# λ-Statistical Convergence of Sequences of Functions in

Authors: Vatan Karakaya; Necip Şimşek; Müzeyyen Ertürk; Faik Gürsoy
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We studyλ-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.


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## Body

## 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 programming [5], 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 Atanassov [8] in 1983.In the literature,t-norm and t-conorm were defined by Schweizer and Sklar [9]. 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 authors [13–17], 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 conceptλ-statistical convergence for sequences of functions 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 1.1 (see [12]).
Let* be a continuous t-norm, let ⋄ be a continuous t-conorm, and X be a linear space over the field IF (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 intuitionistic fuzzy normed space and (μ,ν) is called an intuitionistic fuzzy norm. For every x,y∈X and s,t>0,(i)
μ(x,t)+ν(x,t)≤1,(ii)
μ(x,t)>0,(iii)
μ(x,t)=1⇔x=0,(iv)
μ(ax,t)=μ(x,t/|a|) for each a≠0,(v)
μ(x,t)*μ(y,s)≤μ(x+y,t+s),(vi)
μ(x,·):(0,∞)→[0,1] is continuous,(vii)
limt→∞μ(x,t)=1 and limt→0μ(x,t)=0,(viii)
ν(x,t)<1,(ix)
ν(x,t)=0⇔x=0,(x)
ν(ax,t)=ν(x,t/|a|) for each a≠0,(xi)
ν(x,t)⋄ν(y,s)≥ν(x+y,t+s),(xii)
ν(x,·):(0,∞)→[0,1] is continuous,(xiii)
limt→∞ν(x,t)=0 and limt→0ν(x,t)=1.Definition 1.2 (see [19, 24]).
LetK⊂ℕ and Kn={k∈K:k≤n}. Then, the natural density is defined by δ(K)=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 N(ε) has asymptotic density zero, where
(1.1)N(ε)={k∈ℕ:|xk-L|≥ε}.
This case is stated by st-limx=L.Definition 1.3 (see [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,
(1.2)δ({k∈ℕ:μ′(fk(x)-f(x),t)≤1-εorν′(fk(x)-f(x),t)≥ε})=0,
then we say that the sequence (fk) is pointwise statistically convergent to f with respect to intuitionistic fuzzy norm (μ,ν) and we write it stμ,ν-fk→f.Definition 1.4 (see [20]).
Letλ=(λn) be a nondecreasing sequence of positive numbers tending to ∞ such that
(1.3)λn+1≤λn,λ1=0.
Let K⊂ℕ. The number
(1.4)δλ(K)=limn→∞1λn|{k∈In:k∈K}|
is said to be λ-density of K, where In=[n-λn+1,n].
Ifλn=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 N(ε) has λ-density zero, where
(1.5)N(ε)={k∈ℕ:|xk-L|≥ε}.
This case is stated by stλ-limx=L.Definition 1.5 (see [23]).
Let(X,μ,ν,*,⋄) be an intuitionistic fuzzy normed space. Then, a sequence (xk) is said to be λ-statistically convergent to L∈X with respect to intuitionistic fuzzy norm (μ,ν) provided that for every ε>0 and t>0,
(1.6)δλ({k∈ℕ:μ(xk-L,t)≤1-εorν(xk-L,t)≥ε})=0,
or equivalently
(1.7)δλ({k∈ℕ:μ(xk-L,t)>1-ε,ν(xk-L,t)<ε})=1.
This case is stated by stμ,νλ-limx=L.


## 2.λ-Statistical Convergence of Sequence of Functions in Intuitionistic Fuzzy Normed Spaces

In this section, we define pointwiseλ-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,
(2.1)δλ({k∈ℕ:μ′(fk(x)-f(x),t)≤1-εorν′(fk(x)-f(x),t)≥ε})=0,
or equivalently
(2.2)δλ({k∈ℕ:μ′(fk(x)-f(x),t)>1-ε,ν′(fk(x)-f(x),t)<ε})=1,
then we say that the sequence (fk) is pointwise λ-statistically convergent with respect to intuitionistic fuzzy norm (μ,ν) and we write it stμ,νλ-fk→f.This means that for everyδ>0, there is integer N such that, for all n>N(ε,δ,t,x) and for every ε>0,
(2.3)1λn|{k∈In:μ′(fk(x)-f(x),t)≤1-εorν′(fk(x)-f(x),t)≥εforeachx∈X}|<δ.Remark 2.2.
