various generalizations of metric spaces and fixed point theorems

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Revista de la Real Academia deCiencias Exactas, Fisicas y Naturales.Serie A. Matematicas ISSN 1578-7303Volume 109Number 1 RACSAM (2015) 109:175-198DOI 10.1007/s13398-014-0173-7

Various generalizations of metric spacesand fixed point theorems

Tran Van An, Nguyen Van Dung, ZoranKadelburg & Stojan Radenović

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RACSAM (2015) 109:175–198DOI 10.1007/s13398-014-0173-7

ORIGINAL PAPER

Various generalizations of metric spaces and fixed pointtheorems

Tran Van An · Nguyen Van Dung · Zoran Kadelburg ·Stojan Radenovic

Received: 12 October 2013 / Accepted: 20 May 2014 / Published online: 13 June 2014© Springer-Verlag Italia 2014

Abstract There have been many attempts to generalize the definition of a metric space inorder to obtain possibilities for more general fixed point results. In this paper, we give asurvey of recent results on reducing fixed point theorems on generalized metric spaces tofixed point theorems on metric spaces and then investigate this fact in other generalizedmetric spaces. We show that many generalized metric spaces are topologically equivalent tocertain metric spaces or to previously generalized metric spaces. Also, the fixed point theoryin these generalized metric spaces may be a consequence of the fixed point theory in certainmetric spaces or in previously generalized metric spaces.

Keywords Fixed point · Metric space · Generalized metric space

Mathematics Subject Classification (2010) Primary 47H10 · 54H25;Secondary 54D99 · 54E99

T. Van AnDepartment of Mathematics, Vinh University, Vinh, Nghe An, Viet Name-mail: [email protected]

N. Van DungFaculty of Mathematics and Information Technology Teacher Education,Dong Thap University, Cao Lanh, Dong Thap 871200, Viet Name-mail: [email protected]; [email protected]

Z. Kadelburg (B)Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbiae-mail: [email protected]

S. RadenovicFaculty of Mechanical Engineering, University of Belgrade,Kraljice Marije 16, 11120 Beograd, Serbiae-mail: [email protected]

Author's personal copy

176 T. Van An et al.

1 Introduction

The celebrated Banach contraction principle is one of the cornerstones in the developmentof nonlinear analysis [10]. This result can be stated as follows.

Theorem 1.0.1 [29] Let (X, d) be a complete metric space and T be a self-map on X suchthat

d(T x, T y) ≤ kd(x, y) (1.1)

for some k ∈ [0, 1) and all x, y ∈ X. Then T has a unique fixed point x∗ ∈ X andlimn→∞ T nx = x∗ for all x ∈ X.

This principle was extended and improved in many ways and various fixed point theoremswere obtained. Two usual ways of extending and improving the Banach contraction principleare

1. to extend the contraction condition (1.1) to more general contraction conditions, and2. to replace the complete metric space (X, d) by certain generalized metric spaces.

In the first mentioned direction, the first results were due to Edelstein [61,62], Rakotch[117], Kannan [85], Bianchini [33] and some others. Related results were collected andcompared in the well-known paper of Rhoades [119], and later one of Collaço and Silva[44]. In order to prove that a fixed point theorem is a proper generalization of the Banachcontraction principle, the authors usually show the Banach contraction principle to be a directconsequence of their result and construct a map T : X → X to which the Banach contractionprinciple is not applicable while the new one is. For more details on these counter-examples,see the proofs of [119, Theorem 1] and [44, Section 3].

In the second mentioned direction, many types of generalized metric spaces were intro-duced by modifying the metric axioms, such as 2-metric spaces [65], partial metric spaces[100], G-metric spaces [109], cone metric spaces [71] and many others. For a generalizedmetric, authors usually show that every metric is a generalized metric and that there existsa generalized metric which is not a metric on the same set. However, the correspondinggeneralized metric space often appears to be metrizable and the contraction conditions maybe preserved under certain transforms.

In particular, some authors showed recently that fixed point results on some generalizedmetric spaces may be derived from certain results in standard metric spaces, for example, see[20,51,52,63,67,79,84,92]. However, it should be noted, and this applies to all mentionedmethods, it is not true that all generalized fixed point results become trivial in this way.Namely, these results are based on some contractive conditions, and it is not always clearwhether the given condition remains valid when one considers the problem in the associatedmetric space.

In recent times, some new types of generalized metric spaces were introduced and variousspaces constructed as hybrids of the previous types were considered such as D∗-metricspaces [131], cone rectangular metric spaces [26], topological vector space valued conemetric spaces or tvs-cone metric spaces [31,83], generalized cone metric spaces or tvs-G-cone metric spaces [30], metric-type spaces [89,90], complex valued metric spaces [27],partial cone metric spaces [99], cone symmetric spaces [116], G P-metric spaces, partial-G-metric spaces [125,142], S-metric spaces [130], metric-like spaces [18], quasi b-metricspaces [132], G∗-metric spaces [120], tvs-cone b-metric spaces [114], partial rectangularmetric spaces [134], quasi-partial metric spaces [87], Gb-metric spaces [11], b-metric-likespaces [14], g-3ps [38], G pb -metric spaces [72], non-Archimedean cone metric spaces [74],


Generalized metric spaces and fixed point theorems 177

complex valued b-metric spaces [102], quasi-metric-like spaces [143], M-metric spaces [22],quaternion-valued metric spaces [12], partial b-metric spaces [135] and others. The key inconstructing these spaces is to weaken axioms of metric spaces or certain generalized metricspaces. These papers usually have similar presentations—the authors introduce the notionof a generalized metric space and then state some fixed point theorems in such spaces. Thetopological properties of new generalized metric spaces are often not stated and the relationsbetween fixed point theorems in these generalized metric spaces and previously generalizedmetric spaces are often not considered.

In this paper, we give a survey of recent results on reducing fixed point theoremson generalized metric spaces to fixed point theorems on metric spaces and then investi-gate this fact in other generalized metric spaces. We show that many generalized met-ric spaces are topologically equivalent to certain metric spaces or to previously general-ized metric spaces. Also, fixed point theorems in these generalized metric spaces may bea consequence of fixed point theorems in certain metric spaces or in previously gener-alized metric spaces. Proofs and examples are given in the situations where the respec-tive results could not be found in the literature or when some additional information ispresented.

The main results of the paper are presented in Sects. 2 and 3. Section 2 is devoted to conemetric, partial metric and related spaces, Sect. 3 is devoted to 2-metric and its modifications.The conclusion of the paper is presented in Sect. 4.

2 Cone metric spaces and partial metric spaces

In this section, we discuss two well-known generalizations of a metric which attracted manyauthors, a cone metric and a partial metric. We also mention about weak partial metricspaces, topological vector space valued cone metric spaces or tvs-cone metric spaces, conerectangular metric spaces, generalized cone metric spaces, complex valued metric spaces,cone b-metric spaces, cone symmetric spaces, tvs-cone b-metric spaces, non-Archimedeancone metric spaces, complex valued b-metric spaces and quaternion-valued metric spaces.

