Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 157467, 19 pages
doi:10.1155/2012/157467
Research Article
Statistical Convergence of Sequences of Functions
in Intuitionistic Fuzzy Normed Spaces
Vatan Karakaya,1 Necip Şimşek,2
Müzeyyen Ertürk,3 and Faik Gürsoy3
1
Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler,
34210 Istanbul, Turkey
2
Department of Mathematics, Istanbul Ticaret University, Uskudar, 34672 Istanbul, Turkey
3
Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler,
34220 Istanbul, Turkey
Correspondence should be addressed to Vatan Karakaya, vkkaya@yildiz.edu.tr
Received 29 June 2012; Revised 16 September 2012; Accepted 27 October 2012
Academic Editor: Ljubisa Kocinac
Copyright q 2012 Vatan Karakaya et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The purpose of this work is to investigate types of convergence of sequences of functions in
intuitionistic fuzzy normed spaces and some properties related with these concepts.
1. Introduction and Preliminaries
The fuzzy theory has emerged as one of the most active area of research in many branches
of mathematics and engineering. This new theory was introduced by Zadeh 1 in 1965
and since then a large number of research papers have appeared by using the concept of
fuzzy set/numbers and fuzzification of many classical theories have also been made. It has
also very useful application in various fields, for example, population dynamics 2, chaos
control 3, computer programming 4, nonlinear dynamical systems 5, fuzzy physics 6,
fuzzy topology 7, 8, and so forth. The notion of intuitionistic fuzzy sets, a generalization of
fuzzy sets, was introduced by Atanassov 9 in 1986 and later there has been much progress
in the study of intuitionistic fuzzy sets by many authors including Mursaleen et al. 10,
Mursaleen et al. 11, and Yılmaz 12. Using the idea of intuitionistic fuzzy sets, Park 13
defined the notion of intuitionistic fuzzy metric spaces with the help of the continuous tnorms and the continuous t-conorms as a generalization of fuzzy metric spaces due to George
and Veeramani 14. Samanta and Jebril 15 introduced the definitions of intuitionistic fuzzy
continuity and sequential intuitionistic fuzzy continuity and proved that they are equivalent.
2
Abstract and Applied Analysis
A few of the algebraic and topological properties of intuitionistic fuzzy continuity and
uniformly intuitionistic fuzzy continuity were investigated by Dinda and Samanta 16. On
the other hand, the Fast 17 introduced the concept of statistical convergence for real number
sequences. Different types of statistical convergence of sequences of real functions and related
notions were first studied in 18, and some important results and references on statistical
convergence and function sequences can be found in 19–23.
In this paper, primarily following the line of 18, we define statistical convergence
of sequences of functions in intuitionistic fuzzy normed space IFNS for short, and we
investigate some properties related with these concepts. To explain main problems of this
work, we have to give some definitions and literature acknowledgment. In 24, Schweizer
and Sklar introduced a continuous t-norm and a continuous t-conorm. Afterward Saadati and
Park 8 introduced the following definitions by using concepts mentioned above.
We now first recall some basic notions of intuitionistic fuzzy normed spaces.
Definition 1.1 see 8. Let ∗ be a continuous t-norm, a continuous t-conorm, and X a linear
space over the intuitionistic fuzzy field R or C. If μ and ν are fuzzy sets on X × 0, ∞
satisfying the following conditions, the five-tuple X, μ, ν, ∗, is said to be an IFNS and μ, ν
is called an intuitionistic fuzzy norm IFN for short. For every x, y ∈ X and s, t > 0,
i μx, t νx, t ≤ 1,
ii μx, t > 0,
iii μx, t 1 ⇔ x 0,
iv μax, t μx, t/|a| for each a /
0,
v μx, t ∗ μy, s ≤ μx y, t s,
vi μx, · : 0, ∞ → 0, 1 is continuous,
vii limt → ∞ μx, t 1 and limt → ∞ μx, t 0,
viii νx, t < 1,
ix νx, t 0 ⇔ x 0,
x νax, t νx, t/|a| for each a /
0,
xi νx, t νy, s ≥ νx y, t s,
xii νx, · : 0, ∞ → 0, 1 is continuous,
xiii limt → ∞ νx, t 0 and limt → ∞ νx, t 1.
For an IFNS, we further assume that X, μ, ν, ∗, satisfy the following axiom: see
16
xiv
aaa
a∗aa
∀a ∈ 0, 1.
1.1
Definition 1.2 see 8. Let X, μ, ν, ∗, be a intuitionistic fuzzy normed space. A subset A
of X is said to be IF-bounded if there exist t > 0 and 0 < r < 1 such that μx, t > 1 − r and
νx, t < r for each x ∈ A.