Letfk:(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.
Letfk:(X,μ,ν,*,⋄)→(Y,μ′,ν′,*,⋄) be sequences of functions. Then for every ε>0 and t>0, the following statements are equivalent. (i)
Considerstμ,νλ-fk→f.(ii)
For eachx∈X(2.4)δλ{k∈ℕ:μ′(fk(x)-f(x),t)≤1-ε}=δλ{k∈ℕ:ν′(fk(x)-f(x),t)≥ε}=0.(iii)
For eachx∈X(2.5)δλ{k∈ℕ:μ′(fk(x)-f(x),t)>1-ε,ν′(fk(x)-f(x),t)<ε}=1.(iv)
For eachx∈X(2.6)δλ{k∈ℕ:μ′(fk(x)-f(x),t)>1-ε}=δλ{k∈ℕ:ν′(fk(x)-f(x),t)<ε}=1.(v)
For eachx∈X(2.7)stλ-limμ′(fk(x)-f(x),t)=1,stλ-limν′(fk(x)-f(x),t)=0.Example 2.4.
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, consider
(2.8)μ(x,t)=tt+|x|,ν(x,t)=|x|t+|x|.
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
(2.9)fk(x)={xk+1ifk∈[n-λn+1,n],x∈[0,12)0ifk∉[n-λn+1,n],x∈[0,12)xk+12ifk∈[n-λn+1,n],x∈[12,1)1ifk∉[n-λn+1,n],x∈[12,1)2ifx=1.(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
(2.10)K(ε,t)={k∈In:μ(fk(x)-f(x),t)≤1-εorν(fk(x)-f(x),t)≥ε},
hence we have
(2.11)K(ε,t)={k∈In:tt+|fk(x)-0|≤1-εor|fk(x)-0|t+|fk(x)-0|≥ε}={k∈In:|fk(x)|≥εt1-ε}={k∈In:fk(x)=xk+1},|K(ε,t)|≤λn.
Thus, for each x∈[0,1/2), since
(2.12)δλ(K(ε,t))=limn→∞|K(ε,t)|λn=limn→∞λnλn=0,fk is λ-statistically convergent to 0 with respect to intuitionistic fuzzy norm (μ,ν).If we takex∈[1/2,1), then we have
(2.13)K′(ε,t)={k∈In:tt+|fk(x)-1|≤1-εor|fk(x)-1|t+|fk(x)-1|≥ε}={k∈In:|fk(x)-1|≥εt1-ε}={k∈In:fk(x)=xk+12},|K′(ε,t)|≤λn.
Thus x∈[1/2,1), then fk is λ-statistically convergent to 1 with respect to intuitionistic fuzzy norm (μ,ν).If we takex=1, it can be seen easily that
(2.14)K′′(ε,t)={k∈In:tt+|fk(x)-2|≤1-εor|fk(x)-2|t+|fk(x)-2|≥ε}={k∈In:0≥εt1-ε}=⊘,|K′′(ε,t)|=0,δλ(K′′(ε,t))=limn→∞|K′′(ε,t)|λn=limn→∞0λn=0.
Thus, at x=1, fk is λ-statistically convergent to 2 with respect to intuitionistic fuzzy norm (μ,ν).Consequently, sincefk(x) is λ-statistically convergent to different points with respect to intuitionistic fuzzy norm (μ,ν) for each x∈X, (fk) is pointwise λ-statistically intuitionistic fuzzy convergent on [0,1].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 on X to a function f with respect to (μ,ν), then (fk) is pointwise λ-statistically convergent with respect to intuitionistic fuzzy norm (μ,ν).Proof.
