# On the Distance between Three Arbitrary Points

Authors: Parin Chaipunya; Poom Kumam
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We point out some equivalence between the results in  (Sedghi et al., 2012) and (Khamsi, 2010). Then, we introduce the notion of a general distance between three arbitrary points and study some of its properties. In the final section, some fixed point results are proposed.


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## Body

## 1. Introduction

The literature of a distance for any triple of points in a space was first considered during the sixties by Gähler [1, 2]. It is known as a 2-metric, the concept of which was later extended by Dhage [3] into a D-metric. Both notions are in no easy ways related to the classical concept of a metric. This led to the G-metric due to Mustafa and Sims [4].On the other hands, Sedghi et al. [5] had introduced the notion of a D*-metric, which has later been generalized by Sedghi et al. [6] into an S-metric.These developments confirm that this kind of measurement is recently of many mathematicians’ interests. One of the area that exploited these establishments largely is the fixed point theory, especially the ones involving some generalized contractions; see, for example, Sedghi et al. [5], Sedghi et al. [6], Mustafa et al. [7], Mustafa and Sims [8], Aydi et al. [9], Alghamdi and Karapinar [10], Chandok et al. [11], and Abbas et al. [12].In this present paper, we divide our interests into three parts. Firstly, we give a remark on the existing fixed point result endowed in aS-metric space. Secondly, we propose and study a very general principle in measuring the distance between three arbitrary points called a g-3ps. Thirdly, we construct some fixed point theorems by utilizing the g-3ps and its properties.


## 2. Preliminaries

This section is devoted to the recollection of important definitions and lemmas. We start with a sequence of definitions ofG-, D*-, and S-metric spaces.Definition 1 (see [4]).
LetX be a nonempty set. A function G:X×X×X→ℝ+ is said to be a G-metric if the following conditions are satisfied:(1)
forx,y,z∈X,   G(x,y,z)=0 if x=y=z;(2)
forx,y∈X with x≠y,   G(x,x,y)>0;(3)
forx,y,z∈X with y≠z,   G(x,x,y)≤G(x,y,z);(4)
forx,y,z∈X, G(x,y,z)=G(π(x,y,z)), where π(x,y,z) is any permutation of (x,y,z)∈X×X×X;(5)
forx,y,z,v∈X,   G(x,y,z)≤G(x,v,v)+G(v,y,z).
The pair(X,G) is called a G-metric space. Moreover, if G(x,x,y)=G(y,y,x) for all x,y∈X, then G is said to be symmetric.Definition 2 (see [5]).
LetX be a nonempty set. A function D*:X×X×X→ℝ+ is said to be a D*-metric if the following conditions are satisfied:(1)
forx,y,z∈X, D*(x,y,z)=0 if and only if x=y=z;(2)
forx,y,z∈X, D*(x,y,z)=D*(π(x,y,z)), where π(x,y,z) is any permutation of (x,y,z)∈X×X×X;(3)
forx,y,z,v∈X, D*(x,y,z)≤D*(x,y,v)+D*(v,z,z).
The pair(X,D*) is called a D*-metric space.Definition 3 (see [6]).
LetX be a nonempty set. A function S:X×X×X→ℝ+ is said to be a S-metric if the following conditions are satisfied:(1)
forx,y,z∈X, S(x,y,z)=0 if and only if x=y=z;(2)
forx,y,z,v∈X, S(x,y,z)≤S(x,x,v)+S(y,y,v)+S(z,z,v).
The pair(X,S) is called a S-metric space.Lemma 4 (see [6]).
Let(X,S) be a S-metric space, then S(x,x,y)=S(y,y,x) for all x,y∈X.It can be seen that each symmetricG-metric is a D*-metric and that each D*-metric is a S-metric. In case of nonsymmetric G-metric, the concepts of G-metric and S-metric are independent.Definition 5 (see [6]).
Let(X,S) be a S-metric space. For r>0 and x∈X we define the open ball BS(x;r) as follows:
(1)BS(x;r)={y∈X;S(x,x,y)<r}.
As in [6], one may consider the topology τ for X which is generated from the base containing all open balls in X. Some concepts are also introduced.Definition 6 (see [6]).
Let(X,S) be a S-metric space. A sequence (xn) in X is called(i)
Cauchy if for any ϵ>0, we may find N∈ℕ such that
(2)n∈ℕ,n≥N⟹S(xm,xm,xn)<ϵ;(ii)
convergent if there is a point x∈X in which
(3)limn→∞S(x,x,xn)=0.
Moreover, if every Cauchy sequence inX is also convergent, X is said to be complete.


