# Study of Metric Space and Its Variants

Authors: Surjeet Singh Chauhan (Gonder); Prachi Garg; Kamleshwar Thakur
Journal: Journal of Mathematics
 (2022)

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## Abstract

The objective of this paper is to present a comparative study of metric space and its variants. This study will provide the structure, gap analysis, and application of metric space and its variants from 1906 to 2021.


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## Body

## 1. Introduction

Metric space plays an important role to solve various mathematical problems. A wide range of metric spaces offers a powerful method to learn the optimization and approximation theory, variational inequalities, and several other problems. The word metric originated from the word meter (measure—a class of functions, which consider as generalizing the notion of distance between each pair of elements) [1–4]. The French mathematician Frechet first considered these functions, to generalize the notion of distances and apply them to arbitrary sets. He introduced the definition of metric on a set in his doctoral dissertation, “Less Espaces Abstraint” [5].A functiond:X2⟶R (where X≠∅) is s.t.b. a distance function (or metric) on X, if it meets the subsequent requirements:(1)
dℵ1″,ℵ2″≥0 if ℵ1″≠ℵ2″ and ℵ1″=ℵ2″⇔dℵ1″,ℵ2″=0,(2)
dℵ1″,ℵ2″=dℵ2″,ℵ1″,(3)
dℵ1″,ℵ3″+dℵ3″,ℵ2″≥dℵ1″,ℵ2″, for all ℵ1″,ℵ2″,ℵ3″∈X.So the function“d” composed with a set “X” is titled as metric space, which is denoted by X,d.Example 1.
LetX=R with dℵ1″,ℵ2″=ℵ1″−ℵ2″, for all ℵ1″,ℵ2″∈X.
Now it is easy to get thatX,d is a metric space.Example 2.
LetX=R2 with dℵ1″,ℵ2″,ℷ1″,ℷ2″=ℵ1″−ℷ1″2+ℵ2″−ℷ2″2, for all ℵ1″,ℵ2″,ℷ1″,ℷ1″∈X.
Now it is easy to get thatX,d is a metric space.
Now, some basic results on metric space and their applications are in this manner.


### 1.1. Picard’s Convergence Theorem [2]

For nonlinear equations, Picard proved the theorem for showing the existence of the solutions.Theorem 1.
Letψ:u,v⟶ctsR and ψ:u,v⟶diffR. If ∃μ<1, such that(1)ψ′ℷ″≤μ,∀ℷ″ϵu,v.
Then, the sequenceℷn″ in u,v is defined by(2)ℷn+1″=ψℷn″,∀n≥0 converges to a solution of the equation ψℷ″=ℷ″.
The iterative sequenceℷn″ is termed as Picard’s iterative sequence.


### 1.2. Banach’s Fixed-Point Theorem [2]

The existence of a solution for the integral equation was established by Banach in 1922, which is known as Banach’s fixed-point theorem.Theorem 2.
Letℵ,d be a complete metric space, and T:ℵ⟶ℵ be a contractive mapping (i.e., ∃θϵ0,1 such that dTℷ1″,Tℷ2″≤θdℷ1″,ℷ2″,∀ℷ1″,ℷ2″ϵℵ. Then, the following axioms hold:(i)
T has a unique fixed point zϵℵ;(ii)
∀ℷ0″ϵℵ, the sequence ℷn″ is defined by ℷn+1″=Tℷn″(iii)
∀n≥0 converges to the fixed point zof Ti.e.Tz=z
Metric space is used in various fields such as real-life situations, convergence, and quantum mechanics. With the help of metric space, those functions that satisfy metric space properties can be used to determine the distance between two points, like in quantum mechanics, that is, the conservation laws naturally lead to metric spaces, which are related to the set of physical quantities. The onion-shell geometry is used in such all metric spaces where max and min distances between the states can geometrically interpret. Also, convergence is depending on the choice of metric but not an inherent function of real numbers [1–4].


## 1.1. Picard’s Convergence Theorem [2]

For nonlinear equations, Picard proved the theorem for showing the existence of the solutions.Theorem 1.
Letψ:u,v⟶ctsR and ψ:u,v⟶diffR. If ∃μ<1, such that(1)ψ′ℷ″≤μ,∀ℷ″ϵu,v.
Then, the sequenceℷn″ in u,v is defined by(2)ℷn+1″=ψℷn″,∀n≥0 converges to a solution of the equation ψℷ″=ℷ″.
The iterative sequenceℷn″ is termed as Picard’s iterative sequence.


## 1.2. Banach’s Fixed-Point Theorem [2]

The existence of a solution for the integral equation was established by Banach in 1922, which is known as Banach’s fixed-point theorem.Theorem 2.
Letℵ,d be a complete metric space, and T:ℵ⟶ℵ be a contractive mapping (i.e., ∃θϵ0,1 such that dTℷ1″,Tℷ2″≤θdℷ1″,ℷ2″,∀ℷ1″,ℷ2″ϵℵ. Then, the following axioms hold:(i)
T has a unique fixed point zϵℵ;(ii)
∀ℷ0″ϵℵ, the sequence ℷn″ is defined by ℷn+1″=Tℷn″(iii)
∀n≥0 converges to the fixed point zof Ti.e.Tz=z
Metric space is used in various fields such as real-life situations, convergence, and quantum mechanics. With the help of metric space, those functions that satisfy metric space properties can be used to determine the distance between two points, like in quantum mechanics, that is, the conservation laws naturally lead to metric spaces, which are related to the set of physical quantities. The onion-shell geometry is used in such all metric spaces where max and min distances between the states can geometrically interpret. Also, convergence is depending on the choice of metric but not an inherent function of real numbers [1–4].


## 2. Extensions of Metric Space to Other Spaces

Recently, several authors have incorporated some generalization/extension of metric space in various ways and prepared a comparative study of metric space and its variants to explore the gap analysis and their applications, starting from 1906 to 2020.


### 2.1. Period-I (1906–1950)

#### 2.1.1. Quasi-Metric Spaces [6,7]

Wilson [6] introduced the metric space without symmetric condition, which is termed as quasi-metric space.A functiond:X2⟶R+∪0, (where X≠∅) is well known as a quasi-metric if it satisfied the following succeeding conditions:(1)
dℷ1″,ℷ1″=0,∀ℷ1″∈X;(2)
dℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″, for all ℷ1″,ℷ2″,ℷ3″∈X.Then, quasi-metricd together with set X is titled as quasi-metric space and represented by X,d.Example 3.
LetR be set of all real numbers and ρ>0. Let d:R×R⟶R+∪0 be a mapping, which is defined as follows:(3)dℷ1″,ℷ2″=ℷ2″−ℷ1,″ifℷ2″≥ℷ1″,ρℷ1″−ℷ2″,ifℷ2″<ℷ1″.
It is easy to get thatdis a quasi-metric on R.Remark 1.
(1)
Nearly all “fixed-point theorems” used for″ψF contraction” in quasi-metric space (q.m.s) have been proved.(2)
Quasi-Metric Space is applied forproving Hamilton–Jacobi equation’s existence and uniqueness, iterated function systems, fractal theory, shape-memory alloys, in the rate-independent models for plasticity, etc [7].
A mappingT:X⟶X is called as the following:(1)
“ForwardψF contraction (resp. backward ψF contraction)” if quasi-metric space X,d and functions F∈F and ψ∈Ψ satisfy the following conditions,(4)Tℷ1″≠Tℷ2″⇒FdTℷ1″,Tℷ2″≤ψdℷ1″,ℷ2″,resp.Tℷ1″≠Tℷ2″⇒FdTℷ1″,Tℷ2″≤ψdℷ2″,ℷ1″.(Remark: ifT is a “backward ψF contraction” then T2 be a forward ψF2 contraction.)(2)
“Forward Picard operator (backward Picard operator)” if quasi-metric spaceX,d has a unique fixed point ζ∈X and Tnℷ1″⟶fζ for every ℷ1″∈X.


#### 2.1.2. Probabilistic Metric Space [8–10]

The generalized probabilistic theory was proposed by Menger, which played a significant part in the development of metric space. In particular, he suggested replacing the numberdp,q with the real function Fpq, whose function is Fpqℷ″, for any real number ℷ″. It is defined as the probability that is the distance between p&q is less than x.A functionF:X2⟶R+∪0, (where X≠∅), is s.t.b. probabilistic metric space, if met the following requirements:(1)
Fa,bℷ1″=1,∀ℷ″>0⇒a=b∀a,b∈X;(2)
Fa,bℷ1″=Fb,aℷ1″,∀a,b∈X;(3)
Fa,cℷ1″=1,Fb,cℷ2″=1;(4)
⇒Fa,bℷ1″,ℷ2″=1,∀a,b,c∈Xandℷ1″,ℷ2″∈0,∞.Then, probabilistic metric space is represented asX,F.Remark 2.
With the help of probabilistic metric space, the existence of the random fixed-point theorem was solved. Also, constrained file migration and metrical task systems use a randomized algorithm that is based on the probabilistic metric approximation.
A mappingF:X⟶X is titled as follows:(1)
B–contraction if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈X and all θ>0 satisfy the following implication:(5)Ffℷ1″,fℷ2″σθ≥Fℷ1″,ℷ2″θ.(2)
C–contraction if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈Xandθ>0 satisfy the following implication:(6)Fℷ1″,ℷ2″θ>1−θ⇒Ffℷ1″,fℷ2″σθ>1−σθ.
As a generalization of the concept of “C–contraction,” Pap and Hadzic made known to the concept of Ψ,C-contraction.
A mappingF:X⟶2X is called Ψ,C-contraction (where Ψ:R+⟶R+,R+=0,∞), if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈Xand all θ>0 satisfy the following implication:(7)Fℷ1″,ℷ2″θ>1−θ⇒∀σ′ϵFℷ1″∃σ″ϵFℷ2″,such that(8)Fσ,,σ,,Ψx>1−Ψx.


### 2.2. Period-II (1951–2000)

#### 2.2.1. Two-Metric Space [11–15]

Gahler [15] first attempted to generalize the ordinary distance function to measure the distance between three points. Two-metric space is somewhere dissimilar to the metric space, which provides a unique nonlinear structure.A mappingd:X3⟶R is s.t.b. 2-metric on XX≠∅, if satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″,a≠0,∀ℷ1″,ℷ2″″∈Xandℷ1″≠ℷ2″,∃apointa∈X.(2)
dℷ1″,ℷ2″,ℷ3″=0, only if two or more points are the same.(3)
dℷ1″,ℷ2″,ℷ3″″=dℷ1″,ℷ3″,ℷ3″=dℷ3″,ℷ2″,ℷ1″,∀ℷ1″,ℷ2″,ℷ3″∈X.(4)
dℷ1″,ℷ2″,ℷ3″≤dℷ1″,ℷ2″,ℷ4″+dℷ1″,ℷ4″,ℷ3″″+dℷ4″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X.Then, 2-metricd together with set X is titled as 2-metric space and represented by X,d.Remark 3.
(1)
The “Baire category theorem” and “Cantor intersection theorem” are used for solving problems of fixed point [12].(2)
In 2-metric space, Lebesgue integrable function is used as a summable function for each compactR+ [13].
A mappingT:X,φ⟶X,φ is called contraction in 2-metric space if satisfies the following conditions, φTℷ1″,Tℷ2″,ρ<φℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X where ℷ1″≠ℷ2″≠ρ, and φTℷ1″,Tℷ2″,ρ=0,if any two of ℷ1″,ℷ2″ and ρ are equal.
A mappingT:X⟶X is entitled as “A-contraction” in 2-metric space if satisfies the succeeding conditions, dTℷ1″,Tℷ2″,ρ≤τmaxdTℷ1″,ℷ1″,ρ+dTℷ2″,ℷ2″,ρ,dTℷ2″,ℷ2″,ρ+dℷ1″,ℷ2″,ρ,dTℷ1″,ℷ1″,ρ+dℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X,τ∈0,1/2 [14].


#### 2.2.2. Fuzzy Metric Space [17–19]

After probabilistic metric space, author replaced randomness with fuzziness and presented an innovative theory, called fuzzy metric space (f.m.s.). The idea of probabilistic metric space (p.m.s.) has been expanded to fuzzy metric space. The purpose of introducing the fuzzy metric space was to allocate the non-negative fuzzy number to the distance among two points, which is the most appropriate way to describe the fuzzy metric space.LetM is a “fuzzy set” on X×X×0;∞, that is s.t.b. fuzzy metric (f.m.) on XX≠∅,∗ is a continuous t-norm), if met the subsequent axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X(1)
Mℷ1″,ℷ2″,0=0;(2)
Mℷ1″,ℷ2″,t=1∀t>0ℷ1″=ℷ2″;(3)
Mℷ1″,ℷ2″,t=Mℷ2″,ℷ1″,t;(4)
Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s≤Mℷ1″,ℷ2″,t+s;(5)
Mℷ1″,ℷ2″,·:0;∞⟶0;1 is left continuous, ℷ1″,ℷ2″,ℷ3″∈X and t,s>0.Then, fuzzy metricM together with setX is titled as fuzzy metric space and symbolized as (X;M;∗.Example 4.
LetX=R and M=expℷ1″−ℷ2″/t−1, ∀ℷ1″,ℷ2″∈X, and t∈0,∞.(1)
m∗n=mn;(2)
m∗n=minm,n.
It is easy to get thatM is a fuzzy metric on X=R with a continuous t-norm ∗.Remark 4.
(1)
Gregori and Sapena expanded the “Banach contraction mapping” to the “fuzzy contractive mapping” in the complete fuzzy metric space.(2)
Chos et al. proved Banach’s contraction and Edelstein mapping in fuzzy metric space.(3)
N. Holland proved Baire’s theorem in fuzzy metric space.
A mappingT:X⟶X is known as the following:(1)
“Banach fuzzy contraction” if fuzzy metric spaceX,M,∗ satisfies the subsequent conditions, MTℷ1″,Tℷ2″,kρ≥Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″∈X,k∈0,1, and all ρ>0.(2)
“Edelstein fuzzy contraction” if fuzzy metric spaceX,d,∗ satisfies the subsequent conditions, MTℷ1″,Tℷ2″,ρ>Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X,ℷ1″≠ℷ2″, and all ρ>0.


#### 2.2.3.b-Metric Space [20–22]

In theb-metric space, K (relaxed triangle inequality) was introduced. If β=1, then b-metric and metric space are the same so as compared with the metric space, and b-metric space is the weaker notion.A mappingd:X×Xinto0,∞ is s.t.b. b-metric on XX≠∅ if met the subsequent axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″ (symmetry);(3)
There existsβ≥1 satisfying(9)dℷ1″,ℷ2″≤βℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″ℷ3″∈X.Then,b-metric d together with set X is titled as b-metric space and represented as ordered pair X,d.Example 5.
LetX=N and function fn=––log2n and gn=2n−2fnKfn+2fn−nKfn−1 and a function d fromX⟶Xonto0,∞ by(10)dx,y=gx–y.
It is easy to show thatX,d is a b-metric space.Remark 5.
With the help of b-metric space, the presence of a unique solution for a nonlinear fractional differential equation is given with an integral boundary condition. Biology, physical science, and economics such real-life problems eventually lead to the linear fractional differential equation and integral equation [20–22]. Further extension of b-metric space can be seen in the paper of Chauhan and Kaur [23].


#### 2.2.4.D-Metric Space [24–26]

Since 2-metric space was not continuous and convergent for all three points, so Dhage [24] introduced the new generalized class of the ordinary metric space. The convergence of D-metric defines a Hausdorff topology and (sequentially) continuity for all three points.A mappingD:X3⟶0,∞ is s.t.b. D-metric on XX≠∅, if it meets the following conditions given below:(1)
Dℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″, (coincidence)(2)
Dℷ1″,ℷ2″,ℷ3″=Dpℷ1″,ℷ2″,ℷ3″, where p is a permutation of ℷ1″,ℷ2″,ℷ3″ (symmetry), and(3)
Dℷ”1,ℷ”2,ℷ”3≤Dℷ”1,ℷ”2,ℷ”4+Dℷ”1,ℷ”4,ℷ”3+Dℷ”4,ℷ”2,ℷ”3,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X (tetrahedral inequality).Then,D-metric d together with set X is titled as D-metric space and represented as ordered pair X,d.Example 6.
LetX=R+and D:X3⟶R+ be definite as(11)Dℷ1″,ℷ2″,ℷ3″=maxℷ1″−ℷ2″,ℷ2″−ℷ3″,ℷ3″−ℷ1″,∀ℷ1″,ℷ2″,ℷ3″∈X.
It is easy to show thatX,D is a D-metric space.Remark 6.
(1)
The analysis is obtained from the results of nonlinear self-mapping, which satisfies the particular form (completeness and boundedness) of contraction condition inD-metric space, to prove the existence of a unique fixed point [25].(2)
Singh et al. introduced semicompatibility inD-metric space for “fixed-point theorems” with the help of the orbit concept [26].
EveryD-continuous function T:X,D⟶X,D has a fixed point in X if D-metric space has a D-fixed-point property.
EveryD-weakly continuous function T:X,D⟶X,D has a fixed point in X if and only if D-metric space has a D-weakly fixed-point property.


#### 2.2.5. Partial Metric Space [27–29]

The partial metric space is a component of dataflow networks, denotational semantic study. Compared with the standard metric, the main difference is that there is no need for the self-distance of arbitrary variables is to be zero.A functionp:P2⟶R+ is s.t.b. partial metric on PP≠∅, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″∈P:(1)
ℷ1″=ℷ2″⇔pℷ1″,ℷ1″=pℷ1″,ℷ2″=pℷ2″,ℷ2″;(2)
pℷ1″,ℷ1″≤pℷ1″,ℷ2″;(3)
pℷ1″,ℷ2″=pℷ2″,ℷ1″;(4)
pℷ1″,ℷ2″≤pℷ1″,ℷ3″+pℷ3″,ℷ2″–pℷ3″,ℷ3″.Then, partial metricp together with the set P is titled as partial metric space and represented as ordered pair P,p.Example 7.
LetX=R+=0,∞ and p:X×X⟶R+ be a function defined by pℷ1″,ℷ2″=maxℷ1″,ℷ2″, for all ℷ1″,ℷ2″∈X.
Then, it is easy to see thatX,p is a partial metric, but it is not a metric. Indeed, for any x>0, we have px,x=x≠0.Remark 7.
Through the experience of computer science, the idea of nonzero self-distance for metric space was inspired. Also, it gave an extension of the “Banach contraction principle” in the complete partial metric space [30].
A mappingT:X⟶X is titled as:(1)
Contraction for every partial metric spaceX,p if ∃k∈0,1 so that ∀ℷ1″,ℷ2″∈X satisfies the following condition:(12)pTℷ1″,Tℷ2″≤kpℷ1″,ℷ2″.(2)
Asymptotic regular for every partial metric spaceX,p if at point ℷ1″∈X satisfies the following condition:(13)limn⟶∞dTnℷ1″,Tn+1ℷ1″=0.


#### 2.2.6. Rectangular Metric Space [31, 32]

The rectangular metric space was introduced with the most general triangle inequality in which four points are implemented instead of three points.A generalized metric (rectangular metric) is a functiond:R2⟶0,∞ on RR≠∅, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈R with ℷ1″≠ℷ2″ and ℷ3″,ℷ4″∉ℷ1″,ℷ2″:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ4″+dℷ4″,ℷ2″ (quadrilateral inequality).Then, rectangular metricd together with the set R is titled as partial metric space and represented as ordered pair R,d.Example 8.
LetR1=0,2,R2=1,1/2,1/3,…,R=R1∪R2. define d:R×R⟶0,∞ as follows:(14)dℷ1″,ℷ2″=0,ℷ1″=ℷ2″,1,ℷ1″≠ℷ2″,ℷ1″,ℷ2″⊂R1orℷ1″,ℷ2″⊂R2ℷ2″,ℷ1″∈R1,ℷ2″∈R2,,ℷ1″,ℷ2″∈R1,ℷ1″∈R2.
Then,R,d is a complete rectangular metric space.Remark 8.
Budhia et al. contributed to some innovative fixed-point results in the “rectangular metric space” using a fractional-order functional differential equation.
A mappingT:X⟶X is called as the following:(1)
“Caccioppoli-type fixed-point theorem” for complete metric spaceX,d if for every n∈N,∀ℷ1″,ℷ2″∈X where δn>0 distinct from ℷ1″,ℷ2″ and 0<δ0<1 fulfils the subsequent condition:(15)dTnℷ1″,Tnℷ2″≤δndℷ1″,Tℷ1″+dℷ2″,Tℷ2″.Then, if series∑i=1∞δn is convergent, then T has a unique fixed point in X.(2)
“Kannan fixed-point theorem” for complete metric spaceX,d if for every n∈N,∀ℷ1″,ℷ2″∈X and 0<δ0<1/2 fulfils the subsequent condition:(16)dTℷ1″,Tℷ2″≤δndℷ1″,Tℷ1″+dℷ2″,Tℷ2″.
IfX is T-orbitally complete in X,d, then T has a unique fixed point in X.


### 2.3. Period-III (2001–2010)

#### 2.3.1. Fuzzy Quasi-Metric Space [33]

The associated impression of fuzzy metric space was expanded with a quasi-metric sense and then made known to the fuzzy quasi-metric space.LetM is a fuzzy set mapping on X×X×0,∞, that is, s.t.b. fuzzy quasi-metric (f.q.m.) on XX≠∅,∗-≠∅continuoust−norm, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″∈X,s,t>0.(1)
Mℷ1″,ℷ2″,0=0;(2)
ℷ1″=ℷ2″ if Mℷ1″,ℷ2″,t=Mℷ2″,ℷ1″,t=1;(3)
Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s≤Mℷ1″,ℷ2″,t+s;(4)
ℷ1″,ℷ2″:0,∞⟶0,1 is left continuous.Then, fuzzy quasi-metricM together with the set X is titled as partial metric space and represented as ordered pair X,M.Example 9.
LetX,d be a fuzzy quasi-metric space and be a usual multiplication ∀ℷ1″,ℷ2″∈0,1, and M is a fuzzy set mapping on X×X×0,∞ by(17)Mdℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″.
Then,X,Md is a fuzzy quasi-metric space.Remark 9.
(1)
Gregori and Romaguera proved a quasi-metrizable topology, which is generated via fuzzy quasi-metric space.(2)
In the fuzzy quasi-metric space (f.q.m.s.), the “contraction principle” was demonstrated with an appliance to the word domain.
A mappingT:X⟶X is titled as “B-contraction” on fuzzy quasi-metric space X,M,∗ that fulfils the following circumstances, MTℷ1″,Tℷ2″,kρ≥Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″∈X,k∈0,1,and all ρ>0, where k is called the contraction constant of T [34].


#### 2.3.2.G-Metric Space [35, 36]

In the G-metric space, there is a modification that the distance of three variables is equal to zero if all the variables are equal, and if one of them is different, then the distance is always positive.A mappingG:X3⟶R+ is s.t.b. G-metric (more specifically generalized metric) on XX≠∅, if it satisfied the subsequent axioms:(1)
Gℷ1″,ℷ2″,ℷ3″=0 if ℷ1″=ℷ2″=ℷ3″;(2)
Gℷ1″,ℷ2″,ℷ3″>0∀ℷ1″,ℷ2″,ℷ3″∈X,withℷ1″≠ℷ2″;(3)
Gℷ1″,ℷ1″,ℷ3″≤Gℷ1″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″∈Xwithℷ1″≠ℷ2″;(4)
Gℷ1″,ℷ2″,ℷ3″=Gℷ1″,ℷ3″,ℷ2″=Gℷ2″,ℷ3″,ℷ1″=…, (symmetry);(5)
Gℷ1″,ℷ2″,ℷ3″≤Gℷ1″,ℷ4″,ℷ4″+Gℷ4″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X (rectangle inequality).Then,G-metric G together with the set X is titled as G-metric space and represented as ordered pair X,G.Example 10.
Let (R,d be the usual metric space. We define Gs by(18)Gℷ1″,ℷ2″,ℷ3″=dℷ1″,ℷ2″+dℷ2″,ℷ3″+dℷ1″,ℷ3″,for all ℷ1″,ℷ2″,ℷ3″∈ R. Then, it is easy to see that R,G is a G-metric space.Remark 10.
(1)
InG-metric space, a theorem was introduced for the solution of integral equations.(2)
Existence and uniqueness of the “fixed-point theorem” in generalized metric space (orG-metric space) are proved.


#### 2.3.3.D∗-Metric Space [37]

When most of theD-metric space theorems were invalid, then the description of D-metric space had been possibly updated by Sedghi et al. [37] and introduced D∗-metric space. With some modifications, the basic properties of D∗-metric space (D∗-m.s.) were introduced.A mappingD∗:X3⟶0,∞ is s.t.b. generalized metric (or D∗-metric) on X≠∅, if fulfilled the subsequent conditions:∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X:(1)
D∗ℷ1″,ℷ2″,ℷ3″≥0;(2)
D∗ℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″;(3)
D∗ℷ1″,ℷ2″,ℷ3″=D∗pℷ1″,ℷ2″,ℷ3″, (symmetry) where p is a permutation function;(4)
D∗ℷ1″,ℷ2″,ℷ3″≤D∗ℷ1″,ℷ2″,ℷ4″+D∗ℷ4″,ℷ3″,ℷ3″.Then,D∗-metric D∗ together with the set X is titled as D∗-metric space and represented as ordered pair X,D∗.Example 11.
LetX=R, then we define(19)D∗ℷ1″,ℷ2″,ℷ3″=0,ifℷ1″=ℷ2″=ℷ3″maxℷ1″,ℷ2″,ℷ3″,o.w.
Then, it is easy to check thatX,D∗ is D∗-metric space.Remark 11.
A “fixed-point theorem” was developed for a class of mapping in completeD∗—metric space together with the condition of weakly commuting mappings.
A mappingS,T:X⟶ctsX,M:X⟶X in complete D∗-metric space satisfies the following conditions:(1)
M,S and M,T are weakly commuting pairs s.t.b. MX⊂SX∩TX;(2)
∃ϕ∈Φ such that for all ℷ1″,ℷ2″∈X,(20)D∗Mℷ1″,Mℷ2″,Mℷ3″≤ϕD∗Sℷ1″,Tℷ2″,Tℷ3″,D∗Sℷ1″,Mℷ1″,Mℷ1″,D∗Sℷ1″,Mℷ2″,Mℷ2″,D∗Tℷ2″,Mℷ1″,Mℷ1″,D∗Tℷ2″,Mℷ2″,Mℷ2″.
Then,M,S and T have a common unique fixed point in X.


#### 2.3.4. Cone Metric Space [38–42]

The cone metric space was introduced when the real numbers were replaced by ordering Banach’s space.LetE stand for “real Banach space” and C,E.A functiond:X2⟶E (where X≠∅ ) is s.t.b. a cone metric on X, if it meets the subsequent requirements:(1)
dℵ1″,ℵ2″>0,∀ℵ”1,ℵ”2∈X and ℵ”1=ℵ”2⇔dℵ”1,ℵ”2=0;(2)
dℵ1″,ℵ2″=dℵ2″,ℵ1″, ∀ℵ”1,ℵ”2∈X;(3)
dℵ1″,ℵ3″+dℵ3″,ℵ2″≥dℵ1″,ℵ2″, for all ℵ1″,ℵ2″,ℵ3″∈X.Then, function“d” composed of =set “X” is titled as a cone metric space and represented as X,d.Example 12.
LetE=R2,P=ℵ1″,ℵ2″∈E|ℵ1″,ℵ2″≥0⊂R2,X=Randd:X×X⟶E such that(21)dℵ1″,ℵ2″=ℵ1″−ℵ2″,αℵ1″−ℵ2″,where α≥0 is a constant. Then, X,d is a cone metric space.Remark 12.
A famous “fixed-point theorem” was proven for a nonordinary cone metric space, with the “generalized contractive condition” intended for a sequence of self-maps. As a consequence, in the cone metric space a generalized version of “Banach’s contraction principle” was proved [41].Theorem 3.
(1)
LetX,d be a complete cone metric space, and P be a normal cone with normal constant k. It is supposed that the mapping T:X⟶X fulfils the following condition:(22)dTℵ1″,Tℵ2″≤kdℵ1″,ℵ2″,∀ℵ1″,ℵ2″∈X,wherek∈0,1 is a constant.
Then,T has a unique fixed point in X. For any ℵ″∈X, iterative sequence Tnℵ″ converges to the fixed point.Theorem 4.
LetX,d be a sequentially compact cone metric space, and P be a regular cone.
It is supposed that the mappingT:X⟶X fulfils the following contractive condition:(23)dTℵ1″,Tℵ2″<dℵ1″,ℵ2″,∀ℵ1″,ℵ2″∈X,ℵ1″≠ℵ2″.
Then,T has a unique fixed point in X.Theorem 5.
LetX,d be a complete cone metric space, and P a normal cone with normal constant K. It is supposed that the mapping T:X⟶X,T has a unique fixed point in X if fulfils the contractive condition:(24)dTℵ1″,Tℵ2″≤kdTℵ1″,ℵ1″+dTℵ2″,ℵ2″,∀ℵ1″,ℵ2″∈X,where k∈0,1/2 is a constant. For any ℵ″∈X, iterative sequence Tnℵ″ converges to the fixed point.


#### 2.3.5. Multiplicative Metric Space [43, 44]

Initially, the usual metric withR+ was not complete. To overcome this problem, Bashirov et al. proposed the concept of multiplicative metric space with a multiplicative value of x∈R+ and multiplicative distance function d.A mappingd:X2⟶R+ is s.t.b. multiplicative metric on XX≠∅, when it meets the following conditions:(1)
dℷ1″,ℷ2″≥1,∀ℷ1″,ℷ2″∈X;(2)
dℷ1″,ℷ2″=1⇔ℷ1″=ℷ2″;(3)
ℷ”1,ℷ”2=dℷ”2,ℷ”1,∀ℷ1″,ℷ2″∈X;(4)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″.dℷ3″,ℷ2″ for all ℷ1″,ℷ2″,ℷ3″∈X (multiplicative triangle inequality).Then, multiplicative metricd together with the set X is titled as multiplicative metric space and represented as ordered pair X,d.Example 13.
LetR+nbe the set of all n-tuples of non-negative real numbers. Let d:R+n×R+n⟶R be a mapping defined as follows:(25)dℷ1″,ℷ2″=ℷ1,1″ℷ2,1″∙ℷ1,2″ℷ2,2″⋯ℷ1,n″ℷ2,n″,where ℷ1″=ℷ1,1″,ℷ1,2″,⋯,ℷ1,n″, ℷ2″=ℷ2,1″,ℷ2,2″,⋯,ℷ2,n″∈R+n, and ∙:R+⟶R are defined by the following:(26)ℷ1″=ℷ1″,ifℷ1″≥1,1ℷ1″,ifℷ1″<1.
It is easy to get thatR+n,d is a multiplicative metric space.Remark 13.
We gave the description of multiplicative contraction mapping and “fixed-point theorems” on a complete multiplicative metric space.
A mappingT:X⟶X is called as follows:(1)
“Multiplicative contraction” of multiplicative metric space if∃τ∈0,1,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(27)dTℷ1″,Tℷ2″≤dℷ1″,ℷ2″τ.(2)
“Multiplicative Kannan’s contraction” of multiplicative metric space if∃τ∈0,1/2,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(28)dTℷ1″,Tℷ2″≤dℷ1″,Tℷ1″dℷ2″,Tℷ2″τ.(3)
“Multiplicative Chatterjea’s” contraction of multiplicative metric space if∃τ∈0,1/2,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(29)dTℷ1″,Tℷ2″≤dℷ1″,Tℷ2″dℷ2″,Tℷ1″τ.


#### 2.3.6. Vector Metric Space [45, 46]

Cevik and Altun were made known to the vector metric space in which metric space takes the value on a Riesz space, i.e., Riesz space replaced from ℝ.The functiond:X×X⟶E is s.t.b. vector metric on XwhereX≠∅,E-Riesz space) if it satisfied the following:(1)
dℷ1″,ℷ2″=0 if ℷ1″=ℷ2″,(2)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″∀ℷ1″,ℷ2″,ℷ3″∈X.Then, vector metricd together with the set X is titled as vector metric space and termed as triple X,d,E (briefly X with the default parameters omitted).Example 14.
Let a Riesz space“E” and a mapping d:E×E⟶E are defined by the following:(30)dℷ1″,ℷ2″=ℷ1″−ℷ2″.
It is easy to show a vector metric space.Remark 14.
Cevik and Altun gave “Banach’s contraction principle” in the vector metric space.
LetV be an E-complete vector metric space (v.m.s.), (where “E is Archimedean”).
The mappingT:V⟶V satisfies the contractive condition dTℷ1″,Tℷ2″≤θdℷ1″,ℷ2″,∀ℷ2″,ℷ2″∈V, where θ∈0,1 is a constant.
Then,T has a unique fixed point in V and iterative sequence ℷn″, ∀ℷ0″∈V, defined through ℷn″=Tℷn″−1 for n∈N,E-converges on the fixed point of T.


