International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 146
ISSN 2229-5518
111/1/2013
†Mehmet KIR and ‡Mehmet ACIKGOZ
†Atatürk University, Faculty of Science, Department of Mathematics, 25000 Erzurum, TURKEY
‡University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY
mehmet_040465@yahoo.com; acikgoz@gantep.edu.tr
The fundamental aim of this paper is to consider and introduce fuzzy ϕ-n-normed space, where ϕ function is introduced originally by Golet in [6].
Key Words and Phrases. 2-normed spaces, n-normed space, fuzzy normed space.
The concept of 2-normed space was firstly introduced by Gahler in [1]. Gunawan and Mashadi introduced the concept of n-normed space in [2]. Bag and Samanta also introduced and developed the fuzzy normed space in [3], [4] which is a vital and important to study in fuzzy systems. In [5], Narayanan and Vijayabalaji gave fuzzy n-normed space.
In the value of dimensions n=1,2, Golet considered a generalization of fuzzy normed space in
[6] using a real valued function ϕ as well.
In the present paper we give a generalization both Golet and Narayanan. Finally, we
ıntroduced the concept of fuzzy ϕ-n-normed space as a generalization of fuzzy n-normed space.
Definition 1 [6] Let ϕ be a function defined on the real field R into itself with the following properties:
(a ) ϕ (−t)=ϕ (t), for any t∈R
1
(a ) ϕ (1)=1,
2
(a )
ϕ is strict increasing and continuous on (0,∞ ,)
3
(a ) t→0limϕ (t)=0 and t→∞limϕ (t)=∞.
4
Now, we will give some examples on the above definition:
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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 147
ISSN 2229-5518
p
ϕ (t)= |t|; ϕ (t)= |t| ,p∈R+ ϕ (t)=
2t2n
|t|+1. (1)
Each of equation (?) hold true the conditions of the definition ?.
Definition 2 [6] A t-norm ∗ is a two place function ∗: [0,1 ×]
[0,1 →]
[0,1 ]
which is
associative, commutative, non decreasing in each place and such that a∗1=1 for all a∈ [0,1 .]
The most used t-norms in fuzzy metric spaces theory are following;
a)a∗b=a.b, b)a∗b=min {a,b
}, c)a∗b=max {a+b−1,0 . }
Definition 3 [6] By an operation ∘ on R+ we mean a two place
function ∘: [0,∞ × [0,∞ → [0,∞ which is associative, commutative, non decreasing in each
place and such that a∘0=a, for all a∈ [0,∞ . The most used operations on R+ are following;
1
1)∘ (s,t )=s+t, 2)∘ (s,t =) max {s,t }, 3)∘ (s,t )= (sn+tn) .
In this part, we give the definition of generalized fuzzy ϕ-n-normed space.
Definition 4 Let n∈N and X be a real vector space of dimension d≥n.A real valued
function ∥.,...,.
∥ on Xn satisfying the following
1 ∥ 1 n ∥ 1 n
n ) x ,...,x
=0 if and only if x ,...,x
are linearly dependent;
2 ∥ 1 n ∥
n ) x ,...,x
is invariant under permutation;
3 ∥ 1
n−1 n∥
∥ 1 n−1 n∥
n ) x ,...x
,cx
=ϕ(c)
x ,...x ,x
for all c∈R,
4 ∥ 1
n−1
∥ ∥ 1
n−1
∥ ∥ 1
n−1 ∥
n ) x ,...,x
,y+z ≤
x ,...,x
,y +
x ,...,x
,z ,
is called an ϕ−n−norm on X and the pair (X, ∥.,...,.
∥ ) is called ϕ-n-normed space.
Corollary 1 Taking ϕ (t)= |t| in the definition ?, reduces to the definition of Gunawan and Mashadi in [2] and if we take n=1 and n=2 we obtain one by one definition of ϕ−normed space and ϕ−2−normed space given by Golet in [6],[7]. Also if we take ϕ (t)= |t| and n=2 then we obtain definition Gahler’s in [1].
