# Some New Observations on Wijsmanℐ2-Lacunary Statistical Convergence of Double Set Sequences in Intuitionistic Fuzzy Metric Spaces

Authors: Ömer Kişi
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we investigate the notions of the Wijsmanℐ2-statistical convergence, Wijsman ℐ2-lacunary statistical convergence, Wijsman strongly ℐ2-lacunary convergence, and Wijsman strongly ℐ2-Cesàro convergence of double sequence of sets in the intuitionistic fuzzy metric spaces (briefly, IFMS). Also, we give the notions of Wijsman strongly ℐ2∗-lacunary convergence, Wijsman strongly ℐ2-lacunary Cauchy, and Wijsman strongly ℐ2∗-lacunary Cauchy set sequence in IFMS and establish noteworthy results.


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## Body

## 1. Introduction and Background

Statistical convergence was firstly examined by Henry Fast [1]. This notion was redefined for double sequences by Mursaleen and Edely [2]. As a consequence of the study of ideal convergence defined by Kostyrko et al. [3], there has been valuable research to discover summability works of the classical theories. Das et al. [4] rethought ℐ-convergence of double sequences and worked some features of this convergence. Ideal convergence became a noteworthy topic in summability theory after the studies of [5–11].Fridy and Orhan [12] examined the notion of lacunary statistical convergence by using lacunary sequence. The publication of the paper affected deeply all the scientific fields. Çakan and Altay [13] demonstrated multidimensional similarities of the conclusions given by Fridy and Orhan [12]. Some works in lacunary statistical convergence can be found in [13–17].Theory of fuzzy sets (FSs) was firstly given by Zadeh [18]. This work affected deeply all the scientific fields. The theory of FSs has submitted to employ imprecise, vagueness, and inexact data [18]. FSs have been widely implemented in different disciplines and technologies. The theory of FSs cannot always cope with the lack of knowledge of membership degrees. That is why Atanassov [19] investigated the theory of IFS which is the extension of the theory of FSs. Kramosil and Michalek [20] investigated fuzzy metric space (FMS) utilizing the concepts fuzzy and probabilistic metric space. The FMS as a distance between two points to be a nonnegative fuzzy number was examined by Kaleva and Seikkala [21]. George and Veeramani [22] gave some qualifications of FMS. Some basic features of FMS were given, and significant theorems were proved in [23]. Moreover, FMS has used in practical research studies, for example, decision-making, fixed point theory, and medical imaging. Park [24] generalized FMSs and defined IF metric space (IFMS). Park utilized George and Veeramani’s [22] opinion of using t-norm and t-conorm to the FMS meantime describing IFMS and investigating its fundamental properties. The concept of IF-normed spaces (IFNS for shortly) was given by Lael and Nourouzi [25]. In order to have a different topology from the topology generated by the F-norm μ, the condition μ+v≤1 was omitted from Park’s definition. Statistical convergence, ideal convergence, and different features of sequences in INFS were examined by several authors [26–29].Recently, convergence of sequences of sets was studied by several authors. Nuray and Rhoades [30] presented the idea of statistical convergence of set sequences and established some essential theorems. Ulusu and Nuray [31] examined the lacunary statistical convergence of sequence of sets. Convergence for sequences of sets became a notable topic in summability theory after the studies of (see [32–38]).Lacunary statistical convergence and lacunary strongly convergence for sequence of sets in IFMS were worked by Kisi [39]. Further, Wijsman ℐ-convergence and Wijsman ℐ∗-convergence for sequence of sets in IFMS were investigated by Esi et al. [40].Throughout this work, we indicateℐ2 to be the admissible ideal in ℕ×ℕ, θ2=ju,ks to be a double lacunary sequence, X,η,ν,∗,⋄ to be the IFMS, and FFwq to be nonempty closed subsets of X.


## 2. Main Results

Definition 1.
A sequenceFwq of nonempty closed subsets of X is known as Wijsman ℐ2-statistical convergent to F or SℐW2η,ν-convergent to F with regard to IFM η,ν, if for every ξ∈0,1, σ>0, for each x∈X, and for all p>0,(1)s,t∈ℕ×ℕ:1stw,q:w≤s,q≤t,ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≥σ∈ℐ2.
We demonstrate this symbolically byFwqSℐW2η,ν⟶F or Fwq⟶FSℐW2η,ν. The set of all Wijsman ℐ2-statistical convergent sequences in IFMS is indicated by SℐW2η,ν.Example 1.
LetX=ℝ2 and double sequence Fwq be determined as follows:(2)Fwq≔d,e∈ℝ2:d+w2+e+q2=1,ifwandqare square integers,1,1,otherwise.
