# Best Proximity Points of Multivalued Hardy-Roger’s Type (Cyclic) Contractive Mappings ofb-Metric Spaces

Authors: Basit Ali; Arshad Ali Khan; Azhar Hussain
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, we introduce a new type of generalized multivalued Hardy and Roger’s type proximal contractive and proximal cyclic contractive mappings ofb-metric spaces and develop some results for the existence of best proximity point(s). Moreover, we obtain some results for the existence and uniqueness of best proximity points for single-valued mappings. Examples are given to explain the main results.


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## Body

## 1. Introduction

The metric fixed point theory plays a very fundamental role in many fields of mathematics especially in nonlinear analysis and some related disciplines. The fundamental tool of this theory is the Banach contraction principle (shortly BCP) [1] which states that if a self-mapping T:P⟶P of a complete metric space P,ϱ with metric ϱ satisfies(1)ϱTp1,Tp2≤kϱp1,p2,for all p1,p2∈P, and for some k∈0,1, T has a unique fixed point, that is, there exists a point p∈P, such that(2)ϱp,Tp=0.A mapping that satisfies (1) is known as Banach contraction. After this remarkable result, many mathematicians contributed for the development of fixed-point theory by producing many results with different generalized contractive mappings in complete metric spaces, for details one can see [2–8] and the references therein. One of the important generalizations of BCP was presented by Edelstein [9] in 1962. Later on, many mathematicians generalized Edelstein’s result, for instance Meir and Keeler [10] in 1969 and Reich [11] in 1971. Reich’s result has been further generalized by Hardy and Roger [12] in 1973 as follows.Theorem 1.
LetP,ϱ be a metric space and T:P⟶P a self-mapping satisfying the following conditions for all p1,p2∈P:(1)
(3)ϱTp1,Tp2≤αϱp1,Tp1+βϱp2,Tp2+γϱp1,Tp2+δϱp2,Tp1+τϱp1,p2,whereα,β,γ,δ,τ are nonnegative reals.SetΩ=α+β+γ+δ+τ. Then,(a)
IfP is complete and Ω<1, then T has a unique fixed point(b)
If (1) is modified as
(1′) for all p1≠p2 implies(4)ϱTp1,Tp2<αϱp1,Tp1+βϱp2,Tp2+γϱp1,Tp2+δϱp2,Tp1+τϱp1,p2,and P is compact, T is continuous, and Ω=1; then, T has a unique fixed point.Nadler [13] in 1969 generalized the BCP in the context of multivalued mappings of complete metric spaces. Later on, Nadler’s result has been generalized by Prolla [14] in 1983.Meanwhile, the metric space has been generalized tob-metric space; by then, the fixed point theory has been further generalized for single-valued and multivalued mappings in the context of b-metric space, for instance, Bakhtin [15] in 1989 and Czerwik [16] in 1993.For nonself mapping,T:R⟶S (R and S are two nonempty sets), such that R∩TR=∅ (empty set); then, it is not possible to find the fixed point of T. The best way to deal with such situation is to explore a point r in R, such that(5)ϱr,Tr=ϱR,S,where(6)ϱR,S=infr∈R,s∈Sϱr,s,and if such a point in R exists, it is called the best proximity point of T. If R=S, then the best proximity point becomes a fixed point. So, best proximity point theory is the proper generalization of fixed-point theory. Fan’s result [17] in 1969 was probably the first attempt in this direction.Later on, many mathematicians extended Fan’s result and developed some best proximity point results. For more details, one can see [18]. Best proximity point theory has been further developed by using different proximal contractions, for more details, one can see references [19–23].Kirk [24] in 2003 introduced cyclic contraction and developed some fixed points results. Later on in 2006, Eldered and Veeramani [25] developed some best proximity point results for cyclic contractions.Basha in 2019 [21] introduced proximal contractive and proximal cyclic contractive mappings and developed some results for the existence and uniqueness of best proximity point.Recently, in 2021, Hiranmoy et al. [26] introduced proximal Kannan-type and proximal cyclic Kannan-type contractive mappings in metric spaces (compare with [21]) and developed some best proximity point results.Motivated by the contractive mappings of Hiranmoy, we introduce the notion of multivalued Hardy and Roger’s type proximal and cyclic proximal contractive mappings and develop some results for the existence of best proximity points inb-metric space. Furthermore, we give some examples to explain the results.


