SOOCHOW JOURNAL OF MATHEMATICS
Volume 32, No. 4, pp. 509-520, October 2006
ON FUZZY REAL-VALUED DOUBLE SEQUENCE SPACES
BY
BINOD CHANDRA TRIPATHY AND AMAR JYOTI DUTTA
Abstract. In this article we introduce the fuzzy real-valued sequence spaces 2 wF ,
B
R
(2 c0 )F , (2 c)B
F , (2 c0 )F , (2 c0 )F and (2 λ∞ )F . We study their different topological
and algebraic properties. We introduce different notions like solidness, symmetricity, convergence free for fuzzy real-valued double sequence spaces. We prove some
inclusion results.
2 cF ,
1. Introduction and Preliminaries
A fuzzy real number X is a fuzzy set on R, i.e. a mapping X : R → I(= [0, 1]),
associating each real number t with its grade of membership X(t).
The α-cut of a fuzzy real number X is denoted by [X] α , 0 < α ≤ 1, where
[X]α = {t ∈ R : X(t) ≥ α}.
A fuzzy real number X is said to be upper semi-continuous if for each >
0, X −1 ([0, a
+ )), for all a ∈ I is open in the usual topology of R.
If there exists t ∈ R such that X(t) = 1, then X is called normal.
V
A fuzzy real number X is said to be convex, if X(t) ≥ X(s) X(r) = min
(X(s), X(r)), where s < t < r.
The class of all upper semi-continuous normal convex fuzzy real numbers is
defined by R(I).
Let X, Y ∈ R(I) and the α-level sets be [X] α = [aα1 , aα2 ], [Y ]α = [bα1 , bα2 ],
α ∈ [0, 1].
Received September 15, 2004; revised May 22, 2006.
AMS Subject Classification. 40A05, 40A30, 40D25.
Key words. fuzzy real numbers, fuzzy real-valued double sequence, solid space, symmetric
space, convergence free, sequence algebra.
509
510
BINOD CHANDRA TRIPATHY AND AMAR JYOTI DUTTA
Then the arithmetic operations on R(I) are defined as follows:
(X ⊕ Y )(t) = sup{X(s) ∧ Y (t − s)},
t ∈ R.
(X Y )(t) = sup{X(s) ∧ Y (s − t)},
t ∈ R.
t
(X ⊗ Y )(t) = sup{X(s) ∧ Y ( )},
t ∈ R.
s
(X/Y )(t) = sup{X(st) ∧ Y (s)},
t ∈ R.
The above operations can be defined in terms of α-level sets as follows:
[X ⊕ Y ]α = [aα1 + bα1 , aα2 + bα2 ],
[X Y ]α = [aα1 − bα2 , aα2 − bα1 ],
[X ⊗ Y ]α = [ min aαi bαj , max aαi bαj ],
i,j∈{1,2}
[X
−1
]α =
[(aα2 )−1 ,
i,j∈{1,2}
(aα1 )−1 ],
aα1 > 0.
The absolute value of X ∈ R(I) is defined as (please refer to Kaleva and Seikkala
[5])
(
max{X(t), X(−t)}, for t ≥ 0,
|X|(t) =
0,
otherwise.
The additive identity and multiplicative identity in R(I) are denoted by 0
and 1 respectively.
Let D be the set of all closed and bounded intervals X = [X L , X R ]. Then
we write
X ≤ Y,
if and only if X L ≤ Y L and X R ≤ Y R ,
and
d(X, Y ) = max{|X L − Y L |, |X R − Y R |}.
Then clearly (D, d) is a complete metric space.
Now we define the metric d : R(I) × R(I) → R by
d(X, Y ) = sup d([X]α , [Y ]α ), for X, Y ∈ R(I).
0≤α≤1
Applying the notion of fuzzy real numbers, fuzzy real-valued sequences were
introduced and were studied by Nanda [8], Tripathy and Nanda [13], Das and
Choudhury [3], Subrahmanyam [11], Nuray and Savas [9] and many others.
ON FUZZY REAL-VALUED DOUBLE SEQUENCE SPACES
511
A fuzzy real-valued sequence is denoted by {X k }, where Xk ∈ R(I), for all
k ∈ N.
A sequence X = {Xk } of fuzzy real numbers is said to be convergent to the
fuzzy number X0 , if for every > 0, there exist k0 ∈ N such that
¯ k , X0 ) < , for all k > k0 .
d(X
The initial works on double sequences of real or complex terms is found in
Bromwich [2]. Hardy [4] introduced the notion of regular convergence for double
sequences of real or complex terms. The works on double sequences was further
investigated by Moricz [6], Basarir and Solankan [1], Moricz and Rhoades [7],
Tripathy [12] and many others.
2. Definition and Background
A fuzzy real-valued double sequence is a double infinite array of fuzzy real
numbers. We denote a fuzzy real-valued double sequence by < X mn >, where
Xmn are fuzzy real numbers for each m, n ∈ N.
We introduce the following definitions for fuzzy real-valued double sequences:
Definition 2.1. A fuzzy real-valued double sequence < X mn > is said to be
convergent in Pringshiem’s sense to the fuzzy real number X, if for every > 0,
there exists m0 = m0 (), n0 = n0 (), such that d(Xmn , X) < for all m ≥ m0
and n ≥ n0 .
Definition 2.2. A fuzzy real-valued double sequence < X mn > is said to be
bounded if
sup d(Xmn , 0̄) < ∞,
m,n
equivalently, if there exists µ ∈ R(I) ∗ , such that |Xmn | ≤ µ for all m, n ∈ N ,
where R(I)∗ denotes the set of all positive fuzzy real numbers.
Definition 2.3. A fuzzy real-valued double sequence < X mn > is said to be
regularly convergent if it is convergent in Pringsheim’s sense and the followings
hold:
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BINOD CHANDRA TRIPATHY AND AMAR JYOTI DUTTA
For any > 0, there exists m0 = m0 (, n) and n0 = n0 (, m) such that
d(Xmn , Ln ) < , for all m ≥ m0 , for some Ln ∈ R(I), for each n ∈ N,
and
d(Xmn , Mm ) < , for all n ≥ n0 , for some Mm ∈ R(I), for each m ∈ N.
Definition 2.4. A fuzzy real-valued double sequence space E F is said to be
normal (or solid) if < Xmn >∈ EF , whenever |Xmn | ≤ |Ymn | for all m, n ∈ N
and < Ymn >∈ EF .
Now we introduce the notion of step spaces for double sequences as follows:
Definition 2.5. Let K = {(ni , ki ) : i ∈ N ; n1 < n2 < n3 < · · · and k1 <
k2 < k3 < · · · } ⊆ N × N and E be a double sequence space. A K-step space of
E is a sequence space λE
K = {< xni ki >∈ 2 w :< xnk >∈ E}.
A canonical pre-image of a sequence < x ni ki >∈ E, is a sequence < ynk >∈
2 w defined as follows :

