BEST SIMULTANEOUS APPROXIMATION IN
FUZZY n-NORMED LINEAR SPACES
B. Surender Reddy*
Department of Mathematics, PGCS, Saifabad, Osmania University, Hyderabad-500004, AP, INDIA E-mail: [email protected]
(Received on: 07-10-11; Accepted on: 20-10-11)
________________________________________________________________________________________________
ABSTRACT
The main aim of this paper is to consider the t-best simultaneous approximation in fuzzy n-normed linear spaces. We develop the theory of 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: t-best simultaneous approximation, t-proximinality, t-Chebyshevity, Quotient spaces.
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. Theory of Fuzzy sets was introduced by Zadeh [20] in 1965. The idea of fuzzy norm was initiated by Katsaras in [11]. Felbin [5] defined a fuzzy norm on a linear space whose associated fuzzy metric is of Kaleva and Seikkala type [10]. Cheng and Mordeson [4] introduced an idea of a fuzzy norm on a linear space whose associated metric is Kramosil and Michalek type [12].
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 [12]. 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 [11], Felbin [5], 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 [6,7]. Following Misiak [14], Malceski [13] and Gunawan [9] developed the theory of n-normed linear space. Narayana and Vijayabalaji [15] introduced the concept of fuzzy n-normed linear space. Vijayabalaji and Thillaigovindan [19] introduced the notion of convergent sequence and Cauchy sequences in fuzzy n-normed linear space and studied the completeness of the fuzzy n-normed linear space. Vaezpour and Karimi [18] introduced the concept of t-best approximation in fuzzy normed linear spaces. Surender Reddy [16] introduced the concept of t-best approximation in fuzzy 2-normed linear spaces. Recently Goudarzi and Vaezpour [8] considered the set of all t-best simultaneous approximation in fuzzy normed linear spaces and used the concept of simultaneous t-proximinality and simultaneous t-Chebyshevity to introduce the theory of t-best simultaneous approximation in quotient spaces. Surender Reddy [17] considered the set of all t-best simultaneous approximation in fuzzy 2-normed linear spaces and used the concept of simultaneous t -proximinality and simultaneous t-Chebyshevity to introduce the theory of t-best simultaneous approximation in quotient spaces.
In this paper, we consider the set of all t-best simultaneous approximation in fuzzy n-normed linear spaces and we 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 conditions1
nN
:x
1,
x
2,...,
x
n=
0
if and only ifx
1,
x
2,...,
x
n are linearly dependent,---
IJMA- 2(11), Nov.-2011, Page: 2090-2100
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).Remark 2.4: From
(
n
−
N
3)
, it follows that in F-n-NLS,)
(
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
∈
,)
(
n
−
N
5 : For alls
,
t
∈
R
,N
(
x
1,
x
2,...,
x
i+
x
′
i,...,
x
n,
s
+
t
)
≥
min
{
N
(
x
1,
x
2,...,
x
i,...,
x
n,
s
),
N
(
x
1,
x
2,...,
x
′
i,...,
x
n,
t
)
}
.Example 2.5: Let
(
X
,
•
,
•
,...,
•
)
be a n-normed linear space. Definen n
n n
x
x
x
m
kt
kt
t
x
x
x
N
,...,
,
)
,
,...,
,
(
2 1 2
1
+
=
, ift
>
0
,t
∈
R
,k
,
m
,
n
∈
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. In particular ifk
=
m
=
n
=
1
we haven 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
.IJMA- 2(11), Nov.-2011, Page: 2090-2100
Definition 2.6: A sequence
{
x
k}
in a fuzzy n-normed linear space(
X
,
N
)
is said to be converges tox
∈
X
if given0
>
t
,0
<
r
<
1
, there exists an integern
0∈
N
such thatr
t
x
x
x
x
x
N
(
1,
2,...,
n−1,
k−
,
)
>
1
−
,∀
k
≥
n
0.Theorem 2.7: In a fuzzy n-normed linear space
(
X
,
N
)
, a sequence{
x
k}
converges tox
∈
X
if and only1
)
,
,
,...,
,
(
lim
1 2 −1−
=
∞
→
N
x
x
x
nx
kx
t
k ,
∀
t
>
0
.Definition 2.8: Let
(
X
,
N
)
be a fuzzy n-normed linear space. Let{
x
k}
be a sequence in X then{
x
k}
is said to be a Cauchy sequence iflim
(
1,
2,...,
−1,
+−
,
)
=
1
∞
→
N
x
x
x
nx
k px
kt
k ,
∀
t
>
0
andp
=
1
,
2
,
3
,...
