# Best Proximity Coincidence Point Results forα,D-Proximal Generalized Geraghty Mappings in JS-Metric Spaces

Authors: Atit Wiriyapongsanon; Phakdi Charoensawan; Tanadon Chaobankoh
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We introduce a type of Geraghty contractions in aJS-metric space X, called α,D-proximal generalized Geraghty mappings. By using the triangular-α,D-proximal admissible property, we obtain the existence and uniqueness theorem of best proximity coincidence points for these mappings together with some corollaries and illustrative examples. As an application, we give a best proximity coincidence point result in X endowed with a binary relation.


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## Body

## 1. Introduction and Preliminaries

LetT:A→B be a map where A and B are two nonempty subsets of a metric space X. It is known that if T is a nonself-map, the equation Tx=x does not always have a solution, and it clearly has no solution when A and B are disjoint. However, it is possible to determine an approximate solution x∗ such that the error is dx∗,Tx∗=dA,B. Such point x∗ is called a best proximity point of T. The best proximity point theorem was first studied in [1]. Then, there has been a wide range of research in this framework. Many researchers have studied and generalized the result in many aspects (for example, see [2–15]). For some recent articles regarding these points, see [16, 17] where Geraghty type mappings were studied and [18] where cyclic and noncyclic nonexpansive mappings were considered.One of the well-known generalizations of the Banach contraction principle is the result given by Geraghty [19] which enriches the principle by considering the class of mappings θ:0,∞→0,1 such that tn→0 when θtn→1. By including 1 in the ranges of those mappings θ, Ayari [20] provided a new result on the existence and uniqueness of the best proximity point for α-proximal Geraghty mappings.The concept of the best proximity coincidence point, which is an extension of a best proximity point problem, was mentioned in [21] (see also [22]) where some results of mappings in generalized metric spaces were presented. A point a is called a best proximity coincidence point of the pair g,T, where g is a self-map on A, if dga,Ta=dA,B. Clearly, if g is the identity map, then each best proximity coincidence point of the pair g,T is a best proximity point for T.A large number of results concerning these point problems in various metric spaces have been investigated since then. Hussain and the coauthors contributed several interesting results and generalizations in [23–25], including the recent article [26] where best proximity point results for Suzuki-Edelstein proximal contractions were studied. (See also, [27–31] for his work.)Zhang and Su [32] weakened the P-property, called the weak P-property, and improved a best proximity point theorem for Geraghty nonself-contractions. In 2018, Komal et al. [33] obtained best proximity coincidence point theorems for α-Geraghty contractions g,T in metric spaces by using the weak P-property where g is an isometry.The concept of generalized metric spaces (orJS-metric spaces) was introduced in [34] in 2015. It is a generalization of standard metric spaces covering many topological structures.LetX be a nonempty set, and let D:X×X→0,∞ be a function. For each x∈X, we set
(1)CD,X,x=xn⊆X:limn→∞Dxn,x=0.Definition 1 (see [34]).
A functionD:X×X→0,∞is called a generalized metric onXif it satisfies the following conditions.
(D1)For any x,y∈X, Dx,y=0 implies x=y.
(D2)For any x,y∈X, Dx,y=Dy,x.
(D3)There exists a constant CX>0 such that
(2)Dx,y≤Cxlimsupn→∞Dxn,y,whenever x,y∈X and xn∈CD,X,x.In this case, we say thatX,D is a generalized metric space. It is, however, usually called a JS-metric space.Remark 2.
We note that, in general, results of best proximity points using the weakP-property in usual metric spaces might not be attained in the setting ofJS-metric spaces. For example,Dx,xis not necessarily equal to0,andDxn,ynmight not converge toDx,ywhenxn→xandyn→y.LetX≔X,D be a JS-metric space. We now discuss the convergence and the continuity in these spaces.Definition 3 (see [34]).
Letxnbe a sequence inX. The sequencexnis said toD-converge tox∈Xifxn∈CD,X,x.Moreover,xnis called aD-Cauchy sequence iflimm,n→∞Dxn,xm=0.Finally,X,Dis said to beD-complete if eachD-Cauchy sequence inXis aD-convergent sequence inX.Proposition 4 (see [34]).
For anyx,y∈X, ifxn∈CD,X,x∩CD,X,y, thenx=y.Definition 5 (see [34]).
