Volume 2012, Article ID 926193,14pages doi:10.1155/2012/926193
Research Article
λ-Statistical Convergence of Sequences of
Functions in Intuitionistic Fuzzy Normed Spaces
Vatan Karakaya,
1Necip S¸ims¸ek,
2M ¨uzeyyen Ert ¨urk,
3and Faik G ¨ursoy
31Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, 34210 Istanbul, Turkey
2Department of Mathematics, Istanbul Commerce University, Uskudar, 34672 Istanbul, Turkey
3Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey
Correspondence should be addressed to M ¨uzeyyen Ert ¨urk,merturk3263@gmail.com Received 18 July 2012; Accepted 17 September 2012
Academic Editor: Manuel Sanchis
Copyrightq 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.
We study λ-statistically convergent sequences of functions in intuitionistic fuzzy normed spaces.
We define concept of λ-statistical pointwise convergence and λ-statistical uniform convergence in intuitionistic fuzzy normed spaces and we give some basic properties of these concepts.
1. Introduction and Some Definitions
Fuzzy logic was introduced by Zadeh 1. Since then, the importance of fuzzy logic has come increasingly to the present. There are many applications of fuzzy logic in the field of science and engineering, for example, population dynamics 2, chaos control 3, 4, computer programming5, nonlinear dynamical systems 6, fuzzy topology 7, and so forth. The concept of intuitionistic fuzzy set, as a generalization of fuzzy logic, was introduced by Atanassov8 in 1983.
In the literature, t-norm and t-conorm were defined by Schweizer and Sklar9. The norms on intuitionistic fuzzy sets are introduced firstly in 10. Recently Park 11 has introduced the concept of intuitionistic fuzzy metric space and in 12, Saadati and Park introduced intuitionistic fuzzy normed spaces and concept of convergence of a sequence in intuitionistic fuzzy normed spaces. In light of these developments, intuitionistic fuzzy analogues of many concepts in classical analysis were studied by many authors13–17, and so forth.
The concept of statistical convergence was introduced by Fast 18 and Steinhaus
19 independently. Mursaleen defined λ-statistical convergence in 20. Also the concept of statistical convergence was studied in intuitionistic fuzzy normed space in 21. Quite recently, Karakaya et al. 22 defined and studied statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces. Mohiuddine and Lohani 23 defined and studied λ-statistical convergence in intuitionistic fuzzy normed spaces.
In this paper, we will study concept λ-statistical convergence for sequences of func- tions and investigate some basic properties related to the concept in intuitionistic fuzzy normed space.
We first recall some basic notions and definitions of intuitionistic fuzzy normed spaces.
Definition 1.1see 12. Let ∗ be a continuous t-norm, let be a continuous t-conorm, and X be a linear space over the field IFR or C. If μ and ν are fuzzy sets on X ×0, ∞ satisfying the following conditions, the five-tupleX, μ, ν, ∗, is said to be an intuitionistic fuzzy normed space andμ, ν is called an intuitionistic fuzzy norm. 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 → 0μx, 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 → 0νx, t 1.
Definition 1.2see 19,24. Let K ⊂ N and Kn {k ∈ K : k ≤ n}. Then, the natural density is defined by δK limn → ∞|Kn|/n, where |Kn| denotes the cardinality of Kn. A sequence x xk is said to be statistically convergent to the number L if for every ε > 0, the set Nε
has asymptotic density zero, where
Nε {k ∈ N : |xk− L| ≥ ε}. 1.1
This case is stated by st− lim x L.
Definition 1.3see 22. Let X, μ, ν, ∗, and Y, μ , ν , ∗, be two intuitionistic fuzzy normed linear spaces and let fk:X, μ, ν, ∗, → Y, μ , ν , ∗, be sequences of functions. If for each x ∈ X and for all ε > 0, t > 0,
δ
k ∈ N : μ
fkx − fx, t
≤ 1 − ε or ν
fkx − fx, t
≥ ε
0, 1.2
then we say that the sequencefk is pointwise statistically convergent to f with respect to intuitionistic fuzzy normμ, ν and we write it stμ,ν− fk → f.