Ifℐ2=ℐ2δ (ℐ2δ is the class of K⊂ℕ×ℕ with density of K equal to 0), then the sequence Fwq is Wijsman ℐ2-statistical convergent to F=1,1 with regard to IFM η,ν.Definition 2.
A sequenceFwq is Wijsman strong ℐ2-Cesàro summable to F or C1ℐW2η,ν-summable to F with regard to IFM η,ν, if for every ξ∈0,1, for each x∈X, and for all p>0,(3)s,t∈ℕ×ℕ:1st∑w,q=1,1s,tηx,Fwq,p−ηx,F,p≤1−ξor1st∑w,q=1,1s,tνx,Fwq,p−νx,F,p≥ξ∈ℐ2.
We writeFwq⟶C1ℐW2η,νF or Fwq⟶FC1ℐW2η,ν.Example 2.
LetX=ℝ2 and double sequence Fwq be determined as follows:(4)Fwq≔d,e∈ℝ2:d+12+e2=1wq;ifwandqare square integers,0,1;otherwise.
Ifℐ2=ℐ2f (ℐ2f is the class of finite subsets of ℕ×ℕ), then sequence Fwq is Wijsman strong ℐ2-Cesàro summable to F=0,1 with regard to IFM η,ν.Definition 3.
The sequenceFwq is known as Wijsman ℐ2-lacunary statistically convergent to F or Sθ2ℐW2η,ν-convergent to F with regard to IFM η,ν, if for every ξ∈0,1, σ>0, for each x∈X, and for all p>0,(5)u,s∈ℕ×ℕ:1husw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≥σ∈ℐ2.
In that case, we writeFwq⟶Sθ2ℐW2η,νF or Fwq⟶FSθ2ℐW2η,ν.Example 3.
LetX=ℝ2 and double sequence Fwq be determined as follows:(6)Fwq≔d,e∈ℝ2:d−w2+e+q2=1,ifw,q∈Ius;wandqare square integers,−1,1,otherwise.
If we takeℐ2=ℐ2δ, then the sequence Fwq is Wijsman ℐ2-lacunary statistical convergent to F=−1,1 with regard to IFM η,ν.Definition 4.
A double sequenceFwq is Wijsman strong ℐ2-lacunary summable to F or Nθ2ℐW2η,ν-summable to F with regard to IFM η,ν, if for every ξ∈0,1, for all p>0, and for each x∈X,(7)u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p≤1−ξor1hus∑w,q∈Iusνx,Fwq,p−νx,F,p≥ξ∈ℐ2.
We shall writeFwq→Nθ2ℐW2η,νF or Fwq⟶FNθ2ℐW2η,ν.Example 4.
LetX=ℝ2 and double sequence Fwq be determined as follows:(8)Fwq≔d,e∈ℝ2:d2+e−12=1wq;ifw,q∈Ius;wandqare square integers,1,0;otherwise.
If we takeℐ2=ℐ2f, then the sequence Fwq is Wijsman strong ℐ2-lacunary summable to F=1,0 with regard to IFM η,ν.Theorem 1.
Letθ2=ju,ks be a double lacunary sequence. Then,(9)Fwq⟶FNθ2ℐW2η,ν⟹Fwq⟶FSθ2ℐW2η,ν.Proof.
Letξ∈0,1 and Fwq⟶FNθ2ℐW2η,ν. At that time, for every x∈X, we acquire(10)∑w,q∈Iusηx,Fwq,p−ηx,F,porνx,Fwq,p−νx,F,p≥∑w,q∈Iusηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξηx,Fwq,p−ηx,F,porνx,Fwq,p−νx,F,p≥ξ.w,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ,and so(11)1ξhus∑w,q∈Iusηx,Fwq,p−ηx,F,porνx,Fwq,p−νx,F,p≥1husw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ.
Then, for anyσ>0 and for every x∈X,(12)u,s∈ℕ×ℕ:1husw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≥σ⊆u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p≤1−ξ.σor1hus∑w,q∈Iusνx,Fwq,p−νx,F,p≥ξ.σ∈ℐ2.
This proof is concluded.Theorem 2.
Letθ2=ju,ks be a double lacunary sequence. Then, Fwq is bounded (Fwq∈L∞2X) and(13)Fwq⟶FSθ2ℐW2η,ν⟹Fwq⟶FNθ2ℐW2η,ν.
The set of all bounded double sequences of sets in IFMS is indicated byL∞2X.Proof.
