Operator Fuzzy 2-Normed linear space

(1)

Operator Fuzzy 2-Normed linear space

Ghanshyam Lal¹, Divya Mishra¹, Anil Agrawal¹ , Parijat Sinha² ¹Department of Mathematics, M.G.C.G. University, Satna (INDIA) ²Department of Mathematics, V.S.S.D. College, Kanpur,

Abstract: In this paper we have introduced operator fuzzy 2-normed linear space. We have defined Cauchy sequence, convergent sequence and some results on it and also defined complete operator fuzzy 2-normed linear space.

Keywords: Convergent sequence, Cauchy sequence, Completeness.

1. Introduction: The concept of fuzzy set was introduced by Zadeh [11] in 1965. In 1984, Katsaras [6] first introduced the notion of fuzzy norm on a linear space. Felbin [3], in 1992, introduced the idea of fuzzy norm in a different way by assigning a fuzzy real number to each element of the linear space so that the corresponding metric associated this fuzzy norm is of Kaleva type [5] fuzzy metric. She also introduced an idea of fuzzy bounded linear operator, the norm of which is a fuzzy number. In 2003, Xiao and Zhu [10] redefined in a more general setting the Idea of Felbin’s [3] definition of fuzzy norm of a linear operator from a normed linear space to another fuzzy normed linear space.

A satisfactory theory of 2-norm on a linear space has been introduced and developed by Gähler [4]. In 2009, Sundaram and Beaula[9] defined the concept of fuzzy 2-normed linear space and introduced fuzzy 2-linear operator.

In the present paper, we have introduced the concept of operator fuzzy 2-normed linear space and established some results.

2. Preliminaries:

In this section some definition and preliminary results are given which will be used in this paper.

Definition 2.1[8] LetX be a linear space over a field^F. A fuzzy subset N of XX R (R is the set of real numbers) is called a fuzzy 2-norms on X if and only if

1. for all tR, with t0, N



x₁,x₂,t



0.

(2)



x₁,x₂,t



1 if and only if x and ₁ x are linearly ₂

dependent.

3. N



x₁,x₂,t



is invariant under any permutation of x₁,x₂.

4. for all tR, with

 

_^









 

 c

x t x N t cx x N

t 0, ₁, ₂, ₁, ₂, if c0, cF. 5. for all s ,tR. N



x₁,x₂x₂,st



min



N



x₁,x₂,s

 

,N x₁,x₂,t

 

6. N



x₁,x₂,



is non-decreasing function of Rand



, ,



1

lim ₁ ₂

t 



 N x x t

then



X ,N



is called a fuzzy 2-normed linear space.

Example 2.1[8]: Let



^X^, ^.,.



be 2-normed linear space. Define



x x t



N ₁, ₂, =

2 1, x x t

t

 , when t0, tR, x₁,x₂X = 0 , when t0,tR

Then



X ,N



be a fuzzy 2-normed linear space.

Definition 2.2[8]: Let



X ,N



be a fuzzy 2-normed linear space. A sequence

 

x_n inX is said to be convergent to xXif givent 0, 0r1 there exists an integer ⁿ0^^N such that



x x,y,t



1 r,

N _n    nn₀.



X ,N



be a fuzzy 2-normed linear space. A sequence

 

x in_n X is said to be Cauchy sequence if given 0, 0r1, t0there exists an integer n₀N such that



^x ^x ^,^y^,^t



¹ ^r^,

N _n__p  _n   nn₀, p1,2,3,....



^{X ,}^N



be fuzzy 2-normed linear space. It is said to be complete if any Cauchy sequence in X converges to a point inX .

(3)



X, N1



andC,D are subspaces of fuzzy 2-normed linear space



Y, N2



such that



x x x x

 

T x x

 

T x x

 

T x x

  

T x x T ₁ , ₂   ₁, ₂  ₁,   , ₂  , 

and T



x1,x2



T



x1,x2



, where^^,^^

 

⁰^,¹ ^.