Abstract and Applied Analysis
3
Definition 1.3 see 25. Let X, μ, ν, ∗, be a intuitionistic fuzzy metric space. Let A be any
subset of X. Define
φt inf μ x, y, t : x, y ∈ A ,
ψt sup ν x, y, t : x, y ∈ A ,
1.2
i A is said to be q-bounded if limt → ∞ φt 1 and limt → ∞ ψt 0, ii A is said to be
semibounded if limt → ∞ φt k and limt → ∞ ψt 1 − k, 0 < k < 1 iii A is said to be
unbounded if limt → ∞ φt 0 and limt → ∞ ψt 1.
Theorem 1.4 see 25. Let X, μ, ν, ∗, be a intuitionistic fuzzy metric space. A subset of X is IF
bounded if and only if A is q-bounded or semibounded.
Definition 1.5 see 8. Let X, μ, ν, ∗, be an IFNS and xk be a sequence in X. The sequence
xk is said to be convergent to L ∈ X with respect to IFN μ, ν if for every ε > 0 and t > 0,
there exists a positive integer k0 ε such that μxk − L, t > 1 − ε and νxk − L, t < ε whenever
k > k0 . In this case, we write μ, ν − lim xk L as k → ∞.
Definition 1.6 see 16. Let X, μ, ν, ∗, and Y, μ , ν , ∗, be two IFNS. A mapping f from
X, μ, ν, ∗, to Y, μ , ν , ∗, is said to be intuitionistic fuzzy continuous at x0 ∈ X if for any
given ε > 0, there exist δ δa, ε, β βa, ε ∈ 0, 1 such that for all x ∈ X and for all
a ∈ 0, 1,
μx − x0 , δ > 1 − β ⇒ μ fx − fx0 , ε > 1 − a,
νx − x0 , δ < β ⇒ ν fx − fx0 , ε < a.
1.3
Definition 1.7 see 16. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. The
sequence fk is said to be pointwise intuitionistic fuzzy convergent on X to a function f with
respect to μ , ν if for each x ∈ X, the sequence fk x is convergent to fx with respect to
μ , ν .
Definition 1.8 see 16. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions.
The sequence fk is said to be uniformly intuitionistic fuzzy convergent on X to a function f
with respect to μ, ν, if given 0 < r < 1, t > 0, there exist a positive integer k0 k0 r, t such
that ∀x ∈ X and ∀k > k0 ,
μ fk x − fx, t > 1 − r,
ν fk x − fx, t < r.
1.4
Now, we recall the notion of the statistical convergence of sequences in intuitionistic fuzzy
normed spaces.
Definition 1.9 see 26. Let K ⊂ N and Kn {k ∈ K : k ≤ n}. Then the asymptotic density is
defined by δK limn → ∞ |Kn |/n, where |Kn | denotes the cardinality of Kn .
Definition 1.10. Let A be subset of N. If a property P k holds for all k ∈ A with δA 1, we
say that P holds for almost all ka · a · k.
4
Abstract and Applied Analysis
Definition 1.11 see 26. A sequence x xk is said to be statistically convergent to the
number L, or in short st − lim x L, if for every ε > 0, the set Kε has asymptotic density
zero, where
Kε {k ∈ N : |xk − L| ≥ ε},
1.5
that is,
|xk − L| < ε
a · a · k.
1.6
Definition 1.12 see 23. Let X, μ, ν, ∗, be an IFNS. Then, sequence xk is said to be
statistically convergent to L ∈ X with respect to IFN μ, ν provided that for every ε > 0
and t > 0,
δ
k ∈ N : μxk − L, t ≤ 1 − ε or νxk − L, t ≥ ε 0,
1.7
or equivalently
lim
1 k ≤ n : μxk − L, t ≤ 1 − ε or νxk − L, t ≥ ε 0.
n→∞n
1.8
In this case, we write stμ,ν − limxk L.
Definition 1.13 see 23. Let X, μ, ν, ∗, be an IFNS. Sequence xk is said to be statistically
Cauchy with respect to IFN μ, ν provided that for every ε > 0 and t > 0, there exists a
number m ∈ N satisfying
δ
k ∈ N : μxk − xm , t ≤ 1 − ε or νxk − xm , t ≥ ε 0.
1.9
2. Statistical Convergence of Sequences of Functions in
Intuitionistic Fuzzy Normed Spaces
In this section, we define pointwise statistically and uniformly statistically convergent
sequences of functions in intuitionistic fuzzy normed spaces. Also, we give the statistical
analog of the Cauchy convergence criterion for pointwise and uniformly statistical
convergent in intuitionistic fuzzy normed space. Finally, we prove that uniformly statistical
convergence preserves continuity.
Definition 2.1. Let X, μ, ν, ∗, and Y, μ , ν , ∗, be two IFNS and fk : X, μ, ν, ∗, →
Y, μ , ν , ∗, a sequence of functions. We say that a sequence fk pointwise statistically
converges to f with respect to intuitionistic fuzzy norm μ , ν if fk x statistically converges
to fx for each x ∈ X with respect to intuitionistic fuzzy norm μ , ν and we write
stμ,ν − fk → f.
