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BEST SIMULTANEOUS 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: 02-09-11; Accepted on: 16-09-11)
________________________________________________________________________________
ABSTRACT
The main aim of this paper is to consider the t-best simultaneous approximation in fuzzy anti-n-normed linear spaces.
We develop the theory of t-best simultaneous approximation in its quotient spaces. Then we discuss the relationship in t-proximinality and t-Chebyshevity of a given space and its quotient space.Key Words: Fuzzy anti-n-norms, simultaneous approximation, simultaneous t-proximinality, simultaneous t-chebyshevity, Quotient space.
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 theory of fuzzy sets was introduced by Zadeh [24] in 965. The idea of fuzzy norm was initiated by Katsaras in [13]. Felbin [6] defined a fuzzy norm on a linear space whose associated fuzzy metric is of Kaleva and Seikkala type [12]. Cheng and Mordeson [4] introduced an idea of a fuzzy norm on a linear space whose associated metric is Kramosil and Michalek type [14].
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 [14]. 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 [13], Felbin [6], and Bag and Samanta [1]. The concept of n-norm on a linear space has been introduced and developed by Gahler in [7,8]. Following Misiak [16], Malceski [15] and Gunawan and Mashadi [10] developed the theory of n-normed space. Narayana and Vijayabalaji [17] introduced the concept of fuzzy n-normed linear space. Vijayabalaji and Thillaigovindan [23] introduced the notion of and 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]. Vaezpour and Karimi [22], studied on the set of all t-best approximations on fuzzy normed linear spaces. Goudarzi and Vaezpour [9] considered the set of all t-best simultaneous approximation in fuzzy normed linear spaces.
In [11] 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 [18,19] Surender Reddy introduced the notion of convergent sequence and Cauchy sequence in fuzzy 2-normed linear space and fuzzy anti-n-normed linear space. In [20] Surender Reddy studied on the set of all t-best approximations on fuzzy anti-n-normed linear spaces. In [21] Surender Reddy considered the set of all t-best simultaneous approximation in fuzzy anti-2-normed linear spaces.
In the present paper, we consider the set of all t-best simultaneous approximation in fuzzy anti-n-normed linear spaces and use the concept of simultaneous t-proximinality and simultaneous t-Chebyshevity to introduce the theory of t-best simultaneous approximation in quotient spaces.
2. PRELIMINARIES:
Definition 2.1: Let
n
∈
N
and let X be a real linear space of dimension≥
n
. A real valued function•
,
•
,...,
•
onn
n
X
X
X
X
×
×
...
×
=
satisfying the following conditions---
IJMA- 2(10), Oct.-2011, Page: 1885-1897
1
nN
:x
1,
x
2,...,
x
n=
0
if and only ifx
1,
x
2,...,
x
n are linearly dependent,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
,≥
+
+
−
,
,
)
,...,
,
(
x
1x
2x
1x
y
s
t
N
n nmin
{
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
.)
IJMA- 2(10), Oct.-2011, Page: 1885-1897
)
(
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
,F
c
∈
,)
(
a
−
n
−
N
5 : For alls
,
t
∈
R
,≤
+
+
−
,
,
)
,...,
,
(
x
1x
2x
1x
y
s
t
N
n nmax
{
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
−
n
−
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
,≤
+
′
+
,...,
,
)
,...,
,
(
x
1x
2x
x
x
s
t
N
i i nmax
{
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. Definen n
n n
x
x
x
m
kt
x
x
x
k
t
x
x
x
N
,...,
,
,...,
,
)
,
,...,
,
(
2 1 2 1 2
1
+
=
, ift
>
0
,t
∈
R
,k
,
m
,
n
∈
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. In particular ifk
=
m
=
n
=
1
we have
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
, which is called the standard fuzzy anti-n-norm induced by the n-norm•
,
•
,...,
•
.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 thatr
t
x
x
x
x
x
N
(
1,
2,...,
n−1,
k−
,
)
<
,∀
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 ,
IJMA- 2(10), Oct.-2011, Page: 1885-1897
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.Definition 2.13: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. The open ballB
(
x
,
r
,
t
)
and the closed ball]
,
,
[
x
r
t
B
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 2.14: 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
(
∈
r
such thatB
(
x
,
r
,
t
)
⊂
A
for allx
∈
A
andt
>
0
.Definition 2.15: 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,...,
n1,
k, )
0
k→∞
N x x
x
−x
−
x t
=
, for allt
>
0
implies thatx
∈
A
.Corollary 2.16: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. Then N is a continuous function on
X
X
X
R
n
×
×
×
×
...
