# Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

Authors: Vatan Karakaya; Necip Şimşek; Müzeyyen Ertürk; Faik Gürsoy
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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.


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## Body

## 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 t-norms 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. 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)
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→∞μ(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→∞ν(x,t)=1.For an IFNS, we further assume that(X,μ,ν,*,⋄) satisfy the following axiom:(see [16])(xiv)
(1.1)a⋄a=aa*a=a}∀a∈[0,1].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 μ(x,t)>1-r and ν(x,t)<r for each x∈A.Definition 1.3 (see [25]).
Let(X,μ,ν,*,⋄) be a intuitionistic fuzzy metric space. Let A be any subset of X. Define
(1.2)ϕ(t)=inf{μ(x,y,t):x,y∈A},ψ(t)=sup{ν(x,y,t):x,y∈A},
(i) A is said to be q-bounded if limt→∞ϕ(t)=1 and limt→∞ψ(t)=0, (ii) A is said to be semibounded if limt→∞ϕ(t)=k and limt→∞ψ(t)=1-k,0<k<1 (iii) A is said to be unbounded if limt→∞ϕ(t)=0 and limt→∞ψ(t)=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 μ(xk-L,t)>1-ε and ν(xk-L,t)<ε whenever k>k0. In this case, we write (μ,ν)-limxk=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=δ(a,ε),β=β(a,ε)∈(0,1) such that for all x∈X and for all a∈(0,1),
(1.3)μ(x-x0,δ)>1-β⇒μ′(f(x)-f(x0),ε)>1-a,ν(x-x0,δ)<β⇒ν′(f(x)-f(x0),ε)<a.Definition 1.7 (see [16]).
Letfk:(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 f(x) with respect to (μ′,ν′).Definition 1.8 (see [16]).
Letfk:(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,
(1.4)μ′(fk(x)-f(x),t)>1-r,ν′(fk(x)-f(x),t)<r.
Now, we recall the notion of the statistical convergence of sequences in intuitionistic fuzzy normed spaces.Definition 1.9 (see [26]).
LetK⊂ℕ and Kn={k∈K:k≤n}. Then the asymptotic density is defined by δ(K)=limn→∞(|Kn|/n), where |Kn| denotes the cardinality of Kn.Definition 1.10.
LetA be subset of ℕ. If a property P(k) holds for all k∈A with δ(A)=1, we say that P holds for almost all k(a·a·k).Definition 1.11 (see [26]).
A sequencex=(xk) is said to be statistically convergent to the number L, or in short st-limx=L, if for every ε>0, the set K(ε) has asymptotic density zero, where
(1.5)K(ε)={k∈ℕ:|xk-L|≥ε},
that is,
(1.6)|xk-L|<εa·a·k.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,
(1.7)δ({k∈ℕ:μ(xk-L,t)≤1-εorν(xk-L,t)≥ε})=0,
or equivalently
(1.8)limn→∞1n|{k≤n:μ(xk-L,t)≤1-εorν(xk-L,t)≥ε}|=0.
In this case, we write stμ,ν-lim(xk)=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
(1.9)δ({k∈ℕ:μ(xk-xm,t)≤1-εorν(xk-xm,t)≥ε})=0.


## 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 f(x) for each x∈X with respect to intuitionistic fuzzy norm (μ′,ν′) and we write stμ,ν-fk→f.Theorem 2.2.
Letfk:(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.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
(2.1)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
(2.2){k∈ℕ:μ′(fk(x)-f(x),t)≤1-εorν′(fk(x)-f(x),t)≥ε}
has finite numbers of terms. Since density of finite subset of ℕ is 0, hence
(2.3)δ({k∈ℕ:μ′(fk(x)-f(x),t)≤1-εorν′(fk(x)-f(x),t)≥ε})=0.
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
(2.4)μ(x,t)=tt+|x|,ν(x,t)=|x|t+|x|.
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
(2.5)fk(x)={xk2+1ifk=m2(m∈ℕ),x∈[0,12),0ifk≠m2(m∈ℕ),x∈[0,12),0ifk=m2(m∈ℕ),x∈[12,1),xk+12ifk≠m2(m∈ℕ),x∈[12,1),2ifx=1.