## 3. AS-Metric Space as a Metric Type Space

In this section, we shall be giving a small remark on a fixed point theorem due to [6]. According to [6], a self-operator f on a S-metric space (X,S) is called a contraction if it satisfies the following inequality:
(4)S(fx,fx,fy)≤λS(x,x,y),
for all x,y∈X, where 0≤λ<1. The following result was introduced subsequently.Theorem 7 (see [6]).
Let(X,S) be a complete S-metric space and let f be a contraction on X. Then, f has a unique fixed point.To expedite our remark, we shall first recall the notion of a metric type space, which was introduced by Khamsi in [13].Definition 8 (see [13]).
LetX be a nonempty set and let D:X×X→ℝ+ be a function satisfying the following conditions:(1)
forx,y∈X,  D(x,y)=0 if and only if x=y;(2)
forx,y∈X,  D(x,y)=D(y,x);(3)
there exists a constantK>0 such that for x,y,z1,z2,…,zn∈X,
(5)D(x,y)≤K[D(x,z1)+D(z1,z2)+⋯+D(zn,y)].The triple (X,D,K) is called a metric type space.
In particular, ifK≤1, then (X,D) is a metric space. A self-mapping operator f on a metric type space (X,D,K) is called Lipschizian if there exists λ≥0 such that
(6)D(fx,fy)≤λD(x,y),
for all x,y∈X. The smallest λ>0 satisfying such condition is denoted by Lip(f). Moreover, the following fixed point theorem was proposed.Theorem 9 (see [13]).
Let(X,D,K) be a complete metric type space and let f:X→X be an operator such that the composition fn is Lipschizian for each n∈𝒩 and ∑n∈ℕ‍Lip(fn)<∞. Then, f has a unique fixed point.It follows that iff is a Lipschizian operator with Lip(f)<1, then f has a unique fixed point.Now, let(X,S) be a S-metric space. Suppose that a function D:X×X→ℝ+ is given by
(7)DS(x,y):=S(x,x,y),∀x,y∈X,
for x,y∈X, it is obvious that D(x,y)=0 if and only if x=y and D(x,y)=D(y,x) for all x,y∈X. Now, observe for each z1,z2,…,zn∈X that
(8)DS(x,y)=S(x,x,y)≤2S(x,x,z1)+S(y,y,z1)=2DS(x,z1)+DS(z1,y)≤2DS(x,z1)+2DS(z1,z2)+DS(z2,y)⋮≤2DS(x,z1)+2DS(z1,z2)+⋯+2DS(zn-1,zn)+DS(zn,y)≤2[DS(x,z1)+DS(z1,z2)ccccc+⋯+DS(zn-1,zn)+DS(zn,y)].Thus,(X,DS,2) is a metric type space. Moreover, the balls BS(x;r) and BDS(x;r) coincide.Notice that we may rewrite the inequality (4) as follows:
(9)DS(fx,fy)≤λDS(x,y),
for all x,y∈X with the same λ. Also notice that the definition of Cauchyness, convergence and completeness in a S-metric space (X,S) may be rewriten in terms of metric type spaces. So, these notions are transferred to the corresponding metric type space (X,DS,2).Now, iff is an operator satisfying (9), then each fn, where n∈ℕ, is Lipschizian with Lip(fn)=λn. Therefore, Theorem 7 is obtained via Theorem 9.Even though we set a new condition for an operatorf, where 0≤λ<1, to be
(10)S(fx,fy,fz)≤λS(x,y,z),
for each x,y,z∈X, the unique fixed point can still be obtained via Theorem 9, anyway. Note that not only the mentioned theorem, but also many theorems in the literature may be proved via this concept in metric type spaces. We shall give some results which seem more general than the setting of Theorem 7 but however equivalent.Beforehand, we give the following useful lemma without proof since it is straight forward.Lemma 10.
Let(X,S) be a S-metric space and let x,y∈X. Then, the following inequalities hold:(i)
S(x,y,y)≤S(x,x,y)=S(y,y,x);(ii)
S(x,y,x)≤S(x,x,y)=S(y,y,x).Theorem 11.
Let(X,S) be a complete S-metric space and let f:X→X be an operator such that there exists a sequence (λn) of nonnegative reals satisfying the condition:
(11)S(fx,fy,fz)≤∑n∈ℕ‍λnS(πn(x,y,z)),
for all x,y,z∈X, where Λ:=∑n∈ℕ‍λn<1 and for each n∈ℕ, πn is a fixed permutation in X3. Then, f has a unique fixed point.Proof.
We shall show that (11) implies that f is Lipschizian with Lip(f)<1 in metric type space (X,DS,2). For each x,y∈X, it is easy to verify that S(πn(x,x,y))≤S(x,x,y) no matter which permutations are defined. Thus, we obtain that
(12)DS(fx,fy)=S(fx,fx,fy)≤∑n∈ℕ‍λnS(πn(x,x,y))≤ΛS(x,x,y)=ΛDS(x,y).
Apply Theorem9 (or Theorem 7) to obtain the desired result.


