# Coincidence Best Proximity Point Results in Branciari Metric Spaces with Applications

Authors: Naeem Saleem; Manuel De la Sen; Sadia Farooq
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This paper is aimed at studying the uniqueness of coincidence best proximity point forϑ,α+,g-proximal contractions in complete Branciari metric space. Throughout this article, discontinuity of the Branciari metric space is used and we obtained the desired results without assuming it as a continuous. Some examples are provided to validate the results proved herein. As an application, we derive the best proximity point results in the setup of complete Branciari metric space endowed with graph. Further, our results extend and generalize the existing ones in literature.


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## Body

## 1. Introduction and Preliminaries

LetF:X⟶X be a mapping, where X be any nonempty set. “An element q∗∈X is a fixed point of F if q∗ satisfies the equation Fq∗=q∗ (known as a fixed point equation) or dq∗,Fq∗=0." A collection of all “fixed points” of F will be represented as FX, that is,
(1)FX=q∗∈X:dq∗,Fq∗=0.In this direction, Banach [1] gives the existence and uniqueness of the “fixed point” of the self mapping F, if mapping F is a contraction and X,d is a complete, but it becomes more interesting, if F is a nonself mapping then it is not necessary that the operator equation Fq∗=q∗ has a solution. In this situation, we can find a point q∗∈X which is closest to Fq∗ and we have the following minimization/optimization problem
(2)minq∈Xdq,Fq.Now, considerX=X,d be a metric space, Q and P are nonempty subsets of X, and consider a mapping F:Q⟶P, we can find a point q∗ in Q such that dq∗,Fq∗ is minimum. In other words, we have to minimize dq∗,Fq∗ for all q∗ in Q and Fq∗ in P. It is important to see that the minq∈Qdq,Fq=dQ,P, where dQ,P=infdq,p:q∈Q,p∈P which cannot be further reduced. If such point q∗ in Q exists then q∗ is called an “approximate fixed point” of F [2].Later, several authors studied the results dealing with “approximate fixed points” in different spaces (for detail, see [3–14]).The best proximity point of the mappingF:Q⟶P is actually “a point q∗∈X such that dq∗,Fq∗=dQ,P.” Note that if Q∩P≠ϕ then dQ,P=0; in this case, every “approximate fixed point” becomes “fixed point” of the mapping F. From this perspective, we can say that “the best proximity point results” are natural generalization of “fixed point results.”The concept of “coincidence best proximity point” was introduced in [5] for a pair of mappings in metric space. “A point q∗∈Q is called the coincidence best proximity point of a pair of mappings F:Q⟶P and g:Q⟶Q if dgq∗,Fq∗=dQ,P.” We denote the set of all “coincidence best proximity points” of a pair of mappings F and g by FgQ, that is,
(3)FgQ=q∗∈Q:dgq∗,Fq∗=dQ,P.This is an extension of a “best proximity point problem.” Ifg is an identity mapping on Q then a “coincidence best proximity point” will reduce to a “best proximity point” of mapping F.Recently, Branciari [15] defined a Branciari (by some author generalized/rectangular) metric space. Branciari metric space generalizes the deterministic metric space in a natural way. Few examples are also provided to show that a Branciari metric space is not a metric space.Definition 1 (see [15]).
LetX,d be a nonempty set. Any mapping d:X×X⟶0,∞ is a Branciari metric on X, if for all q,p∈X,u,v∈X\q,p, and the following conditions are satisfied:
(1)
dq,p=0 if and only if q=p(2)
dq,p=dp,q(3)
dq,p≤dq,u+du,v+dv,p
Then, the pairX,d is called a Branciari metric space.Definition 2 (see [15]).
LetX,d be a Branciari metric space. Then, a sequence qn in X,d is as follows:
(a)
Convergent sequence which converges toq∈X if and only if qn⟶q as n⟶∞. In this case, we can write(4)limn⟶∞qn=q.(b)
Cauchy sequence if and only ifdqn,qm⟶0 as n,m⟶∞.(c)
A Branciari metric spaceX,d is complete if every Cauchy sequence in X converges to some element in X.Lemma 3 (see [16]).
LetX,d be a Branciari metric space and qn be a Cauchy sequence in X such that qm≠qn whenever m≠n. Then, the sequence qn can converge to utmost one point.In a Branciari metric space, if a sequence is both Cauchy and convergent, then pathologies provided in an example [17] cannot happen, as shown in the following Lemma.Lemma 4 (see [18]).
Suppose thatqn is a Cauchy sequence in a Branciari metric space X,d with limn⟶∞dqn,q=0, where q∈X. Then, limn⟶∞dqn,u=dq,u, for all u∈X. In particular, the sequence qn cannot converge to u if u≠q.Recently, Jleli and Samet [4] introduced the concept of ϑ-contraction and proved a fixed point result for such mappings in the setup of Branciari metric spaces.Definition 5 (see [4]).
LetΔϑ be the set of all functions ϑ:0,∞⟶1,∞ satisfying the following conditions:
θ1ϑ is increasing
θ2 for any sequence κn in 0,∞,limn⟶∞κn=0+ if and only if limn⟶∞ϑκn=1
θ3 there exist Y∈0,1 and ℓ∈0,∞ such that limκ⟶0+ϑκ−1/κY=ℓDefinition 6 (see [4]).
LetX,d be a complete Branciari metric space and ϑ∈Δϑ. A mapping F:X⟶X is called ϑ-contraction if
(5)ϑdFq,Fp≤ϑdq,pς,for any q,p∈X, where dFq,Fp>0 and 0≤ς<1.Theorem 7 (see [4]).
LetX,d be a Branciari metric space and F:X⟶X be a ϑ-contraction. Then, F has a unique fixed point in X.Following definitions are also needed in the sequel.Definition 8.
LetX,d be a Branciari metric space and Q and P be two nonempty subsets of X. Define
(6)dQ,P=infdq,p:q∈Q,p∈P,Q0=q∈Q:dq,p=dQ,P,forsomep∈P,P0=q∈P:dq,p=dQ,P,forsomep∈Q.Definition 9 (see [19]).
LetQ0 be a nonempty subset of Q and P0 be a nonempty subset of P; then, the pair Q,P satisfies the “weak P-property,” if
(7)dq1,p1=dQ,Pdq2,p2=dQ,Pimpliesdq1,q2≤dp1,p2,for all q1,q2∈Q0 and p1,p2∈P0.Definition 10.
A mappingg:Q⟶Q is called the following:
(1)
[20] An “isometry,” if(8)dgq,gp=dq,p(2)
[21] An “expansive mapping,” if(9)dgq,gp≥dq,p,for any q,p∈QProposition 11 (see [22]).
A self mappingg:Q⟶Q is said to satisfy αR property if there exist a mapping α:Q×Q⟶−∞,∞) such that
(10)αgq,gp≥0impliesthatαq,p≥0,for all q,p∈Q.Definition 12 (see [3]).
LetX,d be a metric space, Q is a nonempty subset of ,α:Q×Q⟶−∞,∞), and F:Q⟶P is a mapping. Then, Q is said to be α-regular, if qn is a sequence in Q such that αqn,qn+1≥0 and qn⟶q∈Q as n⟶∞ then αqn,q≥0 for all n∈N.Now we have the following definitions:Definition 13.
Ifα:Q×Q⟶−∞,∞) then mapping F:Q⟶P is called “triangular proximal α+-admissible,” if
(T1)(11)αq,p≥0du,Fq=dQ,Pdv,Fp=dQ,Pimplies thatαu,v≥0,
(T2)(12)αw,z≥0αz,r≥0implies thatαw,r≥0,for all q,p,u,v,w,z,r∈Q.Example 1.
