# Some New Aspects in the Intuitionistic Fuzzy and Neutrosophic Fixed Point Theory

Authors: Aftab Hussain; Hamed Al Sulami; Umar Ishtiaq
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this manuscript, we use the concepts of continuoust-norms and continuous t-conorms to introduce some definitions, in which intuitionistic fuzzy rectangular metric spaces, intuitionistic fuzzy rectangular metric-like spaces, intuitionistic fuzzy rectangular b-metric spaces, intuitionistic fuzzy rectangular b-metric-like spaces, neutrosophic rectangular metric spaces, neutrosophic rectangular metric-like spaces, neutrosophic rectangular b-metric spaces, and neutrosophic rectangular b-metric-like spaces are included. Continuous t-norms and continuous t-conorms are used to generalize the probability distribution of triangular inequalities in metric space axioms. Nontrivial examples, some fixed point results, and an application to the integral equation are imparted in this manuscript.


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## Body

## 1. Introduction

Fuzzy set (FS) presented by Zadeh [1] is a useful tool for those situations in which the data are imprecise and the idea of degree of membership is involved in FS theory. Intuitionistic fuzzy sets (IFSs) introduced by Atanassov [2] are the generalization of the FS, in which degrees of membership and nonmembership are involved. Smarandache [3] presented the idea of neutrosophic sets (NSs) that are the generalization of the IFS, in which degrees of membership, nonmembership, and uncertainty are involved.By combining the concepts of FS and metric spaces, fuzzy metric spaces (FMSs) were presented by Kramosil and Michalek [4]. Kaleva and Seikkala [5] coined FMS in which they defined a distance between two points to be a nonnegative fuzzy number, and Garbiec [6] presented the fuzzy interpretation of the Banach contraction principle in the FMS. Park [7] presented the intuitionistic fuzzy metric space (IFMS), in which he used George and Veeramani’s [8] approach of applying continuous t-norm (CTN) and continuous t-conorm (CTCN) to the FMS. Kirişci and Şimşek [9] presented the notion of neutrosophic metric space (NMS), in which they used the idea of NS and probabilistic metric spaces. FMS deals with membership functions, and IFMS deals with membership and nonmembership functions. NMS is a generalization of the IFMS that deals with membership, nonmembership, and inconsistent functions. Altun et al. [10] and Aslantas et al. [11] proved some interesting results for cyclic p-contractions and KW-type nonlinear contractions. Al-Omeri et al. [12, 13] proved several neutrosophic fixed point results and generalized theorems in the sense of neutrosophic cone metric spaces.Javed et al. [14] presented the idea of fuzzy b-metric-like spaces (FBMLSs) and proved several fixed point results. Mehmood et al. [15] presented the concept of fuzzy rectangular b-metric spaces (FRBMSs) and proved the Banach contraction principle in the sense of FRBMS. For some necessary definitions and related fixed point results, see [16–19].In this manuscript, we generalized the concepts used in [14, 15]. The main objectives of this manuscript are as follows:(i)
To present different notions in the intuitionistic fuzzy and neutrosophic fixed point theory(ii)
To prove certain fixed point theorems(iii)
To enhance the existing literature of the FMS and fuzzy fixed point theoryThis study is organized with some basic notions of FRBMS, FBMLS, IFMS, and NMS. The notions of intuitionistic fuzzy rectangular metric spaces (IFRMSs), intuitionistic fuzzy rectangular metric-like spaces (IFRMLSs), intuitionistic fuzzy rectangularb-metric spaces (IFRBMSs), intuitionistic fuzzy rectangular b-metric-like spaces (IFRBMLSs), neutrosophic rectangular metric spaces (NRMSs), neutrosophic rectangular metric-like spaces (NRMLSs), neutrosophic rectangular b-metric spaces (NRBMSs), and neutrosophic rectangular b-metric-like spaces (NRBMLSs) are discussed in detail, and several fixed point results, nontrivial examples, and an application to the integral equation are imparted. At the end, conclusion is given for the examined results.


## 2. Preliminaries

In this section, some basic definitions are imparted that are helpful to understand the main section.Definition 1 (see [7]).
A binary operation∗: [0, 1] × [0, 1] ⟶ [0, 1] is called a CTN if it meets the following assertions:C1.ζ∗b=b∗ζ,∀ζ,b∈0,1C2.∗ is continuousC3.ζ∗1=ζ,∀ζ∈0,1C4.ζ∗b∗c=ζ∗b∗c,∀ζ,b,c∈0,1C5. Ifζ≤c and b≤σ, with ζ,b,c,σ∈0,1, then ζ∗b≤c∗σDefinition 2 (see [7]).
A binary operation○: [0, 1] × [0, 1] ⟶ [0, 1] is called a CTCN if it meets the following assertions:T1.ζ○b=b○ζ,for allζ,b∈0,1T2.○ is continuousT3.ζ○0=0,for allζ∈0,1T4.ζ○b○c=ζ○b○c,for allζ,b,c∈0,1T5. Ifζ≤c and b≤σ, with ζ,b,c,σ∈0,1, then ζ○b≤c○σDefinition 3 (see [19]).
Let a setΕ≠∅ and ϑ∈Ε. A NS G in Ε is categorized by a truth membership function BGϑ, an indeterminacy membership function DGϑ, and a falsity membership function QGϑ. The functions BGϑ,DGϑ, and QGϑ are real standard or nonstandard subsets of ]0−,1+[; that is, BGϑ:E⟶0−,1+,DGϑ:E⟶0−,1+[ and QGϑ:E⟶0−,1+. So,(1)0−≤supBGϑ+supDGϑ+supQGϑ≤3+.Definition 4 (see [14]).
LetE be a nonempty set. A triplet E,Fb,∗ is called a FBMLS if ∗ is a CTN and Fb is a FS on E×E×0,∞ and fulfills the following assertions for all ϑ,δ,g∈E and τ,z>0:A1.Fbϑ,δ,τ>0A2.Fbϑ,δ,τ=1; then, ϑ=δA3.Fbϑ,δ,τ=Fbδ,ϑ,τA4.Fbϑ,g,bτ+z≥Fbϑ,δ,τ∗Fbδ,g,zA5.Fbϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Fbϑ,δ,τ=1Definition 5 (see [15]).
