# Common Best Proximity Point Theorems in JS-Metric Spaces Endowed with Graphs

Authors: Chaiporn Thangthong; Phakdi Charoensawan; Supreedee Dangskul; Narawadee Phudolsitthiphat
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce a notion ofG-proximal edge preserving and dominating G-proximal Geraghty for a pair of mappings, which will be used to present some existence and uniqueness results for common best proximity points. Here, the mappings are defined on subsets of a JS-metric space endowed with a directed graph. An example is also provided to support the results. Moreover, we apply our result to a similar setting, where the JS-metric space is endowed with a binary relation.


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## Body

## 1. Introduction

Problems concerning objects that remain unchanged have been of great interest in sciences. Geneticists, for instance, have discovered that gene mutations may be delayed by raising the number of DNAs kept unaltered under exposure and hence cancer prevention. Mathematicians may interpret such objects in their own environment as points kept fixed under self-mapping, known as fixed points.Theory of fixed points and their related notions have been expansively explored, not only in pure mathematics itself but also in real-world problems, where optimal solutions are sought. There have been a large number of publications that contribute to the subject in various approaches (see [1, 2] for some fixed point theorems and see [3–9] for results regarding best proximity points and pairs, to mention but a few). The reader may also be referred to [10] for some applications of the theory in economics.Best proximity and common best proximity points, in particular, have become one of the most studied topics in the field of fixed point theory. These notions generalize fixed points and allow us to deal with nonself-mappings. Several settings and techniques have been used in order to determine which circumstances a best (common best) proximity point can be guaranteed. Hussain and his coauthors are amongst those who have actively contributed results to this research area (see [11, 12], where different kinds of contractions are employed; see [13, 14], where generalized notions of metric spaces are considered; and see [15] for best proximity results in nonlinear dynamical systems). There are many more results on common best proximity points in the literature (see [16–20] for some of the key works).One of the most popular research approaches in the theory of fixed points is to appropriately adjust mappings that control the distance between two points. A well-known result by S. Banach, known as Banach contraction principle, gives rise to a variety of modifications. Instead of contractions, in [21], Ayari considered a new class of mappings containing all θ:0,+∞→0,1 with property that
(1)limn→+∞θtn=1implieslimn→+∞tn=0,which generalizes contractions and also Geraghty’s work [22]. With this generalization, existence and uniqueness results for best proximity points in a closed subset of a complete metric space can be established. A recent work by Khemphet et al. [20] also benefits from this class of mappings—a notion of dominating proximal generalized Geraghty property of a pair of mappings is presented for some existence and uniqueness results of common best proximity coincidence points in complete metric spaces, improving Chen’s work [19].One may try to control the distance between two points in a metric space using directed graphs. This idea was first introduced by Jachymski [23]. Given a directed graph G=V,E, the set of edges E is contained in the Cartesian product V×V. If V is also a metric space, one could impose some conditions on the distance between points x and y, provided that x,y∈E. There have been a number of articles employing this graph-like theoretic approach (see [24, 25] for some fixed point theorems, see [26–28] for some results on Hilbert spaces, and see [29–31] for some assertions regarding common fixed points, coincidence points, and best proximity points).This paper is aimed at establishing some common best proximity point theorems (Theorems12 and 13) on a more general setting, the so-called JS-metric spaces, introduced in [32]. Our space will also be endowed with a directed graph G. More specifically, the main theorems rely on two key assumptions that a pair of mappings is G-proximal edge preserving and dominating G-proximal Geraghty. To the best of our knowledge, this study approach has not yet been investigated.The paper is organized as follows. Section2 collects basic definitions and facts regarding JS-metric spaces, common proximity points, and spaces endowed with directed graphs. Section 3 presents our main results and some example. Section 4 provides some consequences of our main theorems by concretely selecting Geraghty-like functions. Last but not least, Section 5 is devoted for an application of our results for a pair of mappings between subsets of a JS-metric space endowed with a binary relation.


