Available online through
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International Journal of Mathematical Archive- 2 (10), Oct. – 2011 2036
SOME RESULTS ON
t
-BEST APPROXIMATION IN FUZZY
ANTI-
n
-NORMED LINEAR SPACES
B. Surender Reddy*
Department of Mathematics, PGCS, Saifabad, Osmania University, Hyderabad-500004, AP, INDIA E-mail: [email protected]
(Received on: 05-10-11; Accepted on: 20-10-11)
________________________________________________________________________________________________
ABSTRACT
The main aim of this paper to give the set of all t-best approximations on fuzzy anti-n-normed linear spaces and prove
some theorems in the sense of Vaezpour and Karimi [21].Key Words: n-normed space, fuzzy anti-n-normed space, t-best approximation.
2000 AMS Subject Classification No: 46A30, 46S40, 46A70, 54A40.
________________________________________________________________________________________________
1. INTRODUCTION:
Fuzzy set theory is a useful tool to describe situations in which the data are imprecise or vague. Fuzzy sets handle such situation by attributing a degree to which a certain object belongs to a set. The idea of fuzzy norm was initiated by Katsaras in [12]. Felbin [6] defined a fuzzy norm on a linear space whose associated fuzzy metric is of Kaleva and Seikkala type [11]. Cheng and Mordeson [4] introduced an idea of a fuzzy norm on a linear space whose associated metric is Kramosil and Michalek type [13].
Bag and Samanta in [1] gave a definition of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [13]. They also studied some properties of the fuzzy norm in [2] and [3]. Bag and Samanta discussed the notion of convergent sequence and Cauchy sequence in fuzzy normed linear space in [1]. They also made in [3] a comparative study of the fuzzy norms defined by Katsaras [12], Felbin [6], and Bag and Samanta [1]. The concept of 2-norm and n-norm on a linear space has been introduced and developed by Gahler in [7,8]. Following Misiak [15], Malceski [14] and Gunawan [9] developed the theory of n-normed space. Narayana and Vijayabalaji [16] introduced the concept of fuzzy n-normed linear space. Vijayabalaji and Thillaigovindan [22] introduced the notion of convergent sequence and Cauchy sequence in fuzzy n-normed linear space and studied the completeness of the fuzzy n-normed linear space. Many authors studied on fuzzy n-normed linear space [5]. Recently, Vaezpour and Karimi [21], studied on the set of all t-best approximations on fuzzy normed spaces and proved several theorems pertaining to this set.
In [10] Iqbal H. Jebril and Samanta introduced fuzzy norm on a linear space depending on the idea of fuzzy anti-norm was introduced by Bag and Samanta [3] and investigated their important properties. In [17,18] Surender Reddy introduced the notion of convergent sequence and Cauchy sequence in fuzzy 2-normed linear space and fuzzy anti-n-normed linear spaces. Recently Surender Reddy [19,20] studied on the set of all t-best approximations on fuzzy anti-normed linear spaces and fuzzy anti-2-anti-normed linear spaces.
In the present paper, we give the set of all t-best approximations on fuzzy anti-n-normed spaces and prove some theorems in the sense of Vaezpour and Karimi [21].
2. PRELIMINARIES:
Definition 2.1: Let
n
∈
N
and let X be a real linear space of dimension≥
n
. A real valued function• •
, ,...,
•
on nn
X
X
X
X
×
×
...
×
=
satisfying the following conditions1
nN
:x
1,
x
2,...,
x
n=
0
if and only ifx
1,
x
2,...,
x
n are linearly dependent,---
IJMA- 2(10), Oct.-2011, Page: 2036-2044
2
nN
:x
1,
x
2,...,
x
n is invariant under any permutation ofx
1,
x
2,...,
x
n,3
nN
:x
1,
x
2,...,
x
n−1,
α
x
n=
α
x
1,
x
2,...,
x
n−1,
x
n , for everyα
∈
R
,4
nN
:x
1,
x
2,...,
x
n−1,
y
+
z
≤
x
1,
x
2,...,
x
n−1,
y
+
x
1,
x
2,...,
x
n−1,
z
for ally
,
z
,
x
1,
x
2,...,
x
n−1∈
X
,then the function
•
,
•
,...,
•
is called an n-norm on X and the pair(
X
,
•
,
•
,...,
•
)
is called n-normed linear space.Example 2.2: A trivial example of an n-normed linear space is
X
=
R
n equipped with the following Euclidean n-norm.