Definition 1.4 see 20. Let λ λn be a nondecreasing sequence of positive numbers tending to∞ such that
λn1≤ λn, λ1 0. 1.3
Let K⊂ N. The number
δλK lim
n → ∞
1
λn|{k ∈ In: k∈ K}| 1.4
is said to be λ-density of K, where In n − λn 1, n.
If λn n, then λ-density is reduced to asymptotic density. A sequence x xk is said to be λ-statistically convergent to the number L if for every ε > 0, the set Nε has λ-density zero, where
Nε {k ∈ N : |xk− L| ≥ ε}. 1.5
This case is stated by stλ− lim x L.
Definition 1.5 see 23. Let X, μ, ν, ∗, be an intuitionistic fuzzy normed space. Then, a sequencexk is said to be λ-statistically convergent to L ∈ X with respect to intuitionistic fuzzy normμ, ν provided that for every ε > 0 and t > 0,
δλ
k ∈ N : μxk− L, t ≤ 1 − ε or νxk− L, t ≥ ε
0, 1.6
or equivalently
δλ
k ∈ N : μxk− L, t > 1 − ε, νxk− L, t < ε
1. 1.7
This case is stated by stλμ,ν− lim x L.
2. λ-Statistical Convergence of Sequence of Functions in Intuitionistic Fuzzy Normed Spaces
In this section, we define pointwise λ-statistical and uniformly λ-statistical convergent sequences of functions in intuitionistic fuzzy normed spaces. Also, we give the λ-statistical analog of the Cauchy convergence criterion for pointwise and uniformly λ-statistical convergent in intuitionistic fuzzy normed space. We investigate relations of these concepts with continuity. Let us start definition of pointwise λ-statistical convergence in intuitionistic fuzzy normed spaces.
Definition 2.1. Let X, μ, ν, ∗, and Y, μ , ν , ∗, be two intuitionistic fuzzy normed linear spaces over the same field IF and let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be sequences of functions. If for each x∈ X and for all ε > 0, t > 0,
δλ
k ∈ N : μ
fkx − fx, t
≤ 1 − ε or ν
fkx − fx, t
≥ ε
0, 2.1
or equivalently
δλ
k ∈ N : μ
fkx − fx, t
> 1 − ε, ν
fkx − fx, t
< ε
1, 2.2
then we say that the sequence fk is pointwise λ-statistically convergent with respect to intuitionistic fuzzy normμ, ν and we write it stλμ,ν− fk → f.
This means that for every δ > 0, there is integer N such that, for all n > Nε, δ, t, x
and for every ε > 0,
1 λn
k ∈ In : μ
fkx − fx, t
≤ 1 − ε or ν
fkx − fx, t
≥ ε for each x ∈ X
< δ. 2.3
Remark 2.2. Let fk :X, μ, ν, ∗, → Y, μ , ν , ∗, be sequences of functions. If λn n, since λ-density is reduced to asymptotic density, then fk is pointwise statistically convergent on X with respect to μ, ν, that is, stμ,ν− fk → f.
Lemma 2.3. Let fk:X, μ, ν, ∗, → Y, μ , ν , ∗, be sequences of functions. Then for every ε > 0 andt > 0, the following statements are equivalent.
i Consider stλμ,ν− fk → f.
ii For each x ∈ X
δλ
k ∈ N : μ
fkx − fx, t
≤ 1 − ε
δλ
k ∈ N : ν
fkx − fx, t
≥ ε
0. 2.4
iii For each x ∈ X
δλ
k ∈ N : μ
fkx − fx, t
> 1 − ε, ν
fkx − fx, t
< ε
1. 2.5
iv For each x ∈ X
δλ
k ∈ N : μ
fkx − fx, t
> 1 − ε
δλ
k ∈ N : ν
fkx − fx, t
< ε
1. 2.6
v For each x ∈ X
stλ− lim μ
fkx − fx, t
1, stλ− lim ν
fkx − fx, t
0. 2.7
Example 2.4. LetR, | · | 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, consider
μx, t t
t |x|, νx, t |x|
t |x|. 2.8
In this case R, μ, ν, ∗, is intuitionistic fuzzy normed space. Also, 0, 1, μ, ν, ∗, is intuitionistic fuzzy normed space. Let fk : 0, 1 → R be sequences of functions whose terms are given by
fkx
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
xk 1 if k ∈ n −
λn 1, n , x ∈
0,1
2
0 if k /∈ n −
λn 1, n , x ∈
0,1
2
xk1
2 if k∈ n −
λn 1, n , x ∈
1 2, 1
1 if k /∈ n −
λn 1, n , x ∈
1 2, 1
2 if x 1.