A functionf:X→Xis said to beD-continuous at a pointx0∈Xif for anyxn∈CD,X,x0,fxn∈CD,X,fx0.In addition,fis said to beD-continuous onXif it isD-continuous at each point inX.The concept ofα-admissible mapping was introduced by Samet et al. [35] in 2012. The notion of triangular α-admissible mappings was then given by Karapinar [36]. Recently, Khemphet [37] extended the concept as follows.Definition 6 (see [37]).
LetX,Dbe a generalized metric space, and letfandgbe self-mappings onX. Given thatα:X×X→0,∞is a function,fis said to be triangular-α,D-admissible w.r.t.gif, for allx,y,z∈X, the following conditions hold.(i)
Ifαgx,gy≥1, then αfx,fy≥1 and Dgx,gy<∞.(ii)
Ifαx,z≥1 and αz,y≥1, then αx,y≥1.In this article, we introduce a type of Geraghty contractions which will be calledα,D-proximal generalized Geraghty mappings. These maps are motivated by the work of Khemphet [37]. Using the weak P-property in the setting of JS-metric space, we establish a result on the existence and uniqueness of the best proximity coincidence point for these mappings. Examples showing the validity of the main result and some corollaries are listed. Finally, by applying our main result, we obtain a best proximity coincidence point result in X endowed with a binary relation. Note that some other results of best proximity points in X endowed with binary relations can be deduced from our result.


## 2. Main Results

Throughout this article, letX≔X,D be a JS-metric space, and let A and B be nonempty disjoint subsets of X. Also, we require the following notations:
(3)DA,B≔infDa,b:a∈A,b∈B,A0≔a∈A:thereexistsb∈BsuchthatDa,b=DA,B,B0≔b∈B:thereexistsa∈AsuchthatDa,b=DA,B.Clearly, if one ofA0 and B0 is nonempty, then so is the other.Definition 7 (see [21]).
LetT:A→BandS:A→Abe mappings. An elementx∗∈Ais said to be a best proximity coincidence point of the pairS,TifDSx∗,Tx∗=DA,B. The set of all best proximity coincidence points of the pairS,Tis denoted byBCS,T.Definition 8 (see [32]).
Suppose thatA0is nonempty. The pairA,Bis said to have the weakP-property if and only ifDx1,y1=Dx2,y2=DA,BimpliesDx1,x2≤Dy1,y2, wherex1,x2∈A0andy1,y2∈B0.Definition 9.
LetT:A→BandS:A→Abe mappings. The pairS,Tis said to be triangular-α,D-proximal admissible if the following conditions hold.(i)
IfαSx1,Sx2≥1 and DSu1,Tx1=DSu2,Tx2=DA,B, then αSu1,Su2≥1andDSu1,Su2<∞forallx1,x2,u1,u2∈A.(ii)
Ifαx,z≥1 and αz,y≥1, then αx,y≥1, for all x,y,z∈X.We consider the class of mappingsΘ which is a slight generalization of the well-known class of 0,1-valued functions introduced by Geraghty [19]:
(4)Θ≔θ:0,∞→0,1:θtn→1impliestn→0.Now, we introduce a class of our contractions as follows.Definition 10.
LetT:A→BandS:A→Abe mappings. Given thatα:X×X→0,∞is a function, the pairS,Tis said to be anα,D-proximal generalized Geraghty mapping if the following conditions hold.(i)
S,T is triangular-α,D-proximal admissible.(ii)
There isθ∈Θ such that for all x,y,u,v∈A, if DSu,Tx=DSv,Ty=DA,B and αSx,Sy≥1, then(5)αSx,SyDTx,Ty≤θMx,y,u,vMx,y,u,v,where Mx,y,u,v=maxDSx,Sy,DSx,Su,DSy,Sv.We first give a useful lemma.Lemma 11.
Letα:X×X→0,∞be a function. LetT:A→BandS:A→Abe two mappings such thatS,Tis anα,D-proximal generalized Geraghty mapping, and letA,Bhave the weakP-property. IfαSx,Sy≥1for allx,y∈BCS,T,thenSx=Sy.Proof.
Letx,y∈BCS,T, we have that
(6)DSx,Tx=DSy,Ty=DA,B.
