# Lacunaryℐ-Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces

Authors: Mualla Birgül Huban
Journal: Journal of Mathematics
 (2021)

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## Abstract

The concepts of invariant convergence, invariant statistical convergence, lacunary invariant convergence, and lacunary invariant statistical convergence for set sequences were introduced by Pancaroğlu and Nuray (2013). We know that ideal convergence is more general than statistical convergence for sequences. This has motivated us to study the lacunaryℐ-invariant convergence of sequence of sets in intuitionistic fuzzy metric spaces (briefly, IFMS). In this study, we examine the notions of lacunary ℐ-invariant convergence Wℐσθη,ν (Wijsman sense), lacunary ℐ∗-invariant convergence Wℐσθ∗η,ν (Wijsman sense), and q-strongly lacunary invariant convergence WNσθη,νq (Wijsman sense) of sequences of sets in IFMS. Also, we give the relationships among Wijsman lacunary invariant convergence, WNσθη,νq, Wℐσθη,ν, and Wℐσθ∗η,ν in IFMS. Furthermore, we define the concepts of Wℐσθη,ν-Cauchy sequence and Wℐσθ∗η,ν-Cauchy sequence of sets in IFMS. Furthermore, we obtain some features of the new type of convergences in IFMS.


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## Body

## 1. Introduction and Background

Fast [1] investigated the concept of statistical convergence. The publication of the study is affected deeply all the scientific fields. Nuray and Ruckle [2] redefined this concept which is known as generalized statistical convergence. A lot of development has been made in area about statistical convergence. Kostyrko et al. [3] defined ideal convergence, as a generalization of statistical convergence and worked some features of this convergence. Ideal convergence became a remarkable topic in summability theory after the studies of [4–8]. Fridy and Orhan [9] worked the notion of lacunary statistical convergence by using lacunary sequence.Various authors involving Raimi [10], Schaefer [11], and Mursaleen [12] worked invariant convergent sequences. Nuray et al. [13] investigated ℐσ-convergence with the help of σ-uniform density. Mursaleen [14] put forward the idea of strongly σ-convergence. Savaş and Nuray [15] presented the opinion of σ-statistical convergence and lacunary σ-statistical convergence and proved some correlation theorems. Nuray and Ulusu [16] defined lacunary ℐ-invariant convergence and lacunary ℐ-invariant Cauchy sequence of real numbers.After the original study of Zadeh [17], a huge number of research works have appeared on fuzzy theory and its applications as well as fuzzy analogues of the classical theories. Fuzzy sets (FSs) have been extensively applied in different disciplines and technologies. The theory of intuitionistic fuzzy sets (IFS) was presented by Atanassov [18]. The fuzzy sets and intuitionistic fuzzy sets have been widely used to solve many complex problems connected to different areas, especially in decision-making [19–22]. Kramosil and Michalek [23] worked fuzzy metric space (FMS) using the concepts fuzzy and probabilistic metric space. Park [24] rethought FMSs and investigated intuitionistic fuzzy metric space (IFMS). Park utilized George and Veeramani’s [25] opinion of using t-norm and t-conorm to the FMS meantime describing IFMS and investigating its fundamental properties. In [26], motivated by Park’s definition of an IF-metric, Lael and Nourouzi first defined an IF-normed space and then investigated, among other results, the fundamental theorems: open mapping, closed graph, and