### 2.4. Period-IV (2011–2021)

#### 2.4.1. Complex-Valued Metric Space [47, 48]

The explanation of the complex-valued metric spaces (c.m.s.) occurred when replaced range ℂ (with a certain order) form rangeR (with a normal order), i.e., the generalization of the contractive condition in metric space made much of the concept available. There was a considerable amount of content available in generalized metric spaces. In order to satisfy a rational inequality, complex-valued metric space was thus introduced and some results were also developed for mapping.A mappingd:X2⟶C is s.t.b. complex-valued metric space (c.m.s.) on X(X≠∅, if fulfilled the subsequent condition:(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈XAnd dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″, for allℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈X.Then, complex-valued metricd together with the set X is titled as complex-valued metric space and represented as ordered pair X,d.Example 15.
LetX=C be a set of complex number. We define d:C×C⟶C by(31)dℷ1″,ℷ2″=ℷ1,1″−ℷ2,1″+iℷ1,2″+ℷ2,2″,where ℷ1″=ℷ1,1″+ℷ1,2″ and ℷ2″=ℷ2,1″+ℷ2,2″.
Then,C,d is a complex-valued metric space.Remark 15.
In the complex-valued metric space (c.m.s.), it provides the extension of “Banach’s fixed-point theorem,” which helps to give a common unique solution to the Urysohn integral equations.(i)
xσ=∫abK1σ,τ,xτdτ+gσ,(ii)
xσ=∫abK2σ,τ,xτdτ+hσ, where σ∈a,b,x,g,h∈X.
A mappingT,f:X⟶X be two functions in metric space X,d. A mapping T is called f−contraction if ∃φ∈0,1 such that(32)dfTℷ1″,fTℷ2″≤φdfℷ1″,fℷ2″,∀ℷ1″,ℷ2″∈X.


#### 2.4.2. Coneb-Metric Space [49]

The generalization of cone metric space (C.M.S.) together with b-metric space is termed as cone b-metric space.A mappingd:X2⟶E is s.t.b. cone b-metric on X(where X≠∅,s≥1∈R+) if for all C1,C2,C3∈X met the following conditions:(1)
dC1,C2>0 with C1≠C2 and 𝑑C1,C2=0 if and only if C1=C2;(2)
C1,C2=dC2,C1;(3)
dC1,C2≤sdC1,C3+dC3,C2.Then, coneb-metric d together with the setX is titled as cone b-metric space and represented as an ordered pair X,d.Example 16.
LetE=l1,P=un∈E/un≥0,∀n≥1, and d:X×X⟶E such that(33)d∗C1,C2=dC1,C22n,forn≥1.
Then,X,d is a cone b-metric space with coefficient r=21/p>1.Remark 16.
N. Sharma proved contractive mapping for “fixed-point theorem” with rational expression and without normality condition, in cone b-metric space.Theorem 6.
Suppose the mappingT:X⟶X and let X,d be a complete cone b-metric space with the coefficient s≥1 that satisfies the following condition, then(34)dTC1,TC2≤αdC1,TC1dC2,TC2dC1,C2+βdC1,C2,for C1,C2∈X and α,β∈0,1 with α+β<1. Then, T has a unique fixed point in X.


#### 2.4.3.S-Metric Space [50, 51]

The philosophy of S-metric spaces is the modification of D-metric and G-metric spaces in which the 3-dimensional metric space was made known by Sedghi et al. InS-metric space, the distance between all three distinct points is always greater than or equal to zero.A functionS:X3⟶0,∞ is s.t.b. S-metric space on X≠∅ if it fulfils the following axioms: ∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X,(1)
Sℷ1″,ℷ2″,ℷ3″≥0;(2)
Sℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″;(3)
Sℷ1″,ℷ2″,ℷ3″≤Sℷ1″,ℷ2″,ℷ4″+Sℷ1″,ℷ2″,ℷ4″+Sℷ1″,ℷ2″,ℷ4″.Then,S-metric S together with the set X is titled as S-metric space and represented as an ordered pair X,S.Example 17.
LetX=Rn and , a norm on X, then(35)Sℷ1″,ℷ2″,ℷ3″=ℷ2″+ℷ3″−2ℷ1″+ℷ2″+ℷ3″,is an S-metric on X.Remark 17.
Ozgur and Tas characterized the thought of a fixed circle and explored the existence and uniqueness of the “fixed-point theorem” for the self-mapping on S-metric space. Likewise, a few instances of self-mappings having fixed circles are given with a mathematical perspective.
Some new contractive mapping and the concept ofCs and Ls mapping on S-metric space were established.Theorem 7.
LetX;S be a complete S-metric space, and A;B:X⟶X be the mappings that fulfil the subsequent conditions:(1)
AX⊆BX and either AXorBX is a closed subset of X,(2)
The pairA;B is weakly compatible,(3)
SAℷ1″;Aℷ2″;Aℷ3″≤φmaxSBℷ1″;Bℷ2″;Bℷ3″;k1SBℷ3″;Aℷ1″;Aℷ3″;k2SBℷ3″;Aℷ2″;Aℷ3″ for all ℷ1″,ℷ2″,ℷ3″∈X and 0<k1;k2<1, where φ∈∅.
Then, the mapsA and B have a unique common fixed point. If B is continuous at the fixed point p, then A is also continuous at p.


#### 2.4.4. Metric-Like Space [29]

Metric-like space (m.l.s.) is a generalization of partial metric space, in which all axioms are the same as the usual metric except first, that isσx,x may be positive for x∈X.A functionσ:X2⟶R+∪0iss.t.b.metric−likespaceonX≠∅, if it fulfilled the subsequent axioms: for any ℷ1″,ℷ2″∈X,(1)
σℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(2)
σℷ1″,ℷ2″=σℷ2″,ℷ1″;(3)
σℷ1″,ℷ2″≤σℷ1″,ℷ3″+σℷ3″,ℷ2″.Then, metric-likeσ together with the set X is titled as metric-like space and represented as an ordered pair X,σ.Example 18.
LetX=0,1 and let(36)σℷ1″,ℷ2″=2,ifℷ1″=ℷ2″=0,1,otherwise.
It is easy to getX,σ that is metric-like space.Remark 18.
(1)
As an application, A. Harandi unified and generalized some important results of the “fixed-point theory.”(2)
Metric-like space considered new Geraghty-type mappings and proposed “fixed-point theorems” for such space [52].
A topologyτσ for each metric-like σ was generated, which is a family of open balls. Bσx,ε=y∈X:σx,y–σx,x<ε,∀x∈X and ε>0.
Then, a sequencexn converges to a point x∈X if limn⟶∞σxn,x=σx,x.


#### 2.4.5.b-Metric-Like Space [53, 54]

Theb-metric-like space (b-m.l.s.) is a generalized version of metric-like space in which coefficient of metric space was initiated in the triangle inequality.A mappingσ:X2⟶R+, (where X≠∅) is s.t.b. b-metric-like on X if ∀ℷ1″,ℷ2″,ℷ3″∈X and constant s≥1 fulfils the following axioms:(1)
σℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(2)
σℷ1″,ℷ2″=σℷ2″,ℷ1″;(3)
σℷ1″,ℷ2″≤sσℷ1″,ℷ3″+σℷ3″,ℷ2″.Then,b-metric-like σ together with the set X is titled as b-metric-like space and represented as an ordered pair X,σ.Example 19.
LetX=R+,ρ>1 a constant, and σb:X×X⟶R+ be defined by(37)σbℷ1″,ℷ2″=ℷ1″+ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,σb is a b-metric-like space with coefficient =2ρ−1.Remark 19.
Some results ofb-metric space are related to the fixed point and give nontrivial examples to support the concept such as finding the solution of the electric circuit equation with the help of second-order differential equations.
A mappingT:X⟶X is called(1)
Banach G-contraction ifT preserves edges of G, i.e., ∀ℷ”1,ℷ”2∈X;(38)ℷ1″,ℷ2″∈EG⇒Tℷ1″,Tℷ2″∈EG.(2)
Orbitally continuous if givenℷ1″∈X and any sequence kn of positive integers,(39)Tknℷ1″⟶ℷ2″asn⟶∞⇒TTknℷ1″⟶Tℷ2″asn⟶∞.


#### 2.4.6. Quasi-b-Metric Space [53–55]

The class of ordinary quasi-metric space was extended to form quasi-b-metric space.A functiond:X2⟶R+ is entitled as quasi-b-metric on X ≠∅ and constant s≥1, if it satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″≥0, ∀ℷ1″,ℷ2″∈X;(2)
dℷ1″,ℷ2″=0 ⇔ ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″≤sdℷ1″,ℷ3″+dℷ3″,ℷ2″, ∀ℷ1″,ℷ2″″,ℷ3″∈X.Then, quasi-b-metric d together with the set X is titled as quasi-b-metric space and represented as an ordered pair X,d.Example 20.
LetX=lp, where1≤p<∞, be defined by(40)lp=ℷ″1nn≥1⊆R:∑n=1∞ℷ″1np<∞.
We defined:X×X⟶R+ by(41)dℷ1″,ℷ2″=0,ifℷ1″≤ℷ2″,∑n=1∞ℷ″1np1/p,ifℷ1″≥ℷ2″.
It is easy to get thatX,d is a quasi-b-metric space with constant p≥1.Remark 20.
(1)
It is proved that some “fixed-point theorems” for a nondecreasing mapping satisfied the nonlinear contraction.(2)
Partially well-ordered complete quasi-b-metric space was used to solve the fractional differential equations [55].
A mappingT:X⟶X is called α-admissible for nonempty set X, if ∀ℷ1″,ℷ2″∈X, where α:X×X⟶0,∞ be a mapping, then we have(42)αℷ1″,ℷ2″≥1⇒αTℷ1″,Tℷ2″≥1.


#### 2.4.7. Soft Metric Space [56, 57]

When difficulties occur due to inadequacy of the parameterization tool of the theory, then the new mathematical tool “the soft set” theory was introduced by Molodtsov, for dealing with uncertainties and inherent difficulties of theories. Das and Samanta initiated the analysis of soft metric space (s.m.s.) created on soft-point soft sets, for implementations of soft set theory in real-life problems and other domains for better performance.LetA be a nonempty subset of parameters, and ℵ be the absolute soft set, i.e., Fλ=ℷ″,∀λ∈A, where F,A=ℷ″.A mappingd:SPℷ″×SPℷ″⟶RA∗ is called soft metric on ℵ if, for any ℷ″ax,ℷ″by,ℷ″cz∈SPℷ″ (equivalently, ℷ″ax,ℷ″by,ℷ″cz∈ℷ″), satisfied the following conditions:(1)
dℷ″ax,ℷ″by≥0(2)
dℷ″ax,ℷ″by=0 ⇔ ℷ″ax=ℷ″by(3)
dℷ″ax,ℷ″by=dℷ″by,ℷ″ax(4)
dℷ″ax,ℷ″by≤dℷ″ax,ℷ″cz+dℷ″cz,ℷ″by.The soft metric space is symbolized byℷ″,d,A.Example 21.
Letℷ″=A=1/n:n∈N, and the mapping d:SPℷ″×SPℷ″⟶RA∗ be defined by(43)dℷ″ax,ℷ″by=x¯−y¯+a¯−b¯,∀ℷ″ax,ℷ″by∈SPℷ″.
Then, it is easy to show a soft metric space onℷ″.Remark 21.
The soft metric space is used to prove the soft version of the “contraction principle” in complete soft metric space, which provides applications related to dynamic programming.
A soft mappingf:ℷ″⟶ℷ″ and let ℷ″,d,A be a complete soft metric space with finite set A that satisfies(44)dfℷ″ax,fℷ″by≤τdℷ″ax,fℷ″axdℷ″by,fℷ″by,∀ℷ″ax,ℷ″by∈SPℷ″,where 0≤τ≤1/2, and then f has unique fixed point.


#### 2.4.8. Partialb-Metric Space [58, 59]

S. Shukla introduced partialb-metric space and the generalization of partial metric space (p.m.s.) together with b-metric space.A mappingb:X2⟶R+ is s.t.b. partial b-metric (p.b-m.) on X≠∅, if it satisfied the subsequent axioms: (for all ℷ1″,ℷ2″,ℷ3″∈X),(1)
bℷ1″,ℷ1″=bℷ1″,ℷ2″=bℷ2″,ℷ2″⇔ℷ1″=ℷ2″;(2)
bℷ1″,ℷ1″≤bℷ1″,ℷ2″;(3)
bℷ1″,ℷ2″=bℷ2″,ℷ1″;(4)
∃ a real number s≥1 such that(45)bℷ1″,ℷ2″≤sbℷ1″,ℷ3″+bℷ3″,ℷ2″−bℷ3″,ℷ3″.Then, partialb-metric b together with the set X is titled as partial b-metric space and represented as an ordered pair X,b, and s is the coefficient of X,b.Example 22.
LetX=R+,ρ>1 a constant, and b:X×X⟶R+ be defined by(46)bℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,σb is a partial b-metric space with coefficient s=2ρ>1.Remark 22.
Analog of “Kannan’s type fixed-point theorem” and “Banach’s contraction theorem” was proved in the partialb-metric space.Theorem 8.
Suppose the mappingT:X⟶X and let X,b be a complete partial b-metric space with the coefficient s≥1 that satisfies the following condition:(47)bTℷ1″,Tℷ2″≤βbℷ1″,ℷ2″.
For allℷ1″,ℷ2″∈X and where β∈0,1, then T has a unique fixed point u∈X and bu,u=0.


#### 2.4.9. Fuzzy Cone Metric Space [60]

The class of fuzzy metric space (f.m.s.) was extended and generalized into fuzzy cone metric space.A mappingd:X2⟶E∗I is s.t.b. fuzzy cone metric (f.c.m.) on X≠∅, if it satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″≥0;∀ℷ1″,ℷ2″∈X and dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″∀ℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″⊕dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈X.Then, fuzzy cone metricd together with the set X is titled as fuzzy cone metric space and represented as an ordered pair X,d.Example 23.
LetP be any cone and X=N and ℷ1″∗ℷ2″=ℷ1″ℷ2″ and d:X2×intP⟶0,1 be defined by(48)dℷ1″,ℷ2″,t=ℷ1″ℷ2″,,ifℷ1″≤ℷ2″,ℷ2″ℷ1″,ifℷ1″≥ℷ1″.∀ℷ1″,ℷ2″∈Xand t≫0. Then, it is easy to show that it is fuzzy cone metric space.Remark 23.
(1)
In fuzzy cone metric space (f.c.m.s.), some of the basic properties for innumerable types of contraction mappings and for “fixed-point theorems” were established.(2)
Chet et al. provide an application of nonlinear integral equation in fuzzy cone metric space for the existence and uniqueness of the solution.
A mappingT:X⟶X is titled as fuzzy cone contraction on fuzzy cone metric space X,d,∗ if ∃k∈0,1 satisfies the following:(49)1dTℷ1″,Tℷ2″,t−1≤k1dℷ1″,ℷ2″,t−1,∀ℷ”1,ℷ”2∈X and t≫0. t is called the contraction constant of T.


#### 2.4.10. Fuzzy Coneb-Metric Space [61]

The definition of fuzzy coneb-metric space (f.c.b-m.s.), inspired by cone b-metric space (c.b-m.s.) and fuzzy metric space, was made known by T. Bag.A mappingd:X2⟶E∗Iis s.t.b. fuzzy cone b-metric (where X≠∅ and s≥1 be a real number) if it fulfils the following axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈X and dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤sdℷ1″,ℷ3″⊕dℷ3″,ℷ2″.Then, fuzzy coneb-metric d together with the set X is titled as fuzzy cone b-metric space and represented as an ordered pair X,d.Remark 24.
Some results of “Banach’s contraction mapping” for the “fixed-point theorem” were established in fuzzy coneb-metric space.Theorem 9.
A mappingT:X⟶X on a complete fuzzy cone b-metric space X,d has a unique fixed point in X if for complete fuzzy cone b-metric space X,d with the coefficient s≥1fulfils the subsequent condition:(50)dTℷ1″,Tℷ2″≤βdℷ1″,ℷ2″,
for allℷ1″,ℷ2″∈X and whereβ∈0,1.


#### 2.4.11. Fuzzy Soft Metric Space [62–64]

For the fuzzy soft metric space (f.s.m.s.), first fuzzy soft point was introduced in which problems deal with ambiguous data in several fields and that was not effectively demonstrated in mathematics. There are various kinds of mathematical instruments to deal with uncertainties, i.e., the fuzzy set and the soft set that helps to solve the problem difficulty in all areas. In the fuzzy soft metric space (f.s.m.s.), the fuzzy soft open ball and also the closed ball were proposed.LetX be the absolute fuzzy soft set,(51)A∙Ei.e.,Xe=τ,∀e∈E.A∗ be the set of all non-negative fuzzy soft real numbers.A mappingd:FSCX×FSCX⟶RA∗ is s.t.b. fuzzy soft metric (f.s.m.s.) on the soft setX, if it holds the subsequent axioms:(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈FSCX;(2)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″=dℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈FSCX;(4)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈FSCX.Then, fuzzy soft metricd together with the set X is titled as fuzzy soft metric space and represented as an ordered pair X,d.Remark 25.
Fuzzy soft metric is applied on soft open balls and soft closed balls.Theorem 10.
A mappingTρ:E,d⟶E,d is called fuzzy soft contractive mapping on fuzzy soft metric space E,d if ∃two fuzzy soft number ψwith0≤ψ≤1 such that(52)dTρℷ1e″,Tρℷ″2e≤ψdℷ1e″,ℷ″2e,∀ℷ1e″,ℷ″2e∈FSCX.Theorem 11.
A mappingT,ρ:E,d⟶E,d is called fuzzy softα−ψ-contractive mapping on fuzzy soft metric space E,d if ∃ two fuzzy soft functions α,ϕ:FSCE×FSCE⟶RE∗ and ψ∈Ψ such that(53)α,ϕℷ1e″,ℷ″2edT,ρℷ1e″,T,ρℷ″2e≤ψdℷ1e″,ℷ″2e,∀ℷ1e″,ℷ″2e∈FSCX.


#### 2.4.12. M-Metric Space [65, 66]

In partial metric space, the self-distance ofx is less than the distance between two points so another variable is free to take any value. Thus, M-metric space improved the triangle inequality by replacing it with the minimum of self-distance ofxandy, also improved the triangle inequality.A functionm:X2⟶R+ is s.t.b. an m-metric (where X≠∅), if it satisfied the following conditions:(1)
mℷ1″,ℷ1″=mℷ2″,ℷ2″=mℷ1″,ℷ2″if ℷ1″=ℷ2″;(2)
mℷ1″,ℷ2″≤mℷ1″,ℷ2″;(3)
mℷ1″,ℷ2″=mℷ2″,ℷ1″;(4)
mℷ1″,ℷ2″–mℷ1″,ℷ2″≤mℷ1″,ℷ3″–mℷ1″,ℷ3″+mℷ3″,ℷ2″–mℷ3″,ℷ2″.Then,M-metric m together with the set X is titled as M-metric space and represented as an ordered pair X,m.where(54)mℷ1″,ℷ2″=minmℷ1″,ℷ1″,mℷ2″,ℷ2″,Mℷ1″,ℷ2″=maxmℷ1″,ℷ1″,mℷ2″,ℷ2″.Example 24.
LetX=0,∞, then(55)mℷ1″,ℷ2″=ℷ1″+ℷ2″2.
It is easy to get thatm is an m-metric space on X.Remark 26.
M. Asadi et al. generalized the contraction mapping on M-metric space for getting the “fixed-point theorem.”
A mappingT:X⟶X has a fixed a unique fixed point on a complete M-metric space X,m if satisfies the following condition:(56)∃k∈0,1suchthatmTℷ1″,Tℷ2″≤kmℷ1″,ℷ2″∀ℷ1″,ℷ2″∈X,or(57)∃k∈0,12suchthatmTℷ1″,Tℷ2″≤kmℷ1″,Tℷ1″,mℷ2″,Tℷ2″∀ℷ1″,ℷ2″∈X.


#### 2.4.13. 2-Partial Rectangular Metric Space [67]

Partial rectangular metric space (p.m.s.) is the generalized version of rectangular metric space (r.m.s.) and the extension of partial metric space (p.m.s.).A functionρ:X2⟶R is s.t.b. partial rectangular metric (p.r.m.) on X≠∅}) if satisfied the subsequent axioms:(1)
ρℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈X;(2)
ℷ1″=ℷ2″⇔ρℷ1″,ℷ2″=ρℷ1″,ℷ1″=ρℷ2″,ℷ2″,∀ℷ1″,ℷ2″∈X;(3)
ρℷ1″,ℷ1″≤ρℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X;(4)
ℷ1″,ℷ2″=ρℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈X;(5)
ρℷ1″,ℷ2″≤ρℷ1″,ℷ3″+ρℷ3″,ℷ4″+ρℷ4″,ℷ2″−ρℷ3″,ℷ3″−ρℷ4″,ℷ4″, for all ℷ1″,ℷ2″∈X and for all distinct points ℷ3″,ℷ4″∈X|ℷ1″,ℷ2″.Then, partial rectangular metricρ together with the set X is titled as partial rectangular metric space and represented as an ordered pair X,ρ.Example 25.
LetX=0,b,φ≥b≥3 define a mapping ρ:X2⟶R by(58)ρℷ1″,ℷ2″=ℷ1″,ifℷ1″=ℷ2″,3b+ℷ1″+ℷ2″2,ifℷ1″,ℷ2″∈1,2,ℷ1″≠ℷ2″b+ℷ1″+ℷ2″2,otherwise.,
Then, it is easy to see thatX,d is the partial rectangular metric space.Remark 27.
Y. Cheng and Y. C. Shiab presented quasi-type contraction for providing the “fixed-point theorem.”
A mappingT:X⟶X(1)
is calledquasi-contraction for complete partial rectangular metric spaceX,ρ with constant k∈0,1 so that ∀ℷ1″,ℷ2″∈X satisfies the following condition:(59)ρTℷ1″,Tℷ2″≤kmaxρℷ1″,ℷ2″,ρℷ1″,Tℷ1″,ρℷ2″,Tℷ2″,ρℷ1″,Tℷ2″,ρℷ2″,Tℷ1″.(2)
isquasi-contraction for complete partial rectangular metric space X,ρ with constant k. Then, T has a unique fixed point ϑ∈X and ρϑ,ϑ=0.


#### 2.4.14. Fuzzyb-Metric Space [68]

Partial rectangular metric space (p.m.s.) is the generalized version of rectangular metric space (r.m.s.) and the extension of partial metric space (p.m.s.).Let≠∅) and b≥1∈R and ∗ be a continuous t-norm. A fuzzy set M in X×X×0,∞ is termed as fuzzy b-metric on X if ∀f1″,f2″,f3″∈X, which fulfilled the subsequent axioms:(1)
Mℷ3″,ℷ2″,0=0;(2)
Mℷ3″,ℷ2″,t=1,∀t>0 ⇔ ℷ3″=ℷ2″;(3)
Mℷ1″,ℷ2″,t=Mℷ1″,ℷ1″,t;(4)
Mℷ1″,ℷ2″,t+s≥Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s,∀t,s≥0;(5)
IfMℷ1″,ℷ2″,.:0,∞⟶0,1 is left continuous, andlimt⟶∞Mℷ1″,ℷ2″,t=1.Then, fuzzyb-metric M together with the set X is titled as fuzzy b-metric space and represented as quadruple X,M,∗,b.Example 26.
LetX,d,k be b-metric space. Let Md:X×X×0,∞⟶0,1 be defined by(60)Mdℷ1″,ℷ2″,thinspptt+dℷ1″,ℷ2″,ift>0,0,ift=0.
Then,X,Md,∧,k be fuzzy b-metric space.Remark 28.
(1)
Abbas et al. proposed the definition of “ψ-contraction” and monotone “ψ-contraction” in fuzzy b-metric space.(2)
“Banach’s contraction principle” is extended as an application to the linear equation.
Let a mappingF:X⟶X be correspondence in fuzzy b-metric space such that ∀ℷ1″,ℷ2″∈X and F1∈Fℷ1″,F2∈Fℷ2″ satisfy the following condition:(61)MF1,F2,t≤Mℷ1″,ℷ2″,tk,where k∈0,1 and t∈R. Then, F has unique fixed point.
Chauhan and Utreja [69] proved one important result on fuzzy 2-metric space on the same line.


#### 2.4.15. Fuzzy Quasi-b-Metric Space [68]

Nadaban extended the perception of fuzzy quasi-metric space (f.q.m.s.) andb-metric space (b-m.s.).LetM is a fuzzy set in X×X×0,∞ (where X≠∅, ∗ is a continuous t-norm, and k≥1 is a real number) is called fuzzy quasi-b-metric space if it satisfied the subsequent condition: (∀ℷ1″,ℷ2″,ℷ3″∈X, we have the following:(1)
Mℷ1″,ℷ2″,0=0;(2)
Mℷ”1,ℷ”2,t=Mℷ”2,ℷ”1,t=1,∀t>0,⇔ℷ1″=ℷ2″;(3)
Mℷ1″,ℷ2″,kt+s≥Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s,∀t,s≥0;(4)
Mℷ1″,ℷ2″,·:0,∞⟶0,1 is left continuous, and limt⟶∞Mℷ1″,ℷ2″,t=1.Then, fuzzy quasi-b-metric M together with the set X is titled as fuzzy quasi-b-metric space and represented as quadruple X,M,∗,k.Example 27.
LetX,d,k be quasi-b-metric space. Let Md:X×X×0,∞⟶0,1 be defined by(62)Mdℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″,ift>0,0,ift=0.
Then,X,Md,∧,k be fuzzy quasi-b-metric space.Remark 29.
Fuzzy quasi-b-metric space (f.q.b-m.s.) is used to apply in the control theory and convex optimization.
LetX,Md,∧,k be fuzzy quasi-b-metric space and(63)Mℷ1″,ℷ2″,t=minMdℷ1″,ℷ2″,t,Mdℷ2″,ℷ1″,t.
Then,X,M,∧,k is a fuzzy b-metric space.


#### 2.4.16.Mb-Metric Space [70–72]

TheMb-metric space is an enlargement of M-metric space by introducing the coefficient of the Mb-metric space (real number s≥1) in triangle equality. The concept of an extension was based on the simulation functions made known by Khojasteh et al.A functionmb:X2⟶R+ is s.t.b. Mb-metric space on X≠∅, if it satisfied the subsequent conditions: ∀ℷ1″,ℷ2″,ℷ3″∈X, we have(1)
mbℷ1″,ℷ1″=mbℷ2″,ℷ2″=mbℷ1″,ℷ2″ if ℷ1″=ℷ2″;(2)
mbℷ1″,ℷ2″≤mbℷ1″,ℷ2″;(3)
mbℷ1″,ℷ2″=mbℷ2″,ℷ1″;(4)
∃a real number s≥1 s.t.b. ∀ℷ1″,ℷ2″,ℷ3″∈X, we have(64)mbℷ1″,ℷ2″–mbℷ1″,ℷ2″≤smbℷ1″,ℷ3″−mbℷ1″,ℷ3″+mbℷ3″,ℷ2″–mbℷ3″,ℷ2″−mbℷ3″,ℷ3″.Then,Mb-metric Mb together with the set X is titled as Mb-metric space and represented as quadruple X,Mb.The numbers is termed as the coefficient of the Mb-metric space X,Mb.(65)mbℷ1″,ℷ2″=minmbℷ1″,ℷ1″,mbℷ2″,ℷ2″,Mbℷ1″,ℷ2″=maxmbℷ1″,ℷ1″,mbℷ2″,ℷ2″.Example 28.
LetX=R+,ρ>1 be a constant, and Mb:X×X⟶R+ be defined by(66)Mbℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,Mb is an Mb-metric space with coefficient s=2ρ>1.Remark 30.
We demonstrate the existence of the “fixed-point theorem” with the assistance of″Zmb contraction” in complete Mb-metric space.Theorem 12.
Suppose the mappingT:X⟶X and let X,Mb be a complete Mb-metric space with the coefficient s≥1 that satisfies the following condition:(67)MbTℷ1″,Tℷ2″≤βbℷ1″,ℷ2″,
for allℷ1″,ℷ2″∈X and where β∈0,1. Then, T has a unique fixed point u∈X and Mbu,u=0.


#### 2.4.17. Orthogonal Metric Space [73, 74]

Gordji et al. defined a binary relation on the set.Letd be a usual metric on X≠∅ with an orthogonal set X,⊥. Then, metric with an orthogonal set is termed as orthogonal metric space and represented as X,⊥,d (O-metric space).Example 29.
LetX=0,∞, and we define ℷ1″⊥ℷ2″,if ℷ1″ℷ2″≤ℷ1″orℷ1″ℷ2″≤ℷ2″. Then, ∀ℷ1″∈X, 0⊥ℷ1″. So X,⊥ is an orthogonal set, and if X,d be an usual metric, then X,⊥,d is orthogonal metric space.Remark 31.
Gordji et al. gave a generalized version of “Banach’s contraction principle” with the orthogonal set, which stands for finding the existence and uniqueness of first-order ordinary differential equations and nonlinear fractional differential equations that are allied with Caputo fractional derivative.
A mappingT:X⟶X is called the following:(1)
Orthogonal contraction of orthogonal metric space isX,⊥,d if ∃τ∈0,1satisfies the subsequent condition:(68)dTℷ1″,Tℷ2″≤τdℷ1″,ℷ2″.For any two orthogonally related elements,ℷ1″,ℷ2″∈X.(2)
Orthogonalp-contraction of orthogonal metric space is X,⊥,d and p:X2⟶0,∞ if ∃τ∈0,1satisfies the following condition:(69)pTℷ1″,Tℷ2″≤τpℷ1″,ℷ2″.∀ℷ1″,ℷ2″∈X with ℷ1″⊥ℷ2″.
In the same line, Khalil et al. [75] proved some theorems on orthogonal partial-metric spaces with an application.


#### 2.4.18. Rectangularb-Metric Space [22, 76, 77]

George et al. made known the theory of rectangularb-metric space (r.b-m.s.), which is an extension of rectangular metric space (r.m.s.) and the generalization of b-metric space (b-m.s.).A mappingd:X2⟶0,∞ is s.t.b. rectangular b-metric on X(where X≠∅), if fulfils the following axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
ℷ1″,ℷ2″=dℷ2″,ℷ1″,∀r1′,r2′∈X;(3)
∃a real number s≥1 s.t.b.(70)dℷ1″,ℷ2″≤sdℷ1″,ℷ3″+dℷ3″,ℷ4″+dℷ4″,ℷ2″,∀ℷ1″,ℷ2″∈Xandℷ3″,ℷ4″∈X|ℷ1″,ℷ2″,ℷ1″≠ℷ1″.Then, rectangularb-metric d together with the set X is titled as rectangular b-metric space and represented as an ordered pair X,d with coefficient s.Example 30.
LetX=N define d:X2⟶X by(71)dℷ1″,ℷ2″=0,ifℷ1″=ℷ2″,4ρ,ifℷ1″,ℷ2″∈1,2andℷ1″≠ℷ2″,ρ,ifℷ1″orℷ2″∉1,2andℷ1″≠ℷ2″,where ρ>0 is a constant. Then, X,d is rectangular b-metric space with coefficient s=4/3>1.Remark 32.
George et al. gave an analog of “Kannan’s fixed-point theorem” and “Banach’s contraction principle” for generalizing several outcomes in fixed-point theory.
A mappingT:X⟶X has a unique fixed point on a complete rectangular b-metric space X,d with coefficient s>1, if satisfies the following condition:(72)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″,where k∈0,1/s∀ℷ1″,ℷ2″∈X.or(73)dTℷ1″,Tℷ2″≤kdℷ1″,Tℷ1″,dℷ2″,Tℷ2″,where k∈0,1/s+1, ∀ℷ1″,ℷ2″∈X.


