Dongseung Kang

Heejeong Koh

Journal of Inequalities and Applications

Springer - Publisher Connector

Koh and Kang Journal of Inequalities and Applications 2013, 2013:507
http://www.journalofinequalitiesandapplications.com/content/2013/1/507
RESEARCH Open Access
On the Mazur-Ulam problem in
non-Archimedean fuzzy 2-normed spaces
Heejeong Koh and Dongseung Kang*
*Correspondence:
dskang@dankook.ac.kr
Department of Mathematics
Education, Dankook University, 126,
Jukjeon, Suji, Yongin, Gyeonggi
448-701, South Korea
Abstract
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: Primary 46S10; secondary 47S10; 26E30; 12J25
Keywords: Mazur-Ulam theorem; non-Archimedean ﬁeld; non-Archimedean fuzzy
normed spaces; isometry; fuzzy 2-isometry
1 Introduction
A mapping f : X −→ Y is called an isometry if f satisﬁes
dY
(
f (x), f (y)
)
= dX(x, y)
for all x, y ∈ X, where dX(·, ·) and dY (·, ·) denote the metrics in the spaces X and Y , respec-
tively.
The theory of isometric mappings originated in the classical paper [] by Mazur and
Ulam in .
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, x → f (x) – f () 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 [, ] introduced a new approach for a theory of -norm and n-norm on a lin-
ear space. Chu [] studied the Mazur-Ulam theorem in linear -normed spaces. Recently,
Moslehian and Sadeghi [] introduced theMazur-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 partic-
ular, Amyari and Sadeghi [] proved Mazur-Ulam theorem under the condition of strict
convexity in non-Archimedean -normed spaces.
In , 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
©2013 Koh and Kang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 2 of 6
http://www.journalofinequalitiesandapplications.com/content/2013/1/507
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 -normed space over
a linear ordered non-Archimedean ﬁeld and prove thatMazur-Ulam theoremholds under
some conditions in the non-Archimedean fuzzy -normed space.
Deﬁnition . A non-Archimedean ﬁeld is a ﬁeldK equipped with a (valuation) function
from K into [,∞) satisfying the following properties:
() |a| ≥  and equality holds if and only if a = ,
() |ab| = |a||b|,
() |a + b| ≤ max{|a|, |b|}
for all a,b ∈K.
Clearly, || = |–| =  and |n| ≤  for all n ∈ N. An example of a non-Archimedean val-
uation is the function | · | taking everything except  into  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 . Let X be a linear space over a ﬁeld K with a non-Archimedean valuation
| · |. A function ‖ ·‖ : X×X −→ [,∞) is said to be a non-Archimedean -norm if it satisﬁes
the following properties:
() ‖x, y‖ =  if and only if x, y are linearly dependent,
() ‖x, y‖ = ‖y,x‖,
() ‖cx, y‖ = |c|‖x, y‖,
() ‖x, y + z‖ ≤ max{‖x, y‖,‖x, z‖}
for all x, y, z ∈ X and c ∈K. Then (X,‖ · ‖) is called a non-Archimedean -normed space.
Deﬁnition . Let X be a linear space over a ﬁeld K with a non-Archimedean valuation
| · |. A function N : X ×R −→ [, ] is said to be a non-Archimedean fuzzy -norm on X
if for all x, y ∈ X and all s, t ∈R,
(N) N(x, y, t) =  for t ≤ ,
(N) for t > , N(x, y, t) =  if and only if x and y are linearly dependent,
(N) N(x, y, t) =N(y,x, t),
(N) N(x, cy, t) =N(y,x, t|c| ) for c 	= ,
(N) N(x, y + z,max{s, t})≥ min{N(x, y, s),N(x, z, t)},
(N) N(x, y,∗) is a nondecreasing function of R and limt→∞ N(x, y, t) = .
The pair (X,N) is called a non-Archimedean fuzzy -normed space.
The property (N) implies that N(–x, y, t) = N(x, y, t) for all x, y ∈ X and t > . It is easy
to show that (N) is equivalent to the following condition:
N(x, y + z, t)≥ min{N(x, y, t),N(x, z, t)} for all x, y, z ∈ X and t ∈R.