## 2. Preliminaries

Throughout this article,R,R+, N, N1, and ℘P denote the set of reals, nonnegative reals, positive integers, nonnegative integers, and collection of nonempty subsets of P, respectively.Definition 2.
LetP be a nonempty set and b≥1 a real number. The mapping ϱb:P×P⟶0,∞ is a b-metric and P,ϱb is called b-metric space if ϱb satisfies the following axioms:(b1)ϱbp1,p2=0 if and only if p1=p2(b2)ϱbp1,p2=ϱbp2,p1(b3)ϱbp1,p2≤bϱbp1,p3+ϱbp3,p2, for all p1,p3,p2∈P.
Throughout this paper,ϱ and ϱb denote metric and b-metric, respectively. Now, suppose that R and S are two nonempty subsets of P,ϱb. Define(7)ϱbR,S=infϱbr,s:r∈R,s∈S,R0=r∈R:ϱbr,s=ϱbR,Sforsomes∈S,S0=s∈S:ϱbr,s=ϱbR,Sforsomer∈R.Definition 3.
Ab-metric space P,ϱb is boundedly compact if every bounded sequence in P has a convergent subsequence (compare with [27]).Definition 4 (see [26]).
LetR and S be two nonempty subsets of P,ϱ. A mapping T:R⟶S is said to be a proximal Kannan-type contractive mapping if(8)ϱr1,Tr3=ϱR,Sϱr2,Tr4=ϱR,Sr3≠r4impliesϱr1,r2<12ϱr1,r3+ϱr2,r4,ϱr1,Tr3=ϱR,Sϱr2,Tr4=ϱR,Sr3=r4impliesϱr1,r2≤12ϱr1,r3+ϱr2,r4,hold for all r1,r2,r3,r4∈R.Definition 5.
LetR and S be two nonempty subsets of P,ϱb. Then, a mapping T:R∪S⟶R∪S is said to be cyclic if TR⊂S and TS⊂R (compare with [26]).Definition 6 (see [26]).
LetR and S be two nonempty subsets of P,ϱ. A cyclic mapping T:R∪S⟶R∪S is said to be a proximal cyclic Kannan-type contractive mapping if(9)ϱr1,Tr3=ϱR,Sϱr2,Tr4=ϱR,Sϱr3,r4>ϱR,Simpliesϱr1,r2<12ϱr1,r3+ϱr2,r4,ϱr1,Tr3=ϱR,Sϱr2,Tr4=ϱR,Sϱr3,r4=ϱR,Simpliesϱr1,r2=ϱR,S,that hold for all r1,r2,r3,r4∈R.In the following, we introduce a compact weak proximal pair inb-metric space.Definition 7.
LetR and S be two nonempty subsets of P,ϱb. The pair R,S is said to be a compact weak proximal pair if for bounded sequences rn in R and sn in S with ϱbrn,sn⟶ϱbR,S as n⟶∞, the sequences rn and sn have convergent subsequences in R and S, respectively (compare with [26]).Remark 8.
Note that ifR=S in above definition, then R,R is a compact weak proximal pair if and only if R is boundedly compact.Now, we present a lemma in the context ofb-metric space (analogous to [[26], Lemma 2.2]) that will be used in the sequel to prove our main results.Lemma 9.
LetR and S be two nonempty subsets of P,ϱb, such that at least one of R and S is bounded, and R,S is a compact weak proximal pair. Then, R0≠∅, and hence, so is S0.Proof.
As(10)ϱbR,S=infϱbr,s:r∈R,s∈S,so for each n∈N, there exists rn∈R and sn∈S, such that(11)ϱbR,S≤ϱbrn,sn<ϱbR,S+1n.
Therefore, the sequenceϱbrn,sn converges to ϱbR,S. Now, we assume that R is bounded. So, there exists a positive real number K, such that ϱbrn,rm≤K for all n,m∈N, so we have(12)ϱbsn,sm≤bϱbsn,rn+ϱbrn,sm≤bϱbsn,rn+bϱbrn,rm+ϱbrm,sm,which implies(13)ϱbsn,sm<bϱbR,S+1+bK+ϱbR,S+1.