Presume thatFwq⟶FSθ2ℐW2η,ν and Fwq∈L∞2. At this point, there is an H>0 such that(14)ηx,Fwq,p−ηx,F,p≥1−Horνx,Fwq,p−νx,F,p≤H,for every x∈X and all w,q∈ℕ. Given ξ∈0,1, we obtain(15)1hus∑w,q∈Iusηx,Fwq,p−ηx,F,porνx,Fwq,p−νx,F,p=1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p≤1−ξ/2orνx,Fwq,p−νx,F,p≥ξ/2ηx,Fwq,p−ηx,F,porνx,Fwq,p−νx,F,p+1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p>1−ξ/2orνx,Fwq,p−νx,F,p<ξ/2ηx,Fwq,p−ηx,F,porνx,Fwq,p−νx,F,p≤Hhusw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξ2orνx,Fwq,p−νx,F,p≥ξ2+ξ2.
As a consequence, for eachx∈X, we get(16)u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p≤1−ξor1hus∑w,q∈Iusνx,Fwq,p−νx,F,p≥ξ⊆u,s∈ℕ×ℕ:1husw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξ2orνx,Fwq,p−νx,F,p≥ξ2≥ξ2H∈ℐ2.
This proof is concluded.Corollary 1.
We have the following result:(17)Sθ2ℐW2η,ν∩L∞2X=Nθ2ℐW2η,ν∩L∞2X.Theorem 3.
Ifliminfuqu>1 and liminfsqs>1, then Fwq⟶FSℐW2η,ν implies Fwq⟶FSθ2ℐW2η,ν.Proof.
Presume thatliminfuqu>1 and liminfsqs>1. Then, there are ζ>0,ψ>0 such that(18)qu≥1+ζandqs≥1+ψ,for sufficiently large u, s which gives that(19)husjuks≥ζψ1+ζ1+ψ.
Assume thatFwq⟶FSℐW2η,ν. For each ξ∈0,1, for all p>0, and for each x∈X, we have(20)1juksw,q:w≤ju,q≤ks,ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≥1juksw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ=husjuks1husw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≥ζψ1+ζ1+ψ1husw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ.
Thus, for anyσ>0,(21)u,s∈ℕ×ℕ:1husw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≥σ⊆u,s∈ℕ×ℕ:1juksw,q:w≤ju,q≤ks,ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≥ζψσ1+ζ1+ψ.
Hence, by our supposition, the set on the right side belongs toℐ2, and clearly the set on the left side belongs to ℐ2. As a result, we obtain Fwq⟶FSθ2ℐW2η,ν.Theorem 4.
Iflimsupuqu<∞ and limsupsqs<∞, then Fwq⟶FSθ2ℐW2η,ν implies Fwq⟶FSℐW2η,ν.Proof.
Presume thatlimsupuqu<∞ and limsupsqs<∞. Then, there are P,R>0 such that qu<P and qs<R for all u and s. Assume that Fwq⟶FSθ2ℐW2η,ν and let(22)Hus≔w,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ.
SinceFwq⟶FSθ2ℐW2η,ν, it holds for each ξ∈0,1, σ>0, for every x∈X, and for all p>0,(23)u,s∈ℕ×ℕ:1husw,q∈Ius:ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≥σ=u,s∈ℕ×ℕ:Hushus≥σ∈ℐ2.
So, we can select positive integersu0,s0∈ℕ such that Hus/hus<σ for all u≥u0,s≥s0. Now, take(24)T≔maxHus:1≤u≤u0,1≤s≤s0,and let m and n be integers providing ju−1<m≤ju and ks−1<n≤ks. Then, for every ξ>0 and each x∈X, we get(25)1mnw,q:w≤m,q≤n,ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ≤1ju−1ks−1w,q:w≤ju,q≤ks,ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ=1ju−1ks−1H11+H12+H21+H22+⋯+Hu0s0+⋯+Hus≤u0s0ju−1ks−1max1≤w≤u01≤q≤s0Hwq+1ju−1ks−1hu0s0+1Hu0s0+1hu0s0+1+hu0+1s0Hu0+1s0hu0+1s0+hu0+1s0+1Hu0+1s0+1hu0+1s0+1+⋯+husHushus≤u0s0Tju−1ks−1+1ju−1ks−1supu>u0s>s0Hushus∑u,sw≥u0q≥s0hwq≤u0s0Tju−1ks−1+σju−ju0ks−ks0ju−1ks−1≤u0s0Tju−1ks−1+σquqs≤u0s0Tju−1ks−1+σPR.
Sinceju−1ks−1⟶∞ as m,n⟶∞, it concludes that for each x∈X,(26)1mnw,q:w≤m,q≤n,ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ⟶0,and as a result for any σ1>0, the set(27)m,n∈ℕ×ℕ:1mn|w,q:w≤m,q≤n,ηx,Fwq,p−ηx,F,p≤1−ξorνx,Fwq,p−νx,F,p≥ξ|≥σ1∈ℐ2.