uniform boundedness in IF-normed spaces. In order to have a different topology from the topology generated by the F-norm ψ, the condition ψ+ϕ≤1 was omitted from Park’s definition. Statistical convergence, ideal convergence, and different features of sequences in INFS were examined by several authors [27–31]. For the extraction of information by reflecting and modeling the hesitancy present in real-life situation, intuitionistic fuzzy set theory has been playing a significant role. The implementation of IF sets in place of fuzzy sets means the introduction of another degree of freedom into set description. IF fixed point theory has become a subject of great interest for expert in fixed point theory because this branch of mathematics has covered new possibilities for summability theory.Convergence of sequences of sets has been examined by several authors. Nuray and Rhoades [32] presented a new convergence concept for sequences of sets called Wijsman statistical convergence. Ulusu and Nuray [33] examined the lacunary statistical convergence of sequence of sets. Kişi and Nuray [34] investigated ideal convergence for sequences of sets (Wijsman sense) and established some essential theorems. Convergence for sequences of sets became a notable topic in summability theory after the studies of [35–40].Lacunary statistical convergence and lacunary strongly convergence for sequence of sets in IFMS were examined by Kişi [41]. Furthermore, Wijsman ℐ-convergence and Wijsman ℐ∗-convergence for sequence of sets in IFMS were investigated by Esi et al. [42].The purpose of this study is to present some recent development in IFMS. The aim of the study is to examine some features of this new kind of convergence in IFMS. Also, it is demonstrated that the new kind of convergence in IFMS is generally different from the known convergence in classical metric space. However; it is indicated that if certain conditions are met, every classical metric space can be a IFMS at the same time.Throughout this work, we indicateℐ to be the admissible ideal in ℕ, θ be a lacunary sequence, X,ψ,ϕ,∗,⋄ to be the IFMS, and Υ, Yk to be nonempty closed subsets of X.


## 2. Main Results

Definition 1.
A sequenceYk of nonempty closed subsets of X is called to be lacunary invariant convergent (Wijsman sense) to Υ with regards to IFM ψ,ϕ, if for every ξ∈0,1, for each α∈X and for all τ>0, such that(1)limr⟶∞1hr∑k∈Irψα,Yσkm,τ−ψα,Y,τ=1,(2)limr⟶∞1hr∑k∈Irϕα,Yσkm,τ−ϕα,Y,τ=0,uniformly in m.Definition 2.
A sequenceYk of nonempty closed subsets of X is known as lacunary ℐ-invariant convergent or Wℐσθη,ν-convergent (Wijsman sense) to Y with regards to IFM ψ,ϕ, if for every ξ∈0,1, for each α∈X and for all τ>0, the set(3)Pξ,α,τ=k∈ℕ:ψα,Yk,τ−ψα,Y,τ≤1−ξorϕα,Yk,τ−ϕα,Y,τ≥ξ∈ℐσθ,that is, VθPξ,α,τ=0. We demonstrate this symbolically by Yk⟶YWℐσθψ,ϕ.Theorem 1.
LetYk be a bounded sequence. If Yk is Wℐσθψ,ϕ-convergent to Υ, then Yk is lacunary invariant convergent (Wijsman sense) to Υ with regards to IFM ψ,ϕ.Proof.
Letm∈ℕ be arbitrary and ξ∈0,1. For each α∈X and for all τ>0, we estimate(4)sm,r,α≔1hr∑k∈Irψα,Yσkm,τ−ψα,Y,τ.
Then, for eachα∈X and for all τ>0, we get(5)sm,r,α≤s1m,r,α+s2m,r,α,where(6)s1m,r,α≔1hr∑k∈Ir,ψα,Yσkm,τ−ψα,Y,τ≥1−ξψα,Yσkm,τ−ψα,Y,τ,(7)s2m,r,α≔1hr∑k∈Ir,ηα,Yσkm,τ−ηα,Y,τ<1−ξψα,Yσkm,τ−ψα,Y,τ.
For everym≥1 and for every α∈X, it is obvious that s2m,r,α<1−ξ. Since Yk is bounded sequence, there is a M>0, such that(8)ψα,Yσkm,τ−ψα,Y,τ≤M,k∈Ir,m≥1,and so, we have(9)s1m,r,α=1hr∑k∈Ir,ψα,Yσkm,τ−ψα,Y,τ≥1−ξψα,Yσkm,τ−ψα,Y,τ≤Mhrk∈Ir:ψα,Yσkm,τ−ψα,Y,τ≥1−ξ≤Mmaxmk∈Ir:ψα,Yσkm,τ−ψα,Y,τ≥1−ξhr=MSrhr.
