# On Some Coincidence Best Proximity Point Results

Authors: Naeem Saleem; Haroon Ahmad; Hassen Aydi; Yaé Ulrich Gaba
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we discuss some (coincidence) best proximity point results for generalized proximal contractions andλ−μ-proximal Geraghty contractions in controlled metric type spaces. To clarify our study, various examples are given and some conclusions are drawn.


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## Body

## 1. Introduction and Preliminaries

To solve the equationTˇp=p (Tˇ is a mapping defined on a subset of a metric space, a simplified linear space, or a topological vector space), fixed point theory is an important tool. A nonself-mapping Tˇ:J⟶K may not have a fixed point. From this perspective, the best approximation theorem and the best proximity point are relevant. A classical best approximation theorem was due to Fan [1], i.e., if J is a nonempty compact convex subset of a Hausdorff locally convex topological vector space X with a seminorm p and Tˇ:J⟶X is a continuous mapping, then there is an element p in J satisfying the condition that Ψp,Tˇp=ΨTˇp,J. Many subsequent extensions and variations of Fan’s theorem have occurred, including references [2, 3].However, even though the best approximation theorems provide an approximate solution to the equationTˇp=p, they do not provide an ideal approximate solution. Moreover, the theorem of the best proximity point specifies adequate criteria for the presence of an element p to reduce the error Ψp,Tˇp. For a nonself-mapping Tˇ:J⟶K, Ψp,Tˇp is at least ΨJ,K for all p in J, then the best proximity point theorem establishes a globally optimal solution of error Ψp,Tˇp by constraining an approximate solution p of the equation Tˇp=p to the condition that Ψp,Tˇp=ΨJ,K. Such an ideal approximate solution Tˇp=p is the best proximity point of the nonself-mapping Tˇ:J⟶K. For sure, the best proximity point hypotheses are a logical augmentation of fixed point hypotheses, on the grounds that the best proximity point is a fixed point in the light of self-mappings.The best proximity point hypotheses have been demonstrated in [4]. Anuradha and Veeramani have tested the proximal pointwise contractions for the presence of a best proximity point [2]. Generally, several best proximity point theorems were analyzed for multiple variants of contractions in [5–14]. A best proximity point theorem for contraction mappings was presented in [15]. Some interesting common best proximity theorems have been discussed in [7, 15].Nadler [16] was the first who generalized the Banach contraction principle for multivaluated mappings. Later, several works appeared in this direction. For more details, see [17–20]. The best proximity point hypotheses for different sorts of multivalued mappings have likewise been obtained in [21, 22].Recently, the authors in [23] introduced a controlled metric type space in which the function of extended b-metric spaces was substituted by a function αp,q depending on the parameters of the left-hand side of the triangular inequality. The primary goal of this article is to include the best proximity point theorems for generalized and modified proximal contractions in the context of complete controlled type metric spaces, thus providing an optimal approximate solution to the equation Tˇp=p. It is acknowledged that the previous best proximity point theorems include the well-known Banach contraction principle and some of its generalizations.First, we state the following useful definitions in the sequel.Definition 1.
(see [6]). Let X,Ψ be a metric space having a pair of nonempty subsets J,K such that J0 is nonempty. The pair J,K has the P-property if and only if(1)Ψp1,q1=ΨJ,KΨp2,q2=ΨJ,K,impliesΨp1,p2=Ψq1,q2,where p1,p2∈J0 and q1,q2∈K0.Definition 2.
(see [24]). Let X,Ψ be a metric space having a pair of nonempty subsets J and K. Let Tˇ:J⟶K and μ:J×J⟶0,∞. The mapping Tˇ is said to be μ-proximal admissible if(2)μp1,p2≥1Ψu1,Tˇp1=ΨJ,KΨu2,Tˇp2=ΨJ,K,impliesμu1,u2≥1,for all p1,p2,u1,u2∈J.Definition 3.
(see [25]). Let ℬX represent the closed and bounded subsets of X. Let H be the Pompeiu–Hausdroff metric induced by metric Ψ defined by(3)HJ,K=maxsupa∈JDa,K,supb∈KDb,J,for J,K⊆ℬX, where(4)Da,K=infΨa,b:b∈K.Definition 4.
(see [23]). Let X be a nonempty set, and consider α:X×X⟶1,∞ as a function. Let Ψ:X×X⟶0,∞ satisfying(1)
Ψp1,p2=0 if and only if p1=p2(2)
Ψp1,p2=Ψp2,p1(3)
Ψp1,p2≤αp1,p3Ψp1,p3+αp3,p2Ψp3,p2, for all p1,p2,p3∈X, then X,Ψ is called a controlled metric type space
From now on,X,Ψ is a controlled metric type space.Definition 5.
(see [23]). A sequence pn in a controlled metric type space X,Ψ converges to some p in X if for each positive ε, there is some positive Nε such that Ψpn,p<ε for each n≥Nε. It can be written as(5)limn⟶∞pn=p.Definition 6.
(see [23]). The sequence pn in a controlled metric type space X,Ψ is said to be a Cauchy sequence, if for every ε>0, Ψpn,pm<ε for all m,n≥Nε, where Nε∈ℕ.Definition 7.
(see [23]). A controlled metric type space X,Ψ is said to be complete if every Cauchy sequence is convergent in X.Definition 8.
(see [23]). Let p∈X and ε>0.(1)
The open ballKp,ε is defined as follows:(6)Kp,ε=q∈X,Ψp,q<ε.(2)
The mappingTˇ:X⟶X is said continuous at p∈X if for all ε>0, there exists δ>0 such that(7)TˇKp,δ⊆KTˇp,ε.
Clearly, ifTˇ is continuous at p in the controlled metric type space X,Ψ, then pn⟶p implies that Tˇpn⟶Tˇp as n⟶∞.Definition 9.
(see [26]). Define the function ℋ:ℒX×ℒX⟶0,∞ by(8)ℋJ,K=maxsupa∈JDa,K,supb∈KDb,J,if the maximum exists,∞,otherwise,for J,K⊆ℒX (it represents the set of closed subsets of X), where(9)Da,K=infΨa,b:b∈K,forK⊂X.
LetJ and K be two nonempty subsets of X. Define(10)J0=p∈J:Ψp,q=ΨJ,Kfor someq∈K,K0=q∈K:Ψp,q=ΨJ,Kfor somep∈J,where(11)ΨJ,K=infΨp,q:p∈J,q∈Kdistance of a setJto a setK,and we will denote(12)D∗p,q=Ψp,q−ΨJ,K,for allp∈J,q∈K.Theorem 1 (see [26]).
The functionℋ:ℒX×ℒX⟶0,∞ is a generalized Pompeiu–Hausdroff controlled metric space on ℒX.Remark 1.
(see [26]). Let ℒX,H be a generalized Pompeiu–Hausdroff-controlled metric type space. Then, the following assertions hold (for all bounded and closed subsets J,K,C,andD of X):(1)
HC,D=0 is equivalent to C=D(2)
HC,D=HD,C(3)
HJ,C≤maxsupa∈Jαa,b,αb,JHJ,K+maxαb,C,supc∈Cαc,b,HK,CTheorem 2 (see [26]).
IfX,Ψ is a complete controlled metric space with limn,m⟶∞αpn,pmk<1, for all pn,pm∈X, where k≥1, then ℒX,ℋ is complete.


