# Best Proximity Point Theorems forG,D-Proximal Geraghty Maps in JS-Metric Spaces

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

---

## Abstract

We studyG,D-proximal Geraghty contractions in a JS-metric space X endowed with graph G. We obtain some best proximity theorems for such contractions. An example and several consequences are given. As a consequence of our results, we also provide the best proximity point results in X endowed with a binary relation.


---

## Body

## 1. Introduction and Preliminaries

LetA and B be nonempty subsets of a metric space X,d and T:A⟶B be a map. 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 the best proximity point of T. In the case that T is a self-mapping, the best proximity point is simply a fixed 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 [2–22].In 2011, Raj [23] introduced the notion of P-property and subsequently obtained a best proximity point result for a weakly contractive nonself map T:A⟶B. Best proximity point theorems for subsets of X having the P-property were also studied in great details in [24–26]. Zhang and Su [27] weakened the P-property, called the weak P-property, as well as improved the best proximity point theorem for Geraghty nonself contractions, see also [28].Fixed-point theorems concerning a metric space endowed with graphG, which generalizes the Banach contraction principle, were proposed by Jachymski [29]. Klanarong and Suantai [30] recently presented the notion of a G-proximal generalized contraction. Several best proximity point results for these mappings were obtained. The concept of generalized metric spaces, also called JS-metric spaces, was introduced in [31]. It is a generalization of standard metric spaces covering many topological structures. Since then, many researchers have worked with these concepts and gave a large number of results, see [32–36], for example.In this paper, we introduce a type of contractions calledG,D-proximal, Geraghty mappings. These maps are defined on subsets A and B of a JS-metric space X which is endowed with graph G. Then, we establish a result on the existence and uniqueness of the best proximity point for these mappings. An example showing the validity of the main result is illustrated, and several corollaries are listed. Finally, by applying our main result, we obtain a best proximity point result in X endowed with a symmetric binary relation.


## 2. Preliminaries and Definitions

LetX be a nonempty set, and let D:X×X⟶0,∞ be a function. For each x∈X, define
(1)CD,X,x=xn⊆X:limn→∞Dxn,x=0.In 2015, Jleli and Samet [31] introduced a generalization of metric space as follows.Definition 1 (see [31]).
LetX be a nonempty set. A function D:X×X⟶0,∞ is said to be a generalized metric on X if the following conditions hold:
(D1) For any x,y∈X, if Dx,y=0, then x=y
(D2) For any x,y∈X, Dx,y=Dy,x
(D3) There exists C>0 such that for any x,y∈X(2)Dx,y≤Climsupn→∞Dxn,y,where xn∈CD,X,x.In this case, we say thatX,D is a generalized metric space, also known as a JS-metric space.Later, Khemphet [37] modified the condition D3, which will be denoted by D3∗, as follows: “there exists C>0 such that for any x,y∈X, Dx,y≤Climsupn→∞Dxn,yn, where xn∈CD,X,x and yn∈CD,X,y.”Clearly,D3∗ is stronger than D3. For convenience, when D3 is replaced by D3∗, the JS-metric space X,D will be called a JS∗-metric space.Now, letX≔X,D be a JS-metric space if not otherwise specified. We are ready to discuss convergence and continuity in these spaces.Definition 2 (see [31]).
Letxn be a sequence in X. The sequence xn is said to D-converge to x∈X if xn∈CD,X,x. Moreover, xn is called a D-Cauchy sequence if limm,n→∞Dxn,xm=0. Finally, X,D is said to be D-complete if each D-Cauchy sequence in X is a D-convergent sequence in X.Any convergent sequence in aJS-metric space converges to a unique point.Proposition 3 (see [31]).
Letxn be a sequence in X. For any x,y∈X, if xn∈CD,X,x∩CD,X,y, then x=y.Definition 4 (see [37]).
A functionf:X⟶X is said to be D-continuous at a point x0∈X if for any xn∈CD,X,x0, fxn∈CD,X,fx0. In addition, f is said to be D-continuous on X if it is D-continuous at each x in X.Definition 5.
AJS-metric space X,D is said to be endowed with a graph G=VG,EG; if the set of vertices (denoted by VG) is X, the set of edges (denoted by EG) contains the diagonal of X×X but parallel edges.We say thatG is transitive if for all x,y,z∈X, x,z,andz,y∈EG⇒x,y∈EG.Definition 6. (see [38]).
LetX,D be endowed with a graph G. A function f:X,D⟶X,D is said to be G-continuous at x∈X if for each xn∈CD,X,x with xn,xn+1∈EG for all n∈ℕ,fxn∈CD,X,fx.
LetA and B be nonempty subsets of X. 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.Definition 7 (see [39]).
LetT:A⟶B be a mapping. An element x∗∈A is said to be a best proximity point of T if Dx∗,Tx∗=DA,B. We denote the set of all best proximity points of T by BT.Definition 8 (see [27]).
LetA0 be nonempty. Then, the pair A,B is said to have the weak P-property if
(4)Dx1,y1=Dx2,y2=DA,B⇒Dx1,x2≤Dy1,y2,where x1,x2∈A and y1,y2∈B.Definition 9.
Letx∈X. A mapping T:A⟶B is said to be G,D-proximal if x1,x2∈EG and Du1,Tx1=Du2,Tx2=dA,B⇒u1,u2∈EG and Du1,u2<∞ for all x1,x2,u1,u2∈A.


