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AIMS Mathematics

2023, Volume 8, Issue 2: 2532-2555. doi: 10.3934/math.2023131
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Research article Special Issues

Common fixed point, Baire's and Cantor's theorems in neutrosophic 2-metric spaces

  • 1.
    Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore, Pakistan
  • 2.
    Department of Math &amp, Stats, International Islamic University Islamabad, Pakistan
  • 3.
    Department of Mathematics, University of Management and Technology, Lahore, Pakistan
  • 4.
    Department of Mathematics, COMSATS University Islamabad-Sahiwal Campus, Sahiwal, Pakistan
  • 5.
    Department of Mathematics, Cankaya University, 06790 Etimesgut, Ankara, Turkey
  • 6.
    Department of Mathematics, King Abdulaziz University, Saudi Arabia
  • 7.
    Department of Medical Research, China Medical University, Taichung 40402, Taiwan

  • These fundamental Theorems of classical analysis, namely Baire's Theorem and Cantor's Intersection Theorem in the context of Neutrosophic 2-metric spaces, are demonstrated in this article. Naschie discussed high energy physics in relation to the Baire's Theorem and the Cantor space in descriptive set theory. We describe, how to demonstrate the validity and uniqueness of the common fixed-point theorem for four mappings in Neutrosophic 2-metric spaces.

    Citation: Umar Ishtiaq, Khaleel Ahmad, Muhammad Imran Asjad, Farhan Ali, Fahd Jarad. Common fixed point, Baire's and Cantor's theorems in neutrosophic 2-metric spaces[J]. AIMS Mathematics, 2023, 8(2): 2532-2555. doi: 10.3934/math.2023131

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  • These fundamental Theorems of classical analysis, namely Baire's Theorem and Cantor's Intersection Theorem in the context of Neutrosophic 2-metric spaces, are demonstrated in this article. Naschie discussed high energy physics in relation to the Baire's Theorem and the Cantor space in descriptive set theory. We describe, how to demonstrate the validity and uniqueness of the common fixed-point theorem for four mappings in Neutrosophic 2-metric spaces.


    1.   Introduction

    Among the other improvements of the Zadeh-originally-proposed theory of fuzzy sets [1], progress has been made in discovering the fuzzy counterparts of the classical set theory. In actuality, over the past forty years, the fuzzy theory has developed into a topic of active study. Numerous scientific and technical fields have used it, including population dynamics [2], chaos control [3], computer programming [4], non-linear dynamical systems [5], and medicine [6]. Naschie [7,8,9,10,11,12,13] described the relationship of fuzzy Kähler interpolation of e∞ to the latest work on Cosmo-topology and the Poincaré dodecahedral conjecture and gave different applications and results of e∞ –theory from nanotechnology to brain research. This is, where the most fascinating application of fuzzy topology in quantum physics arises. Atanassov [14,15] presented the idea of intuitionistic fuzzy sets, and Oker [16] explored it in more details. We refer to [17,18] for intuitionistic fuzzy topological features. Recently, Park [19] presented the idea of intuitionistic fuzzy metric spaces. Kirişci and Simsek [20] introduced Neutrosophic metric spaces (NMSs). The concepts of intuitionistic fuzzy 2-normed spaces and intuitionistic fuzzy 2-metric spaces were introduced in [21,22], respectively. Schweizer and Sklar [23] worked on statistical metric spaces and Gähler [24] did work on 2-metric spaces. Certainly, there are some circumstances when the conventional metric is ineffective, and in these circumstances the intuitionistic fuzzy metric notion seems to be more appropriate. In other words, we may handle these circumstances by simulating the imperfection of the norm in some circumstances. In intuitionistic fuzzy 2-metric spaces (IF2MS), Mursaleen and Lohani [25] demonstrate Baire's and Cantor's Theorems ((B&C)-Theorems). On IF2MS Bakry [26] established the Common fixed-point Theorem, see [27,28,29] for more details.

    The main objectives of this manuscript are:

    ● To describe the notion of Neutrosophic 2-metric spaces (N2MSs), which would offer a more practical tool to address the inexactness of the metric or 2-metric in particular circumstances.

    ● To present (B&C)-Theorems.

    ● In N2MS, we establish the common fixed-point theorem.

    This article has four parts, in first section we will discuss some relevant definitions and examples. In second section, we will introduce the definition of N2MSs and prove some theorems in sense of N2MSs, in third section, we will prove (B&C)-Theorem in the sense of N2MSs, and in last section, we find common fixed point for contraction mappings in the context of N2MS.

    2.   Preliminaries

    In this section, we provide some basic definitions that are helpful for readers to understand the main section.

    Definition 2.1. [23] A binary operation ∗:[0,1]×[0,1]→[0,1] be a continuous t-norm if it met the conditions listed below:

    (a) ∗ is associative and commutative;

    (b) ∗ is continuous;

    (c) τ∗1=τ  for  all  τ∈[0,1];

    (d) τ∗σ≤c∗d  whenever  τ≤c  and  σ≤d  for  each  τ,σ,c,d∈[0,1].

    Definition 2.2. A binary operation ◊:[0,1]×[0,1]→[0,1] be continuous t-conorm if it satisfies the above circumstance (a), (b), (d) and

    (c')  τ◊0=τ    for   all  τ∈[0,1].

    Definition 2.3. [20] Let Ξ≠∅. Given a six tuple (Ξ,Ψ,Φ,ψ, *, ◊),  where * is a CTN, ◊ is a CTCN, Ψ,Φ  and  ψ are neutrosophic sets on Ξ×Ξ×(0,∞). If (Ξ,Ψ,Φ,ψ, *, ◊) satisfies the following conditions that are given below for all ϖ,θ,ω,∈Ξ  and  t,s>0:

    (N1) Ψ(ϖ,θ,t)+Φ(ϖ,θ,t)+ψ(ϖ,θ,t)≤3,

    (N2) 0≤Ψ(ϖ,θ,t)≤1,

    (N3) Ψ(ϖ,θ,t)=1  if  and  only  if  ϖ=θ,

    (N4) Ψ(ϖ,θ,t)=Ψ(θ,ϖ,t),

    (N5) Ψ(ϖ,ω,t+s)≥Ψ(ϖ,θ,t)*Ψ(θ,ω,s),

    (N6) Ψ(ϖ,θ,⋅):[0,∞)→[0,1] is continuous,

    (N7) limt→∞Ψ(ϖ,θ,t)=1,

    (N8) 0≤Φ(ϖ,θ,t)≤1,

    (N9) Φ(ϖ,θ,t)=0  if  and  only  if  ϖ=θ,

    (N10) Φ(ϖ,θ,t)=Φ(θ,ϖ,t),

    (N11) Φ(ϖ,ω,t+s)≤Φ(ϖ,θ,t)◊Φ(θ,ω,s),

    (N12) Φ(ϖ,θ,⋅):[0,∞)→[0,1] is continuous,

    (N13) limt→∞Φ(ϖ,θ,t)=0,

    (N14) 0≤ψ(ϖ,θ,t)≤1,

    (N15) ψ(ϖ,θ,t)=0  if  and  only  if  ϖ=θ,

    (N16) ψ(ϖ,θ,t)=ψ(θ,ϖ,t),

    (N17) ψ(ϖ,ω,(t+s))≤ψ(ϖ,θ,t)◊ψ(θ,ω,s),

    (N18) ψ(ϖ,θ,⋅):[0,∞)→[0,1] is continuous,

    (N19) limt→∞ψ(ϖ,θ,t)=0,

    (N20) if t≤0, then Ψ(ϖ,θ,t)=0,Φ(ϖ,θ,t)=1,ψ(ϖ,θ,t)=1.

    Then, (Ψ,Φ,ψ) is a neutrosophic metric and (Ξ,Ψ,Φ,ψ, *, ◊) is a NMS.

    Definition 2.4. [25] The 5-tuple (Ξ,Ψ,Φ,∗,◊) is said to be an IF2MS if Ξ is any non-empty set, ∗ is a continuous t-norm, ◊ is a continuous t-conorm, and Ψ,Φ are fuzzy sets on Ξ×Ξ×Ξ×(0,∞), the following criteria, as listed below, must be met, for each ϖ,θ,ω,w∈Ξ and s,t>0:

    (a) Ψ(ϖ,θ,ω,t)+Φ(ϖ,θ,ω,t)≤1,

    (b) Given distinct elements ϖ,θ of Ξ, there exist an element ωofΞ such that Ψ(ϖ,θ,ω,t)>0,

    (c) Ψ(ϖ,θ,ω,t)=1 if at least two of ϖ,θ,ω are equal,

    (d) Ψ(ϖ,θ,ω,t)=Ψ(ϖ,ω,θ,t)=Ψ(θ,ω,ϖ,t) for all ϖ,θ,ω in Ξ,

    (e) Ψ(ϖ,θ,w,t)∗Ψ(ϖ,w,ω,s)∗Ψ(w,θ,ω,r)≤Ψ(ϖ,θ,ω,t+s+r) for all ϖ,θ,ω,w∈Ξ,

    (f) Ψ(ϖ,θ,ω,⋅):(0,∞)→(0,1] is continuous,

    (g) Φ(ϖ,θ,ω,t)<1,

    (h) Φ(ϖ,θ,ω,t)=0 if at least two of ϖ,θ,ω are equal,

    (i) Φ(ϖ,θ,ω,t)=Φ(ϖ,ω,θ,t)=Φ(θ,ω,ϖ,t) for all ϖ,θ,ω in Ξ,

    (j) Φ(ϖ,θ,w,t)◊Φ(ϖ,w,ω,s)◊Φ(w,θ,ω,r)≥Φ(ϖ,θ,ω,t+s+r),

    (k) Φ(ϖ,θ,ω,⋅):(0,∞)→(0,1] is continuous.

    Now, we define the notion of N2MSs and several topological notions in the context of N2MSs.

    Definition 2.5. The 6-tuple (Ξ,Ψ,Φ,ψ,∗,◊) is said to be a N2MS if Ξ is any non-empty set, ∗ is a continuous t-norm, ◊ is a continuous t-conorm, and Ψ,Φ,ψ are neutrosophic sets on Ξ×Ξ×Ξ×(0,∞), satisfying the following conditions for each ϖ,θ,ω,w∈Ξ and s,t>0:

    (N2M1) Ψ(ϖ,θ,ω,t)+Φ(ϖ,θ,ω,t)+ψ(ϖ,θ,ω,t)≤3,

    (N2M2) Given different elements ϖ,θ of Ξ, there exist an element ωofΞ such that Ψ(ϖ,θ,ω,t)>0,

    (N2M3) Ψ(ϖ,θ,ω,t)=1 if at least two of ϖ,θ,ω are equal,

    (N2M4) Ψ(ϖ,θ,ω,t)=Ψ(ϖ,ω,θ,t)=Ψ(θ,ω,ϖ,t) for all ϖ,θ,ω in Ξ,

    (N2M5)

    Ψ(ϖ,θ,w,t)∗Ψ(ϖ,w,ω,s)∗Ψ(w,θ,ω,r)≤Ψ(ϖ,θ,ω,t+s+r)

    for all ϖ,θ,ω,w  in  Ξ,

    (N2M6) Ψ(ϖ,θ,ω⋅):(0,∞)→(0,1] is continuous,

    (N2M7) Φ(ϖ,θ,ω,t)≤1,

    (N2M8) Φ(ϖ,θ,ω,t)=0 if at least two of ϖ,θ,ω are equal,

    (N2M9) Φ(ϖ,θ,ω,t)=Φ(ϖ,ω,θ,t)=Φ(θ,ω,ϖ,t) for all ϖ,θ,ω in Ξ,

    (N2M10) Φ(ϖ,θ,w,t)◊Φ(ϖ,w,ω,s)◊Φ(w,θ,ω,r)≥Φ(ϖ,θ,ω,t+s+r),

    (N2M11) Φ(ϖ,θ,ω,⋅):(0,∞)→(0,1] is continuous,

    (N2M12) ψ(ϖ,θ,ω,t)≤1,

    (N2M13) ψ(ϖ,θ,ω,t)=0 if at least two of ϖ,θ,ω are equal,

    (N2M14) ψ(ϖ,θ,ω,t)=ψ(ϖ,ω,θ,t)=ψ(θ,ω,ϖ,t) for all ϖ,θ,ω in Ξ,

    (N2M15) ψ(ϖ,θ,w,t)◊ψ(ϖ,w,ω,s)◊ψ(w,θ,ω,r)≥ψ(ϖ,θ,ω,t+s+r),

    (N2M16) ψ(ϖ,θ,ω,⋅):(0,∞)→(0,1] is continuous.