.3. t-BEST SIMULTANEOUS APPROXIMATION:
Definition 3.1: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. A subset A of X is called F-bounded if there exists0
>
t
and0
<
r
<
1
such thatN
(
x
1,
x
2,...,
x
n−1,
x
,
t
)
<
r
,∀
x
∈
A
.Definition 3.2: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space, W be a subset of Xand M be a F-bounded subsetin X. For
t
>
0
, we define)
,
,
,...,
,
(
sup
inf
)
,
,
(
M
W
t
N
x
1x
2x
1m
w
t
d
nM m W w
−
=
−∈
∈ .
An element
w
0∈
W
is called a t-best simultaneous approximation to M from W if fort
>
0
,)
,
,
,...,
,
(
sup
)
,
,
(
M
W
t
N
x
1x
2x
1m
w
0t
d
nM m
−
=
−∈
.
The set of all t-best simultaneous approximations to M from W will be denoted by
S
Wt(
M
)
and we have,)}
,
,
(
)
,
,
,...,
,
(
sup
:
{
)
(
M
w
W
N
x
1x
2x
1m
w
t
d
M
W
t
S
nM m t
W
=
∈
−−
=
∈
Definition 3.3: Let W be a subset of a fuzzy anti-n-normed linear space
(
X
,
N
)
then W is called a simultaneous t -proximinal subset of X if for each F-bounded set M in X, there exists at least one t-best simultaneous approximation from W to M. Also W is called a simultaneous t-Chebyshev subset of X if for each F-bounded set M in X, there exists a unique t-best simultaneous approximation from W to M.Definition 3.4: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space. A subset E of X is said to be convex ifE
y
x
+
∈
−
λ
)
λ
1
(
wheneverx
,
y
∈
E
and0
<
λ
<
1
.IJMA- 2(10), Oct.-2011, Page: 1885-1897
Theorem 3.6: Suppose that W is a subset of a fuzzy anti-n-normed linear space
(
X
,
N
)
and M is F-bounded in X.Then
S
Wt(
M
)
is a F-bounded subset of X and if W is convex and is a closed subset of X thenS
Wt(
M
)
is closed and isconvex for each F-bounded subspace M of X.
Proof: Since M is F-bounded, there exists
t
>
0
and0
<
r
<
1
such thatN
(
x
1,
x
2,...,
x
n−1,
x
,
t
)
<
r
, for allM
x
∈
. Ifw
S
t(
M
