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On the Mazur-Ulam problem in non-Archimedean fuzzy 2-normed spaces
Journal of Inequalities and Applicationsvolume 2013, Article number: 507 (2013)
We study the notion of non-Archimedean fuzzy 2-normed space over a non-Archimedean field and prove that the Mazur-Ulam theorem holds under some conditions in the non-Archimedean fuzzy 2-normed space.
MSC:46S10, 47S10, 26E30, 12J25.
A mapping is called an isometry if f satisfies
for all , where and denote the metrics in the spaces X and Y, respectively.
The theory of isometric mappings originated in the classical paper  by Mazur and Ulam in 1932.
Mazur-Ulam 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, is linear, which amounts to the definition that f is affine.
The Mazur-Ulam theorem is not true for a normed complex vector space. In addition, the onto assumption is also essential. Without this assumption, Baker  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 2-norm and n-norm on a linear space. Chu  studied the Mazur-Ulam theorem in linear 2-normed spaces. Recently, Moslehian and Sadeghi  introduced the Mazur-Ulam theorem in the non-Archimedean strictly convex normed spaces. Moreover, Mirmostafaee and Moslehian  introduced a non-Archimedean fuzzy norm on a linear space over a non-Archimedean field. In particular, Amyari and Sadeghi  proved Mazur-Ulam theorem under the condition of strict convexity in non-Archimedean 2-normed spaces.
In 1984, Katsaras  and Wu and Fang  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 n-normed linear spaces were studied by Narayanan and Vijayabalaji; see .
In this paper, we investigate the notion of non-Archimedean fuzzy 2-normed space over a linear ordered non-Archimedean field and prove that Mazur-Ulam theorem holds under some conditions in the non-Archimedean fuzzy 2-normed space.
Definition 1.1 A non-Archimedean field is a field equipped with a (valuation) function from into satisfying the following properties:
and equality holds if and only if ,
for all .
Clearly, and for all . An example of a non-Archimedean valuation is the function taking everything except 0 into 1 and ; see . We call it a non-Archimedean trivial valuation. Also, the most important examples of non-Archimedean spaces are p-adic numbers; see .
Definition 1.2 Let X be a linear space over a field with a non-Archimedean valuation . A function is said to be a non-Archimedean 2-norm if it satisfies the following properties:
if and only if x, y are linearly dependent,
for all and . Then is called a non-Archimedean 2-normed space.
Definition 1.3 Let X be a linear space over a field with a non-Archimedean valuation . A function is said to be a non-Archimedean fuzzy 2-norm on X if for all and all ,
(N1) for ,
(N2) for , if and only if x and y are linearly dependent,
(N4) for ,
(N6) is a nondecreasing function of ℝ and .
The pair is called a non-Archimedean fuzzy 2-normed space.
The property (N4) implies that for all and . It is easy to show that (N5) is equivalent to the following condition:
Example 1.4 Let be a non-Archimedean 2-normed space. Define
where . Then is a non-Archimedean fuzzy 2-normed space.
Definition 1.5 A non-Archimedean fuzzy 2-normed space is said to be strictly convex if and imply and .
Definition 1.6 Let and be two non-Archimedean fuzzy 2-normed spaces. We call a fuzzy 2-isometry if for all and .
Definition 1.7 Let X be a non-Archimedean fuzzy 2-normed space, and let a, b, c be mutually disjoint elements of X. Then a, b and c are said to be collinear if for some real number r.
We denote the set of all elements of whose norms are 1 by , that is,
2 Main results
Lemma 2.1 Let be a non-Archimedean fuzzy 2-normed space over a linear ordered non-Archimedean field . Then
Proof Let and let . Without loss of generality, we may assume . Then
Thus for all . □
Lemma 2.2 Let be a non-Archimedean fuzzy 2-normed space over a linear ordered non-Archimedean field with , and let and . Suppose that X is strictly convex. Then is the unique element of X such that
where a, b and α are collinear.
Proof Let and . By Lemma 2.1, we have
Hence we have , 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 and .
Similarly, we have
We note that
The previous note implies that
The strict convexity of X implies that . Then there exist elements and in ℤ such that and . Since , we know that . Without loss of generality, we let and with . If , then
Hence . This is a contradiction. Thus , that is, . This implies that . Therefore the proof is completed. □
Theorem 2.3 Let X and Y be non-Archimedean fuzzy 2-normed spaces over a linear ordered non-Archimedean field with . Let X and Y be strict convexities. Suppose that is a fuzzy 2-isometry satisfying that , and are collinear when a, b and c are collinear. Then is additive.
Proof Let . Since f is a fuzzy 2-isometry, so is g. It is easy to show that if a, b and c are collinear, then , and are collinear. Since is a fuzzy 2-isometry, we have
Similarly, we get . Hence
By the uniqueness of Lemma 2.2, we have for all . Thus is additive, as desired. □
Example 2.4 Let , where . Suppose that the field has a non-Archimedean trivial valuation . Then , that is, .
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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.