    Here, the functions Ψ(ϖ,θ,ω,t), Φ(ϖ,θ,ω,t) and ψ(ϖ,θ,ω,t) denotes the degree of nearness, the degree of non-nearness and the degree of naturalness between ϖ,θ and ω with respect to t, respectively.

    Example 2.1. Let (Ξ,d) be a 2-metric space. Suppose τ∗σ=τ⋅σ and τ◊σ=max{τ,σ} for all τ,σ∈[0,1] and let Ψ,Φ  and  ψ be neutrosophic sets on Ξ3×(0,∞), defined by

    Ψ(ϖ,θ,ω,t)=tt+md(ϖ,θ,ω),Φ(ϖ,θ,ω,t)=md(ϖ,θ,ω)t+md(ϖ,θ,ω)

    and

    ψ(ϖ,θ,ω,t)=md(ϖ,θ,ω)t,

    for all m∈R+. Then, (Ξ,Ψ,Φ,ψ,∗,◊) is an N2MS.

    Proof. Conditions (N2M1)−(N2M4),(N2M6)−(N2M9),(N2M10)−(N2M14)  and  (N2M16) are trivial, here we examine (N2M5),(N2M10)  and  (N2M15).

    N2M5. From the definition of 2-metric space, we have

    d(ϖ,θ,ω)≤d(ϖ,θ,w)+d(ϖ,w,ω)+d(w,θ,ω).

    Therefore,

    tsr  md(ϖ,θ,ω)
    ≤(rts+rs2+r2s)md(ϖ,θ,w)+(rts+rt2+tr2)md(ϖ,w,ω)
    +(t2s+ts2+rts)md(w,θ,ω)
    ⇒tsrmd(ϖ,θ,ω)
    ≤(t+s+r)rsmd(ϖ,θ,w)+(s+t+r)rtmd(ϖ,w,ω)+(t+s+r)tsmd(w,θ,ω)
    ⇒tsr(t+s+r)+tsrmd(ϖ,θ,ω)
    ≤tsr(t+s+r)+(t+s+r)rsmd(ϖ,θ,w)+(s+t+r)rtmd(ϖ,w,ω)
    +(t+s+r)tsmd(w,θ,ω)
    ⇒tsr[(t+s+r)+md(ϖ,θ,ω)]
    ≤(t+s+r)[tsr+rsmd(ϖ,θ,w)+rtmd(ϖ,w,ω)+tsmd(w,θ,ω)].

    That is,

    ⇒tsr[(t+s+r)+md(ϖ,θ,ω)]
    ≤(t+s+r)[tsr+rsmd(ϖ,θ,w)+rtmd(ϖ,w,ω)+tsmd(w,θ,ω)+rm2d(ϖ,θ,w)d(ϖ,w,ω)+tm2d(w,θ,ω)d(ϖ,w,ω)+sm2d(w,θ,ω)d(ϖ,θ,w)+m3d(ϖ,θ,w)d(ϖ,w,ω)d(w,θ,ω)]
    ⇒tsr[(t+s+r)+md(ϖ,θ,ω)]
    ≤(t+s+r)[(t+md(ϖ,θ,w))(s+md(ϖ,w,ω))(r+md(w,θ,ω))]
    ⇒(t+s+r)(t+s+r)+md(ϖ,θ,ω)
    ≥tsr(t+md(ϖ,θ,w))(s+md(ϖ,w,ω))(r+md(w,θ,ω))
    ⇒(t+s+r)(t+s+r)+md(ϖ,θ,ω)
    ≥tt+md(ϖ,θ,w)⋅ss+md(ϖ,w,ω)⋅rr+md(w,θ,ω).

    We have continuous t-norm τ∗σ=τσ. Hence

    Ψ(ϖ,θ,w,t)∗Ψ(ϖ,w,ω,s)∗Ψ(w,θ,ω,r)≤Ψ(ϖ,θ,ω,t+s+r).

    N2M10. Φ(ϖ,θ,w,t)◊Φ(ϖ,w,ω,s)◊Φ(w,θ,ω,r)≥Φ(ϖ,θ,ω,t+s+r). Observe the fact that

    md(ϖ,θ,ω)
    ≤[t+s+r+md(ϖ,θ,ω)]max{md(ϖ,θ,w)t+md(ϖ,θ,w),md(ϖ,w,ω)s+md(ϖ,w,ω),md(w,θ,ω)r+dm(w,θ,ω)}.

    This implies

    md(ϖ,θ,ω)t+s+r+md(ϖ,θ,ω)≤max{md(ϖ,θ,w)t+md(ϖ,θ,w),md(ϖ,w,ω)s+md(ϖ,w,ω),md(w,θ,ω)r+md(w,θ,ω)}.

    Then

    Φ(ϖ,θ,ω,t+s+r)≤max{Φ(ϖ,θ,w,t),Φ(ϖ,w,ω,s),Φ(w,θ,ω,r)}.

    Hence,

    Φ(ϖ,θ,w,t)◊Φ(ϖ,w,ω,s)◊Φ(w,θ,ω,r)≥Φ(ϖ,θ,ω,t+s+r).

    N2M15. ψ(ϖ,θ,w,t)◊ψ(ϖ,w,ω,s)◊ψ(w,θ,ω,r)≥ψ(ϖ,θ,ω,t+s+r). Observe that,

    md(ϖ,θ,ω)≤[t+s+r+md(ϖ,θ,ω)]max{md(ϖ,θ,w)t,md(ϖ,w,ω)s,md(w,θ,ω)r}.

    This implies

    md(ϖ,θ,ω)t+s+r+md(ϖ,θ,ω)≤max{md(ϖ,θ,w)t,md(ϖ,w,ω)s,md(w,θ,ω)r}.

    Then

    ψ(ϖ,θ,ω,t+s+r)≤max{ψ(ϖ,θ,w,t),ψ(ϖ,w,ω,s),ψ(w,θ,ω,r)}.

    Hence

    ψ(ϖ,θ,w,t)◊ψ(ϖ,w,ω,s)◊ψ(w,θ,ω,r)≥ψ(ϖ,θ,ω,t+s+r).

    Therefore, (Ξ,Ψ,Φ,ψ,∗,◊) is an N2MS.

    Definition 2.6. Suppose (Ξ,Ψ,Φ,ψ,∗,◊) is a N2MS. Suppose r∈(0,1), t>0 and ϖ∈Ξ. The set B(ϖ,r,t)={θ∈Ξ:Ψ(ϖ,θ,ω,t)>1−r,Φ(ϖ,θ,ω,t)<r  and  ψ(ϖ,θ,ω,t)<r,  for   all  ω∈Ξ} is called the open ball with center ϖ and radius r with respect to t.

    Example 2.2. Let Ξ={1,2,3} and (Ξ,d) be a 2-metric space defined by d(ϖ,θ,ω)=|ϖ−θ−ω|. Suppose τ∗σ=τ⋅σ and τ◊σ=max{τ,σ} for all τ,σ∈[0,1] and let Ψ,Φ  and  ψ be neutrosophic sets on Ξ3×(0,∞), defined by

    Ψ(ϖ,θ,ω,t)=tt+|ϖ−θ−ω|,Φ(ϖ,θ,ω,t)=|ϖ−θ−ω|t+|ϖ−θ−ω|

    and

    ψ(ϖ,θ,ω,t)=|ϖ−θ−ω|t.

    Then, (Ξ,Ψ,Φ,ψ,∗,◊) is a N2MS.

    Let the center ϖ=1, radius r=0.6,t=6, then B(1,0.6,6)={θ∈Ξ:Ψ(ϖ,θ,  ω,t)>0.4,  Φ(ϖ,θ,ω,t)<0.6   and  ψ(ϖ,θ,ω,t)<0.6, for   all  ω∈Ξ} is an open ball.

    Definition 2.7. Suppose (Ξ,Ψ,Φ,ψ,∗,◊) is a N2MS. Then, a set U⊂Ξ is open set if each of its points is the center of some open ball contained in U. The open set in a N2MS (Ξ,Ψ,Φ,ψ,∗,◊) is represented by U.

    Definition 2.8. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS. A subset Ω of Ξ is said to be ΦΨψ2 – bounded if there exits t>0 and r∈(0,1) such that Ψ(ϖ,θ,ω,t)>1−r,Φ(ϖ,θ,ω,t)<r  and  ψ(ϖ,θ,ω,t)<r for all ϖ,θ∈Ω, and for all ω∈Ξ.

    Definition 2.9. Assume (Ξ,Ψ,Φ,ψ,∗,◊) is a N2MS. A sequence (ϖn) in Ξ is a Cauchy if for each ϵ>0 and each t>0, there exists n°∈N such that Ψ(ϖn,ϖm,ω,t)>1−r, Φ(ϖn,ϖm,ω,t)<r  and  ψ(ϖn,ϖm,ω,t)<r for all n,m≥n° for all ω∈Ξ.

    Definition 2.10. Suppose (Ξ,Ψ,Φ,ψ,∗,◊) is a N2MS. A sequence ϖ=(ϖk) is convergent to l∈Ξ, with respect to the N2MS if, for every ϵ>0 and t>0, there exist k0∈N such that Ψ(ϖk,l,ω,t)>1−ϵ,Φ(ϖk,l,ω,t)<ϵ  and  ψ(ϖk,l,ω,t)<ϵ for all k≥k0 and for all ω∈Ξ. In this case, we write (Ψ,Φ,ψ)2−limϖ=l or ϖk(Ψ,Φ,ψ)2→l as k→∞.

    Definition 2.11. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS. Define τ(Ψ,Φ,ψ)2 ={Ω⊂Ξ:  for  each  ϖ∈Ξ,  there exist  t>0    and  r∈(0,1)such that  B(ϖ,r,t)⊂Ξ}. Then, τ(Ψ,Φ,ψ)2 is a topology on (Ξ,Ψ,Φ,ψ,∗,◊).

    Definition 2.12. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS. If each Cauchy sequence converges with respect to τ(Ψ,Φ,ψ)2 it is said to be complete.

    Definition 2.13. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS. A collection (Fn)n∈N of non-empty sets is said to be have the neutrosophic diameter zero if for each r∈(0,1), and each t>0, there exists n°∈N such that Ψ(ϖ,θ,ω,t)>1−r, Φ(ϖ,θ,ω,t)<r and ψ(ϖ,θ,ω,t)<r for all ϖ,θ∈Fn° and for all ω∈Ξ.

    Definition 2.14. Let Ξ be any non-empty set and (Y,Ψ,Φ,ψ,∗,◊) be a N2MS. Then, a sequence (fn) of functions from ΞtoY is assumed to converge uniformly to a function f from ΞtoY if given t>0 and r∈(0,1), there exists n°∈N such that

    Ψ(fn(ϖ),f(ϖ),ω,t)>1−r,Φ(fn(ϖ),f(ϖ),ω,t)<r and ψ(fn(ϖ),f(ϖ),ω,t)<r

    for all n≥n° and for all ϖ,ω∈Ξ.

    3.   Baire's and Cantor's intersection theorems

    In this section, we establish Baire's Theorem and Cantor's Intersection Theorem in the context of N2MS.

    Theorem 3.1. Every open ball B(ϖ,r,t) in N2MS is an open set.

