# Equivalent Cauchy Sequences inq1,q2-Quasi Metric-Like Space and Applications to Fixed-Point Theory

Authors: Eriola Sila; Sidite Duraj; Elida Hoxha
Journal: Journal of Mathematics
 (2021)

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## Abstract

This paper introduces the concept ofq1,q2-quasi metric-like space X,p and delves into some topological properties of it. A necessary and sufficient condition related to equivalent Cauchy sequences and Cauchy sequences in X,p have been proved. Furthermore, the results of the fixed point are shown in the setting of q1,q2-quasi metric-like space as applications of conditions of equivalent Cauchy sequences. Besides, some instances are inclined to epitomize the examined consequences.


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## Body

## 1. Introduction and Preliminaries

The theory of fixed point has a crucial role not only in scientific and engineering applications [1–5] but also in theory.The development of this theory is accomplished in two directions, either by generalizing the Banach contraction or by giving new metric type spaces. There are a lot of generalizations of the metric space based on reducing or modifying the metric axioms.Some generalized metric spaces which are obtained by the above changes are,b-metric space [6–8], quasi-metric spaces, partial-metric space [9], partial-metric spaces [10, 11], dislocated metric spaces [12, 13], rectangular partial metric spaces [14], and so on.In 2012, Harandi [15] introduced the notion of metric-like space which was given as dislocated metric space in [12, 13].Later, in 2016, Arutyunov and Greshnov [16] introduced the notion of q1,q2 quasi-metric space and examined their properties. Moreover, they proved some results on fixed points in this space. In addition, they extended their results in [17, 18].Henceforth, many mathematicians have researched several masterwork involving fixed points for self-mappings in generalized metric spaces.Necessarily, a technical tool in the proof of fixed-point theorems is the Cauchy and the equivalent Cauchy sequences.In 1983, Leader [19] obtained a necessary and sufficient condition as a characterization of equivalent Cauchy sequences in the metric space. Nevertheless, the generalizations of the consequences are found in spaces, [20, 21], obtaining fixed-point results for the contractive function of Mein-Keeler type.As a consequence, this invigorated us to introduce the notion of theq1,q2-quasi metric-like space. Further, there are shown some features of topology induced by a q1,q2-quasi metric-like space.The highlight of this paper is Theorem3, which gives a necessary and sufficient condition that two sequences are equivalent Cauchy in the q1,q2-quasi metric-like space. An application of this result is Theorem 1, which extends the existing results for the Mein-Keeler contractions in the q1,q2-quasi metric-like space. One example is given to authenticate the effectiveness and applicability of our results.Amini [15] defined the metric-like space as the forthcoming definition.Definition 1 (see [15]).
LetX be a nonempty set. A metric like in X is a mapping σ:X×X⟶R+ such that for all x,y,andz∈X, it satisfies the following:(i)
σ1σx,y=0 implies x=y(ii)
σ2σx,y=σy,x(iii)
σ3σx,y≤σx,z+σz,y
The pairX,σ is called a metric-like space.
Alghamdi et al. [22] extended the concept of the metric-like space to b metric-like space as follows.Definition 2 (see [22]).
LetX be a nonempty set and s≥1 is a given real number. A function σb:X×X⟶R+ is b-metric like if for all x,y,andz∈X, the following conditions are accomplished:(i)
σb1σx,y=0 implies x=y(ii)
σb2σx,y=σy,x(iii)
σb3σx,y≤sσx,z+σz,y
The pairX,σb is called b metric-like space. The number s is called the coefficient of X,σb.
This definition was generalized by Arutyunov and Greshov [16], who gave the concepts of q1,q2-metric space and q1,q2 quasi-metric space as in Definitions 3 and 4.Definition 3 (see [16]).
LetX be a nonempty set, q1andq2 be two positive numbers, and ρx:X×X⟶R+ be a given function satisfying the following conditions for all x,y,andz∈X:(i)
ρx1ρxx,y=0 if and only if x=y(ii)
ρx2ρxx,y=ρxy,x(iii)
ρx3ρxx,y≤q1ρxx,z+q2ρxz,y
The functionρx is called q1,q2-metric. The pair X,ρx is called q1,q2-metric space.Definition 4 (see [16]).
LetX be a nonempty set, q1andq2 two positive numbers, and ρx:X×X⟶R+ be a given function that satisfies the following conditions:(i)
ρx1ρxx,y=ρxy,x=0 if and only if x=y, for each xandy∈X(ii)
ρx2ρxx,y≤q1ρxx,z+q2ρxz,y, for every x,y,andz∈X
The functionρx is called q1,q2-quasi metric, and the pair X,ρx is called q1,q2 quasi-metric space.
These definitions motivated us to define a new space and give some topological aspects of it.