 xnk , if (n, k) ∈ K,
ynk =
 0,
otherwise.
Definition 2.6. A fuzzy real-valued double sequence space E F is said to be
monotone if EF contains the canonical pre-images of all its step spaces.
Definition 2.7. A fuzzy real-valued double sequence space E F is said to
be symmetric if < Xπ(m),π(n) >∈ EF , whenever < Xmn >∈ EF , where π is a
permutation of N .
Definition 2.8. A fuzzy real-valued double sequence space E F is said to be
a sequence algebra if < Xmn ⊗ Ymn >∈ EF , whenever < Xmn >, < Ymn >∈ EF .
Definition 2.9. A fuzzy real-valued double sequence space E F is said to
be convergence free, if < Xmn >∈ EF , whenever < Ymn >∈ EF and Ymn = 0̄
implies Xmn = 0̄.
B
R
R
Throughout the article 2 wF , (2 `∞ )F , 2 cF , (2 c0 )F , 2 cB
F , (2 c0 )F , 2 cF , (2 c0 )F denote the classes of all, bounded, convergent in Pringsheim’s sense, null in Pring-
ON FUZZY REAL-VALUED DOUBLE SEQUENCE SPACES
513
sheim’s sense, bounded and convergent, bounded and null, regularly convergent
and regularly null fuzzy real-valued double sequence spaces.
3. Main Results
In this section we prove the results of this article.
Theorem 1. The space (2 `∞ )F is a complete metric space.
Proof. Let < X i > be a Cauchy sequence in (2 `∞ )F . Define the metric D
on (2 `∞ )F as,
D(X, Y ) = sup d(Xmn , Ymn ).
m,n
Then for a given > 0, there exists k0 > 0, such that
i
j
D(Xmn
, Xmn
) < , for all i, j ≥ k0 .
i
j
⇒ sup d(Xmn
, Xmn
) < , for all i, j ≥ k0 .
m,n
i , X j ) < , for all i, j ≥ k . That is < X i
This implies d(Xmn
mn
0
mn > is a Cauchy
sequence in R(I), for all m, n ∈ N.
i
i
This implies that < Xmn
> converges in R(I). Let limi→∞ Xmn
= Xmn , for
all m, n ∈ N . Hence for each > 0, there exists k 0 = k0 (m, n), such that
i
sup d(Xmn
, Xmn ) < , for all i ≥ k0 .
m,n
Now to show that < Xmn >∈ (2 `∞ )F . We have
i
i
sup d(Xmn , 0) ≤ sup d(Xmn , Xmn
) + sup d(Xmn
, 0)
m,n
m,n
m,n
≤ + M < ∞.
Hence < Xmn >∈ (2 `∞ )F .
Thus the space (2 `∞ )F is a complete metric space.
R
R
B
Theorem 2. The spaces 2 cB
F , (2 c0 )F , 2 cF , (2 c0 )F are complete metric spaces.
Proof. We shall prove it for (2 c0 )B
F and for the other cases it can be proven
similarly.
514
BINOD CHANDRA TRIPATHY AND AMAR JYOTI DUTTA
i
Let < X i > be a Cauchy sequence in (2 c0 )B
F . Then < X > is a Cauchy
sequence in (2 `∞ )F also, which implies that < X i > converges in (2 `∞ )F . Let
limi→∞ X i = X, say.
We have to prove that X =< Xmn > converges to 0. Since limi→∞ X i = X,
so for a given > 0, we have p0 ∈ N such that
i
d(Xmn , Xmn
) < , for all i ≥ p0 .
(1)
Again since < X i >∈ (2 c0 )B
F , so for > 0, we have for each i ∈ N , m 0 =
m0 (i) and n0 = n0 (i) in N and p0 such that
i
d(Xmn
, 0) < , for all m ≥ m0 and n ≥ n0 .
(2)
Now for a fixed i ≥ p0 , we have m0 = m0 (i) and n0 = n0 (i) in N and p0 such
that for all m ≥ m0 and n ≥ n0 , we have
i
i
d(Xmn , 0) ≤ d(Xmn , Xmn
) + d(Xmn
, 0)
< 2, by (1) and (2).
Hence < Xmn >∈ (2 c0 )B
F.
Therefore the space (2 c0 )B
F is a complete metric space.
Theorem 3. A bounded double sequence < X mn > of fuzzy real numbers
is convergent to L, implies it is Cesàro summable to L. But the converse is not
necessarily always true.
Proof. Consider the bounded double sequence of fuzzy real numbers
< Xmn >, which converges to L. Then for a given > 0, there exist m 0 , n0 ∈ N ,
such that
d(Xmn , L) < , for all m ≥ m0 and n ≥ n0 .
Since < Xmn >∈ (2 `∞ )F , so there exists M such that supm,n d(Xmn , L) < M.
ON FUZZY REAL-VALUED DOUBLE SEQUENCE SPACES
515
Now we have,
1 XX
d(Xmn , L)
p,q→∞ pq
m≤p,n≤q
1
1 X X
d(Xmn , L) + lim
≤ lim
p,q→∞ pq
p,q→∞ pq
lim
m≤m0 ,n≤q
p
X
1
p,q→∞ pq
m=m
+ lim
q
X
X
X
d(Xmn , L)
m0 +1≤m≤p,n≤n0
d(Xmn , L)
0 +1,n=n0 +1
1
= lim
{m0 qM + (p − m0 )n0 M + (p − m0 )(q − n0 )}
p,q→∞ pq
= 0.
Which shows that the double sequence < X mn > is Cesàro summable to L.
The converse is not necessarily true. This is clear from the following example.
Example 1. Consider the double sequence of fuzzy real numbers < X mn >,
defined by