Therefore,rn and sn are bounded sequences. So by compact weak proximality of the pair R,S, there exist subsequences rnk of rn and snk of sn, such that rnk converges to r⋆∈R and snk converges to s⋆∈S. Therefore,(14)ϱbrnk,snk⟶ϱbr⋆,s⋆ask⟶∞.
Thus, we have(15)ϱbr⋆,s⋆=ϱbR,S.
So,r⋆∈R0 and s⋆∈S0. Hence, R0≠∅ and S0≠∅. Similarly, if S is bounded, then R0≠∅ and S0≠∅.Theorem 10 (see [26]).
LetR and S be two nonempty subsets of P,ϱ, such that at least one of R and S is bounded, and R,S is a compact weak proximal pair. Let T:R⟶S be a proximal Kannan-type contractive mapping and assume that(i)
TR0⊂S0(ii)
Ifrn and sn are two bounded sequences in R and S, respectively, such that ϱrn,sn converges to ϱR,S, then limn⟶∞ϱrn,rn+1=0.
Then,T has a unique best proximity point in R.Theorem 11 (see [26]).
LetR and S be two nonempty subsets of P,ϱ, such that at least one of R and S is bounded, and R,S is a compact weak proximal pair. Let T:R∪S⟶R∪S be a proximal cyclic Kannan-type contractive mapping and assume that the following conditions hold:(i)
TR0⊂S0 and TS0⊂R0(ii)
Ifrn and sn are two bounded sequences in R and S, respectively, such that ϱrn,sn converges to ϱR,S, then limn⟶∞ϱrn,rn+1=0.
Then, the following conditions hold:(a)
There existr∈R and s∈S, such that ϱr,Tr=ϱR,S and ϱs,Ts=ϱR,S(b)
Ifr∈R and s∈S, such that ϱr,Tr=ϱR,S and ϱs,Ts=ϱR,S, then ϱr,s=ϱR,S.Now, we introduce the notions of a new type of generalized multivalued Hardy and Roger’s proximal contractive and proximal cyclic contractive mappings.Definition 12.
LetR and S be two nonempty subsets of P,ϱb. A mapping T:R⟶℘S is said to be a new type of generalized multivalued Hardy and Roger’s proximal contractive mapping if(16)ϱbr1,Tr3=ϱbR,Sϱbr2,Tr4=ϱbR,Sr3≠r4,implies,ϱbr1,r2<αϱbr1,r3+βϱbr2,r4+γb2ϱbr3,r4+δbϱbr1,r4+τbϱbr2,r3,ϱbr1,Tr3=ϱbR,Sϱbr2,Tr4=ϱbR,Sr3=r4,impliesϱbr1,r2≤αϱbr1,r3+βϱbr2,r3+δbϱbr1,r4+τbϱbr2,r3,which hold for all r1,r2,r3,r4∈R, where(17)α+β+γ+2τ=1,β≠1,γ+δ+τ<1,α,β,γ,δ,τ∈R+.Remark 13.
If in the Definition12, we replace T:R⟶℘S by T:R⟶S, then T is said to be a new type of generalized Hardy and Roger’s proximal contractive mapping.Definition 14.
LetR and S be two nonempty subsets of P,ϱb. A multivalued cyclic mapping T:R∪S⟶℘R∪℘S is said to be a new type of generalized multivalued Hardy and Roger’s proximal cyclic contractive mapping if(18)ϱbr1,Tr3=ϱbR,Sϱbr2,Tr4=ϱbR,Sϱbr3,r4>ϱbR,S,implies,ϱbr1,r2<αϱbr1,r3+βϱbr2,r4+γb2ϱbr3,r4+δbϱbr1,r4+τbϱbr2,r3,ϱbr1,Tr3=ϱbR,Sϱbr2,Tr4=ϱbR,Sϱbr3,r4=ϱbR,Simpliesϱbr1,r2=ϱbR,S,which hold for all r1,r2,r3,r4∈R, where(19)α+β+γ+2τ=1,α≠1,β≠1,γ+δ+τ<1,α,β,γ,δ,τ∈R+.Remark 15.
If in the Definition14, we replace T:R∪S⟶℘R∪℘S by T:R∪S⟶R∪S, then T is said to be a new type of generalized Hardy and Roger’s proximal cyclic contractive mapping.