Hence, we obtain(10)limr⟶∞1hr∑k∈Irψα,Yσkm,τ−ψα,Y,τ=1.
Similarly, we have(11)limr⟶∞1hr∑k∈Irϕα,Yσkm,τ−ϕα,Y,τ=0.
Hence,Yk is lacunary invariant convergent to Y (Wijsman sense) with regards to IFM ψ,ϕ.Definition 3.
LetX,ψ,ϕ,∗,⋄ be a separable IFMS, and ℐ be a proper ideal in ℕ. The sequence Yk is known as lacunary ℐ∗-invariant convergent or Wℐσθ∗ψ,ϕ-convergent (Wijsman sense) to Y with regards to IFM ψ,ϕ, if there is a set(12)M=m=mj:mj<mj+1,j∈ℕ∈ℱℐσθ.such that for each α∈X and for all τ>0,(13)limr⟶∞ψα,Ymk,τ=ψα,Y,τ,(14)limr⟶∞ϕα,Ymk,τ=ϕα,Y,τ.
In that case, we writeYk⟶YWℐσθ∗ψ,ϕ.Theorem 2.
If a sequenceYk is Wℐσθ∗ψ,ϕ-convergent to Y, then Yk is Wℐσθη,ν-convergent to Y with regards to IFM ψ,ϕ.Proof.
Presume thatYk⟶YWℐσθ∗ψ,ϕ. Then, M=m=mj:mj<mj+1,j∈ℕ∈ℱℐσθ (i.e., ℕ\M=H(say)∈ℐσθ), such that(15)limr⟶∞ψα,Υmk,τ=ψα,Υ,τ,(16)limr⟶∞ϕα,Υmk,τ=ϕα,Υ,τ.
But then, for eachξ∈0,1 and τ>0, there is N>0, such that(17)ψα,Ymk,τ−ψα,Y,τ>1−ξ,(18)ϕα,Ymk,τ−ϕα,Y,τ<ξ.for all k>N. Since(19)mk∈M:ψα,Ymk,τ−ψα,Y,τ≤1−ξorϕα,Ymk,τ−ϕα,Y,τ≥ξis included in m1<m2<⋯<mN−1 and the ideal ℐσθ is admissible, we get(20)mk∈M:ψα,Ymk,τ−ψα,Y,τ≤1−ξorϕα,Ymk,τ−ϕα,Y,τ≥ξ∈ℐσθ.
Hence,(21)k∈ℕ:ψα,Yk,τ−ψα,Y,τ≤1−ξorϕα,Yk,τ−ϕα,Y,τ≥ξ⊆H∪m1<m2<⋯<mN−1∈ℐσθ,for all ξ∈0,1 and τ>0. Therefore, we conclude that Yk⟶YWℐσθψ,ϕ.Theorem 3.
Let the idealℐσθ fulfill the property AP. If Yk is a sequence in X, such that Yk⟶YWℐσθψ,ϕ, then Yk⟶YWℐσθ∗ψ,ϕ.Proof.
Assume thatℐσθ provides the feature AP and Yk⟶YWℐσθψ,ϕ. Then, for every ξ∈0,1, for each α∈X and for all τ>0,(22)Pξ,α,τ=k∈ℕ:ψα,Yk,τ−ψα,Y,τ≤1−ξorϕα,Yk,p−ϕα,Y,τ≥ξ∈ℐσθ.
We define the setQn for n∈ℕ and τ>0 as(23)Qn≔k∈ℕ:1−1n≤ψα,Yk,τ−ψα,Y,τ<1−1n+1or1n+1<ϕα,Yk,τ−ϕα,Y,τ≤1n.