From the assumption,αSx,Sx≥1,αSy,Sy≥1, and αSx,Sy≥1. Since αSx,Sy≥1, S,T is triangular-α,D-proximal admissible and (6), we have that DSx,Sy<∞. Also, since DSx,Tx=DSx,Tx=DA,B,αSx,Sx≥1 and S,T is triangular-α,D-proximal admissible, then DSx,Sx<∞. Similarly, we can show that DSy,Sy<∞.
Note that(7)Mx,x,x,x=maxDSx,Sx,DSx,Sx,DSx,Sx=DSx,Sx<∞.
SinceS,T is an α,D-proximal generalized Geraghty mapping, and A,B has the weak P-property,
(8)DSx,Sx≤αSx,SxDSx,Sx≤αSx,SxDTx,Tx≤θDSx,SxDSx,Sx,for some θ∈Θ. From the property of θ, we can conclude that DSx,Sx=0. Similarly, we also have that DSy,Sy=0. Then,
(9)Mx,y,x,y≔maxDSx,Sy,DSx,Sx,DSy,Sy=DSx,Sy<∞.
SinceαSx,Sy≥1, we have that
(10)DSx,Sy≤αSx,SyDSx,Sy≤αSx,SyDTx,Ty≤θMx,y,x,yMx,y,x,y=θDSx,SyDSx,Sy,
for someθ∈Θ. Thus, DSx,Sy=0 which implies that Sx=Sy.Theorem 12.
Let∅≠A0⊆SA0,and letSA0,DbeD-complete. Given thatα:X×X→0,∞is a function, and letT:A→BandS:A→Abe mappings such thatS,Tis anα,D-proximal generalized Geraghty mapping. Suppose that the following conditions hold.(i)
TA0⊆B0 and the pair A,B has the weak P-property.(ii)
There existx,y∈A0 such that DSx,Ty=DA,B,αSy,Sx≥1 and DSy,Sx<∞.(iii)
ForSxn∈CD,SA0,Sx∗ such that αSxn,Sxn+1≥1 for all n∈ℕ, there is a subsequence Sxnk with αSxnk,Sx∗≥1 for all k∈ℕ.Then, there existsx∗∈A0 such that DSx∗,Tx∗=DA,B. Moreover, if αSx∗,Sy∗≥1 for all x∗,y∗∈BCS,T and S is injective, then S,T has a unique best proximity coincidence point.Proof.
From (ii), there existx0,x1∈A0 such that
(11)DSx1,Tx0=DA,B,αSx0,Sx1≥1,DSx0,Sx1<∞.
SinceTA0⊆B0, A0⊆SA0, and S,T is triangular-α,D-proximal admissible, there exists x2∈A0 such that
(12)DSx2,Tx1=DA,B,αSx1,Sx2≥1,DSx1,Sx2<∞.
Continuing in this way, we obtain a sequenceSxn⊆SA0 such that for all n∈ℕ,(13)DSxn,Txn−1=DA,B=DSxn+1,Txn,αSxn−1,Sxn≥1,DSxn−1,Sxn<∞.
Using the weakP-property to (13), for n and n+1, we have that
(14)DSxn,Sxn+1≤DTxn−1,Txnforalln∈ℕ.
If there existsn0∈ℕ∪0 such that Sxn0=Sxn0+1, then from (13),
(15)DSxn0+1,Txn0=DSxn0,Txn0=DA,B.
Now suppose thatSxn≠Sxn+1 for all n∈ℕ∪0. By the definition of D, DSxn,Sxn+1≠0. We will first show that limn→∞DSxn−1,Sxn=0. Let n∈ℕ. Since S,T is an α,D-proximal generalized Geraghty mapping together with (13) and (14), we obtain that
(16)DSxn,Sxn+1≤DTxn−1,Txn≤αSxn−1,SxnDTxn−1,Txn≤θMxn−1,xn,xn,xn+1Mxn−1,xn,xn,xn+1≤Mxn−1,xn,xn,xn+1,where
(17)Mxn−1,xn,xn,xn+1=maxDSxn−1,Sxn,DSxn,Sxn+1.
IfMxn−1,xn,xn,xn+1=DSxn,Sxn+1, then by (16),
(18)DSxn,Sxn+1≤θDSxn,Sxn+1DSxn,Sxn+1≤DSxn,Sxn+1.
SinceDSxn,Sxn+1>0 for all n≥0,
(19)1≤θDSxn,Sxn+1≤1,and thus,
(20)limn→∞θDSxn,Sxn+1=1.