ConsiderX=È with the usual metric d:X×X⟶È defined by dq,p=∣q−p∣. Let Q and P be subsets of X defined by
(13)Q=16,36,56,76,96,116,136,P=0,26,46,66.
It is easy to see thatdQ,P=1/6. Let F:Q⟶P be the mapping defined as
(14)Fq=66,ifq=360,else.
Also, defineα:Q×Q⟶−∞,∞) by
(15)αq,p=p−qifp>qq−pifq≥pforallq,p∈Q.Case 1.
If we takeq∈1/6,5/6,7/6,9/6,11/6,13/6,p=3/6,v=7/6, and u=1/6 in Q, we get
(16)αq,p≥0,du,Fq=dQ,P,dv,Fp=dQ,P,which implies that
(17)αu,v≥0.
If we takew,z,κ∈Q, we get
(18)αw,z≥0αz,κ≥0impliesαw,κ≥0.Case 2.
If we takeq∈1/6,5/6,7/6,9/6,11/6,13/6,p=3/6,v=5/6, and u=1/6 in Q, we get
(19)αq,p≥0,du,Fq=dQ,P,dv,Fp=dQ,P,which implies that
(20)αu,v≥0.
If we takew,z,κ∈Q, we get
(21)αw,z≥0αz,κ≥0impliesαw,κ≥0.Definition 14.
IfF:Q⟶P and g:Q⟶Q then pair of mappings F,g satisfies
(1)
ϑ,α+,g-proximal contraction, if(22)αq,p≥0dgu,Fq=dQ,Pdgv,Fp=dQ,Pimpliesαq,p+ϑdFq,Fp≤ϑdq,pς1ϑdgu,gvς2(2)
ϑ,α+,g-generalized proximal contraction, if(23)αq,p≥0dgu,Fq=dQ,Pdgv,Fp=dQ,Pimpliesαq,p+ϑdgu,gv≤ϑdq,pς1ϑdu,vς2,where ,p,u,v∈Q,ϑ∈Δϑ, and ς1,ς2≥0 with 0≤ς1+ς2<1Definition 15.
IfF:Q⟶P and g:Q⟶Q then pair of mappings F,g satisfies
(1)
ϑ,g-proximal contraction, if(24)dgu,Fq=dQ,Pdgv,Fp=dQ,PimpliesϑdFq,Fp≤ϑdq,pς1ϑdgu,gvς2(2)
“ϑ,g-generalized proximal contraction,” if(25)dgu,Fq=dQ,Pdgv,Fp=dQ,Pimpliesϑdgu,gy≤ϑdq,pς1ϑdu,vς2,where ,p,u,v∈Q,ϑ∈Δϑ, and ς1,ς2≥0 with 0≤ς1+ς2<1Definition 16.
LetF:Q⟶P be a mapping satisfying
(1)
ϑ,α+-proximal contraction, if(26)αq,p≥0du,Fq=dQ,Pdv,Fp=dQ,Pimpliesαq,p+ϑdFq,Fp≤ϑdq,pς1ϑdu,vς2(2)
ϑ,α+-generalized proximal contraction, if(27)αq,p≥0du,Fq=dQ,Pdv,Fp=dQ,Pimpliesαq,p+ϑdu,v≤ϑdq,pς1ϑdu,vς2,where q,p,u,v∈Q,ϑ∈Δϑ, and ς1,ς2≥0 with0≤ς1+ς2<1.Definition 17.
LetF:Q⟶P be a mapping satisfying
(1)
`ϑ-proximal contraction, if(28)du,Fq=dQ,Pdv,Fp=dQ,PimpliesϑdFq,Fp≤ϑdq,pς1ϑdu,vς2(2)
ϑ-generalized proximal contraction, if(29)du,Fq=dQ,Pdv,Fp=dQ,Pimpliesϑdu,v1−ς2≤ϑdq,pς1,where ,p,u,v∈Q,ϑ∈Δϑ, and ς1,ς2≥0with 0≤ς1+ς2<1.Remark 18.
By takingg=IQ (identity, mapping over Q) then every ϑ,α+,g-proximal contraction will reduce to ϑ,α+-proximal contraction and ϑ,α+,g-generalized proximal contraction will reduce to ϑ,α+-generalized proximal contraction.Remark 19.
By takingαq,p=0, every ϑ,α+,g-proximal contraction will reduce to ϑ,g-proximal contraction and ϑ,α+,g-generalized proximal contraction will reduce to ϑ,g-generalized proximal contraction.Remark 20.
By takingg=IQ (identity, mapping over Q) and αq,p=0 then every ϑ,α+,g-proximal contraction will reduce to ϑ-proximal contraction and ϑ,α+,g-generalized proximal contraction will reduce to ϑ-generalized proximal contraction.Note that now and onwards in this article, we assumed thatQ and P are nonempty, distinct, and disjoint subsets of a complete Branciari metric space X,d and X=X,d will represent a “complete Branciari metric space” until otherwise it is stated. Also, note that if Q,P is nonempty, weakly compact, and convex pair in Banach space X, then Q0 and P0 are nonempty ([23]).


## 2. Main Results

Now we state and prove our main result which runs as follows.Theorem 21.
Letα:Q×Q⟶−∞,∞), F,g be a pair of mappings satisfying ϑ,α+,g-proximal contraction, where F:Q⟶P be a triangular proximal α+-admissible and g:Q⟶Q be a one to one expansive mapping satisfying αR property. Moreover, if Q0 is a nonempty α-regular closed set in X,FQ0⊆P0,Q0⊆gQ0, and Q,P satisfy the weak P-property. Then, there exists a coincidence best proximity point of the pair of mappings F,g provided that there exists q0,q1∈Q0 such that
(30)dgq1,Fq0=dQ,Pandαq0,q1≥0.
Moreover, ifαq,p≥0, for every q,p∈Q such that dgq,Fq=dQ,P=dgp,Fp then q∗ is the unique coincidence best proximity point of the pair of mappings F,g.Proof.
Letq0,q1∈Q0 such that dgq1,Fq0=dQ,P and αq0,q1≥0. As Fq0∈FQ0⊆P0 and gq1∈Q0⊆gQ0, there exists q2∈Q0 such that dgq2,Fq1=dQ,P. Since F is a triangular proximal α+-admissible, we have αgq1,gq2≥0. Since g satisfies αR property, hence αq1,q2≥0. Similarly, by FQ0⊆P0 and Q0⊆gQ0, there exists a point q3∈Q0 such that dgq3,Fq2=dQ,P. Since F is a triangular proximal α+-admissible, this further implies that αgq2,gq3≥0. Following the same arguments, we have αq2,q3≥0. Continuing this way, we can obtain a sequence qn in Q0 such that
(31)dgqn,Fqn−1=dQ,P=dgqn+1,Fqn,αqn,qn+1≥0,foralln∈ℕ∪0.
Ifqn≠qn+1, for all n∈ℕ∪0, the pair of mappings F,g is ϑ,α+,g-proximal contraction, and we have
(32)αqn,qn−1+ϑdFqn−1,Fqn≤ϑdqn−1,qnς1ϑdgqn+1,gqnς2.
Since,F is a triangular proximal α+-admissible, we obtain that
(33)ϑdFqn−1,Fqn≤ϑdqn−1,qnς1ϑdgqn+1,gqnς2.
Since pairQ,P satisfies the weak P-property and g is one to one on Q0 then we have
(34)ϑdgqn,gqn+1≤ϑdFqn−1,Fqn≤ϑdqn−1,qnς1ϑdgqn+1,gqnς2,which further implies that
(35)ϑdgqn,gqn+11−ς2≤ϑdqn−1,qnς1,since g is an expansive and ϑ is an increasing mapping; hence,
(36)ϑdqn,qn+11−ς2≤ϑdgqn,gqn+11−ς2≤ϑdqn−1,qnς1.