LetE be a nonempty set. A triplet E,Rb,∗ is called a FRMS if ∗ is a CTN and Rb is a FS on E×E×0,∞ and fulfills the following assertions for all ϑ,δ,g∈E and τ,z,w>0:(1)
Rbϑ,δ,0=0(2)
Rbϑ,δ,τ=1 if and only if ϑ=δ(3)
Rbϑ,δ,τ=Rbδ,ϑ,τ(4)
Rbϑ,g,τ+z+w≥Rbϑ,δ,τ∗Rbδ,u,z+Rbu,g,w for all distinct δ,u∈E\ϑ,g(5)
Rbϑ,δ,.:0,∞⟶0,1 is left continuous, and limτ⟶∞Rbϑ,δ,τ=1Definition 6 (see [7]).
TakeΕ≠∅. Let ∗ be a CTN, ○ be a CTCN, and F,V be FSs on E×E×0,∞. If E,F,V,∗,○ verifies the following assertions for all ϑ,δ∈Eandz,τ>0,F1.Fϑ,δ,τ+Vϑ,δ,τ≤1F2.Fϑ,δ,τ>0F3.Fϑ,δ,τ=1⟺ϑ=δF4.Fϑ,δ,τ=Fδ,ϑ,τF5.Fϑ,g,τ+z≥Fϑ,δ,τ∗Fδ,g,zF6.Fϑ,δ,z:0,∞⟶0,1 is continuous, and limτ⟶∞Fϑ,δ,τ=1for allτ>0F7.Vϑ,δ,τ>0F8.Vϑ,δ,τ=0⟺zϑ=δF9.Vϑ,δ,τ=Vδ,ϑ,τF10.Vϑ,g,τ+z≤Vϑ,δ,τ○Vδ,g,zF11.Vϑ,δ,z:0,∞⟶0,1 is continuous, and limτ⟶∞Vϑ,δ,τ=0for allτ>0Then,E,F,V,∗,○ is an IFMSDefinition 7 (see [8]).
LetE≠∅, ∗ be a CTN, and ○ be a CTCN. L,W,andQ are NSs on E×E×0,∞ which are said to be a neutrosophic metric on E if for all ϑ,δ,g∈E, the following circumstances fulfill:S1.Lϑ,δ,τ+Wϑ,δ,τ+Qϑ,δ,τ≤3for allτ∈ℝ+S2.Lϑ,δ,τ>0for allτ>0S3.Lϑ,δ,τ=1for allτ>0if  and only ifϑ=δS4.Lϑ,δ,τ=Lδ,ϑ,τfor allτ>0S5.Lϑ,g,τ+z≥Lϑ,δ,τ∗Lδ,g,zfor allτ,z>0S6.Lϑ,δ,z:0,∞⟶0,1 is continuous, and limτ⟶∞Lϑ,δ,τ=1for allτ>0S7.Wϑ,δ,τ<1for allτ>0S8.Wϑ,δ,τ=0for allτ>0if and only ifϑ=δS9.Wϑ,δ,τ=Wδ,ϑ,τfor allτ>0S10.Wϑ,g,τ+z≤Wϑ,δ,τ○Wδ,g,zfor allτ,z>0S11.Wϑ,δ,z:0,∞⟶0,1 is continuous, and limτ⟶∞Wϑ,δ,τ=0for allτ>0S12.Qϑ,δ,τ<1for allτ>0S13.Qϑ,δ,τ=0for allτ>0if and only ifϑ=δS14.Qϑ,δ,τ=Qδ,ϑ,τfor allτ>0S15.Qϑ,g,τ+z≤Qϑ,δ,τ○Qδ,g,zfor allτ,z>0S16.Qϑ,δ,z:0,∞⟶0,1 is continuous, and limτ⟶∞Qϑ,δ,τ=0for allτ>0Then,E,L,W,Q,∗,○ is called a NMS


## 3. Main Results

In this section, we present some new notions as generalizations of intuitionistic fuzzy and neutrosophic metric spaces; also, some fixed point results are proved.Definition 8.
LetΕ be a nonempty set. A five-tuple E,Bi,Di,∗,○ is called an IFRMS if ∗ is a CTN, ○ is a CTCN, and BiandDi are two FSs on E×E×0,∞ which fulfill the following assertions for all ϑ,δ,g∈E and τ,z,w>0:R1.Biϑ,δ,τ+Diϑ,δ,τ≤1R2.Biϑ,δ,0=0R3.Biϑ,δ,τ=1 if and only if ϑ=δR4.Biϑ,δ,τ=Biδ,ϑ,τR5.Biϑ,g,τ+z+w≥Biϑ,δ,τ∗Biδ,u,z+Biu,g,w for all distinct δ,u∈E\ϑ,gR6.Biϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Biϑ,δ,τ=1R7.Diϑ,δ,0=1R8.Diϑ,δ,τ=0 if and only if ϑ=δR9.Diϑ,δ,τ=Diδ,ϑ,τR10.Diϑ,g,τ+z+w≤Diϑ,δ,τ○Diδ,u,z+Diu,g,w for all distinct δ,u∈E\ϑ,gR11.Diϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Diϑ,δ,τ=0Example 1.
LetE,d be a rectangular metric space, define Bi,Di:E×E×0,∞⟶0,1 by(2)Biϑ,δ,τ=ττ+dϑ,δ,Diϑ,δ,τ=1−ττ+dϑ,δfor allϑ,δ∈Eandτ>0,and let ∗ be a CTN and ○ be a CTCN on E. Then, it is easy to see that E,Bi,Di,∗,○ is an IRFMS.Definition 9.
LetE be a nonempty set. A five-tuple E,Bb,Db,∗,○ is called an IFRBMS if there is b≥1,∗ is a CTN, ○ is a CTCN, and BbandD are two FSs on E×E×0,∞ verifying the following assertions for all ϑ,δ,g∈E and τ,z,w>0:I.Bbϑ,δ,τ+Dbϑ,δ,τ≤1II.Bbϑ,δ,0=0III.Bbϑ,δ,τ=1 if and only if ϑ=δIV.Bbϑ,δ,τ=Bbδ,ϑ,τV.Bbϑ,g,bτ+z+w≥Bbϑ,δ,τ∗Bbδ,u,z+Bbu,g,w for all distinct δ,u∈E\ϑ,gVI.Bbϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Bbϑ,δ,τ=1VII.Dbϑ,δ,0=1VIII.Dbϑ,δ,τ=0 if and only if ϑ=δIX.Dbϑ,δ,τ=Dbδ,ϑ,τX.Dbϑ,g,bτ+z+w≤Dbϑ,δ,τ○Dbδ,u,z+Dbu,g,w for all distinct δ,u∈E\ϑ,gXI.Dbϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Dbϑ,δ,τ=0Example 2.