)
W∈
, then
sup
N
(
x
1,
x
2,...,
x
n1,
m
w
,
t
)
d
(
M
,
W
,
t
)
M m=
−
− ∈ .Now, for all
m
∈
M
andw
∈
S
Wt(
M
)
,)
2
,
,
,...,
,
(
)
2
,
,
,...,
,
(
x
1x
2x
1w
t
N
x
1x
2x
1w
m
m
t
N
n−=
n−−
+
≤
max{
N
(
x
1,
x
2,...,
x
n−1,
w
−
m
,
t
),
N
(
x
1,
x
2,...,
x
n−1,
m
,
t
)}
sup
max{
N
(
x
1,
x
2,...,
x
n1,
w
m
,
t
),
N
(
x
1,
x
2,...,
x
n1,
m
,
t
)}
M m − − ∈−
≤
max{
sup
N
(
x
1,
x
2,...,
x
1,
w
m
,
t
),
sup
N
(
x
1,
x
2,...,
x
n 1,
m
,
t
)}
M m n M m − ∈ − ∈−
≤
≤
max{
d
(
M
,
W
,
t
),
r
}
≤
r
0, for some0
<
r
0<
1
.Then
S
t(
M
)
W is F-bounded. Suppose that W is convex and is a closed subset of X. We show that
S
(
M
)
tW is convex
and closed. Let
x
,
y
∈
S
Wt(
M
)
and0
<
λ
<
1
. Since W is convex, there existsz
λ∈
W
such thaty
x
z
λ=
λ
+
(
1
−
λ
)
, for each0
<
λ
<
1
. Now fort
>
0
we have,)
,
,
,...,
,
(
sup
)
,
)
)
1
(
(
,
,...,
,
(
sup
N
x
1x
2x
1x
y
m
t
N
x
1x
2x
n1z
m
t
M m n M m
−
=
−
−
+
− ∈ − ∈ λλ
λ
≥
d
(
M
,
W
,
t
)
.On the other hand, for a given
t
>
0
, take the natural number n such thatt
>
n1, we have)
,
)
(
,
,...,
,
(
sup
)
,
)
)
1
(
(
,
,...,
,
(
sup
N
x
1x
2x
1x
y
m
t
N
x
1x
2x
n1x
y
y
m
t
M m n M m
−
+
−
=
−
−
+
− ∈ − ∈λ
λ
λ
sup
max{
(
1,
2,...,
1,
,
1
),
(
1,
2,...,
1,
,
1
)}
n
t
m
y
x
x
x
N
n
y
x
x
x
x
N
n nM m
−
−
−
≤
− − ∈λ
max{
(
1,
2,...,
1,
,
1
),
sup
(
1,
2,...,
1,
,
1
)}
n
t
m
y
x
x
x
N
n
y
x
x
x
x
N
n M mn
−
−
−
≤
−∈ −
λ
lim
sup
(
1,
2,...,
1,
,
1
)
d
(
M
,
W
,
t
)
n
t
m
y
x
x
x
N
n M m n=
−
−
≤
− ∈ ∞ → .So
S
Wt(
M
)
is convex. Finally let{
w
n}
⊂
S
Wt(
M
)
and suppose{
w
n}
converges to some w in X. Since{
w
n}
⊂
W
and W is closed sow
∈
W
. Therefore by Corollary 2.16, fort
>
0
we have
sup
N
(
x
1,
x
2,...,
x
1,
m
w
,
t
)
sup
N
(
x
1,
x
2,...,
x
1,
lim
w
nm
,
t
)
n n M m n M m−
=
−
∞ → − ∈ − ∈
lim
sup
N
(
x
1,
x
2,...,
x
n 1,
w
nm
,
t
)
d
(
M
,
W
,
t
)
Mm
n
−
=
=
−∈ ∞
→ .