## 4. A General Distance between Three Arbitrary Points

In this section, we shall be dealing with a new concept of a general distance between three arbitrary points (org-3ps).To be able to define theg-3ps, we first consider a nonempty set X together with a function g:X×X×X→ℝ+ for which g(x,y,z)=0 if and only if x=y=z. Given x∈X and r>0, we define an open ball in the usual sense:
(13)Bg(x;r)∶={y∈X;g(x,x,y)<r}.
To be natural, we say that a subset A⊂X is bounded if supx,y,z∈Ag(x,y,z)<∞. Certainly, the assertion “g(x,y,z)=0if and only ifx=y=z” is not enough to guarantee that open balls in X are bounded. We shall illustrate in the following.Example 12.
LetX∶=[0,1] and let g:X×X×X→ℝ+ be a function given by
(14)g(x,y,z)∶={0,ifx=y=z,|(x+y2)-z|,ifx,y∈ℚ∩Xbutz∈ℚ∁∩X,|(x+y2)-z|,ifx,y∈ℚ∁∩Xbutz∈ℚ∩X,1|((x+y)/2)-z|,ifx,y,z∈ℚ∩Xand[x≠yory≠z],1|((x+y)/2)-z|,ifx,y,z∈ℚ∁∩Xand[x≠yory≠z],1,otherwise.
It is clear that g(x,y,z)=0 if and only if x=y=z.
Letx∈X and let r>0. Note that if x∈ℚ∩X, then
(15)ℚ∁∩(x-r,x+r)∩X⊂Bg(x;r).
Let (zn) be a sequence in ℚ∁∩(x-r,x+r)∩X such that |x-zn|<r/(1+nr) for each n∈ℕ. Since |zn-zn+1|<2r/(1+nr), we have g(zn,zn,zn+1)>(1/2r)+(n/2) for all n∈ℕ. The same conclusion can be deduced also when x∈ℚ∁. Therefore, Bg(x;r) is not bounded at each x∈X and r>0.
This is not quite natural and does not meet the requirements we would like to have. So, we may add one more assumption at this stage and define theg-3ps space as follows.Definition 13.
LetX be a nonempty set. A function g:X×X×X→ℝ+ is said to be a g-3ps if the following conditions are satisfied:(g1)
forx,y,z∈X, g(x,y,z)=0 if and only if x=y=z;(g2)
there exists somer0>0 such that the balls Bg(x;r0) are bounded for all x∈X.The pair (X,g) is called a g-3psspace.
Next, we shall give a characterization of ag-3ps space.Lemma 14.
LetX be a nonempty set and let g:X×X×X→ℝ+ be a function satisfying (g1). Then, the following are equivalent:(i)
g satisfies (g2);(ii)
there exist some constantsδ,η>0 such that for any x,u,v,w∈X, one has
(16)g(x,x,u)+g(x,x,v)+g(x,x,w)<δ⟹g(u,v,w)<η.Proof.
[(i)⇒ (ii)] Assume that (i) holds. Set δ∶=r/2 and let x,u,v,w∈X arbitrarily. If g(x,x,u)+g(x,x,v)+g(x,x,w)<δ, then u,v,w∈Bg(x;r). Thus, setting η:=1+supx∈Xsupa,b,c∈Bg(x;r)g(a,b,c)<∞ and it follows that g(u,v,w)<η.
[(ii)⇒ (i)] Assume that (ii) holds. Let x∈X and suppose that u,v,w∈Bg(x,δ/3). Therefore, we have
(17)g(x,x,u)+g(x,x,v)+g(x,x,w)<δ.
From (ii), we obtain that g(u,v,w)<η. Thus, supa,b,c∈Bg(X;δ/3)g(a,b,c)≤η. Since x∈X is arbitrary, the balls Bg(x;δ/3) are bounded for every x∈X.Remark 15.
Suppose that(X,G) is a G-metric space and x,u,v,w∈X. Then, we have
(18)G(u,v,w)≤G(u,x,x)+G(x,v,w)=G(x,x,u)+G(v,x,w)≤G(x,x,u)+G(v,x,x)+G(x,x,w)=G(x,x,u)+G(x,x,v)+G(x,x,w).
Thus, we can choose any δ>0 and let η∶=δ to fulfill the assumptions in Lemma 14. Hence, every G-metric space is in turn a g-3ps space.Remark 16.
Suppose that(X,S) is a S-metric space and x,u,v,w∈X. Then, we have
(19)S(u,v,w)≤S(u,u,x)+S(v,v,x)+S(w,w,x)=S(x,x,u)+S(x,x,v)+S(x,x,w).
The same argument is to be considered as in the previous remark. So, a S-metric space is a g-3ps space. This immediately implies that a D*-metric space is also a g-3ps space.