Example . Let (X,‖·, ·‖) be a non-Archimedean -normed space. Deﬁne
N(x, y, t) =
⎧⎨
⎩
t
t+‖x,y‖ when t > , t ∈R,
 when t ≤ ,
where x, y ∈ X. Then (X,N) is a non-Archimedean fuzzy -normed space.
Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 3 of 6
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Deﬁnition . A non-Archimedean fuzzy -normed 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 . Let (X,N) and (Y ,N) be two non-Archimedean fuzzy -normed spaces.
We call f : (X,N) −→ (Y ,N) a fuzzy -isometry if N(a – c,b – c, t) = N(f (a) – f (c), f (b) –
f (c), t) for all a,b, c ∈ X and t > .
Deﬁnition . Let X be a non-Archimedean fuzzy -normed space, and let a,b, c be mu-
tually disjoint elements of X. Then a, b and c are said to be collinear if b – c = r(a – c) for
some real number r.
We denote the set of all elements of K whose norms are  by C , that is,
C = {r ∈K||r| = }.
2 Main results
Lemma . Let (X,N) be a non-Archimedean fuzzy -normed space over a linear ordered
non-Archimedean ﬁeld K. Then
N(x, y, t) =N(x, y + rx, t) for all r ∈K.
Proof Let x, y ∈ X and let r ∈K. Without loss of generality, we may assume t > . Then
N(x, y + rx, t)≥ min{N(x, y, t),N(x, rx, t)} =N(x, y, t).
Conversely,
N(x, y, t) = N(x, y + rx – rx, t)≥ min{N(x, y + rx, t),N(x, rx, t)}
= N(x, y + rx, t).
Thus N(x, y, t) =N(x, y + rx, t) for all r ∈K. 
Lemma . Let (X,N) be a non-Archimedean fuzzy -normed space over a linear ordered
non-Archimedean ﬁeld K with C = {n|n ∈ Z}, and let a,b, c ∈ X and t > . Suppose that X
is strictly convex. Then α = a+b is the unique element of X such that
N(a – c,a – α, t) =N(b – α,b – c, t) =N(a – c,b – c, t),
where a, b and α are collinear.
Proof Let α = a+b ∈ X and t > . By Lemma ., we have
N(a – c,a – α, t) = N
(
a – c,a – a + b , t
)
= N
(
a – c, a – b , t
)
Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 4 of 6
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= N
(
a – c,a – b, ||t)
= N(a – c,a – b, t)
= N(a – c,b – c, t).
Similarly,
N(b – α,b – c, t) = N
(
b – a + b ,b – c, t
)
=N(b – a,b – c, t)
= N(a – c,b – c, t).
Hence we haveN(a– c,a–α, t) =N(a– c,b– c, t) =N(b–α,b– c, t), that is, the existence
part holds. To show the uniqueness part, assume that β is an element of X such that
N(a – c,a – β , t) =N(b – β ,b – c, t) =N(a – c,b – c, t),
where a, b and β are collinear. Since a, b and β are collinear, there exists a real number s
such that
β = sa + ( – s)b.
We may assume s 	=  and s 	= .
N(a – c,b – c, t) = N(a – c,a – β , t) =N
(
a – c,a –
(
sa + ( – s)b
)
, t
)
= N
(
a – c,a – b, t| – s|
)
= N
(
a – c,b – c, t| – s|
)
.
Similarly, we have
N(a – c,b – c, t) =N
(
a – c,b – c, t|s|
)
,
that is,
N(a – c,b – c, t) =N
(
a – c,b – c, t| – s|
)
=N
(
a – c,b – c, t|s|
)
.
We note that
N
(
a – c + a – c,b – c,max
{ t
|s| ,
t
| – s|
})
≥ min
{
N
(
a – c,b – c, t|s|
)
,N
(
a – c,b – c, t| – s|
)}
=N
(
a – c,b – c, t|s|
)
=N
(
a – c,b – c, t| – s|
)
,
Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 5 of 6
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and
N
(
a – c + a – c,b – c,max
{ t
|s| ,
t
| – s|
})
=N
(
(a – c),b – c,max
{ t
|s| ,
t
| – s|
})
=N
(
a – c,b – c,max
{ t
|s| ,
t
| – s|
})
.
The previous note implies that
N(a – c,b – c, t) =N
(
a – c,b – c, t|s|
)
=N
(
a – c,b – c, t| – s|
)
.