## 2. (q1, q2)-Quasi Metric-Like Space and Its Topology

In this section, there is definition ofq1,q2-quasi metric-like space. Furthermore, the topology induced by q1,q2-quasi metric-like space is given and its properties are proved.Definition 5.
LetX be a set and q1andq2 be two positive numbers. The function p:X×X⟶R+ is called q1,q2-quasi metric-like if for all x,y,andz∈X, and the following conditions are satisfied:(i)
(p1) px,y=py,x=0 implies x=y(ii)
(p2) px,y≤q1px,y+q2px,y
The pairX,p is called the q1,q2-quasi metric-like space.Remark 1.
Ifq1=q2=1, as a result, the concept of q1,q2-quasi metric-like space coincides with the concept of quasi metric-like space [22].
Ifq1=q2>1, consequently, the concept of q1,q2-quasi metric-like space coincides with the concept of b-quasi metric-like space [16].
IfX,p is a q1,q2-quasi metric-like space and maxq1,q2=s>1, then X,p is a b-quasi metric-like space [22].
Ifmaxq1,q2=s≤1, as a consequence X,p is a quasi metric-like space [22].
The above facts prove that theq1,q2-quasi metric-like space is a generalization of space mentioned in [16, 22].
IfX,p is a q1,q2-quasi metric-like space, then for xandy∈X and px,y=py,x=0 implies x=y. However, the converse may not be concordant, and px,x may be positive for x∈X.Example 1.
LetX=0,1, q1=3,q2=2 and p:X×X⟶R+ be defined by(1)px,y=2,x=y,12,x<y,14,x>y.
This prompts the coupleX,p to be a q1,q2-quasi metric-like space, and px,x=2, for every x∈X.
The following definitions are given to introduce the topology induced by aq1,q2-quasi metric-like space.Definition 6.
The setBra,ε=x∈X:px,a<ε is called a right open ball with center a and radius ε>0The setBla,ε=x∈X:pa,x<ε is called a left open ball with center a and radius ε>0
It is clear-cut from Definitions5 and 6 that the center of a ball may not be an element of it. For example, in the q1,q2-quasi metric-like space of Example 2, the element 0is not in Br0,1/2 because p0,0=2>1/2.
Furthermore,a right open ballBra,ε is different from a left open ball Bla,ε. For example, in the q1,q2-quasi metric-like space given in Example 2, there is(2)Br0,13=x∈0,1:px,0<13.
Since for eachx∈0,1,px,0=1/4<1/3, there is implied Br0,1/3=X=0,1.
Additionally,Bl0,1/3=x∈0,1:p0,x<1/3, and since for every x∈X=0,1, p0,x=1/2>1/3. This leads to Bl0,1/3=ϕ.Definition 7.
The setB′ra,ε=x∈X:px,a≤ε is called a right closed ball with center a and radius ε>0The setB′la,ε=x∈X:pa,x≤ε is called a left closed ball with center a and radius ε>0Proposition 1.
LetX,p be a q1,q2-quasi metric-like space. The family τr=ϕ,X,G⊂X,for everya∈G,there existsBra,x⊂G, is a topology in X,p. It is called the right topology in X obtained from q1,q2-quasi metric-like p.
The familyτl=ϕ,X,G⊂X,for everya∈G,there existsBla,x⊂G, is a topology in X,p. It is called left topology obtained from q1,q2-quasi metric-like p.
All following propositions related to right topology can be proved using analog methods for left topology induced byq1,q2-quasi metric-like p.Definition 8.
A setA⊂X is said to be right (left) open if for every point a∈A, there exists a right (left) open ball Bra,ε⊂A,Bla,ε⊂A (this means A∈τrA∈τl).Definition 9.
A setA⊂X is said to be right (left) closed if Ac∈τrAc∈τl.Proposition 2.
Everyq1,q2-quasi metric-like space X,τr is T1.Proof.
LetX,τr be a q1,q2-quasi metric-like space. To prove that space X,τr is T1, it needs to show that, for every x∈X, the set x is right closed or the set xc is a right open.
Indeed, for everya∈xc=X−x, px,a>0. Taking ε=px,a>0, there exists Bra,ε/2⊂X−x. This is true because for every z∈Bra,ε/2, it yields pz,a<ε/2<ε=px,a. It means that z≠x; consequently, z∈X−x.Proposition 3.
The topologyτr satisfies the first axiom of countability.Proof.
The setBra,1/nn∈N forms a countable base of a.Proposition 4.
LetX,p be a q1,q2-quasi metric-like space. The family τ=τr∩τl is a topology in X,p.Proof.
Constructing the familyτ=ϕ,X,G⊂X:for eacha∈G,there existBra,ε1andBla,ε2such thata∈Bra,ε1∩Bla,ε2⊂G, we assure that it is topology.
The topologyτ is called the topology induced by q1,q2-quasi metric-like p.Proposition 5.