Let(fk) be pointwise intuitionistic fuzzy convergent in X. In this case, (fk(x)) is convergent with respect to (μ′,ν′) for each x∈X. Then for every ε>0 and t>0, there is number k0∈ℕ such that
(2.15)μ′(fk(x)-f(x),t)>1-ε,ν′(fk(x)-f(x),t)<ε
for all k≥k0 and for each x∈X. Hence for each x∈X, the set
(2.16){k∈ℕ:μ′(fk(x)-f(x),t)≤1-εorν′(fk(x)-f(x),t)≥ε}
has finite numbers of terms. Since finite subset of ℕ has λ-density 0, hence
(2.17)δλ({k∈ℕ:μ′(fk(x)-f(x),t)≤1-εorν′(fk(x)-f(x),t)≥ε})=0.
That is, stμ,νλ-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 stμ,νλ-fk→f and stμ,νλ-gk→g, then stμ,νλ-(αfk+βgk)→αf+βg where α,β∈ IF (ℝorℂ).Proof.
The proof is clear forα=0 and β=0. Now let α≠0 and β≠0. Since stμ,νλ-fk→f and stμ,νλ-gk→g, for each x∈X if we define
(2.18)A1={k∈ℕ:μ′(fk(x)-f(x),t2|α|)≤1-εorν′(fk(x)-f(x),t2|α|)≥ε},A2={k∈ℕ:μ′(gk(x)-g(x),t2|β|)≤1-εorν′(gk(x)-g(x),t2|β|)≥ε},
then δλ(A1)=0 and δλ(A2)=0. Since δλ(A1)=0 and δλ(A2)=0, if we state A by (A1∪A2), then
(2.19)δλ(A)=0.
Hence A1∪A2≠ℕ and there exists k0∈ℕ such that
(2.20)μ′(fk0(x)-f(x),t2|α|)>1-ε,ν′(fk0(x)-f(x),t2|α|)<ε,μ′(gk0(x)-g(x),t2|β|)>1-ε,ν′(gk0(x)-g(x),t2|β|)<ε.
Let
(2.21)B={k∈ℕ:μ′((αfk+βgk)(x)-(αf(x)+βg(x)),t)>1-ε,ν′((αfk+βgk)(x)-(αf(x)+βg(x)),t)<ε}.
We will show that for each x∈X(2.22)Ac⊂B.
Let k0∈Ac. In this case
(2.23)μ′(fk0(x)-f(x),t2|α|)>1-ε,ν′(fk0(x)-f(x),t2|α|)<ε,μ′(gk0(x)-g(x),t2|β|)>1-ε,ν′(gk0(x)-g(x),t2|β|)<ε.
Using those mentioned previously, we have
(2.24)μ′((αfk0+βgk0)(x)-(αf(x)+βg(x)),t)≥μ′(αfk0(x)-αf(x),t2)*μ′(βgk0(x)-βg(x),t2)=μ′(fk0(x)-f(x),t2|α|)*μ′(gk0(x)-g(x),t2|β|)>(1-ε)*(1-ε)=(1-ε),ν′((αfk0+βgk0)(x)-(αf(x)+βg(x)),t)≤ν′(αfk0(x)-αf(x),t2)⋄ν′(βgk0(x)-βg(x),t2)=ν′(fk0(x)-f(x),t2|α|)⋄ν′(gk0(x)-g(x),t2|β|)<ε⋄ε=ε.
This implies that
(2.25)Ac⊂B.
Since Bc⊂A and δλ(A)=0, hence
(2.26)δλ(Bc)=0.
That is
(2.27)δλ({k∈ℕ:μ′((αfk+βgk)(x)-(αf(x)+βg(x)),t)≤1-ε,ν′((αfk+βgk)(x)-(αf(x)+βg(x)),t)≥ε})=0,stμ,νλ-(αfk+βgk)→αf+βg.Definition 2.7.
Letfk:(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=N(x,ε,t) such that
(2.28)δλ({k∈ℕ:μ′(fk(x)-fN(x),t)≤1-εorν′(fk(x)-fN(x),t)≥εforeachx∈X})=0.Theorem 2.8.
Letfk:(X,μ,ν,*,⋄)→(Y,μ′,ν′,*,⋄) be a sequence of functions. If (fk) is a pointwise λ-statistical convergent sequence with respect to intuitionistic fuzzy norm (μ,ν), then (fk) is a pointwise λ-statistical Cauchy sequence with respect to intuitionistic fuzzy norm (μ,ν).Proof.