2.1 Cone metric spaces

The notion of a metric space was introduced by Fréchet in 1905 [64]. Possibly the oldestgeneralization of a metric was given in 1934 by Kurepa [97] who used an ordered normedspace instead of the set of real numbers as the codomain of a metric. Later, such spaces wereused by several mathematicians under various names such as pseudo-metric spaces, K -metricspaces, generalized metric spaces, vector-valued metric spaces, abstract metric spaces, seehistorical surveys in [115,141].

These spaces were re-introduced in 2007 by Huang and Zhang [71] under the name ofcone metric spaces. A novelty was the use of a relation � which could be defined under thesupposition that the underlying cone had a nonempty interior and then such cones are usuallycalled solid. Thus, convergent sequences and completeness could be defined in a new way.

Let (E, ‖·‖) be a real Banach space with the zero vector θ . A proper nonempty and closedsubset P of E is called a cone if P + P ⊂ P , λP ⊂ P for λ ≥ 0 and P ∩ (−P) = {θ}. Ifthe cone P has a nonempty interior int P then it is called solid. Each cone induces a partialordering on E , denoted by �, and defined by x � y ⇐⇒ y − x ∈ P . The notation x ≺ yindicates that x � y and x �= y, and the notation x � y indicates that y − x ∈ int P .



The cone P is called normal if there exists K ≥ 1 such that for all x, y ∈ E , we have thatθ � x � y implies ‖x‖ ≤ K‖y‖. The least positive number K satisfying the above is calledthe normal constant of P .

We state a well-known example of a non-normal cone, see [140].

Example 2.1.1 Let E = C1R[0, 1], with ‖x‖ = ‖x‖∞ + ‖x ′‖∞, P = {x ∈ E : x(t) ≥

0, t ∈ [0, 1]}. This cone is solid but non-normal. Indeed, for xn(t) = tn

n and yn(t) = 1n , we

have θ � xn � yn , and limn→∞ yn = θ , but ‖xn‖ = maxt∈[0,1]∣∣∣

tn

n

∣∣∣ + maxt∈[0,1] |tn−1| =

1n + 1 > 1; hence {xn} does not converge to zero. It follows that P is a non-normal cone.

An attempt was made by Asadi et al. in [23, Theorem 2.1] to prove that each cone canbe made normal by renorming of the given Banach space. However, this cannot be true, asshown in [120].

Definition 2.1.2 [71, Definition 1] Let X be a non-empty set, (E, P) be a Banach space witha cone and let d : X × X → E satisfy the following

1. θ � d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y,2. d(x, y) = d(y, x) for all x, y ∈ X ,3. d(x, y) � d(x, z)+ d(z, y) for all x, y, z ∈ X .

Then d is called a cone metric on X and the pair (X, d) is called a cone metric space.

Definition 2.1.3 [71, Definitions 2–4] Let (X, d) be a cone metric space.

1. A sequence {xn} in X is called convergent to x ∈ X , written as limn→∞ xn = x , if foreach c ∈ E with θ � c, there exists n0 such that d(xn, x) � c for all n ≥ n0.

2. A sequence {xn} in X is called Cauchy if for each c ∈ E with θ � c there exists n0 suchthat d(xn, xm) � c for all n,m ≥ n0.

3. (X, d) is called complete if every Cauchy sequence in (X, d) is a convergent sequence.

Some fixed point theorems on cone metric spaces were proved in [71] with the assumptionof normality of the underlying cone. In [118], Rezapour and Hamlbarani obtained general-izations of some of the results of [71] omitting the assumption of normality. After that, ahuge amount of fixed point theorems on cone metric spaces were stated, for example see[1,2,75,77,80,115] and the references therein.

In 2009, Beg et al. [31] introduced the notion of a topological vector space valued conemetric space, tvs-cone metric space for short, by replacing the real Banach space E inDefinition 2.1.2 by a topological vector space. Then they obtained some common fixed pointresults on such spaces. This approach was used also in other papers [50,83,114].

But, there were also attempts to reduce these results to their metric counterparts. In 2010,Feng and Mao [63] introduced a metric on a cone metric space and then proved that a completecone metric space is always a complete metric space.

Theorem 2.1.4 [63, Lemma 2.1, Theorem 2.2] Let (X, d) be a cone metric space. Then

D(x, y) = inf{‖u‖ : u ∈ P, u � d(x, y)

}

for all x, y ∈ X, is a metric on X. Moreover, the metric space (X, D) is complete if and onlyif the cone metric space (X, d) is complete.



By using Theorem 2.1.4, Feng and Mao verified that a contractive map on the conemetric space (X, d) is a contractive map on the metric space (X, D), see the proof of [63,Theorem 2.4]. The paper of Feng and Mao may be the first one in the attempts to obtain fixedpoint theorems on generalized metric spaces from fixed point theorems on metric spaces. Itsuggested the idea that the same is possible for a lot of other fixed point results on generalizedmetric spaces.

In 2010, Du [51] investigated the equivalence of vectorial versions of fixed point theoremsin tvs-cone metric spaces and scalar versions of fixed point theorems in metric spaces asfollows.

Let (X, d) be a tvs-cone metric space over a solid cone P and let e ∈ int P . Defineξe : X → R by

ξe(x) = inf{ r ∈ R : x ∈ re − P }.This function is usually called a scalarization function.

Theorem 2.1.5 [51, Theorem 2.1] Let (X, d) be a tvs-cone metric space. Then dξ : X×X →[0,+∞) defined by dξ = ξe ◦ d is a metric.

Theorem 2.1.6 [51, Theorem 2.2] Let (X, d) be a tvs-cone metric space, x ∈ X and {xn} asequence in X. Let dξ be the same as in Theorem 2.1.5. Then the following statements hold:

1. If {xn} tvs-cone converges to x ∈ X, then limn→∞ dξ (xn, x) = 0.2. If {xn} is a tvs-cone Cauchy sequence in (X, d), then {xn} is a Cauchy sequence in(X, dξ ).

3. If (X, d) is tvs-cone complete, then (X, dξ ) is a complete metric space.

By using Theorem 2.1.6, Du showed that the Banach contraction principle and someother fixed point results in metric spaces and in tvs-cone metric spaces are equivalent, seethe proofs of [51, Theorem 2.3] and [51, Remark B]. It may be noted that his proof containeda gap, but it was later repaired by several authors (see, e.g., [84]). His method has becomeknown as the scalarization method.

In 2010, Amini-Harandi and Fakhar [19], motivated by the scalarization method, took anew approach to fixed point theory on cone metric spaces. Using the notion of a dual coneP∗ for the given cone P in a Banach space, they constructed a metric on X equivalent tothe given cone metric d , see [19, Lemma 2.1]. Then they used this to obtain a cone metricversion of Geraghty’s fixed point result from [66] as well as of several other fixed point andcommon fixed point results on cone metric spaces in which the cone need not be normal.These results improved and generalized many results from [1,4,46,71,80], see the details in[19, Theorems 2.2, 2.5, 2.7, Remark 2.6].

Later, in 2011, another approach was given by Kadelburg et al. [84] who used Minkowskifunctionals. Recall the following notions.

