#### 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.

#### 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) . 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” .

A function (where ) is s.t.b. a distance function (or metric) on , if it meets the subsequent requirements:(1) if and (2),(3), for all

So the function composed with a set is titled as metric space, which is denoted by .

Example 1. Let with , for all .
Now it is easy to get that is a metric space.

Example 2. Let with , for all .
Now it is easy to get that is a metric space.
Now, some basic results on metric space and their applications are in this manner.

##### 1.1. Picard’s Convergence Theorem 

For nonlinear equations, Picard proved the theorem for showing the existence of the solutions.

Theorem 1. Let and . If , such thatThen, the sequence in is defined by converges to a solution of the equation .
The iterative sequence is termed as Picard’s iterative sequence.

##### 1.2. Banach’s Fixed-Point Theorem 

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. Let be a complete metric space, and be a contractive mapping (i.e., such that . Then, the following axioms hold:(i) has a unique fixed point (ii) the sequence is defined by (iii) converges to the fixed point of 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 .

#### 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  introduced the metric space without symmetric condition, which is termed as quasi-metric space.

A function (where ) is well known as a quasi-metric if it satisfied the following succeeding conditions:(1)(2)(3) for all

Then, quasi-metric together with set is titled as quasi-metric space and represented by .

Example 3. Let be set of all real numbers and . Let be a mapping, which is defined as follows:It is easy to get that is a quasi-metric on .

Remark 1. (1)Nearly all “fixed-point theorems” used for 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 .A mapping is called as the following:(1)“Forward contraction (resp. backward contraction)” if quasi-metric space and functions and satisfy the following conditions,(Remark: if is a “backward contraction” then be a forward contraction.)(2)“Forward Picard operator (backward Picard operator)” if quasi-metric space has a unique fixed point and for every

###### 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 number with the real function , whose function is , for any real number . It is defined as the probability that is the distance between is less than

A function (where ), is s.t.b. probabilistic metric space, if met the following requirements:(1);(2);(3);(4).

Then, probabilistic metric space is represented as .

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 mapping is titled as follows:(1)–contraction if be a probabilistic metric space and such that and all satisfy the following implication:(2)–contraction if be a probabilistic metric space and such that satisfy the following implication:As a generalization of the concept of “–contraction,” Pap and Hadzic made known to the concept of -contraction.
A mapping is called -contraction (where ), if be a probabilistic metric space and such that and all satisfy the following implication:such that

##### 2.2. Period-II (1951–2000)
###### 2.2.1. Two-Metric Space [11–15]

Gahler  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 mapping is s.t.b. 2-metric on if satisfied the subsequent axioms:(1).(2), only if two or more points are the same.(3).(4)

Then, 2-metric together with set is titled as 2-metric space and represented by .

Remark 3. (1)The “Baire category theorem” and “Cantor intersection theorem” are used for solving problems of fixed point .(2)In 2-metric space, Lebesgue integrable function is used as a summable function for each compact .A mapping is called contraction in -metric space if satisfies the following conditions, where , and if any two of and are equal.
A mapping is entitled as “-contraction” in 2-metric space if satisfies the succeeding conditions, max .

###### 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.

Let is a “fuzzy set” on , that is s.t.b. fuzzy metric (f.m.) on ∗ is a continuous t-norm), if met the subsequent axioms: ((1);(2);(3);(4);(5) is left continuous, and

Then, fuzzy metric together with set is titled as fuzzy metric space and symbolized as (.

Example 4. Let and , , and (1);(2).It is easy to get that is a fuzzy metric on 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 mapping is known as the following:(1)“Banach fuzzy contraction” if fuzzy metric space satisfies the subsequent conditions, , and all .(2)“Edelstein fuzzy contraction” if fuzzy metric space satisfies the subsequent conditions, , and all .

###### 2.2.3. -Metric Space [20–22]

In the -metric space, (relaxed triangle inequality) was introduced. If , then b-metric and metric space are the same so as compared with the metric space, and -metric space is the weaker notion.

A mapping into is s.t.b. -metric on if met the subsequent axioms:(1)(2) (symmetry);(3)There exists satisfying

Then, -metric together with set is titled as -metric space and represented as ordered pair .

Example 5. Let and function and and a function from byIt is easy to show that is a -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 . Further extension of b-metric space can be seen in the paper of Chauhan and Kaur .

###### 2.2.4. -Metric Space [24–26]

Since 2-metric space was not continuous and convergent for all three points, so Dhage  introduced the new generalized class of the ordinary metric space. The convergence of -metric defines a Hausdorff topology and (sequentially) continuity for all three points.

A mapping is s.t.b. -metric on if it meets the following conditions given below:(1) (coincidence)(2) where is a permutation of (symmetry), and(3) (tetrahedral inequality).

Then, -metric together with set is titled as -metric space and represented as ordered pair .

Example 6. Let and be definite asIt is easy to show that is a -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 in -metric space, to prove the existence of a unique fixed point .(2)Singh et al. introduced semicompatibility in -metric space for “fixed-point theorems” with the help of the orbit concept .Every -continuous function has a fixed point in if -metric space has a -fixed-point property.
Every -weakly continuous function has a fixed point in if and only if -metric space has a -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 function is s.t.b. partial metric on if it satisfied the subsequent axioms: (1);(2);(3);(4)

Then, partial metric together with the set is titled as partial metric space and represented as ordered pair .