=
=
nn n
n
n n
ij E
n
x
x
x
x
x
x
x
x
x
abs
x
x
x
x
...
...
...
...
...
...
...
)
det(
,...,
,
2 1
2 22
21
1 12
11
2
1 ,
where
x
i=
(
x
i1,
x
i2,...,
x
in)
∈
R
n for eachi
=
1
,
2
,...,
n
.Definition 2.3: Let X be a linear space over a real field F. A fuzzy subset N of
X
X
X
R
n×
×
×
×
...
is called a fuzzyn-norm on X if the following conditions are satisfied for all
x
1,
x
2,...,
x
n,
y
∈
X
)
(
n
−
N
1 For allt
∈
R
witht
≤
0
,N
(
x
1,
x
2,...,
x
n,
t
)
=
0
,)
(
n
−
N
2 : For allt
∈
R
witht
>
0
,N
(
x
1,
x
2,...,
x
n,
t
)
=
1
if and only ifx
1,
x
2,...,
x
n are linearly dependent,)
(
n
−
N
3 :N
(
x
1,
x
2,...,
x
n,
t
)
is invariant under any permutation ofx
1,
x
2,...,
x
n,)
(
n
−
N
4 : For allt
∈
R
witht
>
0
,(
1,
2,...,
1,
,
)
(
1,
2,...,
1,
,
)
c
t
x
x
x
x
N
t
cx
x
x
x
N
n− n=
n− n , ifc
≠
0
,c
∈
F
,)
(
n
−
N
5 : For alls
,
t
∈
R
,N
(
x
1,
x
2,...,
x
n−1,
x
n+
y
,
s
+
t
)
≥
min
{
N
(
x
1,
x
2,...,
x
n−1,
x
n,
s
),
N
(
x
1,
x
2,...,
x
n−1,
y
,
t
)
}
,)
(
n
−
N
6 :N
(
x
1,
x
2,...,
x
n,
t
)
is a non-decreasing function oft
∈
R
andlim
(
1,
2,...,
,
)
=
1
∞→
N
x
x
x
nt
t .
Then the pair
(
X
,
N
)
is called a fuzzy n-normed linear space (briefly F-n-NLS).Example 2.4: Let
(
X
,
•
,
•
,...,
•
)
be a n-normed linear space. Define
n n
x
x
x
t
t
t
x
x
x
N
,...,
,
)
,
,...,
,
(
2 1 2
1
+
=
, ift
>
0
,t
∈
R
,x
1,
x
2,...,
x
n∈
X
,
=
0
, ift
≤
0
,t
∈
R
,x
1,
x
2,...,
x
n∈
X
. Then(
X
,
N
)
is a fuzzy n-normed linear space.Definition 2.5: Let X be a linear space over a real field F. A fuzzy subset N of
X
X
X
R
n
×
×
×
×
...
is called a fuzzyanti-n-norm on X if the following conditions are satisfied for all
x
1,
x
2,...,
x
n,
y
∈
X
.)
(
a
−
n
−
N
1 For allt
∈
R
witht
≤
0
,N
(
x
1,
x
2,...,
x
n,
t
)
=
1
,)
(
a
−
n
−
N
2 : For allt
∈
R
witht
>
0
,N
(
x
1,
x
2,...,
x
n,
t
)
=
0
if and only ifx
1,
x
2,...,
x
n are linearly dependent,)
(
a
−
n
−
N
3 :N
(
x
1,
x
2,...,
x
n,
t
)
is invariant under any permutation ofx
1,
x
2,...,
x
n,)
(
a
−
n
−
N
4 : For allt
∈
R
witht
>
0
,(
1,
2,...,
1,
,
)
(
1,
2,...,
1,
,
)
c
t
x
x
x
x
N
t
cx
x
x
x
N
n− n=
n− n , ifc
≠
0
,© 2011, IJMA. All Rights Reserved 2038
)
(
a
−
n
−
N
5 : For alls
,
t
∈
R
,N
(
x
1,
x
2,...,
x
n−1,
x
n+
y
,
s
+
t
)
≤
max
{
N
(
x
1,
x
2,...,
x
n−1,
x
n,
s
),
N
(
x
1,
x
2,...,
x
n−1,
y
,
t
)
}
,)
(
a
−
n
−
N
6 :N
(
x
1,
x
2,...,
x
n,
t
)
is a non-increasing function oft
∈
R
andlim
(
1,
2,...,
,
)
=
0
∞→
N
x
x
x
nt
t .
