Some results on wijsman ideal convergence in intuitionistic fuzzy metric spaces

(1)

Research Article

Some Results on Wijsman Ideal Convergence in Intuitionistic

Fuzzy Metric Spaces

Ayhan Esi

,

1

Vakeel A. Khan

,

2

Mobeen Ahmad

,

2

and Masood Alam

3

1_{Malatya Turgut Ozal University Engineering Faculty Dept. of Basic Engineering Sciences Malatya, Malatya 44040, Turkey} 2_{Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India}

3_{Department of Mathematics and IT Center for Preparatory Studies, Sultan Qaboos University, P. O. Box 162 PC 123 Muscat, Oman}

Correspondence should be addressed to Ayhan Esi; aesi23@hotmail.com Received 20 June 2020; Accepted 15 October 2020; Published 11 November 2020 Academic Editor: Syed Abdul Mohiuddine

Copyright © 2020 Ayhan Esi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the present work, we study and extend the notion of WijsmanJ –convergence and Wijsman J∗–convergence for the sequence of closed sets in a more general setting, i.e., in the intuitionistic fuzzy metric spaces (brieﬂy, IFMS). Furthermore, we also examine the concept of WijsmanJ∗–Cauchy and J –Cauchy sequence in the intuitionistic fuzzy metric space and observe some properties.

1. Introduction

In 1951, Fast [1] initiated the theory of statistical conver-gence. It is an extremely eﬀective tool to study the conver-gence of numerical problems in sequence spaces by the idea of density. Statistical convergence of the sequence of sets was examined by Nuray and Rhoades [2]. Ulusu and Nuray [3] studied the Wijsman lacunary statistical convergence sequence of sets and connected with the Wijsman statistical convergence. Esi et al. [4] introduced the Wijsman λ-statis-tical convergence of interval numbers. Kostyrko et al. [5] generalized the statistical convergence and introduced the notion of idealJ –convergence. Salát et al. [6, 7] investigated it from the sequence space viewpoint and associated with the summability theory. Further, it was analyzed by Khan et al. [8] with the help of a bounded operator. In 2008, Das et al. [9] analyzed J and J∗–convergence for double sequences. Kisi and Nuray [10] initiated new convergence deﬁnitions for the sequence of sets. Furthermore, Gümüş [11] studied the Wijsman ideal convergent sequence of sets using the Orlicz function.

In 1965, Zadeh [12] started the fuzzy sets theory. This theory has proved its usefulness and ability to solve many problems that classical logic was unable to handle. Karmosil et al. [13] introduced the fuzzy metric space, which has the most signiﬁcant applications in quantum particle physics.

Afterward, numerous researchers have studied the concept of fuzzy metric spaces in different ways. George et al. [14, 15] modified the notion of fuzzy metric space and deter-mined a Hausdorff topology for fuzzy metric spaces. Atanas-sov [16] generalized the fuzzy sets and introduced the notion of intuitionistic fuzzy sets in 1986. Park [17] examined the notion of IFMS, and Saadati and Park [18] further analyzed the intuitionistic fuzzy topological spaces. Moreover, statisti-cal convergence, ideal convergence, and different properties of sequences in intuitionistic fuzzy normed spaces were examined by Mursaleen et al. [19–21]. Also one can refer to Sengül and Et [22], Sengül et al. [23], Et and Yilmazer [24], Mohiuddine and Alamri [25], and Mohiuddine et al. [26, 27].

2. Preliminaries

We recall some concepts and results which are needed in sequel.

Deﬁnition 1 [5]. A family of subsets J ⊆ 2ℕ_{is known as an}

ideal in a non-empty setℕ , if (1) ∅∈J ,

(2) for anyC, D ∈ J ⇒ C ∪ D ∈ J , (3) for anyC ∈ J and D ⊆ C, ⇒D ∈ J .

Volume 2020, Article ID 7892913, 8 pages https://doi.org/10.1155/2020/7892913

(2)

Remark 2 [5]. An ideal J is known as non-trivial if ℕ ∉ J . A nontrivial idealJ is known as admissible if ffng: n ∈ ℕg

∈ J .

Deﬁnition 3 [5]. A nonempty subset F ⊆ 2ℕis known asﬁlter inℕ if

(1) for every∅∉ F,

(2) for everyC, D ∈ F ⇒ C ∩ D ∈ F,

(3) for everyC ∈ F with C ⊆ D, one obtain D ∈ F:

Proposition 4 [5]. For every ideal J , there is a ﬁlter FðJ Þ

associated withJ deﬁned as follows:

F Jð Þ = K ⊆ ℕ : K = ℕ \ A, for some A ∈ Jf g: ð1Þ Deﬁnition 5 [5]. Let fC₁, C1, ⋯g be a mutually disjoint sequence of sets ofJ . Then, there is sequence of sets fD₁, D1, ⋯g so that ∪∞j=1Dj∈ J and each symmetric diﬀerence

Cj△Djðj = 1, 2, ⋯Þ is ﬁnite. In this case, admissible ideal J

is known as property ðAPÞ.