.Definition 2.9: A fuzzy n-normed linear space
(
X
,
N
)
is said to be complete if every Cauchy sequence in X is convergent.Definition 2.10: A complete fuzzy n-normed linear space
(
X
,
N
)
is called a fuzzy n-Banach space.Definition 2.11: Let
(
X
,
N
)
be a fuzzy 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
)
>
1
−
r
}
B
[
x
,
r
,
t
]
=
{
y
∈
X
:
N
(
x
1,
x
2,...,
x
n−1,
x
−
y
,
t
)
≥
1
−
r
}
Definition 2.12: Let
(
X
,
N
)
be a fuzzy 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 2.13: Let
(
X
,
N
)
be a fuzzy 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,
−
,
)
=
1
∞
→
N
x
x
x
nx
kx
t
k , for all
t
>
0
implies thatx
∈
A
.Corollary 2.14: Let
(
X
,
N
)
be a fuzzy n-normed linear space. Then N is a continuous function onR
X
X
X
n
×
×
×
×
...
.3. t-BEST SIMULTANEOUS APPROXIMATION:
Definition 3.1: Let
(
X
,
N
)
be a fuzzy 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
)
>
1
−
r
, for allx
∈
A
.Definition 3.2: Let
(
X
,
N
)
be a fuzzy n-normed linear space, W be a subset of X and M be a F-bounded subset in X.For
t
>
0
, we define)
,
,
,...,
,
(
inf
sup
)
,
,
(
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
,)
,
,
,...,
,
(
inf
)
,
,
(
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)}
,
,
(
)
,
,
,...,
,
(
inf
:
{
)
(
M
w
W
N
x
1x
2x
1m
w
t
d
M
W
t
S
nM m t
W
=
∈
−−
=
∈
IJMA- 2(11), Nov.-2011, Page: 2090-2100
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 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
.Lemma 3.5: Every open ball in a fuzzy n-normed linear space
(
X
,
N
)
is convex.Theorem 3.6: Suppose that W is a subset of a fuzzy n-normed linear space
(
X
,
N
)
and M is F-bounded in X. Then)
(
M
S
tW is a F-bounded subset of X and if W is convex and is a closed subset of X then
S
(
M
)
tW is closed and is
convex 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
)
>
1
−
r
, for allM
x
∈
. Ifw
∈
S
Wt(
M
)
, then
inf
N
(
x
1,
x
2,...,
x
n 1,
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−−
+
≥
min{
N
(
x
1,
x
2,...,
x
n−1,
w
−
m
,
t
),
N
(
x
1,
x
2,...,
x
n−1,
m
,
t
)}
inf
min{
N
(
x
1,
x
2,...,
x
n1,
w
m
,
t
),
N
(
x
1,
x
2,...,
x
n1,
m
,
t
)}
M
m∈ −
−
−≥
min{
inf
N
(
x
1,
x
2,...,
x
1,
w
m
,
t
),
inf
N
(
x
1,
x
2,...,
x
n 1,
m
,
t
)}
M m n
M
m∈ −
−
∈ −≥
≥
min{
d
(
M
,
W
,
t
),
(
1
−
r
)}
≥
(
1
−
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,)
,
,
,...,
,
(
inf
)
,
)
)
1
(
(
,
,...,
,
(
inf
N
x
1x
2x
1x
y
m
t
N
x
1x
2x
n1z
m
t
M m n M m
−
=
−
−
+
− ∈ −∈
λ
λ
λ
≤
d
(
M
,
W
,
t
)
.On the other hand, for a given
t
>
0
, take the natural number n such thatt
>
n1, we have)
,
)
(
,
,...,
,
(
inf