Theorem 2.2. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. If fk is pointwise
intuitionistic fuzzy convergent on X with respect to μ, ν, then stμ,ν − fk → f. But the converse of
this is not true.
Abstract and Applied Analysis
5
Proof. Let fk be pointwise intuitionistic fuzzy convergent on X. In this case the sequence
fk x is convergent with respect to μ , ν for each x ∈ X. Then for each ε > 0 and t > 0,
there is number k0 ε ∈ N such that
μ fk x − fx, t > 1 − ε,
ν fk x − fx, t < ε
2.1
for all k ≥ k0 and for each x ∈ X. Hence for each x ∈ X the set
k ∈ N : μ fk x − fx, t ≤ 1 − ε or ν fk x − fx, t ≥ ε
2.2
has finite numbers of terms. Since density of finite subset of N is 0, hence
δ
k ∈ N : μ fk x − fx, t ≤ 1 − ε or ν fk x − fx, t ≥ ε 0.
2.3
That is, stμ,ν − fk → f.
Example 2.3. Let R, | · | denote the space of real numbers with the usual norm, and let a ∗ b a · b and a b min{a b, 1} for a, b ∈ 0, 1. For all x ∈ R and every t > 0, we consider
μx, t t
,
t |x|
νx, t |x|
.
t |x|
2.4
In this case, R, μ, ν, ∗, is an IFNS also, 0, 1, μ, ν, ∗, is intuitionistic fuzzy normed
space.. Let fk : 0, 1 → R be a sequence of functions whose terms are given by
⎧
1
⎪
k2
2
⎪
1
if
k
m
∈
N,
x
∈
0,
,
x
m
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
1
⎪
2
⎪
,
0
if k /
m m ∈ N, x ∈ 0,
⎪
⎪
⎪
2
⎪
⎨
fk x 1
2
⎪
,
1
,
0
if
k
m
∈
N,
x
∈
m
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
1
1
⎪
k
2
⎪
⎪
if
k
m
,
1
,
x
∈
N,
x
∈
m
/
⎪
⎪
2
2
⎪
⎪
⎩
2
if x 1.
2.5
6
Abstract and Applied Analysis
Then sequence fk is pointwise statistically intuitionistic convergent on 0, 1 with respect to
μ, ν. Indeed, for x ∈ 0, 1/2, since
Kn ε, t k ≤ n : μ fk x − fx, t ≤ 1 − ε or ν fk x − fx, t ≥ ε ,
fk x − 0
t
≤ 1 − ε or
≥ε
Kn ε, t k ≤ n :
t fk x − 0
t fk x − 0
εt
k ≤ n : fk x ≥
1−ε
2
k ≤ n : fk x xk 1
2.6
k ≤ n : k m2 , m ∈ N ,
we have
√n
1 2
δKn ε, t k ≤ n : k m , m ∈ N ≤
,
n
n
2.7
which yields limn → ∞ δKn ε, t 0. Thus, for each x ∈ 0, 1/2, sequence fk is statistically
convergent to 0 with respect to intuitionistic fuzzy norm μ, ν.
If we take x ∈ 1/2, 1, then we have
fk x − 1/2
t
≤ 1 − ε or
≥ε
k≤n:
t fk x − 1/2
t fk x − 1/2
εt
1
k ≤ n : fk x − ≥
2
1−ε
εt
1
2 k ≤ n : k m2 , fk x − ≥
∪ k ≤ n : k
m
,
/
fk x −
2
1−ε
εt
1 1 2 k
≥
x
k ≤ n : k m2 ∪ k ≤ n : k −
m
:
/
2 2 1 − ε
Kn ε, t
k ≤ n : k m2 ∪ k ≤ n : k /
m2 : xk ≥
εt
.
1−ε
εt
1 ≥
2 1 − ε
2.8
Therefore, density of Kn ε, t is 0 and for each x ∈ 1/2, 1, sequence fk is statistically
convergent to 1/2 with respect to IFN μ, ν. If we take x 1, it can be seen easily that fk is
Abstract and Applied Analysis
7
intuitionistic fuzzy convergent to 2. Hence fk is intuitionistic fuzzy statistically convergent
to 2. That is
⎧
⎪
⎪
⎪0,
⎪
⎪
⎪
⎨
1
stμ,ν − fk x ,
⎪
⎪
2
⎪
⎪
⎪
⎪
⎩
2,
1
x ∈ 0,
2
1
x∈
,1
2
2.9
x 1.
Since fk x is statistically convergent to different points with respect to intuitionistic fuzzy
norm μ, ν for each x ∈ X, it can be seen that fk is pointwise statistically intuitionistic fuzzy
convergent on 0, 1.