Suppose thatstμ,νλ-fk→f and let ε>0,t>0. For a given 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 Axc(ε,t) by
(2.29){k∈ℕ:μ′(fk(x)-f(x),t2)≤1-εorν′(fk(x)-f(x),t2)≥ε},{k∈ℕ:μ′(fk(x)-f(x),t2)>1-ε,ν′(fk(x)-f(x),t2)<ε}
for each x∈X. Then, we have
(2.30)δλ(Ax(ε,t))=0,
which implies that
(2.31)δλ(Axc(ε,t))=1.
Let N∈Axc(ε,t). Then
(2.32)μ′(fN(x)-f(x),t2)>1-ε,ν′(fN(x)-f(x),t2)<ε.
We want to show that there exists a number N=N(x,ε,t) such that
(2.33)δλ({k∈ℕ:μ′(fk(x)-fN(x),t)≤1-sorν′(fk(x)-fN(x),t)≥sforeachx∈X})=0.
Therefore, define for each x∈X,
(2.34)Bx(ε,t)={k∈ℕ:μ′(fk(x)-fN(x),t)≤1-sorν′(fk(x)-fN(x),t)≥s}.
We have to show that
(2.35)Bx(ε,t)⊂Ax(ε,t).
Suppose that
(2.36)Bx(ε,t)⊈Ax(ε,t).
In this case Bx(ε,t) has at least one different element which Ax(ε,t) does not have. Let k∈Bx(ε,t)∖Ax(ε,t) Then we have
(2.37)μ′(fk(x)-fN(x),t)≤1-s,μ′(fk(x)-f(x),t2)>1-ε,
in particular μ′(fN(x)-f(x),t/2)>1-ε. In this case
(2.38)1-s≥μ′(fk(x)-fN(x),t)≥μ′(fk(x)-f(x),t2)*μ′(fN(x)-f(x),t2)≥(1-ε)*(1-ε)>1-s,
which is not possible. On the other hand
(2.39)ν′(fk(x)-fN(x),t)≥s,ν′(fk(x)-f(x),t)<ε,
in particular ν′(fN(x)-f(x),t)<ε. In this case
(2.40)s≤ν′(fk(x)-fN(x),t)≤ν′(fk(x)-f(x),t2)⋄ν′(fN(x)-f(x),t2)<ε⋄ε<s,
which is not possible. Hence Bx(ε,t)⊂Ax(ε,t). Therefore, by δλ(Ax(ε,t))=0,δλ(Bx(ε,t))=0. That is, (fk) is a pointwise λ-statistical Cauchy sequence with respect to intuitionistic fuzzy norm (μ,ν).In the following, we introduce uniformlyλ-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,
(2.41)δλ({k∈ℕ:μ′(fk(x)-f(x),t)≤1-εorν′(fk(x)-f(x),t)≥ε})=0,
or equivalently
(2.42)δλ({k∈ℕ:μ′(fk(x)-f(x),t)>1-ε,ν′(fk(x)-f(x),t)<ε})=1,
then we say that the sequence fk is uniformly λ-statistically convergent with respect to intuitionistic fuzzy norm (μ,ν) and we write it stμ,νλ-fk⇉f.Remark 2.10.
Ifstμ,νλ-fk⇉f, then stμ,νλ-fk→f.Lemma 2.11.
Letfk:(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)
Considerstμ,νλ-fk⇉f.(ii)
For allx∈X(2.43)δλ{k∈ℕ:μ′(fk(x)-f(x),t)≤1-ε}=δλ{k∈ℕ:ν′(fk(x)-f(x),t)≥ε}=0.(iii)
For allx∈X(2.44)δλ{k∈ℕ:μ′(fk(x)-f(x),t)>1-ε,ν′(fk(x)-f(x),t)<ε}=1.(iv)
For allx∈X(2.45)δλ{k∈ℕ:μ′(fk(x)-f(x),t)>1-ε}=δλ{k∈ℕ:ν′(fk(x)-f(x),t)<ε}=1.(v)
For allx∈X(2.46)stλ-limμ′(fk(x)-f(x),t)=1,stλ-limν′(fk(x)-f(x),t)=0.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
(2.47)fk(x)={xk+1,ifn-λn+1≤k≤n0,otherwise.