By the definition ofθ, limn→∞DSxn,Sxn+1=0.
IfMxn−1,xn,xn,xn+1=DSxn−1,Sxn, we again have that
(21)DSxn,Sxn+1≤θDSxn−1,SxnDSxn−1,Sxn≤DSxn−1,Sxn.
Sincen is arbitrary, DSxn,Sxn+1 is nonnegative and nonincreasing. Therefore, DSxn,Sxn+1 converges to s≥0. Suppose on the contrary that s>0. From (21),
(22)DSxn,Sxn+1DSxn−1,Sxn≤θDSxn−1,Sxn≤1.
It follows thatlimn→∞θDSxn−1,Sxn=1. Since θ∈Θ, we have that limn→∞DSxn−1,Sxn=0 which is a contradiction. Thus, s must be 0 and that
(23)limn→∞DSxn−1,Sxn=0.
Next, we shall show thatSxn is a D-Cauchy sequence. Suppose that this is not the case. Then, there exists ε>0 such that for any k∈ℕ, there are subsequences Sxnk and Sxmk of Sxn satisfying DSxnk,Sxmk≥ε for mk≥nk≥k.
SinceS,T is triangular-α,D-proximal admissible, it is easy to see that
(24)αSxn,Sxm≥1andDSxn,Sxm<∞whenm≥nforallm,n∈ℕ.
It follows from (13) and (24) that for any k∈ℕ,(25)αSxnk−1,Sxmk−1≥1andDSxnk,Txnk−1=DA,B=DSxmk,Txmk−1.
SinceS,T is an α,D-proximal generalized Geraghty mapping and A,B has the weak P-property, we obtain that
(26)DSxnk,Sxmk≤DTxnk−1,Txmk−1≤αSxnk−1,Sxmk−1DTxnk−1,Txmk−1≤θMxnk−1,xmk−1,xnk,xmkMxnk−1,xmk−1,xnk,xmk,where
(27)Mxnk−1,xmk−1,xnk,xmk=maxDSxnk−1,Sxmk−1,DSxnk−1,Sxnk,DSxmk−1,Sxmk.
IfMxnk−1,xmk−1,xnk,xmk is either DSxnk−1,Sxnk or DSxmk−1,Sxmk, then, by (23), limk→∞DSxnk,Sxmk=0. This contradicts the assumption that Sxn is not D-Cauchy. Thus, Mxnk−1,xmk−1,xnk,xmk=DSxnk−1,Sxmk−1.
As a consequence,(28)DSxnk,Sxmk≤θDSxnk−1,Sxmk−1DSxnk−1,Sxmk−1.
By repeating the same steps, it follows that(29)DSxnk−i,Sxmk−i≤θDSxnk−i−1,Sxmk−i−1DSxnk−i−1,Sxmk−i−1,where i=0,1,2,⋯,nk−1. Therefore,
(30)DSxnk,Sxmk≤∏i=1nkθDSxnk−i,Sxmk−iDSx0,Sxmk−nk.
Letik∈1,2,⋯,nk such that
(31)θDSxnk−ik,Sxmk−ik=maxθDSxnk−i,Sxmk−i:1≤i≤nk.
Define(32)η=limsupk→∞θDSxnk−ik,Sxmk−ik.
Ifη<1, limk→∞DSxnk,Sxmk=0 which is impossible. Thus, η=1. Without loss of generality, we may assume that limk→∞θDSxnk−ik,Sxnk+mk−ik=1. By the definition of θ, limk→∞DSxnk−ik,Sxnk+mk−ik=0. Then, there exists k0∈ℕ such that
(33)DSxnk0−ik0,Sxnk0+mk0−ik0<ε2.
Now,(34)ε≤DSxnk0,Sxmk0≤∏j=1ik0θDSxnk0−j,Sxmk0−jDSxnk0−ik0,Sxmk0−ik0<ε2,
which is a contradiction. Therefore,Sxn is a D-Cauchy sequence.
SinceSA0,D is D-complete, there exists x∗∈A0 such that
(35)limn→∞DSxn,Sx∗=0.
Equivalently,(36)Sxn∈CD,SA0,Sx∗.
SinceA0⊆SA0 and TA0⊆B0, it follows that there exists a∈A0 such that
(37)DSa,Tx∗=DA,B.
