On the Mazur-Ulam problem in non-Archimedean fuzzy 2-normed spaces
© Koh and Kang; licensee Springer. 2013
Received: 19 November 2012
Accepted: 9 September 2013
Published: 8 November 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.
KeywordsMazur-Ulam theorem non-Archimedean field non-Archimedean fuzzy normed spaces isometry fuzzy 2-isometry
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.
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 .
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.
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.
2 Main results
Thus for all . □
where a, b and α are collinear.
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.
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, .
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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