## 2. Coincidence Best Proximity Points for Generalized Proximal Contractions

In this section, we will discuss some best proximity point theorems using the multivalued concept on a controlled metric spaceX,Ψ.From now and onward,J and K are nonempty subsets of a controlled metric type space X,Ψ (until otherwise stated). Define α:X×X⟶1,∞ by α∗p,J=infαp,a,for alla∈J and α∗J,K=infαa,b,for alla∈Jandb∈K, where α:X×X⟶1,∞ and J and K are nonempty subsets of X.Definition 10.
(see [26]). A mapping Tˇ:X⟶ℬX is continuous in a controlled metric type space X,Ψ at p∈X if for all ε>0, there exists δ>0 such that(13)TˇKp,δ⊆KTˇp,ε,where Kp,ε is given as(14)Kp,ε=q∈X,Ψp,q<ε.
Clearly, ifTˇ is continuous at p, then pn⟶p implies that Tˇpn⟶Tˇp as n⟶∞.
We introduce the following.Definition 11.
LetX,Ψ be a controlled metric type space having two nonempty subsets J and K. Let Tˇ:J⟶K be a mapping. A point p∈J is said to be a best proximity point of the mapping Tˇ if(15)Ψp,Tˇp=ΨJ,K.Definition 12.
LetX,Ψ be a controlled metric type space having two nonempty subsets J and K. A nonempty set J is said to be approximately compact with respect to K if every sequence pn in J satisfying the condition that Dq,pn⟶Dq,J for some q in K has a convergent subsequence.Definition 13.
GivenTˇ:J⟶ℬK and g⌣:J⟶J. A pair of mappings g⌣,Tˇ is said to be a β-generalized proximal contraction if there exists a real number β∈0,1 such that(16)Dg⌣u1,Tˇp1=ΨJ,KDg⌣u2,Tˇp2=ΨJ,K,impliesℋTˇu1,Tˇu2≤βℋTˇp1,Tˇp2,for all u1,u2,p1,andp2 in J.Definition 14.
A mappingTˇ:J⟶ℬK is said to be a βTˇ-generalized proximal contraction if there exists β∈0,1 such that(17)Du1,Tˇp1=ΨJ,KDu2,Tˇp2=ΨJ,K,impliesℋTˇu1,Tˇu2≤βℋTˇp1,Tˇp2,for all u1,u2,p1,andp2 in J.
Note that, if we takeg⌣=IJ (the identity mapping on J), then every β-generalized proximal contraction will reduce to a βTˇ-generalized proximal contraction.Definition 15.
LetX,Ψ be a controlled metric type space having two nonempty subsets J and K. Let Tˇ:J⟶K and g⌣:J⟶J be mappings. A point p∈J is said to be a coincidence best proximity point of the pair of mappings g⌣,Tˇ if(18)Ψg⌣p,Tˇp=ΨJ,K.Remark 2.
If we takeg⌣=IJ (the identity mapping over J), then every coincidence best proximity point becomes a best proximity point of the mapping Tˇ.
IfJ∩K≠∅ or ΨJ,K=0, then every best proximity point will reduce to a fixed point of the mapping Tˇ.
Our first main result is stated as follows:Theorem 3.
LetX,Ψ be a controlled metric type space having two nonempty subsets J and K. Let Tˇ:J⟶ℬK and g⌣:J⟶J be one-to-one and continuous mappings. Assume that K is a closed subset and J is approximately compact with respect to K with TˇJ0⊆K0 and J0⊆g⌣J0. Further, assume that the pair g⌣,Tˇ is a β-generalized proximal contraction such that(19)supm≥1limi⟶∞maxsupqi∈Tˇpiαqi,qi+1,αqi+1,Tˇpimaxsupqi∈Tˇpiαqi,qm,αqm,Tˇpi<1k,and limn⟶∞α∗g⌣pn,Tˇpn−1=1, where k∈0,1. Then, there exists a coincidence best proximity point of the pair g⌣,Tˇ.Proof.
Letp0 be an arbitrary element in J0. Since TˇJ0 is contained in K0 and J0 is contained in g⌣J0, there exists an element p1 in J0 such that(20)Dg⌣p1,Tˇp0=ΨJ,K.
Again, sinceTˇp1 is an element of TˇJ0 which is contained in K0 and J0 is contained in g⌣J0, it follows that there is an element p2 in J0 such that(21)Dg⌣p2,Tˇp1=ΨJ,K.
This process can be continued by selectingpn in J0 satisfying the condition as follows:(22)Dg⌣pn,Tˇpn−1=ΨJ,K.
Having selectedpn satisfying the condition, there exists an element pn+1 in J0 satisfying(23)Dg⌣pn+1,Tˇpn=ΨJ,K,for every integer n≥0.
Since the pairg⌣,Tˇ is a β-generalized proximal contraction, by using equations (22) and (23), we obtain(24)ℋTˇpn+1,Tˇpn≤βℋTˇpn,Tˇpn−1,for eachn≥1.
We deduce that(25)ℋTˇpn+1,Tˇpn≤βnℋTˇp1,Tˇp0,for eachn≥0.
Now, we have to prove thatTˇpn is a Cauchy sequence, for all natural numbers n,m∈ℕ with n<m,(26)ℋTˇpn,Tˇpm≤maxsupqn∈Tˇpnαqn,qn+1,αqn+1,TˇpnℋTˇpn,Tˇpn+1+maxsupqn+1∈Tˇpn+1αqn+1,qm,αqm,Tˇpn+1ℋTˇpn+1,Tˇpm≤maxsupqn∈Tˇpnαqn,qn+1,αqn+1,TˇpnℋTˇpn,Tˇpn+1+maxsupqn+1∈Tˇpn+1αqn+1,qn+2,αqn+2,Tˇpn+1maxsupqn+1∈Tˇpn+1αqn+1,qm,αqm,Tˇpn+1ℋTˇpn+1,Tˇpn+2+maxssupqn+1∈Tˇpn+1αqn+1,qm,αqm,Tˇpn+1maxsupqn+2∈Tˇpn+2αqn+2,qm,αqm,Tˇpn+2ℋTˇpn+2,Tˇpm≤maxsupqn∈Tˇpnαqn,qn+1,αqn+1,TˇpnℋTˇpn,Tˇpn+1+∑i=n+1m−2∏j=n+1imaxsupqj∈Tˇpjαqj,qm,αqm,Tˇpjmaxsupqi∈Tˇpiαqi,qi+1,αqi+1,TˇpiℋTˇpi,Tˇpi+1+∏k=n+1m−1maxsupqnk∈Tˇpkαqk,qm,αqm,TˇpkℋTˇpm−1,Tˇpm≤maxsupqn∈Tˇpnαqn,qn+1,αqn+1,TˇpnβnℋTˇp0,Tˇp1+∑i=n+1m−2∏j=n+1imaxsupqj∈Tˇpjαqj,qm,αqm,Tˇpjmaxsupqi∈Tˇpiαqi,qi+1,αqi+1,TˇpiβiℋTˇp0,Tˇp1+∏k=n+1m−1maxsupqnk∈Tˇpkαqk,qm,αqm,Tˇpkβm−1ℋTˇp0,Tˇp1≤maxsupqn∈Tˇpnαqn,qn+1,αqn+1,TˇpnβnℋTˇp0,Tˇp1+∑i=n+1m−2∏j=n+1imaxsupqj∈Tˇpjαqj,qm,αqm,Tˇpjmaxsupqi∈Tˇpiαqi,qi+1,αqi+1,TˇpiβiℋTˇp0,Tˇp1+∏k=n+1m−1maxsupqnk∈Tˇpkαqk,qm,αqm,Tˇpkmaxsupqm−1∈Tˇpm−1αqm−1,qm,αqm,Tˇpm−1βm−1ℋTˇp0,Tˇp1=maxsupqn∈Tˇpnαqn,qn+1,αqn+1,TˇpnβnℋTˇp0,Tˇp1+∑i=n+1m−2∏j=n+1imaxsupqj∈Tˇpjαqj,qm,αqm,Tˇpjmaxsupqi∈Tˇpiαqi,qi+1,αqi+1,TˇpiβiℋTˇp0,Tˇp1≤maxsupqn∈Tˇpnαqn,qn+1,αqn+1,TˇpnβnℋTˇp0,Tˇp1+∑i=n+1m−2∏j=0imaxsupqj∈Tˇpjαqj,qm,αqm,Tˇpjmaxsupqi∈Tˇpiαqi,qi+1,αqi+1,TˇpiβiℋTˇp0,Tˇp1.