## 2. Preliminaries

### 2.1. JS-Metric Spaces

In [32], a weaker notion of metrics was introduced by Jleli and Samet, known as JS-metrics. These generalized metrics lack the triangle inequality, which somewhat ruins the intuition of distance. It turns out, however, that various types of topological spaces are JS-metric spaces.LetX be a nonempty set, and let D:X×X→0,+∞ be a function. For each x∈X, let us define
(2)CD,X,x=xn⊆X:limn→+∞Dxn,x=0to be the set of sequences in X that converges to x, with respect to D.Definition 1 (see [32]).
LetX be a nonempty set. A function D:X×X→0,+∞ is called JS-metric on a set X if it satisfies the following conditions:
JS1 For any x,y∈X, Dx,y=0 implies x=y
JS2 For any x,y∈X, Dx,y=Dy,x
JS3 There is a constant CX>0 such that
(3)Dx,y≤CXlimsupn→+∞Dxn,y,whenever x,y∈X and xn∈CD,X,x.
The pairX,D is called a JS-metric space.
ConditionsJS1 and JS3 imply the following fact.Proposition 2 (see [33]).
LetX,D be a JS-metric space and x∈X. If CD,X,x≠∅, then Dx,x=0.
As in a metric space, convergence and completeness for a JS-metric space can be defined in a similar way.Definition 3 (see [32]).
LetX,Dbe a JS-metric space andxna sequence inX.
(i)
xnD-converges to x if xn∈CD,X,x(ii)
xn is called a D-Cauchy sequence if limm,n→+∞Dxm,xn=0(iii)
The spaceX,D is said to be complete if every D-Cauchy sequence is D-convergent
In a metric space, triangle inequality forces any convergent sequence to have a unique limit. Here, the conditionJS3 plays that role.Proposition 4 (see [32]).
LetX,Dbe a JS-metric space, and letxnbe a sequence inX. For anyx,y∈X, ifxn∈CD,X,x∩CD,X,y, thenx=y.
Pointwise continuity can then be defined, using convergence of sequences.Definition 5.
LetX,Dbe a JS-metric space. A mappingP:X→Xis said to be continuous at a pointx0∈Xifxn∈CD,X,x0impliesPxn∈CD,X,Px0.In addition,Pis said to be continuous if it is continuous at eachxinX.


### 2.2. Common Proximity

Throughout the paper, for nonempty subsetsA and B of a JS-metric space X,D, the following notations will be used:
(4)DA,B≔infDa,b:a∈A,b∈B,A0≔a∈A:thereexistsb∈BsuchthatDa,b=DA,B,B0≔b∈B:thereexistsa∈AsuchthatDa,b=DA,B.Notice thatA0 and B0 can be empty or even undefined. If DA,B=+∞ and Da,b=+∞, then it is unclear whether a∈A0. Throughout the paper, the distance DA,B will be always assumed to be finite.For nonempty subsetsA and B of a JS-metric space X,D, let us recall that a best proximity point of a mapping T:A→B is a point x∗∈A with Dx∗,Tx∗=DA,B. If DA,B=0, then x∗ becomes a fixed point. The term “common” comes into play when we deal with a pair of mappings.Definition 6 (see [18]).
LetP,Q:A→Bbe mappings. An elementx∗∈Ais said to be a common best proximity point of the pairP,Qif(5)Dx∗,Qx∗=DA,B=Dx∗,Px∗.
The set of common best proximity points ofP and Q is denoted by CPP,Q.
Observe also that ifDA,B=0, then x∗ becomes a common fixed point of the pair P,Q.Definition 7 (see [17]).
LetP,Q:A→B be mappings. A pair P,Q is said to commute proximally if for each x,u,v∈A,
(6)Du,Qx=DA,B=Dv,PximpliesPu=Qv.


### 2.3. Graph Endowment

Given a nonempty setX, a directed graph G=VG,EG will be constructed as follows.
(i)
The set of verticesVG is the set X itself(ii)
The set of edgesEG⊆X×X contains all the loops; that is, x,x∣x∈X⊆EG(iii)
EG contains no parallel edgesWe say thatX is said to be endowed with a directed graphG=VG,EG.LetX,D be a JS-metric space endowed with a directed graph G=VG,EG, denoted by X,D,G. We next introduce a notion of continuity with respect to the graph G.Definition 8 (see [23]).
A mappingP:X→Xis calledG-continuous atx0∈Xif for any sequencexninXwithxn,xn+1∈EGandxn∈CD,X,x0, it follows thatPxn∈CD,X,Px0.
Notice thatG-continuity is a weaker notion than usual continuity, with respect to the JS-metric. In other words, any continuous mapping on a space endowed with a directed graph G is G-continuous. A counterexample of the converse is easily found, when G has an isolated vertex, for example.
Let us now introduce some more terminology used in our main results.Definition 9.
LetA,Bbe nonempty subsets ofX,D,GandP,Q:A→Bbe mappings. A pairP,Qis said to beG-proximal edge preserving if the following hold:(i)
IfPx,Py∈EG, Qx,Qy∈EG(ii)
For anyx,u,v∈A with Px,Qx∈EG and(7)Du,Qx=DA,B=Dv,Px,it follows that v,u∈EGDefine the classΓ of functions to be
(8)Γ=γ:0,+∞→0,1∣limn→+∞γtn=1implieslimn→+∞tn=0.This extends the class of functions in [19, 21]. Notice that any function in γ∈Γ has a property that
(9)γt=1impliest=0.Definition 10.
LetA,Bbe nonempty subsets ofX,D,GandP,Q:A→Bbe mappings. A pairP,Qis said to be dominatingG-proximal Geraghty if there exists a functionγ∈Γsuch that for eachx1,x2,u1,u2,v1,v2∈Asatisfying(10)Px1,Qx1∈EG,Px2,Qx2∈EG,Du1,Qx1=Du2,Qx2=DA,B=Dv1,Px1=Dv2,Px2,it follows that
(11)Du1,u2≤γMv1,v2,u1,u2Mv1,v2,u1,u2,where Mv1,v2,u1,u2=maxDv1,v2,Dv1,u1,Dv2,u2.