Let P be a solid cone in a topological vector space E . If V is an absolutely convex andabsorbing subset of E , its Minkowski functional qV is defined by

E � x �→ qV (x) = inf{ λ > 0 : x ∈ λV }.It is a semi-norm on E . If V is an absolutely convex neighborhood of θ in E , then qV iscontinuous and

{x ∈ E : qV (x) < 1} = int V ⊂ V ⊂ V = {x ∈ E : qV (x) ≤ 1}.



Let e ∈ int P . Then [−e, e] = (P − e) ∩ (e − P) = {z ∈ E : −e � z � e} is an absolutelyconvex neighborhood of θ ; its Minkowski functional q[−e,e] will be denoted by qe. It is anincreasing function on P . If the cone P is solid and normal, qe is a norm in E .

Theorem 2.1.7 [84, Theorem 3.2] Let (X, d) be a tvs-cone metric space over a solid coneP and let e ∈ intP. Let qe be the corresponding Minkowski functional of [−e, e]. Thendq = qe ◦ d is a metric on X. Moreover:

1. limn→∞ xn = x in (X, d) if and only if limn→∞ xn = x in (X, dq).2. A sequence {xn} is a d-Cauchy sequence if and only if it is a dq-Cauchy sequence.3. (X, d) is complete if and only if (X, dq) is complete.

Hence, topologies induced on X by d and dq are equivalent. Thus, again, a lot of theknown fixed point and common fixed point results in tvs-cone metric spaces can be reducedto their metric counterparts.

However, it should be noted, and this applies to all mentioned methods, that it is not truethat all results in cone metric spaces become trivial in this way. Namely, these results arebased on some contractive conditions, and it is not always clear whether the given conditionremains valid when we consider the problem in the associated metric space. For example, in2013, Khamsi and Wojciechowski [91] continued investigations in this direction, showingthat the classical Caristi point theorem [37] can be obtained in the case that the Minkowskifunctional is additive.

We conclude this subsection stating yet another attempt of this kind. In 2011, Khani andPourmahdian [92] made an explicit construction of a standard metric D on a given conemetric space (X, d), see [92, Theorem 3.4]. Moreover, [92, Theorem 3.5] shows that havingcontraction maps f1, . . . , fn on (X, d) of certain type, the metric D can be defined as tomake f1, . . . , fn contractions on (X, D) as well.

2.2 Complex valued metric spaces

In [27], Azam et al. introduced and studied complex valued metric spaces and establishedsome fixed point results for maps satisfying a rational inequality. The idea of complex valuedmetric spaces is simply to replace R with the usual order by C with certain order. The authorsasserted that the idea of complex valued metric spaces can be exploited to define complexnormed metric spaces and complex valued Hilbert spaces.

Definition 2.2.1 [27, Definition 1] Let the set C of complex numbers be equipped with thepartial order � defined by

x � y if and only if Re x ≤ Re y and Im x ≤ Im y. (2.1)

and d : X × X → C such that

1. 0 � d(x, y) for all x, y ∈ X and d(x, y) = 0 if and only if x = y.2. d(x, y) = d(y, x) for all x, y ∈ X .3. d(x, y) � d(x, z)+ d(z, y) for all x, y, z ∈ X .

Then d is called a complex valued metric on X and the pair (X, d) is called a complex valuedmetric space.

Azam et al. also proved some properties of a complex valued metric space and obtainedsufficient conditions for the existence of common fixed points of a pair of maps satisfying



contractive type conditions. After that, some authors considered fixed point theorems on acomplex valued metric space, for example see [123,136,137] and some others.

However, most of these results are just consequences of the known ones and hence areredundant. This was explicitly stated in 2012 by Sastry et al. [128].

Theorem 2.2.2 [128, Theorem 2.1] Let (X, D) be a complex valued metric space. Then

d(x, y) = max{

Re D(x, y), Im D(x, y)}

for all x, y ∈ X is a metric on X. Moreover, topologies induced by D and d are the same.

Again, the only possibility to obtain a nontrivial result in this way is to have contractiveconditions which use the algebraic structure of the set C, for example see [3,36,39].

There are a lot of other generalizations of metric spaces, with the same idea as conemetric spaces and complex valued metric spaces, such as cone rectangular metric spaces[26], generalized cone metric spaces [30], cone b-metric spaces [73], cone symmetric spaces[116], tvs-cone b-metric spaces [114], non-Archimedean cone metric spaces [74], complexvalued b-metric spaces [102], quaternion-valued metric spaces [12]. Also, similar techniquesas the above may be used to investigate these spaces such as using b-metric [28,45] toinvestigate cone b-metric, complex valued b-metric spaces and tvs-cone b-metric; using non-Archimedean metric to investigate non-Archimedean cone metric; using metric to investigatequaternion-valued metric and generalized cone metric; using symmetric to investigate conesymmetric; using rectangular metric [34] to investigate cone rectangular metric. In particular,relations between cone b-metric spaces and b-metric spaces were proved in [52,95].

2.3 Partial metric spaces

In 1992, Matthews [100] introduced the notion of a partial metric space as a part of thestudy of denotational semantics of dataflow networks. The main difference comparing tothe standard metric is that the self-distance of an arbitrary point need not be equal to zero.Originally, this notion was motivated by the experience of computer science. As discussedbelow, the authors showed also how a mathematics of nonzero self-distance for metric spacescan been established, and leads to interesting research in the foundations of topology [35].

Definition 2.3.1 [100, Definition 3.1] Let X be a non-empty set and p : X × X → [0,+∞)

satisfy the following

1. The small self-distance: 0 ≤ p(x, x) ≤ p(x, y) for all x, y ∈ X ,2. The equality implies indistancy: If x = y then d(x, y) = 0,3. The indistancy implies equality: If p(x, x) = p(x, y) = p(y, y) then x = y for all

x, y ∈ X ,4. The symmetry: p(x, y) = p(y, x) for all x, y ∈ X ,5. The triangularity: p(x, z) ≤ p(x, y)+ p(y, z)− p(y, y) for all x, y, z ∈ X .

Then p is called a partial metric and the pair (X, p) is called a partial metric space.

Remark 2.3.2 [88, Remark 1.5] Let (X, p) be a partial metric space. Then p(x, y) = 0implies x = y. The reverse implication does not hold. The simplest example of a partialmetric space with non-zero self-distances is the set X = [0,+∞)with p(x, y) = max{x, y}for x, y ∈ X .

The following is an example of a metric space that is the reason why nonzero self-distancewas worth considering [35, Section 2]. Let Sω be the set of all infinite sequences x = {xn}



over a set S. For all such sequences x and y, let dS(x, y) = 2−k , where k is the largestnumber, possibly ∞, such that xi = yi for each i < k. Thus dS(x, y) is defined to be 1 over2 to the power of the length of the longest initial sequence common to both x and y. It canbe shown that (Sω, dS) is a metric space. How might computer scientists view this metricspace? To be interested in an infinite sequence x they would want to know how to compute it,that is, how to write a computer program to print out, on either a screen or paper, the valuesx0, then x1, then x2 and so on. As x is an infinite sequence, its values cannot be printed outin any finite amount of time, and so computer scientists are interested in how the sequence xis formed from its parts, the finite sequences {}, {x0}, {x0, x1}, {x0, x1, x2} and so on. Aftereach value xk is printed, the finite sequence {x0, . . . , xk} represents that part of the infinitesequence produced so far. Each finite sequence is thus thought of in computer science asbeing a partially computed version of the infinite sequence x , which is totally computed.Suppose now that the above definition of dS is extended to S∗, the set of all finite sequencesover S. Then axioms (1), (2), (3) and (4) in Definition 2.3.1 still hold.