The strict convexity of X implies that |s| = |– s| = . Then there exist elements t and t in
Z such that  – s = t and s = t . Since t + t = , we know that t, t < . Without loss
of generality, we let  – s = –n and s = –n with n ≥ n. If n  n, then
 = –n + –n = –n
(
 + n–n
)
.
Hence n = +n–n . This is a contradiction. Thus n = n, that is, s =  . This implies that
β = a+b = α. Therefore the proof is completed. 
Theorem . Let X and Y be non-Archimedean fuzzy -normed spaces over a linear or-
dered non-Archimedean ﬁeldK with C = {n|n ∈ Z}. Let X and Y be strict convexities. Sup-
pose that f : X −→ Y is a fuzzy -isometry satisfying that f (a), f (b) and f (c) are collinear
when a, b and c are collinear. Then f (x) – f () is additive.
Proof Let g(x) = f (x) – f (). Since f is a fuzzy -isometry, 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 −→ Y is a fuzzy
-isometry, we have
N
(
g(a) – g(c), g(a) – g
(a + b

)
, t
)
=N
(
a – c,a – a + b , t
)
=N(a – c,a – b, t) =N(a – c,b – c, t)
=N
(
g(a) – g(c), g(b) – g(c), t
)
.
Similarly, we get N(g(b) – g( a+b ), g(b) – g(c), t) =N(g(a) – g(c), g(b) – g(c), t). Hence
N
(
g(a) – g(c), g(a) – g
(a + b

)
, t
)
= N
(
g(b) – g
(a + b

)
, g(b) – g(c), t
)
= N
(
g(a) – g(c), g(b) – g(c), t
)
.
By the uniqueness of Lemma ., we have g( a+b ) =
g(a)+g(b)
 for all a,b ∈ X. Thus f (x) – f ()
is additive, as desired. 
Example . Let K = Z, where Z = {, , }. Suppose that the ﬁeld K has a non-
Archimedean trivial valuation | · |. Then || = , that is, C = {n|n ∈ Z}.
Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 6 of 6
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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.
Received: 19 November 2012 Accepted: 9 September 2013 Published: 08 Nov 2013
References
1. Mazur, S, Ulam, S: Sur les transformation isométriques d’espaces vectoriels normés. C. R. Math. Acad. Sci. Paris 194,
946-948 (1932)
2. Baker, JA: Isometries in normed spaces. Am. Math. Mon. 78, 655-658 (1971)
3. Gähler, S: Linear 2-normierte Raume. Math. Nachr. 28, 1-43 (1964)
4. Gähler, S: Untersuchungen über verallgemeinertem-metrische Räume, I, II, III. Math. Nachr. 40, 165-189 (1969)
5. Chu, H-Y: On the Mazur-Ulam problem in linear 2-normed spaces. J. Math. Anal. Appl. 327, 1041-1045 (2007)
6. Moslehian, MS, Sadeghi, G: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal. 69,
3405-3408 (2008)
7. Mirmostafaee, AK, Moslehian, MS: Stability of additive mappings in non-Archimedean fuzzy normed spaces. Fuzzy
Sets Syst. 160, 1643-1652 (2009)
8. Amyari, M, Sadeghi, G: Isometries in non-Archimedean 2-normed spaces. Preprint
9. Katsaras, AK: Fuzzy topological vector spaces II. Fuzzy Sets Syst. 12, 143-154 (1984)
10. Wu, C, Fang, J: Fuzzy generalization of Kolmogoroﬀs theorem. J. Harbin Inst. Technol. 1, 1-7 (1984)
11. Narayanan, A, Vijayabalaji, S: Fuzzy n-normed linear spaces. Int. J. Math. Math. Sci. 2005(24), 3963-3977 (2005)
12. Narici, L, Beckenstein, E: Strange terrain-non-Archimedean spaces. Am. Math. Mon. 88, 667-676 (1981)
10.1186/1029-242X-2013-507