## 2.1. JS-Metric Spaces

In [32], a weaker notion of metrics was introduced by Jleli and Samet, known as JS-metrics. These generalized metrics lack the triangle inequality, which somewhat ruins the intuition of distance. It turns out, however, that various types of topological spaces are JS-metric spaces.LetX be a nonempty set, and let D:X×X→0,+∞ be a function. For each x∈X, let us define
(2)CD,X,x=xn⊆X:limn→+∞Dxn,x=0to be the set of sequences in X that converges to x, with respect to D.Definition 1 (see [32]).
LetX be a nonempty set. A function D:X×X→0,+∞ is called JS-metric on a set X if it satisfies the following conditions:
JS1 For any x,y∈X, Dx,y=0 implies x=y
JS2 For any x,y∈X, Dx,y=Dy,x
JS3 There is a constant CX>0 such that
(3)Dx,y≤CXlimsupn→+∞Dxn,y,whenever x,y∈X and xn∈CD,X,x.
The pairX,D is called a JS-metric space.
ConditionsJS1 and JS3 imply the following fact.Proposition 2 (see [33]).
LetX,D be a JS-metric space and x∈X. If CD,X,x≠∅, then Dx,x=0.
As in a metric space, convergence and completeness for a JS-metric space can be defined in a similar way.Definition 3 (see [32]).
LetX,Dbe a JS-metric space andxna sequence inX.
(i)
xnD-converges to x if xn∈CD,X,x(ii)
xn is called a D-Cauchy sequence if limm,n→+∞Dxm,xn=0(iii)
The spaceX,D is said to be complete if every D-Cauchy sequence is D-convergent
In a metric space, triangle inequality forces any convergent sequence to have a unique limit. Here, the conditionJS3 plays that role.Proposition 4 (see [32]).
LetX,Dbe a JS-metric space, and letxnbe a sequence inX. For anyx,y∈X, ifxn∈CD,X,x∩CD,X,y, thenx=y.
Pointwise continuity can then be defined, using convergence of sequences.Definition 5.
LetX,Dbe a JS-metric space. A mappingP:X→Xis said to be continuous at a pointx0∈Xifxn∈CD,X,x0impliesPxn∈CD,X,Px0.In addition,Pis said to be continuous if it is continuous at eachxinX.


## 2.2. Common Proximity

Throughout the paper, for nonempty subsetsA and B of a JS-metric space X,D, the following notations will be used:
(4)DA,B≔infDa,b:a∈A,b∈B,A0≔a∈A:thereexistsb∈BsuchthatDa,b=DA,B,B0≔b∈B:thereexistsa∈AsuchthatDa,b=DA,B.Notice thatA0 and B0 can be empty or even undefined. If DA,B=+∞ and Da,b=+∞, then it is unclear whether a∈A0. Throughout the paper, the distance DA,B will be always assumed to be finite.For nonempty subsetsA and B of a JS-metric space X,D, let us recall that a best proximity point of a mapping T:A→B is a point x∗∈A with Dx∗,Tx∗=DA,B. If DA,B=0, then x∗ becomes a fixed point. The term “common” comes into play when we deal with a pair of mappings.Definition 6 (see [18]).
LetP,Q:A→Bbe mappings. An elementx∗∈Ais said to be a common best proximity point of the pairP,Qif(5)Dx∗,Qx∗=DA,B=Dx∗,Px∗.
The set of common best proximity points ofP and Q is denoted by CPP,Q.
Observe also that ifDA,B=0, then x∗ becomes a common fixed point of the pair P,Q.Definition 7 (see [17]).
LetP,Q:A→B be mappings. A pair P,Q is said to commute proximally if for each x,u,v∈A,
(6)Du,Qx=DA,B=Dv,PximpliesPu=Qv.