)
,
)
)
1
(
(
,
,...,
,
(
inf
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
−
+
−
inf
min{
(
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
min{
(
1,
2,...,
1,
,
1
),
inf
(
1,
2,...,
1,
,
1
)}
n
t
m
y
x
x
x
N
n
y
x
x
x
x
N
n M mn
−
−
−
=
−∈ −
λ
lim
inf
(
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. SinceW
w
n}
⊂
{
and W is closed sow
∈
W
. Therefore by Corollary 2.14, fort
>
0
we have
inf
N
(
x
1,
x
2,...,
x
1,
m
w
,
t
)
inf
N
(
x
1,
x
2,...,
x
1,
lim
w
nm
,
t
)
n n M m n M
m∈ −
−
=
∈ − →∞−
lim
inf
N
(
x
1,
x
2,...,
x
n1,
w
nm
,
t
)
d
(
M
,
W
,
t
)
M m
n
−
=
=
−∈ ∞
IJMA- 2(11), Nov.-2011, Page: 2090-2100 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
d
=
,∀
λ
∈
C
,(iii)
S
M
x
S
tM
x
W t
x
W+
(
+
)
=
(
)
+
,∀
x
∈
X
, (iv)S
λλWt(
λ
M
)
=
λ
S
Wt(
M
)
+
x
,∀
λ
∈
C
,Proof: (i)
d
(
M
x
,
W
x
,
t
)
sup
inf
N
(
x
1,
x
2,...,
x
n 1,
(
m
x
)
(
w
x
),
t
)
M m W w
+
−
+
=
+
+
−∈ ∈
sup
inf
N
(
x
1,
x
2,...,
x
n 1,
m
w
,
t
)
d
(
M
,
W
,
t
)
M m W w
=
−
=
−∈ ∈
(ii) Clearly equality holds for
λ
=
0
, so suppose thatλ
≠
0
. Then,
d
(
M
,
W
,
t
)
sup
inf
N
(
x
1,
x
2,...,
x
n 1,
(
m
w
),
t
)
M m W w
−
=
−∈ ∈
λ
λ
λ
sup
inf
(
1,
2,...,
1,
,
)
(
,
,
)
λ
λ
t
W
M
d
t
w
m
x
x
x
N
nM m W w
=
−
=
−∈ ∈
(iii)
x
W
S
t(
M
x
)
x
W
+
∈
+
+ if and only if,)
,
,
(
)
,
,
,...,
,
(
inf
N
x
1x
2x
n 1m
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,
)
,
,
(
)
,
,
,...,
,
(
inf
N
x
1x
2x
n 1m
w
t
d
M
W
t
M m
=
−
− ∈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
)
inf
N
(
x
1,
x
2,...,
x
n1,
y
0m
,
t
)
M
m
λ
λ
λ
λ
λ
λ
λ
−
=
−∈
inf
N
(
x
1,
x
2,...,
x
n 1,
y
0m
,
t
)
M 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
)
,
,
,...,
,
(
inf
)
,
,
(
M
W
t
N
x
1x
2x
1y
0m
t
d
nM m
−
=
−∈
λ
or equivalently(
)
0
S
M
y
tW
∈
λ
and the proof is completed.Corollary 3.8: Let A be a nonempty subset of a fuzzy 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 simultaneousα
t
-proximinal (respectively simultaneous
α
t
-Chebyshev), for eachα
∈
C
.Corollary 3.9: Let A be a nonempty subspace of a fuzzy 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
,IJMA- 2(11), Nov.-2011, Page: 2090-2100
(iii)
d
(
A
,
α
M
,
α
t
)
=
d
(
A
,
M
,
t
)
, for0
≠
α
∈
C
,(iv)
S
Aαt(
α
M
)
=
α
S
At(
M
)
, for0
≠
α
∈
C
.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 n-normed linear space, M be a linear manifold in X and letQ
:
X
→
X
M
be the natural map
Qx
=
x
+
M
. We define}
:
)
,
,
,...,
,
(
sup{
)
,
,
,...,
,
(
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 n-normed linear space
(
X
,
N
)
andN
(
x
1,
x
2,...,
x
n−1,
x
+
M
,
t
)
is defined as above then