By (13) and (iii), there is a subsequence Sxnk of Sxn such that αSxnk,x∗≥1 for all k∈ℕ. From (13), we have that
(38)DSxnk+1,Txnk=DA,Bforallk∈ℕ.
By the weakP-property, (37) and (38), we obtain that DSxnk+1,Sa≤DTxnk,Tx∗.
SinceαSxnk,x∗≥1 and S,T is an α,D-proximal generalized Geraghty mapping,
(39)DSxnk+1,Sa≤DTxnk,Tx∗≤αSxnk,Sx∗DTxnk,Tx∗≤θMxnk,x∗,xnk+1,aMxnk,x∗,xnk+1,a≤Mxnk,x∗,xnk+1,a,forallk≥1,
where(40)Mxnk,x∗,xnk+1,a=maxDSxnk,Sx∗,DSxnk,Sxnk+1,DSx∗,Sa.
By (23) and (35), we immediately have that
(41)limk→∞Mxnk,x∗,xnk+1,a=DSx∗,Sa≥0.
IfDSx∗,Sa>0, by letting k→∞ in (39),
(42)1≤limk→∞θMxnk,x∗,xnk+1,a≤1.
We subsequently have that(43)limk→∞θMxnk,x∗,xnk+1,a=1.
By the property ofθ,
(44)limk→∞Mxnk,x∗,xnk+1,a=DSx∗,Sa=0,which is a contradiction. It follows that DSx∗,Sa must be equal to 0, and thus Sx∗=Sa. Therefore, from (37), there exists x∗∈A such that
(45)DSx∗,Tx∗=DA,B.
Suppose further thatx∗,y∗∈BCS,T and αx∗,y∗≥1. By Lemma 11, Sx∗=Sy∗. Since S is injective, x∗=y∗. The proof is now completed.Example 13.
LetX=−3,3 be equipped with the JS-metric D given by
(46)Dx,y=x+y,x≠0andy≠0,x2,y=0,y2,x=0.ChooseA=−2,0 and B=0,1. Let T:A→B be a mapping defined by
(47)Tx=−x3,forallx∈A,and let a mapping S:A→A be defined by
(48)Sx=x2,forallx∈A.It is not difficult to see thatDA,B=0 and A,B has the weak P-property. Next, define the map α:X×X→0,∞ by
(49)αx,y=1,ifx≠0ory=0,0,otherwise,for all x,y∈X. Since A0=0=B0, then TA0=0⊆B0=0 and A0=0⊆SA0=0. Also, there is 0∈A0 satisfying
(50)DS0,T0=D0,0=0=DA,B,α0,0≥1.We will first show thatS,T is triangular-α,D-proximal admissible.Letx1,x2,u1,u2∈A such that αSx1,Sx2≥1 and
(51)DSu1,Tx1=DSu2,Tx2=DA,B.Then,Sx1≠0orSx2=0 and
(52)Du12,−x13=Du22,−x23=0.Assume thatαSu1,Su2≠1, then u1/2=Su1=0 and u2/2=Su2≠0.SinceαSx1,Sx2≥1, we consider the following two cases.Case 1.
IfSx2≠0, then Sx1≠0, and thus,
(53)−x16=D0,−x13=0.Thenx1=0. This implies that Sx1=0 which is impossible.Case 2.
IfSx2=0=x2/2, then
(54)DSu2,−x23=Du22,0=u24=0.This implies thatu2=0 and Su2=0 which is impossible. Thus, αSu1,Su2≥1.Next, assume thatαx,z≥1 and αz,y≥1. Then, we can see that y=0 if z=0, and x≠0 if z≠0. Hence, x≠0 or y=0, and thus, αx,y≥1. This means that S,T is triangular-α,D-proximal admissible.We note that there is a mapθ∈Θ defined by θt=2/3.Now, forx,y satisfying αSx,Sy≥1, we have that Sx≠0 or Sy=0. We consider the following two cases.Case 1.
IfSy=0, then y=0 and
(55)αSx,SyDTx,Ty=αSx,0DTx,T0=D−x3,0=−x6=23x4=23DSx,Sy≤θMx,y,u,vMx,y,u,v.Case 2.