Theorem 3.7: The following assertions are hold for
t
>
0
,(i)
d
(
M
+
x
,
W
+
x
,
t
)
=
d
(
M
,
W
,
t
)
,∀
x
∈
X
,(ii)
(
,
,
)
(
,
,
)
λ
λ
λ
M
W
t
d
M
W
t
IJMA- 2(10), Oct.-2011, Page: 1885-1897
(iii)
S
Wt+x(
M
+
x
)
=
S
Wt(
M
)
+
x
,∀
x
∈
X
,(iv)
S
λλWt(
λ
M
)
=
λ
S
Wt(
M
)
+
x
,∀
λ
∈
C
,Proof: (i)
d
(
M
x
,
W
x
,
t
)
inf
sup
N
(
x
1,
x
2,...,
x
n1,
(
m
x
)
(
w
x
),
t
)
Mm W w
+
−
+
=
+
+
−∈ ∈
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
m
w
,
t
)
d
(
M
,
W
,
t
)
Mm W w
=
−
=
−∈ ∈
(ii) Clearly equality holds for
λ
=
0
, so suppose thatλ
≠
0
. Then,
d
(
M
,
W
,
t
)
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
(
m
w
),
t
)
Mm W w
−
=
−∈
∈
λ
λ
λ
inf
sup
(
1,
2,...,
1,
,
)
(
,
,
)
λ
λ
t
W
M
d
t
w
m
x
x
x
N
nM m W
w
−
=
=
−∈ ∈
(iii)
x
+
W
∈
S
Wt+x(
M
+
x
)
if and only if,)
,
,
(
)
,
,
,...,
,
(
sup
N
x
1x
2x
n1m
x
w
x
t
d
M
x
W
x
t
x M x m
+
+
=
−
−
+
− +∈ +
and by (i), the above equality holds if and only if,
sup
N
(
x
1,
x
2,...,
x
n 1,
m
w
,
t
)
d
(
M
,
W
,
t
)
Mm
=
−
− ∈for all
w
∈
W
and this shows thatw
∈
S
Wt(
M
)
. Sox
+
w
∈
S
Wt(
M
)
+
x
.(iv)
y
0∈
S
λλWt(
λ
M
)
if and only ify
0∈
λ
W
and,
d
(
W
,
M
,
t
)
sup
N
(
x
1,
x
2,...,
x
n1,
y
0m
,
t
)
Mm
λ
λ
λ
λ
λ
λ λ
−
=
−∈
sup
(
,
,...,
,
0,
)
1 2
1
m
t
y
x
x
x
N
nM m
−
=
−∈
λ
But by (ii), we have
d
(
λ
M
,
λ
W
,
λ
t
)
=
d
(
W
,
M
,
t
)
. So we havey
∈
W
λ
0
and
)
,
,
,...,
,
(
sup
)
,
,
(
01 2
1
m
t
y
x
x
x
N
t
W
M
d
nM m
−
=
−∈
λ
or equivalently
y
0S
t(
M
)
W∈
λ
and the proof is completed.Corollary 3.8: Let A be a nonempty subset of a fuzzy anti-n-normed linear space
(
X
,
N
)
then the following statements are hold.(i) A is simultaneous t-proximinal (respectively simultaneous t-Chebyshev) if and only if A+y is simultaneous t-proximinal (respectively simultaneous t-Chebyshev), for each
y
∈
X
,(ii) A is simultaneous t-proximinal (respectively simultaneous t-Chebyshev) if and only if
α
A
is simultaneoust
α
-proximinal (respectively simultaneousα
t
-Chebyshev), for eachα
∈
C
.Corollary 3.9: Let Abe a nonempty subspace of a fuzzy anti-n-normed linear space X and M be a F-bounded subset of X. Then for
t
>
0
,(i)
d
(
A
,
M
+
y
,
t
)
=
d
(
A
,
M
,
t
)
,∀
y
∈
A
, (ii)S
tA(
M
+
y
)
=
S
At(
M
)
+
y
,∀
y
∈
A
,(iii)
d
(
A
,
α
M
,
α
t
)
=
d
(
A
,
M
,
t
)
, for0
≠
α
∈
C
,(iv)
S
(
M
)
S
t(
M
)
A tA
α
α
α
IJMA- 2(10), Oct.-2011, Page: 1885-1897
4. SIMULTANEOUS t-PROXIMINALITY AND SIMULTANEOUS t-CHEBYSHEVITY IN QUOTIENT SPACES:
In this section we give characterization of simultaneous t-proximinality and simultaneous t-Chebyshevity in quotient spaces.