Lemma 6 [28]. Suppose J be an admissible ideal alongside

property ðAPÞ. Let a countable collection of subsets fCkg∞k=1

of positive integer ℕ in such a way that C_k∈ FðJ Þ. Then, there exists a setC ⊂ ℕ such that C \ C_k isﬁnite for all k ∈

ℕ and C ∈ FðJ Þ.

Deﬁnition 7 [29]. Let ðM, dÞ be a metric space and fCkg be a

sequence of nonempty closed subsets ofM which is said to be Wijsman convergent to the closedC of M, if

lim

k→∞d x, Cð kÞ = d x, Cð Þfor every ∈ M: ð2Þ

In other words, W − lim

k→∞Ck= C.

In 2012, Nuray and Rhoades [2] initiated the theory of Wijsman statistical convergence for a sequence of sets. Furthermore, Kisi and Nuray [10] extended it into J -convergence.

Deﬁnition 8 [10]. Suppose ðM, dÞ is a metric space. A non-empty closed subset fCkg of M is known as Wijsman J –

convergent to a closed setC, if for every x ∈ M, one has

k ∈ ℕ : d x, Cj ð kÞ− d x, Cð Þj≥ ϵ

f g∈ J : ð3Þ

Hence, one writesJ_W− lim

k→∞Ck= C.

Deﬁnition 9 [10]. Suppose ðM, dÞ is a metric space. A non-empty closed subset fC_kg of M is known as Wijsman J – Cauchy if for each x ∈ M , there exists a positive integer

m = mðϵÞ so that the set k ∈ ℕ : d x, Cð kÞ− d x, Cp

_≥ϵ

∈ J , for all p ≥ m: ð4Þ

Deﬁnition 10 [10]. Suppose ðM, dÞ is a separable metric space and fCkg, C is nonempty closed subsets of M. A

sequence fC_kg is known as Wijsman J∗ –convergent to C if and only if ∃P ∈ FðJ Þ and P = fp = ðpj< pj+1, j ∈ ℕÞg

⊂ ℕ in such a manner that lim

k→∞d x, Cmk

= d x, Cð Þ, for every x ∈ M: ð5Þ One writesJ∗_W− lim

k→∞Ck= C.

Deﬁnition 11 [10]. Suppose ðM, dÞ is a separable metric space and J is an admissible ideal. A sequence fC_kg of nonempty closed subsets of M is known as the Wijsman J∗ _{–Cauchy sequence if there exists P ∈ FðJ Þ, where P =} fp = ðpj< pj+1, i ∈ ℕÞg in such a way that subsequence

CP= fCpkg is Wijsman Cauchy in M, i.e.,

lim k,l→∞∣d x, Cmk − d x, Cpl ∣ = 0: ð6Þ

Remark 12 [10]. In general, the Wijsman topology is not ﬁrst-countable, if sequence of nonempty sets fCkg is

Wijsman convergent to set C, then every subsequence of fC_kg may not be convergent to C. Every subsequence of the convergent sequence fC_kg converges to the same limit provided thatM is a separable metric space.

Definition 13 [17]. Let M be a nonempty set, η and φ be fuzzy sets on M2_{× ð0,∞Þ, ∗ be a continuous t-norm, and ⋄ be a} continuous t -conorm. Then, the five-tuple ðM, η, φ,∗,⋄Þ is known as an intuitionistic fuzzy metric space (for short, IFMS) if it fulfills the subsequent conditions for all s, t > 0 and for every y, z, w ∈ M:

(a)ηðy, z, sÞ + φðy, z, sÞ ≤ 1, (b)ηðy, z, sÞ > 0,

(c)ηðy, z, sÞ = 1 if and only if y = z, (d) ηðy, z, sÞ = ηðz, y, sÞ,

(e) ηðy, z, sÞ ∗ ηðz, w, tÞ ≤ ηðy, w, s + tÞ, (f) ηðy, z,:Þ: ð0,∞Þ → 0, 1 is continuous, (g) φðy, z, sÞ < 1,

(h) φðy, z, sÞ = 0 if and only if y = z, (i) φðy, z, sÞ = φðz, y, sÞ,

(j) φðy, z, sÞ⋄φðz, w, tÞ ≥ φðy, w, s + tÞ, (k) φðy,:Þ: ð0,∞Þ → 0, 1 is continuous.