#### 2.4.19. Extendedb-Metric Space [78, 79]

The coefficient ofb-metric space was replaced with the function ″e” and formed extended b-metric space (e.b-m.s.), which is the most universal form of b-metric space.A mappingd:X2⟶0,∞ is s.t.b. an extended b-metric on M with mapping e (where X≠∅ and e:X2⟶1,∞), if it fulfils the following axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″ for all ℷ1″,ℷ2″∈M;(3)
dℷ1″,ℷ2″≤eℷ1″,ℷ2″dℷ1″,ℷ3″+dℷ3″,ℷ2″ for all ℷ1″,ℷ2″,ℷ3″∈M.Then, extendedb-metric d together with the set X is titled as extended b-metric space and represented as an ordered pair X,d.Example 31.
LetX=1,2,3. We define e:X2⟶0,∞ and de:X2⟶0,∞ as(74)eℷ1″,ℷ2″=1+ℷ1″+ℷ3″,de1,1=de2,2=de3,3=0,de1,2=de2,1=80,de1,3=de3,1=1000,de2,3=de3,2=600.
Hence,X,dis an extended b-metric space.Remark 33.
(1)
Mitrovic et al. proved single-valued mapping and multivalued mapping (m.v.m.) in extendedb-metric space.(2)
Kamran et al. gave the existence of the unique solution as an application to the Fredholm integral equation.


#### 2.4.20. Extended Fuzzyb-Metric Space [80]

For setting the most general class of fuzzyb-metric space, we extended and generalized into such space.A fuzzy setMα in X×X×0,∞ (where X≠∅, α:X2⟶1,∞ and ∗ -continuous t-norm) is called extended fuzzy b-metric (e.f.b-m.s.) on X if satisfied the following axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X)(1)
Mαℷ1″,ℷ2″,0=0;(2)
Mαℷ1″,ℷ2″,t=1,∀t>0 ⇔ ℷ1″=ℷ2″;(3)
Mαℷ1″,ℷ2″,t=Mαℷ2″,ℷ1″,t;(4)
Mαℷ1″,ℷ2″,αℷ1″,ℷ2″t+s≥Mαℷ1″,ℷ3″,t∗Mαℷ3″,ℷ2″,s,∀t,s≥0;(5)
Mαℷ1″,ℷ2″,.:0,∞⟶0,1 is left continuous.Then, extended fuzzyb-metric Mα together with the set X is titled as extended fuzzy b-metric space and represented as X,Mα,∗,αℷ1″,ℷ2″.Example 32.
LetX=1,2,3 and let db:X×X⟶R by dℷ”1,ℷ”2=ℷ”1−ℷ”22, then it is easy to get that X,d is a b-metric space. We define the mapping α:X×X⟶1,∞,(75)αℷ1″,ℷ2″=1+ℷ1″+ℷ2″.
LetMα:X×X×0,∞⟶0,1 be given by(76)Mαℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″,ift>0.0,ift=0,and continuous t-norm ∗=∧; then, ℷ1″∗ℷ2″=ℷ″1ℷ2″=minℷ1″,ℷ2″.
Then,X,Md,∧,k be fuzzy quasi-b-metric space.Remark 34.
Mehmood et al. established a new kind of “Banach-type fixed-point theorem” with the help of a fuzzy set in the most universal class of fuzzyb-metric space.Theorem 13.
(Banach’s contraction theorem in fuzzy extended b-metric spaces).
LetX,Mα,∗ be a G-complete extended fuzzy b-metric space with the mapping α:X×X⟶1,∞ such that(77)limt⟶∞Mαℷ1″,ℷ2″,t=1.
Letg:X⟶X be a mapping, which satisfies the condition(78)Mαgℷ1″,gℷ2″,kt≥Mαℷ1″,ℷ2″,t,∀ℷ1″,ℷ2″∈X,where k∈0,1. Furthermore, g has a unique fixed point if for an arbitrary b0∈X, and n;q∈N, we have(79)αbn,bn+q<1k,where bn=gnb0.


#### 2.4.21. ExtendedMb-Metric Space [70]

The coefficient ofMb-metric space was replaced with the function θ and formed a new generalized version of Mb-metric space, titled as extended Mb-metric space.A functionmθ:X2⟶0,∞ is titled as extended Mb-metric (e.Mb-m) on X (where θ:X2⟶1,∞ be a function) if fulfilled the subsequent conditions:(1)
mθE1,E1=mθE2,E2=mθE1,E2 if E1=E2;(2)
mθE1,E2≤mθE1,E2;(3)
mθE1,E2=mθE2,E1;(4)
mθE1,E2−mθE1,E2≤θE1,E2mθE1,E3−mθE1,E3+mθE3,E2−mθE3,E2, ∀E1,E2,E3∈X.Then, extendedMb-metric mθ together with the set X is titled as extended Mb-metric space and represented as the ordered pair X,mθ.Example 33.
LetX=Cα,β,R, and define mθ:X2⟶0,∞ and θ:X2⟶1,∞ by(80)mθE1t,E2tsupt∈α,βE1t−E2t2,θE1t,E2t=E1t+E2t+2.
Then,X,mθ is an extended Mb-metric space.Remark 35.
N. Mlaiki et al. provide several fixed-point results on expandedMb-metric space (Mb-m.s.) using the techniques of the classical fixed-point theorems such as “Kannan’s fixed-point theory,” the “Banach fixed-point theory,” and many more.Theorem 14.
LetX,mθ be an extended Mb-metric space and mθb:X2⟶0,∞ be a function defined as(81)mθbE1,E2=mθE1,E2−2mθE1,E2+MθE1,E2,for all x,y∈X. Then, mθbis an extended b-metric and the pair X,mθb is an extended b-metric space.


#### 2.4.22. Soft S-Metric Space [81]

Aras et al. gave the theory of softS-metric space in which S-metric was further defined on soft points of soft sets and introduced some of their characteristics as well.LetS be the absolute soft set, E be the ≠∅ set of parameters; SPS be a collection of all soft points ofS, and RE∗ be the set of all non-negative soft real numbers.A mappings:SPS×SPS×SPS⟶RE∗ is s.t.b. soft S-metric on S (the soft set), if it fulfils the subsequent axioms, for each soft points ℷ1a″,ℷ2b″,ℷ3c″,ℷ4d″∈SPS,(1)
Sℷ1a″,ℷ2b″,ℷ3c″≥0;(2)
Sℷ1a″,ℷ2b″,ℷ3c″=0⇔ℷ1a″=ℷ2b″=ℷ3c″;(3)
Sℷ1a″,ℷ2b″,ℷ3c″≤Sℷ1a″,ℷ1a″,ℷ4d″+Sℷ2b″,ℷ2b″,ℷ4d″+Sℷ3c″,ℷ3c″,ℷ4d″.Then, softS-metric s together with the set S is titled as soft S-metric space and represented as S,s,E.Example 34.
LetE=N be a parameter set, and X=R be defined as(82)Sℷ1a″,ℷ2b″,ℷ3c″=a−c+b−c+ℷ1″−ℷ3″+ℷ2″−ℷ3″.
It is easy to get a softS-metric space.Remark 36.
The idea of “soft contractive mapping” on softS-metric space (s.S-m.s.) was established, and some significant concepts were given, for instance, completeness, Cauchy sequence, soft continuous mapping on soft S−metric space (s.S-m.s.), and also with the help of “soft contractive mapping” on soft S−metric space (s.S-m.s.), and some of the “fixed-point theorems” have been demonstrated.
A mappingρ1,ρ2:S,s,E⟶S,s,E has a unique soft fixed point on a soft sequential compact soft S-metric space S,s,E if satisfies the following condition:(83)Sρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ρ1,ρ2ℷ2b″≤αρ1,ρ2ℷ1a″,
for everyℷ1a″,ℷ2b″∈SPS and ℷ1a″≠ℷ2b″.
A mappingρ1,ρ2:S,s,E⟶S,s,Ehas a unique soft fixed point on a complete soft S-metric space S,s,E if satisfies the following contraction condition:(84)Sρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ρ1,ρ2ℷ2b″≤αSρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ℷ1a″,Sρ1,ρ2ℷ2b″,ρ1,ρ2ℷ2b″,ℷ2b″,
for everyℷ1a″,ℷ2b″∈SPS, where α∈0,1/2 is a soft constant real number. Then, ρ1,ρ2 have a unique soft fixed point in SPS.


#### 2.4.23.g-Metric Space [82]

Due to today’s large and complex data collection, we developed the notion of distance function that is calledg-metric in which we extended the ordinary distance function of metric (two points in between) and G-metric (three points in between). The g-metric is the distance function of degree n, i.e., for n+1 points.A functiong:Xn⟶R+ is s.t.b. g-metric with order n on X≠∅ (where Xn=∏i=1nX), if it fulfils the following axioms:(1)
Positive definiteness(85)gx0,xn=0⇔x0=···=xn.(2)
Permutation invariant(86)gx0,xn=gxτ0,xτn.For any permutationτon0,1,n,(3)
Monotonicity(87)gx0,xn≤gy0,yn,∀x0,xn,y0,yn∈Xn+1,with(88)xi:i=0,n⊊yi:i=0,n.(4)
Triangle inequality(89)∀x0,..,xs,y0,..,yt,w∈Xwiths+t+1=n,gx0,..,xs,y0,..,yt≤gx0,..,xs,w,..,w+gy0,..,yt,w,..,w.Then,g-metric g together with the setX is titled as g-metric space and represented as an ordered pair X,g.Example 35.
LetX≠∅ be defined d:Xn⟶R+ by(90)dx0,xn=0,ifx0=...=xn1,otherwise,∀x0,xn∈X.
It is easy to getX,d that is a g-metric space.Remark 37.
Several “fixed-point theorems” in theg-metric space (g-m.s.) were generalized, which is considered as an oversimplification of “Banach’s contractive mapping” concerning g-metric space.
A mappingT:X⟶X has a unique fixed point on a complete g-metric space X,g if satisfies the condition: ∃k∈0,1 such that gTℷ1″,Tℷ2″…,Tℷn″≤kgℷ1″,ℷ2″,…,ℷn″,∀ℷ1″,ℷ2″,…,ℷn″∈X.


#### 2.4.24.F-Metric Space [83, 84]

F-metric is a generalization of metric space in which we defined the natural topology τF with the different topological properties.The mappingD:X2⟶0,+∞ (where X≠∅) is s.t.b. F-metric, if fulfils the following axioms:We suppose that∃f,g∈F×0,+∞ such that(1)
ℷ1″,ℷ2″∈X×X,Dℷ1″,ℷ2″=0 if ℷ1″=ℷ2″;(2)
ℷ1″,ℷ2″=Dℷ2″,ℷ1″;∀ℷ1″,ℷ2″∈X×X;(3)
∀ℷ1″,ℷ2″∈X×X,∀N∈N,N≥2,∀uii=1N⊂X with u1,uN=ℷ1″,ℷ2″; we have Dℷ1″,ℷ2″>0(91)⇒fDℷ1″,ℷ2″≤f∑i=1N−1Dui,ui+1+g.Then,F-metric D together with the set X is titled as F-metric space and represented as an ordered pair X,D.Example 36.
LetX=R be defined d:X2⟶R+ by(92)dℷ1″,ℷ2″=ℷ1″−ℷ2″2,ifℷ1″,ℷ2″∈0,3×0,3ℷ1″−ℷ2″,ifℷ1″,ℷ2″∉0,3×0,3,∀ℷ1″,ℷ2″∈X,
withft=lnt and ρ=lnt.It is easy to get X,d that is a F-metric space.Remark 38.
Jelli and Samet proved the modified version of the “Banach contraction principle.” Afterward, Alansari et al. established some circumstances for the existence of a solution in the “fuzzy fixed-point theorem” used for solving the fuzzy Cauchy problem.
A mappingT:X⟶X has a unique fixed point on a complete F-metric space X,D if satisfies the following condition: ∃k∈0,1 such that DTℷ1″,Tℷ2″≤kDℷ1″,ℷ2″, ∀ℷ1″,ℷ2″∈X.


#### 2.4.25. Controlled Metric Type Space [85]

The controlled metric is an extension ofb-metric in which we introduced a control function αx,z in the b-triangle inequality.The mappingd:X2⟶0,∞ is s.t.b. controlled metric type (where φ:X×X⟶1,∞and X≠∅), if fulfils the following axioms:(1)
dC1,C2=0 if C1=C2;(2)
dC1,C2=dC2,C1;(3)
dC1,C2≤φC1,C3dC1,C3+φC3,C2dC3,C2, for all C1,C2,C3∈X.Then, controlled metricd together with the set X is titled as controlled metric space and represented as an ordered pair X,d.Example 37.
LetX=0,1,2 and a function d be given as(93)d0,0=d1,1=d2,2=0,d0,1=d1,0=1,d0,2=d2,0=12,d1,2=d2,1=25.
We takeβ:X×X⟶0,∞ to be symmetric and is defined by the following:(94)β0,0=β1,1=β2,2=β0,2=1,β1,2=54,β0,1=1110.
It is easy to get thatd that is a controlled metric space.Remark 39.
Mlaki et al. proved the contraction principle on the controlled metric type space.
A mappingT:X⟶X on a complete controlled metric space is X,d and satisfies the following condition:(95)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,where ∃k∈0,1. For ℷ0″∈X, we take ℷ1n″=Tnx0. Suppose that(96)supm≥1limi⟶∞βℷi+1″,ℷi+2″βℷi″,ℷi+1″βℷi+1″,ℷm″<1k.
In additional, we assume that∀ℷ1″∈X and we have(97)limn⟶∞βℷ1n″,ℷ1″andlimn⟶∞βℷ1″,ℷ1n″.
Then,T has a fixed a unique fixed point.


#### 2.4.26. RectangularM-Metric Space [86]

RectangularM-metric space is an extension and generalization of the class of M-metric space in addition to partial rectangular metric space.The mappingmr:X×X⟶0,∞ is s.t.b. rectangular M-metric space (where X≠∅) if satisfied the following conditions: ∀R1,N2∈X(1)
mrR1,N2=mrR1,N2=MrR1,N2 if R1=N2;(2)
mrR1,N2≤mrR1,N2;(3)
mrR1,N2=mrR2,N1;(4)
mrR1,N2−mrR1,N2≤mrR1,N3−mrR1,N3+mrR3,N4−mrR3,N4+mrR4,N2−mrR4,N2, ∀R3,N4∈X|R1,N2.Then, rectangularM-metric mr together with the set X is titled as rectangular M-metric space and represented as an ordered pair X,mr.where(98)mrR1,N2=minmrR1,N1,mrR2,N2,MrR1,N2=maxmrR1,N1,mrR2,N2.Example 38.
Xβ=R1∈C:argR1=β∪0, for a fixed β,0≤β≤2π. If we define self-mapping mr on Xβ, it is given by(99)mrR1,N2=R1+R22,∀R1,R1∈Xβ.
Then, it is easy to getXβ,mr that is a rectangular M-metric space.Remark 40.
N.Y. Ozgur et al. used different contraction principles for proving some fixed-point theorems with various techniques, which generalized some established fixed-point outcomes, for instance, the “Kannan fixed-point theorem,” “Banach contraction principle” and the “Chatterjea fixed-point theorem.” Also, we provide relevance in the direction of the fixed-circle problem.
A mappingT:X⟶X on a rectangular M-metric space is X,d and satisfies the contraction condition if ∃k∈0,1 such that(100)mrTℷ1″,Tℷ2″≤kmrℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,
and consider the sequenceℷ1n″n≥0 defined by ℷ1n+1″=Tℷ1n″. If ℷ1n″⟶uasn⟶∞, then Tℷ1n″⟶Tu as n⟶∞.


#### 2.4.27. RectangularMb-Metric Space [71]

Inspired by the theory of rectangularM-metric space (M-m.s.) and Mb-metric space (Mb-m.s.), Asim et al. established the rectangular Mb-metric space to get well and enlarge the existing results of the literature.A mappingmrmb:X2⟶R+ is s.t.b. a rectangular Mb-metric with coefficient s≥1, if mrmb satisfies the following (∀R1,N2,R3∈X and all distinct R4,N5∈X|R1,N2:(1)
mrmbR1,N1=mrmbR1,N2=mrmbR2,N2 if and only ifR1=N2;(2)
mrmbR1,N1≤mrmbR1,N2;(3)
mrmbR1,N2=mrmbR2,N1(4)
mrmbR1,N2−mrmbR1,N1≤smrmbR1,N3−mrmbR1,N3+mrmbR3,N4−mrmbR3,N4+mrmbR4,N2−mrmbR4,N2–mrmbR3,N3−mrmbR4,N4.Then, rectangularMb-metric mrmb together with the set X is titled as rectangular Mb-metric space and represented as an ordered pair X,mrmb.Example 39.
LetX=R+,ρ>1 a constant, and mrmb:X×X⟶R+be defined by(101)mrmbℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,mrmb is a rectangular Mb-metric space with coefficient s=3ρ−1>1.Remark 41.
In rectangularMb-metric space, M. Asim et al. proved an equivalence of the “Banach contraction principle.”
A mappingT:X⟶X on a complete rectangular Mb-metric space is X,d with coefficient s≥1 and satisfies the following contraction condition:(102)mrmbTℷ1″,Tℷ2″≤kmrmbℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X.where k∈0,1/s. Then, T has a unique fixed point ℷ1″ such that mrmbℷ1″,ℷ1″=0.


#### 2.4.28.R-Metric Space [87]

In theR-metric space, additionally, an arbitrary relation R is defined between the points on a nonempty set.SupposeX,d is a metric space (m.s.) and R be a relation on X≠∅, then triple X,d,R or X is called R-metric space.Example 40.
LetX=0,∞ equipped with Euclidean metric. We define xRy if ℷ1″ℷ2″≤ℷ1″∨ℷ2″, where ℷ1″∨ℷ2″=ℷ1″orℷ2″.
Then,X,d,R is a R-metric space.Remark 42.
It was difficult to discover the exact value of the equilibrium point, so Khalehoghli et al. tried to provide a structural method for determining the exact value of the fixed point, which is useful to solve a problem related to economics and game theory.
A mappingT:X⟶X is called R-contraction with Lipschitz constant 0<k<0 if ∀ℷ1″,ℷ2″∈X such that ℷ1″Rℷ2″, then we have the following:(103)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″.Theorem 15.
A mappingT:X⟶X on a complete metric space X,d has a unique fixed point in X if for some β∈0,1 in a complete metric satisfies the following condition dTℷ1″,Tℷ2″≤βdℷ1″,ℷ2″, for allℷ1″,ℷ2″∈X.


#### 2.4.29. SoftD-Metric Space [88]

Aras et al. provided the theory of softD-metric space (s.D-m.s.), which was further defined on soft points of soft sets and defined the soft ∙−distance on complete soft D-metric by using the designation of soft D-metric.A mappingD:SPX×SPX×SPX⟶RE is s.t.b. soft D−metric on X (where X≠∅) if D fulfils the subsequent conditions, for each soft points ℷ1″,ℷ2″,ℷ3″,ℷ4″∈SPX,(1)
Dℷ1″,ℷ2″,ℷ3″≥0⇔ℷ1″=ℷ2″=ℷ3″;coincidence;(2)
Dℷ1″,ℷ2″,ℷ3″=Dℷ2″,ℷ1″,ℷ3″=Dℷ1″,ℷ3″,ℷ2″=…symmetry;(3)
Dℷ1″,ℷ2″,ℷ3″≤Dℷ1″,ℷ2″,ℷ4″+Dℷ1″,ℷ4″,ℷ3″+Dℷ4″,ℷ2″,ℷ3″.Then, softD-metric D together with the set X is titled as soft D-metric space and represented as X,D,E.Example 41.
LetX be a nonempty set, and E be the nonempty set of parameters. If we define a mapping D:SPX×SPX×SPX⟶RE by(104)Dℷa″,ℷb″,ℷc″=0,ℷa″=ℷb″=ℷc″1,otherwise,∀ℷa″,ℷb″,ℷc″∈SPX,
then it is easy to show thatD is a soft D-metric space.Remark 43.
(1)
“Fixed-point theorem” of soft continuous mappings on softD-metric space has been proven.(2)
It is given the “fixed-point theorem” using the principle of softΔ−distance.Theorem 16.
LetX,D,E be a complete D-metric space and ∙−distance of X.f,g:X,D,E⟶X,D,E be a soft mapping. Let Xbe ∙ bounded. Suppose that ∃ is a soft real number r∈RE,0≤r<1, such that(105)f,gℷ1″,f,g2ℷ1″,f,gℷ2″≤rℷ1″,f,gℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈SPX. Then, ∃ℷ3″∈SPX such that ℷ3″=f,gℷ3″. In addition to ℷ4″=f,gℷ4″, then ℷ4″,ℷ4″,ℷ4″=0.


#### 2.4.30. Controlled Rectangularb-Metric Space [89]

N. Mlaiki et al. established the generalization of rectangularb-metric space by altering the rectangular equality.A functionb:X4∙1,∞, and db:X2∙0,∞ is s.t.b. controlled rectangular b-metric space on Xwhere X≠∅) if all distinct C1,C2,C3,C4∈X; we have(1)
dbC1,C2=0 if C1=C2;(2)
dbC1,C2=dbC2,C1;(3)
dbC1,C2≤bC1,C2,C3,C4dbC1,C3+dbC3,C4+dbC4,C2.Then, controlled rectangularb-metric db together with the set X is titled as controlled rectangular b-metric space and represented as X,db.Example 42.
LetX=Y∪Z, where Y=1/m,wherem∈N, and Z be a set of positive integers. We define db:X2∙0,∞ by(106)dbC1,C2=0,iffC1=C2,2β,ifC1,C2∈Y,β2,otherwise,where β is a constant bigger than 0 and b:X4∙1,∞ by(107)bC1,C2,C3,C4=maxC1,C2,C3,C4+2β.Remark 44.
Controlled rectangularb-metric space is used to prove some “fixed-point theorems.”
This theorem is proved, as an application.
For any natural numberκ≥3, the equation sκ+1=κ4–1sκ+1+κ4shas a unique real solution.
A mappingT:X⟶X on a controlled rectangular b-metric space is X,d and satisfies the contraction condition if ∃k∈0,1 such that(108)dbTℷ1″,Tℷ2″≤kdbℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,and supm>1limn⟶∞bℷ1″,ℷ2″,ℷ3″,ℷ4″≤1/k. Then, T has a unique fixed point.


#### 2.4.31. Controlled Fuzzy Metric Space

Controlled fuzzy metric space (c.f.m.s.) is the extension of fuzzy metric space (f.m.s.) with generalized triangle inequality and the generalization of fuzzy b-metric space (f.b-m.s.).Letλ:X2⟶1,∞,∗ is a continuous t-norm, and Mλ is a fuzzy set on X2×0,∞ (where X≠∅) is s.t.b. controlled fuzzy metric, if fulfilled the subsequent conditions, (∀C1,C2,C3∈X,s,t>0) such that(1)
MλC1,C2,0=0;(2)
MλC1,C2,t=1 ⇔C1=C2(3)
MλC1,C2,t=MλC2,C1,t(4)
MλC1,C2,t+s≥MλC1,C3,t/λC1,C3∗MλC3,C2,s/λC1,C3;(5)
MλC1,C2,⋅:0,∞⟶0,1 is continuous.Then, controlled fuzzy metricMλ together with the set X is titled as controlled fuzzy metric space and represented as X,Mλ,∗.Example 43.
LetX=A∪C, where A=0,1 and C=N|0,1. We define Mλ:X×X×0,∞⟶0,1 as(109)MλC1,C2,t=1,ifC1=C2,tt+1/C2,ifC1∈AandC2∈C,tt+1/C1,ifC1∈CandC2∈A,tt+1,ifotherwise,
with a continuous t-norm ∗ such thatta∗tb=ta.tb. We define λ:X2⟶1,∞ as(110)λC1,C2=1,ifC1,C2∈A,maxC1,C2,ifotherwise.
It is easy to getX,Mλ,∗ that is a controlled metric space.Remark 45.
M. S. Sezen et al. proved a “Banach contraction principle” and introduced a new concept of “fixed-point theorem” for several self-mappings, which satisfied fuzzy “𝜓-contraction” condition.Theorem 17.
A mappingT:X⟶X on a controlled fuzzy metric space X,Mλ,∗ has a unique fixed point in X if satisfies the following condition:(111)MλTC1,TC2,t≥βMλC1,C2,t,
for allC1,C2∈X and β>0.


#### 2.4.32. CAT(0) Space [90, 91]

A metric spaceX is said to be a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. Complete CAT(0) spaces are often called Hadamard spaces.Definition of (CAT(0) space): let (X,d) be a geodesic space. It is a CAT (0) space if, for any geodesic triangle, ∆ ⊂X and x, y ∈ ∆, d(x, y) ≤ d (x¯, y¯) where ¯x, y¯ ∈ ∆¯.Amnuaykarn et al. [90] proved some theorems on the existence of best proximity points of multivalued mappings in CAT(0) spaces.Ugwunnadi and Okeke [91] studied the approximation of common solutions for a finite family of generalized demimetric mappings and monotone inclusion problems in CAT(0) spaces.


## 2.1. Period-I (1906–1950)

### 2.1.1. Quasi-Metric Spaces [6,7]

Wilson [6] introduced the metric space without symmetric condition, which is termed as quasi-metric space.A functiond:X2⟶R+∪0, (where X≠∅) is well known as a quasi-metric if it satisfied the following succeeding conditions:(1)
dℷ1″,ℷ1″=0,∀ℷ1″∈X;(2)
dℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″, for all ℷ1″,ℷ2″,ℷ3″∈X.Then, quasi-metricd together with set X is titled as quasi-metric space and represented by X,d.Example 3.
LetR be set of all real numbers and ρ>0. Let d:R×R⟶R+∪0 be a mapping, which is defined as follows:(3)dℷ1″,ℷ2″=ℷ2″−ℷ1,″ifℷ2″≥ℷ1″,ρℷ1″−ℷ2″,ifℷ2″<ℷ1″.
It is easy to get thatdis a quasi-metric on R.Remark 1.
(1)
Nearly all “fixed-point theorems” used for″ψF contraction” in quasi-metric space (q.m.s) have been proved.(2)
Quasi-Metric Space is applied forproving Hamilton–Jacobi equation’s existence and uniqueness, iterated function systems, fractal theory, shape-memory alloys, in the rate-independent models for plasticity, etc [7].
A mappingT:X⟶X is called as the following:(1)
“ForwardψF contraction (resp. backward ψF contraction)” if quasi-metric space X,d and functions F∈F and ψ∈Ψ satisfy the following conditions,(4)Tℷ1″≠Tℷ2″⇒FdTℷ1″,Tℷ2″≤ψdℷ1″,ℷ2″,resp.Tℷ1″≠Tℷ2″⇒FdTℷ1″,Tℷ2″≤ψdℷ2″,ℷ1″.(Remark: ifT is a “backward ψF contraction” then T2 be a forward ψF2 contraction.)(2)
“Forward Picard operator (backward Picard operator)” if quasi-metric spaceX,d has a unique fixed point ζ∈X and Tnℷ1″⟶fζ for every ℷ1″∈X.


### 2.1.2. Probabilistic Metric Space [8–10]

The generalized probabilistic theory was proposed by Menger, which played a significant part in the development of metric space. In particular, he suggested replacing the numberdp,q with the real function Fpq, whose function is Fpqℷ″, for any real number ℷ″. It is defined as the probability that is the distance between p&q is less than x.A functionF:X2⟶R+∪0, (where X≠∅), is s.t.b. probabilistic metric space, if met the following requirements:(1)
Fa,bℷ1″=1,∀ℷ″>0⇒a=b∀a,b∈X;(2)
Fa,bℷ1″=Fb,aℷ1″,∀a,b∈X;(3)
Fa,cℷ1″=1,Fb,cℷ2″=1;(4)
⇒Fa,bℷ1″,ℷ2″=1,∀a,b,c∈Xandℷ1″,ℷ2″∈0,∞.Then, probabilistic metric space is represented asX,F.Remark 2.
With the help of probabilistic metric space, the existence of the random fixed-point theorem was solved. Also, constrained file migration and metrical task systems use a randomized algorithm that is based on the probabilistic metric approximation.
A mappingF:X⟶X is titled as follows:(1)
B–contraction if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈X and all θ>0 satisfy the following implication:(5)Ffℷ1″,fℷ2″σθ≥Fℷ1″,ℷ2″θ.(2)
C–contraction if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈Xandθ>0 satisfy the following implication:(6)Fℷ1″,ℷ2″θ>1−θ⇒Ffℷ1″,fℷ2″σθ>1−σθ.
As a generalization of the concept of “C–contraction,” Pap and Hadzic made known to the concept of Ψ,C-contraction.
A mappingF:X⟶2X is called Ψ,C-contraction (where Ψ:R+⟶R+,R+=0,∞), if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈Xand all θ>0 satisfy the following implication:(7)Fℷ1″,ℷ2″θ>1−θ⇒∀σ′ϵFℷ1″∃σ″ϵFℷ2″,such that(8)Fσ,,σ,,Ψx>1−Ψx.


## 2.1.1. Quasi-Metric Spaces [6,7]

Wilson [6] introduced the metric space without symmetric condition, which is termed as quasi-metric space.A functiond:X2⟶R+∪0, (where X≠∅) is well known as a quasi-metric if it satisfied the following succeeding conditions:(1)
dℷ1″,ℷ1″=0,∀ℷ1″∈X;(2)
dℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″, for all ℷ1″,ℷ2″,ℷ3″∈X.Then, quasi-metricd together with set X is titled as quasi-metric space and represented by X,d.Example 3.
LetR be set of all real numbers and ρ>0. Let d:R×R⟶R+∪0 be a mapping, which is defined as follows:(3)dℷ1″,ℷ2″=ℷ2″−ℷ1,″ifℷ2″≥ℷ1″,ρℷ1″−ℷ2″,ifℷ2″<ℷ1″.
It is easy to get thatdis a quasi-metric on R.Remark 1.
(1)
Nearly all “fixed-point theorems” used for″ψF contraction” in quasi-metric space (q.m.s) have been proved.(2)
Quasi-Metric Space is applied forproving Hamilton–Jacobi equation’s existence and uniqueness, iterated function systems, fractal theory, shape-memory alloys, in the rate-independent models for plasticity, etc [7].
A mappingT:X⟶X is called as the following:(1)
“ForwardψF contraction (resp. backward ψF contraction)” if quasi-metric space X,d and functions F∈F and ψ∈Ψ satisfy the following conditions,(4)Tℷ1″≠Tℷ2″⇒FdTℷ1″,Tℷ2″≤ψdℷ1″,ℷ2″,resp.Tℷ1″≠Tℷ2″⇒FdTℷ1″,Tℷ2″≤ψdℷ2″,ℷ1″.(Remark: ifT is a “backward ψF contraction” then T2 be a forward ψF2 contraction.)(2)
“Forward Picard operator (backward Picard operator)” if quasi-metric spaceX,d has a unique fixed point ζ∈X and Tnℷ1″⟶fζ for every ℷ1″∈X.


## 2.1.2. Probabilistic Metric Space [8–10]

The generalized probabilistic theory was proposed by Menger, which played a significant part in the development of metric space. In particular, he suggested replacing the numberdp,q with the real function Fpq, whose function is Fpqℷ″, for any real number ℷ″. It is defined as the probability that is the distance between p&q is less than x.A functionF:X2⟶R+∪0, (where X≠∅), is s.t.b. probabilistic metric space, if met the following requirements:(1)
Fa,bℷ1″=1,∀ℷ″>0⇒a=b∀a,b∈X;(2)
Fa,bℷ1″=Fb,aℷ1″,∀a,b∈X;(3)
Fa,cℷ1″=1,Fb,cℷ2″=1;(4)
⇒Fa,bℷ1″,ℷ2″=1,∀a,b,c∈Xandℷ1″,ℷ2″∈0,∞.Then, probabilistic metric space is represented asX,F.Remark 2.
With the help of probabilistic metric space, the existence of the random fixed-point theorem was solved. Also, constrained file migration and metrical task systems use a randomized algorithm that is based on the probabilistic metric approximation.
A mappingF:X⟶X is titled as follows:(1)
B–contraction if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈X and all θ>0 satisfy the following implication:(5)Ffℷ1″,fℷ2″σθ≥Fℷ1″,ℷ2″θ.(2)
C–contraction if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈Xandθ>0 satisfy the following implication:(6)Fℷ1″,ℷ2″θ>1−θ⇒Ffℷ1″,fℷ2″σθ>1−σθ.
As a generalization of the concept of “C–contraction,” Pap and Hadzic made known to the concept of Ψ,C-contraction.
A mappingF:X⟶2X is called Ψ,C-contraction (where Ψ:R+⟶R+,R+=0,∞), if X,F be a probabilistic metric space and ∃σ∈0,1 such that ∀ℷ1″,ℷ2″∈Xand all θ>0 satisfy the following implication:(7)Fℷ1″,ℷ2″θ>1−θ⇒∀σ′ϵFℷ1″∃σ″ϵFℷ2″,such that(8)Fσ,,σ,,Ψx>1−Ψx.


