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On the MazurUlam problem in nonArchimedean fuzzy 2normed spaces
Journal of Inequalities and Applications volume 2013, Article number: 507 (2013)
Abstract
We study the notion of nonArchimedean fuzzy 2normed space over a nonArchimedean field and prove that the MazurUlam theorem holds under some conditions in the nonArchimedean fuzzy 2normed space.
MSC:46S10, 47S10, 26E30, 12J25.
1 Introduction
A mapping f:X\u27f6Y is called an isometry if f satisfies
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.
MazurUlam 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 MazurUlam 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 2norm and nnorm on a linear space. Chu [5] studied the MazurUlam theorem in linear 2normed spaces. Recently, Moslehian and Sadeghi [6] introduced the MazurUlam theorem in the nonArchimedean strictly convex normed spaces. Moreover, Mirmostafaee and Moslehian [7] introduced a nonArchimedean fuzzy norm on a linear space over a nonArchimedean field. In particular, Amyari and Sadeghi [8] proved MazurUlam theorem under the condition of strict convexity in nonArchimedean 2normed 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 nnormed linear spaces were studied by Narayanan and Vijayabalaji; see [11].
In this paper, we investigate the notion of nonArchimedean fuzzy 2normed space over a linear ordered nonArchimedean field and prove that MazurUlam theorem holds under some conditions in the nonArchimedean fuzzy 2normed space.
Definition 1.1 A nonArchimedean 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=ab,

(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 nonArchimedean valuation is the function \cdot  taking everything except 0 into 1 and 0=0; see [12]. We call it a nonArchimedean trivial valuation. Also, the most important examples of nonArchimedean spaces are padic numbers; see [7].
Definition 1.2 Let X be a linear space over a field \mathcal{K} with a nonArchimedean valuation \cdot . A function \parallel \cdot \parallel :X\times X\u27f6[0,\mathrm{\infty}) is said to be a nonArchimedean 2norm 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 nonArchimedean 2normed space.
Definition 1.3 Let X be a linear space over a field \mathcal{K} with a nonArchimedean valuation \cdot . A function N:{X}^{2}\times \mathbb{R}\u27f6[0,1] is said to be a nonArchimedean fuzzy 2norm 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 nonArchimedean fuzzy 2normed 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:
Example 1.4 Let (X,\parallel \cdot ,\cdot \parallel ) be a nonArchimedean 2normed space. Define
where x,y\in X. Then (X,N) is a nonArchimedean fuzzy 2normed space.
Definition 1.5 A nonArchimedean fuzzy 2normed 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 nonArchimedean fuzzy 2normed spaces. We call f:(X,N)\u27f6(Y,N) a fuzzy 2isometry if N(ac,bc,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 nonArchimedean fuzzy 2normed space, and let a, b, c be mutually disjoint elements of X. Then a, b and c are said to be collinear if bc=r(ac) for some real number r.
We denote the set of all elements of \mathcal{K} whose norms are 1 by \mathcal{C}, that is,
2 Main results
Lemma 2.1 Let (X,N) be a nonArchimedean fuzzy 2normed space over a linear ordered nonArchimedean field \mathcal{K}. Then
Proof Let x,y\in X and let r\in \mathcal{K}. Without loss of generality, we may assume t>0. Then
Conversely,
Thus N(x,y,t)=N(x,y+rx,t) for all r\in \mathcal{K}. □
Lemma 2.2 Let (X,N) be a nonArchimedean fuzzy 2normed space over a linear ordered nonArchimedean 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
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
Similarly,
Hence we have N(ac,a\alpha ,t)=N(ac,bc,t)=N(b\alpha ,bc,t), that is, the existence part holds. To show the uniqueness part, assume that β is an element of X such that
where a, b and β are collinear. Since a, b and β are collinear, there exists a real number s such that
We may assume s\ne 0 and s\ne 1.
Similarly, we have
that is,
We note that
and
The previous note implies that
The strict convexity of X implies that s=1s=1. Then there exist elements {t}_{1} and {t}_{2} in ℤ such that 1s={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 1s={2}^{{n}_{1}} and s={2}^{{n}_{2}} with {n}_{1}\ge {n}_{2}. If {n}_{1}\u2a88{n}_{2}, then
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 nonArchimedean fuzzy 2normed spaces over a linear ordered nonArchimedean field \mathcal{K} with \mathcal{C}=\{{2}^{n}n\in \mathbb{Z}\}. Let X and Y be strict convexities. Suppose that f:X\u27f6Y is a fuzzy 2isometry 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 2isometry, 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\u27f6Y is a fuzzy 2isometry, we have
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
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 nonArchimedean trivial valuation \cdot . Then 2=1, that is, \mathcal{C}=\{{2}^{n}n\in \mathbb{Z}\}.
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Acknowledgements
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. The present research was conducted by the research fund of Dankook university in 2011.
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Koh, H., Kang, D. On the MazurUlam problem in nonArchimedean fuzzy 2normed spaces. J Inequal Appl 2013, 507 (2013). https://doi.org/10.1186/1029242X2013507
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DOI: https://doi.org/10.1186/1029242X2013507