2.9
fk is pointwise λ-statistically convergent on 0, 1 with respect to intuitionistic fuzzy norm
μ, ν. It is fact that for each x ∈ 0, 1/2, and since
Kε, t
k ∈ In: μ
fkx − fx, t
≤ 1 − ε or ν
fkx − fx, t
≥ ε
, 2.10
hence we have
Kε, t
k ∈ In: t
t fkx − 0 ≤ 1 − ε or fkx − 0
t fkx − 0 ≥ ε
k ∈ In:fkx ≥ εt 1− ε
k ∈ In: fkx xk 1 ,
|Kε, t| ≤ λn.
2.11
Thus, for each x∈ 0, 1/2, since
δλKε, t lim
n → ∞
|Kε, t|
λn lim
n → ∞
λn
λn 0, 2.12
fkis λ-statistically convergent to 0 with respect to intuitionistic fuzzy normμ, ν.
If we take x∈ 1/2, 1, then we have
K ε, t
k ∈ In: t
t fkx − 1 ≤ 1 − ε or fkx − 1
t fkx − 1 ≥ ε
k ∈ In:fkx − 1 ≥ εt1− ε
k ∈ In: fkx xk 1 2
,
K ε, t ≤ λn.
2.13
Thus x∈ 1/2, 1, then fkis λ-statistically convergent to 1 with respect to intuitionistic fuzzy normμ, ν.
If we take x 1, it can be seen easily that
K ε, t
k ∈ In: t
t fkx − 2 ≤ 1 − ε or fkx − 2
t fkx − 2 ≥ ε
k ∈ In: 0≥ εt 1− ε
K ε, t 0, ,
δλ
K ε, t
lim
n → ∞
|K ε, t|
λn lim
n → ∞
0 λn 0.
2.14
Thus, at x 1, fkis λ-statistically convergent to 2 with respect to intuitionistic fuzzy norm
μ, ν.
Consequently, since fkx is λ-statistically convergent to different points with respect to intuitionistic fuzzy normμ, ν for each x ∈ X, fk is pointwise λ-statistically intuitionistic fuzzy convergent on0, 1.
Theorem 2.5. Let X, μ, ν, ∗, be an intuitionistic fuzzy normed space and let fk:X, μ, ν, ∗, →
Y, μ , ν , ∗, be sequences of functions. If sequence fk is pointwise intuitionistic fuzzy convergent onX to a function f with respect to μ, ν, then fk is pointwise λ-statistically convergent with respect to intuitionistic fuzzy normμ, ν.
Proof. Let fk be pointwise intuitionistic fuzzy convergent in X. In this case, fkx is convergent with respect toμ , ν for each x ∈ X. Then for every ε > 0 and t > 0, there is number k0∈ N such that
μ
fkx − fx, t
> 1 − ε, ν
fkx − fx, t
< ε 2.15
for all k≥ k0and for each x∈ X. Hence for each x ∈ X, the set
k ∈ N : μ
fkx − fx, t
≤ 1 − ε or ν
fkx − fx, t
≥ ε
2.16
has finite numbers of terms. Since finite subset ofN has λ-density 0, hence
δλ
k ∈ N : μ
fkx − fx, t
≤ 1 − ε or ν
fkx − fx, t
≥ ε
0. 2.17
That is, stλμ,ν− fk → f.
Theorem 2.6. Let fk and gk be two sequences of functions from intuitionistic fuzzy normed space
X, μ, ν, ∗, to Y, μ , ν , ∗, . If stλμ,ν− fk → f and stλμ,ν− gk → g, then stλμ,ν− αfk βgk → αf βg where α, β ∈ IF R or C.