Denoted by𝒰 the family of all open balls in X. Throughout this paper, we shall assume that 𝒯∶=𝒯(𝒰) represents the topology having 𝒰 as its subbase. Also, we write 𝒰* to denote the base generated by 𝒰.Remark 17.
The topology here is defined using different idea from those given in symmetric spaces or in semimetric spaces (see e.g., [14–17]).Proposition 18.
The topology𝒯 for a g-3ps space (X,g) is T1-separable.Proof.
Letx,y∈X with x≠y. So, we have g(x,x,y)=r1 and g(y,y,x)=r2, for some r1,r2>0. Observe that y∉Bg(x;r1/2) and x∉Bg(y;r2/2). The desired result is then followed.We shall now explicate an example of ag-3ps space. In particular, this next example will even show that a g-3ps space is no need to be T2-separable.Example 19.
LetX=[0,1] and define a function g:X×X×X→ℝ+ in the following:
(20)g(x,y,z)∶={0,ifx=y=z,z,ifx=y≠z,z≠0,1,otherwise.
It is clear that any subset in this space is always bounded. Hence, (X,g) is a g-3ps space. Observe that for x∈X and r>0, we have
(21)Bg(x;r)={X∩[{x}∪(0,r)],ifr≤1,X,ifr>1.
Therefore, any two balls intersect one another, implying that X is not T2-separable.
We next introduce a new concept of convergence and compare it with the classical topological ones.Definition 20.
Let(X,g) be a g-3ps space. A sequence (xn) in X is said to be(1)
Cauchy if for any ϵ>0, there exists N∈ℕ such that
(22)m,n≥N⟹g(xm,xm,xn)<ϵ;(2)
g-convergent if we can find a point x∈X in which for any ϵ>0, there exists N∈ℕ satisfying
(23)n≥N⟹g(x,x,xn)<ϵ.
In this case, we say that(xn)  g-converges to x and write xn→gx.Remark 21.
Given ag-3ps space (X,g), a sequence (xn) in X  g-converges to x∈X if and only if for any set U∈𝒰 with U∋x, there exists N∈ℕ such that
(24)n≥N⟹xn∈U.Lemma 22.
Let(X,g) be a g-3ps space and let (xn) be a sequence in X. Then, the following are equivalent:(i)
(xn) converges to x∈X in the topology 𝒯;(ii)
for any neighborhoodV∈𝒯 of x, on can find N∈ℕ such that
(25)n≥N⟹xn∈V;(iii)
for any setU∈𝒰* containing x, we can find N∈ℕ such that
(26)n≥N⟹xn∈U.Proof.
(i)⇔ (ii) is by definition. So, we only need to show that (ii) ⇔ (iii).
[(ii)⇒ (iii)] Since 𝒰*∈𝒯, we again apply (ii) to obtain our desired result.
[(iii)⇒ (ii)] By the definition of 𝒯, for every z∈F∈𝒯, we can find E∈𝒰* in which x∈E⊂F. Suppose that V∈𝒯 is a neighborhood of x, then we can find some U0∈𝒰* such that x∈U0⊂V. Applying (ii), we obtain that xn∈V for every n≥N for some fixed N∈ℕ.We conclude the following lemma immediately from Remark21 and Lemma 22.Lemma 23.
Let(X,g) be a g-3ps space and let (xn) be a sequence in X. If (xn) converges to some point x∈X in the topology 𝒯, then it also g-converges to x.This lemma shows that the concept ofg-convergence is weaker than convergence in topology. However, in case when (X,g) is a G-metric space or when (X,g) is a S-metric space, the two concepts coincide. Along this paper, we shall deal with this new kind of convergence, rather than those topological ones.Definition 24.
Ag-3ps space (X,g) is called(i)
g-Hausdorff if every g-convergent sequence g-converges to at most one point;(ii)
Σ-complete if every sequence (xn) satisfying ∑n∈ℕ‍g(xn,xn,xn+1)<∞  g-converges;(iii)
Cauchy-complete if every Cauchy sequence g-converges.Remark 25.
Ag-3ps space (X,g) is g-Hausdorff if and only if for any two distinct points x,y∈X, there exist two disjoint sets U,V∈𝒰 such that U∋x and V∋y.