LetE,d be a rectangular b-metric space, and define Bb,Db:E×E×0,∞⟶0,1 by(3)Bbϑ,δ,τ=ττ+dϑ,δ,Dbϑ,δ,τ=dϑ,δτ+dϑ,δfor allϑ,δ∈Eandτ>0,with CTN ζ∗b=minζ,b and CTCN ζ○b=maxζ,b. Then, it is easy to see that E,Bb,Db,∗○ is an IFRBMS.Example 3.
LetE,d be a rectangular b-metric space, and define Bb,Db:E×E×0,∞⟶0,1 by(4)Bbϑ,δ,τ=e−dϑ,δ/τ,Dbϑ,δ,τ=1−e−dϑ,δ/τfor allϑ,δ∈Eandτ>0,with CTN ζ∗b=minζ,b and CTCN ζ○b=maxζ,b. Then, it is easy to see that E,Bb,Db,∗○ is an IFRBMS.Remark 1.
The above Examples2 and 3 are also an IFRBMS with CTN ζ∗b=ζ�b and CTCN ζ○b=maxζ,b.Remark 2.
Every IFRMS is an IFRBMS, but the converse may not be true.Definition 10.
LetE be a nonempty set. A five-tuple E,Bl,Dl,∗,○ is called an IFRBMLS if there is b≥1,∗ is a CTN, ○ is a CTCN, and BlandDl are two FSs on E×E×0,∞ fulfilling the following assertions for all ϑ,δ,g∈E and τ,z,w>0:A.Blϑ,δ,τ+Dlϑ,δ,τ≤1B.Blϑ,δ,0=0C.Blϑ,δ,τ=1 implies ϑ=δD.Blϑ,δ,τ=Blδ,ϑ,τE.Blϑ,g,bτ+z+w≥Blϑ,δ,τ∗Blδ,u,z+Blu,g,w for all distinct δ,u∈E\ϑ,gF.Blϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Blϑ,δ,τ=1G.Dlϑ,δ,0=1H.Dlϑ,δ,τ=0 implies if ϑ=δI.Dlϑ,δ,τ=Dlδ,ϑ,τJ.Dlϑ,g,bτ+z+w≤Dlϑ,δ,τ○Dlδ,u,z+Dlu,g,w for all distinct δ,u∈E\ϑ,gK.Dlϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Dlϑ,δ,τ=0Definition 11.
In the above Definition10, if we take b=1, then it becomes an IFRMLS.Example 4.
DefineBl,Dl:E×E×0,∞⟶0,1 by(5)Blϑ,δ,τ=ττ+maxϑ,δp,Dlϑ,δ,τ=maxϑ,δpτ+maxϑ,δpfor allϑ,δ∈Eandτ>0,with CTN ζ∗b=minζ,b and CTCN ζ○b=maxζ,b. Then, it is easy to see that E,Bl,Dl,∗○ is an IFRBMLS, and if we take p=1, then it becomes an IFRMLS.Example 5.
DefineBl,Dl:E×E×0,∞⟶0,1 by(6)Blϑ,δ,τ=e−maxϑ,δp/τ,Dlϑ,δ,τ=1−e−maxϑ,δp/τfor allϑ,δ∈E,p≥1,andτ>0,with CTN ζ∗b=minζ,b and CTCN ζ○b=maxζ,b. Then, it is easy to see that E,Bl,Dl,∗○ is an IFRBMLS.Remark 3.
The above Examples4 and 5 are also an IFRBMLS with CTN ζ∗b=ζ�b and CTCN ζ○b=maxζ,b.Remark 4.
In an IFRBMLS, the self-distance may not be equal to 1 and 0.
For this, consider the above Example5; then, we have(7)Blϑ,ϑ,τ=e−maxϑ,ϑp/τ=e−ϑp/τ≠1,Dlϑ,ϑ,τ=1−e−maxϑ,ϑp/τ=1−e−ϑp/τ≠0.Remark 5.
Every IFRBMS is an IFRBMLS, but the converse may not be true.Remark 6.
IFRBMLS may not be continuous.Example 6.
LetE=0,∞,Blϑ,δ,τ=e−dϑ,δ/τ, and Dlϑ,δ,τ=1−e−dϑ,δ/τ for all ϑ,δ∈E,τ>0, and(8)dϑ,δ=0,ifϑ=δ,2ϑ+δ2,ifϑ,δ∈0,1,12ϑ+δ2,otherwise.
If we define CTN byζ∗b=ζ�b and CTCN by ζ○b=maxζ,b, then E,Bl,Dl,∗,○ is an IFRBMLS. Now, to illustrate continuity, we have(9)limn⟶∞Bl0,1−1n,τ=limn⟶∞e−21−1/n2/τ=e−2/τ=Bl0,1,τ,limn⟶∞Dl0,1−1n,τ=1−limn⟶∞e−21−1/n2/τ=1−e−2/τ=Dl0,1,τ.
However,(10)limn⟶∞Bl1,1−1n,τ=limn⟶∞e−22−1/n2/τ=e−8/τ≠1=Bl1,1,τ,limn⟶∞Dl1,1−1n,τ=1−limn⟶∞e−22−1/n2/τ=1−e−8/τ≠0=Dl1,1,τ.
Hence,E,Bl,Dl,Q,∗,○ is not continuous.Definition 12.