Then the pair
(
X
,
N
)
is called a fuzzy anti-n-normed linear space (briefly Fa-n-NLS).Remark 2.6:
From
(
a
−
2
−
N
3)
, it follows that in Fa-n-NLS,)
(
a
−
n
−
N
4 : For allt
∈
R
witht
>
0
,N
(
x
1,
x
2,...,
cx
i,...,
x
n,
t
)
=
(
1,
2,...,
,...,
,
)
c
t
x
x
x
x
N
i n , ifc
≠
0
,F
c
∈
,)
(
a
−
n
−
N
5 : For alls
,
t
∈
R
,N
(
x
1,
x
2,...,
x
i+
x
′
i,...,
x
n,
s
+
t
)
≤
max
{
N
(
x
1,
x
2,...,
x
i,...,
x
n,
s
),
N
(
x
1,
x
2,...,
x
′
i,...,
x
n,
t
)
}
.Example 2.7: Let
(
X
,
•
,
•
,...,
•
)
be a n-normed linear space. Define
n n n
x
x
x
t
x
x
x
t
x
x
x
N
,...,
,
,...,
,
)
,
,...,
,
(
2 1
2 1 2
1
+
=
, ift
>
0
,t
∈
R
,x
1,
x
2,...,
x
n∈
X
,
=
1
, ift
≤
0
,t
∈
R
,x
1,
x
2,...,
x
n∈
X
. Then(
X
,
N
)
is a fuzzy anti-n-normed linear space.Definition 2.8: A sequence
{
x
k}
in a fuzzy anti-n-normed linear space(
X
,
N
)
is said to be converges tox
∈
X
if givent
>
0
,0
<
r
<
1
, there exists an integern
0∈
N
such that
N
(
x
1,
x
2,...,
x
n−1,
x
k−
x
,
t
)
<
r
,∀
k
≥
n
0.Theorem 2.9: In a fuzzy anti-n-normed linear space
(
X
,
N
)
, a sequence{
x
k}
converges tox∈
X
if and only0
)
,
,
,...,
,
(
lim
1 2 −1−
=
∞
→
N
x
x
x
nx
kx
t
k ,
∀
t
>
0
.Definition 2.10: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. Let{
x
k}
be a sequence in X then{
x
k}
is said tobe a Cauchy sequence if
lim
(
1,
2,...,
−1,
+−
,
)
=
0
∞→
N
x
x
x
nx
k px
kt
k ,
∀
t
>
0
andp
=
1
,
2
,
3
,...
.Definition 2.11: A fuzzy anti-n-normed linear space
(
X
,
N
)
is said to be complete if every Cauchy sequence in X is convergent.Definition 2.12: A complete fuzzy anti-n-normed linear space
(
X
,
N
)
is called a fuzzy anti-n-Banach space.3. MAIN RESULTS:
Definition 3.1: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. The open ball)
,
,
(
x
r
t
B
and the closed ballB
[
x
,
r
,
t
]
with the centerx
∈
X
and radius0
<
r
<
1
,t
>
0
are defined as follows:
B
(
x
,
r
,
t
)
=
{
y
∈
X
:
N
(
x
1,
x
2,...,
x
n−1,
x
−
y
,
t
)
<
r
}
B
[
x
,
r
,
t
]
=
{
y
∈
X
:
N
(
x
1,
x
2,...,
x
n−1,
x
−
y
,
t
)
≤
r
}
Definition 3.2: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. A subset A of X is said to be open if there exists)
1
,
0
(
∈
IJMA- 2(10), Oct.-2011, Page: 2036-2044
Definition 3.3: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. A subset A of X is said to be closed if for anysequence
{
x
k}
in A converges tox
∈
A
.i.e.,
lim
(
1,
2,...,
−1,
−
,
)
=
0
∞→
N
x
x
x
nx
kx
t
k , for all
t
>
0
implies thatx
∈
A
.Definition 3.4: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. A subset B of X is said to be closure ofA
⊂
B