Then sequence (fk) is pointwise statistically intuitionistic convergent on [0,1] with respect to (μ,ν). Indeed, for x∈⌈0,1/2), since
(2.6)Kn(ε,t)={k≤n:μ(fk(x)-f(x),t)≤1-εorν(fk(x)-f(x),t)≥ε},Kn(ε,t)={k≤n:tt+|fk(x)-0|≤1-εor|fk(x)-0|t+|fk(x)-0|≥ε}={k≤n:|fk(x)|≥εt1-ε}={k≤n:fk(x)=xk2+1}={k≤n:k=m2,m∈ℕ},
we have
(2.7)δ(Kn(ε,t))=1n|{k≤n:k=m2,m∈ℕ}|≤nn,
which yields limn→∞δ(Kn(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 takex∈[1/2,1), then we have
(2.8)Kn′(ε,t)={k≤n:tt+|fk(x)-(1/2)|≤1-εor|fk(x)-(1/2)|t+|fk(x)-(1/2)|≥ε}={k≤n:|fk(x)-12|≥εt1-ε}={k≤n:k=m2,|fk(x)-12|≥εt1-ε}∪{k≤n:k≠m2,|fk(x)-12|≥εt1-ε}={k≤n:k=m2}∪{k≤n:k≠m2:|xk+12-12|≥εt1-ε}={k≤n:k=m2}∪{k≤n:k≠m2:|xk|≥εt1-ε}.
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 intuitionistic fuzzy convergent to 2. Hence (fk) is intuitionistic fuzzy statistically convergent to 2. That is
(2.9)stμ,ν-fk(x)={0,x∈⌈0,12)12,x∈⌈12,1)2,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.
Letfk:(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)-f(x),t)≤1-ε}=δ{k∈ℕ:ν′(fk(x)-f(x),t)≥ε}=0 for each x∈X, for each ε>0 and t>0,(iii)
δ{k∈ℕ:μ′(fk(x)-f(x),t)>1-ε and ν′(fk(x)-f(x),t)<ε}=1 for each x∈X, for each ε>0 and t>0,(iv)
st-limμ′(fk(x)-f(x),t)=1 and st-limν′(fk(x)-f(x),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 forα=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
(2.10)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
(2.11)δ(A1)=0,δ(A2)=0.
Sinceδ(A1)=0andδ(A2)=0, if we state A by (A1∪A2) then
(2.12)δ(A)=0.
Hence, A1∪A2≠ℕ and there exists ∃m∈ℕ such that
(2.13)μ′(fm(x)-f(x),t2|α|)>1-ε,ν′(fm(x)-f(x),t2|α|)<ε,μ′(gm(x)-g(x),t2|β|)>1-ε,ν′(gm(x)-g(x),t2|β|)<ε.
Let
(2.14)B={k∈ℕ:μ′((αfk+βgk)(x)-(αf(x)+βg(x)),t)>1-ε,aν′((αfk+βgk)(x)-(αf(x)+βg(x)),t)<ε}.
We will show that for each x∈X(2.15)Ac⊂B.
Let m∈Ac. In this case,
(2.16)μ′(fm(x)-f(x),t2|α|)>1-ε,ν′(fm(x)-f(x),t2|α|)<ε,μ′(gm(x)-g(x),t2|β|)>1-ε,ν′(gm(x)-g(x),t2|β|)<ε.
Using those above, we have
(2.17)μ′((αfm+βgm)(x)-(αf+βg)(x),t)≥μ′(αfm(x)-αf(x),t2)*μ′(βgm(x)-βg(x),t2)=μ′(fm(x)-f(x),t2|α|)*μ′(gm(x)-g(x),t2|β|)>(1-ε)*(1-ε)=(1-ε),ν′((αfm+βgm)(x)-(αf+βg)(x),t)≤ν′(αfm(x)-αf(x),t2)*ν′(βgm(x)-βg(x),t2)=ν′(fm(x)-f(x),t2|α|)*ν′(gm(x)-g(x),t2|β|)<ε⋄ε=ε.
This implies that(2.18)Ac⊂B.
Since Bc⊂A and δ(A)=0, hence
(2.19)δ(Bc)=0,
that is(2.20)δ({k∈ℕ:μ′((αfk+βgk)(x)-(αf+βg)(x),t)≤1-ε,asdaν′((αfk+βgk)(x)-(αf+βg)(x),t)≥ε})=0
which means(2.21)stμ,ν-(αfk+βgk)→αf+βg.Definition 2.6.
Letfk:(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=N(ε,t,x) such that
(2.22)δ({k∈ℕ:μ′(fk(x)-fN(x),t)≤1-εorν′(fk(x)-fN(x),t)≥εforeachx∈X})=0.