## 2.2. Period-II (1951–2000)

### 2.2.1. Two-Metric Space [11–15]

Gahler [15] first attempted to generalize the ordinary distance function to measure the distance between three points. Two-metric space is somewhere dissimilar to the metric space, which provides a unique nonlinear structure.A mappingd:X3⟶R is s.t.b. 2-metric on XX≠∅, if satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″,a≠0,∀ℷ1″,ℷ2″″∈Xandℷ1″≠ℷ2″,∃apointa∈X.(2)
dℷ1″,ℷ2″,ℷ3″=0, only if two or more points are the same.(3)
dℷ1″,ℷ2″,ℷ3″″=dℷ1″,ℷ3″,ℷ3″=dℷ3″,ℷ2″,ℷ1″,∀ℷ1″,ℷ2″,ℷ3″∈X.(4)
dℷ1″,ℷ2″,ℷ3″≤dℷ1″,ℷ2″,ℷ4″+dℷ1″,ℷ4″,ℷ3″″+dℷ4″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X.Then, 2-metricd together with set X is titled as 2-metric space and represented by X,d.Remark 3.
(1)
The “Baire category theorem” and “Cantor intersection theorem” are used for solving problems of fixed point [12].(2)
In 2-metric space, Lebesgue integrable function is used as a summable function for each compactR+ [13].
A mappingT:X,φ⟶X,φ is called contraction in 2-metric space if satisfies the following conditions, φTℷ1″,Tℷ2″,ρ<φℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X where ℷ1″≠ℷ2″≠ρ, and φTℷ1″,Tℷ2″,ρ=0,if any two of ℷ1″,ℷ2″ and ρ are equal.
A mappingT:X⟶X is entitled as “A-contraction” in 2-metric space if satisfies the succeeding conditions, dTℷ1″,Tℷ2″,ρ≤τmaxdTℷ1″,ℷ1″,ρ+dTℷ2″,ℷ2″,ρ,dTℷ2″,ℷ2″,ρ+dℷ1″,ℷ2″,ρ,dTℷ1″,ℷ1″,ρ+dℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X,τ∈0,1/2 [14].


### 2.2.2. Fuzzy Metric Space [17–19]

After probabilistic metric space, author replaced randomness with fuzziness and presented an innovative theory, called fuzzy metric space (f.m.s.). The idea of probabilistic metric space (p.m.s.) has been expanded to fuzzy metric space. The purpose of introducing the fuzzy metric space was to allocate the non-negative fuzzy number to the distance among two points, which is the most appropriate way to describe the fuzzy metric space.LetM is a “fuzzy set” on X×X×0;∞, that is s.t.b. fuzzy metric (f.m.) on XX≠∅,∗ is a continuous t-norm), if met the subsequent axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X(1)
Mℷ1″,ℷ2″,0=0;(2)
Mℷ1″,ℷ2″,t=1∀t>0ℷ1″=ℷ2″;(3)
Mℷ1″,ℷ2″,t=Mℷ2″,ℷ1″,t;(4)
Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s≤Mℷ1″,ℷ2″,t+s;(5)
Mℷ1″,ℷ2″,·:0;∞⟶0;1 is left continuous, ℷ1″,ℷ2″,ℷ3″∈X and t,s>0.Then, fuzzy metricM together with setX is titled as fuzzy metric space and symbolized as (X;M;∗.Example 4.
LetX=R and M=expℷ1″−ℷ2″/t−1, ∀ℷ1″,ℷ2″∈X, and t∈0,∞.(1)
m∗n=mn;(2)
m∗n=minm,n.
It is easy to get thatM is a fuzzy metric on X=R with a continuous t-norm ∗.Remark 4.
(1)
Gregori and Sapena expanded the “Banach contraction mapping” to the “fuzzy contractive mapping” in the complete fuzzy metric space.(2)
Chos et al. proved Banach’s contraction and Edelstein mapping in fuzzy metric space.(3)
N. Holland proved Baire’s theorem in fuzzy metric space.
A mappingT:X⟶X is known as the following:(1)
“Banach fuzzy contraction” if fuzzy metric spaceX,M,∗ satisfies the subsequent conditions, MTℷ1″,Tℷ2″,kρ≥Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″∈X,k∈0,1, and all ρ>0.(2)
“Edelstein fuzzy contraction” if fuzzy metric spaceX,d,∗ satisfies the subsequent conditions, MTℷ1″,Tℷ2″,ρ>Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X,ℷ1″≠ℷ2″, and all ρ>0.


### 2.2.3.b-Metric Space [20–22]

In theb-metric space, K (relaxed triangle inequality) was introduced. If β=1, then b-metric and metric space are the same so as compared with the metric space, and b-metric space is the weaker notion.A mappingd:X×Xinto0,∞ is s.t.b. b-metric on XX≠∅ if met the subsequent axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″ (symmetry);(3)
There existsβ≥1 satisfying(9)dℷ1″,ℷ2″≤βℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″ℷ3″∈X.Then,b-metric d together with set X is titled as b-metric space and represented as ordered pair X,d.Example 5.
LetX=N and function fn=––log2n and gn=2n−2fnKfn+2fn−nKfn−1 and a function d fromX⟶Xonto0,∞ by(10)dx,y=gx–y.
It is easy to show thatX,d is a b-metric space.Remark 5.
With the help of b-metric space, the presence of a unique solution for a nonlinear fractional differential equation is given with an integral boundary condition. Biology, physical science, and economics such real-life problems eventually lead to the linear fractional differential equation and integral equation [20–22]. Further extension of b-metric space can be seen in the paper of Chauhan and Kaur [23].


### 2.2.4.D-Metric Space [24–26]

Since 2-metric space was not continuous and convergent for all three points, so Dhage [24] introduced the new generalized class of the ordinary metric space. The convergence of D-metric defines a Hausdorff topology and (sequentially) continuity for all three points.A mappingD:X3⟶0,∞ is s.t.b. D-metric on XX≠∅, if it meets the following conditions given below:(1)
Dℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″, (coincidence)(2)
Dℷ1″,ℷ2″,ℷ3″=Dpℷ1″,ℷ2″,ℷ3″, where p is a permutation of ℷ1″,ℷ2″,ℷ3″ (symmetry), and(3)
Dℷ”1,ℷ”2,ℷ”3≤Dℷ”1,ℷ”2,ℷ”4+Dℷ”1,ℷ”4,ℷ”3+Dℷ”4,ℷ”2,ℷ”3,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X (tetrahedral inequality).Then,D-metric d together with set X is titled as D-metric space and represented as ordered pair X,d.Example 6.
LetX=R+and D:X3⟶R+ be definite as(11)Dℷ1″,ℷ2″,ℷ3″=maxℷ1″−ℷ2″,ℷ2″−ℷ3″,ℷ3″−ℷ1″,∀ℷ1″,ℷ2″,ℷ3″∈X.
It is easy to show thatX,D is a D-metric space.Remark 6.
(1)
The analysis is obtained from the results of nonlinear self-mapping, which satisfies the particular form (completeness and boundedness) of contraction condition inD-metric space, to prove the existence of a unique fixed point [25].(2)
Singh et al. introduced semicompatibility inD-metric space for “fixed-point theorems” with the help of the orbit concept [26].
EveryD-continuous function T:X,D⟶X,D has a fixed point in X if D-metric space has a D-fixed-point property.
EveryD-weakly continuous function T:X,D⟶X,D has a fixed point in X if and only if D-metric space has a D-weakly fixed-point property.


### 2.2.5. Partial Metric Space [27–29]

The partial metric space is a component of dataflow networks, denotational semantic study. Compared with the standard metric, the main difference is that there is no need for the self-distance of arbitrary variables is to be zero.A functionp:P2⟶R+ is s.t.b. partial metric on PP≠∅, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″∈P:(1)
ℷ1″=ℷ2″⇔pℷ1″,ℷ1″=pℷ1″,ℷ2″=pℷ2″,ℷ2″;(2)
pℷ1″,ℷ1″≤pℷ1″,ℷ2″;(3)
pℷ1″,ℷ2″=pℷ2″,ℷ1″;(4)
pℷ1″,ℷ2″≤pℷ1″,ℷ3″+pℷ3″,ℷ2″–pℷ3″,ℷ3″.Then, partial metricp together with the set P is titled as partial metric space and represented as ordered pair P,p.Example 7.
LetX=R+=0,∞ and p:X×X⟶R+ be a function defined by pℷ1″,ℷ2″=maxℷ1″,ℷ2″, for all ℷ1″,ℷ2″∈X.
Then, it is easy to see thatX,p is a partial metric, but it is not a metric. Indeed, for any x>0, we have px,x=x≠0.Remark 7.
Through the experience of computer science, the idea of nonzero self-distance for metric space was inspired. Also, it gave an extension of the “Banach contraction principle” in the complete partial metric space [30].
A mappingT:X⟶X is titled as:(1)
Contraction for every partial metric spaceX,p if ∃k∈0,1 so that ∀ℷ1″,ℷ2″∈X satisfies the following condition:(12)pTℷ1″,Tℷ2″≤kpℷ1″,ℷ2″.(2)
Asymptotic regular for every partial metric spaceX,p if at point ℷ1″∈X satisfies the following condition:(13)limn⟶∞dTnℷ1″,Tn+1ℷ1″=0.


### 2.2.6. Rectangular Metric Space [31, 32]

The rectangular metric space was introduced with the most general triangle inequality in which four points are implemented instead of three points.A generalized metric (rectangular metric) is a functiond:R2⟶0,∞ on RR≠∅, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈R with ℷ1″≠ℷ2″ and ℷ3″,ℷ4″∉ℷ1″,ℷ2″:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ4″+dℷ4″,ℷ2″ (quadrilateral inequality).Then, rectangular metricd together with the set R is titled as partial metric space and represented as ordered pair R,d.Example 8.
LetR1=0,2,R2=1,1/2,1/3,…,R=R1∪R2. define d:R×R⟶0,∞ as follows:(14)dℷ1″,ℷ2″=0,ℷ1″=ℷ2″,1,ℷ1″≠ℷ2″,ℷ1″,ℷ2″⊂R1orℷ1″,ℷ2″⊂R2ℷ2″,ℷ1″∈R1,ℷ2″∈R2,,ℷ1″,ℷ2″∈R1,ℷ1″∈R2.
Then,R,d is a complete rectangular metric space.Remark 8.
Budhia et al. contributed to some innovative fixed-point results in the “rectangular metric space” using a fractional-order functional differential equation.
A mappingT:X⟶X is called as the following:(1)
“Caccioppoli-type fixed-point theorem” for complete metric spaceX,d if for every n∈N,∀ℷ1″,ℷ2″∈X where δn>0 distinct from ℷ1″,ℷ2″ and 0<δ0<1 fulfils the subsequent condition:(15)dTnℷ1″,Tnℷ2″≤δndℷ1″,Tℷ1″+dℷ2″,Tℷ2″.Then, if series∑i=1∞δn is convergent, then T has a unique fixed point in X.(2)
“Kannan fixed-point theorem” for complete metric spaceX,d if for every n∈N,∀ℷ1″,ℷ2″∈X and 0<δ0<1/2 fulfils the subsequent condition:(16)dTℷ1″,Tℷ2″≤δndℷ1″,Tℷ1″+dℷ2″,Tℷ2″.
IfX is T-orbitally complete in X,d, then T has a unique fixed point in X.


## 2.2.1. Two-Metric Space [11–15]

Gahler [15] first attempted to generalize the ordinary distance function to measure the distance between three points. Two-metric space is somewhere dissimilar to the metric space, which provides a unique nonlinear structure.A mappingd:X3⟶R is s.t.b. 2-metric on XX≠∅, if satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″,a≠0,∀ℷ1″,ℷ2″″∈Xandℷ1″≠ℷ2″,∃apointa∈X.(2)
dℷ1″,ℷ2″,ℷ3″=0, only if two or more points are the same.(3)
dℷ1″,ℷ2″,ℷ3″″=dℷ1″,ℷ3″,ℷ3″=dℷ3″,ℷ2″,ℷ1″,∀ℷ1″,ℷ2″,ℷ3″∈X.(4)
dℷ1″,ℷ2″,ℷ3″≤dℷ1″,ℷ2″,ℷ4″+dℷ1″,ℷ4″,ℷ3″″+dℷ4″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X.Then, 2-metricd together with set X is titled as 2-metric space and represented by X,d.Remark 3.
(1)
The “Baire category theorem” and “Cantor intersection theorem” are used for solving problems of fixed point [12].(2)
In 2-metric space, Lebesgue integrable function is used as a summable function for each compactR+ [13].
A mappingT:X,φ⟶X,φ is called contraction in 2-metric space if satisfies the following conditions, φTℷ1″,Tℷ2″,ρ<φℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X where ℷ1″≠ℷ2″≠ρ, and φTℷ1″,Tℷ2″,ρ=0,if any two of ℷ1″,ℷ2″ and ρ are equal.
A mappingT:X⟶X is entitled as “A-contraction” in 2-metric space if satisfies the succeeding conditions, dTℷ1″,Tℷ2″,ρ≤τmaxdTℷ1″,ℷ1″,ρ+dTℷ2″,ℷ2″,ρ,dTℷ2″,ℷ2″,ρ+dℷ1″,ℷ2″,ρ,dTℷ1″,ℷ1″,ρ+dℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X,τ∈0,1/2 [14].


## 2.2.2. Fuzzy Metric Space [17–19]

After probabilistic metric space, author replaced randomness with fuzziness and presented an innovative theory, called fuzzy metric space (f.m.s.). The idea of probabilistic metric space (p.m.s.) has been expanded to fuzzy metric space. The purpose of introducing the fuzzy metric space was to allocate the non-negative fuzzy number to the distance among two points, which is the most appropriate way to describe the fuzzy metric space.LetM is a “fuzzy set” on X×X×0;∞, that is s.t.b. fuzzy metric (f.m.) on XX≠∅,∗ is a continuous t-norm), if met the subsequent axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X(1)
Mℷ1″,ℷ2″,0=0;(2)
Mℷ1″,ℷ2″,t=1∀t>0ℷ1″=ℷ2″;(3)
Mℷ1″,ℷ2″,t=Mℷ2″,ℷ1″,t;(4)
Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s≤Mℷ1″,ℷ2″,t+s;(5)
Mℷ1″,ℷ2″,·:0;∞⟶0;1 is left continuous, ℷ1″,ℷ2″,ℷ3″∈X and t,s>0.Then, fuzzy metricM together with setX is titled as fuzzy metric space and symbolized as (X;M;∗.Example 4.
LetX=R and M=expℷ1″−ℷ2″/t−1, ∀ℷ1″,ℷ2″∈X, and t∈0,∞.(1)
m∗n=mn;(2)
m∗n=minm,n.
It is easy to get thatM is a fuzzy metric on X=R with a continuous t-norm ∗.Remark 4.
(1)
Gregori and Sapena expanded the “Banach contraction mapping” to the “fuzzy contractive mapping” in the complete fuzzy metric space.(2)
Chos et al. proved Banach’s contraction and Edelstein mapping in fuzzy metric space.(3)
N. Holland proved Baire’s theorem in fuzzy metric space.
A mappingT:X⟶X is known as the following:(1)
“Banach fuzzy contraction” if fuzzy metric spaceX,M,∗ satisfies the subsequent conditions, MTℷ1″,Tℷ2″,kρ≥Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″∈X,k∈0,1, and all ρ>0.(2)
“Edelstein fuzzy contraction” if fuzzy metric spaceX,d,∗ satisfies the subsequent conditions, MTℷ1″,Tℷ2″,ρ>Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″,ρ∈X,ℷ1″≠ℷ2″, and all ρ>0.


## 2.2.3.b-Metric Space [20–22]

In theb-metric space, K (relaxed triangle inequality) was introduced. If β=1, then b-metric and metric space are the same so as compared with the metric space, and b-metric space is the weaker notion.A mappingd:X×Xinto0,∞ is s.t.b. b-metric on XX≠∅ if met the subsequent axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″ (symmetry);(3)
There existsβ≥1 satisfying(9)dℷ1″,ℷ2″≤βℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″ℷ3″∈X.Then,b-metric d together with set X is titled as b-metric space and represented as ordered pair X,d.Example 5.
LetX=N and function fn=––log2n and gn=2n−2fnKfn+2fn−nKfn−1 and a function d fromX⟶Xonto0,∞ by(10)dx,y=gx–y.
It is easy to show thatX,d is a b-metric space.Remark 5.
With the help of b-metric space, the presence of a unique solution for a nonlinear fractional differential equation is given with an integral boundary condition. Biology, physical science, and economics such real-life problems eventually lead to the linear fractional differential equation and integral equation [20–22]. Further extension of b-metric space can be seen in the paper of Chauhan and Kaur [23].


## 2.2.4.D-Metric Space [24–26]

Since 2-metric space was not continuous and convergent for all three points, so Dhage [24] introduced the new generalized class of the ordinary metric space. The convergence of D-metric defines a Hausdorff topology and (sequentially) continuity for all three points.A mappingD:X3⟶0,∞ is s.t.b. D-metric on XX≠∅, if it meets the following conditions given below:(1)
Dℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″, (coincidence)(2)
Dℷ1″,ℷ2″,ℷ3″=Dpℷ1″,ℷ2″,ℷ3″, where p is a permutation of ℷ1″,ℷ2″,ℷ3″ (symmetry), and(3)
Dℷ”1,ℷ”2,ℷ”3≤Dℷ”1,ℷ”2,ℷ”4+Dℷ”1,ℷ”4,ℷ”3+Dℷ”4,ℷ”2,ℷ”3,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X (tetrahedral inequality).Then,D-metric d together with set X is titled as D-metric space and represented as ordered pair X,d.Example 6.
LetX=R+and D:X3⟶R+ be definite as(11)Dℷ1″,ℷ2″,ℷ3″=maxℷ1″−ℷ2″,ℷ2″−ℷ3″,ℷ3″−ℷ1″,∀ℷ1″,ℷ2″,ℷ3″∈X.
It is easy to show thatX,D is a D-metric space.Remark 6.
(1)
The analysis is obtained from the results of nonlinear self-mapping, which satisfies the particular form (completeness and boundedness) of contraction condition inD-metric space, to prove the existence of a unique fixed point [25].(2)
Singh et al. introduced semicompatibility inD-metric space for “fixed-point theorems” with the help of the orbit concept [26].
EveryD-continuous function T:X,D⟶X,D has a fixed point in X if D-metric space has a D-fixed-point property.
EveryD-weakly continuous function T:X,D⟶X,D has a fixed point in X if and only if D-metric space has a D-weakly fixed-point property.


## 2.2.5. Partial Metric Space [27–29]

The partial metric space is a component of dataflow networks, denotational semantic study. Compared with the standard metric, the main difference is that there is no need for the self-distance of arbitrary variables is to be zero.A functionp:P2⟶R+ is s.t.b. partial metric on PP≠∅, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″∈P:(1)
ℷ1″=ℷ2″⇔pℷ1″,ℷ1″=pℷ1″,ℷ2″=pℷ2″,ℷ2″;(2)
pℷ1″,ℷ1″≤pℷ1″,ℷ2″;(3)
pℷ1″,ℷ2″=pℷ2″,ℷ1″;(4)
pℷ1″,ℷ2″≤pℷ1″,ℷ3″+pℷ3″,ℷ2″–pℷ3″,ℷ3″.Then, partial metricp together with the set P is titled as partial metric space and represented as ordered pair P,p.Example 7.
LetX=R+=0,∞ and p:X×X⟶R+ be a function defined by pℷ1″,ℷ2″=maxℷ1″,ℷ2″, for all ℷ1″,ℷ2″∈X.
Then, it is easy to see thatX,p is a partial metric, but it is not a metric. Indeed, for any x>0, we have px,x=x≠0.Remark 7.
Through the experience of computer science, the idea of nonzero self-distance for metric space was inspired. Also, it gave an extension of the “Banach contraction principle” in the complete partial metric space [30].
A mappingT:X⟶X is titled as:(1)
Contraction for every partial metric spaceX,p if ∃k∈0,1 so that ∀ℷ1″,ℷ2″∈X satisfies the following condition:(12)pTℷ1″,Tℷ2″≤kpℷ1″,ℷ2″.(2)
Asymptotic regular for every partial metric spaceX,p if at point ℷ1″∈X satisfies the following condition:(13)limn⟶∞dTnℷ1″,Tn+1ℷ1″=0.


## 2.2.6. Rectangular Metric Space [31, 32]

The rectangular metric space was introduced with the most general triangle inequality in which four points are implemented instead of three points.A generalized metric (rectangular metric) is a functiond:R2⟶0,∞ on RR≠∅, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈R with ℷ1″≠ℷ2″ and ℷ3″,ℷ4″∉ℷ1″,ℷ2″:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ4″+dℷ4″,ℷ2″ (quadrilateral inequality).Then, rectangular metricd together with the set R is titled as partial metric space and represented as ordered pair R,d.Example 8.
LetR1=0,2,R2=1,1/2,1/3,…,R=R1∪R2. define d:R×R⟶0,∞ as follows:(14)dℷ1″,ℷ2″=0,ℷ1″=ℷ2″,1,ℷ1″≠ℷ2″,ℷ1″,ℷ2″⊂R1orℷ1″,ℷ2″⊂R2ℷ2″,ℷ1″∈R1,ℷ2″∈R2,,ℷ1″,ℷ2″∈R1,ℷ1″∈R2.
Then,R,d is a complete rectangular metric space.Remark 8.
Budhia et al. contributed to some innovative fixed-point results in the “rectangular metric space” using a fractional-order functional differential equation.
A mappingT:X⟶X is called as the following:(1)
“Caccioppoli-type fixed-point theorem” for complete metric spaceX,d if for every n∈N,∀ℷ1″,ℷ2″∈X where δn>0 distinct from ℷ1″,ℷ2″ and 0<δ0<1 fulfils the subsequent condition:(15)dTnℷ1″,Tnℷ2″≤δndℷ1″,Tℷ1″+dℷ2″,Tℷ2″.Then, if series∑i=1∞δn is convergent, then T has a unique fixed point in X.(2)
“Kannan fixed-point theorem” for complete metric spaceX,d if for every n∈N,∀ℷ1″,ℷ2″∈X and 0<δ0<1/2 fulfils the subsequent condition:(16)dTℷ1″,Tℷ2″≤δndℷ1″,Tℷ1″+dℷ2″,Tℷ2″.
IfX is T-orbitally complete in X,d, then T has a unique fixed point in X.


## 2.3. Period-III (2001–2010)

### 2.3.1. Fuzzy Quasi-Metric Space [33]

The associated impression of fuzzy metric space was expanded with a quasi-metric sense and then made known to the fuzzy quasi-metric space.LetM is a fuzzy set mapping on X×X×0,∞, that is, s.t.b. fuzzy quasi-metric (f.q.m.) on XX≠∅,∗-≠∅continuoust−norm, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″∈X,s,t>0.(1)
Mℷ1″,ℷ2″,0=0;(2)
ℷ1″=ℷ2″ if Mℷ1″,ℷ2″,t=Mℷ2″,ℷ1″,t=1;(3)
Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s≤Mℷ1″,ℷ2″,t+s;(4)
ℷ1″,ℷ2″:0,∞⟶0,1 is left continuous.Then, fuzzy quasi-metricM together with the set X is titled as partial metric space and represented as ordered pair X,M.Example 9.
LetX,d be a fuzzy quasi-metric space and be a usual multiplication ∀ℷ1″,ℷ2″∈0,1, and M is a fuzzy set mapping on X×X×0,∞ by(17)Mdℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″.
Then,X,Md is a fuzzy quasi-metric space.Remark 9.
(1)
Gregori and Romaguera proved a quasi-metrizable topology, which is generated via fuzzy quasi-metric space.(2)
In the fuzzy quasi-metric space (f.q.m.s.), the “contraction principle” was demonstrated with an appliance to the word domain.
A mappingT:X⟶X is titled as “B-contraction” on fuzzy quasi-metric space X,M,∗ that fulfils the following circumstances, MTℷ1″,Tℷ2″,kρ≥Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″∈X,k∈0,1,and all ρ>0, where k is called the contraction constant of T [34].


### 2.3.2.G-Metric Space [35, 36]

In the G-metric space, there is a modification that the distance of three variables is equal to zero if all the variables are equal, and if one of them is different, then the distance is always positive.A mappingG:X3⟶R+ is s.t.b. G-metric (more specifically generalized metric) on XX≠∅, if it satisfied the subsequent axioms:(1)
Gℷ1″,ℷ2″,ℷ3″=0 if ℷ1″=ℷ2″=ℷ3″;(2)
Gℷ1″,ℷ2″,ℷ3″>0∀ℷ1″,ℷ2″,ℷ3″∈X,withℷ1″≠ℷ2″;(3)
Gℷ1″,ℷ1″,ℷ3″≤Gℷ1″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″∈Xwithℷ1″≠ℷ2″;(4)
Gℷ1″,ℷ2″,ℷ3″=Gℷ1″,ℷ3″,ℷ2″=Gℷ2″,ℷ3″,ℷ1″=…, (symmetry);(5)
Gℷ1″,ℷ2″,ℷ3″≤Gℷ1″,ℷ4″,ℷ4″+Gℷ4″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X (rectangle inequality).Then,G-metric G together with the set X is titled as G-metric space and represented as ordered pair X,G.Example 10.
Let (R,d be the usual metric space. We define Gs by(18)Gℷ1″,ℷ2″,ℷ3″=dℷ1″,ℷ2″+dℷ2″,ℷ3″+dℷ1″,ℷ3″,for all ℷ1″,ℷ2″,ℷ3″∈ R. Then, it is easy to see that R,G is a G-metric space.Remark 10.
(1)
InG-metric space, a theorem was introduced for the solution of integral equations.(2)
Existence and uniqueness of the “fixed-point theorem” in generalized metric space (orG-metric space) are proved.


### 2.3.3.D∗-Metric Space [37]

When most of theD-metric space theorems were invalid, then the description of D-metric space had been possibly updated by Sedghi et al. [37] and introduced D∗-metric space. With some modifications, the basic properties of D∗-metric space (D∗-m.s.) were introduced.A mappingD∗:X3⟶0,∞ is s.t.b. generalized metric (or D∗-metric) on X≠∅, if fulfilled the subsequent conditions:∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X:(1)
D∗ℷ1″,ℷ2″,ℷ3″≥0;(2)
D∗ℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″;(3)
D∗ℷ1″,ℷ2″,ℷ3″=D∗pℷ1″,ℷ2″,ℷ3″, (symmetry) where p is a permutation function;(4)
D∗ℷ1″,ℷ2″,ℷ3″≤D∗ℷ1″,ℷ2″,ℷ4″+D∗ℷ4″,ℷ3″,ℷ3″.Then,D∗-metric D∗ together with the set X is titled as D∗-metric space and represented as ordered pair X,D∗.Example 11.
LetX=R, then we define(19)D∗ℷ1″,ℷ2″,ℷ3″=0,ifℷ1″=ℷ2″=ℷ3″maxℷ1″,ℷ2″,ℷ3″,o.w.
Then, it is easy to check thatX,D∗ is D∗-metric space.Remark 11.
A “fixed-point theorem” was developed for a class of mapping in completeD∗—metric space together with the condition of weakly commuting mappings.
A mappingS,T:X⟶ctsX,M:X⟶X in complete D∗-metric space satisfies the following conditions:(1)
M,S and M,T are weakly commuting pairs s.t.b. MX⊂SX∩TX;(2)
∃ϕ∈Φ such that for all ℷ1″,ℷ2″∈X,(20)D∗Mℷ1″,Mℷ2″,Mℷ3″≤ϕD∗Sℷ1″,Tℷ2″,Tℷ3″,D∗Sℷ1″,Mℷ1″,Mℷ1″,D∗Sℷ1″,Mℷ2″,Mℷ2″,D∗Tℷ2″,Mℷ1″,Mℷ1″,D∗Tℷ2″,Mℷ2″,Mℷ2″.
Then,M,S and T have a common unique fixed point in X.


### 2.3.4. Cone Metric Space [38–42]

The cone metric space was introduced when the real numbers were replaced by ordering Banach’s space.LetE stand for “real Banach space” and C,E.A functiond:X2⟶E (where X≠∅ ) is s.t.b. a cone metric on X, if it meets the subsequent requirements:(1)
dℵ1″,ℵ2″>0,∀ℵ”1,ℵ”2∈X and ℵ”1=ℵ”2⇔dℵ”1,ℵ”2=0;(2)
dℵ1″,ℵ2″=dℵ2″,ℵ1″, ∀ℵ”1,ℵ”2∈X;(3)
dℵ1″,ℵ3″+dℵ3″,ℵ2″≥dℵ1″,ℵ2″, for all ℵ1″,ℵ2″,ℵ3″∈X.Then, function“d” composed of =set “X” is titled as a cone metric space and represented as X,d.Example 12.
LetE=R2,P=ℵ1″,ℵ2″∈E|ℵ1″,ℵ2″≥0⊂R2,X=Randd:X×X⟶E such that(21)dℵ1″,ℵ2″=ℵ1″−ℵ2″,αℵ1″−ℵ2″,where α≥0 is a constant. Then, X,d is a cone metric space.Remark 12.
A famous “fixed-point theorem” was proven for a nonordinary cone metric space, with the “generalized contractive condition” intended for a sequence of self-maps. As a consequence, in the cone metric space a generalized version of “Banach’s contraction principle” was proved [41].Theorem 3.
(1)
LetX,d be a complete cone metric space, and P be a normal cone with normal constant k. It is supposed that the mapping T:X⟶X fulfils the following condition:(22)dTℵ1″,Tℵ2″≤kdℵ1″,ℵ2″,∀ℵ1″,ℵ2″∈X,wherek∈0,1 is a constant.
Then,T has a unique fixed point in X. For any ℵ″∈X, iterative sequence Tnℵ″ converges to the fixed point.Theorem 4.
LetX,d be a sequentially compact cone metric space, and P be a regular cone.
It is supposed that the mappingT:X⟶X fulfils the following contractive condition:(23)dTℵ1″,Tℵ2″<dℵ1″,ℵ2″,∀ℵ1″,ℵ2″∈X,ℵ1″≠ℵ2″.
Then,T has a unique fixed point in X.Theorem 5.
LetX,d be a complete cone metric space, and P a normal cone with normal constant K. It is supposed that the mapping T:X⟶X,T has a unique fixed point in X if fulfils the contractive condition:(24)dTℵ1″,Tℵ2″≤kdTℵ1″,ℵ1″+dTℵ2″,ℵ2″,∀ℵ1″,ℵ2″∈X,where k∈0,1/2 is a constant. For any ℵ″∈X, iterative sequence Tnℵ″ converges to the fixed point.


### 2.3.5. Multiplicative Metric Space [43, 44]

Initially, the usual metric withR+ was not complete. To overcome this problem, Bashirov et al. proposed the concept of multiplicative metric space with a multiplicative value of x∈R+ and multiplicative distance function d.A mappingd:X2⟶R+ is s.t.b. multiplicative metric on XX≠∅, when it meets the following conditions:(1)
dℷ1″,ℷ2″≥1,∀ℷ1″,ℷ2″∈X;(2)
dℷ1″,ℷ2″=1⇔ℷ1″=ℷ2″;(3)
ℷ”1,ℷ”2=dℷ”2,ℷ”1,∀ℷ1″,ℷ2″∈X;(4)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″.dℷ3″,ℷ2″ for all ℷ1″,ℷ2″,ℷ3″∈X (multiplicative triangle inequality).Then, multiplicative metricd together with the set X is titled as multiplicative metric space and represented as ordered pair X,d.Example 13.
LetR+nbe the set of all n-tuples of non-negative real numbers. Let d:R+n×R+n⟶R be a mapping defined as follows:(25)dℷ1″,ℷ2″=ℷ1,1″ℷ2,1″∙ℷ1,2″ℷ2,2″⋯ℷ1,n″ℷ2,n″,where ℷ1″=ℷ1,1″,ℷ1,2″,⋯,ℷ1,n″, ℷ2″=ℷ2,1″,ℷ2,2″,⋯,ℷ2,n″∈R+n, and ∙:R+⟶R are defined by the following:(26)ℷ1″=ℷ1″,ifℷ1″≥1,1ℷ1″,ifℷ1″<1.
It is easy to get thatR+n,d is a multiplicative metric space.Remark 13.
We gave the description of multiplicative contraction mapping and “fixed-point theorems” on a complete multiplicative metric space.
A mappingT:X⟶X is called as follows:(1)
“Multiplicative contraction” of multiplicative metric space if∃τ∈0,1,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(27)dTℷ1″,Tℷ2″≤dℷ1″,ℷ2″τ.(2)
“Multiplicative Kannan’s contraction” of multiplicative metric space if∃τ∈0,1/2,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(28)dTℷ1″,Tℷ2″≤dℷ1″,Tℷ1″dℷ2″,Tℷ2″τ.(3)
“Multiplicative Chatterjea’s” contraction of multiplicative metric space if∃τ∈0,1/2,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(29)dTℷ1″,Tℷ2″≤dℷ1″,Tℷ2″dℷ2″,Tℷ1″τ.


