# Some Results on Wijsman Ideal Convergence in Intuitionistic Fuzzy Metric Spaces

Authors: Ayhan Esi; Vakeel A. Khan; Mobeen Ahmad; Masood Alam
Journal: Journal of Function Spaces
 (2020)

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## Abstract

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 Wijsman J∗–Cauchy and J–Cauchy sequence in the intuitionistic fuzzy metric space and observe some properties.


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## Body

## 1. Introduction

In 1951, Fast [1] initiated the theory of statistical convergence. It is an extremely effective tool to study the convergence 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-statistical convergence of interval numbers. Kostyrko et al. [5] generalized the statistical convergence and introduced the notion of ideal J–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 determined a Hausdorff topology for fuzzy metric spaces. Atanassov [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, statistical 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 subsetsJ⊆2ℕ is known as an ideal in a non-empty set ℕ, if(1)
∅∈J,(2)
for anyC,D∈J⇒C∪D∈J,(3)
for anyC∈J and D⊆C,⇒D∈J.Remark 2 [5].
An idealJ is known as non-trivial if ℕ∉J. A nontrivial ideal J is known as admissible if n:n∈ℕ∈J.Definition 3 [5].
A nonempty subsetF⊆2ℕ is known as filter in ℕ 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 idealJ, there is a filter FJ associated with J defined as follows:
(1)FJ=K⊆ℕ:K=ℕ\A,forsomeA∈J.Definition 5 [5].
LetC1,C1,⋯ be a mutually disjoint sequence of sets of J. Then, there is sequence of sets D1,D1,⋯ so that ∪j=1∞Dj∈J and each symmetric difference Cj△Djj=1,2,⋯ is finite. In this case, admissible ideal J is known as property AP.Lemma 6 [28].
SupposeJ be an admissible ideal alongside property AP. Let a countable collection of subsets Ckk=1∞ of positive integer ℕ in such a way that Ck∈FJ. Then, there exists a set C⊂ℕ such that C\Ck is finite for all k∈ℕ and C∈FJ.Definition 7 [29].
LetM,d be a metric space and Ck be a sequence of nonempty closed subsets of M which is said to be Wijsman convergent to the closed CofM, if
(2)limk→∞dx,Ck=dx,Cforevery∈M.In other words,W−limk→∞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].
SupposeM,d is a metric space. A nonempty closed subset Ck of M is known as Wijsman J–convergent to a closed set C, if for every x∈M, one has
(3)k∈ℕ:dx,Ck−dx,C≥ϵ∈J.
Hence, one writesJW−limk→∞Ck=C.Definition 9 [10].
SupposeM,dis a metric space. A nonempty closed subsetCkofMis known as WijsmanJ–Cauchy if for eachx∈M, there exists a positive integerm=mϵso that the set(4)k∈ℕ:dx,Ck−dx,Cp≥ϵ∈J,forallp≥m.Definition 10 [10].
SupposeM,d is a separable metric space and Ck,C is nonempty closed subsets of M. A sequence Ck is known as Wijsman J∗–convergent to C if and only if ∃P∈FJ and P=p=pj<pj+1,j∈ℕ⊂ℕ in such a manner that
(5)limk→∞dx,Cmk=dx,C,foreveryx∈M.
One writesJW∗−limk→∞Ck=C.Definition 11 [10].
SupposeM,d is a separable metric space and J is an admissible ideal. A sequence Ck of nonempty closed subsets of M is known as the Wijsman J∗–Cauchy sequence if there exists P∈FJ, where P=p=pj<pj+1,i∈ℕ in such a way that subsequence CP=Cpk is Wijsman Cauchy in M, i.e.,
(6)limk,l→∞∣dx,Cmk−dx,Cpl∣=0.Remark 12 [10].
In general, the Wijsman topology is not first-countable, if sequence of nonempty setsCk is Wijsman convergent to set C, then every subsequence of Ck may not be convergent to C. Every subsequence of the convergent sequence Ck converges to the same limit provided that M is a separable metric space.Definition 13 [17].
LetM be a nonempty set, η and φ be fuzzy sets on M2×0,∞, ∗ be a continuous t-norm, and ⋄ be a continuous t-conorm. Then, the five-tuple M,η,φ,∗,⋄ 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)
ηy,z,s+φy,z,s≤1,(b)
ηy,z,s>0,(c)
ηy,z,s=1 if and only if y=z,(d)
ηy,z,s=ηz,y,s,(e)
ηy,z,s∗ηz,w,t≤ηy,w,s+t,(f)
ηy,z,.:0,∞→0,1 is continuous,(g)
φy,z,s<1,(h)
φy,z,s=0 if and only if y=z,(i)
φy,z,s=φz,y,s,(j)
φy,z,s⋄φz,w,t≥φy,w,s+t,(k)
φy,.:0,∞→0,1 is continuous.
In such situation,η,φ is called the intuitionistic fuzzy metric (briefly, IFM).Example 14 [17].
SupposeM,d is a metric space. Define c⋄d=minc+d,1 and c∗d=cd for all c,d∈0,1, and suppose η and φ are fuzzy sets on M2×0,∞ defined as
(7)ηy,z,s=ss+dy,z,φy,z,s=dy,zs+dy,z.
ThenM,η,φ,∗,⋄ is an IFMS.Definition 15 [18].
LetM,η,φ,∗,⋄ be an IFMS and C be a nonempty subset of M. For all s>0 and x∈M, we define
(8)ηx,C,s=supηx,y,s:y∈Cand
(9)φx,C,s=infφx,y,s:y∈C,where ηx,C,s and φx,C,s 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].
LetM,η,φ,∗,⋄ be an IFMS. A sequence x=xk is convergent to ξ if for any 0<ϵ<0ands>0 there exists k0∈ℕ in such a way that
(10)ηxk,ξ,s>1−ϵandφxk,ξ,s<ϵforallk≥k0.Definition 17 [20].
An IFMSM,η,φ,∗,⋄ is known as separable if it contains a countable dense subset, i.e., there is a countable set xk along with subsequent property: for any s>0 and for all ξ∈M, there is at least one xn in order that
(11)ηxn,ξ,s≥1−ϵandφxn,ξ,s≤ϵ,foreachϵ∈0,1.