That is, there exists a number N=N(ε,t,x) for each x∈X such that
(2.23)μ′(fk(x)-fN(x),t)>1-ε,ν′(fk(x)-fN(x),t)<εfora·a·k.Theorem 2.7.
Letfk:(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 thatstμ,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 Axc(ε,t) by
(2.24){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
(2.25)δ(Ax(ε,t))=0,
which implies that
(2.26)δ(Axc(ε,t))=1.
Let N∈Axc(ε,t). Then
(2.27)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
(2.28)δ({k∈ℕ:μ′(fk(x)-fN(x),t)≤1-sorν′(fk(x)-fN(x),t)≥sforeachx∈X})=0.
Therefore, define for each x∈X,
(2.29)Bx(ε,t)={k∈ℕ:μ′(fk(x)-fN(x),t)≤1-sorν′(fk(x)-fN(x),t)≥s}.
We have to show that
(2.30)Bx(ε,t)⊂Ax(ε,t).
Suppose that
(2.31)Bx(ε,t)⊈Ax(ε,t).
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
(2.32)μ′(fk(x)-fN(x),t)≤1-s,μ′(fk(x)-f(x),t2)>1-ε,
in particularly μ′(fN(x)-f(x),t/2)>1-ε. In this case,
(2.33)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
(2.34)ν′(fk(x)-fN(x),t)≥s,ν′(fk(x)-f(x),t)<ε,
in particularly ν′(fN(x)-f(x),t)<ε. In this case,
(2.35)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 δ(Ax(ε,t))=0,δ(Bx(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 definitionDefinition 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, δ(M)=1 and ∃k0=k0(ε,t)∈M϶   ∀k>k0 and k∈M and ∀x∈X,
(2.36)μ′(fk(x)-f(x),t)>1-ε,ν′(fk(x)-f(x),t)<ε.Lemma 2.9.
Letfk:(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)-f(x),t)≤1-ε}=δ{k∈ℕ:ν′(fk(x)-f(x),t)≥ε}=0 for all x∈X, for every ε>0 and t>0.(iii)
δ{k∈ℕ:μ′(fk(x)-f(x),t)>1-ε and ν′(fk(x)-f(x),t)<ε}=1 for all x∈X, for every ε>0 and t>0.(iv)
st-limμ′(fk(x)-f(x),t)=1 and st-limν′(fk(x)-f(x),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
(2.37)st-liminfμ′(fk(x)-f(x),t)=1,st-limsupν′(fk(x)-f(x),t)=0,
where the supremum and infimum are taken over all x∈X.Proof.
Suppose thatstμ,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 infμ′(fk(x)-f(x),t)=rk and supν′(fk(x)-f(x),t)=1-rk for each k∈ℕ and for each t>0,(0<rk<1). By using (iv) of Lemma 2.9, we get
(2.38)st-liminfμ′(fk(x)-f(x),t)=1,st-limsupν′(fk(x)-f(x),t)=0,
where the supremum and infimum are taken over all x∈X.
Conversely, suppose that(2.39)st-liminfμ′(fk(x)-f(x),t)=1,st-limsupν′(fk(x)-f(x),t)=0,
where the supremum and infimum are taken over all x∈X. Since
(2.40)infμ′(fk(x)-f(x),t)∈(0,1],supν′(fk(x)-f(x),t)∈[0,1),
from definition statistical convergence, for a·a·k, for every ε>0 and t>0(2.41)|infμ′(fk(x)-f(x),t)-1|<ε,|supν′(fk(x)-f(x),t)-0|<ε.
For all x∈X(2.42)μ′(fk(x)-f(x),t)≥infμ′(fk(x)-f(x),t)⇒-μ′(fk(x)-f(x),t)<-infμ′(fk(x)-f(x),t)⇒1-μ′(fk(x)-f(x),t)<1-infμ′(fk(x)-f(x),t)sinceμ′(fk(x)-f(x),t)∈(0,1]∀x∈X⇒|1-μ′(fk(x)-f(x),t)|<|1-infμ′(fk(x)-f(x),t)|⇒|1-μ′(fk(x)-f(x),t)|<|1-infμ′(fk(x)-f(x),t)|<ε⇒|1-μ′(fk(x)-f(x),t)|<εa·a·k,ν′(fk(x)-f(x),t)≤supν′(fk(x)-f(x),t)⇒|ν′(fk(x)-f(x),t)-0|≤|supν′(fk(x)-f(x),t)-0|<ε⇒|ν′(fk(x)-f(x),t)-0|<εa·a·k.