3. Operator fuzzy 2-normed linear space:

Definition 3.1: LetX and Ybe two fuzzy 2-normed linear spaces, T:ABCD be an operator. ^Tis said to be fuzzy 2-bounded if and only if there existt0 and 0r1

Such that^N



^Tx1^,^Tx2^,^t



^r^N



^x1^,^x2^,^t



.

Remark 3.1: LetX and Y be two fuzzy 2-normed linear spaces and^L



^X^,^Y



to be the set of all fuzzy 2- bounded linear operators fromX intoY .

Remark 3.2: L



X,Y



become a linear space if we define the sum of two operators



X Y



L T

T₁, ₂ , is



T1T2



x1,x2



T1



x1,x2



T2



x1,x2



and the product

Tof TL



X,Y



and a scalarR by

 

T x1,x2



T



x1,x2



.

Definition 3.2 ( operator fuzzy 2-normed linear space): LetL



X,Y



be a linear space over



R C



K or . Let N be a fuzzy subset of L



X,Y

 

L X,Y



R is called fuzzy 2-norm on



X Y



L , if and only if for allT₁,T₂,T₂L



X,Y



andcK

(N1) for all tR, with t 0, N



T₁,T₂,t



0.

(N₂) for all tR, with t 0, N



T₁,T₂,t



1 if and only if T and ₁ T are linearly ₂ dependent.

(N₃) N



T₁,T₂,t



is invariant under any permutation of T₁,T₂

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(N4) for all tR, with

 

_^









 

 c

T t T N t cT T N

t 0, ₁, ₂, ₁, ₂, if c0, cK. (N5) for all s ,tR. N



T₁,T₂T₂,st



min



N



T₁,T₂,s

 

,N T₁,T₂,t

 

(N6) N



T₁,T₂,



is non-decreasing function of Rand



^, ^,



¹

lim ₁ ₂

t 



 N T T t

Then



L



X,Y



,N



is called a operator fuzzy 2-normed linear space.

Example 3.1: Let



^L



^X^,Y



^,^.,.



be 2-normed linear space. Define



T T t



N ₁, ₂, =

2 1,T T t

t

 , when t0, tR, T₁,T₂L



X,Y



= 0 , when t0,tR

Then



L



X,Y



,N



be an operator fuzzy 2-normed linear space.

Proof: Now we have to show that N



T₁,T₂,t



is a fuzzy 2-norm on^X. (N1) t Rwith t0, we have by definition N



T₁,T₂,t



0.

(N2) tRwitht0,

 

₁ ₂ ₁ ₂

2 1 2

1 1 , 0 ,

1 , ,

, T T T T

T T t t t

T T

N    

 

 are linearly

independent.

(N3) As T₁,T₂ is invariant under any permutation ofT₁,T₂, it follows that N



T₁,T₂,t



is invariant under any permutation ofT₁,T₂.

(N4)tRwith t 0and ^c

 

0 ^K,we get

 

_^









 



 

 ^

c T t T N T c T

t c t

T T c t

t cT

T t t t cT T

N , ,

, , . , ,

, ₁ ₂

2 1 2

1 2

1 .

(Na5) For alls,tR andT₁,T₂,T₂L



X,Y



,

We have to show thatN



T₁,T₂T₂,st



min



N



T₁,T₂,s

 

,N T₁,T₂,t

 

.

(5)

(c) st 0,s0,t0,s0,t 0 Then in these cases the relation is obvious.

If (d) s0,t0,st0 then assume that

 

2 2 1 2

2

1, , ,

T T T t s

t t s

s T T T

N    

 

 



2 1 2

1,T T,T T

t s

 



 

If

2 1 2

1, t T,T

t T

T s

s

 

 

Then 0

,

, ₂ ₁ ₂

1

 

 

 t T T

t T

T s

s

s



t T1^,T2

 

t s T1^,T2



⁰ sT₁,T₂ tT₁,T₂ 0

So



, ,



,



⁰

, ,

, ₁ ₂ ₁ ₂ ₁ ₂

2 1 2 1 2

1 2

1 2 1

 

 