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Definition 5 Let X be a linear space over real field R of dimension d≥n and let N be a mapping
defined on nX×...×X× [0,∞ with values into [0,1
x,y,x ,...,x ∈X and all s,t∈ [0,∞
] satisfying the following conditions: for all
1 n
fn ) N(x
,...,x ,0)=0,
1 1 n
fn ) for all t>0, N(x
,...,x ,t)=1 if and only if x ,...,x
are linearly dependent,
2 1 n 1 n
fn ) N(x
,...,x ,t) is invariant under any permutation of x ,...,x ,
3 1 n
1 n
t
fn ) N(x
,...,cx
,t)=N(x ,...,x ,
) , ϕ (c)≠0∈R,
4 1 n
1 n ϕ (c)
fn ) N(x
,...,x ,⋅) is nondecreasing function on [0,∞ and
5 1 n
n→∞ limN(x ,...,x ,t)=1
1 n
fn )
6
N(x+y,...,x ,t∘s)≥N(x,...,x ,t)∗N(y,...,x ,s) ,
n n n
The triple (X,N,∗
) is called generalized fuzzy ϕ-n-normed space.
Corollary 2 Substituting ∘ (s,t =) s+t , a∗b=min {a,b
} and ϕ (t)= |t| in the definition ?,
then (X,N,∗
) is called fuzzy n-normed space which is defined by Narayanan and Vijayabaliji
in [5] and for n=1 and n=2 Golet ıntroduced fuzzy ϕ−2−normed and fuzzy ϕ−normed space in
[6],[7]. Also for n=1 ıf we take ∘ (s,t =) s+t , a∗b=min {a,b
} and ϕ (t)= |t| then we obtain
definition of fuzzy normed space which is given by T. Bag and S.K. Samanta in [3],[4].
Example 1 Let (X, ∥.,...,.
∥ ) be a ϕ-n-normed space. For all x ,...,x ∈X , t∈ [0,∞
1 n
t
,...,x ,t)=
N(x1 n
t+ ∥x ,...,x ∥
1 n
then (X,N,∗
) is generalized fuzzy ϕ-n-normed space. In the case of
∘ (s,t =) s+t , a∗b=min {a,b } then we
called (X,N,∗
) as the standard fuzzy ϕ-n-normed space. Also if we take ϕ (t)= |t| then (X,N,∗
) is called
standard fuzzy n-normed space.
Solution 1 It is clear that fn ), fn ), fn ), fn ) and fn ) satisfy for all x ,...,x ∈X . So, it is
1 2 3 5 6 1 n
necessary to show the following:
fn ) For c∈R and ϕ (c)≠0, we see that
t
,...,cx
,t) =
N(x1 n
t+ ∥x ,...,cx ∥
1 n
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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 149
ISSN 2229-5518
t
t+ϕ (c) ∥x ,...,x ∥
1 n
t
t
ϕ (c)
ϕ (c) ∥ 1 n ∥
+ x ,...,x
,...,x ,
t
).
n ϕ (c)
Corollary 3 Let (X,N,∗
property ϕ (−t)=ϕ (t)
) be an generalized fuzzy ϕ-n-normed space by the
for all x,y,x ,...,x ∈X and t∈ [0,∞ we can write
2 n
,...,x ,t)=N(x−y,x ,...,x ,t)
N(y−x,x2 n 2 n
Corollary 4 Let (X,N,∗
) be an generalized fuzzy ϕ-n-normed space by fn ) and fn ) for all
3 4
x ,x ,...,x ∈X , t∈ [0,∞ and c∈R with ϕ (c)≠0 we can write
1 2 n
,cx
...,cx
,t) = N(x
,cx
...,cx ,
t
)
N(cx1 2, n
1 2,
n ϕ (c)
,x ...,cx ,
t
)
n ϕ2 (c)
,x ...,x ,
t
)
n ϕn (c)
,cx
...,cx ,t)=N(x ,x
...,x ,
t
)
Thus we obtain N(cx1 2, n
1 2,
n ϕn (c)
Theorem 1 Let (X,N,∗
) be an generalized fuzzy ϕ-n-normed space with assumption
fn ) N(x
,...,x ,t)>0 , for all t>0 implies x ,...,x
are linearly dependent. Define
7 1 n 1 n
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ISSN 2229-5518
∥x1 n ∥
{ 1 n }
,...,x
=inf
ϕ (α)
t:N(x ,...,x ,t)≥ϕ (α)
,α∈ (0,1 )
Then ∥.,...,.
∥ :α∈ (0,1 )
ϕ (α)
is an ascending family of ϕ-n-normed space on X.They
are called ϕ (α)-norm on X corresponding to the fuzzy ϕ-n-norm on X.
Remark 1 For the case of ∘ (s,t =) s+t , a∗b=min {a,b
by Narayanan and Vijayabaliji in [5].
} and ϕ (t)= |t|
the proof was given
[Sorry. Ignored \begin{proof} ... \end{proof}] Acknowledgement 1 The authors is thankful to S. Araci for his valuable comments and suggestions.
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