Proof. The proof is clear for α 0 and β 0. Now let α / 0 and β / 0. Since stλμ,ν− fk → f and stλμ,ν− gk → g, for each x ∈ X if we define
A1
k ∈ N : μ
fkx − fx, t 2|α|
≤ 1 − ε or ν
fkx − fx, t 2|α|
≥ ε
,
A2
k ∈ N : μ
gkx − gx, t 2β
≤ 1 − ε or ν
gkx − gx, t 2β
≥ ε
,
2.18
then δλA1 0 and δλA2 0. Since δλA1 0 and δλA2 0, if we state A by A1∪ A2, then
δλA 0. 2.19
Hence A1∪ A2/ N and there exists k0∈ N such that
μ
fk0x − fx, t 2|α|
> 1 − ε, ν
fk0x − fx, t 2|α|
< ε,
μ
gk0x − gx, t 2β
> 1 − ε, ν
gk0x − gx, t 2β
< ε.
2.20
Let
B
k ∈ N : μ
αfk βgk
x −
αfx βgx , t
> 1 − ε, ν
αfk βgk
x −
αfx βgx , t
< ε
. 2.21
We will show that for each x∈ X
Ac⊂ B. 2.22
Let k0∈ Ac. In this case
μ
fk0x − fx, t 2|α|
> 1 − ε, ν
fk0x − fx, t 2|α|
< ε,
μ
gk0x − gx, t 2β
> 1 − ε, ν
gk0x − gx, t 2β
< ε.
2.23
Using those mentioned previously, we have
μ
αfk0 βgk0
x −
αfx βgx , t
≥ μ
αfk0x − αfx, t 2
∗ μ
βgk0x − βgx,t 2
μ
fk0x − fx, t 2|α|
∗ μ
gk0x − gx, t 2β
> 1 − ε ∗ 1 − ε
1 − ε, ν
αfk0 βgk0
x −
αfx βgx , t
≤ ν
αfk0x − αfx,t 2
ν
βgk0x − βgx,t 2
ν
fk0x − fx, t 2|α|
ν
gk0x − gx, t 2β
< ε ε
ε.
2.24
This implies that
Ac⊂ B. 2.25
Since Bc⊂ A and δλA 0, hence
δλBc 0. 2.26
That is
δλ
k ∈ N : μ
αfk βgk
x −
αfx βgx , t
≤ 1 − ε, ν
αfk βgk
x −
αfx βgx , t
≥ ε
0,
stλμ,ν−
αfk βgk
−→ αf βg.
2.27
Definition 2.7. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be sequences of functions. fk is a pointwise λ-statistical Cauchy sequence in intuitionistic fuzzy normed space provided that for every ε > 0 and t > 0 there exists a number N Nx, ε, t such that
δλ
k ∈ N : μ
fkx − fNx, t
≤ 1 − ε or ν
fkx − fNx, t
≥ ε for each x ∈ X
0. 2.28
Theorem 2.8. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. If fk is a pointwiseλ-statistical convergent sequence with respect to intuitionistic fuzzy norm μ, ν, then fk is a pointwiseλ-statistical Cauchy sequence with respect to intuitionistic fuzzy norm μ, ν.