 i, for m = n and n = i2 , i ∈ N ,
Xmn =
 0, otherwise.
Which is Cesàro summable to 0, since
1 XX
1
lim
d(Xmn , 0) = lim 2 2 {1 + 2 + 3 + · · · + n}
p,q→∞ pq
p,q→∞ n n
1 n(n + 1)
= lim 4 {
}
n→∞ n
2
n+1
= 0.
= lim
n→∞ 2n3
Which implies that < Xmn > is Cesàro summable to 0. But the sequence is
unbounded.
R
Theorem 4. The spaces (2 c0 )F , (2 c0 )B
F , (2 c0 )F , (2 `∞ )F are solid whereas the
R
spaces 2 cF , 2 cB
F , 2 cF are not solid.
Proof. Let < Xmn > and < Ymn > be two fuzzy real-valued double sequences, such that
|Xmn | ≤ |Ymn |,
for all m, n ∈ N .
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BINOD CHANDRA TRIPATHY AND AMAR JYOTI DUTTA
R
Let < Ymn >∈ Z, for Z = (2 c0 )F , (2 c0 )B
F , (2 c0 )F , (2 `∞ )F . Then the result
follows from the following inequality
d(Xmn , 0) ≤ d(Ymn , 0), for all m, n ∈ N.
The rest of the theorem follows from the following example.
Example 2. Consider the fuzzy real-valued double sequence < X mn >
defined by

