# On the Fuzzy Convergence

Authors: Abdul Hameed Q. A. Al-Tai
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

The aim of this paper is to introduce and study the fuzzy neighborhood, the limit fuzzy number, the convergent fuzzy sequence, the bounded fuzzy sequence, and the Cauchy fuzzy sequence on the base which is adopted by Abdul Hameed (every real numberr is replaced by a fuzzy number r¯ (either triangular fuzzy number or singleton fuzzy set (fuzzy point))). And then, we will consider that some results respect effect of the upper sequence on the convergent fuzzy sequence, the bounded fuzzy sequence, and the Cauchy fuzzy sequence.


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## Body

## 1. Introduction

Zadeh [1] introduced the concept of fuzzy set in 1965. Kramosil and Michálek [2] introduced the concept of fuzzy metric space using continuous t-norms in 1975.Matloka [3] introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties in 1986. Sequences of fuzzy numbers also were discussed by Nanda [4], Kwon [5], Esi [6], and many others.In 2010, Al-Tai defined the fuzzy metric space, the fuzzy sequence, and many other concepts, on other base, that the family of fuzzy real numbers isR̿=Z̿∪Q̿∪Q̿′. Z̿ is the family of fuzzy integer numbers, where every r̅∈Z̿(r∈Z) is a singleton fuzzy set (fuzzy point) (see [7–12]). Q̿andQ̿′ are the family of fuzzy rational numbers and the family of fuzzy irrational numbers, respectively, where every r̅∈Q̿(r∈Q) or r̅′∈Q̿′(r′∈Q′) is a triangular fuzzy number [13] and by using the representation Theorem (the resolution principle) [14].In this research, we will consider the fuzzy neighborhood, the limit fuzzy number, the convergent fuzzy sequence, the bounded fuzzy sequence, and the Cauchy fuzzy sequence, and then we will introduce some results about properties of the above concepts on the base which depended by Al-Tai to define the fuzzy metric space.Definition 1.1.
Let(X̿,d̿) be a fuzzy metric space. If x̅∈X̿, then a neighborhood of x̅ is a family of fuzzy numbers N̿ε̅(x̅) consisting of all fuzzy numbers y̅∈X̿, such that d̅(x̅,y̅)<ε̅(ε̅>0̅). The fuzzy number ε̅ will be called the fuzzy radius of N̿ε̅(x̅).Definition 1.2.
A fuzzy numberx̅ is a limit fuzzy number of the family of the fuzzy numbers E̿, if every fuzzy neighborhood of  x̅ contains a fuzzy number y̅≠x̅, such that y̅∈E̿.Definition 1.3 (Al-Tai [13]).
If we have the family of the fuzzy numbersX̿ and the family of the fuzzy natural numbers N̿, the fuzzy sequence 〈x̅n〉 in X̿ is a fuzzy function from N̿ to X̿.
A fuzzy sequence〈x̅n〉 consists of all ordered tuples (sequences) at degree α, for all α∈]0,1];  〈xj,n,α〉=〈xj,1,α,xj,2,α,…〉∈〈x̅1[α],x̅2[α],…〉=〈x̅1,x̅2,…〉[α]is its α-cut, j∈N. The fuzzy value of the fuzzy function at n̅∈N̿ is denoted by x̅n.Definition 1.4.
A fuzzy sequence〈x̅n〉 in a fuzzy metric space (X̿,d̿) is said to be convergent, if there is a fuzzy number x̅∈X̿, such that for every ε̅>0̅, there is a fuzzy natural number N̅, with n̅≥N̅, which implies that
(1.1)d̅(x̅n,x̅)<ε̅.
That is, for everyεj,α∈ε̅[α]α-cut of ε̅, there is a natural number N, with n≥N, such that
(1.2)d(xj,n,α,xj,α)<εj,α,
where xj,n,α∈x̅n[α]α-cut of x̅n, xj,α∈x̅[α]α-cut of x̅, N∈N̅, and n∈n̅, for all α∈]0,1]. x̅ will be called the fuzzy limit of 〈x̅n〉, and we write
(1.3)lim⁡n→∞⁡x̅n=x̅.
If〈xj,n,α〉 is not convergent at anyα∈]0,1], then 〈x̅n〉 is said to be divergent.Definition 1.5.