Definition 4.1: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space, M be a linear manifold in X and letM
X
X
Q
:
→
be the natural mapQx
=
x
+
M
. We define}
:
)
,
,
,...,
,
(
inf{
)
,
,
,...,
,
(
x
1x
2x
1x
M
t
N
x
1x
2x
1x
y
t
y
M
N
n−+
=
n−+
∈
,t
>
0
Theorem 4.2: If M is a closed subspace of a fuzzy anti-n-normed linear space
(
X
,
N
)
and)
,
,
,...,
,
(
x
1x
2x
1x
M
t
N
n−+
is defined as above then(a) N is a fuzzy anti-n-norm on
X
M
.(b)
N
(
x
1,
x
2,...,
x
n−1,
Qx
,
t
)
≤
N
(
x
1,
x
2,...,
x
n−1,
x
,
t
)
.(c) If
(
X
,
N
)
is a fuzzy anti-n-Banach space then so is(
X
M
,
N
)
.Proof: (a) It is clear that
N
(
x
1,
x
2,...,
x
n−1,
x
+
M
,
t
)
=
1
fort
≤
0
.Let
N
(
x
1,
x
2,...,
x
n−1,
x
+
M
,
t
)
=
0
fort
>
0
. By definition there is a sequence{
x
k}
in M such thatN
(
x
1,
x
2,...,
x
n−1,
x
+
x
k,
t
)
→
0
. Sox
+
x
k→
0
or equivalentlyx
k→
(
−
x
)
and since M is closed soM
x
∈
andx
+
M
=
M
, the zero element ofX
M
. On the other hand we have,)
,
)
(
,
,...,
,
(
)
),
(
)
(
,
,...,
,
(
x
1x
2x
1x
M
y
M
t
N
x
1x
2x
1x
y
M
t
N
n−+
+
+
=
n−+
+
≤
N
(
x
1,
x
2,...,
x
n−1,
(
x
+
m
)
+
(
y
+
n
),
t
)
≤
max{
N
(
x
1,
x
2,...,
x
n−1,
x
+
m
,
t
1),
N
(
x
1,
x
2,...,
x
n−1,
y
+
n
,
t
2)}
for
m
,
n
∈
M
,x
1,
x
2,...,
x
n−1,
x
,
y
∈
X
andt
1+
t
2=
t
. Now if we take infimum on both sides, we have,
≤
+
+
+
−
,
(
)
(
),
)
,...,
,
(
x
1x
2x
1x
M
y
M
t
N
nmax{
N
(
x
1,
x
2,...,
x
n−1,
x
+
M
,
t
1),
N
(
x
1,
x
2,...,
x
n−1,
y
+
M
,
t
2)}
.Also we have,
N
(
x
1,
x
2,...,
x
n−1,
α
(
x
+
M
),
t
)
=
N
(
x
1,
x
2,...,
x
n−1,
α
x
+
M
,
t
)
=
inf{
N
(
x
1,
x
2,...,
x
n−1,
α
x
+
α
y
,
t
)
:
y
∈
M
}
=
inf{
N
(
x
1,
x
2,...,
x
n−1,
x
+
y
,
t
)
:
y
∈
M
}
α
(
1,
2,...,
1,
,
)
α
t
M
x
x
x
x
N
n+
=
−and the remaining properties are obviously true. Therefore N is a fuzzy anti-n-norm on
X
M
.(b) We have,
N
(
x
1,
x
2,...,
x
n−1,
Qx
,
t
)
=
N
(
x
1,
x
2,...,
x
n−1,
x
+
M
,
t
)
=
inf{
N
(
x
1,
x
2,...,
x
n−1,
x
+
y
,
t
)
:
y
∈
M
}
≤
N
(
x
1,
x
2,...,
x
n−1,
x
,
t
)
(c) Let
{
y
k+
M
}
be a Cauchy sequence inX
M
. Then there existsε
k>
0
such thatε
k→
0
andk k
k
n
y
M
y
M
t
x
x
x
IJMA- 2(10), Oct.-2011, Page: 1885-1897
But
N
(
x
1,
x
2,...,
x
n−1,
(
y
1−
y
2)
+
M
,
t
)
≤
ε
1. Therefore,
N
(
x
1,
x
2,...,
x
n−1,
y
1−
(
y
2−
z
2),
t
)
≤
max{
ε
1,
ε
1}
=
ε
1.Now suppose
z
k−1 has been chosen,z
k∈
M
can be chosen such that}
),
,
)
(
,
,...,
,
(
max{
)
),
(
)
(
,