It gives thatFwq⟶FSℐW2η,ν.Theorem 5.
Letθ2 be a double lacunary sequence. If(28)1<liminfuqu<limsupuuq<∞,and1<liminfsqs<limsupssq<∞,then Fwq⟶FSθ2ℐW2η,ν if Fwq⟶FSℐW2η,ν.Proof.
It obvious from Theorem3 and Theorem 4.Theorem 6.
Letℐ2 be a strongly admissible ideal providing feature AP2, θ2∈ℱℐ2. If Fwq∈SℐW2η,ν∩Sθ2ℐW2η,ν, then(29)SℐW2η,ν−limw,q⟶∞Fwq=Sθ2ℐW2η,ν−limw,q⟶∞Fwq.Proof.
Presume thatSℐW2η,ν−limw,q⟶∞Fwq=A and Sθ2ℐW2η,ν−limw,q⟶∞Fwq=B and A≠B. Let(30)0<ξ<12ηx,A,p−ηx,B,p,and0<ξ<12νx,A,p−νx,B,p,for every x∈X. Since ℐ2 provides the feature AP2, then there is M∈ℱℐ2(i.e., ℕ×ℕ╲M∈ℐ2) such that for every x∈X and for m,n∈M,(31)limm,n⟶∞1mnw≤m,q≤n:ηx,Fwq,p−ηx,A,p≤1−ξorνx,Fwq,p−νx,A,p≥ξ=0.
Let(32)T=w≤m,q≤n:ηx,Fwq,p−ηx,A,p≤1−ξ,orνx,Fwq,p−νx,A,p≥ξ,S=w≤m,q≤n:ηx,Fwq,p−ηx,B,p≤1−ξ,orνx,Fwq,p−νx,B,p≥ξ.
Then,mn=T∪S≤T+S. This gives that 1≤T/mn+S/mn. Since S/mn≤1 and limm,n⟶∞T/mn=0, we have to get limm,n⟶∞S/mn=1.
LetM∗=M∩θ2∈ℱℐ2. Then, for every x∈X and for wk,qj∈M∗, the wkqjth term of the statistical limit expression(33)1mnw≤m,q≤n:ηx,Fwq,p−ηx,B,p≤1−ξ,orνx,Fwq,p−νx,B,p≥ξ,is (34)1wkqjw,q∈∪u,s=1,1k,jIus:ηx,Fwq,p−ηx,B,p≤1−ξ,orνx,Fwq,p−νx,B,p≥ξ=1∪u,s=1,1k,jhus∪u,s=1,1k,jvushus,where(35)vus=1husw,q∈Ius:ηx,Fwq,p−ηx,B,p≤1−ξ,orνx,Fwq,p−νx,B,p≥ξ⟶ℐ20,because Sθ2ℐW2η,ν−limw,q⟶∞Fwq=B. Since θ2 is a lacunary sequence, (34) is a regular weighted mean transform of vus’s, and as a result, it is ℐ2-convergent to 0 as k,j⟶∞, and also it has a subsequence which is convergent to 0 since ℐ2 provides the feature AP2. However, since this a subsequence of(36)1mnw≤m,q≤n:ηx,Fwq,p−ηx,B,p≤1−ξ,orνx,Fwq,p−νx,B,p≥ξm,n∈M,we conclude that(37)1mnw≤m,q≤n:ηx,Fwq,p−ηx,B,p≤1−ξ,orνx,Fwq,p−νx,B,p≥ξm,n∈M,which is not convergent to 1. This contradiction indicates that we cannot have A≠B.Theorem 7.
Ifliminfuqu>1 and liminfsqs>1, then C1ℐW2η,ν⊆Nθ2ℐW2η,ν.Proof.
Letliminfuqu>1 and liminfsqs>1. Then, there are ζ,ψ>0 such that qu≥1+ζ and qs≥1+ψ for all uands which gives that(38)jukshus≤1+ζ1+ψζψ,andju−1ks−1hus≤1ζψ.
Presume thatFwq⟶FC1ℐW2η,ν. For each x∈X, we get(39)1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p−1=1hus∑w,q=1,1ju,ksηx,Fwq,p−ηx,F,p−1hus∑w,q=1,1ju−1,ks−1ηx,Fwq,p−ηx,F,p−1=jukshus1juks∑w,q=1,1ju,ksηx,Fwq,p−ηx,F,p−1−ju−1ks−1hus1ju−1ks−1∑w,q=1,1ju−1,ks−1ηx,Fwq,p−ηx,F,p−1.