In such situation, ðη, φÞ is called the intuitionistic fuzzy metric (brieﬂy, IFM).

Example 14 [17]. Suppose ðM, dÞ is a metric space. Deﬁne

c⋄d = min ðc + d, 1Þ and c ∗ d = cd for all c, d ∈ ½0, 1, and

(3)

η y, z, sð Þ = s

s + d y, zð Þ, φ y, z, sð Þ =

d y, zð Þ

s + d y, zð Þ: ð7Þ

Then ðM, η, φ,∗,⋄Þ is an IFMS.

Deﬁnition 15 [18]. Let ðM, η, φ,∗,⋄Þ be an IFMS and C be a nonempty subset ofM. For all s > 0 and x ∈ M, we deﬁne

η x, C, sð Þ = sup η x, y, sf ð Þ: y ∈ Cg ð8Þ and

φ x, C, sð Þ = inf φ x, y, sf ð Þ: y ∈ Cg, ð9Þ whereηðx, C, sÞ and φðx, C, sÞ are the degree of nearness and nonnearness of x to C at s, respectively.

Saadati and Park [18] studied the notion of convergence sequence with respect to IFMS which are deﬁned as follows: Deﬁnition 16 [18]. Let ðM, η, φ,∗,⋄Þ be an IFMS. A sequence

x = ðxkÞ is convergent to ξ if for any 0 < ϵ < 0 and s > 0 there

exists k0∈ ℕ in such a way that

η xð k, ξ, sÞ > 1 − ϵ and φ xð k, ξ, sÞ < ϵ for all k ≥ k0: ð10Þ Deﬁnition 17 [20]. An IFMS ðM, η, φ,∗,⋄Þ is known as sepa-rable if it contains a countable dense subset, i.e., there is a countable set fxkg along with subsequent property: for any s > 0 and for all ξ ∈ M, there is at least one xnin order that

η xð n, ξ, sÞ≥ 1 − ϵ and φ xð n, ξ, sÞ≤ ϵ, for each ϵ ∈ 0, 1ð Þ:

ð11Þ

3. Wijsman

J and J

∗

–convergence in IFMS

Throughout this section, we denoteJ to be the admissible ideal inℕ. We begin with the following deﬁnitions as follows. Deﬁnition 18. Let ðM, η, φ,∗,⋄Þ be an IFMS. A sequence of sets fCkg is said be Wijsman convergent to C if for every ϵ > 0 and s > 0 there exists k0∈ ℕ such that

lim

k→∞η x, Cð k, sÞ = η x, C, sð Þ and limk→∞φ x, Cð k, sÞ

= φ x, C, sð Þ for all k ≥ k0:

ð12Þ The set of all Wijsman limit point of the sequence fCkg is

denoted by LfCkg.

Deﬁnition 19. Let ðM, η, φ,∗,⋄Þ be an IFMS and J be a proper ideal in ℕ. A sequence fC_kg of nonempty closed subsets ofM is known as Wijsman J –convergent to C with respect to IFM ðη, φÞ, if for every 0 < ϵ < 1, for each x ∈ M and for all s > 0 such that

k ∈ ℕ : ∣η x, Cð k, sÞ− η x, C, sð Þ∣ f ≤1 − ϵ or ∣ φ x, Cð k, sÞ− φ x, C, sð Þ∣≥ϵg ∈ J : ð13Þ We write ðη, φÞ − J_W− lim k→∞Ck= C.

Example 20. Suppose ðM, η, φ,∗,⋄Þ is an IFMS and C, fC_kg is nonempty closed subsets ofM. Assume M = ℝ2and fCkg

are sequence deﬁned by Ck= x, y ð Þ∈ ℝ2_{: 0 ≤ x ≤ k, 0 ≤ y ≤}1 k:x, if k ≠ n 2 x, y ð Þ∈ ℝ2_{: x ≥ 0, y = 1, if k = n}2_, 8 < : C = x, yð Þ∈ ℝ2_{: x ≥ 0, y = 0}_: ð14Þ Since lim k→∞ 1 k∣ n ≤ k : ∣η x, yf ðð Þ, Ck, sÞ− η x, yðð Þ, C, sÞ∣ ≤1 − ϵ or ∣ φ x, yðð Þ, Ck, sÞ− φ x, yðð Þ, C, sÞ∣≥ϵg∣ = 0: ð15Þ Therefore, the sequence of sets fC_kg is Wijsman statistical convergent to the setC.