IfX,p is a q1,q2-quasi metric-like space, then the function dq1q2:X×X⟶R+ where dq1q2x,y=maxpx,y,py,x is a q1,q2 metric like.Proof.
This is clear sincedq1q2x,y satisfies also the symmetry condition.
The topology induced byq1,q2 metric like dq1q2is the topology τ=τr∩τl defined above.
In the following paragraphs, there are defined Cauchy sequences and equivalent Cauchy sequences inq1,q2-quasi metric-like space.Definition 10.
LetX,p be a q1,q2-quasi metric-like space and xnn∈N be a sequence in X.(1)
The sequencexnn∈N is called right Cauchy in X if for every ε>0, there exists n0∈N, such that, for every n>m>n0, pxn,xm<ε. It is denoted that limn,m⟶+∞pxn,xm=0.(2)
The sequencexnn∈N is called left Cauchy in X if for every ε>0, there exists n0∈N, such that, for every n>m>n0, pxm,xn<ε (limn,m⟶+∞pxm,xn=0).
The concepts of the right Cauchy sequence and the left Cauchy sequence are independent of each other. This means that a sequence may be right Cauchy but not left Cauchy or vice versa.
The following example illustrates it.Example 2.
LetX=0,3 and px,y=2x+y,x≥y,1,x<y.
The functionpx,y is a q1,q2-quasi metric-like with q1=1 and q2=3.
Nevertheless, it is not a quasi metric-like because forx∈X,x>y,y=2, and z=0, and we attain px,y=2x+y=2x+2, px,z=2x+z=2x,pz,y=1.
As a result,px,y=2x+2≥2x+1=px,z+pz,y.
Taking the sequencexn=1/n+1 in X, for n>m, it yields xn=1/n+1<1/m+1=xm and pxm,xn=2xm+xn=2/m+1+1/n+1<3/m+1.
Consequently,0≤limn,m⟶+∞pxm,xn≤limn,m⟶+∞3/m+1=0 and limn,m⟶+∞pxm,xn=0 which proves that the sequence xnn∈N is left Cauchy.
In addition,limn,m⟶+∞pxn,xm=limn,m⟶+∞1=1. Hereof, it yields that xnn∈N is not a right Cauchy sequence.Definition 11.
LetX,p be a q1,q2-quasi metric-like space and xnn∈N be a sequence in X. The sequence xnn∈N in X is called Cauchy if it is left Cauchy and right Cauchy.Definition 12.
LetX,p be a q1,q2-quasi metric-like space and xnn∈N be a sequence in X.(1)
The sequencexnn∈N is called right convergent to x∈Xif, for every ε>0, there exists n0∈N, such that, for every n>n0, px,xn<ε. It is denoted limn⟶+∞px,xn=0.(2)
The sequencexnn∈N is called left convergent to x∈Xif for every ε>0, there exists n0∈N, such that, for every n>n0, pxn,x<ε (limn⟶+∞pxn,x=0).(3)
The sequencexnn∈N is called convergent to x∈Xif it is left convergent and right convergent to x∈X. Consequently, for every ε>0, there exists n0∈N, such that, for every n>n0, px,xn<ε and pxn,x<ε or limn⟶+∞px,xn=limn⟶+∞pxn,x=0.
SinceX,τr and X,τl could not be T2 space, there is conceded that these spaces do not satisfy the uniqueness of the limit. As a result, a right (left) convergent sequence might converge to more than one limit.
However, spaceX,dq1q2, where dq1q2is the q1,q2 metric like obtained from q1,q2-quasi metric-like p, is T2, and the convergent sequences in X,p converge only to a unique limit.
Note that every convergent sequence inX,p is a Cauchy sequence.
Indeed, ifxnn∈N is a convergent sequence to a point a in X,p, then the following inequalities are veritable: pxn,xm≤q1pxn,a+q2pa,xm and pxm,xn≤q1pxm,a+q2pa,xn.
Consequently,limn,m⟶+∞pxn,xm=limn,m⟶+∞pxm,xn=0.Definition 13.
LetX,p be a q1,q2 metric-like space and xnn∈N and ynn∈N be two sequences in X.(1)
The sequencesxnn∈N and ynn∈N are left equivalents if for every ε>0, there exists n0∈N, such that, for every n>m>n0, pxn,ym<ε. It is denoted limn,m⟶+∞pxn,xm=0.(2)
The sequencesxnn∈N and ynn∈N are right equivalents if for every ε>0, there exists n0∈N, such that, for every n>m>n0, pyn,xm<ε (limn,m⟶+∞pyn,xm=0).(3)
The sequencesxnn∈N and ynn∈N are equivalents if they are left equivalent and right equivalent or limn,m⟶+∞pxn,xm=limn,m⟶+∞pxm,xn=0.Definition 14.
LetX,p be a q1,q2 metric-like space and xnn∈N and ynn∈N be two sequences in X.(1)
The sequencesxnn∈N and ynn∈N are left equivalent Cauchy if they are left equivalent and left Cauchy(2)
The sequencesxnn∈N and ynn∈N are right equivalent Cauchy if they are right equivalent and right Cauchy(3)
The sequencesxnn∈N and ynn∈N are equivalent Cauchy if they are left equivalent Cauchy and right equivalent Cauchy