## 3. Best Proximity Points Results for a New Type of Multivalued Hardy and Roger’s Proximal Contractive Mappings inb-metric Space

The following is our main result of this section.Theorem 16.
LetR and S be two nonempty subsets of P,ϱb, such that at least one of R and S is bounded and R,S is a compact weak proximal pair. Let T:R⟶℘S be a new type of generalized multivalued Hardy and Roger’s proximal contractive mapping. Further assume that(i)
For eachr∈R0, Tr⊂S0(ii)
Ifrn and sn are two bounded sequences in R and S, respectively, such that ϱbrn,sn converges to ϱbR,S, then limn⟶∞ϱbrn,rn+1=0.
Then,T has a best proximity point.Proof.
Lemma9 implies R0≠∅. Let r0∈R0; then, Tr0⊂S0. We can pick an element s1∈Tr0⊂S0, so that there exists r1∈R, such that(20)ϱbr1,s1=ϱbR,S.
Continuing this way, we can construct sequencesrn in R and sn in Trn−1, such that(21)ϱbrn,sn=ϱbR,S,for all n∈N. Therefore,(22)ϱbR,S≤ϱbrn,Trn−1≤ϱbrn,sn=ϱbR,S,that is,(23)ϱbrn,Trn−1=ϱbR,S.
Ifrn=rn−1 for some n∈N, then rn−1 is the best proximity point of T, and the proof is completed. So, we may assume that rn≠rn−1 for all n∈N. Now, we show that rn and sn are bounded sequences. As we have(24)ϱbrn,Trn−1=ϱbR,S,ϱbrn+1,Trn=ϱbR,S,rn≠rn−1,so by the given condition, we obtain(25)ϱbrn,rn+1<αϱbrn,rn−1+βϱbrn+1,rn+γb2ϱbrn−1,rn,+δbϱbrn,rn+τbϱbrn+1,rn−1≤αϱbrn,rn−1+βϱbrn+1,rn+γϱbrn−1,rn+τϱbrn−1,rn+ϱbrn,rn+1,which implies(26)1−β+τϱbrn,rn+1<α+γ+τϱbrn−1,rn.
If1−β+τ=0, then β+τ=1, so (17) implies α=γ=τ=0, and so β=1, a contradiction. Therefore, 1−β+τ≠0 and 1−β+τ=α+γ+τ. Hence, we get(27)ϱbrn,rn+1<ϱbrn−1,rn,that is,(28)ϱbrn,rn+1<ϱbrn−1,rn<···<ϱbr0,r1=Ksay.
As(29)ϱbrn,Trn−1=ϱbR,S,ϱbrm,Trm−1=ϱbR,S,so, if rn−1=rm−1, then we have(30)ϱbrn,rm≤bϱbrn,rm−1+ϱbrm−1,rm,≤bbϱbrn,rn−1+ϱbrn−1,rm−1+ϱbrm−1,rm,≤b2+bK.
Ifrn−1≠rm−1, then(31)ϱbrn,rm<αϱbrn,rn−1+βϱbrm,rm−1+γb2ϱbrn−1,rm−1,+δbϱbrn,rm−1+τbϱbrm,rn−1≤α+βK+γb2bϱbrn−1,rn+bϱbrn,rm+ϱbrm,rm−1+δϱbrn,rm+ϱbrm,rm−1+τϱbrm,rn+ϱbrn,rn−1,which implies(32)1−γ+δ+τϱbrn,rm<α+β+2γ+δ+τK,and by (2), γ+δ+τ<1. Therefore,(33)ϱbrn,rm<α+β+γ+δ+τ1−γ+δ+τK.
Hence,rn is a bounded sequence. Furthermore, for all n,m∈N, we have(34)ϱbsn,sm≤bϱbsn,rn+ϱbrn,sm≤bϱbR,S+b2ϱbrn,rm+ϱbrm,sm≤b+b2ϱbR,S+b2α+β+γ+δ+τ1−γ+δ+τK.
Therefore,sn is also a bounded sequence. From (69), it is clear that ϱbrn,rn+1 is a decreasing sequence of nonnegative real numbers and hence convergent. Using hypothesis (ii), ϱbrn,rn+1 converges to 0. Now, by compact weak proximality of the pair R,S, there exist two subsequences rnk of rn and snk of sn, such that rnk converges to some r⋆∈R and snk converges to some s⋆∈S. Consequently,(35)ϱbrnk,snk⟶ϱbr⋆,s⋆ask⟶∞.