(m+1)t−2
2

for m+1
≤ t ≤ 1,

m−1 ,


2m
Xmm (t) = (m+1)t−2m
, for 1 ≤ t ≤ m+1
,
1−m



 0,
otherwise.
Xmn = 1, for m 6= n.
R
Then limm,n→∞ Xmn = 1. Hence < Xmn >∈ Z, for Z = 2 cF , 2 cB
F , 2cF .
Define < Ymn > by
Ymm = Xmm ,

 1,
for m 6= n and m odd, for all n ∈ N ,
Ymn =
 2−1 , otherwise.
Then |Ymn | ≤ |Xmn | for all m, n ∈ N . But < Ymn >∈
/ Z for Z = 2cF , 2cB
F,
R
2cF .
R
Hence the spaces 2 cF , 2 cB
F and 2 cF are not solid.
Theorem 5. The spaces (2 c0 )R
F and (2 `∞ )F are symmetric but the spaces
B , ( c )B and cR are not symmetric.
c
,
(
c
)
,
c
2 F
2 0 F 2 F
2 0 F
2 F
Proof. The sequence space (2 `∞ )F is symmetric is a routine verification.
Now consider the sequence space (2 c0 )R
F . Let > 0 be given, then there exist
m0 = m0 () and n0 = n0 () such that
d(Xmn , 0) < , for all m ≥ m0 and n ≥ n0 .
(3)
Let < Ymn > be an arrangement of < Xmn >. Then
Xij = Ymi nj ,
for all i, j ∈ N.
(4)
ON FUZZY REAL-VALUED DOUBLE SEQUENCE SPACES
517
Let p = max {m1 , m2 , . . . , mm0 } and q = max {n1 , n2 , . . . , nn0 }.
Then for all m > p and n > q we have d(Ymn , 0) < , by (3) and (4).
Hence < Ymn >∈ (2 c0 )R
F.
Thus (2 c0 )R
F is symmetric.
B
R
The spaces 2 cF , (2 c0 )F , 2 cB
F , (2 c0 )F and 2 cF are not symmetric follows from
the following example.
Example 3. Consider the sequence < X mn >, defined by
for all n ∈ N and

1+(1+t)n

, for −(1 + n1 ) ≤ t ≤ 0,


 1+n
1
X1n (t) = 1+(1−t)n
1+n , for 0 ≤ t ≤ (1 + n ),



 0,
otherwise,
Xmn = 0, for m 6= 1 and all n ∈ N.
B
R
Then < Xmn >∈ Z, for Z = 2cF , (2 c0 )F , 2 cB
F , (2 c0 )F and 2 cF .
Consider its rearrangement < Ymn > defined by