After simplification, we have(37)ϑdqn,qn+1≤ϑdqn−1,qnς1/1−ς2.
Further, we can write as(38)ϑdqn,qn+1≤ϑdqn−2,qn−1ς1/1−ς22≤ϑdq0,q1ς1/1−ς2n,which implies that
(39)ϑdqn,qn+1≤ϑdq0,q1vn,foralln∈ℕ,
wherev=ς1/1−ς2<1. Taking limit n⟶∞ in the above inequality, we have
(40)limn⟶∞ϑdqn,qn+1=1,which implies that
(41)limn⟶∞dqn,qn+1=0.
Sinceϑ∈Δϑ then there exist 0<Y<1 and 0<ℓ≤∞, such that the following limit holds true:
(42)limn⟶∞ϑdqn,qn+1−1dqn,qn+1Y=ℓ.
Assume thatℓ<∞ and C=ℓ/2. Thus, there exist n0∈ℕ, such that
(43)ϑdqn,qn+1−1dqn,qn+1Y−ℓ≤C,foralln≥n0.
Hence, we have(44)ϑdqn,qn+1−1dqn,qn+1Y≥ℓ−C=ℓ2=C,foralln≥n0.
Further, we can write as(45)ndqn,qn+1Y≤ωDϑdqn,qn+1−1,foralln≥n0,where d=1/C. If ℓ=∞ then there exists n0∈ℕ, such that
(46)ϑdqn,qn+1−1dqn,qn+1Y≥C,foralln≥n0,which implies that
(47)ndqn,qn+1Y≤ωDϑdqn,qn+1−1,foralln≥n0,where D=1/C. Hence, in all cases, there exist D>0 and n0∈ℕ, such that
(48)ndqn,qn+1Y≤ωDϑdqn,qn+1−1,foralln≥n0.
From inequalities (39) and (48), we have
(49)ndqn,qn+1Y≤ωDϑdq0,q1vn−1,foralln≥n0.
Then, by taking limit asn⟶∞ on both sides of the above inequality, we obtain
(50)ndqn,qn+1Y=0.
Thus, there existsn0∈ℕ such that
(51)dqn,qn+1≤1n1/Y,foralln≥n0.
Now suppose thatqn=qm for all n,m∈ℕ and n=m. If qn≠qn+2, for all n∈ℕ∪0 then
(52)dgqn,Fqn−1=dgqn+2,Fqn+1=dQ,P.
SinceF is a triangular proximal α+-admissible αqn−1,qn≥0 and αqn,qn+1≥0, by (T2) of Definition 13, we have
(53)αqn−1,qn+1≥0.
Again, since the pair of mappingsF,g is ϑ,α+,g-proximal contraction, we have
(54)ϑdFqn−1,Fqn+1≤αqn−1,qn+1+ϑdFqn−1,Fqn+1≤ϑdqn−1,qn+1ς1ϑdgqn+2,gqnς2.
The pairQ,P satisfies the weak P-property; g is one to one on Q0, and ϑ is increasing; then,
(55)ϑdgqn,gqn+2≤ϑdFqn−1,Fqn+1.
Then, from inequalities (54) and (55), we have
(56)ϑdgqn,gqn+2≤ϑdqn−1,qn+1ς1ϑdgqn+2,gqnς2,which further implies that
(57)ϑdgqn+2,gqn1−ς2≤ϑdqn−1,qn+1ς1.
Sinceg is an expansive mapping and ϑ is an increasing mapping, then we have
(58)ϑdqn+2,qn1−ς2≤ϑdgqn+2,gqn1−ς2≤ϑdqn−1,qn+1ς1.
After further simplifications, we have(59)ϑdqn+2,qn≤ϑdqn−1,qn+1ς1/1−ς2≤ϑdqn−2,qnς1/1−ς22≤ϑdq0,q2ς1/1−ς2n,which implies that
(60)ϑdqn+2,qn≤ϑdq0,q2ς1/1−ς2n,foralln∈ℕ,where v=ς1/1−ς2<1. Taking the limit n⟶∞ in above inequality, we have
(61)limn⟶∞ϑdqn+2,qn=1.
Similarly, from conditionϑ2, we have
(62)limn⟶∞dqn,qn+2=0.
Similarly, from conditionϑ3, there exists n1∈ℕ, such that
(63)dqn,qn+2≤1n1/Y,foralln≥n1.Now we have the following cases:Case 1.
Ifm>2 and m is odd. Consider m=2κ+1,κ≥1, using (51), and we obtain
(64)dqn,qn+m≤dqn,qn+1+dqn+1,qn+2+dqn+2,qn+m≤dqn,qn+1+dqn+1,qn+2+dqn+2,qn+3+dqn+3,qn+4+dqn+4,qn+m≤dqn,qn+1+dqn+1,qn+2+dqn+2,qn+3+⋯+dqn+m−1,qn+m≤1n1/Y+1n+11/Y+1n+21/Y+⋯+1n+m−11/Ydqn,qn+m≤∑i=n∞1i1/Y,for all m≥n≥N, where N=maxn0,n1.Case 2.
Ifm>2 is even. Consider m=2κ,κ≥2, using (51) and (63), and we obtain
(65)dqn,qn+m≤dqn,qn+2+dqn+2,qn+3+dqn+3,qn+m≤dqn,qn+2+dqn+2,qn+3+dqn+3,qn+4+dqn+4,qn+5+dqn+5,qn+m≤dqn,qn+2+dqn+2,qn+3+dqn+3,qn+4+⋯+dqn+m−1,qn+m≤1n1/Y+1n+21/Y+⋯+1n+m−11/Y≤∑i=n∞1i1/Y,for all m≥n≥N, where N=maxn0,n1. Thus, by combining all the cases, we have
(66)dqn,qn+m≤∑i=n∞1i1/Y,forallm≥n≥N,where N=maxn0,n1. Then, by P-series test, ∑i=n∞1/i1/Y converges as 1/Y>1. We deduce that qn is a Cauchy sequence in Q0⊆Q⊂X. By the completeness of space X and Q0 is closed, there exists q∗∈Q0 such that qn⟶q∗∈Q0. Since Q0⊆gQ0, we have q∗=gq∗. Since Q0 is α-regular then αqn,q∗≥0. Since q∗∈Q0,FQ0⊆P0 then Fq∗∈P0; thus, there exists a point z∈Q0 such that z=q∗ and
(67)dgz,Fq∗=dQ,P.
Since the pair of mappingsF,g is ϑ,α+,g-proximal contraction and by using weak P-property, we obtain
(68)ϑdgqn+1,gz≤ϑdFqn,Fq∗≤αqn,q∗+ϑdFqn,Fq∗≤ϑdqn,q∗ς1ϑdgqn+1,gzς2.Further, the above inequality can be written as follows:(69)ϑdgqn+1,gz1−ς2≤ϑdqn,q∗ς1.Sinceg is an expansive mapping, we have
(70)ϑdqn+1,z1−ς2≤ϑdgqn+1,gz1−ς2≤ϑdq∗,qnς1<ϑdq∗,qn1−ς2.This implies that(71)ϑdqn+1,z<ϑdq∗,qn.Asϑ is increasing, we have
(72)dqn+1,z<dq∗,qn.Then, by rectangular property, (51) and (72), we have
(73)dq,z≤dq∗,qn+dqn,qn+1+dqn+1,z≤dq∗,qn+1n1/Y+dq∗,qn.Taking limitn⟶∞ in the above inequality, we conclude that z=q∗. Hence,
(74)dgq∗,Fq∗=dQ,P.Thus,q∗ is a coincidence best proximity point of pair of mappings F,g.Uniqueness. Now we have to show that q∗ is a unique coincidence best proximity point of pair of mappings F,g. Suppose that q∗ and w∗ be two coincidence best proximity points of a pair of mappings F,g, that is,
(75)dgq∗,Fq∗=dQ,P=dgw∗,Fw∗.Sinceαq,p≥0, for every q,p∈Q, and by using properties of F and g and reasoning as above, we obtain that
(76)ϑdgq∗,gw∗≤ϑdFq∗,Fw∗≤αq∗,w∗+ϑdFq∗,Fw∗≤ϑdq∗,w∗ς1ϑdgq∗,gw∗ς2ϑdgq∗,gw∗1−ς2≤ϑdq∗,w∗ς1<ϑdq∗,w∗1−ς2.Further, we have(77)ϑdq∗,w∗1−ς2≤ϑdgq∗,gw∗1−ς2<ϑdq∗,w∗1−ς2,which is a contradiction. Therefore, q∗=w∗. Hence, pair of mappings F,g has a unique coincidence best proximity point.Ifg is an isometry in Theorem 21 then it yields the following theorem.Theorem 22.