Lemma 2.4. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be sequence of functions. Then the following
statements are equivalent:
i stμ,ν − fk → f,
ii δ{k ∈ N : μ fk x − fx, t ≤ 1 − ε} δ{k ∈ N : ν fk x − fx, t ≥ ε} 0 for each
x ∈ X, for each ε > 0 and t > 0,
iii δ{k ∈ N : μ fk x − fx, t > 1 − ε and ν fk x − fx, t < ε} 1 for each x ∈ X, for
each ε > 0 and t > 0,
iv st − lim μ fk x − fx, t 1 and st − lim ν fk x − fx, t 0 for each x ∈ X and
t > 0.
Theorem 2.5. Let fk and gk be two sequences of functions from X, μ, ν, ∗, to Y, μ , ν , ∗, .
If stμ,ν − fk → f and stμ,ν − gk → g, then stμ,ν − αfk βgk → αf βg where α, β ∈ R or C.
Proof. The proof is clear for α 0 and β 0. Now let α /
0 and β /
0. Since stμ,ν − fk → f and
stμ,ν − gk → g, for each x ∈ X, if we define
t
t
≤ 1 − ε or ν fk x − fx,
≥ε ,
A1 k ∈ N : μ fk x − fx,
2|α|
2|α|
t
t
A2 k ∈ N : μ gk x − gx, ≤ 1 − ε or ν gk x − gx, ≥ ε ,
2β
2β
2.10
then
δA1 0 ,
δA2 0.
2.11
Since δA1 0 and δA2 0, if we state A by A1 ∪ A2 then
δA 0.
2.12
8
Abstract and Applied Analysis
N and there exists ∃m ∈ N such that
Hence, A1 ∪ A2 /
t
> 1 − ε,
μ fm x − fx,
2|α|
t
μ gm x − gx, > 1 − ε,
2β
t
ν fm x − fx,
< ε,
2|α|
t
ν gm x − gx, < ε.
2β
2.13
Let
B k ∈ N : μ αfk βgk x − αfx βgx , t > 1 − ε,
ν αfk βgk x − αfx βgx , t < ε .
2.14
We will show that for each x ∈ X
Ac ⊂ B.
2.15
Let m ∈ Ac . In this case,
t
> 1 − ε,
μ fm x − fx,
2|α|
t
μ gm x − gx, > 1 − ε,
2β
t
ν fm x − fx,
< ε,
2|α|
t
ν gm x − gx, < ε.
2β
2.16
Using those above, we have
μ
αfm βgm
t
t
∗ μ βgm x − βgx,
x − αf βg x, t ≥ μ αfm x − αfx,
2
2
t
t
μ fm x − fx,
∗ μ gm x − gx, 2|α|
2β
> 1 − ε ∗ 1 − ε
1 − ε,
t
t
ν αfm βgm x − αf βg x, t ≤ ν αfm x − αfx,
∗ ν βgm x − βgx,
2
2
t
t
ν fm x − fx,
∗ ν gm x − gx, 2|α|
2β
< εε
ε.
2.17
Abstract and Applied Analysis
9
This implies that
Ac ⊂ B.
2.18
δB c 0,
2.19
k ∈ N : μ αfk βgk x − αf βg x, t ≤ 1 − ε,
ν αfk βgk x − αf βg x, t ≥ ε 0
2.20
stμ,ν − αfk βgk −→ αf βg.
2.21
Since Bc ⊂ A and δA 0, hence
that is
δ
which means
Definition 2.6. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. The sequence
fk is a pointwise statistically Cauchy sequence in IFNS provided that for each ε > 0 and
t > 0 there exists N Nε, t, x such that
δ
k ∈ N : μ fk x − fN x, t ≤ 1 − ε or ν fk x − fN x, t ≥ ε for each x ∈ X 0.
2.22
That is, there exists a number N Nε, t, x for each x ∈ X such that
μ fk x − fN x, t > 1 − ε, ν fk x − fN x, t < ε
for a · a · k.
2.23
Theorem 2.7. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. If fk is a
pointwise statistically convergent sequence with respect to intuitionistic fuzzy norm μ, ν, then fk is a pointwise statistically Cauchy sequence with respect to intuitionistic fuzzy norm μ, ν.
Proof. Suppose that stμ,ν − fk → f and let ε > 0, t > 0. For given each ε > 0, choose s > 0 such
that 1 − ε ∗ 1 − ε > 1 − s and ε ε < s. If we state, respectively, Ax ε, t and Acx ε, t by
t
t
k ∈ N : μ fk x − fx,
≤ 1 − ε or ν fk x − fx,
≥ε ,
2
2
t
t
k ∈ N : μ fk x − fx,
> 1 − ε, ν fk x − fx,
<ε ,
2
2
2.24
for each x ∈ X. Then, we have
δAx ε, t 0,
2.25
10
Abstract and Applied Analysis
which implies that
δAcx ε, t 1.