,...,
,
(
x
1x
2x
n−1y
k−1+
z
k−1−
y
k+
z
kt
≤
N
x
1x
2x
n−1y
k−1−
y
k+
M
t
k−1N
ε
andtherefore,
N
(
x
1,
x
2,...,
x
n−1,
(
y
k−1+
z
k−1)
−
(
y
k+
z
k),
t
)
≤
max{
ε
k−1,
ε
k−1}
=
ε
k−1.Thus,
{
y
k+
z
k}
is Cauchy sequence in X. Since X is complete, there is any
0 in X such thaty
k+
z
k→
y
0 in X. On the other handy
k+
M
=
Q
(
y
k+
z
k)
→
Q
(
y
0)
=
y
0+
M
. Therefore every Cauchy sequence{
y
k+
M
}
is convergent inX
M
and soX
M
is complete and(
X
M
,
N
)
is a fuzzy anti-n-Banach space.Definition 4.3: Let A be a nonempty set in a fuzzy anti-n-normed linear space
(
X
,
N
)
. Forx
∈
X
andt
>
0
, we shall denote the set of all elements of t-best approximation to x from A byP
At(
x
)
;i.e.,
P
At(
x
)
=
{
y
∈
A
:
d
(
A
,
x
,
t
)
=
N
(
x
1,
x
2,...,
x
n−1,
y
−
x
,
t
)}
.where,
d
(
A
,
x
,
t
)
inf{
N
(
x
1,
x
2,...,
x
1,
y
x
,
t
)
:
y
A
}
inf
N
(
x
1,
x
2,...,
x
n1,
y
x
,
t
)
Ay
n
−
∈
=
−
=
−∈
− .
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.Lemma 4.4: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space and M be a t-proximinal subspace of X. For eachnonempty F-bounded set S in X and
t
>
0
,
d
(
S
,
M
,
t
)
sup
inf
N
(
x
1,
x
2,...,
x
n1,
s
m
,
t
)
Mm S s
−
=
−∈ ∈
Proof: Since M is t-proximinal it follows that for each
s
∈
S
there existsm
s∈
P
Mt(
S
)
such that fort
>
0
,)
,
,
,...,
,
(
inf
)
,
,
,...,
,
(
x
1x
2x
1s
m
t
N
x
1x
2x
1s
m
t
N
nM m s
n−
−
=
∈ −−
.So,
d
(
S
,
M
,
t
)
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
Ss M m
−
=
−∈ ∈
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
s,
t
)
Ss
−
≤
−∈
sup
inf
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
Mm S s
−
≤
−∈ ∈
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
d
(
S
,
M
,
t
)
Ss M m
=
−
≤
−∈ ∈
This implies that,
d
(
S
,
M
,
t
)
sup
inf
N
(
x
1,
x
2,...,
x
n1,
s
m
,
t
)
Mm S s
−
=
−∈ ∈
.
Example 4.5: Let
(
X
=
R
2,
•
,
•
,...,
•
)
be a anti-n-normed linear space and consider
(
X
,
N
)
as its standardinduced fuzzy anti-n-normed linear space (Example 2.7). A nonempty subset S of X is F-bounded if and only if S is
bounded in
(
X
,
•
,
•
,...,
•
)
. If we takeM
=
R
we can easily prove that M is proximinal in(
X
,
•
,
•
,...,
•
)
.Lemma 4.6: Let
(
X
,
N
)
be a fuzzy anti-n-normed linear space, M be a t-proximinal subspace of X and S be an arbitrary subset of X then the following assertions are equivalent:(i) S is a F-bounded subset of X.