### 2.3.6. Vector Metric Space [45, 46]

Cevik and Altun were made known to the vector metric space in which metric space takes the value on a Riesz space, i.e., Riesz space replaced from ℝ.The functiond:X×X⟶E is s.t.b. vector metric on XwhereX≠∅,E-Riesz space) if it satisfied the following:(1)
dℷ1″,ℷ2″=0 if ℷ1″=ℷ2″,(2)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″∀ℷ1″,ℷ2″,ℷ3″∈X.Then, vector metricd together with the set X is titled as vector metric space and termed as triple X,d,E (briefly X with the default parameters omitted).Example 14.
Let a Riesz space“E” and a mapping d:E×E⟶E are defined by the following:(30)dℷ1″,ℷ2″=ℷ1″−ℷ2″.
It is easy to show a vector metric space.Remark 14.
Cevik and Altun gave “Banach’s contraction principle” in the vector metric space.
LetV be an E-complete vector metric space (v.m.s.), (where “E is Archimedean”).
The mappingT:V⟶V satisfies the contractive condition dTℷ1″,Tℷ2″≤θdℷ1″,ℷ2″,∀ℷ2″,ℷ2″∈V, where θ∈0,1 is a constant.
Then,T has a unique fixed point in V and iterative sequence ℷn″, ∀ℷ0″∈V, defined through ℷn″=Tℷn″−1 for n∈N,E-converges on the fixed point of T.


## 2.3.1. Fuzzy Quasi-Metric Space [33]

The associated impression of fuzzy metric space was expanded with a quasi-metric sense and then made known to the fuzzy quasi-metric space.LetM is a fuzzy set mapping on X×X×0,∞, that is, s.t.b. fuzzy quasi-metric (f.q.m.) on XX≠∅,∗-≠∅continuoust−norm, if it satisfied the subsequent axioms: ∀ℷ1″,ℷ2″,ℷ3″∈X,s,t>0.(1)
Mℷ1″,ℷ2″,0=0;(2)
ℷ1″=ℷ2″ if Mℷ1″,ℷ2″,t=Mℷ2″,ℷ1″,t=1;(3)
Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s≤Mℷ1″,ℷ2″,t+s;(4)
ℷ1″,ℷ2″:0,∞⟶0,1 is left continuous.Then, fuzzy quasi-metricM together with the set X is titled as partial metric space and represented as ordered pair X,M.Example 9.
LetX,d be a fuzzy quasi-metric space and be a usual multiplication ∀ℷ1″,ℷ2″∈0,1, and M is a fuzzy set mapping on X×X×0,∞ by(17)Mdℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″.
Then,X,Md is a fuzzy quasi-metric space.Remark 9.
(1)
Gregori and Romaguera proved a quasi-metrizable topology, which is generated via fuzzy quasi-metric space.(2)
In the fuzzy quasi-metric space (f.q.m.s.), the “contraction principle” was demonstrated with an appliance to the word domain.
A mappingT:X⟶X is titled as “B-contraction” on fuzzy quasi-metric space X,M,∗ that fulfils the following circumstances, MTℷ1″,Tℷ2″,kρ≥Mℷ1″,ℷ2″,ρ,∀ℷ1″,ℷ2″∈X,k∈0,1,and all ρ>0, where k is called the contraction constant of T [34].


## 2.3.2.G-Metric Space [35, 36]

In the G-metric space, there is a modification that the distance of three variables is equal to zero if all the variables are equal, and if one of them is different, then the distance is always positive.A mappingG:X3⟶R+ is s.t.b. G-metric (more specifically generalized metric) on XX≠∅, if it satisfied the subsequent axioms:(1)
Gℷ1″,ℷ2″,ℷ3″=0 if ℷ1″=ℷ2″=ℷ3″;(2)
Gℷ1″,ℷ2″,ℷ3″>0∀ℷ1″,ℷ2″,ℷ3″∈X,withℷ1″≠ℷ2″;(3)
Gℷ1″,ℷ1″,ℷ3″≤Gℷ1″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″∈Xwithℷ1″≠ℷ2″;(4)
Gℷ1″,ℷ2″,ℷ3″=Gℷ1″,ℷ3″,ℷ2″=Gℷ2″,ℷ3″,ℷ1″=…, (symmetry);(5)
Gℷ1″,ℷ2″,ℷ3″≤Gℷ1″,ℷ4″,ℷ4″+Gℷ4″,ℷ2″,ℷ3″,∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X (rectangle inequality).Then,G-metric G together with the set X is titled as G-metric space and represented as ordered pair X,G.Example 10.
Let (R,d be the usual metric space. We define Gs by(18)Gℷ1″,ℷ2″,ℷ3″=dℷ1″,ℷ2″+dℷ2″,ℷ3″+dℷ1″,ℷ3″,for all ℷ1″,ℷ2″,ℷ3″∈ R. Then, it is easy to see that R,G is a G-metric space.Remark 10.
(1)
InG-metric space, a theorem was introduced for the solution of integral equations.(2)
Existence and uniqueness of the “fixed-point theorem” in generalized metric space (orG-metric space) are proved.


## 2.3.3.D∗-Metric Space [37]

When most of theD-metric space theorems were invalid, then the description of D-metric space had been possibly updated by Sedghi et al. [37] and introduced D∗-metric space. With some modifications, the basic properties of D∗-metric space (D∗-m.s.) were introduced.A mappingD∗:X3⟶0,∞ is s.t.b. generalized metric (or D∗-metric) on X≠∅, if fulfilled the subsequent conditions:∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X:(1)
D∗ℷ1″,ℷ2″,ℷ3″≥0;(2)
D∗ℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″;(3)
D∗ℷ1″,ℷ2″,ℷ3″=D∗pℷ1″,ℷ2″,ℷ3″, (symmetry) where p is a permutation function;(4)
D∗ℷ1″,ℷ2″,ℷ3″≤D∗ℷ1″,ℷ2″,ℷ4″+D∗ℷ4″,ℷ3″,ℷ3″.Then,D∗-metric D∗ together with the set X is titled as D∗-metric space and represented as ordered pair X,D∗.Example 11.
LetX=R, then we define(19)D∗ℷ1″,ℷ2″,ℷ3″=0,ifℷ1″=ℷ2″=ℷ3″maxℷ1″,ℷ2″,ℷ3″,o.w.
Then, it is easy to check thatX,D∗ is D∗-metric space.Remark 11.
A “fixed-point theorem” was developed for a class of mapping in completeD∗—metric space together with the condition of weakly commuting mappings.
A mappingS,T:X⟶ctsX,M:X⟶X in complete D∗-metric space satisfies the following conditions:(1)
M,S and M,T are weakly commuting pairs s.t.b. MX⊂SX∩TX;(2)
∃ϕ∈Φ such that for all ℷ1″,ℷ2″∈X,(20)D∗Mℷ1″,Mℷ2″,Mℷ3″≤ϕD∗Sℷ1″,Tℷ2″,Tℷ3″,D∗Sℷ1″,Mℷ1″,Mℷ1″,D∗Sℷ1″,Mℷ2″,Mℷ2″,D∗Tℷ2″,Mℷ1″,Mℷ1″,D∗Tℷ2″,Mℷ2″,Mℷ2″.
Then,M,S and T have a common unique fixed point in X.


## 2.3.4. Cone Metric Space [38–42]

The cone metric space was introduced when the real numbers were replaced by ordering Banach’s space.LetE stand for “real Banach space” and C,E.A functiond:X2⟶E (where X≠∅ ) is s.t.b. a cone metric on X, if it meets the subsequent requirements:(1)
dℵ1″,ℵ2″>0,∀ℵ”1,ℵ”2∈X and ℵ”1=ℵ”2⇔dℵ”1,ℵ”2=0;(2)
dℵ1″,ℵ2″=dℵ2″,ℵ1″, ∀ℵ”1,ℵ”2∈X;(3)
dℵ1″,ℵ3″+dℵ3″,ℵ2″≥dℵ1″,ℵ2″, for all ℵ1″,ℵ2″,ℵ3″∈X.Then, function“d” composed of =set “X” is titled as a cone metric space and represented as X,d.Example 12.
LetE=R2,P=ℵ1″,ℵ2″∈E|ℵ1″,ℵ2″≥0⊂R2,X=Randd:X×X⟶E such that(21)dℵ1″,ℵ2″=ℵ1″−ℵ2″,αℵ1″−ℵ2″,where α≥0 is a constant. Then, X,d is a cone metric space.Remark 12.
A famous “fixed-point theorem” was proven for a nonordinary cone metric space, with the “generalized contractive condition” intended for a sequence of self-maps. As a consequence, in the cone metric space a generalized version of “Banach’s contraction principle” was proved [41].Theorem 3.
(1)
LetX,d be a complete cone metric space, and P be a normal cone with normal constant k. It is supposed that the mapping T:X⟶X fulfils the following condition:(22)dTℵ1″,Tℵ2″≤kdℵ1″,ℵ2″,∀ℵ1″,ℵ2″∈X,wherek∈0,1 is a constant.
Then,T has a unique fixed point in X. For any ℵ″∈X, iterative sequence Tnℵ″ converges to the fixed point.Theorem 4.
LetX,d be a sequentially compact cone metric space, and P be a regular cone.
It is supposed that the mappingT:X⟶X fulfils the following contractive condition:(23)dTℵ1″,Tℵ2″<dℵ1″,ℵ2″,∀ℵ1″,ℵ2″∈X,ℵ1″≠ℵ2″.
Then,T has a unique fixed point in X.Theorem 5.
LetX,d be a complete cone metric space, and P a normal cone with normal constant K. It is supposed that the mapping T:X⟶X,T has a unique fixed point in X if fulfils the contractive condition:(24)dTℵ1″,Tℵ2″≤kdTℵ1″,ℵ1″+dTℵ2″,ℵ2″,∀ℵ1″,ℵ2″∈X,where k∈0,1/2 is a constant. For any ℵ″∈X, iterative sequence Tnℵ″ converges to the fixed point.


## 2.3.5. Multiplicative Metric Space [43, 44]

Initially, the usual metric withR+ was not complete. To overcome this problem, Bashirov et al. proposed the concept of multiplicative metric space with a multiplicative value of x∈R+ and multiplicative distance function d.A mappingd:X2⟶R+ is s.t.b. multiplicative metric on XX≠∅, when it meets the following conditions:(1)
dℷ1″,ℷ2″≥1,∀ℷ1″,ℷ2″∈X;(2)
dℷ1″,ℷ2″=1⇔ℷ1″=ℷ2″;(3)
ℷ”1,ℷ”2=dℷ”2,ℷ”1,∀ℷ1″,ℷ2″∈X;(4)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″.dℷ3″,ℷ2″ for all ℷ1″,ℷ2″,ℷ3″∈X (multiplicative triangle inequality).Then, multiplicative metricd together with the set X is titled as multiplicative metric space and represented as ordered pair X,d.Example 13.
LetR+nbe the set of all n-tuples of non-negative real numbers. Let d:R+n×R+n⟶R be a mapping defined as follows:(25)dℷ1″,ℷ2″=ℷ1,1″ℷ2,1″∙ℷ1,2″ℷ2,2″⋯ℷ1,n″ℷ2,n″,where ℷ1″=ℷ1,1″,ℷ1,2″,⋯,ℷ1,n″, ℷ2″=ℷ2,1″,ℷ2,2″,⋯,ℷ2,n″∈R+n, and ∙:R+⟶R are defined by the following:(26)ℷ1″=ℷ1″,ifℷ1″≥1,1ℷ1″,ifℷ1″<1.
It is easy to get thatR+n,d is a multiplicative metric space.Remark 13.
We gave the description of multiplicative contraction mapping and “fixed-point theorems” on a complete multiplicative metric space.
A mappingT:X⟶X is called as follows:(1)
“Multiplicative contraction” of multiplicative metric space if∃τ∈0,1,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(27)dTℷ1″,Tℷ2″≤dℷ1″,ℷ2″τ.(2)
“Multiplicative Kannan’s contraction” of multiplicative metric space if∃τ∈0,1/2,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(28)dTℷ1″,Tℷ2″≤dℷ1″,Tℷ1″dℷ2″,Tℷ2″τ.(3)
“Multiplicative Chatterjea’s” contraction of multiplicative metric space if∃τ∈0,1/2,∀ℷ1″,ℷ2″∈X fulfils the subsequent condition is as follows:(29)dTℷ1″,Tℷ2″≤dℷ1″,Tℷ2″dℷ2″,Tℷ1″τ.


## 2.3.6. Vector Metric Space [45, 46]

Cevik and Altun were made known to the vector metric space in which metric space takes the value on a Riesz space, i.e., Riesz space replaced from ℝ.The functiond:X×X⟶E is s.t.b. vector metric on XwhereX≠∅,E-Riesz space) if it satisfied the following:(1)
dℷ1″,ℷ2″=0 if ℷ1″=ℷ2″,(2)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″∀ℷ1″,ℷ2″,ℷ3″∈X.Then, vector metricd together with the set X is titled as vector metric space and termed as triple X,d,E (briefly X with the default parameters omitted).Example 14.
Let a Riesz space“E” and a mapping d:E×E⟶E are defined by the following:(30)dℷ1″,ℷ2″=ℷ1″−ℷ2″.
It is easy to show a vector metric space.Remark 14.
Cevik and Altun gave “Banach’s contraction principle” in the vector metric space.
LetV be an E-complete vector metric space (v.m.s.), (where “E is Archimedean”).
The mappingT:V⟶V satisfies the contractive condition dTℷ1″,Tℷ2″≤θdℷ1″,ℷ2″,∀ℷ2″,ℷ2″∈V, where θ∈0,1 is a constant.
Then,T has a unique fixed point in V and iterative sequence ℷn″, ∀ℷ0″∈V, defined through ℷn″=Tℷn″−1 for n∈N,E-converges on the fixed point of T.


## 2.4. Period-IV (2011–2021)

### 2.4.1. Complex-Valued Metric Space [47, 48]

The explanation of the complex-valued metric spaces (c.m.s.) occurred when replaced range ℂ (with a certain order) form rangeR (with a normal order), i.e., the generalization of the contractive condition in metric space made much of the concept available. There was a considerable amount of content available in generalized metric spaces. In order to satisfy a rational inequality, complex-valued metric space was thus introduced and some results were also developed for mapping.A mappingd:X2⟶C is s.t.b. complex-valued metric space (c.m.s.) on X(X≠∅, if fulfilled the subsequent condition:(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈XAnd dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″, for allℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈X.Then, complex-valued metricd together with the set X is titled as complex-valued metric space and represented as ordered pair X,d.Example 15.
LetX=C be a set of complex number. We define d:C×C⟶C by(31)dℷ1″,ℷ2″=ℷ1,1″−ℷ2,1″+iℷ1,2″+ℷ2,2″,where ℷ1″=ℷ1,1″+ℷ1,2″ and ℷ2″=ℷ2,1″+ℷ2,2″.
Then,C,d is a complex-valued metric space.Remark 15.
In the complex-valued metric space (c.m.s.), it provides the extension of “Banach’s fixed-point theorem,” which helps to give a common unique solution to the Urysohn integral equations.(i)
xσ=∫abK1σ,τ,xτdτ+gσ,(ii)
xσ=∫abK2σ,τ,xτdτ+hσ, where σ∈a,b,x,g,h∈X.
A mappingT,f:X⟶X be two functions in metric space X,d. A mapping T is called f−contraction if ∃φ∈0,1 such that(32)dfTℷ1″,fTℷ2″≤φdfℷ1″,fℷ2″,∀ℷ1″,ℷ2″∈X.


### 2.4.2. Coneb-Metric Space [49]

The generalization of cone metric space (C.M.S.) together with b-metric space is termed as cone b-metric space.A mappingd:X2⟶E is s.t.b. cone b-metric on X(where X≠∅,s≥1∈R+) if for all C1,C2,C3∈X met the following conditions:(1)
dC1,C2>0 with C1≠C2 and 𝑑C1,C2=0 if and only if C1=C2;(2)
C1,C2=dC2,C1;(3)
dC1,C2≤sdC1,C3+dC3,C2.Then, coneb-metric d together with the setX is titled as cone b-metric space and represented as an ordered pair X,d.Example 16.
LetE=l1,P=un∈E/un≥0,∀n≥1, and d:X×X⟶E such that(33)d∗C1,C2=dC1,C22n,forn≥1.
Then,X,d is a cone b-metric space with coefficient r=21/p>1.Remark 16.
N. Sharma proved contractive mapping for “fixed-point theorem” with rational expression and without normality condition, in cone b-metric space.Theorem 6.
Suppose the mappingT:X⟶X and let X,d be a complete cone b-metric space with the coefficient s≥1 that satisfies the following condition, then(34)dTC1,TC2≤αdC1,TC1dC2,TC2dC1,C2+βdC1,C2,for C1,C2∈X and α,β∈0,1 with α+β<1. Then, T has a unique fixed point in X.


### 2.4.3.S-Metric Space [50, 51]

The philosophy of S-metric spaces is the modification of D-metric and G-metric spaces in which the 3-dimensional metric space was made known by Sedghi et al. InS-metric space, the distance between all three distinct points is always greater than or equal to zero.A functionS:X3⟶0,∞ is s.t.b. S-metric space on X≠∅ if it fulfils the following axioms: ∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X,(1)
Sℷ1″,ℷ2″,ℷ3″≥0;(2)
Sℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″;(3)
Sℷ1″,ℷ2″,ℷ3″≤Sℷ1″,ℷ2″,ℷ4″+Sℷ1″,ℷ2″,ℷ4″+Sℷ1″,ℷ2″,ℷ4″.Then,S-metric S together with the set X is titled as S-metric space and represented as an ordered pair X,S.Example 17.
LetX=Rn and , a norm on X, then(35)Sℷ1″,ℷ2″,ℷ3″=ℷ2″+ℷ3″−2ℷ1″+ℷ2″+ℷ3″,is an S-metric on X.Remark 17.
Ozgur and Tas characterized the thought of a fixed circle and explored the existence and uniqueness of the “fixed-point theorem” for the self-mapping on S-metric space. Likewise, a few instances of self-mappings having fixed circles are given with a mathematical perspective.
Some new contractive mapping and the concept ofCs and Ls mapping on S-metric space were established.Theorem 7.
LetX;S be a complete S-metric space, and A;B:X⟶X be the mappings that fulfil the subsequent conditions:(1)
AX⊆BX and either AXorBX is a closed subset of X,(2)
The pairA;B is weakly compatible,(3)
SAℷ1″;Aℷ2″;Aℷ3″≤φmaxSBℷ1″;Bℷ2″;Bℷ3″;k1SBℷ3″;Aℷ1″;Aℷ3″;k2SBℷ3″;Aℷ2″;Aℷ3″ for all ℷ1″,ℷ2″,ℷ3″∈X and 0<k1;k2<1, where φ∈∅.
Then, the mapsA and B have a unique common fixed point. If B is continuous at the fixed point p, then A is also continuous at p.


### 2.4.4. Metric-Like Space [29]

Metric-like space (m.l.s.) is a generalization of partial metric space, in which all axioms are the same as the usual metric except first, that isσx,x may be positive for x∈X.A functionσ:X2⟶R+∪0iss.t.b.metric−likespaceonX≠∅, if it fulfilled the subsequent axioms: for any ℷ1″,ℷ2″∈X,(1)
σℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(2)
σℷ1″,ℷ2″=σℷ2″,ℷ1″;(3)
σℷ1″,ℷ2″≤σℷ1″,ℷ3″+σℷ3″,ℷ2″.Then, metric-likeσ together with the set X is titled as metric-like space and represented as an ordered pair X,σ.Example 18.
LetX=0,1 and let(36)σℷ1″,ℷ2″=2,ifℷ1″=ℷ2″=0,1,otherwise.
It is easy to getX,σ that is metric-like space.Remark 18.
(1)
As an application, A. Harandi unified and generalized some important results of the “fixed-point theory.”(2)
Metric-like space considered new Geraghty-type mappings and proposed “fixed-point theorems” for such space [52].
A topologyτσ for each metric-like σ was generated, which is a family of open balls. Bσx,ε=y∈X:σx,y–σx,x<ε,∀x∈X and ε>0.
Then, a sequencexn converges to a point x∈X if limn⟶∞σxn,x=σx,x.


### 2.4.5.b-Metric-Like Space [53, 54]

Theb-metric-like space (b-m.l.s.) is a generalized version of metric-like space in which coefficient of metric space was initiated in the triangle inequality.A mappingσ:X2⟶R+, (where X≠∅) is s.t.b. b-metric-like on X if ∀ℷ1″,ℷ2″,ℷ3″∈X and constant s≥1 fulfils the following axioms:(1)
σℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(2)
σℷ1″,ℷ2″=σℷ2″,ℷ1″;(3)
σℷ1″,ℷ2″≤sσℷ1″,ℷ3″+σℷ3″,ℷ2″.Then,b-metric-like σ together with the set X is titled as b-metric-like space and represented as an ordered pair X,σ.Example 19.
LetX=R+,ρ>1 a constant, and σb:X×X⟶R+ be defined by(37)σbℷ1″,ℷ2″=ℷ1″+ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,σb is a b-metric-like space with coefficient =2ρ−1.Remark 19.
Some results ofb-metric space are related to the fixed point and give nontrivial examples to support the concept such as finding the solution of the electric circuit equation with the help of second-order differential equations.
A mappingT:X⟶X is called(1)
Banach G-contraction ifT preserves edges of G, i.e., ∀ℷ”1,ℷ”2∈X;(38)ℷ1″,ℷ2″∈EG⇒Tℷ1″,Tℷ2″∈EG.(2)
Orbitally continuous if givenℷ1″∈X and any sequence kn of positive integers,(39)Tknℷ1″⟶ℷ2″asn⟶∞⇒TTknℷ1″⟶Tℷ2″asn⟶∞.


### 2.4.6. Quasi-b-Metric Space [53–55]

The class of ordinary quasi-metric space was extended to form quasi-b-metric space.A functiond:X2⟶R+ is entitled as quasi-b-metric on X ≠∅ and constant s≥1, if it satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″≥0, ∀ℷ1″,ℷ2″∈X;(2)
dℷ1″,ℷ2″=0 ⇔ ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″≤sdℷ1″,ℷ3″+dℷ3″,ℷ2″, ∀ℷ1″,ℷ2″″,ℷ3″∈X.Then, quasi-b-metric d together with the set X is titled as quasi-b-metric space and represented as an ordered pair X,d.Example 20.
LetX=lp, where1≤p<∞, be defined by(40)lp=ℷ″1nn≥1⊆R:∑n=1∞ℷ″1np<∞.
We defined:X×X⟶R+ by(41)dℷ1″,ℷ2″=0,ifℷ1″≤ℷ2″,∑n=1∞ℷ″1np1/p,ifℷ1″≥ℷ2″.
It is easy to get thatX,d is a quasi-b-metric space with constant p≥1.Remark 20.
(1)
It is proved that some “fixed-point theorems” for a nondecreasing mapping satisfied the nonlinear contraction.(2)
Partially well-ordered complete quasi-b-metric space was used to solve the fractional differential equations [55].
A mappingT:X⟶X is called α-admissible for nonempty set X, if ∀ℷ1″,ℷ2″∈X, where α:X×X⟶0,∞ be a mapping, then we have(42)αℷ1″,ℷ2″≥1⇒αTℷ1″,Tℷ2″≥1.


### 2.4.7. Soft Metric Space [56, 57]

When difficulties occur due to inadequacy of the parameterization tool of the theory, then the new mathematical tool “the soft set” theory was introduced by Molodtsov, for dealing with uncertainties and inherent difficulties of theories. Das and Samanta initiated the analysis of soft metric space (s.m.s.) created on soft-point soft sets, for implementations of soft set theory in real-life problems and other domains for better performance.LetA be a nonempty subset of parameters, and ℵ be the absolute soft set, i.e., Fλ=ℷ″,∀λ∈A, where F,A=ℷ″.A mappingd:SPℷ″×SPℷ″⟶RA∗ is called soft metric on ℵ if, for any ℷ″ax,ℷ″by,ℷ″cz∈SPℷ″ (equivalently, ℷ″ax,ℷ″by,ℷ″cz∈ℷ″), satisfied the following conditions:(1)
dℷ″ax,ℷ″by≥0(2)
dℷ″ax,ℷ″by=0 ⇔ ℷ″ax=ℷ″by(3)
dℷ″ax,ℷ″by=dℷ″by,ℷ″ax(4)
dℷ″ax,ℷ″by≤dℷ″ax,ℷ″cz+dℷ″cz,ℷ″by.The soft metric space is symbolized byℷ″,d,A.Example 21.
Letℷ″=A=1/n:n∈N, and the mapping d:SPℷ″×SPℷ″⟶RA∗ be defined by(43)dℷ″ax,ℷ″by=x¯−y¯+a¯−b¯,∀ℷ″ax,ℷ″by∈SPℷ″.
Then, it is easy to show a soft metric space onℷ″.Remark 21.
The soft metric space is used to prove the soft version of the “contraction principle” in complete soft metric space, which provides applications related to dynamic programming.
A soft mappingf:ℷ″⟶ℷ″ and let ℷ″,d,A be a complete soft metric space with finite set A that satisfies(44)dfℷ″ax,fℷ″by≤τdℷ″ax,fℷ″axdℷ″by,fℷ″by,∀ℷ″ax,ℷ″by∈SPℷ″,where 0≤τ≤1/2, and then f has unique fixed point.


### 2.4.8. Partialb-Metric Space [58, 59]

S. Shukla introduced partialb-metric space and the generalization of partial metric space (p.m.s.) together with b-metric space.A mappingb:X2⟶R+ is s.t.b. partial b-metric (p.b-m.) on X≠∅, if it satisfied the subsequent axioms: (for all ℷ1″,ℷ2″,ℷ3″∈X),(1)
bℷ1″,ℷ1″=bℷ1″,ℷ2″=bℷ2″,ℷ2″⇔ℷ1″=ℷ2″;(2)
bℷ1″,ℷ1″≤bℷ1″,ℷ2″;(3)
bℷ1″,ℷ2″=bℷ2″,ℷ1″;(4)
∃ a real number s≥1 such that(45)bℷ1″,ℷ2″≤sbℷ1″,ℷ3″+bℷ3″,ℷ2″−bℷ3″,ℷ3″.Then, partialb-metric b together with the set X is titled as partial b-metric space and represented as an ordered pair X,b, and s is the coefficient of X,b.Example 22.
LetX=R+,ρ>1 a constant, and b:X×X⟶R+ be defined by(46)bℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,σb is a partial b-metric space with coefficient s=2ρ>1.Remark 22.
Analog of “Kannan’s type fixed-point theorem” and “Banach’s contraction theorem” was proved in the partialb-metric space.Theorem 8.
Suppose the mappingT:X⟶X and let X,b be a complete partial b-metric space with the coefficient s≥1 that satisfies the following condition:(47)bTℷ1″,Tℷ2″≤βbℷ1″,ℷ2″.
For allℷ1″,ℷ2″∈X and where β∈0,1, then T has a unique fixed point u∈X and bu,u=0.


### 2.4.9. Fuzzy Cone Metric Space [60]

The class of fuzzy metric space (f.m.s.) was extended and generalized into fuzzy cone metric space.A mappingd:X2⟶E∗I is s.t.b. fuzzy cone metric (f.c.m.) on X≠∅, if it satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″≥0;∀ℷ1″,ℷ2″∈X and dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″∀ℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″⊕dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈X.Then, fuzzy cone metricd together with the set X is titled as fuzzy cone metric space and represented as an ordered pair X,d.Example 23.
LetP be any cone and X=N and ℷ1″∗ℷ2″=ℷ1″ℷ2″ and d:X2×intP⟶0,1 be defined by(48)dℷ1″,ℷ2″,t=ℷ1″ℷ2″,,ifℷ1″≤ℷ2″,ℷ2″ℷ1″,ifℷ1″≥ℷ1″.∀ℷ1″,ℷ2″∈Xand t≫0. Then, it is easy to show that it is fuzzy cone metric space.Remark 23.
(1)
In fuzzy cone metric space (f.c.m.s.), some of the basic properties for innumerable types of contraction mappings and for “fixed-point theorems” were established.(2)
Chet et al. provide an application of nonlinear integral equation in fuzzy cone metric space for the existence and uniqueness of the solution.
A mappingT:X⟶X is titled as fuzzy cone contraction on fuzzy cone metric space X,d,∗ if ∃k∈0,1 satisfies the following:(49)1dTℷ1″,Tℷ2″,t−1≤k1dℷ1″,ℷ2″,t−1,∀ℷ”1,ℷ”2∈X and t≫0. t is called the contraction constant of T.


### 2.4.10. Fuzzy Coneb-Metric Space [61]

The definition of fuzzy coneb-metric space (f.c.b-m.s.), inspired by cone b-metric space (c.b-m.s.) and fuzzy metric space, was made known by T. Bag.A mappingd:X2⟶E∗Iis s.t.b. fuzzy cone b-metric (where X≠∅ and s≥1 be a real number) if it fulfils the following axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈X and dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤sdℷ1″,ℷ3″⊕dℷ3″,ℷ2″.Then, fuzzy coneb-metric d together with the set X is titled as fuzzy cone b-metric space and represented as an ordered pair X,d.Remark 24.
Some results of “Banach’s contraction mapping” for the “fixed-point theorem” were established in fuzzy coneb-metric space.Theorem 9.
A mappingT:X⟶X on a complete fuzzy cone b-metric space X,d has a unique fixed point in X if for complete fuzzy cone b-metric space X,d with the coefficient s≥1fulfils the subsequent condition:(50)dTℷ1″,Tℷ2″≤βdℷ1″,ℷ2″,
for allℷ1″,ℷ2″∈X and whereβ∈0,1.


### 2.4.11. Fuzzy Soft Metric Space [62–64]

For the fuzzy soft metric space (f.s.m.s.), first fuzzy soft point was introduced in which problems deal with ambiguous data in several fields and that was not effectively demonstrated in mathematics. There are various kinds of mathematical instruments to deal with uncertainties, i.e., the fuzzy set and the soft set that helps to solve the problem difficulty in all areas. In the fuzzy soft metric space (f.s.m.s.), the fuzzy soft open ball and also the closed ball were proposed.LetX be the absolute fuzzy soft set,(51)A∙Ei.e.,Xe=τ,∀e∈E.A∗ be the set of all non-negative fuzzy soft real numbers.A mappingd:FSCX×FSCX⟶RA∗ is s.t.b. fuzzy soft metric (f.s.m.s.) on the soft setX, if it holds the subsequent axioms:(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈FSCX;(2)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″=dℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈FSCX;(4)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈FSCX.Then, fuzzy soft metricd together with the set X is titled as fuzzy soft metric space and represented as an ordered pair X,d.Remark 25.
Fuzzy soft metric is applied on soft open balls and soft closed balls.Theorem 10.
A mappingTρ:E,d⟶E,d is called fuzzy soft contractive mapping on fuzzy soft metric space E,d if ∃two fuzzy soft number ψwith0≤ψ≤1 such that(52)dTρℷ1e″,Tρℷ″2e≤ψdℷ1e″,ℷ″2e,∀ℷ1e″,ℷ″2e∈FSCX.Theorem 11.
A mappingT,ρ:E,d⟶E,d is called fuzzy softα−ψ-contractive mapping on fuzzy soft metric space E,d if ∃ two fuzzy soft functions α,ϕ:FSCE×FSCE⟶RE∗ and ψ∈Ψ such that(53)α,ϕℷ1e″,ℷ″2edT,ρℷ1e″,T,ρℷ″2e≤ψdℷ1e″,ℷ″2e,∀ℷ1e″,ℷ″2e∈FSCX.