2.26
Let N ∈ Acx ε, t. Then
t
> 1 − ε,
μ fN x − fx,
2
t
ν fN x − fx,
< ε.
2
2.27
We want to show that there exists a number N Nx, ε, t such that
δ
k ∈ N : μ fk x − fN x, t ≤ 1 − s or ν fk x − fN x, t ≥ s for each x ∈ X 0.
2.28
Therefore, define for each x ∈ X,
Bx ε, t k ∈ N : μ fk x − fN x, t ≤ 1 − s or ν fk x − fN x, t ≥ s .
2.29
We have to show that
Bx ε, t ⊂ Ax ε, t.
2.30
⊆Ax ε, t.
Bx ε, t/
2.31
Suppose that
In this case, Bx ε, t has at least one different element which Ax ε, t does not has. Let k ∈
Bx ε, t \ Ax ε, t. Then we have
μ fk x − fN x, t ≤ 1 − s,
t
> 1 − ε,
μ fk x − fx,
2
2.32
in particularly μ fN x − fx, t/2 > 1 − ε. In this case,
t
t
∗ μ fN x − fx,
1 − s ≥ μ fk x − fN x, t ≥ μ fk x − fx,
2
2
2.33
≥ 1 − ε ∗ 1 − ε > 1 − s,
which is not possible. On the other hand
ν fk x − fN x, t ≥ s,
ν fk x − fx, t < ε,
2.34
Abstract and Applied Analysis
11
in particularly ν fN x − fx, t < ε. In this case,
t
t
s ≤ ν fk x − fN x, t ≤ ν fk x − fx,
ν fN x − fx,
2
2
2.35
< εε <s
which is not possible. Hence Bx ε, t ⊂ Ax ε, t. Therefore, by δAx ε, t 0, δBx ε, t 0.
That is, fk is a pointwise statistical Cauchy sequence with respect to intuitionistic fuzzy
norm μ, ν.
Afterward this step, we introduce a uniformly statistical convergence of sequences of
function in an IFNS. To do this, we need the following definition
Definition 2.8. Let X, μ, ν, ∗, and Y, μ , ν , ∗, be two intuitionistic fuzzy normed linear
space over the same field IF and fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions.
fk converges uniform statistically to f with respect to μ, ν ⇔ ∀ε > 0, ∃M ⊂ N, δM 1
and ∃k0 k0 ε, t ∈ M ∀k > k0 and k ∈ M and ∀x ∈ X,
μ fk x − fx, t > 1 − ε,
ν fk x − fx, t < ε.
2.36
Lemma 2.9. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. Then the following
statements are equivalent:
i stμ,ν − fk ⇒ f.
ii δ{k ∈ N : μ fk x − fx, t ≤ 1 − ε} δ{k ∈ N : ν fk x − fx, t ≥ ε} 0 for all
x ∈ X, for every ε > 0 and t > 0.
iii δ{k ∈ N : μ fk x − fx, t > 1 − ε and ν fk x − fx, t < ε} 1 for all x ∈ X, for
every ε > 0 and t > 0.
iv st − lim μ fk x − fx, t 1 and st − lim ν fk x − fx, t 0 for all x ∈ X and
t > 0.
Proposition 2.10. Let the sequence fk and f be bounded functions from X, μ, ν, ∗, to
Y, μ , ν , ∗, . fk is intuitionistic fuzzy uniformly statistically convergent to f if and only if
st − lim inf μ fk x − fx, t 1,
st − lim sup ν fk x − fx, t 0,
2.37
where the supremum and infimum are taken over all x ∈ X.
Proof. Suppose that stμ,ν − fk ⇒ f on X. Since fk and f are bounded in X, μ, ν, ∗, for
each k ∈ N, by using Definition 1.3 and Theorem 1.4, we have inf μ fk x − fx, t rk and
sup ν fk x − fx, t 1 − rk for each k ∈ N and for each t > 0, 0 < rk < 1. By using iv of
Lemma 2.9, we get
st − lim inf μ fk x − fx, t 1,
st − lim sup ν fk x − fx, t 0,
where the supremum and infimum are taken over all x ∈ X.
2.38
12
Abstract and Applied Analysis
Conversely, suppose that
st − lim inf μ fk x − fx, t 1,
st − lim sup ν fk x − fx, t 0,
2.39
where the supremum and infimum are taken over all x ∈ X. Since
inf μ fk x − fx, t ∈ 0, 1,
sup ν fk x − fx, t ∈ 0, 1,
2.40
from definition statistical convergence, for a · a · k, for every ε > 0 and t > 0
inf μ fk x − fx, t − 1 < ε,
sup ν fk x − fx, t − 0 < ε.