The family of the fuzzy numbersE̿ in the fuzzy metric space (X̿,d̿) is bounded, if there is a fuzzy real number r̅>0̅ and a fuzzy number x̅∈X̿, such that
(1.4)d̅(y̅,x̅)<r̅
for all y̅∈X̿. That is, if  rα∈r̅[α]α-cut of r̅ and xα∈x̅[α]α-cut of x̅∈X̅, then
(1.5)d(xα,yα)<rα,
where yα∈y̅[α]α-cut of all y̅∈X̿, for all α∈]0,1].Definition 1.6.
The fuzzy sequence〈x̅n〉 in a fuzzy metric space (X̿,d̿) is said to be bounded if its fuzzy range is bounded.Definition 1.7.
A fuzzy sequence〈x̅n〉 in a fuzzy metric space (X̿,d̿) is said to be a Cauchy fuzzy sequence if for every ε̅>0̅, there is a fuzzy natural number N̅, such that
(1.6)d̅(x̅n,x̅m)<ε̅,
whenever n̅≥N̅ and m̅≥N̅. In other words, every xj,n,α∈x̅n[α]α-cut of x̅n is a Cauchy sequence; that is, for every εj,α∈ε̅[α]α-cut of ε̅, there is a natural number N, such that
(1.7)d(xj,n,α,xj,m,α)<εj,α
for every n≥N,m≥N, where xj,n,α∈x̅n[α]α-cut of x̅n, xj,m,α∈x̅m[α]α-cut of x̅m, n∈n̅, m∈m̅, and N∈N̅, for all α∈]0,1].Examples 1.8.
(1)〈1/(n+1),1/n,1/(n-1)〉 converges to 0̅, since for all 〈xn,α〉∈[(n+α)/n(n+1),(n-α)/n(n-1)]α-cut of 〈1/(n+1),1/n,1/(n-1)〉, for all εα>0,∃N∈N̅∈N̿:dα(xn,α,0)<εα, for all n>N, for all α∈]0,1].
Therefore,(1.8)lim⁡n→∞⁡〈1n+1,1n,1n-1〉=0̅.
(2)〈1/en+1,1/en,1/en-1〉 converges to 0̅, since for all 〈xn,α〉∈[((e-1)α+1)/en+1,((1-e)α+e)/en]α-cut of 〈1/en+1,1/en,1/en-1〉, for all εα>0,∃N∈N̅∈N̿:dα(xn,α,0)<εα, for all n>N, for all α∈]0,1].
Therefore,(1.9)lim⁡n→∞〈1en+1,1en,1en-1〉=0̅.
(3)sin̅(n̅·(x1,x,x2))/n̅=(sin(nx1),sin(nx),sin(nx2))/((n)1,n,(n)2) converges to 0̅, where α-cut of (sin(nx1),sin(nx),sin(nx2)) is [(sin(nx)-sin(nx1))α+sin(nx1),(sin(nx)-sin(nx2))α+sin(nx2)]=[a1(α),a2(α)].
Andα-cut of ((n)1,n,(n)2) is
(1.10)[(n-(n)1)α+(n)1,(n-(n)2)α+(n)2]=[b1(α),b2(α)].
Since(1.11)∀〈xn,α〉∈[min⁡⁡(a1(α)b2(α),a1(α)b1(α),a2(α)b2(α),a2(α)b1(α)),max⁡⁡(a1(α)b2(α),a1(α)b1(α),a2(α)b2(α),a2(α)b1(α))],0∉[b1(α),b2(α)],α-cut of (sin(nx1),sin(nx),sin(nx2))/((n)1,n,(n)2), for all εα>0,∃N∈N̅∈N̿:dα(xn,α,0)<εα, for all n>N, for all α∈]0,1].
Therefore,(1.12)lim⁡n→∞(sin⁡⁡(nx1),sin⁡⁡(nx),sin⁡⁡(nx2))((n)1,n,(n)2)=0̅.
(4)〈(1/2,1/3,1/4)〉 converges to (1/2,1/3,1/4).
(5)〈n̅〉 is divergent and unbounded.
(6)cos̅(n̅·(x1,x,x2))=(cos(nx1),cos(nx),cos(nx2)) is divergent, since for all 〈xn,α〉∈[(cos(nx)-cos(nx1))α+cos(nx1),(cos(nx)-cos(nx2))α+cos(nx2)]α-cut of cos̅(n̅·(x1,x,x2)) is divergent. And it is bounded.Theorem 1.9.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is bounded for all α∈]0,1], then 〈x̅n〉 is bounded.Proof.