Now, deﬁne the set S as

S ϵð Þ = k ∈ ℕ : ∣η x, yf ðð Þ, Ck, sÞ− η x, yðð Þ, C, sÞ∣

≤1 − ϵ or ∣ φ x, yðð Þ, Ck, sÞ− φ x, yðð Þ, C, sÞ∣≥ϵg:

ð16Þ If we assumeJ = J_d, then the Wijsman statistical con-vergence coincides with the Wijsman ideal concon-vergence. Therefore, k ∈ ℕ : ∣η x, yðð Þ, Ck, sÞ− η x, yðð Þ, C, sÞ∣ f ≤1 − ϵ or ∣ φ x, yðð Þ, Ck, sÞ− φ x, yðð Þ, C, sÞ∣≥ϵg = k ∈ ℕ : k = n 2⊂ Jd: ð17Þ Deﬁnition 21. Let ðM, η, φ,∗,⋄Þ be a separable IFMS and J be an admissible ideal in ℕ. A sequence fC_kg of nonempty closed subsets of M is known as Wijsman J –Cauchy with respect to IFM ðη, φÞ, if for each 0 < ϵ < 1, for each x ∈ M and for all s > 0, ∃ l = lðϵÞ such that

k ∈ ℕ : ∣η x, Cð k, sÞ− η x, Cð l, sÞ∣

f

≤1 − ϵ or ∣ φ x, Cð k, sÞ− φ x, Cð l, tÞ∣≥ϵg ∈ J :

ð18Þ Deﬁnition 22. Let ðM, η, φ,∗,⋄Þ be a separable IFMS and fCkg be any nonempty closed subset of M. The sequence

fCkg is known as Wijsman J∗–Cauchy with respect to IFM

ðη, φÞ, if there exists P = fp = ðpjÞ: pj< pj+1, j ∈ ℕg ⊂ ℕ and P ∈ FðJ Þ with the result that the subsequence CP= fCp_kg

(4)

lim k,l→∞∣η x, Cpk, s − η x, Cpl, s ∣ = 1 ð19Þ and lim k,l→∞∣φ x, Cpk, s − φ x, Cp_l, s ∣ = 0: ð20Þ

Deﬁnition 23. Let ðM, η, φ,∗,⋄Þ be a separable IFMS and J be an proper ideal inℕ. Let fC_kg be nonempty closed subsets of M. The sequence fCkg is known as Wijsman J∗

–conver-gent to C with respect to ðη, φÞ, if there exists P ∈ FðJ Þ, where P = fp = ðpjÞ: pj< pj+1, j ∈ ℕg ⊂ ℕ such that for each s > 0, we have lim k→∞η x, Cpk, s = η x, C, sð Þ, ð21Þ and lim k→∞φ x, Cpk, s = φ x, C, sð Þ: ð22Þ In such case, we write ðη, φÞ − J∗_W− limC_k= C. In the following theorem, we prove that every Wijsman J –convergent implies the Wijsman J –Cauchy condition in IFMS:

Theorem 24. Let ðM, η, φ,∗,⋄Þ be a separable IFMS and let

J be an arbitrary admissible ideal. Then, every Wijsman J –convergent sequence of closet sets fCkg is Wijsman J –

Cauchy with respect to IFM ðη, φÞ. Proof. Suppose ðη, φÞ − J_W− lim

k→∞Ck= C. Then, for every

0 < ϵ < 1, for all s > 0 and x ∈ X, the set

U ϵ, sð Þ = k ∈ ℕ : ∣η x, Cf ð k, sÞ− η x, C, sð Þ∣

≤1 − ϵ or ∣ φ x, Cð k, sÞ− φ x, C, sð Þ∣≥ϵg

ð23Þ belongs to J . Since J is an admissible ideal, then there exists k0∈ ℕ with the result that k0∉ Uðϵ, sÞ. Now, sup-pose that V ϵ, sð Þ = k ∈ ℕ : ∣η x, Cð k, sÞ− η x, Ck0, s ∣ ≤ 1 − 2ϵð Þ or ∣ φ x, Cð k, sÞ− φ x, Ck0, s ∣≥2ϵg: ð24Þ Considering the inequality

η x, Cð k, sÞ− η x, Ck0, s _{≤ η x,C}_j _ð _k_{, s}_Þ_{− η x, C, s}_ð _Þ_j + η x, Ck0, s − η x, C, sð Þ _, ð25Þ and ∣φ x, Cð k, sÞ− φ x, Ck0, s ∣ ≤ ∣φ x, Cð k, sÞ− φ x, C, sð Þ∣ + φ x, Ck0, s − φ x, C, sð Þ _: ð26Þ