## 3. Main Results

### 3.1. A Necessary and Sufficient Condition for Equivalent Cauchy Sequences in theq1,q2 Metric-Like Space

Leader [19] has given the following necessary and sufficient condition for the characterization of equivalent Cauchy sequences in the metric space.Theorem 1.
(see [19]). Two sequences xnn∈N and ynn∈N in metric space X,d are equivalent Cauchy if and only if for every ε>0, there exists δ∈0,+∞ and r∈N such that dxi,yj<ε+δ, implying dxi+r,yj+r<ε, for all iandj∈N.Takingyn=xn for every n∈N, Theorem 3.2.7 in [19], it gives a necessary and sufficient condition that a sequence is Cauchy in the metric space.Besides, Pasicki [20] and Hoxha and Duraj [21] have given necessary and sufficient conditions in order for a sequence xnn∈N to be Cauchy in dislocated spaces and quasi-dislocated spaces.In this section, we derive a result for equivalent Cauchy sequences and Cauchy sequences in theq1,q2 metric-like space.Theorem 2.
LetX,p be a q1,q2 metric-like space and xnn∈N and ynn∈N be two sequences in X. If the sequences xnn∈N and ynn∈N are equivalent and convergent to a and b, respectively, then a=b.Proof.
Taking limits in the following inequalities:(3)pa,b≤q1pa,xn+q2pxn,b≤q1pa,xn+q1q2pxn,yn+q22pyn,b,pb,a≤q1pb,yn+q2pyn,b≤q1pa,yn+q1q2pyn,xn+q22pxn,b,where pa,b=0=pb,a is yielded. Consequently, a=b.Theorem 3.
LetX,p be a q1,q2 metric-like space where q1>0andq2≥1and xnn∈N and ynn∈N be two sequences in X.
The sequencesxnn∈N and ynn∈N are equivalent Cauchy if and only if for every ε>0, there exists δ>0 and r∈N, such that (1) 1/q22maxpxi,yj,pyj,xi<ε+δ, implying maxqxi+r,yj+r,qyj+r,xi+r<ε, for all iandj∈N.Proof.
Denotepkn=maxpxi,yj,pyi,xj:n≤i,j≤n+k. The set pkn:n∈N, where k is fixed, is lower bound from 0. As a result, infpkn:n∈N=a≥0.
Ifa>0, then for ε=a in (1), there exist δ>0 and r∈N, such that 1/q22maxpxi,yj,pyi,xj<ε+δ=a+δ, implying maxqxi+r,yj+r,qyj+r,xi+r<ε=a.
From the characteristic property of inferior forδ>0, there exists n∈N, such that a≤pkn≤a+δ. Besides, maxpxi,yj,pyi,xj:n≤i,j≤n+k<a+δ.
Accordingly,
1/q22maxpxi,yj,pyi,xj<ε+δ=a+δ, and from (1), maxqxi+r,yj+r,qyj+r,xi+r<ε=a is yielded for every n≤iandj≤n+k meaning that pkn+r<a which is absurd.
As a result,(4)a=0,orinfpkn:n∈N=0.
For everyiandj≥n,(5)pxi,yj≤q1pxi,yn+r+q2pyn+r,yj≤q1pxi,yn+r+q1q2pyn+r,xn+r+q22pxn+r,yj.
Takingε>0, there exist δ>0 and r∈N which satisfies condition (1). Using (4), for these ε>0 and δ>0, there exists n∈N, such that(6)prn<minε,q22δq11+q2.
Suppose that there existsj≥n,(7)maxqxn+r,yj,qyj,xn+r≥ε,where j is the smallest index that satisfies this inequality.
Consequently,(8)maxpxn+r,yi,pyi,xn+r<ε,for everyn<i<j.
From (6), there is obtained prn<ε, and maxpxi,yj,pyi,xj:n≤i,j≤n+r<ε.
Ifmaxpxn+r,yj,pyj,xn+r≥ε then j>n+r. As a result, n<j−r<j Using this fact and (8), there is resulted pxn+r,yj−r<ε and pyj−r,xn+r<ε:(9)pxn,yj−r≤q1pxn,yn+q2pyn,yj−r≤q1pxn,yn+q1q2pyn,xn+r+q22pxn+r,yj−r≤q1prn+q1q2prn+q22ε≤q11+q2prn+q22ε.
Using (6), it results in(10)1q22pxn,yj−r≤q11+q2q22prn+ε<δ+ε.
Additionally,1/q22pyj−r,xn<δ+ε and 1/q22maxpxn,yj−r,pyj−r,xn<δ+ε. Using (1), is obtained, which contradicts (7).
Hence, for everyi,j≥n,maxqxn+r,yj,qyj,xn+r<ε. Considering (5) for every i,j≥n, we obtain(11)pxi,yj≤q1ε+q1q2ε+q22ε=q1+q1q2+q22ε.
Moreover,pyj,xi≤q1+q1q2+q22ε, resulting in(12)limi,j⟶+∞pxi,yj=limi,j⟶+∞pyj,xi=0.
Subsequently, the sequencesxii∈N and yjj∈N are equivalent.
The next step is to show that each of them is Cauchy.
Indeed,pxi,xj≤q1pxi,yj+q2pyj,xj and pyi,yj≤q1pyi,xi+q2pxi,yj.
Taking limits in the above inequalities results inlimi,j⟶+∞pxi,xj=limi,j⟶+∞pyi,yj=0.
Using the same method, it can be proved thatlimi,j⟶+∞pxj,xi=limi,j⟶+∞pyj,yi=0.
Thus, the sequencesxii∈N and yjj∈N are Cauchy.
Conversely, ifxnn∈N and ynn∈N are two equivalent Cauchy sequences, using Definitions 10, 11, and 12, then for every ε>0, there exist n0∈N and r∈N, such that maxqxi+r,yj+r,qyj+r,xi+r<ε, for iandj≥n0.
This assures that condition (1) is accomplished forδ=+∞.Corollary 1.
LetX,p be a q1,q2-quasi metric-like space, where q1>0andq2≥1. The sequence xnn∈N in X,p is Cauchy if and only if for every ε>0, there exist δ>0 and r∈N,such that 1/q22maxpxi,xj,pxj,xi<ε+δ implying maxpxi+r,xj+r,pxj+r,xi+r<ε.Proof.
Replacingyj=xi in Theorem 3, the corollary is true.Remark 2.
Corollary1 generalizes Lemma 2.3 and Corollary 6.9 in [20] for the q1,q2-quasi metric-like space.