LetE be a nonempty set. A six-tuple E,Be,De,Qe,∗,○ is called a NRMS if ∗ is a CTN, ○ is a CTCN, and Be,De,andQe are three NSs on E×E×0,∞ fulfilling the following assertions for all ϑ,δ,g∈E and τ,z,w>0:(i)
Beϑ,δ,τ+Deϑ,δ,τ+Qϑ,δ,τ≤3(ii)
Beϑ,δ,0=0(iii)
Beϑ,δ,τ=1 if and only if ϑ=δ(iv)
Beϑ,δ,τ=Beδ,ϑ,τ(v)
Beϑ,g,τ+z+w≥Beϑ,δ,τ∗Beδ,u,z+Beu,g,w for all distinct δ,u∈E\ϑ,g(vi)
Beϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Beϑ,δ,τ=1(vii)
Deϑ,δ,0=1(viii)
Deϑ,δ,τ=0 if and only if ϑ=δ(ix)
Deϑ,δ,τ=Deδ,ϑ,τ(x)
Deϑ,g,τ+z+w≤Deϑ,δ,τ○Deδ,u,z+Deu,g,w for all distinct δ,u∈E\ϑ,g(xi)
Deϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Deϑ,δ,τ=0(xii)
Qeϑ,δ,0=1(xiii)
Qeϑ,δ,τ=0 if and only if ϑ=δ(xiv)
Qeϑ,δ,τ=Qeδ,ϑ,τ(xv)
Qeϑ,g,τ+z+w≤Qeϑ,δ,τ○Qeδ,u,z+Qeu,g,w for all distinct δ,u∈E\ϑ,g(xvi)
Qeϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Qeϑ,δ,τ=0Example 7.
LetE,d be a rectangular metric space, and define Be,D,Qe:E×E×0,∞⟶0,1 by(11)Beϑ,δ,τ=ττ+dϑ,δ,Deϑ,δ,τ=1−ττ+dϑ,δ,Qeϑ,δ,τ=dϑ,δτ,for allϑ,δ∈Eandτ>0, with CTN ζ∗b=minζ,b and CTCN ζ○b=maxζ,b. Then, it is easy to see that E,Be,De,Qe,∗○ is a NRMS.Definition 13.
LetE be a nonempty set. A six-tuple E,B,D,Q,∗,○ is called a NRBMS if there is b≥1,∗ is a CTN, ○ is a CTCN, and B,D, andQ are three NSs on E×E×0,∞ fulfilling the following assertions for all ϑ,δ,g∈E and τ,z,w>0:(a)
Bϑ,δ,τ+Dϑ,δ,τ+Qϑ,δ,τ≤3(b)
Bϑ,δ,0=0(c)
Bϑ,δ,τ=1 if and only if ϑ=δ(d)
Bϑ,δ,τ=Bδ,ϑ,τ(e)
Bϑ,g,bτ+z+w≥Bϑ,δ,τ∗Bδ,u,z+Bu,g,w for all distinct δ,u∈E\ϑ,g(f)
Bϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Bϑ,δ,τ=1(g)
Dϑ,δ,0=1(h)
Dϑ,δ,τ=0 if and only if ϑ=δ(i)
Dϑ,δ,τ=Dδ,ϑ,τ(j)
Dϑ,g,bτ+z+w≤Dϑ,δ,τ○Dδ,u,z+Du,g,w for all distinct δ,u∈E\ϑ,g(k)
Dϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Dϑ,δ,τ=0(l)
Qϑ,δ,0=1(m)
Qϑ,δ,τ=0 if and only if ϑ=δ(n)
Qϑ,δ,τ=Qδ,ϑ,τ(o)
Qϑ,g,bτ+z+w≤Qϑ,δ,τ○Qδ,u,z+Qu,g,w for all distinct δ,u∈E\ϑ,g(p)
Qϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Qϑ,δ,τ=0Example 8.
LetE,d be a rectangular b-metric space, and define B,D,Q:E×E×0,∞⟶0,1 by(12)Bϑ,δ,τ=ττ+dϑ,δ,Dϑ,δ,τ=1−ττ+dϑ,δ,Qϑ,δ,τ=dϑ,δτ,for allϑ,δ∈Eandτ>0, with CTN ζ∗b=minζ,b and CTCN ζ○b=maxζ,b. Then, it is easy to see that E,B,D,Q,∗○ is a NRBMS.Remark 7.
The above Example6 is also a NRBMS with CTN ζ∗b=ζ�b and CTCN ζ○b=maxζ,b.Remark 8.
Every NRMS is a NRBMS, but the converse may not be true.Definition 14.
LetE,B,D,Q,∗,○ be a NRBMS, and assume ϑn to be a sequence in E. Then,(i)
ϑn is named to be a convergent sequence if there exists ϑ∈E such that(13)limn⟶∞Bϑn,ϑ,τ=1,limn⟶∞Dϑn,ϑ,τ=0,limn⟶∞Qϑn,ϑ,τ=0for allτ>0.(ii)
ϑn is named to be a Cauchy sequence if(14)limn⟶∞Bϑn,ϑn+q,τ=1,limn⟶∞Dϑn,ϑn+q,τ=0,limn⟶∞Qϑn,ϑn+q,τ=0.(iii)
If every Cauchy sequence is convergent inE, then E,B,D,Q,∗,○ is said to be a complete NRBMS.Definition 15.
LetE be a nonempty set. A six-tuple E,Bh,Dh,Qh,∗,○ is called a NRBMLS if there is b≥1,∗ is a CTN, ○ is a CTCN, and Bh,Dh,andQh are three NSs on E×E×0,∞ fulfilling the following assertions for all ϑ,δ,g∈E and τ,z,w>0:L1.Bhϑ,δ,τ+Dhϑ,δ,τ+Qhϑ,δ,τ≤3L2.Bhϑ,δ,0=0L3.Bhϑ,δ,τ=1 implies if ϑ=δL4.Bhϑ,δ,τ=Bhδ,ϑ,τL5.Bhϑ,g,bτ+z+w≥Bhϑ,δ,τ∗Bhδ,u,z+Bhu,g,w for all distinct δ,u∈E\ϑ,gL6.Bhϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Bhϑ,δ,τ=1L7.Dhϑ,δ,0=1L8.Dhϑ,δ,τ=0 implies ϑ=δL9.Dhϑ,δ,τ=Dhδ,ϑ,τL10.Dhϑ,g,bτ+z+w≤Dhϑ,δ,τ○Dhδ,u,z+Dhu,g,w for all distinct δ,u∈E\ϑ,gL11.Dhϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Dhϑ,δ,τ=0L12.Qhϑ,δ,0=1L13.Qhϑ,δ,τ=0 implies ϑ=δL14.Qhϑ,δ,τ=Qhδ,ϑ,τL15.Qhϑ,g,bτ+z+w≤Qhϑ,δ,τ○Qhδ,u,z+Qhu,g,w for all distinct δ,u∈E\ϑ,gL16.Qhϑ,δ,.:0,∞⟶0,1 is continuous, and limτ⟶∞Qhϑ,δ,τ=0Definition 16.