Clearly,Q1,Q2,… is countable and belongs to ℐσθ and Qi∩Qj= for i≠j. By the feature AP, there is a sequence of Fnn∈ℕ, such that the symmetric differences QjΔFj are finite sets for j∈ℕ and F=∪j=1∞Fj∈ℐσθ. Now, to conclude the proof, it is enough to show that for M=ℕ\F and for each α∈X, we get(24)limr⟶∞ψα,Yk,τ=ψα,Y,τ,(25)limr⟶∞ϕα,Yk,τ=ϕα,Y,τ,for k∈M. Let ρ>0. Select n∈ℕ, such that 1/n+1<ρ. For each α∈X, we acquire(26)k∈ℕ:ψα,Yk,τ−ψα,Y,τ≤1−ρorϕα,Yk,τ−ϕα,Y,τ≥ρ⊂k∈ℕ:ψα,Yk,τ−ψα,Y,τ≤1−1n+1orϕα,Yk,τ−ϕα,Y,τ≥1n+1⊂∪j=1n+1Qj.
SinceQjΔFjj=1,2,…,n+1 are finite sets, there is a k0∈ℕ, such that(27)∪j=1n+1Fj∩k∈ℕ:k>k0=∪j=1n+1Qj∩k∈ℕ:k>k0.
Ifk>k0 and k∉F, then(28)k∉∪j=1n+1Fjand by27k∉∪j=1n+1Qj.
Hence, for eachk>k0 and k∈M, we have(29)ψα,Yk,τ−ψα,Y,τ>1−ρandϕα,Yk,τ−ϕα,Y,τ<ρ.
Sinceρ>0 is arbitrary, we obtain Tk⟶TWℐσθ∗ψ,ϕ.Definition 4.
A sequenceYk is known as lacunary ℐ-invariant Cauchy sequence or Wℐσθψ,ϕ-Cauchy sequence (Wijsman sense) with regards to IFM ψ,ϕ if for each ξ∈0,1, for each α∈X and for all τ>0, there is N=Nε,x∈ℕ, such that(30)Aξ,α,τ=k∈ℕ:ψα,Yk,τ−ψα,YN,τ≤1−ξorϕα,Yk,τ−ϕα,YN,τ≥ξ∈ℐσθ,that is, VθAξ,α,τ=0.Definition 5.
A sequenceYk is known as lacunary ℐ∗-invariant Cauchy sequence or Wℐσθ∗η,ν-Cauchy sequence (Wijsman sense) with regards to IFM ψ,ϕ provided that there is a set(31)M=m=mj:mj<mj+1,j∈ℕ∈ℱℐσθ,such that(32)limk,l⟶∞ψα,Ymk,τ−ψα,Yml,τ=1,(33)limk,l⟶∞ϕα,Ymk,τ−ϕα,Yml,τ=0,for each α∈X and for all τ>0.
We give following theorems which indicate relationships amongWℐσθψ,ϕ-convergence, Wℐσθψ,ϕ-Cauchy sequence, and Wℐσθ∗ψ,ϕ-Cauchy sequence with regards to IFM ψ,ϕ.Theorem 4.
If a sequenceYk is Wℐσθψ,ϕ-convergent, then Yk is Wℐσθψ,ϕ-Cauchy sequence with regards to IFM ψ,ϕ.Proof.
Presume thatYk⟶ΥWℐσθψ,ϕ. Then, for every ξ∈0,1, for each α∈X and for all τ>0, the set(34)Pξ,α,τ=k∈ℕ:ψα,Yk,τ−ψα,Y,τ≤1−ξorϕα,Yk,τ−ϕα,Y,τ≥ξ,belongs to ℐσθ. Since ℐσθ is an admissible ideal, then there is k0∈ℕ with the result that k0∉Pξ,α,τ. Now, assume that(35)Rξ,α,τ=k∈ℕ:ψα,Yk,τ−ψα,Yk0,τ≤1−2ξorϕα,Yk,τ−ϕα,Yk0,τ≥2ξ.