Letα:Q×Q⟶−∞,∞), F,g be a pair of mappings satisfying ϑ,α+,g-proximal contraction, where F:Q⟶P be a triangular proximal α+-admissible and g:Q⟶Q be a one to one isometry mapping satisfying αR property. Moreover, if Q0 is a nonempty α-regular closed set in X,FQ0⊆P0,Q0⊆gQ0, and Q,P satisfy the weak P-property. Then, there exists a coincidence best proximity point of the pair of mappings F,g provided that there exists q0,q1∈Q0 such that
(78)dgq1,Fq0=dQ,Pandαq0,q1≥0.
Moreover, ifαq,p≥0, for every q,p∈Q such that dgq,Fq=dQ,P=dgp,Fp then q∗ is the unique coincidence best proximity point of the pair of mappings F,g.Proof.
The result follows from Theorem21 by choosing g as an isometry mapping instead of expansive mapping, and the remaining proof follows under the same lines.Corollary 23.
LetF:Q⟶P be a ϑ,g-proximal contraction and g:Q⟶Q be a one to one expansive mapping satisfying qR property. Moreover, if Q0 is a nonempty closed set in X, FQ0⊆P0, Q0⊆gQ0, and Q,P satisfy the weak P-property. Then, there exists a unique coincidence best proximity point of pair of mappings F,g provided that there exists q0,q1∈Q0 such that
(79)dgq1,Fq0=dQ,P.Proof.
The result follows from Theorem21 by choosing αq,p=0, and the remaining proof follows under the same lines.Corollary 24.
Letα:Q×Q⟶−∞,∞) and F:Q⟶P be a triangular proximal α+-admissible and ϑ,α+-proximal contraction. Moreover, if Q0 is a nonempty α-regular closed set, FQ0⊆P0 and Q,P satisfy the weak P-property. Then, there exists a best proximity point of F provided that there exists q0,q1∈Q0 such that
(80)dq1,Fq0=dQ,Pandαq0,q1≥0.
Moreover, ifαq,p≥0, for every q,p∈Q such that dq,Fq=dQ,P=dp,Fp then q∗ is a unique best proximity point of F.Proof.
The result follows from Theorem21 by choosing g=IQ, and the remaining proof follows under the same lines.Corollary 25.
LetF:Q⟶P be a ϑ-proximal contraction. Moreover, if Q0 is a nonempty closed set, FQ0⊆P0 and Q,P satisfy the weak P-property. Then, there exists a unique best proximity point of F provided that there exists q0,q1∈Q0 such that
(81)dq1,Fq0=dQ,P.Proof.
The result follows from Theorem21 by choosing g=IQ and αq,p=0, and the remaining proof follows under the same lines.To support the Corollary25, we provide the following example.Example 2.
LetX=0,1,2,3,4,5,6,7,8 and d:X×X⟶0,∞) which is defined as
(82)dq,p=dp,qanddq,q=0,forallq,p∈X,d5,7=1,d2,6=d1,3=2,d0,2=3,d3,7=d2,4=d3,5=4,d0,4=d1,5=5,d2,7=d4,7=d6,1=d6,5=d6,7=d2,5=7,d4,6=d0,8=d2,8=d4,8=d6,8=d1,8=d4,1=8,d0,1=d2,3=d4,5=d6,7=d0,5=d0,7=d2,1=7,d0,6=d1,7=d6,3=d4,3=d0,3=d3,8=d5,8=d7,8=8.
Then,X,d is a Branciari metric space. Since
(83)d0,6≤d0,2+d2,6then X,d is not a metric space.
Suppose thatQ=2,4,6 and P=1,3,7 are subsets of Branciari metric space X,d. Note that dQ,P=7,Q=Q0, and P=P0. It is easy to see that Q,P satisfy the weak P-property. Define a mapping F:Q⟶P as follows:
(84)F2=3,F4=1,F6=3.
Clearly,F has no fixed point; also, note that FQ0⊆P0. Define a function ϑ:0,∞⟶1,∞ as follows:
(85)ϑκ=eκ.
Hence,F satisfies the conditions of ϑ-proximal contractive mapping. Further, by taking u=2,v=6,q=6,p=4∈Q,ς1=1/2, and ς2≥0 in such a way that 0≤ς1+ς2<1, we have
(86)d2,F6=d6,F4=7=dQ,P.
This implies thatF satisfies
(87)ϑdFq,Fp≤ϑdq,p1/2ϑdu,vς2.
Hence,F is ϑ-proximal contraction for all u,v,q,p∈Q. Thus, all conditions of Corollary 25 hold true; after simple calculation, we can find q=2 is a unique best proximity point of F.Remark 26.
If we takeX=Q=P and ς2=0 in the Example 2, we obtained the example of the main Theorem (1.7) in [4].Theorem 27.
Letα:Q×Q⟶−∞,∞), F,g be a pair of mappings satisfying ϑ,α+,g-generalized proximal contraction, where F:Q⟶P be a triangular proximal α+-admissible and g:Q⟶Q be a one to one expansive mapping satisfying αR property. Moreover, if Q0 is a nonempty α-regular closed set in X,FQ0⊆P0,Q0⊆gQ0, and Q,P satisfy the weak P-property. Then, there exists a coincidence best proximity point of the pair of mappings F,g provided that there exists q0,q1∈Q0 such that
(88)dgq1,Fq0=dQ,Pandαq0,q1≥0.
Moreover, ifαq,p≥0, for every q,p∈Q such that dgq,Fq=dQ,P=dgp,Fp then q∗ is the unique coincidence best proximity point of the pair of mappings F,g.Proof.
Following the arguments similar to those given in the proof of Theorem21, we obtain a sequence qn in Q0 such that
(89)dgqn,Fqn−1=dQ,P,dgqn+1,Fqn=dQ,P,αqn+1,qn≥0,foralln∈ℕ∪0.
Ifqn≠qn+1, for all n∈ℕ∪0, also the pair of mappings F,g is ϑ,α+,g-generalized proximal contraction and g is an expansive mapping, we have
(90)ϑdqn,qn+1≤ϑdgqn,gqn+1≤αqn,qn−1+ϑdgqn,gqn+1≤ϑdqn−1,qnς1ϑdqn+1,qnς2.
Further, the above inequality becomes(91)ϑdqn,qn+11−ς2≤ϑdqn,qn−1ς1.
Again by using the arguments similar to those given in the proof of Theorem21, we obtain
(92)dqn,qn+1≤1n1/Y,foralln≥n0.