In the above definition, if we takeb=1, then it becomes a NRMLS.Example 9.
DefineBh,Dh,Qh:E×E×0,∞⟶0,1 by(15)Bhϑ,δ,τ=ττ+ϑ+δp,Dhϑ,δ,τ=1−ττ+ϑ+δp,Qhϑ,δ,τ=ϑ+δpτ,for allϑ,δ∈E,p≥1,andτ>0, with CTN ζ∗b=minζ,b and CTCN ζ○b=maxζ,b. Then, it is easy to see that E,Bh,Dh,Qh,∗○ is a NRBMLS, and if we take p=1, then it becomes a NRMLS.Remark 9.
The above Example9 is also a NRBMLS with CTN ζ∗b=ζ�b and CTCN ζ○b=maxζ,b.Remark 10.
Every NRBMS is a NRBMLS, but the converse may not be true.Remark 11.
From Remark4 and Example 9, it is clear that, in the NRBMLS, the self-distances Bhϑ,ϑ,τ≠1,Dhϑ,ϑ,τ≠0,andQhϑ,ϑ,τ≠0.Remark 12.
It is clear from Example6 that the NRBMLS may not be continuous.Definition 17.
LetE,Bh,Dh,Qh,∗,○ be a NRBMLS, and assume ϑn to be a sequence in E. Then,(i)
ϑn is named to be a convergent sequence if there exists ϑ∈E such that(16)limn⟶∞Bhϑn,ϑ,τ=Bhϑ,ϑ,τ,limn⟶∞Dhϑn,ϑ,τ=Dhϑ,ϑ,τ,limn⟶∞Qhϑn,ϑ,τ=Qhϑ,ϑ,τfor allτ>0.(ii)
ϑn is named to be a Cauchy sequence if(17)limn⟶∞Bhϑn,ϑn+q,τ,limn⟶∞Dhϑn,ϑn+q,τ,limn⟶∞Qhϑn,ϑn+q,τ,exist and are finitefor allτ>0.(iii)
If every Cauchy sequence is convergent inE, then E,Bh,Dh,Qh,∗,○ is said to be a complete NRBMLS such that(18)limn⟶∞Bhϑn,ϑ,τ=Bhϑ,ϑ,τ=limn⟶∞Bhϑn,ϑn+q,τ,limn⟶∞Dhϑn,ϑ,τ=Dhϑ,ϑ,τ=limn⟶∞Dhϑn,ϑn+q,τ,limn⟶∞Qhϑn,ϑ,τ=Qhϑ,ϑ,τ=limn⟶∞Qhϑn,ϑn+q,τ,for allτ>0andq≥1.Definition 18.
LetE,Bh,Dh,Qh,∗,○ be a NRBMLS. For, ϑ∈E,θ∈0,1,andτ>0, we define the open ball as(19)Bϑ,x,τ=δ∈E:Bhϑ,δ,τ>1−x,Dhϑ,δ,τ<x,Qhϑ,δ,τ<xcenterϑ,radiusxwith respecttoτ.Lemma 1.
LetE,B,D,Q,∗,○ be a NRBMS and(20)Bϑ,δ,kτ≥Bϑ,δ,τ,Dϑ,δ,kτ≤Dϑ,δ,τ,Qϑ,δ,kτ≤Qϑ,δ,τ,for all ϑ,δ∈E,0<k<1,andτ>0;then,ϑ=δ.Proof.
It is immediate from (f), (k), and (p).Theorem 1 (Banach contraction theorem in neutrosophic rectangularb-metric spaces).
LetE,B,D,Q,∗,○ be a NRBMS with b≥1 such that(21)limτ⟶∞Bϑ,δ,τ=1,limτ⟶∞Dϑ,δ,τ=0,limτ⟶∞Qϑ,δ,τ=0for allϑ,δ∈E.LetΨ:E⟶E be a mapping satisfying(22)BΨϑ,Ψδ,kτ≥Bϑ,δ,τ,DΨϑ,Ψδ,kτ≤Dϑ,δ,τ,QΨϑ,Ψδ,kτ≤Qϑ,δ,τ,for all ϑ,δ∈Eandk∈0,1/b. Then, Ψ has a unique fixed point.Proof.
Fix an arbitrary pointζ0∈E, and for n=0,1,2,…, start an iterative process ζn+1=Ψζn. Successively applying inequality (22), we get for all n,τ>0,(23)Bζn,ζn+1,τ≥Bζ0,ζ1,τkn,Dζn,ζn+1,τ≤Dζ0,ζ1,τkn,Qζn,ζn+1,τ≤Qζ0,ζ1,τkn.
SinceE,B,D,Q,∗,○ is a NRBMS for the sequence ζn, write τ=τ/3+τ/3+τ/3 and use the rectangular inequalities given in (e), (j), and (o) on Bζn,ζn+p,τ,Dζn,ζn+p,τ, andQζn,ζn+p,τ in the following cases.Case 1.
