# New Results onI2-Statistically Limit Points and I2-Statistically Cluster Points of Sequences of Fuzzy Numbers

Authors: Ö. Kişi; M. B. Huban; M. Gürdal
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, some existing theories on convergence of fuzzy number sequences are extended toI2-statistical convergence of fuzzy number sequence. Also, we broaden the notions of I-statistical limit points and I-statistical cluster points of a sequence of fuzzy numbers to I2-statistical limit points and I2-statistical cluster points of a double sequence of fuzzy numbers. Also, the researchers focus on important fundamental features of the set of all I2-statistical cluster points and the set of all I2-statistical limit points of a double sequence of fuzzy numbers and examine the relationship between them.


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## Body

## 1. Introduction

The theory of statistical convergence reverts to the first edition of monograph of Zygmund [1]. Statistical convergence of number sequences was given by Fast [2] and then was reissued by Schoenberg [3] independently for real and complex sequences. This conception was studied for the double sequences by Mursaleen and Edely [4]. Fridy [5] considered statistical limit points and statistical cluster points of real number sequences. When we focus on the statistical convergence in the literature, we meet Fridy [6], Temizsu and Mikail [7], Braha et al. [8], Nuray and Ruckle [9], Das et al. [10], and so many other researchers (see [11–13]).The concept of ideal convergence was given by Kostyrko et al. [14] which generalizes and combines different concepts of convergence of sequences containing usual convergence and statistical convergence. Das et al. [15] presented the concept of I-convergence of double sequences in a metric space. In [16], Savas and Das extended the conception of ideal convergence as studied by Kostyrko et al. [14] to I-statistical convergence and examined remarkable basic features of it. For different studies on these topics, we refer to [17–23].The theory of fuzzy sets was firstly given by Zadeh [24]. Matloka [25] identified the convergence of a sequence of fuzzy numbers. Nanda [26] worked on the sequences of fuzzy numbers and displayed that the set of all convergent sequences of fuzzy numbers generates a complete metric space. Nuray and Savas [27] generalized ordinary convergence and defined statistically Cauchy and statistical convergent sequences of fuzzy number. Later on, it was studied and advanced by Aytar and Pehlivan [28] and many others. Aytar [29] worked on the conception of statistical limit points and cluster points for sequences of fuzzy number. Kumar et al. [30, 31] worked I-convergence, I-limit points, and I-cluster point for sequence of fuzzy numbers. The concepts of I-statistically convergence for sequences of fuzzy numbers were established by Debnath and Debnath [32]. Later on, I-statistically limit points and I-statistically cluster points of sequences of fuzzy numbers were studied by Tripathy et al. [33].In this paper, we examine some essential features ofI2-statistically convergent sequence of fuzzy numbers and describe I2-statistical limit point and I2-statistical cluster point for fuzzy number sequences.