    Proof. Consider B(ϖ,r,t) be an open ball with center ϖ and radius r. Assume θ∈B(ϖ,r,t). Therefore, Ψ(ϖ,θ,ω,t)>1−r,Φ(ϖ,θ,ω,t)<r,  and  ψ(ϖ,θ,ω,t)<r for each ω∈Ξ. There exists t3∈(0,t) such that Ψ(ϖ,θ,w,t3)>1−r,Φ(ϖ,θ,w,t3)<r,  and  ψ(ϖ,θ,w,t3)<r, due to Ψ(ϖ,θ,ω,t)>1−r. If we take r0=Ψ(ϖ,θ,w,t3), then for r0>1−r,ϵ∈(0,1) will exist such that r0>1−ϵ>1−r. Given r0 and ϵ such that r0>1−ϵ. Then, {ri}6i=1∈(0,1) such that r0∗r1∗r2>1−ϵ,(1−r0)◊(1−r3)◊(1−r4)≤ϵ, and (1−r0)◊(1−r5)◊(1−r6)≤ϵ. Choose r7=max{ri}6i=1. Consider the open ball B(θ,1−r7,t3). We will show that B(θ,1−r7,t3)⊂B(ϖ,r,t). If we take v∈B(θ,1−r7,t3), then Ψ(ϖ,w,ω,t3)>r7, Φ(ϖ,w,ω,t3)<r7,  and  ψ(ϖ,w,ω,t3)<r7 and Ψ(w,θ,ω,t3)>r7, Φ(w,θ,ω,t3)<r7,  and   ψ(w,θ,ω,t3)<r7. Then,

    Ψ(ϖ,θ,ω,t)≥Ψ(ϖ,θ,w,t3)∗Ψ(ϖ,w,ω,t3)∗Ψ(w,θ,ω,t3)
    ≥r0∗r7∗r7≥r0∗r1∗r2≥1−ϵ>1−r,
    Φ(ϖ,θ,ω,t)≥Φ(ϖ,θ,w,t3)∗Φ(ϖ,w,ω,t3)∗Φ(w,θ,ω,t3)
    ≤(1−r0)◊(1−r7)◊(1−r7)
    ≤(1−r0)◊(1−r1)◊(1−r2)≤ϵ<r,
    ψ(ϖ,θ,ω,t)≥ψ(ϖ,θ,w,t3)∗ψ(ϖ,w,ω,t3)∗ψ(w,θ,ω,t3)
    ≤(1−r0)◊(1−r7)◊(1−r7)
    ≤(1−r0)◊(1−r1)◊(1−r2)≤ϵ<r.

    It shows that v∈B(ϖ,r,t) and B(θ,1−r7,t3)⊂B(ϖ,r,t).

    Theorem 3.2 Every N2MS is Hausdorff.

    Proof. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS. Let ϖ and θ be any distinct points in Ξ. Then, 0<Ψ(ϖ,θ,ω,t)<1,0<Φ(ϖ,θ,ω,t)<1,  and  0<ψ(ϖ,θ,ω,t)<1 for every ω∈Ξ. Put r1=Ψ(ϖ,θ,ω1,t),1−r2=Φ(ϖ,θ,ω1,t),  and  1−r3=ψ(ϖ,θ,ω1,t), r4=Ψ(ϖ,θ,w,t3), 1−r5=Φ(ϖ,θ,w,t3), 1−r6=ψ(ϖ,θ,w,t3) and r=max{r1,1−r2,1−r3,r4,1−r5,1−r6}. For each r°∈(r,1) there exist r7  and  r8 such that r4∗r7∗r7≥r° and (1−r5)∗(1−r8)∗(1−r8)≤1−r°. Put r9=max{r7,r8} and consider the open balls B(ϖ,1−r9,t3)  and  B(θ,1−r9,t3). Then, clearly

    yB(ϖ,1−r9,t3)∩B(θ,1−r9,t3)=∅.

    If there is w∈B(ϖ,1−r9,t3)∩B(θ,1−r9,t3)=∅. Then,

    r1=Ψ(ϖ,θ,ω1,t)≥Ψ(ϖ,w,ω1,t3)∗Ψ(w,θ,ω1,t3)∗Ψ(ϖ,θ,w,t3)
    ≥r4∗r9∗r9≥r4∗r7∗r7≥r°>r1

    and similarly, 1−r2<1−r2, which is a contradiction. Hence, (Ξ,Ψ,Φ,ψ,∗,◊) is Hausdorff.

    Definition 3.1. Suppose Ω⊆⋃U∈OU, a collection O of open sets is called an open cover of Ω. A subspace Ω of a NM2S is compact, if every open cover of Ω has a finite subcover.

    Theorem 3.3. Every compact subset Ω of a N2MS (Ξ,Ψ,Φ,ψ,∗,◊) is ΦΨψ2 –bounded.

    Proof. Let Ω be a compact subset of a N2MS (Ξ,Ψ,Φ,ψ,∗,◊). Suppose the open cover {O(ϖ,ϵ,t):ϖ∈Ω} for t>0,ϵ∈(0,1). Since Ω is compact, then there exist ϖ1,ϖ2,⋯,ϖn∈Ω such that Ω⊆⋃nk=1O(ϖk,ϵ,t). For some ϖ,ω∈Ω,there  existk,m≤nsuch  thatϖ∈O(ϖk,ϵ,t) and ω∈O(ϖm,ϵ,t). Then, we get

    {Ψ(ϖ,ϖk,w,t)>1−ϵ,Φ(ϖ,ϖk,w,t)<ϵ,ψ(ϖ,ϖk,w,t)<ϵ,Ψ(ω,ϖm,w,t)>1−ϵ,Φ(ω,ϖm,w,t)<ϵ,ψ(ω,ϖm,w,t)<ϵ,

    for each ω∈Ξ. Let

    {ρ=min{Ψ(ϖk,ϖm,w,t):1≤k,m≤n},σ=max{Φ(ϖk,ϖm,w,t):1≤k,m≤n},γ=max{ψ(ϖk,ϖm,w,t):1≤k,m≤n}.

    Hence, for 0<ξ1,ξ2,ξ3<1, we have

    Ψ(ϖ,ω,w,5t)≥Ψ(ϖ,ϖk,w,t)∗Ψ(ϖ,ω,ϖk,t)∗Ψ(ϖk,ω,w,3t)≥Ψ(ϖ,ϖk,w,t)∗Ψ(ϖ,ω,ϖk,t)∗Ψ(ϖk,ϖm,w,t)∗Ψ(ϖk,ω,ϖm,t)∗Ψ(ϖm,ω,w,t)≥(1−ϵ)∗(1−ϵ)∗ρ∗ρ∗(1−ϵ)>1−ξ1,
    Φ(ϖ,ω,w,5t)≤Φ(ϖ,ϖk,w,t)◊Φ(ϖ,ω,ϖk,t)◊Φ(ϖk,ω,w,3t)≤Φ(ϖ,ϖk,w,t)◊Φ(ϖ,ω,ϖk,t)◊Φ(ϖk,ϖm,w,t)◊Φ(ϖk,ω,ϖm,t)◊Φ(ϖm,ω,w,t)≤ϵ◊ϵ◊σ◊σ◊ϵ<ξ2,
    ψ(ϖ,ω,w,5t)≤ψ(ϖ,ϖk,w,t)◊ψ(ϖ,ω,ϖk,t)◊ψ(ϖk,ω,w,3t)≤ψ(ϖ,ϖk,w,t)◊ψ(ϖ,ω,ϖk,t)◊ψ(ϖk,ϖm,w,t)◊ψ(ϖk,ω,ϖm,t)◊ψ(ϖm,ω,w,t)≤ϵ◊ϵ◊σ◊σ◊ϵ<ξ3,

    If we take ξ=max{ξ1,ξ2,ξ3} and t0=3t, we have Ψ(ϖ,ω,w,t0)>1−ξ,Φ(ϖ,ω,w,t0)<ξ and ψ(ϖ,ω,w,t0)<ξ for all ϖ,ω∈Ω. Hence, Ω is ΦΨψ2 –bounded.

    Remark 3.1. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS induced by a 2-metric d on Ξ. Then, Ω⊂Ξ is ΦΨψ2 – bounded if and only if it is bounded.

    Remark 3.2. In a N2MS every compact set is closed and bounded.

    Theorem 3.4. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS and τ(Ψ,Φ,ψ)2 be the topology on Ξ induced by the (Ψ,Φ,ψ)2. Then, for a sequence (ϖn) such that ϖn(Ψ,Φ,ψ)2→ϖ if and only if Ψ(ϖn,ϖ,ω,t)→1 and Φ(ϖn,ϖ,ω,t)→0 and ψ(ϖn,ϖ,ω,t)→0 as n→∞ for all ω∈Ξ and t>0.

    Proof. Let t>0. Suppose that ϖn(Ψ,Φ,ψ)2→ϖ. If 0<ϵ<1, then there exist N∈N with ϖn∈O(ϖ,ϵ,t), for all n≥N. Therefore, 1−Ψ(ϖn,ϖ,w,t)<ϵ,Φ(ϖn,ϖ,w,t)<ϵ and ψ(ϖn,ϖ,w,t)<ϵ. So, we can write Ψ(ϖn,ϖ,w,t)→1,Φ(ϖn,ϖ,w,t)→0, and ψ(ϖn,ϖ,w,t)→0 as n→∞.

    Conversely, Ψ(ϖn,ϖ,w,t)→1,Φ(ϖn,ϖ,w,t)→0 and ψ(ϖn,ϖ,w,t)→0 as n→∞, for all t>0. Then, for 0<ϵ<1, there exist N∈N such that 1−Ψ(ϖn,ϖ,w,t)<ϵ,Φ(ϖn,ϖ,w,t)<ϵ and ψ(ϖn,ϖ,w,t)<ϵ for all N∈N. Then, Ψ(ϖn,ϖ,w,t)>1−ϵ,Φ(ϖn,ϖ,w,t)<ϵ and ψ(ϖn,ϖ,w,t)<ϵ for all N∈N. Then, ϖn∈O(ϖ,ϵ,t), for all n≥N. This completes the proof.

    Theorem 3.5. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a NS2MS such that every Cauchy sequence in Ξ has a convergent subsequence. Then, (Ξ,Ψ,Φ,ψ,∗,◊) is complete.

    Proof. Let the sequence (ϖn) be a Cauchy and let (ϖni) be a subsequence of (ϖn) and ϖni→ϖ. Let t>0 and μ∈(0,1). Consider 0<ϵ<1 such that (1−ϵ)∗(1−ϵ)∗(1−ϵ)≥1−μ, ϵ◊ϵ◊ϵ≤μ. Since (ϖn) is a Cauchy sequence, there exist N∈N such that Ψ(ϖm,ϖn,w,t3)>1−ϵ,Φ(ϖm,ϖn,w,t3)<ϵ and ψ(ϖm,ϖn,w,t3)<ϵ for all m,n≥N. Since ϖni→ϖ, there is positive integer ip such that ip>N, Ψ(ϖip,ϖ,w,t3)>1−ϵ,Φ(ϖip,ϖ,w,t3)<ϵ and ψ(ϖip,ϖ,w,t3)<ϵ. Therefore, if n≥N,

    Ψ(ϖn,ϖ,w,t)≥Ψ(ϖip,ϖ,w,t3)∗Ψ(ϖn,ϖip,w,t3)∗Ψ(ϖn,ϖ,ϖip,t3)
    >(1−ϵ)∗(1−ϵ)∗(1−ϵ)≥1−μ,
    Φ(ϖn,ϖ,w,t)≤Φ(ϖip,ϖ,w,t3)◊Φ(ϖn,ϖip,w,t3)◊Φ(ϖn,ϖ,ϖip,t3)<ϵ◊ϵ◊ϵ≤μ,
    ψ(ϖn,ϖ,w,t)≤ψ(ϖip,ϖ,w,t3)◊ψ(ϖn,ϖip,w,t3)◊ψ(ϖn,ϖ,ϖip,t3)<ϵ◊ϵ◊ϵ≤μ.

    Thus, we have ϖn→ϖ. This completes the proof.

    Theorem 3.6. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS and let Ω be subset of Ξ with the subspace N2MS (ΨΩ,ΦΩ,ψΩ)2=(Ψ|Ω3×(0,∞),Φ|Ω3×(0,∞),ψ|Ω3×(0,∞))2. Then, (Ω,ΨΩ,ΦΩ,ψΩ,∗,◊) is complete if and only if Ω is closed subset of Ξ.

    Proof. Assume Ω is a closed subset of Ξ and let ϖn be a Cauchy sequence in (Ω,ΨΩ,ΦΩ,ψΩ,∗,◊). Then, (ϖ)n be a Cauchy sequence in Ξ and hence there is a point ϖ∈Ξ such that ϖn(Ψ,Φ,ψ)2→ϖ. Then, ϖ∈−Ω=Ω and thus (ϖ)n converges in Ω. Hence (Ω,ΨΩ,ΦΩ,ψΩ,∗,◊) is complete.