SinceFwq⟶FC1ℐW2η,ν, then for each(40)1juks∑w,q=1,1ju,ksηx,Fwq,p−ηx,F,p−1⟶ℐ20,and,1ju−1ks−1∑w,q=1,1ju−1,ks−1ηx,Fwq,p−ηx,F,p−1⟶ℐ20.
Hence, when the above equality is examined, for everyx∈X, we have(41)1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p−1⟶ℐ20.
Similarly, we obtain(42)1hus∑w,q∈Iusνx,Fwq,p−νx,F,p⟶ℐ20.
That is,Fwq⟶FNθ2ℐW2η,ν. As a result, we obtain C1ℐW2η,ν⊆Nθ2ℐW2η,ν.Theorem 8.
Ifliminfuqu=1 and liminfsqs=1, then Nθ2ℐW2η,ν⊆C1ℐW2η,ν.Proof.
Takeliminfuqu=1, liminfsqs=1, and Fwq∈Nθ2ℐW2η,ν. Then, for every p>0, we acquire(43)Hus=1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p⟶ℐ21,Hus′=1hus∑w,q∈Iusνx,Fwq,p−νx,F,p⟶ℐ20,as u,s⟶∞. Then, for ξ>0, there are u0,s0∈ℕ such that Hus<1+ξ for all u>u0, s>s0. Also, we can find K>0 such that Hus<K and Hus′<K, u,s=1,2,…. Let mandn be an integer with ju−1<m≤ju and ks−1≤n≤ks. Then,(44)1mn∑w,q=1,1m,nηx,Fwq,p−ηx,F,p≤1ju−1ks−1∑w,q=1,1ju,ksηx,Fwq,p−ηx,F,p,=1ju−1ks−1∑w,q∈I11ηx,Fwq,p−ηx,F,p+⋯+∑w,q∈Iusηx,Fwq,p−ηx,F,p=sup1≤u≤u0,1≤s≤s0Husju0ks0ju−1ks−1+hu0+1s0+1ju−1ks−1Hu0+1s0+1+⋯+husju−1ks−1Hus<Kju0ks0ju−1ks−1+1+ξjuks−ju0ks0ju−1ks−1.
Sinceju−1ks−1⟶∞ as m,n⟶∞, it follows that 1/mn∑w,q=1m,nηx,Fwq,p−ηx,F,p⟶ℐ21. Similarly, we can show that 1/mn∑w,q=1m,nνx,Fwq,p−νx,F,p⟶ℐ20. Hence, Fwq∈C1ℐW2η,ν.Theorem 9.
IfFwq∈Nθ2ℐW2η,ν∩C1ℐW2η,ν, then Nθ2ℐW2η,ν−limFwq=C1ℐW2η,ν−limFwq.Proof.
LetFwq⟶F1Nθ2ℐW2η,ν and Fwq⟶F2C1ℐW2η,ν. Assume r∈ℕ and ξ>0 in such way that r>2/ξ. Then, for any p>0, there are u0,s0∈ℕ such that(45)1hus∑w,q∈Iusηx,Fwq,p2−ηx,F1,p2>1−1r,and1hus∑w,q∈Iusνx,Fwq,p2−νx,F1,p2<1r,for all u>u0 and s>s0. Also, there are m0,n0∈ℕ such that(46)1mn∑w,q=1,1m,nηx,Fwq,p2−ηx,F2,p2>1−1r,and1mn∑w,q=1,1m,nνx,Fwq,p2−νx,F2,p2<1r,for all m>m0 and n>n0. Take r1=maxu0,m0 and r2=maxs0,n0. Then, we take k,t∈ℕ such that(47)ηx,Fkt,p2−ηx,F1,p2≥1hus∑w,q∈Iusηx,Fwq,p2−ηx,F1,p2>1−1r,ηx,Fkt,p2−ηx,F2,p2≥1mn∑w,q=1,1m,nηx,Fwq,p2−ηx,F2,p2>1−1r.
Therefore, we get(48)ηx,F1,p−ηx,F2,p≤ηx,Fkt,p2−ηx,F1,p2+ηx,Fkt,p2−ηx,F2,p2>1−1r+1−1r>1−ξ.
Sinceξ>0 is arbitrary, we get ηx,F1,p−ηx,F2,p=1 for all p>0, which yields F1=F2.Throughout the following definitions and theorems, we considerX,η,ν,∗,⋄ to be a separable IFMS and ℐ2 to be a strongly admissible ideal.Definition 5.
The sequenceFwq is strongly ℐ2-lacunary Cauchy sequence (Wijsman sense) if for each ξ∈0,1, for each x∈X, and for all p>0, there are s=sξ,x,t=tξ,x∈ℕ such that(49)Aξ,x,p=u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,Fst,p≤1−ξor,1hus∑w,q∈Iusνx,Fwq,p−νx,Fst,p≥ξ∈ℐ2.Theorem 10.
