({f}_{1},{f}_{2},\cdot ) is a nondecreasing 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 2normed linear space. Assume that
(2N8) 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 \u2022,\u2022\parallel}_{\alpha}:\alpha \in (0,1)\} is an ascending family of 2norms on \mathrm{\Im}(X). These 2norms are called α2norms on \mathrm{\Im}(X) corresponding to the 2fuzzy 2norm on X.
3 On the MazurUlam theorem
Recently, Alaca [25] introduced the concept of 2isometry which is suitable to represent the notion of areapreserving mappings in fuzzy 2normed linear spaces as follows.
For f,g,h\in \mathrm{\Im}(X) and \alpha ,\beta \in (0,1), {\parallel fh,gh\parallel}_{\alpha} is called an area of f, g and h. We call Ψ a 2isometry if {\parallel fh,gh\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 MazurUlam theorem has been obtained in [25] as follows.
Theorem 3.1 [25]
Assume that \mathrm{\Im}(X) and \mathrm{\Im}(Y) are fuzzy 2normed linear spaces. If \mathrm{\Psi}:\mathrm{\Im}(X)\to \mathrm{\Im}(Y) is a 2isometry 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 2isometry in the fuzzy 2normed 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 2fuzzy 2normed linear space or \mathrm{\Im}(X) is a fuzzy 2normed linear space.
Lemma 3.1 [25]
For all f,g\in \mathrm{\Im}(X), \alpha \in (0,1) and \lambda \in \mathbb{R}. Then
Lemma 3.2 Let f,g,h\in \mathrm{\Im}(X) and \alpha \in (0,1). Then v=\frac{f+g}{2} is the unique element of \mathrm{\Im}(X) satisfying
with {\parallel fh,gh\parallel}_{\alpha}\ne 0 and v\in \{kf+(1k)g:k\in \mathbb{R}\}.
Proof From Lemma 3.1, it is obvious that v=\frac{f+g}{2} satisfies
with {\parallel fh,gh\parallel}_{\alpha}\ne 0 and v\in \{kf+(1k)g:k\in \mathbb{R}\}.
For the uniqueness of v, assume that u\in \mathrm{\Im}(X) also satisfies
with {\parallel fh,gh\parallel}_{\alpha}\ne 0 and u\in \{kf+(1k)g:k\in \mathbb{R}\}. Let u=kf+(1k)g for some k\in \mathbb{R}. From Lemma 3.1, we have
and
Since {\parallel fh,gh\parallel}_{\alpha}\ne 0, we have 1=21k=2k. 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 2normed linear spaces. If \mathrm{\Psi}:\mathrm{\Im}(X)\to \mathrm{\Im}(Y) is a 2isometry, then Ψ is affine.
Proof Let \mathrm{\Phi}(f)=\mathrm{\Psi}(f)\mathrm{\Psi}(0). Obviously, \mathrm{\Phi}(0)=0 and Φ is a 2isometry. Now, we prove that Φ is linear.
Firstly, we show that Φ is additive. For f,g,h\in \mathrm{\Im}(X), \alpha ,\beta \in (0,1) with {\parallel fh,gh\parallel}_{\alpha}\ne 0, {\parallel \mathrm{\Phi}(f)\mathrm{\Phi}(h),\mathrm{\Phi}(g)\mathrm{\Phi}(h)\parallel}_{\beta}\ne 0 and from Lemma 3.1, we have
Similarly,
And
So, we get
for some k\in \mathbb{R} by Definition 2.7. That is,
Thus, from Lemma 3.2,
for all f,g\in \mathrm{\Im}(X).
Since \mathrm{\Phi}(0)=0, we have
and
It follows that Φ is additive.
Secondly, we show that \mathrm{\Phi}(rf)=r\mathrm{\Phi}(f) for every r\in \mathbb{R}, f\in \mathrm{\Im}(X) and \alpha ,\beta \in (0,1). Let r\in {\mathbb{R}}^{+} and f\in \mathrm{\Im}(X) and \alpha ,\beta \in (0,1). Since \mathrm{\Phi}(0)=0 and Φ is a 2isometry, we have
So, \mathrm{\Phi}(rf)=s\mathrm{\Phi}(f) for some s\in \mathbb{R} from Definition 2.7. As dim\mathrm{\Im}(X)>1, there exists a g\in \mathrm{\Im}(X) such that {\parallel f,g\parallel}_{\alpha}\ne 0. It is easy to see that
So, s=r or s=r. If s=r, then
So, r1=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 MazurUlam theorem has been proved under much weaker conditions than the main result of Alaca [25] in the framework of 2fuzzy 2normed linear spaces.
Open problem How can obtain some results for the Aleksandrov problem in fuzzy 2normed 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.
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Park, C., Alaca, C. MazurUlam theorem under weaker conditions in the framework of 2fuzzy 2normed linear spaces. J Inequal Appl 2013, 78 (2013). https://doi.org/10.1186/1029242X201378
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DOI: https://doi.org/10.1186/1029242X201378