    Conversely, let (Ω,ΨΩ,ΦΩ,ψΩ,∗,◊) is a complete and Ω is not closed. Let ϖ∈−Ω∖Ω. Then, there is a sequence (ϖ)n of points in Ω that converges to ϖ and thus (ϖ)n is a Cauchy sequence. Thus, for each 0<ϵ<1 and each >0, there is k°∈N such that Ψ(ϖk,ϖl,θ,t)>1−ϵ, Φ(ϖk,ϖl,θ,t)<ϵ and ψ(ϖk,ϖl,θ,t)<ϵ for all k,l≥k° and for all θ∈Ξ. Since, (ϖ)n is a sequence in Ω,

    Ψ(ϖk,ϖl,θ,t)=ΨΩ(ϖk,ϖl,θ,t),Φ(ϖk,ϖl,θ,t)=ΦΩ(ϖk,ϖl,θ,t)

    and

    ψ(ϖk,ϖl,θ,t)=ψΩ(ϖk,ϖl,θ,t)

    for all θ∈Ξ. Therefore (ϖ)n is a Cauchy sequence in Ω. Since (Ξ,Ψ,Φ,ψ,∗,◊) is complete, there is a ω∈Ω such that ϖn→ω. That is, for each 0<ϵ<1 and each t>0, there is k°∈N such that Ψ(ϖl,ω,θ,t)>1−ϵ, Φ(ϖl,ω,θ,t)<ϵ and ψ(ϖl,ω,θ,t)<ϵ for all l≥k° and for all θ∈Ξ. But since (ϖ)n is a sequence in Ω and ω∈Ω, Ψ(ϖl,ω,θ,t)=ΨΩ(ϖl,ω,θ,t), Φ(ϖl,ω,θ,t)=ΦΩ(ϖl,ω,θ,t) and ψ(ϖl,ω,θ,t)=ψΩ(ϖl,ω,θ,t) we see that (ϖ)n converges in (Ω,Ψ,Φ,ψ,∗,◊) to both ϖ and ω. Since, ϖ∉Ω and ω∈Ω, ϖ≠ω, that results in a contradiction.

    Lemma 3.1. Let (Ξ,Ψ,Φ,ψ,∗,◊) be a N2MS. If t>0 and r,s∈(0,1) such that (1−s)∗(1−s)∗α≥(1−r), s◊s◊α'≤r, then −B(ϖ,s,t3)⊂B(ϖ,r,t), where α=Ψ(ϖ,θ,w,t3), α'=Φ(ϖ,θ,w,t3) and α'=ψ(ϖ,θ,w,t3).

    Theorem 3.7. A subset Ω of a N2MS (Ξ,Ψ,Φ,ψ,∗,◊) is nowhere dense if and only if every non-empty open set in Ξ contains an open ball whose closure is disjoint from Ω.

    Proof. Let U be a non-empty open subset of Ξ. Then, there exist a non-empty open subset of V⊂U and V∩−Ω≠∅. Let ϖ∈V. Then, there exist r∈(0,1) and t>0 such that B(ϖ,r,t)⊂V. Choose s∈(0,1) such that (1−s)∗(1−s)∗α≥(1−r) and s◊s◊α'≤r for some fixed α∈(0,1). By Lemma 3.1, we have −B(ϖ,s,t3)⊂B(ϖ,r,t). Thus −B(ϖ,s,t3)⊂U and −B(ϖ,s,t3)∩Ω=∅.

    Conversely suppose Ω is not nowhere dense. Then, int −Ω≠∅, so there exists a non-empty open set U such that U⊂−Ω. Let B(ϖ,r,t) be an open ball such that B(ϖ,r,t)⊂U. Then, −B(ϖ,r,t)∩Ω≠∅. This is a contradiction.

    Theorem 3.8. Let (Un:n∈N) be a sequence of dense open subsets of a complete N2MS (Ξ,Ψ,Φ,ψ,∗,◊). Then, ∩n∈NUn is also dense in (Ξ,Ψ,Φ,ψ,∗,◊).

    Proof. Let V be a non-empty open set of Ξ. Since, U1 is dense in Ξ,V∩U1≠∅. Let ϖ1∈V∩U1. Since V∩U1 is open, there exists r1∈(0,1) and t1>0 such that B(ϖ1,r1,t1)⊂V∩U1. Choose r'1<r1 and t'1=m  in  (t1,1) such that −B(ϖ1,r'1,t'1)⊂V∩U1. Since, U2 is dense in Ξ. By Theorem 3.1 B(ϖ1,r'1,t'1)∩U2≠∅. Let ϖ2∈B(ϖ1,r'1,t'1)∩U2. Since, B(ϖ1,r'1,t'1)∩U2 is open, there exists r2∈(0,12) and t2>0 such that B(ϖ2,r2,t2)⊂B(ϖ1,r'1,t'1)∩U2. Choose r'2<r2 and t'2=m  in  (t2,12) such that −B(ϖ2,r'2,t'2)⊂B(ϖ1,r'1,t'1)∩U2. Continuing in this manner, we obtain a sequence (ϖ)n in Ξ and a sequence (t'n) such that

    0<t'n<1n  and  −B(ϖn+1,r'n+1,t'n+1)⊂B(ϖn,r'n,t'n)∩Un+1.

    Now, it's simple to observe that (ϖ)n is a Cauchy sequence. Since Ξ is complete, there exists ϖ∈Ξ such that ϖn(Ψ,Φ,ψ)2→ϖ. Since ϖk∈B(ϖn,r'n,t'n) for k≥n, we obtain ϖ∈−B(ϖn,r'n,t'n). Hence ϖ∈−B(ϖn,r'n,t'n) ⊂B(ϖn−1,r'n−1,t'n−1)∩Un for all n. Therefore, V∩(∩n∈NUn≠∅. Hence ∩n∈NUn is dense in (Ξ,Ψ,Φ,ψ,∗,◊).

    Remark 3.3. A non-empty subset F of a N2MS Ξ has Neutrosophic diameter zero if and only if F is a singleton set.

    Theorem 3.9. A N2MS (Ξ,Ψ,Φ,ψ,∗,◊) is complete if and only if every nested sequence (Fn)nϵN of non-empty closed sets with Neutrosophic diameter zero have non-empty intersection.

    Proof. Firstly, we prove that (Ξ,Ψ,Φ,ψ,∗,◊) is complete under the given hypothesis. Let \left({\varpi }_{n}\right) be a Cauchy sequence in \varXi . Set {B}_{n} = \left({\varpi }_{k}:k\ge n\right) and {F}_{n} = {\stackrel{-}{B}}_{n} , then we claim that {F}_{n} has Neutrosophic diameter zero. For given s\in \left({\rm{0, 1}}\right) and t > 0, we choose r\in \left({\rm{0, 1}}\right) such that \left(1-r\right)*\left(1-r\right)*\left(1-r\right)*\left(1-r\right)*\left(1-r\right) > 1-s and r◊r◊r◊r◊r < s. Since \left({\varpi }_{n}\right) is a Cauchy, there exists {n}_{\circ }\in \mathbb{N} such that \varPsi \left({\varpi }_{n}, {\varpi }_{m}, \omega, \frac{t}{9}\right) > 1-r , \varPhi \left({\varpi }_{n}, {\varpi }_{m}, \omega, \frac{t}{9}\right) < r and \psi \left({\varpi }_{n}, {\varpi }_{m}, \omega, \frac{t}{9}\right) < r for all n, m\ge {n}_{\circ } and for all \omega \in \varXi. Therefore \varPsi \left(\varpi, \theta, \omega, \frac{t}{9}\right) > 1-r , \varPhi \left(\varpi, \theta, \omega, \frac{t}{9}\right) < r and \psi \left(\varpi, \theta, \omega, \frac{t}{9}\right) < r for all \varpi, \theta \in {B}_{{n}_{\circ }} and for all \omega \in \varXi. Let \varpi, \theta \in {F}_{{n}_{\circ }} . Then, there exist sequences ({\varpi }_{n}^{\text{'}} ) and ({\theta }_{n}^{\text{'}} ) in {B}_{{n}_{\circ }} such that {\varpi }_{n}^{\text{'}}\stackrel{{\left(\varPsi, \varPhi, \psi \right)}_{2}}{\to }\varpi and {\theta }_{n}^{\text{'}}\stackrel{{\left(\varPsi, \varPhi, \psi \right)}_{2}}{\to }\theta . Hence, {\varpi }_{n}^{\text{'}}\in \mathbb{B}\left(\varpi, r, \frac{t}{9}\right) and {\theta }_{n}^{\text{'}}\in \mathbb{B}\left(\theta, r, \frac{t}{9}\right) for sufficiently large n . Now {\varpi }_{n}^{\text{'}}\in \mathbb{B}\left(\varpi, r, \frac{t}{9}\right) implies that \varPsi \left(\varpi, {\varpi }_{n}^{\text{'}}, \omega, \frac{t}{9}\right) > 1-r , \varPhi \left(\varpi, {\varpi }_{n}^{\text{'}}, \omega, \frac{t}{9}\right) < r and \psi \left(\varpi, {\varpi }_{n}^{\text{'}}, \omega, \frac{t}{9}\right) < r for all \omega \in \varXi, therefore in particular for some \theta \in \varXi we have \varPsi \left(\varpi, {\varpi }_{n}^{\text{'}}, \theta, \frac{t}{9}\right) > 1-r , \varPhi \left(\varpi, {\varpi }_{n}^{\text{'}}, \theta, \frac{t}{9}\right) < r and \psi \left(\varpi, {\varpi }_{n}^{\text{'}}, \theta, \frac{t}{9}\right) < r, similarly for {\theta }_{n}^{\text{'}}\in \mathbb{B}\left(\theta, r, \frac{t}{9}\right) . Now, we have

    \varPsi \left(\varpi , \theta , \omega , t\right)\ge \varPsi \left({\varpi }_{n}^{\text{'}}, \theta , \omega , \frac{t}{3}\right)*\varPsi \left(\varpi , {\varpi }_{n}^{\text{'}}, \omega , \frac{t}{3}\right)*\varPsi \left(\varpi , {\theta , \varpi }_{n}^{\text{'}}, \frac{t}{3}\right),
    \varPsi \left({\varpi }_{n}^{\text{'}}, \theta , \omega , \frac{t}{3}\right)\ge \varPsi \left({\theta }_{n}^{\text{'}}, \theta , \omega , \frac{t}{9}\right)*\varPsi \left({\varpi }_{n}^{\text{'}}, {\theta }_{n}^{\text{'}}, \omega , \frac{t}{9}\right)*\varPsi \left({\varpi }_{n}^{\text{'}}, {\theta , \theta }_{n}^{\text{'}}, , \frac{t}{9}\right) .

    Hence

    \varPsi \left(\varpi , \theta , \omega , t\right)\ge \left(1-r\right)*\left(1-r\right)*\left(1-r\right)*\left(1-r\right)*\left(1-r\right) > 1-s ,
    \varPhi \left(\varpi , \theta , \omega , t\right)\le r◊r◊r◊r◊r < s,

    and similarly

    \psi \left(\varpi , \theta , \omega , t\right)\le r◊r◊r◊r◊r < s,

    for all \varpi, \theta \in {F}_{{n}_{\circ }} and for all \omega \in \varXi . Thus \left({F}_{n}\right) has Neutrosophic diameter zero and hence by hypothesis {\cap }_{n\in \mathbb{N}}{F}_{n} is non-empty. Take \varpi \in {\cap }_{n\in \mathbb{N}}{F}_{n}. We see that {\varpi }_{n}\stackrel{{(\varPsi, \varPhi, \psi)}_{2}}{\to }\varpi . Then, for each t\to 0 , there exist {n}_{\circ }\in \mathbb{N} such that \varPsi \left({\varpi }_{n}, \varpi, \omega, t\right) > 1-r , \varPhi \left({\varpi }_{n}, \varpi, \omega, t\right) < r \ \ {\rm{and}} \ \ \psi \left({\varpi }_{n}, \varpi, \omega, t\right) < r for all n\ge {n}_{\circ } and for all \omega \in \varXi. Therefore, for each t > 0, \varPsi \left({\varpi }_{n}, \varpi, \omega, t\right)\to 1 , \varPhi \left({\varpi }_{n}, \varpi, \omega, t\right)\to 0 and \psi \left({\varpi }_{n}, \varpi, \omega, t\right)\to 0 as n\to \infty for all \omega \in \varXi. Hence, (\varXi, \varPsi, \varPhi, \psi, *, ◊) is complete.