Letxj,α, be a supremum of a sequence 〈xj,n,α〉,j=1,2,…, for all α∈]0,1]. We have from the hypothesis, the upper sequence 〈x1,n,α〉 is bounded for all α∈]0,1]; that is, there is a real number r1,α∈r̅[α]α-cut of r̅∈R̿, such that
(1.13)d(x1,i,α,0)<r1,α
for all x1,i,α∈〈x1,n,α〉, i∈N, for all α∈]0,1].
Whenj=2,
(1.14)d(x2,i,α,0)≤d(x2,i,α,x1,α)+d(x1,α,0)<r2,α′+r1,α=r2,α,
where r2,α′=max{d(x2,i,α,x1,α),x2,i,α∈〈x2,n,α〉}. That is, there is a real number r2,α∈r̅[α]α-cut of r̅∈R̿, such that
(1.15)d(x2,i,α,0)<r2,α
for all x2,i,α∈〈x2,n,α〉, i∈N, for all α∈]0,1].
Whenj=3,
(1.16)d(x3,i,α,0)≤d(x3,i,α,x2,α)+d(x2,α,0)<r3,α′+r2,α=r3,α,
where r3,α′=max{d(x3,i,α,x2,α),x3,i,α∈〈x3,n,α〉}. That is, there is a real number r3,α∈r̅[α]α-cut of r̅∈R̿, such that
(1.17)d(x3,i,α,0)<r3,α
for all x3,i,α∈〈x3,n,α〉, i∈N, for all α∈]0,1].
And so on, whenj=m+1,
(1.18)d(xm+1,i,α,0)≤d(xm+1,i,α,xm,α)+d(xm,α,0)<rm+1,α′+rm,α=rm+1,α,
where rm+1,α′=max{d(xm+1,i,α,xm,α),xm+1,i,α∈〈xm+1,n,α〉}. That is, there is a real number rm+1,α∈r̅[α]α-cut of  r̅∈R̿, such that
(1.19)d(xm+1,i,α,0)<rm+1,α
for all xm+1,i,α∈〈xm+1,n,α〉,i∈N,m∈N, for all α∈]0,1]. By Definitions 1.5 and 1.6, 〈x̅n〉 is bounded.Theorem 1.10.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is bounded for all α∈]0,1], then 〈x̅n〉 is convergent.Proof.
Suppose that the upper sequence〈x1,n,α〉 of the α-cut of 〈x̅n〉 is bounded for all α∈]0,1]. By Theorem 1.9, 〈x̅n〉 is bounded; that is, every sequence 〈xj,n,α〉,j=2,3,…, of the α-cut of 〈x̅n〉is bounded for all α∈]0,1], but every sequence 〈xj,n,α〉,j=1,2,…,  of the α-cut of 〈x̅n〉 is monotonic, for all α∈]0,1]; then, every sequence 〈xj,n,α〉,j=1,2,…, of the α-cut of 〈x̅n〉is convergent, for all α∈]0,1] (see [15, Theorem 3.14]).  By Definition 1.4, 〈x̅n〉 is  convergent.Theorem 1.11.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is convergent for all α∈]0,1], then 〈x̅n〉 is convergent.Proof.
Suppose that the upper sequence〈x1,n,α〉 of the α-cut of 〈x̅n〉 is convergent for all α∈]0,1], so it is bounded (see [15, Theorem 3.2-c]). By Theorem 1.10, 〈x̅n〉is convergent.Theorem 1.12.
Every convergent fuzzy sequence〈x̅n〉 in a fuzzy metric space (X̿,d̿) is bounded.Proof.
Let〈x̅n〉 be a convergent fuzzy sequence in a fuzzy metric space (X̿,d̿). By Definition 1.4, every sequence 〈xj,n,α〉,j=1,2,…, of the α-cut of 〈x̅n〉 is convergent for all α∈]0,1]; that is, the upper sequence 〈x1,n,α〉 of 〈x̅n〉 is bounded for all α∈]0,1] (see [15, Theorem 3.2-c]). By Theorem 1.9, 〈x̅n〉 is bounded.Theorem 1.13.
Every bounded fuzzy sequence〈x̅n〉 in the fuzzy metric space (R̿,d̿) is convergent.Proof.
Let〈x̅n〉 be a bounded fuzzy sequence in a (R̿,d̿). By Definition 1.5, we get that the upper sequence 〈x1,n,α〉 of 〈x̅n〉 is bounded for all α∈]0,1]. By Theorem 1.10, 〈x̅n〉 is convergent.Theorem 1.14.
