Lacunary I-Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces

*e 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. *is has motivated us to study the lacunaryI-invariant convergence of sequence of sets in intuitionistic fuzzymetric spaces (briefly, IFMS). In this study, we examine the notions of lacunaryI-invariant convergence (WI σθ ) (Wijsman sense), lacunary I ∗-invariant convergence (WI σθ ) (Wijsman sense), and q-strongly lacunary invariant convergence ([WN σθ ]q) (Wijsman sense) of sequences of sets in IFMS. Also, we give the relationships among Wijsman lacunary invariant convergence, [WN σθ ]q, WI (η,]) σθ , and WI ∗(η,]) σθ in IFMS. Furthermore, we define the concepts of WI (η,]) σθ -Cauchy sequence and WI ∗(η,]) σθ -Cauchy sequence of sets in IFMS. Furthermore, we obtain some features of the new type of convergences in IFMS.


Introduction and Background
Fast [1] investigated the concept of statistical convergence. e 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][5][6][7][8]. Fridy and Orhan [9] worked the notion of lacunary statistical convergence by using lacunary sequence.
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. e theory of intuitionistic fuzzy sets (IFS) was presented by Atanassov [18]. e fuzzy sets and intuitionistic fuzzy sets have been widely used to solve many complex problems connected to different areas, especially in decision-making [19][20][21][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 IFmetric, 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][28][29][30][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. e 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][36][37][38][39][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]. e purpose of this study is to present some recent development in IFMS. e 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.
roughout this work, we indicate I to be the admissible ideal in N, θ be a lacunary sequence, (X, ψ, ϕ, * , ◇) to be the IFMS, and Υ, Y k to be nonempty closed subsets of X.

Main Results
Definition 1. A sequence Y k 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 uniformly in m.

Definition 2.
A sequence Y k of nonempty closed subsets of X is known as lacunary I-invariant convergent or WI σθ -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 Proof. Let m ∈ N be arbitrary and ξ ∈ (0, 1). For each α ∈ X and for all τ > 0, we estimate en, for each α ∈ X and for all τ > 0, we get where 2

Journal of Mathematics
For every m ≥1 and for every α ∈ X, it is obvious that (8) and so, we have Hence, we obtain Similarly, we have Hence, Y k is lacunary invariant convergent to Y (Wijsman sense) with regards to IFM (ψ, ϕ). □ Definition 3. Let (X, ψ, ϕ, * , ◇) be a separable IFMS, and I be a proper ideal in N. e sequence Y k is known as lacunary I * -invariant convergent or WI * (ψ,ϕ) σθ such that for each α ∈ X and for all τ > 0, In that case, we write Y k ⟶ Y(WI * (ψ,ϕ) σθ ).
But then, for each ξ ∈ (0, 1) and for all k > N. Since is included in m 1 < m 2 < · · · < m N−1 and the ideal I σθ is admissible, we get Hence, for all ξ ∈ (0, 1) and τ > 0. erefore, we conclude that Proof. Assume that I σθ provides the feature (AP) and en, for every ξ ∈ (0, 1), for each α ∈ X and for all τ > 0, We define the set Q n for n ∈ N and τ > 0 as Clearly, Q 1 , Q 2 , . . . is countable and belongs to I σθ and Q i ∩ Q j � for i ≠ j. By the feature (AP), there is a sequence of F n n∈N , such that the symmetric differences Q j ΔF j are finite sets for j ∈ N and F � ( ∪ ∞ j�1 F j ) ∈ I σθ . Now, to conclude the proof, it is enough to show that for M � N\F and for each α ∈ X, we get for k ∈ M. Let ρ > 0. Select n ∈ N, such that (1/n + 1) < ρ. For each α ∈ X, we acquire Since Q j ΔF j (j � 1, 2, . . . , n + 1) are finite sets, there is a Hence, for each k > k 0 and k ∈ M, we have Since ρ > 0 is arbitrary, we obtain T k ⟶ T(W I * (ψ,ϕ) σθ that is, V θ (A(ξ, α, τ)) � 0.
Proof. Assume that sequence Y k is WI * (ψ,ϕ) σθ -Cauchy with regards to IFM (ψ, ϕ). en, for each α ∈ X and for each ξ for all k > k 0 . Now, assume that H � N\M. Obviously, H ∈ I σθ and (46) As a consequence, for all τ > 0 and for each ξ ∈ (0, 1), one can identify N � N(ξ), such that Q(ξ, α, τ) ∈ I σθ , that is, sequence Y k is WI (ψ,ϕ) σθ -Cauchy with regards to IFM (ψ, ϕ). □ Lemma 1 (see [4]). Assume I be an admissible ideal with the feature (AP). Let there be a countable collection of subsets Y k ∞ k�1 of N in such a way that Y k ∈ F(I). As a result, there is a set Y ⊂ N, such that Y\Y k is finite for all k ∈ N and Y ∈ F(I). Proof. e direct part has been proved in eorem 5. Now, assume that the sequence is WI (ψ,ϕ) σθ -Cauchy sequence with regards to IFM (ψ, ϕ).

Journal of Mathematics
For each α ∈ X. Hence, we obtain Uniformly in m. As a result, we get (iii) It is clear from the consequence of (i) and (ii) □ Definition 7. A sequence Y k is named to be lacunary invariant statistical convergent or WS θ σ (ψ, ϕ)-convergent (Wijsman sense) to Y, if for each ξ ∈ (0, 1), for each α ∈ X, and for all τ > 0, uniformly in m.

Conclusion
Fuzzy set theory is based on the assumption that reasoning is not crystal clear. is 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. is study aims to find out the use of the notion of lacunary I-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. e elements of IFMS have been studied. e 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.