However, if x is a finite sequence then dS(x, x) = 2−k for some number k < ∞, which isnot 0, since x j = x j can only hold if x j is defined. Thus axiom (2) in Definition 2.3.1 does nothold for finite sequences. This raises an intriguing contrast between 20th century mathematics,of which the theory of metric spaces is our working example, and the contemporary experienceof computer science. The truth of the statement x = x is surely unchallenged in mathematics,while in computer science its truth can only be asserted to the extent to which x is computed.The article of Matthews [100] and some others [35,101] showed that rather than collapsing,the theory of metric spaces was actually expanded and enriched by the generalization ofdropping the requirement for equality to imply indistancy.

Definition 2.3.3 [35, Definitions 4–6], [121, Definition 2.1] Let (X, p) be a partial metricspace.

1. A sequence {xn} is called convergent to x ∈ X , written as limn→∞ xn = x , iflimn→∞ p(xn, x) = p(x, x).

2. A sequence {xn} is called Cauchy if limn,m→∞ p(xn, xm) exists and is finite.3. (X, p) is called complete if each Cauchy sequence {xn} in X is convergent to some x ∈ X

and limm,n→∞ p(xn, xm) = p(x, x).4. A sequence {xn} is called 0-Cauchy if limn,m→∞ p(xn, xm) = 0. The space (X, p) is 0-

complete if each 0-Cauchy sequence {xn} converges to some x ∈ X such that p(x, x) = 0.

Lemma 2.3.4 [88, page 2] Let (X, p) be a partial metric space. Then the functionsdp, dm : X × X → [0,+∞) given by

dp(x, y) = 2p(x, y)− p(x, x)− p(y, y),

dm(x, y) = max{p(x, y)− p(x, x), p(x, y)− p(y, y)}for all x, y ∈ X, are metrics on X.

Lemma 2.3.5 [88, Lemma 1.4] Let (X, p) be a partial metric space and dp be defined as inLemma 2.3.4. Then we have

1. A sequence {xn} is Cauchy in a partial metric space (X, p) if and only if it is Cauchy inthe metric space (X, dp).

2. The partial metric space (X, p) is complete if and only if the metric space (X, dp) iscomplete. Moreover

limn→∞ dp(xn, x) = 0 ⇐⇒ p(x, x) = lim

n→∞ p(xn, x) = limn,m→∞ p(xn, xm).



In 1992, Matthews [100] proved the partial metric version of Banach contraction principle.Later, many authors studied partial metric spaces and their topological properties, as well asfixed point results in these spaces, see [68,93,96,113,121] and many others.

In 2008, Rus [124] investigated the basic problems of metric fixed point theory in the caseof partial metric spaces—existence and uniqueness, the convergence of successive approx-imations, data dependence, fibre contraction principle, well-posedness of the fixed pointproblem and limit shadowing property. Rus posed the following questions for the metric dp

defined in Lemma 2.3.4:

1. If T : X → X is a generalized contraction with respect to a partial metric p, whichcondition T satisfies with respect to dp?

2. How to use fixed point theorems in a metric space (X, dp) to give analogous fixed pointresults in a partial metric space?

In 2013, Samet et al. [127] presented new fixed point theorems on complete metric spacesto answer the above questions. As an application of the results, the authors showed that theMatthews fixed point theorem in partial metric spaces is a particular case of certain result oncomplete metric spaces. We present a sample of these results.

Theorem 2.3.6 [127, Theorem 2.1] Let (X, d) be a complete metric space, ϕ : X →[0,+∞) be a lower semi-continuous function and T : X → X be a map. Suppose thatfor any 0 < a < b < +∞, there exists 0 < γ (a, b) < 1 such that for all x, y ∈ X,

a ≤ d(x, y)+ ϕ(x)+ ϕ(y) ≤ b

�⇒ d(T x, T y)+ ϕ(T x)+ ϕ(T y) ≤ γ (a, b)[

d(x, y)+ ϕ(x)+ ϕ(y)]

.

Then T has a unique fixed point x∗ in X. Moreover, we have ϕ(x∗) = 0.

Taking ϕ(x) = p(x, x) in the previous result, the following fixed point result in a partialmetric space immediately follows.

Corollary 2.3.7 [127, Corollary 3.1] Let (X, p) be a complete partial metric space andT : X → X be a map. Suppose that for any 0 < a < b < +∞ there exists 0 < γ (a, b) < 1such that for all x, y ∈ X,

a ≤ p(x, y) ≤ b �⇒ p(T x, T y) ≤ γ (a, b)p(x, y).

Then T has a unique fixed point x∗ ∈ X. Moreover, p(x∗, x∗) = 0.

Obviously, Matthews fixed point theorem follows as a special case of Corollary 2.3.7. In acompletely similar manner, other fixed point results such as Kannan’s, Reich’s, Chatterjea’sresults in partial metric spaces can be derived, see [127, Theorems 2.3–2.5 and Corollaries3.3–3.5]. As a continuation of [127], Jleli et al. [78] established some new fixed point theoremson metric spaces and obtained analogous fixed point theorems on partial metric spaces.

In 2013, Haghi et al. [67] reproved the following result which was first presented in [69],where τd and τdp are topologies induced by d and dp respectively.

Theorem 2.3.8 [69, Propositions 4.8.8 and 4.8.10], [67, Proposition 2.1] Let (X, p) be apartial metric space. Then the function d : X × X → [0,+∞) defined by d(x, y) = 0whenever x = y and d(x, y) = p(x, y) whenever x �= y, is a metric on X such thatτdp ⊂ τd . Moreover, (X, d) is complete if and only if (X, p) is 0-complete.



By using Theorem 2.3.8, Haghi et al. proved that several fixed point generalizations topartial metric spaces can be obtained from the corresponding results in metric spaces, suchas: [17, Theorem 1] from [32, Theorem 2], [7, Theorem 6] from the main result of [60], [6,Theorem 5] from [13, Theorem 1.3], [43, Theorem 2.1] from [13, Theorem 2.4], [17, Theorem2] from the main result of [42], [5, Theorem 8] from [41, Theorem 3.1], [43, Theorem 2.1]from [13, Theorem 2.4]. For more details, see the proofs of [67, Theorems 2.3–2.8].

2.4 Weak partial metric spaces

In 1999, by omitting the small self-distance axiom of partial metric, see Definition 2.3.1(1),Heckmann [68] defined a weak partial metric as a generalization of partial metric. Afterthat, some fixed point theorems on weak partial metric spaces were stated [16,59]. In a weakpartial metric space, the convergence of a sequence, Cauchy sequence, completeness andcontinuity of a map are defined as in a partial metric space [16], see Definition 2.3.3.