Ifp is odd, then p=2m+1 where m∈1,2,3,…, and we have(24)Bζn,ζn+2m+1,τ≥Bζn,ζn+1,τ3b∗Bζn+1,ζn+2,τ3b∗Bζn+2,ζn+2m+1,τ3b≥Bζn,ζn+1,τ3b∗Bζn+1,ζn+2,τ3b∗Bζn+2,ζn+3,τ3b2∗Bζn+3,ζn+4,τ3b2∗Bζn+4,ζn+2m+1,τ3b2≥Bζn,ζn+1,τ3b∗Bζn+1,ζn+2,τ3b∗Bζn+2,ζn+3,τ3b2∗Bζn+3,ζn+4,τ3b2∗Bζn+4,ζn+5,τ3b3∗⋯∗Bζn+2m,ζn+2m+1,τ3bm,Dζn,ζn+2m+1,τ≤Dζn,ζn+1,τ3b○Dζn+1,ζn+2,τ3b○Dζn+2,ζn+2m+1,τ3b≤Dζn,ζn+1,τ3b○Dζn+1,ζn+2,τ3b○Dζn+2,ζn+3,τ3b2○Dζn+3,ζn+4,τ3b2○Dζn+4,ζn+2m+1,τ3b2≤Dζn,ζn+1,τ3b○Dζn+1,ζn+2,τ3b○Dζn+2,ζn+3,τ3b2○Dζn+3,ζn+4,τ3b2○Dζn+4,ζn+5,τ3b3○⋯○Dζn+2m,ζn+2m+1,τ3bm,Qζn,ζn+2m+1,τ≤Qζn,ζn+1,τ3b○Qζn+1,ζn+2,τ3b○Qζn+2,ζn+2m+1,τ3b≤Qζn,ζn+1,τ3b○Qζn+1,ζn+2,τ3b○Qζn+2,ζn+3,τ3b2○Qζn+3,ζn+4,τ3b2○Qζn+4,ζn+2m+1,τ3b2≤Qζn,ζn+1,τ3b○Qζn+1,ζn+2,τ3b○Qζn+2,ζn+3,τ3b2○Qζn+3,ζn+4,τ3b2○Qζn+4,ζn+5,τ3b3○⋯∗○ζn+2m,ζn+2m+1,τ3bm.
Using (23) in the above inequalities, we deduce(25)Bζn,ζn+2m+1,τ≥Bζ0,ζ1,τ3bkn∗Bζ0,ζ1,τ3bkn+1∗Bζ0,ζ1,τ3b2kn+2∗Bζ0,ζ1,τ3b2kn+3∗Bζn+4,ζn+5,τ3b3kn+4∗⋯∗Bζ0,ζ1,τ3bmkn+m≥Bζ0,ζ1,τ3bkn∗Bζ0,ζ1,τ3bkkn∗Bζ0,ζ1,τ3bk2kn∗Bζ0,ζ1,τ3bk2kn+1∗Bζn+4,ζn+5,τ3bk3kn+1∗⋯∗Bζ0,ζ1,τ3bkmkn+m,Dζn,ζn+2m+1,τ≤Dζ0,ζ1,τ3bkn○Dζ0,ζ1,τ3bkn+1○Dζ0,ζ1,τ3b2kn+2○Dζ0,ζ1,τ3b2kn+3○Dζn+4,ζn+5,τ3b3kn+4○⋯○Dζ0,ζ1,τ3bmkn+m≤Dζ0,ζ1,τ3bkn○Dζ0,ζ1,τ3bkkn○Dζ0,ζ1,τ3bk2kn○Dζ0,ζ1,τ3bk2kn+1○Dζn+4,ζn+5,τ3bk3kn+1○⋯○Dζ0,ζ1,τ3bkmkn+m,Qζn,ζn+2m+1,τ≤Qζ0,ζ1,τ3bkn○Qζ0,ζ1,τ3bkn+1○Qζ0,ζ1,τ3b2kn+2○Qζ0,ζ1,τ3b2kn+3○Qζn+4,ζn+5,τ3b3kn+4○⋯○Qζ0,ζ1,τ3bmkn+m≤Qζ0,ζ1,τ3bkn○Qζ0,ζ1,τ3bkkn○Qζ0,ζ1,τ3bk2kn○Qζ0,ζ1,τ3bk2kn+1○Qζn+4,ζn+5,τ3bk3kn+1○⋯○Qζ0,ζ1,τ3bkmkn+m.Case 2.
Ifp is even, then p=2m;m∈1,2,3,…; then, we have(26)Bζn,ζn+2m,τ≥Bζn,ζn+1,τ3b∗Bζn+1,ζn+2,τ3b∗Bζn+2,ζn+2m,τ3b≥Bζn,ζn+1,τ3b∗Bζn+1,ζn+2,τ3b∗Bζn+2,ζn+3,τ3b2∗Bζn+3,ζn+4,τ3b2∗Bζn+4,ζn+2m,τ3b2≥Bζn,ζn+1,τ3b∗Bζn+1,ζn+2,τ3b∗Bζn+2,ζn+3,τ3b2∗Bζn+3,ζn+4,τ3b2∗Bζn+4,ζn+5,τ3b3∗⋯∗Bζn+2m−4,ζn+2m−3,τ3bm−1∗Bζn+2m−3,ζn+2m−2,τ3bm−1∗Bζn+2m−2,ζn+2m,τ3bm−1,Dζn,ζn+2m,τ≤Dζn,ζn+1,τ3b○Dζn+1,ζn+2,τ3b○Dζn+2,ζn+2m,τ3b≤Dζn,ζn+1,τ3b○Dζn+1,ζn+2,τ3b○Dζn+2,ζn+3,τ3b2○Dζn+3,ζn+4,τ3b2○Dζn+4,ζn+2m,τ3b2≤Dζn,ζn+1,τ3b○Dζn+1,ζn+2,τ3b○Dζn+2,ζn+3,τ3b2○Dζn+3,ζn+4,τ3b2○Dζn+4,ζn+5,τ3b3○⋯○Dζn+2m−4,ζn+2m−3,τ3bm−1○Dζn+2m−3,ζn+2m−2,τ3bm−1○Dζn+2m−2,ζn+2m,τ3bm−1,Qζn,ζn+2m,τ≤Qζn,ζn+1,τ3b○Qζn+1,ζn+2,τ3b○Qζn+2,ζn+2m,τ3b≤Qζn,ζn+1,τ3b○Qζn+1,ζn+2,τ3b○Qζn+2,ζn+3,τ3b2○Qζn+3,ζn+4,τ3b2○Qζn+4,ζn+2m,τ3b2≤Qζn,ζn+1,τ3b○Qζn+1,ζn+2,τ3b○Qζn+2,ζn+3,τ3b2○Qζn+3,ζn+4,τ3b2○Qζn+4,ζn+5,τ3b3○⋯○ζn+2m−4,ζn+2m−3,τ3bm−1○Qζn+2m−3,ζn+2m−2,τ3bm−1○Qζn+2m−2,ζn+2m,τ3bm−1.