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Koh and Kang Journal of Inequalities and Applications 2013, 2013:507http://www.journalofinequalitiesandapplications.com/content/2013/1/507RESEARCH Open AccessOn the Mazur-Ulam problem innon-Archimedean fuzzy 2-normed spacesHeejeong Koh and Dongseung Kang**Correspondence:dskang@dankook.ac.krDepartment of MathematicsEducation, Dankook University, 126,Jukjeon, Suji, Yongin, Gyeonggi448-701, South KoreaAbstractWe study the notion of non-Archimedean fuzzy 2-normed space over anon-Archimedean ﬁeld and prove that the Mazur-Ulam theorem holds under someconditions in the non-Archimedean fuzzy 2-normed space.MSC: Primary 46S10; secondary 47S10; 26E30; 12J25Keywords: Mazur-Ulam theorem; non-Archimedean ﬁeld; non-Archimedean fuzzynormed spaces; isometry; fuzzy 2-isometry1 IntroductionA mapping f : X −→ Y is called an isometry if f satisﬁesdY(f (x), f (y))= dX(x, y)for all x, y ∈ X, where dX(·, ·) and dY (·, ·) denote the metrics in the spaces X and Y , respec-tively.The theory of isometric mappings originated in the classical paper [] by Mazur andUlam in .Mazur-Ulam theorem Every isometry f of a normed real linear space X onto a normedreal linear space is a linear mapping up to translation, that is, x → f (x) – f () 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 anisometry from a normed real linear space into a strictly convex normed real linear spaceis aﬃne.Gähler [, ] introduced a new approach for a theory of -norm and n-norm on a lin-ear space. Chu [] studied the Mazur-Ulam theorem in linear -normed spaces. Recently,Moslehian and Sadeghi [] introduced theMazur-Ulam theorem in the non-Archimedeanstrictly convex normed spaces. Moreover, Mirmostafaee and Moslehian [] introduced anon-Archimedean fuzzy norm on a linear space over a non-Archimedean ﬁeld. In partic-ular, Amyari and Sadeghi [] proved Mazur-Ulam theorem under the condition of strictconvexity in non-Archimedean -normed spaces.In , Katsaras [] and Wu and Fang [] introduced the notion of fuzzy norm, andalso Wu and Fang gave the generalization of the Kolmogoroﬀ normalized theorem for a©2013 Koh and Kang; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 2 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/507fuzzy topological linear space. In addition, fuzzy n-normed linear spaces were studied byNarayanan and Vijayabalaji; see [].In this paper, we investigate the notion of non-Archimedean fuzzy -normed space overa linear ordered non-Archimedean ﬁeld and prove thatMazur-Ulam theoremholds undersome conditions in the non-Archimedean fuzzy -normed space.Deﬁnition . A non-Archimedean ﬁeld is a ﬁeldK equipped with a (valuation) functionfrom K into [,∞) satisfying the following properties:() |a| ≥  and equality holds if and only if a = ,() |ab| = |a||b|,() |a + b| ≤ max{|a|, |b|}for all a,b ∈K.Clearly, || = |–| =  and |n| ≤  for all n ∈ N. An example of a non-Archimedean val-uation is the function | · | taking everything except  into  and || = ; see []. Wecall it a non-Archimedean trivial valuation. Also, the most important examples of non-Archimedean spaces are p-adic numbers; see [].Deﬁnition . Let X be a linear space over a ﬁeld K with a non-Archimedean valuation| · |. A function ‖ ·‖ : X×X −→ [,∞) is said to be a non-Archimedean -norm if it satisﬁesthe following properties:() ‖x, y‖ =  if and only if x, y are linearly dependent,() ‖x, y‖ = ‖y,x‖,() ‖cx, y‖ = |c|‖x, y‖,() ‖x, y + z‖ ≤ max{‖x, y‖,‖x, z‖}for all x, y, z ∈ X and c ∈K. Then (X,‖ · ‖) is called a non-Archimedean -normed space.Deﬁnition . Let X be a linear space over a ﬁeld K with a non-Archimedean valuation| · |. A function N : X ×R −→ [, ] is said to be a non-Archimedean fuzzy -norm on Xif for all x, y ∈ X and all s, t ∈R,(N) N(x, y, t) =  for t ≤ ,(N) for t > , N(x, y, t) =  if and only if x and y are linearly dependent,(N) N(x, y, t) =N(y,x, t),(N) N(x, cy, t) =N(y,x, t|c| ) for c 	= ,(N) N(x, y + z,max{s, t})≥ min{N(x, y, s),N(x, z, t)},(N) N(x, y,∗) is