## 5. Some Fixed Point Theorems

Under this section, we propose some fixed point theorems in the framework of ag-3ps space.We first introduce a lemma which will be used in our main theorems.Lemma 26.
Let(X,g) be a g-3ps space. Suppose that f:X→X be an operator such that
(27)g(fx,fy,z)≤λg(π(x,y,z)),
for all x,y,z∈X, where 0≤λ<1 and π is a fixed permutation on X3. Then, the following hold for every x∈X:(i)
∑n∈ℕ‍g(fnx,fnx,fn+1x)<∞;(ii)
∑n∈ℕ‍g(fn+1x,fn+1x,fnx)<∞.Proof.
Letx∈X be arbitrary. We shall consider the permutation π case-by-case.
(i)Case I:  π(x,y,z):=(x,y,z) or π(x,y,z):=(y,x,z).
Observe that(28)g(fnx,fnx,fn+1x)≤λg(fn-1x,fn-1x,fnx)≤λ2g(fn-2x,fn-2x,fn-1x)⋮≤λng(x,x,fx).
(ii)Case II:  π(x,y,z):=(x,z,y).
Observe that(29)g(fnx,fnx,fn+1x)≤λg(fn-1x,fnx,fn-1x)≤λ2g(fn-2x,fn-2x,fn-1x)⋮≤λnΓ1,
where ∞>Γ1≥max{g(x,x,fx),g(x,fx,x)}.
(iii)Case III:  π(x,y,z):=(y,z,x).
Observe that(30)g(fnx,fnx,fn+1x)≤λg(fn-1x,fnx,fn-1x)≤λ2g(fn-1x,fn-2x,fn-2x)≤λ3g(fn-3x,fn-3x,fn-2x)⋮≤λnΓ2,
where ∞ > Γ2 ≥ max{g(x,x,fx),f(x,fx,x),f(fx,x,x)}.
(iv)Case IV:  π(x,y,z):=(z,x,y) or π(x,y,z):=(z,y,x).
Observe that(31)g(fnx,fnx,fn+1x)≤λg(fnx,fn-1x,fn-1x)≤λ2g(fn-2x,fn-2x,fn-1x)⋮≤λnΓ1.
In each case, we may conclude that(32)∑n∈ℕ‍g(fnx,fnx,fn+1x)≤Γ2∑n∈ℕ‍λn<∞.
Similarly, we may prove that(33)∑n∈ℕ‍g(fn+1x,fn+1x,fnx)≤Γ3∑n∈ℕ‍λn<∞,
where∞>Γ3≥ max{g(fx,fx,x),g(fx,x,fx),g(x,fx,fx)}.Now, we consider our fixed point results, which exploited the above lemma.Theorem 27.
Let(X,g) be a g-Hausdorff Σ-complete g-3ps space. Suppose that f:X→X is a g-sequentially continuous operator (i.e., xn→gx⇒fxn→gfx, for every sequence (xn) in X) satisfying
(34)g(fx,fy,z)≤λg(π(x,y,z)),
for all x,y,z∈X, where 0≤λ<1 and π is a fixed permutation on X3. Then, f has exactly one fixed point.Proof.
Letx∈X be arbitrary. By (i) in Lemma 26, we have
(35)∑n∈ℕ‍g(fnx,fnx,fn+1x)<∞.
Since X is Σ-complete, (fnx)g-converges to some point x*∈X. Now, since f is g-sequentially continuous, ffnx=fn+1x→gfx*. Since X is g-Hausdorff, we have fx*=x*.
Assume thaty*∈X is also a fixed point of f. Observe that
(36)g(x*,x*,y*)=g(fx*,fx*,fy*)≤λg(x*,x*,y*).
The only possibility of the value of g(x*,x*,y*) allows us to conclude that x*=y*. So, the theorem is proved.Theorem 28.
Let(X,g) be a g-Hausdorff Cauchy-complete g-3ps space. Suppose that f:X→X is a g-sequentially continuous operator satisfying
(37)g(fx,fy,fz)≤λg(π(x,y,z)),
for all x,y,z∈X, where 0≤λ≤δ/3η and δ,η>0 are given as in Lemma 14. Then, f has exactly one fixed point.Proof.