## 3. WijsmanJ and J∗–convergence in IFMS

Throughout this section, we denoteJ to be the admissible ideal in ℕ. We begin with the following definitions as follows.Definition 18.
LetM,η,φ,∗,⋄ be an IFMS. A sequence of sets Ck is said be Wijsman convergent to C if for every ϵ>0 and s>0 there exists k0∈ℕ such that
(12)limk→∞ηx,Ck,s=ηx,C,sandlimk→∞φx,Ck,s=φx,C,sforallk≥k0.
The set of all Wijsman limit point of the sequenceCk is denoted by LCk.Definition 19.
LetM,η,φ,∗,⋄ be an IFMS and J be a proper ideal in ℕ. A sequence Ck of nonempty closed subsets of M is known as Wijsman J–convergent to C with respect to IFM η,φ, if for every 0<ϵ<1, for each x∈M and for all s>0 such that
(13)k∈ℕ:∣ηx,Ck,s−ηx,C,s∣≤1−ϵor∣φx,Ck,s−φx,C,s∣≥ϵ∈J.
We writeη,φ−JW−limk→∞Ck=C.Example 20.
SupposeM,η,φ,∗,⋄ is an IFMS and C,Ck is nonempty closed subsets of M. Assume M=ℝ2 and Ck are sequence defined by
(14)Ck=x,y∈ℝ2:0≤x≤k,0≤y≤1k.x,ifk≠n2x,y∈ℝ2:x≥0,y=1,ifk=n2,C=x,y∈ℝ2:x≥0,y=0.
Since(15)limk→∞1k∣n≤k:∣ηx,y,Ck,s−ηx,y,C,s∣≤1−ϵor∣φx,y,Ck,s−φx,y,C,s∣≥ϵ∣=0.Therefore, the sequence of setsCk is Wijsman statistical convergent to the set C.Now, define the setS as
(16)Sϵ=k∈ℕ:∣ηx,y,Ck,s−ηx,y,C,s∣≤1−ϵor∣φx,y,Ck,s−φx,y,C,s∣≥ϵ.If we assumeJ=Jd, then the Wijsman statistical convergence coincides with the Wijsman ideal convergence. Therefore,
(17)k∈ℕ:∣ηx,y,Ck,s−ηx,y,C,s∣≤1−ϵor∣φx,y,Ck,s−φx,y,C,s∣≥ϵ=k∈ℕ:k=n2⊂Jd.Definition 21.
LetM,η,φ,∗,⋄ be a separable IFMS and J be an admissible ideal in ℕ. A sequence Ck of nonempty closed subsets of M is known as Wijsman J–Cauchy with respect to IFM η,φ, if for each 0<ϵ<1, for each x∈M and for all s>0,∃l=lϵ such that
(18)k∈ℕ:∣ηx,Ck,s−ηx,Cl,s∣≤1−ϵor∣φx,Ck,s−φx,Cl,t∣≥ϵ∈J.Definition 22.
LetM,η,φ,∗,⋄ be a separable IFMS and Ck be any nonempty closed subset of M. The sequence Ck is known as Wijsman J∗–Cauchy with respect to IFM η,φ, if there exists P=p=pj:pj<pj+1,j∈ℕ⊂ℕ and P∈FJ with the result that the subsequence CP=Cpk is Wijsman Cauchy in M, i.e.
(19)limk,l→∞∣ηx,Cpk,s−ηx,Cpl,s∣=1and
(20)limk,l→∞∣φx,Cpk,s−φx,Cpl,s∣=0.Definition 23.