## 2. Preliminaries

First, we emphasize some properties of double sequences which are not satisfied by a (single) sequence. This provides a proper motivation for studying double sequences.The essential deficiency of this kind of convergence is that a convergent sequence does not require to be bounded. Hardy [34] defined the concept of regular sense, which does not have this shortcoming, for double sequence. In regular convergence, both the row-index and the column-index of the double sequence need to be convergent besides the convergent in Pringsheim’s sense.The notion of Cesàro summable double sequences was described by [35]. Note that if a bounded sequence xmn is statistically convergent then it is also Cesàro summable but not contrariwise.Letxmn=−1m, ∀n; then, limp,r∑m=1p∑n=1rxmn=0, but apparently x is not statistically convergent.The convergence of double sequences plays a significant part not only in pure mathematics but also in other subjects of science including computer science, biological science, and dynamical systems, as well. Also, the double sequence can be used in convergence of double trigonometric series and in the opening series of double functions and in the making differential solution.Now, we remember some notions and fundamental definitions required in this study.We signify byD the set of all bounded and closed intervals on ℝ, i.e.,
(1)D=M⊂ℝ:M=M¯,M¯.ForM,N∈D, we describe M≤N iff M¯≤N¯ and M¯≤N¯ and d¯=maxM¯−N¯,M¯−N¯. D,d¯ forms a complete metric space.Definition 1.
A fuzzy number is a functionX from ℝ to 0,1, which satisfies the subsequent conditions:
(i)
X is normal(ii)
X is fuzzy convex(iii)
X is upper semicontinuous(iv)
The closure of the setx∈ℝ:Xx>0 is compact
The features (i)-(iv) give that for eachα∈0,1, the α-level set,
(2)Xα=x∈ℝ:Xx≥α=X¯α,X¯α,is a nonempty compact convex subset of ℝ. The 0-level set is the class of the strong 0-cut, i.e., clx∈ℝ:Xx≥0. The set of all fuzzy numbers is indicated by Lℝ. Consider a map d¯X,Y=supα∈0,1dXα,Yα. Lℝ,d¯ also forms a complete metric space [36].Definition 2 (see [25]).
A sequenceXk of fuzzy numbers is named to be convergent to a fuzzy number X0 if for each ε>0 there is m>0 such that d¯Xk,X0<ε for every k≥m. We write limk⟶∞Xk=X0.Definition 3 (see [25]).
A fuzzy numberX0 is known as a limit point of a sequence of fuzzy number Xk on condition that there is a subsequence of Xk that converges to X0.Definition 4 (see [27]).
A sequenceXk of fuzzy numbers is named to be statistically convergent to a fuzzy number X0 if for each ε>0 the set
(3)Aε=k∈ℕ:d¯Xk,X0≥ε,has natural density zero. We write St−limk⟶∞Xk=X0.Definition 5 (see [30]).
TakeI as a nontrivial ideal. A sequence Xk of fuzzy numbers is known as I-convergent to a fuzzy number X0 provided that each ξ>0(4)Aξ=k∈ℕ:d¯Xk,X0≥ξ∈I.
We writeI−limk⟶∞Xk=X0.Definition 6 (see [31]).
A fuzzy numberX0 is known as ideal limit point of a sequence of fuzzy number Xk provided that there is a subset M=k1<k2<⋯⊂ℕ such that M∉I and limXkn=X0.Definition 7 (see [31]).
A fuzzy numberX0 is known as ideal cluster point of a sequence of fuzzy number Xk provided that for each ξ>0 the set k∈ℕ:d¯Xk,X0<ξ∉I.
The set of allI-limit points and I-cluster points of the sequence X is shown by IΛX and IΓX, respectively.
Natural density of a subsetK of ℕ×ℕ is demonstrated by
(5)dK=limm,n⟶∞Km,nm.n,where Km,n=j,k∈ℕ×ℕ:j≤m,k≤n.A nontrivial idealI2 of ℕ×ℕ is known as strongly admissible if i×ℕ and ℕ×i belong to I2 for each i∈ℕ.It is the proof that a strongly admissible ideal is admissible also.Throughout the paper, we considerI2 as a strongly admissible ideal in ℕ×ℕ.