2.41
For all x ∈ X
μ fk x − fx, t ≥ inf μ fk x − fx, t
⇒ − μ fk x − fx, t < − inf μ fk x − fx, t
⇒ 1 − μ fk x − fx, t < 1 − inf μ fk x − fx, t
since μ fk x − fx, t ∈ 0, 1 ∀ x ∈ X
⇒ 1 − μ fk x − fx, t < 1 − inf μ fk x − fx, t 2.42
⇒ 1 − μ fk x − fx, t < 1 − inf μ fk x − fx, t < ε
⇒ 1 − μ fk x − fx, t < ε a · a · k,
ν fk x − fx, t ≤ sup ν fk x − fx, t
⇒ ν fk x − fx, t − 0 ≤ sup ν fk x − fx, t − 0 < ε
⇒ ν fk x − fx, t − 0 < ε a · a · k.
Therefore, st − lim μ fk x − fx, t 1 and st − lim ν fk x − fx, t 0 for all x ∈ X and
t > 0. From Lemma 2.9, we get stμ,ν − fk ⇒ f.
Example 2.11. Let R, μ, ν, ∗, be as Example 2.3. Consider fk : 0, 1 → R be sequence of
functions whose terms are given by
fk xk 1,
2,
if k /
m2
m ∈ N
otherwise.
2.43
Then, for every 0 < ε < 1 and for every t > 0, we define
Kn k ≤ n : μ fk x − fx, t ≤ 1 − ε or ν fk x − fx, t ≥ ε .
2.44
Abstract and Applied Analysis
13
For all x ∈ X, we have
δKn δ
k ≤ n : μ fk x − 1, t ≤ 1 − ε or ν fk x − 1, t ≥ ε 0.
2.45
Since
stμ,ν − fk x 1,
2.46
for all x ∈ X, the sequence fk is uniformly statistically intuitionistic fuzzy convergent to 1
on 0, 1.
Example 2.12. Let R, μ, ν, ∗, be as Example 2.3. Consider fk : −1, 1 → R be sequence of
functions whose terms are given by
⎧
2
⎪
⎨ x− 1 ,
fk k
⎪
⎩2,
if k / m2
m ∈ N
2.47
otherwise.
Since stμ,ν − fk x x2 for all x ∈ −1, 1,fk is uniformly statistically intuitionistic fuzzy
convergent to x2 on −1, 1. We can show this using Proposition 2.10 as the following, since
fk is bounded functions sequence on −1, 1.
We want to find
inf μ fk x − fx, t t
,
k → ∞ x∈−1,1
t |−2x/k 1/k2 |
−2x/k 1/k2 ·
lim sup ν fk x − fx, t k → ∞ x∈−1,1
t |−2x/k 1/k2 |
lim
2.48
Firstly, we need to find limk → ∞ inf μfk x − fx, t and limk → ∞ sup νfk x − fx, t for
over all x ∈ −1, 1/2k and for over all x ∈ 1/2k, 1, respectively. In case of x ∈ −1, 1/2k,
we have −2x/k 1/k2 ≥ 0. Since
0≤−
1
1
1
1
2
2x
2
2x
2 ≤ 2 ⇒ t ≤ t −
2 ≤t 2
k
k k
k
k k
k
k
⇒
1
1
≤
2
t 2/k 1/k t − 2x/k 1/k2 ⇒
t
t
≤
≤ 1,
2
t 2/k 1/k t − 2x/k 1/k2 ≤
1
t
2.49
we get
lim
inf
μ fk x − x2 , t lim
k → ∞ x∈−1,1/2k
inf
k → ∞ x∈−1,1/2k t
t
− 2x/k 1/k2 t
lim
1.
k → ∞ t 2/k 1/k 2 2.50
14
Abstract and Applied Analysis
On the other hand, we have
−2x/k 1/k2
2/k 1/k2
,
≤
t
t − 2x/k 1/k2 2.51
−2x/k 1/k2
lim sup ν fk x − x , t lim sup
k → ∞ x∈−1,1/2k
k → ∞ x∈−1,1/2k t − 2x/k 1/k 2 2/k 1/k2
0.
lim
k→∞
t
2.52
and so
2
In case of x ∈ 1/2k, 1, we have −2x/k 1/k2 < 0. Since
−
1
1
1
1
2
2x
2x
2
2 ≤ 0 ⇒ 0 ≤
− 2 < − 2
<−
k k2
k
k
k k
k
k
⇒ t < t 1
1
2x
2
− 2 ≤t − 2
k
k k
k
1
1
1
<
≤
⇒
t 2/k − 1/k2 t 2x/k − 1/k2 t
⇒
2.53
t
t
≤
< 1,
t 2/k − 1/k2 t 2x/k − 1/k2 we get
lim
inf
k → ∞ x∈1/2k,1
μ fk x − x2 , t lim
inf
k → ∞ x∈1/2k,1 t
t
2x/k − 1/k2 t
1.
k → ∞ t 2/k − 1/k 2 2.54
lim
On the other hand, we have
2x/k − 1/k2
2/k − 1/k2
,
≤
t
t 2x/k − 1/k2 2.55
2x/k − 1/k2
lim sup ν fk x − x , t lim sup
k → ∞ x∈1/2k,1
k → ∞ x∈1/2k,1 t 2x/k − 1/k 2 2/k − 1/k2
0.
lim
k→∞
t
2.56
and so
2
Abstract and Applied Analysis
15
That is, the sequence fk is uniformly statistically intuitionistic fuzzy convergent to x2 on
−1, 1.