Now suppose thatqn=qm for every n,m∈ℕ such that n=m. If qn≠qn+2, for all n∈ℕ∪0, since the pair of mappings F,g is ϑ,α+,g-generalized proximal contraction and g is an expansive mapping, we have
(93)ϑdqn,qn+2≤ϑdgqn,gqn+2≤ϑdqn−1,qn+1ς1ϑdqn+2,qnς2.
Further, the above inequality becomes(94)ϑdqn,qn+21−ς2≤ϑdqn−1,qn+1ς1.
By using the arguments similar to those given in the proof of Theorem21, we deduce that qn is a Cauchy sequence in Q0⊆Q⊂X and
(95)dgz,Fq∗=dQ,P.
Since the pair of mappingsF,g is ϑ,α+,g-generalized proximal contraction and g is an expansive mapping, we have
(96)ϑdqn,z≤ϑdgqn,gz≤αqn,q∗+ϑdgqn,gz≤ϑdqn−1,q∗ς1ϑdz,qnς2ϑdqn,z1−ς2≤ϑdqn−1,q∗ς1<ϑdqn−1,q∗1−ς2.By using the arguments similar to those given in the proof of Theorem21, we conclude that q∗ is a coincidence best proximity point of pair of mappings F,g.Uniqueness. Now we have to show that q∗ is a unique coincidence best proximity point of pair of mappings F,g. Suppose that q∗ and w∗ be two coincidence best proximity points of a pair of mappings F,g, that is,
(97)dgq∗,Fq∗=dQ,P=dgw∗,Fw∗.Sinceαq,p≥0, for every q,p∈Q and by using properties of F and g and reasoning as above, we obtain that
(98)ϑdq∗,w∗≤ϑdgq∗,gw∗≤αq∗,w∗+ϑdgq∗,gw∗≤ϑdq∗,w∗ς1ϑdq∗,w∗ς2=ϑdq∗,w∗ς1+ς2<ϑdq∗,w∗,which is a contradiction. Therefore,q∗=w∗. Hence, pair of mappings F,g has a unique coincidence best proximity point.Ifg is an isometry then the preceding theorem yields the following Theorem.Theorem 28.
Letα:Q×Q⟶−∞,∞), F,g be a pair of mappings satisfying ϑ,α+,g-generalized proximal contraction, where F:Q⟶P be a triangular proximal α+-admissible and g:Q⟶Q be a one to one isometry mapping satisfying αR property. Moreover, if a Q0 is nonempty α-regular closed set in X,FQ0⊆P0,Q0⊆gQ0, and Q,P satisfy the weak P-property. Then, there exists a coincidence best proximity point of the pair of mappings F,g provided that there exists q0,q1∈Q0 such that
(99)dgq1,Fq0=dQ,Pandαq0,q1≥0.Moreover, ifαq,p≥0, for every q,p∈Q such that dgq,Fq=dQ,P=dgp,Fp then q∗ is the unique coincidence best proximity point of the pair of mappings F,g.Proof.
The result follows from Theorem27 by choosing g as an isometry mapping instead of an expansive mapping, and the remaining proof follows under the same lines.Corollary 29.
LetF,g be a pair of mappings satisfying ϑ,α+,g-generalized proximal contraction and g:Q⟶Q be an expansive mapping satisfying αR property. Moreover, if Q0 is nonempty closed set in X,FQ0⊆P0,Q0⊆gQ0, and Q,P satisfy the weak P-property. Then, there exists a unique coincidence best proximity point of the pair of mappings F,g provided that there exists q0,q1∈Q0 such that
(100)dgq1,Fq0=dQ,P.Proof.
The result follows from Theorem27 by choosing αq,p=0, and the remaining proof follows under the same lines.Corollary 30.
Letα:Q×Q⟶−∞,∞), F:Q⟶P be a ϑ,α+-generalized proximal contraction. Moreover, if Q0 is a nonempty α-regular closed set in X,FQ0⊆P0 and Q,P satisfy the weak P-property. Then, there exists a best proximity point of mapping F provided that there exists q0,q1∈Q0 such that
(101)dq1,Fq0=dQ,Pandαq0,q1≥0.
Moreover, ifαq,p≥0, for every q,p∈Q such that dq,Fq=dQ,P=dp,Fp then q∗ is the unique best proximity point of the mapping F.Proof.
The result follows from Theorem27 by choosing g=IQ, and the remaining proof follows under the same lines.Corollary 31.
LetF:Q⟶P be a ϑ-generalized proximal contraction. Moreover, if Q0 is a nonempty closed set, FQ0⊆P0 and Q,P satisfy the weak P-property. Then, there exists a unique best proximity point of F provided that there exists q0,q1∈Q0 such that
(102)dq1,Fq0=dQ,P.Proof.
The result follows from Theorem27 by choosing g=IQ and αq,p=0, and the remaining follows under the same lines.To support the Corollary31, we provide the following example.Example 3.
LetX=0,1,2,3,4,5,6,7,8 and d:X×X⟶0,∞) which is defined as
(103)dq,p=dp,qanddq,q=0,forallq,p∈X,where
(104)d5,7=1,d2,6=d1,3=2,d0,2=3,d3,7=d2,4=d3,5=4,d0,4=d1,5=5,d2,7=d6,1=d6,5=d6,7=d2,5=7,d4,6=d0,8=d4,7=d2,8=d4,8=d6,8=d1,8=d4,1=8,d0,1=d2,3=d4,5=d6,7=d0,5=d0,7=d2,1=7,d0,6=d1,7=d6,3=d4,3=d0,3=d3,8=d5,8=d7,8=8.
Then,X,d is a Branciari metric space. Since
(105)d0,6≤d0,2+d2,6,then X,d is not a metric space. Suppose that Q=2,4,6 and P=3,5,7. Note that dQ,P=7,Q=Q0, and P=P0. Define a mapping F:Q⟶P as
(106)F2=3,F4=7,F6=3.
Clearly,F has no fixed point; also, note that FQ0⊆P0. Define a function ϑ:0,∞⟶1,∞ as
(107)ϑκ=eκ.
Hence,F satisfies the conditions of ϑ-generalized proximal contractive mapping. Further, by taking u=2,v=6,q=6,p=4∈Q,ς1=1/2, and ς2≥0 in such a way that 0≤ς1+ς2<1, we have
(108)du,Fq=d2,F6=d6,F4=dv,Fp=7.
Further,F satisfies
(109)ϑdu,v≤ϑdq,pς1ϑdu,vς2.Hence,F is ϑ-generalized proximal contraction for all u,v,q,p∈Q. Thus, all conditions of Corollary 31 hold true; after simple calculation, we can find q=2 is a unique best proximity point of F.


## 3. Application to Coincidence Point and Fixed Point Theory

If we takeQ=P=X, then from Definition 13, triangular proximal α+-admissible implies
(110)du,Fq=dQ,P=0impliesu=Fq,dv,Fp=dQ,P=0impliesv=Fp,which becomes(P1)(111)αq,p≥0impliesαFq,Fp≥0,(P2)(112)αq,z≥0αz,p≥0impliesαq,p≥0,forallq,p∈X.Remark 32.
Note that, for self mapping, every triangular proximalα+-admissible self mapping is triangular α+-admissible mapping.Remark 33.
IfF,g:X⟶X then ϑ,g-proximal contraction becomes
(113)ϑdFq,Fp≤ϑdq,pς1ϑdFq,Fpς2,and ϑ,g-generalized proximal contraction becomes
(114)ϑdFq,Fp≤ϑdq,pς1ϑdv,uς2,where ϑ∈Δϑ,ς∈0,1,gu=Fq, and gv=Fp, for all q,p∈X.Definition 34.
A self mappingF satisfying inequality (113) is called ϑ,F-contraction, and the mapping which satisfies inequality (114) is called ϑ,F-generalized contraction.Remark 35.