### 2.4.12. M-Metric Space [65, 66]

In partial metric space, the self-distance ofx is less than the distance between two points so another variable is free to take any value. Thus, M-metric space improved the triangle inequality by replacing it with the minimum of self-distance ofxandy, also improved the triangle inequality.A functionm:X2⟶R+ is s.t.b. an m-metric (where X≠∅), if it satisfied the following conditions:(1)
mℷ1″,ℷ1″=mℷ2″,ℷ2″=mℷ1″,ℷ2″if ℷ1″=ℷ2″;(2)
mℷ1″,ℷ2″≤mℷ1″,ℷ2″;(3)
mℷ1″,ℷ2″=mℷ2″,ℷ1″;(4)
mℷ1″,ℷ2″–mℷ1″,ℷ2″≤mℷ1″,ℷ3″–mℷ1″,ℷ3″+mℷ3″,ℷ2″–mℷ3″,ℷ2″.Then,M-metric m together with the set X is titled as M-metric space and represented as an ordered pair X,m.where(54)mℷ1″,ℷ2″=minmℷ1″,ℷ1″,mℷ2″,ℷ2″,Mℷ1″,ℷ2″=maxmℷ1″,ℷ1″,mℷ2″,ℷ2″.Example 24.
LetX=0,∞, then(55)mℷ1″,ℷ2″=ℷ1″+ℷ2″2.
It is easy to get thatm is an m-metric space on X.Remark 26.
M. Asadi et al. generalized the contraction mapping on M-metric space for getting the “fixed-point theorem.”
A mappingT:X⟶X has a fixed a unique fixed point on a complete M-metric space X,m if satisfies the following condition:(56)∃k∈0,1suchthatmTℷ1″,Tℷ2″≤kmℷ1″,ℷ2″∀ℷ1″,ℷ2″∈X,or(57)∃k∈0,12suchthatmTℷ1″,Tℷ2″≤kmℷ1″,Tℷ1″,mℷ2″,Tℷ2″∀ℷ1″,ℷ2″∈X.


### 2.4.13. 2-Partial Rectangular Metric Space [67]

Partial rectangular metric space (p.m.s.) is the generalized version of rectangular metric space (r.m.s.) and the extension of partial metric space (p.m.s.).A functionρ:X2⟶R is s.t.b. partial rectangular metric (p.r.m.) on X≠∅}) if satisfied the subsequent axioms:(1)
ρℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈X;(2)
ℷ1″=ℷ2″⇔ρℷ1″,ℷ2″=ρℷ1″,ℷ1″=ρℷ2″,ℷ2″,∀ℷ1″,ℷ2″∈X;(3)
ρℷ1″,ℷ1″≤ρℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X;(4)
ℷ1″,ℷ2″=ρℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈X;(5)
ρℷ1″,ℷ2″≤ρℷ1″,ℷ3″+ρℷ3″,ℷ4″+ρℷ4″,ℷ2″−ρℷ3″,ℷ3″−ρℷ4″,ℷ4″, for all ℷ1″,ℷ2″∈X and for all distinct points ℷ3″,ℷ4″∈X|ℷ1″,ℷ2″.Then, partial rectangular metricρ together with the set X is titled as partial rectangular metric space and represented as an ordered pair X,ρ.Example 25.
LetX=0,b,φ≥b≥3 define a mapping ρ:X2⟶R by(58)ρℷ1″,ℷ2″=ℷ1″,ifℷ1″=ℷ2″,3b+ℷ1″+ℷ2″2,ifℷ1″,ℷ2″∈1,2,ℷ1″≠ℷ2″b+ℷ1″+ℷ2″2,otherwise.,
Then, it is easy to see thatX,d is the partial rectangular metric space.Remark 27.
Y. Cheng and Y. C. Shiab presented quasi-type contraction for providing the “fixed-point theorem.”
A mappingT:X⟶X(1)
is calledquasi-contraction for complete partial rectangular metric spaceX,ρ with constant k∈0,1 so that ∀ℷ1″,ℷ2″∈X satisfies the following condition:(59)ρTℷ1″,Tℷ2″≤kmaxρℷ1″,ℷ2″,ρℷ1″,Tℷ1″,ρℷ2″,Tℷ2″,ρℷ1″,Tℷ2″,ρℷ2″,Tℷ1″.(2)
isquasi-contraction for complete partial rectangular metric space X,ρ with constant k. Then, T has a unique fixed point ϑ∈X and ρϑ,ϑ=0.


### 2.4.14. Fuzzyb-Metric Space [68]

Partial rectangular metric space (p.m.s.) is the generalized version of rectangular metric space (r.m.s.) and the extension of partial metric space (p.m.s.).Let≠∅) and b≥1∈R and ∗ be a continuous t-norm. A fuzzy set M in X×X×0,∞ is termed as fuzzy b-metric on X if ∀f1″,f2″,f3″∈X, which fulfilled the subsequent axioms:(1)
Mℷ3″,ℷ2″,0=0;(2)
Mℷ3″,ℷ2″,t=1,∀t>0 ⇔ ℷ3″=ℷ2″;(3)
Mℷ1″,ℷ2″,t=Mℷ1″,ℷ1″,t;(4)
Mℷ1″,ℷ2″,t+s≥Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s,∀t,s≥0;(5)
IfMℷ1″,ℷ2″,.:0,∞⟶0,1 is left continuous, andlimt⟶∞Mℷ1″,ℷ2″,t=1.Then, fuzzyb-metric M together with the set X is titled as fuzzy b-metric space and represented as quadruple X,M,∗,b.Example 26.
LetX,d,k be b-metric space. Let Md:X×X×0,∞⟶0,1 be defined by(60)Mdℷ1″,ℷ2″,thinspptt+dℷ1″,ℷ2″,ift>0,0,ift=0.
Then,X,Md,∧,k be fuzzy b-metric space.Remark 28.
(1)
Abbas et al. proposed the definition of “ψ-contraction” and monotone “ψ-contraction” in fuzzy b-metric space.(2)
“Banach’s contraction principle” is extended as an application to the linear equation.
Let a mappingF:X⟶X be correspondence in fuzzy b-metric space such that ∀ℷ1″,ℷ2″∈X and F1∈Fℷ1″,F2∈Fℷ2″ satisfy the following condition:(61)MF1,F2,t≤Mℷ1″,ℷ2″,tk,where k∈0,1 and t∈R. Then, F has unique fixed point.
Chauhan and Utreja [69] proved one important result on fuzzy 2-metric space on the same line.


### 2.4.15. Fuzzy Quasi-b-Metric Space [68]

Nadaban extended the perception of fuzzy quasi-metric space (f.q.m.s.) andb-metric space (b-m.s.).LetM is a fuzzy set in X×X×0,∞ (where X≠∅, ∗ is a continuous t-norm, and k≥1 is a real number) is called fuzzy quasi-b-metric space if it satisfied the subsequent condition: (∀ℷ1″,ℷ2″,ℷ3″∈X, we have the following:(1)
Mℷ1″,ℷ2″,0=0;(2)
Mℷ”1,ℷ”2,t=Mℷ”2,ℷ”1,t=1,∀t>0,⇔ℷ1″=ℷ2″;(3)
Mℷ1″,ℷ2″,kt+s≥Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s,∀t,s≥0;(4)
Mℷ1″,ℷ2″,·:0,∞⟶0,1 is left continuous, and limt⟶∞Mℷ1″,ℷ2″,t=1.Then, fuzzy quasi-b-metric M together with the set X is titled as fuzzy quasi-b-metric space and represented as quadruple X,M,∗,k.Example 27.
LetX,d,k be quasi-b-metric space. Let Md:X×X×0,∞⟶0,1 be defined by(62)Mdℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″,ift>0,0,ift=0.
Then,X,Md,∧,k be fuzzy quasi-b-metric space.Remark 29.
Fuzzy quasi-b-metric space (f.q.b-m.s.) is used to apply in the control theory and convex optimization.
LetX,Md,∧,k be fuzzy quasi-b-metric space and(63)Mℷ1″,ℷ2″,t=minMdℷ1″,ℷ2″,t,Mdℷ2″,ℷ1″,t.
Then,X,M,∧,k is a fuzzy b-metric space.


### 2.4.16.Mb-Metric Space [70–72]

TheMb-metric space is an enlargement of M-metric space by introducing the coefficient of the Mb-metric space (real number s≥1) in triangle equality. The concept of an extension was based on the simulation functions made known by Khojasteh et al.A functionmb:X2⟶R+ is s.t.b. Mb-metric space on X≠∅, if it satisfied the subsequent conditions: ∀ℷ1″,ℷ2″,ℷ3″∈X, we have(1)
mbℷ1″,ℷ1″=mbℷ2″,ℷ2″=mbℷ1″,ℷ2″ if ℷ1″=ℷ2″;(2)
mbℷ1″,ℷ2″≤mbℷ1″,ℷ2″;(3)
mbℷ1″,ℷ2″=mbℷ2″,ℷ1″;(4)
∃a real number s≥1 s.t.b. ∀ℷ1″,ℷ2″,ℷ3″∈X, we have(64)mbℷ1″,ℷ2″–mbℷ1″,ℷ2″≤smbℷ1″,ℷ3″−mbℷ1″,ℷ3″+mbℷ3″,ℷ2″–mbℷ3″,ℷ2″−mbℷ3″,ℷ3″.Then,Mb-metric Mb together with the set X is titled as Mb-metric space and represented as quadruple X,Mb.The numbers is termed as the coefficient of the Mb-metric space X,Mb.(65)mbℷ1″,ℷ2″=minmbℷ1″,ℷ1″,mbℷ2″,ℷ2″,Mbℷ1″,ℷ2″=maxmbℷ1″,ℷ1″,mbℷ2″,ℷ2″.Example 28.
LetX=R+,ρ>1 be a constant, and Mb:X×X⟶R+ be defined by(66)Mbℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,Mb is an Mb-metric space with coefficient s=2ρ>1.Remark 30.
We demonstrate the existence of the “fixed-point theorem” with the assistance of″Zmb contraction” in complete Mb-metric space.Theorem 12.
Suppose the mappingT:X⟶X and let X,Mb be a complete Mb-metric space with the coefficient s≥1 that satisfies the following condition:(67)MbTℷ1″,Tℷ2″≤βbℷ1″,ℷ2″,
for allℷ1″,ℷ2″∈X and where β∈0,1. Then, T has a unique fixed point u∈X and Mbu,u=0.


### 2.4.17. Orthogonal Metric Space [73, 74]

Gordji et al. defined a binary relation on the set.Letd be a usual metric on X≠∅ with an orthogonal set X,⊥. Then, metric with an orthogonal set is termed as orthogonal metric space and represented as X,⊥,d (O-metric space).Example 29.
LetX=0,∞, and we define ℷ1″⊥ℷ2″,if ℷ1″ℷ2″≤ℷ1″orℷ1″ℷ2″≤ℷ2″. Then, ∀ℷ1″∈X, 0⊥ℷ1″. So X,⊥ is an orthogonal set, and if X,d be an usual metric, then X,⊥,d is orthogonal metric space.Remark 31.
Gordji et al. gave a generalized version of “Banach’s contraction principle” with the orthogonal set, which stands for finding the existence and uniqueness of first-order ordinary differential equations and nonlinear fractional differential equations that are allied with Caputo fractional derivative.
A mappingT:X⟶X is called the following:(1)
Orthogonal contraction of orthogonal metric space isX,⊥,d if ∃τ∈0,1satisfies the subsequent condition:(68)dTℷ1″,Tℷ2″≤τdℷ1″,ℷ2″.For any two orthogonally related elements,ℷ1″,ℷ2″∈X.(2)
Orthogonalp-contraction of orthogonal metric space is X,⊥,d and p:X2⟶0,∞ if ∃τ∈0,1satisfies the following condition:(69)pTℷ1″,Tℷ2″≤τpℷ1″,ℷ2″.∀ℷ1″,ℷ2″∈X with ℷ1″⊥ℷ2″.
In the same line, Khalil et al. [75] proved some theorems on orthogonal partial-metric spaces with an application.


### 2.4.18. Rectangularb-Metric Space [22, 76, 77]

George et al. made known the theory of rectangularb-metric space (r.b-m.s.), which is an extension of rectangular metric space (r.m.s.) and the generalization of b-metric space (b-m.s.).A mappingd:X2⟶0,∞ is s.t.b. rectangular b-metric on X(where X≠∅), if fulfils the following axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
ℷ1″,ℷ2″=dℷ2″,ℷ1″,∀r1′,r2′∈X;(3)
∃a real number s≥1 s.t.b.(70)dℷ1″,ℷ2″≤sdℷ1″,ℷ3″+dℷ3″,ℷ4″+dℷ4″,ℷ2″,∀ℷ1″,ℷ2″∈Xandℷ3″,ℷ4″∈X|ℷ1″,ℷ2″,ℷ1″≠ℷ1″.Then, rectangularb-metric d together with the set X is titled as rectangular b-metric space and represented as an ordered pair X,d with coefficient s.Example 30.
LetX=N define d:X2⟶X by(71)dℷ1″,ℷ2″=0,ifℷ1″=ℷ2″,4ρ,ifℷ1″,ℷ2″∈1,2andℷ1″≠ℷ2″,ρ,ifℷ1″orℷ2″∉1,2andℷ1″≠ℷ2″,where ρ>0 is a constant. Then, X,d is rectangular b-metric space with coefficient s=4/3>1.Remark 32.
George et al. gave an analog of “Kannan’s fixed-point theorem” and “Banach’s contraction principle” for generalizing several outcomes in fixed-point theory.
A mappingT:X⟶X has a unique fixed point on a complete rectangular b-metric space X,d with coefficient s>1, if satisfies the following condition:(72)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″,where k∈0,1/s∀ℷ1″,ℷ2″∈X.or(73)dTℷ1″,Tℷ2″≤kdℷ1″,Tℷ1″,dℷ2″,Tℷ2″,where k∈0,1/s+1, ∀ℷ1″,ℷ2″∈X.


### 2.4.19. Extendedb-Metric Space [78, 79]

The coefficient ofb-metric space was replaced with the function ″e” and formed extended b-metric space (e.b-m.s.), which is the most universal form of b-metric space.A mappingd:X2⟶0,∞ is s.t.b. an extended b-metric on M with mapping e (where X≠∅ and e:X2⟶1,∞), if it fulfils the following axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″ for all ℷ1″,ℷ2″∈M;(3)
dℷ1″,ℷ2″≤eℷ1″,ℷ2″dℷ1″,ℷ3″+dℷ3″,ℷ2″ for all ℷ1″,ℷ2″,ℷ3″∈M.Then, extendedb-metric d together with the set X is titled as extended b-metric space and represented as an ordered pair X,d.Example 31.
LetX=1,2,3. We define e:X2⟶0,∞ and de:X2⟶0,∞ as(74)eℷ1″,ℷ2″=1+ℷ1″+ℷ3″,de1,1=de2,2=de3,3=0,de1,2=de2,1=80,de1,3=de3,1=1000,de2,3=de3,2=600.
Hence,X,dis an extended b-metric space.Remark 33.
(1)
Mitrovic et al. proved single-valued mapping and multivalued mapping (m.v.m.) in extendedb-metric space.(2)
Kamran et al. gave the existence of the unique solution as an application to the Fredholm integral equation.


### 2.4.20. Extended Fuzzyb-Metric Space [80]

For setting the most general class of fuzzyb-metric space, we extended and generalized into such space.A fuzzy setMα in X×X×0,∞ (where X≠∅, α:X2⟶1,∞ and ∗ -continuous t-norm) is called extended fuzzy b-metric (e.f.b-m.s.) on X if satisfied the following axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X)(1)
Mαℷ1″,ℷ2″,0=0;(2)
Mαℷ1″,ℷ2″,t=1,∀t>0 ⇔ ℷ1″=ℷ2″;(3)
Mαℷ1″,ℷ2″,t=Mαℷ2″,ℷ1″,t;(4)
Mαℷ1″,ℷ2″,αℷ1″,ℷ2″t+s≥Mαℷ1″,ℷ3″,t∗Mαℷ3″,ℷ2″,s,∀t,s≥0;(5)
Mαℷ1″,ℷ2″,.:0,∞⟶0,1 is left continuous.Then, extended fuzzyb-metric Mα together with the set X is titled as extended fuzzy b-metric space and represented as X,Mα,∗,αℷ1″,ℷ2″.Example 32.
LetX=1,2,3 and let db:X×X⟶R by dℷ”1,ℷ”2=ℷ”1−ℷ”22, then it is easy to get that X,d is a b-metric space. We define the mapping α:X×X⟶1,∞,(75)αℷ1″,ℷ2″=1+ℷ1″+ℷ2″.
LetMα:X×X×0,∞⟶0,1 be given by(76)Mαℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″,ift>0.0,ift=0,and continuous t-norm ∗=∧; then, ℷ1″∗ℷ2″=ℷ″1ℷ2″=minℷ1″,ℷ2″.
Then,X,Md,∧,k be fuzzy quasi-b-metric space.Remark 34.
Mehmood et al. established a new kind of “Banach-type fixed-point theorem” with the help of a fuzzy set in the most universal class of fuzzyb-metric space.Theorem 13.
(Banach’s contraction theorem in fuzzy extended b-metric spaces).
LetX,Mα,∗ be a G-complete extended fuzzy b-metric space with the mapping α:X×X⟶1,∞ such that(77)limt⟶∞Mαℷ1″,ℷ2″,t=1.
Letg:X⟶X be a mapping, which satisfies the condition(78)Mαgℷ1″,gℷ2″,kt≥Mαℷ1″,ℷ2″,t,∀ℷ1″,ℷ2″∈X,where k∈0,1. Furthermore, g has a unique fixed point if for an arbitrary b0∈X, and n;q∈N, we have(79)αbn,bn+q<1k,where bn=gnb0.


### 2.4.21. ExtendedMb-Metric Space [70]

The coefficient ofMb-metric space was replaced with the function θ and formed a new generalized version of Mb-metric space, titled as extended Mb-metric space.A functionmθ:X2⟶0,∞ is titled as extended Mb-metric (e.Mb-m) on X (where θ:X2⟶1,∞ be a function) if fulfilled the subsequent conditions:(1)
mθE1,E1=mθE2,E2=mθE1,E2 if E1=E2;(2)
mθE1,E2≤mθE1,E2;(3)
mθE1,E2=mθE2,E1;(4)
mθE1,E2−mθE1,E2≤θE1,E2mθE1,E3−mθE1,E3+mθE3,E2−mθE3,E2, ∀E1,E2,E3∈X.Then, extendedMb-metric mθ together with the set X is titled as extended Mb-metric space and represented as the ordered pair X,mθ.Example 33.
LetX=Cα,β,R, and define mθ:X2⟶0,∞ and θ:X2⟶1,∞ by(80)mθE1t,E2tsupt∈α,βE1t−E2t2,θE1t,E2t=E1t+E2t+2.
Then,X,mθ is an extended Mb-metric space.Remark 35.
N. Mlaiki et al. provide several fixed-point results on expandedMb-metric space (Mb-m.s.) using the techniques of the classical fixed-point theorems such as “Kannan’s fixed-point theory,” the “Banach fixed-point theory,” and many more.Theorem 14.
LetX,mθ be an extended Mb-metric space and mθb:X2⟶0,∞ be a function defined as(81)mθbE1,E2=mθE1,E2−2mθE1,E2+MθE1,E2,for all x,y∈X. Then, mθbis an extended b-metric and the pair X,mθb is an extended b-metric space.


### 2.4.22. Soft S-Metric Space [81]

Aras et al. gave the theory of softS-metric space in which S-metric was further defined on soft points of soft sets and introduced some of their characteristics as well.LetS be the absolute soft set, E be the ≠∅ set of parameters; SPS be a collection of all soft points ofS, and RE∗ be the set of all non-negative soft real numbers.A mappings:SPS×SPS×SPS⟶RE∗ is s.t.b. soft S-metric on S (the soft set), if it fulfils the subsequent axioms, for each soft points ℷ1a″,ℷ2b″,ℷ3c″,ℷ4d″∈SPS,(1)
Sℷ1a″,ℷ2b″,ℷ3c″≥0;(2)
Sℷ1a″,ℷ2b″,ℷ3c″=0⇔ℷ1a″=ℷ2b″=ℷ3c″;(3)
Sℷ1a″,ℷ2b″,ℷ3c″≤Sℷ1a″,ℷ1a″,ℷ4d″+Sℷ2b″,ℷ2b″,ℷ4d″+Sℷ3c″,ℷ3c″,ℷ4d″.Then, softS-metric s together with the set S is titled as soft S-metric space and represented as S,s,E.Example 34.
LetE=N be a parameter set, and X=R be defined as(82)Sℷ1a″,ℷ2b″,ℷ3c″=a−c+b−c+ℷ1″−ℷ3″+ℷ2″−ℷ3″.
It is easy to get a softS-metric space.Remark 36.
The idea of “soft contractive mapping” on softS-metric space (s.S-m.s.) was established, and some significant concepts were given, for instance, completeness, Cauchy sequence, soft continuous mapping on soft S−metric space (s.S-m.s.), and also with the help of “soft contractive mapping” on soft S−metric space (s.S-m.s.), and some of the “fixed-point theorems” have been demonstrated.
A mappingρ1,ρ2:S,s,E⟶S,s,E has a unique soft fixed point on a soft sequential compact soft S-metric space S,s,E if satisfies the following condition:(83)Sρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ρ1,ρ2ℷ2b″≤αρ1,ρ2ℷ1a″,
for everyℷ1a″,ℷ2b″∈SPS and ℷ1a″≠ℷ2b″.
A mappingρ1,ρ2:S,s,E⟶S,s,Ehas a unique soft fixed point on a complete soft S-metric space S,s,E if satisfies the following contraction condition:(84)Sρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ρ1,ρ2ℷ2b″≤αSρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ℷ1a″,Sρ1,ρ2ℷ2b″,ρ1,ρ2ℷ2b″,ℷ2b″,
for everyℷ1a″,ℷ2b″∈SPS, where α∈0,1/2 is a soft constant real number. Then, ρ1,ρ2 have a unique soft fixed point in SPS.


### 2.4.23.g-Metric Space [82]

Due to today’s large and complex data collection, we developed the notion of distance function that is calledg-metric in which we extended the ordinary distance function of metric (two points in between) and G-metric (three points in between). The g-metric is the distance function of degree n, i.e., for n+1 points.A functiong:Xn⟶R+ is s.t.b. g-metric with order n on X≠∅ (where Xn=∏i=1nX), if it fulfils the following axioms:(1)
Positive definiteness(85)gx0,xn=0⇔x0=···=xn.(2)
Permutation invariant(86)gx0,xn=gxτ0,xτn.For any permutationτon0,1,n,(3)
Monotonicity(87)gx0,xn≤gy0,yn,∀x0,xn,y0,yn∈Xn+1,with(88)xi:i=0,n⊊yi:i=0,n.(4)
Triangle inequality(89)∀x0,..,xs,y0,..,yt,w∈Xwiths+t+1=n,gx0,..,xs,y0,..,yt≤gx0,..,xs,w,..,w+gy0,..,yt,w,..,w.Then,g-metric g together with the setX is titled as g-metric space and represented as an ordered pair X,g.Example 35.
LetX≠∅ be defined d:Xn⟶R+ by(90)dx0,xn=0,ifx0=...=xn1,otherwise,∀x0,xn∈X.
It is easy to getX,d that is a g-metric space.Remark 37.
Several “fixed-point theorems” in theg-metric space (g-m.s.) were generalized, which is considered as an oversimplification of “Banach’s contractive mapping” concerning g-metric space.
A mappingT:X⟶X has a unique fixed point on a complete g-metric space X,g if satisfies the condition: ∃k∈0,1 such that gTℷ1″,Tℷ2″…,Tℷn″≤kgℷ1″,ℷ2″,…,ℷn″,∀ℷ1″,ℷ2″,…,ℷn″∈X.


### 2.4.24.F-Metric Space [83, 84]

F-metric is a generalization of metric space in which we defined the natural topology τF with the different topological properties.The mappingD:X2⟶0,+∞ (where X≠∅) is s.t.b. F-metric, if fulfils the following axioms:We suppose that∃f,g∈F×0,+∞ such that(1)
ℷ1″,ℷ2″∈X×X,Dℷ1″,ℷ2″=0 if ℷ1″=ℷ2″;(2)
ℷ1″,ℷ2″=Dℷ2″,ℷ1″;∀ℷ1″,ℷ2″∈X×X;(3)
∀ℷ1″,ℷ2″∈X×X,∀N∈N,N≥2,∀uii=1N⊂X with u1,uN=ℷ1″,ℷ2″; we have Dℷ1″,ℷ2″>0(91)⇒fDℷ1″,ℷ2″≤f∑i=1N−1Dui,ui+1+g.Then,F-metric D together with the set X is titled as F-metric space and represented as an ordered pair X,D.Example 36.
LetX=R be defined d:X2⟶R+ by(92)dℷ1″,ℷ2″=ℷ1″−ℷ2″2,ifℷ1″,ℷ2″∈0,3×0,3ℷ1″−ℷ2″,ifℷ1″,ℷ2″∉0,3×0,3,∀ℷ1″,ℷ2″∈X,
withft=lnt and ρ=lnt.It is easy to get X,d that is a F-metric space.Remark 38.
Jelli and Samet proved the modified version of the “Banach contraction principle.” Afterward, Alansari et al. established some circumstances for the existence of a solution in the “fuzzy fixed-point theorem” used for solving the fuzzy Cauchy problem.
A mappingT:X⟶X has a unique fixed point on a complete F-metric space X,D if satisfies the following condition: ∃k∈0,1 such that DTℷ1″,Tℷ2″≤kDℷ1″,ℷ2″, ∀ℷ1″,ℷ2″∈X.


### 2.4.25. Controlled Metric Type Space [85]

The controlled metric is an extension ofb-metric in which we introduced a control function αx,z in the b-triangle inequality.The mappingd:X2⟶0,∞ is s.t.b. controlled metric type (where φ:X×X⟶1,∞and X≠∅), if fulfils the following axioms:(1)
dC1,C2=0 if C1=C2;(2)
dC1,C2=dC2,C1;(3)
dC1,C2≤φC1,C3dC1,C3+φC3,C2dC3,C2, for all C1,C2,C3∈X.Then, controlled metricd together with the set X is titled as controlled metric space and represented as an ordered pair X,d.Example 37.
LetX=0,1,2 and a function d be given as(93)d0,0=d1,1=d2,2=0,d0,1=d1,0=1,d0,2=d2,0=12,d1,2=d2,1=25.
We takeβ:X×X⟶0,∞ to be symmetric and is defined by the following:(94)β0,0=β1,1=β2,2=β0,2=1,β1,2=54,β0,1=1110.
It is easy to get thatd that is a controlled metric space.Remark 39.
Mlaki et al. proved the contraction principle on the controlled metric type space.
A mappingT:X⟶X on a complete controlled metric space is X,d and satisfies the following condition:(95)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,where ∃k∈0,1. For ℷ0″∈X, we take ℷ1n″=Tnx0. Suppose that(96)supm≥1limi⟶∞βℷi+1″,ℷi+2″βℷi″,ℷi+1″βℷi+1″,ℷm″<1k.
In additional, we assume that∀ℷ1″∈X and we have(97)limn⟶∞βℷ1n″,ℷ1″andlimn⟶∞βℷ1″,ℷ1n″.
Then,T has a fixed a unique fixed point.


### 2.4.26. RectangularM-Metric Space [86]

RectangularM-metric space is an extension and generalization of the class of M-metric space in addition to partial rectangular metric space.The mappingmr:X×X⟶0,∞ is s.t.b. rectangular M-metric space (where X≠∅) if satisfied the following conditions: ∀R1,N2∈X(1)
mrR1,N2=mrR1,N2=MrR1,N2 if R1=N2;(2)
mrR1,N2≤mrR1,N2;(3)
mrR1,N2=mrR2,N1;(4)
mrR1,N2−mrR1,N2≤mrR1,N3−mrR1,N3+mrR3,N4−mrR3,N4+mrR4,N2−mrR4,N2, ∀R3,N4∈X|R1,N2.Then, rectangularM-metric mr together with the set X is titled as rectangular M-metric space and represented as an ordered pair X,mr.where(98)mrR1,N2=minmrR1,N1,mrR2,N2,MrR1,N2=maxmrR1,N1,mrR2,N2.Example 38.
Xβ=R1∈C:argR1=β∪0, for a fixed β,0≤β≤2π. If we define self-mapping mr on Xβ, it is given by(99)mrR1,N2=R1+R22,∀R1,R1∈Xβ.
Then, it is easy to getXβ,mr that is a rectangular M-metric space.Remark 40.
N.Y. Ozgur et al. used different contraction principles for proving some fixed-point theorems with various techniques, which generalized some established fixed-point outcomes, for instance, the “Kannan fixed-point theorem,” “Banach contraction principle” and the “Chatterjea fixed-point theorem.” Also, we provide relevance in the direction of the fixed-circle problem.
A mappingT:X⟶X on a rectangular M-metric space is X,d and satisfies the contraction condition if ∃k∈0,1 such that(100)mrTℷ1″,Tℷ2″≤kmrℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,
and consider the sequenceℷ1n″n≥0 defined by ℷ1n+1″=Tℷ1n″. If ℷ1n″⟶uasn⟶∞, then Tℷ1n″⟶Tu as n⟶∞.


### 2.4.27. RectangularMb-Metric Space [71]

Inspired by the theory of rectangularM-metric space (M-m.s.) and Mb-metric space (Mb-m.s.), Asim et al. established the rectangular Mb-metric space to get well and enlarge the existing results of the literature.A mappingmrmb:X2⟶R+ is s.t.b. a rectangular Mb-metric with coefficient s≥1, if mrmb satisfies the following (∀R1,N2,R3∈X and all distinct R4,N5∈X|R1,N2:(1)
mrmbR1,N1=mrmbR1,N2=mrmbR2,N2 if and only ifR1=N2;(2)
mrmbR1,N1≤mrmbR1,N2;(3)
mrmbR1,N2=mrmbR2,N1(4)
mrmbR1,N2−mrmbR1,N1≤smrmbR1,N3−mrmbR1,N3+mrmbR3,N4−mrmbR3,N4+mrmbR4,N2−mrmbR4,N2–mrmbR3,N3−mrmbR4,N4.Then, rectangularMb-metric mrmb together with the set X is titled as rectangular Mb-metric space and represented as an ordered pair X,mrmb.Example 39.
LetX=R+,ρ>1 a constant, and mrmb:X×X⟶R+be defined by(101)mrmbℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,mrmb is a rectangular Mb-metric space with coefficient s=3ρ−1>1.Remark 41.
In rectangularMb-metric space, M. Asim et al. proved an equivalence of the “Banach contraction principle.”
A mappingT:X⟶X on a complete rectangular Mb-metric space is X,d with coefficient s≥1 and satisfies the following contraction condition:(102)mrmbTℷ1″,Tℷ2″≤kmrmbℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X.where k∈0,1/s. Then, T has a unique fixed point ℷ1″ such that mrmbℷ1″,ℷ1″=0.


### 2.4.28.R-Metric Space [87]

In theR-metric space, additionally, an arbitrary relation R is defined between the points on a nonempty set.SupposeX,d is a metric space (m.s.) and R be a relation on X≠∅, then triple X,d,R or X is called R-metric space.Example 40.
LetX=0,∞ equipped with Euclidean metric. We define xRy if ℷ1″ℷ2″≤ℷ1″∨ℷ2″, where ℷ1″∨ℷ2″=ℷ1″orℷ2″.
Then,X,d,R is a R-metric space.Remark 42.
It was difficult to discover the exact value of the equilibrium point, so Khalehoghli et al. tried to provide a structural method for determining the exact value of the fixed point, which is useful to solve a problem related to economics and game theory.
A mappingT:X⟶X is called R-contraction with Lipschitz constant 0<k<0 if ∀ℷ1″,ℷ2″∈X such that ℷ1″Rℷ2″, then we have the following:(103)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″.Theorem 15.
A mappingT:X⟶X on a complete metric space X,d has a unique fixed point in X if for some β∈0,1 in a complete metric satisfies the following condition dTℷ1″,Tℷ2″≤βdℷ1″,ℷ2″, for allℷ1″,ℷ2″∈X.


### 2.4.29. SoftD-Metric Space [88]

Aras et al. provided the theory of softD-metric space (s.D-m.s.), which was further defined on soft points of soft sets and defined the soft ∙−distance on complete soft D-metric by using the designation of soft D-metric.A mappingD:SPX×SPX×SPX⟶RE is s.t.b. soft D−metric on X (where X≠∅) if D fulfils the subsequent conditions, for each soft points ℷ1″,ℷ2″,ℷ3″,ℷ4″∈SPX,(1)
Dℷ1″,ℷ2″,ℷ3″≥0⇔ℷ1″=ℷ2″=ℷ3″;coincidence;(2)
Dℷ1″,ℷ2″,ℷ3″=Dℷ2″,ℷ1″,ℷ3″=Dℷ1″,ℷ3″,ℷ2″=…symmetry;(3)
Dℷ1″,ℷ2″,ℷ3″≤Dℷ1″,ℷ2″,ℷ4″+Dℷ1″,ℷ4″,ℷ3″+Dℷ4″,ℷ2″,ℷ3″.Then, softD-metric D together with the set X is titled as soft D-metric space and represented as X,D,E.Example 41.
LetX be a nonempty set, and E be the nonempty set of parameters. If we define a mapping D:SPX×SPX×SPX⟶RE by(104)Dℷa″,ℷb″,ℷc″=0,ℷa″=ℷb″=ℷc″1,otherwise,∀ℷa″,ℷb″,ℷc″∈SPX,
then it is easy to show thatD is a soft D-metric space.Remark 43.
(1)
“Fixed-point theorem” of soft continuous mappings on softD-metric space has been proven.(2)
It is given the “fixed-point theorem” using the principle of softΔ−distance.Theorem 16.
LetX,D,E be a complete D-metric space and ∙−distance of X.f,g:X,D,E⟶X,D,E be a soft mapping. Let Xbe ∙ bounded. Suppose that ∃ is a soft real number r∈RE,0≤r<1, such that(105)f,gℷ1″,f,g2ℷ1″,f,gℷ2″≤rℷ1″,f,gℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈SPX. Then, ∃ℷ3″∈SPX such that ℷ3″=f,gℷ3″. In addition to ℷ4″=f,gℷ4″, then ℷ4″,ℷ4″,ℷ4″=0.