 

 

 

 



T T t T T T T t s

T T t T T s T

T t

t T

T T T t s

t s

So

0

, ,

, ₂ ₁ ₂ ₁ ₂

1

 

 

 



T T t

t T

T T T t s

t s

¹, ² ¹, ² t T¹,T² t T

T T T t s

t s

 

 



 

Similarly, if

¹, ² s T¹,T² s T

T t

t

 

 

(6)

¹, ² ¹, ² s T¹,T² s T

T T T t s

t s

 

 



Thus

N



T₁,T₂ T₂,st



min



N



T₁,T₂,s

 

,NT₁,T₂,t

 

(N6) If t₁t₂ 0then we have N



T₁,T₂,t₁



 N



T₁,T₂,t₂



0,T₁,T₂L



X,Y



if 0t₁ t₂ then

 



1 1 2



2 1 2

 ^

¹ ² ¹

^ ^

¹ ² ²

^

1 2 2 1 2

1 1

1 2

2 0 , , , ,

, ,

, N T T t N T T t

T T t T T t

t t T T T

T t

t T

T t

t   



 

 

 .

Thus N



T1^,T2^,t



¹ is a non- decreasing function. Again

 

1

lim , ,

, lim

2 1 2

1 

 







 t T T

t t T T N

t

t , for allT₁,T₂L



X,Y



.

HenceN



T₁,T₂,t



is an operator fuzzy 2-normed onL



X,Y



.

Definition 3.3: A sequence

 

T in operator fuzzy 2-normed linear space _n



L



X,Y



,N



is said to be convergent to TL



X,Y



if for given t0, 0r1 there exists an integer ⁿ0^^N

such that N



T_n T,T,t



1r, nn₀.

Theorem 3.1: Let



L



X,Y



,N



be fuzzy 2-normed linear space of operators, then a sequence

 

T_n is converges to T if and only ifN



T_n T,T,t



1 as n i.e., lim



 , ,



1



 N T_n T T t

n .

Proof: For all t0 and suppose

 

T converges to_n T. Then for a givenr,0r 1

(7)



T_n T,T,t



1rfor n>n0. Hence



T T,T,t



1

N _n asn .

Conversely, if for eacht0,N



T_n T,T,t



1 asn . Then for everyr,0r 1, there exists an integer n₀N such that 1N



T_n T,T,t



rfor all nn₀Thus



T T T t



r

N _n  , , 1 , for all nn₀

Hence

 

T converges to_n T.

Theorem 3.2: If a sequence

 

T in fuzzy 2-normed linear space of operators_n



L



X,Y



,N



is convergent then its limit is unique.

Proof: Let

 

T be a convergent sequence. If the sequence_n

 

T converges to_n T and₁ T in₂



^X ^Y



L , .

If lim  ₁ lim



 ₁, ,



1







 T T N T_n T T t

n n n

If lim  ₂ lim



 ₂, ,



1







 T T NT_n T T t

n n n

Now, N



T₁T₂,T,t



N



T₁ T_n T_n T₂,T,t 2t 2



min



N

 

T_n T₁



,T,t 2

 

,N T_n T₂,T,t 2

 

min



N

 

T_n T₁



,T,t 2

 

,N T_n T₂,T,t 2

 

Taking the limit, we have



^, ^,



^min



^lim

  

^, ^, ²



^,^lim



^, ^, ²

 

¹

lim ₁ ₂    ₁   ₂  











 N T T T t N T T T t N T_n T T t

n n n

n



1 2, ,



1 1 2 0 1 2

limNT T T t T T T T

n        



 ,

(8)

Theorem 3.3: In operator fuzzy 2-normed linear space



L



X,Y



,N



every subsequence of convergent sequence converges to the limit of the sequence.