    Conversely, suppose that \left(\varXi, \varPsi, \varPhi, \psi, *, ◊\right) is complete and ({{F}_{n})}_{n\in \mathbb{N}} is nested sequence of non-empty closed sets with Neutrosophic diameter zero. For each n\in \mathbb{N} , choose a point {\varpi }_{n}\in {F}_{n}. We claim that \left({\varpi }_{n}\right) is a Cauchy sequence. Since \left({F}_{n}\right) has Neutrosophic diameter zero, for r\in \left({\rm{0, 1}}\right) and t > 0, there exists {n}_{\circ }\in \mathbb{N} such that \varPsi \left(\varpi, \theta, \omega, t\right) > 1-r, \varPhi \left(\varpi, \theta, \omega, t\right) < r, and \psi \left(\varpi, \theta, \omega, t\right) < r for all n\ge {n}_{\circ }, \varpi, \theta \in {F}_{n}, and \omega \in \varXi. Since \left({F}_{n}\right) is nested sequence, \varPsi \left({\varpi }_{n}, {\theta }_{n}, \omega, t\right) > 1-r , \varPhi \left({\varpi }_{n}, {\theta }_{n}, \omega, t\right) < r, and \psi \left({\varpi }_{n}, {\theta }_{n}, \omega, t\right) < r for all n, m\in {n}_{\circ } and for all \omega \in \varXi. Hence \left({\varpi }_{n}\right) is a Cauchy sequence. Since (\varXi, \varPsi, \varPhi, \psi, *, ◊) is complete, {\varpi }_{n}\stackrel{{(\varPsi, \varPhi, \psi)}_{2}}{\to }\varpi for some \varpi \in \varXi. Therefore \varpi \in {\stackrel{-}{F}}_{n} = {F}_{n} for every n , and hence {\varpi \in \cap }_{n\in \mathbb{N}}{F}_{n} .

    Remark 3.4. The element {\varpi \in \cap }_{n\in \mathbb{N}}{F}_{n} is unique.

    Note that the topologies induced by the standard N2MS and the corresponding 2-metric are same. So, we have the following result.

    Corollary 3.1. A 2-metric space (\varXi, d) is complete if and only if every nested sequence {\left({F}_{n}\right)}_{n\in \mathbb{N}} of non-empty closed sets with diameter tending to zero have non-empty intersection.

    4.   Common fixed-point results in N2MS

    Lemma 4.1. If \left(\varXi, \varPsi, \varPhi, \psi, *, ◊\right) be a N2MS. Then, \varPsi \left(\varpi, \theta, \omega, t\right) is non-decreasing \varPhi \left(\varpi, \theta, \omega, t\right) non-increasing and \psi \left(\varpi, \theta, \omega, t\right) is non-increasing for all \varpi, \theta, \omega \in \varXi.

    Proof. Let s, t > 0 be any points such that t > s.t = s+\frac{t-s}{2}+\frac{t-s}{2} . Hence, we have

    \varPhi \left(\varpi , \theta , \omega , t\right) = \varPhi \left(\varpi , \theta , \omega , s+\frac{t-s}{2}+\frac{t-s}{2}\right)
    \le \varPhi \left(\varpi , \theta , \omega , s\right)◊\varPhi \left(\varpi , \omega , \omega , \frac{t-s}{2}\right)◊\varPhi \left(\omega , \theta , \omega , \frac{t-s}{2}\right) = \varPhi \left(\varpi , \theta , \omega , s\right)

    and

    \psi \left(\varpi , \theta , \omega , t\right) = \psi \left(\varpi , \theta , \omega , s+\frac{t-s}{2}+\frac{t-s}{2}\right)
    \le \psi \left(\varpi , \theta , \omega , s\right)◊\psi \left(\varpi , \omega , \omega , \frac{t-s}{2}\right)◊\psi \left(\omega , \theta , \omega , \frac{t-s}{2}\right) = \psi \left(\varpi , \theta , \omega , s\right).

    Similarly, \varPsi \left(\varpi, \theta, \omega, t\right) > \varPsi \left(\varpi, \theta, \omega, s\right).

    From Lemma 4.1, let \left(\varXi, \varPsi, \varPhi, \psi, *, ◊\right) be a N2MS with the following conditions:

    \underset{t\to \infty }{{\rm{lim}}}\varPsi \left(\varpi, \theta, \omega, t\right) = 1 , \underset{t\to \infty }{{\rm{l}}{\rm{i}}{\rm{m}}}\varPhi \left(\varpi, \theta, \omega, t\right) = 0 \text{ and } \underset{t\to \infty }{{\rm{l}}{\rm{i}}{\rm{m}}}\psi \left(\varpi, \theta, \omega, t\right) = 0.

    Lemma 4.2. Let \left(\varXi, \varPsi, \varPhi, \psi, *, ◊\right) be a N2MS. If there exists q\in \left({\rm{0, 1}}\right) such that \varPsi \left(\varpi, \theta, \omega, qt+0\right)\ge \varPsi (\varpi, \theta, \omega, t) , \varPhi \left(\varpi, \theta, \omega, qt+0\right)\le \varPhi (\varpi, \theta, \omega, t) and \psi \left(\varpi, \theta, \omega, qt+0\right)\le \psi \left(\varpi, \theta, \omega, t\right) for all \varpi, \theta, \omega \in \varXi with \omega \ne \varpi, \omega \ne \theta and t > 0 . Then, \varpi = \theta.

    Proof. Since

    \varPsi \left(\varpi , \theta , \omega , t\right)\ge \varPsi \left(\varpi , \theta , \omega , qt+0\right)\ge \varPsi \left(\varpi , \theta , \omega , t\right) ,
    \varPhi \left(\varpi , \theta , \omega , t\right)\le \varPhi \left(\varpi , \theta , \omega , qt+0\right)\le \varPhi (\varpi , \theta , \omega , t) ,

    and

    \psi \left(\varpi , \theta , \omega , t\right)\le \psi \left(\varpi , \theta , \omega , qt+0\right)\le \psi (\varpi , \theta , \omega , t)

    for all t > 0, \varPsi (\varpi, \theta, \omega, .) , \varPhi (\varpi, \theta, \omega, .) and \psi (\varpi, \theta, \omega) are constant. Since \underset{t\to \infty }{{\rm{lim}}}\varPsi \left(\varpi, \theta, \omega, t\right) = 1, \underset{t\to \infty }{{\rm{lim}}}\varPhi \left(\varpi, \theta, \omega, t\right) = 0 and \underset{t\to \infty }{{\rm{lim}}}\psi \left(\varpi, \theta, \omega, t\right) = 0. Then, \varPsi \left(\varpi, \theta, \omega, t\right) = 1 , \varPhi \left(\varpi, \theta, \omega, t\right) = 0 and \psi \left(\varpi, \theta, \omega, t\right) = 0 . Consequently, for all t > 0 . Hence \varpi = \theta because \omega \ne \varpi, \omega \ne \theta.

    Lemma 4.3. Let (\varXi, \varPsi, \varPhi, \psi, *, ◊) be a N2MS and let \underset{t\to \infty }{{\rm{lim}}}\varpi {\text{­}}_{n} = \varpi, \underset{t\to \infty }{{\rm{lim}}}{\theta }_{n} = \theta. Then, the following conditions are satisfied for all \tau \in \varXi and t\ge 0:

    (1)

    \underset{n\to \infty }{{\rm{lim}}}{\rm{inf}}\varPsi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)\ge \varPsi \left(\varpi , \theta , \tau , t\right), \underset{n\to \infty }{{\rm{lim}}}{\rm{sup}}\varPhi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)\le \varPhi \left(\varpi , \theta , \tau , t\right)

    and

    \underset{n\to \infty }{{\rm{lim}}}{\rm{sup}}\psi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)\le \psi (\varpi , \theta , \tau , t). (2)
    \varPsi \left(\varpi , \theta , \tau , t\right)\ge \underset{n\to \infty }{{\rm{lim}}}{\rm{S}}{\rm{u}}{\rm{p}}\varPsi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right), \varPhi \left(\varpi , \theta , \tau , t+0\right)\le \underset{n\to \infty }{{\rm{lim}}}{\rm{inf}}\varPhi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)

    and

    \psi \left(\varpi , \theta , \tau , t+0\right)\le \underset{n\to \infty }{{\rm{lim}}} \ \ {\rm{i}}{\rm{n}}{\rm{f}}\psi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right).

    Proof. For all \tau \in \varXi and t\ge 0, we have

    \varPsi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)\ge \varPsi \left({\varpi }_{n}, {\theta }_{n}, \varpi , {t}_{1}\right)*\varPsi \left({\varpi }_{n}, \varpi , \tau , {t}_{2}\right)*\varPsi \left(\varpi , {\theta }_{n}, \tau , t\right), {t}_{1}+{t}_{2} = 0
    \ge \varPsi \left({\varpi }_{n}, {\theta }_{n}, \varpi , {t}_{1}\right)*\varPsi \left({\varpi }_{n}, \varpi , \tau , {t}_{2}\right)*\varPsi \left(\varpi , {\theta }_{n}, \theta , {t}_{3}\right)
    *\varPsi \left(\varpi , \theta , \tau , {t}_{4}\right)*\varPsi \left(\theta , {\theta }_{n}, \tau , t\right), {t}_{3}+{t}_{4} = 0

    which implies \underset{n\to \infty }{{\rm{lim}}}{\rm{inf}}\varPsi \left({\varpi }_{n}, {\theta }_{n}, \tau, t\right)\ge 1*1*1*\varPsi \left(\varpi, \theta, \tau, t\right)*1 = \varPsi (\varpi, \theta, \tau, t), also

    \varPhi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)\le \varPhi \left({\varpi }_{n}, {\theta }_{n}, \varpi , {t}_{1}\right)◊\varPhi \left({\varpi }_{n}, \varpi , \tau , {t}_{2}\right)◊\varPhi \left(\varpi , {\theta }_{n}, \tau , t\right), {t}_{1}+{t}_{2} = 0
    \le \varPhi \left({\varpi }_{n}, {\theta }_{n}, \varpi , {t}_{1}\right)◊\varPhi \left({\varpi }_{n}, \varpi , \tau , {t}_{2}\right)◊\varPhi \left(\varpi , {\theta }_{n}, \theta , {t}_{3}\right)
    ◊\varPhi \left(\varpi , \theta , \tau , {t}_{4}\right)◊\varPhi \left(\theta , {\theta }_{n}, \tau , t\right), {t}_{3}+{t}_{4} = 0

    which implies

    \underset{n\to \infty }{{\rm{lim}}}{\rm{S}}{\rm{u}}{\rm{p}}\varPhi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)\le 0◊0◊0◊\varPhi \left(\varpi , \theta , \tau , t\right)◊0 = \varPhi (\varpi , \theta , \tau , t)

    and

    \psi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)\le \psi \left({\varpi }_{n}, {\theta }_{n}, \varpi , {t}_{1}\right)◊\psi \left({\varpi }_{n}, \varpi , \tau , {t}_{2}\right)◊\psi \left(\varpi , {\theta }_{n}, \tau , t\right), {t}_{1}+{t}_{2} = 0
    \le \psi \left({\varpi }_{n}, {\theta }_{n}, \varpi , {t}_{1}\right)◊\psi \left({\varpi }_{n}, \varpi , \tau , {t}_{2}\right)◊\psi \left(\varpi , {\theta }_{n}, \theta , {t}_{3}\right)
    ◊\psi \left(\varpi , \theta , \tau , {t}_{4}\right)◊\psi \left(\theta , {\theta }_{n}, \tau , t\right), {t}_{3}+{t}_{4} = 0

    which implies \underset{n\to \infty }{{\rm{lim}}}{\rm{S}}{\rm{u}}{\rm{p}}\psi \left({\varpi }_{n}, {\theta }_{n}, \tau, t\right)\le 0◊0◊0◊\psi \left(\varpi, \theta, \tau, t\right)◊0 = \psi (\varpi, \theta, \tau, t).

    (2) Let \epsilon > 0 be given. For all \tau \in \varpi and t > 0, we have

    \varPsi \left(\varpi , \theta , \tau , t+2\epsilon \right)\ge \varPsi \left(\varpi , \theta , {\varpi }_{n}, \frac{\epsilon }{2}\right)*\varPsi \left(\varpi , {\varpi }_{n}, \tau , \frac{\epsilon }{2}\right)*\varPsi ({\varpi }_{n}, \theta , \tau , t+\epsilon )
    \ge \varPsi \left(\varpi , \theta , {\varpi }_{n}, \frac{\epsilon }{2}\right)*\varPsi \left(\varpi , {\varpi }_{n}, \tau , \frac{\epsilon }{2}\right)*\varPsi \left({\varpi }_{n}, \theta , {\theta }_{n}, \frac{\epsilon }{2}\right)*\varPsi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)*\varPsi \left({\theta }_{n}, \theta , \tau , \frac{\epsilon }{2}\right).

    Consequently,

    \varPsi \left(\varpi , \theta , \tau , t+2\epsilon \right)\ge \underset{n\to \infty }{{\rm{lim}}}{\rm{sup}}\varPsi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right).

    Letting \epsilon \to 0, we have

    \varPsi \left(\varpi , \theta , \tau , t+0\right)\ge \underset{n\to \infty }{{\rm{lim}}}{\rm{sup}}\varPsi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right).