Consequently,(36)ϱbr⋆,s⋆=ϱbR,S.
Thus,r⋆∈R0, which implies Tr⋆⊂S0. For each ν∈Tr⋆, there exists υ∈R, such that ϱbυ,ν=ϱbR,S. Now,(37)ϱbR,S≤ϱbυ,Tr⋆≤ϱbυ,ν=ϱbR,S,which implies(38)ϱbυ,Tr⋆=ϱbR,S.
Moreover, we have(39)ϱbrnk+1,r⋆≤bϱbrnk+1,rnk+ϱbrnk,r⋆.
Lettingk⟶∞, we get(40)limk⟶∞rnk+1=r⋆,then using the facts(41)ϱbrnk+1,Trnk=ϱbR,S,ϱbυ,Tr⋆=ϱbR,S,we get(42)ϱbrnk+1,υ≤αϱbrnk+1,rnk+βϱbυ,r⋆+γb2ϱbrnk,r⋆+δbϱbrnk+1,r⋆+τbϱbυ,rnk≤αϱbrnk+1,rnk+βϱbυ,r⋆+γϱbrnk,r⋆+δϱbrnk+1,r⋆+τϱbυ,rnk.
Lettingk⟶∞, we get(43)1−β+τϱbυ,r⋆≤0.
It impliesυ=r⋆. Thus, we have ϱbr⋆,Tr⋆=ϱbR,S, that is, r⋆ is a best proximity point of T. This completes the proof.
Now, we give an example to explain our claim.Example 17.
LetP=R, R=1,2, and S=1/4,1/2. Consider(44)ϱbp1,p2=p1−p22,for all p1,p2∈R. Then, ϱb is a b-metric on P with b=2. It implies ϱbR,S=1/4, R0=1, and S0=1/2; now, define a mapping T:R⟶℘S as follows:(45)Tr=12n:1≤n≤r.
It implies for eachr∈R0, Tr⊂S0. Now, we check T is a new type of generalized multivalued Hardy and Roger’s proximal contractive mapping. Let(46)r1,r2,r3,r4∈R.
Then, we discuss two possible cases.Case 1: if(47)r1,r2,r3,r4>1,then(48)ϱbr1,Tr3≠dR,S,ϱbr2,Tr4≠dR,S.Case 2: if(49)r1=r2=r3=r4=1,then(50)ϱbr1,Tr3=dR,S,ϱbr2,Tr4=dR,S,r3=r4.
It implies(51)ϱbr1,r2=0≤αϱbr1,r3+βϱbr2,r4+δbϱbr1,r4+τbϱbr2,r3,where(52)α+β+γ+2τ=1,α≠1,β≠1,γ+δ+τ<1,α,β,γ,δ,τ∈R+. So, all axioms of Theorem 16 are satisfied. Hence, T has the best proximity points set 1.Theorem 18.
LetR and S be two nonempty subsets of P,ϱb, such that at least one of R and S is bounded, and R,S is a compact weak proximal pair. Let T:R⟶S be a new type of generalized Hardy and Roger’s proximal contractive mapping and assume that(i)
For eachr∈R0, Tr∈S0(ii)
Ifrn and sn are two bounded sequences in R and S, respectively, such that ϱbrn,sn converges to ϱbR,S, then limn⟶∞ϱbrn,rn+1=0
Then,T has a unique best proximity point.Proof.
Existence of best proximity point follows from Theorem16. Now, to prove the uniqueness, consider r1 and r2 be two distinct best proximity points of T. Then, we have(53)ϱbr1,Tr1=ϱbR,S,ϱbr2,Tr2=ϱbR,S,r1≠r2.
It implies(54)ϱbr1,r2<αϱbr1,r1+βϱbr2,r2+γb2ϱbr1,r2+δbϱbr1,r2+τbϱbr1,r2,≤αϱbr1,r1+βϱbr2,r2+γϱbr1,r2+δϱbr1,r2+τϱbr1,r2,so(55)ϱbr1,r2<γ+δ+τϱbr1,r2<ϱbr1,r2,a contradiction as γ+δ+τ<1. Hence, T has a unique best proximity point.Corollary 19.
If we take in Theorem18b=1 and α=β=1/2,γ=δ=τ=0, then we get Theorem 10.