## 3. Main Results

In this section, we assume thatX is a JS-metric space (or JS∗-metric space when specified) endowed with a transitive graph G=VG,EG. Let A and B be nonempty subsets of X for which A0 is nonempty.The class of functions(5)B≔β:0,∞⟶0,1:βtn⟶1impliestn⟶0was used as an important tool in [24]. It is clearly a generalization of the well-known class of 0,1-valued functions introduced by Geraghty [40].We now introduce a new type of Geraghty contractions.Definition 10.
A mappingT:A⟶B is said to be a G,D-proximal Geraghty mapping if the following hold:
(i)
T is G,D-proximal(ii)
For allx,y∈A such that x,y∈EG, there exists β∈B such that
(6)DTx,Ty≤βDx,yDx,yLemma 11.
LetT:A⟶B be a G,D-proximal Geraghty mapping and the pair A,B have the weak P-property. Then, for any x,y∈BT,
(1)
Dx,x=0(2)
Ifx,y∈EG, then x=yProof.
(1)
Letx∈BT. Then, dx,Tx=dA,B. Since x,x∈EG, we have
(7)Dx,x≤DTx,Tx≤βDx,xDx,x≤Dx,x<∞We can easily check that Dx,x=0.(2)
Letx,y∈BT such that x,y∈EG. From (6), Dx,x=Dy,y=0. By assumptions, dx,Tx=dy,Ty=dA,B and
(8)Dx,y=DTx,Ty≤βDx,yDx,y≤Dx,y<∞Similarly, Dx,y=0 and so x=y.Theorem 12.
LetX be a JS∗-metric space and A0 be D-complete. Let T:A⟶B be a G,D-proximal Geraghty map. Suppose that the following conditions hold:
(i)
TA0⊆B0 and A,B has the weak P-property(ii)
There existx,y∈A0 such that dx,Ty=dA,B, y,x∈EG, and Dy,x<∞(iii)
T is G-continuous and for C>0 such that Dx,Tx≤Climsupn→∞Dxn,Txn−1, there exists λ≥1 such that Cλ≤1Then, there existsx∗∈BT. Moreover, T has a unique best proximity point if x∗,y∗∈EG for all x∗,y∗∈BT.Proof.
From (ii), there existx0,x1∈A0 such that
(9)Dx1,Tx0=DA,B,x0,x1∈EG,Dx0,x1<∞.
Sincex1∈A0,Tx1∈TA0⊆B0. Then, there exists x2∈A such that dx2,Tx1=dA,B and so x2∈A0.
SinceT is G,D-proximal, we finally have that
(10)Dx2,Tx1=DA,B,x1,x2∈EG,Dx1,x2<∞.
Continuing this process, we obtain a sequencexn⊆A0 such that
(11)Dxn,Txn−1=DA,B,xn−1,xn∈EG,Dxn−1,xn<∞foralln≥1.
By using the weakP-property with (11), we have that
(12)Dxn,xn+1≤DTxn−1,Txnforalln≥1.
If there existsn0∈ℕ such that xn0=xn0−1, then
(13)Dxn0,Txn0−1=Dxn0,Txn0=DA,B.
Now, suppose thatxn≠xn−1 for all n≥1. We shall prove that xn is a Cauchy sequence. We first show that limn→∞Dxn−1,xn=0.
Sincexn−1,xn∈EG and T is a G,D-proximal, Geraghty mapping, then there exists β∈B such that
(14)Dxn,xn+1≤DTxn−1,Txn≤βDxn−1,xnDxn−1,xn≤Dxn−1,xnforalln≥1.
Clearly,Dxn−1,xn is nonincreasing. Thus, limn→∞Dxn,xn−1=r≥0. Suppose that r>0 and let n⟶∞ in (14). Then,
(15)1≤limn→∞βDxn−1,xn≤1.
It follows thatlimn→∞βDxn−1,xn=1. By the definition of β, limn→∞Dxn,xn−1=r=0 which is a contradiction. Thus, limn→∞Dxn−1,xn must be 0.
Now, suppose thatxn is not a Cauchy sequence. Then, there exists ε>0 such that for all k∈ℕ, there are subsequences xnk and xmk of xn such that Dxnk,xmk≥ε for k∈ℕ.
SinceG is transitive, xnk,xmk∈EG for all k∈ℕ. Since T is a G,D-proximal Geraghty mapping and (12),
(16)Dxnk,xmk≤DTxnk−1,Txmk−1≤βDxnk−1,xmk−1Dxnk−1,xmk−1.
Consequently,(17)Dxnk,xmk≤∏i=1nkβDxnk−i,xmk−iDx0,xmk−nk.
Forik∈1,2,⋯,nk, we have that
(18)βDxnk−ik,xmk−ik=maxβDxnk−i,xmk−i:1≤i≤nk.