IfSy≠0, then Sx≠0, and thus, x≠0 and y≠0. We obtain that
(56)αx,yDTx,Ty=DTx,Ty=D−x3,−y3=−x3−y3≤23x+y=23DSx,Sy≤θMx,y,u,vMx,y,u,v.Therefore,S,T is an α,D-proximal generalized Geraghty mapping.Finally, we will show that assumption (iii) in Theorem12 holds. Let a∈A0 and Sxn∈CD,SA0,Sa such that αSxn,Sxn+1≥1 for all n∈ℕ. Then,
(57)Sxn≠0orSxn+1=0foreachn∈ℕ.IfSxn≠0 for all n∈ℕ, then αSxn,Sa≥1 for all n∈ℕ. Assume that there exists n0∈ℕ such that Sxn0=0. By (57), Sxk=0 for all k≥n0. Suppose that Sa≠0. Then,
(58)DSxk,Sa=D0,a=a2≠0forallk≥n0.This contradicts with the fact thatSxn∈CD,SA0,Sa. Thus, Sa=0 and so αSxn,Sa≥1. We also have that SA0,D is D-complete. Therefore, by Theorem 12, S,T has a best proximity coincidence point, which is 0.Example 14.
LetX=ℝ2be equipped with theJS-metricDgiven by(59)Dx1,y1,x2,y2=x1−x2+y1−y2,x1,x2≠0,0,y1,y2≠0,0,x1−x2,y1,y2=0,0,y1−y22,x1,x2=0,0.We consider the disjoint subsetsA and B of X given by A=−1,y;0≤y≤1 and B=1,y;0≤y≤1. We can check that DA,B=2 and the pair A,B has the weak P-property.LetT:A→B be a map defined by
(60)T−1,y=1,ln1+y,forall−1,y∈A,and let S:A→A be a map defined by
(61)S−1,y=−1,y,forall−1,y∈A.Then, we consider a mapα:X×X→0,∞ given by
(62)αx1,y1,x2,y2=1,ifx1≤x2,y1≥y2,0,otherwise,for all x=x1,y1, y=x2,y2∈X.Next, we will show thatS,T is triangular-α,D-proximal admissible. Let x,y,u,v∈A such that x=−1,x^,y=−1,y^,u=−1,u^, and v=−1,v^ satisfying αSx,Sy≥1 and
(63)DSu,Tx=DSv,Ty=DA,B.Consequently,x^≥y^ and
(64)D−1,u^,−1,ln1+x^=D−1,v^,−1,ln1+y^=2.It follows thatu^=ln1+x^ and v^=ln1+y^. Since x^≥y^, then u^≥v^, and thus, αSu,Sv≥1.Assume thatαx,y≥1 and αy,u≥1. Then, we can see that x^≥y^ and y^≥u^. Therefore, x^≥u^, and thus, αx,u≥1. This means that S,T is triangular-α,D-proximal admissible.We choose a mapθ∈Θ which is defined by
(65)θt=1,t=0,ln1+tt,t>0.Letx,y,u,v∈A such that x=−1,x^,y=−1,y^,u=−1,u^, and v=−1,v^ satisfying αSx,Sy≥1. If x=y, then we are done. Suppose that x≠y. It follows that Dx,y>0, and so, Mx,y,u,v>0. Thus,
(66)αSx,SyDTx,Ty=DTx,Ty=D1,ln1+x^,1,ln1+y^=∣ln1+x^−ln1+y^∣=ln1+y^+x^−y^1+y^≤ln1+∣x^−y^∣=ln1+Dx,y≤ln1+Mx,y,u,v=ln1+Mx,y,u,vMx,y,u,vMx,y,u,v=θMx,y,u,vMx,y,u,v.Therefore,S,T is an α,D-proximal generalized Geraghty mapping.SinceA0=A=−1,y;0≤y≤1andB0=B=1,y;0≤y≤1,(67)TA0=1,y;0≤y≤ln2⊆B0,A0=−1,0⊆SA0=A0.Also,SA0,D is D-complete, and there is −1,0∈A0 satisfying
(68)DS−1,0,T−1,0=D−1,0,1,0=2=DA,B,αS−1,0,T−1,0=α−1,0,1,0≥1.We have left to that show assumption (iii) in Theorem12 holds. Let a=−1,a^∈A0 and Sxn∈CD,SA0,Sa such that αSxn,Sxn+1=α−1,yn,−1,yn+1≥1 for all n∈ℕ. Then, yn≥yn+1 for all n. Since SA0=A0=−1,y;0≤y≤1 and yn is nonincreasing which Sxn∈CD,SA0,Sa=CD,A0,a. It follows that yn≥a^ for all n∈ℕ. Then, αSxn,Sa≥1 for all n∈ℕ. Therefore, by Theorem 12, S,T has a best proximity coincidence point.Next, we present a corollary of our result. The following definition is required.Definition 15.