    Also, we have

    \varPhi \left(\varpi , \theta , \tau , t+2\epsilon \right)\le \varPhi \left(\varpi , \theta , {\varpi }_{n}, \frac{\epsilon }{2}\right)◊\varPhi \left(\varpi , {\varpi }_{n}, \tau , \frac{\epsilon }{2}\right)◊\varPhi ({\varpi }_{n}, \theta , \tau , t+\epsilon )
    \ge \varPhi \left(\varpi , \theta , {\varpi }_{n}, \frac{\epsilon }{2}\right)◊\varPhi \left(\varpi , {\varpi }_{n}, \tau , \frac{\epsilon }{2}\right)◊\varPhi \left({\varpi }_{n}, \theta , {\theta }_{n}, \frac{\epsilon }{2}\right)◊\varPhi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)
    ◊\varPhi \left({\theta }_{n}, \theta , \tau , \frac{\epsilon }{2}\right)◊\varPhi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)◊\varPhi \left({\theta }_{n}, \theta , \tau , \frac{\epsilon }{2}\right).

    Consequently,

    \varPhi \left(\varpi , \theta , \tau , t+2\epsilon \right)\le \underset{n\to \infty }{{\rm{lim}}}{\rm{inf}}\varPhi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right).

    Letting \epsilon \to 0, we have

    \varPhi \left(\varpi , \theta , \tau , t+0\right)\le \underset{n\to \infty }{{\rm{lim}}}{\rm{inf}}\varPhi ({\varpi }_{n}, {\theta }_{n}, \tau , t).

    and

    \psi \left(\varpi , \theta , \tau , t+2\epsilon \right)\le \psi \left(\varpi , \theta , {\varpi }_{n}, \frac{\epsilon }{2}\right)◊\psi \left(\varpi , {\varpi }_{n}, \tau , \frac{\epsilon }{2}\right)◊\psi ({\varpi }_{n}, \theta , \tau , t+\epsilon )
    \ge \psi \left(\varpi , \theta , {\varpi }_{n}, \frac{\epsilon }{2}\right)◊\psi \left(\varpi , {\varpi }_{n}, \tau , \frac{\epsilon }{2}\right)◊\psi \left({\varpi }_{n}, \theta , {\theta }_{n}, \frac{\epsilon }{2}\right)◊\psi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)
    ◊\psi \left({\theta }_{n}, \theta , \tau , \frac{\epsilon }{2}\right)◊\psi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right)◊\psi \left({\theta }_{n}, \theta , \tau , \frac{\epsilon }{2}\right).

    Consequently,

    \psi \left(\varpi , \theta , \tau , t+2\epsilon \right)\le \underset{n\to \infty }{{\rm{lim}}}{\rm{inf}}\psi \left({\varpi }_{n}, {\theta }_{n}, \tau , t\right).

    Letting \epsilon \to 0, we have

    \psi \left(\varpi , \theta , \tau , t+0\right)\le \underset{n\to \infty }{{\rm{lim}}}{\rm{inf}}\psi ({\varpi }_{n}, {\theta }_{n}, \tau , t).

    Lemma 4.4. Let \left(\varXi, \varPsi, \varPhi, \psi, *, ◊\right) be a N2MS and let \varOmega and B continuous self-mappings of \varXi \ \ {\rm{and}} \ \ [\varOmega, B] are compatible. Let {\varpi }_{n} be a sequence in X such that \varOmega {\varpi }_{n}\to \omega and B{\varpi }_{n}\to \omega. Then, \varOmega B{\varpi }_{n}\to B\omega.

    Proof. Since \varOmega, B are compatible maps, \varOmega B{\varpi }_{n}\to \varOmega \omega, B\varOmega {\varpi }_{n}\to B\omega and so, \varPsi \left(\varOmega B{\varpi }_{n}, \varOmega \omega, \tau, \frac{t}{3}\right)\to 1, \varPhi \left(B\varOmega {\varpi }_{n}, B\omega, \tau, \frac{t}{3}\right)\to 0 \ \ {\rm{and}} \ \ \psi \left(B\varOmega {\varpi }_{n}, B\omega, \tau, \frac{t}{3}\right)\to 0 for all \tau \in \varXi and t > 0.

    \varPsi \left(\varOmega B{\varpi }_{n}, B\omega , \tau , t\right)\ge \varPsi \left(\varOmega B{\varpi }_{n}, B\omega , B\varOmega {\varpi }_{n}, \frac{t}{3}\right)*\varPsi \left(\varOmega B{\varpi }_{n}, B\varOmega {\varpi }_{n}, \tau , \frac{t}{3}\right)*\varPsi \left(B\varOmega {\varpi }_{n}, B\omega , \tau , \frac{t}{3}\right)
    \ge \varPsi \left(B\varOmega {\varpi }_{n}, B\omega , \varOmega B{\varpi }_{n}, \frac{t}{3}\right)*\varPsi \left(B\varOmega {\varpi }_{n}, \varOmega B{\varpi }_{n}, \tau , \frac{t}{3}\right)*\varPsi \left(B\varOmega {\varpi }_{n}, B\omega , \tau , \frac{t}{3}\right)\to 1.

    Also, we have

    \varPhi \left(\varOmega B{\varpi }_{n}, B\omega , \tau , t\right)\le \varPhi \left(\varOmega B{\varpi }_{n}, B\omega , B\varOmega {\varpi }_{n}, \frac{t}{3}\right)◊\varPhi \left(\varOmega B{\varpi }_{n}, B\varOmega {\varpi }_{n}, \tau , \frac{t}{3}\right)◊\varPhi \left(B\varOmega {\varpi }_{n}, B\omega , \tau , \frac{t}{3}\right)
    \le \varPhi \left(B\varOmega {\varpi }_{n}, B\omega , \varOmega B{\varpi }_{n}, \frac{t}{3}\right)◊\varPhi \left(B\varOmega {\varpi }_{n}, \varOmega B{\varpi }_{n}, \tau , \frac{t}{3}\right)◊\varPhi \left(B\varOmega {\varpi }_{n}, B\omega , \tau , \frac{t}{3}\right)\to 0,

    for all \tau \in \varXi \ \ {\rm{and}} \ \ t > 0, and

    \psi \left(\varOmega B{\varpi }_{n}, B\omega , \tau , t\right)\le \psi \left(\varOmega B{\varpi }_{n}, B\omega , B\varOmega {\varpi }_{n}, \frac{t}{3}\right)◊\psi \left(\varOmega B{\varpi }_{n}, B\varOmega {\varpi }_{n}, \tau , \frac{t}{3}\right)◊\psi \left(B\varOmega {\varpi }_{n}, B\omega , \tau , \frac{t}{3}\right)
    \le \psi \left(B\varOmega {\varpi }_{n}, B\omega , \varOmega B{\varpi }_{n}, \frac{t}{3}\right)◊\psi \left(B\varOmega {\varpi }_{n}, \varOmega B{\varpi }_{n}, \tau , \frac{t}{3}\right)◊\psi \left(B\varOmega {\varpi }_{n}, B\omega , \tau , \frac{t}{3}\right)\to 0,

    for all \tau \in \varXi \ \ {\rm{and}} \ \ t > 0. Hence, \varOmega B{\varpi }_{n}\to B\omega.

    Theorem 4.1. Let (\varXi, \varPsi, \varPhi, \psi, *, ◊) be a complete N2MS with continuous t-norm * and continuous t-conorm ◊ . Let S and T be continuous self-mapping of \varXi . Then, S and T have a unique common fixed point in \varXi if and only if there exist two self-mappings \varOmega, B of \varXi satisfying

    (1) \varOmega \varXi \subset T\varXi , B\varXi \subset S\varXi ,

    (2) {\rm{T}}{\rm{h}}{\rm{e}} \ \ {\rm{p}}{\rm{a}}{\rm{i}}{\rm{r}}\left\{\varOmega, S\right\} \ \ {\rm{and}} \ \ \left\{B, T\right\} are compatible

    (3) There exists q\in \left({\rm{0, 1}}\right) such that for every \varpi, \theta, \tau \in \varXi and t > 0,

    \varPsi \left(\varOmega \varpi , B\theta , \tau , qt\right)\ge {\rm{min}}\left\{\varPsi \left(S\varpi , T\varpi , \tau , t\right), \varPsi \left(\varOmega \varpi , S\varpi , \tau , t\right), \varPsi \left(B\theta , T\theta , \tau , t\right), \varPsi \left(\varOmega \varpi , B\theta , \tau , qt\right)\right\},
    \varPhi \left(\varOmega \varpi , B\theta , \tau , qt\right)\le {\rm{max}}\left\{\varPhi \left(S\varpi , T\varpi , \tau , t\right), \varPhi \left(\varOmega \varpi , S\varpi , \tau , t\right), \varPhi \left(B\theta , T\theta , \tau , t\right), \varPhi \left(\varOmega \varpi , B\theta , \tau , qt\right)\right\},
    \psi \left(\varOmega \varpi , B\theta , \tau , qt\right)\le {\rm{max}}\left\{\psi \left(S\varpi , T\varpi , \tau , t\right), \psi \left(\varOmega \varpi , S\varpi , \tau , t\right), \psi \left(B\theta , T\theta , \tau , t\right), \psi \left(\varOmega \varpi , B\theta , \tau , qt\right)\right\}.

    Then, \varOmega, B, S \ \ {\rm{and}} \ \ T have a unique common fixed point in \varXi.

    Proof. Suppose S \ \ {\rm{and}} \ \ T have a (unique) common fixed point say \omega \in \varXi. Define \varOmega :\varXi \to \varXi be \varOmega \varpi = \omega for all \varpi \in \varXi, \ \ {\rm{and}} \ \ B:\varXi \to \varXi be B\varpi = \omega for all \varpi \in \varXi. Then, one can see that (1)–(3) are satisfied.

    Conversely, assume that there exist two self-mapping \varOmega, B{\rm{o}}{\rm{f}}\varpi satisfying condition (1)–(3). From condition (1) we can construct two sequences {\varpi }_{n} and {\theta }_{n} of \varXi such that {\theta }_{2n-1} = T{\varpi }_{2n-1} and {\theta }_{2n-1} = S{\varpi }_{2n} = B{\varpi }_{2n-1} for n = {\rm{1, 2}}, 3, \dots, putting \varpi = {\varpi }_{2n} and \varpi = {\varpi }_{2n+1} in condition (3), we have that for all \tau \in \varXi and t > 0,

    \varPsi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+2}, \tau , qt\right) = \varPsi \left(\varOmega {\varpi }_{2n}, B{\varpi }_{2n+1}, \tau , qt\right)
    \ge {\rm{min}}\left\{\varPsi \right(S{\varpi }_{2n}, T{\varpi }_{2n+1}, \tau , t), \varPsi (\varOmega {\varpi }_{2n}, S{\varpi }_{2n}, \tau , t),
    \varPsi \left(B{\varpi }_{2n+1}, T{\varpi }_{2n+1}, \tau , t\right), \varPsi (\varOmega {\varpi }_{2n}, T{\varpi }_{2n+1}, \tau , t)\}
    \ge {\rm{min}}\left\{\varPsi \right(\theta {\varpi }_{2n}, \theta {\varpi }_{2n+1}, \tau , qt), \varPsi (\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+1}, \tau , qt\left)\right\},
    \varPhi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+2}, \tau , qt\right) = \varPhi \left(\varOmega {\varpi }_{2n}, B{\varpi }_{2n+1}, \tau , qt\right)
    \le {\rm{max}}\left\{\varPhi \right(S{\varpi }_{2n}, T{\varpi }_{2n+1}, \tau , t), \varPhi (\varOmega {\varpi }_{2n}, S{\varpi }_{2n}, \tau , t),
    \varPhi \left(B{\varpi }_{2n+1}, T{\varpi }_{2n+1}, \tau , t\right), \varPhi (\varOmega {\varpi }_{2n}, T{\varpi }_{2n+1}, \tau , t)\}
    \le {\rm{max}}\left\{\varPhi \right(\theta {\varpi }_{2n}, \theta {\varpi }_{2n+1}, \tau , qt), \varPhi (\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+1}, \tau , qt\left)\right\},

    and

    \psi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+2}, \tau , qt\right) = \psi \left(\varOmega {\varpi }_{2n}, B{\varpi }_{2n+1}, \tau , qt\right)
    \le \mathit{max}\left\{\psi \right(S{\varpi }_{2n}, T{\varpi }_{2n+1}, \tau , t), \psi (\varOmega {\varpi }_{2n}, S{\varpi }_{2n}, \tau , t),
    \psi \left(B{\varpi }_{2n+1}, T{\varpi }_{2n+1}, \tau , t\right), \psi (\varOmega {\varpi }_{2n}, T{\varpi }_{2n+1}, \tau , t)\}
    \le \mathit{max}\left\{\psi \right(\theta {\varpi }_{2n}, \theta {\varpi }_{2n+1}, \tau , qt), \psi (\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+1}, \tau , qt\left)\right\}.