Ifα:X×X⟶−∞,∞ and a self mapping F on X is ϑ,α+-proximal contraction and ϑ,α+-generalized proximal contraction then αq,p≥0 implies that
(115)αq,p+ϑdFq,Fp≤ϑdq,pς1ϑdFq,Fpς2,where ϑ∈Δϑ,ς∈0,1,u=Fq, and v=Fp, for all q,p∈X.Definition 36.
A self mappingF on X and α:X×X⟶−∞,∞ satisfying inequality (115) is called ϑ,α+,F-contraction.Corollary 37.
LetX be a complete Branciari metric space. Let F:X⟶X be a ϑ,g-proximal contraction. Then, the pair of mappings F,g has a coincidence point.Proof.
LetQ=P=X. We show that F satisfies ϑ,F-contraction. Then, we have
(116)dgu,Fq=dQ,P=0,dgv,Fp=dQ,P=0,for all q,p,u,v∈X, since dQ,P=0 implies that gu=Fq and gv=Fp and since F satisfies condition (113). So,
(117)ϑdFq,Fp≤ϑdq,pς1ϑdgq,gpς2=ϑdq,pς1ϑdFq,Fpς2,which implies that
(118)ϑdFq,Fp≤ϑdq,pς1ϑdFq,Fpς2,which further implies that F is ϑ,F-contraction. As dgq,Fq=dQ,P=0 then q is a coincidence point of F and g.Corollary 38.
LetX be a complete Branciari metric space and pair of mapping F,g:X⟶X be a ϑ,g-generalized proximal contraction. Then, pair of mappings F,g has a coincidence point.Proof.
LetQ=P=X. We show that F satisfies ϑ,F-generalized contraction. Then, we have
(119)dgu,Fq=dQ,P=0,dgv,Fp=dQ,P=0,for all q,p,u,v∈X, since dQ,P=0 implies that gu=Fq and gv=Fp and since F satisfies condition (114). So,
(120)ϑdFq,Fp=ϑdgu,gv≤ϑdq,pς1ϑdv,uς2,≤ϑdq,pς1ϑdu,vς2,which implies that
(121)ϑdFq,Fp≤ϑdq,pς1ϑdu,vς2,which further implies that F is ϑ,F-generalized contraction. As dgq,Fq=dQ,P=0 then q is a coincidence point of F and g.Ifg is an identity mapping then we will have the following corollary.Corollary 39.
LetX be a complete Branciari metric space, α:X×X⟶−∞,∞, and F:X⟶X be a ϑ,α+,F-contraction. If qn is a sequence in X such that αqn,qn+1≥0 and qn⟶q∈X as n⟶∞ then αqn,q≥0 for all n∈ℕ. Then, there exists a fixed point of F provided that there exists q0∈X such that αq0,Fq0≥0. Moreover, if αq,p≥0, for all q,p∈FX then q∗ is the unique fixed point of the mapping F.Proof.
LetQ=P=X. We show that F satisfies ϑ,α+-proximal contraction and ϑ,α+-generalized proximal contraction. Then, we have
(122)αq,p≥0,du,Fq=dQ,P=0,dv,Fp=dQ,P=0,for all q,p,u,v∈X, since dQ,P=0 implies that u=Fq and v=Fp and since F satisfies condition (115). So,
(123)αq,p+ϑdFq,Fp=αq,p+ϑdFq,Fp≤ϑdq,pς1ϑdFq,Fpς2=ϑdq,pς1ϑdu,vς2,which implies that
(124)αq,p+ϑdFq,Fp≤ϑdq,pς1ϑdu,vς2,which further implies that F is ϑ,α+-proximal contraction. Also, we have
(125)αq,p+ϑdu,v≤αq,p+ϑdFq,Fp≤ϑdq,pς1ϑdFq,Fpς2≤ϑdq,pς1ϑdu,vς2,which implies that
(126)αq,p+ϑdu,v≤ϑdq,pς1ϑdu,vς2,which further implies that F is ϑ,α+-generalized proximal contraction. Since F is triangular proximal α+-admissible, by Remark 32, F is a triangular α+-admissible mapping. If there exists q∈X such that αq,Fq≥0 and dq,Fq=dQ,P=0, it gives that q is a fixed point of F.Uniqueness. Let αq,p≥0 for all q,p∈FX. Now we will show that q∗ is a unique fixed point of F. On contrary, suppose that w∗ be another fixed point of mapping F with q∗=w∗. Hence,
(127)dgw∗,Fw∗=dQ,P.Then, by using the properties ofF, we obtain that
(128)ϑdq∗,w∗=ϑdFq∗,Fw∗≤αq∗,w∗+ϑdFq∗,Fw∗≤ϑdq∗,w∗ς1ϑdFq∗,Fw∗ς2=ϑdq∗,w∗ς1ϑdq∗,w∗ς2=ϑdq∗,w∗ς1+ς2<ϑdq∗,w∗,which is a contradiction. Therefore,q∗=w∗. Hence, the mapping F has a unique fixed point.Remark 40.
By takingαq,p=0 and ς2=0, for all q,p∈X in the above Corollary 39, we obtain the main Theorem (1.7) in [4] as Corollary.


## 4. Application to Graph Theory

LetX be a nonempty set and define a set Δ=q,p∈X×X,q,p∈X. A graph g is a pair V,E, where V=Vg is a set of vertices coinciding with X and E=Eg is a set of its edges such that Δ⊂Eg. Moreover, we suppose that the graph g is without parallel edges. By reversing the direction of edges in g, we obtain a graph g−1 whose edge set and vertex set are defined as follows:
(129)Eg−1=q,p∈X2:p,q∈EgandVg−1=Vg.Consider the graphg~ comprising of all vertices and edges of g and g−1, that is,
(130)Eg~=Eg∪Eg−1.We denote the undirected graph byg~ obtained by ignoring the direction of edges of g.Definition 41 (see [24]).
(1)
A subgraphH of a graph g is consisting upon a subset of edges of graph g and associated vertices.(2)
Letq and p be two vertices in a graph g. A path of length n (where n∈ℕ∪0) in g from q to p is a sequence qii=0n of n+1 distinct vertices such that q0=q,qn=p, and qi,qi+1∈Eg for i=1,2,⋯,n.(3)
A graphg is called a connected graph if there exists a path between any two vertices of graph g, and it is said to be a weakly connected graph if g~ is connected.(4)
A path is called elementary if no vertices appear more than one time in it.
Throughout this section, we suppose thatX,d be a Branciari metric space and a graph g may be transformed to a weighed graph by appointing to each edge the distance given by the Branciari metric between its vertices. In order to apply the rectangular inequality to the vertices of the graph g, we will consider the graph of length greater than 2, which signifies that between two vertices, we will obtain a path between two vertices.Definition 42.
LetQ,P be a pair of nonempty subsets of a Branciari metric space X and g be a directed graph without parallel edges such that Vg=X. A mapping F:Q⟶P is said to be ϑ,g-proximal contraction if for all q,p∈Q,q≠p with q,p∈Eg such that
(131)du,Fq=dQ,Pdv,Fp=dQ,PimpliesϑdFq,Fp≤ϑdq,pς1ϑdv,uς2,u,v∈Eg,where ,p,u,v∈Q,ϑ∈Δϑ, and ς1,ς2≥0 with 0<ς1+ς2<1.Definition 43.
LetQ,P be a pair of nonempty subsets of a Branciari metric space X and g be a directed graph without parallel edges such that Vg=X. A mapping F:Q⟶P is said to be ϑ,g-generalized proximal contraction if for all q,p∈Q,q≠p with q,p∈Eg such that
(132)du,Fq=dQ,Pdv,Fp=dQ,Pimpliesϑdu,v≤ϑdq,pς1ϑdv,uς2,u,v∈Eg,where ,p,u,v∈Q,ϑ∈Δϑ, and ς1,ς2≥0 with 0<ς1+ς2<1.Corollary 44.
