Filomat 26:4 (2012), 673–681
DOI 10.2298/FIL1204673M
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Statistical convergence of double sequences in fuzzy normed spaces
S.A. Mohiuddine
a
, H. S¸ evli
b
, M. Cancan
c
a
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
b
Department of Mathematics, Istanbul Commerce University, Uskudar, Istanbul, Turkey
c
Department of Mathematics, Yuzuncu Yil University, 65080 Van, Turkey
Abstract. In this paper, we study the concepts of statistically convergent and statistically Cauchy double
sequences in the framework of fuzzy normed spaces which provide better tool to study a more general class
of sequences. We also introduce here statistical limit point and statistical cluster point for double sequences
in this framework and discuss the relationship between them.
1. Introduction and preliminaries
By modifying own studies on fuzzy topological vector spaces, Katsaras [13] first introduced the notion
of fuzzy seminorm and norm on a vector space and later on Felbin [7] gave the concept of a fuzzy normed
space (for short, FNS) by applying the notion fuzzy distance of Kaleva and Seikala [11] on vector spaces.
Further, Xiao and Zhu [29] improved a bit the Felbin’s definition of fuzzy norm of a linear operator between
FNSs. Recently, Bag and Samanta [2] has given another notion of boundedness in FNS and introduced
another type of boundedness of operators. With the novelty of their approach they can introduce the fuzzy
dual spaces and some important analogues of fundamental theorems in classical functional analysis [3].
In many branches of science and engineering we often come across double sequences, i.e. sequences
of matrices and certainly there are situations where either the idea of ordinary convergence does not work
or the underlying space does not serve our purpose. Therefore to deal with such situations we have
to introduce some new type of measures which can provide a better tool and a suitable framework. In
particular, we are interested to put forward our studies to deal with the sequences of chaotic behaviour.
The idea of statistical convergence was introduced by Fast [6] and Steinhaus [28] independently in the
same year 1951 and since then several generalizations and application of this concept have been investigated
by various authors, e.g. [9], [12], [20], [21], [22], [24] and [25]. Recently, fuzzy version of this concept were
discussed in [15], [16], [18], [19] and [27].
In this paper we shall study the concept of convergence, statistical convergence and statistically Cauchy
for double sequences in the framework of fuzzy normed spaces. Finally, Section 3 is devoted to introduce
limit point, thin subsequence, non-thin subsequence, statistical limit point and statistical cluster point of
double sequences in fuzzy normed spaces and find relations among these concepts.
Firstly, we recall some notations and basic definitions used in this paper.
2010 Mathematics Subject Classification. Primary 40A05; Secondary 40D05, 46S40
Keywords. Fuzzy normed space, statistical convergence, statistically Cauchy, statistical limit point, statistical cluster point
Received: 26 December 2010; Accepted: 04 December 2011
Communicated by Dragan S. Djordjevi
´
c
Corresponding author: S.A. Mohiuddine
Email addresses: mohiuddine@gmail.com (S.A. Mohiuddine), hsevli@yahoo.com (H. S¸evli), m
−
cencen@yahoo.com (M. Cancan)
S.A. Mohiuddine et al. / Filomat 26:4 (2012), 673–681 674
According to Mizumoto and Tanaka [14], a fuzzy number is a mapping x : R → [0, 1] over the set R
of all real numbers. A fuzzy number x is convex if x(t) ≥ min{x(s), x(r)} where s ≤ t ≤ r. If there exists a
t
0
∈ R such that x(t
0
) = 1, then x is called normal. For 0 < α ≤ 1, α-level set of an upper semi continuous
convex normal fuzzy number η (denoted by [η]
α
) is a closed interval [a
α
, b
α
], where a
α
= −∞ and b
α
= +∞
admissible. When a
α
= −∞, for instance, then [a
α
, b
α
] means the interval (−∞, b
α
]. Similar is the case when
b
α
= +∞. A fuzzy number x is called non-negative if x(t) = 0, for all t < 0. We denoted the set of all convex,
normal, upper semicontinuous fuzzy real numbers by L(R) and the set of all non-negative, convex, normal,
upper semicontinuous fuzzy real numbers by L(R
∗
). Given a number r ∈ R, we define a corresponding
fuzzy number
˜
r by
˜
r(t) =
{
1 if t = r,
0 otherwise.
As α-level sets of a convex fuzzy number is an interval, there is a debate in the nomenclature of fuzzy
numbers/fuzzy real numbers. In [5], Dubois and Prade suggested to call this as fuzzy interval.