Proof: Let

 

T be a sequence converges to_n T and let

 

Tn_k be a subsequence of

 

T_n converges toT . Since ₀

 

Tn_k is converges, then by theorem (3.1), ^lim



 0^, ^,



¹



 NT T T t

nk

n ,

Now, consider



^, ^,



¹ ^min



^lim



^, ^, ²



^,^lim

^

^, ^, ²

^ 

¹

lim    ₀       ₀  











 NT T T T T t NT T T t N T_n T T t

n n n n

n n

n n^k ^k

2 1 ,

0, 



 



   t T T T

N _n , which is contradiction, because the convergent point is unique. Hence

T T₀  .

Definition3.4 : A sequence

 

T in operator fuzzy 2-normed linear space is said to be Cauchy _n sequence if t 0, ^limN



T_n_p ^T_n^,T^^,t



^¹

n

, p1,2,3,...

Theorem 3.4: In an operator fuzzy 2-normed linear space



L



X,Y



,N



every convergent sequence is a Cauchy sequence.

Proof: Let

 

T be a convergent sequence in the operator fuzzy 2-normed linear space_n

 



L X,Y ,N



which converges toT . For tRandp1,2,3,..., we have



^T ^T ^,^T^,^t

 

^N^T ^T ^T ^T ^,^T ^,^t ² ^t ²



N _n__p  _n   _n__p    _n  

^min



^N



^Tn_p ^T^,^T^,^t ²



^,^N



^T^Tn^,^T^,^t ²

 

^min



^N



^Tn_p^T^,^T^,^t ²



^,^N



^Tn^T^,^T^,^t ²

 

(9)



^, ^,



^min



^lim



^, ^, ²



^,^lim



^, ^, ²

 

¹

lim        



 



 



 NT T T t NT T T t N T_n T T t

p n n n

n p n n



^, ^,



¹

lim _   



 NT_n _p T_n T t

n , for all tRandp1,2,3,...

Thus

 

T is a Cauchy sequence in_n



L



X,Y



,N



.

Theorem 3.5: Let



^L



^X^,^Y



^,^N



be an operator fuzzy 2-normed linear space, such that every Cauchy sequence in



^L



^X^,^Y



^,^N



has a convergent subsequence. Then



^L



^X^,^Y



^,^N



^is

complete.

Proof: Let

 

T be a Cauchy sequence in_n



^L



^X^,^Y



^,^N



^and

 

Tnk be a subsequence of the sequence

 

T converges to_n TL



X,Y



.Since

 

T be a Cauchy sequence in_n L



X,Y



, for

0

t we have 1

,2 ,

lim 



 



  





T t T T N n nK

n .

Again , since

 

Tn_k converges toT we have, 1 ,2 ,

lim 



 



  





T t T T N nK

n

Now,



^T ^T ^T ^t



^N



^T ^T ^T ^T ^T ^t



N n , ,  n  nK  nK  , ,

^min



^N



^Tn^Tn_k^,^T^,^t ²

 

^,^N^Tn_k ^T^,^T^,^t ²

 

Taking the limit, we have lim



 , ,



1



 N T_n T T t

n

Then

 

T converges to T in _n



L



X,Y



,N



and hence



L



X,Y



,N



is complete.

Theorem 3.6 : Let N be fuzzy 2-norm on



L



X,Y



,N



then

(10)



T₁,T₂,t



is a non-decreasing function with respect to t for each

 



L X Y N



T

T₁, ₂ , , .

(ii) N



T₁,T₂ T₃,t



N



T₁,T₃ T₂,t



Proof (i) letts thenkst 0 and we have

N



T1^,T2^,t



^min



N



T1^,T2^,t



^,¹



min



N



T₁,T₂,t

 

,N T₁,0,k

 

min



N



T₁,T₂,t

 

,NT₁,0,st

 

N



T₁,T₂ 0,tst



N



T₁,T₂,s



Hence forts,^N



^T1^,^T2^,^t



 ^N



^T1^,^T2^,^s



. This proves (i).

To prove (ii) we have, ^N



^T₁^,^T₂ ^T₃^,^t



 ^N



^T₁^,

 

¹ ^T₃ ^T₂



^,^t



_^











 

 ₁, ₃ , t1 T T T N

 N



T₁,T₃ T₂,t



.

************************

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