Setη=limsupk→∞βDxnk−ik,xmk−ik. If η<1, then limk→∞Dxnk,xmk=0 which is a contradiction. If η=1, without loss of generality, we may assume that limk→∞βDxnk−ik,xnk+mk−ik=1. Then, we have that
(19)limk→∞Dxnk−ik,xnk+mk−ik=0.
This implies that there existsk0∈ℕ such that
(20)Dxnk0−ik0,xnk0+mk0−ik0<ε2.
Thus,(21)ε≤Dxnk0,xnk0+mk0≤∏j=1ik0βDxnk0−j,xnk0+mk0−jDxnk0−ik0,xnk0+mk0−ik0<ε2,which is a contradiction. Therefore, xn is a D-Cauchy sequence in A0.
SinceA0 is a D-complete, there exists x∗∈A0 such that limn→∞Dxn,x∗=0 and so xn∈CD,A0,x∗. Since T is G-continuous, Txn∈CD,A0,Tx∗.
ByD3∗ and (iii), there exist C>0 and λ≥1 such that
(22)DA,B≤λDA,B≤λDx∗,Tx∗≤Cλlimsupn→∞Dxn,Txn−1≤DA,B.
It follows thatDA,B=Dx∗,Tx∗. Suppose that x∗,y∗∈BT such that x∗,y∗∈EG. By Lemma 11, x∗=y∗. The proof is now completed.Theorem 13.
LetA0 be D-complete and T:A⟶B be a G,D-proximal Geraghty map. Suppose that the following conditions hold:
(i)
TA0⊆B0 and A,B has the weak P-property(ii)
There existx,y∈A0 such that dx,Ty=dA,B, y,x∈EG, and Dy,x<∞(iii)
Forxn∈CD,A0,x∗, if xn,xn+1∈EG for all n∈ℕ, then there exists a subsequence xnk with xnk,x∗∈EG for all k∈ℕThen, there existsx∗∈BT. Moreover, T has a unique best proximity point if x∗,y∗∈EG for all x∗,y∗∈BT.Proof.
From the proof of Theorem12, by using the assumptions (i)-(ii), we obtain a sequence xn∈A0 such that
(23)limn→∞Dxn,x∗=0forsomex∗∈A0.
Equivalently,xn∈CD,A0,x∗. Since x∗∈A0 and TA0⊆B0, Tx∗∈B0. It follows that there exists a∈A such that
(24)Da,Tx∗=DA,B.
By (11) and (iii), there exists a subsequence xnk of xn such that xnk,x∗∈EG for all k∈ℕ. Then, from (11),
(25)Dxnk+1,Txnk=DA,Bforallk∈ℕ.
By the weakP-property of A,B, (24) and (25), we have that Dxnk+1,a≤DTxnk,Tx∗.
Sincexnk,x∗∈EG and T is a G,D-proximal Geraghty mapping, we obtain that
(26)Dxnk+1,a≤DTxnk,Tx∗≤βDxnk,x∗Dxnk,x∗≤Dxnk,x∗<∞foralln≥1.
Takingk⟶∞ in (26), limk→∞Dxnk,a=0.
Therefore,xnk∈CD,A0,x∗∩CD,A0,a. It follows from Proposition 3 that x∗=a. From (24), there exists x∗∈A such that Dx∗,Tx∗=DA,B. Finally, if x∗,y∗∈BT such that x∗,y∗∈EG. By Lemma 11, we have that x∗=y∗. The proof is now completed.Example 1.
LetX=ℝ be equipped with a JS-metric D given by
(27)Dx,y=x+y,x≠0andy≠0,x2,y=0,y2,x=0.
LetA=−5,0 and B=0,10. We can see that the pair A,B have the weak P-property. Also, we have that
(28)A0=−5,0,B0=0,5andA0isD‐complete.
LetT:A⟶B be a mapping defined by Tx=−x/10,forallx∈A. Then,
(29)TA0=0,12⊆B0=0,5.
Define a graphG=VG,EG by VG=X and EG=x,y:x≠0ory=0. Clearly, G is transitive and there is 0∈A0 such that
(30)D0,T0=D0,0=0=DA,Band0,0∈EG.
We first check thatT is G,D-proximal. Let x1,x2,u1,u2∈A such that x1,x2∈EG and
(31)Du1,Tx1=Du2,Tx2=DA,B.
Thus,(32)Du1,−x110=Du2,−x210=0.
Now, suppose thatu1,u2∉EG. Then, u1=0 and u2≠0. Since x1,x2∈EG, we consider the following two cases.
Ifx2≠0, then, x1≠0 and D0,−x1/10=Du2,−x2/10=0. It follows that −x1/20=u2+−x2/10=0. Thus, x1=0 which is a contradiction.
Ifx2=0, then D0,−x1/10=Du2,0=0 and so −x1/20=u2/2=0. This implies that u2=0 which is a contradiction.
Therefore,u1,u2∈EG and so T is G,D-proximal.