Every Wijsman stronglyℐ2-lacunary convergent sequence of closed sets Fwq is Wijsman strongly ℐ2-lacunary Cauchy with regard to IFM η,ν.Proof.
LetFwq⟶Nθ2ℐW2η,νF. At that case, for each ξ∈0,1, for every x∈X, and for all p>0,(50)Aξ,x,p=u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p≤1−ξor,1hus∑w,q∈Iusνx,Fwq,p−νx,F,p≥ξ∈ℐ2.
Sinceℐ2 is a strongly admissible ideal, the set(51)Acξ,x,p=u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p>1−ξand1hus∑w,q∈Iusνx,Fwq,p−νx,F,p<ξ,is nonempty and belongs to ℱℐ2. So, we select positive integers uands such that u,s∉Aξ,x,p, and we get(52)1hus∑w0,q0∈Iusηx,Fw0q0,p−ηx,F,p>1−ξ,and1hus∑w0,q0∈Iusνx,Fw0q0,p−νx,F,p<ξ.
Now, presume that(53)Bξ,x,p=u,s∈ℕ×ℕ:1hus∑w,q,w0,q0∈Iusηx,Fwq,p−ηx,Fw0q0,p≤1−2ξor1hus∑w,q,w0,q0∈Iusνx,Fwq,p−νx,Fw0q0,p≥2ξ.
Consider the inequality(54)1hus∑w,q,w0,q0∈Iusηx,Fwq,p−ηx,Fw0q0,p≤1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p+1hus∑w0,q0∈Iusηx,Fw0q0,p−ηx,F,p,1hus∑w,q,w0,q0∈Iusνx,Fwq,p−νx,Fw0q0,p≤1hus∑w,q∈Iusνx,Fwq,p−νx,F,p+1hus∑w0,q0∈Iusνx,Fw0q0,p−νx,F,p.
Notice that ifu,s∈Bξ,x,p, therefore,(55)1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p+1hus∑w0,q0∈Iusηx,Fw0q0,p−ηx,F,p≤1−2ξ,1hus∑w,q∈Iusνx,Fwq,p−νx,F,p+1hus∑w0,q0∈Iusνx,Fw0q0,p−νx,F,p≥2ξ.
From another point of view, sinceu,s∉Aξ,x,p, we get(56)1hus∑w0,q0∈Iusηx,Fw0q0,p−ηx,F,p>1−ξ,and1hus∑w0,q0∈Iusνx,Fw0q0,p−νx,F,p<ξ.
We reach that(57)1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p≤1−ξ,or1hus∑w,q∈Iusνx,Fwq,p−νx,F,p≥ξ.
Hence,u,s∈Aξ,x,p. This gives that Bξ,x,p⊂Aξ,x,p∈ℐ2 for every ξ∈0,1 and for each x∈X. Therefore, Bξ,x,p∈ℐ2, so the sequence is Wijsman strongly ℐ2-lacunary sequence.Definition 6.
The sequenceFwq is Wijsman strongly ℐ2∗-lacunary convergent to F iff there is a set M=w,q∈ℕ×ℕ such that M′=u,s∈ℕ×ℕ:w,q∈Ius∈ℱℐ2 for each x∈X,(58)limu,s⟶∞1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p=1,limu,s⟶∞1hus∑w,q∈Iusνx,Fwq,p−νx,F,p=0.
In this case, we writeFwq⟶FNθ2ℐW2∗η,ν.Theorem 11.
If the sequenceFwq is Wijsman strongly ℐ2∗-lacunary convergent to F, then Fwq is Wijsman strongly ℐ2-lacunary convergent to F.Proof.
Presume thatFwq⟶FNθ2ℐW2∗η,ν. Then, there is a set M=w,q∈ℕ×ℕ such that(59)M′=u,s∈ℕ×ℕ:w,q∈Ius∈ℱℐ2,for each x∈X,(60)1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p>1−ξ,1hus∑w,q∈Iusνx,Fwq,p−νx,F,p<ξ.for every ξ>0 and for all w,q≥k0=k0ξ,x∈ℕ. Hereby, for each ξ>0 and x∈X, we get(61)Pξ,x,p=u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p≤1−ξor,1hus∑w,q∈Iusνx,Fwq,p−νx,F,p≥ξ,⊂H∪M′∩1,2,…,k0−1×ℕ∪ℕ×1,2,…,k0−1,for ℕ×ℕ╲M′=H∈ℐ2. Since ℐ2 is an admissible ideal, we obtain(62)H∪M′∩1,2,…,k0−1×ℕ∪ℕ×1,2,…,k0−1∈ℐ2,and so Pξ,x,p∈ℐ2. Hence, Fwq⟶FNθ2ℐW2η,ν.Theorem 12.