Example 7. Let and be a function defined by , for all
Then, it is easy to see that is a partial metric, but it is not a metric Indeed, for any we have

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 .
A mapping is titled as:(1)Contraction for every partial metric space if so that satisfies the following condition:(2)Asymptotic regular for every partial metric space if at point satisfies the following condition:

###### 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 function on if it satisfied the subsequent axioms: with and :(1)(2);(3) (quadrilateral inequality).

Then, rectangular metric together with the set is titled as partial metric space and represented as ordered pair .

Example 8. Let define as follows:Then, 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 mapping is called as the following:(1)“Caccioppoli-type fixed-point theorem” for complete metric space if for every where distinct from and fulfils the subsequent condition:Then, if series is convergent, then has a unique fixed point in .(2)“Kannan fixed-point theorem” for complete metric space if for every and fulfils the subsequent condition:If is -orbitally complete in , then has a unique fixed point in .

##### 2.3. Period-III (2001–2010)
###### 2.3.1. Fuzzy Quasi-Metric Space 

The associated impression of fuzzy metric space was expanded with a quasi-metric sense and then made known to the fuzzy quasi-metric space.

Let is a fuzzy set mapping on , that is, s.t.b. fuzzy quasi-metric (f.q.m.) on X,∗- if it satisfied the subsequent axioms: .(1)(2) if ;(3)(4) is left continuous.

Then, fuzzy quasi-metric together with the set is titled as partial metric space and represented as ordered pair .

Example 9. Let be a fuzzy quasi-metric space and be a usual multiplication , and is a fuzzy set mapping on byThen, 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 mapping is titled as “-contraction” on fuzzy quasi-metric space that fulfils the following circumstances, and all , where is called the contraction constant of .

###### 2.3.2. -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 mapping is s.t.b. -metric (more specifically generalized metric) on X, if it satisfied the subsequent axioms:(1) if (2)(3)(4) (symmetry);(5) (rectangle inequality).

Then, -metric together with the set is titled as -metric space and represented as ordered pair .

Example 10. Let ( be the usual metric space. We define byfor all R. Then, it is easy to see that is a -metric space.

Remark 10. (1)In G-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 (or G-metric space) are proved.

###### 2.3.3. -Metric Space 

When most of the D-metric space theorems were invalid, then the description of D-metric space had been possibly updated by Sedghi et al.  and introduced -metric space. With some modifications, the basic properties of -metric space (-m.s.) were introduced.

A mapping is s.t.b. generalized metric (or -metric) on , if fulfilled the subsequent conditions:(1)(2)(3), (symmetry) where p is a permutation function;(4)

Then, -metric together with the set is titled as -metric space and represented as ordered pair .

Example 11. Let , then we defineThen, it is easy to check that is -metric space.

Remark 11. A “fixed-point theorem” was developed for a class of mapping in complete —metric space together with the condition of weakly commuting mappings.
A mapping in complete -metric space satisfies the following conditions:(1) and are weakly commuting pairs s.t.b. (2) such that for all Then, and have a common unique fixed point in .

###### 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.

Let stand for “real Banach space” and ,.

A function (where ) is s.t.b. a cone metric on , if it meets the subsequent requirements:(1) and (2), ;(3), for all

Then, function composed of =set is titled as a cone metric space and represented as .

Example 12. Let such thatwhere is a constant. Then, 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 .

Theorem 3. (1)Let be a complete cone metric space, and be a normal cone with normal constant . It is supposed that the mapping fulfils the following condition:where is a constant.Then, has a unique fixed point in . For any , iterative sequence converges to the fixed point.

Theorem 4. Let be a sequentially compact cone metric space, and be a regular cone.
It is supposed that the mapping fulfils the following contractive condition:Then, has a unique fixed point in

Theorem 5. Let be a complete cone metric space, and a normal cone with normal constant K. It is supposed that the mapping has a unique fixed point in if fulfils the contractive condition:where is a constant. For any iterative sequence converges to the fixed point.

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

Initially, the usual metric with was not complete. To overcome this problem, Bashirov et al. proposed the concept of multiplicative metric space with a multiplicative value of and multiplicative distance function

A mapping is s.t.b. multiplicative metric on when it meets the following conditions:(1)(2);(3) ;(4) for all (multiplicative triangle inequality).

Then, multiplicative metric together with the set is titled as multiplicative metric space and represented as ordered pair .

Example 13. Let be the set of all -tuples of non-negative real numbers. Let be a mapping defined as follows:where , , and are defined by the following:It is easy to get that 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 mapping is called as follows:(1)“Multiplicative contraction” of multiplicative metric space if fulfils the subsequent condition is as follows:(2)“Multiplicative Kannan’s contraction” of multiplicative metric space if fulfils the subsequent condition is as follows:(3)“Multiplicative Chatterjea’s” contraction of multiplicative metric space if fulfils the subsequent condition is as follows:

###### 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 function is s.t.b. vector metric on where -Riesz space) if it satisfied the following:(1) if ,(2).

Then, vector metric together with the set is titled as vector metric space and termed as triple (briefly with the default parameters omitted).

Example 14. Let a Riesz space and a mapping are defined by the following:It is easy to show a vector metric space.

Remark 14. Cevik and Altun gave “Banach’s contraction principle” in the vector metric space.
Let be an -complete vector metric space (v.m.s.), (where “ is Archimedean”).
The mapping satisfies the contractive condition where is a constant.
Then, has a unique fixed point in and iterative sequence , , defined through for -converges on the fixed point of .

##### 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 range (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 mapping is s.t.b. complex-valued metric space (c.m.s.) on (X if fulfilled the subsequent condition:(1)And (2), for all