Assume that(27)Sm−2=∑i=n+1m−2∏j=0imaxsupqj∈Tˇpjαqj,qm,αqm,Tˇpjmaxsupqi∈Tˇpiαqi,qi+1,αqi+1,Tˇpiβi.
Then, we obtain(28)ℋTˇpn,Tˇpm≤ℋTˇp0,Tˇp1βnmaxsupqn∈Tˇpnαqn,qn+1,αqn+1,Tˇpn+Sm−1−Sn.
Using the ratio test, we have(29)ai=∏j=0imaxsupqj∈Tˇpjαqj,qm,αqm,Tˇpjmaxsupqi∈Tˇpiαqi,qi+1,αqi+1,Tˇpiβi,where ai+1/ai<1/k. Taking limit as n,m⟶∞, we obtain(30)limn⟶∞ℋTˇpn,Tˇpm=0.
That is,Tˇpn is a Cauchy sequence in the complete generalized Pompeiu–Hausdroff controlled metric type space ℬX,ℋ; hence, it converges to some q in K (as the set K is closed). Therefore,(31)Ψq,J≤Ψq,g⌣pn≤α∗q,Tˇpn−1Dq,Tˇpn−1+α∗Tˇpn−1,g⌣pnDTˇpn−1,g⌣pn=α∗q,Tˇpn−1Dq,Tˇpn−1+α∗Tˇpn−1,g⌣pnΨJ,K.
Takinglimn⟶∞ on both sides of the above inequality, we have(32)limn⟶∞Ψq,g⌣pn≤limn⟶∞α∗q,Tˇpn−1Dq,Tˇpn−1+α∗Tˇpn−1,g⌣pnΨJ,K≤ΨJ,K≤Ψq,J.
Therefore,Ψq,g⌣pn⟶Ψq,J. In view of the fact that J is approximately compact with respect to K, g⌣pn has a subsequence g⌣pnk converging to some z=g⌣p∈J for some p∈J0. Thus,(33)Ψz,q=limk⟶∞Dg⌣pnk,Tˇpnk−1=ΨJ,K.
Therefore,z is a member of J0. Since J0 is contained in g⌣J0 and z=g⌣p for some p in J0, g⌣pnk⟶g⌣p and g⌣ is a one-to-one continuous mapping, so pnk⟶p. Since Tˇ is continuous, it can be concluded that Tˇpnk⟶Tˇp. This implies that(34)Dg⌣p,Tˇp=limk⟶∞Dg⌣pnk,Tˇpnk−1=ΨJ,K.
That is,p is a coincidence best proximity point of the pair g⌣,Tˇ.
To prove the uniqueness of the coincidence best proximity point of the pair of mappingsg⌣,Tˇ, suppose that there is another coincidence best proximity point q≠p of the pair g⌣,Tˇ. We have(35)Dg⌣p,Tˇp=ΨJ,K,Dg⌣q,Tˇq=ΨJ,K.
As the mappingTˇ is one-to-one on the set J and p≠q, one has ℋTˇp,Tˇq>0. Since the pair g⌣,Tˇ is a β-generalized proximal contraction, one can write(36)0<ℋTˇp,Tˇq≤βℋTˇp,Tˇq<ℋTˇp,Tˇq.
It is a contradiction.Corollary 1.
LetTˇ:J⟶ℬK and α∗:X×X⟶1,∞ be mappings, where K is a closed subset and J is approximately compact with respect to K with TˇJ0⊆K0. Suppose that Tˇ is a continuous and βTˇ-generalized proximal contraction such that(37)supm≥1limi⟶∞α∗pi,pi+1α∗pi,pm<1k,limn⟶∞α∗pn,Tˇpn−1=1,wherek∈0,1,then there exists a unique best proximity point of Tˇ.Proof.
If we take identity mappingg⌣=IJ (g⌣ is identity on J), the remaining proof is same as in Theorem 3Definition 16.
LetTˇ:J⟶K and g⌣:J⟶J. A pair of mappings g⌣,Tˇ is said to be a β-modified proximal contraction if there exists β∈0,1 such that(38)Ψg⌣u1,Tˇp1=ΨJ,KΨg⌣u2,Tˇp2=ΨJ,K,impliesΨTˇu1,Tˇu2≤βΨTˇp1,Tˇp2,for all u1,u2,p1,andp2 in J.Definition 17.
A mappingTˇ:J⟶K is said to be a βTˇ-modified proximal contraction if there exists β∈0,1 such that(39)Ψu1,Tˇp1=ΨJ,KΨu2,Tˇp2=ΨJ,K,impliesΨTˇu1,Tˇu2≤βΨTˇp1,Tˇp2,for all u1,u2,p1,andp2 in J.
Note that if we takeg⌣=IJ (the identity mapping on J), then every β-modified proximal contraction is a βTˇ-modified proximal contraction.Theorem 4.
LetTˇ:J⟶K and g⌣:J⟶J be two continuous and one-to-one mappings, where K is a closed subset and J is approximately compact with respect to K with TˇJ0⊆K0 and J0⊆g⌣J0. If the pair g⌣,Tˇ is a β-modified proximal contraction and(40)supm≥1limi⟶∞αpi,pi+1αpi,pm<1k,limn⟶∞αg⌣pn,Tˇpn−1=1,wherek∈0,1,then there exists a unique coincidence best proximity point of the pair g⌣,Tˇ.Proof.
Letp0 be an arbitrary element in J0. Since TˇJ0 is contained in K0 and J0 is contained in g⌣J0, there exists an element p1 in J0 such that(41)Ψg⌣p1,Tˇp0=ΨJ,K.
SinceTˇp1 is an element of TˇJ0 which is contained in K0 and J0 is contained in g⌣J0, it follows that there exists an element p2 in J0 such that(42)Ψg⌣p2,Tˇp1=ΨJ,K.
By continuing this process, we can construct a sequencepn in J0, satisfying the condition as follows:(43)Ψg⌣pn,Tˇpn−1=ΨJ,K.
Having chosenpn in J0, there exists an element pn+1 in J0, such that(44)Ψg⌣pn+1,Tˇpn=ΨJ,K,for every positive integer n. Since the pair g⌣,Tˇ is a β-modified proximal contraction from equations (43) and (44), we obtain(45)ΨTˇpn+1,Tˇpn≤βΨTˇpn,Tˇpn−1.
Recursively, we have(46)ΨTˇpn+1,Tˇpn≤βnΨTˇp1,Tˇp0.