Remark 2.13. If stμ,ν − fk ⇒ f, then stμ,ν − fk → f. But the converse of this is not true.
We prove this with the following example.
Example 2.14. Let us define the sequence of functions
fk x ⎧
⎨0
⎩
k n2
k x
1 k 3 x2
2
otherwise,
2.57
on 0, 1. This sequence of functions is pointwise statistically intuitionistic fuzzy convergent
to 0 indeed, for x 1/k, stμ,ν − fk 1/k 1 and for x 0, stμ,ν − fk 0 0. But, it is
not uniformly statistical intuitionistic fuzzy convergent. Since the sequence of functions is
bounded on 0, 1, we can use Proposition 2.10 to prove our claim. Let us take infumum for
μfk x − 0, t μk 2 x/1 k3 x2 , t t/t k2 x/1 k3 x2 and supremum for νfk x −
0, t νk2 x/1 k 3 x2 , t k2 x/1 k3 x2 /t k2 x/1 k 3 x2 , over all x ∈ 0, 1.
Firstly, we try to find supx∈0,1 k2 x/1 k3 x2 and infx∈0,1 k2 x/1 k3 x2 . For this, we
have
k2 x
1 k 3 x2
k 2 1 − k 3 x2
1 n3 x2 2
Since fk 0 0, fk 1 k2 /1 k3 and fk k−3/2 √
k/2 and infx∈0,1 k2 x/1 k3 x2 0. Then,
0 ⇒ x k−3/2 .
2.58
√
k/2, we get supx∈0,1 k2 x/1 k3 x2 √
√
k2 x
k
k
k2 x
⇒ t ≤
≤t
3
2
3
2
2
2
1k x
1k x
⇒
⇒
0≤
t
t
1
√
1
≤
2 x/1 k 3 x2 t
k
k/2
t
√
t
,
≤
2 x/1 k 3 x2 t
k
k/2
k x
k x
⇒ t 0 ≤ t 1 k 3 x2
1 k 3 x2
2
2
⇒
1
t
k2 x/1
k3 x2 ≤
1
t
√
k2 x/ 1 k3 x2
k2 x/ 1 k3 x2
k
≤
,
⇒
≤
t
2t
t k2 x/1 k3 x2 2.59
16
Abstract and Applied Analysis
so, we get
lim inf μ
k→∞
k2 x
,t
1 k 3 x2
lim inf
k→∞
t
t
lim
√
t k2 x/1 k3 x2 k → ∞ t k/2
2t
1,
√ 0/
k → ∞ 2t k
√
k2 x/ 1 k3 x2
k2 x
k
∞/
0.
lim
, t lim sup
lim sup v
k→∞
k→∞
1 k 3 x2
t k2 x/1 k3 x2 k → ∞ 2t
lim
2.60
Therefore, we conclude that fk does not intuitionistic fuzzy uniformly statistical convergent
to 0.
Theorem 2.15. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. If fk is
uniformly intuitionistic fuzzy convergent on X to a function f with respect to μ, ν, then stμ,ν −fk ⇒
f. But the converse of this is not true.
Proof. Let fk be uniformly intuitionistic fuzzy convergent on X to a function f. In this case,
given 0 < ε < 1, t > 0, there exist a positive integer k0 k0 r, t such that ∀x ∈ X and ∀k > k0 ,
μ fk x − fx, t > 1 − ε,
ν fk x − fx, t < ε.
2.61
μ fk x − fx, t ≤ 1 − ε,
ν fk x − fx, t ≥ ε,
2.62
That is, for k ≤ k0
is satisfied and these k’s are finite. Since finite set has 0-density, density of complement of
finite set is 1. If complement of this finite set is stated by M, for every ε > 0, there exist
M ⊂ N, δM 1 and ∃k0 k0 ε, t ∈ M such that ∀k > k0 and k ∈ M and ∀x ∈ X,
μ fk x − fx, t > 1 − ε,
ν fk x − fx, t < ε.
2.63
This shows that stμ,ν − fk ⇒ f.
Definition 2.16. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions.The
sequence fk is a uniformly statistically Cauchy sequence in intuitionistic fuzzy normed
space provided that for every ε > 0 and t > 0, there exists a number N Nε, t such that
δ
k ∈ N : μ fk x − fN x, t ≤ 1 − ε or ν fk x − fN x, t ≥ ε ∀ x ∈ X 0.