LetT:A→BandS:A→Abe mappings. Letα:X×X→0,∞be a function. Then, the pairS,Tis said to be anα,D-proximal mapping if the following conditions hold.(i)
The pairS,T is triangular-α,D-proximal admissible.(ii)
There existsk∈0,1 such that for all x,y,u,v∈X, if DSu,Tx=DSv,Ty=DA,B and αSx,Sy≥1, then(69)DTx,Ty≤kDx,y.By puttingθt=k, where k∈0,1 in Theorem 12, we have the following result.Corollary 16.
LetA0⊆SA0andSA0,DbeD-complete. Given thatα:X×X→0,∞is a function, and letT:A→BandS:A→Abe mappings such thatS,Tis anα,D-proximal mapping. Suppose that the following conditions hold.(i)
TA0⊆B0 and the pair A,B has the weak P-property.(ii)
There existx,y∈A0 such that DSx,Ty=DA,B and αSy,Sx≥1 and DSy,Sx<∞.(iii)
ForSxn∈CD,SA0,Sx∗, if αSxn,Sxn+1≥1 for all n∈ℕ, then there is a subsequence Sxnk with αSxnk,Sx∗≥1 for all k∈ℕ.Then, there existsx∗∈A such that DSx∗,Tx∗=DA,B. Moreover, if αSx∗,Sy∗≥1 for all x∗,y∗∈BCS,T, then S,T has a unique best proximity coincidence point.


## 3. Consequence

We will apply our result on the best proximity coincidence point on aJS-metric space endowed with a binary relation R.LetT:A→B and S:A→A be mappings. The pair S,T is said to be R,D-proximally comparative if SxRSy and DSu1,Tx=DSu2,Ty=DA,B⇒Su1RSu2 and DSu1,Su2<∞ for all x,y,u1,u2∈A.Definition 17.
LetT:A→BandS:A→Abe mappings. The pairS,Tis said to be anR,D-proximally comparative generalized Geraghty mapping if the following hold.(1)
The pairS,T is R,D-proximally comparative.(2)
There existsθ∈Θ such that for all x,y,u,v∈A, if DSu,Tx=DSv,Ty=DA,B and SxRSy, then(70)DTx,Ty≤θMx,y,u,vMx,y,u,v,where Mx,y,u,v=maxDSx,Sy,DSx,Su,DSy,Sv.Corollary 18.
LetX be endowed with a symmetric, transitive binary relation R.LetT:A→BandS:A→Abe mappings such that∅≠A0⊆SA0andSA0,DbeD-complete. IfS,Tis anR,D-proximally comparative generalized Geraghty mapping and the following conditions hold:(i)
TA0⊆B0 and the pair A,B has the weak P-property;(ii)
there existx,y∈A0 such that DSx,Ty=DA,B and SyRSx and DSy,Sx<∞;(iii)
forSxn∈CD,SA0,Sx∗, if SxnRSxn+1 for all n∈ℕ, then there is a subsequence Sxnk with SxnkRSx∗ for all k∈ℕ,
then there existsx∗∈A0 such that DSx∗,Tx∗=DA,B. Moreover, if Sx∗RSy∗ for all x∗,y∗∈BCS,T and S is injective, then S,T has a unique best proximity coincidence point.Proof.
Define(71)αx,y=1,ifxRy,0,otherwise,
for allx,y∈X. We can see that the hypotheses of Theorem 12 hold which imply that there is x∗∈A such that DSx∗,Tx∗=DA,B. Let x∗,y∗∈BCS,T. Then, Sx∗RSy∗ which implies that αSx∗,Sy∗≥1. Again, by Theorem 12, x∗=y∗.


## 4. Conclusion and Open Questions

We have introduced new classes of Geraghty’s type mappings calledα,D-proximal generalized Geraghty mappings. Then, we investigated some conditions for this type of mappings to have a best proximity coincidence point in JS-metric spaces using the weak P-property. The question is whether one can extend Theorem 12 to the framework of common best proximity point in a JS-metric space X. Can we also extend the result when X is other generalized metric spaces?



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*Source: 8832662-2020-11-23.xml*