### 2.4.30. Controlled Rectangularb-Metric Space [89]

N. Mlaiki et al. established the generalization of rectangularb-metric space by altering the rectangular equality.A functionb:X4∙1,∞, and db:X2∙0,∞ is s.t.b. controlled rectangular b-metric space on Xwhere X≠∅) if all distinct C1,C2,C3,C4∈X; we have(1)
dbC1,C2=0 if C1=C2;(2)
dbC1,C2=dbC2,C1;(3)
dbC1,C2≤bC1,C2,C3,C4dbC1,C3+dbC3,C4+dbC4,C2.Then, controlled rectangularb-metric db together with the set X is titled as controlled rectangular b-metric space and represented as X,db.Example 42.
LetX=Y∪Z, where Y=1/m,wherem∈N, and Z be a set of positive integers. We define db:X2∙0,∞ by(106)dbC1,C2=0,iffC1=C2,2β,ifC1,C2∈Y,β2,otherwise,where β is a constant bigger than 0 and b:X4∙1,∞ by(107)bC1,C2,C3,C4=maxC1,C2,C3,C4+2β.Remark 44.
Controlled rectangularb-metric space is used to prove some “fixed-point theorems.”
This theorem is proved, as an application.
For any natural numberκ≥3, the equation sκ+1=κ4–1sκ+1+κ4shas a unique real solution.
A mappingT:X⟶X on a controlled rectangular b-metric space is X,d and satisfies the contraction condition if ∃k∈0,1 such that(108)dbTℷ1″,Tℷ2″≤kdbℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,and supm>1limn⟶∞bℷ1″,ℷ2″,ℷ3″,ℷ4″≤1/k. Then, T has a unique fixed point.


### 2.4.31. Controlled Fuzzy Metric Space

Controlled fuzzy metric space (c.f.m.s.) is the extension of fuzzy metric space (f.m.s.) with generalized triangle inequality and the generalization of fuzzy b-metric space (f.b-m.s.).Letλ:X2⟶1,∞,∗ is a continuous t-norm, and Mλ is a fuzzy set on X2×0,∞ (where X≠∅) is s.t.b. controlled fuzzy metric, if fulfilled the subsequent conditions, (∀C1,C2,C3∈X,s,t>0) such that(1)
MλC1,C2,0=0;(2)
MλC1,C2,t=1 ⇔C1=C2(3)
MλC1,C2,t=MλC2,C1,t(4)
MλC1,C2,t+s≥MλC1,C3,t/λC1,C3∗MλC3,C2,s/λC1,C3;(5)
MλC1,C2,⋅:0,∞⟶0,1 is continuous.Then, controlled fuzzy metricMλ together with the set X is titled as controlled fuzzy metric space and represented as X,Mλ,∗.Example 43.
LetX=A∪C, where A=0,1 and C=N|0,1. We define Mλ:X×X×0,∞⟶0,1 as(109)MλC1,C2,t=1,ifC1=C2,tt+1/C2,ifC1∈AandC2∈C,tt+1/C1,ifC1∈CandC2∈A,tt+1,ifotherwise,
with a continuous t-norm ∗ such thatta∗tb=ta.tb. We define λ:X2⟶1,∞ as(110)λC1,C2=1,ifC1,C2∈A,maxC1,C2,ifotherwise.
It is easy to getX,Mλ,∗ that is a controlled metric space.Remark 45.
M. S. Sezen et al. proved a “Banach contraction principle” and introduced a new concept of “fixed-point theorem” for several self-mappings, which satisfied fuzzy “𝜓-contraction” condition.Theorem 17.
A mappingT:X⟶X on a controlled fuzzy metric space X,Mλ,∗ has a unique fixed point in X if satisfies the following condition:(111)MλTC1,TC2,t≥βMλC1,C2,t,
for allC1,C2∈X and β>0.


### 2.4.32. CAT(0) Space [90, 91]

A metric spaceX is said to be a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. Complete CAT(0) spaces are often called Hadamard spaces.Definition of (CAT(0) space): let (X,d) be a geodesic space. It is a CAT (0) space if, for any geodesic triangle, ∆ ⊂X and x, y ∈ ∆, d(x, y) ≤ d (x¯, y¯) where ¯x, y¯ ∈ ∆¯.Amnuaykarn et al. [90] proved some theorems on the existence of best proximity points of multivalued mappings in CAT(0) spaces.Ugwunnadi and Okeke [91] studied the approximation of common solutions for a finite family of generalized demimetric mappings and monotone inclusion problems in CAT(0) spaces.


## 2.4.1. Complex-Valued Metric Space [47, 48]

The explanation of the complex-valued metric spaces (c.m.s.) occurred when replaced range ℂ (with a certain order) form rangeR (with a normal order), i.e., the generalization of the contractive condition in metric space made much of the concept available. There was a considerable amount of content available in generalized metric spaces. In order to satisfy a rational inequality, complex-valued metric space was thus introduced and some results were also developed for mapping.A mappingd:X2⟶C is s.t.b. complex-valued metric space (c.m.s.) on X(X≠∅, if fulfilled the subsequent condition:(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈XAnd dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″, for allℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈X.Then, complex-valued metricd together with the set X is titled as complex-valued metric space and represented as ordered pair X,d.Example 15.
LetX=C be a set of complex number. We define d:C×C⟶C by(31)dℷ1″,ℷ2″=ℷ1,1″−ℷ2,1″+iℷ1,2″+ℷ2,2″,where ℷ1″=ℷ1,1″+ℷ1,2″ and ℷ2″=ℷ2,1″+ℷ2,2″.
Then,C,d is a complex-valued metric space.Remark 15.
In the complex-valued metric space (c.m.s.), it provides the extension of “Banach’s fixed-point theorem,” which helps to give a common unique solution to the Urysohn integral equations.(i)
xσ=∫abK1σ,τ,xτdτ+gσ,(ii)
xσ=∫abK2σ,τ,xτdτ+hσ, where σ∈a,b,x,g,h∈X.
A mappingT,f:X⟶X be two functions in metric space X,d. A mapping T is called f−contraction if ∃φ∈0,1 such that(32)dfTℷ1″,fTℷ2″≤φdfℷ1″,fℷ2″,∀ℷ1″,ℷ2″∈X.


## 2.4.2. Coneb-Metric Space [49]

The generalization of cone metric space (C.M.S.) together with b-metric space is termed as cone b-metric space.A mappingd:X2⟶E is s.t.b. cone b-metric on X(where X≠∅,s≥1∈R+) if for all C1,C2,C3∈X met the following conditions:(1)
dC1,C2>0 with C1≠C2 and 𝑑C1,C2=0 if and only if C1=C2;(2)
C1,C2=dC2,C1;(3)
dC1,C2≤sdC1,C3+dC3,C2.Then, coneb-metric d together with the setX is titled as cone b-metric space and represented as an ordered pair X,d.Example 16.
LetE=l1,P=un∈E/un≥0,∀n≥1, and d:X×X⟶E such that(33)d∗C1,C2=dC1,C22n,forn≥1.
Then,X,d is a cone b-metric space with coefficient r=21/p>1.Remark 16.
N. Sharma proved contractive mapping for “fixed-point theorem” with rational expression and without normality condition, in cone b-metric space.Theorem 6.
Suppose the mappingT:X⟶X and let X,d be a complete cone b-metric space with the coefficient s≥1 that satisfies the following condition, then(34)dTC1,TC2≤αdC1,TC1dC2,TC2dC1,C2+βdC1,C2,for C1,C2∈X and α,β∈0,1 with α+β<1. Then, T has a unique fixed point in X.


## 2.4.3.S-Metric Space [50, 51]

The philosophy of S-metric spaces is the modification of D-metric and G-metric spaces in which the 3-dimensional metric space was made known by Sedghi et al. InS-metric space, the distance between all three distinct points is always greater than or equal to zero.A functionS:X3⟶0,∞ is s.t.b. S-metric space on X≠∅ if it fulfils the following axioms: ∀ℷ1″,ℷ2″,ℷ3″,ℷ4″∈X,(1)
Sℷ1″,ℷ2″,ℷ3″≥0;(2)
Sℷ1″,ℷ2″,ℷ3″=0⇔ℷ1″=ℷ2″=ℷ3″;(3)
Sℷ1″,ℷ2″,ℷ3″≤Sℷ1″,ℷ2″,ℷ4″+Sℷ1″,ℷ2″,ℷ4″+Sℷ1″,ℷ2″,ℷ4″.Then,S-metric S together with the set X is titled as S-metric space and represented as an ordered pair X,S.Example 17.
LetX=Rn and , a norm on X, then(35)Sℷ1″,ℷ2″,ℷ3″=ℷ2″+ℷ3″−2ℷ1″+ℷ2″+ℷ3″,is an S-metric on X.Remark 17.
Ozgur and Tas characterized the thought of a fixed circle and explored the existence and uniqueness of the “fixed-point theorem” for the self-mapping on S-metric space. Likewise, a few instances of self-mappings having fixed circles are given with a mathematical perspective.
Some new contractive mapping and the concept ofCs and Ls mapping on S-metric space were established.Theorem 7.
LetX;S be a complete S-metric space, and A;B:X⟶X be the mappings that fulfil the subsequent conditions:(1)
AX⊆BX and either AXorBX is a closed subset of X,(2)
The pairA;B is weakly compatible,(3)
SAℷ1″;Aℷ2″;Aℷ3″≤φmaxSBℷ1″;Bℷ2″;Bℷ3″;k1SBℷ3″;Aℷ1″;Aℷ3″;k2SBℷ3″;Aℷ2″;Aℷ3″ for all ℷ1″,ℷ2″,ℷ3″∈X and 0<k1;k2<1, where φ∈∅.
Then, the mapsA and B have a unique common fixed point. If B is continuous at the fixed point p, then A is also continuous at p.


## 2.4.4. Metric-Like Space [29]

Metric-like space (m.l.s.) is a generalization of partial metric space, in which all axioms are the same as the usual metric except first, that isσx,x may be positive for x∈X.A functionσ:X2⟶R+∪0iss.t.b.metric−likespaceonX≠∅, if it fulfilled the subsequent axioms: for any ℷ1″,ℷ2″∈X,(1)
σℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(2)
σℷ1″,ℷ2″=σℷ2″,ℷ1″;(3)
σℷ1″,ℷ2″≤σℷ1″,ℷ3″+σℷ3″,ℷ2″.Then, metric-likeσ together with the set X is titled as metric-like space and represented as an ordered pair X,σ.Example 18.
LetX=0,1 and let(36)σℷ1″,ℷ2″=2,ifℷ1″=ℷ2″=0,1,otherwise.
It is easy to getX,σ that is metric-like space.Remark 18.
(1)
As an application, A. Harandi unified and generalized some important results of the “fixed-point theory.”(2)
Metric-like space considered new Geraghty-type mappings and proposed “fixed-point theorems” for such space [52].
A topologyτσ for each metric-like σ was generated, which is a family of open balls. Bσx,ε=y∈X:σx,y–σx,x<ε,∀x∈X and ε>0.
Then, a sequencexn converges to a point x∈X if limn⟶∞σxn,x=σx,x.


## 2.4.5.b-Metric-Like Space [53, 54]

Theb-metric-like space (b-m.l.s.) is a generalized version of metric-like space in which coefficient of metric space was initiated in the triangle inequality.A mappingσ:X2⟶R+, (where X≠∅) is s.t.b. b-metric-like on X if ∀ℷ1″,ℷ2″,ℷ3″∈X and constant s≥1 fulfils the following axioms:(1)
σℷ1″,ℷ2″=0⇒ℷ1″=ℷ2″;(2)
σℷ1″,ℷ2″=σℷ2″,ℷ1″;(3)
σℷ1″,ℷ2″≤sσℷ1″,ℷ3″+σℷ3″,ℷ2″.Then,b-metric-like σ together with the set X is titled as b-metric-like space and represented as an ordered pair X,σ.Example 19.
LetX=R+,ρ>1 a constant, and σb:X×X⟶R+ be defined by(37)σbℷ1″,ℷ2″=ℷ1″+ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,σb is a b-metric-like space with coefficient =2ρ−1.Remark 19.
Some results ofb-metric space are related to the fixed point and give nontrivial examples to support the concept such as finding the solution of the electric circuit equation with the help of second-order differential equations.
A mappingT:X⟶X is called(1)
Banach G-contraction ifT preserves edges of G, i.e., ∀ℷ”1,ℷ”2∈X;(38)ℷ1″,ℷ2″∈EG⇒Tℷ1″,Tℷ2″∈EG.(2)
Orbitally continuous if givenℷ1″∈X and any sequence kn of positive integers,(39)Tknℷ1″⟶ℷ2″asn⟶∞⇒TTknℷ1″⟶Tℷ2″asn⟶∞.


## 2.4.6. Quasi-b-Metric Space [53–55]

The class of ordinary quasi-metric space was extended to form quasi-b-metric space.A functiond:X2⟶R+ is entitled as quasi-b-metric on X ≠∅ and constant s≥1, if it satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″≥0, ∀ℷ1″,ℷ2″∈X;(2)
dℷ1″,ℷ2″=0 ⇔ ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″≤sdℷ1″,ℷ3″+dℷ3″,ℷ2″, ∀ℷ1″,ℷ2″″,ℷ3″∈X.Then, quasi-b-metric d together with the set X is titled as quasi-b-metric space and represented as an ordered pair X,d.Example 20.
LetX=lp, where1≤p<∞, be defined by(40)lp=ℷ″1nn≥1⊆R:∑n=1∞ℷ″1np<∞.
We defined:X×X⟶R+ by(41)dℷ1″,ℷ2″=0,ifℷ1″≤ℷ2″,∑n=1∞ℷ″1np1/p,ifℷ1″≥ℷ2″.
It is easy to get thatX,d is a quasi-b-metric space with constant p≥1.Remark 20.
(1)
It is proved that some “fixed-point theorems” for a nondecreasing mapping satisfied the nonlinear contraction.(2)
Partially well-ordered complete quasi-b-metric space was used to solve the fractional differential equations [55].
A mappingT:X⟶X is called α-admissible for nonempty set X, if ∀ℷ1″,ℷ2″∈X, where α:X×X⟶0,∞ be a mapping, then we have(42)αℷ1″,ℷ2″≥1⇒αTℷ1″,Tℷ2″≥1.


## 2.4.7. Soft Metric Space [56, 57]

When difficulties occur due to inadequacy of the parameterization tool of the theory, then the new mathematical tool “the soft set” theory was introduced by Molodtsov, for dealing with uncertainties and inherent difficulties of theories. Das and Samanta initiated the analysis of soft metric space (s.m.s.) created on soft-point soft sets, for implementations of soft set theory in real-life problems and other domains for better performance.LetA be a nonempty subset of parameters, and ℵ be the absolute soft set, i.e., Fλ=ℷ″,∀λ∈A, where F,A=ℷ″.A mappingd:SPℷ″×SPℷ″⟶RA∗ is called soft metric on ℵ if, for any ℷ″ax,ℷ″by,ℷ″cz∈SPℷ″ (equivalently, ℷ″ax,ℷ″by,ℷ″cz∈ℷ″), satisfied the following conditions:(1)
dℷ″ax,ℷ″by≥0(2)
dℷ″ax,ℷ″by=0 ⇔ ℷ″ax=ℷ″by(3)
dℷ″ax,ℷ″by=dℷ″by,ℷ″ax(4)
dℷ″ax,ℷ″by≤dℷ″ax,ℷ″cz+dℷ″cz,ℷ″by.The soft metric space is symbolized byℷ″,d,A.Example 21.
Letℷ″=A=1/n:n∈N, and the mapping d:SPℷ″×SPℷ″⟶RA∗ be defined by(43)dℷ″ax,ℷ″by=x¯−y¯+a¯−b¯,∀ℷ″ax,ℷ″by∈SPℷ″.
Then, it is easy to show a soft metric space onℷ″.Remark 21.
The soft metric space is used to prove the soft version of the “contraction principle” in complete soft metric space, which provides applications related to dynamic programming.
A soft mappingf:ℷ″⟶ℷ″ and let ℷ″,d,A be a complete soft metric space with finite set A that satisfies(44)dfℷ″ax,fℷ″by≤τdℷ″ax,fℷ″axdℷ″by,fℷ″by,∀ℷ″ax,ℷ″by∈SPℷ″,where 0≤τ≤1/2, and then f has unique fixed point.


## 2.4.8. Partialb-Metric Space [58, 59]

S. Shukla introduced partialb-metric space and the generalization of partial metric space (p.m.s.) together with b-metric space.A mappingb:X2⟶R+ is s.t.b. partial b-metric (p.b-m.) on X≠∅, if it satisfied the subsequent axioms: (for all ℷ1″,ℷ2″,ℷ3″∈X),(1)
bℷ1″,ℷ1″=bℷ1″,ℷ2″=bℷ2″,ℷ2″⇔ℷ1″=ℷ2″;(2)
bℷ1″,ℷ1″≤bℷ1″,ℷ2″;(3)
bℷ1″,ℷ2″=bℷ2″,ℷ1″;(4)
∃ a real number s≥1 such that(45)bℷ1″,ℷ2″≤sbℷ1″,ℷ3″+bℷ3″,ℷ2″−bℷ3″,ℷ3″.Then, partialb-metric b together with the set X is titled as partial b-metric space and represented as an ordered pair X,b, and s is the coefficient of X,b.Example 22.
LetX=R+,ρ>1 a constant, and b:X×X⟶R+ be defined by(46)bℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,σb is a partial b-metric space with coefficient s=2ρ>1.Remark 22.
Analog of “Kannan’s type fixed-point theorem” and “Banach’s contraction theorem” was proved in the partialb-metric space.Theorem 8.
Suppose the mappingT:X⟶X and let X,b be a complete partial b-metric space with the coefficient s≥1 that satisfies the following condition:(47)bTℷ1″,Tℷ2″≤βbℷ1″,ℷ2″.
For allℷ1″,ℷ2″∈X and where β∈0,1, then T has a unique fixed point u∈X and bu,u=0.


## 2.4.9. Fuzzy Cone Metric Space [60]

The class of fuzzy metric space (f.m.s.) was extended and generalized into fuzzy cone metric space.A mappingd:X2⟶E∗I is s.t.b. fuzzy cone metric (f.c.m.) on X≠∅, if it satisfied the subsequent axioms:(1)
dℷ1″,ℷ2″≥0;∀ℷ1″,ℷ2″∈X and dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″∀ℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″⊕dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈X.Then, fuzzy cone metricd together with the set X is titled as fuzzy cone metric space and represented as an ordered pair X,d.Example 23.
LetP be any cone and X=N and ℷ1″∗ℷ2″=ℷ1″ℷ2″ and d:X2×intP⟶0,1 be defined by(48)dℷ1″,ℷ2″,t=ℷ1″ℷ2″,,ifℷ1″≤ℷ2″,ℷ2″ℷ1″,ifℷ1″≥ℷ1″.∀ℷ1″,ℷ2″∈Xand t≫0. Then, it is easy to show that it is fuzzy cone metric space.Remark 23.
(1)
In fuzzy cone metric space (f.c.m.s.), some of the basic properties for innumerable types of contraction mappings and for “fixed-point theorems” were established.(2)
Chet et al. provide an application of nonlinear integral equation in fuzzy cone metric space for the existence and uniqueness of the solution.
A mappingT:X⟶X is titled as fuzzy cone contraction on fuzzy cone metric space X,d,∗ if ∃k∈0,1 satisfies the following:(49)1dTℷ1″,Tℷ2″,t−1≤k1dℷ1″,ℷ2″,t−1,∀ℷ”1,ℷ”2∈X and t≫0. t is called the contraction constant of T.


## 2.4.10. Fuzzy Coneb-Metric Space [61]

The definition of fuzzy coneb-metric space (f.c.b-m.s.), inspired by cone b-metric space (c.b-m.s.) and fuzzy metric space, was made known by T. Bag.A mappingd:X2⟶E∗Iis s.t.b. fuzzy cone b-metric (where X≠∅ and s≥1 be a real number) if it fulfils the following axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈X and dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈X;(3)
dℷ1″,ℷ2″≤sdℷ1″,ℷ3″⊕dℷ3″,ℷ2″.Then, fuzzy coneb-metric d together with the set X is titled as fuzzy cone b-metric space and represented as an ordered pair X,d.Remark 24.
Some results of “Banach’s contraction mapping” for the “fixed-point theorem” were established in fuzzy coneb-metric space.Theorem 9.
A mappingT:X⟶X on a complete fuzzy cone b-metric space X,d has a unique fixed point in X if for complete fuzzy cone b-metric space X,d with the coefficient s≥1fulfils the subsequent condition:(50)dTℷ1″,Tℷ2″≤βdℷ1″,ℷ2″,
for allℷ1″,ℷ2″∈X and whereβ∈0,1.


## 2.4.11. Fuzzy Soft Metric Space [62–64]

For the fuzzy soft metric space (f.s.m.s.), first fuzzy soft point was introduced in which problems deal with ambiguous data in several fields and that was not effectively demonstrated in mathematics. There are various kinds of mathematical instruments to deal with uncertainties, i.e., the fuzzy set and the soft set that helps to solve the problem difficulty in all areas. In the fuzzy soft metric space (f.s.m.s.), the fuzzy soft open ball and also the closed ball were proposed.LetX be the absolute fuzzy soft set,(51)A∙Ei.e.,Xe=τ,∀e∈E.A∗ be the set of all non-negative fuzzy soft real numbers.A mappingd:FSCX×FSCX⟶RA∗ is s.t.b. fuzzy soft metric (f.s.m.s.) on the soft setX, if it holds the subsequent axioms:(1)
dℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈FSCX;(2)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(3)
dℷ1″,ℷ2″=dℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈FSCX;(4)
dℷ1″,ℷ2″≤dℷ1″,ℷ3″+dℷ3″,ℷ2″,∀ℷ1″,ℷ2″,ℷ3″∈FSCX.Then, fuzzy soft metricd together with the set X is titled as fuzzy soft metric space and represented as an ordered pair X,d.Remark 25.
Fuzzy soft metric is applied on soft open balls and soft closed balls.Theorem 10.
A mappingTρ:E,d⟶E,d is called fuzzy soft contractive mapping on fuzzy soft metric space E,d if ∃two fuzzy soft number ψwith0≤ψ≤1 such that(52)dTρℷ1e″,Tρℷ″2e≤ψdℷ1e″,ℷ″2e,∀ℷ1e″,ℷ″2e∈FSCX.Theorem 11.
A mappingT,ρ:E,d⟶E,d is called fuzzy softα−ψ-contractive mapping on fuzzy soft metric space E,d if ∃ two fuzzy soft functions α,ϕ:FSCE×FSCE⟶RE∗ and ψ∈Ψ such that(53)α,ϕℷ1e″,ℷ″2edT,ρℷ1e″,T,ρℷ″2e≤ψdℷ1e″,ℷ″2e,∀ℷ1e″,ℷ″2e∈FSCX.


## 2.4.12. M-Metric Space [65, 66]

In partial metric space, the self-distance ofx is less than the distance between two points so another variable is free to take any value. Thus, M-metric space improved the triangle inequality by replacing it with the minimum of self-distance ofxandy, also improved the triangle inequality.A functionm:X2⟶R+ is s.t.b. an m-metric (where X≠∅), if it satisfied the following conditions:(1)
mℷ1″,ℷ1″=mℷ2″,ℷ2″=mℷ1″,ℷ2″if ℷ1″=ℷ2″;(2)
mℷ1″,ℷ2″≤mℷ1″,ℷ2″;(3)
mℷ1″,ℷ2″=mℷ2″,ℷ1″;(4)
mℷ1″,ℷ2″–mℷ1″,ℷ2″≤mℷ1″,ℷ3″–mℷ1″,ℷ3″+mℷ3″,ℷ2″–mℷ3″,ℷ2″.Then,M-metric m together with the set X is titled as M-metric space and represented as an ordered pair X,m.where(54)mℷ1″,ℷ2″=minmℷ1″,ℷ1″,mℷ2″,ℷ2″,Mℷ1″,ℷ2″=maxmℷ1″,ℷ1″,mℷ2″,ℷ2″.Example 24.
LetX=0,∞, then(55)mℷ1″,ℷ2″=ℷ1″+ℷ2″2.
It is easy to get thatm is an m-metric space on X.Remark 26.
M. Asadi et al. generalized the contraction mapping on M-metric space for getting the “fixed-point theorem.”
A mappingT:X⟶X has a fixed a unique fixed point on a complete M-metric space X,m if satisfies the following condition:(56)∃k∈0,1suchthatmTℷ1″,Tℷ2″≤kmℷ1″,ℷ2″∀ℷ1″,ℷ2″∈X,or(57)∃k∈0,12suchthatmTℷ1″,Tℷ2″≤kmℷ1″,Tℷ1″,mℷ2″,Tℷ2″∀ℷ1″,ℷ2″∈X.


## 2.4.13. 2-Partial Rectangular Metric Space [67]

Partial rectangular metric space (p.m.s.) is the generalized version of rectangular metric space (r.m.s.) and the extension of partial metric space (p.m.s.).A functionρ:X2⟶R is s.t.b. partial rectangular metric (p.r.m.) on X≠∅}) if satisfied the subsequent axioms:(1)
ρℷ1″,ℷ2″≥0,∀ℷ1″,ℷ2″∈X;(2)
ℷ1″=ℷ2″⇔ρℷ1″,ℷ2″=ρℷ1″,ℷ1″=ρℷ2″,ℷ2″,∀ℷ1″,ℷ2″∈X;(3)
ρℷ1″,ℷ1″≤ρℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X;(4)
ℷ1″,ℷ2″=ρℷ2″,ℷ1″,∀ℷ1″,ℷ2″∈X;(5)
ρℷ1″,ℷ2″≤ρℷ1″,ℷ3″+ρℷ3″,ℷ4″+ρℷ4″,ℷ2″−ρℷ3″,ℷ3″−ρℷ4″,ℷ4″, for all ℷ1″,ℷ2″∈X and for all distinct points ℷ3″,ℷ4″∈X|ℷ1″,ℷ2″.Then, partial rectangular metricρ together with the set X is titled as partial rectangular metric space and represented as an ordered pair X,ρ.Example 25.
LetX=0,b,φ≥b≥3 define a mapping ρ:X2⟶R by(58)ρℷ1″,ℷ2″=ℷ1″,ifℷ1″=ℷ2″,3b+ℷ1″+ℷ2″2,ifℷ1″,ℷ2″∈1,2,ℷ1″≠ℷ2″b+ℷ1″+ℷ2″2,otherwise.,
Then, it is easy to see thatX,d is the partial rectangular metric space.Remark 27.
Y. Cheng and Y. C. Shiab presented quasi-type contraction for providing the “fixed-point theorem.”
A mappingT:X⟶X(1)
is calledquasi-contraction for complete partial rectangular metric spaceX,ρ with constant k∈0,1 so that ∀ℷ1″,ℷ2″∈X satisfies the following condition:(59)ρTℷ1″,Tℷ2″≤kmaxρℷ1″,ℷ2″,ρℷ1″,Tℷ1″,ρℷ2″,Tℷ2″,ρℷ1″,Tℷ2″,ρℷ2″,Tℷ1″.(2)
isquasi-contraction for complete partial rectangular metric space X,ρ with constant k. Then, T has a unique fixed point ϑ∈X and ρϑ,ϑ=0.


## 2.4.14. Fuzzyb-Metric Space [68]

Partial rectangular metric space (p.m.s.) is the generalized version of rectangular metric space (r.m.s.) and the extension of partial metric space (p.m.s.).Let≠∅) and b≥1∈R and ∗ be a continuous t-norm. A fuzzy set M in X×X×0,∞ is termed as fuzzy b-metric on X if ∀f1″,f2″,f3″∈X, which fulfilled the subsequent axioms:(1)
Mℷ3″,ℷ2″,0=0;(2)
Mℷ3″,ℷ2″,t=1,∀t>0 ⇔ ℷ3″=ℷ2″;(3)
Mℷ1″,ℷ2″,t=Mℷ1″,ℷ1″,t;(4)
Mℷ1″,ℷ2″,t+s≥Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s,∀t,s≥0;(5)
IfMℷ1″,ℷ2″,.:0,∞⟶0,1 is left continuous, andlimt⟶∞Mℷ1″,ℷ2″,t=1.Then, fuzzyb-metric M together with the set X is titled as fuzzy b-metric space and represented as quadruple X,M,∗,b.Example 26.
LetX,d,k be b-metric space. Let Md:X×X×0,∞⟶0,1 be defined by(60)Mdℷ1″,ℷ2″,thinspptt+dℷ1″,ℷ2″,ift>0,0,ift=0.
Then,X,Md,∧,k be fuzzy b-metric space.Remark 28.
(1)
Abbas et al. proposed the definition of “ψ-contraction” and monotone “ψ-contraction” in fuzzy b-metric space.(2)
“Banach’s contraction principle” is extended as an application to the linear equation.
Let a mappingF:X⟶X be correspondence in fuzzy b-metric space such that ∀ℷ1″,ℷ2″∈X and F1∈Fℷ1″,F2∈Fℷ2″ satisfy the following condition:(61)MF1,F2,t≤Mℷ1″,ℷ2″,tk,where k∈0,1 and t∈R. Then, F has unique fixed point.
Chauhan and Utreja [69] proved one important result on fuzzy 2-metric space on the same line.


## 2.4.15. Fuzzy Quasi-b-Metric Space [68]

Nadaban extended the perception of fuzzy quasi-metric space (f.q.m.s.) andb-metric space (b-m.s.).LetM is a fuzzy set in X×X×0,∞ (where X≠∅, ∗ is a continuous t-norm, and k≥1 is a real number) is called fuzzy quasi-b-metric space if it satisfied the subsequent condition: (∀ℷ1″,ℷ2″,ℷ3″∈X, we have the following:(1)
Mℷ1″,ℷ2″,0=0;(2)
Mℷ”1,ℷ”2,t=Mℷ”2,ℷ”1,t=1,∀t>0,⇔ℷ1″=ℷ2″;(3)
Mℷ1″,ℷ2″,kt+s≥Mℷ1″,ℷ3″,t∗Mℷ3″,ℷ2″,s,∀t,s≥0;(4)
Mℷ1″,ℷ2″,·:0,∞⟶0,1 is left continuous, and limt⟶∞Mℷ1″,ℷ2″,t=1.Then, fuzzy quasi-b-metric M together with the set X is titled as fuzzy quasi-b-metric space and represented as quadruple X,M,∗,k.Example 27.
LetX,d,k be quasi-b-metric space. Let Md:X×X×0,∞⟶0,1 be defined by(62)Mdℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″,ift>0,0,ift=0.
Then,X,Md,∧,k be fuzzy quasi-b-metric space.Remark 29.
Fuzzy quasi-b-metric space (f.q.b-m.s.) is used to apply in the control theory and convex optimization.
LetX,Md,∧,k be fuzzy quasi-b-metric space and(63)Mℷ1″,ℷ2″,t=minMdℷ1″,ℷ2″,t,Mdℷ2″,ℷ1″,t.
Then,X,M,∧,k is a fuzzy b-metric space.