Proposition 2.4.1 [16, Proposition 2.3] Let (X, p) be a weak partial metric space. Then dwdefined by

dw(x, y) = p(x, y)− min{

p(x, x), p(y, y)}

for all x, y ∈ X, is a metric on X.

Lemma 2.4.2 [16, Lemma 2.4] Let (X, p) be a weak partial metric space.

1. {xn} is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequence in the metricspace (X, dw).

2. (X, p) is complete if and only if (X, dw) is complete.

Lemma 2.4.2 was used to prove some fixed point theorems on weak partial metric spaces,see the proofs of [16, Theorems 3.1 and 3.6]. But this lemma seems to be useful only forlinear conditions.

From [68, Subsection 2.5], we have the following result.

Theorem 2.4.3 Let (X, p) be a weak partial metric space and

dp(x, y) = max{

p(x, y), p(x, x), p(y, y)}

for all x, y ∈ X. Then we have

1. dp is a partial metric space on X.2. limn→∞ xn = x in (X, p) if and only if limn→∞ xn = x in (X, dp).3. (X, p) is 0-complete if and only if (X, dp) is complete.

By using Theorem 2.4.3, we see that fixed point theorems on weak partial metric spacesmay be obtained from fixed point theorems on partial metric spaces. Then, by Theorem2.3.8, fixed point theorems on weak partial metric spaces may be obtained from fixed pointtheorems on metric spaces.

There are other generalizations of metric spaces, with the same idea as partial metricspaces and cone metric spaces, such as partial cone metric spaces [99], partial rectangularmetric spaces [134], quasi-partial metric spaces [87], partial b-metric spaces [107,135].Also, similar techniques as the above may be used to investigate these spaces, such as usingrectangular metric (the other name of generalized metric space in the sense of Branciari[34]), to investigate partial rectangular metric; using metric to investigate partial cone metric



and quasi-partial metric; using b-metric to investigate partial b-metric. In particular, therelation between partial rectangular metric spaces and rectangular metric spaces was statedin [55]; the relation between b-metric spaces and partial b-metric spaces was stated in [54].For two-variable generalizations of a metric such as metric-like spaces [18], quasi b-metricspaces [132], b-metric-like spaces [14], quasi-metric-like spaces [143], M-metric spaces[22], quaternion-valued metric spaces [12], the relations with metric spaces or with knowngeneralized metric spaces may be stated. In particular, the relation between metric-like spacesand metric spaces was stated in [57]; the relations between rectangular metric spaces andmetric spaces were considered in [82,94].

3 2-metric and its modifications

In this section, we concentrate on three-variable generalizations of a metric including a2-metric and a G-metric. Also, we give some critical remarks on other three-variable gener-alizations of a metric space such as D∗-metric spaces, G P-metric spaces, S-metric spaces,G∗-metric spaces, Gb-metric spaces, g-3ps, G pb -metric spaces.

3.1 2-metric spaces

Recall that a metric measures the distance between two points. In 1960’s, Gähler [65] intro-duced the notion of 2-metric, simply taking as inspiration the area determined by three points.

Definition 3.1.1 [65] Let X be a non-empty set and d : X×X×X → R satisfy the following:

1. For all pairs of distinct points x, y ∈ X , there exists a point z ∈ X such that d(x, y, z) �= 0,2. If at least two of three points x, y, z are the same, then d(x, y, z) = 0,3. The symmetry: d(x, y, z) = d(x, z, y) = d(y, x, z) = d(y, z, x) = d(z, x, y) =

d(z, y, x) for all x, y, z ∈ X ,4. The rectangle inequality: d(x, y, z) ≤ d(x, y, t)+d(y, z, t)+d(z, x, t) for all x, y, z, t ∈

X .

Then d is called a 2-metric on X and (X, d) is called a 2-metric space.

It is easy to see that every 2-metric is non-negative and every 2-metric space contains atleast three distinct points.

Definition 3.1.2 [76] Let (X, d) be a 2-metric space.

1. A sequence {xn} is called convergent to x ∈ X , written as limn→∞ xn = x , if for alla ∈ X , limn→∞ d(xn, x, a) = 0.

2. A sequence {xn} is called a Cauchy sequence if for all a ∈ X , limn,m→∞ d(xn, xm, a) = 0.3. (X, d) is called complete if each Cauchy sequence is a convergent sequence.

Note that a 2-metric space is not metrizable in general. For example, see the 2-metric space(R, σ ) in [65, page 145]. Many fixed point theorems on 2-metric spaces were obtained, forexample see [98,111] and references therein. For a map T : X → X , contraction conditionsused in these papers are usually of the form d(T x, T y, a) ≤ kd(x, y, a), for all x, y, a ∈ X ,and its modifications. In 1986, Hsiao [70], and then, in 1988, Cho et al. [40] showed that thesekinds of contraction conditions do not have a wide range of applications, since they implycollinearity of the sequence of iterates starting with some point, see the following results.



Lemma 3.1.3 [40, Lemma 1.2] Let (X, d) be a 2-metric space and S, T, fi : X → X bemaps, for all i ∈ N, such that

1. fi (X) ⊂ S(X) ∩ T (X) for all i ∈ N,2. There exists φ : [0,+∞) → [0,+∞) that is non-decreasing, upper semi-continuous andφ(t) < t for all t > 0 satisfying

d( fi x, f j y, a) ≤ φ(

max{

d(Sx, T y, a), d(Sx, fi x, a), d(T y, f j y, a),

d(T y, fi x, a), d(Sx, f j y, a)})

for all x, y, a ∈ X, i �= j ∈ N.

For any x0 ∈ x, take T x1 = f1x0, Sx2 = f2x1 and

y2n+1 = T x2n+1 = f2n+1x2n and y2n = Sx2n = f2n x2n−1

for all n ∈ N. Then d(yi , y j , yk) = 0 for all i, j, k ∈ N.

Definition 3.1.4 [40, Definition 1.6] Let (X, d) be a 2-metric space and S, T : X → X betwo maps. The pair (S, T ) is called asymptotically regular at x0 in X if for each a ∈ X ,limn→∞ d(yn, yn+1, a) = 0, where {yn} is defined as in Lemma 3.1.3.

Lemma 3.1.5 [40, Lemma 1.3] Let (X, d) be a 2-metric space and S, T : X → X beasymptotically regular at x0. Then the sequence {yn} defined as in Lemma 3.1.3 is a Cauchysequence.

In 2013, the following results were proved by Dung et al. [58].

Theorem 3.1.6 [58, Theorem 2.1] Let (X, d) be a 2-metric space and T, F : X → X betwo maps. If d(T x, Fy, x) = d(T x, Fy, y) = 0 for all x, y ∈ X, then T x is a fixed point ofT and Fy is a fixed point of F for all x, y ∈ X.

Corollary 3.1.7 [58, Corollary 2.2] Let (X, d) be a 2-metric space and T : X → X be amap. If d(T x, T y, x) = 0 for all x, y ∈ X, then T x is a fixed point of T for all x ∈ X.