Observe that if k ∈ Vðϵ, sÞ, therefore η x, Cð k, sÞ− η x, C, sð Þ j j + η x, C_k 0, s − η x, C, sð Þ _{≤ 1−2ϵ}ð Þ, ð27Þ and ∣φ x, Cð k, sÞ− φ x, C, sð Þ∣ + ∣φ x, Ck0, s − φ x, C, sð Þ∣ ≥ 2ϵ: ð28Þ From another point of view, since k0∉ Uðϵ, sÞ, we obtain η x, Ck0, s − η x, C, sð Þ _{> 1 − ϵ and ∣φ x, C}_k 0, s − φ x, C, sð Þ∣ < ϵ: ð29Þ We achieve that η x, Cð _k, sÞ− η x, C, sð Þ j j≤ 1 − ϵ or ∣φ x, Cð _k, sÞ− φ x, C, sð Þ∣ ≥ ϵ: ð30Þ Hence, k ∈ Uðϵ, sÞ. This implies that Uðϵ, sÞ ⊂ Vðϵ, sÞ ∈ J for every 0 < ϵ < 1 and for all s > 0 and x ∈ M. There-fore, Vðϵ, sÞ ∈ J , so the sequence is fCkg which is

Wijsman J –Cauchy.

Theorem 25. Let ðM, η, φ,∗,⋄Þ be a separable IFMS and let J

be an admissible ideal. Then, every Wijsman J∗–Cauchy sequence of closed sets is Wijsman J –Cauchy.

Proof. Suppose that sequence fCkg is Wijsman J∗–Cauchy

with respect to IFM ðη, φÞ. Then, for each x ∈ M and for each 0 < ϵ < 1, there exists P ∈ FðJ Þ, where P = fðp_jÞ: p_j<

p_j+1, j ∈ ℕg in such a way that

η x, Cpk, s − η x, Cpl, s ≤ 1 − ϵ, ð31Þ and φ x, Cpk, s − φ x, Cpl, s ≥ ϵ, ∀k, l > k0= k0ð Þϵ : ð32Þ Suppose N = NðϵÞ = pk0+1. Therefore, for eachϵ > 0, one obtains η x, Cpk, s − η x, Cð N, sÞ ≤ 1 − ϵ, ð33Þ

(5)

and φ x, Cpk, s − φ x, Cð N, sÞ ≥ ϵ for all k > k0: ð34Þ

Now, suppose that K = ℕ \ P. Clearly, K ∈ J and

Q ϵ, sð Þ = k ∈ ℕ ∣ η x, Cf ð k, sÞ− η x, Cð N, sÞ∣ ≤ 1 − ϵ or ∣φ x, Cð k, sÞ− φ x, Cð N, sÞ∣ ≥ ϵ ⊂ K ∪ p1, p2, ⋯, pk0 n o ∈ J : ð35Þ Hence, for all s > 0 and for each 0 < ϵ < 1, one can deter-mine N = NðϵÞ so that Qðϵ, sÞ ∈ J , that is, sequence fCkg is

WijsmanJ –Cauchy.

Theorem 26. Let J be an admissible ideal including property

(AP) and ðM, η, φ,∗,⋄Þ be a separable IFMS. Then, the notion of Wijsman J∗–Cauchy sequence of sets coincides with Wijsman J –Cauchy with respect to ðη, φÞ and vice-versa. Proof. The direct part is already proven in Theorem 25.

Now, suppose that sequence fC_kg is Wijsman J – Cauchy sequence with respect to IFM ðη, φÞ. Then by deﬁni-tion, if for every 0 < ϵ < 1, for each x ∈ X and for all s > 0, there exists a m = mðϵÞ such that

B ϵ, sð Þ = k ∈ ℕ η x, Cf j ð k, sÞ− η x, Cð n, sÞj

≤ 1 − ϵ or φ x, Cj ð k, sÞ− φ x, Cð n, sÞj≥ ϵg ∈ I:

ð36Þ Now, suppose that

Pjðϵ, sÞ = k ∈ ℕ η x, Cð k, sÞ− η x, Cmj, s n > 1 − 1 j or ∣ x, Cð k, sÞ− φ x, Cmj, s ∣<1 j , ð37Þ where mj= mð1/jÞ, j = 1, 2, 3, ⋯. Obviously, for j = 1, 2, 3 ⋯ , Pjðϵ, sÞ ∈ FðJ Þ. Using Lemma 6, there exists P ⊂ ℕ so that P ∈ FðJ Þ and P \ Pjareﬁnite for all j.