2.64
Theorem 2.17. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be sequence of functions. If fk is a
uniformly statistically convergent sequence with respect to intuitionistic fuzzy norm μ, ν, then fk is uniformly statistically Cauchy sequence with respect to intuitionistic fuzzy norm μ, ν.
Abstract and Applied Analysis
17
Proof. Suppose that stμ,ν − fk ⇒ f. In this case, ∀ε > 0, there exists M ⊂ N, δM 1 and
k0 k0 ε, t ∈ M such that ∀k > k0 , k ∈ M and ∀x ∈ X,
t
> 1 − ε,
μ fk x − fx,
2
t
ν fk x − fx,
< ε.
2
2.65
Choose N Nε, t ∈ M, N > k0 . So, μ fN x−fx, t/2 > 1−ε and ν fN x−fx, t/2 < ε.
We investigate N Nε, t such that
δ
k ∈ N : μ fk x − fN x, t ≤ 1 − ε or ν fk x − fN x, t ≥ ε ∀ x ∈ X 0,
2.66
or
δ K δ k ∈ N : μ fk x − fN x, t > 1 − ε or ν fk x − fN x, t < ε ∀ x ∈ X 1.
2.67
For every k ∈ M, we have
μ fk x − fN x, t μ fk x − fx fx − fN x, t
t
t
≥ μ fk x − fx,
∗ μ fx − fN x,
2
2
> 1 − ε ∗ 1 − ε
1 − ε,
ν fk x − fN x, t ν fk x − fx fx − fN x, t
t
t
≤ ν fk x − fx,
∗ ν fx − fN x,
2
2
2.68
> εε
ε.
Since δM 1, fk is a uniformly statistically Cauchy sequence in intuitionistic fuzzy
normed space.
Theorem 2.18. Let X, μ, ν, ∗, and Y, μ , ν , ∗, be two IFNS and the mapping fk :
X, μ, ν, ∗, → Y, μ , ν , ∗, be the intuitionistic fuzzy continuous on X of sequence of functions.
If stμ,ν − fk ⇒ f, the mapping f : X → Y is the intuitionistic fuzzy continuous on X.
Proof. Let x0 ∈ X be an arbitrary point. By the intuitionistic fuzzy continuity of fk ’s, for every
ε > 0 and t > 0 there exists δ δx0 , ε, t/3 > 0 such that
t
> 1 − ε,
μ fk x0 − fk x,
3
ν
t
fk x0 − fk x,
3
< ε,
2.69
18
Abstract and Applied Analysis
for every k ∈ N and all x such that μx0 − x, t > 1 − δ and νx0 − x, t < δ. Let x ∈ Bx0 , δ, t
be fixed Bx0 , δ, t stands for an open ball in X, μ, ν, ∗, with center x0 and radius δ. Since
stμ,ν − fk ⇒ f on X, for all x ∈ X, if we state, respectively, A and B by these sets
t
t
k ∈ N : μ fk x − fx,
≤ 1 − ε or ν fk x − fx,
≥ ε ∀x ∈ X ,
3
3
t
t
B k ∈ N : μ fk x0 − fx0 ,
≤ 1 − ε or ν fk x0 − fx0 ,
≥ ε ∀x ∈ X
3
3
A
2.70
then, δA 0 and δB 0, hence δA ∪ B 0 and A ∪ B is different from N. Thus, there
exists m ∈ N such that
t
μ fm x − fx,
>1−ε ,
3
t
μ fm x0 − fx0 ,
> 1 − ε,
3
t
ν fm x − fx,
< ε,
3
t
ν fm x0 − fx0 ,
< ε.
3
2.71
Now, we will show that f is intuitionistic fuzzy contunious at x0 . Using Definition 1.1, we
have
μ fx − fx0 , t μ fx − fm x fm x − fm x0 fm x0 − fx0 , t
t
t
t
≥ μ fx − fm x,
∗ μ fm x − fm x0 ,
∗ μ fm x0 − fx0 ,
3
3
3
> 1 − ε ∗ 1 − ε ∗ 1 − ε
1 − ε,
ν fx − fx0 , t ν fx − fm x fm x − fm x0 fm x0 − fx0 , t
t
t
t
≤ ν fx − fm x,
∗ ν fm x − fm x0 ,
∗ ν fm x0 − fx0 ,
3
3
3
< εεε
ε.
2.72
Thus, the proof is completed.
Acknowledgments
This work is supported by The Scientific and Technological Research Council of Turkey
TÜBİTAK under the Project number 110T699. We have benefited a lot from the report of the
anonymous referee. So, the authors are thankful for his/her valuable comments and careful
corrections on the first draft of this paper which improved the presentation and readability.
Abstract and Applied Analysis
19
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