International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 146

ISSN 2229-5518

111/1/2013

ON THE GENERALIZED FUZZY n-NORMED SPACES INCLUDING ϕ FUNCTION

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

Abstract

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].

2000 Mathematics Subject Classification. 46A16, 46A19, 54D35.

Key Words and Phrases. 2-normed spaces, n-normed space, fuzzy normed space.

1 Introduction

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)(t), for any tR

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:

IJSER © 2013 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 147

ISSN 2229-5518

p

ϕ (t)= |t|; ϕ (t)= |t| ,pR+ 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 a1=1 for all a[0,1 .]
The most used t-norms in fuzzy metric spaces theory are following;
a)ab=a.b, b)ab=min {a,b

}, c)ab=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 a0=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) .

2 Main Results

In this part, we give the definition of generalized fuzzy ϕ-n-normed space.

Definition 4 Let nN 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 cR,

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].

IJSER © 2013 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 148

ISSN 2229-5518

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)≠0R,

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 ,ts)≥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 , ab=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 , ab=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 , ab=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 cR and ϕ (c)≠0, we see that

t

,...,cx
,t) =
N(x1 n
t+ x ,...,cx

1 n

IJSER © 2013 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 149

ISSN 2229-5518

Error! Bookmark not defined. =

t

t+ϕ (c) x ,...,x

1 n

Error! Bookmark not defined. =

t


t

ϕ (c)
ϕ (c) 1 n
+ x ,...,x
,...,x ,

t

).

Error! Bookmark not defined. = N(x1

n ϕ (c)
Corollary 3 Let (X,N,
property ϕ (−t)(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 cR 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

)

Error! Bookmark not defined. = N(x1 2,

n ϕ2 (c)

Error! Bookmark not defined.Error! Bookmark not defined. .

Error! Bookmark not defined.Error! Bookmark not defined. .

Error! Bookmark not defined.Error! Bookmark not defined. .

,x ...,x ,

t

)

Error! Bookmark not defined. = N(x1 2,

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

IJSER © 2013 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 150

ISSN 2229-5518

x1 n

{ 1 n }

,...,x
=inf

ϕ (α)

t:N(x ,...,x ,t)≥ϕ (α)
(0,1 )
Then .,...,.

(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 , ab=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.

References

[]

S. Gähler, Lineare 2-normierte Räume, Diese Nachr. 28, 1-43 (1965).

[] H. Gunawan and M. Mashadi, On n-Normed Spaces, IJMMS, 27:10(2001),631-639.

[] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear space, J. Fuzzy Math. 11 (3) (2003),687-705.

[] T. Bag and S.K. Samanta, Fixed point theorems on fuzzy normed linear space,Inf. Sci. 176 (2006)

2910-2931.

[] AL. Narayanan and S. Vijayabalaji, Fuzzy n-Normed Space, IJMMS, 2005:24 (2005) 3963-3977.

[] I. Golet, On Generalized fuzzy normed spaces and coincidence point theorems, Fuzzy Sets and Systems 161 (2010) 1138-1144.

[] I. Golet , On Generalized fuzzy normed space, International Mathematical Forum, 4, (2009), no. 25,

1237-1242.

[] M. Acikgöz, N. Aslan and S. Araci, The Generalization of Appollonious identity to linear n-normed space, Int. J. Comtempt. Math. Sciences, vol.5, no.24 (2010) 1187-1192.

[] M. Kır and M. Acikgoz, A study involving the completion of a quasi-2-normed space, International Journal of Analysis, Volume 2013 (2013), Article ID 512372, 4 pages.

[] M.Kır, A new approach to the Accretive Operators arising from 2-Banach, Gen. Math. Notes, vol. 16, no.2,

(2013)

IJSER © 2013 http://www.ijser.org