Proof. Suppose that stλμ,ν− fk → f and let ε > 0, t > 0. For a given ε > 0, choose s > 0 such that1 − ε ∗ 1 − ε > 1 − s and ε ε < s. If we state, respectively, Axε, t and Acxε, t by
k ∈ N : μ
fkx − fx, t 2
≤ 1 − ε or ν
fkx − fx, t 2
≥ ε
,
k ∈ N : μ
fkx − fx, t 2
> 1 − ε, ν
fkx − fx, t 2
< ε
2.29
for each x∈ X. Then, we have
δλAxε, t 0, 2.30
which implies that
δλAcxε, t 1. 2.31
Let N∈ Acxε, t. Then
μ
fNx − fx, t 2
> 1 − ε, ν
fNx − fx, t 2
< ε. 2.32
We want to show that there exists a number N Nx, ε, t such that
δλ
k ∈ N : μ
fkx − fNx, t
≤ 1 − s or ν
fkx − fNx, t
≥ s for each x ∈ X
0. 2.33
Therefore, define for each x∈ X, Bxε, t
k ∈ N : μ
fkx − fNx, t
≤ 1 − s or ν
fkx − fNx, t
≥ s
. 2.34
We have to show that
Bxε, t ⊂ Axε, t. 2.35
Suppose that
Bxε, t/⊆Axε, t. 2.36
In this case Bxε, t has at least one different element which Axε, t does not have. Let k ∈ Bxε, t \ Axε, t Then we have
μ
fkx − fNx, t
≤ 1 − s, μ
fkx − fx, t 2
> 1 − ε, 2.37
in particular μ fNx − fx, t/2 > 1 − ε. In this case
1− s ≥ μ
fkx − fNx, t
≥ μ
fkx − fx, t 2
∗ μ
fNx − fx, t 2
≥ 1 − ε ∗ 1 − ε > 1 − s,
2.38
which is not possible. On the other hand ν
fkx − fNx, t
≥ s, ν
fkx − fx, t
< ε, 2.39
in particular ν fNx − fx, t < ε. In this case s ≤ ν
fkx − fNx, t
≤ ν
fkx − fx, t 2
ν
fNx − fx, t 2
< ε ε < s,
2.40
which is not possible. Hence Bxε, t ⊂ Axε, t. Therefore, by δλAxε, t 0, δλBxε, t 0. That is,fk is a pointwise λ-statistical Cauchy sequence with respect to intuitionistic fuzzy normμ, ν.
In the following, we introduce uniformly λ-statistical convergence in intuitionistic fuzzy normed spaces.
Definition 2.9. Let X, μ, ν, ∗, and Y, μ , ν , ∗, be two intuitionistic fuzzy normed linear spaces over the same field IF and let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. If for all x∈ X and for all ε > 0, t > 0,
δλ
k ∈ N : μ
fkx − fx, t
≤ 1 − ε or ν
fkx − fx, t
≥ ε
0, 2.41
or equivalently
δλ
k ∈ N : μ
fkx − fx, t
> 1 − ε, ν
fkx − fx, t
< ε
1, 2.42
then we say that the sequence fk is uniformly λ-statistically convergent with respect to intuitionistic fuzzy normμ, ν and we write it stλμ,ν− fk⇒ f.
Remark 2.10. If stλμ,ν− fk⇒ f, then stλμ,ν− fk → f.
Lemma 2.11. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be a sequence of functions. Then for every ε > 0 and t > 0, the following statements are equivalent:
i Consider stλμ,ν− fk⇒ f.
ii For all x ∈ X
δλ
k ∈ N : μ
fkx − fx, t
≤ 1 − ε
δλ
k ∈ N : ν
fkx − fx, t
≥ ε
0. 2.43
iii For all x ∈ X
δλ
k ∈ N : μ
fkx − fx, t
> 1 − ε, ν
fkx − fx, t
< ε
1. 2.44
iv For all x ∈ X
δλ
k ∈ N : μ
fkx − fx, t
> 1 − ε
δλ
k ∈ N : ν
fkx − fx, t
< ε
1. 2.45
v For all x ∈ X
stλ− lim μ
fkx − fx, t
1, stλ− lim ν
fkx − fx, t
0. 2.46
Example 2.12. Let R, μ, ν, ∗, be as Example 2.4. Let fk : 0, 1 → R be a sequence of functions whose terms are given by
fkx
xk 1, if n −
λn 1 ≤ k ≤ n
0, otherwise. 2.47
Since Kε, t {k ∈ In: μfkx − fx, t ≤ 1 − ε or νfkx − fx, t ≥ ε}, hence we have
Kε, t
k ∈ In: t
t fkx − 0 ≤ 1 − ε or fkx − 0
t fkx − 0 ≥ ε
k ∈ In:fkx ≥ εt1− ε
k ∈ In: fkx xk 1 ,
|Kε, t| ≤ λn.
2.48
Thus, for all x∈ 0, 1, since
δλKε, t lim
n → ∞
|Kε, t|
λn lim
n → ∞
λn
λn 0, 2.49
fkis uniformly λ-statistically convergent to 0 with respect to intuitionistic fuzzy normμ, ν.