Using (23) in the above inequalities, we deduce(27)Bζn,ζn+2m,τ≥Bζ0,ζ1,τ3bkn∗Bζ0,ζ1,τ3bkn+1∗Bζ0,ζ1,τ3b2kn+2∗Bζ0,ζ1,τ3b2kn+3∗Bζn+4,ζn+5,τ3b3kn+4∗⋯∗Bζ0,ζ1,τ3bm−1kn+2m−2≥Bζ0,ζ1,τ3bkn∗Bζ0,ζ1,τ3bkkn∗Bζ0,ζ1,τ3bk2kn∗Bζ0,ζ1,τ3bk2kn+1∗Bζn+4,ζn+5,τ3bk3kn+1∗⋯∗Bζ0,ζ1,τ3bkm−1kn+m−1,Dζn,ζn+2m,τ≤Dζ0,ζ1,τ3bkn○Dζ0,ζ1,τ3bkn+1○Dζ0,ζ1,τ3b2kn+2○Dζ0,ζ1,τ3b2kn+3○Dζn+4,ζn+5,τ3b3kn+4○⋯○Dζ0,ζ1,τ3bm−1kn+2m−2≤Dζ0,ζ1,τ3bkn○Dζ0,ζ1,τ3bkkn○Dζ0,ζ1,τ3bk2kn○Dζ0,ζ1,τ3bk2kn+1○Dζn+4,ζn+5,τ3bk3kn+1○⋯○Dζ0,ζ1,τ3bkm−1kn+m−1,Qζn,ζn+2m,τ≤Qζ0,ζ1,τ3bkn○Qζ0,ζ1,τ3bkn+1○Qζ0,ζ1,τ3b2kn+2○Qζ0,ζ1,τ3b2kn+3○Qζn+4,ζn+5,τ3b3kn+4○⋯○Qζ0,ζ1,τ3bm−1kn+2m−2≤Qζ0,ζ1,τ3bkn○Qζ0,ζ1,τ3bkkn○Qζ0,ζ1,τ3bk2kn○Qζ0,ζ1,τ3bk2kn+1○Qζn+4,ζn+5,τ3bk3kn+1○⋯○Qζ0,ζ1,τ3bkm−1kn+m−1.
Therefore, from Cases1 and 2 and together with (21), it follows that, for all p∈1,2,3,…, we have(28)limn⟶∞Bζn,ζn+p,τ=1∗1∗⋯∗1=1,limn⟶∞Dζn,ζn+p,τ=0○0○⋯○0=0,limn⟶∞Dζn,ζn+p,τ=0○0○⋯○0=0.
Hence,ζn is a Cauchy sequence. Since E,B,D,Q,∗,○ is a complete NRBMS, there exists u∈Esuch thatlimn⟶∞ζn=u.
Now, we examine thatu is a fixed point of Ψ.(29)Bu,Ψu,τ≥Bu,ζn,τ3b∗Bζn,ζn+1,τ3b∗Bζn+1,Ψu,τ3b≥Bu,ζn,τ3b∗BΨζn−1,Ψζn,τ3b∗BΨζn,Ψu,τ3b≥Bu,ζn,τ3b∗Bζn−1,ζn,τ3bk∗Bζn,u,τ3bk⟶1∗1∗1=1asn⟶∞,Du,Ψu,τ≤Du,ζn,τ3b○Dζn,ζn+1,τ3b○Dζn+1,Ψu,τ3b≤Du,ζn,τ3b○DΨζn−1,Ψζn,τ3b○DΨζn,Ψu,τ3b≤Du,ζn,τ3b○Dζn−1,ζn,τ3bk○Dζn,u,τ3bk⟶0○0○0=0asn⟶∞,Qu,Ψu,τ≤Qu,ζn,τ3b○Qζn,ζn+1,τ3b○Qζn+1,Ψu,τ3b≤Qu,ζn,τ3b○QΨζn−1,Ψζn,τ3b○QΨζn,Ψu,τ3b≤Qu,ζn,τ3b○Qζn−1,ζn,τ3bk○Qζn,u,τ3bk⟶0○0○0=0asn⟶∞,which shows that u is a fixed point of Ψ.
Now, we show the uniqueness.
Assumev is another fixed point of Ψ for some v∈E; then,(30)Bv,u,τ=BΨv,Ψu,τ≥Bv,u,τk=BΨv,Ψu,τk≥Bv,u,τk2≥⋯≥Bv,u,τkn⟶1asn⟶∞,Dv,u,τ=DΨv,Ψu,τ≤Dv,u,τk=DΨv,Ψu,τk≤Dv,u,τk2≤⋯≤Dv,u,τkn⟶0asn⟶∞,Qv,u,τ=QΨv,Ψu,τ≤Qv,u,τk=QΨv,Ψu,τk≤Qv,u,τk2≤⋯≤Qv,u,τkn⟶0asn⟶∞.
Thus,u=v. Hence, the fixed point is unique.Remark 13.
In Theorem1, if we take b=1, then it will become a Banach contraction theorem in the sense of NRMS.Example 10.
LetE=0,1, and define B,D,Q:E×E×0,∞⟶0,1 by(31)Bϑ,δ,τ=ττ+|ϑ−δ|2,Dϑ,δ,τ=1−ττ+|ϑ−δ|2,Qϑ,δ,τ=|ϑ−δ|2τ,for allϑ,δ∈Eandτ>0, with CTN ζ∗b=ζ.b and CTCN ζ○b=maxζ,b. Then, it is easy to see that E,B,D,∗○ is a complete NRBMS.
DefineΨ:E⟶EbyΨϑ=1−2−ϑ/3. Then,(32)BΨϑ,Ψδ,kτ=B1−2−ϑ3,1−2−δ3,kτ=kτkτ+1−2−ϑ/3−1−2−δ/32=9kτ9kτ+|2−ϑ−2−δ|2≥9kτ9kτ+|ϑ−δ|2≥ττ+|ϑ−δ|2=Bϑ,δ,τ,DΨϑ,Ψδ,kτ=D1−2−ϑ3,1−2−δ3,kτ=1−kτkτ+1−2−ϑ/3−1−2−δ/32=1−9kτ9kτ+|2−ϑ−2−δ|2≤1−9kτ9kτ+|ϑ−δ|2≤1−ττ+|ϑ−δ|2=Dϑ,δ,τ,QΨϑ,Ψδ,kτ=Q1−2−ϑ3,1−2−δ3,kτ=1−2−ϑ/3−1−2−δ/32kτ=2−ϑ−2−δ29kτ≤ϑ−δ29kτ≤ϑ−δ2τ=Qϑ,δ,τ,for all ϑ,δ∈E, where k∈1/2,1. Thus, all the conditions of Theorem 1 are satisfied, and hence, 0 is a unique fixed point of Ψ.Corollary 1.