    Which implies that

    \varPsi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+2}, \tau , qt\right)\ge \varPsi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+1}, \tau , qt\right)
    \varPhi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+2}, \tau , qt\right)\le \varPhi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+1}, \tau , qt\right),

    and

    \psi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+2}, \tau , qt\right)\le \psi \left(\theta {\varpi }_{2n+1}, \theta {\varpi }_{2n+1}, \tau , qt\right).

    By using Lemma 4.1 and letting \varpi = {\varpi }_{2n+1} and \theta = {\varpi }_{2n+1} in condition (3), we have that

    \varPsi \left({\theta }_{2n+2}, {\theta }_{2n+3}, \tau , qt\right)\ge \varPsi \left({\theta }_{2n+1}, {\theta }_{2n+1}, \tau , t\right)
    \varPhi \left({\theta }_{2n+2}, {\theta }_{2n+3}, \tau , qt\right)\ge \varPhi \left({\theta }_{2n+1}, {\theta }_{2n+1}, \tau , t\right),

    and

    \psi \left({\theta }_{2n+2}, {\theta }_{2n+3}, \tau , qt\right)\ge \psi \left({\theta }_{2n+1}, {\theta }_{2n+1}, \tau , t\right)

    for all \tau \in \varXi and t > 0.

    In general, we obtain that for all \tau \in \varXi and t > 0 and n = {\rm{1, 2}}, 3, \cdots , we have

    \varPsi \left({\theta }_{n}, {\theta }_{n+1}, \tau , qt\right)\ge \varPsi \left({\theta }_{n-1}, {\theta }_{n}, \tau , t\right) ,
    \varPhi \left({\theta }_{n}, {\theta }_{n+1}, \tau , qt\right)\le \varPhi \left({\theta }_{n-1}, {\theta }_{n}, \tau , t\right),

    and

    \psi \left({\theta }_{n}, {\theta }_{n+1}, \tau , qt\right)\le \psi \left({\theta }_{n-1}, {\theta }_{n}, \tau , t\right).

    Thus, for all \tau \in \varXi and t > 0 and n = {\rm{1, 2}}, 3, \cdots , we have

    \varPsi \left({\theta }_{n}, {\theta }_{n+1}, \tau , t\right)\ge \varPsi \left({\theta }_{0}, {\theta }_{1}, \tau , \frac{t}{{q}^{n}}\right). (4.1)
    \varPhi \left({\theta }_{n}, {\theta }_{n+1}, \tau , t\right)\le \varPhi \left({\theta }_{0}, {\theta }_{1}, \tau , \frac{t}{{q}^{n}}\right) . (4.2)
    \psi \left({\theta }_{n}, {\theta }_{n+1}, \tau , t\right)\le \psi \left({\theta }_{0}, {\theta }_{1}, \tau , \frac{t}{{q}^{n}}\right). (4.3)

    We now show that \left\{{\theta }_{n}\right\} is a Cauchy sequence in \varXi , let m > n. Then, for all \tau \in \varXi and t > \varpi, we have

    \varPsi \left({\theta }_{m}, {\theta }_{n}, \tau , t\right)\ge \varPsi \left({\theta }_{m}, {\theta }_{n}, {\theta }_{n+1}, \frac{t}{3}\right)*\varPsi \left({\theta }_{n+1}, {\theta }_{n}, \tau , \frac{t}{3}\right)*\varPsi \left({\theta }_{m}, {\theta }_{n+1}, \tau , \frac{t}{3}\right)
    \ge \varPsi \left({\theta }_{m}, {\theta }_{n}, {\theta }_{n+1}, \frac{t}{3}\right)*\varPsi \left({\theta }_{n+1}, {\theta }_{n}, \tau , \frac{t}{3}\right)*\varPsi \left({\theta }_{m}, {\theta }_{n+1}, {\theta }_{n+2}, \frac{t}{3}\right)
    *\varPsi \left({\theta }_{n+2}, {\theta }_{n+1}, \tau , \frac{t}{{3}^{2}}\right)*\varPsi \left({\theta }_{m}, {\theta }_{n+2}, \tau , \frac{t}{{3}^{2}}\right)*\cdots *\varPsi \left({\theta }_{m}, {\theta }_{m-1, }\tau , \frac{t}{{3}^{m-n}}\right),
    \varPhi \left({\theta }_{m}, {\theta }_{n}, \tau , t\right)\le \varPhi \left({\theta }_{m}, {\theta }_{n}, {\theta }_{n+1}, \frac{t}{3}\right)◊\varPhi \left({\theta }_{n+1}, {\theta }_{n}, \tau , \frac{t}{3}\right)◊\varPhi \left({\theta }_{m}, {\theta }_{n+1}, \tau , \frac{t}{3}\right)
    \le \varPhi \left({\theta }_{m}, {\theta }_{n}, {\theta }_{n+1}, \frac{t}{3}\right)◊\varPhi \left({\theta }_{n+1}, {\theta }_{n}, \tau , \frac{t}{3}\right)◊\varPhi \left({\theta }_{m}, {\theta }_{n+1}, {\theta }_{n+2}, \frac{t}{3}\right)
    ◊\varPhi \left({\theta }_{n+2}, {\theta }_{n+1}, \tau , \frac{t}{{3}^{2}}\right)◊\varPhi \left({\theta }_{m}, {\theta }_{n+2}, \tau , \frac{t}{{3}^{2}}\right)◊\cdots ◊\varPhi \left({\theta }_{m}, {\theta }_{m-1, }\tau , \frac{t}{{3}^{m-n}}\right),

    and

    \psi \left({\theta }_{m}, {\theta }_{n}, \tau , t\right)\le \psi \left({\theta }_{m}, {\theta }_{n}, {\theta }_{n+1}, \frac{t}{3}\right)◊\psi \left({\theta }_{n+1}, {\theta }_{n}, \tau , \frac{t}{3}\right)◊\psi \left({\theta }_{m}, {\theta }_{n+1}, \tau , \frac{t}{3}\right)
    \le \psi \left({\theta }_{m}, {\theta }_{n}, {\theta }_{n+1}, \frac{t}{3}\right)◊\psi \left({\theta }_{n+1}, {\theta }_{n}, \tau , \frac{t}{3}\right)◊\psi \left({\theta }_{m}, {\theta }_{n+1}, {\theta }_{n+2}, \frac{t}{3}\right)
    ◊\psi \left({\theta }_{n+2}, {\theta }_{n+1}, \tau , \frac{t}{{3}^{2}}\right)◊\psi \left({\theta }_{m}, {\theta }_{n+2}, \tau , \frac{t}{{3}^{2}}\right)◊\cdots ◊\psi \left({\theta }_{m}, {\theta }_{m-1, }\tau , \frac{t}{{3}^{m-n}}\right).

    Letting m, n\to \infty, we have

    \underset{n\to \infty }{\mathit{lim}}\varPsi \left({\theta }_{m}, {\theta }_{n}, \tau , t\right) = 1, \underset{n\to \infty }{\mathit{lim}}\varPhi \left({\theta }_{m}, {\theta }_{n}, \tau , t\right) = 0, and\underset{n\to \infty }{\mathit{lim}}\psi \left({\theta }_{m}, {\theta }_{n}, \tau , t\right) = 0.

    Thus, \left\{{\theta }_{n}\right\} is a Cauchy sequence in \varXi. It follows from completeness of \varXi that there exists \omega \in \varXi such that

    \underset{n\to \infty }{{\rm{lim}}}{\theta }_{n} = \omega , \underset{n\to \infty }{{\rm{lim}}}{\theta }_{2n-1} = \underset{n\to \infty }{{\rm{lim}}}{\varpi }_{2n-1} = \underset{n\to \infty }{{\rm{lim}}}{\varOmega \varpi }_{2n-2} = \omega ,

    and

    \underset{n\to \infty }{{\rm{lim}}}{\theta }_{2n} = \underset{n\to \infty }{{\rm{lim}}}S{\varpi }_{2n} = \underset{n\to \infty }{{\rm{lim}}}B{\varpi }_{2n-1} = \omega .

    From Lemma 4.4, we have

    \varOmega S{\varpi }_{2n+1} = S\omega \text{ and } BT{\varpi }_{2n+1} = T\omega. (4.4)

    Meanwhile, for all \tau \in \varXi with \tau \ne S\omega \ \ {\rm{and}} \ \ \tau \ne T\omega and t > 0 , we have

    \varPsi \left(\varOmega S{\varpi }_{2n+1}, BT{\varpi }_{2n+1}, \tau , qt\right)\ge \mathit{min}\left\{\varPsi \right(SS{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t),
    \varPsi \left(\varOmega S{\varpi }_{2n+1}, SS{\varpi }_{2n+1}, \tau , t\right), \varPsi \left(BT{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, q\tau , t\right), \varPsi (\varOmega S{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t)\},
    \varPhi \left(\varOmega S{\varpi }_{2n+1}, BT{\varpi }_{2n+1}, \tau , qt\right)\le \mathit{max}\left\{\varPhi \right(SS{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t),
    \varPhi \left(\varOmega S{\varpi }_{2n+1}, SS{\varpi }_{2n+1}, \tau , t\right), \varPhi \left(BT{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, q\tau , t\right), \varPhi (\varOmega S{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t)\},

    and

    \psi \left(\varOmega S{\varpi }_{2n+1}, BT{\varpi }_{2n+1}, \tau , qt\right)\le \mathit{max}\left\{\psi \right(SS{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t),
    \psi \left(\varOmega S{\varpi }_{2n+1}, SS{\varpi }_{2n+1}, \tau , t\right), \psi \left(BT{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, q\tau , t\right), \psi (\varOmega S{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t)\}.

    Taking limit as n\to \infty and using (4.4), we have for all \tau \in \varXi with \tau \ne S\omega and \tau \ne T\omega \ \ {\rm{and}} \ \ t > 0.

    \varPsi \left(S\omega , T\omega , \tau , qt+0\right)\ge \mathit{min}\{\varPsi \left(S\omega , T\omega , \tau , t\right), \varPsi (S\omega , S\omega , \tau , t),
    \varPsi \left(T\omega , T\omega , \tau , t\right), \varPsi (S\omega , S\omega , \tau , t)\} = \varPsi \left(S\omega , T\omega , \tau , t\right),
    \varPhi \left(S\omega , T\omega , \tau , qt+0\right)\le \mathit{max}\{\varPhi \left(S\omega , T\omega , \tau , t\right), \varPhi (S\omega , S\omega , \tau , t),
    \varPhi \left(T\omega , T\omega , \tau , t\right), \varPhi (S\omega , S\omega , \tau , t)\} = \varPhi \left(S\omega , T\omega , \tau , t\right),

    and

    \psi \left(S\omega , T\omega , \tau , qt+0\right)\le \mathit{max}\{\psi \left(S\omega , T\omega , \tau , t\right), \psi (S\omega , S\omega , \tau , t),
    \psi \left(T\omega , T\omega , \tau , t\right), \psi (S\omega , S\omega , \tau , t)\} = \psi \left(S\omega , T\omega , \tau , t\right).

    By Lemma 4.2, we have

    S\omega = T\omega . (4.5)

    From condition (3), we get for all \tau \in \varXi with \tau \ne \varOmega \omega, \tau \ne T\omega and t > 0

    \varPsi \left(\varOmega \omega , BT{\varpi }_{2n+1}, \tau , qt\right)\ge \mathit{min}\left\{\varPsi \right(S\omega , TT{\varpi }_{2n+1}, \tau , t), \varPsi ({\varOmega }_{\omega }, {S}_{\omega }, \tau , t),
    \varPsi \left(BT{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t\right), \varPsi (\varOmega \omega , TT{\varpi }_{2n+1}, \tau , t)\},
    \varPhi \left(\varOmega \omega , BT{\varpi }_{2n+1}, \tau , qt\right)\le \mathit{max}\left\{\varPhi \right(S\omega , TT{\varpi }_{2n+1}, \tau , t), \varPhi ({\varOmega }_{\omega }, {S}_{\omega }, \tau , t),
    \varPhi \left(BT{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t\right), \varPhi (\varOmega \omega , TT{\varpi }_{2n+1}, \tau , t)\},

    and

    \psi \left(\varOmega \omega , BT{\varpi }_{2n+1}, \tau , qt\right)\le \mathit{max}\left\{\psi \right(S\omega , TT{\varpi }_{2n+1}, \tau , t), \psi ({\varOmega }_{\omega }, {S}_{\omega }, \tau , t),
    \psi \left(BT{\varpi }_{2n+1}, TT{\varpi }_{2n+1}, \tau , t\right), \psi (\varOmega \omega , TT{\varpi }_{2n+1}, \tau , t)\}.