Thinking the inequality(36)ψα,Yk,τ−ψα,Yk0,τ≤ψα,Yk,τ−ψα,Y,τ+ψα,Yk0,τ−ψα,Y,τ,(37)ϕα,Yk,τ−ϕα,Yk0,τ≤ϕα,Yk,τ−ϕα,Y,τ+ϕα,Yk0,τ−ϕα,Y,τ.
Observe that ifk∈Rξ,α,τ, therefore,(38)ψα,Yk,τ−ψα,Y,τ+ψα,Yk0,τ−ψα,Y,τ≤1−2ξ,(39)ϕα,Yk,τ−ϕα,Y,τ+ϕα,Yk0,p−ϕα,Y,τ≥2ξ.
From another standpoint, sincek0∉Pξ,α,τ, we obtain(40)ψα,Yk0,τ−ψα,Y,p>1−ξandϕα,Yk0,τ−ϕα,Y,τ<ξ.
We reach that(41)ψα,Yk,p−ψα,Y,p≤1−ξorϕα,Yk,τ−ϕα,Y,τ≥ξ.
Hence,k∈Pξ,α,τ. This gives that Rξ,α,τ⊂Pξ,α,τ∈ℐσθ for every ξ∈0,1 and x∈X. Therefore, Rξ,α,τ∈ℐσθ, so Yk is Wℐσθψ,ϕ-Cauchy sequence with regards to IFM ψ,ϕ.Theorem 5.
LetX,ψ,ϕ,∗,⋄ be a separable IFMS and ℐ be an admissible ideal. If a sequence Yk is Wℐσθ∗ψ,ϕ-Cauchy sequence, then Yk is Wℐσθψ,ϕ-Cauchy sequence with regards to IFM ψ,ϕ.Proof.
Assume that sequenceYk is Wℐσθ∗ψ,ϕ-Cauchy with regards to IFM ψ,ϕ. Then, for each α∈X and for each ξ∈0,1, there is M∈ℱℐσθ, where M=m=mj:mj<mj+1,j∈ℕ, such that(42)ψα,Ymk,τ−ψα,Yml,τ≤1−ξ,(43)ϕα,Ymk,τ−ϕα,Yml,τ≥ξ,∀k,l>k0=k0ξ.
Presume thatN=Nξ=mk0+1. Therefore, for each ξ∈0,1, one gets(44)ψα,Ymk,τ−ψα,YN,τ≤1−ξ,(45)ϕα,Ymk,τ−ϕα,YN,τ≥ξ.for all k>k0. Now, assume that H=ℕ\M. Obviously, H∈ℐσθ and(46)Qξ,α,τ=k∈ℕ:ψα,Yk,τ−ψα,YN,τ≤1−ξorϕα,Yk,τ−ϕα,YN,τ≥ξ⊂H∪m1,m2,…,mk0∈ℐσθ.
As a consequence, for allτ>0 and for each ξ∈0,1, one can identify N=Nξ, such that Qξ,α,τ∈ℐσθ, that is, sequence Yk is Wℐσθψ,ϕ-Cauchy with regards to IFM ψ,ϕ.Lemma 1 (see [4]).
Assumeℐ be an admissible ideal with the feature AP. Let there be a countable collection of subsets Ykk=1∞ of ℕ in such a way that Yk∈ℱℐ. As a result, there is a set Y⊂ℕ, such that Y\Yk is finite for all k∈ℕ and Y∈ℱℐ.Theorem 6.
Letℐσθ provides the feature AP. Then, the notions Wℐσθ∗ψ,ϕ-Cauchy sequence and Wℐσθψ,ϕ-Cauchy sequence with regards to IFM ψ,ϕ coincide.Proof.
The direct part has been proved in Theorem5.
Now, assume that the sequence isWℐσθψ,ϕ-Cauchy sequence with regards to IFM ψ,ϕ. Then, for each ξ∈0,1, for each x∈X, and for all τ>0, there is a N=Nε,x∈ℕ, such that(47)Aξ,α,τ=k∈ℕ:ψα,Yk,τ−ψα,YN,τ≤1−ξorϕα,Yk,τ−ϕα,YN,p≥ξ∈ℐσθ.