if for anyx
∈
B
, there exists a sequence{
x
k}
in A such thatlim
(
1,
2,...,
−1,
−
,
)
=
0
∞
→
N
x
x
x
nx
kx
t
k , for all
t
>
0
. We denote the set B byA
.Definition 3.5: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. A subset A of X is said to be compact if for anysequence
{
x
k}
in A has a sequence converging to an element of A.Lemma 3.6: If
(
X
,
N
)
be a fuzzy anti-n-normed linear space then(i) the function
(
x
,
y
)
→
x
+
y
is continuous. (ii) the function(
α
,
x
)
→
α
x
is continuous.Proof: (i) If
x
k→
x
andy
k→
y
then ask
→
∞
,0 )} 2 , , ,..., , ( ), 2 , , ,..., , ( max{ ) ), ( ) ( , ,..., ,
( 1 2 −1 + − + ≤ 1 2 −1 − 1 2 −1 − → t y y x x x N t x x x x x N t
y x y x x x x
N n k k n k n k
(ii) If
x
k→
x
,α
k→
α
andα
k≠
0
then
N
(
x
1,
x
2,...,
x
n−1,
α
kx
k−
α
x
,
t
)
=
N
(
x
1,
x
2,...,
x
n−1,
α
k(
x
k−
x
)
+
x
(
α
k−
α
),
t
)
)}
2
),
(
,
,...,
,
(
),
2
),
(
,
,...,
,
(
max{
N
x
1x
2x
n 1α
kx
k−
x
t
N
x
1x
2x
n1x
α
k−
α
t
≤
− −
)}
2
,
,
,...,
,
(
),
2
,
,
,...,
,
(
max{
1 2 1 1 2 1α
α
α
−
−
≤
− −k n
k k
n
t
x
x
x
x
N
t
x
x
x
x
x
N
→
0
ask
→
∞
.Definition 3.7: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of X.Let
d
(
A
,
x
,
t
)
=
inf{
N
(
x
1,
x
2,...,
x
n−1,
x
−
y
,
t
)
:
y
∈
A
}
, wherex
∈
X
,t
>
0
. An elementy
0∈
A
is said to bea t-best approximation of x from A if
N
(
x
1,
x
2,...,
x
n−1,
y
0−
x
,
t
)
=
d
(
A
,
x
,
t
)
.Definition 3.8: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of X. Forx
∈
X
,0
>
t
, we shall denote the set of all elements of t-best approximation of x from A byP
At(
x
)
and is defined as
P
At(
x
)
=
{
y
∈
A
:
d
(
A
,
x
,
t
)
=
N
(
x
1,
x
2,...,
x
n−1,
y
−
x
,
t
)}
.If each
x
∈
X
has at least (respectively exactly) one t-best approximation in A then A is called a t-proximinal (respectively t-chebyshev) set.Definition 3.9: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of X. Fort
>
0
, A is said to be t-boundedly compact if for eachx
∈
X
and0
<
r
<
1
,B
[
x
,
r
,
t
]
∩
A
is a compact subset of X.Definition 3.10: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of Xthen(i)
d
(
A
+
y
,
x
+
y
,
t
)
=
d
(
A
,
x
,
t
)
, for allx
,
y
∈
X
andt
>
0
, (ii)P
At(
x
+
y
)
=
P
At(
x
)
+
y
, for allx
,
y
∈
X
andt
>
0
,(iii)
(
,
,
)
(
,
,
)
α
α
α
A
x
t
d
A
x
t
© 2011, IJMA. All Rights Reserved 2040
(iv)
P
ααAt(
α
x
)
=
α
P
At(
x
)
, for allx
∈
X
,t
>
0
andα
∈
R
−
{
0
}
,(v) A is t-proximinal (respectively t-chebyshev) if and only if A+y is t-proximinal (respectively t-chebyshev), for any given
y
∈
X
,(vi) A is t-proximinal (respectively t-chebyshev) if and only if