### 3.2. Fixed-Point Results in theq1,q2-Quasi Metric-Like Space

Many authors have given fixed-point theorems for the contractive function of the Mein-Keeler type in different spaces [23–25]. In this section, we acquire a fixed-point result related to the Mein-Keeler type contraction in the q1,q2-quasi metric-like space.Theorem 4.
LetX,p be a q1,q2-quasi metric-like space where q1>0andq2≥1. The following propositions are equivalent.(1)
The mappingT:X⟶X has a fixed point, and for every x∈X, the iterative sequences Tnxn∈N converge to the fixed point of T(2)
For everyxandy∈X and for each ε>0, there exist δ>0 and r∈N, such that 1/q22maxpTix,Tjy;pTjy,Tix<ε+δ implying
maxpTi+rx,Tj+ry,pTj+ry,Ti+rx<ε.Proof.
The first step is to prove that condition (1) implies condition (2).
Letz∈X be the unique fixed point of mapping T, Tz=z, and for every xandy∈X, the sequences Tixi∈N and Tiyi∈N converge to z∈X.
Using the definition of convergence inX, it yielded the following:(13)limi⟶+∞pTix,z=limi⟶+∞pz,Tix=limi⟶+∞pTiy,z=limi⟶+∞pz,Tiy=0.
From the definition ofq1,q2-quasi metric-like, it is obtained that(14)pTix,Tiy≤q1pTix,z+q2pz,Tiy,pTiy,Tix≤q1pTiy,z+q2pz,Tix.
Taking limit wheni⟶+∞ and using (5), it is taken that(15)limi⟶+∞pTix,Tiy=limi⟶+∞pTiy,Tix=0.
Significantly, the sequencesTixi∈N and Tiyi∈N are equivalent.
Since the sequencesTixi∈N and Tiyi∈Nare convergent, therefore they are Cauchy. Consequently, the sequences Tixi∈N and Tiyi∈N are equivalent Cauchy. From Theorem 3, condition (2) is yielded.
The next step is to prove that condition (2) implies condition (1).
Letxbe a point in X and xi=Tixi∈N be an iterative sequence in X. If there exists i0∈N such that xi0=xi0+1=Txi0, then xi0 is a fixed point of T.
Suppose that, for alli∈N, xi≠xi+1. Since T satisfies condition (2) and replacing x=y in (2), we obtain that, for every ε>0, there exist δ>0 and r∈N, such that(16)1q22maxpTix,Tjx;pTjx,Tix<ε+δ,implying maxpTi+rx,Tj+rx,pTj+rx,Ti+rx<ε,(17)or1q22maxpxi+1,xj+1;pxj+1,xi+1<ε+δ,implying maxpxi+r+1,xj+r+1,pxj+r+1,xi+r+1<ε.
Consequently, the sequencexii∈N satisfies the condition of Corollary 1. So, xii∈N is the Cauchy sequence in X. Since X,p is complete, then there exists z∈X, such that limi⟶+∞xi+1=limi⟶+∞Tix=z.
AsT satisfies condition (2), then for xandz∈X, the sequences Tixi∈N and Tizi∈N are equivalent Cauchy.
Due tolimi⟶+∞Tix=z from Theorem 2, limi⟶+∞Tix=limi⟶+∞Tiz=z is yielded.
Using condition (2) forxandz∈X, the following results are obtained.
For everyε>0, there exist δ>0 and r∈N, such that 1/q22maxpTix,Tjz;pTjz,Tix<ε+δ, implying maxpTi+rx,Tj+rz,pTj+rz,Ti+rx<ε (6).
Sincelimi⟶+∞Tix=z, for ε+δ>0, there exists p∈N, such that, for every i>p,pTix,z<ε+δ and pz,Tix<ε+δ are obtained.
Consequently,maxpTix,z,pz,Tix<ε+δ.
Furthermore, due to the fact thatq2≥1, it is implied that(18)1q22maxpTix,z;pz,Tix<ε+δ.
Considering (6), it is yielded for ε>0, there exists p∈N and r∈N such that, for every i>p it results in(19)maxpTi+rx,Trz,pTrz,Tix<ε.
This means thatlimi⟶+∞pTi+rx,Trz=limi⟶+∞pTrz,Ti+rx=0.
In addition,limi⟶+∞pTi+rx,Trz=pz,Trz and limi⟶+∞pTrz,Ti+rx=pTrz,z=0.
Consequently,Trz=z.
SinceTr+1z=TTrz=Tz and limi⟶+∞Tiz=z, then zis the fixed point of T, and for every x∈X, the iterative sequences Tixi∈N converge to the fixed point of T.
An example to illustrate the validity of Theorem1 is introduced below.Example 3.
LetX=0,1,2 and p:X×X⟶R+, where p0,0=0,p0,1=1,p0,2=1,p1,1=3,p1,0=2,p1,2=1,p2,2=6,p2,1=5,andp2,0=4.
The coupleX,p is a q1,q2-quasi metric-like where q1=5/4>1 and q2=3/2>1.
Define the functionT:X⟶X, such that T0=T1=0 and T2=1.
Fori=j=0, the function T accomplishes condition 2 of Theorem 1.Case 1.
x=0 and y=1 such that(20)px,y=p0,1=1,py,x=p1,0=2.
SinceT0=0,T20=TT0=T0=0 and T1=0,T21=TT1=T0=0, Tr0=0 and Tr1=0 are obtained for each r∈N.
For every0<ε<8/9, there exist δ>8/9−εandr∈N, such that(21)1q22maxp0,1,p1,0=19/4max1,2=49⋅2=89<ε+δ,and maxpTr0,Tr1,pTr1,Tr0=max0,0=0<ε.Case 2.
x=0 and y=2 such that(22)px,y=p0,2=1,py,x=p2,0=4,where Tr0=0, for each r∈N. Since Ty=T2=1andT2y=T22=TT2=T1=0, Tr2=0 is obtained as a result, for each r≥2.
For every0<ε<16/9, there exist δ>16/9−ε>0andr∈Nwherer≥2, such that(23)1q22maxp0,1,p1,0=49max1,4=49⋅4=169<ε+δ,and maxpTrx,Try,pTry,Trx=0<ε.Case 3.
x=1 and y=2 such that(24)px,y=p1,2=1,py,x=p2,1=5,and Tr1=0, for each r∈N, and Tr2=0, for each r≥2.
For every0<ε<2, there exist δ>20/9−ε>0andr∈Nwherer≥2, such that(25)1q22maxpx,y,py,x=49max1,5=49⋅5=209<ε+δ,and maxpTrx,Try,pTry,Trx=0<ε.
It is discernible that the functionT accomplishes the conditions of Theorem 1.
Thus, it has a fixed pointx=0. Since pTnx,0=p0,0=0 and p0,Tnx=p0,0=0 for every n≥2, then for each x∈X, the sequence Tnxn∈N is constant and converges to 0.Remark 3.
Condition (2) in Theorem1 is the contractive condition of the Meir-Keeler type [26]. As a result, Theorem 1 is an application of Mein-Keeler contraction in the metric spaces mentioned in [16, 20, 22], which are generalized by the q1,q2-quasi metric-like space.