Letℐ2 be a strongly admissible ideal involving feature AP2. Then, Fwq⟶FNθ2ℐW2η,ν implies Fwq⟶FNθ2ℐW2∗η,ν.Definition 7.
The sequenceFwq is known as Wijsman strongly ℐ2∗-lacunary Cauchy sequence if for each ξ∈0,1, for all x∈X, and for all p>0, and there is a set M=w,q∈ℕ×ℕ such that M′=u,s∈ℕ×ℕ:w,q∈Ius∈ℱℐ2 and N=Nε,x∈ℕ such that(63)1hus∑w,q,s,t∈Iusηx,Fwq,p−ηx,Fst,p>1−ξ,and1hus∑w,q,s,t∈Iusνx,Fwq,p−νx,Fst,p<ξ,for every w,q,s,t≥N.Theorem 13.
Every Wijsman stronglyℐ2∗-lacunary Cauchy sequence of closed sets is Wijsman strongly ℐ2-lacunary Cauchy in IFMS with regard to η,ν.Proof.
If the hypothesis is provided, then for eachξ∈0,1, for each x∈X, and for all p>0, there is a set M=w,q∈ℕ×ℕ such that(64)M′=u,s∈ℕ×ℕ:w,q∈Ius∈ℱℐ2,and N=Nε,x∈ℕ such that(65)1hus∑w,q,s,t∈Iusηx,Fwq,p−ηx,Fst,p>1−ξ,and1hus∑w,q,s,t∈Iusνx,Fwq,p−νx,Fst,p<ξ,for each w,q,s,t≥N. Let H=ℕ×ℕ╲M′. It is clear that H∈ℐ2 and(66)Pξ,x,p=u,s∈ℕ×ℕ:1hus∑w,q,s,t∈Iusηx,Fwq,p−ηx,Fst,p>1−ξ,or1hus∑w,q,s,t∈Iusνx,Fwq,p−νx,Fst,p<ξ,,⊂H∪M′∩1,2,…,N−1×ℕ∪ℕ×1,2,…,N−1.
Asℐ2 be a strongly admissible ideal, then(67)H∪M′∩1,2,…,N−1×ℕ∪ℕ×1,2,…,N−1∈ℐ2.
Therefore, we obtainPξ,x,p∈ℐ2; that is, Fwq is strongly ℐ2-lacunary Cauchy sequence (Wijsman sense) with regard to η,ν.Theorem 14.
Letℐ2 be an admissible ideal involving property AP2. Then, the concept of Wijsman strongly ℐ2-lacunary Cauchy sequence of sets coincides with Wijsman strongly ℐ2∗-lacunary Cauchy sequence of sets.Proof.
If a set sequence is stronglyℐ2∗-lacunary Cauchy sequence, then it is strongly ℐ2-lacunary Cauchy sequence according to Theorem 13, where ℐ2 need not to have the feature AP2.
Now, it is adequate to demonstrate that a sequenceFwq in X is a strongly ℐ2∗-lacunary Cauchy sequence under assumption that it is a strongly ℐ2-lacunary Cauchy sequence. Let Fwq be a Wijsman strongly lacunary Cauchy sequence. In this case, for each ξ∈0,1, for all x∈X, there is a number s=sε,x,t=tε,x∈ℕ such that(68)Aξ,x,p=u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,Fst,p≤1−ξor1hus∑w,q∈Iusνx,Fwq,p−νx,Fst,p≥ξ∈ℐ2.