In some cases, fixed point theorems on 2-metric spaces with contraction conditions of theform d(T x, T y, a) ≤ kd(x, y, a), for all x, y, a ∈ X , and its modifications can be obtainedeasily from 2-metric axioms by choosing a = x and applying Theorem 3.1.6 and Corollary3.1.7, see details in [58].

Further, the basic philosophy is that since 2-metric measures area, a contraction shouldtend the space towards a configuration of zero area, which is to say to a line. In 2012,Aliouche and Simpson [15] introduced the notion of the fixed line of a map T : X → Xwhich is contractive in the sense that d(T x, T y, T z) ≤ kd(x, y, z) for some k ∈ [0, 1)and all x, y, z ∈ X . This contraction condition is different from the kinds of contractionconditions considered by many authors as recalled previously, in that the map T is applied toall three variables. Also in [15], Aliouche and Simpson considered bounded 2-metric spacessatisfying an additional axiom and showed that a contractive map has either a fixed point ora fixed line.

Theorem 3.1.8 [15, Theorem 6.2] Let (X, ϕ) be a compact and bounded transitive 2-metricspace, and F : X → X be a map such that d(Fx, Fy, Fz) ≤ kd(x, y, z) for some k ∈ [0, 1)and all x, y, z ∈ X. Then, either F has a fixed point, or there is a line Y fixed in the sensethat F(Y ) ⊂ Y and Y is the only line containing F(Y ).



The paper of Aliouche and Simpson seems to open a new way to investigate fixed pointresults on 2-metric spaces. Following the extensive literature in this subject, more compli-cated situations involving several compatible maps and additional terms in the contractivitycondition may also be envisioned [15].

3.2 D-metric spaces, G-metric spaces and quasi-metric spaces

The first modification of a 2-metric was a D-metric which was introduced in 1984 by Dhage[47–49]. While for a 2-metric d , d(x, y, z) can be thought as a generalization of the area ofa triangle with vertices at x, y, z in R

2, then, for a D-metric D, D(x, y, z) can be treated asa generalization of the perimeter of this triangle.

But in 2003, Mustafa and Sims [108] stated some remarks concerning D-metric spacesand presented some examples which showed that many of the basic claims concerning thetopological structure of D-spaces were incorrect, thus nullifying many of the results claimedfor D-spaces. After that, in 2005, various concepts of open balls in D-metric spaces werestudied in the case of certain D-metric spaces and many results in the literature on such ballswere shown to be false by Naidu et al. [112]. In 2006, to overcome the flaws of D-metricspaces which were pointed out, Mustafa and Sims [109] introduced the notion of a G-metricspace by modifying axioms of a D-metric.

Definition 3.2.1 [109, Definition 3] Let X be a nonempty set and G : X × X × X → [0,∞)

be a function such that, for all x, y, z, a ∈ X ,

1. G(x, y, z) = 0 if x = y = z,2. 0 < G(x, x, y) if x �= y ∈ X ,3. G(x, x, y) ≤ G(x, y, z) if y �= z,4. G(x, y, z) = G(x, z, y) = G(y, x, z) = G(y, z, x) = G(z, x, y) = G(z, y, x),5. G(x, y, z) ≤ G(x, a, a)+ G(a, y, z).

Then G is called a G-metric on X and the pair (X,G) is called a G-metric space.

Definition 3.2.2 [110, Definitions 1.3, 1.5] Let (X,G) be a G-metric space.

1. A sequence {xn} is called convergent to x ∈ X , written as limn→∞ xn = x , if, for eachε > 0, there exists n0 ∈ N such that G(xm, xn, x) < ε for m, n > n0.

2. A sequence {xn} is called Cauchy if, for each ε > 0, there exists n0 ∈ N such thatG(xl , xm, xn) < ε for l,m, n > n0.

3. The space (X,G) is called complete if each Cauchy sequence in (X,G) is convergent.

This notion attracted many authors and various kinds of fixed point theorems for mapson G-metric spaces were proved, for example see [24,25,103,104,110,133] and referencestherein.

A G-metric space (X,G) is called symmetric if G(x, x, y) = G(x, y, y)holds for arbitraryx, y ∈ X . One of important differences between symmetric and asymmetric cases was pointedout already in [109, Section 4]. Namely, if (X,G) is a G-metric space, then

G∗(A,C, E) = G(a, c, e)+ G(b, d, f ), A = (a, b),C = (c, d), E = (e, f ) ∈ X2,

(3.1)defines a G-metric on X2 if and only if the G-metric G is symmetric. As a sample, we justnote that, as a consequence of the previous fact, the so-called coupled fixed point resultsin G-metric spaces are easy to obtain in the symmetric case, see [81, Theorems 4.1–4.3]



and [9, Section 4], while in the asymmetric case more involved procedure has to be used, see[8, Theorems 1–3 and Example 2].

In 2012, Jleli and Samet [79] showed that most of the obtained fixed point theorems onG-metric spaces can be deduced immediately from fixed point theorems on metric spaces orquasi-metric spaces. The similar results can be found in [20,126]. Note that the notion of aquasi-metric in [79] was used widely in asymmetric topology, see [122,129] for example.

Definition 3.2.3 [79, Definition 2.1] Let X be a non-empty set and d : X × X → [0,+∞)

satisfy the following:

1. d(x, y) = 0 if and only if x = y,2. d(x, y) ≤ d(x, z)+ d(z, y) for all x, y ∈ X .

Then d is called a quasi-metric and the pair (X, d) is called a quasi-metric space.

Definition 3.2.4 [79, Definitions 2.2–2.6] Let (X, d) be a quasi-metric space.

1. A sequence {xn} is called convergent to x ∈ X , written as limn→∞ xn = x , if

limn→∞ d(xn, x) = lim

n→∞ d(x, xn) = 0.

2. A sequence {xn} is called left-Cauchy if for each ε > 0 there exists n(ε) such thatd(xn, xm) < ε for all n ≥ m > n(ε).

3. A sequence {xn} is called right-Cauchy if for each ε > 0 there exists n(ε) such thatd(xn, xm) < ε for all m ≥ n > n(ε).

4. A sequence {xn} is called Cauchy if for each ε > 0 there exists n(ε) such that d(xn, xm) <

ε for all m, n > n(ε), that is, limn,m→∞ d(xn, xm) = limn,m→∞ d(xm, xn) = 0.5. (X, d) is called left-complete if each left-Cauchy sequence in (X, d) is convergent.6. (X, d) is called right-complete if each right-Cauchy sequence in (X, d) is convergent.7. (X, d) is called complete if each Cauchy sequence in (X, d) is convergent.

Obviously, a sequence {xn} is Cauchy if and only if it is left-Cauchy and right-Cauchy. Thecompleteness of a quasi-metric space introduced in [79] is well-known as Smyth completeness[122,138]. Moreover, if (X, d) is a quasi-metric space, one can construct in a natural waythe topology τd on X which has a base of open balls

{

Bd(x, r) : x ∈ X, r > 0}

whereBd(x, r) = {y ∈ X : d(x, y) < r}. Customary, a topological space (X, τ ) is said to bequasi-metrizable if there exists a quasi-metric d on X such that τ = τd and the terms inquasi-metric spaces may be refereed to convergence with respect to τd .