Now, we have to prove thatTˇpn is a Cauchy sequence. For all natural numbers n,m∈ℕ with n<m, we have(47)ΨTˇpn,Tˇpm≤αTˇpn,Tˇpn+1ΨTˇpn,Tˇpn+1+αTˇpn+1,TˇpmΨTˇpn+1,Tˇpm≤αTˇpn,Tˇpn+1ΨTˇpn,Tˇpn+1+αTˇpn+1,TˇpmαTˇpn+1,Tˇpn+2ΨTˇpn+1,Tˇpn+2+αTˇpn+1,TˇpmαTˇpn+2,TˇpmΨTˇpn+2,Tˇpm≤αTˇpn,Tˇpn+1ΨTˇpn,Tˇpn+1+αTˇpn+1,TˇpmαTˇpn+1,Tˇpn+2ΨTˇpn+1,Tˇpn+2+αTˇpn+1,TˇpmαTˇpn+2,TˇpmαTˇpn+2,Tˇpn+3ΨTˇpn+2,Tˇpn+3+αTˇpn+1,TˇpmαTˇpn+2,TˇpmαTˇpn+3,TˇpmΨTˇpn+3,Tˇpm≤αTˇpn,Tˇpn+1ΨTˇpn,Tˇpn+1+∑i=n+1m−2∏j=n+1iαTˇpj,TˇpmαTˇpi,Tˇpi+1ΨTˇpi,Tˇpi+1+∏k=n+1m−1αTˇpk,TˇpmΨTˇpm−1,Tˇpm≤αTˇpn,Tˇpn+1βnΨTˇp0,Tˇp1+∑i=n+1m−2∏j=n+1iαTˇpj,TˇpmαTˇpi,Tˇpi+1βiΨTˇp0,Tˇp1+∏k=n+1m−1αTˇpk,Tˇpmβm−1ΨTˇp0,Tˇp1≤αTˇpi,Tˇpi+1βnΨTˇp0,Tˇp1+∑i=n+1m−2∏j=n+1iαTˇpj,TˇpmαTˇpi,Tˇpi+1βiΨTˇp0,Tˇp1+∏k=n+1m−1αTˇpk,TˇpmαTˇpm−1,Tˇpmβm−1ΨTˇp0,Tˇp1=αTˇpn,Tˇpn+1βnΨTˇp0,Tˇp1+∑i=n+1m−1∏j=n+1iαTˇpj,TˇpmαTˇpi,Tˇpi+1βiΨTˇp0,Tˇp1≤αTˇpn,Tˇpn+1βnΨTˇp0,Tˇp1+∑i=n+1m−1∏j=0iαTˇpj,TˇpmαTˇpi,Tˇpi+1βiΨTˇp0,Tˇp1.
Assume that(48)Sl=∑i=0l∏j=0iαTˇpj,TˇpmαTˇpi,Tˇpi+1βi.
It follows that(49)ΨTˇpn,Tˇpm≤ΨTˇp0,Tˇp1βnαTˇpn,Tˇpn+1+Sm−1−Sn.
Using the ratio test, we have(50)ai=∏j=0iαTˇpj,TˇpmαTˇpi,Tˇpi+1βi,whereai+1ai<1k.
By applying limitm,n⟶∞ in inequality (49), we get(51)limn⟶∞ΨTˇpn,Tˇpm=0,which shows that Tˇpn is a Cauchy sequence; hence, it is convergent to some q in K (as the set K is closed). Therefore,(52)Ψq,J≤Ψq,g⌣pn≤αq,Tˇpn−1Ψq,Tˇpn−1+αTˇpn−1,g⌣pnΨTˇpn−1,g⌣pn=αq,Tˇpn−1Ψq,Tˇpn−1+αTˇpn−1,g⌣pnΨJ,K.
Taking limitn⟶∞ on both sides of the above inequality, we have(53)limn⟶∞Ψq,g⌣pn≤limn⟶∞αq,Tˇpn−1Ψq,Tˇpn−1+αTˇpn−1,g⌣pnΨJ,K≤ΨJ,K≤Ψq,J.
Therefore,Ψq,g⌣pn⟶Ψq,J. In view of the fact that J is approximately compact with respect to K, g⌣pn has a subsequence g⌣pnk converging to some z=g⌣p∈J for some p∈J0. It follows that(54)Ψz,q=limk⟶∞Ψg⌣pnk,Tˇpnk−1=ΨJ,K.
Therefore,z is an element of J0. Since J0 is contained in g⌣J0, we have z=g⌣p for some p in J0. As g⌣pnk⟶g⌣p and g⌣ is a one-to-one continuous mapping, pnk⟶p. Since Tˇ is continuous, it can be concluded that Tˇpnk⟶Tˇp. Hence,(55)Ψg⌣p,Tˇp=limk⟶∞Ψg⌣pnk,Tˇpnk−1=ΨJ,K.
To prove the uniqueness, suppose thatq is another coincidence best proximity point of the pair g⌣,Tˇ such that p≠q. Then,(56)Ψg⌣p,Tˇp=ΨJ,K,Ψg⌣q,Tˇq=ΨJ,K.
Since the pairg⌣,Tˇ is a β-modified proximal contraction, we have(57)0<ΨTˇp,Tˇq≤βΨTˇp,Tˇq<ΨTˇp,Tˇq,which is a contradiction (as Tˇ is one-to-one mapping on J). Hence, the pair g⌣,Tˇ has a unique coincidence best proximity point.Corollary 2.
LetTˇ:J⟶K be a given continuous mapping, where K is a closed subset and J is approximately compact with respect to K with TˇJ0⊆K0. If Tˇ is a βTˇ-modified proximal contraction and suppose that(58)supm≥1limi⟶∞αpi,pi+1αpi,pm<1k,limn⟶∞αpn,Tˇpn−1=1,wherek∈0,1,then there exists a unique best proximity point of Tˇ.Proof.
If we takeg⌣=IJ (the identity mapping over the set J), the remaining proof is same as Theorem 4.Example 1.
LetX=0,1,2,3,4,5. Consider the function Ψ given as Ψp,p=0 and Ψp,q=Ψq,p, where
Takeα:X×X⟶1,∞ to be symmetric which is defined as αp,q=19p+21q. It is easy to see that X,Ψ is a controlled metric type space. Take J=0,1,2 and K=3,4,5. Obviously, ΨJ,K=1/15, J0=J, and K0=K. Now, consider Tˇ:J⟶K as follows:(59)Tˇp=3,ifp=0,1,5,ifp=2.
Clearly,TˇJ0⊆K0. Define g⌣:J⟶J by(60)g⌣p=0,ifp=0,1,ifp=2,2,ifp=1.
We getJ0⊆g⌣J0. Now, we have to show that the pair g⌣,Tˇ satisfies(61)Ψg⌣0,Tˇ1=Ψ0,3=ΨJ,K,Ψg⌣1,Tˇ2=Ψ2,5=ΨJ,K,where u1=0,u2=1,p1=1, and p2=2. Since the pair g⌣,Tˇ is a β-modified proximal contraction:(62)ΨTˇ0,Tˇ1≤βΨTˇ1,Tˇ2,for every β∈0,1, the pair g⌣,Tˇ is a β-modified proximal contraction. Hence, 0 is the unique coincidence best proximity point of Tˇ and g⌣.Definition 18.
LetTˇ:J⟶K and g⌣:J⟶J. A pair of mappings g⌣,Tˇ is said to be a β− proximal contraction if there exists β∈0,1 such that(63)Ψg⌣u1,Tˇp1=ΨJ,KΨg⌣u2,Tˇp2=ΨJ,K,impliesΨg⌣u1,g⌣u2≤βΨp1,p2,for all u1,u2,p1,andp2 in J.Definition 19.