A partial ordering ≼ on L(R) is defined by u ≼ v if and only if u
−
α
≤ v
−
α
and u
+
α
≤ v
+
α
for all α ∈ [0, 1],
where [u]
α
= [u
−
α
, u
+
α
] and [v]
α
= [v
−
α
, v
−
α
]. The strict inequality in L(R) is defined by u ≺ v if and only if
u
−
α
< v
−
α
and u
+
α
< v
+
α
for all α ∈ [0, 1]. For k > 0, ku is defined as ku(t) = u(t/k) and (0u)(t) is defined to be
˜
0(t).
According to Mizumoto and Tanaka [14], the arithmetic operations ⊕, ⊖, ⊗ on L(R) ×L(R ) are defined
by
(x ⊕ y)(t) = sup
s∈R
min{x(s), y(t −s)}, (x ⊖ y)(t) = sup
s∈R
min{x(s), y(s −t)}, and
(x ⊗ y)(t) = sup
s∈R,s,0
min{x(s), y(t/s)},
for all t ∈ R.
Let u, v ∈ L(R). Define
D(u, v) = sup
α∈[0,1]
max{|u
−
α
− v
−
α
|, |u
+
α
− v
+
α
|},
then D is called the supremum metric on L(R). Let (u
n
) ⊂ L(R) and u ∈ L(R). We say that a sequence
(u
n
) converges to u in the metric D (for short, D-converges to u), written as u
n
D
→ u or (D)- lim
n→∞
u
n
= u if
lim
n→∞
D(u
n
, u) = 0.
In [7] Felbin introduced the concept of fuzzy normed linear space by applying the notion fuzzy distance
of Kaleva and Seikkala [11] on vector spaces. Recently, S¸enc¸imen and Pehlivan [27] gave a slightly simplified
version of this FNS as follows.
Let X is a vector space over R, ∥.∥ : X → L
∗
(R), mapping L, R : [0, 1] × [0, 1] → [0, 1] be symmetric,
non-decreasing in both arguments and satisfy L(0, 0) = 0 and R(1, 1) = 1.
Write [∥x∥]
α
= [∥x∥
−
α
, ∥x∥
+
α
] for x ∈ X and 0 ≤ α ≤ 1. Suppose that for all x ∈ X, x , θ, inf
α∈[0,1]
∥x∥
−
α
> 0,
where θ is the zero vector of X.
The quadruple (X, ∥.∥) is said to be fuzzy normed space (for short FNS) if the following conditions are
satisfied for every x, y ∈ X and s, t ∈ R:
(i) ∥x∥ =
˜
0 if and only if x = θ,
(ii) ∥αx∥ = |α|∥x∥, α ∈ R,
(iii) ∥x + y∥(s + t) ≥ L(∥x∥(s), ∥y∥(t)) whenever s ≤ ∥x∥
−
1
, t ≤ ∥y∥
−
1
and s + t ≤ ∥x + y∥
−
1
,
(iv) ∥x + y∥(s + t) ≤ R(∥x∥(s), ∥y∥(t)) whenever s ≥ ∥x∥
−
1
, t ≥ ∥y∥
−
1
and s + t ≥ ∥x + y∥
−
1
,
In this case ∥.∥ is called a fuzzy norm.
In the sequel we take L(x, y) = min(x, y) and R(x, y) = max(x, y) for all x, y ∈ [0, 1].
S.A. Mohiuddine et al. / Filomat 26:4 (2012), 673–681 675
Example 1.1. Let (X, ∥.∥
C
) be a ordinary normed linear space. Then a fuzzy norm ∥.∥on Xcan be obtained as
∥x∥(t) =
0 if 0 ≤ t ≤ a∥x∥
C
or t ≥ b∥x∥
C
,
t
(1−a)∥x∥
C
−
a
1−a
if a∥x∥
C
≤ t ≤ ∥x∥
C
,
−t
(b−1)∥x∥
C
+
b
b−1
if ∥x∥
C
≤ t ≤ b∥x∥
C
,
(1.1.1)
where ∥x∥
C
is the ordinary norm of x(, θ), 0 < a < 1 and 1 < b < ∞. For x = θ, define ∥x∥ =
˜
0. Hence (X, ∥.∥)
is a FNS. This fuzzy norm is called triangular fuzzy norm.
Let us consider the topological structure of a FNS (X, ∥.∥). For any ϵ > 0,α ∈ [0, 1] and x ∈ X, the (ϵ, α )-
neighborhood of x is the set
N
x
(ϵ, α) := {y ∈ X : ∥x − y∥
+
α
< ϵ}.
2. Statistically convergent and statistically Cauchy double sequences
Before proceeding further, we should recall some of the basic concepts on statistical convergence.
Let K be a subset of N, the set of natural numbers. Then the asymptotic density of K, denoted by δ(K) (see
[8],[28]), is defined as
δ(K) = lim
n
1
n
|{k ≤ n : k ∈ K}|,
where the vertical bars denote the cardinality of the enclosed set.