α
A
isα
t
-proximinal (respectivelyα
t
-chebyshev),for any given each
α
∈
R
−
{
0
}
.Proof: (i) For
x
,
y
∈
X
andt
>
0
,}
:
)
),
(
)
(
,
,...,
,
(
inf{
)
,
,
(
A
y
x
y
t
N
x
1x
2x
1z
y
x
y
t
z
A
d
+
+
=
n−+
−
+
∈
=
inf{
N
(
x
1,
x
2,...,
x
n−1,
z
−
x
,
t
)
:
z
∈
A
}
=
d
(
A
,
x
,
t
)
. (ii) On using (i),y
0∈
P
At+y(
x
+
y
)
if and only ify
0∈
A
+
y
and)
,
,
,...,
,
(
)
,
,
(
A
y
x
y
t
N
x
1x
2x
1x
y
y
0t
d
+
+
=
n−+
−
if and only ify
0−
y
∈
A
and)
),
(
,
,...,
,
(
)
,
,
(
A
x
t
N
x
1x
2x
1x
y
0y
t
d
=
n−−
−
if and only ify
0y
P
t(
x
)
A
∈
−
i.e.,
y
0∈
P
At(
x
)
+
y
.(iii) We have
d
(
α
A
,
α
x
,
t
)
=
inf{
N
(
x
1,
x
2,...,
x
n−1,
α
x
−
α
z
,
t
)
:
z
∈
A
}
=
inf{
N
(
x
1,
x
2,...,
x
n−1,
α
(
x
−
z
),
t
)
:
z
∈
A
}
inf{
(
1,
2,...,
1,
,
)
:
}
(
,
,
)
α
α
t
x
A
d
A
z
t
z
x
x
x
x
N
n−
∈
=
=
− .(iv) On using (iii), it follows that
y
0∈
P
ααAt(
α
x
)
if and only ify
0∈
α
A
and)
,
,
,...,
,
(
)
,
,
(
A
x
t
N
x
1x
2x
1x
y
0t
d
α
α
=
n−α
−
if and only ify
∈
A
α
0and
)
,
,
(
)
,
,
,...,
,
(
01 2
1
t
d
A
x
t
y
x
x
x
x
N
n−−
=
α
. However, this is equivalent to(
)
0
P
x
y
tA
∈
α
.i.e.,
y
0P
t(
x
)
A
α
∈
.(v) The proof of (v) is an immediate consequence of (ii). (vi) The proof of (vi) follows from (iv).
Corollary 3.11: Let M is a non empty subset of X then
(i)
d
(
M
,
x
+
y
,
t
)
=
d
(
M
,
x
,
t
)
, for allt
>
0
x
∈
X
andy
∈
M
, (ii)P
Mt(
x
+
y
)
=
P
Mt(
x
)
+
y
, for allt
>
0
x
∈
X
andy
∈
M
,(iii)
d
(
M
,
α
x
,
α
t
)
=
d
(
M
,
x
,
t
)
, for allt
>
0
,x
∈
X
andα
∈
R
−
{
0
}
,(iv)
P
Mαt(
α
x
)
=
α
P
Mt(
x
)
, for allt
>
0
,x
∈
X
andα
∈
R
−
{
0
}
.Proof: The proof of (i) and (ii) follows from theorem 2(i) and 2(ii) and the fact that if M is a subspace and
y
∈
M
thenM
+
y
=
M
.The proof of (iii) and (iv) follows from theorem 2(iii) and 2(iv) and the fact that if M is a subspace and
α
≠
0
thenM
M
=
α
.Definition 3.12: For
x
∈
X
,0
<
r
<
1
,t
>
0
,
S
[
x
,
r
,
t
]
=
{
y
∈
X
:
N
(
x
1,
x
2,...,
x
n−1,
x
−
y
,
t
)
=
r
}
ande
tA(
x
)
=
d
(
A
,
x
,
t
)
.Theorem 3.13: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space,A
⊂
X
,x
∈
X
A
andt
>
0
then we haveP
At(
x
)
=
A
∩
B
[
x
,
e
tA(
x
),
t
]
=
A
∩
S
[
x
,
e
tA(
x
),
t
]
(1)Proof: This inclusions
IJMA- 2(10), Oct.-2011, Page: 2036-2044
are obvious by the definitions of
P
At(
x
)
ande
tA(
x
)
.Conversely, let
y
∈
A
∩
B
[
x
,
e
At(
x
),
t
]
, then we havey
∈
A
and
N
(
x
1,
x
2,...,
x
n−1,
y
−
x
,
t
)
≤
e
At(
x
)
=
d
(
A
,
x
,
t
)
≤
N
(
x
1,
x
2,...,
x
n−1,
y
−
x
,
t
)
.Therefore
y
∈
A
andN
(
x
1,
x
2,...,
x
n−1,
y
−
x
,
t
)