A mappingTˇ:J⟶K is said to be a βTˇ-proximal contraction if there exists β∈0,1 such that(64)Ψu1,Tˇp1=ΨJ,KΨu2,Tˇp2=ΨJ,K,impliesΨu1,u2≤βΨp1,p2,for all u1,u2,p1,andp2 in J.
Note that, if we takeg⌣=IJ, then every β-proximal contraction is a βTˇ-proximal contraction.Theorem 5.
LetTˇ:J⟶K and g⌣:J⟶J be continuous mappings, where K is a closed subset and J is approximately compact with respect to K with TˇJ0⊆K0 and J0⊆g⌣J0. Suppose that g⌣ is an expansive mapping and the pair g⌣,Tˇ is a β-proximal contraction such that(65)supm≥1limi⟶∞αpi,pi+1αpi,pm<1k,limn⟶∞αg⌣pn,Tˇpn−1=1,wherek∈0,1,then there exists a unique coincidence best proximity point of the pair g⌣,Tˇ.Proof.
Letp0 be an arbitrary element in J0. Since TˇJ0 is contained in K0 and J0 is contained in g⌣J0, there exists an element p1 in J0 such that(66)Ψg⌣p1,Tˇp0=ΨJ,K.
Again, sinceTˇp1 is an element of TˇJ0 which is contained in K0 and J0 is contained in g⌣J0, it follows that there is an element p2 in J0, such that(67)Ψg⌣p2,Tˇp1=ΨJ,K.
This process can be continued by selectingpn in J0 so that(68)Ψg⌣pn+1,Tˇpn=ΨJ,K.
Since the pairg⌣,Tˇ is a β-proximal contraction, we have(69)Ψg⌣pn+1,g⌣pn≤βΨpn,pn−1.
Asg⌣ is an expansive mapping, one writes(70)Ψpn+1,pn≤Ψg⌣pn+1,g⌣pn≤βnΨp1,p0,so we have(71)Ψpn+1,pn≤βnΨp1,p0.
We claim thatpn is a Cauchy sequence. For all natural numbers n,m∈ℕ with n<m, we have(72)Ψpn,pm≤αpn,pn+1Ψpn,pn+1+αpn+1,pmΨpn+1,pm≤αpn,pn+1Ψpn,pn+1+αpn+1,pmαpn+1,pn+2Ψpn+1,pn+2+αpn+1,pmαpn+2,pmΨpn+2,pm≤αpn,pn+1Ψpn,pn+1+αpn+1,pmαpn+1,pn+2Ψpn+1,pn+2+αpn+1,pmαpn+2,pmαpn+2,pn+3Ψpn+2,pn+3+αpn+1,pmαpn+2,pmαpn+3,pmΨpn+3,pm≤αpn,pn+1Ψpn,pn+1+∑i=n+1m−2∏j=n+1iαpj,pmαpi,pi+1Ψpi,pi+1+∏k=n+1m−1αpk,pmΨpm−1,pm≤αTˇpn,Tˇpn+1βnΨp0,p1+∑i=n+1m−2∏j=n+1iαpj,pmαpi,pi+1βiΨp0,p1+∏k=n+1m−1αpk,pmβm−1Ψp0,p1≤αpn,pn+1βnΨp0,p1+∑i=n+1m−2∏j=n+1iαpj,pmαpi,pi+1βiΨp0,p1+∏k=n+1m−1αpk,pmαpm−1,pmβm−1Ψp0,p1=αpn,pn+1βnΨp0,p1+∑i=n+1m−1∏j=n+1iαpj,pmαpi,pi+1βiΨp0,p1≤αpn,pn+1βnΨp0,p1+∑i=n+1m−1∏j=0iαpj,pmαpi,pi+1βiΨp0,p1.
Assume that(73)Sl=∑i=0l∏j=0iαpj,pmαpi,pi+1βi.
Then, we obtain(74)Ψpn,pm≤Ψp0,p1βnαpn,pn+1+Sm−1−Sn.
Using the ratio test, we have(75)ai=∏j=0iαpj,pmαpi,pi+1βi,whereai+1ai<1k.
By taking limit asn,m⟶∞, (74) becomes(76)limn⟶∞Ψpn,pm=0.
Therefore,pn is a Cauchy sequence in the complete controlled metric type space X,Ψ; hence, it is convergent to some p in J (as set J is closed). Since g⌣ and Tˇ are continuous, we have(77)Ψg⌣p,Tˇp=limn⟶∞Ψg⌣pn+1,Tˇpn=ΨJ,K.
Hence,p is the unique coincidence best proximity point of the pair g⌣,Tˇ. To prove the uniqueness, suppose that q is another coincidence best proximity point of the pair g⌣,Tˇ such that p≠q. Then,(78)Ψg⌣p,Tˇp=ΨJ,K,Ψg⌣q,Tˇq=ΨJ,K.
Since the pairg⌣,Tˇ is a β-modified proximal contraction, we have(79)Ψp,q≤Ψg⌣p,g⌣q≤βΨp,q<Ψp,q,which is a contradiction. Hence, the pair g⌣,Tˇ has a unique coincidence best proximity point.Corollary 3.
LetTˇ:J⟶K be a continuous mapping, where K is closed subset and J is approximately compact with respect to K with TˇJ0⊆K0. If Tˇ is a βTˇ-proximal contraction and suppose that(80)supm≥1limi⟶∞αpi,pi+1αpi,pm<1k,limn⟶∞αpn,Tˇpn−1=1,wherek∈0,1,then there exists a best proximity point of Tˇ.Proof.
If we take identity mappingg⌣=IJ, the remaining proof is same as Theorem 5.


## 3. Coincidence Best Proximity Points for Geraghty Type Proximal Contractive Mappings

First, we need to define a generalized Geraghty type proximal contractive mapping.From now and onward,F is a class of all nondecreasing functions λ:0,∞⟶0,1 such that for any bounded sequence tn of positive real numbers, λtn⟶1 implies tn⟶0.Definition 20.
LetX,Ψ be a controlled metric type space having a pair of nonempty subsets J,K such that J0 is nonempty. Then, a pair J,K has the P-property if and only if(81)Ψp1,q1=ΨJ,KΨp2,q2=ΨJ,KimpliesΨp1,p2=Ψq1,q2.Definition 21.
LetTˇ:J⟶ℬK, g⌣:J⟶J are mappings. A pair g⌣,Tˇ is said to be a λ−μ-proximal Geraghty contraction if μ:J×J⟶0,∞ is such that(82)μp,q≥1Dg⌣u,Tˇp=ΨJ,KDg⌣v,Tˇq=ΨJ,Kimplies thatμp,qℋTˇp,Tˇq≤λMu,v,p,qMu,v,p,q,where(83)Mu,v,p,q=maxΨg⌣p,g⌣q,Dg⌣p,Tˇp−α∗g⌣q,TˇpΨJ,Kα∗g⌣p,g⌣q,D∗g⌣u,Tˇp,Dg⌣u,Tˇq−α∗g⌣v,TˇqΨJ,Kα∗g⌣u,g⌣v,for all u,v,p,q∈J, where λ∈F.Definition 22.
A mappingTˇ:J⟶ℬK is said to be a λ−μTˇ-proximal Geraghty contraction if μ:J×J⟶0,∞ is such that(84)μp,q≥1Du,Tˇp=ΨJ,KDv,Tˇq=ΨJ,K,implies thatμp,qℋTˇp,Tˇq≤λMu,v,p,qMu,v,p,q,where(85)Mu,v,p,q=maxΨp,q,Ψp,Tˇp−α∗q,TˇpΨJ,Kα∗p,q,Ψ∗u,Tˇp,Ψu,Tˇq−α∗v,TˇqΨJ,Kα∗u,v,for all u,v,p,q∈J, where λ∈F.