## 3.1. A Necessary and Sufficient Condition for Equivalent Cauchy Sequences in theq1,q2 Metric-Like Space

Leader [19] has given the following necessary and sufficient condition for the characterization of equivalent Cauchy sequences in the metric space.Theorem 1.
(see [19]). Two sequences xnn∈N and ynn∈N in metric space X,d are equivalent Cauchy if and only if for every ε>0, there exists δ∈0,+∞ and r∈N such that dxi,yj<ε+δ, implying dxi+r,yj+r<ε, for all iandj∈N.Takingyn=xn for every n∈N, Theorem 3.2.7 in [19], it gives a necessary and sufficient condition that a sequence is Cauchy in the metric space.Besides, Pasicki [20] and Hoxha and Duraj [21] have given necessary and sufficient conditions in order for a sequence xnn∈N to be Cauchy in dislocated spaces and quasi-dislocated spaces.In this section, we derive a result for equivalent Cauchy sequences and Cauchy sequences in theq1,q2 metric-like space.Theorem 2.
LetX,p be a q1,q2 metric-like space and xnn∈N and ynn∈N be two sequences in X. If the sequences xnn∈N and ynn∈N are equivalent and convergent to a and b, respectively, then a=b.Proof.
Taking limits in the following inequalities:(3)pa,b≤q1pa,xn+q2pxn,b≤q1pa,xn+q1q2pxn,yn+q22pyn,b,pb,a≤q1pb,yn+q2pyn,b≤q1pa,yn+q1q2pyn,xn+q22pxn,b,where pa,b=0=pb,a is yielded. Consequently, a=b.Theorem 3.
LetX,p be a q1,q2 metric-like space where q1>0andq2≥1and xnn∈N and ynn∈N be two sequences in X.
The sequencesxnn∈N and ynn∈N are equivalent Cauchy if and only if for every ε>0, there exists δ>0 and r∈N, such that (1) 1/q22maxpxi,yj,pyj,xi<ε+δ, implying maxqxi+r,yj+r,qyj+r,xi+r<ε, for all iandj∈N.Proof.
Denotepkn=maxpxi,yj,pyi,xj:n≤i,j≤n+k. The set pkn:n∈N, where k is fixed, is lower bound from 0. As a result, infpkn:n∈N=a≥0.
Ifa>0, then for ε=a in (1), there exist δ>0 and r∈N, such that 1/q22maxpxi,yj,pyi,xj<ε+δ=a+δ, implying maxqxi+r,yj+r,qyj+r,xi+r<ε=a.
From the characteristic property of inferior forδ>0, there exists n∈N, such that a≤pkn≤a+δ. Besides, maxpxi,yj,pyi,xj:n≤i,j≤n+k<a+δ.
Accordingly,
1/q22maxpxi,yj,pyi,xj<ε+δ=a+δ, and from (1), maxqxi+r,yj+r,qyj+r,xi+r<ε=a is yielded for every n≤iandj≤n+k meaning that pkn+r<a which is absurd.
As a result,(4)a=0,orinfpkn:n∈N=0.
For everyiandj≥n,(5)pxi,yj≤q1pxi,yn+r+q2pyn+r,yj≤q1pxi,yn+r+q1q2pyn+r,xn+r+q22pxn+r,yj.
Takingε>0, there exist δ>0 and r∈N which satisfies condition (1). Using (4), for these ε>0 and δ>0, there exists n∈N, such that(6)prn<minε,q22δq11+q2.
Suppose that there existsj≥n,(7)maxqxn+r,yj,qyj,xn+r≥ε,where j is the smallest index that satisfies this inequality.
Consequently,(8)maxpxn+r,yi,pyi,xn+r<ε,for everyn<i<j.
From (6), there is obtained prn<ε, and maxpxi,yj,pyi,xj:n≤i,j≤n+r<ε.
Ifmaxpxn+r,yj,pyj,xn+r≥ε then j>n+r. As a result, n<j−r<j Using this fact and (8), there is resulted pxn+r,yj−r<ε and pyj−r,xn+r<ε:(9)pxn,yj−r≤q1pxn,yn+q2pyn,yj−r≤q1pxn,yn+q1q2pyn,xn+r+q22pxn+r,yj−r≤q1prn+q1q2prn+q22ε≤q11+q2prn+q22ε.
Using (6), it results in(10)1q22pxn,yj−r≤q11+q2q22prn+ε<δ+ε.
Additionally,1/q22pyj−r,xn<δ+ε and 1/q22maxpxn,yj−r,pyj−r,xn<δ+ε. Using (1), is obtained, which contradicts (7).
Hence, for everyi,j≥n,maxqxn+r,yj,qyj,xn+r<ε. Considering (5) for every i,j≥n, we obtain(11)pxi,yj≤q1ε+q1q2ε+q22ε=q1+q1q2+q22ε.
Moreover,pyj,xi≤q1+q1q2+q22ε, resulting in(12)limi,j⟶+∞pxi,yj=limi,j⟶+∞pyj,xi=0.
Subsequently, the sequencesxii∈N and yjj∈N are equivalent.
The next step is to show that each of them is Cauchy.
Indeed,pxi,xj≤q1pxi,yj+q2pyj,xj and pyi,yj≤q1pyi,xi+q2pxi,yj.
Taking limits in the above inequalities results inlimi,j⟶+∞pxi,xj=limi,j⟶+∞pyi,yj=0.
Using the same method, it can be proved thatlimi,j⟶+∞pxj,xi=limi,j⟶+∞pyj,yi=0.
Thus, the sequencesxii∈N and yjj∈N are Cauchy.
Conversely, ifxnn∈N and ynn∈N are two equivalent Cauchy sequences, using Definitions 10, 11, and 12, then for every ε>0, there exist n0∈N and r∈N, such that maxqxi+r,yj+r,qyj+r,xi+r<ε, for iandj≥n0.
This assures that condition (1) is accomplished forδ=+∞.Corollary 1.
LetX,p be a q1,q2-quasi metric-like space, where q1>0andq2≥1. The sequence xnn∈N in X,p is Cauchy if and only if for every ε>0, there exist δ>0 and r∈N,such that 1/q22maxpxi,xj,pxj,xi<ε+δ implying maxpxi+r,xj+r,pxj+r,xi+r<ε.Proof.
Replacingyj=xi in Theorem 3, the corollary is true.Remark 2.
Corollary1 generalizes Lemma 2.3 and Corollary 6.9 in [20] for the q1,q2-quasi metric-like space.