Let〈x̅n〉 be a fuzzy sequence in a fuzzy metric space (X̿,d̿); then 〈x̅n〉 converges to x̅∈X̿  if and only if every fuzzy neighborhood of x̅ contains all but finitely many of the terms of 〈x̅n〉.Proof.
Suppose thatN̿ε̅(x̅) is a fuzzy neighborhood of  x̅. For ε̅>0̅, d̅(y̅,x̅)<ε̅, y̅∈X̿, imply y̅∈N̿ε̅(x̅). We have from the hypothesis 〈x̅n〉 converges to x̅, so for same ε̅>0̅, there exists N̅∈N̿, such that n̅≥N̅, implying d̅(x̅n,x̅)<ε̅. That is, x̅n∈N̿ε̅(x̅).
Conversely, suppose that every fuzzy neighborhood ofx̅ contains all but finitely many of 〈x̅n〉. Fix ε̅>0̅, and let N̿ε̅(x̅) be the set of all y̅∈X̿, such that d̅(y̅,x̅)<ε̅.
By assumption, there existsN̅∈N̿ corresponding to N̿ε̅(x̅), such that x̅n∈N̿ε̅(x̅) if  n̅≥N̅. Thus d̅(x̅n,x̅)<ε̅ if n̅≥N̅. Hence 〈x̅n〉 converges to x̅.Theorem 1.15.
In any fuzzy metric space(X̿,d̿), every convergent fuzzy sequence is a Cauchy  fuzzy sequence.Proof.
Let〈x̅n〉 be a convergent fuzzy sequence in a fuzzy metric space (X̿,d̿). By Definition 1.4, every sequence 〈xj,n,α〉,j=1,2,…, of the α-cut of 〈x̅n〉 is convergent for all α∈]0,1]; then every sequence 〈xj,n,α〉,j=1,2,…, of the α-cut of 〈x̅n〉 is a Cauchy sequence for all α∈]0,1] (see [15, Theorem 3.11-a]); by Definition 1.7, 〈x̅n〉 is a Cauchy fuzzy sequence.Theorem 1.16.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is bounded for all α∈]0,1],then 〈x̅n〉 is a Cauchy fuzzy sequence.Proof.
Suppose that the upper sequence〈x1,n,α〉 of the α-cut of 〈x̅n〉 is bounded for all α∈]0,1]. By Theorem 1.10, 〈x̅n〉 is convergent. By Theorem 1.15, 〈x̅n〉 is a Cauchy fuzzy sequence.Theorem 1.17.
In the fuzzy metric space(R̿,d̿), every bounded fuzzy sequence is a Cauchy fuzzy sequence.Proof.
Let〈x̅n〉 be a bounded fuzzy sequence in (R̿,d̿). By Theorem 1.13, 〈x̅n〉 is a convergent fuzzy sequence. By Theorem 1.15, 〈x̅n〉 is a Cauchy fuzzy sequence.Theorem 1.18.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is convergent for all α∈]0,1], then 〈x̅n〉 is a bounded fuzzy sequence.Proof.
Suppose that the upper sequence〈x1,n,α〉 of the α-cut of 〈x̅n〉 is convergent for all α∈]0,1]. By Theorem 1.11, 〈x̅n〉 is a convergent fuzzy sequence. By Theorem 1.12, 〈x̅n〉 is a bounded fuzzy sequence.Theorem 1.19.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is convergent for all α∈]0,1], then 〈x̅n〉 is a Cauchy fuzzy sequence.Proof.
Suppose that the upper sequence〈x1,n,α〉 of the α-cut of 〈x̅n〉 is convergent for all α∈]0,1]. By Theorem 1.11, 〈x̅n〉 is a convergent fuzzy sequence. By Theorem 1.15, 〈x̅n〉 is a Cauchy fuzzy sequence.Theorem 1.20.
Every Cauchy fuzzy sequence inR̿n, the n-dimensional fuzzy Euclidean space, is convergent.Proof.
Let〈x̅m〉 be a Cauchy fuzzy sequence in R̿n. By Definitions 1.7 and 1.3, x̅m consists of all Cauchy sequences at degreeα, 〈xm,α〉=〈x1,α,x2,α,…〉∈〈x̅1,x̅2,…〉[α]α-cut of 〈x̅m〉 for all α∈]0,1], where every element xi,α∈〈xm,α〉 is (xi,1,α,xi,2,α,…,xi,n,α)∈(x̅i,1,x̅i,2,…,x̅i,n)[α]α-cut of x̅i=(x̅i,1,x̅i,2,…,x̅i,n) for all α∈]0,1] (Al-Tai [13]). By Theorem 3.11-c (see [15]), 〈xm,α〉 is a convergent sequence for all α∈]0,1]. By Definition 1.4, 〈x̅m〉 is a convergent fuzzy sequence.Theorem 1.21.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is a Cauchy sequence for all α∈]0,1]; then 〈x̅n〉 is convergent.Proof.