## 2.3. Graph Endowment

Given a nonempty setX, a directed graph G=VG,EG will be constructed as follows.
(i)
The set of verticesVG is the set X itself(ii)
The set of edgesEG⊆X×X contains all the loops; that is, x,x∣x∈X⊆EG(iii)
EG contains no parallel edgesWe say thatX is said to be endowed with a directed graphG=VG,EG.LetX,D be a JS-metric space endowed with a directed graph G=VG,EG, denoted by X,D,G. We next introduce a notion of continuity with respect to the graph G.Definition 8 (see [23]).
A mappingP:X→Xis calledG-continuous atx0∈Xif for any sequencexninXwithxn,xn+1∈EGandxn∈CD,X,x0, it follows thatPxn∈CD,X,Px0.
Notice thatG-continuity is a weaker notion than usual continuity, with respect to the JS-metric. In other words, any continuous mapping on a space endowed with a directed graph G is G-continuous. A counterexample of the converse is easily found, when G has an isolated vertex, for example.
Let us now introduce some more terminology used in our main results.Definition 9.
LetA,Bbe nonempty subsets ofX,D,GandP,Q:A→Bbe mappings. A pairP,Qis said to beG-proximal edge preserving if the following hold:(i)
IfPx,Py∈EG, Qx,Qy∈EG(ii)
For anyx,u,v∈A with Px,Qx∈EG and(7)Du,Qx=DA,B=Dv,Px,it follows that v,u∈EGDefine the classΓ of functions to be
(8)Γ=γ:0,+∞→0,1∣limn→+∞γtn=1implieslimn→+∞tn=0.This extends the class of functions in [19, 21]. Notice that any function in γ∈Γ has a property that
(9)γt=1impliest=0.Definition 10.
LetA,Bbe nonempty subsets ofX,D,GandP,Q:A→Bbe mappings. A pairP,Qis said to be dominatingG-proximal Geraghty if there exists a functionγ∈Γsuch that for eachx1,x2,u1,u2,v1,v2∈Asatisfying(10)Px1,Qx1∈EG,Px2,Qx2∈EG,Du1,Qx1=Du2,Qx2=DA,B=Dv1,Px1=Dv2,Px2,it follows that
(11)Du1,u2≤γMv1,v2,u1,u2Mv1,v2,u1,u2,where Mv1,v2,u1,u2=maxDv1,v2,Dv1,u1,Dv2,u2.