(a) N is a fuzzy 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 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
)
=
0
fort
≤
0
.Let
N
(
x
1,
x
2,...,
x
n−1,
x
+
M
,
t
)
=
1
fort
>
0
. By definition there is a sequence{
x
k}
in M such that1
)
,
,
,...,
,
(
x
1x
2x
−1x
+
x
t
→
N
n k . Sox
+
x
k→
0
or equivalentlyx
k→
(
−
x
)
and since M is closed sox
∈
M
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
)
≥
min{
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 supremum on both sides, we have)
),
(
)
(
,
,...,
,
(
x
1x
2x
1x
M
y
M
t
N
n−+
+
+
≥
min{
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
)
=
sup{
N
(
x
1,
x
2,...,
x
n−1,
α
x
+
α
y
,
t
)
:
y
∈
M
}
=
sup{
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 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
)
=
sup{
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(11), Nov.-2011, Page: 2090-2100
N
(
x
1,
x
2,...,
x
n−1,
y
1−
(
y
2−
z
2),
t
)
≥
min{
N
(
x
1,
x
2,...,
x
n−1,
(
y
1−
y
2)
+
M
,
t
),
(
1
−
ε
1)}
.But
N
(
x
1,
x
2,...,
x
n−1,
(
y
1−
y
2)
+
M
,
t
)
≥
(
1
−
ε
1)
. Therefore,
N
(
x
1,
x
2,...,
x
n−1,
y
1−
(
y
2−
z
2),
t
)
≥
min{(
1
−
ε
1),
(
1
−
ε
1)}
=
(
1
−
ε
1)
.Now suppose
z
k−1 has been chosen,z
k∈
M
can be chosen such that)}
1
(
),
,
)
(
,
,...,
,
(
min{
)
),
(
)
(
,
,...,
,
(
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
and therefore,
)
1
(
)}
1
(
),
1
min{(
)
),
(
)
(
,
,...,
,
(
x
1x
2x
n−1y
k−1+
z
k−1−
y
k+
z
kt
≥
−
k−1−
k−1=
−
k−1N
ε
ε
ε
.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 n-Banach space.Definition 4.3: Let A be a nonempty set in a fuzzy n-normed linear space
(
X
,
N
)
. Forx
∈
X
andt
>
0
, we shalldenote the set of all elements of t-best approximation to x from A by
P
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
)
sup{
N
(
x
1,
x
2,...,
x
1,
y
x
,
t
)
:
y
A
}
sup
N
(
x
1,
x
2,...,
x
n1,
y
x
,
t
)
A y
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 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
)
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
M m 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
,)
,
,
,...,
,
(
sup
)
,
,
,...,
,
(
x
1x
2x
1s
m
t
N
x
1x
2x
1s
m
t
N
nM m s
n
−
=
−−
∈
− .
So,
d
(
S
,
M
,
t
)
sup
inf
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
S s M m
−
=
−∈ ∈
inf
N
(
x
1,
x
2,...,
x
n 1,
s
m
s,
t
)
S s
−
≥
−∈
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
M m S s
−
=
−∈ ∈
sup
inf
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
d
(
S
,
M
,
t
)
S s M m
=
−
≥
−∈ ∈
This implies that,
d
(
S
,
M
,
t
)
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
M m S s
−
=
−∈
∈ .