=
d
(
A
,
x
,
t
)
, which implies thaty
P
t(
x
)
A
∈
. So,)
(
]
),
(
,
[
x
e
x
t
P
x
B
A
∩
tA⊂
At . Hence by (2) we have (1) which completes the proof.Remark 3.14: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of X,x
∈
X
A
and0
>
t
then we have
A
∩
B
(
x
,
e
At(
x
),
t
)
=
Φ
, (3) because, ify
0∈
A
∩
B
(
x
,
e
tA(
x
),
t
)
thend
(
A
,
x
,
t
)
≤
N
(
x
1,
x
2,...,
x
n−1,
x
−
y
0,
t
)
<
d
(
A
,
x
,
t
)
which is impossible.Corollary 3.15: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of X,x
∈
X
A
withP
At(
x
)
≠
Φ
and0
<
r
<
1
such that,
Φ
≠
A
∩
B
[
x
,
r
,
t
]
⊆
S
[
x
,
r
,
t
]
(4) Then we haver
=
e
tA(
x
)
, and we can writeA
∩
B
[
x
,
r
,
t
]
=
P
At(
x
)
.Proof: If
r
<
e
tA(
x
)
then by the definition ofe
At(
x
)
we haveA
∩
B
[
x
,
r
,
t
]
=
Φ
, which contradicts (4). Ifr
>
e
tA(
x
)
, sinceP
At(
x
)
≠
Φ
, then by (1) we have)
,
,
(
]
),
(
,
[
)
(
x
A
B
x
e
x
t
A
B
x
r
t
P
At=
∩
tA⊆
∩
≠
Φ
, which contradicts (4), and this completes the proof.Definition 3.16: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space,0
<
r
<
1
andt
>
0
.We shall say that a set
A
⊂
X
supports the cellB
[
x
,
r
,
t
]
, or that A is a support set of the cellB
[
x
,
r
,
t
]
, if we have1
)
],
,
,
[
,
(
A
B
x
r
t
t
=
d
andA
∩
B
(
x
,
r
,
t
)
=
Φ
.Theorem 3.17: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of X andA
X
x
∈
,a
0∈
A
andt
>
0
. We havea
0∈
P
At(
x
)
if and only if the set A supports the cell]
),
,
,
,...,
,
(
,
[
x
N
x
1x
2x
1a
0x
t
t
B
B
=
n−−
.Proof: Assume that
a
0∈
P
At(
x
)
. HenceN
(
x
1,
x
2,...,
x
n−1,
a
0−
x
,
t
)
=
d
(
A
,
x
,
t
)
. Then by (3), we haveΦ
=
−
∩
B
(
x
,
N
(
x
1,
x
2,...,
x
−1,
a
0x
,
t
),
t
)
A
n , on the other hand, since]
),
,
,
,...,
,
(
,
[
1 2 1 00
A
B
x
N
x
x
x
a
x
t
t
a
∈
∩
n−−
, we haved
(
A
,
B
,
t
)
=
1
. Consequently, the set A supports the cellB. Conversely, suppose
a
0∉
P
At(
x
)
, henceN
(
x
1,
x
2,...,
x
n−1,
a
0−
x
,
t
)
>
d
(
A
,
x
,
t
)
and let0
<
ε
<
1
such thatε
+
>
−
−
,
,
)
(
,
,
)
,...,
,
(
x
1x
2x
1a
0x
t
d
A
x
t
N
n . Then there exists ana
∈
A
such that)
,
,
,...,
,
(
)
,
,
(
)
,
,
,...,
,
(
x
1x
2x
1a
0x
t
d
A
x
t
N
x
1x
2x
1a
x
t
N
n−−
>
+
ε
>
n−−
, hence)
),
,
,
,...,
,
(
,
(
x
N
x
1x
2x
1a
0x
t
t
B
a
∈
n−−
. Consequently, A does not support the cell B.Remark 3.18: We recall that a set A in a topological space
τ
is said to be countably compact, if every countable opencover of A has a finite subcover, or, which is equivalent, if for every decreasing sequence
A
1⊃
A
2⊃
...
of non-voidclosed subset of A we have
≠
Φ
∞= n n
A
1
.