## 3. Existence Theorems

Throughout this section, letX,D,G be a complete JS-metric space endowed with a directed graph G, A,B be nonempty subsets of X, and P,Q:A→B be mappings. The following assumptions will be imposed.(A0)A0,B0≠∅(A1)A0 is bounded with respect to D and closed, in the sense that any convergent sequence in A0 has its limit in A0(A2)QA0⊆B0(A3) For anyu,v∈A, if there exist x,y∈A such that
(12)Du,Qx=DA,B=Dv,Py,then Du,v<+∞(A4) The pairP,Q commutes proximally(A5) The pairP,Q is dominating G-proximal GeraghtyLemma 11.
IfPu=Qufor someu∈A0, thenCPP,Q≠∅.Proof.
Letu∈A0 such that Pu=Qu. By (A2), there exists x∗∈A0 such that
(13)Dx∗,Qu=DA,B=Dx∗,Pu.
Since the pairP,Q commutes proximally, we have Px∗=Qx∗. Again, the assumption (A2) implies that
(14)Dz∗,Qx∗=DA,B=Dz∗,Px∗,for some z∗∈A0. From (13) and (14), the assumption (A3) yields Dx∗,x∗<+∞, Dz∗,z∗<+∞, and Dx∗,z∗<+∞. Since Pu,Qu=Pu,Pu∈EG, it follows from (13) and (A5) that
(15)Dx∗,x∗≤γMx∗,x∗,x∗,x∗Mx∗,x∗,x∗,x∗=γDx∗,x∗Dx∗,x∗≤Dx∗,x∗.
That is,γDx∗,x∗=1, and hence, Dx∗,x∗=0. Similarly, by (14) and Px∗,Qx∗=Px∗,Px∗∈EG, we also get Dz∗,z∗=0.
Now, observe that (14) will prove the lemma if x∗=z∗. In fact, the equality can be achieved by a similar argument above. Since
(16)Mx∗,z∗,x∗,z∗=maxDx∗,z∗,Dx∗,x∗,Dz∗,z∗=Dx∗,z∗,Dx∗,z∗≤γMx∗,z∗,x∗,z∗Mx∗,z∗,x∗,z∗=γDx∗,z∗Dx∗,z∗≤Dx∗,z∗,it follows that γDx∗,z∗=1 yielding Dx∗,z∗=0. Thus, x∗∈CPP,Q.Theorem 12.
Assume (A0)–(A5) as before. In addition, if the following are satisfied(i)
QA0⊆PA0(ii)
there existsx0∈A0 such that Px0,Qx0∈EG(iii)
P,Q is G-proximal edge preserving(iv)
P and Q are G-continuous,then CPP,Q≠∅. Moreover, if Px,Qx∈EG, for all x∈CPP,Q, then the pair P,Q has a unique common best proximity point.Proof.
Letx0∈A0 be such that Px0,Qx0∈EG. From the assumption QA0⊆PA0 and the G-proximal edge preserving property of P,Q, we can construct a sequence xn in A0 satisfying
(17)Qxn=Pxn+1,Pxn,Qxn∈EG,for all integers n≥0. For each n, since QA0⊆B0, there exists an element un∈A0 such that
(18)Dun,Qxn=DA,B=Dun,Pxn+1.
Ifun0=un0+1 for some n0, we then have
(19)Dun0+1,Qxn0+1=DA,B=Dun0,Pxn0+1.
It follows from (A4) thatQun0=Pun0+1=Pun0. Applying Lemma 11 yields CPP,Q≠∅.
Let us assumeun≠un+1 for all integers n≥0. From (18), we get
(20)Dun,Qxn=Dun+1,Qxn+1=DA,B=Dun−1,Pxn=Dun,Pxn+1,and (A3) gives Dun,un+1<+∞ for all n. Since P,Q is dominating G-proximal Geraghty, we have that
(21)Dun,un+1≤γMun−1,un,un,un+1Mun−1,un,un,un+1≤Mun−1,un,un,un+1=maxDun−1,un,Dun,un+1,implying that Dun,un+1≤Dun−1,un for all integers n≥1. Let Dun,un+1 converge to a real number r≥0. So does Mun−1,un,un,un+1.
Ifr were positive, we would obtain
(22)1=limn→+∞Dun,un+1Mun−1,un,un,un+1≤limn→+∞γMun−1,un,un,un+1≤1,which leads to a contradiction as follows:
(23)limn→+∞γMun−1,un,un,un+1=1implieslimn→+∞Mun−1,un,un,un+1=0.
Therefore,limn→+∞Dun,un+1=0.
Let us next show thatun is a D-Cauchy sequence. Suppose that this is not the case. Then, we can construct subsequences unk and umk of un satisfying
(24)Dunk,umk≥ε,mk>nk≥k,for some ε>0. Notice that Dunk,umk<+∞ for all positive integers k, by (18) and (A3), and Pxnk,Qxnk∈EG and Pxmk,Qxmk belong to EG, by (17). Thus,
(25)Dunk,Qxnk=Dumk,Qxmk=DA,B=Dunk−1,Pxnk=Dumk−1,Pxmkfor all k. The dominating G-proximal Geraghty property of P,Q implies
(26)Dunk,umk≤γMunk−1,umk−1,unk,umkMunk−1,umk−1,unk,umk,where
(27)Munk−1,umk−1,unk,umk=maxDunk−1,umk−1,Dunk−1,unk,Dumk−1,umk.
Observe thatMxnk−1,xmk−1,xnk,xmk is clearly neither Dunk−1,unk nor Dumk−1,umk for sufficiently large k, as limn→+∞Dun,un+1=0. Without loss of generality, we may assume
(28)Mxnk−1,xmk−1,xnk,xmk=Dunk−1,umk−1,implying
(29)Dunk,umk≤γDunk−1,umk−1Dunk−1,umk−1for all k. Moreover, by induction, we obtain
(30)Dunk−i,umk−i≤γDunk−i−1,umk−i−1Dunk−i−1,umk−i−1,where i=0,1,2,⋯,nk−1. Therefore,
(31)Dunk,umk≤∏i=1nkγDunk−i,umk−iDu0,umk−nk.
Define(32)γDunk−ik,umk−ik=maxγDunk−i,umk−i:1≤i≤nk,η=limsupk→+∞γDunk−ik,umk−ik.
Note that0≤η≤1. If η<1, then limk→+∞∏i=1nkγDunk−i,umk−i=0. If η=1, then γ forces Dunk−ik,unk+mk−ik to possess a subsequence converging to 0 as k→+∞. Both cases above contradict the fact that Dunk,umk≥ε for all k.