If we takeg⌣=IJ (the identity mapping over J), then every λ−μ-proximal Geraghty contraction will reduce to a λ−μ-generalized proximal Geraghty contraction.Theorem 6.
LetTˇ:J⟶ℬK, g⌣:J⟶J, and μ:J×J⟶0,+∞ be mappings, where J is a closed subset and the pair J,K satisfies the P-property with TˇJ0⊆K0 and J0⊆g⌣J0. If a pair of continuous mappings g⌣,Tˇ is a λ−μ-proximal Geraghty contraction, where Tˇ is μ-proximal admissible, then there exist elements p0,p1∈J0 such that Dg⌣p1,Tˇp0=ΨJ,K and μp0,p1≥1. If pn is a sequence in J such that μpn,pn+1≥1 and suppose that(86)supm≥1limi⟶∞α∗pi,pi+1α∗pi,pm<1k,wherek∈0,1,then the pair g⌣,Tˇ has a unique coincidence best proximity pointp∗∈J.Proof.
From the given condition, there existp0,p1∈J0 such that Dg⌣p1,Tˇp0=ΨJ,K and μp0,p1≥1. As TˇJ0⊆K0, there exists p2∈J0 such that Dg⌣p2,Tˇp1=ΨJ,K. As Tˇ is μ-proximal admissible, μp0,p1≥1,(87)Dg⌣p1,Tˇp0=ΨJ,K,Dg⌣p2,Tˇp1=ΨJ,K,using the P-property Ψg⌣p1,g⌣p2=ℋTˇp0,Tˇp1. Since the pair g⌣,Tˇ is a λ−μ-proximal Geraghty contraction with μp1,p2≥1, we have(88)Ψg⌣p1,g⌣p2≤λMp0,p1,p1,p2Mp0,p1,p1,p2,where(89)Mp0,p1,p1,p2≤maxΨg⌣p0,g⌣p1,Dg⌣p0,Tˇp0−α∗g⌣p1,Tˇp0ΨJ,Kα∗g⌣p0,g⌣p1,D∗g⌣p1,Tˇp0,Dg⌣p1,Tˇp1−α∗g⌣p2,Tˇp1ΨJ,Kα∗g⌣p1,g⌣p2≤maxΨg⌣p0,g⌣p1,α∗g⌣p0,g⌣p1Ψg⌣p0,g⌣p1+α∗g⌣p1,Tˇp0Dg⌣p1,Tˇp0α∗g⌣p0,g⌣p1−α∗g⌣p1,Tˇp0ΨJ,Kα∗g⌣p0,g⌣p1,Dg⌣p1,Tˇp0−ΨJ,K,α∗g⌣p1,g⌣p2Ψg⌣p1,g⌣p2α∗g⌣p1,g⌣p2+α∗g⌣p2,Tˇp1Dg⌣p2,Tˇp1−α∗g⌣p2,Tˇp1ΨJ,Kα∗g⌣p1,g⌣p2≤maxΨg⌣p0,g⌣p1,Ψg⌣p0,g⌣p1,0,Ψg⌣p1,g⌣p2,and we have(90)Mp0,p1,p1,p2≤maxΨg⌣p0,g⌣p1,Ψg⌣p1,g⌣p2.
If(91)maxΨg⌣p0,g⌣p1,Ψg⌣p1,g⌣p2=Ψg⌣p1,g⌣p2,then inequality (88) becomes(92)Ψg⌣p1,g⌣p2≤λΨg⌣p1,g⌣p2Ψg⌣p1,g⌣p2,which is a contradiction. So, we can conclude that(93)Ψg⌣p1,g⌣p2≤λΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1.
Further, by the fact thatTˇJ0⊆K0, there exists p3∈J0 such that Dg⌣p3,Tˇp2=ΨJ,K. As Tˇ is μ-proximal admissible mapping, where μp2,p3≥1,(94)Dg⌣p2,Tˇp1=ΨJ,K,Dg⌣p3,Tˇp2=ΨJ,K,using the P-Property, we have Ψg⌣p2,g⌣p3=ℋTˇp1,Tˇp2. Since the pair g⌣,Tˇ is a λ−μ-proximal Geraghty mapping with μp2,p3≥1, one writes(95)Ψg⌣p2,g⌣p3≤λMp2,p3,p1,p2Mp2,p3,p1,p2,where(96)Mp2,p3,p1,p2≤maxΨg⌣p1,g⌣p2,Dg⌣p1,Tˇp1−α∗g⌣p2,Tˇp1ΨJ,Kα∗g⌣p1,g⌣p2,D∗g⌣p2,Tˇp1,Dg⌣p2,Tˇp2−α∗g⌣p3,Tˇp2ΨJ,Kα∗g⌣p2,g⌣p3≤maxΨg⌣p1,g⌣p2,α∗g⌣p1,g⌣p2Ψg⌣p1,g⌣p2+α∗g⌣p2,Tˇp1Dg⌣p2,Tˇp1α∗g⌣p1,g⌣p2−α∗g⌣p2,Tˇp1ΨJ,Kα∗g⌣p1,g⌣p2,Dg⌣p2,Tˇp1−ΨJ,K,α∗g⌣p2,g⌣p3Ψg⌣p2,g⌣p3α∗g⌣p2,g⌣p3+α∗g⌣p3,Tˇp2Dg⌣p3,Tˇp2−α∗g⌣p3,Tˇp2ΨJ,Kα∗g⌣p2,g⌣p3≤maxΨg⌣p1,g⌣p2,Ψg⌣p1,g⌣p2,0,Ψg⌣p2,g⌣p3,and we have(97)Mp2,p3,p1,p2≤maxΨg⌣p1,g⌣p2,Ψg⌣p2,g⌣p3.
If(98)maxΨg⌣p1,g⌣p2,Ψg⌣p2,g⌣p3=Ψg⌣p2,g⌣p3,then inequality (95) becomes(99)Ψg⌣p2,g⌣p3≤λΨg⌣p2,g⌣p3Ψg⌣p2,g⌣p3,which is a contradiction. Thus,(100)Ψg⌣p2,g⌣p3≤λΨg⌣p1,g⌣p2Ψg⌣p1,g⌣p2.
Similarly, we can construct a sequencepn⊆J0, where μpn,pn+1≥1 for all n∈ℕ∪0,(101)Dg⌣pn,Tˇpn−1=ΨJ,K,Dg⌣pn+1,Tˇpn=ΨJ,K,using the P-property Ψg⌣pn,g⌣pn+1=ℋTˇpn,Tˇpn−1. Since the pair g⌣,Tˇ is a λ−μ-proximal Geraghty contraction with μpn,pn+1≥1, we get(102)Ψg⌣pn,g⌣pn+1≤λMpn,pn+1,pn−1,pnMpn,pn+1,pn−1,pn,where(103)Mpn,pn+1,pn−1,pn≤maxΨg⌣pn−1,g⌣pn,Dg⌣pn−1,Tˇpn−1−α∗g⌣pn,Tˇpn−1ΨJ,Kα∗g⌣pn−1,g⌣pn,D∗g⌣pn,Tˇpn−1,Dg⌣pn,Tˇpn−α∗g⌣pn+1,TˇpnΨJ,Kα∗g⌣pn,g⌣pn+1≤maxΨg⌣pn−1,g⌣pn,α∗g⌣pn−1,g⌣pnΨg⌣pn−1,g⌣pn+α∗g⌣pn,Tˇpn−1Dg⌣pn,Tˇpn−1α∗g⌣pn−1,g⌣pn−α∗g⌣pn,Tˇpn−1ΨJ,Kα∗g⌣pn−1,g⌣pnDg⌣pn,Tˇpn−1−ΨJ,Kα∗g⌣pn,g⌣pn+1Ψg⌣pn,g⌣pn+1+α∗gpn+1,TˇpnDg⌣pn+1,Tˇpnα∗g⌣pn,g⌣pn+1−α∗g⌣pn+1,TˇpnΨJ,Kα∗g⌣pn,g⌣pn+1≤maxΨg⌣pn−1,g⌣pn,Ψg⌣pn−1,g⌣pn,0,Ψg⌣pn,g⌣pn+1.