Example 4.5: Let
(
X
=
R
n,
•
,
•
,...,
•
)
be n-normed linear space and consider
(
X
,
N
)
as its standard inducedfuzzy n-normed linear space (Example 2.5). 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 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(11), Nov.-2011, Page: 2090-2100
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,
,
)
>
1
−
, for allx
∈
S
. But,r
t
x
x
x
x
N
t
y
x
x
x
x
N
t
M
x
x
x
x
N
n nM y
n
+
=
−+
≥
−≥
−
∈
−
,
,
)
sup
(
,
,...,
,
,
)
(
,
,...,
,
,
)
1
,...,
,
(
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
)
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
M m s
n
−
=
−−
∈
− (1)
Now from Lemma 4.4, we conclude that for
t
>
0
,
inf
N
(
x
1,
x
2,...,
x
1,
s
m
,
t
)
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
s
m
,
t
)
M m S s s n S s
−
=
−
− ∈ ∈ − ∈
sup
inf
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
≥
−
− ∈(
,
,...,
,
,
)
inf
1 2 1 andt
>
0
there existsm
r∈
M
such that
inf
(
1,
2,...,
1,
−
,
)
≥
inf
(
1,
2,...,
−1,
−
,
)
−
≥
0
∈ −
∈S
N
x
x
x
ns
m
rt
s SN
x
x
x
ns
m
st
r
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
,
,
,...,
,
(
min{
N
x
1x
2x
n−1s
−
m
rt
N
x
1x
2x
n−1m
rt
≥
)}
2
,
,
,...,
,
(
),
2
,
,
,...,
,
(
min{
inf
N
x
1x
2x
n 1s
m
rt
N
x
1x
2x
n1m
rt
Ss∈ −
−
−≥
)}
2
,
,
,...,
,
(
),
)
2
,
,
,...,
,
(
(inf
min{
N
x
1x
2x
n 1s
m
st
r
N
x
1x
2x
n1m
rt
S s − − ∈
−
−
≥
)}
2
,
,
,...,
,
(
),
)
2
,
,
,...,
,
(
sup
(inf
min{
N
x
1x
2x
n1s
m
t
r
N
x
1x
2x
n 1m
rt
M m S s − − ∈ ∈
−
−
=
)}
2
,
,
,...,
,
(
),
)
2
,
,
,...,
,
(
(inf
min{
N
x
1x
2x
n 1s
M
t
r
N
x
1x
2x
n 1m
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 greater than or equal to(
1
−
r
0)
and this completes the proof.Lemma 4.7: Let M be a t-proximinal subspace of a fuzzy n-normed linear space
(
X
,
N
)
andW
⊇
M
a subspace of X. Let K be F-bounded in X. Ifw
0S
t(
K
)
W
∈
, then)
(
0
M
S
K
M
w
+
∈
Wt M .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
tM W
∉
+
. Thus there existsw
′
∈
M
such that fort
>
0
,
inf
N
(
x
1,
x
2,...,
x
1,
k
(
w
M
),
t
)
inf
N
(
x
1,
x
2,...,
x
n 1,
k
(
w
0M
),
t
)
K k n K k
+
−
>
+
′
−
− ∈ − ∈
inf
N
(
x
1,
x
2,...,
x
n1,
k
w
0,
t
)
d
(
K
,
W
,
t
)
K
k
−
=
≥
−∈ (3)
On the other hand for each
k
∈
K
and fort
>
0
,
N
(
x
1,
x
2,...,
x
1,
k
(
w
M
),