LetM,η,φ,∗,⋄ be a separable IFMS and J be an proper ideal in ℕ. Let Ck be nonempty closed subsets of M. The sequence Ck is known as Wijsman J∗–convergent to C with respect to η,φ, if there exists P∈FJ, where P=p=pj:pj<pj+1,j∈ℕ⊂ℕ such that for each s>0, we have
(21)limk→∞ηx,Cpk,s=ηx,C,s,and
(22)limk→∞φx,Cpk,s=φx,C,s.
In such case, we writeη,φ−JW∗−limCk=C.In the following theorem, we prove that every WijsmanJ–convergent implies the Wijsman J–Cauchy condition in IFMS:Theorem 24.
LetM,η,φ,∗,⋄ be a separable IFMS and let J be an arbitrary admissible ideal. Then, every Wijsman J–convergent sequence of closet sets Ck is Wijsman J–Cauchy with respect to IFM η,φ.Proof.
Supposeη,φ−JW−limk→∞Ck=C. Then, for every 0<ϵ<1, for all s>0 and x∈X, the set
(23)Uϵ,s=k∈ℕ:∣ηx,Ck,s−ηx,C,s∣≤1−ϵor∣φx,Ck,s−φx,C,s∣≥ϵbelongs to J. Since J is an admissible ideal, then there exists k0∈ℕ with the result that k0∉Uϵ,s. Now, suppose that
(24)Vϵ,s=k∈ℕ:∣ηx,Ck,s−ηx,Ck0,s∣≤1−2ϵor∣φx,Ck,s−φx,Ck0,s∣≥2ϵ.
Considering the inequality(25)ηx,Ck,s−ηx,Ck0,s≤ηx,Ck,s−ηx,C,s+ηx,Ck0,s−ηx,C,s,and
(26)∣φx,Ck,s−φx,Ck0,s∣≤∣φx,Ck,s−φx,C,s∣+φx,Ck0,s−φx,C,s.
Observe that ifk∈Vϵ,s, therefore
(27)ηx,Ck,s−ηx,C,s+ηx,Ck0,s−ηx,C,s≤1−2ϵ,and
(28)∣φx,Ck,s−φx,C,s∣+∣φx,Ck0,s−φx,C,s∣≥2ϵ.
From another point of view, sincek0∉Uϵ,s, we obtain
(29)ηx,Ck0,s−ηx,C,s>1−ϵand∣φx,Ck0,s−φx,C,s∣<ϵ.
We achieve that(30)ηx,Ck,s−ηx,C,s≤1−ϵor∣φx,Ck,s−φx,C,s∣≥ϵ.
Hence,k∈Uϵ,s. This implies that Uϵ,s⊂Vϵ,s∈J for every 0<ϵ<1 and for all s>0 and x∈M. Therefore, Vϵ,s∈J, so the sequence is Ck which is Wijsman J–Cauchy.Theorem 25.
LetM,η,φ,∗,⋄ 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 sequenceCk is Wijsman J∗–Cauchy with respect to IFM η,φ. Then, for each x∈M and for each 0<ϵ<1, there exists P∈FJ, where P=pj:pj<pj+1,j∈ℕ in such a way that
(31)ηx,Cpk,s−ηx,Cpl,s≤1−ϵ,and
(32)φx,Cpk,s−φx,Cpl,s≥ϵ,∀k,l>k0=k0ϵ.
SupposeN=Nϵ=pk0+1. Therefore, for each ϵ>0, one obtains
(33)ηx,Cpk,s−ηx,CN,s≤1−ϵ,and
(34)φx,Cpk,s−φx,CN,s≥ϵforallk>k0.
Now, suppose thatK=ℕ\P. Clearly, K∈J and
(35)Qϵ,s=k∈ℕ∣ηx,Ck,s−ηx,CN,s∣≤1−ϵor∣φx,Ck,s−φx,CN,s∣≥ϵ⊂K∪p1,p2,⋯,pk0∈J.
Hence, for alls>0 and for each 0<ϵ<1, one can determine N=Nϵ so that Qϵ,s∈J, that is, sequence Ck is Wijsman J–Cauchy.Theorem 26.