We have shown thatun is a D-Cauchy sequence. By the assumption (A1), we have limn→+∞un=u for some u∈A0. Notice that, from (18), we may write
(33)Dun+1,Qxn+1=DA,B=Dun,Pxn+1for all integers n≥0. Since the pair P,Q commutes proximally and is G-proximal edge preserving, we have
(34)Pun+1=Qun,un,un+1∈EG,for all n. The G-continuity of P and Q implies
(35)Pu=limn→+∞Pun=limn→+∞Qun−1=Qu.
Lemma11 then guarantees a common best proximity point.
For uniqueness, we assume that any common proximity point ofP,Q satisfies the property Px,Qx∈EG. Let x∗,y∗∈CPP,Q. Then,
(36)Dx∗,Px∗=Dy∗,Py∗=DA,B=Dx∗,Qx∗=Dy∗,Qy∗.
As seen in the proof of Lemma11, we have Dx∗,x∗=Dy∗,y∗=0 and
(37)Mx∗,y∗,x∗,y∗=maxDx∗,y∗,Dx∗,x∗,Dy∗,y∗=Dx∗,y∗<+∞.
SinceP,Q is dominating G-proximal Geraghty, we obtain
(38)Dx∗,y∗≤γDx∗,y∗Dx∗,y∗≤Dx∗,y∗.
The inequalities above together with the property ofγ yield Dx∗,y∗=0, as required.
The following is a modification of Theorem12. Note that the G-continuity of P and Q is dropped and replaced by (iv), which helps facilitate the existence of a common best proximity point.Theorem 13.
Assume (A0)–(A5) and (i), (ii), and (iii) as in Theorem12. In addition, if the following holds(iv)
For any sequencexn in A that D-converges to x∈A and satisfies Qxn,Pxn+1∈EG, there exists subsequence xnk of xn such that(39)Dx,Qxnk=DA,B=Dx,Pxnk,then CPP,Q≠∅. Moreover, if Px,Qx∈EG, for all x∈CPP,Q, then P,Q has a unique common best proximity point.Proof.
The conditionsi and ii are used to construct xn and un, as in the proof of Theorem 12. Let us assume that limn→+∞un=u for some u∈A0. By (18), we have
(40)Dun,Qxn=DA,B=Dun−1,Pxn,implying, since P,Q commutes proximally, that
(41)Pun=Qun−1for all integers n≥1. Since Qun,Pun+1=Qun,Qun∈EG and the condition (iv), there exists subsequence unk of un such that
(42)Du,Qunk=DA,B=Du,Punk.
This, again, yieldsPu=Qu. Lemma 11 shows CPP,Q≠∅.
The uniqueness part is shown in the same fashion as in Theorem12.Example 14.
LetX=ℝ3 be equipped with the JS-metric D given by
(43)Dx1,y1,z1,x2,y2,z2=x1−x2+y1−y2+z1−z23,y1,y2=0,0orz1,z2=0,0,x1−x2+y1−y2+z1−z2,otherwise.
LetA=x,1,1:0≤x≤3 and B=x,−1,1:0≤x≤3. It is easy to see that DA,B=2.
Define the mappingsP,Q:A→B by
(44)Px,1,1=x,−1,1,Qx,1,1=ln2+x,−1,1for all x,1,1∈A. Notice that P and Q are continuous.Let(45)EG=x,y,z,u,v,w∈ℝ2×ℝ2x≥uandy≤vandz≥w.We will show thatP,Q is G-proximal edge preserving.
(1)
Letx∗,1,1,y∗,1,1∈A and(46)Px∗,1,1,Py∗,1,1=x∗,−1,1,y∗,−1,1∈EG,we have x∗≥y∗ and ln2+x∗≥ln2+y∗. Thus,
(47)Qx∗,1,1,Qy∗,1,1=ln2+x∗,−1,1,ln2+y∗,−1,1∈EG.(2)
Letx,u,v∈A. Observe that they must have the following forms:(48)x=x^,1,1,u=u^,1,1,v=v^,1,1,such that Px,Qx∈EG and
(49)Du,Qx=DA,B=Dv,Px.Thus,x^≥ln2+x^ and
(50)Du^,1,1,ln2+x^,−1,1=2=Dv^,1,1,x^,−1,1.We haveu^=ln2+x^,v^=x^ implying that v^≥u^; that is, v,u∈EG.To show that the pairP,Q is dominating G-proximal Geraghty, define the mapping γ:0,+∞→0,1 by
(51)γt=1,t=0,ln1+tt,0<t,0,t=+∞.Then,γ∈Γ.Letx1,x2,u1,u2,v1,v2∈A. Notice that they must have the following forms:
(52)x1=x^1,1,1,x2=x^2,1,1,u1=u^1,1,1,u2=u^2,1,1,v1=v^1,1,1,v2=v^2,1,1,such that Px1,Qx1,Px2,Qx2∈EG and
(53)Du1,Qx1=Du2,Qx2=DA,B=Dv1,Px1=Dv2,Px2.Thus,x^1≥ln2+x^1,x^2≥ln2+x^2 and
(54)u^1=ln2+x^1,u^2=ln2+x^2,v^1=x^1,v^2=x^2.To obtain the inequality (11), if u1=u2 or v1=v2, then we are done. Assume that u1≠u2. Then, u^1,u^2,v^1,v^2 are all distinct. As a consequence, Mv1,v2,u1,u2>0. Thus, we have that
(55)Du1,u2=u^1−u^2=ln2+v^1−ln2+v^2=ln2+v^2+v^1−v^22+v^2≤ln1+v^1−v^2≤ln1+Mv1,v2,u1,u2=ln1+Mv1,v2,u1,u2Mv1,v2,u1,u2Mv1,v2,u1,u2=γMv1,v2,u1,u2Mv1,v2,u1,u2.Therefore, the pairP,Q is dominating G-proximal Geraghty.Next, consider, by the definition ofA0 and B0, that A0=A and B0=B. Additionally,
(56)QA0=x,−1,1:ln2≤x≤ln5⊆x,−1,1:0≤x≤3=B0=PA0.Now, it remains to show thatP,Q commutes proximally. Let x,u,v∈A be such that
(57)Du,Qx=DA,B=Dv,Px.Consequently,x=x^,1,1,u=u^,1,1,v=v^,1,1, where u^=ln2+x^ and v^=x^. Thus,
(58)Qv=ln2+v^,−1,1=ln2+x^,−1,1=u^,−1,1=Pu.Thus,P,Q commutes proximally.Finally, by Theorem12, we can conclude that there is a unique common best proximity point of the pair P,Q. In fact, the point 0,1,1 is the unique common best proximity point of P,Q.