Suppose that the upper sequence〈x1,n,α〉 of the α-cut of 〈x̅n〉 is a Cauchy sequence for all α∈]0,1], so it is convergent (see [15, Theorem 3.11-c]). By Theorem 1.11, 〈x̅n〉 is convergent.Theorem 1.22.
Every Cauchy fuzzy sequence in the fuzzy metric space(R̿,d̿) is bounded.Proof.
Let〈x̅n〉 be a Cauchy fuzzy sequence in (R̿,d̿). By Theorem 1.20, 〈x̅n〉 is a convergent fuzzy sequence. By Theorem 1.12, 〈x̅n〉 is bounded.Theorem 1.23.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is a Cauchy sequence for all α∈]0,1], then 〈x̅n〉 is bounded.Proof.
Suppose that the upper sequence〈x1,n,α〉 of the α-cut of 〈x̅n〉 is a Cauchy sequence for all α∈]0,1]. By Theorem 1.21, 〈x̅n〉 is convergent. By Theorem 1.12, 〈x̅n〉 is bounded.Theorem 1.24.
Let〈x̅n〉 be a fuzzy sequence in the fuzzy metric space (R̿,d̿). If the upper sequence of the α-cut of 〈x̅n〉 is a Cauchy sequence for all α∈]0,1], then 〈x̅n〉 is Cauchy.Proof.
Suppose that the upper sequence〈x1,n,α〉 of the α-cut of 〈x̅n〉 is a Cauchy sequence for all α∈]0,1]; then it is convergent (see [15, Theorem 3.11-c]). By Theorem 1.19, 〈x̅n〉 is Cauchy.Theorem 1.25.
The limit fuzzy number of a convergent fuzzy sequence〈x̅n〉 in a fuzzy metric space (X̿,d̿) is unique.Proof.
Let〈x̅n〉 be a fuzzy sequence in a fuzzy metric space (X̿,d̿) which converges to x̅∈(X̿,d̿). By Definition 4 and Theorem 3.2-b [15], we have that every sequence 〈xn,α〉 of the α-cut of 〈x̅n〉 has a unique limit point xα∈x̅[α]α-cut of x̅ for all α∈]0,1], and hence, 〈x̅n〉 converges to a unique limit fuzzy number x̅.Theorem 1.26.
If〈y̅n〉 is a convergent fuzzy sequence in the fuzzy metric space (R̿,d̿), then a fuzzy sequence 〈x̅n〉 is convergent for all n̅≥N̅, as 0̅≤〈x̅n〉≤〈y̅n〉.Proof.
Let〈y1,n,α〉 be the upper sequence of the α-cut of 〈y̅n〉 for all α∈]0,1]. Since 0̅≤〈x̅n〉≤〈y̅n〉, then 0≤〈x1,n,α〉≤〈yn,α〉 [16], where 〈x1,n,α〉 is an upper sequence of the α-cut of 〈x̅n〉 for all α∈]0,1]. By Theorem 7.4.4 (Comparison Test) [17], 〈x1,n,α〉is convergent for all α∈]0,1]. By Theorem 1.11, 〈x̅n〉 is convergent.


## 2. Conclusion

Fuzzy neighborhood, limit fuzzy point, convergent fuzzy sequence, bounded fuzzy sequence, and Cauchy fuzzy sequence can be defined on the dependent base by Al-Tai, these concepts help us to show that the fuzzy sequence in(R̿,d̿) is convergent, if an upper sequence of the α-cut of it is convergent, bounded, or Cauchy for all α∈]0,1]. The fuzzy sequence in (R̿,d̿) is bounded, if an upper sequence of the α-cut of it is convergent, bounded, or Cauchy for all α∈]0,1]. The fuzzy sequence in (R̿,d̿) is Cauchy, if an upper sequence of the α-cut of it is convergent, bounded, or, Cauchy for all α∈]0,1]. The limit fuzzy number of a convergent fuzzy sequence is unique. And other results about above concepts.



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*Source: 147130-2011-11-24.xml*