## 4. Best Proximity Points Results for a New Type of Multivalued Hardy and Roger’s Proximal Cyclic Contractive Mappings inb-metric Space

In this section, we consider new type of multivalued Hardy and Roger’s proximal cyclic contractive mapping for the existence of best proximity points.Theorem 20.
LetR and S be two nonempty subsets of P,ϱb, such that at least one of R and S is a bounded subset of P and R,S is a compact weak proximal pair. Let T:R∪S⟶℘R∪℘S be a new type of generalized multivalued Hardy and Roger’s proximal cyclic contractive mapping and assume that(i)
For eachr∈R0, Ts⊂S0, and for each s∈S0,Ts⊂R0, and(ii)
Ifrn and sn are two bounded sequences in R and S, respectively, such that ϱbrn,sn converges to ϱbR,S, then limn⟶∞ϱbrn,rn+1=0.
Then, there existr∈R and s∈S, such that ϱbr,Tr=ϱbR,S and ϱbs,Ts=ϱbR,S, and furthermore, ϱbr,s=ϱbR,S.Proof.
SinceR,S is a compact weakly proximal pair and at least one of R and S is bounded, so by Lemma 9, it follows that R0≠0 and S0≠0. Let r0∈R0 and s0∈S0 imply Tr0⊂S0 and Ts0⊂R0, so there exists r1∈R and s1∈S, such that(56)ϱbr1,ν1=ϱbR,S,ν1∈Tr0. Continuing this way, we construct sequences rn in R, sn in S, νn in Trn−1, and υn in Tsn−1, such that(57)ϱbrn,νn=ϱbR,S,and(58)ϱbsn,υn=ϱbR,S.
It implies(59)ϱbR,S≤ϱbrn,Trn−1≤ϱbrn,νn=ϱbR,S,and(60)ϱbR,S≤ϱbsn,Tsn−1≤ϱbsn,υn=ϱbR,S,for all n∈N. Thus, we have(61)ϱbrn,Trn−1=ϱbR,S,ϱbsn,Tsn−1=ϱbR,S.
First, we assume thatR is bounded. Then, there exists a positive real number K, such that ϱbrn,rm≤K for all n,m∈N. Therefore, for all n,m∈N, we have(62)ϱbνn,νm≤bϱbνn,rn+ϱbrn,νm,≤bϱbR,S+bϱbrn,rm+ϱbrm,νm,≤bϱbR,S+bK+bϱbR,S,implies(63)ϱbνn,νm≤b+b2ϱbR,S+b2K.
Therefore,νn is bounded. Also, T is cyclic, so for each s∈S, Ts⊂R, and so υn∈R, for all n∈N. Therefore, there exists a positive real number K1, such that ϱbυn,υm≤K1. It implies υn is bounded, so(64)ϱbsn,sm≤bϱbsn,υn+ϱbυn,sm,≤bϱbR,S+bϱbυn,υm+ϱbυm,sm,≤b+b2ϱbR,S+b2K1.
It impliessn is bounded. Thus, rn, sn, νn, and υn all are bounded sequences. On a similar line, we can prove rn, sn, νn, and υn are bounded whenever S is bounded. Since R,S is a compact weak pair, therefore, there exist subsequences rnk, snk, νnk, and υnk of rn, sn, νn, and υn, respectively, such that rnk⟶r⋆∈R, snk⟶s⋆∈S, νnk⟶ν⋆∈S, and υnk⟶υ⋆∈R, as k⟶∞. First, we show that ϱbr⋆,Tr⋆=ϱbR,S. As we have(65)ϱbrn,Trn−1=ϱbR,S,and(66)ϱbrn+1,Trn=ϱbR,S,so if ϱbrn−1,rn>ϱbR,S, then we have(67)ϱbrn,rn+1<αϱbrn,rn−1+βϱbrn+1,rn+γb2ϱbrn−1,rn+δbϱbrn,rn+τbϱbrn+1,rn−1,≤αϱbrn,rn−1+βϱbrn+1,rn+γϱbrn−1,rn+τϱbrn−1,rn+ϱbrn,rn+1≤α+γϱbrn−1,rn+βϱbrn,rn+1+τϱbrn−1,rn+ϱbrn,rn+1.
So,(68)1−β+τϱbrn,rn+1<α+γ+τϱbrn−1,rn.