A number sequence x = (x
k
) is said to be statistically convergent to the number ℓ if for each ϵ > 0, the set
K(ϵ) = {k ≤ n : |x
k
− ℓ| > ϵ} has asymptotic density zero, i.e.
lim
n
1
n
|{k ≤ n : |x
k
− ℓ| > ϵ}| = 0.
In this case we write st- lim x = ℓ.
Notice that every convergent sequence is statistically convergent to the same limit, but its converse need
not be true.
A double sequence x = (x
jk
) is said to be Prin1sheim
′
s conver1ent (or P-conver1ent) if for given ϵ > 0 there
exists an integer N such that |x
jk
− ℓ| < ϵ whenever j, k > N. We shall write this as
lim
j,k→∞
x
jk
= ℓ,
where j and k tending to infinity independent of each other (cf.[23]).
Let K ⊆ N ×N be a two-dimensional set of positive integers and let K(m, n) be the numbers of (j, k) in K
such that j ≤ m and k ≤ n. Then the two-dimensional analogue of natural density can be defined as follows
[17].
The lower asymptotic density of the set K ⊆ N ×N is defined as
δ
2
(K) = lim inf
m,n
K(m, n)
mn
.
In case the sequence (K(m, n)/mn) has a limit in Pringsheim’s sense then we say that K has a double natural
density and is defined as
lim
m,n
K(m, n)
mn
= δ
2
(K).
For example, let K = {(i
2
, j
2
) : i, j ∈ N}. Then
δ
2
(K) = lim
m,n
K(m, n)
mn
≤ lim
m,n
√
m
√
n
mn
= 0,
S.A. Mohiuddine et al. / Filomat 26:4 (2012), 673–681 676
i.e. the set K has double natural density zero, while the set {(i, 2j) : i, j ∈ N} has double natural density 1/2.
Note that, if we set m = n , we have a two dimensional natural density due two Christopher [4].
A real double sequence x = (x
jk
) is said to be statistically convergent [17] to the number ℓ if for each ϵ > 0,
the set
{(j, k), j ≤ m and k ≤ n :| x
jk
− ℓ |≥ ϵ}
has double natural density zero. In this case we write st
2
- lim
j,k
x
jk
= ℓ.
Now we study the concept of convergence, statistical convergence and statistically Cauchy for double
sequences in fuzzy normed spaces. We define the following:
Definition 2.1. Let (X, ∥.∥) be a FNS. Then a double sequence (x
jk
) is said to be conver1ent to x ∈ X with
respect to the fuzzy norm on X if for every ϵ > 0 there exists a number N = N(ϵ) such that
D(∥x
jk
− x∥,
˜
0) < ϵ for all j, k ≥ N.
In this case we write x
jk
FN
−→ x. This means that for every ϵ > 0 there exists a number N = N(ϵ) such that
sup
α∈[0,1]
∥x
jk
− x∥
+
α
= ∥x
jk
− x∥
+
0
< ϵ
for all j, k ≥ N. In terms of neighborhoods, we have x
jk
FN
−→ x provided that for any ϵ > 0 there exists a
number N = N(ϵ) such that x
jk
∈ N
x
(ϵ, 0) whenever j, k ≥ N.
Definition 2.2. Let (X, ∥.∥) be a FNS. We say that a double sequence (x
jk
) is said to be statistically convergent
to x ∈ X with respect to the fuzzy norm on X if for every ϵ > 0,
δ
2
({(j, k) ∈ N × N : D(∥x
jk
− x∥,
˜
0) ≥ ϵ}) = 0.
This implies that for each ϵ > 0, the set
K(ϵ) := {(j, k) ∈ N × N : ∥x
jk
− x∥
+
0
≥ ϵ}
has natural density zero; namely, for each ϵ > 0, ∥x
jk
− x∥
+
0
< ϵ for almost all j, k. In this case we write
st
2
(FN)- lim ∥x
jk
− x∥ =
˜
0 or x
jk
st
2
(FN)
−→ x.
In terms of neighborhoods, we have x
jk
st
2
(FN)
−→ x if for every ϵ > 0,
δ
2
({(j, k) ∈ N × N : x
jk
< N
x
(ϵ, 0)}) = 0,
i.e., for each ϵ > 0, (x
jk
) ∈ N
x
(ϵ, 0) for almost all j, k.
A useful interpretation of the above definition is the following:
x
jk
st
2
(FN)
−→ x iff st
2
(FN)- lim ∥x
jk
− x∥
+
0
= 0.