## 2.4.16.Mb-Metric Space [70–72]

TheMb-metric space is an enlargement of M-metric space by introducing the coefficient of the Mb-metric space (real number s≥1) in triangle equality. The concept of an extension was based on the simulation functions made known by Khojasteh et al.A functionmb:X2⟶R+ is s.t.b. Mb-metric space on X≠∅, if it satisfied the subsequent conditions: ∀ℷ1″,ℷ2″,ℷ3″∈X, we have(1)
mbℷ1″,ℷ1″=mbℷ2″,ℷ2″=mbℷ1″,ℷ2″ if ℷ1″=ℷ2″;(2)
mbℷ1″,ℷ2″≤mbℷ1″,ℷ2″;(3)
mbℷ1″,ℷ2″=mbℷ2″,ℷ1″;(4)
∃a real number s≥1 s.t.b. ∀ℷ1″,ℷ2″,ℷ3″∈X, we have(64)mbℷ1″,ℷ2″–mbℷ1″,ℷ2″≤smbℷ1″,ℷ3″−mbℷ1″,ℷ3″+mbℷ3″,ℷ2″–mbℷ3″,ℷ2″−mbℷ3″,ℷ3″.Then,Mb-metric Mb together with the set X is titled as Mb-metric space and represented as quadruple X,Mb.The numbers is termed as the coefficient of the Mb-metric space X,Mb.(65)mbℷ1″,ℷ2″=minmbℷ1″,ℷ1″,mbℷ2″,ℷ2″,Mbℷ1″,ℷ2″=maxmbℷ1″,ℷ1″,mbℷ2″,ℷ2″.Example 28.
LetX=R+,ρ>1 be a constant, and Mb:X×X⟶R+ be defined by(66)Mbℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,Mb is an Mb-metric space with coefficient s=2ρ>1.Remark 30.
We demonstrate the existence of the “fixed-point theorem” with the assistance of″Zmb contraction” in complete Mb-metric space.Theorem 12.
Suppose the mappingT:X⟶X and let X,Mb be a complete Mb-metric space with the coefficient s≥1 that satisfies the following condition:(67)MbTℷ1″,Tℷ2″≤βbℷ1″,ℷ2″,
for allℷ1″,ℷ2″∈X and where β∈0,1. Then, T has a unique fixed point u∈X and Mbu,u=0.


## 2.4.17. Orthogonal Metric Space [73, 74]

Gordji et al. defined a binary relation on the set.Letd be a usual metric on X≠∅ with an orthogonal set X,⊥. Then, metric with an orthogonal set is termed as orthogonal metric space and represented as X,⊥,d (O-metric space).Example 29.
LetX=0,∞, and we define ℷ1″⊥ℷ2″,if ℷ1″ℷ2″≤ℷ1″orℷ1″ℷ2″≤ℷ2″. Then, ∀ℷ1″∈X, 0⊥ℷ1″. So X,⊥ is an orthogonal set, and if X,d be an usual metric, then X,⊥,d is orthogonal metric space.Remark 31.
Gordji et al. gave a generalized version of “Banach’s contraction principle” with the orthogonal set, which stands for finding the existence and uniqueness of first-order ordinary differential equations and nonlinear fractional differential equations that are allied with Caputo fractional derivative.
A mappingT:X⟶X is called the following:(1)
Orthogonal contraction of orthogonal metric space isX,⊥,d if ∃τ∈0,1satisfies the subsequent condition:(68)dTℷ1″,Tℷ2″≤τdℷ1″,ℷ2″.For any two orthogonally related elements,ℷ1″,ℷ2″∈X.(2)
Orthogonalp-contraction of orthogonal metric space is X,⊥,d and p:X2⟶0,∞ if ∃τ∈0,1satisfies the following condition:(69)pTℷ1″,Tℷ2″≤τpℷ1″,ℷ2″.∀ℷ1″,ℷ2″∈X with ℷ1″⊥ℷ2″.
In the same line, Khalil et al. [75] proved some theorems on orthogonal partial-metric spaces with an application.


## 2.4.18. Rectangularb-Metric Space [22, 76, 77]

George et al. made known the theory of rectangularb-metric space (r.b-m.s.), which is an extension of rectangular metric space (r.m.s.) and the generalization of b-metric space (b-m.s.).A mappingd:X2⟶0,∞ is s.t.b. rectangular b-metric on X(where X≠∅), if fulfils the following axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
ℷ1″,ℷ2″=dℷ2″,ℷ1″,∀r1′,r2′∈X;(3)
∃a real number s≥1 s.t.b.(70)dℷ1″,ℷ2″≤sdℷ1″,ℷ3″+dℷ3″,ℷ4″+dℷ4″,ℷ2″,∀ℷ1″,ℷ2″∈Xandℷ3″,ℷ4″∈X|ℷ1″,ℷ2″,ℷ1″≠ℷ1″.Then, rectangularb-metric d together with the set X is titled as rectangular b-metric space and represented as an ordered pair X,d with coefficient s.Example 30.
LetX=N define d:X2⟶X by(71)dℷ1″,ℷ2″=0,ifℷ1″=ℷ2″,4ρ,ifℷ1″,ℷ2″∈1,2andℷ1″≠ℷ2″,ρ,ifℷ1″orℷ2″∉1,2andℷ1″≠ℷ2″,where ρ>0 is a constant. Then, X,d is rectangular b-metric space with coefficient s=4/3>1.Remark 32.
George et al. gave an analog of “Kannan’s fixed-point theorem” and “Banach’s contraction principle” for generalizing several outcomes in fixed-point theory.
A mappingT:X⟶X has a unique fixed point on a complete rectangular b-metric space X,d with coefficient s>1, if satisfies the following condition:(72)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″,where k∈0,1/s∀ℷ1″,ℷ2″∈X.or(73)dTℷ1″,Tℷ2″≤kdℷ1″,Tℷ1″,dℷ2″,Tℷ2″,where k∈0,1/s+1, ∀ℷ1″,ℷ2″∈X.


## 2.4.19. Extendedb-Metric Space [78, 79]

The coefficient ofb-metric space was replaced with the function ″e” and formed extended b-metric space (e.b-m.s.), which is the most universal form of b-metric space.A mappingd:X2⟶0,∞ is s.t.b. an extended b-metric on M with mapping e (where X≠∅ and e:X2⟶1,∞), if it fulfils the following axioms:(1)
dℷ1″,ℷ2″=0⇔ℷ1″=ℷ2″;(2)
dℷ1″,ℷ2″=dℷ2″,ℷ1″ for all ℷ1″,ℷ2″∈M;(3)
dℷ1″,ℷ2″≤eℷ1″,ℷ2″dℷ1″,ℷ3″+dℷ3″,ℷ2″ for all ℷ1″,ℷ2″,ℷ3″∈M.Then, extendedb-metric d together with the set X is titled as extended b-metric space and represented as an ordered pair X,d.Example 31.
LetX=1,2,3. We define e:X2⟶0,∞ and de:X2⟶0,∞ as(74)eℷ1″,ℷ2″=1+ℷ1″+ℷ3″,de1,1=de2,2=de3,3=0,de1,2=de2,1=80,de1,3=de3,1=1000,de2,3=de3,2=600.
Hence,X,dis an extended b-metric space.Remark 33.
(1)
Mitrovic et al. proved single-valued mapping and multivalued mapping (m.v.m.) in extendedb-metric space.(2)
Kamran et al. gave the existence of the unique solution as an application to the Fredholm integral equation.


## 2.4.20. Extended Fuzzyb-Metric Space [80]

For setting the most general class of fuzzyb-metric space, we extended and generalized into such space.A fuzzy setMα in X×X×0,∞ (where X≠∅, α:X2⟶1,∞ and ∗ -continuous t-norm) is called extended fuzzy b-metric (e.f.b-m.s.) on X if satisfied the following axioms: (∀ℷ1″,ℷ2″,ℷ3″∈X)(1)
Mαℷ1″,ℷ2″,0=0;(2)
Mαℷ1″,ℷ2″,t=1,∀t>0 ⇔ ℷ1″=ℷ2″;(3)
Mαℷ1″,ℷ2″,t=Mαℷ2″,ℷ1″,t;(4)
Mαℷ1″,ℷ2″,αℷ1″,ℷ2″t+s≥Mαℷ1″,ℷ3″,t∗Mαℷ3″,ℷ2″,s,∀t,s≥0;(5)
Mαℷ1″,ℷ2″,.:0,∞⟶0,1 is left continuous.Then, extended fuzzyb-metric Mα together with the set X is titled as extended fuzzy b-metric space and represented as X,Mα,∗,αℷ1″,ℷ2″.Example 32.
LetX=1,2,3 and let db:X×X⟶R by dℷ”1,ℷ”2=ℷ”1−ℷ”22, then it is easy to get that X,d is a b-metric space. We define the mapping α:X×X⟶1,∞,(75)αℷ1″,ℷ2″=1+ℷ1″+ℷ2″.
LetMα:X×X×0,∞⟶0,1 be given by(76)Mαℷ1″,ℷ2″,t=tt+dℷ1″,ℷ2″,ift>0.0,ift=0,and continuous t-norm ∗=∧; then, ℷ1″∗ℷ2″=ℷ″1ℷ2″=minℷ1″,ℷ2″.
Then,X,Md,∧,k be fuzzy quasi-b-metric space.Remark 34.
Mehmood et al. established a new kind of “Banach-type fixed-point theorem” with the help of a fuzzy set in the most universal class of fuzzyb-metric space.Theorem 13.
(Banach’s contraction theorem in fuzzy extended b-metric spaces).
LetX,Mα,∗ be a G-complete extended fuzzy b-metric space with the mapping α:X×X⟶1,∞ such that(77)limt⟶∞Mαℷ1″,ℷ2″,t=1.
Letg:X⟶X be a mapping, which satisfies the condition(78)Mαgℷ1″,gℷ2″,kt≥Mαℷ1″,ℷ2″,t,∀ℷ1″,ℷ2″∈X,where k∈0,1. Furthermore, g has a unique fixed point if for an arbitrary b0∈X, and n;q∈N, we have(79)αbn,bn+q<1k,where bn=gnb0.


## 2.4.21. ExtendedMb-Metric Space [70]

The coefficient ofMb-metric space was replaced with the function θ and formed a new generalized version of Mb-metric space, titled as extended Mb-metric space.A functionmθ:X2⟶0,∞ is titled as extended Mb-metric (e.Mb-m) on X (where θ:X2⟶1,∞ be a function) if fulfilled the subsequent conditions:(1)
mθE1,E1=mθE2,E2=mθE1,E2 if E1=E2;(2)
mθE1,E2≤mθE1,E2;(3)
mθE1,E2=mθE2,E1;(4)
mθE1,E2−mθE1,E2≤θE1,E2mθE1,E3−mθE1,E3+mθE3,E2−mθE3,E2, ∀E1,E2,E3∈X.Then, extendedMb-metric mθ together with the set X is titled as extended Mb-metric space and represented as the ordered pair X,mθ.Example 33.
LetX=Cα,β,R, and define mθ:X2⟶0,∞ and θ:X2⟶1,∞ by(80)mθE1t,E2tsupt∈α,βE1t−E2t2,θE1t,E2t=E1t+E2t+2.
Then,X,mθ is an extended Mb-metric space.Remark 35.
N. Mlaiki et al. provide several fixed-point results on expandedMb-metric space (Mb-m.s.) using the techniques of the classical fixed-point theorems such as “Kannan’s fixed-point theory,” the “Banach fixed-point theory,” and many more.Theorem 14.
LetX,mθ be an extended Mb-metric space and mθb:X2⟶0,∞ be a function defined as(81)mθbE1,E2=mθE1,E2−2mθE1,E2+MθE1,E2,for all x,y∈X. Then, mθbis an extended b-metric and the pair X,mθb is an extended b-metric space.


## 2.4.22. Soft S-Metric Space [81]

Aras et al. gave the theory of softS-metric space in which S-metric was further defined on soft points of soft sets and introduced some of their characteristics as well.LetS be the absolute soft set, E be the ≠∅ set of parameters; SPS be a collection of all soft points ofS, and RE∗ be the set of all non-negative soft real numbers.A mappings:SPS×SPS×SPS⟶RE∗ is s.t.b. soft S-metric on S (the soft set), if it fulfils the subsequent axioms, for each soft points ℷ1a″,ℷ2b″,ℷ3c″,ℷ4d″∈SPS,(1)
Sℷ1a″,ℷ2b″,ℷ3c″≥0;(2)
Sℷ1a″,ℷ2b″,ℷ3c″=0⇔ℷ1a″=ℷ2b″=ℷ3c″;(3)
Sℷ1a″,ℷ2b″,ℷ3c″≤Sℷ1a″,ℷ1a″,ℷ4d″+Sℷ2b″,ℷ2b″,ℷ4d″+Sℷ3c″,ℷ3c″,ℷ4d″.Then, softS-metric s together with the set S is titled as soft S-metric space and represented as S,s,E.Example 34.
LetE=N be a parameter set, and X=R be defined as(82)Sℷ1a″,ℷ2b″,ℷ3c″=a−c+b−c+ℷ1″−ℷ3″+ℷ2″−ℷ3″.
It is easy to get a softS-metric space.Remark 36.
The idea of “soft contractive mapping” on softS-metric space (s.S-m.s.) was established, and some significant concepts were given, for instance, completeness, Cauchy sequence, soft continuous mapping on soft S−metric space (s.S-m.s.), and also with the help of “soft contractive mapping” on soft S−metric space (s.S-m.s.), and some of the “fixed-point theorems” have been demonstrated.
A mappingρ1,ρ2:S,s,E⟶S,s,E has a unique soft fixed point on a soft sequential compact soft S-metric space S,s,E if satisfies the following condition:(83)Sρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ρ1,ρ2ℷ2b″≤αρ1,ρ2ℷ1a″,
for everyℷ1a″,ℷ2b″∈SPS and ℷ1a″≠ℷ2b″.
A mappingρ1,ρ2:S,s,E⟶S,s,Ehas a unique soft fixed point on a complete soft S-metric space S,s,E if satisfies the following contraction condition:(84)Sρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ρ1,ρ2ℷ2b″≤αSρ1,ρ2ℷ1a″,ρ1,ρ2ℷ1a″,ℷ1a″,Sρ1,ρ2ℷ2b″,ρ1,ρ2ℷ2b″,ℷ2b″,
for everyℷ1a″,ℷ2b″∈SPS, where α∈0,1/2 is a soft constant real number. Then, ρ1,ρ2 have a unique soft fixed point in SPS.


## 2.4.23.g-Metric Space [82]

Due to today’s large and complex data collection, we developed the notion of distance function that is calledg-metric in which we extended the ordinary distance function of metric (two points in between) and G-metric (three points in between). The g-metric is the distance function of degree n, i.e., for n+1 points.A functiong:Xn⟶R+ is s.t.b. g-metric with order n on X≠∅ (where Xn=∏i=1nX), if it fulfils the following axioms:(1)
Positive definiteness(85)gx0,xn=0⇔x0=···=xn.(2)
Permutation invariant(86)gx0,xn=gxτ0,xτn.For any permutationτon0,1,n,(3)
Monotonicity(87)gx0,xn≤gy0,yn,∀x0,xn,y0,yn∈Xn+1,with(88)xi:i=0,n⊊yi:i=0,n.(4)
Triangle inequality(89)∀x0,..,xs,y0,..,yt,w∈Xwiths+t+1=n,gx0,..,xs,y0,..,yt≤gx0,..,xs,w,..,w+gy0,..,yt,w,..,w.Then,g-metric g together with the setX is titled as g-metric space and represented as an ordered pair X,g.Example 35.
LetX≠∅ be defined d:Xn⟶R+ by(90)dx0,xn=0,ifx0=...=xn1,otherwise,∀x0,xn∈X.
It is easy to getX,d that is a g-metric space.Remark 37.
Several “fixed-point theorems” in theg-metric space (g-m.s.) were generalized, which is considered as an oversimplification of “Banach’s contractive mapping” concerning g-metric space.
A mappingT:X⟶X has a unique fixed point on a complete g-metric space X,g if satisfies the condition: ∃k∈0,1 such that gTℷ1″,Tℷ2″…,Tℷn″≤kgℷ1″,ℷ2″,…,ℷn″,∀ℷ1″,ℷ2″,…,ℷn″∈X.


## 2.4.24.F-Metric Space [83, 84]

F-metric is a generalization of metric space in which we defined the natural topology τF with the different topological properties.The mappingD:X2⟶0,+∞ (where X≠∅) is s.t.b. F-metric, if fulfils the following axioms:We suppose that∃f,g∈F×0,+∞ such that(1)
ℷ1″,ℷ2″∈X×X,Dℷ1″,ℷ2″=0 if ℷ1″=ℷ2″;(2)
ℷ1″,ℷ2″=Dℷ2″,ℷ1″;∀ℷ1″,ℷ2″∈X×X;(3)
∀ℷ1″,ℷ2″∈X×X,∀N∈N,N≥2,∀uii=1N⊂X with u1,uN=ℷ1″,ℷ2″; we have Dℷ1″,ℷ2″>0(91)⇒fDℷ1″,ℷ2″≤f∑i=1N−1Dui,ui+1+g.Then,F-metric D together with the set X is titled as F-metric space and represented as an ordered pair X,D.Example 36.
LetX=R be defined d:X2⟶R+ by(92)dℷ1″,ℷ2″=ℷ1″−ℷ2″2,ifℷ1″,ℷ2″∈0,3×0,3ℷ1″−ℷ2″,ifℷ1″,ℷ2″∉0,3×0,3,∀ℷ1″,ℷ2″∈X,
withft=lnt and ρ=lnt.It is easy to get X,d that is a F-metric space.Remark 38.
Jelli and Samet proved the modified version of the “Banach contraction principle.” Afterward, Alansari et al. established some circumstances for the existence of a solution in the “fuzzy fixed-point theorem” used for solving the fuzzy Cauchy problem.
A mappingT:X⟶X has a unique fixed point on a complete F-metric space X,D if satisfies the following condition: ∃k∈0,1 such that DTℷ1″,Tℷ2″≤kDℷ1″,ℷ2″, ∀ℷ1″,ℷ2″∈X.


## 2.4.25. Controlled Metric Type Space [85]

The controlled metric is an extension ofb-metric in which we introduced a control function αx,z in the b-triangle inequality.The mappingd:X2⟶0,∞ is s.t.b. controlled metric type (where φ:X×X⟶1,∞and X≠∅), if fulfils the following axioms:(1)
dC1,C2=0 if C1=C2;(2)
dC1,C2=dC2,C1;(3)
dC1,C2≤φC1,C3dC1,C3+φC3,C2dC3,C2, for all C1,C2,C3∈X.Then, controlled metricd together with the set X is titled as controlled metric space and represented as an ordered pair X,d.Example 37.
LetX=0,1,2 and a function d be given as(93)d0,0=d1,1=d2,2=0,d0,1=d1,0=1,d0,2=d2,0=12,d1,2=d2,1=25.
We takeβ:X×X⟶0,∞ to be symmetric and is defined by the following:(94)β0,0=β1,1=β2,2=β0,2=1,β1,2=54,β0,1=1110.
It is easy to get thatd that is a controlled metric space.Remark 39.
Mlaki et al. proved the contraction principle on the controlled metric type space.
A mappingT:X⟶X on a complete controlled metric space is X,d and satisfies the following condition:(95)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,where ∃k∈0,1. For ℷ0″∈X, we take ℷ1n″=Tnx0. Suppose that(96)supm≥1limi⟶∞βℷi+1″,ℷi+2″βℷi″,ℷi+1″βℷi+1″,ℷm″<1k.
In additional, we assume that∀ℷ1″∈X and we have(97)limn⟶∞βℷ1n″,ℷ1″andlimn⟶∞βℷ1″,ℷ1n″.
Then,T has a fixed a unique fixed point.


## 2.4.26. RectangularM-Metric Space [86]

RectangularM-metric space is an extension and generalization of the class of M-metric space in addition to partial rectangular metric space.The mappingmr:X×X⟶0,∞ is s.t.b. rectangular M-metric space (where X≠∅) if satisfied the following conditions: ∀R1,N2∈X(1)
mrR1,N2=mrR1,N2=MrR1,N2 if R1=N2;(2)
mrR1,N2≤mrR1,N2;(3)
mrR1,N2=mrR2,N1;(4)
mrR1,N2−mrR1,N2≤mrR1,N3−mrR1,N3+mrR3,N4−mrR3,N4+mrR4,N2−mrR4,N2, ∀R3,N4∈X|R1,N2.Then, rectangularM-metric mr together with the set X is titled as rectangular M-metric space and represented as an ordered pair X,mr.where(98)mrR1,N2=minmrR1,N1,mrR2,N2,MrR1,N2=maxmrR1,N1,mrR2,N2.Example 38.
Xβ=R1∈C:argR1=β∪0, for a fixed β,0≤β≤2π. If we define self-mapping mr on Xβ, it is given by(99)mrR1,N2=R1+R22,∀R1,R1∈Xβ.
Then, it is easy to getXβ,mr that is a rectangular M-metric space.Remark 40.
N.Y. Ozgur et al. used different contraction principles for proving some fixed-point theorems with various techniques, which generalized some established fixed-point outcomes, for instance, the “Kannan fixed-point theorem,” “Banach contraction principle” and the “Chatterjea fixed-point theorem.” Also, we provide relevance in the direction of the fixed-circle problem.
A mappingT:X⟶X on a rectangular M-metric space is X,d and satisfies the contraction condition if ∃k∈0,1 such that(100)mrTℷ1″,Tℷ2″≤kmrℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,
and consider the sequenceℷ1n″n≥0 defined by ℷ1n+1″=Tℷ1n″. If ℷ1n″⟶uasn⟶∞, then Tℷ1n″⟶Tu as n⟶∞.


## 2.4.27. RectangularMb-Metric Space [71]

Inspired by the theory of rectangularM-metric space (M-m.s.) and Mb-metric space (Mb-m.s.), Asim et al. established the rectangular Mb-metric space to get well and enlarge the existing results of the literature.A mappingmrmb:X2⟶R+ is s.t.b. a rectangular Mb-metric with coefficient s≥1, if mrmb satisfies the following (∀R1,N2,R3∈X and all distinct R4,N5∈X|R1,N2:(1)
mrmbR1,N1=mrmbR1,N2=mrmbR2,N2 if and only ifR1=N2;(2)
mrmbR1,N1≤mrmbR1,N2;(3)
mrmbR1,N2=mrmbR2,N1(4)
mrmbR1,N2−mrmbR1,N1≤smrmbR1,N3−mrmbR1,N3+mrmbR3,N4−mrmbR3,N4+mrmbR4,N2−mrmbR4,N2–mrmbR3,N3−mrmbR4,N4.Then, rectangularMb-metric mrmb together with the set X is titled as rectangular Mb-metric space and represented as an ordered pair X,mrmb.Example 39.
LetX=R+,ρ>1 a constant, and mrmb:X×X⟶R+be defined by(101)mrmbℷ1″,ℷ2″=maxℷ1″,ℷ2″ρ+ℷ1″−ℷ2″ρ,∀ℷ1″,ℷ2″∈X.
Then,X,mrmb is a rectangular Mb-metric space with coefficient s=3ρ−1>1.Remark 41.
In rectangularMb-metric space, M. Asim et al. proved an equivalence of the “Banach contraction principle.”
A mappingT:X⟶X on a complete rectangular Mb-metric space is X,d with coefficient s≥1 and satisfies the following contraction condition:(102)mrmbTℷ1″,Tℷ2″≤kmrmbℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X.where k∈0,1/s. Then, T has a unique fixed point ℷ1″ such that mrmbℷ1″,ℷ1″=0.


## 2.4.28.R-Metric Space [87]

In theR-metric space, additionally, an arbitrary relation R is defined between the points on a nonempty set.SupposeX,d is a metric space (m.s.) and R be a relation on X≠∅, then triple X,d,R or X is called R-metric space.Example 40.
LetX=0,∞ equipped with Euclidean metric. We define xRy if ℷ1″ℷ2″≤ℷ1″∨ℷ2″, where ℷ1″∨ℷ2″=ℷ1″orℷ2″.
Then,X,d,R is a R-metric space.Remark 42.
It was difficult to discover the exact value of the equilibrium point, so Khalehoghli et al. tried to provide a structural method for determining the exact value of the fixed point, which is useful to solve a problem related to economics and game theory.
A mappingT:X⟶X is called R-contraction with Lipschitz constant 0<k<0 if ∀ℷ1″,ℷ2″∈X such that ℷ1″Rℷ2″, then we have the following:(103)dTℷ1″,Tℷ2″≤kdℷ1″,ℷ2″.Theorem 15.
A mappingT:X⟶X on a complete metric space X,d has a unique fixed point in X if for some β∈0,1 in a complete metric satisfies the following condition dTℷ1″,Tℷ2″≤βdℷ1″,ℷ2″, for allℷ1″,ℷ2″∈X.


## 2.4.29. SoftD-Metric Space [88]

Aras et al. provided the theory of softD-metric space (s.D-m.s.), which was further defined on soft points of soft sets and defined the soft ∙−distance on complete soft D-metric by using the designation of soft D-metric.A mappingD:SPX×SPX×SPX⟶RE is s.t.b. soft D−metric on X (where X≠∅) if D fulfils the subsequent conditions, for each soft points ℷ1″,ℷ2″,ℷ3″,ℷ4″∈SPX,(1)
Dℷ1″,ℷ2″,ℷ3″≥0⇔ℷ1″=ℷ2″=ℷ3″;coincidence;(2)
Dℷ1″,ℷ2″,ℷ3″=Dℷ2″,ℷ1″,ℷ3″=Dℷ1″,ℷ3″,ℷ2″=…symmetry;(3)
Dℷ1″,ℷ2″,ℷ3″≤Dℷ1″,ℷ2″,ℷ4″+Dℷ1″,ℷ4″,ℷ3″+Dℷ4″,ℷ2″,ℷ3″.Then, softD-metric D together with the set X is titled as soft D-metric space and represented as X,D,E.Example 41.
LetX be a nonempty set, and E be the nonempty set of parameters. If we define a mapping D:SPX×SPX×SPX⟶RE by(104)Dℷa″,ℷb″,ℷc″=0,ℷa″=ℷb″=ℷc″1,otherwise,∀ℷa″,ℷb″,ℷc″∈SPX,
then it is easy to show thatD is a soft D-metric space.Remark 43.
(1)
“Fixed-point theorem” of soft continuous mappings on softD-metric space has been proven.(2)
It is given the “fixed-point theorem” using the principle of softΔ−distance.Theorem 16.
LetX,D,E be a complete D-metric space and ∙−distance of X.f,g:X,D,E⟶X,D,E be a soft mapping. Let Xbe ∙ bounded. Suppose that ∃ is a soft real number r∈RE,0≤r<1, such that(105)f,gℷ1″,f,g2ℷ1″,f,gℷ2″≤rℷ1″,f,gℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈SPX. Then, ∃ℷ3″∈SPX such that ℷ3″=f,gℷ3″. In addition to ℷ4″=f,gℷ4″, then ℷ4″,ℷ4″,ℷ4″=0.


## 2.4.30. Controlled Rectangularb-Metric Space [89]

N. Mlaiki et al. established the generalization of rectangularb-metric space by altering the rectangular equality.A functionb:X4∙1,∞, and db:X2∙0,∞ is s.t.b. controlled rectangular b-metric space on Xwhere X≠∅) if all distinct C1,C2,C3,C4∈X; we have(1)
dbC1,C2=0 if C1=C2;(2)
dbC1,C2=dbC2,C1;(3)
dbC1,C2≤bC1,C2,C3,C4dbC1,C3+dbC3,C4+dbC4,C2.Then, controlled rectangularb-metric db together with the set X is titled as controlled rectangular b-metric space and represented as X,db.Example 42.
LetX=Y∪Z, where Y=1/m,wherem∈N, and Z be a set of positive integers. We define db:X2∙0,∞ by(106)dbC1,C2=0,iffC1=C2,2β,ifC1,C2∈Y,β2,otherwise,where β is a constant bigger than 0 and b:X4∙1,∞ by(107)bC1,C2,C3,C4=maxC1,C2,C3,C4+2β.Remark 44.
Controlled rectangularb-metric space is used to prove some “fixed-point theorems.”
This theorem is proved, as an application.
For any natural numberκ≥3, the equation sκ+1=κ4–1sκ+1+κ4shas a unique real solution.
A mappingT:X⟶X on a controlled rectangular b-metric space is X,d and satisfies the contraction condition if ∃k∈0,1 such that(108)dbTℷ1″,Tℷ2″≤kdbℷ1″,ℷ2″,∀ℷ1″,ℷ2″∈X,and supm>1limn⟶∞bℷ1″,ℷ2″,ℷ3″,ℷ4″≤1/k. Then, T has a unique fixed point.


## 2.4.31. Controlled Fuzzy Metric Space

Controlled fuzzy metric space (c.f.m.s.) is the extension of fuzzy metric space (f.m.s.) with generalized triangle inequality and the generalization of fuzzy b-metric space (f.b-m.s.).Letλ:X2⟶1,∞,∗ is a continuous t-norm, and Mλ is a fuzzy set on X2×0,∞ (where X≠∅) is s.t.b. controlled fuzzy metric, if fulfilled the subsequent conditions, (∀C1,C2,C3∈X,s,t>0) such that(1)
MλC1,C2,0=0;(2)
MλC1,C2,t=1 ⇔C1=C2(3)
MλC1,C2,t=MλC2,C1,t(4)
MλC1,C2,t+s≥MλC1,C3,t/λC1,C3∗MλC3,C2,s/λC1,C3;(5)
MλC1,C2,⋅:0,∞⟶0,1 is continuous.Then, controlled fuzzy metricMλ together with the set X is titled as controlled fuzzy metric space and represented as X,Mλ,∗.Example 43.
LetX=A∪C, where A=0,1 and C=N|0,1. We define Mλ:X×X×0,∞⟶0,1 as(109)MλC1,C2,t=1,ifC1=C2,tt+1/C2,ifC1∈AandC2∈C,tt+1/C1,ifC1∈CandC2∈A,tt+1,ifotherwise,
with a continuous t-norm ∗ such thatta∗tb=ta.tb. We define λ:X2⟶1,∞ as(110)λC1,C2=1,ifC1,C2∈A,maxC1,C2,ifotherwise.
It is easy to getX,Mλ,∗ that is a controlled metric space.Remark 45.
M. S. Sezen et al. proved a “Banach contraction principle” and introduced a new concept of “fixed-point theorem” for several self-mappings, which satisfied fuzzy “𝜓-contraction” condition.Theorem 17.
A mappingT:X⟶X on a controlled fuzzy metric space X,Mλ,∗ has a unique fixed point in X if satisfies the following condition:(111)MλTC1,TC2,t≥βMλC1,C2,t,
for allC1,C2∈X and β>0.


## 2.4.32. CAT(0) Space [90, 91]

A metric spaceX is said to be a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. Complete CAT(0) spaces are often called Hadamard spaces.Definition of (CAT(0) space): let (X,d) be a geodesic space. It is a CAT (0) space if, for any geodesic triangle, ∆ ⊂X and x, y ∈ ∆, d(x, y) ≤ d (x¯, y¯) where ¯x, y¯ ∈ ∆¯.Amnuaykarn et al. [90] proved some theorems on the existence of best proximity points of multivalued mappings in CAT(0) spaces.Ugwunnadi and Okeke [91] studied the approximation of common solutions for a finite family of generalized demimetric mappings and monotone inclusion problems in CAT(0) spaces.


## 3. Conclusions

In this review paper, it is perfectly understood that the metric space is exceptionally valuable in science and real-life scenarios. Being closely related to the concept of distance, it has many applications in the field of convergence, quantum mechanics, the field of navigation, etc.In the beginning, metric space was introduced with the concept of ordinary distance function between two points. Later on, its variants were established in the form of generalized, extended, and/or combinational metric spaces with different properties. Some of the properties are metrics, which consider distance function betweenn points, constant function/control function αx,y in triangle inequality, metric without symmetric property, an arbitrary relation, change triangle inequality into quadrilateral inequality, fuzziness between the points on a nonempty set and function between the points on a nonempty set, the concept of nonzero self-distance, an orthogonal property, and many more on the set of Rn, complex space, Riesz space, ordered Banach’s space, etc.Variants of metric spaces were introduced from an implementation perspective to find the existence and uniqueness of the “fixed-point theorem,” “Banach contraction theorem,” “extension of Banach contraction principle,” and “Kannan fixed-point theorem,” “fuzzy fixed-point theorem” with the support of some of the “contraction principle,” for instance,A-contraction, Z-contraction mapping, soft set-valued maps, simulation function, F-contraction mapping, and Zm-contraction mapping. Consequently, originated “fixed-point theorem” for single-valued mapping and multivalued mapping, which is further connected to find the simple and efficient solution for a Volterra integral equation, and mixed Volterra–Fredholm type integral equation, helpful in developing some computational methods for answering the system of nonlinear matrix equations, in the proof of Picard–Lindelof theorem about finding the existence and uniqueness of certain ordinary differential equations, solution of initial value problems, and so on. Since metric space plays an important role in various real-life scenarios, these are some of the applications; we can still find many more applications based on variants of metric spaces.



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