LetJ be an admissible ideal including property (AP) and M,η,φ,∗,⋄ be a separable IFMS. Then, the notion of Wijsman J∗–Cauchy sequence of sets coincides with Wijsman J–Cauchy with respect to η,φ and vice-versa.Proof.
The direct part is already proven in Theorem25.
Now, suppose that sequenceCk is Wijsman J–Cauchy sequence with respect to IFM η,φ. Then by definition, if for every 0<ϵ<1, for each x∈X and for all s>0, there exists a m=mϵ such that
(36)Bϵ,s=k∈ℕηx,Ck,s−ηx,Cn,s≤1−ϵorφx,Ck,s−φx,Cn,s≥ϵ∈I.
Now, suppose that(37)Pjϵ,s=k∈ℕηx,Ck,s−ηx,Cmj,s>1−1jor∣x,Ck,s−φx,Cmj,s∣<1j,where mj=m1/j, j=1,2,3,⋯. Obviously, for j=1,2,3⋯, Pjϵ,s∈FJ. Using Lemma 6, there exists P⊂ℕ so that P∈FJ and P\Pj are finite for all j.
Now, we prove that(38)limk,l→∞∣ηx,Ck,s−ηx,Cl,s∣=1,and
(39)limk,l→∞∣φx,Ck,s−φx,Cl,s∣=0.
To show the above equations, letϵ>0, and r∈ℕ such that r>2/ϵ. If k,l∈P, then P\Pj is a finite set; therefore, there exists w=wr in order that
(40)ηx,Ck,s−ηx,Clr,s>1−1r,ηx,Cl,s−ηx,Clr,s>1−1r,and
(41)φx,Ck,s−φx,Clr,s<1r,φx,Cl,s−φx,Clr,s<1r,for all k,l>wr. Then, the above inequalities follow that for k,l>wr(42)ηx,Ck,s−ηx,Cl,s≤ηx,Ck,s−ηx,Clr,s+ηx,Cl,s−ηx,Clr,s>1−1r+1−1r>1−ϵ,and
(43)φx,Ck,s−φx,Cl,s≤φx,Ck,s−φx,Clj,s+φx,Cl,s−φx,Clj,s<1r+1r<ϵ.
Therefore, for eachϵ>0,∃w=wϵ and k,l∈P∈FI, we achieve
(44)k∈ℕ:∣ηx,Ck,s−ηx,Cl,s∣≤1−ϵor∣φx,Ck,s−φx,Cl,s∣≥ϵ∈J.
This proves that the sequenceCk is a Wijsman J∗–Cauchy.Theorem 27.
LetM,η,φ,∗,⋄ be a separable IFMS and let J be an admissible ideal. Then
(45)η,φ−JW∗−limk→∞Ck=Cimplies that sequence Ck is a Wijsman J–Cauchy sequence with respect to IFM η,φ.Proof.
Suppose thatη,φ−JW∗−limk→∞Ck=C. Then, there exists P=p=pj:pj<pj+1,j∈ℕ⊂ℕ with P∈FJ so that CP=Cpk(46)limk→∞ηx,Cpk,s=ηx,C,s,and
(47)limk→∞φx,Cpk,s=φx,C,s,for any ϵ>0 and k,l>k0.
Supposer∈ℕ and ϵ>0 in such a way that r>2/ϵ. If k,l∈P, then P\Pj is a finite set; therefore, there exists kr=k so that
(48)ηx,Cpk,s−ηx,Cpl,s≤ηx,Cpk,s−x,C,s+ηx,Cpl,s−ηx,C,s>1−1r+1−1r>1−ϵ,and
(49)φx,Cpk,s−φx,Cpl,s<φx,Cpk,s−φx,C,s+φx,Cpl,s−φx,C,s<1r+1r<ϵ.
Therefore,(50)limk,l→∞∣ηx,Cpk,s−ηx,Cpl,s∣=1,and
(51)limk,l→∞∣φx,Cpk,s−φx,Cpl,s∣=0.
Hence, sequenceCk is Wijsman J–Cauchy with respect to IFM η,φ.