Now, presume that(48)Qjξ,α,τ=k∈ℕ:ψα,Yk,p−ψα,Ymj,τ>1−1jandϕα,Yk,τ−ϕα,Ymj,τ<1j,where mj=m1/j, j=1,2,…. Clearly, for j=1,2,…, Qjξ,α,τ∈ℱℐσθ. Using Lemma 1, there is Q⊂ℕ, so that Q∈ℱℐσθ and Q\Qj are finite for all j.
Now, we denote that(49)limk,l⟶∞ψα,Yk,τ−ψα,Yl,τ=1,(50)limk,l⟶∞ϕα,Yk,τ−ϕα,Yl,τ=0.
To demonstrate the above equations, letξ∈0,1 and r∈ℕ, such that r>2/ξ. If k,l∈Q, then Q\Qj is a finite set; so, there is v=vr in order that(51)ψα,Yk,τ−ψα,Ylr,τ>1−1r,ψα,Yl,τ−ψα,Ylr,τ>1−1r,(52)ϕα,Yk,p−ϕα,Ylr,p<1r,ϕα,Yl,τ−ϕα,Ylr,τ<1r,for all k,l>vr. From the above inequalities, we obtain(53)ψα,Yk,τ−ψα,Yl,τ≤ψα,Yk,τ−ψα,Ylr,τ+ψα,Yl,τ−ψα,Ylr,τ>1−1r+1−1r>1−ξ,(54)ϕα,Yk,τ−ϕα,Yl,τ≤ϕα,Yk,τ−ϕα,Ylr,τ+ϕα,Yl,p−ϕα,Ylr,p<1r+1r<ξ,for all k,l>vr.
Therefore, for everyξ∈0,1, ∃v=vξ, and k,l∈Q∈ℱℐσθ, we acquire(55)k∈ℕ:ψα,Yk,τ−ψα,Yl,τ≤1−ξorϕα,Yk,τ−ϕα,Yl,τ≥ξ∈ℐσθ.
This give that the sequenceYk is Wℐσθ∗ψ,ϕ-Cauchy.Definition 6.
The sequenceYk is named to be q-strongly lacunary invariant convergent (Wijsman sense) to Y, provided that for each α∈X and for all τ>0,(56)limr⟶∞1hr∑k∈Irψα,Υσkm,τ−ψα,Υ,τq=1,(57)limr⟶∞1hr∑k∈Irϕα,Υσkm,τ−ϕα,Υ,τq=0,uniformly in m, where 0<q<∞. We indicate this symbolically by Yk⟶YWNσθψ,ϕq.Theorem 7.
Letℐσθ⊂2ℕ be an admissible ideal and 0<q<∞.(i)
IfYk⟶YWNσθψ,ϕq, then Yk⟶YWℐσθψ,ϕ(ii)
IfYk is bounded and Yk⟶YWℐσθψ,ϕ, then Yk⟶YWNσθψ,ϕq(iii)
IfYk∈L∞, then Yk⟶YWℐσθψ,ϕ iff Yk⟶YWNσθψ,ϕqProof.