t
)
sup
N
(
x
1,
x
2,...,
x
n 1,
k
(
w
m
),
t
)
M m
n
−
′
+
=
−−
′
+
IJMA- 2(11), Nov.-2011, Page: 2090-2100
Then for each
0
<
ε
<
1
andk
∈
K
there existsm
k∈
M
such that fort
>
0
,
N
(
x
1,
x
2,...,
x
n−1,
k
−
(
w
′
+
m
k),
t
)
≥
N
(
x
1,
x
2,...,
x
n−1,
k
−
(
w
′
+
M
),
t
)
−
ε
. Sincew
′
+
m
k∈
M
we conclude that
d
(
K
,
W
,
t
)
inf
N
(
x
1,
x
2,...,
x
n 1,
k
(
w
m
k),
t
)
K k
+
′
−
≥
−∈
≥
−−
′
+
−
ε
∈
(
,
,...,
,
(
),
)
inf
N
x
1x
2x
n 1k
w
M
t
K k
Thus,
d
(
K
,
W
,
t
)
inf
N
(
x
1,
x
2,...,
x
n 1,
k
(
w
M
),
t
)
K k
+
′
−
≥
−∈ (4)
By (3) and (4) we get,
d
(
K
,
W
,
t
)
inf
N
(
x
1,
x
2,...,
x
n 1,
k
(
w
M
),
t
)
d
(
K
,
W
,
t
)
K k
>
+
′
−
≥
−∈ ,
and this is a contradiction. Therefore
w
0M
S
t(
K
M
)
M W
∈
+
and the proof is completed.Corollary 4.8: Let M be a t-proximinal subspace of a fuzzy n-normed linear space
(
X
,
N
)
andW
⊇
M
a subspaceX. If W is simultaneous t-proximinal then
W
M
is a simultaneous t-proximinal subspace ofX
M
.Corollary 4.9: Let M be a t-proximinal subspace of a fuzzy n-normed linear space
(
X
,
N
)
andW
⊇
M
a subspace X. If W is simultaneous t-proximinal then for each F-bounded set K in X,
Q
(
S
Wt(
K
))
⊆
S
Wt M(
K
M
)
.Theorem 4.10: Let M be a t-proximinal subspace of a fuzzy n-normed linear space
(
X
,
N
)
andW
⊇
M
subspace of X. If K is F-bounded set in X such thatw
0+
M
∈
S
Wt M(
K
M
)
andm
0S
t(
K
w
0)
M
−
∈
, then)
(
00
m
S
K
w
+
∈
Wt .Proof: In view of Lemma 4.4, for
t
>
0
we have,)
,
)
(
,
,...,
,
(
inf
sup
)
,
)
(
,
,...,
,
(
inf
N
x
1x
2x
1k
w
0m
0t
N
x
1x
2x
n1k
w
0m
t
Kk M m n
K k
−
−
=
−
−
−∈ ∈ −
∈
inf
sup
N
(
x
1,
x
2,...,
x
n 1,
k
(
w
0m
),
t
)
M m K k
+
−
=
−∈ ∈
inf
N
(
x
1,
x
2,...,
x
n 1,
k
(
w
0M
),
t
)
K k
+
−
=
−∈
inf
N
(
x
1,
x
2,...,
x
n 1,
k
(
w
M
),
t
)
K k
+
−
≥
−∈
∀
w
∈
W
inf
N
(
x
1,
x
2,...,
x
n 1,
k
w
,
t
)
K k
−
≥
−∈
∀
w
∈
W
.Hence,
inf
N
(
x
1,
x
2,...,
x
1,
k
(
w
0m
0),
t
)
inf
N
(
x
1,
x
2,...,
x
n 1,
k
w
,
t
)
K k n
K
k∈ −
−
+
≥
∈ −−
∀
w
∈
W
.But
w
0+
m
0∈
W
. Thenw
0+
m
0∈
S
Wt(
K
)
and so the proof is completed.Theorem 4.11: Let M be a t-proximinal subspace of a fuzzy n-normed linear space
(
X
,
N
)
andW
⊇
M
a simultaneous t-proximinal subspace of X. Then for each F-bounded set K in X,
Q
(
S
(
K
))
S
t(
K
M
)
M W t
W
=
Proof: By Corollary 4.9, we obtain