Since K(ε,t)={k∈In:μ(fk(x)-f(x),t)≤1-εorν(fk(x)-f(x),t)≥ε}, hence we have
(2.48)K(ε,t)={k∈In:tt+|fk(x)-0|≤1-εor|fk(x)-0|t+|fk(x)-0|≥ε}={k∈In:|fk(x)|≥εt1-ε}={k∈In:fk(x)=xk+1},|K(ε,t)|≤λn.
Thus, for all x∈[0,1], since
(2.49)δλ(K(ε,t))=limn→∞|K(ε,t)|λn=limn→∞λnλn=0,fk is uniformly λ-statistically convergent to 0 with respect to intuitionistic fuzzy norm (μ,ν).Definition 2.13.
Letfk:(X,μ,ν,*,⋄)→(Y,μ′,ν′,*,⋄) be sequences of functions. The sequence (fk) 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=N(ε,t) such that
(2.50)δλ({k∈ℕ:μ′(fk(x)-fN(x),t)≤1-εorν′(fk(x)-fN(x),t)≥ε∀x∈X})=0.Definition 2.14.
Let(X,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′(f(x0)-f(x),t)>1-ε and ν′(f(x0)-f(x),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, x0 and 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 stμ,νλ-fk→f on X where functions fk:(X,μ,ν,*,⋄)→(Y,μ′,ν′,*,⋄),k∈ℕ are intuitionistic fuzzy equicontinuous on X and f:X→Y. Then f is continuous on X.Proof.
Letx0∈X be an arbitrary point. By the intuitionistic fuzzy equicontinuity of fk’s, for every ε>0 and t>0 there exist δ=δ(x0,ε,t/3)>0 such that
(2.51)μ′(fk(x0)-fk(x),t3)>1-ε,ν′(fk(x0)-fk(x),t3)<ε
for every k∈ℕ and all x such that μ′(x0-x,t)>1-δ and ν′(x0-x,t)<δ. Let x∈B(x0,δ,t) (B(x0,δ,t) stands for an open ball in (X,μ,v,*,⋄) with center x0 and radius δ) be fixed. Since stμ,νλ-fk→f on X, for each x∈X, if we state, respectively, Ax(ε,t) and Bx(ε,t) by the sets
(2.52)Ax(ε,t)={k∈ℕ:μ′(fk(x)-f(x),t3)≤1-εorν′(fk(x)-f(x),t3)≥εforeachx∈X},Bx(ε,t)={k∈ℕ:μ′(fk(x0)-f(x0),t3)≤1-εorν′(fk(x0)-f(x0),t3)≥εforeachx∈X},
then δλ(Ax(ε,t))=0 and δλ(Bx(ε,t))=0; hence δλ(Ax(ε,t)∪Bx(ε,t))=0 and Ax(ε,t)∪Bx(ε,t) is different from ℕ. Thus, there exists ∃m∈ℕ such that
(2.53)μ′(fm(x)-f(x),t3)>1-ε,ν′(fm(x)-f(x),t3)<ε,μ′(fm(x0)-f(x0),t3)>1-ε,ν′(fm(x0)-f(x0),t3)<ε.
Now, we will show that f is intuitionistic fuzzy continuous at x0. Since stμ,νλ-fk→f and for every k∈ℕfk’s are continuous, fm is also continuous for m∈ℕ, we have
(2.54)μ′(f(x)-f(x0),t)=μ′(f(x)-fm(x)+fm(x)-fm(x0)+fm(x0)-f(x0),t)≥μ′(f(x)-fm(x),t3)*μ′(fm(x)-fk(x0),t3)*μ'(fm(x0)-f(x0),t3)>(1-ε)*(1-ε)*(1-ε)=1-ε,ν′(f(x)-f(x0),t)=ν′(f(x)-fm(x)+fm(x)-fm(x0)+fm(x0)-f(x0),t)≤ν′(f(x)-fm(x),t3)⋄ν′(fm(x)-fm(x0),t3)⋄ν′(fm(x0)-f(x0),t3)<ε⋄ε⋄ε=ε.
Thus, the proof is completed.



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*Source: 926193-2012-10-16.xml*