## 4. Some Special Cases

Recall that(59)Γ=γ:0,+∞→0,1limn→+∞γtn=1implieslimn→+∞tn=0.In this section, we present some existence results where functions inΓ are concretely chosen. These results are direct consequences of Theorems 12 and 13.First of all, letX,D,G be a complete JS-metric space endowed with a directed graph, A,B be nonempty subsets of X, and P,Q:A→B be mappings.Corollary 15.
Assume (A0)–(A4) and (i), (ii), and (iii) as in Theorem12. In addition, if the following are satisfied(i)
either of the following holds(a)
P and Q are G-continuous(b)
For any sequencexn in A that D-converges to x∈A and satisfies Qxn,Pxn+1∈EG, there exists subsequence xnk of xn such that(60)Dx,Qxnk=DA,B=Dx,Pxnk(ii)
there existsk∈0,1 such that for any x1,x2,u1,u2,v1,v2∈A with Pxi,Qxi∈EG and(61)Du1,Qx1=Du2,Qx2=DA,B=Dv1,Px1=Dv2,Px2,it follows
(62)Du1,u2≤kMv1,v2,u1,u2;then, CPP,Q≠∅. Moreover, if Px,Qx∈EG, for all x∈CPP,Q, then P,Q has a unique common best proximity point.Proof.
Defineγ:0,+∞→0,1 by γt=k for some k∈0,1. Clearly, γ∈Γ, and hence, (A5) is satisfied.Corollary 16.
Assume (A0)–(A4) and (i), (ii), (iii), and (iv) as in Corollary15. In addition, if the following holds(i)
for anyx1,x2,u1,u2,v1,v2∈A with Pxi,Qxi∈EG and(63)Du1,Qx1=Du2,Qx2=DA,B=Dv1,Px1=Dv2,Px2,it follows
(64)Du1,u2≤Mv1,v2,u1,u21+Mv1,v2,u1,u2;then CPP,Q≠∅. Moreover, if Px,Qx∈EG, for all x∈CPP,Q, then P,Q has a unique common best proximity point.Proof.
Defineψ:0,+∞→0,1 by ψt=1/1+t for all t∈0,+∞ and ψ+∞=0. For any sequence tn with limn→∞ψtn=1, we easily have limn→∞tn=0. Thus, ψ∈Γ, and hence, (A5) is satisfied.