© 2011, IJMA. All Rights Reserved 2042
Proof: For all
n
∈
N
,1
1
)
,
,
(
)
,
,
(
1
0
<
+
+
−
<
n
t
x
A
d
t
x
A
d
. Put
+
+
−
∩
=
t
n
t
x
A
d
t
x
A
d
x
B
A
A
nt,
1
)
,
,
(
)
,
,
(
1
,
,(
n
=
1
,
2
,...)
.Since for every
n
∈
N
,(
,
,
)
1
1
1
)
,
,
(
d
A
x
t
n
t
x
A
d
>
+
−
, obviously 1t⊃
2t⊃
...
A
A
and eachA
nt≠
Φ
. Hencethere exists
a
nt∈
A
such that
(
,
,...,
,
,
)
1
1
1
)
,
,
(
N
x
1x
2x
1a
x
t
n
t
x
A
d
t n n−
>
+
−
− .It follows that
a
nt∈
A
nt. Now, since eachA
nt isτ
-countably compact andτ
-closed, we conclude that there exists ant n n
A
a
∞ =∈
10 . Then we have
+
−
≤
−
≤
−1
1
1
)
,
,
(
)
,
,
,...,
,
(
)
,
,
(
1 2 1 0n
t
x
A
d
t
x
a
x
x
x
N
t
x
A
d
n ,(
n
=
1
,
2
,...)
,hence
)
(
0
P
x
a
∈
At which completes the proof.Definition 3.20: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of X. An elementA
y
0∈
is said to be an F-best approximation ofx
∈
X
from A if it is a t-best approximation of x from A, for every0
>
t
, i.e.,(
)
) , 0 (
0
P
x
y
Att∈ ∞
∈
.The set of all elements of F-best approximations of
x
∈
X
from A is denoted byFP
A(
x
)
and is defined as)
(
)
(
) , 0 (x
P
x
FP
Att A
∞ ∈
=
.If each
x
∈
X
has at least (respectively exactly) one F-best approximation in A then A is called a F-proximinal (respectively F-chebyshev) set.Example 3.21: Let
X
=
R
3. DefineN
:
X
×
X
×
X
×
[
0
,
∞
)
→
[
0
,
1
]
byt
x
x
x
t
x
x
x
N
(
1,
2,
3,
)
=
1,
2,
3 , ift
>
0
,t
∈
R
,x
1,
x
2,
x
3∈
X
,
=
1
, ift
≤
0
,t
∈
R
,x
1,
x
2,
x
3∈
X
,where = ≤ ≤
=
3 1 3 1 3 21
,
,
min
j ij
i
x
x
x
x
. Then(
X
,
N
)
is a fuzzy anti-3-normed linear space.Let
A
=
{(
a
,
b
,
c
)
∈
R
3:
a
2+
b
2≤
1
,
0
≤
c
≤
a
2+
b
2}
and
x
1=
(
1
,
0
,
0
)
,x
2=
(
0
,
1
,
0
)
,x
=
(
0
,
0
,
4
)
are in X. Leta
0=
(
0
,
−
1
,
1
)
anda
1=
(
0
,
1
,
1
)
are in A, Then for everyt
>
0
,
t
t
x
x
N
t
x
a
x
x
N
(
1,
2,
0−
,
)
=
(
1,
2,
(
0
,
−
1
,
1
)
−
(
0
,
0
,
4
),
)
=
1
t
t
x
x
N
t
x
a
x
x
N
(
1,
2,
1−
,
)
=
(
1,
2,
(
0
,
1
,
1
)
−
(
0
,
0
,
4
),
)
=
1
.On the other hand
}
:
)
),
4
,
0
,
0
(
,
,
(
inf{
)
),
4
,
0
,
0
(
,
(
)
,
,
(
A
x
t
d
A
t
N
x
1x
2u
t
u
A
d
=
=
−
∈
inf{
(
,
,
(
,
,
)
(
0
,
0
,
4
),
)
:
2 21
,
2
1
−
+
≤
IJMA- 2(10), Oct.-2011, Page: 2036-2044
=
(
+
+
+
+
+
+
−
)
t
x
x
x
x
x
x
x
x
x
,
,
4
min
inf
11 12 13 21 22 23 31 32 33
t
1
=
So, for every
t
>
0
,a
0=
(
0
,
−
1
,
1
)
anda
1=
(
0
,
1
,
1
)
are t-best approximations of(
0
,
0
,
4
)
from A. Hence)
1
,
1
,
0
(
0
=
−
a
anda
1=
(
0
,
1
,
1
)
are F-best approximations ofx
=
(
0
,
0
,
4
)
from A. Therefore A is not an F -chebyshev set.Example 3.22: Let
X
=
R
3. DefineN
:
X
×
X
×
X
×
R
→
[
0
,
1
]
by
3 2 1
3 2 1 3
2 1
,
,
,
,
)
,
,
,
(
x
x
x
t
x
x
x
t
x
x
x
N
+
=
, ift
>
0
,t
∈
R
,x
1,
x
2,
x
3∈
X
,
=
1
, ift
≤
0
,t
∈
R
,x
1,
x
2,
x
3∈
X
,where
= ≤ ≤
=
3
1 3 1 3 2
1
,
,
min
j ij
i
x
x
x
x
. Then(
X
,
N
)
is a fuzzy anti-3-normed linear space.Let
A
=
{(
x
,
y
,
z
)
∈
R
3:
x
2+
y
2+
z
2≥
1
}
.Then, for every
a
=
(
x
,
y
,
z
)
∈
R
3 wherex
2+
y
2+
z
2<
1
, there exists a uniquea
0=
(
x
0,
y
0,
z
0)
∈
A
(especially in∂
A
) which is an F-best approximation of a from A.So A is an F-proximinal set.
Remark 3.23: For an arbitrary set
A
⊂
X
we shall denote by∂
A
the boundary of A, and byΜ
A the set of all elements of the F-best approximation of the elementsx
∈
X
from A.i.e.,
M
FP
A(
x
)
X x A
∈
=
.Theorem 3.24: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space andA
is a non empty subset of X, and A be a F-best proximinal set in X then
∂
A
⊂
Μ
A.Proof: If
∂
A
=
Φ
, the proof is obvious. If∂
A
≠
Φ
, leta
0∈
∂
A
,0
<
ε
<
1
andt
>
0
be arbitrary. Then thereexists
0
<
ε
′
<
1
such thatε
′
<
ε
and the cell)
2
,
,
(
a
0t
B
ε
′
contains at least one elementx
∈
X
A
. Let)
(
)
(
x
FP
Ax
A∈
π
(it exists, since by hypothesis, A is F-proximinal). Then we have,)}
2
),
(
,
,...,
,
(
),
2
,
,
,...,
,
(
max{
)
),
(
,
,...,
,
(
x
1x
2x
1a
0x
t
N
x
1x
2x
1a
0x
t
N
x
1x
2x
1x
x
t
N
n−−
π
A≤
n−−
n−−
π
A
)}
2
,
,
,...,
,
(
),
2
,
,
,...,
,
(
max{
N
x
1x
2x
n 1a
0−
x
t
N
x
1x
2x
n 1A
−
x
t
=
− −
)}
2
,
,
,...,
,
(
),
2
,
,
,...,
,
(
max{
N
x
1x
2x
n1a
0−
x
t
N
x
1x
2x
n1a
0−
x
t
≤
− −
≤
max{
ε
′
,
ε
′
}
=
ε
′
<
ε
© 2011, IJMA. All Rights Reserved 2044
ACKNOWLEDGEMENT:
The author is grateful to the referee for careful reading of the article and suggested improvements.
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[1] T. Bag and T.K. Samanta, Finite dimensional fuzzy normed linear spaces, The Journal of Fuzzy Mathematics, Vol. 11(3) (2003), 687-705.
[2] T. Bag and T.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems, Vol. 151 (2005), 513-547.
[3] T. Bag and T.K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy Sets and Systems, Vol. 159 (2008), 670-684.
[4] S.C. Cheng and J.N. Mordesen, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Cal. Math. Soc., Vol. 86 (1994), 429-436.
[5] H. Efe and C. Yildiz, some results in fuzzy compact linear operators, Journal of computational Analysis Applications, Vol. 12(1-B) (2010), 251-262.
[6] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems, Vol. 48 (1992), 239-248.
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