Now, we prove that lim

k,l→∞∣η x, Cð k, sÞ− η x, Cð l, sÞ∣ = 1, ð38Þ and

lim

k,l→∞∣φ x, Cð k, sÞ− φ x, Cð l, sÞ∣ = 0: ð39Þ To show the above equations, letϵ > 0, and r ∈ ℕ such that r > 2/ϵ. If k, l ∈ P, then P \ Pj is aﬁnite set; therefore,

there exists w = wðrÞ in order that

η x, Cð k, sÞ− η x, Clr, s _{> 1 −}1 r, η x, Cð l, sÞ− η x, Clr, s _{> 1 −}1 r, ð40Þ and φ x, Cð k, sÞ− φ x, Clr, s _<1 r, φ x, Cð l, sÞ− φ x, Clr, s _<1 r, ð41Þ

for all k, l > wðrÞ. Then, the above inequalities follow that for

k, l > wðrÞ η x, Cð k, sÞ− η x, Cð l, sÞ j j≤ η x, Cð k, sÞ− η x, Clr, s + η x, Cð l, sÞ− η x, Clr, s _{> 1 −}1 r + 1 −1 r > 1 − ϵ, ð42Þ and φ x, Cð k, sÞ− φ x, Cð l, sÞ j j≤ φ x, Cð k, sÞ− φ x, Clj, s + φ x, Cð l, sÞ− φ x, Clj, s <1 r + 1 r < ϵ: ð43Þ Therefore, for eachϵ > 0, ∃w = wðϵÞ and k, l ∈ P ∈ FðIÞ, we achieve

k ∈ ℕ : ∣η x, Cð k, sÞ− η x, Cð l, sÞ∣≤1

f

− ϵ or ∣ φ x, Cð k, sÞ− φ x, Cð l, sÞ∣≥ϵg ∈ J :

ð44Þ This proves that the sequence fCkg is a Wijsman J∗–

Cauchy.

Theorem 27. Let ðM, η, φ,∗,⋄Þ be a separable IFMS and let J

be an admissible ideal. Then

η, φ

ð Þ− J∗

W− lim_k→∞Ck= C ð45Þ

implies that sequence fCkg is a Wijsman J –Cauchy sequence

with respect to IFM ðη, φÞ.

Proof. Suppose that ðη, φÞ − J∗_W− lim_k→∞C_k= C. Then, there exists P = fp = ðpjÞ: pj< pj+1, j ∈ ℕg ⊂ ℕ with P ∈ F

ðJ Þ so that CP= fCpkg

lim

k→∞η x, Cpk, s

(6)

and

lim

k→∞φ x, Cpk, s

= φ x, C, sð Þ, ð47Þ for any ϵ > 0 and k, l > k₀.

Suppose r ∈ ℕ and ϵ > 0 in such a way that r > 2/ϵ. If k,

l ∈ P, then P \ Pjis aﬁnite set; therefore, there exists kðrÞ = k so that η x, Cpk, s − η x, Cpl, s ≤ η x, Cpk, s − x, C, sð Þ + η x, Cpl, s − η x, C, sð Þ > 1 − 1 r + 1 −1 r > 1 − ϵ, ð48Þ and φ x, Cpk, s − φ x, Cpl, s < φ x, Cpk, s − φ x, C, sð Þ + φ x, Cp_l, s − φ x, C, sð Þ <1 r + 1 r < ϵ: ð49Þ Therefore, lim k,l→∞∣η x, Cpk, s − η x, Cpl, s ∣ = 1, ð50Þ and lim k,l→∞∣φ x, Cpk, s − φ x, Cpl, s ∣ = 0: ð51Þ

Hence, sequence fCkg is Wijsman J –Cauchy with

respect to IFM ðη, φÞ.

4. Wijsman

J –cluster points and Wijsman J –

limit points in IFMS

Throughout this section, we denoteJ to be the proper ideal inℕ and deﬁne Wijsman J –cluster and J –limit points of the sequence of sets in intuitionistic fuzzy metric space and obtain some results.

Deﬁnition 28. Let ðM, η, φ,∗,⋄Þ be a separable IFMS. An ele-mentC ∈ M is known as the Wijsman J -cluster point of fCkg if and only if for any x ∈ M and for all ϵ, s > 0, one has

k ∈ ℕ : η x, Cj ð k, sÞ− η x, C, sð Þj < 1

f

− ϵ or φ x, Cj ð k, sÞ− φ x, C, sð Þj > ϵg ∉ J :

ð52Þ We denoteJðη,φÞ_W ðΓ_fC

kgÞ as the collection of all Wijsman

J -cluster points.