## 5. Application on a JS-Metric Space Endowed with an Arbitrary Relation

In this section, it is shown that our result gives rise to a common best proximity point theorem for a mapping on a JS-metric space endowed with a binary relationR on X denoted by X,D,R. To begin with, let us introduce some terminology.Definition 17.
LetA,Bbe nonempty subsets ofX,D,Randx0∈X. A mappingP:A→Bis calledR-continuous atx0if for any sequencexninAthatD-converges tox0andxnRxn+1for alln, the sequencePxnD-converges toPx0.Definition 18.
LetA,Bbe nonempty subsets ofX,D,RandP,Q:A→Bbe mappings. A pairP,Qis said to beR-proximally comparative preserving if the following assertions hold:(i)
IfPxRPy, then QxRQy(ii)
For anyx,u,v∈A such that PxRQx and(65)Du,Qx=DA,B=Dv,Px,it follows vRu.Definition 19.
LetA,Bbe nonempty subsets ofX,D,RandP,Q:A→Bbe mappings. A pairP,Qis said to be dominatingR-proximally comparative Geraghty if there exists a functionγ∈Γsuch that for anyx1,x2,u1,u2,v1,v2∈AsatisfyingPx1RQx1, Px2RQx2, and(66)Du1,Qx1=Du2,Qx2=DA,B=Dv1,Px1=Dv2,Px2,it follows that
(67)Du1,u2≤γMv1,v2,u1,u2Mv1,v2,u1,u2,where Mv1,v2,u1,u2=maxDv1,v2,Dv1,u1,Dv2,u2.Corollary 20.
LetA,Bbe nonempty subsets of a complete JS-metric spaceX,D,Rand letP,Q:A→Bbe mappings. Suppose that the pairP,Qis dominatingR-proximally comparative Geraghty. Assume thatA0andB0are nonempty such thatA0is closed and bounded. If the following assertions hold:(i)
QA0⊆B0 and QA0⊆PA0(ii)
P and Q commute proximally(iii)
For eachu,v∈A, if there exist x,y∈A such that(68)Du,Qx=DA,B=Dv,Py,then Du,v<+∞(iv)
There existsx0∈A0 such that Px0RQx0(v)
P,Q is R-proximally comparative preserving(vi)
Suppose that one of the following holds(a)
P and Q are R-continuous(b)
For sequencexn in A that D-converges to x∈A and satisfies QxnRPxn+1, there exists a subsequence xnk of xn such that(69)Dx,Qxnk=DA,B=Dx,Pxnk,then CPP,Q≠∅. Moreover, if PxRQx for all x∈CPP,Q, then the pair P,Q has a unique common best proximity point.Proof.
We define a directed graphG=VG,EG with VG=X and
(70)EG=x,y∈X×X:xRy.
In order to apply Theorem12 or 13, all the hypotheses must hold.
(1)
We will show thatP,Q is dominating G-proximal Geraghty. To this end, let x1,x2,u1,u2,v1,v2∈A be such that(71)Px1,Qx1∈EG,Px2,Qx2∈EG,Du1,Qx1=Du2,Qx2=DA,B=Dv1,Px1=Dv2,Px2.
Then, we havePx1RQx1 and Px2RQx2. Since P,Q is dominating R-proximally comparative Geraghty, the pair P,Q is dominating G-proximally Geraghty.
(2)
The condition (iv) implies that there existsx0∈A0 such that(72)Px0,Qx0∈EG(3)
LetP,Q be R-proximally comparative preserving. We will show that P,Q is G-proximal edge preserving. Notice that Px,Py∈EG implies PxRPy. Since P,Q is R-proximally comparative preserving, we get QxRQy implying that Qx,Qy∈EG
Letx,u,v∈A such that Px,Qx∈EG and
(73)Du,Qx=DA,B=Dv,Px.
By the definition ofEG, we have PxRQx. Since P,Q is R-proximally comparative preserving, we obtain vRu. This yields v,u∈EG.
(4)
The condition (a) implies thatP and Q are G-continuous on A. Applying Theorem 12, we achieve CPP,Q≠∅
Suppose the condition (b) holds. Letxn be a sequence in A that D-converges to x∈A and satisfy Qxn,Pxn+1∈EG. Then, QxnRPxn+1. By (b), there exists a subsequence xnk of xn such that
(74)Dx,Qxnk=DA,B=Dx,Pxnk.
Theorem13 now applies and gives CPP,Q≠∅.
Finally, ifPxRQx for all x∈CPP,Q, then Px,Qx∈EG for all x∈CPP,Q. Therefore, P,Q has a unique common best proximity point.


## 6. Conclusion

This work has proposed some existence and uniqueness theorems for common best proximity points of any two mappings in a JS-metric space endowed with a directed graphG. The results obtained were mainly due to the assumptions that the pair of mappings is G-proximal edge preserving and dominated G-proximal Geraghty. The results have been further applied to a situation where the JS-metric space enjoys a binary relation.



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*Source: 5524494-2021-05-11.xml*