IJMA- 2(10), Oct.-2011, Page: 1885-1897
Proof: Suppose that S be a F-bounded subset of X. Then there exist
t
>
0
,0
<
r
<
1
such that,r
t
x
x
x
x
N
(
1,
2,...,
n−1,
,
)
<
, for allx
∈
S
. But,
N
x
x
x
x
M
t
N
x
x
x
nx
y
t
N
x
x
x
nx
t
r
M y
n
+
=
−+
≤
−≤
∈
−
,
,
)
inf
(
,
,...,
,
,
)
(
,
,...,
,
,
)
,...,
,
(
1 2 1 1 2 1 1 2 1 .So,
(
i
)
(
ii
)
is proved. Now to prove that(
ii
)
(
i
)
. LetS
M
be a F-bounded subset ofX
M
. Since M is t-proximinal, then for each
s
∈
S
there existsm
s∈
M
such thatm
s∈
P
Mt(
S
)
. So for eachs
∈
S
,
N
(
x
1,
x
2,...,
x
1,
s
m
,
t
)
inf
N
(
x
1,
x
2,...,
x
n1,
s
m
,
t
)
Mm s
n−
−
=
∈ −−
(1)Now from Lemma 4.4, we conclude that for
t
>
0
,
sup
N
(
x
1,
x
2,...,
x
1,
s
m
,
t
)
sup
inf
N
(
x
1,
x
2,...,
x
n1,
s
m
,
t
)
M m S s s n S s−
=
−
− ∈ ∈ − ∈
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
S s M m−
=
− ∈ ∈ .Then for
0
<
r
<
1
such thatN
x
x
x
ns
m
st
r
S s
≤
−
− ∈)
,
,
,...,
,
(
sup
1 2 1 andt
>
0
there existsm
r∈
M
such that,
sup
(
1,
2,...,
1,
−
,
)
≤
sup
(
1,
2,...,
−1,
−
,
)
−
≤
0
∈ − ∈
r
t
m
s
x
x
x
N
t
m
s
x
x
x
N
n sS s r n S s .
So by (1), for all
s
∈
S
we have,)
,
,
,...,
,
(
)
,
,
,...,
,
(
x
1x
2x
1s
t
N
x
1x
2x
1s
m
m
t
N
n−=
n−−
r+
r
)}
2
,
,
,...,
,
(
),
2
,
,
,...,
,
(
max{
N
x
1x
2x
n−1s
−
m
rt
N
x
1x
2x
n−1m
rt
≤
)}
2
,
,
,...,
,
(
),
2
,
,
,...,
,
(
max{
sup
N
x
1x
2x
n1s
m
rt
N
x
1x
2x
n1m
rt
S s − − ∈
−
≤
)}
2
,
,
,...,
,
(
),
)
2
,
,
,...,
,
(
(sup
max{
N
x
1x
2x
n 1s
m
st
r
N
x
1x
2x
n1m
rt
S
s∈ − −
−
−
≤
)}
2
,
,
,...,
,
(
),
)
2
,
,
,...,
,
(
inf
(sup
max{
N
x
1x
2x
n 1s
m
t
r
N
x
1x
2x
n1m
rt
M m S s − − ∈ ∈
−
−
=
)}
2
,
,
,...,
,
(
),
)
2
,
,
,...,
,
(
(sup
max{
N
x
1x
2x
n 1s
M
t
r
N
x
1x
2x
n1m
rt
S s − − ∈
−
+
≤
. (2)Since
S
M
is F-bounded, by its definition we can find0
<
r
0<
1
such that in the right hand side of (2) be less than or equal tor
0 and this completes the proof.Lemma 4.7: Let M be a t-proximinal subspace of a fuzzy anti-n-normed linear space
(
X
,
N
)
andW
⊇
M
a subspace of X. Let K be F-bounded in X. Ifw
0∈
S
Wt(
K
)
, then
w
0M
S
t(
K
M
)
MW
∈
+
.Proof: Since K is F-bounded by Lemma 4.6,
K
M
is F-bounded inX
M
. Assume thatw
0∈
S
Wt(
K
)
and)
(
0
M
S
K
M
w
+
∉
Wt M . Thus there existsw
′
∈
M
such that fort
>
0
,)
),
(
,
,...,
,
(
sup
)
),
(
,
,...,
,
(
sup
N
x
1x
2x
1k
w
M
t
N
x
1x
2x
n1k
w
0M
t
K k n K k
+
−
<
+
′
−
− ∈ − ∈