1+(1+t)n

, for −(1 + n1 ) ≤ t ≤ 0,


 1+n
1
Ynn (t) = 1+(1−t)n
1+n , for 0 ≤ t ≤ (1 + n ),



 0,
otherwise.
Ymn = 0, for m 6= n.
B
R
Then < Ymn >∈
/ Z, for Z = 2 cF , (2 c0 )F , 2 cB
F , (2 c0 )F and 2 cF . Hence the
B
R
sequence spaces 2 cF , (2 c0 )F , 2 cB
F , (2 c0 )F and 2 cF are not symmetric.
R
R
B
Theorem 6. The spaces (2 `∞ )F , 2 cF , (2 c0 )F , 2 cB
F , (2 c0 )F , 2 cF , (2 c0 )F are
not convergence free.
Proof. The spaces are not convergence free follows from the following example.
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BINOD CHANDRA TRIPATHY AND AMAR JYOTI DUTTA
Example 4. Consider the sequence < X mn > defined by

1

(1 + mt), for − m
≤ t ≤ 0,



1
Xmm (t) = (1 − mt), for 0 ≤ t ≤ m
,




0,
otherwise.
Xmn = 0, for m 6= n.
B
R
Thus < Xmn >∈ Z, for Z = (2 `∞ )F , 2 cF , (2 c0 )F , 2 cB
F , (2 c0 )F , 2 cF and
(2 c0 )R
F.
Consider the sequence < Ymn > defined by


(1 + mt ), for −m ≤ t ≤ 0,



Ymm (t) = (1 − mt ), for 0 ≤ t ≤ m,



 0,
otherwise.
Ymn = 0, for m 6= n.
B
R
R
Thus < Ymn >∈
/ Z, for Z = (2 `∞ )F , 2 cF , (2 c0 )F , 2 cB
F , (2 c0 )F , 2 cF and (2 c0 )F .
Hence these fuzzy real-valued double sequence spaces are not convergence
free.
B
R
R
Theorem 7. The spaces (2 `∞ )F , 2 cF , (2 c0 )F , 2 cB
F , (2 c0 )F , 2 cF and (2 c0 )F
are sequence algebra.
Proof. We prove it for the space (2 c0 )F and for the other spaces it can be
proven using similar techniques.
Let < Xmn >, < Ymn >∈ (2 c0 )F . Let > 0 be given. Without loss of
generality let < 1. Then we can find m 1 , n1 , m2 , n2 ∈ N such that
d(Xmn , 0) < , for all m ≥ m1 and n ≥ n1 ,
(5)
d(Ymn , 0) < , for all m ≥ m2 and n ≥ n2 ,
(6)
and
Let m3 = max (m1 , m2 ) and n3 = max (n1 , n2 ). Then for all m ≥ m3 and
n ≥ n3 we have by (5) and (6),
d(Xmn ⊗ Ymn , 0) < .
ON FUZZY REAL-VALUED DOUBLE SEQUENCE SPACES
519
Hence X ⊗ Y =< Xmn ⊗ Ymn >∈ (2 c0 )F . Thus (2 c0 )F is a sequence algebra.
The proof of the following result follows from the definitions.
Proposition 8. (i) (2 c0 )F ⊂ 2 cF .
(ii) ZFR ⊂ ZFB ⊂ ZF , for Z = 2 c and 2 c0 .
(iii) ZFR ⊂ ZFB ⊂ (2 `∞ )F , for Z = 2 c and 2 c0 .
All the above inclusion relations are strict.
Acknowledgment
The authors thank the referees for their comments and suggestions.
References
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[4] G. H. Hardy, On the convergence of certain multiple series, Proc. Camb. Phil. Soc.,
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[7] F. Moricz and B.E. Rhoades, Almost convergence of double sequences and strong regularity
of summability matrices, Math. Proc. Camb. Phil. Soc., 104(1988), 129-136.
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[9] F. Nuray and E. Savas, Statistical convergence of sequences of fuzzy real numbers, Math.
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[11] P. V. Subrahmanyam, Cesàro summability of fuzzy real numbers, Jour. Analysis, 7(1999),
159-168.
[12] B. C. Tripathy, Statistically convergent double sequences, Tamkang J. Math., 34(2003), 231237.
[13] B. K. Tripathy and S. Nanda, Absolute value of fuzzy real numbers and fuzzy sequnce spaces,
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520
BINOD CHANDRA TRIPATHY AND AMAR JYOTI DUTTA
Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim
Boragaon, Garchuk, Guwahati-781 035, India.
E-mail: [email protected]; [email protected]
Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim
Boragaon, Garchuk, Guwahati-781 035, India.
E-mail: amar [email protected]