Deﬁnition 29. Let ðM, η, φ,∗,⋄Þ be a separable IFMS. An ele-ment C ∈ M is known as Wijsman J –limit point of sequence fCkg of nonempty closed subsets of M provided

that P = fp = ðpjÞ: pj< pj+1, j ∈ ℕg ⊂ ℕ in such a way that P

∉ J , and for any x ∈ M and s > 0, we obtain lim

k→∞η x, Cð k, sÞ = η x, C, sð Þ and limk→∞φ x, Cð k, sÞ = φ x, C, sð Þ:

ð53Þ We denoteJðη,φÞ_W ðΛ_fC

kgÞ as the collection of all Wijsman

J –limit points.

Theorem 30. Let ðM, η, φ,∗,⋄Þ be a separable IFMS. Then, for

any sequence sets, fCkg ⊂ M, J ðη,φÞ

W ðΛfCkgÞ ⊂ J

ðη,φÞ W ðΓfCkgÞ.

Proof. Suppose C ∈ Jðη,φÞ_W ðΛ_fC

kgÞ. Then, there exists P =

fp1< p2<⋯g ⊂ ℕ such that P = fp = ðpjÞ: pj< pj+1, j ∈ ℕg

∉ J and for all s > 0 and x ∈ M, we have lim k→∞η x, Cpk, s = η x, C, sð Þ, ð54Þ and lim k→∞φ x, Cpk, s = φ x, C, sð Þ: ð55Þ According to Equations (54) and (55), there exists

k0∈ ℕ so that for each ϵ > 0 and for any x ∈ X and k > k0 η x, Cp_k, s − η x, C, sð Þ > 1 − ϵ, ð56Þ and φ x, Cpk, s − φ x, C, sð Þ < ϵ: ð57Þ Hence, k ∈ ℕ : ∣η x, Cpk, s − η x, C, sð Þ∣>1 n − ϵ,∣φ x, Cpk, s − φ x, C, sð Þ∣<ϵo⊇ P \ p1, p1, _,pk0 n o : ð58Þ Then, the right-hand side of (58) does not belong to J , and then k ∈ ℕ : η x, Cpk, s − η x, C, sð Þ > 1 n − ϵ, φ x, Cpk, s − φ x, C, sð Þ < ϵo∉ J , ð59Þ

which means that C ∈ Jðη,φÞ_W ðΓ_fC

kgÞ.

Theorem 31. Let ðM, η, φ,∗,⋄Þ be a separable IFMS. Then,

for any sequence fCkg ⊂ M, J ðη,φÞ

(7)

Proof. LetC ∈ Jðη,φÞ_W ðΓ_fC

kgÞ. Then, for each ϵ > 0 and for all s > 0 and for each x ∈ M, one has

k ∈ ℕ : η x, Cj ð k, sÞ− η x, C, sð Þj < 1 f − ϵ or φ x, Cj ð k, sÞ− φ x, C, sð Þj > ϵg ∉ J : ð60Þ Suppose Qk= n k ∈ ℕ : η x, Cj ð k, sÞ− η x, C, sð Þj > 1 −1 k, φ x, Cj ð k, sÞ− φ x, C, sð Þj < 1 k o , ð61Þ

for k ∈ ℕ. fQkg∞k=1is a descending sequence of subsets ofℕ.

Hence, Q = fk = ðkiÞ: ki< ki+1, i ∈ ℕg ∉ J so that

lim k→∞η x, Cki, s = η x, C, sð Þ, ð62Þ and lim k→∞φ x, Cki, s = φ x, C, sð Þ, ð63Þ which means thatC ∈ L_fC_kg.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conﬂict of interest.

Authors’ Contributions

All authors contributed equally and signiﬁcantly in writing this article.

References

[1] H. Fast,“Sur la convergence statistique,” Colloquium mathe-maticae, vol. 2, no. 3-4, pp. 241–244, 1951.

[2] F. Nuray and B. E. Rhoades, “Statistical convergence of sequence sets,” Fasciculi Mathematici, vol. 49, pp. 87–99, 2012. [3] U. Ulusu and F. Nuray, “Lacunary statistical convergence of sequence of sets,” Progress in Applied Mathematics, vol. 4, no. 2, pp. 99–109, 2012.

[4] A. Esi, N. L. Braha, and A. Rushiti,“Wijsman $\lambda-$sta-tistical convergence of interval numbers,” Boletim da Socie-dade Paranaense de Matemática, vol. 35, no. 2, pp. 9–18, 2017. [5] P. Kostyrko, Šalát, and Wilczyński, “I-convergence,” Real

Analysis Exchange, vol. 26, no. 2, pp. 669–686, 2000. [6] T. Salát, B. C. Tripathy, and M. Ziman, On some properties of

I-convergence, vol. 28, Tatra Mountains Mathematical Publica-tions, 2004.