## 3. Main Results

In this study, the researchers focus on remarkable features of the set of allI2-statistical cluster points and the set of all I2-statistical limit points of fuzzy number sequences. We examine interrelationship between them.Theorem 8.
IfXkl be a double sequence of fuzzy numbers such that I2−stlimXkl=X0, then, X0 identified uniquely.Proof.
Presume thatI2−stlimXkl=X0 and I2−stlimXkl=Y0, where X0≠Y0. For any ε,δ>0, we get
(6)K1=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,X0≥ε<δ∈FI2,(7)K2=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,Y0≥ε<δ∈FI2.
Therefore,K1∩K2≠∅, since K1∩K2∈FI2. Let i,j∈K1∩K2 and take ε≔d¯X0,Y0/3>0 such that we have
(8)1ijk≤i,l≤j:d¯Xkl,X0≥ε<δ,and it goes along with
(9)1ijk≤i,l≤j:d¯Xkl,Y0≥ε<δ,i.e., for maximum k≤i, l≤j, we have d¯Xkl,X0<ε and d¯Xkl,Y0<ε for a very small δ>0. Thus, we have to acquire
(10)k≤i,l≤j:d¯Xkl,X0<ε∩k≤i,l≤j:d¯Xkl,Y0<ε≠∅,a contradiction, as the nbd of X0 and Y0 are disjoint. Hence, X0 is uniquely identified.Theorem 9.
LetXkl be a fuzzy numbers sequence then stlimXkl=X0 implies I2−stlimXkl=X0.Proof.
LetstlimXkl=X0. Then, for each ε>0, the set
(11)Kε=k≤s,l≤w:d¯Xkl,X0≥ε,has natural density zero, i.e.,
(12)lims,w⟶∞1swk≤s,l≤w:d¯Xkl,X0≥ε=0.
Therefore, for everyε>0 and δ>0,
(13)Tε,δ=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,X0≥ε≥δ,is a finite set and so Tε,δ∈I2, where I2 is an admissible ideal. Hence, we get I2−stlimXkl=X0.Theorem 10.
TakeXkl as a sequence of fuzzy numbers. I2−limXkl=X0 implies I2−stlimXkl=X0.Proof.
The proof of this theorem is clear.But the reverse is not true. For instance, take forI2 the class I2f of all finite subsets of ℕ×ℕ, the fuzzy number Xkl, where
(14)Xklp≔n+m+pn+m,−n−m≤p≤0,n+m−pn+m0≤p≤n+m,for k=n2,l=m2, n,m∈ℕ, and
(15)Xklp≔1+pnm,−1nm≤p≤0,1−pnm0≤p≤1nm,for k≠n2,l≠m2, n,m∈ℕ. Then, Xkl is I2-statistically convergent, but not I2-convergent.Theorem 11.
LetXkl and Ykl be two fuzzy numbers sequence. Then,
(i)
I2−stlimXkl=X0, c∈ℝ implies I2−stlimcXkl=cX0(ii)
I2−stlimXkl=X0, I2−stlimYkl=Y0 implies I2−stlimXkl+Ykl=X0+Y0Proof. (i) Forc=0, there is nothing to prove.
So, presume thatc≠0. Now
(16)1swk≤s,l≤w:d¯cXkl,cX0≥ε=1swk≤s,l≤w:cd¯Xkl,X0≥ε≤1swk≤s,l≤w:d¯Xkl,X0≥εc<δ.
Therefore, we obtain(17)s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯cXkl,cX0≥ε<δ∈FI2,i.e., I2−stlimcXkl=cX0.
(ii) We have(18)K1=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,X0≥ε2<δ2∈FI2,(19)K2=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Ykl,Y0≥ε2<δ2∈FI2.
Since,K1∩K2≠∅, therefore, for all s,w∈K1∩K2, we get
(20)1swk≤s,l≤w:d¯Xkl+Ykl,X0+Y0≥ε≤1swk≤s,l≤w:d¯Xkl,X0≥ε2+1swk≤s,l≤w:d¯Ykl,Y0≥ε2<δ,i.e.,
(21)s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl+Ykl,X0+Y0≥ε<δ∈FI2.
Hence, we haveI2−stlimXkl+Ykl=X0+Y0.Definition 12.
An elementX0∈Lℕ is called to be an I2-statistical limit point of a fuzzy number sequence X=Xkl provided that for each ε>0 there is a set
(22)M=k1,l1<k2,l2<⋯<kr,ls<⋯⊂ℕ×ℕ,such that M∉I2 and st−limXkr,ls=X0.
I2−SΛX indicates the set of all I2-statistical limit point of a fuzzy number sequence Xkl.Theorem 13.
TakeXkl as a sequence of fuzzy numbers. If I2−stlimXkl=X0, then I2−SΛX=X0.Proof.
SinceI2−stlimXkl=X0, for each ε,δ>0, the set
(23)K=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,X0≥ε≥δ∈I2,where I2 is an admissible ideal.