Therefore, st-limμ′(fk(x)-f(x),t)=1 and st-limν′(fk(x)-f(x),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
(2.43)fk={xk+1,ifk≠m2(m∈ℕ)2,otherwise.
Then, for every 0<ε<1 and for every t>0, we define
(2.44)Kn={k≤n:μ(fk(x)-f(x),t)≤1-εorν(fk(x)-f(x),t)≥ε}.
For all x∈X, we have
(2.45)δ(Kn)=δ({k≤n:μ(fk(x)-1,t)≤1-εorν(fk(x)-1,t)≥ε})=0.
Since
(2.46)stμ,ν-fk(x)=1,
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
(2.47)fk={(x-1k)2,ifk≠m2(m∈ℕ)2,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(2.48)limk→∞infx∈[-1,1]μ(fk(x)-f(x),t)=tt+|-(2x/k)+(1/k2)|,limk→∞supx∈[-1,1]ν(fk(x)-f(x),t)=|-(2x/k)+1/k2|t+|-(2x/k)+(1/k2)|·
Firstly, we need to find limk→∞infμ(fk(x)-f(x),t) and limk→∞supν(fk(x)-f(x),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
(2.49)0≤-2xk+1k2≤2k+1k2⇒t≤t-2xk+1k2≤t+2k+1k2⇒1t+(2/k)+(1/k2)≤1t-(2x/k)+(1/k2)≤1t⇒tt+(2/k)+(1/k2)≤tt-(2x/k)+(1/k2)≤1,
we get
(2.50)limk→∞infx∈[-1,1/2k]μ(fk(x)-x2,t)=limk→∞infx∈[-1,1/2k]tt-(2x/k)+(1/k2)=limk→∞tt+(2/k)+(1/k2)=1.
On the other hand, we have
(2.51)-(2x/k)+(1/k2)t-(2x/k)+(1/k2)≤(2/k)+(1/k2)t,
and so
(2.52)limk→∞supx∈[-1,1/2k]ν(fk(x)-x2,t)=limk→∞supx∈[-1,1/2k]-(2x/k)+(1/k2)t-(2x/k)+(1/k2)=limk→∞(2/k)+(1/k2)t=0.
In case of x∈(1/2k,1],we have -(2x/k)+(1/k2)<0. Since
(2.53)-2k+1k2<-2xk+1k2≤0⇒0≤2xk-1k2<2k-1k2⇒t<t+2xk-1k2≤t+2k-1k2⇒1t+(2/k)-(1/k2)<1t+(2x/k)-(1/k2)≤1t⇒tt+(2/k)-(1/k2)≤tt+(2x/k)-(1/k2)<1,
we get
(2.54)limk→∞infx∈(1/2k,1]μ(fk(x)-x2,t)=limk→∞infx∈(1/2k,1]tt+(2x/k)-(1/k2)=limk→∞tt+(2/k)-(1/k2)=1.
On the other hand, we have
(2.55)(2x/k)-(1/k2)t+(2x/k)-(1/k2)≤(2/k)-(1/k2)t,
and so
(2.56)limk→∞supx∈(1/2k,1]ν(fk(x)-x2,t)=limk→∞supx∈(1/2k,1](2x/k)-(1/k2)t+(2x/k)-(1/k2)=limk→∞(2/k)-(1/k2)t=0.
That is, the sequence (fk) is uniformly statistically intuitionistic fuzzy convergent to x2 on [-1,1].Remark 2.13.
Ifstμ,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(2.57)fk(x)={0k=n2k2x1+k3x2otherwise,
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 μ(fk(x)-0,t)=μ(k2x/(1+k3x2),t)=t/(t+(k2x)/(1+k3x2)) and supremum for ν(fk(x)-0,t)=(ν(k2x/(1+k3x2),t))=(k2x/(1+k3x2)/(t+(k2x)/(1+k3x2)), over all x∈[0,1]. Firstly, we try to find supx∈[0,1](k2x/(1+k3x2)) and infx∈[0,1](k2x/(1+k3x2)). For this, we have
(2.58)(k2x1+k3x2)′=k2(1-k3x2)(1+n3x2)2=0⇒x=k-3/2.