## 3.2. Fixed-Point Results in theq1,q2-Quasi Metric-Like Space

Many authors have given fixed-point theorems for the contractive function of the Mein-Keeler type in different spaces [23–25]. In this section, we acquire a fixed-point result related to the Mein-Keeler type contraction in the q1,q2-quasi metric-like space.Theorem 4.
LetX,p be a q1,q2-quasi metric-like space where q1>0andq2≥1. The following propositions are equivalent.(1)
The mappingT:X⟶X has a fixed point, and for every x∈X, the iterative sequences Tnxn∈N converge to the fixed point of T(2)
For everyxandy∈X and for each ε>0, there exist δ>0 and r∈N, such that 1/q22maxpTix,Tjy;pTjy,Tix<ε+δ implying
maxpTi+rx,Tj+ry,pTj+ry,Ti+rx<ε.Proof.
The first step is to prove that condition (1) implies condition (2).
Letz∈X be the unique fixed point of mapping T, Tz=z, and for every xandy∈X, the sequences Tixi∈N and Tiyi∈N converge to z∈X.
Using the definition of convergence inX, it yielded the following:(13)limi⟶+∞pTix,z=limi⟶+∞pz,Tix=limi⟶+∞pTiy,z=limi⟶+∞pz,Tiy=0.
From the definition ofq1,q2-quasi metric-like, it is obtained that(14)pTix,Tiy≤q1pTix,z+q2pz,Tiy,pTiy,Tix≤q1pTiy,z+q2pz,Tix.
Taking limit wheni⟶+∞ and using (5), it is taken that(15)limi⟶+∞pTix,Tiy=limi⟶+∞pTiy,Tix=0.
Significantly, the sequencesTixi∈N and Tiyi∈N are equivalent.
Since the sequencesTixi∈N and Tiyi∈Nare convergent, therefore they are Cauchy. Consequently, the sequences Tixi∈N and Tiyi∈N are equivalent Cauchy. From Theorem 3, condition (2) is yielded.
The next step is to prove that condition (2) implies condition (1).
Letxbe a point in X and xi=Tixi∈N be an iterative sequence in X. If there exists i0∈N such that xi0=xi0+1=Txi0, then xi0 is a fixed point of T.
Suppose that, for alli∈N, xi≠xi+1. Since T satisfies condition (2) and replacing x=y in (2), we obtain that, for every ε>0, there exist δ>0 and r∈N, such that(16)1q22maxpTix,Tjx;pTjx,Tix<ε+δ,implying maxpTi+rx,Tj+rx,pTj+rx,Ti+rx<ε,(17)or1q22maxpxi+1,xj+1;pxj+1,xi+1<ε+δ,implying maxpxi+r+1,xj+r+1,pxj+r+1,xi+r+1<ε.
Consequently, the sequencexii∈N satisfies the condition of Corollary 1. So, xii∈N is the Cauchy sequence in X. Since X,p is complete, then there exists z∈X, such that limi⟶+∞xi+1=limi⟶+∞Tix=z.
AsT satisfies condition (2), then for xandz∈X, the sequences Tixi∈N and Tizi∈N are equivalent Cauchy.
Due tolimi⟶+∞Tix=z from Theorem 2, limi⟶+∞Tix=limi⟶+∞Tiz=z is yielded.
Using condition (2) forxandz∈X, the following results are obtained.
For everyε>0, there exist δ>0 and r∈N, such that 1/q22maxpTix,Tjz;pTjz,Tix<ε+δ, implying maxpTi+rx,Tj+rz,pTj+rz,Ti+rx<ε (6).
Sincelimi⟶+∞Tix=z, for ε+δ>0, there exists p∈N, such that, for every i>p,pTix,z<ε+δ and pz,Tix<ε+δ are obtained.
Consequently,maxpTix,z,pz,Tix<ε+δ.
Furthermore, due to the fact thatq2≥1, it is implied that(18)1q22maxpTix,z;pz,Tix<ε+δ.
Considering (6), it is yielded for ε>0, there exists p∈N and r∈N such that, for every i>p it results in(19)maxpTi+rx,Trz,pTrz,Tix<ε.