Assume thatI2−SΛX involves Y0 different from X0, i.e., Y0∈I2−SΛX. So, there is a M⊂ℕ×ℕ such that M∉I2 and st−limXkr,ls=Y0.
Let(24)P=s,w∈M:1swk≤s,l≤w:d¯Xkl,Y0≥ε≥δ.
SoP is a finite set and therefore P∈I2. So
(25)Pc=s,w∈M:1swk≤s,l≤w:d¯Xkl,Y0≥ε<δ∈FI2.
Again let(26)K1=s,w∈M:1swk≤s,l≤w:d¯Xkl,X0≥ε≥δ.
SoK1⊂K∈I2, i.e., K1c∈FI2. Therefore, K1c∩Pc≠∅, since K1c∩Pc∈FI2.
Leti,j∈K1c∩Pc and take ε≔d¯X0,Y0/3>0, so
(27)1ijk≤i,l≤j:d¯Xkl,X0≥ε<δand,(28)1ijk≤i,l≤j:d¯Xkl,Y0≥ε<δ,i.e., for maximum k≤i,l≤j will satisfy d¯Xkl,X0<ε and d¯Xkl,Y0<ε for a very small δ>0. Thus, we have to obtain
(29)k≤i,l≤j:d¯Xkl,X0<ε∩k≤i,l≤j:d¯Xkl,Y0<ε≠∅,a contradiction, as the nbd of X0 and Y0 are disjoint. Hence, I2−SΛX=X0.Definition 14.
An elementX0∈Lℝ is known as I2-statistical cluster point of a fuzzy number sequence X=Xkl if for each ε>0 and δ>0, the set
(30)s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,X0≥ε<δ∉I2.
I2−SΓX demonstrates the set of all I2-statistical cluster point of a fuzzy number sequence Xkl.Theorem 15.
For any sequenceXkl of fuzzy numbers I2−SΓX is closed.Proof.
Let the fuzzy numberY0 be a limit point of the set I2−SΓX. Then, for any ε>0,
(31)I2−SΓX∩BY0,ε≠∅,where
(32)BY0,ε=W∈Lℝ:d¯W,Y0<ε.
LetZ0∈I2−SΓX∩BY0,ε and select ε1>0 such that BZ0,ε1⊆BY0,ε. Then, we get
(33)k≤s,l≤w:d¯Xkl,Z0≥ε1⊇k≤s,l≤w:d¯Xkl,Y0≥ε,which implies that
(34)1swk≤s,l≤w:d¯Xkl,Z0≥ε1≥1swk≤s,l≤w:d¯Xkl,Y0≥ε.
Now, for anyδ>0, we obtain
(35)s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,Z0≥ε1<δ,(36)⊆s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,Y0≥ε<δ.
SinceZ0∈I2−SΓX, we have
(37)s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,Y0≥ε<δ∉I2,i.e., Y0∈I2−SΓX. This concludes the proof.Theorem 16.
For any fuzzy number sequenceXkl,
(38)I2−SΛX⊆I2−SΓX.Proof.
LetX0∈I2−SΛX. In that case, there is a set
(39)M=k1,l1<k2,l2<⋯<kr,ls<⋯∉I2,such that st−limXkr,ls=X0. So, we have
(40)limk,l⟶∞1klkr≤k,ls≤l:d¯Xkrls,X0≥ε=0.
Takeδ>0, so there is n0∈ℕ such that for s,w>n0, we have
(41)1swkr≤s,ls≤w:d¯Xkrls,X0≥ε<δ.
Let(42)K=s,w∈ℕ×ℕ:1swkr≤s,ls≤w:d¯Xkrls,X0≥ε<δ.
Also, we have(43)K⊃M\k1,l1,k2,l2,⋯,kn0,ln0.
Considering thatI2 is an admissible ideal and M∉I2, therefore, K∉I2. Hence, according to the definition of I2-statistical cluster point X0∈I2−SΓX, this finalizes the proof.Theorem 17.
IfXkl and Ykl are two sequences of fuzzy numbers such that
(44)k,l∈ℕ×ℕ:Xkl≠Ykl∈I2,then
(45)I2−SΛX=I2−SΛY.(46)I2−SΓX=I2−SΓY.Proof. (i) LetX0∈I2−SΛX.
So, according to the definition, there is a set(47)M=k1,l1<k2,l2<⋯<kr,ls<⋯⊂ℕ×ℕ,such that M∉I2 and st−limXkr,ls=X0. Since
(48)k,l∈M:Xkl≠Ykl⊆k,l∈ℕ×ℕ:Xkl≠Ykl∈I2,(49)M′=k,l∈M:Xkl=Ykl∉I2andM′⊆M.
So, we havest−limYkr′,ls′=X0. This denotes that X0∈I2−SΛY and therefore I2−SΛX⊆I2−SΛY. By symmetry, I2−SΛY⊆I2−SΛX. Hence, we obtain I2−SΛX=I2−SΛY.
(ii) LetX0∈I2−SΓX. So, by the definition for each ε>0, we have
(50)K=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Xkl,X0≥ε<δ∉I2.
Let(51)L=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Ykl,X0≥ε<δ.
We have to prove thatL∉I2. Presume that L∈I2, So
(52)Lc=s,w∈ℕ×ℕ:1swk≤s,l≤w:d¯Ykl,X0≥ε≥δ∈FI2.
By hypothesis,(53)P=k,l∈ℕ×ℕ:Xkl=Ykl∈FI2.
Therefore,Lc∩P∈FI2. Also, it is clear that Lc∩P⊆Kc∈FI2, i.e., K∈I2, this is a contradiction. Therefore, L∉I2 and thus the desired result was achieved.



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