If1−β+τ=0, then β+τ=1, so (19) implies α=γ=δ=τ=0, and β=1 is a contradiction. Therefore,(69)ϱbrn,rn+1<ϱbrn−1,rn.
If(70)ϱbrn−1,rn=ϱbR,S,then(71)ϱbrn,rn+1=ϱbrn−1,rn.
Therefore,(72)ϱbrn,rn+1≤ϱbrn−1,rn,for all n∈N, and hence, the sequence ϱbrn,rn+1 is a convergent sequence of real numbers. By hypothesis (ii), it follows that ϱbrn,rn+1 converges to 0. Now,(73)ϱbrnk+1,r⋆≤bϱbrnk+1,rnk+ϱbrnk,r⋆⟶0ask⟶∞.
Therefore,limk⟶∞rnk+1=r⋆. Again, we have ϱbrnk,νnk⟶ϱbr⋆,ν⋆ as k⟶∞, and hence, we get(74)ϱbr⋆,ν⋆=ϱbR,S.
So,r⋆∈R0 implies Tr⋆⊂S0. Thus, there exists υ∈R, such that(75)ϱbυ,ν=ϱbR,S,where ν∈Tr⋆. It implies(76)ϱbυ,Tr⋆=ϱbR,S.If(77)ϱbrnk,r⋆=ϱbR,S,only for finitely many k, then we can exclude those rnk from rnk and then assume(78)ϱbrnk,r⋆>ϱbR,Sfor all k. If(79)ϱbrnk,r⋆=ϱbR,S,for infinitely many k, then we can extract a subsequencernkl from rnk, such that(80)ϱbrnkl,r⋆=ϱbR,S,for all l. This gives(81)liml⟶∞ϱbrnkl,r⋆=ϱbR,SimpliesϱbR,S=0.
From the relations(82)ϱbυ,Tr⋆=ϱbR,S,ϱbrnkl+1,Trnkl=ϱbR,S,and(83)ϱbrnkl,r⋆=ϱbR,S,we get(84)ϱbrnkl+1,υ=ϱbR,S=0,for all l. Taking limit as l⟶∞, we get(85)ϱbr⋆,υ=0,so r⋆=υ. Therefore, we have(86)ϱbr⋆,Tr⋆=ϱbR,S.
Next, we assume that(87)ϱbrnkl,r⋆>ϱbR,S,for all k; then, from relations(88)ϱbυ,Tr⋆=ϱbR,S,ϱbrnkl+1,Trnkl=ϱbR,S,we get(89)ϱbrnkl+1,υ<αϱbυ,r⋆+βϱbrnkl+1,rnkl+γb2ϱbr⋆,rnkl+δbϱbrnkl+1,r⋆+τbϱbυ,rnkl.
Taking limit ask⟶∞ in above, we get(90)ϱbυ,r⋆≤αϱbυ,r⋆+τbϱbυ,r⋆,ϱbυ,r⋆≤αϱbυ,r⋆+τϱbυ,r⋆.
It implies(91)1−α+τϱbυ,r⋆≤0.
Ifα+τ=1, then (19) implies β=γ=τ=0 which implies α=1, a contradiction, so(92)ϱbυ,r⋆=0impliesυ=r⋆.
Hence,(93)ϱbr⋆,Tr⋆=ϱbR,S.
Similarly, we can prove(94)ϱbs⋆,Ts⋆=ϱbR,S.
Now, letr∈R,s∈S, such that(95)ϱbr,Tr=ϱbR,S,and(96)ϱbs,Ts=ϱbR,S.
Ifϱbr,s>ϱbR,S, then(97)ϱbr,s<αϱbr,r+βϱbs,s+γb2ϱbr,s+δbϱbr,s+τbϱbr,s,so(98)ϱbr,s<γϱbr,s+δϱbr,s+τϱbr,s.
It implies(99)1−γ+δ+τϱbr,s<0,which further implies(100)ϱbr,s<0,a contradiction. So, ϱbr,s=ϱbR,S. This completes the proof.Theorem 21.
LetR and S be two nonempty subsets of P,ϱb, such that at least one of R and S is a bounded subset of P and R,S is a compact weak proximal pair. Let T:R∪S⟶R∪S be a new type of generalized Hardy and Roger’s proximal cyclic contractive mapping and assume that(i)
For eachr∈R0, Tr∈S0, and for each s∈S0, Ts∈R0,(ii)
Ifrn and sn are two bounded sequences in R and S, respectively, such that ϱbxn,yn converges to ϱbR,S, then limn⟶∞ϱbrn,rn+1=0.
Then, there existr∈R and s∈S, such that ϱbr,Tr=ϱbR,S and ϱbs,Ts=ϱbR,S. Furthermore, ϱbr,s=ϱbR,S.Proof.
Following Theorem20, we can get the required result.Corollary 22.
If we takeb=1, α=β=1/2, and γ=δ=τ=0 in Theorem 21, we get Theorem 11.