After simplification, we have(104)Mpn,pn+1,pn−1,pn≤maxΨg⌣pn−1,g⌣pn,Ψg⌣pn,g⌣pn+1.
If(105)maxΨg⌣pn−1,g⌣pn,Ψg⌣pn,g⌣pn+1=Ψg⌣pn,g⌣pn+1,then inequality (102) becomes(106)Ψg⌣pn,g⌣pn+1≤λΨg⌣pn,g⌣pn+1Ψg⌣pn,g⌣pn+1,which is a contradiction. So, we conclude that(107)Ψg⌣pn,g⌣pn+1≤λΨg⌣pn−1,g⌣pnΨg⌣pn−1,g⌣pn.
Further,(108)Ψg⌣pn,g⌣pn+1≤λΨg⌣pn−1,g⌣pnΨg⌣pn−1,g⌣pn≤Ψg⌣pn−1,g⌣pn,which shows that Ψg⌣pn,g⌣pn+1 is a decreasing sequence. Since λ∈F, from (108), we have(109)λΨg⌣pn,g⌣pn+1≤λΨg⌣pn−1,g⌣pn,λΨg⌣pn−1,g⌣pn≤λΨg⌣pn−2,g⌣pn−1.
Continuing on the same lines, we can write(110)λΨg⌣pn−1,g⌣pn≤λΨg⌣pn−2,g⌣pn−1≤⋯≤λΨg⌣p0,g⌣p1.
Using inequality (107),(111)Ψg⌣pn−1,g⌣pn≤λΨg⌣pn−2,g⌣pn−1Ψg⌣pn−2,g⌣pn−1.
From inequalities (107) and (111), we have(112)Ψg⌣pn,g⌣pn+1≤λΨg⌣pn−1,g⌣pnΨg⌣pn−1,g⌣pn≤λΨg⌣pn−1,g⌣pnλΨg⌣pn−2,g⌣pn−1Ψg⌣pn−2,g⌣pn−1.
Following on similar lines, we have(113)Ψg⌣pn,g⌣pn+1≤λΨg⌣pn−1,g⌣pnλΨg⌣pn−2,g⌣pn−1,…,λΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1=λnΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1.
We deduce(114)Ψg⌣pn,g⌣pn+1≤λnΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1.
From (108), suppose that Ψg⌣pn−1,g⌣pn>0, so we can conclude(115)Ψg⌣pn,g⌣pn+1Ψg⌣pn−1,g⌣pn≤λΨg⌣pn−1,g⌣pn≤1,for alln≥1.
Letl=limn⟶+∞Ψg⌣pn−1,g⌣pn. Using equation (108) and letting n⟶+∞, we obtain that(116)ll=1≤limn⟶+∞λΨg⌣pn−1,g⌣pn≤1.
Thus,limn⟶+∞Ψg⌣pn−1,g⌣pn=1. Using the definition of λ, we conclude that(117)limn⟶+∞Ψg⌣pn−1,g⌣pn=0.
Now, we have to show thatg⌣pn is a Cauchy sequence. For all natural numbers n,m∈ℕ with n<m, we have(118)Ψg⌣pn,g⌣pm≤α∗g⌣pn,g⌣pn+1Ψg⌣pn,g⌣pn+1+α∗g⌣pn+1,g⌣pmΨg⌣pn+1,g⌣pm≤α∗g⌣pn,g⌣pn+1Ψg⌣pn,g⌣pn+1+α∗g⌣pn+1,g⌣pmα∗g⌣pn+1,g⌣pn+2Ψg⌣pn+1,g⌣pn+2+α∗g⌣pn+1,g⌣pmα∗g⌣pn+2,g⌣pmΨg⌣pn+2,g⌣pm≤α∗g⌣pn,g⌣pn+1Ψg⌣pn,g⌣pn+1+α∗g⌣pn+1,g⌣pmα∗g⌣pn+1,g⌣pn+2Ψg⌣pn+1,g⌣pn+2+α∗g⌣pn+1,g⌣pmα∗gpn+2,g⌣pmα∗g⌣pn+2,g⌣pn+3Ψg⌣pn+2,g⌣pn+3+α∗g⌣pn+1,g⌣pmα∗g⌣pn+2,g⌣pmα∗g⌣pn+3,g⌣pmΨg⌣pn+3,g⌣pm≤α∗g⌣pn,g⌣pn+1Ψg⌣pn,g⌣pn+1+∑i=n+1m−2∏j=n+1iα∗g⌣pj,g⌣pmα∗g⌣pi,g⌣pi+1Ψg⌣pi,g⌣pi+1+∏k=n+1m−1α∗g⌣pk,g⌣pmΨg⌣pm−1,g⌣pm≤α∗g⌣pn,g⌣pn+1λnΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1+∑i=n+1m−2∏j=n+1iα∗g⌣pj,g⌣pmα∗g⌣pi,g⌣pi+1λiΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1+∏k=n+1m−1α∗g⌣pk,g⌣pmλm−1Ψg⌣p0,g⌣p1Ψg⌣p0,g⌣p1≤α∗g⌣pn,g⌣pn+1λnΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1+∑i=n+1m−2∏j=n+1iα∗g⌣pj,g⌣pmα∗g⌣pi,g⌣pi+1λΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1+∏k=n+1m−1α∗g⌣pk,g⌣pmα∗g⌣pm−1,g⌣pmλm−1Ψg⌣p0,g⌣p1Ψg⌣p0,g⌣p1=α∗g⌣pn,g⌣pn+1λnΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1+∑i=n+1m−1∏j=n+1iα∗g⌣pj,g⌣pmα∗g⌣pi,g⌣pi+1λiΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1≤α∗g⌣pn,g⌣pn+1λnΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1+∑i=n+1m−1∏j=0iα∗g⌣pj,g⌣pmα∗g⌣pi,g⌣pi+1λiΨg⌣p0,g⌣p1Ψg⌣p0,g⌣p1.
Assume that(119)Sl=∑i=0l∏j=0iα∗g⌣pj,g⌣pmα∗g⌣pi,g⌣pi+1λiΨg⌣p0,g⌣p1.
Then, we obtain(120)Ψg⌣pn,g⌣pm≤Ψg⌣p0,g⌣p1λnΨg⌣p0,g⌣p1α∗g⌣pn,g⌣pn+1+Sm−1−Sn.
Using the ratio test, we have(121)ai=∏j=0iα∗g⌣pj,g⌣pmα∗g⌣pi,g⌣pi+1λiΨg⌣p0,g⌣p1,whereai+1ai<1k.
Taking limit asn,m⟶∞, inequality (120) becomes(122)limn,m⟶∞Ψg⌣pn,g⌣pm=0.