Letx∈X. From Lemma 26, we also have g(fnx,fnx,fn+1x)→0 and g(fn+1x,fn+1x,fnx)→0. So, we may choose N∈ℕ such that
(38)g(fNx,fNx,fN+1x)<δ3,g(fN+1x,fN+1x,fNx)<δ3.
Consequently, we obtain that
(39)g(fNx,fNx,fN+1x)<η.
We will show that g(fNx,fNx,fN+nx)<η for all n∈ℕ via the mathematical induction. Now, we assume that
(40)g(fNx,fNx,fN+n0x)<η,
for some n0∈ℕ. Then, observe that
(41)2g(fN+1x,fN+1x,fNx)+g(fN+1x,fN+1x,fN+n0+1x)≤2g(fN+1x,fN+1x,fNx)+λg(fNx,fNx,fN+n0x)<2g(fN+1x,fN+1x,fNx)+λη<δ.
Hence, from Lemma 14, we have g(fNx,fNx,fN+n0+1x)<η. Therefore, we have g(fNx,fNx,fN+nx)<η for all n∈ℕ.
Letϵ>0 be given and let m,n∈ℕ with N≤m<n. Thus, we may write
(42)m=N+ℓ+p,n=N+ℓ+q,
for some ℓ,p,q∈ℕ. Note that p<q. It follows that
(43)g(fmx,fmx,fnx)=g(fN+ℓ+px,fN+ℓ+px,fN+ℓ+qx)≤λg(fN+ℓ+p-1x,fN+ℓ+p-1x,≤λgccfN+ℓ+q-1x)⋮≤λℓλpg(fNx,fNx,fN+q-px)<λℓλpη.
Since λ≤δ/3η<1, if M:=N+ℓ is chosen large enough so that λℓη<ϵ, then we ended up with g(fmx,fmx,fnx)<ϵ. Therefore, (fnx) is Cauchy. By mean of the Cauchy-completeness of X, it converges to some point x*∈X. Since f is g-sequentially continuous, we have ffnx→gfx*. Moreover, since X is g-Hausdorff, fx*=x*.
Assume thaty*∈X is also a fixed point of f. Observe that
(44)g(x*,x*,y*)=g(fx*,fx*,fy*)≤λg(x*,x*,y*).
This implies that x*=y*. Therefore, the uniqueness is proved.Example 29.
LetX=[0,1] and define a function g:X×X×X→ℝ+ by
(45)g(x,y,z)∶={0,ifx=y=z,18,ifx,y,zarenotpairwiseequalandx,y,z∈{0,1},12,ifx,y,zarenotpairwiseequaland{0,1}⊂{x,y,z}≠{0,1},18,otherwise.
It is easy to verify that (X,g) is a g-3ps space which is g-Hausdorff and Σ-complete. Note also that g is neither a G-metric nor a S-metric.
Now, let us consider the mapf:X→X given by
(46)fx∶={1,ifx=0orx=1,0,otherwise.
Obviously, f is g-sequentially continuous.
Our results (Theorems27 and 28) then guarantee the existence and uniqueness of the fixed point f1=1.


## 6. Conclusions

In this work, we pointed out that the results in [6] are obtainable through a metric type space. In addition, a S-metric generalizes a G-metric only in the case when G is symmetric. We then fill this gap by introducing a new space, namely, a g-3ps space, which covers a S-metric space and also a G-metric space in which the symmetric is absent. We also study the underlying topology for this new space in the new direction, totally different from those studied in symmetric and semimetric spaces. We lastly give the sufficient conditions for a fixed point to exist and to be unique.



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