Note that st
2
(FN)- lim ∥x
jk
− x∥
+
0
= 0 implies that
st
2
(FN )- lim ∥x
jk
− x∥
−
α
= st
2
(FN )- lim ∥x
jk
− x∥
+
α
= 0
for each α ∈ [0, 1], since
0 ≤ ∥x
jk
− x∥
−
α
≤ ∥x
jk
− x∥
+
α
≤ ∥x
jk
− x∥
+
0
holds for every j, k ∈ N and for each α ∈ [0, 1]. Hence the result.
Remark 2.1. If a double sequence (x
jk
) in a fuzzy normed space (X, ∥.∥) is convergent then it is also statisti-
cally convergent but converse need not be true, which can be seen by the following example.
S.A. Mohiuddine et al. / Filomat 26:4 (2012), 673–681 677
Example 2.1. Let (R
m
, ∥.∥) be a FNS and x = (x
jk
)
m
j,k=1
∈ R
m
be a fixed nonzero vector, where the fuzzy norm
on R
m
is defined as in (1.1.1) such that ∥x∥
C
=
(
m
∑
n=1
m
∑
j=1
|x
nj
|
2
)
1/2
. Now we define a double sequence (x
nj
) in
R
m
as
x
nj
=
{
x ; if n = j = k
2
, k ∈ N
θ ; otherwise.
Then we see that for any ϵ satisfying 0 < ϵ ≤ b∥x∥
C
where 1 < b < ∞, we have
K(ϵ) = {(j, k) ∈ N × N : ∥x
nj
− θ∥
+
0
≥ ϵ} = {(1 , 1), (4 , 4), (9 , 9), ···}.
Hence δ
2
(K(ϵ)) = 0. If we choose ϵ > b∥x∥
C
then K(ϵ) = ∅ and hence δ
2
(∅) = 0, that is (x
nj
)
st
2
(FN)
−→ θ. However
(x
nj
) is not convergent in (R
m
, ∥.∥).
Definition 2.3. Let (X, ∥.∥) be a FNS. Then a double sequence (x
jk
) is said to be statistically Cauchy with
respect to the fuzzy norm on X if for every ϵ > 0 there exist N = N(ϵ) and M = M(ϵ) such that for all j, p ≥ N;
k, q ≥ M
δ
2
({(j, k) ∈ N × N, j ≤ n and k ≤ m : ∥x
jk
− x
pq
∥
+
0
≥ ϵ}) = 0.
Theorem 2.1. Let (x
jk
) and (y
jk
) be a double sequences in a FNS (X, ∥.∥) such that x
jk
st
2
(FN)
−→ x and y
jk
st
2
(FN)
−→ y for all
x, y ∈ X . Then we have the following:
(i) (x
jk
+ y
jk
)
st
2
(FN)
−→ x + y,
(ii) αx
jk
st
2
(FN)
−→ αx, α ∈ R,
(iii) st
2
(FN )- lim ∥x
jk
∥ = ∥x∥.
Proof. (i) Suppose that x
jk
st
2
(FN)
−→ x and y
jk
st
2
(FN)
−→ x. Since ∥.∥
+
0
is a norm in the usual sense, we get
∥(x
jk
+ y
jk
) − (x + y)∥
+
0
≤ ∥x
jk
− x∥
+
0
+ ∥y
jk
− y∥
+
0
(2.1.1)
for all j, k ∈ N. Write
K(ϵ) := {(j, k) ∈ N × N : ∥(x
jk
+ y
jk
) − (x + y)∥
+
0
≥ ϵ},
K
1
(ϵ) := {(j, k) ∈ N ×N : ∥x
jk
− x∥
+
0
≥ ϵ/2},
K
2
(ϵ) = {(j, k) ∈ N × N : ∥y
jk
− y∥
+
0
≥ ϵ/2}.
From (2.1.1) that K(ϵ) ⊆ K
1
(ϵ) ∪ K
2
(ϵ). Now by assumption we have δ
2
(K
1
(ϵ)) = δ
2
(K
2
(ϵ)) = 0. This yields
δ
2
(K(ϵ)) = 0, i.e., (i) holds.
(ii) is obvious.
(iii) Since ∥.∥
−
α
and ∥.∥
+
α
are norms in the usual sense, we have
0 ≤ |∥x
jk
∥
−
α
− ∥x∥
−
α
| ≤ ∥x
jk
− x∥
−
α
and
0 ≤ |∥x
jk
∥
+
α
− ∥x∥
+
α
| ≤ ∥x
jk
− x∥
+
α
for all α ∈ [0, 1]. Therefore
0 ≤ max{|∥x
jk
∥
−
α
− ∥x∥
−
α
|, |∥x
jk
∥
+
α
− ∥x∥
+
α
|} ≤ ∥x
jk
− x∥
+
α
for all α ∈ [0, 1]. Taking supremum over α ∈ [0, 1], we get
0 ≤ D(∥x
jk
∥, ∥x∥) ≤ ∥x
jk
− x∥
+
0
.