We consider a constant mapβt=1/10∈B.
Letx,y∈EG. Then, x≠0 or y=0.
Ify=0, then
(33)DTx,Ty=DTx,T0=D−x10,0=−x20=110x2≤110Dx,y.
Ifx≠0, then
(34)DTx,Ty=D−x10,−y10=−x10+−y10=110x+y≤110Dx,y.
Thus,T is a G,D-proximal Geraghty map.
Finally, we will show that the condition (iii) in Theorem13 holds. Let zn∈CD,A0,a for some a∈A0 such that zn,zn+1∈EG for all n∈ℕ. Then,
(35)zn≠0orzn+1=0foreachn∈ℕ.
Ifzn≠0 for all n∈ℕ, then zn,a∈EG for all n∈ℕ. Assume that there exists an n0∈ℕ such that zn0=0. By (35), zk=0 for all k≥n0. Suppose that a≠0. Then,
(36)Dzk,a=D0,a=a2≠0forallk≥n0.
This contradicts to the fact thatzn∈CD,A0,a. Thus, a=0 and so zn,a∈EG. Therefore, 0∈BT by Theorem 13.We next present some consequences from our main results.Definition 14.
A mappingT:A⟶B is said to be a G,D-proximal contraction if the following hold:
(i)
T is G,D-proximal(ii)
For allx,y∈A such that x,y∈EG, there exists k∈0,1 such that
(37)DTx,Ty≤kDx,y
We immediately have the following corollaries.Corollary 15.
LetX be a JS∗-metric space and A0 be D-complete. Let T:A⟶B be a G,D-proximal contraction. Suppose that the following conditions hold:
(i)
TA0⊆B0 and A,B has the weak P-property(ii)
There existx,y∈A0 such that dx,Ty=dA,B, y,x∈EG, and Dy,x<∞(iii)
T is G-continuous and for C>0 such that Dx,Tx≤Climsupn→∞Dxn,Txn−1, Cλ≤1 for some λ≥1Then, there existsx∗∈BT. Moreover, if x∗,y∗∈EG for all x∗,y∗∈BT, T has a unique best proximity point.Corollary 16.
LetA0 be D-complete and T:A⟶B be a G,D-proximal contraction. Suppose that the following conditions hold:
(i)
TA0⊆B0 and A,B has the weak P-property(ii)
There existx,y∈A0 such that dx,Ty=dA,B, y,x∈EG, and Dy,x<∞(iii)
For any sequencexn in CD,A0,x∗, if xn,xn+1∈EG for all n∈ℕ, then there exists a subsequence xnk with xnk,x∗∈EG for all k∈ℕThen, there existsx∗∈BT. Moreover, if x∗,y∗∈EG for all x∗,y∗∈BT, T has a unique best proximity point.Definition 17.
A mappingT:A⟶B is said to be a G,D-proximal, R-type mapping if the following hold:
(i)
T is G,D-proximal(ii)
For allx,y∈A such that x,y∈EG,
(38)DTx,Ty≤Dx,yDx,y+1
Applyingβt=1/t+1 in Theorems 12 and 13, we obtain two corollaries as follows.Corollary 18.
LetX be a JS∗-metric space and A0 be D-complete. Let T:A⟶B be a G,D-proximal, R-type mapping. Suppose that the following conditions hold:
(i)
TA0⊆B0 and A,B has the weak P-property(ii)
There existx,y∈A0 such that dx,Ty=dA,B, y,x∈EG, and Dy,x<∞(iii)
T is G-continuous and for C>0 such that Dx,Tx≤Climsupn→∞Dxn,Txn−1, Cλ≤1 for some λ≥1Then, there existsx∗∈BT. Moreover, if x∗,y∗∈EG for all x∗,y∗∈BT, T has a unique best proximity point.Corollary 19.
LetA0 be D-complete and T:A⟶B be a G,D-proximal, R-type mapping. Suppose that the following conditions hold:
(i)
TA0⊆B0 and A,B has the weak P-property(ii)
There existx,y∈A0 such that dx,Ty=dA,B, y,x∈EG, and Dy,x<∞(iii)
If for any sequencexn in CD,A0,x∗, if xn,xn+1∈EG for all n∈ℕ, then there is a subsequence xnk with xnk,x∗∈EG for all k∈ℕThen, there existsx∗∈BT. Moreover, if x∗,y∗∈EG for all x∗,y∗∈BT, T has a unique best proximity point.