a nondecreasing function of R and limt→∞ N(x, y, t) = .The pair (X,N) is called a non-Archimedean fuzzy -normed space.The property (N) implies that N(–x, y, t) = N(x, y, t) for all x, y ∈ X and t > . It is easyto show that (N) is equivalent to the following condition:N(x, y + z, t)≥ min{N(x, y, t),N(x, z, t)} for all x, y, z ∈ X and t ∈R.Example . Let (X,‖·, ·‖) be a non-Archimedean -normed space. DeﬁneN(x, y, t) =⎧⎨⎩tt+‖x,y‖ when t > , t ∈R, when t ≤ ,where x, y ∈ X. Then (X,N) is a non-Archimedean fuzzy -normed space.Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 3 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/507Deﬁnition . A non-Archimedean fuzzy -normed space is said to be strictly convexif 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 ands = t.Deﬁnition . Let (X,N) and (Y ,N) be two non-Archimedean fuzzy -normed spaces.We call f : (X,N) −→ (Y ,N) a fuzzy -isometry if N(a – c,b – c, t) = N(f (a) – f (c), f (b) –f (c), t) for all a,b, c ∈ X and t > .Deﬁnition . Let X be a non-Archimedean fuzzy -normed space, and let a,b, c be mu-tually disjoint elements of X. Then a, b and c are said to be collinear if b – c = r(a – c) forsome real number r.We denote the set of all elements of K whose norms are  by C , that is,C = {r ∈K||r| = }.2 Main resultsLemma . Let (X,N) be a non-Archimedean fuzzy -normed space over a linear orderednon-Archimedean ﬁeld K. ThenN(x, y, t) =N(x, y + rx, t) for all r ∈K.Proof Let x, y ∈ X and let r ∈K. Without loss of generality, we may assume t > . ThenN(x, y + rx, t)≥ min{N(x, y, t),N(x, rx, t)} =N(x, y, t).Conversely,N(x, y, t) = N(x, y + rx – rx, t)≥ min{N(x, y + rx, t),N(x, rx, t)}= N(x, y + rx, t).Thus N(x, y, t) =N(x, y + rx, t) for all r ∈K. Lemma . Let (X,N) be a non-Archimedean fuzzy -normed space over a linear orderednon-Archimedean ﬁeld K with C = {n|n ∈ Z}, and let a,b, c ∈ X and t > . Suppose that Xis strictly convex. Then α = a+b is the unique element of X such thatN(a – c,a – α, t) =N(b – α,b – c, t) =N(a – c,b – c, t),where a, b and α are collinear.Proof Let α = a+b ∈ X and t > . By Lemma ., we haveN(a – c,a – α, t) = N(a – c,a – a + b , t)= N(a – c, a – b , t)Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 4 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/507= N(a – c,a – b, ||t)= N(a – c,a – b, t)= N(a – c,b – c, t).Similarly,N(b – α,b – c, t) = N(b – a + b ,b – c, t)=N(b – a,b – c, t)= N(a – c,b – c, t).Hence we haveN(a– c,a–α, t) =N(a– c,b– c, t) =N(b–α,b– c, t), that is, the existencepart holds. To show the uniqueness part, assume that β is an element of X such thatN(a – c,a – β , t) =N(b – β ,b – c, t) =N(a – c,b – c, t),where a, b and β are collinear. Since a, b and β are collinear, there exists a real number ssuch thatβ = sa + ( – s)b.We may assume s 	=  and s 	= .N(a – c,b – c, t) = N(a – c,a – β , t) =N(a – c,a –(sa + ( – s)b), t)= N(a – c,a – b, t| – s|)= N(a – c,b – c, t| – s|).Similarly, we haveN(a – c,b – c, t) =N(a – c,b – c, t|s|),that is,N(a – c,b – c, t) =N(a – c,b – c, t| – s|)=N(a – c,b – c, t|s|).We note thatN(a – c + a – c,b – c,max{ t|s| ,t| – s|})≥ min{N(a – c,b – c, t|s|),N(a – c,b – c, t| – s|)}=N(a – c,b – c, t|s|)=N(a – c,b – c, t| – s|),Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 5 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/507andN(a – c + a – c,b – c,max{ t|s| ,t| – s|})=N((a – c),b – c,max{ t|s| ,t| – s|})=N(a – c,b – c,max{ t|s| ,t| – s|}).The previous note implies thatN(a – c,b – c, t) =N(a – c,b – c, t|s|)=N(a – c,b – c, t| – s|).The strict convexity of X implies that |s| = |– s| = . Then there exist elements t and t inZ such that  – s = t and s = t . Since t + t = , we know that t, t < . Without lossof generality, we let  – s = –n and s = –n with n ≥ n. If n  n, then = –n + –n = –n( + n–n).Hence n = +n–n . This is a contradiction. Thus n = n, that is, s =  . This implies thatβ = a+b = α. Therefore the proof is completed. Theorem . Let X and Y be non-Archimedean fuzzy -normed spaces over a linear or-dered non-Archimedean ﬁeldK with C = {n|n ∈ Z}. Let X and Y be strict convexities. Sup-pose that f : X −→ Y is a fuzzy -isometry satisfying that f (a), f (b) and f (c) are collinearwhen a, b and c are collinear. Then f (x) – f () is additive.Proof Let g(x) = f (x) – f (). Since f is a fuzzy -isometry, so is g . It is easy to show that ifa, b and c are collinear, then g(a), g(b) and g(c) are collinear. Since g : X −→ Y is a fuzzy-isometry, we haveN(g(a) – g(c), g(a) – g(a + b), t)=N(a – c,a – a + b , t)=N(a – c,a – b, t) =N(a – c,b – c, t)=N(g(a) – g(c), g(b) – g(c), t).Similarly, we get N(g(b) – g( a+b ), g(b) – g(c), t) =N(g(a) – g(c), g(b) – g(c), t). HenceN(g(a) – g(c), g(a) – g(a + b), t)= N(g(b) – g(a + b), g(b) – g(c), t)= N(g(a) – g(c), g(b) – g(c), t).By the uniqueness of Lemma ., we have g( a+b ) =g(a)+g(b) for all a,b ∈ X. Thus f (x) – f ()is additive, as desired. Example . Let K = Z, where Z = {, , }. Suppose that the ﬁeld K has a non-Archimedean trivial valuation | · |. Then || = , that is, C = {n|n ∈ Z}.Koh and Kang Journal of Inequalities and Applications 2013, 2013:507 Page 6 of 6http://www.journalofinequalitiesandapplications.com/content/2013/1/507AcknowledgementsThe authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. Thepresent research was conducted by the research fund of Dankook university in 2011.Received: 19 November 2012 Accepted: 9 September 2013 Published: 08 Nov 2013References1. Mazur, S, Ulam, S: Sur les transformation isométriques d’espaces vectoriels normés. C. R. Math. Acad. Sci. Paris 194,946-948 (1932)2. Baker, JA: Isometries in normed spaces. Am. Math. Mon. 78, 655-658 (1971)3. Gähler, S: Linear 2-normierte Raume. Math. Nachr. 28, 1-43 (1964)4. Gähler, S: Untersuchungen über verallgemeinertem-metrische Räume, I, II, III. Math. Nachr. 40, 165-189 (1969)5. Chu, H-Y: On the Mazur-Ulam problem in linear 2-normed spaces. J. Math. Anal. Appl. 327, 1041-1045 (2007)6. Moslehian, MS, Sadeghi, G: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal. 69,3405-3408 (2008)7. Mirmostafaee, AK, Moslehian, MS: Stability of additive mappings in non-Archimedean fuzzy normed spaces. FuzzySets Syst. 160, 1643-1652 (2009)8. Amyari, M, Sadeghi, G: Isometries in non-Archimedean 2-normed spaces. Preprint9. Katsaras, AK: Fuzzy topological vector spaces II. Fuzzy Sets Syst. 12, 143-154 (1984)10. Wu, C, Fang, J: Fuzzy generalization of Kolmogoroﬀs theorem. J. Harbin Inst. Technol. 1, 1-7 (1984)11. Narayanan, A, Vijayabalaji, S: Fuzzy n-normed linear spaces. Int. J. Math. Math. Sci. 2005(24), 3963-3977 (2005)12. Narici, L, Beckenstein, E: Strange terrain-non-Archimedean spaces. Am. Math. Mon. 88, 667-676 (1981)10.1186/1029-242X-2013-507Cite this article as: Koh and Kang: On the Mazur-Ulam problem in non-Archimedean fuzzy 2-normed spaces. Journalof Inequalities and Applications 2013, 2013:507

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On the Mazur-Ulam problem in non-Archimedean fuzzy 2-normed spaces

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