    Taking limit as n\to \infty , using condition (3), and Lemma 4.3, we have for all \tau \in \varXi,

    \varPsi \left(\varOmega \omega , T\omega , \tau , qt+0\right)\ge \mathit{min}\{\varPsi \left(S\omega , T\omega , \tau , t\right), \varPsi \left(\varOmega \omega , S\omega , \tau , t\right),
    \varPsi \left(T\omega , T\omega , \tau , t\right), \varPsi (\varOmega \omega , T\omega , \tau , t)\} = \varPsi \left(\varOmega \omega , T\omega , \tau , t\right),
    \varPhi \left(\varOmega \omega , T\omega , \tau , qt+0\right)\le \mathit{max}\{\varPhi \left(S\omega , T\omega , \tau , t\right), \varPhi \left(\varOmega \omega , S\omega , \tau , t\right),
    \varPhi \left(T\omega , T\omega , \tau , t\right), \varPhi (\varOmega \omega , T\omega , \tau , t)\} = \varPhi \left(\varOmega \omega , T\omega , \tau , t\right),

    and

    \psi \left(\varOmega \omega , T\omega , \tau , qt+0\right)\le \mathit{max}\{\psi \left(S\omega , T\omega , \tau , t\right), \psi \left(\varOmega \omega , S\omega , \tau , t\right),
    \psi \left(T\omega , T\omega , \tau , t\right), \psi (\varOmega \omega , T\omega , \tau , t)\} = \psi \left(\varOmega \omega , T\omega , \tau , t\right).

    By Lemma 4.2, we have

    \varOmega \omega = T\omega . (4.6)

    For all \tau \in \varXi with \tau \ne \varOmega \omega \ \ {\rm{and}} \ \ \tau \ne B\omega, \ \ {\rm{and}} \ \ t > 0, we have

    \varPsi \left(\varOmega \omega , B\omega , \tau , qt\right)\ge \mathit{min}\{\varPsi \left(S\omega , T\omega , \tau , t\right), \varPsi \left(\varOmega \omega , S\omega , \tau , t\right),
    \varPsi \left(B\omega , T\omega , \tau , t\right), \varPsi (\varOmega \omega , T\omega , \tau , t)\},
    \ge \mathit{min}\left\{\varPsi \right(T\omega , T\omega , \tau , t), \varPsi \left(\varOmega \omega , T\omega , \tau , t\right), \varPsi \left(B\omega , \varOmega \omega , \tau , t\right), \varPsi (T\omega , T\omega , \tau , t)\}
    = \varPsi \left(\varOmega \omega , B\omega , \tau , t\right),
    \varPhi \left(\varOmega \omega , B\omega , \tau , qt\right)\le \mathit{max}\{\varPhi \left(S\omega , T\omega , \tau , t\right), \varPhi \left(\varOmega \omega , S\omega , \tau , t\right),
    \varPhi \left(B\omega , T\omega , \tau , t\right), \varPhi (\varOmega \omega , T\omega , \tau , t)\},
    \le \mathit{max}\left\{\varPhi \right(T\omega , T\omega , \tau , t), \varPhi \left(\varOmega \omega , T\omega , \tau , t\right), \varPhi \left(B\omega , \varOmega \omega , \tau , t\right), \varPhi (T\omega , T\omega , \tau , t)\}
    = \varPhi \left(\varOmega \omega , B\omega , \tau , t\right),

    and

    \psi \left(\varOmega \omega , B\omega , \tau , qt\right)\le \mathit{max}\{\psi \left(S\omega , T\omega , \tau , t\right), \psi \left(\varOmega \omega , S\omega , \tau , t\right),
    \psi \left(B\omega , T\omega , \tau , t\right), \psi (\varOmega \omega , T\omega , \tau , t)\}
    \le \mathit{max}\left\{\psi \right(T\omega , T\omega , \tau , t), \psi \left(\varOmega \omega , T\omega , \tau , t\right), \psi \left(B\omega , \varOmega \omega , \tau , t\right), \psi (T\omega , T\omega , \tau , t)\}
    = \psi \left(\varOmega \omega , B\omega , \tau , t\right).

    By Lemma 4.2, \varOmega \omega = B\omega and (4.7), it follows that \varOmega \omega = B\omega = S\omega = T\omega. For all \tau \in \varXi with \tau \ne B\omega and \tau \ne \omega, and t > 0

    \varPsi \left(\varOmega {\varpi }_{2n}, B\omega , \tau , qt\right)\ge \mathit{min}\{\varPsi \left(S{\varpi }_{2n}, T\omega , \tau , t\right), \varPsi (\varOmega {\varpi }_{2n}, S{\varpi }_{2n}, \tau , t)
    \varPsi \left(B\omega , T\omega , \tau , t\right), \varPsi (\varOmega {\varpi }_{2n}, {T}_{\omega }, \tau , t)\},
    \varPhi \left(\varOmega {\varpi }_{2n}, B\omega , \tau , qt\right)\le \mathit{max}\{\varPhi \left(S{\varpi }_{2n}, T\omega , \tau , t\right), \varPhi (\varOmega {\varpi }_{2n}, S{\varpi }_{2n}, \tau , t)
    \varPhi \left(B\omega , T\omega , \tau , t\right), \varPhi (\varOmega {\varpi }_{2n}, {T}_{\omega }, \tau , t)\},

    and

    \psi \left(\varOmega {\varpi }_{2n}, B\omega , \tau , qt\right)\le \mathit{max}\{\psi \left(S{\varpi }_{2n}, T\omega , \tau , t\right), \psi (\varOmega {\varpi }_{2n}, S{\varpi }_{2n}, \tau , t)
    \psi \left(B\omega , T\omega , \tau , t\right), \psi (\varOmega {\varpi }_{2n}, {T}_{\omega }, \tau , t)\}.

    Taking limit as n\to \infty and using (4.3) and Lemma 4.3, we have for all \tau \in \varXi we \tau \ne B\omega, \tau \ne \omega and t > 0

    \varPsi \left(\omega , B\omega , \tau , qt+0\right)\ge \mathit{min}\left\{\varPsi \right(\omega , T\omega , \tau , t), \varPsi \left(\omega , \omega , \omega , t\right), \varPsi \left(B\omega , B\omega , \tau , t\right), \varPsi (\omega , T\omega , \tau , t)\}
    \ge \varPsi \left(\omega , T\omega , \tau , t\right)\ge \varPsi \left(\omega , B\omega , \tau , t\right),
    \varPhi \left(\omega , B\omega , \tau , qt+0\right)\le \mathit{max}\left\{\varPhi \right(\omega , T\omega , \tau , t), \varPhi \left(\omega , \omega , \omega , t\right), \varPhi \left(B\omega , B\omega , \tau , t\right), \varPhi (\omega , T\omega , \tau , t)\}
    \le \varPhi \left(\omega , T\omega , \tau , t\right)\le \varPhi \left(\omega , B\omega , \tau , t\right),

    and

    \psi \left(\omega , B\omega , \tau , qt+0\right)\le \mathit{max}\left\{\psi \right(\omega , T\omega , \tau , t), \psi \left(\omega , \omega , \omega , t\right), \psi \left(B\omega , B\omega , \tau , t\right), \psi (\omega , T\omega , \tau , t)\}
    \le \psi \left(\omega , T\omega , \tau , t\right)\le \psi \left(\omega , B\omega , \tau , t\right).

    So, we have

    \varPsi \left(\omega , B\omega , \tau , qt\right)\ge \varPsi (\omega , B\omega , \tau , t) , \varPhi \left(\omega , B\omega , \tau , qt\right)\le \varPhi (\omega , B\omega , \tau , t)

    and

    {\rm{\Phi }}\left({\rm{ \mathsf{ ω} }}, {\rm{B}}{\rm{ \mathsf{ ω} }}, {\rm{ \mathsf{ τ} }}, {\rm{q}}{\rm{t}}\right)\le {\rm{\Phi }}\left({\rm{ \mathsf{ ω} }}, {\rm{B}}{\rm{ \mathsf{ ω} }}, {\rm{ \mathsf{ τ} }}, {\rm{t}}\right),

    here B\omega = \omega . Thus \omega = \varOmega \omega = B\omega = S\omega = T\omega \ \ {\rm{and}} \ \ {\rm{s}}{\rm{o}}, \omega is a common fixed point of \varOmega, B, C \ \ {\rm{and}} \ \ T. For uniqueness, let w be another common fixed point of \varOmega, B, S, T for all \tau \in \varXi with \tau \ne \omega, \tau \ne w \ \ {\rm{and}} \ \ t > 0 , we have

    \varPsi \left(\omega , w, \tau , qt\right) = \varPsi \left(\varOmega \omega , Bw, \tau , qt\right)
    \ge \mathit{min}\{\varPsi \left(S\omega , Tw, \tau , t\right), \varPsi \left(\varOmega \omega , S\omega , \tau , t\right), \varPsi \left(Bw, Tw, \tau , t\right), \varPsi (\varOmega \omega , Tw, \tau , t)\}
    \ge \mathit{min}\{\varPsi \left(\omega , w, \tau , t\right), \varPsi \left(\omega , w, \tau , t\right), \varPsi \left(w, w, \tau , t\right), \varPsi \left(\omega , w, \tau , t\right)\}\ge \varPsi \left(\omega , w, \tau , t\right),
    \varPhi \left(\omega , w, \tau , qt\right) = \varPhi \left(\varOmega \omega , Bw, \tau , qt\right)
    \le \mathit{max}\{\varPhi \left(S\omega , Tw, \tau , t\right), \varPhi \left(\varOmega \omega , S\omega , \tau , t\right), \varPhi \left(Bw, Tw, \tau , t\right), \varPhi (\varOmega \omega , Tw, \tau , t)\}
    \le \mathit{max}\{\varPhi \left(\omega , w, \tau , t\right), \varPhi \left(\omega , w, \tau , t\right), \varPhi \left(w, w, \tau , t\right), \varPhi \left(\omega , w, \tau , t\right)\}\le \varPhi \left(\omega , w, \tau , t\right),

    and

    \psi \left(\omega , w, \tau , qt\right) = \psi \left(\varOmega \omega , Bw, \tau , qt\right)
    \le \mathit{max}\{\psi \left(S\omega , Tw, \tau , t\right), \psi \left(\varOmega \omega , S\omega , \tau , t\right), \psi \left(Bw, Tw, \tau , t\right), \psi (\varOmega \omega , Tw, \tau , t)\}
    \le \mathit{max}\{\psi \left(\omega , w, \tau , t\right), \psi \left(\omega , w, \tau , t\right), \psi \left(w, w, \tau , t\right), \psi \left(\omega , w, \tau , t\right)\}\le \psi \left(\omega , w, \tau , t\right).

    Which implies that

    \varPsi \left(\omega , w, \tau , qt\right)\ge \varPsi \left(\omega , w, \tau , t\right), \varPhi \left(\omega , w, \tau , qt\right)\le \varPhi \left(\omega , w, \tau , t\right),

    and

    \psi \left(\omega , w, \tau , qt\right)\le \psi \left(\omega , w, \tau , t\right),

    hence \omega = w. This completes the proof.

    5.   Conclusions

    The N2MS idea, which is an extension of the NMS, was investigated in this article because it offers a greater context for dealing with the ambiguity and uncertainty in natural problems that arises in many fields of research and engineering. In this new setting, we constructed the Baire's and Cantor's Theorems, which could be very helpful tools in the advancement of fuzzy set theory. We derived the common fixed-point theorem with respect to N2MS. This work can easily be extending in the context of neutrosophic b-2-metric spaces, neutrosophic controlled 2-metric spaces and neutrosophic partial 2-metric spaces.

    Conflict of interest

    There are no competing interests declared by the authors.



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Article outline
Abstract
   Introduction
   Preliminaries
   Baire's and Cantor's intersection theorems
   Common fixed-point results in N2MS
   Conclusions
Conflict of interest
References

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