## 5. Some Fixed Points Results

In this section, we derive some fixed points results from our main results.Theorem 23.
LetP,ϱb be a boundedly compact b-metric space; then,(i)
A mappingT:P⟶℘P, such that for q1,q2∈P, there exist p1∈Tq1 and p2∈Tq2, such that(101)ϱbp1,p2<αϱbp1,q1+βϱbp2,q2+γb2ϱbq1,q2+δbϱbp1,q2+τbϱbp2,q1,s.Forq1≠q2And(102)ϱbp1,p2≤αϱbp1,q1+βϱbp2,q2+δbϱbp1,q2+τbϱbp2,q1,Forq1=q2, where(103)α+β+γ+2τ=1,β≠1,γ+δ+τ<1,α,β,γ,δ,τ∈R+.(ii)
Ifpn is a bounded sequence in P, then limn⟶∞pn,pn+1=0.Then,FixT (set of fixed points of T) is nonemptyTheorem 24.
LetP,ϱb be a boundedly compact b-metric space; then,(i)
A mappingT:P⟶P, such that for all q1,q2∈P with q1≠q2,(104)ϱbTq1,Tq2<αϱbTq1,q1+βϱbTq2,q2+γb2ϱbq1,q2+δbϱbTq1,q2+τbϱbTq2,q1,where(105)α+β+γ+2τ=1,β≠1,γ+δ+τ<1,α,β,γ,δ,τ∈R+.(ii)
Ifpn is a bounded sequence in P, then limn⟶∞pn,pn+1=0
Then,FixT is singleton.


## 6. Conclusion

We presented a new type of generalized multivalued Hardy and Roger’s proximal contractive and proximal cyclic contractive mappings inb-metric spaces and developed results for the existence of best proximity points. Furthermore, we have derived results for the existence and uniqueness of best proximity points for new type of generalized Hardy and Roger’s proximal contractive and proximal cyclic contractive mappings. Our results are the generalization of the results already existing in literature.



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*Source: 1139731-2022-01-24.xml*