(i)
IfYk⟶YWNσθψ,ϕq, then for every ξ∈0,1, for each α∈X and for all τ>0, we obtain(58)∑k∈Irψα,Yσkm,τ−ψα,Y,τqorϕα,Yσkm,τ−ϕα,Y,τq≥∑k∈Ir,ψα,Yσkm,τ−ψα,Y,τ≤1−ξorϕα,Yσkm,τ−ϕα,Y,τ≥ξψα,Yσkm,τ−ψα,Y,τqorϕα,Yσkm,τ−ϕα,Y,τq≥ξqk∈Ir:ψα,Yσkm,τ−ψα,Y,τ≤1−ξorϕα,Yσkm,τ−ϕα,Y,τ≥ξ≥ξqmaxmk∈Ir:ψα,Yσkm,τ−ψα,Y,τ≤1−ξorϕα,Yσkm,τ−ϕα,Y,τ≥ξ.And so,(59)1hr∑k∈Irψα,Yσkm,τ−ψα,Y,τqorϕα,Yσkm,τ−ϕα,Y,τq≥ξqmaxmk∈Ir:ψα,Yσkm,τ−ψα,Y,τ≤1−ξorϕα,Yσkm,τ−ϕα,Y,τ≥ξhr=ξqSrhr.For everym=1,2,…. This gives that limr⟶∞Sr/hr=0, and hence, Tk⟶TWℐσθψ,ϕ.(ii)
Presume thatYk∈L∞, then Yk⟶YWℐσθψ,ϕ. Let ξ>0. By supposition, we get VθPξ,α,τ=0. Since Yk is bounded, there is M>0, such that for each α∈X and for all k, m. Then, we acquire(60)1hr∑k∈Irψα,Yσkm,τ−ψα,Y,τqorϕα,Yσkm,τ−ϕα,Y,τq=1hr∑k∈Ir,ψα,Yσkm,τ−ψα,Y,τ≤1−ξorϕα,Yσkm,τ−ϕα,Y,τ≥ξψα,Yσkm,τ−ψα,Y,τqorϕα,Yσkm,τ−ϕα,Y,τq+1hr∑k∈Ir,ψα,Yσkm,τ−ψα,Y,τ>1−ξorϕα,Yσkm,τ−ϕα,Y,τ<ξψα,Yσkm,τ−ψα,Y,τqorϕα,Yσkm,τ−ϕα,Y,τq≤Mmaxmk∈Ir:ψα,Yσkm,τ−ψα,Y,τ≤1−ξorϕα,Yσkm,τ−ϕα,Y,τ≥ξhr+ξq≤MSrhr+ξq,For eachα∈X. Hence, we obtain(61)limr⟶∞1hr∑k∈Irψα,Υσkm,τ−ψα,Υ,τq=1,(62)limr⟶∞1hr∑k∈Irϕα,Υσkm,τ−ϕα,Υ,τq=0,Uniformly inm. As a result, we get Yk⟶YWNσθψ,ϕq.(iii)
It is clear from the consequence of (i) and (ii)Definition 7.
A sequenceYk is named to be lacunary invariant statistical convergent or WSσθψ,ϕ-convergent (Wijsman sense) to Y, if for each ξ∈0,1, for each α∈X, and for all τ>0,(63)limr⟶∞1hrk∈Ir:ψα,Υσkm,τ−ψα,Υ,τ≤1−ξorϕα,Υσkm,τ−ϕα,Υ,τ≥ξ=0,uniformly in m.Theorem 8.
A sequenceYk is WSσθψ,ϕ-convergent to Y iff it is Wℐσθψ,ϕ-convergent to Y.


## 3. Conclusion

Fuzzy set theory is based on the assumption that reasoning is not crystal clear. This theory has a significant role in the areas of technology and science. Intuitionistic fuzzy set has many application areas; for example, sale analysis, new product marketing, and financial services. This study aims to find out the use of the notion of lacunaryℐ-invariant convergence of sequence of sets for demonstrating some results in the area of intuitionistic fuzzy metric space. With the help of its applications, we give the notions of Wijsman lacunary ℐ-invariant convergent, Wijsman lacunary ℐ∗-invariant convergent, and Wijsman q-strongly lacunary invariant convergent sequences in IFMS and acquired meaningful results for these notions. Also, we have examined the notions of Wijsman lacunary ℐ-invariant Cauchy and Wijsman ℐ∗-invariant Cauchy sequence in IFMS. The elements of IFMS have been studied. The results acquired here are more common than corresponding results for metric spaces. It is expected that new results will help to understand deeply the concept of this new type of convergence on IFMS. It could also be possible to work with the opinion of “Lacunary ℐ-invariant convergence of sequence of sets in probabilistic metric space” utilizing intuitionistic probability theory in the prospective studies.



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