This implies thatg⌣pn is a Cauchy sequence in the complete controlled metric type space X,Ψ; hence, it is convergent and suppose that it converges to some p∗ in J0⊆J (as set J is closed), which assures that the sequence pn⊆J0 since pn⟶p∗. As g⌣,Tˇ is a pair of continuous mappings, one writes(123)Dg⌣p∗,Tˇp∗=ΨJ,K.
Therefore,p∗ is a coincidence best proximity point of the pair g⌣,Tˇ.
For uniqueness, suppose that there are two distinct coincidence best proximity points ofg⌣,Tˇ such that p∗≠q∗. Thus, s=Ψp∗,q∗>0. Since Ψg⌣p∗,Tˇp∗=Ψg⌣q∗,Tˇq∗=ΨJ,K, using the P-property, we conclude that s=ℋTˇp∗,Tˇq∗. Since the pair g⌣,Tˇ is a λ−μ-proximal Geraghty contraction, we obtain s≤λss. Thus, λs≥1. Since λs≥1, we conclude that λs=1 and therefore s=0, which is contradiction.Example 2.
LetX=0,1,2,3,4,5 be endowed with the function Ψ given as Ψp,q=Ψq,p and Ψp,p=0, where
Takeα:X×X⟶1,∞ to be symmetric and defined as αp,q=16p+18q. It is easy to see that X,Ψ is controlled type metric space. Suppose J=0,1,2 and K=3,4,5. After a simple calculation, ΨJ,K=1/10, the P-property is satisfied, J0=J, and K0=K. Consider(124)Tˇp=3,ifp=2,3,4,ifp=0,1,g⌣p=0,ifp=0,1,ifp=2,2,ifp=1.
Clearly,TˇJ0⊆K0 and J0⊆g⌣J0. Now, we have to show that the pair g⌣,Tˇ is a λ−μ-proximal Geraghty contraction:(125)μp,qℋTˇp,Tˇq≤λMu,v,p,qMu,v,p,q,for all u,v,p,q∈J and for the function μ:J×J⟶0,+∞ is defined by(126)μp,q=Ψp,q+1.
Hence,(127)Dg⌣0,Tˇ2=D1,3=ΨJ,K,Dg⌣2,Tˇ1=D1,3,4=ΨJ,K.
After simple calculations,ℋTˇp,Tˇq=ℋ3,3,4=0, μp,q=Ψ3,3,4+1=1, and(128)M0,2,2,1=maxΨg⌣2,g⌣1,Dg⌣2,Tˇ2−α∗g⌣1,Tˇ21/10α∗g⌣2,g⌣1D∗g⌣0,Tˇ2,Dg⌣0,Tˇ1−α∗g⌣2,Tˇ11/10α∗g⌣0,g⌣2=maxΨ1,2,D1,3−α∗2,31/10α∗1,2D∗0,3,D0,3,4−α∗1,3,41/10α∗0,1=max14,−2381560,0,−69180=14.
Now, we have to show that the pairg⌣,Tˇ is a λ−μ-proximal Geraghty contraction:(129)10≤λMu,v,p,q14≤λMu,v,p,q14,and for every λ:0,∞⟶0,1, the pair g⌣,Tˇ is a λ−μ-proximal Geraghty contraction. Hence, 0 is the unique coincidence point of the pair of mappings g⌣,Tˇ.Corollary 4.
LetTˇ:J⟶ℬK be a continuous mapping, where J is a closed subset and the pair J,K satisfies the P-property with TˇJ0⊆K0. If Tˇ is a λ−μTˇ-proximal Geraghty contraction, where Tˇ is μ-proximal admissible, then there exist elements p0,p1∈J0 such that Dp1,Tˇp0=ΨJ,K and μp0,p1≥1. If pn is a sequence in J such that μpn,pn+1≥1 and suppose that(130)supm≥1limi⟶∞α∗pi,pi+1α∗pi,pm<1k,wherek∈0,1,then Tˇ has a unique best proximity pointp∗∈J.Proof.
If we takeg⌣=IJ (an identity mapping over J), the remaining proof is same as Theorem 6.Definition 23.
LetTˇ:J⟶K, g⌣:J⟶J, and μ:J×J⟶0,+∞ be mappings. A pair of mappings g⌣,Tˇ is said to be a λ−μ-modified proximal Geraghty contraction if(131)μp,q≥1Ψg⌣u,Tˇp=ΨJ,KΨg⌣v,Tˇq=ΨJ,K,impliesμp,qΨTˇp,Tˇq≤λMu,v,p,qMu,v,p,q,where(132)Mu,v,p,q=maxΨg⌣p,g⌣q,Ψg⌣p,Tˇp−αg⌣q,TˇpΨJ,Kαg⌣p,g⌣q,Ψ∗g⌣u,Tˇp,Ψg⌣u,Tˇq−αg⌣v,TˇqΨJ,Kαg⌣u,g⌣v,for all u,v,p,q∈J, where λ∈F.Definition 24.
LetTˇ:J⟶K and μ:J×J⟶0,+∞ be mappings. A mapping Tˇ is said to be a λ−μTˇ-modified proximal Geraghty contraction if(133)μp,q≥1Ψu,Tˇp=ΨJ,KΨv,Tˇq=ΨJ,K,impliesμp,qΨTˇp,Tˇq≤λMu,v,p,qMu,v,p,q,where(134)Mu,v,p,q=maxΨp,q,Ψp,Tˇp−αq,TˇpΨJ,Kαp,q,Ψ∗u,Tˇp,Ψu,Tˇq−αv,TˇqΨJ,Kαu,v,for all u,v,p,q∈J, where λ∈F.
Note that, if we takeg⌣=IJ (an identity mapping over J), then every λ−μ-modified proximal Geraghty contraction will reduce to a λ−μTˇ-modified proximal Geraghty contraction.Theorem 7.
LetTˇ:J⟶K and g⌣:J⟶J be continuous mappings, where J is closed subset and the pair J,K satisfies the P-property with TˇJ0⊆K0 and J0⊆g⌣J0. If the pair of mappings g⌣,Tˇ is a λ−μ-modified proximal Geraghty contraction, where Tˇ is a μ-proximal admissible, then there exist elements p0,p1∈J0 such that Ψg⌣p1,Tˇp0=ΨJ,K and μp0,p1≥1. If pn is a sequence in J such that μpn,pn+1≥1 and suppose that(135)supm≥1limi⟶∞αpi,pi+1αpi,pm<1k,wherek∈0,1,then the pair g⌣,Tˇ has a unique coincidence best proximity pointp∗∈J.Proof.
It is a simple consequence of Theorem6.Corollary 5.
LetTˇ:J⟶K be a continuous mapping, where J is closed subset and the pair J,K satisfies the P-property with TˇJ0⊆K0. If Tˇ is a λ−μTˇ-modified proximal Geraghty contraction, where Tˇ is a μ-proximal admissible, then there exist elements p0,p1∈J0 such that Ψg⌣p1,Tˇp0=ΨJ,K and μp0,p1≥1. If pn is a sequence in J such that μpn,pn+1≥1 and suppose that(136)supm≥1limi⟶∞αpi,pi+1αpi,pm<1k,wherek∈0,1,then the pair g⌣,Tˇ has a unique best proximity pointp∗∈J.Proof.
If we takeg⌣=IJ, the remaining proof is same as Theorem 7.


## 4. Conclusion

In our paper, we ensured the existence of some best proximity point results via the multivalued concept in controlled metric spaces. To our knowledge, we are the first who worked on best proximity points for the class of multivalued mappings in this setting. We open the door for new perspectives when dealing with new generalized multivalued (proximal) contractions.



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*Source: 8005469-2021-12-22.xml*