This means thatlimi⟶+∞pTi+rx,Trz=limi⟶+∞pTrz,Ti+rx=0.
In addition,limi⟶+∞pTi+rx,Trz=pz,Trz and limi⟶+∞pTrz,Ti+rx=pTrz,z=0.
Consequently,Trz=z.
SinceTr+1z=TTrz=Tz and limi⟶+∞Tiz=z, then zis the fixed point of T, and for every x∈X, the iterative sequences Tixi∈N converge to the fixed point of T.
An example to illustrate the validity of Theorem1 is introduced below.Example 3.
LetX=0,1,2 and p:X×X⟶R+, where p0,0=0,p0,1=1,p0,2=1,p1,1=3,p1,0=2,p1,2=1,p2,2=6,p2,1=5,andp2,0=4.
The coupleX,p is a q1,q2-quasi metric-like where q1=5/4>1 and q2=3/2>1.
Define the functionT:X⟶X, such that T0=T1=0 and T2=1.
Fori=j=0, the function T accomplishes condition 2 of Theorem 1.Case 1.
x=0 and y=1 such that(20)px,y=p0,1=1,py,x=p1,0=2.
SinceT0=0,T20=TT0=T0=0 and T1=0,T21=TT1=T0=0, Tr0=0 and Tr1=0 are obtained for each r∈N.
For every0<ε<8/9, there exist δ>8/9−εandr∈N, such that(21)1q22maxp0,1,p1,0=19/4max1,2=49⋅2=89<ε+δ,and maxpTr0,Tr1,pTr1,Tr0=max0,0=0<ε.Case 2.
x=0 and y=2 such that(22)px,y=p0,2=1,py,x=p2,0=4,where Tr0=0, for each r∈N. Since Ty=T2=1andT2y=T22=TT2=T1=0, Tr2=0 is obtained as a result, for each r≥2.
For every0<ε<16/9, there exist δ>16/9−ε>0andr∈Nwherer≥2, such that(23)1q22maxp0,1,p1,0=49max1,4=49⋅4=169<ε+δ,and maxpTrx,Try,pTry,Trx=0<ε.Case 3.
x=1 and y=2 such that(24)px,y=p1,2=1,py,x=p2,1=5,and Tr1=0, for each r∈N, and Tr2=0, for each r≥2.
For every0<ε<2, there exist δ>20/9−ε>0andr∈Nwherer≥2, such that(25)1q22maxpx,y,py,x=49max1,5=49⋅5=209<ε+δ,and maxpTrx,Try,pTry,Trx=0<ε.
It is discernible that the functionT accomplishes the conditions of Theorem 1.
Thus, it has a fixed pointx=0. Since pTnx,0=p0,0=0 and p0,Tnx=p0,0=0 for every n≥2, then for each x∈X, the sequence Tnxn∈N is constant and converges to 0.Remark 3.
Condition (2) in Theorem1 is the contractive condition of the Meir-Keeler type [26]. As a result, Theorem 1 is an application of Mein-Keeler contraction in the metric spaces mentioned in [16, 20, 22], which are generalized by the q1,q2-quasi metric-like space.


## 4. Conclusions

This scrutiny elucidates theq1,q2-quasi metric-like space and enunciates some topological properties of it. Some compelling results are acquired correlated with equivalent Cauchy sequence and Cauchy sequences (Theorem 3, Corollary 1), by generalizing Lemma 2.3 and Corollary 6.9 in [20] for q1,q2-quasi metric-like spaces.In addition, the existence of fixed-point theorem for the Meir-Keeler type contractions inq1,q2-quasi metric-like spaces is attained. Moreover, some of the provided examples ascertain the application of the fixed-point theorem and the topological attributions. Accordingly, all obtained results are veridical in q1,q2 quasi-metric spaces, b-quasi metric-like spaces, q1,q2-metric spaces since the q1,q2-quasi metric-like space generalizes the cited spaces.



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*Source: 5542787-2021-05-06.xml*