Definition 2.13. Let fk : X, μ, ν, ∗, → Y, μ , ν , ∗, be sequences of functions. The sequencefk is a uniformly λ-statistical 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
δλ
k ∈ N : μ
fkx − fNx, t
≤ 1 − ε or ν
fkx − fNx, t
≥ ε ∀x ∈ X
0. 2.50
Definition 2.14. LetX, μ, ν, ∗, and Y, μ , ν , ∗, be two intuitionistic fuzzy normed spaces, and let F be a family of functions from X to Y . The family F is intuitionistic fuzzy equicontinuous at a point x0 ∈ X if for every ε > 0 and t > 0, there exists a δ > 0 such that μ fx0 − fx, t > 1 − ε and ν fx0 − fx, t < ε for all f ∈ F and all x such that μ x0− x, t > 1 − δ and ν x0− x, t < δ. The family is intuitionistic fuzzy equicontinuous if it is equicontinuous at each point of X.For continuity, δ may depend on ε, x0and f; for equicontinuity, δ must be independent of f.
Theorem 2.15. Let X, μ, ν, ∗, , Y, μ , ν , ∗, be intuitionistic fuzzy normed space. Assume that stλμ,ν− fk → f on X where functions fk:X, μ, ν, ∗, → Y, μ , ν , ∗, , k ∈ N are intuitionistic fuzzy equicontinuous onX and f : X → Y. Then f is continuous on X.
Proof. Let x0 ∈ X be an arbitrary point. By the intuitionistic fuzzy equicontinuity of fk’s, for every ε > 0 and t > 0 there exist δ δx0, ε, t/3 > 0 such that
μ
fkx0 − fkx,t 3
> 1 − ε, ν
fkx0 − fkx, t 3
< ε 2.51
for every k∈ N and all x such that μ x0− x, t > 1 − δ and ν x0− x, t < δ. Let x ∈ Bx0, δ, t
Bx0, δ, t stands for an open ball in X, μ, v, ∗, with center x0and radius δ be fixed. Since stλμ,ν− fk → f on X, for each x ∈ X, if we state, respectively, Axε, t and Bxε, t by the sets
Axε, t
⎧⎪
⎪⎨
⎪⎪
⎩
k ∈ N : μ
fkx − fx, t 3
≤ 1 − ε or ν
fkx − fx, t 3
≥ ε for each x ∈ X
⎫⎪
⎪⎬
⎪⎪
⎭,
Bxε, t
⎧⎪
⎪⎨
⎪⎪
⎩
k ∈ N : μ
fkx0 − fx0,t 3
≤ 1 − ε or ν
fkx0 − fx0,t 3
≥ ε for each x ∈ X
⎫⎪
⎪⎬
⎪⎪
⎭,
2.52
then δλAxε, t 0 and δλBxε, t 0; hence δλAxε, t∪Bxε, t 0 and Axε, t∪Bxε, t
is different from N. Thus, there exists ∃m ∈ N such that
μ
fmx − fx, t 3
> 1 − ε, ν
fmx − fx, t 3
< ε,
μ
fmx0 − fx0,t 3
> 1 − ε, ν
fmx0 − fx0,t 3
< ε.
2.53
Now, we will show that f is intuitionistic fuzzy continuous at x0. Since stλμ,ν− fk → f and for every k∈ N fk’s are continuous, fmis also continuous for m∈ N, we have
μ
fx − fx0, t
μ
fx − fmx fmx − fmx0 fmx0 − fx0, t
≥ μ
fx − fmx,t 3
∗ μ
fmx − fkx0,t 3
∗ μ
fmx0 − fx0,t 3
> 1 − ε ∗ 1 − ε ∗ 1 − ε
1 − ε, ν
fx − fx0, t
ν
fx − fmx fmx − fmx0 fmx0 − fx0, t
≤ ν
fx − fmx,t 3
ν
fmx − fmx0,t 3
ν
fmx0 − fx0,t 3
< ε ε ε
ε.
2.54
Thus, the proof is completed.
Acknowledgment
This work is supported by The Scientific and Technological Research Council of Turkey
TUBITAK under the project no. 110T699.
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