Let(69)Pjξ,x,p=u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,Fsjtj,p>1−1jor1hus∑w,q∈Iusνx,Fwq,p−νx,Fsjtj,p<1j,where sj=s1/j and tj=t1/j, j=1,2,…. Clearly, for j=1,2,…, Pjξ,x,p∈ℱℐ2. Since ℐ2 has the property AP2, then by Theorem 3.3 in [9], there is P⊂ℕ×ℕ so that P∈ℱℐ2 and P╲Pj is finite for all j. Now, we demonstrate that(70)limw,q,s,t⟶∞1hus∑w,q∈Ius,s,t∈Iusηx,Fwq,p−ηx,Fst,p=1,ℐ2 and (71)limw,q,s,t⟶∞1hus∑w,q∈Ius,s,t∈Iusνx,Fwq,p−νx,Fst,p=0,for each x∈X and w,q,s,t∈P. To show these, let ξ∈0,1 and r∈ℕ such that r>2/ξ. If w,q,s,t∈P, then P╲Pr is a finite set; therefore, there is u=ur so that(72)1hus∑w,q∈Iusηx,Fwq,p−ηx,Fsrtr,p>1−1r,1hus∑s,t∈Iusηx,Fst,p−ηx,Fsrtr,p>1−1r,1hus∑w,q∈Iusνx,Fwq,p−νx,Fsrtr,p<1r,1hus∑s,t∈Iusνx,Fst,p−νx,Fsrtr,p<1r,for all w,q,s,t>ur. So, it follows that(73)1hus∑w,q∈Iusηx,Fwq,p−ηx,Fst,p≤1hus∑w,q∈Iusηx,Fwq,p−ηx,Fsrtr,p+1hus∑s,t∈Iusηx,Fst,p−ηx,Fsrtr,p>1−1r+1−1r>1−ξ,1hus∑w,q∈Iusνx,Fwq,p−νx,Fst,p≤1hus∑w,q∈Iusνx,Fwq,p−νx,Fsrtr,p+1hus∑s,t∈Iusνx,Fst,p−νx,Fsrtr,p<1r+1r<ξ.
Therefore, for eachξ∈0,1, ∃u=uξ, and w,q,s,t∈P∈ℱℐ2, we get(74)u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,Fst,p≤1−ξor1hus∑w,q∈Iusνx,Fwq,p−νx,Fst,p≥ξ∈ℐ2.
This implies thatFwq is Wijsman strongly ℐ2∗-lacunary Cauchy.Definition 8.
A sequenceFwq in IFMS is called to be Wijsman lacunary convergent to F with regard to IFM η,ν if, for every p>0 and ξ∈0,1, there is m0,n0∈ℕ such that(75)1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p>1−ξ,and1hus∑w,q∈Iusνx,Fwq,p−νx,F,p<ξ,for all u≥m0 and s≥n0. We write μ,vθ2−limFwq=F.Definition 9.
TakeX,η,ν,∗,⋄ as a separable IFMS and take Fwq∈X.(a)F∈X is known as Wijsman lacunary ℐ2-limit point of Fwq if there is set M=w1,q1<w2,q2<⋯<wu,qs<⋯⊂ℕ×ℕ such that the set(76)M′=u,s∈ℕ×ℕ:wu,qs∈Ius∉ℐ2,and η,vθ2−limFwuqs=F.(b)F∈X is known as Wijsman lacunary ℐ2-cluster point of Fwq if, for every p>0 and ε∈0,1, we get(77)u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p>1−ξand,1hus∑w,q∈Iusνx,Fwq,p−νx,F,p<ξ∉ℐ2.Here,Λη,vℐθ2Fwq denotes the set of all Wijsman lacunary ℐ2-limit points and Γη,vℐθ2Fwq indicates the set of all Wijsman lacunary ℐ2-cluster points in IFMS.Theorem 15.
For each sequenceFwq in IFMS, we have Λη,vℐθ2Fwq⊆Γη,vℐθ2Fwq.Proof.
LetF∈Λη,vℐθ2Fwq. So, there is a set M⊂ℕ×ℕ such that M′∉ℐ2, where M and M′ are as in Definition 9, satisfying η,vθ2−limFwuqs=F. Hence, for every p>0 and ξ∈0,1, there are m0,n0∈ℕ such that(78)1hus∑w,q∈Iusηx,Fwuqs,p−ηx,F,p>1−ξ,and1hus∑w,q∈Iusνx,Fwuqs,p−νx,F,p<ξ,for all u≥m0 and s≥n0. Therefore,(79)B=u,s∈ℕ×ℕ:1hus∑w,q∈Iusηx,Fwq,p−ηx,F,p>1−ξand1hus∑w,q∈Iusνx,Fwq,p−νx,F,p<ξ,⊇M′\w1,q1,w2,q2,…,wm0,qn0.
Now, withℐ2 being admissible, we must have M′\w1,q1,w2,q2,…,wm0,qn0∉ℐ2 and as such B∉ℐ2. Hence, F∈Γη,vℐθ2Fwq.


## 3. Conclusion

In this study, we examined a version of ideal convergence, named Wijsman lacunary ideal convergence of double set sequences, in IFMS. We investigated new convergence concepts for double set sequences in IFMS and obtained some meaningful results. In addition, Wijsman lacunaryℐ2-limit points and Wijsman lacunary ℐ2-cluster points of double set sequences in IFMS were defined. Some of the results presented in this article are analogous to the research studies in the relevant topic, but in most situations, the proofs follow a different approach.



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*Source: 6897038-2021-10-18.xml*