[7] T. Salát, B. C. Tripathy, and M. Ziman, “On I-convergence ﬁeld,” Italian Journal of Pure and Applied Mathematics, vol. 17, pp. 45–54, 2005.

[8] V. A. Khan, K. M. A. S. Alshlool, and S. A. A. Abdullah, “Spaces of ideal convergent sequences of bounded linear oper-ators,” Numerical Functional Analysis and Optimization, vol. 39, no. 12, pp. 1278–1290, 2018.

[9] P. Das, P. Kostyrko, W. Wilczynski, and P. Malik,“I and I∗ -convergence of double sequences,” Mathematica Slovaca, vol. 58, no. 5, pp. 605–620, 2008.

[10] O. Kişi and F. Nuray, “New convergence deﬁnitions for sequences of sets,” Abstract and Applied Analysis, vol. 2013, Article ID 852796, 6 pages, 2013.

[11] H. Gümüş, “On Wijsman ideal convergent set of sequences deﬁned by an Orlicz function,” Filomat, vol. 30, no. 13, pp. 3501–3509, 2016.

[12] L. A. Zadeh,“Fuzzy sets,” Inform Control, vol. 8, pp. 338–353, 1965.

[13] I. Karmosil and J. Michalek,“Fuzzy metric and statistical met-ric spaces,” Kybernetica, vol. 11, no. 5, pp. 333–344, 1975. [14] A. George and P. Veeramani,“On some results in fuzzy metric

spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.

[15] A. George and P. Veeramani,“On some results of analysis for fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 90, no. 3, pp. 365–368, 1997.

[16] K. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.

[17] J. H. Park,“Intuitionistic fuzzy metric space,” Chaos Solitons Fractals, vol. 22, no. 5, pp. 1039–1046, 2004.

[18] R. Saadati and J. H. Park,“On the intuitionistic fuzzy topolog-ical spaces,” Chaos Solitons Fractals, vol. 27, no. 2, pp. 331– 344, 2006.

[19] M. Mursaleen and S. A. Mohiuddine,“Statistical convergence of double sequences in intuitionistic fuzzy normed spaces,” Chaos, Solitons Fractals, vol. 41, no. 5, pp. 2414–2421, 2009. [20] M. Mursaleen, V. Karakaya, and S. A. Mohiuddine,

“Separabil-iSchauder basis, separability, and approximation property in intuitionistic fuzzy normed spacety and approximation prop-erty in intuitionistic fuzzy normed space,” Abstract and Applied Analysis, vol. 2010, Article ID 131868, 14 pages, 2010. [21] M. Mursaleen, S. A. Mohiuddine, and O. H. H. Edely,“On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces,” Computers and Mathematics with Applica-tions, vol. 59, no. 2, pp. 603–611, 2010.

[22] H.Şengül and M. Et, “On I-lacunary statistical convergence of orderα of sequences of sets,” Filomat, vol. 31, no. 8, pp. 2403– 2412, 2017.

[23] H.Şengül, M. Et, and H. Çakallı, “On ðf, IÞ-lacunary statistical convergence of orderα of sequences of sets,” Boletim da Soci-edade Paranaense de Matematica, vol. 38, no. 7, pp. 85–97, 2019.

[24] M. Et and M. C. Yilmazer,“On deferred statistical convergence of sequences of sets,” AIMS MATHEMATICS, vol. 5, no. 3, pp. 2143–2152, 2020.

[25] S. A. Mohiuddine and B. A. S. Alamri,“Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 113, no. 3, pp. 1955–1973, 2019.

[26] S. A. Mohiuddine, A. Asiri, and B. Hazarika,“Weighted statis-tical convergence through diﬀerence operator of sequences of fuzzy numbers with application to fuzzy approximation

(8)

theorems,” International Journal of General Systems, vol. 48, no. 5, pp. 492–506, 2019.

[27] S. A. Mohiuddine, B. Hazarika, and M. A. Alghamdi,“Ideal relatively uniform convergence with Korovkin and Voronovs-kaya types approximation theorems,” Filomat, vol. 33, no. 14, pp. 4549–4560, 2019.

[28] P. Kostyrko, M. Macaj, T. Salat, and M. Sleziak,“I-convergence and extremal I-limit point,” Mathematica Slovaca, vol. 55, no. 4, pp. 443–464, 2005.

[29] M. Baronti and P. Papani, Convergence of Sequence of Sets, Method of Functional Analysis in Approximation Theory, Birkhauser, Basel, 1986.