LetF:Q⟶Paϑ,g-proximal contraction. If FQ0⊆P0 and Q0 is nonempty closed subset in X, and Q,P satisfy the weak P-property. Also, there exists q0,q1∈Q0 such that there exists an elementary path between them and
(133)dq1,Fq0=dQ,P.
Moreover, if there exists a pathviςi,ς=0N⊆Q0 in g between any two elements q and p then F has a unique best proximity point.Proof.
Letq0,q1∈Q0 such that dq1,Fq0=dQ,P. A path v0ii=0N in g is a sequence containing points of Q0.Consequently,v00=q0,v0N=q1andv0i,v0i+1∈Egforall0≤i≤N.Giventhatv01∈Q0,since FQ0⊆P0 and from definition ofQ0, there exists v11∈Q0 such that dv11,Fv01=dQ,P. Similarly, for each i=2,⋯,N, there exists v1i∈Q0 such that dv1i,Fv0i=dQ,P. As v0ii=0N is a path in g then v00,v01=q0,v01∈Eg. From the above argument, we have dq1,Fq0=dQ,P and dv11,Fv01=dQ,P. Since, F is ϑ,g-proximal contraction, it follows that q1,v11∈Eg. In similar manner, we have the following:
(134)v1i−1,v1i∈Eg,foralli=1,2,⋯,N.
Letq2=v1N then v1ii=0N is a path from q1=v10 to q2=v1N. For each i=1,2,3,⋯,N, as v1i∈Q0 and Fv1i∈FQ0⊆P0 then by definition of P0, there exists v2i∈Q0 such that dv2i,Fv1i=dQ,P. In addition, we have dq2,Fq1=dQ,P. As above mentioned, we have
(135)q2,v21∈Egandv2i−1,v2i∈Eg,foralli=1,2⋯,N.
Similarly, byFQ0⊆P0, there exists a point q3∈Q0 where q3=v2N. Then, v2ii=0N is a path from v20=q2 and v2N=q3. Continuing this way, we can obtain a sequence qnn∈ℕ where qn+1∈qngN and dqn+1,Fqn=dQ,P by producing a path vnii=0N from qn=vn0 and qn+1=vnN in such a way that
(136)dvn+1i,Fvni=dQ,P,for all i=0,1,⋯,N,n∈ℕ. Thus, we have
(137)dvni−1,Fvn−1i−1=dQ,P=dvni,Fvn−1i,foralli=1,2,⋯,N,n∈ℕ.
Next, we claim thatdqn,qn+1≤C/n1/Y, where C is a constant. To prove the claim, we need to consider the following two cases where vnii=0,1,⋯,N is a path from qn to qn+1. Note that for all i=0,1,⋯,Nvnii=0,1,⋯,N are different owing to the fact that the considered path vni is elementary. Then, we can apply the rectangular inequality.Case 1.
N=2k+1 (N is odd). For any positive integer n, we get
(138)dqn,qn+1=dvn0,vnN=dvn0,vn2k+1≤dvn0,vn1+dvn1,vn2+dvn2,vn2k+1≤dvn0,vn1+dvn1,vn2+⋯+dvn2k,vn2k+1≤∑i=12k+1dvni−1,vni.
SinceF is ϑ,g-proximal contraction, we have
(139)∑i=12k+1ϑdFvn−1i−1,Fvn−1i≤∑i=12k+1ϑdvn−1i−1,vn−1iς1ϑdvn−1i−1,vn−1iς2.
Since pair of subsetsQ,P satisfies the weak P-property then we have
(140)∑i=12k+1dvni−1,vni≤∑i=12k+1dFvn−1i−1,Fvn−1i,foralln∈ℕ.
Then, from above inequalities, we have(141)∑i=12k+1ϑdvni−1,vni≤∑i=12k+1ϑdvn−1i−1,vn−1iς1ϑdvn−1i−1,vn−1iς2∑i=12k+1ϑdvni−1,vni1−ς2≤∑i=12k+1ϑdvn−1i−1,vn−1iς1.
After simplification, we have(142)∑i=12k+1ϑdvni−1,vni≤∑i=12k+1ϑdvn−1i−1,vn−1iς1/1−ς2.
By induction, it follows that for alln∈ℕ,
(143)∑i=12k+1ϑdvni−1,vni≤∑i=12k+1ϑdv0i−1,v0iς1/1−ς2n,which implies that
(144)∑i=12k+1ϑdvni−1,vni≤∑i=12k+1ϑdv0i−1,v0ivnforalln∈ℕ,where v=ς1/1−ς2<1. Taking limit n⟶∞ in the above inequality, we have
(145)limn⟶∞∑i=12k+1ϑdvni−1,vni=1,which implies that
(146)limn⟶∞∑i=12k+1dvni−1,vni=0.
Sinceϑ∈Δϑ then there exist 0<Y<1 and 0<ℓ≤∞, such that the following limit holds true:
(147)limn⟶∞∑i=12k+1ϑdvni−1,vni−1dvni−1,vniY=ℓ.
Assume thatℓ<∞ and C=ℓ/2. Thus, there exists n0∈ℕ, such that
(148)∑i=12k+1ϑdvni−1,vni−1dvni−1,vniY−ℓ≤C,foralln≥n0.
Hence, we have(149)∑i=12k+1ϑdvni−1,vni−1dvni−1,vniY≥ℓ−C=ℓ2=C,foralln≥n0,and so we obtain
(150)n∑i=12k+1dvni−1,vniY≤ωD∑i=12k+1ϑdvni−1,vni−1,foralln≥n0,where d=1/C. If ℓ=∞ then there exists n0∈ℕ, such that
(151)∑i=12k+1ϑdvni−1,vni−1dvni−1,vniY≥C,foralln≥n0,which implies
(152)n∑i=12k+1dvni−1,vniY≤ωD∑i=12k+1ϑdvni−1,vni−1,foralln≥n0,where d=1/C. Hence, in all cases, there exist d>0 and n0∈ℕ, such that
(153)n∑i=12k+1dvni−1,vniY≤ωD∑i=12k+1ϑdvni−1,vni−1,foralln≥n0.
From inequalities (144) and (153), we have
(154)n∑i=12k+1dvni−1,vniY≤ωD∑i=12k+1ϑdv0i−1,v0ivn−1,foralln≥n0,and taking limit as n⟶∞ on both sides of the above inequality, we obtain
(155)n∑i=12k+1dvni−1,vniY=0.
Thus, there existsn0∈ℕ such that
(156)∑i=12k+1dvni−1,vni≤1n1/Y,foralln≥n0.
By putting inequality (156) in inequality (138), we get
(157)dqn,qn+1≤Cn1/Y,where C=1.Case 2.
N=2k (N is even).
(158)dqn,qn+1=dvn0,vnN=dvn0,vn2k≤dvn0,vn1+dvn1,vn2+dvn2,vn2k≤dvn0,vn1+dvn1,vn2+⋯+dvn2k−2,vn2k=∑i=12kdvni−1,vni−dvn2k−2,vn2k−1−dvn2k−1,vn2k+dvn2k−2,vn2k≤∑i=12kdvni−1,vni+dvn2k−2,vn2k.
By the same arguments used in Case1, we deduce that
(159)∑i=12kdvni−1,vni≤1n1/Y,foralln≥n0.
Indeed, from (136), we have
(160)dvn2k−2,Fvn−12k−2=dvn2k,Fvn−12k=dQ,P.