## 4. Application

In this section, we apply our result on best proximity points on a metric space endowed with binary relation. LetA and B be nonempty subset of a JS-metric space X with a binary relation R, and let T:A⟶B be a nonself mapping. The mapping T is said to be a D-proximally comparative if xRg and Du1,Tx=Du2,Ty=DA,B⇒u1Ru2 and Du1,u2<∞ for all x,y,u1,u2∈A.Definition 20.
The mappingT is said to be D-proximally comparative, Geraghty mapping if the following hold:
(1)
T is a D-proximally comparative(2)
There existsβ∈B such that for all x,y∈A, if xRy, then
(39)DTx,Ty≤βDx,yDx,yCorollary 21.
LetR be symmetric and transitive, and let A0 be D-complete. Let T:A→B be a D-proximally comparative, Geraghty map. Suppose that the following conditions hold:
(i)
TA0⊆B0 and A,B has the weak P-property(ii)
There existx,y∈A0 such that dx,Ty=dA,B, yRx, and Dy,x<∞(iii)
For any sequencexn in CD,A0,x∗, if xnRxn+1 for all n∈ℕ, then there exists a subsequence xnk with xnkRx∗ for all k∈ℕThen, there existsx∗∈BT. Moreover, if x∗,y∗∈EG for all x∗,y∗∈BT, T has a unique best proximity point.Proof.
Define a graphG=VG,EG by VG=X and EG=x,y∈X×X:xRy. Let x1,x2,u1,u2∈A such that x1,x2∈EG and Du1,Tx1=Du2,Tx2=DA,B.
By the definition ofEG, we have that xRy. Since T is a D-proximally comparative, u1Ru2. It follows that u1,u2∈EG. Therefore, T is a G,D-proximal Geraghty mapping. The condition (ii) implies that there exist x0,x1∈A0 such that Dx1,Tx0=DA,B and x0,x1∈EG. Also, the condition (iii) of Theorem 13 follows from the property of EG and the condition (iii). By applying Theorem 13, we have that BT≠∅. Moreover, if x∗,y∗∈BT, then x∗Ry∗ which implies that x∗,y∗∈EG. Again, by Theorem 13, x∗=y∗.Note that whenX is a JS∗-metric space, and the condition (iii) in the above corollary is replaced by the condition (iii) in Theorem 12, the result also follows.



---

*Source: 5681253-2020-12-01.xml*