Theorem 3.2.5 [53, Proposition 3] Let (X, d) be a quasi-metric space and δd(x, y) =max

{

d(x, y), d(y, x)}

for all x, y ∈ X. Then we have

1. [79] (X, δd) is a metric space.2. limn→∞ xn = x in (X, d) if and only if limn→∞ xn = x in (X, δd).3. A sequence {xn} is Cauchy in (X, d) if and only if it is Cauchy in (X, δd).4. The quasi-metric space (X, d) is complete if and only if the metric space (X, δd) is

complete.

Theorem 3.2.6 [79, Theorem 2.2] Let (X,G) be a G-metric space and dG : X × X →[0,+∞) be defined by dG(x, y) = G(x, y, y) for all x, y ∈ X. Then we have

1. (X, dG) is a quasi-metric space.2. limn→∞ xn = x in (X,G) if and only if limn→∞ xn = x in (X, dG).



3. A sequence {xn} is Cauchy in (X,G) if and only if it is Cauchy in (X, dG).4. The G-metric space (X,G) is complete if and only if the metric space (X, dG) is complete.

Theorem 3.2.7 [79, Theorem 2.3] Let (X,G) be a G-metric space and δG : X × X →[0,+∞) be defined by δG(x, y) = max

{

G(x, y, y),G(y, x, x)}

for all x, y ∈ X. Then wehave

1. (X, δG) is a metric space.2. limn→∞ xn = x in (X,G) if and only if limn→∞ xn = x in (X, δG).3. A sequence {xn} is Cauchy in (X,G) if and only if it is Cauchy in (X, δG).4. The G-metric space (X,G) is complete if and only if the metric space (X, δG) is complete.

The following result shows that the product space X × X is also a quasi-metric space.

Proposition 3.2.8 [53, Proposition 1] Let (X, dX ) and (Y, dY ) be two quasi-metric spaces.Then we have

1. d(x, y) = dX (x1, y1) + dY (x2, y2) (for all x = (x1, x2), y = (y1, y2) ∈ X × Y ) is aquasi-metric on X × Y .

2. limn→∞(xn, yn) = (x, y) in (X × Y, d) if and only if limn→∞ xn = x in (X, dX ) andlimn→∞ yn = y in (Y, dY ). In particular, the product topology on X × Y coincides withthe topology induced by d.

3.{

(xn, yn)}

is a Cauchy sequence in (X × Y, d) if and only if it is a Cauchy sequence in(X, dX ) and {yn} is a Cauchy sequence in (Y, dY ).

4. (X × Y, d) is complete if and only if (X, dX ) and (Y, dY ) are complete.

The following notation on quasi-metric spaces is the same as notation on metric spaces[42].

Definition 3.2.9 Let (X, d) be a quasi-metric space and T : X → X be a map. For eachx ∈ X , put

OT (x, n) = {x, T x, . . . , T n x}, OT (x,+∞) = {x, T x, . . . , T n x, . . .}.Then X is called T -orbitally complete if each Cauchy sequence in OT (x,+∞) is convergentin X .

Note that every complete quasi-metric space is T -orbitally complete for each map T : X →X . The following easy example shows that there exists a T -orbitally complete quasi-metricspace which is not complete.

Example 3.2.10 [53, Example 3] Let (X, d) be a quasi-metric space which is not completeand T : X → X be the map defined by T x = x0 for all x ∈ X and some x0 ∈ X . Then(X, d) is a T -orbitally complete quasi-metric space which is not complete.

The following fixed point theorem on quasi-metric spaces was recently presented in [53].

Theorem 3.2.11 [53, Theorem 5] Let (X, d) be a quasi-metric space and T : X → X be amap satisfying the following

1. X is weak T -orbitally complete,2. There exists q ∈ [0, 1) such that for all x, y ∈ X,

d(T x, T y) ≤ q max{

d(x, y), d(y, x), d(x, T x), d(T x, x), d(y, T y), d(x, T y),

d(y, T x), d(T x, y), d(T 2x, x), d(x, T 2x), d(T 2x, T x),

d(T x, T 2x), d(T 2x, y), d(y, T 2x), d(T 2x, T y)}

. (3.2)



Then we have

1. T has a unique fixed point x∗ in X,2. limn→∞ T n x = x∗ for all x ∈ X,

3. max{

d(T n x, x∗), d(x∗, T n x)} ≤ qn

1 − qmax

{

d(x, T x), d(T x, x)}

for all x ∈ X and

n ∈ N.

If d in Theorem 3.2.11 is a metric, then we have the following result.

Corollary 3.2.12 [53, Corollary 1] Let (X, d) be a metric space and T : X → X satisfy thefollowing

1. X is T -orbitally complete.2. There exists q ∈ [0, 1) such that for all x, y ∈ X,


d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x),

d(T 2x, x), d(T 2x, T x), d(T 2x, y), d(T 2x, T y)}

. (3.3)

Then we have

1. T has a unique fixed point x∗ in X.2. limn→∞ T n x = x∗ for all x ∈ X.

3. d(T n x, x∗) ≤ qn

1 − qd(x, T x) for all x ∈ X.

Note that Corollary 3.2.12 is a generalization of the following well-known result of Ciric.

Corollary 3.2.13 [42, Theorem 1] Let (X, d) be a metric space and T : X → X satisfy thefollowing

1. X is T -orbitally complete.2. There exists q ∈ [0, 1) such that for all x, y ∈ X,


d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x)}

. (3.4)

Then we have

1. T has a unique fixed point x∗ in X.2. limn→∞ T n x = x∗ for all x ∈ X.

3. d(T n x, x∗) ≤ qn

1 − qd(x, T x) for all x ∈ X.

In a similar way, we may generalize other fixed point theorems containing at most fivementioned values in the literature to those containing d(T 2x, x), d(T 2x, T x), d(T 2x, y),d(T 2x, T y) in addition.

The following example shows that Corollary 3.2.12 is a proper generalization of Corollary3.2.13.

Example 3.2.14 [53, Example 2.10] Let X = {1, 2, 3, 4, 5} with d defined as

d(x, y) =

⎧

⎪⎪⎨

⎪⎪⎩

0, if x = y,

2, if (x, y) ∈ {

(1, 4), (1, 5), (4, 1), (5, 1)}

,

1, otherwise.



Then (X, d) is a complete metric space. Let T : X → X be defined by

T 1 = T 2 = T 3 = 1, T 4 = 2, T 5 = 3.