LetE,Bb,Db,∗,○ be a IFRBMS with b≥1 such that(33)limτ⟶∞Bbϑ,δ,τ=1,limτ⟶∞Dbϑ,δ,τ=0,for allϑ,δ∈E.LetΨ:E⟶E be a mapping satisfying(34)BbΨϑ,Ψδ,kτ≥Bbϑ,δ,τ,DbΨϑ,Ψδ,kτ≤Dbϑ,δ,τ,for all ϑ,δ∈Eandk∈0,1/b. Then, Ψ has a unique fixed point.Proof.
It is clear from Theorem1.Theorem 2.
LetE,Bh,Dh,Qh,∗,○ be a NRBMLS with b≥1 such that(35)limτ⟶∞Bhϑ,δ,τ=1,limτ⟶∞Dhϑ,δ,τ=0,limτ⟶∞Qhϑ,δ,τ=0for allϑ,δ∈E.LetΨ:E⟶E be a mapping satisfying(36)BhΨϑ,Ψδ,kτ≥Bhϑ,δ,τ,DhΨϑ,Ψδ,kτ≤Dhϑ,δ,τ,QhΨϑ,Ψδ,kτ≤Qhϑ,δ,τ,for all ϑ,δ∈Eandk∈0,1/b. Then, Ψ has a unique fixed point.Proof.
It is easy to show on the lines of Theorems 1 and 2 in [14].


## 4. Application

In this section, we present an application to the integral equation of Theorem1. In particular, we show the existence of the solution of an integral equation of the form(37)ϑj=gj+∫0jFj,r,ϑrdr,for all j∈0,l where l>0. Let C0,l,ℝ be the space of all continuous functions defined on 0,l with CTN ζ∗b=ζ.b and CTCN ζ○b=maxζ,b for all ζ,b∈0,1, and define a complete NRBMS by(38)Bϑ,δ,τ=supj∈0,lττ+ϑj−δj2,Dϑ,δ,τ=supj∈0,lϑj−δj2τ+ϑj−δj2,Qϑ,δ,τ=supj∈0,lϑj−δj2τfor allϑ,δ∈C0,l,ℝandτ>0.Theorem 3.
LetΨ:C0,l,ℝ⟶C0,l,ℝ be the integral operator given by(39)Ψϑj=gj+∫0jFj,r,ϑrdr,g∈C0,l,ℝ,where F∈C0,l×0,l×ℝ,ℝ satisfies the following conditions:(i)
There existsf:0,l×0,l⟶0,+∞ such that, for all r,j∈0,l,fj,r∈L10,l,ℝ and for all ϑ,δ∈C0,l,ℝ, we have(40)Fj,r,ϑr−Fj,r,δr2≤f2j,rϑr−δr2.(ii)
Also,(41)supj∈0,l∫0jf2j,rdr≤k<1.Then, the integral equation has the solutionϑ∗∈C0,l,ℝ.Proof.
For allϑ,δ∈C0,l,ℝ, we have
(42)BΨϑj,Ψδj,kτ=supj∈0,lkτkτ+Ψϑj−Ψδj2≥supj∈0,lkτkτ+∫0jFj,r,ϑr−Fj,r,δr2dr≥supj∈0,lkτkτ+∫0jf2j,rϑr−δr2dr≥kτkτ+ϑr−δr2supj∈0,l∫0jf2j,rdr≥kτkτ+ϑr−δr2≥ττ+ϑr−δr2=Bϑ,δ,τ,DΨϑj,Ψδj,kτ=supj∈0,lΨϑj−Ψδj2kτ+Ψϑj−Ψδj2≤supj∈0,l∫0jFj,r,ϑr−Fj,r,δr2drkτ+∫0jFj,r,ϑr−Fj,r,δr2dr≤supj∈0,l∫0jf2j,rϑr−δr2drkτ+∫0jf2j,rϑr−δr2dr≤ϑr−δr2supj∈0,l∫0jf2j,rdrkτ+ϑr−δr2supj∈0,l∫0jf2j,rdr≤ϑr−δr2kτ+ϑr−δr2≤ϑr−δr2τ+ϑr−δr2=Dϑ,δ,τ,QΨϑj,Ψδj,kτ=supj∈0,lΨϑj−Ψδj2kτ≤supj∈0,l∫0jFj,r,ϑr−Fj,r,δr2drkτ≤supj∈0,l∫0jf2j,rϑr−δr2drkτ≤ϑr−δr2supj∈0,l∫0jf2j,rdrkτ≤ϑr−δr2kτ≤ϑr−δr2τ=Qϑ,δ,τ.
Hence,ϑ∗ is a fixed point of Ψ, which is the solution of integral equation (37).


## 5. Conclusion

The aim of this study is to present the notions of intuitionistic fuzzy rectangular metric spaces, intuitionistic fuzzy rectangular metric-like spaces, intuitionistic fuzzy rectangularb-metric spaces, intuitionistic fuzzy rectangular b-metric-like spaces, neutrosophic rectangular metric spaces, neutrosophic rectangular metric-like spaces, neutrosophic rectangular b-metric spaces, and neutrosophic rectangular b-metric-like spaces and prove the Banach contraction theorem in these spaces, and nontrivial examples and an application to the integral equation are also given to support our results. Due to a diverse range of applications of the metric fixed point theory in mathematics, science, and economics, it is researched widely. Different types of fixed point results for single- and multivalued mappings can be proven in the sense of the above-defined notions in this manuscript. Also, presented notions can be extended in different mathematical structures, i.e., intuitionistic fuzzy controlled rectangular metric spaces, intuitionistic fuzzy triple controlled rectangular metric spaces, neutrosophic extended rectangular metric spaces, etc.



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