## 4. WijsmanJ–cluster points and Wijsman J–limit points in IFMS

Throughout this section, we denoteJ to be the proper ideal in ℕ 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.
LetM,η,φ,∗,⋄ be a separable IFMS. An element C∈M is known as the Wijsman J-cluster point of Ck if and only if for any x∈M and for all ϵ,s>0, one has
(52)k∈ℕ:ηx,Ck,s−ηx,C,s<1−ϵorφx,Ck,s−φx,C,s>ϵ∉J.We denoteJWη,φΓCk as the collection of all Wijsman J-cluster points.Definition 29.
LetM,η,φ,∗,⋄ be a separable IFMS. An element C∈M is known as Wijsman J–limit point of sequence Ck of nonempty closed subsets of M provided that P=p=pj:pj<pj+1,j∈ℕ⊂ℕ in such a way that P∉J, and for any x∈M and s>0, we obtain
(53)limk→∞ηx,Ck,s=ηx,C,sandlimk→∞φx,Ck,s=φx,C,s.We denoteJWη,φΛCk as the collection of all Wijsman J–limit points.Theorem 30.
LetM,η,φ,∗,⋄ be a separable IFMS. Then, for any sequence sets, Ck⊂M, JWη,φΛCk⊂JWη,φΓCk.Proof.
SupposeC∈JWη,φΛCk. Then, there exists P=p1<p2<⋯⊂ℕ such that P=p=pj:pj<pj+1,j∈ℕ∉J and for all s>0 and x∈M, we have
(54)limk→∞ηx,Cpk,s=ηx,C,s,and
(55)limk→∞φx,Cpk,s=φx,C,s.
According to Equations (54) and (55), there exists k0∈ℕ so that for each ϵ>0 and for any x∈X and k>k0(56)ηx,Cpk,s−ηx,C,s>1−ϵ,and
(57)φx,Cpk,s−φx,C,s<ϵ.
Hence,(58)k∈ℕ:∣ηx,Cpk,s−ηx,C,s∣>1−ϵ,∣φx,Cpk,s−φx,C,s∣<ϵ⊇P\p1,p1,,̇pk0.
Then, the right-hand side of (58) does not belong to J, and then
(59)k∈ℕ:ηx,Cpk,s−ηx,C,s>1−ϵ,φx,Cpk,s−φx,C,s<ϵ∉J,which means that C∈JWη,φΓCk.Theorem 31.
LetM,η,φ,∗,⋄ be a separable IFMS. Then, for any sequence Ck⊂M, JWη,φΓCk⊂LCk.Proof.
LetC∈JWη,φΓCk. Then, for each ϵ>0 and for all s>0 and for each x∈M, one has
(60)k∈ℕ:ηx,Ck,s−ηx,C,s<1−ϵorφx,Ck,s−φx,C,s>ϵ∉J.
Suppose(61)Qk=k∈ℕ:ηx,Ck,s−ηx,C,s>1−1k,φx,Ck,s−φx,C,s<1k,for k∈ℕ. Qkk=1∞ is a descending sequence of subsets of ℕ. Hence, Q=k=ki:ki<ki+1,i∈ℕ∉J so that
(62)limk→∞ηx,Cki,s=ηx,C,s,and
(63)limk→∞φx,Cki,s=φx,C,s,which means that C∈LCk.



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