Then (X, d) is T -orbitally complete. We have

d(T x, T y) = d(1, 1) = 0 if x, y ∈ {1, 2, 3},d(T 1, T 4) = d(T 2, T 4) = d(T 3, T 4) = d(1, 2) = 1,

d(T 1, 4) = d(T 2, 4) = d(T 3, 4) = d(1, 4) = 2,

d(T 1, T 5) = d(T 2, T 5) = d(T 3, T 5) = d(1, 3) = 1,

d(T 1, 5) = d(T 2, 5) = d(T 3, 5) = d(1, 5) = 2,

d(T 4, T 5) = d(2, 3) = 1,

d(4, 5) = d(4, T 4) = d(5, T 5) = d(4, T 5) = d(5, T 4) = 1,

d(T 24, 4) = d(T 2, 4) = d(1, 4) = 2, d(T 25, 5) = d(T 3, 5) = d(1, 5) = 2.

The above calculations show that (3.4) does not hold for x = 4, y = 5 but (3.3) holds forall x, y ∈ X and some q ∈ [ 1

2 , 1)

. Then Corollary 3.2.13 is not applicable to T and (X, d)while Corollary 3.2.12 is applicable to T and (X, d).

In 2013, with an attempt to find fixed point theorems that fit the nature of a G-metricspace, Karapinar and Agarwal [86] suggested some results asserting that for their resultsthe techniques used in [79,126] are inapplicable. But in [53], it was shown that most ofthe obtained results on G-metric spaces in [21,86] may be also deduced from certain fixedpoint theorems on quasi-metric spaces, see also [139]. We present the proof of the followingcorollary as a sample.

Corollary 3.2.15 [86, Theorem 3.1] Let (X,G) be a complete G-metric space and T : X →X be a map such that

G(T x, T y, T z) ≤ k M(x, y, z) (3.5)

for all x, y, z ∈ X, where k ∈ [

0, 12

)

and

M(x, y, z) = max

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

G(x, T x, y),G(y, T 2x, T y),G(T x, T 2x, T y),

G(y, T x, T y),G(x, T x, z),G(z, T 2x, T z),

G(T x, T 2x, T z),G(z, T x, T y),G(x, y, z),

G(x, T x, T x),G(y, T y, T y),G(z, T z, T z),

G(z, T x, T x),G(x, T y, T y),G(y, T z, T z).

⎫

⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

Then T has a unique fixed point.

Proof Let dG be the quasi-metric introduced in Theorem 3.2.6. By choosing z = y and usingthe axioms (4) and (5) in Definition 3.2.1, we have



M(x, y, y) = max

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

G(x, T x, y),G(y, T 2x, T y),G(T x, T 2x, T y),

G(y, T x, T y),G(x, T x, y),G(y, T 2x, T y),

G(T x, T 2x, T y),G(y, T x, T y),G(x, y, y),

G(x, T x, T x),G(y, T y, T y),G(y, T y, T y),

G(y, T x, T x),G(x, T y, T y),G(y, T y, T y)

⎫

⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

= max

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

G(x, T x, y),G(y, T y, T 2x),G(T x, T y, T 2x),

G(y, T y, T x),G(x, T x, y),G(y, T y, T 2x),

G(T 2x, T x, T y),G(y, T y, T x),G(x, y, y),

G(x, T x, T x),G(y, T y, T y),G(y, T y, T y),


⎫

⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

≤ max

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

G(x, T x, T x)+ G(T x, T x, y),G(y, T y, T y)+ G(T y, T y, T 2x),

G(T x, T y, T y)+ G(T y, T y, T 2x),

G(y, T y, T y)+ G(T y, T y, T x),G(x, T x, T x)+ G(T x, T x, y),

G(y, T y, T y)+ G(T y, T y, T 2x),

G(T 2x, T x, T x)+ G(T x, T x, T y),G(y, T y, T y)+ G(T y, T y, T x),

G(x, y, y),G(x, T x, T x),G(y, T y, T y),G(y, T y, T y),


⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

= max

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

dG(x, T x)+ dG(y, T x), dG(y, T y)+ dG(T 2x, T y),

dG(T x, T y)+ dG(T 2x, T y), dG(y, T y)+ dG(T x, T y),

dG(x, T x)+ dG(y, T x), dG(y, T y)+ dG(T 2x, T y),

dG(T 2x, T x)+ dG(T y, T x), dG(y, T y)+ dG(T x, T y), dG(x, y),

dG(x, T x), dG(y, T y), dG(y, T y), dG(y, T x), dG(x, T y), dG(y, T y)

⎫

⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

≤ 2 max{dG(x, T x), dG(y, T x), dG(y, T y), dG(T2x, T y),

dG(T x, T y), dG(x, y), dG(x, T y)}.Then (3.5) becomes

dG(T x, T y) ≤ 2k · max{

dG(x, T x), dG(y, T x), dG(y, T y), dG(T2x, T y),

dG(T x, T y), dG(x, y), dG(x, T y)}

.

Since 0 ≤ 2k < 1, we have

dG(T x, T y) ≤ 2k · max{

dG(x, y), dG(x, T x), dG(y, T y),

dG(x, T y), dG(y, T x), dG(T2x, T y)

}

.

By Theorem 3.2.11, we see that T has a unique fixed point. ��There are several other three-variable generalizations of metric spaces such as D∗-metric

spaces [131], G-cone metric spaces [30], G P-metric spaces [142], S-metric spaces [130],G∗-metric spaces [120], Gb-metric spaces [11,106], g-3ps [38], G pb -metric spaces [72],partial G-metric spaces [125], b2-metric spaces [105]. Similar techniques as above may beused to investigate these spaces. In particular, relations among bounded 2-metric spaces,metric-type space in the sense of Khamsi [89] and S-metric spaces [130] were stated in [56].



4 Conclusions

There exist dozens of generalizations of metric spaces and a lot of fixed point results havebeen obtained. However, not all these generalizations always produce results which are reallynew. Very often, a generalized metric space is shown to be topologically equivalent to a metricspace or a known generalized metric space. It does not automatically mean that the obtainedfixed point results on these spaces are redundant, since sometimes it is not easy to transferthe given contractive condition to the new environment. However, often this can be done,and then the results are redundant.

Hence, two important facts must be considered in this procedure:

1. Is the new generalized metric space topologically equivalent to a metric space or to apreviously known generalized metric space?

2. Can fixed point theorems on the new generalized metric space be directly obtained fromfixed point theorems on a metric space or from a known generalized metric space?

Among mentioned generalized metric spaces in this work, a 2-metric space is not topo-logically equivalent to a metric space. Moreover, fixed point theorems on 2-metric spaces,b-metric spaces and generalized metric spaces in the sense of Branciari may not be directlyobtained from the fixed point theorems on certain metric spaces. In the case of other spacesmentioned in this review, one has to be very careful and check whether in each concretesituation indeed new results are obtained.

Acknowledgments The authors sincerely thank referees for their valuable comments on revising the paper.Also, the authors sincerely thank The Dong Thap Seminar on Mathematical Analysis and Applications fordiscussing partly this article and thank Prof. S. Sedghi, Islamic Azad University, Iran; Prof. D. Wardowski,University of Łodz, Poland; Prof. P. Kumam, King Mongkut’s University of Technology Thonburi, Thai-land; Prof. Z. Mustafa, The Hashemite University, Jordan; Prof. E. Karapinar, Atilim University, Turkey fortheir helps. The third and fourth authors are grateful to Ministry of Education, Science and TechnologicalDevelopment of Serbia.

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