Since fk(0)=0,fk(1)=k2/(1+k3) and fk(k-3/2)=k/2, we get supx∈[0,1](k2x/(1+k3x2))=k/2 and infx∈[0,1](k2x/(1+k3x2))=0. Then,
(2.59)k2x1+k3x2≤k2⇒t+k2x1+k3x2≤t+k2⇒1t+(k/2)≤1t+(k2x/(1+k3x2))⇒tt+(k/2)≤tt+(k2x/(1+k3x2)),0≤k2x1+k3x2⇒t+0≤t+k2x1+k3x2⇒1t+(k2x/(1+k3x2))≤1t⇒k2x/(1+k3x2)t+(k2x/(1+k3x2))≤k2x/(1+k3x2)t≤k2t,
so, we get
(2.60)limk→∞infμ(k2x1+k3x2,t)=limk→∞inft(t+k2x/(1+k3x2))=limk→∞tt+k/2=limk→∞2t2t+k=0≠1,limk→∞supv(k2x1+k3x2,t)=limk→∞supk2x/(1+k3x2)t+(k2x/(1+k3x2))=limk→∞k2t=∞≠0.
Therefore, we conclude that (fk) does not intuitionistic fuzzy uniformly statistical convergent to 0.Theorem 2.15.
Letfk :(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.
Letfk 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,
(2.61)μ′(fk(x)-f(x),t)>1-ε,ν′(fk(x)-f(x),t)<ε.
That is, for k≤k0(2.62)μ′(fk(x)-f(x),t)≤1-ε,ν′(fk(x)-f(x),t)≥ε,
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, δ(M)=1 and ∃k0=k0(ε,t)∈M such that ∀k>k0 and k∈M and ∀x∈X,
(2.63)μ′(fk(x)-f(x),t)>1-ε,ν′(fk(x)-f(x),t)<ε.
This shows that stμ,ν-fk⇉f.Definition 2.16.
Letfk:(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=N(ε,t) such that
(2.64)δ({k∈ℕ:μ′(fk(x)-fN(x),t)≤1-εorν′(fk(x)-fN(x),t)≥ε∀x∈X})=0.Theorem 2.17.
Letfk :(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).Proof.
Suppose thatstμ,ν-fk⇉f. In this case, ∀ε>0, there exists M⊂ℕ, δ(M)=1 and k0=k0(ε,t)∈M such that ∀k>k0,k∈M and ∀x∈X,
(2.65)μ′(fk(x)-f(x),t2)>1-ε,ν′(fk(x)-f(x),t2)<ε.
Choose N=N(ε,t)∈M,N>k0. So, μ′(fN(x)-f(x),t/2)>1-ε and ν′(fN(x)-f(x),t/2)<ε. We investigate N=N(ε,t) such that
(2.66)δ({k∈ℕ:μ′(fk(x)-fN(x),t)≤1-εorν′(fk(x)-fN(x),t)≥ε∀x∈X})=0,
or
(2.67)δ(K′)=δ({k∈ℕ:μ′(fk(x)-fN(x),t)>1-εorν′(fk(x)-fN(x),t)<ε∀x∈X})=1.
For every k∈M, we have
(2.68)μ′(fk(x)-fN(x),t)=μ′(fk(x)-f(x)+f(x)-fN(x),t)≥μ′(fk(x)-f(x),t2)*μ′(f(x)-fN(x),t2)>(1-ε)*(1-ε)=(1-ε),ν′(fk(x)-fN(x),t)=ν′(fk(x)-f(x)+f(x)-fN(x),t)≤ν′′(fk(x)-f(x),t2)*ν′(f(x)-fN(x),t2)>ε⋄ε=ε.
Since δ(M)=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.
Letx0∈X be an arbitrary point. By the intuitionistic fuzzy continuity of fk’s, for every ε>0 and t>0 there exists δ=δ(x0,ε,t/3)>0 such that
(2.69)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) be fixed (B(x0,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
(2.70)A={k∈ℕ:μ′(fk(x)-f(x),t3)≤1-εorν′(fk(x)-f(x),t3)≥ε∀x∈X},B={k∈ℕ:μ′(fk(x0)-f(x0),t3)≤1-εorν′(fk(x0)-f(x0),t3)≥ε∀x∈X}
then, δ(A)=0 and δ(B)=0, hence δ(A∪B)=0 and A∪B is different from ℕ. Thus, there exists m∈ℕ such that
(2.71)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 contunious at x0. Using Definition 1.1, we have
(2.72)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)-fm(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: 157467-2012-11-26.xml*