SinceF is ϑ,g-proximal contraction, we have
(161)ϑdFvn−12k−2,Fvn−12k≤ϑdvn−12k−2,vn−12kς1ϑdvn2k−2,vn2kς2.
The pairQ,P satisfies weak P-property; then, we can write
(162)dvn2k−2,vn2k≤dFvn−12k−2,Fvn−12k.
Sinceϑ is increasing then the above inequality becomes
(163)ϑdvn2k−2,vn2k≤ϑdFvn−12k−2,Fvn−12k.
Then, from inequalities (161) and (163), we have
(164)ϑdvn2k−2,vn2k≤ϑdvn−12k−2,vn−12kς1ϑdvn2k−2,vn2kς2.
After further simplifications,(165)ϑdvn2k−2,vn2k1−ς2≤ϑdvn−12k−2,vn−12kς1.
By induction, it follows that for alln∈ℕ,
(166)ϑdvn2k−2,vn2k≤ϑdv02k−2,v02kς1/1−ς2n,foralln∈ℕ,where v=ς1/1−ς2. Taking the limit n⟶∞ in above inequality, we have
(167)limn⟶∞ϑdvn2k−2,vn2k=1.
Similarly, from conditionϑ2, we have
(168)limn⟶∞dvn2k−2,vn2k=0.
Similarly, from conditionϑ3, there exists n1∈ℕ, such that
(169)dvn2k−2,vn2k≤1n1/Y,foralln≥n1.
Then, from inequalities (158), (159), and (169), we have
(170)dqn,qn+1≤Cn1/Y,foralln≥n1.where C=2. Let us prove that qn is a Cauchy sequence. Let n,m∈N such that m>n. We suppose that m is odd (m=2k+1) since the case m=2k is similar. Note that qn=v0n,qn+1=vnN, and v0n≠vnN for all n since the path vnii=0,1,⋯,N is elementary. Then, by using the rectangular metric, we obtain
(171)dqn,qm≤dqn,qn+1+dqn+1,qn+2+dqn+2,qn+3+⋯+dqm−1,qm≤Cn1/Y+Cn+11/Y+Cn+21/Y+⋯+Cn+m−21/Y≤∑i=n∞Ci1/Y,
for allm≥n≥N, where N=maxn0,n1. Hence, we obtain
(172)dqn,qm≤∑i=n∞Ci1/Y,foralln≥N,m∈ℕ.
We deduce thatqn is a Cauchy sequence in Q0⊆Q⊂X. By using the completeness of Branciari metric space X and closeness of Q0, there exists q∗∈Q0 such that qn⟶q∗∈Q0. Since q∗∈Q0,FQ0⊆P0 then Fq∗∈P0, thus there exists a point z∈Q0 such that z=q∗ and
(173)dz,Fq∗=dQ,P.
AsF is ϑ,g-proximal contraction, we obtain
(174)ϑdFq∗,Fqn≤ϑdq∗,qnς1ϑdqn+1,zς2.
Then, by using the weakP-property of subsets Q,P of Branciari metric space,we have
(175)ϑdz,qn+1≤ϑdFq∗,Fqn.
Hence, we have(176)ϑdz,qn+1≤ϑdq∗,qnς1ϑdqn+1,zς2.
Further,(177)ϑdqn+1,z1−ς2≤ϑdq∗,qnς1<ϑdq∗,qn1−ς2.
This implies that(178)ϑdqn+1,z<ϑdq∗,qn.
Asϑ is increasing, we have
(179)dqn+1,z<dq∗,qn.
Then, by rectangular property and using above inequality and inequality (170), we have
(180)dq∗,z≤dq∗,qn+dqn,qn+1+dqn+1,z<dq∗,qn+Cn1/Y+dq∗,qn.
Now, by taking limit asn⟶∞, we have z=q∗.(181)dgq∗,Fq∗=dQ,P.Thus,q∗ is a coincidence best proximity point of pair of mappings F,g.Uniqueness. We now show that q∗ is a unique best proximity point of F. Let w be another best proximity point of F with q∗=w. Hence,
(182)dQ,P=dw,Fw.Then, by using properties ofF and g and reasoning as above, we obtain that
(183)ϑdq∗,w≤ϑdFq∗,Fw≤ϑdq∗,wς1ϑdq∗,wς2≤ϑdq∗,wς1+ς2<ϑdq∗,w,which is a contradiction. Therefore, q∗=w. Hence, F has a unique best proximity point.Corollary 45.
LetF:Q⟶P be a ϑ,g-generalized proximal contraction. Moreover, Q0 is a nonempty closed subset in Branciari metric space X and FQ0⊆P0. If there exists a path zii=0v⊆Q0 in g between two elements q and p then there exists a unique best proximity of F provided that there exists q0,q1∈Q0 such that there is an elementary path in Q0 between them and
(184)dq1,Fq0=dQ,P.Proof.
Letq0,q1∈Q0 such that dq1,Fq0=dQ,P. A path v0ii=0N in g is a sequence containing points of Q0. Following arguments similar to those given in the proof of Corollary 44, we obtain
(185)dvni−1,Fvn−1i−1=dQ,P=dvni,Fvn−1i,foralli=1,2,⋯,N,n∈ℕ∪0.
SinceF is ϑ,g-generalized proximal contraction, we have
(186)ϑdvni−1,vni≤ϑdvn−1i−1,vn−1iς1ϑdvni,vni−1ς2.
Again by using the arguments similar to those given in the proof of Corollary44, we have
(187)dqn,qn+1≤Cn1/Y,foralln≥n0,where C=1. Again by using the arguments similar to those given in the proof of Corollary 44 and since F is ϑ,g-proximal contraction, we have
(188)ϑdvn2k−2,vn2k≤ϑdvn−12k−2,vn−12kς1ϑdvn2k,vn2k−2ς2ϑdvn2k−2,vn2k1−ς2≤ϑdvn−12k−2,vn−12kς1.
By using the arguments similar to those given in the proof of Corollary44, we deduce that qn is a Cauchy sequence in Q0⊆Q⊂X. By using the completeness of space X and Q0 is closed, there exists q∗∈Q0 such that qn⟶q∗∈Q0. Since q∗∈Q0,FQ0⊆P0 then Fq∗∈P0; thus, there exists a point z∈Q0 such that z=q∗ and
(189)dz,Fq∗=dQ,P.
AsF is ϑ,g-generalized proximal contraction, we have
(190)ϑdz,qn+1≤ϑdq∗,qnς1ϑdqn+1,zς2ϑdz,qn+11−ς2≤ϑdq∗,qnς1<ϑdq∗,qn1−ς2.Following arguments similar to those given in the proof of Corollary44, we obtain that q∗ is the best proximity point of the mapping F.Uniqueness. Now we have to show that q∗ is a unique best proximity point of F. Let w be another best proximity point of F with q∗=w. Hence,
(191)dQ,P=dw,Fw.Then, by using properties ofF and g and reasoning as above, we obtain that
(192)ϑdq∗,w≤ϑdq∗,wς1ϑdq∗,wς2≤ϑdq∗,wς1+ς2<ϑdq∗,w,which is a contradiction. Therefore, q∗=w. Hence, F has a unique best proximity point.


## 5. Conclusion

In this paper, we defineϑ,α+,g-proximal contraction and provide the existence results for coincidence best proximity point in Branciari metric space. The important aspect of Branciari metric space is that it is not continuous; we dealt with discontinuity of Branciari metric space and obtained the desired results. As an application, we derive the coincidence point and fixed point results for some self mappings. We also introduce the notion of ϑ,g-proximal contraction and provide an application to graph theory in the setting of Branciari metric space. Some examples are also provided to illustrate the novelty of the result proved herein.



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*Source: 4126025-2020-12-14.xml*